Positive Alpha Generation: Designing Sound Investment Processes (The Wiley Finance Series)

Positive Alpha Generation Designing Sound Investment Processes

Dr. Claude Diderich,

CEFA, FRM, NPDP

A John Wiley and Sons, Ltd., Publication

Positive Alpha Generation Designing Sound Investment Processes

For other titles in the Wiley Finance Series please see www.wiley.com/finance

Positive Alpha Generation Designing Sound Investment Processes

Dr. Claude Diderich,

CEFA, FRM, NPDP

A John Wiley and Sons, Ltd., Publication

c 2009 Copyright


John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777

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Library of Congress Cataloging-in-Publication Data Diderich, Claude G. Positive alpha generation : designing sound investment processes / Dr. Claude G. Diderich. p. cm. – (Wiley finance series) Includes bibliographical references and index. ISBN 978-0-470-06111-4 (cloth : alk. paper) 1. Portfolio management. 2. Investment analysis. 3. Investments–Management. 4. Investment advisors. I. Title. II. Series. HG4529.5.D53 2008 332.6–dc22 2008053934 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-06111-4 Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Acknowledgments First and foremost I would like to thank, in alphabetical order, Stefan Angele, Guido B¨achli, Matthias Hochrein, Maurzio Pedrini, Christoph Schenk, and Walter Sperb for having given me over the years the opportunities and platforms to design and implement various investment processes, from asset and liability management, through fixed income and equities, to multiasset class solutions. Developing and rolling out a new or enhanced investment process is not possible without teamwork. I therefore would like to thank Stefan B¨ar, Patrick Bucher, Luca Diener, Per Erikson, Ian Fishwick, Christian Fitze, Klaus G¨oggelmann, Robin Gottschalk, Bruno Heusser, Tomas Hilfing, Urs Hiller, Gregor Hirt, Markus H¨ubscher, Thomas Isenschmid, Aous Labbane, Giovanni Leonardo, Valerie Nicoud, Bernard Possa, Philipp R¨oh, Scott R¨udlinger, Michael Schmid, Karsten Steinberg, Guy Stern, Eric Suter, Ahmed Talahoui, Anne-Sophie van Royen, Philipp Vorndran, Karin Vrang, Benno Weber, Sabina Weber Sauser, and Gerhard Werginz for having participated at some stage in the implementation of an investment process or associated investment solutions. It is this collaboration that formed the groundwork for this book. I thank also all my friends, including a special journalist from Geneva, for having supported me during the period of writing of this book, and hopefully thereafter. Thanks also to Bill Falloon of John Wiley & Sons, Ltd for making the suggestion to write this book and Caitlin Cornish and her team for taking up this project at John Wiley & Sons, Ltd. Furthermore, I would like to thank the anonymous reviewer for his helpful comments which have allowed me to improve the quality of the text. Finally, I am very grateful to my parents for having taught me, by their example, to pursue one’s own goals and to do whatever one undertakes with enthusiasm. Zurich, Switzerland January 2009

Dr Claude Diderich, CEFA, FRM, NPDP

Contents

Preface

xi

1 Introduction

1

1.1 1.2 1.3 1.4 1.5 Part I

Characteristics of a successful investment process Challenges to be solved Approach taken in this book Structure of the book Notation The Value Chain of Active Investment Management

2 Key Success Factors for Generating Positive Alpha 2.1 2.2 2.3 2.4 2.5 2.6

Key success factors Decomposing return Defining risk The information ratio Fundamental law of active management The process of developing an investment process

3 The Investment Management Value Chain 3.1 3.2 3.3 3.4 Part II

The value chain components Designing a value chain based investment process Implementing the value chain approach Investment processes example Forecasting Markets

4 Judgmental Approaches for Forecasting Markets 4.1 4.2 4.3 4.4

Market efficiencies Understanding asset returns Forecasting asset returns Example

1 2 3 4 7 9 11 11 15 17 18 19 21 23 23 25 28 31 35 37 37 39 44 51

viii

Contents

5 Quantitative Approaches for Forecasting Markets 5.1 5.2 5.3 5.4 5.5 5.6

Building a quantitative forecasting model Defining the model structure Handling data in parameter estimation Testing the model Mitigating model risk Example

6 Taking Investment Decisions 6.1 6.2 6.3 Part III

8

57 60 68 71 74 74 81

Understanding the theory of decision making Building a decision making process Example

82 83 88

Risk Measurement and Management

91

7 Modeling Risk 7.1 7.2 7.3 7.4 7.5 7.6

57

The different dimensions of risk Risk management from an investor’s perspective Risk from an investment manager’s perspective The theory behind modeling market risk The process of developing a risk model Information risk

93 93 95 95 96 99 104

Volatility as a Risk Measure

109

8.1 8.2 8.3 8.4 8.5 8.6

109 110 111 119 122 125

The volatility risk model in theory Selecting data for parameter estimation Estimating the risk model’s parameters Decomposing volatility Additional pitfalls Testing risk models

9 Alternative Risk Measures 9.1 9.2 9.3 9.4 9.5 9.6 Part IV

Framework defining risk Alternative return distributions Exposure based risk models Nonparametric risk models Handling assets with nonlinear payoffs Credit risk models Portfolio Construction

129 130 133 138 140 144 144 147

10 Single Period Mean-Variance Based Portfolio Construction

149

10.1 Developing a modular portfolio construction process 10.2 The mean-variance framework 10.3 The Markowitz mean-variance model

149 152 154

Contents

10.4 10.5 10.6 10.7

ix

Alternative mean-variance based models Models with alternative risk definitions Information risk based models Selecting a portfolio construction approach

11 Single Period Factor Model Based Portfolio Construction 11.1 11.2 11.3 11.4 11.5

Factor models and their relation to risk Portfolio construction exploiting idiosyncratic risk Pure factor model exposure based portfolio construction Factor sensitivity based portfolio construction Combining systematic and specific risk based portfolio construction algorithms

12 Dynamic Portfolio Construction 12.1 Dynamic portfolio construction models 12.2 Dynamic portfolio construction algorithms 12.3 A practical example Part V

Portfolio Implementation

13 Transaction Costs, Liquidity and Trading 13.1 Understanding transaction costs and market liquidity 13.2 The action and context of trading 13.3 Implementation and trading as a module of an investment process value chain 13.4 Equity asset allocation trading approach example 14 Using Derivatives 14.1 Derivative instrument characteristics 14.2 Using derivatives to implement an investment strategy 14.3 Example Part VI

Investment Products and Solutions

15 Benchmark Oriented Solutions 15.1 15.2 15.3 15.4 15.5

Benchmarks Passive benchmark oriented investment solutions Active benchmark oriented investment solutions Core–satellite solutions A sample benchmark oriented solution

16 Absolute Positive Return Solutions 16.1 What absolute positive return can mean 16.2 Satisfying the investor’s expectations 16.3 The relationship between risk and return

161 164 166 170 171 171 172 176 179 180 183 184 186 191 195 197 197 202 205 208 213 213 219 224 231 233 233 238 244 249 251 253 253 254 258

x

Contents

16.4 Long-only forecasting based solutions 16.5 The portable alpha approach 16.6 Combining absolute positive return and benchmark oriented solutions 17 Capital Protection and Preservation Approaches 17.1 17.2 17.3 17.4 17.5

The investor’s utility function Portfolio insurance investment processes Comparing different portfolio insurance investment processes Managing risk Designing a client specific capital protection solution

18 Hedge Funds 18.1 18.2 18.3 18.4 18.5

Success factors of hedge funds Exploitable alpha generating sources Issues specific to hedge funds Developing a hedge fund investment process Hedge funds as an asset class

19 Liability Driven Investing 19.1 19.2 19.3 19.4 Part VII

The concept of liability driven investing Portfolio construction in a liability driven investment context Liability driven investment solutions A process for determining a liability driven investment solution

259 261 265 267 267 268 274 276 277 279 280 282 284 286 289 293 293 295 297 302

Quality Management

303

20 Investment Performance Measurement

305

20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8

Performance measurement dimensions Setting up a performance measurement framework Basics of performance measurement Performance attribution Performance contribution The process behind the process Practical considerations in performance measurement Examples of performance measurement frameworks

305 307 308 316 320 320 323 325

Bibliography

331

Index

341

Preface The foundation for creating sound investment solutions was laid out when Markowitz introduced in 1952 the concept of modern portfolio theory in his seminal paper called ‘Portfolio Selection’. The full significance of the relationship between return and risk, and with it the effects of diversification, was articulated. In the following years numerous models of financial markets, like the capital asset pricing model or the arbitrage pricing theory, were developed. A large number of theoretical as well as practical toolboxes have been introduced that directly or indirectly allow the quality of investment solutions to be enhanced. However, little has been written on combining these investment models into successful investment products and especially customized solutions. In this book I take an engineering approach to portfolio management. I present the different models and toolboxes that have been developed in the past and compare their strengths and weaknesses. I especially focus on the process of combining them in order to build sound investment processes. Both modeling as well as organizational aspects are addressed. Successful investment managers differentiate themselves along four key dimensions: • They deliver the promised performance, whether relative to a benchmark or absolute. • They implement an investment process that is cost efficient in production, but flexible enough to satisfy the specificities of the investor’s needs. • They offer a transparent and trustworthy investment management approach. • They provide innovation in order to understand and best satisfy the investor’s needs. This book shows how an investment manager can provide investment solutions that maximize his or her opportunities along these four dimensions. Delivering the promised investment performance is essentially driven by the investment manager’s forecasting skills. I describe techniques that can be used to develop these skills in a consistent manner and combine them to leverage their impact. I strongly believe that cost will become an even more important differentiating factor in the future, especially with the introduction of new pricing models. A very promising approach to achieve cost efficiency, especially when dealing with a large client base, is the value chain approach or a variation of it, such as value nets. The different constituents of an investment process, called modules, are presented throughout this book based on the value chain approach. From an investor’s perspective, buying an investment solution or entering into a contractual agreement with an investment manager is accepting a promise into the future. Especially when the investor is not the actual owner of the funds, but only the trustee, it becomes very important that the proposed investment solution is transparent to a degree that the investor is capable and willing to

xii

Preface

trust it. This can be achieved through three elements, that is a structured investment process that can easily be communicated and understood, a clear definition of roles and responsibilities, and a quality or performance management approach. Finally, innovation can only be achieved if the investor’s needs are well understood and translated into investment solutions that can be produced with the available skills. I describe in this book numerous techniques that can be used to achieve investment management success. They are based on the experience of colleagues and me through designing and implementing numerous investment processes. However, they are only examples and should be considered as such. It is the role of each investment manager to combine and adapt the proposed techniques so that they fit with individual needs and are based on specific skills. My goal with this book is achieved if I have provided readers with ideas that they had not yet thought of or had thought of from a different perspective and that they believe could be relevant, or at least interesting, to adapt and implement in their own environment.

1 Introduction The success of any actively managed investment solution depends on the investment manager’s capabilities to consistently generate positive alpha. An investment solution represents an actively managed portfolio maximizing the investor’s utility function. It has a medium to long term investment horizon1 and differentiates itself from a short term oriented trading strategy. In this book I define alpha as the total return of an investment solution that can be attributed to the skills of the investment manager. During the lifetime of an investment solution, that is the investment horizon, the investment manager takes so-called investment decisions. These decisions are then transferred into the portfolio such as to generate positive alpha. I call this process the investment process associated with the investment solution. The existence of an investment process is a necessary condition for generating consistently positive alpha. It allows the separation of skills from luck. Defining a successful investment process is not an easy task. It requires the combination of various distinct skill sets. First, the investor’s needs have to be understood and translated into the investment process’s framework. Second, markets have to be forecasted, that is successful investment decisions generated. Third, portfolio management skills are required to transfer or implement the investment decisions along with the investor’s needs into the portfolios. Finally, engineering and project management skills are necessary to build the investment process end to end. There exist many approaches for developing sound investment processes. The goal of this book is to present a set of methods and building blocks aiming at engineering successful investment processes. I present different theoretical concepts and illustrate how they can be applied. Theory is used as a means to an end, that is as a foundation on which a practical investment process is built.

1.1 CHARACTERISTICS OF A SUCCESSFUL INVESTMENT PROCESS From the perspective of an investor, an investment process is successful if it maximizes the utility function. Most of the time this means maximizing the realized alpha or the risk adjusted realized alpha. Other approaches are based on probabilistic definitions of success (Rabin, 2000; Samuelson, 1963; Stutzer, 2003, 2004). In order to achieve this goal, a sound investment process must be designed around four key dimensions: • Investor needs. The investment process must be designed in such a way that the resulting investment solution satisfies the investor’s needs and expectations, especially maximizing the utility function. 1

Usually the time horizon is at least one year, more often three to five years or longer.

2

Positive Alpha Generation: Designing Sound Investment Processes

• Investment decision skills. The investment process must be built around a set of investment decision skills. It is only through superior skills, especially in forecasting market returns, that positive alpha can be generated. • Transfer mechanism. The best forecasts are useless if they cannot be transferred into a portfolio. The transfer mechanism should aim at minimizing information loss, that is translate the investment decision skills as precisely as possible into the portfolios. • Consistency. To provide consistently positive alpha, the investment process must allow for a seamless execution. It must help separate skills from pure luck. High performance along these four dimensions is ultimately the basis for success. Nevertheless, a process is only as good as the people executing it, that is success = process + people

1.2

CHALLENGES TO BE SOLVED

Developing a successful investment process is hard and time consuming. This is especially true as the outcome cannot be directly tested for success. Only time can tell if an investment process is really successful. The two exogenous factors: (i) investor needs and (ii) investment decision skills play a key part in the success. An investment process only provides a platform for the deployment of these needs and skills. Some of the key challenges found during the development and introduction of a new or the refinement of an existing investment process are: • Matching investor needs with investment decision skills. An investor may need a solution investing in BRIC2 countries, but the existing investment strategists do not have any skills in forecasting these markets. Can these skills be built or bought? • Degree of sophistication. Any investment process is based on one or more underlying theories or models, like, for example, the capital asset pricing model (CAPM) (Sharpe, 1964). Depending on the investor’s sophistication and the investment professional’s skills, more or less sophisticated theoretical concepts, models and qualitative techniques can and should be applied. • Process flexibility. On the one hand, a very detailed investment process allows for an efficient and flawless execution as each and every step is clearly defined and documented. On the other hand, allowing for flexibility makes it easier to incorporate unforeseen stages and decisions. A reasonable trade-off must be found. • Change management. The most difficult challenge, when introducing a new investment process, is the people aspect. The reasoning leading to a new or changed investment process must be described. Advantages on an individual basis have to be presented and put along with the associated changes. Changes have to be communicated internally and to existing investors affected. An overview of change management techniques can be found in Holmes and Devane (1999). 2

Brazil, Russia, India and China.

Introduction

3 Investor needs: Marketing, sales

Investment decision skills: Investment strategists

Consistency: Investment process engineer

Transferability: Portfolio manager

Investment process

Investment solution Investment process development Investment solution production

FIGURE 1.1 Interaction between different skills and capabilities during the design of a sound investment process

Figure 1.1 summarizes the four key dimensions on which a sound investment process and ultimately successful investment solutions are designed.

1.3

APPROACH TAKEN IN THIS BOOK

A three step approach is taken in this book. First, the theoretical foundations of each concept are presented. Then, how the theory and models can be used or adapted in practice is shown. Finally, the different concepts are illustrated with real world examples. To avoid unfulfilled expectations, I would like to stress that this book is about the design of the investment process underlying an investment solution and not about developing the required investment decision skills that are required to generate positive alpha. The investment decision skills represent the core capabilities and competitive advantage of each individual investment manager. The interested reader may find insights into building alpha generating skills in Campbell et al. (1997), Cochrane (2005), Darst (2007), Edwards et al. (2007), Grinold and Kahn (2000) and Ingersoll (1987), to name just a few. A simplified version of the standardized business process modeling notation (Object Management Group, 2006) to describe different process stages is used. A complete set of modules that need to be developed when designing a sound investment process is described. The methods presented are structured such that they can be applied directly. I believe that structured design methods are valuable for three reasons: • First, they make the design process explicit and transparent, allowing the design rationale to be understood. • Second, by acting as a checklist of the key steps, they ensure that important issues are not forgotten. • Third, structured approaches are usually self-documenting.

4

Positive Alpha Generation: Designing Sound Investment Processes

Although the methods presented are highly structured, they are not intended to be applied blindly. Each investment process must adapt the presented methods to the specified investor, that is investment solution needs and the available skills. Throughout this book, I use different examples to illustrate different methods and concepts. These example are based on existing or fictious investment processes and are adjusted or simplified. They are not intended to be historically accurate case studies.

1.4 STRUCTURE OF THE BOOK I have structured the book along the different stages of the value chain underlying any investment solution. Each chapter focuses on a different topic and is as such independent. Nevertheless, it is only the interaction between the different modules, that is chapters, that expresses the full complexity of designing sound investment processes. Figure 1.2 shows which characteristics of a successful investment process are covered by each chapter: • Part I focuses on the concepts on which any investment process should be based. Chapter 2 presents the key success factors for generating positive alpha. The risk–return framework used throughout this book is described. Chapter 3 defines the value chain of an investment process and describes it from both a process and an organizational point of view. • Part II concentrates on forecasting market returns. In Chapter 4 the theory and practice for designing qualitative forecasting processes is illustrated. Chapter 5 presents an overview of traditional and nontraditional quantitative forecasting techniques. Chapter 6 is devoted to taking investment decisions through combining information and forecasts. • Part III defines the risk measurement and management frameworks used. Chapter 7 focuses on modeling risk. Chapter 8 describes in detail the notions of volatility and tracking error as the most used risk measures and shows how to forecast or estimate their parameters. Chapter 9 is devoted to surveying other risk measures that are useful within the context of providing investment solutions. • In Part IV of the book the focus is on models translating forecasts into portfolios. Chapter 10 describes classical mean-variance based portfolio construction models. Chapter 11 presents static factor based models whereas Chapter 12 focuses on dynamic, multiperiod models, which are especially relevant in an asset liability management context. • Part V focuses on key issues that have to be addressed at the portfolio implementation level. Chapter 13 concentrates on market liquidity issues and transaction costs whereas Chapter 14 shows how to use derivatives in the context of an investment process. • Part VI is devoted to providing investment solutions. Different types of products and solutions are described. Chapter 15 focuses on traditional benchmark oriented solutions whereas Chapter 16 illustrates the development of absolute positive return investment solutions. Chapter 17 shows how to develop capital protection and preservation approaches and combine them with benchmark or absolute positive return oriented solutions. Chapter 18 shows the similarities and dissimilarities between hedge funds and so-called traditional or long-only investment solutions. Chapter 19 focuses on liability driven investing. • Finally, Part VII focuses on quality management. In Chapter 20 different approaches to performance measurement are presented. Their use as an integral part of an investment process is illustrated.

Introduction

Investor needs

5

Transferability

Investment decision skills

Consistency

Chapter 2 – Key success factors for generating positive alpha

Chapter 3 –The investment management value chain Chapter 4 – Qualitative approaches for forecasting markets Chapter 5 – Quantitative approaches for forecasting markets Chapter 6 – Taking investment decisions Chapter 7 – Modeling risk

Chapter 8 – Volatility as a risk measure

Chapter 9 – Alternative risk measures Chapter 10 – Single period mean-variance based portfolio construction Chapter 11 – Single period factor based portfolio construction Chapter 12 – Dynamic portfolio construction Chapter 13 – Transaction costs, liquidity, and trading Chapter 14 – Using derivatives Chapter 15 – Benchmark oriented solutions Chapter 16 – Absolute positive return solutions Chapter 17 – Capital protection and preservation approaches Chapter 18 – Hedge funds Chapter 19 – Liability driven investing Chapter 20 – Investment performance measurement

FIGURE 1.2 Mapping of the different chapters on to the four key dimensions of a successful investment process

6

Positive Alpha Generation: Designing Sound Investment Processes Table 1.1 Typographical conventions used

Convention

Description

v M xa , ya,b

Vector, represented by a bold lowercase letter Matrix, represented by a bold uppercase letter Element of the vector x, respectively the matrix Y, represented by a lowercase indexed letter Random variable, represented by a calligraphic letter Estimated value of a parameter, denoted by a hat

ᑬ pˆ

Table 1.2

Commonly used functions

Function

Description

v v , M M−1

Arithmetic mean of the elements of vector v, that is Transpose of vector v, respectively matrix M Inverse of matrix M 

2 Frobenius norm, that is i j (mi,j )

||M||f diag(M) trace(M) E[ᑬ], Et [ᑬ] O(f (v)) N(µ, σ ) ᑬ ∼ N(µ, σ )

1 N

·


N

i=1 vi

Matrix containing the diagonal
elements of matrix M Trace of matrix M, that is i mi,i Expected value of the random variable ᑬ, respectively expected value of the random variable ᑬ conditioned on all information available at time t Big O representing the upper bound of magnitude of the function f (v) (Knuth, 1976) Normal distribution with mean µ and variance σ Random variable ᑬ following a normal distribution with mean µ and variance σ

Table 1.3 Constants, parameters and variables Convention

Description

ᑛ A T h H K C Ra,t , Rt ra,t , rt

Set of all considered assets or asset classes Number of assets or asset classes Number of time periods Time horizon between two time periods Investment horizon Number of factors in a linear factor model Constant Discrete total return of asset a or a portfolio between time t − 1 and t Log-return or continuous compounded return of asset a or a portfolio between time t − 1 and t Excess return between asset a and asset b between time t − 1 and t, that is Eb−a,t = Rb,t − Ra,t , respectively eb−a,t = rb,t − ra,t Risk free rate at time t Return of the market portfolio between time t − 1 and t

Eb−a,t , eb−a,t RF,t RM,t

Introduction

7 Table 1.3 (continued)

Convention

Description

RB,t βf,a Rf Ff d, D
σa,b ρa,b σa , σ τP ,B , τ w µa

Return of a given benchmark portfolio between time t − 1 and t Exposure of asset a to factor component f Return of factor component f Sensitivity of factor component f Vector, matrix of data Covariance matrix Covariance between assets a and b Correlation between assets a and b Volatility of asset a or portfolio Tracking error of portfolio versus benchmark Portfolio weights Expected return of asset a

1.5 NOTATION To simplify the description of the models, algorithms, formulas and properties, a common notation is used throughout the book. I rely on frequently used typographical conventions for defining constants, vectors, matrices, as well as random variables, described in Table 1.1. Table 1.2 lists the most common functions used, such as the transpose, norm or the expected value. As most quantitative descriptions are related to assets, their returns over time, their decomposition, as well as their volatilities, a specific notation is introduced, described in Table 1.3.

Part I The Value Chain of Active Investment Management

2 Key Success Factors for Generating Positive Alpha When designing an investment process, the key goal is to define the most adequate process for the target investment solution or product satisfying the investor’s needs. There does not exist a single investment process that fits all investment solutions, but there are three key requirements that any investment process must fulfill in order to be successful. These are the existence of investment opportunities, forecasting skills and a transfer mechanism. Let me first define some terminology. An asset is a financial instrument (physical or contractual) that can be bought or sold in reasonable volume, within a reasonable time frame and at a reasonable price. I define an asset class as a collection of assets with similar properties, like stocks, bonds, but also property, currency forward contracts or total return swaps. A portfolio is defined as a collection of asset classes or assets combined or weighted according to a specified scheme. The scheme defines the assets belonging to the portfolio as well as their relative weights in the portfolio. A German equity portfolio could be composed of 20 companies from the DAX index equally weighted, each taking up 5 %. A key functionality of each investment process is to construct portfolios, that is to define schemes on how assets should be combined to generate positive alpha. I call benchmark a specific portfolio that is used as an input to an investment process. I assume that it is possible to invest in the assets of the benchmark, that is to replicate exactly the benchmark with a portfolio. For example, a benchmark could be defined as 43 % aluminum, 43 % copper, 7 % nickel and 7 % silver, the strategic metals1 contained in the Reuters/Jefferies CRB index. A special case of a benchmark portfolio is the market portfolio, that is all existing tradable securities weighted proportionally to their total value. The market portfolio represents an equilibrium, that is an idealized situation where all market forces are perfectly in balance.

2.1 KEY SUCCESS FACTORS Many different approaches exist for systematically generating positive alpha. All successfull approaches have three properties in common: • They are based on a well defined set of investment opportunities. • They show forecasting skills or capabilities matched to investment opportunities. • They include a methodology that translates the expressed skills on investment opportunities into portfolios, that is a sound transfer mechanism. Although the forecasting skills form the core of active management, they are not sufficient. If the opportunity set is not well defined, opportunities may be missed and the opportunity

1 Strategic metals are metals that are judged essential for defense and for which the US is almost totally dependent upon foreign sources (Sinclair and Parker, 1983).

12

Positive Alpha Generation: Designing Sound Investment Processes

costs paid may by far outweigh the value added. Consider an investment universe aiming at generating alpha through investing in strategic metals. It is not clear whether the investment manager, for example, is allowed to hold any gold or not, as gold is not considered a strategic metal. In addition, the best forecasting skills are worthless if they cannot be transferred into actual value. For example, correctly forecasting the number of rain days in the city of London is only relevant if a reward for that forecast exists.2 Let me now elaborate in more detail on the three key success factors: (i) investment opportunities, (ii) forecasting skills and (iii) transfer mechanisms. 2.1.1 Investment opportunities The investment opportunities available to an investment manager are defined by: (i) an investment universe and (ii) a decomposition of the investment universe into asset classes or assets. The fundamental law of active management states that the expected alpha of a portfolio is, under certain assumptions, proportional to the square root of the number of available investment opportunities (Grinold, 1989). Therefore, having a large number of assets defined increases the opportunities for generating high alpha. If the individual assets are actual securities, I talk about a security selection approach. For example, the investment universe may be the universe of stocks in the S&P 500 index and the assets the individual stocks. If the assets are a combination of homogeneous securities, I talk about an asset allocation approach. For example, consider all developed equity markets as defined by the index MSCI world as the investment universe and consider Europe excluding the UK, UK, North America, Japan, Asia excluding Japan and Australia/New Zealand as asset classes. This forms a possible set of investment opportunities using the asset allocation approach. A different subdivision of the same universe would be to define sectors, like utilities, financial services, etc., as asset classes. If the asset classes are characterized by a set of exogenous properties, I talk about a factor model approach. For example, the investment universe defined by all UK government bonds and the asset classes by buckets of modified durations, modified duration being the exogenous factor, represents a factor model approach. Many of the investment opportunities used by hedge funds, such as, for example, arbitrage strategies, are based on the factor model approach. A style based approach subdivides the investment universe according to specific characteristics of the securities, for example between growth and value stocks, large capitalized and small capitalized stocks. Style based approaches are special cases of factor models where the characteristic defining the style corresponds to the factor of a factor model. In the definition of value and growth used for determining the constituents of the two indices Russell 1000 value and Russell 1000 growth the price-to-book ratio is used as the factor. Finally, if the asset classes are characterized by an individual, a team or a company managing the asset class, I talk about a manager selection approach. Most prominent candidates of this approach are funds of fund based investment 2 There actually exists such a weather derivative financial market. According to the Weather Risk Management Association, the total trade volume of weather contracts was about $45.2 billions of notional in 2005/2006.

Key Success Factors

13

products. The investment opportunities of all these approaches have three properties in common: (1) Homogeneity. The securities forming a given asset class are homogeneous with respect to their expected return and their properties on which the skills or forecasts are based. For example, in an asset allocation approach on equity regions or countries, US securities exhibit similar expected return characteristics, for example, when compared to Japanese equities. (2) Distinguishable. The asset classes exhibit times of significant relative differences in expected return and properties on which the skills or forecasts are based. In an asset allocation based approach, expected returns between US and Japanese equities are potentially significantly different due to the difference in the underlying economy. (3) Mutually exclusive. Individual securities belong to exactly one asset class. A security is classified either as a US based or a Japanese based company. Mutual exclusion is needed to avoid redundancies. A well defined investment universe is ideally constructed of a large number of asset classes. The returns of the asset classes should exhibit low correlations and thus allow a large number of independent forecasts to be expressed, which, by the fundamental law of active management, as I will show in the last section of this chapter, is beneficial to the overall portfolio performance. 2.1.2 Forecasting skills The second important success factor is the ability to formulate forecasts on the opportunity set. I define forecasting as the act of expressing an expectation regarding the total return of one or more asset classes. This expectation is for a specific explicitly or implicitly defined time horizon. Grinold and Kahn (2000) wrote in their seminal work that ‘active management is forecasting. The consensus forecast of expected returns, efficiently implemented, leads to the market or benchmark portfolio. Active managers earn their title by investing in portfolios that differ from their benchmark. As long as they claim to be efficiently investing based on their information, they are at least implicitly forecasting expected returns.’ It is important to have systematic forecasting capabilities on the defined opportunity set. Otherwise expressed, the opportunity set must be matched to the available forecasting skills. It is useless to define, for example, the investment universe in a selection based approach as all Icelandic stocks, if no research capabilities exist about Iceland and its companies. Many different approaches exist for forecasting. The most common approach, but also one of the most difficult ones, is forecasting expected total returns for the individual asset classes. Another approach is to rank the different assets according to their expected returns. Even combining asset classes into a portfolio is a way of forecasting as the relative weight of each asset class expresses a relative preference or expectations. The forecasting methodology should be selected such that the forecasting skills are maximized. The citation from Albert Einstein ‘make it as simple as possible, but not simpler’ should be applied. There is no need to add complexity to the forecasting process if it cannot systematically improve the forecasting skills.

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Positive Alpha Generation: Designing Sound Investment Processes

A forecasting methodology exhibiting the investment manager’s skills needs to adhere to three basic principles: • Each forecast must be quantifiable. • Forecasts must be consistent among each other. • All forecasts must be over an explicitly or implicitly defined time horizon. 2.1.3 Transferability The third success factor for generating positive alpha is the ability to transfer the forecasts on the opportunity set into portfolios. The transfer mechanism must ensure that the performance of the constructed portfolio is consistent with the forecasts and their timing. It must especially be defined such that for all portfolios to be constructed: • each individual investor’s utility function, that is the tradeoff between risk and return, is maximized, • any general and specific requirements and restrictions are taken into account and • all portfolios are constructed fairly, that is no single portfolio is preferred. Usually maximizing the investor’s utility function translates into maximizing the expected return of the portfolio, given the forecasts, such that a certain risk tolerance is not exceeded and all specific requirements and restrictions are met. In addition, market liquidity or other exogenous restrictions may influence the resulting portfolios. It goes without saying that if a given position cannot be implemented in one portfolio because of exogenous restrictions, like for example market liquidity, it must not be implemented in any portfolio in order to assure that all portfolios are treated fairly and no single investor is preferred over others. 2.1.4 A comprehensive example First, I define my investment opportunity set by using as the investment universe the 19 commodities contained in the Reuters/Jefferies CRB index (CRB index). These are aluminum, cocoa, coffee, copper, corn, cotton, crude oil, gold, heating oil, lean hogs, live cattle, natural gas, nickel, orange juice, silver, soybeans, sugar, unleaded gas and wheat. I implement an asset allocation approach and subdivide the investment universe into the six asset classes of grains, industrial metals, soft commodities, precious metals, energy and livestock. It can be shown that the six asset classes satisfy the properties of homogeneity, distinguishability and mutual exclusiveness. Then, I define the forecasting skills as ranking the different asset classes in decreasing order of their expected return, with a time horizon of the date where the future with the shortest time to expiration on the CRB index expires. Usually the time horizon is three months, the time between the expiration dates of two futures. To actually come up with the proposed ranking I use analysis of supply and demand of the individual commodities and aggregate that information for each of the asset classes. Finally, I use a simple transfer mechanism that states to over- and underweight the different asset classes by 5 % for the best and worst asset classes and by 3 % for the second best and second worst asset classes. No relative weights are attributed to the asset classes

Key Success Factors

15

Table 2.1 Ranking of the different asset classes, absolute and relative weight in a sample commodity universe asset allocation based investment process Asset class

Grains Industrial metals Soft commodities Precious metals Energy Livestock

Ranking

Initial weight (%)

New weight (%)

Relative weight (%)

4 3 5 2 1 6

13 13 21 7 39 7

13 13 18 10 44 2

±0 ±0 −3 +3 +5 −5

Source: Reuters/Jefferies CRB index, author’s calculations.

ranked third or fourth. I assume that the portfolio from which I over- and underweight the commodities is initially weighted according to the weighting of the CRB index at the beginning of the month. The relative weight determined for each asset class is then distributed proportionally according to the initial weights on all the securities of that asset class. Table 2.1 shows the resulting asset allocation. This transfer mechanism allows a consistent transfer of the forecasts into the portfolio. Risk is controlled through maximal possible relative weights. In addition, by construction, no single commodity position in the portfolio can become negative.

2.2 DECOMPOSING RETURN Up to now, I have described the key characteristics of a successful investment process to generate positive alpha. However, I have actually not yet given a formal definition of what I understand by positive alpha. I denote by RP ,t+1 the total return of a portfolio P between time t and t + 1. If the time horizon is obvious the notation is simplified to RP . Assume an exogenously defined benchmark portfolio B, that is whose definition is independent of the underlying investment process. Denote by RB the return of the benchmark portfolio. Furthermore, assume that RF is the risk free rate, that is the return that is achievable without bearing any risk of loss. Usually this return is approximated by the LIBOR3 or similar rates. Then the return of any portfolio P can be decomposed into three components, as shown by RP

=

RF + risk free rate

(RB − RF ) risk reward return

+

α skill based return

(2.1)

I define by RB − RF the risk reward return of the portfolio P . It is the excess return over the risk free rate that is due to the selection of the benchmark B and can be interpreted as the risk premium the market pays to an investor for holding a specific portfolio B. The return α is called the residual return, alpha return, or skill return. It is the portion of the return of the portfolio that can be attributed to the investment manager’s skills. It is specific to the portfolio P . Active management is the process, some call it art, to construct portfolios 3

London Interbank Offered Rate – interest rate at which banks lend and borrow funds among themselves.

16

Positive Alpha Generation: Designing Sound Investment Processes

such that, over the specified time period, the realized alpha is positive. Equation (2.1) can be interpreted in an ex-ante, which is a forward looking context, as well as an ex-post, which is a backward looking context. In an ex-ante context, the alpha is the forecasted residual return over a specified period. It represents the expected excess return due to the investment decisions taken. In an ex-post context, it is the realized residual return, positive or negative, due to the investment manager’s skills. 2.2.1 Timing versus selection Some authors decompose the portfolio return RP as RP = RF + β · (RB − RF ) + α. This decomposition can be interpreted as the result of a regression of the portfolio return on the benchmark’s excess return; β is called the market timing return coefficient and α the selection return. However, as both α and β are due to active investment decisions taken, there is no need to distinguish between them when extracting the return portion that is attributable to skill. Indeed, I can write RP = RF + β · (RB − RF ) + α = RF + (RB − RF ) + α − (1 − β) · RB + (1 − β) · RF    α

where the portfolio is underweight in the benchmark by 1 − β and overweight in the risk free asset by the same quantity. Combining these over- and underweights with the selection return defines the α  due to skills. 2.2.2 Benchmarks Up to now, I have assumed that the benchmark is given. As the benchmark, per definition, is a portfolio, its return can also be decomposed according to Equation (2.1). Assume that, for the definition of a benchmark, the used benchmark is the risk free rate. Then equation (2.1) can be written as RB = RF + α

(2.2)

where α represents the return due to the skill of selecting the benchmark. What is important to understand is that the subdivision of return into a risk–reward part and a skill part according to equation (2.1) depends on the context of the decisions taken. At the end, any return in excess of the risk free rate is due to some sort of skill, whether it is through selecting a benchmark portfolio or through under- and overweighting specific asset classes. I consider the process of constructing or defining a benchmark portfolio as a specific investment process. An opportunity set must be selected. The benchmark constituents are selected out of this opportunity set. The transfer mechanism associates weights to the selected constituents and assures the investability of the benchmark portfolio. For example, selecting the Dow Jones Euro STOXX 50 index or the MSCI EMU index as a benchmark for a Euroland equity investment solution is an active decision and has an impact on the absolute performance of the portfolio managed against it. Usually the time horizon for the forecasts underlying the benchmark selection is different and much longer than for the alpha generating forecasts. In traditional benchmark oriented investment solutions RB  α and RB  RF , where  means very superior, as defined in equation (2.1). Therefore the selection of the benchmark

Key Success Factors

17

predominantly determines the overall portfolio return and its sound definition is key. On the other hand, absolute return oriented solutions have α  RF and RB = RF ; thus the investment manager’s skills become predominant. 2.2.3 Market portfolio A special case of a benchmark portfolio exists, called the market portfolio. The market portfolio is a portfolio containing all tradable assets proportional to their market value. Very often, the market portfolio is approximated though an index including all actively traded equity securities, for example the FTSE all world index. If only focusing on a single region or country, the market portfolio is sometimes approximated by a market capitalization weighted index of traded equities in that country or region, for example the S&P 500 index for the US. The market portfolio plays an important role in equilibrium based approaches. It has been extensively studied and is at the basis of many market models, including the capital asset pricing model (Sharpe, 1964) or the Black–Litterman model (Black and Litterman, 1991; Littermann, 2003). The definition of the market equilibrium theory can be seen as the added value through skill in Equation (2.2), the skill being attributed to the underlying equilibrium theory. Different theoretical models lead to different total returns in excess of the risk free rate. Therefore the selection of an underlying model is an active investment decision based on the investment manager’s skill in selecting the most appropriate model underlying an investment solution.

2.3

DEFINING RISK

In the context of an investment process, risk is defined as the uncertainty of achieving an expected outcome, observed through variability from an expected result. This definition allows a one-to-one relationship between risk and a random variable as known from mathematical probability theory. It is important to differentiate between the uncertain future described by the random variable and its modeling, described by the probability distribution function and its parameters defining the random variable. The main consequence is that risk cannot be directly measured or observed. It can only be forecasted or estimated. This estimation is subject to the use of a model and associated assumptions. More precisely, risk is the uncertainty of achieving a certain return. This risk is called market risk. In this book, when not stated otherwise, risk is defined as the volatility of the expected returns assuming that returns can be modeled by a normally distributed random variable and that only the two first moments of the distribution are relevant. As in the decomposition of return, the risk of a portfolio can be subdivided into a benchmark risk and an asset or active risk; the asset risk is called an active risk because it is the consequence of an active investment decision. (σP )2 = (σB )2 + (σα )2 + ρ · σB · σα

(2.3)

where σP denotes the risk or the volatility of the portfolio, σB the benchmark risk, σα the active risk and ρ is the correlation or dependence between the benchmark and the active risk. Very often, the active risk is denoted by τ . Because of the subadditivity4 of coherent risk 4 A function or risk measure is called subadditive if the sum of two elements is always less or equal to the sum of the function’s value at each element.

18

Positive Alpha Generation: Designing Sound Investment Processes

measures, as will be shown in Chapter 7, the risk function is concave. In contrast with the return decomposition in Equation (2.1), the risk free rate does not appear in Equation (2.3) as it does not, by definition, bear any risk. Risk management is not about minimizing risk, but allocating and controlling risk in order to match return expectations. Sound risk management should lead to being able to take more risk, rather than less. Other types of risk exist apart from market risk. These are information risk, credit risk, legal risk, operation risk, liquidity risk or model risk, to name the most important ones. In the context of designing investment processes, I am mainly concerned with market and information risks.5

2.4 THE INFORMATION RATIO It is often requested and useful to asses the alpha generating capability of an investment manager through a single number. As shown in the previous sections, two dimensions exist, that is return and risk, along which an investment manager can position himself. The information ratio, denoted by IR, is a measure that seeks to summarize the mean-variance properties due to skills of an actively managed portfolio. It builds on the paradigm that says that the mean and variance of returns are sufficient statistics to characterize an investment portfolio. Let RP be the return of a portfolio and let RB be the return of the associated benchmark portfolio over the same time horizon. Then I define EP , the excess return of the portfolio, as EP −B = 6
RP − RB . I denote by E P −B the annualized arithmetic average of excess returns of the historical period of the portfolio and by τP the annualized standard deviation of the excess returns over the same period, called residual risk. I define the ex-post information ratio IR as the ratio between the realized excess or residual return and the residual risk, that is IR =


E P −B τP

IR represents the amount of value added (or destroyed) by the investment manager per unit of active risk taken. Formulated differently, it represents the investment manager’s skills in a comparable or normalized form. Grinold and Kahn (2000), as well as other authors, have shown that an investment manager should target an information ratio of around 0.5 over a long time horizon. This does not mean that short term information ratios above 0.5 or even 1.0 are not achievable, but that they are not sustainable, as shown by empirical work. I have argued previously that selection of the benchmark needs skill. Table 2.2 shows the information ratios resulting from the selection of different indices and benchmarks composed of multiple indices. It can be seen that there are differences in information ratios between benchmarks, but they situate themselves overall in the range of 0.2 to 0.7. Benchmark selection seems easier than taking active investment decisions. This is due to the fact that: (a) All asset classes used in the benchmark construction examples exhibit an internal rate of return or risk premium.7 5

Information risk is defined as the risk of information being incorrect or incorrectly interpreted. For the sake of simplicity I use arithmetic averages of excess returns. Goodwin (1998) has shown that the effect of using geometric, continuous compounded or even frequency-converted excess return aggregation matters very little when calculating information ratios. 7 See Chapter 4 for a detailed description of the concept of risk premium. 6

Key Success Factors

19

Table 2.2 Information ratios of different indices or benchmarks when compared to the risk free rate, based on 20 years of monthly historical data Asset class

Index (total return)

US government bonds Currency hedged global government bonds US corporate bonds

Citigroup WGBI ($) all maturities Citigroup WGBI world all maturities/hedged in US$

Citigroup US BIG overall broad investment grade US large cap equities MSCI USA (gross total return) US small cap equities MSCI USA small cap (data since 1995) Hedged global equities MSCI world (hedged in $) 50 % US corporate bonds, 25 % hedged global government bonds, 25 % hedged world equities 50 % hedged global government bonds, 25 % US equities, 25 % hedged world equities 25 % US government bonds, 25 % US equities, 50 % hedged world equities

τˆ (%)

IR

4.6 3.1

0.51 0.69

3.9

0.64

13.6 19.7

0.54 0.44

13.2 4.4

0.20 0.56

6.9

0.52

9.8

0.38

Source: data sourced from Bloomberg Finance LP, author’s calculations.

(b) Risk increases over time with the square root of time under normality assumptions whereas returns are geometrically compounded over time. The ex-ante information ratio is the expected level of residual return per unit of forecasted active risk. Similarly, I define the expected hit ratio (HR) or confidence level of an investment manager over a given time period as the probability to generate a positive alpha or a positive information ratio, that is HR = Pr{IR
0}

(2.4)

Under the assumption that the information ratio is a normally distributed random variable, it is possible to solve Equation (2.4) (Lee, 2000). In summary, the information ratio is a powerful tool for assessing the skills of an active manager. However, no single measure should ever be relied upon exclusively. As with all statistics based on historical data, there is no guaranteed relationship between past and future information ratios, although I would hope for some consistency.

2.5 FUNDAMENTAL LAW OF ACTIVE MANAGEMENT Generating positive alpha is tightly linked to the set of available investment opportunities, the skills available to exploit these opportunities and the degree to which these skills can be transferred into total return. The generalized fundamental law of active management states that, if an investment manager exploits all opportunities in a mean-variance efficient way, then the value added or alpha will be proportional to the square root of the number of exploited opportunities.

20

Positive Alpha Generation: Designing Sound Investment Processes

More formally, let BR denote the breath, that is the number of independent opportunities or forecasts. Let IC be the manager’s information coefficient or skills. It measures the correlation between the forecasts and the actual outcomes. Let TC denote the transfer coefficient, that is the percentage of the actual forecasts that are transferred into actual performance or alpha of the portfolio. The transfer coefficient is a function of (i) the transfer mechanism used, (ii) the restrictions both from the investor and the market, and (iii) the associated costs. Then, the generalized fundamental law of active management states that IR ∼ =

√ BR · IC · TC

(2.5)

assuming the same information coefficient IC for each forecast. Equation (2.5) may be considered abstract at first sight. Consider a set of n forecasts, each correct with a probability p. Denote by α the expected return from each of those forecasts. Then the expected return of an equally weighted portfolio implementing the n forecasts is n · (α/n) · p. The uncertainty or volatility of this portfolio, assuming that all forecasts are independent is given by   n 2  1 1 2 2

·σ ·p =σ ·p· n n i=1

√ and as such the risk adjusted return of the portfolio equals α · n · p. The analogy with Equation (2.5) can easily be seen, where BR = n and IC = p. A detailed analysis of the fundamental law of active management, as well as its derivation, can be found in Grinold (1989) and Grinold and Kahn (2000). If the forecasts or opportunities are not independent of each other, which is the case in most situations, then the fundamental law of active management extends to IR ∼ = TC ·



IC · Q−1 · IC < TC ·

 #(IC) · IC

where IC is a vector of the information coefficients associated with the individual opportunities and Q−1 the inverse of the covariance matrix between the different investment opportunities (Lee, 2000). The key message of the generalized fundamental law of active management is that an investment process should be designed to: (i) maximize the number of independent opportunities, (ii) maximize the skills of forecasting capabilities on each opportunity and (iii) implement a sound and efficient transfer mechanism with a high transfer coefficient. In addition to increasing the expected information ratio, the larger the number of opportunities forecasted, the smaller the overall impact of an individual incorrect forecast, assuming a sound transfer mechanism. This is especially true as the value of the information coefficient IC as well as the transfer coefficient TC are bound above. Increasing the number opportunities, that is increasing BR, is the main possibility to increase the overall information ratio of the portfolio.

Key Success Factors

21

2.6 THE PROCESS OF DEVELOPING AN INVESTMENT PROCESS The first step when starting a project to introduce a new refined process and enhance an existing investment process is to define the overall project goal. The goal specifies the key properties that the investment process must satisfy in order to generate positive alpha consistently. It is defined as a compromise between: • availability or potential availability of skills,8 capabilities and resources, and • investor needs and expectation in the investment solutions produced. It is this goal that will be at the heart of the investment manager’s unique selling proposition or key competitive advantage. For example, an investment manager may define the investment process goal as delivering a highly customized actively managed asset allocation based on investor benchmarks. The unique skills and expertise available are in the area of macroeconomic forecasting. The investment process to be developed must allow for: • a best possible use of the macroeconomic forecasting capabilities, • a high degree of flexibility with respect to asset classes (traditional and nontraditional9 ), benchmarks and restrictions, and • a streamlined implementation with indexed components, like exchange traded funds, passively implemented mutual funds or futures as building blocks. Any investment process is made up of three competencies, as shown in Figure 2.1. Only a well balanced combination of these components will lead to success. I therefore recommend basing the development of the investment process on the following four best practice basic principles: (1) The development project for introducing a new or enhanced investment process must only be started after the project goal has been defined and accepted by the key stakeholders. (2) Portfolio managers responsible for implementing the investment process must be involved in the development. (3) The organization structure must be adapted to the investment process’s structure. (4) The process of developing the new or enhanced investment process must be managed according to project management best practice rules (Office of Government Commerce, 2002; Project Management Institute, 2004). To proceed, the investment process to be developed should be decomposed into modules which: • can be developed independently, • require specific skills, capabilities and resources, and • are connected together through predefined interfaces. 8 The skills may either exist within the company or may be bought on the market. I advise not to outsource the skills on which the investment process is built as this will make it hard to advocate a unique selling proposition based on these skills. 9 Nontraditional asset classes could, for example, be commodities, real estate or inflation linked bonds.

22

Positive Alpha Generation: Designing Sound Investment Processes

Methodologies models

Skills

Successful investment process

People

FIGURE 2.1 Competencies required for successfully developing a new or enhancing an existing investment process

This decomposition approach is detailed in Chapter 3 through the concept of the value chain. I recommend using a top-down approach based on the following steps: 1. 2. 3. 4.

Recursively decompose the investment process in independent modules. Define the goal of each module. Define the interfaces (input and output) between the modules. Define the quality criteria as well as the measurement approaches for each module and for the interfaces. 5. Design solutions for each module. 6. Test the solution for each module independently according to the defined quality criteria. 7. Combine the modules together and perform a so-called integration test. It is important that the solutions developed for each module are objective and include feedback loops to preceding modules, thus ensuring that input quality is guaranteed.

3 The Investment Management Value Chain

delegation of responsibility

Strategy is the art of creating value. The value chain concept is a strategy for organizing complex production processes efficiently. But what distinguishes it from other work process management approaches? The value chain approach is based on subdividing processes into individual sub-systems, so-called modules, including the delegation of the responsibility for the process within each module. A module can be seen as a role and responsibility delegation strategy. This idea is illustrated in Figure 3.1. Modules are interconnected by input–output or producer–consumer streams, so-called interfaces. value chain approach

traditional portfolio management

delegation of roles/modularization

FIGURE 3.1 The value chain concept presented in the two-dimensional space formed by delegation of roles and responsibilities

The value chain approach of an investment process is the foundation of achieving any active investment manager’s goal, that is producing consistent positive alpha for all investors according to their needs and restrictions. Similar to the automobile industry, a well designed investment process should allow the production of highly customized investment solutions (car model, color, interior, etc.), all exhibiting the same high quality (low fuel consumption, stated horsepower, minimal servicing costs, etc.) at a reasonable cost (competitive pricing). An investment manager’s value chain may differ in scope from that of his competitors and represent a potential source of competitive advantage.

3.1

THE VALUE CHAIN COMPONENTS

3.1.1 Modules A module represents a well defined and in itself closed process step that requires specific skills different from those required by other modules. Each module takes an input, transforms it by adding specific value to it and produces a distinct output. This output serves as the input to the following modules in the value chain. Each module provider in the value

24

Positive Alpha Generation: Designing Sound Investment Processes

chain focuses on the core competencies. An investment manager gains competitive advantage by performing the strategically important individual modules cheaper or better than his competitors (Porter, 1985). For example, a module could implement trading individual securities and guaranteeing best execution.1 Modules may be shared by different investment processes, therefore increasing efficiency of the overall investment manager. From a knowledge point of view, each module represents a distinct set of skills. This know-how is represented in my trading module example by a deep knowledge of all available counterparties for a given security transaction as well as current market conditions. In addition to focusing on specialization, the value chain approach allows the complexity of a sophisticated investment process to be significantly reduced. Innovation is boosted because enhancements at the individual module level are possible without affecting the overall investment process. In my trading module example, a new automatic order matching tool can be introduced without affecting the other modules of the investment process. 3.1.2 Interfaces The second important concept of a value chain based investment process is the definition of interfaces between the different modules of the value chain. These interfaces have three distinct and important roles: (1) They define the information exchanges required between modules. (2) They combine the value added by each of the modules. (3) They add implicit quality control steps to the process. Reconnecting with my trading module example, the input interface specifies when, how and what information the trader will receive from the preceding module of the value chain. This could be a specific list of securities to buy and sell within the next two hours with amounts and price limits. As different modules are generally executed by different persons or teams, the information receiving module implicitly or explicitly provides a quality control of the input obtained as its own output quality depends on it. The information exchange may be either push or pull based. In a push based setup, the providing module initiates the information transfer whereas in a pull based setup, the receiving module requests the information. Both approaches have their advantages and disadvantages. I believe that in the context of an investment process, push based communication approaches are the most efficient as information is produced and transferred on an ad hoc basis. 3.1.3 Organizational aspects As with any process implementation, the value chain approach is only as good as the people implementing it. Not only must the underlying processes be organized, subdivided into modules and interfaces defined, but also the organizational structure must be adapted. The organizational structure should represent a balance between the benefits of separation into modules and the integration or combination of value producing activities. The logic 1 Best execution defines the buying or selling of a security at the best possible price in the market at a given moment in time.

The Investment Management Value Chain

25

Head investment management Investment process architect

Head investment strategy

Top down strategy

Head investor servicing

Bottom up research

Switerland Germany Italy United States Japan

Head portfolio construction

Asset allocation

Security selection

Fixed income Module - Market forecasting

Modules - Solution specification - Investor reporting

Trading

Equities

Modules - Risk management and portfolio construction - Implementation and trading

FIGURE 3.2 Sample organizational structure for an investment management department, including modules associated with the individual organizational functions

of the structure should exploit putting activities that have similarities that can be exploited under the responsibility of one manager. There should be a one-to-one relationship between managers and modules. In some setups, it may be useful to combine more than one module under one manager’s responsibility, that is implement a one-to-many relationship between managers and modules. Responsibilities for the individual modules must be delegated to the owners or providers of the modules, that is their managers. An overall body responsible for the interaction between modules must be installed. In contrast to other approaches, the value chain strategy makes rising issues transparent much quicker. It therefore allows issues to be remedied much faster. This increases the overall quality of the investment solutions. Figure 3.2 represents a possible organizational structure for an investment management department.

3.2

DESIGNING A VALUE CHAIN BASED INVESTMENT PROCESS

Two key roles exist in the design of a value chain based investment process, that is the architect, and the module designer. It is the architect’s responsibility to define which modules are required, their semantics, and to provide interface specifications between the modules. The quality of the resulting investment process depends on the quality of the work done by the architect. Each module designer is responsible for proposing and implementing the semantics of his or her module. Usually the module designers are tightly related to the providers of the modules, although this is not mandatory. In addition designers of different modules may or may not be the same person or team. They may even belong to different organizational entities or event companies. Figure 3.3 defines the generic value chain of an investment process.2 It is used to demonstrate how a value chain can be constructed for a particular investment process, reflecting 2 The investment process decomposition only focuses on the process steps that are related to generating positive alpha and delivering it to investors. Any purely operational aspects have been explicitly left out.

26

Positive Alpha Generation: Designing Sound Investment Processes Solution specification Market forecasting

Portfolio construction and risk management

Implementation and trading

Investor reporting

FIGURE 3.3 Individual modules forming an investment process, market forecasting being the only module independent of the investor

its specificities. It also shows how the modules are linked to each other and how these interfaces affect competitive advantages. I believe that this value chain decomposition is efficient and effective for producing all major traditional and most nontraditional investment management products and solutions. There are five generic modules involved in any investment process value chain. Each of these five modules represents a distinct process step requiring specific skills and adding specific value to the investment solution. The implementation of each module represents the distinct characteristics of the produced investment solutions and the investment manager’s strategy3 or competitive advantage. 3.2.1 Solution specification A key task for any service provider is to understand exactly the specific investor needs and translate them into a formal solution specification. This is no different when offering investment management solutions. The solution specification module focuses on understanding the investor’s investment needs, for example the liability structure, restrictions, regulatory and other constraints, risk appetite and ability to take risk, expressed, for example, by the current surplus at risk, and translating them into a formalism that is used as input to the following modules of the investment process. For example, the investor solution could be the specification of an investment universe, a benchmark or a target tracking error. The key skills needed by the module provider are a deep understanding of investor needs and the capabilities to translate them into the implemented investment process framework. It is the application of that skill set than represents the value added by this module. 3.2.2 Market forecasting The second of the two most important modules with respect to value added focuses on forecasting markets. As Grinold and Kahn (2000) state, active management is forecasting. Without consistent forecasting capabilities, no investment process can generate consistent positive alpha that is not based on pure luck. Positive alpha can only be generated by having an opinion on the future that deviates from the market consensus. It is that forecasting capability that adds value to the investment solution. It also represents key differentiation or competitive advantage over other investment managers. A concise understanding of financial markets and their interactions with macroeconomic and microeconomic effects is central to 3 The term ‘strategy’ represents a business decision to perform a given process in a given way. It is not to be confused with an investment strategy, which represents a market forecast transformed into a portfolio context.

The Investment Management Value Chain

27

a high quality implementation of the market forecasting module. I have therefore devoted a whole part of this book to the topic. 3.2.3 Portfolio construction and risk management Modern investment managers not only focus on generating return but also focus on the risk taken to generate these returns. Risk management4 is made up of two major aspects: (1) managing information risk, that is the risk of incorrectly forecasting markets and (2) measuring and monitoring exposure to exogenous risks for which the market is expected to reward the investor with return. In many modern investment processes risk management is used as a tool to transfer market forecasts into asset allocations or portfolio structures. It can be seen as the foundation of a portfolio construction algorithm. To develop, apply and maintain risk models, statistical and econometric skills, as well as a feeling for numbers, are essential. Based on the market forecasts and the solution specification, the two inputs to the portfolio construction and risk management module, a risk managed portfolio is constructed and trades generated. The aim has shifted from return to risk as well as quality management. Model portfolios are constructed and adjusted to the investor’s needs. Ideally the market forecasts are directly transformed into investor specific portfolios and trades. If this is unrealistic, intermediate model portfolios shared by multiple investors are constructed. These construction and compilation processes are ideally risk management driven. Alternatively, risk measurement techniques are used and a feedback loop applied until the constructed portfolio meets its needs. A good high level market understanding (available investment vehicles, liquidity, etc.) as well as knowledge in quality management approaches (six sigma, operational excellence, etc.) form the basic skills needed by a portfolio constructor team. 3.2.4 Implementation and trading The implementation and trading module of the investment process focuses on a best execution of all trades, as determined by the portfolio construction module. The trading operation is separated from portfolio construction in order to avoid determining the portfolio structure by trading related market expectations, which are therefore no longer in line with the overall return and risk profile determined by the preceding modules of the investment process. In addition, it allows a fair treatment of all investors by combining multiple trades into single executions. The key skills needed by providers of this module are knowledge about the markets, counterparty price building and volume impacts on prices. Efficiently executed trades, especially for large volumes or in illiquid securities, add nonnegligible value to the overall investment solution.

4 In the context of an investment process, I use the term risk management to define ex-ante risk management, sometimes called risk budgeting.

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Positive Alpha Generation: Designing Sound Investment Processes

3.2.5 Investor reporting The final module of the investment process is charged with reporting to the investor how the positive alpha was generated or why it was not generated. A key understanding of the whole investment process is necessary as well as skills to translate technical, process and market related information into investor language. The investment solution is only as good as its communication to the investor. This module is in charge of the successful delivery of the produced solution. The presented structure of the investment process may be adapted to the individual products and solutions to be produced. For example, some investment professionals may combine portfolio construction and implementation as well as the trading modules into a single module. Other setups combine the solution providing and portfolio construction modules into a single module. All these variations have their advantages and disadvantages. Each investment manager has to find the combination and interaction of modules that best satisfies his or her unique value proposition and competitive advantage. Nevertheless, I believe that two basic principles exist that should be adhered to: • Each module should focus on a single set of skills with a well defined, measurable output assuring investment process quality. • Feedback loops should be avoided, as they introduce unnecessary overheads and dilute the responsibilities associated with an individual module. It goes without saying that individual modules may and eventually should be outsourced. First, each investment manager should identify core competencies and map them to individual modules. Next, the advantages and disadvantages of outsourcing noncore modules should be analyzed. Outsourcing does not necessarily mean outsourcing to an external provider. It can also be implemented by delegating different modules to other departments within the company, these departments providing the same or similar modules to different investment processes. Such a structure is called a feeder structure. A good example for such an outsourcing is the creation of a dedicated trading desk whose sole role is to provide best trade execution to all investment managers within the company, independent of which products and solutions are produced.

3.3 IMPLEMENTING THE VALUE CHAIN APPROACH Implementing an investment process based on the value chain approach is not without challenges. However, these challenges can be successfully mastered if the value chain approach introduction is accompanied by a change management process. First and foremost, a sense of urgency for change must be staged. Then the key advantages of subdividing the investment process along the value chain must be presented. Finally, the result of introducing a value chain based investment process is only as good as the people introducing the process. A commitment from senior management to the framework is mandatory for success. Traditional investment managers are very often organized along product lines. Dedicated teams are responsible for specific products. No formal interactions between the teams exist. Good results have been obtained by first reorganizing the investment management organization along an investment manager type focused structure, as shown in Figure 3.4. For example, investment managers of type A may be all tactical asset allocation portfolio managers in Frankfurt, investment managers of type B all portfolio managers of institutional

The Investment Management Value Chain

29 Head investment management

Investment managers of type A

Investment managers of type B

Investment managers of type C

1) Investor servicing 2) Market forecasting 3) Portfolio construction 4) Implementation

FIGURE 3.4 Matrix organization structure combining the value chain approach with an investor type focused organizational structure

investors in Zurich, investment managers of type C all portfolio managers of mutual funds in Zurich, and so on. It can easily be seen that different investor types share common requirements. For example, each investor type based team requires market forecasts. Through a matrix structure, shown as dotted lines in Figure 3.4, these commonalities, which map to the value chain modules, can be extracted. Each value chain module should be chaired by a coach with leadership qualities. This coach needs to extract the skills and commonalities from the different investor type teams without formal authority. Once the sensibility of the investment management teams has shifted from the investor type structure to the module or value chain structure, the matrix organizational structure can be switched to have the value chain modules as primary reporting lines. This process usually takes about 6 to 12 months and its success depends on the leadership qualities of the module coaches. It is important to adjust the incentives structure and also the compensation scheme to support the value chain. This means focusing on the quality of the outcome of the individual modules. Although skeptical at first, experience has shown that investment professionals who have switched to a value chain approach do not want to switch back. A major inhibitor of the introduction of a value chain for generating positive alpha is answering the question of investor ownership. In a traditional product or investor type oriented framework, the person or team owning the investor decides on what the investor is offered and delivered. In a value chain based approach, this ownership no longer exists. The solution specification and investor reporting modules interact with the investors. However, it is only the collaboration of the individual modules of the value chain that unleashes the whole value to the investor. The key advantages for the investors as well as the organization introducing the value chain based investment processes are: • Each module focuses on a specific role requiring specific skills. Not only is the work delegated to the module, but also the responsibility for delivering high quality. Conflicts of interests between modules are avoided because of a streamlined association of roles and responsibilities.

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Positive Alpha Generation: Designing Sound Investment Processes

• The module provider has freedom as to what process to implement within the module, given that the interfaces with other modules are adhered to. This allows for greater flexibility and faster cycles of innovation. It accentuates the pace. • The contribution of each module to the overall performance, that is its value added, can be measured by comparing the module’s output to its input. • Resources and skills are optimally leveraged. • The setup allows for an efficient combination of proprietary as well as off-the-shelf modules. • A well implemented value chain investment process presents a barrier to entry in the sense defined by Porter (1985, Chapter 2). The competitive pressure is heightened. Nevertheless, environments also exist in which a value chain based approach is inefficient or ineffective: • A value chain approach is inconsistent with a business model based on individuals, that is so-called star business models. • As the value chain approach focuses on efficiency, it will not unleash its full potential if only a small number of investors need to be serviced or if the solutions provided do not allow leveraging process and skill efficiencies through subdivision into modules. • The initial overhead when setting up a value chain is high due to the requirement to separate independent tasks and their responsibilities into different modules. • As resources and skills are combined in modules, which are then leveraged for different products and solutions, it is more difficult to attribute revenues, especially in a total cost framework, to the different modules of the value chain. Table 3.1 summarizes the key advantages and drawbacks of the three most common organizational structures found. 3.3.1 Value chain versus value net In 2000 Bovet and Martha presented the value net approach as a replacement of the more traditional value chain concept. The value net concept is based on the following five characteristics: (1) Investor-aligned. The presented approach ensures that investor choices trigger the solution production through the solution specification module; that is the investor drives the solution design, rather than the solution design being imposed on the investor. (2) Collaborative and systemic. The value chain approach, as presented, assigns each activity to a specific module, according to the key stills available. If applicable, modules can be outsourced. (3) Agile and scalable. As individual modules only interact through well defined interfaces, reconfiguring different modules can be done in an efficient way, allowing them to be responsive to changes in demand, new products or solutions, rapid growth, or redesign of the individual module providers. (4) Fast flow. The value chain approach allows an order-to-deliver or just-in-time approach, as suggested by the value net framework. (5) Digital. The information flow or interfaces represent the heart of the proposed investment process.

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Table 3.1 Key advantages and drawbacks of the three most common investment management organizational structures Product oriented

Investor type oriented

Value chain oriented

Advantages

– Product oriented – Simple responsibility structure – Star oriented

– Unique point of contact for investor – Simple responsibility structure – No conflicts of interests on products – Investor focused – Clear investor ownership

– Scalable – Consistent performance – Focused on skills – No conflicts of interests – Implicit controlling through interfaces – Responsibilities related to added values

Drawbacks

– Star oriented – Conflicts of interest: products versus investors – Silo approach – No skill leverage – Broad knowledge required – Priority favors performance over investors – No investor ownership – Absence of implicit controlling

– Priority favors investor over performance – No product identity with person – Not salable – Possible inconsistencies between investors – Absence of implicit controlling – Broad knowledge required

– Requires large number of investors – No single person responsible for final output – Communication overhead between modules – No product identity

Therefore, the value chain approach presented in this chapter can also be seen as a value net, providing a powerful engine for generating added values to both the investors as well as the shareholders.

3.4 INVESTMENT PROCESSES EXAMPLE Consider the following investment solution description: We offer highly customized actively managed global equity solutions based on fundamental research and a disciplined and risk-managed investment process. I will now show how to design a value chain based investment process to produce high quality research driven equity solutions efficiently. Before I start, let me define the unique selling proposition or competitive advantage: • The alpha, that is added value, is purely driven by bottom-up stock research, mainly based on a DCF5 model. 5

Discounted cash flow.

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Positive Alpha Generation: Designing Sound Investment Processes

• The stock research is transferred into portfolios through a fully automated sophisticated quantitative model, which not only manages portfolio risk but also ensures best execution for all investors. • The proposed equity investment solutions are highly customizable with respect to: (a) investment universe and (b) target and maximal total portfolio risk. Minahan (2006) describes a variation of the unique selling proposition as the investment manager’s investment philosophy. He argues that each investment manager needs to pass the investment philosophy test, that is be able to express how he or she expects to generate positive alpha. 3.4.1 Modules I first define the individual modules of the value chain based on Figure 3.3: (1) Solution specification. The investor servicing expert defines, together with the investor, the actual investment universe, with respect to regions to be included as well as specific sectors or stocks, like tobacco stocks, to be excluded. A reference portfolio or benchmark is defined. Finally, the investor’s subjective risk perception is translated into quantitative target and maximal tracking error figures. Head investment management Investment process architect

Investment strategy

Investor servicing Solution specification

Investor reporting

Portfolio construction Portfolio management Risk management

Economic strategy

Equity research Equity research America

Macroeconomics team

Portfolio trading

Implementation and trading

Portfolio construction

Equity research Europe Equity research Asia

FIGURE 3.5 Value chain and organization structure of a research driven equity investment process; the organizational structure is shown using rectangular black boxes and lines and the value chain modules and interfaces are in rounded grey boxes and lines

The Investment Management Value Chain

33

(2) Market forecasting. Three regional modules, represented by three teams of buy-side equity research analysts, provide, for each stock in the potential investment universe, a rating, which is either strong sell, sell, neutral, buy or strong buy. This rating is derived from a DCF model that is based on a single set of economic forecasts (interest rates, inflation, growth, etc.) provided by a small module or team of macroeconomists. (3) Risk management and portfolio construction. A centralized team of risk experts uses an in-house developed order aggregation and management system to derive portfolio holdings as well as associated trades for each individual investor based on the single set of stock ratings and the investor’s individual benchmark, investment universe and tracking error. Risk management and portfolio construction are integrated as the module is essentially model driven. (4) Implementation and trading. Dedicated teams of traders in each of the major markets ensure that stocks are bought and sold at best price, guaranteeing fair execution for any investor. In addition, large trade volumes are broken down over time periods in order to minimize market impacts. (5) Investor reporting. In contrast to the highly quantitative modules of the investment, investor reporting is mainly based on a one-to-one discussion of the results obtained with the investor on a very personal basis. The investor is given the feeling that the investment manager only works for him or her. Figure 3.5 illustrates the defined investment process as well as the associated organizational structure. 3.4.2 Interfaces Four key interfaces exist between modules in the described investment process: (1) Information from the macroeconomics module is transferred to the equity research teams as well as the investor servicing specialists, for reporting purposes. This is done by providing monthly macroeconomics research presentations, including specific forecasts for key economic figures, such as interest rates, inflation, growth, etc. (2) Next, research analysts provide their rating for each individual stock covered to the portfolio construction team. In addition, this information is made transparent to the investor servicing team so that they can explain the portfolio holdings as well as the executed trades. (3) Then the portfolio construction and risk management module provides a list of trades to be executed to the trading module. This is done through an order management computer system. (4) Finally, the trading team reports what trades have been executed at what price to the investor servicing team, as well as the resulting portfolio structure. Each receiver of information is required to verify and control the quality of the data obtained and, if necessary, request amendments and additional information. The module providing the information is responsible for the contents provided. The module receiving the information is responsible for the quality control of the information received. This setup separates the roles of producer and controller of information traversing the value chain.

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Positive Alpha Generation: Designing Sound Investment Processes

3.4.3 Organizational structure As suggested previously, the research modules are completely independent of the investor servicing modules as well as the risk management. First, all forecasting modules or capabilities are associated with the head investment strategy. Two separate roles are defined, that is top-down macroeconomics forecasting, in the role of the economic strategy team, and bottom-up equity research, in the role of regional equity research teams. Next, investor servicing is separated from forecasting and portfolio construction as well as trading. This ensures that investor servicing experts can fully focus on the investor relationship and are not distracted, for example, by market forecasting or trading activities. Third, I set up a dedicated team of quantitative and technology experts to run the portfolio construction modules, including risk management and portfolio construction, using the in-house system. Finally, in order to ensure best execution as well as equal treatment of all investors, I have regrouped the implementation and trading module in a separate and dedicated role. This allows, in addition to avoiding any potential conflicts of interest between the forecasting team, the investor servicing experts and the actual market participants, for example, to avoid buying a security because a market participant is putting a large volume on the market without having a positive research view on that stock. The proposed organizational structure is shown in Figure 3.5 as rectangular boxes and black reporting lines.

Part II Forecasting Markets

4 Judgmental Approaches for Forecasting Markets It is an open question whether it is possible to forecast markets systematically. The existence of a large number of active investment managers1 shows that at least there is a belief in the existence of forecasting capabilities. Recent studies (Kosowski et al., 2006; Wermers, 2000) have provided compelling evidence that portfolio managers exist who are capable of generating persistent positive alpha net of costs that is not due to pure luck.2 I distinguish between frameworks and tools for designing a process to forecast market returns and specific forecasting skills, but do not focus on the latter ones. They represent the core capabilities and competitive advantage of each individual investment manager.

4.1

MARKET EFFICIENCIES

The concept of efficient markets is at the heart of studying the existence of forecasting capabilities. The efficient market hypothesis (Fama et al., 1969; Fama, 1970, 1991) states that asset prices are typically in equilibrium; that is assets are fairly priced in the sense that the prices reflect all available information. There are three forms of market efficiencies: • Weak form of efficient markets. All information contained in past asset prices is fully reflected in the current prices. • Semi-strong form of efficient markets. All publicly available information, whether historical price or other information, is fully reflected in the current asset prices. • Strong form of efficient markets. All public and private information is fully reflected in the asset prices. Many empirical studies3 have considered the efficient market hypothesis. Some authors conclude from their analyses of the efficient market hypothesis that it is, in general, impossible for an investment manager to consistently generate positive alpha. However, the market efficiency hypothesis merely states that, in order to generate positive alpha, an investment manager needs to be able to: • forecast information flows, that is have superior information gathering capabilities, and • predict the impact of those information flows on asset prices, that is have superior information processing skills. 1

Beechey et al. (2000) estimate that about 60 % of institutional funds in the United States are invested actively. Kosowski et al. (2006) have studied 1788 US domestic equity mutual funds generating alpha through stock picking. The superior after-cost performance mainly exists for growth oriented funds and occurs in the top 10 % of alpha ranged funds. 3 See Beechey et al. (2000) for a survey of the efficient market hypothesis. 2

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Positive Alpha Generation: Designing Sound Investment Processes

For example, an investment professional could study the global steel industry and come to the conclusion that a consolidation on a global basis is sound. More precisely, it could be assumed that a merger between A and B would create a new company with expected cash flows in excess of the sum of those of A and B. Along the efficient market hypothesis, as soon as the market participants also expect such a merger, the prices of companies A and B will adjust. The professional therefore expects that A and B will announce a merger within the next 12 months. In addition, a forecast is made that this merger will increase the total return of the stock of company A relative to its nearest competitor C. This forecast would, if correct and transferred into a portfolio by a long position in A financed by a short position in C, generate positive alpha. This forecast is consistent with the semi-strong form of efficient markets. Wolfers and Zizewitz (2006) have shown that, under a unique logarithmic utility function, relative asset prices equal the mean belief of investors. This statement is called the no-trade theorem. It means that no trading or alpha generation should occur unless the mean belief of the investors changes. It formalizes the efficient market hypothesis. Each investor trading must thus possess an information advantage, whether through superior information gathering or information processing. 4.1.1 Market inefficiencies A second possibility of generating positive alpha exists. Indeed, markets are not efficient all the time. Inefficiencies exist because of: • Uninformed outsiders or noise traders. These are investors who do not own or have not processed all publicly available information and take suboptimal investment decisions with respect to the no-trade theorem. • Hedgers. These are investors who actively take positions in order to reduce risk, voluntarily giving up alpha or even accepting negative alpha. • Investors with different utility functions. Such investors violate the unique logarithmic utility function assumption of the no-trade theorem, for example, due to – different reference currencies, – alternative investment time horizons or – specific liabilities to be matched. If an investment manager is able to forecast such temporary anomalies consistently, called arbitrage inefficiencies, he or she can consistently generate positive alpha. A possible arbitrage inefficiency forecast could be to expect that the Japanese central bank, in order to keep the yen versus the US dollar exchange rate around 110, will actively trade large amounts of USD/JPY. Taking active long or short positions in the US dollar versus the yen, depending on whether the exchange rate is above or below the target exchange rate of 110, would allow the forecast to be transformed into positive alpha. This forecast is composed of (i) an information forecast, that is the central bank expected intervention, and (ii) an impact of that information forecast on the currency asset price, that is the reversion of the exchange rate to the central bank’s target. Market returns are a zero sum game. The return of the market portfolio is equal to the capitalization weighted sum of all asset returns. Thus, if one asset outperforms the market portfolio, another one underperforms it. If investors hold underperforming assets in

Judgmental Approaches

39

excess to the respective market capitalization, then opportunities exist to generate positive alpha. Interestingly, one of the largest classes of investment products, that is indexed or passive investment products, is investing inefficiently according to the semi-strong form of the market efficiency hypothesis. As described in Chapter 15, a passive investment product replicates the holding, and thus the performance, of a given index or benchmark. This index is an approximation of the market portfolio. Index providers adjust from time to time the constituents and/or the weights of the assets in the index. These changes are made available to investment managers before they become effective. As passive investment products aim to replicate the performance of an index, they will only implement the index changes as of the date when they become effective. This lag in exploitation of publicly available information is suboptimal with respect to the efficient market hypothesis and thus provides alpha generating opportunities for others. These market inefficiencies have led to a whole class of investment products, called enhanced indexing products.

4.2 UNDERSTANDING ASSET RETURNS Asset prices and their changes over time are at the core of forecasting markets. Academic research has focused for more than half a century on defining models of markets and asset prices (Campbell, 2000; Cochrane, 2005). These models are tools for structuring the forecasting processes and are based on a sound foundation of assumptions. They allow streamlining of the process of thought that leads to asset return forecasts. All market and asset pricing models are based on assumptions. These assumptions need to be well understood and consistent with the developed forecasting framework. 4.2.1 Modeling asset prices I am interested in forecasting future returns, respectively future prices of assets. It is therefore natural to start by understanding the structure of current asset prices. In the absence of arbitrage opportunities, a stochastic discount factor (SDF) exists that relates future cash flows of an asset to its current price, for all assets in the economy (Cochrane, 2005). This means that the price Pt of an asset or portfolio at time t is determined by Pt = Et [Mt+1 · Xt+1 ]

(4.1)

where Et [•] is the expectation conditioned on information available at time t, Xt+1 are the random cash flows of asset a at time t + 1 and Mt+1 is the SDF or pricing kernel. The SDF is a random variable that is always positive. The challenge is to understand the economic structure that defines the SDF. If there is no uncertainty, then the SDF is the inverse of the risk free rate, that is Mt+1 = 1/(1 + RF ), which discounts the expected cash flows of tomorrow into values of today. Consider an asset with a price of $110.0 at time t and assume that it will generate a future cash flow of $125 with a probability of 75 % and a cash flow of $85 with a 25 % probability. Then the SDF making the current price consistent with the expected future cash flows is equal to Mt+1 =

$110 = 0.9565 75 % · $125 + 25 % · $85

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Positive Alpha Generation: Designing Sound Investment Processes

4.2.2 Defining the risk premium of an asset I define the risk premium of an asset as the expected excess return over the risk free rate. A risk premium only exists when the return of the asset co-varies with the return of the market portfolio or the aggregated marginal rate of substitution in consumption that makes the risk not diversifiable. Idiosyncratic risk, that is diversifiable risk, uncorrelated with the stochastic discount factor does not generate a risk premium. Let Rt+1 be the return of an asset or portfolio between t and t + 1, and RF,t the risk free rate at time t. Then the risk premium Qt+1 can be derived from Equation (4.1) such that Qt+1 = Et [Rt+1 ] − RF,t = −

covt (Mt+1 , Rt+1 − RF,t ) Et [Mt+1 ]

where covt is the covariance between the SDF and the asset’s excess return conditioned on the information available at time t. It is therefore possible to generate alpha by forecasting the covariance of the portfolio’s excess return with that of the market portfolio. Consider an expected market return of 7.5 %, that is Et [Mt+1 ] = 1/(1 + 7.5 %) ≈ 0.93 and a correlation of 0.75 between the market portfolio and the target portfolio, resulting in covt (Mt+1 , Rt+1 − RF,t ) ≈ −0.05. Then the expected risk premium equals 4.65 %. The SDF is related to but is not identical with the market portfolio. Generating alpha through forecasting risk premiums is called exploiting systematic risk. It is based on two forecasts, that is: • determining the appropriate SDF model and • forecasting the return of and the covariance with the portfolio’s expected excess return of the SDF. Many examples of assets exist where the estimated volatility implied by the SDF, that is the risk premium, is surprisingly large. Indeed, in order to reconcile the return on equity with the one on government bonds, investors must have implausibly high risk aversion according to economic models. Mehra and Prescott (1985) first drew attention to this phenomenon and named it the equity premium puzzle. Some authors explain this phenomenon by a selection bias in the data used for the estimation. Several other plausible explanations have been proposed in the academic literature, but a generally accepted solution remains elusive.

4.2.3 Stochastic discount factor based asset pricing models Most of the practical and qualitative forecasting oriented users ask themselves how the theoretical SDF model can help formulate market forecasts. Some of the most common asset pricing models can be interpreted as special cases or variations of the general SDF framework. Thus, the general idea of risk premiums can be applied. It is therefore important to be aware of the most common SDF based pricing models. Indeed, identifying appropriate SDFs may lead to identifying risk factors rewarded by the market, when forecasted correctly.

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41

Discounted cash flow model If an asset has a constant expected return, then its price is a linear function of its expected future cash flows. Equation (4.1) can be rewritten as Pt =

Et [Pt+1 + Ct+1 ] 1+R

(4.2)

where Ct+1 is cash flows at time t + 1 and R is the expected rate of return, also called the discount factor or cost of capital associated with the asset. Solving Equation (4.2) for Pt gives Pt =

∞  v=t+1

Et [Cv ] (1 + R)v

(4.3)

The discounted cash flow model (DCF) states that the price of any asset at a given point in time equals the discounted value of its expected future cash flows. This model is common in bottom-up equity valuation as well as fixed income forecasting approaches. The assumptions underlying the DCF model is that the associated discount rate is constant. A special case of the DCF model arises when the expected rate at which the cash flows grow over time is constant, that is Et [Ct+1 /Ct ] = 1 + G. In this case, Equation (4.3) simplifies to Et [Ct+1 ] Pt = R−G This model is called the dividend discount model (DDM) and finds wide application in equity valuation and return forecasting models. Capital asset pricing model The capital asset pricing model (CAPM) defines the required rate of return of an asset as a linear function of the market portfolio’s excess return over the risk free rate. It was introduced by Sharpe (1964) and states that the expected return of each asset can be expressed by Et [Rt+1 ] = RF,t + β · (Et [RM,t+1 ] − RF,t ) where RF,t is the risk free rate at time t, Et [RM,t+1 ] is the expected return of the market portfolio described in Chapter 2 and β is the exposure or sensitivity of the asset to the market portfolio. The CAPM assumes that there exits a single market portfolio based on all traded assets weighted according to their market capitalization. It also assumes that all investors have a concave utility function; that is their optimal portfolios are always composed of a linear combination of the risk free asset and the market portfolio. The CAPM can be interpreted as a special case of Equation (4.1). In the context of the CAPM, the risk premium is defined by

vart (RM,t+1 ) covt (Rt+1 , RM,t+1 ) · − or Qt+1 = var(RM,t+1 ) Et [RM,t+1 ] Qt+1 =

covt (Rt+1 , RM,t+1 ) · (Et [RM,t+1 ] − RF,t ) var(RM,t+1 )

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Positive Alpha Generation: Designing Sound Investment Processes

Unfortunately, within the context of SDF, the CAPM can be rejected (Smith et al., 2003) as it ignores significant priced sources of risk. A two-factor SDF model with consumption growth and inflation as the two observable factors is preferred. Nevertheless, the CAPM is often used and is useful as a reasonable first approximation. Arbitrage pricing theory In 1976 Ross introduced the arbitrage pricing theory model (APT). It states that the price of any asset is defined such that no arbitrage is possible. The price of an asset is defined by the price of a portfolio replicating the characteristics of the asset. Each of these characteristics can be modeled by an SDF. Equation (4.1) can be used to derive the expected return as Et [Rt+1 ] = RF,t +



βf · Et [Ff,t+1 ]

f

or loadwhere Rf represents the return associated with factor f of the asset, βf the exposure
ing to that factor and RF,t the risk free rate. The SDF is defined as Mt+1 = f βf · Rf,t+1 ; thus the risk premium is defined by Qt+1 =



βf · covt (Rt+1 , Rf,t+1 )

f

The following factors have been identified as some of the most relevant factors for explaining equity returns: • • • • • •

unexpected changes in inflation, surprises in GDP, indicated by industrial production, changes in investor confidence, surprising shifts in the yield curve, changes of the steepness of the yield curve or oil prices.

The APT assumes that the market participants will exploit any arbitrage opportunity, until the price of any asset will no longer allow for arbitrage. Furthermore, a linear relationship between the return and the different factors as well as the absence of transaction costs is assumed. 4.2.4 Revising the decomposition of return Given an SDF based asset pricing model, the return of a portfolio can be decomposed as R = RF + Q + α

(4.4)

where RF denotes the risk free rate, Q the risk premium of the assets in the portfolio and α the return added by the skills of the investment professional in excess of the risk premiums. Equation (4.4) does not contradict the return decomposition R = RF + (RB − RF ) + α defined in Chapter 2. Indeed, the risk premium Q can be interpreted as the return

Judgmental Approaches

43

due to the skill of developing the underlying model and defining its SDF or pricing kernel. If the benchmark portfolio relates to the market model used, the risk premium is equal to the benchmark portfolio’s excess return over the risk free rate. The third component in Equation (4.4), that is α, is due to exploiting specific or idiosyncratic risk actively. As most modern forecasting models are based on a market model, the portfolio return should be decomposed based on Equation (4.4). Depending on the market model, the risk premium may or may not be constant over time. Forecasting the risk premium relates to forecasting the volatility of the portfolio and its correlation with the SDF underlying the used asset return model. It can be interpreted as the excess return expectations of the market, meaning that Et [R by the market participants] = RF + Q

(4.5)

I distinguish between four types of assets with respect to their risk premium. Greer (1997) has proposed a similar classification. (1) Risk free assets (RF). The expected return of risk free assets for a given time horizon is equal to the realized total return over that same horizon. These assets, by definition, do not exhibit any risk, that is any potential loss. No risk premium has to be paid to the investor for holding the assets. Therefore, RRF = RF and QRF = 0. (2) Capital assets (CA). A capital or internal rate of return asset is a claim against future cash flows. Its price is the present value of the expected future cash flows. As it is only interesting to hold such an asset if future cash flows are expected, its expected total return is positive and larger than the risk free rate. The expected excess return above the risk free rate compensates the owner of the assets for the uncertainty of the future cash flows. I will distinguish between two types of capital assets, that is: • debt assets, where future cash flows are known, the only unknown parameters being whether or not the cash flows are actually paid out, that is whether the issuer of the claim defaults on its obligations or not, and • equity assets, where future cash flows are unknown and determined by the unknown cash flows generated by the issuer of the claim. Therefore Et [RCA ] = RF + QCA and QCA > 0. (3) Consumption and value storing assets (CV). The prices and the total returns of consumption or value storing assets are determined by supply and demand. Two types of such assets exist, that is value storing assets, like gold or art, and consumption assets, like rice or black beans. If supply and demand are in equilibrium over a given time period, no return can be expected. Sometimes the return over such a period is negative due to storage costs. The investor is not compensated for holding consumption or value storing assets without a forecast on future prices. Therefore, Et [RCV ] = 0. (4) Actively managed assets (AM). The total return of actively managed assets is determined by the return of a portfolio of assets, that is by the skill of the investment manager managing the portfolio. It is an open question whether or not the investor is rewarded for holding actively managed assets. An investor holding an actively managed asset expects that Et [RAM ] > RF , that is that he or she is rewarded for taking the risk of selecting an active asset manager. Note the similarity between actively managed assets and equity capital assets. In both cases future cash flows, denoted by alpha in the case of actively managed assets, depend on the skills of managers.

44

Positive Alpha Generation: Designing Sound Investment Processes Table 4.1 Classification of the most common assets according to their expected total returns structure Asset

Nature of total return

Cash Commodities Coupon paying bonds Currency forwards Equity indices Hedge funds Property Stocks

RF CV CA/debt CV CA/equity AM CA CA/equity

Table 4.1 classifies some of the most common assets into these four categories.

4.3 FORECASTING ASSET RETURNS 4.3.1 Approaches I distinguish between two basic approaches for forecasting asset and portfolio returns, that is: • top-down approaches and • bottom-up approaches. Top-down approaches are characterized by the fact that they start with a big picture and then break it down into individual components. A top-down approach example would be to start at the political level, that is the overall direction of a government (socialist, conservative, democratic, etc.). From this tendency, impacts on different areas of the economy would be derived (labor market, housing market, taxes, government spending, etc.). Further down the line, these tendencies would become translated into expected actions (interest rate hike, tax increase by 5 %, flat budget spending, etc.). Expected market data, conditional upon current information, are forecasted. Finally, these forecasts would be compared with the market’s expectations and total return forecasted for the different industry sectors of the market (consumer goods, financial services, industrials, utilities, etc.). Bottom-up approaches start by forecasting individual pieces of information. These pieces of information are then aggregated to derive relative forecasts of market expectations at the asset or portfolio level. A typical example would be an approach based on equity analysts’ research. Individual analysts study different companies and forecast total returns for each of them individually. This could be done through forecasting expected cash flows and discounting them at the respective cost of the capital rate. Next, the individual company forecasts would be aggregated at the industry sector level (consumer goods, financial services, industrials, utilities, etc.). Return expectations could be derived. Different approaches can be used to derive return forecasts for the same assets. The investment management industry shows a trend to identify individual investment products

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45

with the underlying approach used. Investment management houses classify themselves as either being purely top-down or bottom-up oriented. I believe that each investment manager should identify his or her forecasting strengths and use either approach or a combination of both, the goal being to achieve the highest possible consistent information ratio. 4.3.2 The forecasting process A good investment process clearly defines its areas of focus according to the available skill sets. Minahan (2006) calls this area of focus the investment philosophy and the available skills the alpha thesis. Figure 4.1 illustrates the general structure of the forecasting part of an investment process. First, and foremost, market expectations need to be derived. Market expectations express the consensus or market expected returns. They serve as the basis against which information is forecasted. Information only has forecasting power if it differs from information already expected by the market. Indeed, the market efficiency hypothesis states that all available information is already contained in the asset prices. The second step of a forecasting process is to process and forecast information, and represents the value added, that is the application of the investment manager’s skills on the opportunity set. The information is then translated in a third step through either a quantitative or a qualitative subprocess into asset return expectations. Finally, in a step that is very often neglected, the forecasts should be checked for consistency and coherence. Deriving market expectations

Information processing and forecasting

Translating information into returns

Yes

Consistent & coherent

No

FIGURE 4.1 Overall structure of an asset return forecasting process

4.3.3 Deriving market expectations According to the efficient market hypothesis, the asset prices reflect the information processed by the market participants, that is the market’s expectations as indicated in Equation (4.5). Therefore, the first step of any forecasting approach must be to determine and understand the market’s expectations. Consider, for example, the interest rate market. Assume that the ten-year spot rate is at 4.55 %. A naive forecasting approach would be to expect the ten-year spot rate to rise to 4.75 % over the next six months, thus expecting a positive alpha of 15bp by selling a ten-year zero bond against a six-month cash deposit at 1.25 %. Unfortunately, this forecast is not sound as it neglects the expected ten-year spot rate in six months time, that is the six-month forward rate of a ten-year zero bond. If this rate is already 4.94 %, then no positive alpha will be generated by the yield forecast and the associated trades. Deriving market expectations is a task in its own right. Only very rarely can market expectations be derived as easily as with forward interest rates, inflation rates or implied volatilities. Often street consensus is the best available approximation.

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Positive Alpha Generation: Designing Sound Investment Processes

4.3.4 Information processing and forecasting Structure of information Processing and forecasting information is the key element of a sound investment process. Information can be structured according to three dimensions: (1) Publicly available versus private information. The wider information is disseminated, the higher the probability that it is already fully reflected in the asset prices as assumed by the efficient market hypothesis. (2) Backward looking versus forward looking. Backward looking information is derived purely from factual information at the time the forecast is being made. Forward looking information is based on assumptions or hypotheses. Different market participants will make different assumptions and thus come to different conclusions. Assumptions leading to market expected asset prices are called the market assumptions. (3) Objective versus subjective information. Objective information is derived or deduced in a manner that is repeatable and independent of the individual doing the deduction. Subjective information includes a personal and perceptive aspect that makes it unique. The goal of this structuring scheme is twofold. First, it allows verification that information forecasting methods used are of a different nature, diversifying the approaches. This is important in order not to pull all cards from the same deck. Second, it allows methods to be used that tend to be forward looking, subjective and based on private information approaches. The impact of such methods when forecasting correctly is very great as the probability of someone else deriving the same conclusion is small. On the other hand, getting these methods to produce correct forecasts is not easy. Classifying information using different perspectives Information used during the forecasting process can be classified along four dimensions: • Exogenous. Information is associated with its characteristics or source, for example macroeconomics or technical analysis. • Endogenous. Information is classified along its information contents or risk factors, for example economic cycle or liquidity premium. • Asset or asset class based. Information is associated with the underlying asset class, for example German equities or USD bonds. • Decision based. Information is associated with the investment decisions affected, for example JPY bonds outperforming Japanese equities. To obtain a consistent set of forecasts, only the first two dimensions used to classify information should be applied. Indeed, asset or decision based classification schemes may lead to certain information, for example growth expectations, being used for a certain asset or decision and different information, such as inflation expectations, which are inconsistent with growth expectations, being used for different assets or decisions. As shown in Figure 4.2, information can be classified by a combination of exogenous and endogenous dimensions.

Price dynamics

Real interest rates Yield curve steepness

Oil prices Inflation linked bond yields House prices

Equity implied volatility Historical squared returns

Earnings

47

P/E ratio Dividend yield

Producer prices

Changes in earnings estimates

Macroeconomics

Exogenous dimension

Judgmental Approaches

GDP deflator

Consumer prices Producer prices Unemployment rate

Changes in money supply

Inflation pressure

Volatility

Growth

Endogenous dimension

FIGURE 4.2 Example classifying information along both endogenous and exogenous dimensions

Seven key exogenous dimensions can be identified to classify information: (1) Macroeconomics/fundamentals. The effects of macroeconomic changes and their impact on asset returns are analyzed. Central bank policies play an important role. A macroeconomics view is generally long term oriented and forecasts economic cycles and changes in economic cycles. (2) Political. The effects of political and regulatory decisions and interventions are studied. Fiscal policy analysis and forecasting play a key role when taking a view based on a political angle. (3) Earnings. Company earnings are analyzed and aggregated. The earnings perspective is based on two forecasting components, which are (i) forecasting cash flows and (ii) estimating the discounting rate, that is the cost of capital of the analyzed company, industry or region. (4) Valuation. Ratios, like the price to book, price/earnings, that exhibit mean reverting properties are considered. Forecasting in the context of valuation is equivalent to determining (i) the long term mean of the ratio, (ii) whether the current ratio is lower or higher than the long term mean and (iii) whether there are arguments as to why, over the next time period, the current ratio should revert closer to the long term mean. (5) Sentiment. Market behavior is analyzed and extrapolated. Various indicators, like the put/call ratio, are analyzed and their causality on asset returns forecasted. Sentiment views are usually a combination of qualitative analysis determining the sentiment indicators studied combined with a quantitative approach to relate changes in the indicators to forecasted asset returns. (6) Time series analysis. Techniques, like quantitative models or technical analysis indicators, are used to extract pattern like behavior and derive asset return expectations. I usually distinguish between forecasting models, that is models that calculate the expected price or return based on past information, and models that translate time series information into return information.

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Positive Alpha Generation: Designing Sound Investment Processes

(7) Manager selection. The behavior of investment managers and their forecasting capabilities in different market environments are studied. The manager selection view is crucial when forecasting in the context of fund approaches, which are common in the hedge fund industry. Experience and different studies (Naik et al., 2007) show that the most common endogenous dimensions, also called risk factors, are: (1) Economic or business cycle. Information in this category allows the state of the overall economy to be asserted, for example GDP growth or unemployment rates. (2) Inflation pressure. This category includes information related to changes in prices, such as consumer and producer price data. (3) Real growth. Whereas economic cycle information shows the overall state of the economy from a top-down approach, real growth information measures the economy from a bottom-up way. Information classified in this category includes real yields, corporate earnings growth or dividend yields. (4) Risk premium. Information in this category measures the return investors require for holding a certain asset. Typical information in this category is the steepness of the yield curve or the expected excess equity return over the risk free rate. (5) Mean reversion. Different sets of information, especially indicators, admit a long term equilibrium. Information classified as mean reversion indicates the current state versus its long term equilibrium and the expected speed at which the state will move toward the equilibrium. (6) Momentum. Momentum based information relates past performance to expected future performance. Trend indicators fall into this class of information. It is mainly populated with quantitative rather than qualitative information. (7) Volatility. Volatility information measures the uncertainty of information. Quantitative volatility indicators are usually obtained by squaring information.

4.3.5 Translating information into returns Until now, I have expressed the goal of forecasting as deriving total return expectations for the considered assets. There are different ways that can be used to express these expected total returns. It is my experience that, whatever way is used, it should follow Einstein’s quote ‘make it as simple as possible, but not simpler’. Forecasting is already difficult enough, so there is no need for additional complexity in this translation process. Nevertheless, every asset total return formulation approach has to fulfill four characteristics: (1) (2) (3) (4)

It It It It

has to be quantified. has to be objective. has to have a time horizon associated with it. has to be related to the mechanism used for transferring the forecasts into portfolios.

Six categories of return forecasts can be identified, listed in decreasing order of their complexity: • Absolute total return. In the expected total return formulation approaches, the absolute total return, in percentage terms over the considered time horizon, is forecasted. Excess

Judgmental Approaches

•

•

•

•

•

49

returns to the risk free rate, the long-term average or an equilibrium rate, as used in the Black–Litterman model (Litterman, 2003), are also considered to be absolute return approaches. For example, US equities will return 12.4 %, Japanese equities 7.4 % and European equities 9.3 % over the next six months are a set of absolute total return forecasts. Relative total return. The expected relative total return between two assets is forecasted. For example, US equities will outperform European equities by 3.1 % and European equities will outperform Japanese equities by 1.9 % in EUR terms. The absolute level does not matter. Most transfer algorithms do not need absolute levels to function properly. Return ranges. Instead of expressing the expected total return as a point forecast, a continuous range is specified. The range may or may not be associated with a confidence level. For example, with a 95 % probability US equities will have a total return between 10.4 and 14.4 %. If no confidence level is specified, the forecasted ranges define the lower and upper bounds possible. In addition, an associated mean, median or mode may be forecasted. Scores. Scores are an abstract form of total returns. They specify a total ordering between the assets as well as a relative magnitude between the expected returns of the assets: for example US equities 124, Japanese equities 74 and European equities 93, ranking the US ahead of Europe and Europe ahead of Japan. In addition, the outperformance of the US against Europe is expected to be larger than the outperformance of Europe against Japan, because 124 − 93 > 93 − 74. Rankings. Rankings are a special form of scores, defining a total order on the expected asset returns. In addition no statement is made about the magnitude of relative returns between consecutively ranked asset returns. A possible ranking for the three equity markets could, for example, be (1) the US, (2) Europe and (3) Japan. Pairs. Pairs, also called investment or market views, indicate which of a pair of assets outperforms the other. For example, US equities will outperform European equities and European equities will outperform Japanese equities. No explicit forecast is given between US equities and Japanese equities. Pairs are a form of defining a partial order between asset class returns.

Each of these forecasting approaches can be combined by specifying a degree of confidence, also called the expected hit ratio (HR), in the specified forecast. The degree of confidence expresses the probability of the forecast being correct, that is HR = Pr{forecast is correct}

(4.6)

With certain assumptions about the probability distribution of forecasts being correct, Equation (4.6) can be solved and the confidence level calculated. As shown in Chapter 2, if absolute or relative expected returns are forecasted and assuming that these returns follow a normal distribution, HR can be computed explicitly. For example, based on historical asset returns of US equities, HR = Pr{return
12.4 %} = 38.4 % using monthly historical return data of the S&P 500 index data between December 1967 and December 2007 to estimate the mean µ = 7.95 % and standard deviation σ = 15.11 %.

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Positive Alpha Generation: Designing Sound Investment Processes

The second important output of each forecasting approach is the time horizon of each forecast, that is the time-frame over which it will materialize. Time horizons are usually defined explicitly in weeks, months or quarters. However, they may also be defined by an external event, like an index attaining a certain value or interest rates being raised by the European Central Bank (ECB). As most portfolio construction approaches, as presented in Chapter 9 and Chapter 10, consider the set of all forecasts combined, all forecasting time horizons need to be identical. This is in fact not a real restriction, as any forecast over a longer time horizon can be subdivided into identical consecutive forecasts over smaller time horizons. The choice of which forecasting output to chose is determined by: (i) the forecasting skills as defined by the forecasting approach and (ii) the surrounding investment process, that is the transfer or portfolio construction mechanism used. For example, if classical Markowitz portfolio optimization is to be used, absolute return forecasts are required. On the other hand, the optimal risk budgeting approach works well with scores or pairs and associated confidence levels. Optionally, probabilities may be associated with the different scenarios. When using a multiple scenario approach, the different scenarios may be specified explicitly. However, they may also be generated using Monte Carlo simulation techniques. The latter assume that the forecasts are the parameters of random variables rather than actual values. 4.3.6 Checking consistency and coherence The last stage of the forecasting process is to check the output for: • consistency, that is the absence of contradictions, and • coherency, that is the absence of ambiguities from interference. One of the most common checks for finding contradictions is to visualize the partial order introduced by the forecasts formulated as a partial order graph (Davey and Priestley, 2002). Vertices represent asset classes and edges the relationship
between any pair of asset classes. Figure 4.3 shows a sample partial order graph from forecasting currency returns against the USD. Contradictions are identified by edges going from the right to the left in the graph. Ambiguities are indicated by assets that are not connected by edges or a path of edges. The coherence of a set of forecasts can be derived by explicitly formulating all hypotheses and showing the logical deductions applied, using, for example, formal logic theory (Smullyan, 1995) to translate hypotheses step by step into market forecasts. Ambiguities can be determined and corrected. The coherence of a set of forecasts can be tested by studying the negation of the forecasts, that is assuming that the forecasts are incorrect and deducing the assumptions leading to the negated forecasts. Sensitivity analysis is an additional very powerful tool to test for coherence. The significance of changes in the forecast are analyzed with respect to changes in assumptions. If small changes in the assumptions imply large changes in the resulting forecasts, a coherence problem may exist with respect to the derivation of the forecasts. Similarly, trying to answer

Judgmental Approaches

51

GBP

NZD

AUD IV

CHF

IV

IV

IV

USD IV

EUR

IV

IV

JPY

HKD

FIGURE 4.3 Consistent and coherent partial order graph of a set of currency forecasts

the question, ‘What has to change in the assumptions in order for the forecasts to change significantly?’, very often allows additional confidence to be gained in the consistency of the forecasts. Another tool used to analyze coherence, especially with respect to past events, is to calculate the correlation between historical realizations of the forecasts. If the correlation is high, the probability that the forecasts will be correct or wrong at the same time is high. Conversely, if the correlation is negative, history shows that in the past either of the forecasts was correct, but that the probability of both forecasts being correct at the same time is very low. For example, the historical correlation between the forecasts that US equities as well as UK equities will have a positive return over the next month is 0.48. This number is positive, but not very high, meaning that the forecast that both US and UK equities will have either a positive or a negative return is reasonable. However, there is a certain nonnegligible probability that the return of the two markets will move in different directions, but, as analyzed by Boyer et al. (1999), correlations, that is the parameters of the underlying distribution, do change over time. Therefore correlation based coherence results must be used with a certain degree of caution. Consider the currency pairs USD/GBP and CHF/USD. These two currency pairs, using five years of monthly historical returns, have a correlation of −0.70; that is the probability that a portfolio being long USD against GBP and long CHF against USD will exhibit a significant return is small. Indeed, consider a portfolio with a −12.9 %4 long USD position against GBP and an 11.4 % long CHF position against the USD. Over the last five years, the first position resulted in an annualized return contribution of −0.56 % whereas the second currency pair returned one of 0.65 %, based on monthly rebalancing. The net portfolio contribution of these positions is 0.09 %, consistent with the high negative correlation, suggesting that the returns from the two positions cancel themselves out most of the time.

4.4 EXAMPLE To illustrate the presented forecasting concepts, consider a simple forecasting process based purely on fundamental or macroeconomic analysis. First, I define the investment universe as the three major developed economies, that is the United States, modeled by the S&P 500 equity index, the European Union, modeled by the German DAX equity index, and Japan, 4

The weights have been normalized relative to the currency pairs’ volatilities.

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Positive Alpha Generation: Designing Sound Investment Processes

modeled by the Nikkei 225 equity index. The skills, that is the forecasts, are represented by a score for each region, expressing an ordering between regions and relative magnitudes of the expected returns. 4.4.1 Defining the underlying model and assumptions I use a model based on the two indicators GDP growth and inflation. In the context of an SDF model I would define two stochastic discount factors, one based on GDP growth and the second on inflation. This would lead to a two-factor APT type model. As the forecasting process I design is qualitatively oriented, I refrain from formalizing, estimating and backtesting the underlying market model used. I assume that the semi-strong efficient market hypothesis holds; that is I am not focusing on finding market frictions or arbitrage opportunities. I consider a fixed three-month time horizon for the formulated forecasts. 4.4.2 Deriving market expectations I rely on bond yields as well as the TIPS5 market to derive current market expectations for GDP growth and changes in inflation. When no TIPS market exists, as is the case in Japan, I use sell-side consensus forecasts of inflation as a proxy for market expectations. Table 4.2 illustrates a sample set of figures and compares them to sell-side economic consensus forecasts. Table 4.2 Illustrative sample of current, market expected and consensus forecasts for GDP growth and CPI change for a six-month time horizon Region GDP growth United States European Union Japan CPI change United States European Union Japan

Current (%)

Market implied (%)

Consensus (%)

2.1 1.2 3.5

2.36 1.07 n/a

2.35 1.10 3.75

4.3 2.5 0.6

4.32 2.14 n/a

4.35 2.60 0.62

Source: sample data and forecasts.

4.4.3 Processing and forecasting information To derive my forecasts for the two indicators studied, I use a scorecard based approach. I analyze the three factors per region shown in Table 4.3 and forecast their impact on growth and inflation. I denote by ↑ a small and by ↑↑ a large positive expected impact on asset returns. A → denotes that the considered factor will not have an impact on the forecasts. Down arrows ↓ indicate negative expected impacts. 5

Treasury Inflation Protected Securities, that is inflation linked government bonds.

Judgmental Approaches

53

Table 4.3 Macroeconomic indicator

Scorecard used to forecast and process information Growth

Inflation

United States Monetary policy Housing markets Labor markets Consolidated

↓ → ↑↑ →

↓↓ ↑ → ↓

European union Monetary policy Housing markets Labor markets Consolidated

→ → ↓ ↓

↓ → ↑ →

Japan Monetary policy Housing markets Labor markets Consolidated

↑↑ ↑ → ↑↑

→ ↓ → →

Arguments Tightening of the monetary policy is expected based of the Fed’s comments. This will mainly impact short rates and thus have no immediate impact on housing prices. The current strong trend in labor markets will last for another 2 to 3 months until the earnings reporting seasons starts. The ECB will be able to stabilize inflation within the predefined ranges. As the housing market is only loosely coupled to short term rates, the current monetary policy will not impact the housing market. Labor market fiscal policy will have a negative impact on growth, but increase prices. For the first time in several years, the low interest rate policy will have a strong impact on the Japanese economy, without causing any issues on the inflation side. The housing market will be strong and not driven by speculations. No changes on the labor market side are expected.

For each forecast, I indicate the key qualitative arguments that lead to my conclusion. Then I study the resulting scorecard to derive forecasts for growth and inflation for the three markets. This derivation is based on averaging the expected impacts, rounding and adding a subjective bias. The obtained scorecard is shown in Table 4.3. I have used a mix of objective and subjective information. My focus is forward looking as I assume efficient markets. The derivation of the scorecard is purely based on public information and the interpretation thereof. Table 4.4 Rules describing the relationship between changes in GDP and CPI on market returns Growth

Inflation

Score

Growth

Inflation

Score

↑/↓ ↑/↓ ↑↑ / ↓↓ ↑↑ / ↓↓ ↑/↓ ↑↑ / ↓↓

↑/↓ ↑↑ / ↓↓ ↑/↓ ↑↑ / ↓↓ → →

3/−3 6/−6 5/−5 8/−8 2/−1 2/−1

↑/↓ ↑/↓ ↑↑ / ↓↓ ↑↑ / ↓↓ → →

↓/↑ ↓↓ / ↑↑ ↓/↑ ↓↓ / ↑↑ ↑/↓ ↑↑ / ↓↓

0/0 −2/3 2/−2 1/−1 2/−3 3/−4

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Positive Alpha Generation: Designing Sound Investment Processes

4.4.4 Translating information forecasts into asset return forecasts

Tools and models

Experts

Information

The last step of the forecasting process translates the consolidated results from the scorecard into scores for the three equity market regions. To do so, I use the derivation rules shown in Table 4.4 and then compute the following forecasts:

FIGURE 4.4 approach

Information (forward looking, public, subjective)

Current GDP Current CPI TIPS market

Constructing scorecard

Deriving expected GDP and CPI changes

Argumentation

Aggregating scores

Rules

Deriving return expectations

Yes Consistent No

Example forecasting process based on macroeconomics using a scorecard based

• Unites States score: −3 • European Union score: −1 • Japan score: 2

4.4.5 Checking consistency I implement a simple consistency check based on a historical frequency analysis. For each of the possible orders of the three markets, United States, European Union and Japan, I calculate the historical occurrence frequency using rolling semi-annual data between December 1997 and December 2007. In addition, I calculate the frequency of the outperformance of the preferred markets against the market ranked second being higher than the outperformance of the second ranked market against the least preferred one. The results are shown in Table 4.5. Based on the forecasted scores, I see a historical frequency of 13.2 %, respectively 56.3 % when taking into account the relative magnitude. These numbers show that my forecasts are not the most likely to happen. On the other hand, the relative magnitude, assuming a correct order, was above 50 % over the last decade. From these numbers, I conclude that I need to double-check my forecasts, especially those in the Japanese economy, before confirming

Judgmental Approaches

55

the macroeconomics forecast on the world economies. Figure 4.4 summarizes the developed forecasting process. Table 4.5 Historical occurrence frequency of the forecasted order between the United States, European Union and Japan markets Forecasted order

United States–European Union–Japan United States–Japan–European Union European Union–United States–Japan European Union–Japan–United States Japan–European Union–United States Japan–United States–European Union

Frequency of order (%)

Frequency of magnitude (%)6

19.8 12.4 28.9 15.7 13.2 9.9

37.5 53.5 34.6 73.7 56.3 75.0

Source: data sourced from Bloomberg Finance LP, author’s calculations.

6 Frequency with which the outperformance of the preferred market against the market ranked second being higher than the outperformance of the second ranged market against the least preferred one.

5 Quantitative Approaches for Forecasting Markets Econometric theories as well as statistical modeling are at the heart of quantitative forecasting. I distinguish between two types of quantitative models: • unconditioned or pure forecasting models and • conditioned forecasting or explanatory models. Pure forecasting models use time series data up to time t to forecast market returns between time t and t + 1. Examples of such models are moving average based models like GARCH processes. Explanatory models describe a relationship between market returns between time t − 1 and t and data available up to time t. Explanatory forecasting models do not have direct forecasting power. They explain market returns using, for example, volatility, inflation or real interest rates. The most well known explanatory model is the capital asset pricing model (CAPM) described in Chapter 3. The portfolio return is expressed as a function of the risk free rate, the market portfolio return and the exposure to the market portfolio as specified by the β factor exposure. Building a quantitative model is, to a large extent, making educated choices. Which choices lead to the most appropriate forecasting model for a given investment product is in the hands of each investment manager.

5.1 BUILDING A QUANTITATIVE FORECASTING MODEL Building quantitative models is equivalent to trying to approximate reality as illustrated in Figure 5.1. First markets are observed. These observations introduce a first approximation. For example, observing asset prices at the end of each month ignores intra-month changes. Depending on the to-be-built forecasting model, this may be a reasonable approximation. Next, based on the observations, assumptions are made on the structure of the markets and thus on the forecasted returns. For example, it may be assumed that market returns can be explained by two factors, the market portfolio return and the domestic inflation rate. These assumptions introduce a second layer of approximation as asset returns may be influenced by other factors. The assumptions lead to the definition of a formal model. For example, the return of an asset or portfolio may be expressed by a linear combination of the market portfolio return and the domestic inflation rate. The residual or unexplained return is assumed to behave like white noise. Before being able to use a model for forecasting purposes, I estimate the parameters of the designed model. In my example, this would be estimating the expected return contribution due to the market portfolio return and the domestic inflation rate change factors. An ordinary least squares parameter estimation algorithm could be used. The introduction of approximations in the forecasting model is not an inherent disadvantage if the approximations made are reasonable within the context in which the model and the resulting forecasts will be used. For example, if monthly asset returns are to be

58

Positive Alpha Generation: Designing Sound Investment Processes Financial markets

Observations

Structure/ Assumptions

Formal model

Model parameters

Forecasts

Degree of approximation Equities Bonds Currencies

End of day asset prices Index levels Returns

Randomness Multivariate Independence normal random variable

Mean Variance Correlation

Forecasted return

FIGURE 5.1 Approximations introduced when building a quantitative forecasting model

forecasted, it is a reasonable approximation to use only monthly historical return data to estimate the model parameters. 5.1.1 The model building process Building a quantitative forecasting model requires the combination of skills from three key areas: • Theory. A sound theoretical foundation as well as an associated model structure must be selected. • Parameterization. The explanatory variables, often called factors, must be defined. • Estimation. The model parameter values must be calculated or estimated, for example, using historical data. This skill set needs to be applied as illustrated in Figure 5.2. When designing a new model, it is very easy to become overwhelmed by the available tools and techniques. A trade-off between sophistication and the resulting potential model misspecification as well as oversimplification has to be made. The first step in any model developing approach is to define the target or expected result. The assets considered as well as the nature of the forecasts must be defined. Assets may be individual securities, asset classes or risk factors, which can be invested. Possible categories of output are illustrated in Section 4.3.5 of Chapter 4. Due to the nature of quantitative models, more sophisticated categories may be used than in qualitative forecasting models. In addition, the type of model to be developed must be defined as either being unconditioned or conditioned. Purely quantitative forecasting models must be unconditioned. A conditioned model may be adequate if combining a qualitative forecasting approach with a quantitative model. The model may be transparent or opaque (black box). Transparent models allow a nonexpert to understand the functionality of the model. Opaque models act like black boxes where only input and output are visible. Transparent models are usually limited in complexity and thus easier to explain. Next, a framework must be selected, that is an underlying theory chosen and assumptions made. Any model is only as sound as the underlying theory it is built on. For example, a single stock return forecasting model can be built on the model where asset prices are determined as the discounted sum of all expected cash flows (DCF). An assumption could be that the cost of capital used as a discount factor in the DCF model is a mean reverting value with mean equal to the average of the discount factors of all companies in a specific industry. Restrictions to the conditions under which the model is

Quantitative Approaches for Forecasting Markets

Expected results

Estimation algorithm

Underlying theory

59

Assumptions

Markets

Theoretical model

Observed data

Estimable model

Pre-processing

Estimated model

Testing

Forecasting model

FIGURE 5.2 Quantitative model building framework

appropriate, for example only in low interest rate environments, must be stated. Both the underlying theory as well as the assumptions made should be stated explicitly. It has proved very helpful for me to write them down on paper and have them reviewed by a third party. This allows possible inconsistencies as well as missing information to be uncovered at an early stage. The combination of expected results, the underlying theory as well as the assumptions define the so-called theoretical model. It determines the choices made for combining the observed data with the theory selected. The focus is towards a sound representation or approximation of market reality. It is only when translating the theoretical model into an estimable model that practical considerations should be taken into account. Adjustments and trade-offs are made such that, given a set of observed data, the model’s parameter can be estimated. The distinction between a theoretical model and an estimable model was introduced by Haavelmo (1944). Pagan (1990) surveys different methodologies for selecting a theoretical model and deriving an associated estimable model. Ideally, the theoretical and the estimable models should coincide. For example, a theoretical model may be based on real interest rates. The associated estimable model would rely on observable nominal interest rates as well as yields from inflation linked bonds to derive implied real interest rates. Before estimating the parameters of the estimable model, observed data must be selected. This step usually involves compromises between timeliness of the data, amount of information available to allow for a decent quality of the estimation approach and the actual data history and frequency available. The data must be adequate for estimating the model’s parameters. For example, if the parameters of a normally distributed random variable are to be estimated, the historical total return time series must exhibit independence, that is no serial correlation, and be identically distributed. Often it is useful or even necessary to pre-process or transform the time series data used for estimation. For example, in economic

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Positive Alpha Generation: Designing Sound Investment Processes

time series, like jobless claims, seasonality patterns may be removed. Index level time series may be transformed into log return time series to make them stationary and additive. Outliers may be removed. To estimate the empirical model’s parameters, an estimation algorithm needs to be selected. It is a compromise between estimation quality, its complexity and required computing resources. Especially for nonlinear models, for example multilayer neural network models, the calculation time becomes a severe restriction. Often an ordinary least squares approach can be chosen. In some situations more complex models, like a maximum likelihood estimator, for example, may be the most appropriate. If the model is of a nonlinear nature, heuristic search techniques very often are the only alternative. Finally, the observed data, the estimable model and the estimation algorithm are combined to define and compute the estimated model, including all associated parameters. Although it may seem as if the model building process is complete at this stage, it is only really beginning with the backtesting phase. The developed model must be tested using out-of-sample back tests, scenario analysis, extreme event tests, Monte Carlo simulation techniques, statistical tests or any combination of those tests. Well designed tests need to be applied in order to evaluate the quality of the model. Depending on the outcome, adjustments at the different stages of the model building process may become necessary. For example, a dummy parameter may be added to the estimable model to take into account temporary anomalies in the time series to be forecasted, like, for example, higher fall temperatures, when forecasting the expected cash flows and thus total return of refining companies. Finally, the developed model is ready for real world use. The model needs to be continuously monitored. For example, changes in the underlying assumptions may require adjustments over time.

5.2

DEFINING THE MODEL STRUCTURE

5.2.1 Theory and associated assumptions The first stage after having defined the expected results is to define the underlying model structure or theory and specify assumptions. A theory, however rudimentary, is required to serve as a foundation for any modeling of the observed phenomenon of interest. I distinguish between two types of underlying theories: • Economic theories. The economic background of the selected theory dominates. Examples of economy based theories are the intertemporal capital asset pricing model, or a linear factor model with wealth and state variables, like consumption, forecasting changes in the distribution of returns. • Statistical theories. The statistical properties of the selected theory dominate. For example, total returns can be modeled by a normally distributed random variable with constant mean and variance, and thus variance may be decomposed using principal component techniques. Boundary assumptions define under which conditions the theory and the resulting model are applicable. For example, a forecasting model based on the dividend discount model is most effective in mature economies, where cash flows are paid back to investors rather than being reinvested in new growth opportunities. Other assumptions are related to the

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61

structure and nature of observable data, like, for example, serial correlations. Assumptions are like axioms. They cannot be proved within the context of the theory. They can only be asserted at a meta level. Nevertheless, their soundness is key to the practicability of the model relying on them. 5.2.2 Theoretical and estimable model structures Three key observations exist that have to be taken into account when designing a theoretical model that can be transformed efficiently into an estimable model. These are: • The complicated forces that drive financial markets introduce nonlinear dynamics, especially over longer time horizons. Linear approximations may be reasonable over shorter time horizons. Alternatively, nonlinear model types need to be used. • A dominant feature of financial time series is persistence, or long range dependences. It is crucial that this persistence, for example expressed as equilibrium returns, is taken into account if forecasting time horizons are long. On short time horizons, white noise or random walk assumptions may be reasonable. • Quantitative forecasting models, however appealing from a theoretical point of view, ultimately must be judged by their out-of-sample performance in forecasting asset returns. Estimable model structures The vast majority of forecasting models can be described as a function f such that Rt+1 = f (, Rt , . . . , Rt−m , βt ) + εt+1

(5.1)

where  represents a vector of unknown parameters, βt a vector of factor exposures or information available at time t and εt+1 the error term or portion of the asset’s return unexplained by the model. The expected or forecasted return is thus calculated as  Rt , . . . , Rt−m , βt ) E[Rt+1 ] = f (,  are the estimated parameters and E[εi+1 ] = 0. where  Equation (5.1) defines an unconditioned or pure forecasting model as the expected return over the next time period is expressed as a function of information available at time t. In the context of the strong or semi-strong form of the efficient market hypothesis, no pure forecasting model can exist. Pure forecasting models require the existence of arbitrage inefficiencies. A pure return based forecasting model is a model that only relies on historical return data, that is Rt+1 = f (, Rt , . . . , Rt−m ) + εt+1 . The forecasted return over the next time period is a function of past returns. For example, the model in the following equation defines a pure return based forecasting model: E[Rt+1 ] =

1 2

· Rt +

1 4

· Rt−1 +

1 8

· Rt−2 +

1 8

· Rt−3

Technical analysis models are typical pure return based forecasting models. I define a conditioned or explanatory forecasting model as a function f such that Rt = f (, βt ) + εt

(5.2)

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Positive Alpha Generation: Designing Sound Investment Processes

In contrast with an unconditioned forecasting model, the observed total return of today is expressed as a function of information, represented by time series, available as of today. The model explains current returns through the use of current and past data. Conditioned forecasting models are consistent with the efficient market hypothesis. They can be used to forecast markets by supplying them with forecasted information, in the sense of time series. For example, a typical conditioned forecasting model of the S&P 500 index return would be such that RS&P 500,t = f (, GDPt , CPIt ) + εt

Process based models • Auto regressive models • Moving average models • Exponential smoothing • Brownian motion models • Pattern matching models

Conditioned forecasting /explanatory

Unconditioned/ pure forecasting

where  are the parameters, GDPt is the change in gross domestic product between time t − 1 and t and CPIt the change in the consumer price index. If I were able to forecast GDPt and CPIt , either through a quantitative model or a qualitative approach, I could derive a forecast for RS&P 500,t , the return of the S&P 500 index. A conditioned forecasting model can be used to predict the impact of information flows on market prices, that is on the market equilibrium, and thus represent a quantitative component of a qualitative forecasting process, as described in Chapter 4. I classify different quantitative forecasting approaches based on whether they are conditioned or unconditioned and which type of time series data they rely on. Figure 5.3 shows the classification of the most common quantitative forecasting models according to these criteria. Factor based models • Linear factor models • Bayesian models • Kalman filter models • MCMC models • Artificial neural networks

Economic models • Stochastic discount factor models • Capital asset pricing model • Arbitrage pricing model • Equilibrium models

Return time series data

Non return based time series data

FIGURE 5.3 Classification of quantitative forecasting models

5.2.3 Linear models The most common forecasting models are linear models. A linear model is defined by R=

K  f =1

β f · Ff + ε

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63

where f represent the different factors of the model, βf the exposures of the asset or portfolio to that factor, Ff the factor sensitivity and ε the error term, such that ε ∼ N (0, σ ). There are three approaches to define a linear model: • The factor exposures are determined at a meta level based on qualitative analysis, for example using an economic model. The factor sensitivities, on the other hand, are determined using statistical estimation techniques. • The factor sensitivities are defined through a qualitative analysis of the asset’s characteristics. The exposures are derived using statistical techniques. • Both the explanatory variables and the parameters are determined by quantitative techniques. The most common technique used is the principal component analysis (Jackson, 1991). The factor sensitivities may be either constants or random variables. In the latter case, the distribution or the parameters of the distribution have to be determined. Factor based models are typical linear explanatory models. They aim at explaining the highest possible percentage of the total return and its variance by using exogenous factors. The factors are usually observable characteristics of the asset. For example, the industry to which an asset belongs may be such a factor. Another type of factor would be the debt-to-equity ratio of the company. The selection of the factors requires experience in both econometrics (to select factors that can be statistically exploited) and economics (to select factors with relevant information content). Process based models can be seen as a variation of factor models, but the underlying idea is different. In process based models, the data to be forecasted are expected to evolve over time according to a well defined deterministic or stochastic process. The most common deterministic process based models are autoregressive moving average (ARMA) models. An ARMA(p, q) process is expressed by
1 · Ft−1 + 
q · Ft−q
1 · Rt + · · · + 
p · Rt−p+1 + Ft +  E[Rt+1 ] = C +   the moving average  the autoregressive parameters and  where C is a constant,  parameters. Another class of process based models are stochastic process models, like Brownian motion. These models are an extension of deterministic processes adding a stochastic component, usually a normally distributed random variable. An example of a stochastic process is the Cox, Ingersoll and Ross (Cox et al., 1985) short term interest rate forecasting model, that is √ drF = −θ · (rF − µ) · dt + σ · rF · dW where θ , µ and σ are parameters and dW is a Wiener process. 5.2.4 Parameter estimation techniques Once the structure of the forecasting model has been defined, the parameters of the estimable model need to be estimated. The most prominent parameter estimation techniques used in the context of linear models are, in increasing order of their complexity:

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Positive Alpha Generation: Designing Sound Investment Processes

• Ordinary least squares (OLS). The OLS algorithm estimates the model’s parameters such that the squared sum of the error terms is minimized. • Generalized least squares (GLS). The GLS algorithm is a generalization of the OLS approach. Individual observations are weighted and the correlations between the error terms are considered. • Maximum likelihood (ML). The maximum likelihood estimation technique is a statistical method to estimate parameters of the underlying probability distribution of a given data set. It is based on the assumption that a probability distribution function exists that associates a probability to each observation. The parameters are estimated in order to maximize the likelihood function associated with the observations and the probability distribution function. Hamilton (1994) extensively describes these and other estimation techniques with a focus on the mathematical properties. Cochrane (2005) shows how estimation techniques can be used in the context of developing asset pricing models. Granger (1990) reviews various approaches of using econometric techniques in economic forecasting. The choice of which estimation technique to use, given its assumptions are satisfied, is a trade-off between statistical efficiency, the effects of misspecification of the forecasting model, both from an economic as well as a statistical point of view, and the clarity and interpretability of the results. There are situations where it is better to trade efficiency gains for the robustness of simpler estimation techniques. In addition, each estimation technique requires a certain number of assumptions to be satisfied. For example, to be able to use an OLS estimator, the observed data must be such that the error terms from the model are uncorrelated and normally distributed with a zero mean and the same variance σ 2 . Sample averages must converge to population means as the sample size grows for most estimators to be applicable. If the investment manager has prior information about the parameters of the designed model, then Bayesian analysis may be used to enhance the parameter estimation process. In Bayesian statistics, parameters are considered random variables instead of constants. Any prior information available about the parameters can be expressed through the density function of these random variables. Therefore, the estimated value of the parameters can be written as a weighted sum of the parameters estimated using classical estimation techniques and parameters estimated based purely on prior information. Zellner (1971) offers an introduction to Bayesian inference theory in econometrics. Hamilton (1994) provides a detailed mathematical overview of Bayesian analysis. Fama (1965) shows how Bayesian theory can be used to study the behavior of stock market prices. Black and Litterman (1991) and Litterman (2003) extensively use Bayesian statistics to derive their equilibrium based asset allocation model, described in Chapter 10. 5.2.5 Nonlinear models More advanced models take into account the fact that financial time series data, over longer time horizons, exhibits nonlinear properties. Nonlinear models are complex and usually require calculation intense algorithms to be applied. However, they allow much more general data properties to be modeled than are possible with linear models. The question to be answered is whether the additional complexity is worth the additional forecasting precision obtained.

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Hierarchical decision trees Sorensen et al. (1998) proposed a decision tree based forecasting technique that utilizes a set of if-then decision rules, called the CART framework. They observed that a hierarchy exists in the factors forecasting markets. In linear quantitative models, this hierarchy is determined indirectly through regressions. The CART framework examines the relationships on a conditional basis. Rather than being estimated, the relationships and the resulting decision rules are designed by the investment manager. Nonhierarchical forecasting models would, for example, use the difference between the equity earnings yield and the long bond yield as an indicator of whether equities will outperform bonds or bonds will outperform equities. This relative value indicator on its own has limited statistical explanatory power. However, when using this indicator conditioned on the state of the economy, for example in low growth environments, the indicator provides much better explanatory power. Figure 5.4 presents a possible decision tree derived in this framework. The decision tree is constructed so as to maximize the explanatory power of its leaves. The individual decisions are hierarchically ordered according to their relevance for the investment decisions to be taken. The main advantages of the CART framework are its flexibility, its nonlinearity in decisions taken and its conditioning based approach. A major drawback is that the model must be built using economic intuition rather than formal algorithms. Relative performance of large stocks versus small stocks

Shape of the economy low growth

high growth

Risk of recession steep yield curve

Yield curve steepness

flat yield curve

Expected risk premium

Equity earnings yields versus long bond yields

? high

?

low

Relative value of stocks versus bonds Equities outperform bonds

positive Equities outperform bonds

Equity total return versus short term interest rates

negative Bonds outperform equities

FIGURE 5.4 Sample decision tree used to forecast whether equities will outperform bonds or vice versa

Artificial neural networks Artificial neural networks (ANN) are the most common category of nonlinear models used in finance. They are built on an understanding of the human brain (Rosenblatt, 1962; Shadbolt

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Positive Alpha Generation: Designing Sound Investment Processes

x1

R

x2

x3

FIGURE 5.5 Sample three-layer artificial neural network (ANN)

and Talor, 2002). An ANN is made up of one or more layers of perceptrons. Each perceptron implements a linear classification function, that is R=h

 

 Fi · β i

i

where R is the output, F the input, β the parameters of the perceptron and h(·) is typically taken to be a smooth, monotonically increasing function such as the sigmoid function 1/(1 + e−x ). A sample three-layer network is illustrated in Figure 5.5. The parameters F of an ANN are determined by solving a nonlinear optimization problem such that the explanatory power of the data set used to estimate the parameters is maximized. These calculations are called training the network. Combining multiple layers of perceptrons into a neural network model allows approximating nearly any type of nonlinear function. The structure of an ANN is complex and abstracted from any type of economic explanation. This makes ANNs behave like black boxes. The training stage requires a large training data set as well as significant computational resources. Training even a simple perceptron to forecast a data set optimally has been shown to be computationally intractable (Amaldi, 1991), requiring any reasonable training algorithm to be a heuristic approximation algorithm. Neural networks have been applied successfully in various areas of finance (Shadbolt and Talor, 2002; Smith and Gupta, 2002; Swingler, 1996). Pattern matching Pattern matching techniques, also called artificial intelligence (AI) algorithms, try to exhibit a repetitive structure from a data set. Figure 5.6 illustrates a typical time series pattern. This pattern is called the head and shoulder formation in technical analysis (Murphy, 1999). Pattern matching algorithms recognize and learn structures in data sets. Pattern matching algorithms are separated into two modules, the learning module and the forecasting module. In the learning module, the algorithm tries to identify repetitive patterns in data time series. It memorizes these patterns for future reference. When applied in forecasting mode to a data

Quantitative Approaches for Forecasting Markets

67 2.84

exchange rate

y

1.17 2.67

1.17

y

y

2.50 t

t

t

t

10 days 10 days 10 days10 days Source: sample data

FIGURE 5.6 Typical time series structure recognized by a pattern matching algorithm (left figure) and its use to forecast exchange rates by extrapolating current levels (dotted line) (right figure)

set, the model strives to recognize partial patterns in the data set and extrapolates or forecasts the remaining part of the recognized pattern. Figure 5.6 illustrates the forecasting capabilities of a pattern matching algorithm. The current level of the GBP/CHF exchange rate is 2.50, which represents the second bottom of the illustrated pattern. Based on extrapolation, the GBP/USD is forecasted to rise and top at around 2.67 in approximately 10 days. The pattern matching technique evolved from the field of image recognition, a subfield of artificial intelligence in computer science. The developed techniques are very powerful, but unfortunately they require large sets of data and significant computing power for the learning mode. Technical analysis is a simple form of pattern matching performed by humans instead of computer algorithms. Kalman filters A Kalman filter model (Kalman, 1960; Harvey, 1991) describes an algorithm that recursively estimates unobservable time varying parameters of a system. The filter proceeds in two steps. First, the current observations are used to predict the next period’s unobservable variables, that is βt = F · βt−1 + εat

(5.3)

where F is a matrix of parameters and εat a vector representing white noise. Equation (5.3) is called the state equation and represents the prediction phase. Second, the current exogenous data, that is dt , is used to estimate the endogenous variables rt , such that rt = A · dt + H · βt + ε bt

(5.4)

where A and H are matrices of parameters and εbt a vector representing white noise. Equation (5.4) is called the observation equation.

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Positive Alpha Generation: Designing Sound Investment Processes

Kalman filters are very useful in building forecasting models because they allow time varying unobservable parameters to be incorporated (Wells, 1995). This allows, for example, taking into account structural shifts dividing the forecasting period into distinct regimes with different parameter values. In this context, Kalman filters provide three key advantages over traditional models: • First, the explicit presence of time varying parameters allows a more precise detection of structural changes, both in magnitude and timing, in the built model. • Second, Kalman filers can take into account specific levels of integration and co-integration properties. • Third, Chow (1984) has shown that any ordinary regression model can be described as a special case of a Kalman filter model. Consider the one factor CAPM, where Et [Rt+1 ] = RF,t + β · (Et [RM,t+1 ] − RF,t ) as described in Chapter 4. It is well documented that the parameter β is not constant over time, but changes, among others, based on the state of the economy. Using the Kalman filter approach, the CAPM can be modeled by the following state space model: βt = ft−1 · βt−1 Et [Rt+1 ] = RF,t + βt · (Et [RM,t+1 ] − RF,t )

Markov chain Monte Carlo methods for inference and forecasting In financial modeling, the forecasting goal can often be formulated conditional on observed information. In the Bayesian framework, this means determining the distribution of parameters and variables  conditional on observed information D, that is the probability distribution P [|D]. Unfortunately, determining P [|D] is a high-dimensional inference problem for which there usually does not exist an analytical solution. The Markov chain Monte Carlo (MCMC) model is a method for solving this problem using simulation techniques. The key idea is to create a Markov chain whose stationary distribution is given by P [|D] and run simulations sufficiently long until the distribution of the Markov chain is close enough to the stationary distribution sought after. Verhofen (2005) presents an overview of MCMC models in financial econometrics. Geweke (2005), as well as Geweke and Whiteman (2006), present the underlying Bayesian mathematical theory as well as algorithms for construction of the Markov chain.

5.3 HANDLING DATA IN PARAMETER ESTIMATION Handling data time series used to estimate the model parameters or to train the algorithms is a complex, time consuming and low profile task. It even gets worse! It is very often tempting to use data mining techniques to map the estimable model on to the used data rather than the opposite. Available data should support the specialization of the theoretical model into the estimable model but not control it.

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The following steps need to be performed to select and process the data used for parameter estimation. First, the universe of available data needs to be screened to find time series that allow the needed parameters to be estimated and are: • Reliable. The source as well as the quality of the data must be trusted. Examples of reliable sources are stock exchanges, national bureaus of statistics, as well as major data providers like Bloomberg or Thompson Reuters, to name the most important ones. • Consistent. The different data elements have to be observed, derived or calculated using a single consistent methodology. For example, in a consumer price index, the basket used to calculate the index should not change over time or changes only according to well defined rules. In fixed income total return indices, the reinvestment of the coupon payments should be done according to a formulaic approach. In a second step, the data time series should be studied for structural particularities. At this stage it is useful to plot the data using different plotting techniques in order to identify trends, seasonality, volatility clustering, unit root properties, serial or cross-sectional correlations or other properties. Figure 5.7 shows a time series and the associated return figures exhibiting minor volatility clustering. Visualization techniques help to provide a hint on what time series properties to analyze in detail. Detailed analysis using statistical techniques should be applied to confirm the properties identified from the plotted data. Third, the questions of what length of historical data to use and at what frequency it should be must be answered. Answering these questions is driven by two opposing goals. From a statistical point of view, the size of the sample data used to estimate the parameter should be as large as possible. On the other hand, the historical data set used needs to exhibit the properties required by the estimable model. For example, in order to estimate the second moment of a normal random variable, the data used for the estimation must be assumed 6%

Weekly return

4%

2%

0% −2 % −4 % −6 % ‘98

‘99

‘00

‘01

‘02

‘03

‘04

‘05

‘06

‘07

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 5.7 Plot of weekly USD/EUR exchange rate returns between December 1998 and December 2007

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Positive Alpha Generation: Designing Sound Investment Processes

to be independent and identically distributed. This assumption could be verified by using a data frequency that avoids serial correlations, which is, for example, for many index time series the case with monthly data, and use the shortest possible history that still allows for a reasonable estimation quality, ensuring that the identically distributed assumption holds. The estimation algorithms used should be selected so that they can cope with the availability of data for estimations. For example, Ledoit and Wolf (2003) have shown that estimating the covariance matrix of a large number of assets with only limited time series available is improved by combining the classical sample estimator with a one-factor model estimator using statistical shrinkage techniques. Once the time series as well as their properties have been identified and the actual size and frequency determined, the data should be pre-processed in order to: • remove properties from the data that are inconsistent with the estimable model as well as the selected estimation algorithms and • handle missing data as well as outliers. From a practical point of view, it is useful to design the data pre-processing steps such that they can be efficiently repeated. Very often, during the course of the model design, data used are changed, new data sets added, old ones removed and frequencies and time horizons changed. A well designed data handing framework makes the model development process efficient, allowing it to be able to repeat the different steps that lead to the final model decision, making the maintenance and update of the model easier. 5.3.1 Handling missing data More often than expected data are missing. For example, EUR/USD exchange rates did not exist prior to 1999. Daily equity index data may not be available for some countries due to national holidays. Another reason for missing data may be that some information is not available at the needed frequency, like GDP data, which is in many countries only available on a quarterly basis, or hedge fund prices, which are only published on a monthly basis. A data element is said to be missing completely at random if the probability of the data missing is unrelated to its value or the value of any other variable in the data set (Allison, 2001). The technique most often used to handle missing data is the listwise deletion technique. It removes from the data set any data record that has missing information on one or more variables. This method can be used on any data set and does not impact the statistical analysis of the reduced data set. Especially, for any parameter of interest, if the estimates are unbiased for the full data set, that is with no missing data, they will also be unbiased for the listwise deleted data set. The major drawback of listwise deletion is that the standard error generally increases with the number of data records removed. The EM or expectation maximization algorithm (Dempster et al., 1997; Little and Rubin, 1987; Schafer, 1997) is a more advanced technique. The idea is to substitute some reasonable guess for each missing data element. It is based on the maximum likelihood approach and assumes that the data set can be modeled by a multivariate normal random variable. The algorithm is called EM because it consists of the two steps: (i) E, an expectation calculation step, and (ii) M, a maximization step.

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In the E step, regression techniques are used to impute or guess the missing data. The missing data element is taken to be the explained variable and the available data to be the explanatory variables. After all missing data elements have been imputed, the M step consists of calculating the parameters of the multivariate normal distribution using the nonmissing data combined with the imputed data. Special care must be taken when calculating the covariance matrix, as it must be adjusted for the residual covariance from the regression. The EM algorithm iteratively cycles through the E and M steps until the estimated parameters converge.

5.4 TESTING THE MODEL The last stage of the quantitative model development process is to test the model. The goal is to show that the developed model is successfully predicting market returns rather than fitting the data used to estimate its parameters. I distinguish between four categories of model tests: (1) Statistical tests. These include hypothesis testing of the model’s parameters, variance calculations, as well as estimation error statistics reviews. (2) Back tests. These apply the developed model to a data set different from the one used for the estimation process, and calculate different statistics describing the difference between the forecasted and the realized value. Monte Carlo simulated data sets as well as specifically designed scenario data sets may also be used in such back tests. (3) Robustness and assumption tests. These check how the model’s results differ if inputs change or assumptions made are no longer satisfied. (4) Boundary and extreme events tests. These tests verify how the model reacts at boundary conditions as well as in the presence of extreme events, that is situations which are theoretically sound but which differ significantly from observed situations in the past. It is key that a model identifies these boundary and extreme situations and alerts the user to the fact that the forecasted results may be insignificant or incorrect. Statistical tests Statistical tests are based on statistical and econometrics theory. I distinguish between three categories of statistical model tests. These are: • Model parameter significance tests. For example, null hypothesis tests can be applied to test for the significance of the estimated parameters of the model. • Explained variance tests. Explained variance statistics give a rough sense of adequacy of the estimated model. The most common measures are R 2 , the proportion of the sum of squares about the mean that is explained, and the adjusted R 2 , the proportion of the mean sum of squares, that is the variance, that is explained for time series models. Less common are the Akaike information criterion (AIC) (Akaike, 1974), Mallow’s Cp and various elaborations of the AIC (Burnham and Anderson, 2002). • Structure of unexplained residuals. Ideally the unexplained residuals should not exhibit any structure, that is they should be independent and follow a normal distribution with mean zero and identical variance σ 2 . Any other structure would point toward a model underspecification. The Durbin-Watson (Durbin and Watson, 1950, 1951) test may be used to detect the presence of autocorrelation.

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As always, a certain degree of caution is appropriate as the theoretical assessments of the confidences in the model rely on the validity of the theory used to perform the assessments. If the theoretical assumptions are incorrect, for example because of serial correlations of the data, then the test results lose much of their significance. Details about performing statistical tests can, for example, be found in Maindonald and Braun (2003). Back tests Let {r, D} be the data set available to develop and test the model. First I subdivide the data set into two nonoverlapping subsets, the subset {re , De }, called the estimation data set, and the subset {rv , Dv }, called the validation data set. Usually r = {re rv } and D = {De Dv }. I use the estimation data set to estimate the parameters of the model and then apply the model to the validation data set to forecast the expected returns. To determine the quality of the forecasting model, I compute the five statistics shown in Table 5.1 on both the estimation and validation data sets. If only the estimation data set is used to test the quality of the model, I may end up understating the magnitude of the forecast errors that will be made when the model is used to predict the future, because it is possible that the model has been overfitted. Indeed, the model may inadvertently fit some of the noise in the estimation data. Overfitting is especially likely to occur when a model with a large number of parameters has been fitted using a small sample of data. In addition, data mining issues may be uncovered. As the data in the validation data set has not been used during the parameter estimation, the resulting forecasts are honest and their error statistics are representative of errors that will be made in forecasting in the future. In addition, it is possible to determine confidence intervals of the forecasts made based on the standard deviation of the forecast errors at each period. For example, the confidence interval at the 95 % level is roughly equal to the forecast plus-or-minus two times the estimated standard deviation of the forecast error. The confidence interval typically widens as

Table 5.1 Summary statistics used to validate a sample forecasting model Statistic Mean error (ME)

Mean squared error (MSE)

Mean absolute error (MAE)

Mean percentage squared error (MPSE)

Mean absolute percentage error (MAPE)

ME =

n 1  (Rt − Rˆ t ) · n

MSE = MAE =

t=1 n 

1 · n

1 · n

t=1 n 

(Rt − Rˆ t )2 |Rt − Rˆ t |

t=1

 2 n 1  Rt − Rˆ t MPSE = · n Rt t=1   n 1   Rt − Rˆ t  MAPE = ·    Rt  n t=1

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73

the forecast horizon increases under the normality distribution assumption. For random walk models, like Brownian motion models, the size of the confidence interval is proportional to the square root of the forecast horizon. If not enough data are available to define large enough estimation and validation sets, re-sampling techniques may be used. The set of available historical data is subdivided multiple times randomly into an estimation and a validation data set. The results obtained from the different random subsets are studied for forecasting capabilities. Robustness and assumption tests Different approaches exist that can be used to check the robustness of the developed model, including its dependence on the validity of the made assumptions. First-order partial derivations of the forecasting model allow the sensitivity of the model to its various input parameters to be studied. If a small change in a single parameter has a large impact on the forecasted results, there should be a suspicion about the presence of a model misspecification. In linear models, this test is simplified by comparing the size of the estimated parameters against each other. A second test to check the robustness of the model is to replace the actual input parameters during the back test by random variables. The model’s output can then be checked against the variance of the random variable in order to view the significance of small changes in the input parameters on the model’s output. Different economic cycles can be identified and the model’s performance tested during any of them. This will provide information if the model is independent of a specific economic cycle or if, eventually, the model’s parameters need to be estimated separately for different market cycles. The conclusion of these tests could even be that a specific model only works in a certain market environment where additional model assumptions can be identified. Finally, it is important to verify all model assumptions at the meta level. Both statistical tests as well as sound reasoning should be applied. Boundary and extreme event tests There are three approaches that can be used to test how the model will perform when boundary conditions are satisfied or during extreme events: • Screen the historical data available for outliers and test the model on these outliers. • Randomly generate outliers using distributions specifically developed to model extreme events, like generalized Pareto distributions (McNeil et al., 2005) and check the forecasted values for consistency. The question of whether or not it is possible to recognize, based on the model’s forecast, that the input data were outliers should be answered. • Construct hypothetical, but consistent, extreme market movements and test how the model performs under these hypothetical scenarios. It is important to know and understand how the model reacts in such situations, rather than ensure that the forecasts are sound even in these cases. It is usually more important to have a model that provides good forecasting capabilities under normal market conditions than one that gives an average performance in all market environments.

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5.5

MITIGATING MODEL RISK

Forecasting models are approximations of reality, that is a simplified structure. I define model risk as the risk that mistakes or misjudgments are made during the development phase that have an impact on the performance of the model. Model risk occurs at the meta level of a model. There are four key sources of model risk that need to be mitigated: • Incorrect model structure specification. Underlying processes are incorrectly specified. Significant risk factors are ignored. Relationships between variables are incorrect. • Incorrect model usage. The underlying assumptions are invalid in the context where the model is used. • Implementation risk. Decisions taken when implementing the model specification are incorrect. Programming errors find their way into the software. • Calibration errors. Inadequate data sets are used to estimate the model’s parameters. The significance of the estimated parameters is not controlled. Any of these risks by itself, when materializing, may invalidate the developed forecasting model. Gibson (2000) presents a comprehensive compilation on the concept of model risk and the potential pitfalls associated with modeling financial risks. Unfortunately a formal framework does not exist to detect and eliminate model errors. Nevertheless there are a number of ways practitioners can protect themselves against model risk (Dowd, 2005). The use of a structured model development process as described in this chapter requires the following key issues to be addressed: • Key assumptions need to be verified and especially re-verified on a regular basis during the lifetime of the model. • The model structure should be kept as simple as possible. Model features must be offset against possible misspecifications. • The model’s output needs to pass a qualitative relevancy check. • The model should be tested against known problems. • Not only should thorough back tests be performed but stress or extreme event tests should help to find potential model errors. • Any model should be checked by a third party who was not involved in the model’s design. • Do not change a working model. Enhancements or other minor changes are very often the cause of model errors, as they usually are not developed using a thorough process. As stated by Dowd (2005), ‘Model risk is like the proverbial ghost at the banquet – an unwelcome guest, but one that I would be very unwise to ignore.’

5.6

EXAMPLE

The model building process will be illustrated by designing a simple multifactor based pure, that is unconditioned, forecasting model.

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Yield spread over time 4.0 % 3.0 % 2.0 % 1.0 % 0.0 % −1.0 % −2.0 % ‘88 ‘90 ‘92 ‘94 ‘96 ‘98 ‘00 ‘02 ‘04 ‘06 Signal - Change in real yield 6.0 % 5.0 % 4.0 % 3.0 % 2.0 % 1.0 % 0.0 % −1.0 % ‘88 ‘90 ‘92 ‘94 ‘96 ‘98 ‘00 ‘02 ‘04 ‘06

Signal - Change in yield curve steepness 2.0% 1.5% 1.0% 0.5% 0.0% −0.5% −1.0% −1.5% ‘88 ‘90 ‘92 ‘94 ‘96 ‘98 ‘00 ‘02 ‘04 ‘06 Signal - Squared equity return 1.5% 1.0% 0.5% 0.0% ‘88 ‘90 ‘92 ‘94 ‘96 ‘98 ‘00 ‘02 ‘04 ‘06 Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 5.8 Historical yield spread between the ten-year constant maturity US treasury yield and the one-month eurodollar deposit rate and the signals used as explanatory variables between December 1988 and December 2007

5.6.1 Expected result The goal of my model is to forecast the direction of the change in the difference between the ten-year constant maturity US treasury yield and the one-month LIBOR rate. I design a factor model that forecasts the actual value of the expected yield spread change. The spread change therefore represents the expected result. If the yield spread change is positive, that is the spread is narrow, taking a long position in ten-year US treasuries, financed by borrowing at the short term rate, or entering into a long ten-year Treasury note futures contract on the Chicago Board of Trade (CBOT) would lead to the generation of a positive alpha, assuming the forecast to be correct. Figure 5.8 plots the historical changes of the to be forecasted yield spread. 5.6.2 Assumptions I assume that it is possible to explain the change in yield spread, that is the expected result, as a linear function of an observable market data time series. In addition, I expect the developed model to be valid independently of the economic cycle as well as the economic environment of the market. This also means that I assume that changes in the global economic regimes do not have an impact on the developed forecasting model. This assumption would require validation at the latest in the model testing phase.

5.6.3 Underlying theory I base the developed model on the theory that states that the future spread between the long term yields and the short term interest rate, that is the risk free rate, is a function of:

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Positive Alpha Generation: Designing Sound Investment Processes

• long term real yields, which exhibit a mean reversion property; that is if real yields are above their long term average, they tend to drop and if they are below their long term average, they tend to increase, • steepness of the yield curve, that is the spread between long term and short term interest rates, expressing expected growth, and • changes in equity market returns, modeling the degree of market uncertainty or risk.

5.6.4 Theoretical model I introduce the following notation to describe the theoretical as well as estimable model: • st . Difference between the ten year constant maturity US treasury bond yield and the one-month treasury bill rate at time t, • yt . Real yield, calculated as the difference between the ten-year constant maturity US treasury bond yield and the year-on-year change of the seasonally adjusted US consumer price index (CPI) at time t, • rt . Monthly end-of-day S&P 500 price log return between time t − 1 and t. From this underlying theory, I derive the theoretical model such that   st+1 = f st − s t , yt , rt2 where st denotes the one-month change of variable x between time t − 1 and t, s the six-month average between dates t − 6 and t. The future change in yield spread is impacted by the current yield spread compared to a short term six-month moving average. In addition, current real yields are related to the change in yield curve steepness, as real yields relate to economic growth. As a third factor of my model I use squared equity returns as a proxy for uncertainty. 5.6.5 Estimable model Next I transform the theoretical model into the estimable model shown as st+1 = β0 + β1 · (st − s t ) + β2 · yt + β3 · rt2 + εt

(5.5)

I define the model as a linear three-factor model, based on the theoretical model as well as the assumption that it is possible to explain the change in yield spread, that is the expected result as a linear function of observable market data time series. I have chosen this model because it allows a simple illustration to be made of the different steps of the model building process. So-called real world models are usually more complex and model the underlying economic environment in more detail. 5.6.6 Observed data I use the time series and data sources shown in Table 5.2 as approximations of the yield spreads, real yields and equity market performance.

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Table 5.2 Time series used to estimate the parameters of the estimable model, their modeling as well as the used data source Theoretical time series

Observed time series

Yield spread

• 10-year US treasury constant maturity yield • 1-month US treasury bill rate

Long term real yield

• 10-year US treasury constant maturity yield • US city average consumer price index (all items), seasonally adjusted

Equity market return

• Log return of S&P 500 index

5.6.7 Pre-processing Before choosing an estimation algorithm I analyze the historical time series used in the estimable model as illustrated in Figure 5.8. I do not detect any anomalies that need further investigation. In addition there is no data missing. Thus the pre-processing step becomes redundant. Based on the assumption that the parameters of the developed model will remain stable over time, I use data between 1988 and 2002, that is 14 years or 168 observations of historical data, to estimate the model’s parameter. I use the remaining five years of data between 2003 and 2007 to back test the model. 5.6.8 Estimation algorithm As my model is a simple linear regression model and I do not have any additional, so-called prior, information, I use the ordinary least squares parameter estimation algorithm. 5.6.9 Estimated model Using a generic statistical software package, I estimate the parameters as shown in the following equation and Table 5.3 and 5.4: E[st+1 ] ∼ = 0.0012 + 0.3526 · (st − s t ) − 0.0358 · yt + 0.04210 · (rt )2

(5.6)

5.6.10 Testing Finally, and most important, I test the developed model for its validity. I use the four-angle approach presented. The classical null hypothesis tests for significance using the t statistic as well as the R 2 statistics are shown in Table 5.3 and Table 5.4. Except for the intercept, that is the constant α0 , all estimated parameters are statistically significantly different from zero at the 5 %

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Positive Alpha Generation: Designing Sound Investment Processes

Table 5.3 Sample output from a statistical estimation package applying an ordinary least squares regression Coefficients β0 , β1 , β2 , β3 ,

intercept st − s t yt rt2

0.001157 0.352640 −0.035874 0.042095

Standard error 0.000860 0.031336 0.023384 0.066575

t statistic 1.346142 11.253368 −1.534138 0.632297

Lower 95 % −0.000540 0.290768 −0.082044 −0.089353

Upper 95 % 0.002855 0.414512 0.010296 0.173544

Table 5.4 Estimated model parameters as well as statistical tests Regression statistics Multiple R R square Adjusted R square Standard error Observations

0.6690 0.4475 0.4375 0.0024 169

confidence level. The adjusted R 2 gives reasonable confidence that the forecasting model will allow a certain degree of explanatory power. Remember that the quality of the model represents the investment manager’s hopefully unique skills and capabilities, which is not at the core of this book. In a second step, I back test the designed model using sample data between 2003 and 2007. The results are shown in Figure 5.9, the top graph displaying the actual versus the fitted change in spread and the bottom graph displaying whether or not the forecasting direction was correct. The out-of-sample tests show a correctly forecasted direction in 70 % of the tested months during the out-of-sample period. In addition I calculate the summary statistics defined in Table 5.1. The results are shown in Table 5.5 for both the estimation as well as the validation or out-of-sample testing period. All these figures give reasonable confidence that the build model has actual forecasting capabilities. The model is checked for robustness by adding a normally distributed random variable with zero mean and a variance of 0.1 % to the historical explanatory variables and the resulting forecasted result is checked with the actual forecasted results. Although the model Table 5.5

Summary statistics

Statistic

Estimation period

Validation period

Mean Mean Mean Mean Mean

0.00 % 5.55 × 10−6 0.18 % 8.63 1.51

0.05 % 3.59 × 10−6 0.15 % 9.08 1.48

error (ME) squared error (MSE) absolute error (MAE) percentage squared error (MPSE) absolute percentage error (MAPE)

Quantitative Approaches for Forecasting Markets

Actual ‘88

‘89

‘90

‘91

79

Forecast ‘92

‘93

‘94

‘95

‘96

‘97

‘98

‘99

‘00

‘01

‘02

‘03

‘04

‘05

‘06

‘07

‘01

‘02

‘03

‘04

‘05

‘06

‘07

Correctness of directional forecast

correct

wrong ‘88

‘89

‘90

‘91

‘92

‘93

‘94

‘95

‘96

‘97

‘98

‘99

‘00

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 5.9 Actual change in yield spread compared to forecasted yield spread change over the out-of-sample back test period (top figure) and comparison of actual versus forecasted direction of spread change (bottom figure) Table 5.6 Extreme event scenario analysis Spread change (%) −0.02 −0.02 1.00 −2.00 −0.02

Real yield (%)

Risk (%)

Regression result (%)

1.00 10.00 2.95 2.95 2.95

0.16 0.16 0.16 0.16 1.00

0.08 −0.024 0.37 −0.34 0.04

Source: sample data

introduces a slight bias to a lower yield change forecast when randomly adjusting the explanatory variable values, the impact is small. This shows that the model is robust to small changes in the explanatory variables. Finally, I use a scenario analysis to determine the model’s capabilities in extreme situations. The proposed scenarios, as well as the resulting forecasts are shown in Table 5.6. As with any model, using the model in a live environment, that is when managing actual assets against the model’s forecast, the assumptions made during the model building process must not be forgotten. In addition, the model should be verified regularly and the assumptions, especially that the estimated parameters do not change over time, checked.

6 Taking Investment Decisions The investment manager responsible for investment decisions, called the decision maker, is confronted with multiple forecasts for multiple variables and may be a single person, a workgroup or even a team. The different forecasts can result from different individuals, subjective judgments, information sets, assumptions or models used. In both Chapters 4 and 5, I have presented processes and tools to formulate such forecasts. It is the decision maker’s role to analyze the different forecasts and exploit information contained therein to formulate single and hopefully consistent investment decisions. The decisions to be taken all have three characteristics in common (Geweke, 2005): • They must be made based on less than perfect information. • They must be made at a specific date and cannot be postponed. • They are based on multiple sources of information or forecasts, which must be aggregated explicitly or implicitly in the decision. In a purely quantitative investment process, the decision maker is replaced by a decision making algorithm. Nevertheless, the decision making process must follow some basic principles, whether it is judgmental or purely formula driven. Let me start by formulating the goal of a decision maker. A typical decision maker has a utility function U (f , d), which depends upon some uncertain variables f , called the forecasts, and a set of decision variables d. As shown in Figure 6.1, the decision maker tries to determine the decision variables d such as to maximize the utility function U (·), given the forecasts f . The utility function may be the expected return of a normalized long-short portfolio, where d represents the assets to be long or short and f the expected relative return forecasts, based on different views on information or different forecasts. Maximizing the utility function formalizes making a choice when faced with uncertainty, even when being presented complete and reliable information (Saari, 2001b). The utility function maximization may either be based on judgmental decisions or quantitative formulas. It is a way to combine potentially conflicting information into a consistent and summarized form. There are three steps in any such decision making process: 1. Forecasts are determined using the processes described in Chapters 4 and 5. 2. The forecasts as well as the information associated are communicated, that is transferred, to the decision maker. 3. The decision maker takes a decision based on the forecasted information such as to maximize the utility function. It is important to separate the role of the forecasts from the actual investment decision. The decision making process introduces a hierarchy between the forecast providers and the decision makers. Indeed, forecast and information providers serve as facilitators.

82

Positive Alpha Generation: Designing Sound Investment Processes Forecast 1 – Macroeconomics f1(EMU) = 2.4% ± 0.5% f1(USA) = 1.3% ± 0.9% f1(Japan) = 3.7% ± 1.5% f1 Forecast 2 – Political f2(EMU) = 0.7% ± 1.2% f2(USA) = 1.7% ± 1.4% f2(Japan) = 2.4% ± 2.5%

f2

Investment decision max U(f1,f2,f3,d )

d (EMU) = −1.0% d (USA) = −2.0% d (Japan) = +2.0%

f3 Forecast 3 – Company earnings f3(EMU) = 1.4% ± 0.2% f3(USA) = 1.8% ± 0.6% f3(Japan) = 3.3% ± 0.9%

FIGURE 6.1 Sample decision making process based on forecasted return information

Forecast/ Investigate

Communicate

Consistent

Decide

yes

no

FIGURE 6.2 Key steps of forecasting in a decision making process

The decision making process should be accompanied by a consistency check combined with a feedback loop. Such a process structure is illustrated in Figure 6.2. It is important that the underlying organizational structure, as illustrated in Chapter 3, supports the information flow from the forecasters to the decision makers. Forecasters are also known under the name of investigators. They are sometimes also called specialists. Investigators and decision makers may be the same or different persons. Combining them in one person increases the overall commitment and involvement of the investigators in the resulting decisions. Separating the roles and responsibilities on an organizational level enhances specialization, which eventually leads to better quality.

6.1

UNDERSTANDING THE THEORY OF DECISION MAKING

Two schools exist in decision theory, the classical school and Bayesian inference school (Geweke, 2005). Classical decision theory concentrates on the uncertainty about the future conditioned on known information, like a model, parameter values or other exogenous variables. In classical decision theory, the known information is used to decide on the decision variable d. The decisions taken are purely based on known deterministic information.

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The second school, the Bayesian decision theory school, is based on two principles: • All assumptions are explicitly formulated using probability statements about the joint distribution of future relevant events. • All decisions are made conditional on probability weighted future events, called the prior. Although Bayesian decision theory is superior to classical decision theory on paper, its use can be severely limited by the technical obstacles of formulating the prior, that is, the future expectations on which the decision is to be conditioned.

6.2 BUILDING A DECISION MAKING PROCESS Depending on the forecasts available, either classical or Bayesian decision theory is the most appropriate to use. The key of any decision making process is to have well defined interfaces in place that allow efficient information communication as well as a systematic decision making mechanism. 6.2.1 Communicating information Once forecasts have been determined by the investigators, they need to be communicated to the decision maker. The possible communication approaches can be classified along two dimensions, that is: • degree of formalism, how structured the forecasts and associated information are described, and • communication initiator, who is in the lead for the information transfer, the investigator or the decision maker. These two dimensions are illustrated in Figure 6.3. Along the dimension of degree of formalism, different communication tools can be used. The most appropriate medium for informal communication approaches is speech or open discussion supported by a visualization medium,1 like a blackboard. Slides or overhead projector based presentations allow some formalism to be introduced into the communication process. The investigator is required to pre-formalize the key information to be communicated. Nevertheless, the advantage of open information exchange, as in speech or discussion based media, is not completely lost. The most formal communication medium is the written self-contained report. The investigator must address all possible questions that may arise. The feedback from the decision maker to the investigator is removed. There is no one single medium that is best. The medium must be adapted to the overall communication approach selected. Highly formalized communication approaches are most appropriate when large amounts of detailed information are available. Such forecasts could be return expectations based on 1 The visualization medium allows the potential language barriers to be reduced between the investigator and the decision maker.

84

Positive Alpha Generation: Designing Sound Investment Processes Communication medium

pull/active

slide presentation

report

effective

ion

recis

of p evel

l

push/passive

communication initiator

speech

efficient informal

formal degree of formalism

FIGURE 6.3 Forecasting communication approaches

discounted cash flow (DCF) models. Assumptions made on costs of capital, consumption growth, etc., as well as the level of detail of all cash flow forecasts make a formal communication approach preferable. If the decision maker uses a Bayesian decision theory approach, the information used to formulate the prior on which the decisions are conditioned can be easily described. On the other hand, if the forecasts on which the decision is made are of a high level nature, informal communication approaches focused on providing a big picture or key message are more appropriate. A second important dimension in the communication process is whether the forecasts and associated information are transferred using a push/passive or pull/active approach, that is who is the initiator of the communication. In a push type approach, the forecasts are pre-structured by the forecasting provider and the communication is initiated by the investigator, thus giving a push communication approach. As the provider is the expert, push approaches tend to focus on the most relevant information and are very timely. Forecasts are structured by their importance and urgency. This makes push based communication approaches very efficient. Unfortunately, they also have a major disadvantage. The investigator may be biased towards his or her own view on what the decision of the decision maker should be. For example, the earnings investigator in an asset allocation based investment process may provide as input that US equities will have a positive expected return, the main arguments being the expected improvements in consumption rates of private consumers. If the investigator has a subjective view, explicitly or implicitly, that US equities will perform well, potential risks to the forecast may not be communicated, for example that a slowdown in exports will have a significant negative impact on earnings growth and thus bias the probability of positive equity returns in the United States. Pull based approaches rely on the decision maker initiating the communication process. The decision maker asks for the forecasting information he needs to make his investment decision. The potential bias introduced by the forecaster is thus eliminated.2 As the decision maker is in the driving seat, 2

Assuming that the investigator is not withholding relevant information on purpose.

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85

such an approach is highly effective. The focus is on the forecasting information perceived to be relevant to the investment decision. The main drawback of pull based methods is that some available forecasting information may be missed because it is not requested by the decision maker and the timing of information availability is not actively managed. I believe that the most appropriate decision making results are obtained by combining a formal push approach with a subsequent informal pull method. In addition, in order to provide the consistency check feedback loop, the roles of the investigator and the decision maker should temporarily be inverted. The investigator will apply an informal pull approach to question the forecasting information processed by the decision maker. This role exchange process may be iterated until the decision maker is sufficiently confident to make his final investment decision. An information communication setup could be that on a regular basis, for example monthly, the investigators and decision makers meet. Each investigator presents the findings and the resulting forecasts. The decision makers then ask questions and potentially challenge the investigator’s forecasts. At the end, the decision makers, independent of the investigators, make a decision based on the presented and discussed findings. An alternative setup could be that each member of the decision team is allocated one or more investigators. Assuming no conflicts of interest and adequate skills, one person may eventually take on both roles, the investigator and the decision maker. In this setup the information transfer becomes a one-to-one process. After the information transfer has taken place, the decision maker presents the findings from his or her investigator and is challenged. Finally, a decision is taken. The latter setup, although more complex, provides a better buy-in of the decision makers as they have to some degree to represent an investigator. Numerous other setups are possible. It is the investment manager’s overall setup that determines the most adequate approach. 6.2.2 Deciding Once all forecasting information has been transferred from the investigator to the decision maker, a decision has to be taken. Mechanisms I distinguish between three types of mechanisms to take an investment decision based on the communicated information. Democratic. Each decision maker is given a vote to express his or her utility maximizing decision. The votes are then tallied according to a predefined scheme and the final decision derived. The voting mechanism may either be open or secret. Not disclosing the individual’s decisions assures independence for future decisions, whereas full disclosure makes the decision process transparent and thus increases confidence in the decisions. There are many ways to count and aggregate votes. The most common approaches are plurality voting (each participant votes for one single decision and the decision with the largest number of votes wins), antiplurality voting (each participant votes against a single decision and the decision with the smallest number of votes wins), ranking voting (each participant ranks the available decisions), approval voting (each participant votes for one or more decisions and the decision with the largest number of votes wins), cumulative

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Positive Alpha Generation: Designing Sound Investment Processes

voting (each participant associates one or more points to each decision and the decision with the largest number of points wins) or Borda count voting (each participant associates a percentage or weight to each decision and the decision with the highest total weight wins). Unfortunately, under some reasonable restrictions, and as long as more than two decisions are available for voting, it is always possible to construct a vote aggregation approach, which leads to any of the available decisions to be selected as a winner without changing the actual votes (Saari, 2001a). Monarchic. A single decision maker takes the utility maximizing decision, who may or may not take into account all information provided. Even though monarchic mechanisms are among the simplest ones to implement, a distinction should be made between those who seek a certain commitment from the investigators and those who take their decision in an autonomous way. My experience has shown that the former approach assures the highest confidence in the decisions, especially from the professionals involved in the client reporting value chain element. Formulaic. The task of the decision maker is to aggregate all provided information using a formulaic or algorithmic approach. This role is limited to assuring the validity of the information and performing the aggregation. Formulaic approaches are most common in quantitative decision processes. They are also very often used as a concise way to combine qualitative information. In the latter case, it is important to ensure that the combination mechanism is chosen according to a sound process, ideally using an approach similar to that of designing a quantitative forecasting model, as illustrated in Chapter 5. The formulaic approach can be defined: • as a fixed a priori formula, for example using equal weights for all decisions, • based on prior information, as Bayesian decision theory would suggest or • based on historical quality, correctness or performance of the decisions, for example using regression techniques. There is no single best mechanism. Even worse, Arrow (1963) has shown that all nonmonarchic decision processes are plagued with basic flaws; that is the decision outcome more accurately reflects the choice of the decision process rather than the views and preferences of the decision maker. The ideal decision making process does not exist. Therefore, depending on the forecasting as well as information processing capabilities, including the utility function associated with the decision, different approaches are sound. In addition, the organizational structure implemented around the decision making process plays an important role. Consider the following example. The goal of the designed investment process is to avoid losses due to incorrect investment decisions. This goal translates into a utility function that minimizes the probability of an incorrect decision. In addition, consider a seasoned team of full time investment professionals in charge of forecasting and providing information. In this example, a democratic decision mechanism based on a two-thirds majority decision is implemented. Assuming that the probability of a decision being correct is directly related to the number of investment professionals supporting it, the democratic decision mechanism maximizes the defined utility function. The two-thirds majority ensures that sufficient confidence exists before a decision is actually taken.

Taking Investment Decisions Table 6.1

Organizational structures implementing the decision mechanism

Organizational structure

Single person One-dimensional hierarchy Matrix/two-dimensional hierarchy Workgroup Teama Anarchy a

87

Number responsible

Number involved in decision

Number of investigators

1 1 2 1 n 0

1 1 2 n n n

1 n n + n n n n

I use the definitions of workgroup and team given by Katzenbach and Smith (1993).

Organizational structures The organizational structure implementing the decision mechanism plays a key role in its success. Table 6.1 summarizes possible organizational structures as well as the number of involved parties. My experience has shown that the best decision quality is obtained if there exists: • a single person responsible for the decision, • a small team of between five and 10 experts involved in supporting the decision making process and the decision maker, and • investigators who focus on quality rather than quantity. Having a single person responsible for the decision is advantageous as it allows: • a decision to be made even if multiple possible contradicting decisions exist, that is avoiding any incapability to take a decision, and • the avoidance of a situation where all decisions are consensus decisions, that is allowing clearly differentiated views. Furthermore, working and deciding in a team environment allows challenging individual thoughts and thinking ‘out of the box’. This works best in a well balanced environment that supports the free expression of thoughts, that is no possible decision is wrong a priori. As with selecting the right number of factors when building a qualitative forecasting model, a choice must be made between quality and quantity. Table 6.2 summarizes the key characteristics of the different organizational structures as well as their complexity. In many organizations the forecasting and information processing capabilities are regrouped in the research department. Investment management companies using sophisticated investment processes or managing multibillion portfolios have dedicated research teams, called buy-side research. Their sole role is to support the decision making process and therefore the decision makers. Smaller organizations, especially those also providing advisory services, sometimes use the research team’s output for multiple purposes. It is important to note that such a setup may lead to conflict of interests as the research output, that is the forecasts, are used by competing entities. This issue can be addressed by ensuring that the decision maker is not bound to decide solely based on the forecasts from the research teams; that is the research teams only provide a subset of the information f .

88 Table 6.2

Positive Alpha Generation: Designing Sound Investment Processes Key characteristics of organizational structures with respect to decision mechanisms

Organizational structure

Complexity

Preferred mechanism

Efficiency

Key success factors

Single person

Low

Monarchic

High

One-dimensional hierarchy

Low

Monarchic

High

Matrix/twodimensional hierarchy Workgroup

Medium

Democratic

Medium

Medium

Democratic

Medium

Team

High

Democratic

Low

Anarchy

High

Formulaic

Low

High expertise of individual Communication along hierarchy Willingness to listen to input Aggregation expertise of decision maker Team approach responsible for the matrix dimensions Complementary skills Leadership of workgroup leader Compromise seeking Complementary skills Openness for alternative decisions Willingness to support decisions None

6.3

EXAMPLE

Consider the three equity market asset classes Brazil, Russia and China. Consider a team of four seasoned emerging market investment professionals formulating pairwise forecasts between those three markets. A set of sample forecasts is illustrated in Table 6.3. I assume that the investigators are also the decision makers. I allow for omissions. I define the utility function to be maximized by the decision maker as maximizing the number of correct pairwise forecasts, given the input by the investigators. Each investigator provides his pairwise forecasts and specifies a confidence level associated with each of the provided pairs. Possible confidence levels are low and high. In addition, qualitative arguments are provided supporting the individual forecasts, which I Table 6.3 Sample pairwise market forecasts from four investment professionals: Jimmy, Ronald, George and Bill Investigator

Forecast A

Forecast B

Jimmy Ronald George Bill

Brazil outperforms Russia Russia outperforms Brazil China outperforms Russia Russia outperforms Brazil

Brazil outperforms China Brazil outperforms China – Russia outperforms China

Taking Investment Decisions Table 6.4

89

Sample voting decision mechanism based on the forecasts shown in Table 6.3

Decision maker

Jimmy Ronald George Bill Total voting score

Voting weight

Brazil > China

12 8 11 15

0.5 0.5

China > Brazil

Brazil > Russia

Russia > Brazil

China > Russia

Russia < China

0.5 1.0 1.0

10

0

6

1.0 23

11

0.5 7.5

do not illustrate explicitly. This approach allows for the Bayesian decision theory to be used by associating the specified confidence level to the information prior. To transfer the forecasted information to the decision maker, I set up a monthly meeting where each investigator presents his forecasts and underlying arguments through a presentation followed by a question-and-answer session. This approach tries to combine the advantages of a push based communication mechanism with a pull based approach through the question-and-answer session. This setup is very adequate for a small team of experts. As already stated, I rely on a team based organizational structure. Each of the experts has the same decision power and is hierarchically at the same level. In order to come up with an investment decision I use a democratic decision mechanism based on the following skill based voting system. Each investigator is requested to vote for any of the six possible pairs of equity stock markets. The votes are weighted according to a personal voting weight associated with each decision maker. In addition, a low confidence vote counts 0.5 times the voting weight and a high confidence vote counts 1.0. Initially the voting weight is set to 10 for all team members. Each time a decision maker takes a correct decision, his voting weight is increased by 0.5, respectively 1, depending on the confidence of the forecast, up to a maximum of 20. If the decision maker takes an incorrect decision, his score is decreased by 0.5, respectively 1, depending on the confidence of the forecast, down to a minimum of 0. Table 6.4 illustrates possible votes and the associated scores. The consolidated investment decision is derived by considering the pairwise forecasts with the highest scores shown in Table 6.4, until an inconsistency is detected. In the shown example, this leads to the investment decisions that: • Russia outperforms Brazil and • China outperforms Russia.

Part III Risk Measurement and Management

7 Modeling Risk Risk management represents, together with forecasting market returns, one of the key components of a successful investment process. It supports the investment manager in generating alpha, transferring investment skills into the portfolio and avoiding unintended sources of risk. Risk management provides a tool to manage the uncertainty of alpha. It deals with the impact of possible future market outcomes on the portfolio values. There are three viewpoints to investment risk, that is: • the investor’s perspective, • the investment manager’s perspective and • the controller or regulator’s perspective. In designing an investment process the focus has to be set on all three viewpoints. Once the investment process is in place, the key focus must be on the investment manager’s perspective, that is: • manage the risk of incorrect market forecasts or the impact of market outcomes, given active investment decisions, on the portfolio value and • mitigate the impact of unintended or residual risk. The controlling approach to risk allows risks to be monitored that are supposed to be managed by the investment process, like exposure to specific assets, asset classes or asset characteristics. These risks are not actively linked to investment decisions, but are unintended consequences of them. One of the key focuses of the controlling viewpoint to risk is to detect and manage the investment process soundness risk.

7.1

THE DIFFERENT DIMENSIONS OF RISK

When looking at an investment manager’s main objective, that is generating consistently positive, ideally risk adjusted, alpha through implementing a sound investment process combined with specific forecasting skills, eight key dimensions to risk can be identified: (1) Market risk. Market risk represents the uncertainty inherent in any future asset price. It is through relating forecasts to these risks that an investment manager aims at generating positive alpha. In addition, it represents the key risk an investor is faced with when holding assets. (2) Counterparty or credit risk. Counterparty risk is the risk faced when entering into a transaction or agreement that the counterparty will partially or completely fail on its obligations. In some situations, counterparty risk is tightly related to market risk, for example when the issuer of a bond defaults1 the price of the bond drops. Counterparty 1

The issuer files bankruptcy and thus is no longer able to repay the notional of the bond at maturity.

94

(3)

(4)

(5)

(6)

(7)

(8)

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risk may also be market risk independent. For example, a counterparty with which a currency forward agreement was entered into fails to meets its payments has no impact on the currency exchange rate. Liquidity risk. Two types of liquidity risk exist, that is (i) market liquidity risk and (ii) cash liquidity risk. I define market liquidity risk as the risk stemming from holding an asset for which potentially insufficient buyers exist at a given point in time, especially in times of crisis. Alternatively, the buyers may only be willing to buy the asset at a substantial discount of its fair value. I have devoted part of Chapter 13 to describing the details of market liquidity risk and how to manage or mitigate it. The second type of liquidity risk, called cash liquidity risk, is the risk of holding insufficient cash to meet the obligations of the contractual agreements, for example satisfying a margin call as described in Chapter 14. Market and cash liquidity risk can be seen as being the complement or the consequence of each other. Information risk. Positive alpha generation is based on forecasting skills. Forecasting skills exist if the investment manager either (i) has superior information relative to other market participants or (ii) is able to process information in a superior way. Information risk is the risk of information or its processing being incorrect or not forecasting the correct outcome of markets. Information risk is the key risk dimension that an investment manager has to manage. For example, correctly forecasting whether the Roche stock will out- or underperform the Novartis stock is relevant to the investment manager and his or her resulting alpha generating capabilities, rather than the relative performance between those two stocks. Information risk is nevertheless tightly related to market risk. It can even be seen as a consequence of the latter. Investment process risk. Investment process risk is the risk of systematic errors resulting from the design of the investment process. For example, a sector based asset allocation investment process may systematically introduce an unintended country exposure bias to countries with a low credit rating. Investment process risks cannot be handled within the investment process itself. They must be addressed at a meta level, during the design of the investment process or through external controlling mechanisms. Model or modeling risk. Whereas investment process risk is mainly related to the process aspects, model or modeling risk is due to the potential mistakes in the mathematical models used or their implementation. In an investment process it can be found in three areas, that is (i) asset pricing, (ii) risk measurement and (iii) portfolio construction. In Chapter 5 I described how to mitigate model risk when designing a return forecasting model. Those techniques can also be used and adapted to manage risk model risk. Operational risk. I define operational risk as the uncertainty associated with the execution of any operations related to but not included in the investment process. A typical operational risk is the settlement risk, that is the risk that a transaction is not correctly booked. Legal and reputation risk. Legal and reputation risk includes eventualities of claims from investors due to incorrect or seen as incorrect executions of contractual agreements. It

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95

not only includes the risk of mistakes, but also the delivery of poor quality, which results in opportunity costs. In this book, I essentially focus on information risk, market risk and the interaction between the two.

7.2 RISK MANAGEMENT FROM AN INVESTOR’S PERSPECTIVE A rational investor aims at maximizing the personal utility function. In general this utility function can be expressed as a combination of (i) expected return and (ii) aversion to losses. It is a concave function that increases with wealth. In the traditional mean-variance or Markowitz (1952) model, the investor’s utility function U (R) is defined by the following equation: U (R) = E[R] − γ · σ 2 (R)

(7.1)

where E[R] is the expected return, σ 2 (R) the expected variance or volatility of the returns and γ the risk aversion parameter. Grauer and Hakansson (1993) have shown that Equation (7.1) reasonably approximates the utility function of an expected utility maximization investor over a short investment horizon. In addition, investors expect risk management to: • avoid frequent small losses and/or • avoid large but unlikely losses. Usually an investor’s utility function includes a combination of both. Rosengarten and Zangari (2003) describe the implementation of a sound risk management process as the basis for producing portfolio returns that: (i) (ii) (iii) (iv)

meet the investor’s expectations, are derived from the investment manager’s skills, are not the result of luck and are stable, consistent and controlled.

7.3 RISK FROM AN INVESTMENT MANAGER’S PERSPECTIVE Risk, from an investment manager’s perspective, means the uncertainty of excess return or alpha. Lee and Lam (2001) distinguish between two types of risk regarding the uncertainty of alpha: • Statistical risk. Statistical risk is a statistical measure of the uncertainty of the expected alpha, like standard deviation, value at risk or semi-variance. • Information risk. Information risk is the risk related to the quality of the information or information processing advantage over uncertainty, like the confidence level of generating positive alpha.

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Any sound investment process needs to manage information risk actively. Statistical risk measures should be applied. One of the most adequate approaches to managing information risk is using a risk budgeting approach. Based on the investor’s utility function, a certain risk budget, usually defined using statistical risk measures, is given to the investment manager. The investment manager then generates investment decisions. Information risk management aims to allocate portions of the statistical risk budget of the investor to the individual investment decisions. A good information risk measure is the degree of confidence in the investment decision being correct. Information risk management is closely related to the portfolio construction module of the investment process’s value chain. A risk model should be such that it is consistent with the return forecasting model. Only this will allow information risk to be modeled and managed, that is the impact of incorrect investment decisions on the portfolio returns.

7.4 THE THEORY BEHIND MODELING MARKET RISK I will now focus on modeling market risk, which serves as the basis of modern portfolio theory and is tightly related to investment risk. In Chapter 2, I defined risk as the uncertainty of achieving an expected outcome, observed through the variability from an expected return. In the context of an investment process, the risk model measures the uncertainty of achieving a return target, whether in absolute or relative terms, for a given portfolio. Risk is always conditioned on the portfolio holdings. Let w denote the portfolio holdings, either expressed in absolute terms, that is in dollar terms, or in relative terms, that is in percentages. Let γ be a vector of parameters defining ˆ be the investor’s specific utility or risk aversion function, including its time horizon. Let  the estimated or forecasted parameters of the risk model defined. The absolute risk or total ˆ Let b denote a reference portfolio or risk of a portfolio is denoted by a function σ (w, γ, ). benchmark portfolio. The relative risk, active risk or tracking error of a portfolio is denoted ˆ In general, by a function σ (w, b, γ, ). ˆ = σ (w − b,γ, ) ˆ σ (w, b,γ, ) The relative risk is also denoted by ˆ ≡ σ (w, b,γ, ) ˆ τ (w,γ, ) when the associated reference or benchmark portfolio is clear from the context. Let wF be a portfolio investing in a risk free asset over the time horizon in which risk is measured. It is a good practice to define the absolute risk as the risk relative to the risk free portfolio, such that ˆ ≡ σ (w, wF , γ, ) ˆ σ (w,γ, ) ˆ are omitted in the notation of risk and If it is clear from the context the parameters γ and  the absolute risk is denoted by σ (w) and the relative risk√ by τ (w)σ (·) may either be defined as an analytical function, like, for example, σ (w,
) = w ·
· w, or defined through an

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97

ˆ = function σ (w,γ ,) r(t) ← historical return for month t between 1997 and 2007 of the portfolio w r(t) ← remove all nonnegative return values from r(t) sort(r(t)) return abs(r( 10/n))3 where n is the number of remaining values r(t)

FIGURE 7.1 Sample algorithm for calculating the portfolio risk

algorithm,2 as, for example, shown in Figure 7.1. Given a definition of risk and a context, that is an investment process, it is the goal of the risk model definition process to formalize that loss or portfolio value function and test it for soundness. One key aspect when modeling market risk is that there is no one best model. Designing a risk model corresponds to making educated choices conditioned on the use of the risk model in the investment process. Any risk model must be seen as a means to an end. For example, in a Markowitz (1952) mean-variance efficient frontier based investment process, the volatility or standard deviation of the expected returns is a reasonable way of modeling risk. In the case of avoiding low probability extreme losses, a risk model using a generalized extreme value distribution (Embrechts et al., 1997; McNeil et al., 2005) is an appropriate risk model. In fixed income portfolios, duration sensitivity is used as a risk model to assess and manage the exposure of the portfolio’s value to changes in interest rates. 7.4.1 An axiomatic approach to modeling risk Artzner et al. (1997, 1999) defined a set of four desirable properties that any risk model should satisfy. Let ᑰ denote the set of random variables representing the value or return of given portfolios at a given future date T and let ρ(·) define a coherent risk measure. The four properties are: • Monotonicity axiom. More wealth implies more wealth at risk: ∀ᑲ, ᑳ ∈ ᑰ : ᑲ
ᑳ ⇒ ρ(ᑲ)
ρ(ᑳ) • Translation invariance axiom. Risk declines when the proportion of risk free asset increases: ∀ᑲ ∈ ᑰ, f ∈ R+ : ρ(ᑲ + f ) = ρ(ᑲ) − f where f represents the risk free asset. • Positive homogeneity axiom. Wealth at risk is proportional to the overall wealth: ∀ᑲ ∈ ᑰ, λ ∈ R+ : ρ(λ · ᑲ) = λ · ρ(ᑲ) 2 3

An algorithm is a finite sequence of calculation steps for accomplishing some task, like calculating a risk measure.

• denotes the floor, that is the integer rounded down.

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Positive Alpha Generation: Designing Sound Investment Processes

• Subadditivity axiom. The aggregated risk of two assets is always smaller or equal to the sum of the risks of each asset: ∀ᑲ, ᑳ ∈ ᑰ : ρ(ᑲ + ᑳ)  ρ(ᑲ) + ρ(ᑳ) Any risk model satisfying these properties is called a coherent risk measure. Note that the positive homogeneity and subadditivity axioms imply that the risk measure ρ(·) is convex. 7.4.2 Decomposing market risk It is always possible to decompose the portfolio’s market risk into two categories, that is: • systematic risk and • specific risk, given a certain model. This model is in general either a market model, like the CAPM, or a risk factor model, like duration. Systematic risk is often associated with uncertain but expected cash flows generated by the assets in the portfolio. It is a common assumption that the investor will be rewarded with return for taking systematic risk. Specific risk, on the other hand, is not rewarded as it can be removed from the portfolio through the effect of diversification. Furthermore, these two categories can again be subdivided into general uncertainty and event risk. This decomposition is illustrated in Figure 7.2. When designing an investment process, it is important to understand the decomposition of risk and especially for which risk the market is expected to pay a return premium over the considered time horizon. An investment process should focus on: • budgeting risk to assets and subportfolios that can be classified as systematic risk functions and • eliminating specific risk through the effect of diversification. Be aware of the fact that exposure in a portfolio may be classified as contributing to specific or systematic risk depending on the selected model, which again is closely related to the investment decisions taken. For example, if a CAPM is assumed, uncertainties of corporate earnings contribute to the category of specific risk that should be diversified away by holding a large number of securities. On the other hand, if the investment process focuses on forecasting corporate earnings, the forecasted earnings become the driving factor and Portfolio risk Systematic risk Market risk

Specific risk Event risk (terror attacks, liquidity crisis . . .)

Idiosyncratic risk (company specific)

Event risk (bankruptcy ...)

FIGURE 7.2 Decomposition of the portfolio’s market risk for a market model according to its characteristics

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99

make up the portfolio’s systematic risk. Indeed, I would expect to be rewarded with excess return when holding companies with high expected corporate earnings.

7.5 THE PROCESS OF DEVELOPING A RISK MODEL Any risk model can be decomposed into three key components, that is: • the environment in which the model is applicable, • the mathematical model specification and • the parameters of the mathematical model. The three components are closely interrelated. When developing a risk model, I start by formulating the environment in which I want to use the model. This means: (i) defining for which assets and at which level of detail I want to use the risk model, (ii) specifying the surrounding investment process and its needs regarding the risk model’s usability and (iii) determining the availability of data that can be used to estimate the model’s parameters or the skills available to forecast them. Consider an asset allocation based investment process calculating asset class weights such that a defined utility function is maximized and the risk of the asset allocation is not larger than a given constant. From this requirement specification, I can derive the level of detail required as being the asset class level. As the risk model will be used as a component of an optimization procedure, it must allow for a fast evaluation and convexity. Finally, index total return data would be available to estimate the model’s parameters. The second component of structuring the risk model, that is defining the mathematical model, can be subdivided into two subcomponents, that is: (i) defining the assumptions of the model and (ii) formulating the mathematical structure of the model. These tasks are in general preceded by an observation of the real world, that is the assets and portfolio returns whose risk is to be modeled. Consider the asset allocation example. The real world, it can be said, consists of asset class returns and the risk to be modeled to the variability of them. I observe the end of month index level data associated with each asset class and thus derive monthly total returns. Analyzing the structure of these returns, I assume that they follow a multivariate normal distribution. This assumption allows the mathematical structure of the risk model to be defined as σ (w) =

√

w ·
· w

where w is a vector of weights in the modeled asset classes and
the covariance matrix of the multivariate normal distribution. Under the assumption that the covariance matrix
is positive semi-definite, σ (w) is a coherent risk measure and satisfies the properties of efficient evaluation and convexity.

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Once the model is defined, I need to estimate the parameters of the model. In my example, this is the covariance matrix
. Determining a parameter estimation algorithm can be subdivided into two subparts, that is: (i) the definition of the estimation algorithm, usually maximizing a statistical likelihood estimator, and (ii) the selection of the data to be used as input to the estimation algorithm. It is important to select the estimation algorithm based on the available data and its properties. In many cases, the data needs to be pre-processed before being usable. There is a close link between the data used, the assumptions made and the estimation algorithm. As for estimating the parameters of a return forecasting model, as described in Chapter 5, the choice of the estimation technique is a trade-off between statistical efficiency, misspecification of the model, clarity and usability. To estimate the covariance matrix
in my example, I use monthly total return data that I assume to be independent and identically distributed (i.i.d.). I rely on the traditional sample covariance matrix estimator. Under the defined assumptions of i.i.d.,
is an unbiased maximum likelihood estimation estimator (McNeil et al., 2005). I select a data set of 10 years monthly historical total return values assuming that they are independent and follow the same distribution. As can be seen from Figure 7.3, the risk model development process is not a linear process. Interactions between the different process steps need to be considered and iterative updates made. Consider the three key steps of the process of developing a sound risk model in more detail. Investment process environment Mathematical model structure

Assumptions

Mathematical model Real works markets

Observations of the markets

Data

Parameter estimation

Parameter estimation algorithm

FIGURE 7.3 The process of developing a risk model

7.5.1 Investment process environment The first step in designing a risk model is to define the environment in which the risk model will be used, that is the assumptions made by the underlying investment process. The risk model may be used either for measurement purposes or for taking active investment decisions, that is for managing risk. Measurement-only approaches are much simpler to

(i) Risk measurement based investment process

101

Investment decision

(ii) Risk management based investment process

Modeling Risk

Investment decision

Portfolio structure

Risk management

yes OK no

Risk management

Portfolio structure

FIGURE 7.4 Integrating a risk model into an investment process using (i) a feedback loop approach and (ii) a risk management approach

develop as they are only loosely coupled with the remaining components of the investment process. They can be integrated into an investment process through a feedback loop, as shown in Figure 7.4. Due to the fact that the output of a risk measurement approach is a risk figure, more complex calculation methods can be implemented than in a risk management approach. In the latter, the output is not only a risk characterization but also a portfolio structure that is in line with the targeted risk profile. Furthermore, the calculated risk figures may only be needed at the aggregated portfolio level. Historical simulation based risk models can be used. However, more often the calculated risk needs to be decomposed. A distinction can be made between three types of risk decomposition approaches: • Marginal contribution to risk. The sensitivity of the total risk relative to (i) the individual assets or asset classes, (ii) the taken investment decisions or (iii) external factors, like interest rates, is measured. Mathematically speaking, the sensitivity is calculated as the first partial derivative of the risk measure relative to the considered factors. For example, the √ marginal contribution to risk of a volatility based risk model, defined by σ (w) = w ·
· w relative to the asset class a, is calculated as  ∂σ 2 = wa · (σa,b · wb )/σ ∂wa A

b=1

where w is a vector of asset class weights and σa,b the elements of a covariance matrix. • Factor exposure. The exposure to exogenous factors, like interest rates or GDP growth, or endogenous factors, like assets or asset classes, is calculated. Each factor is associated with a risk contribution, allowing a decomposition of the portfolio’s risk relative to the different factors.

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• Total risk of components. The total risk of subcomponents of a portfolio is calculated. The total risk of the portfolio equals the total risk of each subcomponent minus the risk reduction due to the diversification effect of combining multiple assets in the portfolio. For example, in a two-asset portfolio, using volatility as the risk measure gives σP2  portfolio risk

= w12 · σ12 + w22 · σ22 + 2 · w1 · w2 · σ1 · σ2 · ρ1,2          component 1

component 2

diversification effect

where wa is the weight in asset a, σa its volatility or standard deviation and ρa,b the correlation between assets a and b. Both the marginal contribution to risk and the factor exposure risk provide the same type of result. They only distinguish themselves by the approach used. The marginal contribution to risk starts in a top-down way and decomposes the total risk into its individual components. Factor exposure risk uses a bottom-up approach combining the exposure to different factors with their corresponding risk sensitivities. A second important decision to be taken is whether the risk model should be conditional or unconditional. Unconditional or ex-post risk models measure the uncertainty, that is variability, as it has happened or would have happened in the past. Conditional or ex-ante risk models forecast risk conditioned on all information currently available. In a sound investment process, only conditional risk models should be applied. Furthermore, it must be decided whether the risk model: • assumes the current portfolio structure over the time horizon where the risk is forecasted or • assumes a trading strategy, like constant readjustments. Usually a current portfolio structure based approach is appropriate. However, if the time horizon of the risk forecast extends beyond the investment decision time horizon or a predefined portfolio rebalancing horizon, a readjustment strategy should be taken into account. A third approach would be to decompose risk over time such that, for any time period, current holdings can be assumed. The third decision to be taken when designing a risk model is that of time. Assuming a conditioned risk model, the time horizon for which the risk forecast is valid must be defined. Ideally the risk forecasting time horizon is identical with the time horizon of the return forecasts. Alternatively, the time horizon may be fixed to the period over which the portfolio structure remains unchanged. The time horizon may also be defined as the period over which the investor is interested in returns, allowing for intermediate losses to be recovered. When taking a decision about the risk forecasting time horizon it is important to note that return accumulates geometrically over time whereas risk accumulates with the square root of time under the independence and normality assumption. Finally, it is important that the risk modeling approach chosen is consistent with the return forecasting approach, both: • in the mathematical structure and • in the parameter estimation approach selected.

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For example, if equity return expectations are forecasted on a region-by-region basis, then the risk model should also rely on a regional approach, rather than a sector approach. Inconsistencies may also occur if a risk model is estimated based on historical mean returns that significantly differ from the forecasted ones. 7.5.2 Mathematical model A large number of sound mathematical structures exist that can be used to design a market risk model. Figure 7.5 illustrates different approaches that can be taken and a distinction is made between parametric and nonparametric model structures. In parametric models, risk is modeled based on random variables and statistical theory. The random variables are either defined through their whole distribution using, for example, re-sampling techniques to define the distribution. More common are models where a given analytical distribution of the random variables is assumed, such as a Student-t or a normal distribution. The parameters of the distribution, rather than a sampling of the distribution, are used. The parameters are estimated using statistical estimation algorithms. In general, they are based on maximizing a likelihood estimator function. The parameters are assumed to be either constant, conditionally or unconditionally, or expected to follow a well defined process. The parameters of that process are then assumed to be constant. The most common processes used in modeling risk are GARCH-like processes. They allow clustering effects to be taken into account as well as mean reversion properties. A third class of parametric models are so-called factor models. Each factor represents a specific risk characteristic. Combining exposure to the factors with their associated risks allows the risk of a portfolio to be defined.

Full distribution Constant Moments of distribution Process

Parametric

Market factor Factor decomposition

Economic factor

Risk model Ad hoc factor Historical simulation Nonparametric

Scenario analysis Stress testing

FIGURE 7.5 Decomposition of the universe of possible mathematical structures for a market risk model

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In contrast to parametric models, nonparametric models do not require any assumptions to be made about the structure of risk. The risk structure is rather a consequence of the model. The most common nonparametric risk modeling approach is the historical simulation model. The risk is asserted by simulating how a given asset or portfolio would have performed over a given historical time period. In the stress testing as well as the scenario analysis approaches, hypothetical asset returns are used. They are either based on actual historical market scenarios, like the burst of the internet bubble, or on a market model, like a factor model, to which sets of input parameters, so-called stress situations or scenarios, are generated. Again, the resulting portfolio returns are studied. 7.5.3 Parameter estimation Estimating the parameters of a risk model is very similar to estimating the parameters of a return forecasting model, as described in Chapter 5. The key steps to be followed are: • The data used for estimating the model’s parameters must be pre-processed. Spurious outliers should be removed, missing data substituted and disturbing characteristics, like trends, filtered out. • The algorithm chosen to estimate the parameters should represent a sound mix of statistical accuracy, calculation complexity and understandability. It should be adequate in the context in which the model is used. For example, when estimating a covariance matrix to be used as input to an optimizer, it is important that the matrix is not singular and numerically stable. Special estimation algorithms may be used to cope with specific characteristics of the data used, like autocorrelation or heteroskedacity.

7.6

INFORMATION RISK

I define information risk as the risk that investment decisions taken lead to negative alpha contributions in the portfolio. There are three levels of detail at which information risk exists, that is: (1) at the information gathering level through estimating or forecasting the probability of the information being correct or incorrect, (2) at the information processing and aggregation level through estimating or forecasting the probability of the decision taken, that is the resulting forecasting being correct or incorrect, conditioned on the available information, and (3) at the market level through estimating the probability that the investment decision can be mapped to a positive alpha. Figure 7.6 illustrates the three process steps. Information risk is closely related to the fundamental law of active management, as illustrated in Chapter 2. It measures the uncertainty of the information coefficient (IC). In addition it takes into account the interrelationship between the different pieces of information as well as the resulting investment decisions. For example, an investment manager may have better skills at forecasting equity markets

Example

Approaches to be used

Process step

Modeling Risk

105

Information gathering risk

Impact of information risk

Qualitative Judgmental

Bayesian inference Historical data Judgmental Correlations/ causalities

Pr{rates drop} = 60% Pr{GDP>4%} = 54%

Pr{equities outperform bonds | rates drop and GDP > 4%} < 40%

Information processing risk

Statistical Process steps Historical data Market model

Pr{a > 0}

FIGURE 7.6 Approaching information risk

than fixed income markets. The risk of the information coefficient associated with equity decisions is thus smaller than the one for fixed income decisions. An investment manager may have gathered more convincing information that interest rates will rise in the United States than in Japan, leading to a higher expected probability of the forecast being correct for the United States than for Japan. Consider an investment manager taking the following investment decisions: • US equities will outperform USD bonds over the next three months and • European equities will outperform EUR bonds over the next three months. Information risk management must take into account the fact that the first two forecasts are highly interdependent; that is if US equities outperform USD bonds, then the probability is very high that European equities will also outperform EUR bonds. A key issue when modeling information risk is that statistical techniques modeling the probability distribution functions usually fail. Indeed, there is generally insufficient historical data available. This is essentially due to the interdependence of the different information gathered and decisions taken. As the goal of the investment manager is to generate positive alpha consistently, there must be a tight dependence between the realized probabilities and information risk. Furthermore, the information gathering and processing process incorporates explicitly or implicitly a learning process that eventually decreases the information risk probabilities. 7.6.1 Information gathering risk Asserting the risk associated with gathering superior information is highly qualitative and judgmental. It is usually related to the degree of confidence of the investment manager in a given piece of information. It may also be related to the amount of similar information gathered. For example, an investment manager whose selected information set was correct in 55 % of the monthly periods over the last five years may be allocated a hit ratio of 55 %, assuming that his or her capabilities in the future relate to those in the past.

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7.6.2 Information processing risk Similar techniques to those for information gathering risk can be applied. In addition, Bayesian inference theory should be used to derive confidence or risk levels. Historical data can be used to calculate the associated conditional probabilities. For example, over the 20-year time horizon between 1987 and 2007, US large capitalized stocks, modeled by the S&P 500 index, outperformed US small capitalized stocks, modeled by the Russell 2000 index, in 47 % of cases, when interest rates, modeled by the 1-month LIBOR rate, dropped, that is Pr{rUS

large capitalized stocks

> rUS

small capitalized stocks |interest

rates drop} = 47 %

Correlations and causalities among investment decisions must also be taken into account. 7.6.3 Impact of information risk Finally, the information gathering and information processing risks must be related to their impact on the portfolio performance. Statistical models combined with transformation steps as well as market models can be used. For example, consider the investment decision that US large capitalized stocks outperform US small capitalized stocks. This decision can be translated into a long-short portfolio having exactly one unit of risk under a given market risk model, that is

w(US large capitalized stocks) w(US small capitalized stocks)

=

+8.4 % −8.4 %

16 % 14 %

Probability

12 % 10 % 8% 6% 4% 2% 0%

−10% −9% −8 % −7 % −6 % −5 % −4 % −3 % −2 % −1 % 0 % 1 % 2 % 3 % 4 % 5 % 6 % 7 % 8 % 9 % 10 % Alpha Source: data sourced from Bloomberg Finance LP, author's calculations

FIGURE 7.7 Relationship between the expected alpha or excess return between the S&P 500 and the Russell 200 stock market indices between December 1977 and December 2007 and the probability of achieving that expected alpha

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107

The performance or alpha of this portfolio can then be related to the investment decisions. Indeed, Pr{RUS large capitalized stocks >RUS small capitalized stocks } = Pr{α>0}

(7.2)

Based on historical data, the probability shown in Equation (7.2) can be related to the expected alpha using the alpha or excess return distribution between the S&P 500 and the Russell 2000 stock market indices between December 1977 and December 2007, as shown in Figure 7.7.

8 Volatility as a Risk Measure Volatility or the standard deviation of returns is probably the most common risk measure used throughout the investment management industry. The risk of a portfolio is defined as the weighted sum of the variance and correlation of all its assets. It is at the heart of modern portfolio theory (Black and Litterman, 1992; Lintner, 1965; Markowitz, 1952, 1987; Merton, 1990; Sharpe, 1964). Its uses are manifold, for example in: • • • •

constructing an optimized portfolio, calculating optimal hedge ratios, decomposing and attributing risk to assets or investment decisions or determining asset prices.

The goal of the volatility risk measure is to forecast the expected variability of asset or portfolio returns over a given time period in the future using an analytic formulation. In contrast with return expectation forecasting, forecasting volatility is not aiming at generating positive alpha, but rather at controlling the alpha generating outcome. Therefore it should be conservative and unbiased. Furthermore, uncertainty of future returns is assumed to be tightly related to their past variability. In Chapter 7, I presented a process for developing a risk model. Let me show how to derive the volatility risk model using that approach. As I am interested in developing investment processes generating positive alpha, it is important that the developed risk model fits well with the portfolio construction techniques that are developed in Chapters 10, 11 and 12. Within these frameworks, there are two important properties that the risk model needs to satisfy, that is: • it must be defined analytically, to allow for efficient calculation and integration into portfolio optimization models and • it must be defined as a convex function, ensuring the uniqueness of risk minimizing portfolios as well as coherence.

8.1 THE VOLATILITY RISK MODEL IN THEORY Risk is a characteristic that cannot be directly observed in the market, but it can be analyzed using statistical techniques. The volatility risk model assumes that asset returns over a given time horizon H can be modeled as a random variable following a multivariate normal distribution. Volatility risk is therefore defined as the second moment of that distribution, that is its covariance. Let ᑬa denote a random variable representing the
return of asset a and let P represent a portfolio of assets a with weights wa . Then ᑬP = a wa · ᑬa defines the random variable representing the total return of portfolio P . I denote by σ (ᑬP ), or simply σP , the volatility or second moment of the return distribution of portfolio P with weights w. Let (µ,
)

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denote the parameters of the multivariate normal distribution associated with the assets in portfolio P . Then the volatility of portfolio P is defined as σP =



wP ·
· wP

(8.1)

An important property of the volatility risk measure is that it can be calculated using a convex analytical function. This property is especially helpful when calculating optimal asset weights that minimize the volatility risk measure, given a certain number of restrictions.

8.2 SELECTING DATA FOR PARAMETER ESTIMATION Most approaches used to forecast or estimate the parameters
are based on historical observations. Therefore, before presenting different algorithms to estimate the parameters of the volatility risk model, I will take an in-depth look at the data used to estimate them. As discrete total return data are not additive over time, it is a common practice to use logarithmic or continuous compounded total return data in all risk model estimation algorithms. Let Rt denote the discrete return between time t − 1 and t. Then the logarithmic or continuous compounded return rt is defined by rt = ln(1 + Rt ) ⇔ Rt = ert − 1

(8.2)

Then rt,t+T = rt + rt+1 + · · · + rt+T −1 , thus allowing the average return over T time periods
to be  calculated as the arithmetic mean r = 1/T · Tt=1 rt instead of the geometric mean  R = T Tt=1 (1 + Rt ). There is no single best set of return data to be used to estimate the parameters of the risk model, that is the covariance matrix. Selecting the data to be used needs to make a compromise between four objectives: • Estimation quality. The data set used should be as large as possible allowing for the highest estimation quality, that is the smallest standard estimation error. • Timeliness. The data set should be as recent as possible, assuming that the recent past is closely related to the future. Going too far back in history would contaminate the data set used with data from different regimes, thus biasing the risk estimates (Merton, 1980). • Availability. The data must be available and of good quality over the time horizon used. If some data are missing or the histories available have different time horizons, the most appropriate of the following techniques should be used: (i) If a proxy time series exists or can be constructed, it should be used. For example, if euro interest rates are needed for dates before the introduction of the euro in 1998, the German mark rates could be used or, even better, a weighted average of interest rates from the most important countries in the European Union, for example France, Germany and Italy. (ii) If no reasonable proxy time series exists, the EM algorithm described in Chapter 5 should be applied. (iii) Only if the data missing cannot be classified as missing at random should the listwise deletion algorithm be applied.

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Table 8.1 Sample data sets selected to estimate the covariance matrix parameters of the volatility risk model

Assets Number of time series Length of history used Data frequency Total data elements available Number of parameters to estimated mean + variance + correlation

Global asset allocation investment process

German stock security selection investment process

S&P 500, FTSE 100, DAX, CAC 40, Nikkei 225 6 10 years

DAX stocks

Monthly 6 · 10 · 12 = 720

Daily 30 · 130 = 3900

6 + 6 + 6 · (6 − 1)/2 = 27

30 + 30 + 30 · (30 − 1)/2 = 495

30 6 months

• Forecasting horizon. The data frequency should be consistent with the forecasting horizon. Indeed, data may exhibit different properties when observed over different time horizons. For example, stock market return data show serial correlation when observing daily data whereas monthly data no longer exhibit such autocorrelations. Table 8.1 presents two different data sets for estimating the covariance matrix. In the first set the investment universe is composed of a small number of asset classes, as can be found in a global tactical asset allocation based investment process. Assume that the covariance matrix is constant over time and therefore rely on a long history of monthly return data, one month being the risk forecasting horizon. The second set aims at a stock selection investment process for leading German companies. Daily return data are used for all DAX stocks over a six-month time horizon. Here it is assumed that especially the volatility of the individual securities changes over time and that a six-month time horizon allows a reasonably sized data set timely enough to estimate a covariance matrix that is sound to forecast risk over one week.

8.3

ESTIMATING THE RISK MODEL’S PARAMETERS

Defining the volatility based risk model consists in estimating the covariance matrix
, that is the parameter of the assumed multivariate normal distribution of asset returns.

8.3.1 Sample estimator Under the assumption that the historical asset return data set R is an independent set of samples drawn from a single normal distribution, that is are independent and identically distributed (i.i.d. assumption), the covariance matrix
can be estimated as shown in the

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following equation, where ra is the estimated mean of the distribution for asset a: ∀a, b ∈ ᑛ : σˆ a,b =

T  1 · (ra,t − ra ) · (rb,t − rb ) T −1

(8.3)

t=1

It can be shown that, under the i.i.d. assumption, Equation (8.3) represents the best unbiased estimator for
; that is it is an estimator that maximizes the likelihood that the estimated covariance matrix equals the true parameters of the distribution. As shown in Equation (8.3), estimating the covariance matrix also requires the estimation of the mean of the distribution. From a statistical point of view, the best estimator of the mean under the i.i.d. assumption is the sample mean estimator shown in the following equation: T 1  ra = · ra,t (8.4) T t=1

From an economic point of view, the following estimation approaches are reasonable, depending on the context in which the mean is estimated: • Rather than estimate the mean, it is assumed to be zero. This approach, taken by the RiskMetrics model presented later in Chapter 9, is sound if the time horizon of the individual return data elements is small, for example one day, and a conservative stance is taken. It is also a reasonable approach under the assumption that risk should forecast the uncertainty related to having negative returns, that is to deviate from zero rather than from a historical mean. Estimating the covariance matrix simplifies to σˆ a,b =

T  1 · (ra,t · rb,t ) T −1 t=1

Alternatively, the mean can be set to the risk free rate, defining risk as the deviation from the return that can be obtained without taking any risk. • The mean is set to the explicitly forecasted asset returns. The economic idea is that risk should model the uncertainty to deviate from the forecasted returns. This approach especially takes into account risk introduced by return forecasts that differ significantly from the mean of the historical return data. • Similarly, an equilibrium theory can be assumed and the mean estimates can be set equal to the long term equilibrium returns. This is especially useful if a sophisticated model has been developed to determine these long term equilibrium returns. The presented alternatives for obtaining an estimate of the mean of the return distribution results in a biased covariance matrix estimate. However, the introduced statistical bias represents conditioning information and reflects an economic assumption. It should be remembered that the goal of a risk model is not to determine the exact values of the parameters of the distribution, but rather to define the best possible forecast for the parameters of the risk model. Table 8.2 illustrates the differences in volatility, given a specific estimate of the mean. It can be seen that the more the monthly mean return deviates from the historical mean, the larger the annualized volatility. Nevertheless, the differences remain

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113

Table 8.2 Estimated volatility for the DAX index for different estimated mean values using monthly historical index data between December 1997 and December 2007 r DAX (%)

Monthly mean

Annualized volatility (%)

Estimated mean 0.23 Zero mean 0.00 Risk free rate (1-month LIBOR) Forecasted return 1.50

10.34 10.37 10.39 11.25

Source: data sourced from Bloomberg Finance LP, author’s calculations.

small when comparing the volatility based on the estimated mean, the zero mean and the risk free rate. The sample estimator performs poorly when insufficient data are available to estimate all parameters with sufficient precision. Indeed, the estimation of the covariance matrix for A assets requires the estimation of A +  A + A · (A − 1)/2     mean

variance

correlation

independent parameters. This means that the covariance matrix will contain many estimation errors (Jobson and Korkie, 1981) if A is large relative to the sample size T . The most extreme coefficients tend to take extreme values, not because these are the true values but because they contain an extreme amount of error. In a mean-variance portfolio allocation framework, as presented in Chapter 10, this has as a consequence that the largest positions are taken where the coefficients are the most unreliable. Michaud (1991) calls this effect error maximization. If the sample data size T is smaller than the number of assets, then the covariance matrix estimated will always be singular. Furthermore, if the historical time series data available does not satisfy the i.i.d. assumption, the sample estimator will over- or underestimate the true volatility risk. Most financial time series, especially when the time horizon selected is short, exhibit autocorrelation, that is dependences between time horizons, and/or heteroskedasticity, that is nonconstant variance. Alternative estimation techniques then need to be applied. 8.3.2 Factor model estimator The precision of the sample estimator depends on the sample data size available relative to the size of the number of assets modeled being large. If this is not the case, a factor model estimator can help to alleviate this problem by imposing a structure on the historical return data when estimating the covariance matrix. There are many explanatory factor models for market returns. The most common ones are: • the capital asset pricing model, that is the single index factor model proposed by Sharpe (1964) and Lintner (1965) where the market return, for example represented by the index MSCI world free all countries, represents the single factor, • industry classification factor models, that is models where the returns are explained through a reduced set of industry factors, as proposed by Fama and French (1993),

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Positive Alpha Generation: Designing Sound Investment Processes

• an economic factor model, where the returns are expressed as a function of macroeconomic and microeconomic variables and indicators (Elliot et al., 2006), and • the principal component models, that is models where the factors are extracted from the sample covariance matrix using principal component statistical techniques (Jackson, 1991). The first three model types have the advantage of allowing an economic interpretation of the used factors, whereas principal component analysis is a purely statistical approach. The disadvantage of a factor based approach is that there is no consensus on both the number and identity of the factors (Connor and Korajczyk, 1993). Let me define a factor model as ∀a ∈ ᑛ : Ra = βa · f + εa

(8.5)

where f is a vector of historical factor returns, βa the factor exposure of asset a to the individual factors and εa the idiosyncratic error term. In the CAPM single factor model, f = RM , the return of the market portfolio. It is important that the factors are determined such that they capture all cross-sectional risk. This means that E[εa |f] = 0 and cov(E|f) =
ε , where E = (ε1 , . . . , εA ) is the diagonal matrix of all residual returns. Under the factor model in Equation (8.5),
factor = cov(B · f) + cov(E) = B · cov(f) · B +
ε

(8.6)

where B = (β1 , . . . , βA ) is the matrix of all factor exposures, that is the regression coefficients. A natural estimator for
factor is then given by estimating B, cov(f) and
ε using an ordinary least square estimator. Therefore, ε
· B +
ˆ factor = B · cov(f)


(8.7)

ε =
is the sample covariance matrix of the factor returns and
where cov(f) diag(var(ˆε1 ), . . . , var(ˆεA )) is the diagonal matrix with εˆ a = Ra − βa · f being the matrix of residuals. Fan et al. (2006) have compared the sample covariance matrix with a factor model based covariance matrix. The key difference is that
factor is always invertible, whereas
sample is only invertible if the number of assets is smaller than the sample size. They showed that under the Frobenius norm, assuming that the factor model captures all cross-sectional risk, both models converge at the same speed to the actual covariance matrix of the distribution. Interestingly, the inverse of the factor covariance matrix converges faster toward the true inverse than the inverse of the sample covariance matrix (Fan et al., 2006). It is shown in Chapter 10 that the asset allocation in a mean-variance framework depends on the inverse of the covariance matrix rather than on the matrix itself. Hence it is advantageous to use a factor model based covariance matrix estimator, especially for large asset universes. 8.3.3 Shrinkage based models Both the sample as well as the factor based estimation approaches have their advantages and drawbacks. The sample estimator
sample is easy to calculate and has the property of

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115

being unbiased. However, it is prone to estimation error, especially when the sample size is small. On the other hand, the factor based estimator
factor contains relatively few estimation errors, but is prone to misspecification of the underlying factor model. Ledoit and Wolf (2003, 2004a, 2004b) proposed the definition of an estimator for the covariance matrix based on a compromise between the sample covariance matrix and the highly structured factor model covariance matrix. To estimate the factor model, they proposed using the CAPM model. It is important that, whichever factor model is chosen, it involves only a small number of factors, that is a lot of structure, and reflects the important characteristics of the asset returns. Under the assumption that the asset returns are i.i.d., the shrinkage estimator
shrinkage is defined as
shrinkage = γ ·
factor + (1 − γ ) ·
sample

(8.8)

where γ , the shrinkage factor, is a number between zero and one. This estimator is called the shrinkage estimator because the sample covariance matrix is shrunk toward the structured estimator. This technique is well known in statistics (Stein, 1956). Within empirical Bayesian interpretation, the prior is based on the single index model combined with the sample information. To estimate the shrinkage factor, Ledoit and Wolf (2004a) used a maximum likelihood approach, where the likelihood function is defined as    ˆ factor + (1 − γ ) ·
ˆ sample −
 L(γ ) = γ ·
 f

where
is the true covariance matrix. Minimizing L(γ ) leads to γˆ = (pˆ − q)/( ˆ cˆ · T ) being a consistent estimator for the optimal shrinkage factor γ , where pˆ a,b =

qˆa,b =

1 T                       

·

T  

  ra,t − ra · (rb,t − rb ) − sˆa,b

t=1

pˆ a,a

if a = b

T       1  1 · · sˆb,M · sˆM · ra,t − ra + sˆa,M · sˆM · rb,t − rb · 2 T (ˆsM ) t=1           −ˆsa,M · sˆb,M · rM,t − r · rM,t − r M M · ra,t − r a · rb,t − r b  −fˆa,b · sˆa,b

if a = b

2  cˆa,b = fˆa,b − sˆa,b ˆ sample , fˆa,b the elements with sˆa,b representing the elements of the sample covariance matrix
ˆ of the factor covariance matrix
factor , sˆa,M the covariance asset a
and
the market
between
portfolio, s ˆ the variance of the market portfolio, p ˆ = p ˆ , qˆ = a b qˆa,b and M a,b a b

cˆ = a b cˆa,b . ˆ shrinkage is both invertible as well as The resulting estimated covariance matrix
well conditioned. In addition, as a byproduct, the dependence on the factor model is reduced.

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Positive Alpha Generation: Designing Sound Investment Processes

8.3.4 Process based models The covariance matrix estimation approaches presented so far all assume that the covariance matrix does not change over time, at least over the time horizon used for estimating the parameters. In 1986, Bollerslev suggested the use of a GARCH1 model to estimate the covariance matrix. A GARCH model describes the covariance matrix as a process rather than a constant. Let σt denote the volatility at time t. Then the most general form of a GARCH process, denoted by GARCH(p, q), is defined as σt2

=ω+

p  i=1

αi ·

2 rt−i+1

+

q 

2 βi · σt−i

(8.9)

i=1

where ω > 0, αi ≥ 0 and βi ≥ 0 are the parameters of the GARCH(p, q) model to be estimated. The volatility depends on both the realized return time series as well as the previous volatilities. Equation (8.9) contains a persistence factor βi specifying how sticky the volatility is and a reactive factor αi specifying how strong the volatility changes are based on recent events. In practice the GARCH(1, 1) model version is generally used, that is 2 σt2 = ω + α · rt2 + β · σt−1

(8.10)

If the mean is not assumed to be zero, rt2 is to be replaced by (rt − r)2 in both Equations (8.9) and (8.10). A GARCH process is called covariance stationary if α + β < 1. Alternatives to the GARCH based volatility forecasting approach are stochastic volatility (SV) models, reviewed, for example, by Taylor (1994). SV models are defined by their first two moments. Under the assumption of normality, the variance equation of an SV type model can be written as σt2 = σ 2 · eht where σ 2 is a positive scaling factor and ht a stochastic process such that ht = φ · ht−1 + ση · εt , with εt ∼ N (0, 1) where the degree of volatility persistence is measured by the coefficient 0 < φ < 1 and ση a constant. Other definitions for the stochastic process ht are possible. Compared to GARCH models, SV forecasting models allow for greater flexibility in modeling specific properties of volatility.2 However, this flexibility comes at a cost as the ability to estimate the parameters based on a likelihood function is usually intractable. The RiskMetrics approach The RiskMetrics model, initially developed by J.P. Morgan and Reuters (1995), is based on a GARCH(1, 1) type model assuming a zero mean. Rather than estimating the parameters in Equation (8.10) for each asset class, or more precisely for each modeled risk factor, 1 2

Generalized auto-regressive conditionally heteroskedastic. Fortunately, or unfortunately, this improved flexibility is very often not required within an investment process.

Volatility as a Risk Measure

117

RiskMetrics specifies the covariance matrix value as 2 σˆ a,b

= (1 − λ) ·

T 

λt−1 · ra,t · rb,t

(8.11)

t=1

where λ = 0.94 when ra,t are daily returns and λ = 0.97 when ra,t are monthly returns. These constants have been determined empirically during the development of the model. Fixing the parameters is a compromise between simplicity of the model and accuracy. The economic interpretation behind the RiskMetrics approach is that more recent observations contain more information relevant for forecasting future volatility that less recent ones, mainly due to a persistence property observed in volatility, called volatility clustering. A drawback of the RiskMetrics specification is that it implicitly assumes that the term structure of volatility at a given moment in time is flat. Nerlove and Wage (1964) and Theil and Wage (1964) have shown that the specification in Equation (8.11) is optimal if the squared returns can be modeled by a random walk plus white noise. Estimating the process parameters In a GARCH model approach, rather than estimating the parameters, that is the covariance matrix, itself, one has to estimate the parameters of the underlying process. The parameters can be estimated using a standard maximum likelihood statistical approach, where the maximum likelihood is defined as

T 1  rt2 2 L(ω, αi , βi , r) = − · log(σt (ω, αi , β)) + 2 (8.12) 2 σt (ω, αi , β) t=1 Unfortunately, maximizing Equation (8.12) is not as easy as it may look at first glance. Indeed, three major difficulties need to be overcome: • First, the parameters can only be estimated numerically; that is there is no analytical solution to maximizing Equation (8.12). • The second, and much more important, issue is the large number of parameters that have to be estimated. For a simple one-asset GARCH(1,1) process, three parameters have to be estimated. Due to the structure of the likelihood function in Equation (8.12), the maximization problem to be solved converges only very slowly to an optimal solution and is numerically unstable. • Finally, even if it were possible to estimate all the parameters of the GARCH process, there is no assurance in the way the processes are constructed that the resulting covariance matrix would be positive semi-definite, a required property of the covariance matrix in the context of designing a coherent risk model to be used in an investment process. Overcoming the estimation difficulties A possible approach to overcoming the difficulties of estimating the parameters of a GARCH model for forecasting the whole covariance matrix is to decompose it such that
= V · Q · V

(8.13)

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Positive Alpha Generation: Designing Sound Investment Processes

where V is a diagonal matrix of variances and Q the positive semi-definite correlation matrix. First, the correlation matrix is estimated using a sample, factor based or shrinkage based estimator assuming that the correlations change more slowly over time, making the i.i.d. assumption realistic. Then, for each asset, a separate univariate GARCH process is defined and its parameters estimated independently. Finally, the covariance matrix is calculated by combining the correlation matrix with the volatility matrix composed of the independently forecasted volatility. Different approaches that address the dimension problem in estimating GARCH parameters are the dynamic conditional correlation approach (Engle, 2002), the constant conditional correlation approach (Bollerslev, 1990) and the asymmetric GARCH model (Gloster et al., 1993). 8.3.5 Market implied models The market implied estimation technique relies on current asset prices rather than historical return data. Chapter 7 indicates that risk cannot be observed. This is not entirely true. It can, at least partially, be derived from how market participants value derivative securities. Black and Scholes (1973) have shown that the price of an option on an equity or equity index asset is a function of the asset price, its expected cash flows, the risk free rate and its volatility. The following equation illustrates the relation between the price of a European call option C(S, X) and the stock price S, the risk free rate rF , the strike price X, the time to expiration H and the volatility σ , assuming no dividend payments: C(S, X) = S · N (d1 ) − X · e−rF ·H · N (d2 ) where

ln d1 =

S H

(8.14)



σ2 + rF + ·H √ s and d2 = d1 − σ · T √ σ· H

and N (·) is the cumulative distribution function of a standard normal random variable. If it is assumed that both the stock and the option are fairly priced by the market participants, that is S and C(S, X) can be observed with reasonable precision, then Equation (8.14) can be numerically solved for the volatility σ . σ is called the marked implied volatility. It represents the expectation of the market participants of the uncertainty over the option’s live. Figure 8.1 shows the call implied volatility of the DAX index, the Home Depot stock, as well as the value of the VIX index modeling the implied volatility of the SP& 500 index between December 1997 and December 2007. When calculating the implied volatility, it is a common practice to use at the money options with the same maturity as the risk forecasting horizon. In fact, these options satisfy at best the assumptions of the Black and Scholes option pricing model, especially with regard to normality of returns. The market implied estimation approach provides the most adequate risk forecast. Unfortunately, there are not sufficient liquid options on all stocks and indices to make it a viable alternative to estimating parameters of distributions or processes. Another issue is that the Black and Scholes model does not help to derive correlations between assets. Sophisticated Monte Carlo algorithms can nevertheless be used to derive correlations from options on multiple assets, at least in theory. In practice only very few derivatives on pairs of assets

Volatility as a Risk Measure

119

80% 70%

Implied volatility

60% 50% 40% 30% 20% 10% 0% ‘98

‘99

‘00

‘01

DAX implied volatility

‘02

‘03

‘04

‘05

Home Depot implied volatility

‘06

‘07 VIX index

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 8.1 Call option implied volatility over time for the DAX index and the Home Depot stock as calculated by Bloomberg, as well as the VIX index representing the 30-day forward implied volatility of the S&P 500 index as derived by all quoted S&P 500 options on the Chicago Board of Trade between December 1998 and December 2007

exist. In addition, they are in general not very liquid, that is priced with large bid–ask spreads, making the implied correlations very imprecise. The approach separating the forecast of the correlation from that of the volatilities as described for overcoming the estimation difficulties with multivariate GARCH models can also be used in the context of market implied risk forecasting. The correlation between assets is forecasted using estimation techniques based on historical data. The volatilities of the assets are derived from the option market and then combined with the correlation matrix to form the covariance matrix. A major drawback of using market implied estimated volatility is that significant changes in asset prices are correlated with changes in the asset allocation. Constructing portfolios based on implied volatility react fast to changes in market perceived risk, but these changes are pro-cyclical. Figure 8.2 illustrates the construction of a portfolio investing in the S&P 500 index and cash such that the portfolio volatility is equal to 10 % at any given moment in time based on the implied volatility of the S&P 500 index measured by the VIX index and cash with a zero volatility. Significant increases in volatility, usually due to market corrections, are associated with significant sales of risky assets in the portfolio, but only after the volatility has increased. As with all covariance matrix estimation approaches presented, it is the context in which the risk model is used that determines the most appropriate technique to be used.

8.4 DECOMPOSING VOLATILITY Volatility as a risk measure only provides an aggregated figure about the uncertainty of portfolio returns. It is therefore important to be able to decompose the volatility figure and relate its constituents (i) to the individual holdings in the portfolios and (ii) to the

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Positive Alpha Generation: Designing Sound Investment Processes

100%

1800 1600

80%

1400 1200

60%

1000 800

40%

600 400

20%

200 0

D

ec . M 02 ar .0 Ju 3 n. Se 03 p. D 03 ec . M 03 ar .0 Ju 4 n. Se 04 p. D 04 ec . M 04 ar .0 Ju 5 n. Se 05 p. D 05 ec . M 05 ar .0 Ju 6 n. Se 06 p. D 06 ec . M 06 ar .0 Ju 7 n. Se 07 p. D 07 ec .0 7

0%

Exposure to the S&P 500 index for 10% volatility (left axis)

S&P 500 index (right axis)

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 8.2 Asset allocation over time of a portfolio with a fixed volatility of 10 % investing in the S&P 500 index and cash compared to the S&P 500 index level

investment decisions taken. There are two complementary approaches to volatility decomposition, that is: • the total risk approach and • the marginal risk approach. 8.4.1 Total risk approach In the total risk approach, the portfolio is first subdivided into nonoverlapping subportfolios. Then the volatility of each subportfolio is calculated independently. The difference between the volatility of the portfolio and the sum of the volatilities of the subportfolios is called the diversifying risk component. It is always nonpositive and, in most cases, negative. Let P denote a portfolio decomposed into subportfolios Pa . The volatility of portfolio P can then be calculated as σP and the volatility of the subportfolios as σPa . Denote by wa the weight of each subportfolio Pa in portfolio P and ρa,b the correlation between the subportfolios Pa and Pb . Then σP =

 a

wa2 · σP2a +

 a

wa · wb · ρa,b · σPa · σPb

b=a


and σP ≤ a wa · σPa . Table 8.3 illustrates a sample decomposition of a global balanced portfolio using the total risk approach. Both equity asset classes have the largest total risk, as expected. In addition, the diversification effect reduces by 2.15 % the sum of the individual risks.

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121

Table 8.3 Decomposition of the portfolio volatility of a global balanced portfolio according to the total risk and marginal risk approaches Asset class

Weight (%)

–Cash –Domestic bonds –International bonds –Domestic equities –International equities –Global properties –Hedge funds Portfolio Diversification effect

5 30 25 10 20 7 3 100 n/a

Total risk (TR) (%)

Marginal risk (MR) (%)

Marginal contribution to risk (MCR) (%)

0.00 0.19 1.15 0.94 1.88 0.74 0.22 5.12 n/a

−0.01 3.73 22.46 18.34 36.69 14.51 4.27 100 n/a

0.01 0.91 1.64 1.48 2.09 0.88 0.27 5.12 −2.15

Source: data sourced from Bloomberg Finance LP, author’s calculations.

8.4.2 Marginal risk approach The marginal risk approach focuses on determining the sensitivity of the portfolio’s volatility relative to small changes in its allocation. The marginal risk describes by how much the volatility of the portfolio would increase or decrease if the allocation to a single asset or subportfolio changes by a small amount. The marginal risk (MR) can be expressed in absolute terms or as a percentage of the portfolio’s total risk. In the latter case I talk about the marginal contribution to risk (MCR). Let P denote a portfolio decomposed into subportfolios Pa with weights wa . Denote by √
the covariance matrix allowing the total volatility σP to be calculated, that is σP = w ·
· w. The marginal risk MRa for each asset a is calculated as the first partial derivate of the portfolio volatility relative to that asset’s weight, that is ∂σP ∂(w ·
· w)1/2 MRa = = = ∂wa ∂wa

wa ·



√

wb · σa,b

b

w ·
· w

Table 8.3 shows a sample decomposition of the risk according to the marginal risk approach for a sample global balanced portfolio. The largest marginal risk is attributed to international equities, with an exposure of 1.88 % or more than one-third of the total risk. It is also possible to calculate the sensitivity of the portfolio’s volatility to exogenous factors, like interest rates or equity market returns. I then talk about risk sensitivity rather than marginal risk. 8.4.3 Volatility decomposition in the investment process context Let me now focus on how to use the volatility decomposition in the context of an investment process. This process can be subdivided into three stages: 1. First, I decompose the portfolio into subportfolios along the dimensions on which I have formulated investment decisions. Ideally each subportfolio relates to a single investment

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decision. If this is not feasible, the decomposition should be along asset classes or individual assets. 2. In the second step I calculate the volatility of each subportfolio and relate it to the total volatility of the portfolio and especially the other subportfolios. This allows me to – determine if large chunks of risk are allocated to a single or a few subportfolios and – study the diversification effect of the constructed portfolios through the relative size of the diversifying risk component. In the example in Table 8.3, the volatility of the international equity subportfolio is the largest with 2.09 %, but is not significantly larger than the total risk of international bonds or domestic equities, especially when expressed as a function of the allocated weight. In addition, the diversification effect, that is a diversifying risk component of 2.15 %, more than one-third of the total risk, shows that the diversification impact of combining these seven asset classes is significant. 3. Finally, I calculate the marginal risk in absolute and percentage terms of each subportfolio and study the relationship between their weights and marginal risks. My goal is to avoid the overall portfolio volatility being very sensitive to small subportfolios. The MR measure allows me to find where small changes in the weights have the largest impact on portfolio volatility. For example, if I want to reduce the volatility of the portfolio in Table 8.3, I would start by reducing the weight of international equities relative to the cash asset as international equities have the largest MCR.

8.5 ADDITIONAL PITFALLS 8.5.1 Determining and handling autocorrelation One key assumption made when using the sample, the factor or the shrinkage based covariance matrix estimating approach is that, over time, the observed returns are independent from each other and drawn from the same distribution (i.i.d. assumption). Unfortunately, asset returns, especially when observed at short time frequencies, for example daily, exhibit autocorrelation. Figure 8.3 shows daily return data for some assets exhibiting these properties. Both the S&P 500 index and the DAX index exhibit a cluster of high volatility during 2002–2003, whereas, for example, in 2005 the volatility was very low. In addition, the Microsoft stock shows a number of extreme returns, especially on the downside, that are not consistent with the normality assumption. I suggest that a start could be made by analyzing the available historical return data for autocorrelation by visualizing a cross-correlogram. I define the cross-time sample covariance ˆ matrix
(h) by σˆ a,b (h) =

T  1 · (ra,h+t − r a ) · (rb,t − r b ) T −1

(8.15)

t=1

ˆ for 0 ≤ h < T .
(0) is the sample covariance matrix. Figure 8.4 shows the sample cross-time correlation matrix over time for the three asset classes S&P 500, DAX and Nikkei 255. It can be seen that there is little or no cross-correlation between the S&P 500 and Nikkei 225 indices (top right cross-correlogram). On the other hand, there is an autocorrelation with a one-day lag between the Nikkei 225 and the S&P 500 indices, the cross-correlation being larger than 0.54 (left bottom cross-correlogram). This is due to the Japanese market being

Volatility as a Risk Measure 6%

123 6%

S&P 500

4%

4%

2%

2%

0%

0%

−2 %

−2 %

−4 %

−4 %

−6 % 2002

2003

2004

6%

2005

2006

2007

−6 % 2002

4%

2%

2%

0%

0%

−2 %

−2 %

−4 %

−4 % 2003

2004

2003

2004

6%

Microsoft

4%

−6 % 2002

DAX

2005

2006

2007

2005

2006

2007

2006

2007

USD/JPY

−6 % 2002

2003

2004

2005

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 8.3 Historical daily returns between December 2002 and December 2007 for the S&P 500 index, the DAX index, the Microsoft stock price and the USD/JPY exchange rate 0.9

0.9

S&P 500 (t)/S&P 500 (t+h)

0.9

S&P 500/DAX

0.6

0.6

0.6

0.3

0.3

0.3

0.0

0.0

0.0

−0.3

−0.3

−0.3

−0.6

0.9

20 18 16 14 12 10 8 h

6

4

2

0

−0.6

0.9

DAX/S&P 500

20 18 16 14 12 10 8 h

6

4

2

0

−0.6 20 18 16 14 12 10 8 h 0.9

DAX/DAX

0.6

0.6

0.6

0.3

0.3

0.3

0.0

0.0

0.0

−0.3

−0.3

−0.3

−0.6

0.9

20 18 16 14 12 10 8 h

6

4

2

0

−0.6

0.9

Nikkei 225/S&P 500

20 18 16 14 12 10 8 h

6

4

2

0

−0.6

0.9

Nikkei 225/DAX

0.6

0.6

0.6

0.3

0.3

0.3

0.0

0.0

0.0

−0.3

−0.3

−0.3

−0.6

20 18 16 14 12 10 8 h

6

4

2

0

−0.6

20 18 16 14 12 10 8 a

6

4

2

0

S&P 500/Nikkei 225

−0.6

6

4

2

0

4

2

0

2

0

DAX/Nikkei 225

20 18 16 14 12 10 8 h

6

Nikkei 225/Nikkei 225

20 18 16 14 12 10 8 h

6

4

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 8.4 Cross-correlation matrix of daily returns of the S&P 500, the DAX and the Nikkei 225 indices expressed in their respective local currency using six months of daily data between July and December 2007, the second time series being shifted relative to the first one

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Positive Alpha Generation: Designing Sound Investment Processes

open ahead of the S&P 500 index. For most time horizon factors h, the autocorrelation in this example can be safely ignored. The Box–Ljung test statistic BL(h) (West and Cho, 1995) can be used to test formally for autocorrelation. Let BL(h) = T · (T + 2) ·

h  ρˆa,b (h)2 t=1

T −k

Under the null hypothesis that the time series is not autocorrelated with a lag of h, BL(h) follows a chi-square distribution with h degrees of freedom. If the historical return data available do not satisfy the i.i.d. assumption, it may be appropriate to use a covariance matrix estimation approach that takes into account autocorrelations. Over the last 30 years numerous procedures and algorithms have been developed to estimate autocorrelation consistent covariance matrices (Andrews, 1991; De Santis, 2003; MacKinnon and White, 1985; White, 1980). Den Haan and Levin (1997) provide a survey of these techniques. Unfortunately all these techniques require the return data to exhibit some structure, for example to be expressed by a linear factor model. 8.5.2 Scaling risk for different risk forecasting horizons The most commonly used approach to transform volatility or covariance matrix forecasts from one time horizon to another is to apply√the so-called square root of time (SROT) rule, that is multiply the original volatility by T , where T is the difference between the two time horizons. For example, √ the weekly volatility σweekly is transformed into an annual volatility by setting σannual = 52 · σweekly , one year having 52 weeks. It can be shown that the SROT rule is optimal if the data used to estimate the volatility are normally distributed and satisfy the i.i.d. assumption. Unfortunately, this is not always the case. Kaufmann (2004) has studied the quality of the SROT rule under different assumptions and variance estimation techniques. He showed the following properties: • The SROT rule overestimates the volatility if the data used for the estimation are heavily tailed. The heavier tailed the data, the worse is the performance. • For a GARCH process with estimated parameters based on normally distributed data, the SROT rule generally underestimates the volatility. • In the presence of serial correlation, the SROT rule underestimates the true volatility. There are various alternatives to the SROT rule, like taking into account autocorrelations, independent re-sampling or extreme value methods, to name just a few. However, no single alternative performs best in all cases. Nevertheless, it is fair to say that the SROT rule often works reasonably well. Even the Basle Committee on Banking Supervision explicitly permits the SROT rule for transforming a one-day value-at-risk (VaR) into a ten-day VaR (Basle Committee, 1996). There are other areas in the process of building a risk model where the impact of poor choice is much more significant than the issue of scaling risk.

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125

8.6 TESTING RISK MODELS From a process point of view, testing the soundness of a covariance matrix based risk model is no different from testing a quantitative forecasting model as described in Chapter 5. However, in contrast to asset returns, risk cannot be directly observed. Except for the volatility of assets on which options are traded, only single observations of the distribution modeled by the covariance matrix are available. 8.6.1 Statistical tests The statistical quality of the estimation algorithm used is determined by the convergence of   
 (8.16) 
−
 towards zero, where || · || denotes a distance measure, like the Frobenius norm. For example, ˆ is calculated using the sample estimator with a time series of size T for A assets, if  2 " ! A 
 E 
sample −
 = O T Calculating Equation (8.16) for a specific estimation algorithm requires sophisticated statistical skills. A second statistical test is to study whether or not the used estimator is biased, ˆ =
. Ideally any estimator should be unbiased. However, if a that is whether or not E[
] certain bias significantly reduces the estimation error, it may be acceptable. For the sample ˆ sample ] =
. Third, an estimation algorithm should be analyzed for its effiestimator E[
ciency, that is its expected estimation error. Finally, the estimator should be robust, that is not susceptible to the presence of outlying data. 8.6.2 Backtesting Backtesting the performance of a covariance matrix estimation algorithm is very important because risk models are used in a forward looking way rather than in an explanatory approach. A backtesting approach aims to answer the following questions: 1. How well does the model predict the size of losses? 2. How well does the model predict the frequency of losses? 3. How well does the model predict a percentile of the entire distribution? I suggest that the algorithm shown in Figure 8.5 should be implemented for backtesting a risk model. The idea behind this algorithm is to generate portfolio weights and then calculate their volatility and their forward looking return. Various approaches for generating weights can be taken. Ex-post optimal weights with respect to the ex-ante risk measure should be avoided as they tend to generate extreme returns; that is the best possible ones give the realized results of the individual asset classes and so do not effectively test the quality of the risk model. The realized returns should then be consistent with the forecasted volatility. About 95 % of the returns should be no larger in absolute terms than 1.96 · σP and about 99 % should be no larger than 2.58 · σP .

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Positive Alpha Generation: Designing Sound Investment Processes

for t ← 1 to T : ˆ t , using data available 1. Estimate the parameters of the risk model, that is the covariance matrix
up to time t. 2. Determine a portfolio structure wt , for example: – an equally weighted portfolio, – a randomly sampled portfolio weights from a spherical distribution,3 – the actual portfolio held at time t, – a portfolio constructed based on perfect forecasting capabilities. ˆ t. 3. Calculate the risk σˆ t of portfolio wt using
4. Compare the forecasted risk σˆ t to the realized return of portfolio wt . next t

FIGURE 8.5 Algorithm for backtesting a risk model 20 % 15 % 10 % 5% 0% −5 % −10 % −15 % −20 % 1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Source: data sourced from Bloomberg Finance LP, author's calculations

FIGURE 8.6 Backtesting of a global equity asset allocation risk model based on sample covariance matrices between December 1997 and December 2007

Figure 8.6 illustrates the backtesting of the sample covariance matrix estimator applied to a rolling five-year monthly data time horizon and the equally weighted portfolio valued at the end of each month. The investment universe selected is composed of the S&P 500, FTSE-100, DAX, CAC 40 and Nikkei 225 indices, expressed in USD. Both the realized returns as well as the risk at the 95 % and 99 % confidence levels are shown. At the 99 % confidence level (±2.58 · σ ), the back test shows three outliers, one on the positive side and two on the negative side. Theoretically one would expect around two outliers. At the 95 % confidence level (±1.96 · σ ), eight outliers can be observed versus a theoretically expected value of nine. This shows that the sample covariance matrix estimator is efficient at measuring risk. To test formally whether or not the observed frequency of the exceptions is consistent with the theoretical frequency of expected exceptions at a chosen confidence level, the Kupiec 3

A random vector w has a spherical distribution if it is uniformly distributed on the unit sphere {w : w · w = 1}.

Volatility as a Risk Measure

127

test could be applied. Under the null hypothesis that the model is correct, the number of exceptions follows a binomial distribution. Unfortunately this test often fails to exclude the null hypothesis; that is it results in a type II error. 8.6.3 Robustness and assumption tests The robustness of a covariance matrix based risk model depends on the robustness of the covariance matrix itself. Different numerical methods exist for ensuring the robustness of a covariance matrix: • The covariance matrix should be positive (semi-)definite, that is ∀x = 0 : x ·
· x >(≥)0. The positive semi-definiteness property is required for the risk model to be coherent. A matrix is positive (semi-)definite if all its Eigenvalues are positive (nonnegative). Any positive definite matrix X is invertible, that is X−1 exists. Table 8.4 Sample set of realized extreme event scenarios with significant temporary impacts on markets Event description

Time frame

Arab oil embargo, followed by a quadrupling of oil prices and a consequent increase in external debt of non-oil-producing countries South Korea and Thailand stock market crashes due to balance of payment problems Doubling of oil prices leading to balance of payment problems for non-oil-producing countries Inability of Brazil, Argentina and Mexico to meet their foreign debt obligations Black Monday Brazilian stock market crash due to the government’s failure to implement economic reforms Gulf war Unexpected devaluation of the Mexican pesos preceded by a strong increase in US federal fund interest rates Price declines over high interest rates and bad bank loans Russian rouble devaluation and temporary default of Russia on its government debt LTCM crisis initiated by credit spread widening One-third price drops at the New York stock exchange triggered by high trade deficits and proposed takeover related legislations US slips into recession September 11 terror attacks on New York and Washington Worries that Asia’s growth perspectives are overestimated, resulting in an increased level of risk aversion Corporate scandals, including the Worldcom bankruptcy, less than 12 months after the Enron debacle Iraq invasion US subprime crisis US recession fear Global liquidity crises

1973–1974 1979–1980 1980–1981 1980–1982 19 October 1986 1986–1987 August 1990 1994–1995 1996 Fall 1998 September 1998 October 1998 March 2001 September 2001 May 2002 July 2002 January 2003 Second half of 2007 Spring 2008 Fall 2008

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Positive Alpha Generation: Designing Sound Investment Processes

• The conditioning factor, defined by ||A|| · ||A−1 || for a given norm || · ||, should be large, indicating numerical stability. 8.6.4 Boundary and extreme event tests It is generally suggested that the performance of a risk model should be evaluated under extreme scenarios. Table 8.4 lists a sample set of extreme scenarios that can be used. In addition to using actual extreme events, theoretical scenarios should be constructed to test the responsiveness of the covariance matrix to specific factors. When performing extreme event tests, it is important to keep in mind the actual use of the risk model. In general, covariance matrix based risk models are not aiming to perform well under extreme scenarios, but rather to focus on the nonextreme events.

9 Alternative Risk Measures Volatility is the most common risk measure used in the asset management industry. It is expressed as the standard deviation or variance of the expected returns. However, there are many more approaches to defining and calculating asset as well as portfolio risk. Four dimensions along which risk measures can be defined are: • Definition of risk. The meaning of risk is defined in a specific way, for example as the maximal expected loss from the highest portfolio value over the next six months with a probability of 95 %.1 • Distribution function. Different statistical distribution functions of the returns can be defined to model specific characteristics of the asset returns. For example, if the focus is on extreme events, assuming a Pareto distribution is more appropriate than a normal distribution. • Investment decision mechanism. The risk measure needs to be linked to the investment decisions taken. For example, if forecasting interest rate changes, an appropriate risk measure, like duration, expresses the sensitivity of the portfolio’s return to the forecasted interest rate change. • Calculation algorithm. Different algorithms can be applied to calculate the risk or estimate the parameters of the risk model, depending on the available historical data and its structure. If the portfolio contains a large number of assets with a nonlinear payoff, like options, the risk can be calculated through the re-evaluation of the portfolio’s assets under a large number of randomly generated scenarios. The selection of the most appropriate risk measure must always be made within the context of the investment process in which it will be used. Indeed: • the risk measure must be consistent with the investment decision structure, that is the output of the market forecasting value chain module, ideally even supporting the market forecasting module, and • it must fit within the transfer mechanism of the portfolio construction and risk management module used to compile the return forecasts into portfolio structures and trades. Depending on the investment process, it may be useful to apply two different risk models. For example, the portfolio structure of a region based equity investment process is determined using a volatility based risk model. Then the resulting portfolio structure is evaluated against a pre-defined set of historical extreme scenarios. The portfolio returns under these extreme scenarios are studied. If for some scenarios the returns are outside the predefined limits on maximal losses, the portfolio structure is changed until both the volatility as well as the extreme scenario returns are within the fixed boundaries. Table 9.1 illustrates the use of two complementary risk models for a global equity asset allocation investment process. The 1

This risk measure is called peak to bottom or maximal drawdown.

130

Positive Alpha Generation: Designing Sound Investment Processes Table 9.1 Application of two risk models to determine the target asset allocation structure of a global equity asset allocation based portfolio Asset class

Forecast (%)

US equities (S&P 500) UK equities (FTSE 100) German equities (DAX) French equities (CAC 40) Japanese equities (Nikkei 225) – Expected return Volatility based risk model • Volatility

Initial portfolio (%)

Adjusted portfolio (%)

9.61 8.74 15.55 11.07 2.72

36 29 35 0 0 11.40

84 0 12 3 1 10.28

<16.0

16.00

15.15

<12

−14.86

−11.96

<5

−5.63

−4.97

Scenario based risk model • US corporate scandal (08.07–07.08.2002) • Russian rouble devaluation (25.08–24.09.1998)

Source: data sourced from Bloomberg Finance LP, author’s calculations.

initial portfolio is constructed maximizing the expected return, given a 16 % upper bound on the volatility. Then the two additional restrictions requiring a maximal loss of 12 % during the US corporate scandal crisis in 2002 and 5 % during the month of the Russian rouble devaluation crisis in summer 1998 are added. Additional diversification in France and Japan leads to a portfolio with less than a 1 % lower risk and a loss of 11.96 %, respectively 4.97 %, during the two considered crisis scenarios. Both portfolios are mean-variance efficient under their corresponding restrictions.

9.1

FRAMEWORK DEFINING RISK

To allow for a common understanding, it is important to start with a precise definition of the term portfolio risk.2 Possible definitions are: • Volatility. Risk is defined as the second moment of the return distribution underlying the portfolio. Under the assumptions that portfolio returns follow a multivariate normal distribution with parameters (µ,
), the risk of a portfolio is defined as σvola (w) =

√

w ·
· w

• Value at risk (VaR) Risk is defined as the maximal value, in absolute or relative terms, that is lost over a given time horizon with a predefined probability α. Formally, σVaR (w) = max{l ∈ R : Pr[ᑬ > − l]  1 − α} 2

An asset can be considered as a special case of a portfolio.

Alternative Risk Measures

131

where ᑬ is the random variable representing the portfolio value or return. If ᑬ ∼ N (µ,
), √ then σVaR (w) = w · µ − w ·
· w · N −1 (α), where N −1 (α) denotes the α-quantile of the standard normal distribution. If the asset returns follow a Student-t √ distribution with v degrees of freedom, that is T (µ,
, v), then σVaR (w) = w · µ − w ·
· w · Tv−1 (α), where Tv−1 (α) denotes the α-quantile of the Student-t distribution with v degrees of freedom. • Conditional value at risk (CVaR) or expected shortfall. The conditional value at risk is defined as the expected loss, conditioned on the loss being larger than a given threshold. Formally, σCVaR (w) = E[l ∈ R : l
σVaR (w)] Again, under the assumption that the asset returns follow a normal distribution N (µ,
), σCVaR (w) = w · µ +

√

w ·
· w ·

N (N −1 (α)) 1−α

where N (·) is the density function of the standard normal distribution. Similarly, under a Student-t distribution σCVaR (w) = w · µ +

√

w ·
· w ·

Tv (Tv−1 (α)) v + (Tv−1 (α))2 · 1−α v−1

where Tv (·) is the density function of the Student-t distribution. • Peak to bottom or maximal drawdown. The risk is defined as the maximal loss that can occur within a given time period from the highest portfolio value over the whole time horizon. Figure 9.1 illustrates different definitions of a risk measure. All the presented frameworks assume that the portfolio holdings w do not change over the time horizon over which the risk is measured. Depending on how the loss function is defined, σVaR (w) may or may not be a coherent risk measure. Consider two bonds B1 and B2 who repay 100 after one year with a probability of 97 % and zero otherwise. Then σVaR95 % (B1 ) = 0 and σVaR95 % (B2 ) = 0 as their default probability is only 3 %. Now consider a portfolio composed of B1 and B2 . The probability of at least one bond defaulting over the next year is then 1 − 97 % · 97 % = 5.91 %. This means that σVaR95 % (B1 + B2 )
0 and thus σVaR95 % (B1 + B2 )  σVaR95 % (B1 ) + σVaR95 % (B2 ) is not subadditive. Volatility as the most prominent risk measure, assuming a normal or even an elliptic distribution3 of the asset returns, is a coherent risk measure, while σCVaR (w) is always a coherent risk measure. The choice of the applicable definition of risk measure defines one of the key specificities of an investment process, and is therefore a differentiating factor for any investment manager. When selecting a risk definition, the following aspects need to be taken into account: • The risk definition must be consistent with the investor’s preference for risk. Consider the two portfolio return time series in Figure 9.2. Both have the same mean and standard 3 An elliptic distribution is defined as a multivariate affine transformation of a spherical distribution. A spherical distribution is a distribution where the random values are uniformly distributed on a sphere. The normal distribution belongs to the family of elliptic distributions (McNeil et al., 2005).

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Positive Alpha Generation: Designing Sound Investment Processes

25% µ = 0.8 %

Probability

20%

σ vola = 3.9 %

15%

10%

σ VaR 95% = 7.5 %

σCVaR 95% 5%

= E[R | R ≤ −σVaR 95%] = 7.6%

15.0%

13.5%

12.0%

10.5%

9.0%

7.5%

6.0%

4.5%

3.0%

1.5%

0.0%

−1.5%

−3.0%

−4.5%

−6.0%

−7.5%

−9.0%

−10.5%

−12.0%

−13.5%

−15.0%

0%

S&P 500 monthly return Source: data sourced from Bloomberg Finance LP, author's calculations

FIGURE 9.1 Illustration of the value at risk as well as conditional value at risk definitions, given a historical distribution of realized returns based on weekly returns of the S&P 500 index from December 1987 to December 2007 as well as the normal distribution N(µ, σ ). The first figure shown is based on the normal distribution and the second one on the actual data

3% 1% −1 % −3 % −5 % −7 % −9 % Dec.02

Jun.03

Dec.03

Jun.04

Dec.04

Jun.05

Time series A

Dec.05

Jun.06

Dec.06

Jun.07

Dec.07

Time series B Source: author's calculations

FIGURE 9.2 Two portfolio return time series having identical mean and standard deviations

Alternative Risk Measures

133

deviation. Nevertheless, some investors may prefer time series A over B or vice versa. Such subjective preferences need to be considered. • The risk definition must be consistent with the return forecasting approach allowing actual control of the resulting portfolio risk. • The risk definition must allow a sound implementation, that is selection of a risk model, and estimation of the parameters. In practice a compromise between adequacy and overspecification must be made. For example, the added value of precisely modeling the tails of a distribution must be compared with the complexity introduced.

9.2

ALTERNATIVE RETURN DISTRIBUTIONS

In Chapter 8, my main attention was focused on estimating the parameters of a risk model assuming that the asset returns follow a multivariate normal distribution. Unfortunately, this assumption is not always satisfied. Indeed, many financial time series, especially when observed at high frequencies, for example on a weekly or daily basis, exhibit so-called fat tails and asymmetries. Here I distinguish between two categories of alternative return distributions, that is: • those that try to improve the fit of the whole distribution to the observed historical data and • those that only focus on the tails of the distribution, that is on extreme events. 9.2.1 Elliptical distributions The most common class of distribution functions used to fit the whole distribution to the observed historical data is the elliptical distribution class. An elliptical distribution is an affine transformation of a spherical distribution, that is a distribution where the values of the random variable are uniformly distributed on a sphere (McNeil et al., 2005). Any elliptical distribution E(µ,
, ) can be described by three parameters, the location value µ, the dispersion matrix
and the characteristic generator  of the distribution. Any elliptical distribution E(µ,
, ) defines a coherent risk model, given that the variance is finite. The most common elliptical distributions are: • the multivariate normal distribution N (µ,
) = E(µ,
, ), where (u) = exp(− 12 · u), • the Student-t distribution (Gosset, 1908) T (µ,
, ν), which is a suitable alternative distribution of asset returns, where smaller degrees of freedom √ ν imply heavier tails, • the normal mixture distribution, that is M(µ,
, W ) ∼ µ + W ·
· N (0, I), where W
0 is a nonnegative scalar, and • generalized hyperbolic distributions (Barndorff-Nielsen, 1978) with four parameters GH(µ,
, λ, α), where µ is a location parameter,
a dispersion parameter and λ as well as α two shape parameters, for example – the normal inverse Gaussian distribution (Barndoff-Nielsen, 1997), – the hyperbolic distribution (Eberlein and Keller, 1995) or – the variance gamma distribution (Madan and Seneta, 1990). McNeil et al. (2005), as well as Platen and Stahl (2003) compared different elliptical distributions against each other and assert their fitness to model different types of assets, especially for short time horizons. They conclude that elliptical distributions in general

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and the Student-t distribution in particular provide better fits of financial data than the traditionally used normal distribution, especially with respect to appropriately modeling fat tails. However, the enhanced fitness comes at a price. Indeed, the parameter estimation becomes more difficult, due to both the increased number of parameters to estimate and the numerical complexity of finding optimal values for the parameters. Furthermore, the risk of a portfolio can no longer be calculated using an analytical function. Especially the absence of an analytical specification of the risk measure makes it hard to use within a translation algorithm, as shown in Chapters 10, 11 and 12. In addition, the advantages of elliptical distributions in portfolio construction are weakened by the fact that mean-variance efficient portfolios, that is optimal risk minimizing portfolios attaining a certain return, under any elliptical distribution with finite variance are equivalent (Embrechts et al., 2002). Example Let me illustrate the difference between the normal and the Student-t distributions for two specific time series. Figure 9.3 shows the actual distribution (histogram) of the Nasdaq 100 index (left) and the JPY/USD exchange rate (right) compared to the respective normal and Student-t distributions. Both actual distributions exhibit more extreme events than predicted by the normal distribution. The Nasdaq 100 exhibits four weeks with a return of less than −10 %, even one with a return of −25.3 %. On the positive return side, four weeks exhibited returns of more than 10 %. Similar effects can be observed for the JPY/USD exchange rate. In addition, especially the JPY/USD distribution is skewed, an effect not modeled by either the normal or the Student-t distribution. The Student-t distribution with two degrees of freedom improves on modeling fat tails in the analyzed data. Nasdaq 100

JPY/USD

20%

35 %

18%

30 %

16% 25 %

14% 12%

20 %

10% 8%

15 %

6%

10 %

4% 5%

2% 0% −15% −10% −5 % Sample

0%

5%

Normal

10 % 15 % Student

0% −5 %

0% Sample

Normal

5% Student

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 9.3 Comparison of the actual distribution (histogram) with the estimated normal distribution (line) and the Student-t distribution (dashed line) for weekly historical return data between December 1987 and December 2007 for the Nasdaq 100 index (left) and the JPY/USD exchange rate (right)

Alternative Risk Measures

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9.2.2 Extreme event distributions The extreme event theory, also called the extreme value theory (EVT), finds its roots in modeling engineering risks, like floodwalls, or pharmaceutical studies. Rather than studying the whole distribution, EVT concentrates on extreme events, that is the tails of the distributions. There are two categories of EVT distributions, that is: • so-called block maximal models and • threshold exceeding models. The block maximal models are defined by the three-parameter generalized extreme value (GEV) distribution with the density function GEV(x; ξ, µ, σ ), where   

− 1   ξ x − µ     if ξ = 0 and 1 + ξ · x > 0   exp − 1 + ξ · σ GEV (x; ξ, µ, σ ) = 

  x−µ   if ξ = 0  exp − exp − σ where 1 + ξ · (x − µ)/σ > 0; µ is called the location parameter, σ > 0 the scale parameter and ξ the slope or tail index parameter. Embrechts et al. (1997) showed that the tails of any return distribution fall within three categories, named after the mathematicians who first studied them: • Weibull. The tail of the distribution has a finite endpoint. • Gumbel. The tail of the distribution is exponential in form. • Fr´echet. The tail of the distribution declines by a power. The GEV(x; ξ, µ, σ ) distribution is a Fr´echet distribution if ξ > 0, a Gumbel distribution if ξ = 0 and a Weibull distribution if ξ < 0. Figure 9.4 (left) illustrates the density function GEV(x; ξ, µ = 0, σ = 1) for ξ = 0, ξ = 1 and ξ = −1. The second category of extreme event models, the threshold exceeding models, are defined by the generalized Pareto distribution (GPD). The density function of the GPD(x; ξ, β) is defined by 

1   x −ξ   1 − 1 + ξ ·  β GPD(x; ξ, β) =

  x    1 − exp − β

if ξ > 0 and β > 0 and x
0 if ξ < 0 and 0  x  −β/ξ if ξ = 0 and x
0;

where β > 0; β is called the scale and ξ the shape parameter of the distribution. Figure 9.4 (right) shows the density function of the GPD. It can be seen that the smaller the tail index parameter ξ , the more mass of the distributed is concentrated in the extremes; that is the modeled tails of the distribution are fatter when compared to the normal distribution. To estimate the parameters of both the GEV distribution and the GPD, numerical algorithms based on the maximum likelihood approaches need to be used.

136

Positive Alpha Generation: Designing Sound Investment Processes Generalized extreme value distributions

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 −2 −1

Generalized Pareto distributions 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3

4

5

6

7

8

0

1

2

Weibull (−0.5) Gumbel (0.0) Fréchet (0.5)

3

4

5

6

7

8

9

10

Pareto type II (−0.5) Exponential (0.0) Pareto (0.5) Source: sample data, author's calculations

FIGURE 9.4 Density function of the generalized extreme value (GEV) distribution for ξ = 0 (dotted line), ξ = 1 (straight line) and ξ = −1 (dashed line) (left) and density function of the generalized Pareto distribution (GPD) for ξ = 0 (dotted line), ξ = 1 (straight line) and ξ = −1 (dashed line) (right)

There are three key reasons why solely relying on extreme value theory within an investment process is inadequate: • Portfolio risk is not only due to extreme events. Extreme value theory is highly inaccurate to model risk outside the scope of extreme events, that is the tails of the return distribution. • Due to the lack of historical data, it is hard to estimate precisely the parameters of an extreme value theory distribution, especially at high threshold levels. • Extreme value theory does not integrate efficiently with traditional risk measures, especially in the context of defining transfer functions, that is defining the portfolio construction and risk management value chain element. Example Consider an investment in a single stock, like, for example, Apple, Inc. I am interested in modeling the weekly losses in excess of a given threshold u, which I fix at u = −10 %. Out of the 1043 observations between December 1987 and December 2007, that is over a 20-year period, the weekly return exceeded 43 times that threshold. Figure 9.5 displays the weekly stock returns as well as the defined threshold (dotted line). I assume that the excess loss distribution can be modeled by a generalized Pareto distribution (GPD). To estimate the parameters ξ and β I decide to use a maximum likelihood approach. The log-likelihood can be calculated as



Y 1 ya ln L(ξ, β; y) = −Y · ln(β) − 1 + · ln 1 + ξ · ξ β a=1

(9.1)

Alternative Risk Measures

137 Apple, Inc.

50 % 40 % 30 % Weekly return

20 % 10 % 0% −10 % −20 % −30 % −40 % −50 % 1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 9.5 Weekly return of the Apple, Inc. stock price between December 1987 and December 2007, the dotted line indicating the selected threshold level of u = −10 % 100% Theoretical loss Realized loss

Probability of loss

80%

60%

40%

20%

0% −40%

−35 %

−30 %

−25 %

−20 %

−15 %

−10 %

Maximal loss Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 9.6 Probability of a loss not exceeding the specified maximal loss, given the loss is larger than the threshold, fitted GPD (line) and actual realizations (dots)

where y = {ya = ra − u : ra < u} is a vector of the weekly return data exceeding the selected threshold and Y is the size of the vector y. Figure 9.6 shows the empirical distribution of excess returns fitted to a GPD. This graph allows me to get a better understanding of the relationship between the size of a loss and the probability of it occurring, assuming that the loss exceeds the defined threshold. Next, I want to design a bottom-up security selection based equity investment process optimizing the following utility function: • maximize the portfolio return,

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Select a set of admissible security weights wa : loop
1. Calculate the portfolio return rt = t wa · ra,t for all considered time periods in the past. 2. Calculate the threshold level u such that 1 % of the historical returns belongs to the tail of the distribution; that is the number of returns rt being smaller than u does not exceed 1 %. 3. Estimate the parameters ξ and β of a GPD based on all returns rt being smaller than u using the maximum likelihood estimator shown in Equation (9.1). 4. Calculate the maximal loss Lw of the portfolio w given a probability of 1 %, that is a probability of 10 % in the tail using the estimated
GPD. exit loop when Lw  15 % and a wa · σa · IRa is maximal 5. Adjust security weights wa . end loop

FIGURE 9.7 Algorithm for selecting a set of equity securities such that the maximal loss in any week does not exceed 15 % with a probability of 0.1%

• subject to avoiding large losses or, more specifically, ensuring that a loss of more than 15 % in any given week is less than 0.1 %, that is less than once in 20 years. This utility function differs from more common utility functions in the sense that it only focuses on extreme events. I consider that the forecasts are neutral, overweight and strong overweight on a security-by-security basis. This forecast is derived, for example, using a discounted cash flow model. To construct the portfolio I assume that: • The expected return per unit of volatility is IRa = 1 for a strong overweight, IRa = 0.5 for an overweight and IRa = 0 for a neutral equity security. • The risk is modeled using a generalized Pareto distribution with a threshold value set at two standard deviations of the expected portfolio return. • The risk characteristics, especially the extreme risks, of the portfolio can be modeled using 10 years of weekly historical return data. Based on these assumptions I calculate the weights to allocate to the individual equity securities using the algorithm shown in Figure 9.7.

9.3 EXPOSURE BASED RISK MODELS The risk models described so far focus on measuring and modeling the uncertainty inherent in asset returns. In exposure based risk models, the focus is on analyzing the sensitivity of the portfolio returns to endogenous or exogenous factors. The basic idea behind an exposure based risk model is the calculation of the partial derivative of the portfolio’s expected return versus a given risk factor, that is ∂R(w) ∂f

(9.2)

where R(w) is the expected portfolio return and f a factor. Equation (9.2) expresses the change in the portfolio’s return or value, given a change in the underlying factor f . An important caveat of the exposure based risk modeling approach is that it describes the

Alternative Risk Measures

139

expected changes of the portfolio return for small changes in the factor. It does not cope with large changes and therefore does not accurately model extreme events. Exposure based risk models become important especially if the uncertainty of specific factors cannot be modeled by the volatility or nonparametric risk models as presented. 9.3.1 Exposure risk in fixed income portfolios Exposure based risk models are especially common in fixed income investment processes. Indeed, interest rate risk is: (i) an exogenous risk and (ii) cannot effectively be modeled through other risk measures, like volatility. One of the most common models for decomposing the risk of a fixed income portfolio is the three-factor model based on the factors: (i) shift, that is parallel movements of the yield curve, (ii) twist, that is rotations around the center point of the yield curve and (iii) butterfly, that is movement of the extreme short and long positions in a different direction from the center point of the yield curve. These three factors are illustrated in Figure 9.8. The risk of a portfolio is characterized by the exposure of the portfolio to these three factors. The reader who is interested in more detail on fixed income risk models will find specific treatment in Golub and Tilman (2002). Yield curve shift

Yield curve twist

Yield curve butterfly

7%

7%

7%

6%

6%

6%

5%

5%

5%

4%

4%

4%

3%

3%

3%

2%

2%

0

5

10

15

20

25

30

0

5

10

15

20

25

30

2%

0

5

10

15

20

25

30

Source: sample data, author’s calculations

FIGURE 9.8 Fixed income factor model based on structural changes in the yield curve for the shift factor (left), the twist factor (center) and the butterfly factor (right), showing interest rate change (vertical axis) as a function of the time to maturity (horizontal axis)

9.3.2 The RiskMetrics approach RiskMetrics uses a combination of volatility based and factor model based approaches to model portfolio risk. It relies on the value at risk definition of risk, that is the maximal loss with a pre-defined probability over a pre-set time horizon. The RiskMetrics approach proceeds in three steps:

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Positive Alpha Generation: Designing Sound Investment Processes

1. First, the portfolio is decomposed on exposure to generic risk factors and future cash flows. Specific methods are defined for the different instrument types, like bonds, forward rate agreements or interest rate swaps, to name just a few. 2. In the second step, the cash flows are mapped to the so-called RiskMetrics vertices, representing a simplified term structure. 3. Finally, RiskMetrics calculates the portfolio risk by assuming a conditionally multivariate normal distribution for the basic vertices. In the presence of nonlinear instruments, second-order effects are taken into account. RiskMetrics provides estimated parameters of the distributions underlying the vertices on to which the cash flows are mapped. A detailed description, including many examples, can be found in the RiskMetrics (J. P. Morgan and Reuters, 1995) document. Although RiskMetrics can be used in the context of an investment process, it mainly focuses on balance sheet portfolios containing sophisticated fixed income instruments and derivatives with nonlinear payoffs. Its main advantages are: • its conceptual simplicity through mapping any instrument on to basic risk factors or cash flow vertices, • its applicability at the security level and • its generality with respect to instruments and cash flow generating assets.

9.4

NONPARAMETRIC RISK MODELS

A number of risk calculation algorithms exist that rely on actual assets or portfolio returns data rather than on return distributions or their moments. The idea is first to generate a large number of portfolio returns and then calculate the risk values based on this data set, assuming only that the generated return data set represents the true distribution of the returns. For example, the value at risk at the 95 % confidence level can be determined by partitioning the sorted set of return data in a subset containing the 95 % largest returns and a second subset with the remaining 5 %. The value at risk is determined as the boundary between the two sets, that is, for example, the largest value in the 5 % set. Different approaches are used to generate a large number of return data elements. The most common techniques are: • • • • •

historical simulation, bootstrapping or re-sampling, Monte Carlo simulation, asset re-pricing and stress testing.

Each of these techniques has its own characteristics and field of application. 9.4.1 Historical simulation The historical simulation method is the most common nonparametric risk measurement algorithm. It is especially useful in the context of portfolios with a large number of assets and/or assets whose returns cannot be easily modeled using distributions. Consider a set of assets and a matrix R = (ra,t ) of their historical returns. I assume that these historical returns are a fair representation of the expected future returns and their uncertainty. Consider

Alternative Risk Measures

141

Table 9.2 Illustration of the historical simulation approach based on daily return data, where the value at risk and the conditional value at risk are shown for the 95 % confidence level and a one-day time horizon Asset

Holding σVaR95 % σCVaR95 %

Apple, Inc.

IBM

Dell, Inc.

250 −5.13 % −6.21 %

200 −2.20 % −3.71 %

400 −3.11 % −5.72 %

Jan. 2008 call on Microsoft @ 30 5000 −9.25 % −11.2 %

8 7/8 \% US treasury 8/17 1000 −0.72 % −0.96 %

Portfolio

−0.55 % −0.84 %

Source: data sourced from Bloomberg Finance LP, author’s calculations.

a portfolio investing in these assets. Then the historical simulation method proceeds by calculating the historical portfolio returns as r = R · w, where w are the portfolio weights. This time series is then used to calculate the risk value based on a given definition. Table 9.2 illustrates an example of a portfolio consisting of six assets, three stocks, one bond, one currency forward and one call option, over a six-month period between 31 May and 30 November 2007 using daily asset prices. For simplicity, the portfolio holdings are expressed in units rather than in %. Figure 9.9 shows the histogram of the constructed portfolio based on the data in Table 9.2. It can be seen that the actual portfolio distribution exhibits fat tails, when compared to the normal distribution, especially on the negative side of the distribution. They are due to the fact than the technology stocks included in the portfolio exhibit a higher probability mass in the tails than the normal distribution would suggest. In addition, the one-sided fat tail is mainly due to the call option included in the portfolio. The historical simulation algorithm calculates the return of the portfolio for each time period. It is useful when the return distribution cannot be effectively fitted by a known distribution or when specific characteristics of the distribution, which cannot be modeled efficiently by a parametric distribution, are 14% 12%

Probability

10% 8% 6% 4%

s VaR

2% 0%

−1.0%

−0.8 % −0.6 %

−0.4 % −0.2 %

0.0 %

0.2 %

0.4 %

0.6 %

0.8 %

1.0 %

Return Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 9.9 Histogram of the portfolio returns used by the historical simulation algorithm and illustrated in Table 9.2

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Positive Alpha Generation: Designing Sound Investment Processes

relevant. Such an example is the maximal drawdown risk framework definition or a portfolio composed of short equity call options. 9.4.2 Bootstrapping or re-sampling Very often the historical simulation algorithm cannot be applied because insufficient historical return data of the individual assets are available. In this situation the bootstrapping algorithm can be applied. The basic idea of the bootstrapping algorithm is to generate larger return data sets by re-sampling from the original return data set, assuming that the historical return data are independent and drawn from an identical distribution. The historical simulation algorithm can then be applied to the re-sampled return data set and the risk measure derived. The bootstrapping algorithm proceeds as shown in Figure 9.10 to generate a re-sampled data set. for s ← 1 to target re-sampling size 1. Randomly select a subset of data from the original data set of smaller size. 2. Estimate the mean of the subset of data. 3. The calculated mean is an element of the re-sampled set. next s

FIGURE 9.10 Bootstrapping or re-sampling algorithm

Although the basic idea of bootstrapping is fairly simple, its practical application requires thoroughness in the construction of the re-sampled data sets from the original data set, which is step one of the proposed algorithm, in order to avoid estimation biases. Details on the bootstrapping algorithm and its application can be found in Davison and Hinkley (1997). 9.4.3 Monte Carlo simulation Monte Carlo simulation algorithms represent a third category of nonparametric risk calculation algorithms. Rather than rely on actual historical return data, Monte Carlo simulation uses randomly generated return data. The randomly generated return data are then analyzed using the same techniques as those used in the historical simulation algorithm. To generate random returns a distribution or a process model is assumed for the individual assets or its characteristics. For example, bond returns can be modeled by assuming a Cox–Ingersoll–Ross model (Cox et al., 1985) for short term interest rates and a constant liquidity premium for long term bonds over short term ones. Interest rates are then randomly generated based on the selected model and bond prices and returns are calculated. Based on the selected distribution or process, a large number of random return samples are generated. Finally, the returns are weighted according to the weights of the assets in the portfolio to construct a set of portfolio returns. This set is then analyzed using the historical simulation techniques. The difficulty in applying the Monte Carlo simulation algorithm is the definition of the distributions and processes of the underlying assets as well as their interactions and correlations. On the other hand, as the sample set is randomly generated, the small sample size issue of the historical simulation approach is voided. Monte Carlo algorithms are most

Alternative Risk Measures

143

appropriate for modeling assets with specific return generation functions, like many factor based asset pricing models as well as assets whose price depends on the price or return of other assets that can be modeled relatively well. 9.4.4 Asset re-pricing The asset re-pricing algorithm is most appropriate when the assets in the portfolio: (i) do not have a long return history, such as for example futures or options, and (ii) their price or return depends on the price of other assets or observable variables for which a relatively long historical time series exists. An example of such a portfolio would be one of interest rate swaps, the interest rate levels being known with a long time series history. In order to derive a set of portfolio returns, the assets of the portfolio are valued using historical data for the dependent variables. Then the historical simulation techniques are applied. 9.4.5 Stress testing It is often valuable to characterize the portfolio risk under stress scenarios. Stress scenarios are return scenarios that model extreme events. Stress scenarios can be derived using either: (i) specific historical extreme event situations, as for example shown in Table 8.4 in Chapter 8, or (ii) based on the asset re-pricing algorithm using randomly generated extreme movements in the dependence variables, for example a change of 2 % in short term interest rates over a one-week time horizon. These generated scenarios and the portfolio holdings or weights can then be used to construct a return time series that models the behavior of a portfolio under such a stress scenario. Figure 9.11 illustrates the daily return of an equally weighted portfolio composed of the stocks Boeing, Home Depot, J. P. Morgan Chase, Microsoft and Exxon Mobile for a 30-day period starting on 10 September 2001. It can be seen that the stress test shows a maximal drawdown of 18.7 %. In addition, the stress test shows that the portfolio value recovers to 92.1 or a loss of 7.9 % over the considered one-month time period. Nonparametric risk calculation algorithms allow high precision in the calculation of the risk measure under the assumption that the used historical or randomly generated data are a good approximation of the uncertainty of the future. They do not suffer from the drawbacks of an imposed structure, as is the case for parametric models. However, most importantly, they give a very high degree of flexibility and level of detail of the analysis, which can be calculated. On the downside, nonparametric risk models are numerical approaches rather than analytical frameworks, making them hard to use in a transfer algorithm. In addition, they are usually very calculation intense, especially when bootstrapping, Monte Carlo simulation or asset re-pricing algorithms are implemented.

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Positive Alpha Generation: Designing Sound Investment Processes

100

95

90

85

Se p 14 . Se p. 16 Se p. 18 Se p. 20 Se p. 22 Se p. 24 Se p 26 . Se p. 28 Se p. 30 Se p. 02 O ct . 04 O ct . 06 O ct . 08 O ct .

12

10

Se p.

80

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 9.11 Daily returns for an equally weighted portfolio of five major US stocks over a 30-day period starting 10 September 2001, one day before the terror attacks on New York and Washington

9.5 HANDLING ASSETS WITH NONLINEAR PAYOFFS As soon as the portfolio contains derivative instruments, especially those whose returns depend in a nonlinear way on other asset prices, so called nonlinear assets, traditional parametric approaches become useless. In this case, three common approaches exist: • Nonparametric risk models based on the asset re-pricing algorithm are applied. • Asset prices and returns are linearized using, for example, Taylor expansions. The most used approach in this category is the delta normal approach. Asset returns are represented by a first-order linear approximation. Such approximations again allow traditional parametric risk models to be applied. • Assets are decomposed into constituting assets, like, for example, the expected cash flows of fixed income derivatives, which can be modeled using parametric models. I recommend that, whenever feasible from a modeling and calculation perspective, nonparametric approaches are used when the portfolio includes assets with nonlinear payout structures. If this is not possible, linearization approaches, like RiskMetrics (J. P. Morgan and Reuters, 1995), should be used.

9.6

CREDIT RISK MODELS

Although the main focus in this part of the book is on modeling and managing market risk, it is useful to know the three most common credit risk models used to measure and manage credit risk. This is especially important in the context of developing fixed income credit investment processes. Here credit risk is defined as the risk that a company defaults on its financial obligations. There are three common credit risk modeling approaches, that is:

Alternative Risk Measures

145

• the KMV model, • the CreditMetrics approach and • the CreditRisk+ model. The KMV model is based on the relationship between the equity value, the firm value and the debt value in an option pricing framework. Merton (1990, Chapter 11) has shown that the value of a firm is equal to the value of its debt, modeled by a bond, and a call option on the firm’s assets with a strike price equal to its debt level. The KMV model uses this relationship to derive the expected default value. In addition, comparable companies are analyzed to derive a frequency of default. From this information, the company value and its volatility, the probability of default, is calculated. The CreditMetrics model, developed by J. P. Morgan, uses a slightly different approach. It derives the probability of default from a correlation matrix of the firm’s asset prices. It assumes that stock prices are a fair proxy for the firm’s assets. Based on Monte Carlo simulations and the Merton model (1990) of a firm, the end-of-period distribution of values is derived. This value is then used to determine the default probability. The CreditRisk+ model, developed in 1997 by Credit Suisse First Boston, models credit risk of a portfolio based on a set of common risk factors. The probability of default is dependent on the credit rating and the sensitivity to each of the risk factors. The obligors are assumed to be uncorrelated except for the correlations between the common risk factors. Each of the three models has its unique advantages and drawbacks. However, none of them relates the probability of default to the current economic conditions and as such a direct link to market risk is missing.

Part IV Portfolio Construction

10 Single Period Mean-Variance Based Portfolio Construction Portfolio construction is defined as the efficient and effective translation of market forecasts into risk controlled portfolio structures. It combines the investment strategist’s expectations, taking into account uncertainty, with the investor’s needs and restrictions, maximizing the utility function. Good portfolio construction cannot cover up for poor forecasting skills, but poor portfolio construction capabilities can make good forecasting skills fail.

10.1 DEVELOPING A MODULAR PORTFOLIO CONSTRUCTION PROCESS The core component of any portfolio construction process is the translation mechanism that transfers the investment manager’s forecasts into the investor’s portfolio. The translation mechanism can be seen as a compiler that transforms the source code, that is the market forecasts, into machine code, that is the investor’s portfolio, based on the target computer architecture, that is the investor’s needs. The translation function may be quantitative or qualitative. To ensure that every investor portfolio is constructed in the best possible way, the transfer mechanism should be: • objective and • formulated in an algorithmic framework. It addition, due to the multidimensionality of risk, I recommend using a quantitative model as the core of the portfolio construction process. In this chapter different quantitative models are presented. They all have in common the fact that they are context independent; that is they do not rely on an economic interpretation of its inputs. They are purely based on the two concepts return and risk. It is important that the same portfolio construction process can be applied to as large a number of investment products as possible in order to achieve flexibility as well as scalability, a key goal of the value chain approach proposed in Chapter 3. It should also allow most investor restrictions to be taken into account, like the definition of the investment universe, lower and upper bounds of foreign currency exposures, hedging restrictions and maximal duration deviations, to name just a few. 10.1.1 A risk budgeting based modular portfolio construction approach In an ideal world, the translation function within the portfolio construction and risk management module would be defined as a single function. Unfortunately, in most practical investment processes, the dimensionality of both the investment universe as well as the forecasting set is too large to make such an approach viable.

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Positive Alpha Generation: Designing Sound Investment Processes

I suggest implementing a risk budgeting based modular approach. The idea is to differentiate between different sources of alpha. It can be described by the following four steps: 1. Decompose the investment universe such that separate sources of alpha can be applied to each constituent. These sources of alpha should be based as much as possible on independent sets of forecasts. 2. Allocate or budget risk to each alpha source. Ideally, the risk budgeting approach should allocate risk such that the ratio of the incremental impact on performance to the incremental impact on total portfolio risk is the same for all sources of alpha (Winkelmann, 2000). 3. For each alpha source, develop a dedicated portfolio construction translation function. This function or model should be customized to the structure of the forecasting skills. In addition, it has to construct the subportfolio structure such as to consume a given budget of risk. 4. The different portfolio construction models are combined. As the expected performance impact and the interaction between the different translation functions on a risk and return basis is associated with a high degree of uncertainty, in practice the risk budgets associated with each source of alpha are often defined subjectively. In addition, such subjective risk decomposition allows specific characteristics of a product to be defined. Figure 10.1 illustrates a sample risk budgeting approach for a global asset allocation product. The portfolio risk is decomposed into three sources of alpha, that is: – the tactical asset allocation alpha source, – the currency overlay alpha source and – the asset class implementation alpha source. The asset class implementation source is again decomposed into different alpha sources. In addition, the US equities asset class implementation alpha source is subdivided into a style allocation alpha source and a security selection alpha source. Ideally the correlation between alpha sources should tend towards zero in order to maximize the impact of the fundamental law of active management. The risk budgeting approach may be either based: • on total risk per alpha source or • marginal contribution to total portfolio risk. Both approaches are illustrated in Figure 10.1. The total risk based approach has the advantage that a risk budget can be allocated to a given module independent of the other modules. For example, I might decide that the equities Japan alpha source should not exceed an active risk of 1.5 % versus a benchmark or index representing the Japanese equity market, for example the Nikkei 300 index. In addition, it is easier to construct a translation function against a total risk target. On the other hand, only the marginal contribution to risk approach manages the risk at the portfolio level. The risk budgeting process first allocates risk to the individual sources of alpha on a marginal risk basis. Then, in a second stage the marginal risk is translated into a target total risk per alpha module by assuming a fixed

Single Period Mean-Variance Based Portfolio Construction

151

Portfolio Risk budget = 15.5% Marginal contribution to risk = 100% Asset class implementation

Tactical asset allocation

Currency overlay

Risk budget = 12.0% Marginal contribution to risk = 60%

Risk budget = 3.0% Marginal contribution to risk = 20%

Risk budget = 2.0% Marginal contribution to risk = 20%

Asset class: Fixed income Risk budget = 1.50% Marginal contribution to risk = 10% Asset class: Equities Europe Risk budget = 2.50% Marginal contribution to risk = 15%

Equities USA: Style allocation

Asset class: Equities USA

Risk budget = 1.50% Marginal contribution to risk = 5%

Risk budget = 4.50% Marginal contribution to risk = 20%

Equities USA: Security selection

Asset class: Equities Japan

Risk budget = 4.00% Marginal contribution to risk = 15%

Risk budget = 1.50% Marginal contribution to risk = 15%

FIGURE 10.1 Risk budgeting applied to a global asset allocation investment process allocating risk to the different alpha sources

correlation between alpha sources. This total risk is communicated to the investment manager responsible for implementing that alpha source. The risk allocation decision may be defined on: • a static a priori decomposition based on assumptions, as shown for example in Figure 10.1, • an ex-post decomposition based on the realized return of each alpha module, for example its information ratio, or • an ex-ante decomposition where the expected confidence in the individual alpha modules is used. Ideally, for a given investor portfolio, the translation functions for the individual alpha modules are applied using the investor specific utility function, that is risk preference, return expectations and restrictions. In practice, however, especially when some of the alpha sources are outsourced, individual alpha sources are managed independently of specific restrictions against a given fixed total risk budget. Very often the risk budget is even defined by the alpha sources provider rather than the overall investment process owner. This is, for example, the case in a fund-of-fund approach, as illustrated in Table 10.1. First, 4.5 % volatility is allocated to the target asset allocation between the individual fixed income components. Next, risk budgets in terms of target active risk or tracking error are allocated to each asset class. These target risk budgets represent the upper bounds of active risk available to the managers of the individual asset classes. If the individual manager exceeds that target

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Positive Alpha Generation: Designing Sound Investment Processes Table 10.1 Fund-of-fund based fixed income tactical asset allocation investment process

Asset allocation weight (%)

Asset class

Alpha source

100 25 15

n/a Bonds EUR Bonds GBP

30 10 15

Bonds USD Bonds USD Bonds JPY

Asset allocation Yield curve positioning Duration management, including possible net short duration Yield curve positioning Credit exposure Relative value bond/company selection

Target risk/tracking error (%) 4.5 1.5 1.0 0.5 2.0 2.5

Source: sample data

risk, combining the manager’s actively managed portfolio with an indexed portfolio targeting the same market allows an adjustment of the active risk budget allocated.

10.2 THE MEAN-VARIANCE FRAMEWORK The mean-variance framework is one of the most important frameworks for building translation functions. It is based on the assumption that the characteristics of asset and portfolio returns can be completely described by the two parameters: • mean and • variance or standard deviation of a random variable. In order to give an interpretation to these parameters, it is necessary to assume that asset returns follow a multivariate normal distribution. In addition, the framework is built on the assumption that the two parameters are known with certainty. The mean parameter represents the return expectations and the variance and covariance the risk characteristics. No additional modeling or assumptions are necessary. Based on these assumptions, it is possible to define the investor’s utility function as U (w) = R · w − λ · w ·
· w

(10.1)

where w are the weights of the assets in the portfolio, R a vector of expected returns,
the covariance matrix and λ a risk aversion parameter. A portfolio w is called mean-variance efficient if, for a given parameter λ, the utility function U (w) is maximized. The set of all mean-variance efficient portfolios defines the efficient frontier, sometimes called the mean-variance frontier or the mean-variance efficient frontier, as shown in Figure 10.2. By construction, the efficient frontier is a concave curve in the return-risk or mean-variance space. The portfolio w, which minimizes the risk, that is w ·
· w, is called the minimum-variance portfolio. It always exists and is unique. The mean-variance framework is a so-called single period framework. Risk and return are modeled over one time period and the portfolio holdings are assumed not to be adjusted. The time period may be one day, one month or even five years. The portfolio weights are determined once for the beginning of the time period. More advanced models take into

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12%

Return/mean

10% efficient frontier

8%

set of feasible portfolios 6% minimum-variance portfolios 4% 12%

14%

16% 18 % Risk/standard deviation

20 %

22 %

Source: sample data, author's calculations

FIGURE 10.2 Set of all mean-variance efficient portfolios forming the efficient frontier

account multiple time periods and especially mean and variance parameters changing over time. Such approaches are illustrated in Chapter 12. Under the assumption that the only restriction imposed is that the portfolio is fully invested, that is  wa = 1 or w · 1 = 1 a

Tobin (1958) has shown that any mean-variance efficient portfolio can be expressed as a linear combination of two other mean-variance efficient portfolios. This property is called the two-fund separation theorem. The minimal set of portfolios necessary to describe all the portfolios on the efficient frontier is called the set of corner portfolios. The mean-variance framework as a basis for building translation algorithms is most appropriate when the following conditions are satisfied: • The investor’s utility function can be approximated by the quadratic utility function U (w) shown in Equation (10.1). This means that the investor has an increasing absolute risk aversion. • The return characteristics of all assets are similar. They can be described by a normally distributed random variable. • The number of assets is small and the assets are significantly different from each other. This is especially the case in asset allocation based investment processes, but much less in security selection based approaches. • The parameters R and
can be forecasted or estimated with sufficient precision. Such a framework is known as a certainty equivalent framework. This is, for example, the case when purely relying on historical data. • The parameters R and
are constant over the time period considered and no active changes, like cash flows or transactions, are expected to occur during that period.1 1 If cash flows, like dividends, occur, they can be reinvested according to a predefined scheme and as such considered an integral part of the asset return forecast.

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• Risk can be defined by a single number, that is the portfolio variance. • The number of additional restrictions is large and/or they are of a complex nature. Although the mean-variance framework is easy to specify and implement, investment processes exist where using the mean-variance framework will not result in an efficient translation function. This is the case when any of the requirements is not met. For example, applying the mean-variance framework in a bottom-up stock selection investment process on the S&P 500 index universe will fail. Using a mean-variance based translation function will result in strange and highly unstable asset weights when there is a large degree of uncertainty associated with the input parameters. When considering long time horizons over which the forecasts and/or restrictions change in a pre-defined way, multiperiod models are more appropriate. Markowitz (1952, 1987) has shown that mean-variance efficient portfolios for a short time horizon, for example one week or one month, may not be efficient in the long run. Long term efficiency is not necessarily monotonic in portfolio risk. Finally, if the portfolio includes assets that cannot be modeled through normal random variables, the outcome of a mean-variance based translation function will not be consistent with the investor’s utility function.

10.3 THE MARKOWITZ MEAN-VARIANCE MODEL The Markowitz mean-variance model is based on Markowitz’s seminal work published in 1952. He showed that any investor considering return as desirable and variance of return as undesirable will prefer a diversified portfolio over all nondiversified ones. A diversified portfolio is one maximizing the quadratic utility function in Equation (10.1). Interestingly, Roy (1952) independently came up with the same idea as Markowitz. He advised choosing the single portfolio on the mean-variance efficient frontier that maximizes the utility function U (w) =

R · w − d w ·
· w

where d is a disaster level return on which the investor places a high priority that it will not fall below. In 1999, Markowitz acknowledged the work of Roy by writing ‘on the basis of Markowitz (1952), I am often called the father of modern portfolio theory, but Roy can claim an equal share in this honor.’ 10.3.1 The mean-variance model without a risk free asset Let R denote the vector of forecasted returns and
the forecasted covariance matrix for a given set of assets and a given time horizon. The mean variance model without a risk free asset can be formulated as a quadratic optimization program as follows: min w ·
· w

max R · w

max R · w − λ · w ·
· w

R · w = µ w · 1 = 1 risk minimization

s.t.

w ·
· w = σ 2 w · 1 = 1 return maximization

s.t.

w

s.t.

w

⇔

w

⇔

w · 1 = 1

Lagrangian formulation (10.2)

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100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Port. 1/ 13 %

Port. 2/ 14 %

Port. 3/ 15 %

Eq. USA

Port. 4/ 16 %

Eq. UK

Port. 5/ 17 %

Port. 6/ 18 %

Eq. Germany

Port. 7/ 19 %

Port. 8/ 20 %

Eq. France

Port. 9/ 21 %

Port. 10/ 22 %

Port. 11/ 23 %

Eq. Japan

FIGURE 10.3 Example of a discrete subset of mean-variance efficient portfolios in the context of an equity asset portfolio

The constraint w · 1 = 1, where 1 is a vector of ones, is called the budget constraint. It ensures that the portfolio is fully invested. In its most general form, the mean-variance model does not include any positivity constraints of the form w
0. This means that a solution to Equation (10.2) can include net short positions as well as leverage. The three variations of the mean-variance model shown are equivalent. Solving the optimization program by varying the parameters µ, σ or λ results in a set of optimal portfolio weights with associated risk and return making up the efficient frontier. Figure 10.3 shows the corresponding weights for a discrete sample of mean-variance efficient portfolios. Although equivalent from a mathematical point of view, solving the risk minimization or the Lagrangian formulation is much easier numerically than solving the return maximization formulation. This is due to the fact that handling a quadratic constraint, as in the return maximization formulation, is much harder than only handling linear constraints and a quadratic utility function. The most common algorithms used to solve optimization2 problems in Equation (10.2) are variations of the sequential quadratic programming algorithm (Nocedal and Wright, 2006) or interior point algorithms (Ye, 1997). The latter ones are generally preferred, especially when the number of asset classes is large. If the covariance matrix
is singular or near-singular, interior point algorithms perform poorly as they rely on calculating the inverse of
. In that case, less sophisticated iterative search algorithms based on gradient calculations should be preferred. An interesting property of the mean-variance model is that the minimum-variance portfolio w∗ always has a covariance of cov(w, w∗ ) =

1

1 ·
−1 · 1

with any asset or portfolio w (Ingersoll, 1987, Chapter 4).3 2

An optimization problem containing both an objective function and constraints is called an optimization program. If the covariance matrix
is singular, it is always possible to transform the problem such that
is not singular and the optimal solution is preserved. 3

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Figure 10.3 shows the weights for different portfolios lying on the efficient frontier. The mean and covariance matrices used to calculate these portfolios have been estimated using 10 years of historical index return data between December 1987 and December 2007. In addition, the nonnegativity constraint w
0 was added, forbidding any short positions and leverage. It can be seen that, for low risk figures, the efficient portfolios are mainly based on UK equities, the asset with the lowest volatility, that is σUK equities = 13.5 %. The minimum-variance portfolio is not solely composed of UK equities because adding US as well as Japanese equities reduces the risk below that of UK equities. As more risk is added to the portfolio, French equities are added. Finally, the most risky portfolios are essentially composed of German equities, the asset class with the highest estimated mean return (µGerman equities = 10.5 %), but also the highest volatility (σGerman equities = 23.1 %). 10.3.2 The mean-variance model in the presence of a risk free asset If the investor allows investment in a risk free asset,4 the mean-variance model is changed in two ways: • There is no budget constraint on the risky assets in the portfolio, as the cash position makes up for the residual holding. • The constraint on the expected return must be adjusted for the return of the risk free asset. The mean-variance model in the presence of a risk free asset with return RF is defined by max w

s.t.



R

RF

w ·


0



·w

0 · w = σ2 0

(10.3)

Interestingly, the set of mean-variance efficient portfolios maximizing Equation (10.3) for different values of σ 2 is a straight line, as shown in Figure 10.4. Any mean-variance efficient portfolio is a linear combination of the risk free asset and the tangency portfolio. The tangency portfolio is the portfolio on the mean-variance efficient frontier without a risk free asset that is tangent to the straight line passing though the risk free asset portfolio’s representation on the risk-return diagram. In the absence of additional restrictions, such as no short sales, the tangency portfolio is defined by wT =


−1 · (R − RF · 1) 1 ·
−1 · R − (1 ·
−1 · 1) · RF

In practice, the mean-variance model with a risk free asset is rarely used. The risk free asset is generally modeled though a risky asset composed of short term fixed income investments, as for example modeled by the Citigroup World Money Market 1 Month Euro Deposit index. The mean-variance model with cash can be seen as a context independent 4 A risk free asset is an asset that has a guaranteed return over the considered time horizon and no volatility or correlation with any other asset return. For example, a five-year zero coupon bond is a risk free asset for a five-year time horizon.

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157

12%

Return/mean

10% 8%

efficient frontier without a risk free asset

efficient frontier with a risk free asset

6% set of feasible portfolios without a risk free asset

4% 2% risk free asset 0% 0%

2%

4%

6%

8%

10 %

12 %

14 %

Risk/standard deviation Source: sample data, author's calculations

FIGURE 10.4 Mean-variance efficient frontier in the presence of a risk free asset

generalization of the capital asset pricing model (CAPM), where the market portfolio is actually the tangency portfolio. 10.3.3 Inverse optimization Assume that I have determined a solution w to the mean-variance problem in Equation (10.2). Knowing also the covariance matrix
, I can ask myself the question, ‘What would the return expectations have to be in order for my solution w to be an optimal solution?’ Answering this question is called solving the inverse optimization problem. It can be shown that setting R=a·1+b·
·w where a
0 and b > 0 are two constants, answers that question. The expected returns are only determined up to two constants. I call M =
· w the implied return multiples associated with the given mean-variance efficient portfolio w. 10.3.4 Handling restrictions One of the most important practical advantages of the mean-variance model is that it allows to be combined with practically any investor restriction, without making the resulting optimization program unsolvable. Different product types The mean-variance model can be applied to determine optimal portfolio weights, given return and risk forecasts, when optimality is measured against a benchmark b, rather than

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in absolute terms. This model is called the benchmark oriented mean-variance model and is specified by max R · (w − b) w

s.t.

(w − b) ·
· (w − b)  τ 2 w · 1 = 1

(10.4)

where τ is the maximal active risk or tracing error allowed. The equality in the risk constraint has been replaced by an inequality, indicating that the risk is to be considered as an upper bound. A consequence of this is that if a return maximizing solution with less than τ active risk can exist, there is no need to take additional risk just for the sake of taking risk. Indeed, a solution to the problem in Equation (10.4) always is at least as good as a solution to the same problem but with an equality risk constraint of the form (w − b) ·
· (w − b) = τ 2 . Equation (10.4) ignores the total risk of the portfolio and rather focuses solely on the active risk. Active risk based mean-variance models may lead to portfolios that have systematically higher risk than the benchmark and as such are not optimal from an investor’s perspective. This issue can be addressed by restricting the universe of feasible portfolios to those that have a total risk no larger than the benchmark. This can be achieved by adding the restriction w ·
· w  b ·
· b to the optimization problem in Equation (10.4). Jorion (1992) proposes a similar approach based on constraining the tracking error in the traditional mean-variance space. If the goal is to translate the forecasts into an overlay portfolio structure, the mean-variance model is rewritten as max R · w w

s.t.

w ·
· w  τ 2 w · 1 = 0

(10.5)

This variant of the mean-variance model is called the long–short mean-variance model. Although there is not much difference between the formulations in Equations (10.4) and (10.5), most gradient based optimizers have a hard time with the constraint w · 1 = 0. Sophisticated interior point optimization algorithms are not sensitive to that problem. Alternatively, problem (10.5) can be reformulated avoiding the constraint w · 1 = 0. Handling currency restrictions If the currency forecast is separated from the market forecast, two modeling approaches can be applied. First, the currency and market forecasts may be considered as two separate and independent alpha sources using the risk budgeting approach described in Section 10.1.1. The mean-variance framework is applied to each alpha module independently and both are combined using risk budgeting. A more advanced approach is to combine the two alpha modules in a single mean-variance framework. Let Rm denote the forecasted returns, hedged into the portfolio’s reference

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159

currency, of the market asset classes and Rc the forecasted currency profit and losses. Denote by wm the weights of the market asset classes and wc the currency overlay asset weights. Let
be the covariance matrix of both market and currency asset classes. Then, the currency overlay mean-variance model is defined by max w

s.t.



a a wm · ((1 + Rm ) · (1 + Rca ) − 1) + a wm    τ2 (wm wc ) ·
· wc wm · 1 = 1 wc · 1 = 0



wca · Rca

c

(10.6)

If the investor allows only currency hedging, then this can be modeled by adding the following restrictions to Equation (10.6): ∀c ∈ foreign currencies: wcc  0  ∀c ∈ foreign currencies:

a wm
−wcc

currency (a)=c

The first set of inequalities ensures that no foreign currency exposure in excess to the market exposure is held on a currency-by-currency basis. The second set of inequalities ensures that at most the currency exposure held through holding market assets is hedged. Investor restrictions Any investor restriction that can be expressed as a function f (·) of parameters and the weights w can be modeled. The modeling is easiest if the function f (·) is a linear equality or inequality. For example, the investor restriction, requiring that the exposure in developed Asian markets in a country based asset allocation investment process is no larger than 10 %, can be modeled by the equation wJapan + wHong

Kong

+ wSingapore  10 %

The fundamental law of active management suggests that there should be as large an opportunity set as possible, that is a large number of asset classes in the context of a mean-variance model. Unfortunately, it may be impossible to find suitable investment vehicles for the individual asset classes. Rather than reducing the set of asset classes at the forecasting level, I recommend adding restrictions to the mean-variance model that require the optimized solution to be calculated in a consistent way with the available investment vehicles. Assume, for example, that a forecast is formulated for Hong Kong and Singapore, but that the only investment vehicle available invests in developed Asia excluding Japan based on the market capitalization of these two countries, denoted by mHong Kong and mSingapore . Then the investment vehicle restriction can be modeled by adding the following restriction: wHong Kong wSingapore =− mHong Kong mSingapore

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Positive Alpha Generation: Designing Sound Investment Processes

Leverage restrictions Consider a product specification that allows net short positions to be taken, but which requires that the total exposure, calculated according to the commitment method,5 is no larger than 100 %. Such a restriction forbids any leverage, but allows the investment manager to take short position to express market forecasts. This restriction can be formulated as  |wa |  100 % (10.7) a = cash

Note that Equation (10.7) is not a linear inequality. Classical quadratic programming optimization algorithms can no longer be applied as Equation (10.7) is not differentiable in all values of wa . Transaction costs It will be shown in Chapter 13 that transaction costs can be taken into account at different modules of the value chain. One approach is to handle them at the portfolio construction module level and to integrate them into the mean-variance model. Let b
0 denote a vector of assets bought and s
0 a vector of assets sold, where wc denotes the current asset holdings. Denote, furthermore, by t a vector of proportional transaction costs. Then the mean-variance model in the presence of transaction costs can be formulated as max w · R − (b + s) · t w,b,s

s.t.

w ·
· w  τ2 (w · 1)2 w = wc + b − s b
0, s
0 w · 1 + b · t + s · t = 1

(10.8)

Unfortunately the optimization problem (10.8) is no longer a traditional optimization problem. Mitchell and Braun (2004) showed that this problem can be transformed back into a classical quadratic programming problem. 10.3.5 Main drawbacks The mean-variance framework is a very powerful and flexible framework. However, it comes with two major drawbacks: • First, the calculation of mean-variance efficient portfolios, especially in the absence of restrictions, is highly sensitive to the parameter values, especially the return expectations. 5 The commitment method calculates the total exposure as the sum of all exposures excluding cash in absolute value. For example, a portfolio composed of 70 % cash, 30 % US equities and a 10 % short position of Nikkei futures, has a commitment method based exposure of 40 %.

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161

• Second, the more restrictions are added to the mean-variance framework, the more difficult it becomes to solve the resulting optimization problem efficiently and optimally. Detailed expertise, especially in the area of numerical stability in optimization, becomes relevant. In addition, if the return and risk forecasts are not formulated in a consistent framework, mean-variance efficient portfolios very often contain extreme positions in the sense of the investor. In the next sections, I present some of the most common solutions proposed to deal with the first drawback. I refer the reader to Nocedal and Wright (2006) or Ye (1997) for solutions to the second drawback, that is solving the resulting optimization problem.

10.4 ALTERNATIVE MEAN-VARIANCE BASED MODELS 10.4.1 The re-sampled efficient frontier approach Michaud (1989, 1991) stated that the most serious practical limitations of the mean-variance model are its instability and ambiguity. Small changes in input, that is return or risk forecasts, often lead to large changes in the efficient portfolios. He relates this issue to the observation that mean-variance efficient portfolios are ‘estimation error maximization.’ He suggests coping with this issue by calculating the region of statistically equivalent portfolios using a re-sampling procedure of the input parameters. He calls this set the re-sampled efficient frontier. A re-sampled efficient portfolio is defined for a given return target as the portfolio defined by the average weights on each asset for the portfolio based on the re-sampled input for the given return target. The re-sampling simulation procedure is described by the following six-step algorithm: 1. 2. 3. 4. 5. 6.

Generate input data, that is forecast scenarios. Draw a random subset or sample from the input data. Compute the mean and covariance matrix on that subset. Calculate a mean-variance efficient frontier on that subset. Repeat steps 2 through 4 until sufficient mean-variance efficient frontiers are generated. Select the optimal portfolio from the generated set of mean variance efficient portfolios, based on a given rule.

Different selection rules have been proposed by Michaud (1989, 1991), Jorion (1992) and others. Although the re-sampled efficient frontier is generally associated with Michaud, Jobson and Korkie (1981) first suggested a statistical re-sampling perspective. Unfortunately the proposed technique has two major drawbacks. First, the theoretical foundation of the approach is insufficient, especially with respect to selecting the optimal portfolio. Worse, the approach has been protected by a patent in the United States and worldwide patents are pending, making any use of the technique illegal unless a license is obtained. 10.4.2 The min–max approach A different approach to cope with the uncertainty associated with the input parameters is the min–max efficient portfolio model. The idea is to define robustness as finding portfolio

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Table 10.2 Equity asset classes scenario returns (annual) under different economic scenarios Equity market USA United Kingdom Germany France Japan

Scenario A (%) 5.50 4.50 4.00 3.50 1.00

Scenario B (%) 8.00 5.00 6.50 7.50 −1.50

Scenario C (%) 2.00 2.00 7.50 5.50 3.50

Source: sample estimates

weights that have the best performance under the worst case scenario resulting from the uncertainty of the input parameters. Calculating a min–max efficient portfolio can be subdivided into two steps. First, a set of scenarios of input parameters is generated. Different approaches exist, of which the most common ones are: • The investment manager explicitly forecasts a set of values for the parameters rather than a single value. This approach is illustrated in Table 10.2 for forecasting equity asset class returns. It is important that no scenario is dominant, that is one that is the worst scenario for all assets. More generally, the explicit scenarios should be linearly independent. • If the parameters are estimated using historical data, bootstrapping as well as moving average approaches can be used. In the bootstrapping approach, for example, random subsets of the data set are generated and the parameters estimated for each such subset, resulting in different scenarios. This approach is similar to the scenario generation approach in the re-sampled mean-variance efficient frontier model. The critical issue with bootstrapping is to ensure that no bias is introduced through the re-sampling. • A third approach is to assume a distribution of the uncertainty of the parameters. Stated differently, rather than assume that the parameters are constant, they are modeled as random variables with a known distribution. As tempting as this approach may sound, it has two drawbacks. First, the distribution of the parameters must be modeled. The simplest approach is to assume they are independent and normally distributed. It may not model the uncertainty perfectly, but it does not introduce any bias. Second, the parameters of those random variables must be estimated. Hopefully the impact of parameter miss-estimation on to the optimal portfolio weights is reduced, which is, in general, the case. Table 10.2 illustrates a set of return forecasts for five equity asset classes. The scenarios model different possible economic expectations. The second step in determining the min–max efficient portfolio is to calculate portfolio weights that maximize the expected return in the worst case. Let Rs and
s be the scenarios for the expected returns and covariance matrices. Very often scenarios are only generated for the expected returns; that is Rs and a single covariance matrix
are used. The min–max optimal portfolio weights w are defined as the solution to the min–max optimization problem shown in the following equation: max min(Rs · w) s w (10.9) s.t. w ·
s · w  σ 2 , ∀s  w ·1=1

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Expected return (p.a.)

6% 5% 4% 3% 2% 1% 0% 13.0%

13.5 %

14.0 %

14.5 %

15.0 %

15.5 %

16.0%

Risk (p.a.) Min-max efficient portfolio worst case return Min-max efficient portfolio under scenario A Mean-variance efficient portfolio for scenario A under scenario C Mean-variance efficient portfolio for scenario A under scenario A Source: data sourced from Bloomberg Finance LP, author's calculations

FIGURE 10.5 Min–max efficient frontier compared to mean-variance efficient portfolios under different scenarios

The min–max optimization problem can be re-formulated as a quadratic optimization program by introducing a dummy variable γ as max γ w

s.t.

Rs · w
γ , ∀s w ·
s · w  σ 2 , ∀s w · 1 = 1

Figure 10.5 illustrates the min–max efficient frontier and compares it to the mean-variance efficient frontiers calculated under scenario A and evaluated under scenarios A and C. It can be seen that the expected return of the min–max efficient portfolio is lower than the mean-variance efficient portfolio under scenario A, but it is higher than the mean-variance efficient portfolios when evaluated under scenario C. Indeed, the min–max efficient portfolio approach is best suited for risk averse investors, who prefer limiting the downside risk rather than maximizing the upside potential. It minimizes the uncertainty generated by the dispersion in scenarios, thus giving it the alternative name of robust efficient portfolios. In addition, the expected return of min–max efficient portfolios under scenario A, for the lower range of target volatilities, is higher than the worst case expected return. Figure 10.6 illustrates the weights for some of the portfolios on the min–max efficient frontier. It can be seen that the portfolio structures are more diversified than the ones on the mean-variance efficient frontier. All portfolios are at least invested in three of the five asset classes considered in the example, showing the diversifying and uncertainty reducing effect of the min–max approach. It can be shown that the min–max efficient portfolio approach is similar to a Bayesian approach in which no probabilities need to be associated with the scenarios. The min–max

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100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Port. 1/ 13.0%

Port. 2/ 13.5%

Port. 3/ 14.0%

Eq. USA

Port. 4/ 14.5%

Eq. UK

Port. 5/ 15.0%

Port. 6/ 15.5 %

Eq. Germany

Port. 7/ 16.0 %

Port. 8/ 16.5 %

Eq. France

Port. 9/ 17.0 %

Port. 10/ 17.5 %

Eq. Japan

Source: data sourced from Bloomberg Finance LP, author's calculations

FIGURE 10.6 Min–max efficient portfolio structures for different risk targets under the defined scenarios

efficient portfolio approach is best suited when independent sets of parameters can be generated and the investor has a high degree of aversion against the forecasting uncertainty. The information risk, as such, is minimized. Rustem et al. (2000) have shown that the approach is well suited in a fixed income portfolio environment. In addition, the flexibility of the Markowitz mean-variance approach is retained. Diderich and Marty (2001) have shown how the model can be extended to handle transaction costs in the context of balanced portfolios.

10.5 MODELS WITH ALTERNATIVE RISK DEFINITIONS Numerous researchers have studied the structure of optimal portfolios in a risk-return framework, using a different definition of risk. The two most relevant models in the context of an investment process are the mean semi-variance and conditional value-at-risk models. 10.5.1 The mean semi-variance model The mean semi-variance model (Sing and Ong, 2000), also called the downside risk or mean lower partial moment model, was first suggested by Markowitz (1952) in his seminal paper. At that time, he did not follow up on that approach because of its numerical complexity. The basic idea behind the mean semi-variance model is to distinguish between undesirable and desirable uncertainty of expected returns. The mean semi-variance problem can be formulated as a piecewise quadratic optimization problem: max R · w w  wa · wb · σa− · σa− · ρa,b  σ 2 s.t. a w

b

·1=1

(10.10)

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165


where σa− denotes the semi-variance, that is σa− = T1 · Tt=1 (min(Ra,t − ς, 0))2 and ρa,b the correlation between assets a and b; ς is called the return threshold below which the investor expresses a risk aversion. In general, the return threshold equals the average return, zero or the risk free rate. The main advantage of the mean semi-variance is that it focuses on defining risk as returns below a given threshold, rather than return deviation, both negative and positive. It is not used more often because of its numerical complexity and the fact that it is only better than the original mean-variance model if its returns exhibit a nonnormal or, more specifically, an asymmetric distribution. Indeed, if the expected returns follow a multivariate normal distribution, the problems in Equations (10.10) and (10.2) are equivalent. 10.5.2 The conditional value-at-risk model The conditional value-at-risk, or CVaR, model, also known as expected shortfall risk, can be seen as a generalization of the mean semi-variance model. It was made popular by Artzner et al. (1997, 1999) and focuses on maximizing the expected return, subject to the expected loss, conditioned on it being larger than a given threshold. Formally, the CVaR model is specified by the optimization problem: min

σCVaRα (w)

s.t.

R · w
µ w · 1 = 1

w

(10.11)

Recall the definition of σCVaRα (w), that is σCVaRα (w) = E[−R · w| −'R · w
σVaRα (w)] 1 · = w · p(R) · dR 1−α

(10.12)

−r ·w
σVaRα (w)

where p(R) is the probability distribution of the return R. Now, assume that the distribution p(·) is known through its realizations, rather than through a parameterization. This is, for example, the case when relying purely on historical data. In that context, the realizations can be used to discretize the integral in Equation (10.12). Using algebraic techniques, the optimization problem in Equation (10.11) can be written as a linear program: min w,ζ

s.t.



zi

i

∀i : Ri · w
µ w · 1 = 1 ∀i : zi
Ri · w − ζ ∀i : zi
0

(10.13)

where zi and ζ are pseudo-variables. It can be shown that Equation (10.13) is equivalent to the original mean-variance model in Equation (10.2) if and only if the return realizations Ri represent a multivariate normal distribution with mean R and covariance
.

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The CVaR model is especially useful if I have a sample from the return distribution rather than a parameterization of it. In addition, there is no assumption made on the structure of the distribution and therefore the CVaR model allows the optimal to be calculated whatever the return distribution. Finally, from a purely optimization point of view, although significantly larger in dimension, that is in the number of variables,6 the problem is linear rather than quadratic and therefore the interior point algorithm or even the well known simplex algorithm can be used to find the optimal solution efficiently.

10.6 INFORMATION RISK BASED MODELS The mean-variance models presented up to now essentially focused on maximizing the expected return for a given market risk, expressed as volatility σ or tracking error τ . More sophisticated models focus on mitigating the forecasting risk, that is the parameter uncertainty or, expressed more generally, the information risk. These models find their roots in Bayesian theory. 10.6.1 The Black–Litterman model In 1991, Black and Litterman (1991, 1992) introduced a model for combining market equilibrium with tactical views about investment opportunities and transferring them into asset allocations. Although the model is based on market equilibrium theory, using the global CAPM equilibrium as the center of gravity (Jorion, 1986), it was developed as a solution to the instability of the classical mean-variance model. The Black–Litterman model is not, however, a better optimizing framework, but a reformulation of the investor’s utility function. The optimal portfolio weights are calculated as the weighted sum of an equilibrium portfolio and a portfolio resulting from the investment manager’s forecasts. In general, the equilibrium portfolio is chosen to be equal to the market capitalization based portfolio. It may also be set to a given strategic benchmark. The Black–Litterman model assumes that there are two distinct sources of information for forecasting asset returns, that is: • the investment manager’s subjective forecast and • the market’s objective expectation, represented by the market equilibrium. The weighting of the two sources of information is based on the degree of confidence in the former. As in Bayesian theory, on which the model relies, a probability is associated with each available information. The forecasting uncertainty and therefore the information risk is characterized. The Black–Litterman model can be described by the following four-step algorithm: 1. Determine a vector π of equilibrium returns. This vector, for example, can be determined through inverse optimization. Hence, the equilibrium return distributed is specified by N (π, ξ ·
), where
is the covariance matrix based on historical returns and ξ a rescaling or shrinkage factor. In order to be consistent, the inverse optimization model used to determine π should rely on the same covariance matrix
. 6

There is one variable zi per return scenario and one variable wa per asset or asset class.

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2. Forecast the deviation of markets from the equilibrium by expressing market views in terms of a probability distribution. Market views are best expressed as expected relative returns between pairs of asset classes, for example E[equities Germany] − E[equities France] = Q + ε where E[·] is the expected excess return of each asset class, Q is the expected excess return of the forecasted view and ε the error term of the forecast. Then the investment manager’s returns can be expressed as P · E[R] = Q + ε where P is a matrix selecting the active asset classes, Q the vector expected excess returns and ε an unobservable normally distributed random vector with zero mean and diagonal covariance matrix . The error term measures the degree of uncertainty of a particular forecast. 3. Combine the equilibrium return with the forecasted returns using Bayesian theory. The resulting return distribution E[R] is a normal distribution with mean equal to  = ((ξ ·
)−1 + P · −1 · P)−1 · ((ξ ·
)−1 · π + P · −1 · Q) E[R] 4. Solve the classical mean-variance problem in Equation (10.2) using the calculated mean  and the covariance matrix
. Additional restrictions can be added as shown return E[R] in Section 10.3.4. The key advantages of using the Black–Litterman model are the stability of its resulting portfolio weights relative to small changes in forecasts and the fact that the forecasting uncertainty is explicitly considered. Information risk, in addition to market risk, is managed. On the downside, apart from having to express point forecasts as well as confidence levels, the model’s output is highly conditioned on the equilibrium returns used. This means that the equilibrium state must be determined with a high degree of precision. This is especially hard for assets for which no long term equilibrium exists, as their prices are solely dependent on short term supply and demand. 10.6.2 Lee’s optimal risk budgeting model Lee (Lee, 2000; Lee and Lam, 2001) developed the optimal risk budgeting model. It is, similarly to the Black–Litterman model, based on the idea of managing information risk and using Bayesian theory. The investment manager formulates market views by expressing the relative attractiveness between pairs of assets or groups of assets and associates a degree of confidence with each pair. Then, the optimal risk budgeting model translates these market views into portfolio weights by determining an optimal weighting, called the optimal aggressiveness factor, for each market view. The optimal aggressiveness factor expresses the risk budget allocated to each forecast. In contrast to the Black–Litterman approach, the outcome of the optimal risk budgeting model is a portfolio structure rather than return expectations. It can be shown that the resulting weights are mean-variance efficient and thus inverse optimization techniques can be used to determine implied return multiples, if necessary. In addition, no explicit point return forecasts are necessary.

168

Positive Alpha Generation: Designing Sound Investment Processes Equities Japan Equities Germany Equities USA

Equities UK Equities France

FIGURE 10.7 Sample transitivity graph representing the forecasted market views (the strength of the edges represents the associated confidence level)

The optimal risk budgeting approach is defined by the following algorithm: 1. Forecast a set of market views by expressing the relative attractiveness between pairs of assets and associate a confidence level, in percentage terms, to each pair. Only the direction, not the magnitude of the pair forecast, is relevant. Rather than comparing single assets, asset classes or event subportfolios can be used. Figure 10.7 illustrates the transitivity graph of a set of sample forecasts. 2. Transform each market view v into a long–short portfolio7 uv representing that view. The weights of the long–short portfolio are scaled such  that it has exactly 1 % of risk, based on a single covariance matrix. This means that uv ·  · uv = 1 %; uv is called a unit risk portfolio associated with market view v. Table 10.3 illustrates this step for a global asset allocation investment process. For example, the market view that US equities will outperform German equities translates in a portfolio with a 6.6 % long position in US equities and a 6.6 % short position in German equities. This long–short portfolio has a risk of exactly 1 %. It can be seen that the more similar two asset classes involved in a market view, the larger are the weights in the unit risk portfolio. This second step makes the forecasts comparable, assuming that the expected return per unit of risk is constant across all forecasts with the same associated confidence level. Table 10.3 Making market views comparable by constructing scaled long–short portfolios with identical risk representing the forecasts Asset class

USA UK Germany France Japan Confidence

USA versus Germany (%) 6.6 0.0 −6.6 0.0 0.0 52.0

USA versus France (%) 8.2 0.0 0.0 −8.2 0.0 52.0

Germany versus Japan (%) 0.0 0.0 3.9 0.0 −3.9 54

Source: data sourced from Bloomberg Finance LP, author’s calculations 7

A long–short portfolio is a portfolio whose weights sum to zero.

Germany versus UK (%) 0.0 −6.6 6.6 0.0 0.0 52

France versus UK (%) 0.0 −9.4 0.0 9.4 0.0 52

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3. Calculate the expected return per unit of risk for each market view based on the confidence level formulated. This means solving Conf = Pr {IR
0}

(10.14)

where IR is the expected return per unit of risk. If IR follows a normal distribution with known mean and variance, then Equation (10.14) can be solved quite easily. Lee (2000) has shown that, under the normality assumption, Equation (10.14) is equivalent to Conf = Pr{g
−IR|g ∼ N (0, 1)} 4. Compute the optimal aggressiveness factors o or weights for each market view solving the quadratic optimization problem max o · IR o

s.t.

o ·  · o  σ 2 o
0

(10.15)

where IR is a vector of expected returns per unit of risk or information ratios, determined in step 3,  the covariance matrix of the market views and σ the target portfolio risk. The market view covariance matrix represents the covariance between the expected returns of the individual unit risk portfolios. Formally, ωa,b = ua ·
· ub . Table 10.4 illustrates the results of this optimization step. The highest risk budget or weight is associated with the USA versus Germany market view. On the contrary, no risk budget is associated with the USA versus France market view and only a small weight to the France versus UK, as these two market views are somewhat redundant due to the market views USA versus Germany and Germany versus UK. 5. Calculate the optimal portfolio weights as the weighted sum of the unit risk portfolios, that is  w= uv · ov v

Table 10.4 Optimal aggressiveness factors (OAFs) calculated by solving the optimization problem (10.15) Asset class

OAF USA UK Germany France Japan

USA versus Germany 1.63 10.77 % 0.00 % −10.77 % 0.00 % 0.00 %

USA versus France 0.00 0.00 % 0.00 % 0.00 % 0.00 % 0.00 %

Germany versus Japan

Germany versus UK

France versus UK

0.95

0.77

0.19

0.00 % 0.00 % 3.73 % 0.00 % −3.73 %

0.00 % −5.05 % 5.05 % 0.00 % 0.00 %

0.00 % −1.82 % 0.00 % 1.82 % 0.00 %

Source: data sourced from Bloomberg Finance LP, author’s calculations

Optimal weights

10.77 % −6.87 % −1.98 % 1.82 % −3.73 %

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Positive Alpha Generation: Designing Sound Investment Processes

6. Optionally, the optimal implied return multiples can be calculated as R =
· w and used as input to a mean-variance optimization. The main advantages of the optimal risk budgeting approach are: • It allows a one-to-one mapping of the forecasts into portfolio weights. • Information risk is optimized through allocating weights or budgeting market risk to the individual forecasts based on the investment manager’s confidence. • It only relies on the estimated or forecasted covariance matrix as well as the assumption that the expected return per unit of risk of the formulated market views are normally distributed. As the model assumes that the reward or alpha per unit of risk is the same for all market views with the same associated confidence levels, no explicit return forecast in magnitude is required. Through the optimization of information risk, the resulting portfolio is expected to outperform if more than half of the market views are correct, subject to the correlations between market views and the relative confidence levels formulated. Unfortunately the optimal risk budgeting model does not easily allow advanced restrictions to be taken into account. In this case a deviation through implied return multiples is necessary. In addition, the normality assumption of the alphas for each market view is often violated when considering individual assets rather than asset classes.

10.7 SELECTING A PORTFOLIO CONSTRUCTION APPROACH I have presented the most common portfolio construction models based on a two-dimensional risk and return framework. Selecting the most adequate approach for a specific investment process depends on the following three factors: • the investor’s utility function, • the degree of uncertainty in the forecasts, both on the risk and return side, and • the availability of forecasting data. The different approaches may also be combined, for example, by using multiple market view scenarios in the optimal risk budgeting approach combined with a min–max optimization model to take into account investor restrictions. My experience has shown that, from an investment manager’s perspective, focusing on managing information risk rather than market risk provides the most consistent results over time.

11 Single Period Factor Model Based Portfolio Construction Mean-variance based portfolio construction models summarize the key information in two parameters, expected return and volatility. This makes Markowitz type models both simple and elegant to use. Unfortunately, under certain conditions, these advantages turn into drawbacks. This is the case when: • the investment manager is faced with a large number of assets in the investment universe, making the parameter estimation, that is the mean and especially covariance matrix, difficult and error prone and • the characteristics of the assets cannot be summarized into the two parameters expected return and variance. Factor models have been used for a long time to model and decompose asset returns, as described in Chapter 5. At least since the introduction of the capital asset pricing model (Sharpe, 1964) and the Treynor and Black (1973) paper on improving portfolio selection, factor models have been used as tools or models in the portfolio construction and risk management value chain element. Factor model based approaches offer two key advantages over mean-variance based models, that is: • They reduce the dimensionality of the problem to the number of factors plus one, the residual. • They allow the risk taken to be decomposed and therefore allow only certain well characterized risks to be taken. The drawback of factor based portfolio construction models lies in the determination of the factors to use. The factor model approach is common in securities selection investment processes, that is bottom-up based approaches. The single factor market portfolio model is often used. Financial literature provides a large number of multifactor models that can be used in portfolio construction, including the often cited Fama and French (1992) three-factor model. Fixed income investment managers rely heavily on factor models, the most prominent factor model being the duration model. Factor model based portfolio construction has been used much less in asset allocation or for nontraditional asset classes, like commodities, because of the difficulties in determining common and adequate factors.

11.1 FACTOR MODELS AND THEIR RELATION TO RISK Portfolio construction is about maximizing the investor’s utility function, taking into account its multidimensionality. This function is generally a combination of return or wealth expectations and risk aversion. Consider the following factor model:  Ra = RF + βf,a · Rf + εa (11.1) f

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where Ra is the return of asset a, RF the risk free rate, βf,a the factor exposure, Rf the factor return and εa the residual or unexplained return. In some factor models, the factor return is expressed as an excess return over the risk free rate, that is Rf − RF , rather than Rf . Then, the volatility of the asset returns can be calculated as σa2 = β · F · β + σε2a

(11.2)

where σa is the volatility of asset a, F the covariance matrix of the factor returns and σεa the residual risk. If the factors are orthogonal, then the matrix F is a diagonal matrix. The term β · F · β is called the systematic risk. It is also sometimes denoted by market risk or undiversifiable risk, depending on the underlying factor model. Risk σεa is called the specific risk or idiosyncratic risk, as it is specific to asset a (Bali et al., 2005; Campbell et al., 2001). In an equilibrium factor model:  • The investor is rewarded by the return RF + βf,a · Rf for the risk exposure βf,a taken to the individual factors; that is systematic risk is rewarded. • The investor is not rewarded for taking any security specific risk. Only market risk is priced or rewarded in an equilibrium model. However, as described in Chapters 4 and 5, markets are not always in equilibrium. Thus, over short time horizons, E[εa ] = 0 for some assets. These situations are called mispricing opportunities in the context of an equilibrium model. If the investment manager is capable of forecasting the sign and size of E[εa ], positive alpha can be generated by exploiting security specific risk. It follows from this discussion on the risk decomposition in factor models that there are two complementary approaches to generate positive alpha, that is: • exploiting mispriced securities through taking on idiosyncratic risk and • gaining specific factor exposure and as such taking systematic risk.

11.2 PORTFOLIO CONSTRUCTION EXPLOITING IDIOSYNCRATIC RISK The most common portfolio construction approach to exploit idiosyncratic risk through detecting mispriced assets is the Treynor and Black model (Treynor and Black, 1973; Treynor, 2007). A naive portfolio construction model would be to detect mispriced assets, that is assets for which E[εa ] = 0, and equally to over- and underweight these assets. However, this approach is suboptimal as it does not take into account the associated risk. In an optimal portfolio construction algorithm, the exposure to an asset must be proportional to its expected alpha, that is E[εa ], and inversely proportional to its idiosyncratic risk, that is σεa . The Treynor and Black approach can be described by the following five-step algorithm: 1. Define the investment universe and assume that it represents the market or equilibrium portfolio. 2. Estimate the factor model’s parameters.

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3. Focus on a small subset of assets for which forecasting capabilities exist and identify those assets with significant positive or negative expected alpha. 4. Construct an active portfolio by investing in the assets with a nonzero expected alpha maximizing return in a risk controlled framework. 5. Construct an optimal portfolio by mixing the market or equilibrium portfolio from step 1 with the active portfolio from step 3. 11.2.1 The Treynor and Black portfolio construction algorithm First, a factor model, which ideally represents an equilibrium model, must be designed. A possible model for equities is the one-factor capital asset pricing model (CAPM). An alternative and often used model for constructing equity portfolios is the Fama and French (1992) model, adding size and book-to-market as additional factors. A description of equity factor models implemented in commercial software can be found in Fabozzi (1998). One of the most common fixed income factor models is the shift, twist and butterfly three-factor model. A simple alternative is the duration model. Second, the parameters RF and Rf in Equation (11.1) and F as well as σεa in Equation (11.2) have to be estimated or forecasted. As a consequence of the constructed factor model, cov(εa , εb ) ∼ = 0 for all assets a = b. If these conditions are not verified, the factor model should be revised. In the third step, for a subset of securities, where research capabilities and forecasting skills exist, E[εa ] = αa must be forecasted such that E[Ra ] = RF +



βf,a · E[Rf ] + αa

f

In the fourth step, an active portfolio A is constructed such that for each asset a where αa =  0, the position weight equals αa /σε2a wa =  αb /σε2b

(11.3)

b

By construction



wa = 1.1 The characteristics of this constructed active portfolio are

a

expected alpha: factor exposures: expected volatility:


a wa · αa
βf = a wa · βf,a , ∀f
σ 2 = β · F · β + a wa2 · σε2a α=

The fifth and last step consists in combining the active portfolio with the portfolio resulting from an equilibrium situation, called the equilibrium portfolio2 E, that is where all
If b αb /σε2 = 0 for all assets, then forecasts are offsetting. The forecasts need to be revised and adjusted or b forecasts in which there is the least confidence dropped. 2 In the CAPM, the equilibrium portfolio is the market portfolio. 1

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E[εa ] = αa = 0. This combined portfolio P needs to maximize its Sharpe ratio, that is max

E[RP ]−RF σP

s.t.

wA + wE = 1

wA

where wA is the relative weight invested in the active portfolio, wE the relative weight invested in the equilibrium portfolio, RP the return of the combined portfolio and σP its volatility. Formally, this means max wA

s.t.



(1 − wA ) · E[RE ] + wA · E[RA ] − RF (1 − wA )2 · σE2 + wA2 · σE2 + 2 · ρA,E · (1 − wA ) · wA · σE · σA

(11.4)

wA + wE = 1

where RE is the forecasted or estimated return of the equilibrium portfolio and ρA,E the correlation between the equilibrium and the active portfolio. Solving Equation (11.4) results in wA =

ω 1 − (1 − βA ) · ω

(11.5)

where βA is the regression coefficient between the active portfolio A and the equilibrium portfolio E and ω=

αA /σA2 (E[RE ] − RF )/σE2

is the ratio of reward to risk for the active portfolio A and the equilibrium portfolio E. 11.2.2 Example I assume that the capital asset pricing one-factor model is a fair equilibrium model. To illustrate the Treynor and Black algorithm, I consider the Swiss equity market, represented by the SMI index, as the equilibrium or market portfolio for the selected investment universe. First, using five years of weekly return data between 2003 and 2007, I estimate the annualized parameters E[RE ] = 8.76 % and σE = 15.28 %. In addition, I estimate the exposure of each security in the index to the equilibrium portfolio using Equation (11.1) as well as its specific risk using Equation (11.2) and an ordinary least squares estimator. The results are shown in Table 11.1. Next I derive, using techniques described in Chapters 4 and 5, the forecasted E[εa ] = αa for a subset of stocks, as shown in Table 11.2. I then calculate the weights of the assets with nonzero αa forecasts using Equation (11.3), as shown in Table 11.1. It can be seen that the largest expected alpha, that is the expected 2.7 % excess return for the Nobel Biocare stock, results in the second largest active weight of 17.95 %. The largest active position, that is 40.07 %, results from the 1.25 % alpha forecast in the Swisscom security. This is mainly due to Swisscom’s low specific risk of 1.68 %. The negative forecast in the Zurich Financial Services stock results in a negative active position of −8.59 %. Finally, I compute the relative weights between the active and equilibrium portfolios using Equation (11.5), that is wA = 19.99%. The final portfolio, showing only the positions with active forecasts, is given in Table 11.2. Except for the Roche stock, all positive return forecasts resulted in an overweight of the security versus the equilibrium weight.

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Table 11.1 Equilibrium weight, market exposure and specific risk of the 20 stocks in the Swiss Market Index (SMI) as of 31 December 2007 Asset

Equilibrium weight (%)

Market exposure

Specific risk (%)

7.41 0.76 2.00 0.60 4.03 0.24 6.81 2.57 20.58 0.76 15.65 13.85 1.08 1.00 3.02 0.92 2.94 0.82 10.08 4.88

1.64 1.39 1.48 1.88 1.07 1.81 1.56 1.21 0.67 0.88 0.83 0.92 0.93 2.03 1.86 0.32 0.78 0.59 1.09 1.60

4.65 4.18 3.71 3.11 3.12 5.96 3.00 2.67 1.94 3.69 1.74 2.53 3.15 7.29 3.05 1.68 2.88 4.04 1.88 2.81

ABB Adecco Julius Baer Holding Baloise Richemont Clariant Credit Suisse Group Holcim Nestle Noble Biocare Novartis Roche Swatch Swiss Life Swiss Re Swisscom Syngenta Synthes UBS Zurich Financial Services

Source: data sourced from Bloomberg Finance LP, author’s calculations.

Table 11.2 Forecasted αa , active, equilibrium and consolidated weights for the assets with active forecasts as of 31 December 2007 Asset

ABB Julius Baer Holding Credit Suisse Group Holcim Noble Biocare Roche Swatch Swisscom Zurich Financial Services

Alpha (%)

2.25 1.75 1.25 0.25 2.70 0.50 0.75 1.25 −0.75

Active weight (%)

Equilibrium weight (%)

Total portfolio weight (%)

9.42 11.51 12.57 3.18 17.95 7.07 6.83 40.07 −8.59

7.41 2.00 6.81 2.57 0.76 13.85 1.08 0.92 4.88

7.81 3.90 7.96 2.70 4.19 12.49 2.23 8.75 2.19

Source: data sourced from Bloomberg Finance LP, author’s calculations.

11.2.3 Advantages and drawbacks The main advantage of the Treynor and Black algorithm is its simplicity and the ease with which it can be applied to a large portfolio universe. On the downside, there are a number of aspects that need to be remembered:

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• The constructed portfolio is only as good as the underlying equilibrium factor model. • The model is sensitive to the quality of the estimated parameters. Shrinkage techniques can be used to improve the stability of the estimated parameters. • Explicit alpha forecasts are required for at least some securities. This drawback may be eased by introducing techniques used in the Black and Litterman (1991, 1992) or Lee (2000) models described in Chapter 10. • As the model is a static model, hedging considerations as well as learning effects are not taken into account. Cvitanic et al. (2002) proposed an intertemporal version of the Treynor and Black model to address these issues. The fundamental law of active management suggests formulating as many alpha sources as possible in order to maximize the breadth. However, care must be taken not to increase the breadth at the expense of the quality, that is the information coefficient. Indeed, although the breadth is unbounded, it only increases the information ratio by the square root, whereas the information coefficient, bound by one, is linearly related to the information ratio.

11.3 PURE FACTOR MODEL EXPOSURE BASED PORTFOLIO CONSTRUCTION Rather than focusing on idiosyncratic risk taking to generate alpha, pure factor model exposure based portfolio construction algorithms generate alpha through taking active exposure to factors, avoiding any specific risk. Empirical investigations show that the exposure to specific risk can be minimized by holding a diversified portfolio. This is even true when the exact factor model is unknown. Figure 11.1 shows the relationship between risk, expressed as annualized volatility, and a portfolio of randomly selected equally weighted stocks from the S&P 500 index universe as of 31 December 2007 using five years of monthly historical returns to estimate the risk. 23 %

Annualized volatility

21 % 19 % 17 % 15 % 13 % 11 % 9% 7% 5%

0

10

20

30

40 50 60 Number of stocks

70

80

90

100

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 11.1 Total risk of the sample set of equally weighted portfolios composed of randomly selected stocks from the S&P 500 index universe versus the number of stocks selected, calculating volatility using historical monthly asset price data between December 2002 and December 2007

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All assets included in the randomly generated portfolio containing n stocks are also included in the randomly generated portfolio containing n + 1 stocks. The average annualized volatility of the individual stocks included in the randomly generated portfolios is 23.72 %. It can be seen that the volatility of the portfolio containing 100 stocks is less than 10 %. Diversification reduces the portfolio risk by more than half, starting at about 25 securities in the calculated example. Evans and Archer (1968), Solnik (1974) and Statman (1987) have shown that the major benefits of diversification can be obtained with as few as 20 to 40 stocks. More recent studies (Campbell et al., 2001; Malkiel, 2002) suggest that 50 to 200 stocks are necessary to reduce idiosyncratic risk. In addition, Boscaljon et al. (2005) have shown that equally weighted portfolios are more diversified than value weighted, that is market capitalization weighted ones. 11.3.1 Using factor models to reduce the complexity One of the most common usages of factor models is to reduce the complexity of the resulting portfolio construction problem. Consider an investment process based on security selection. Although the mean-variance framework would be theoretically sound, due to the size of the investment universe, it is practically inadequate. In this case, one possible portfolio construction approach is to reduce the problem size to a small number of well defined factors. The mean-variance based portfolio construction approaches can then be applied to the factors rather than the individual assets. The dimension reducing factor model portfolio construction algorithm can be described by the following four steps: 1. Develop a factor model using techniques presented in Chapter 5. The factors should be chosen such that: – it is possible to forecast the associated factor returns, – the number of factors is small compared to the size of the investment universe, usually no more than five to ten, and – the idiosyncratic risk terms of the individual assets are uncorrelated. 2. Forecast the factor returns using a qualitative or quantitative approach. Due to the structure of the individual factors and their interpretation, I recommend using a quantitative approach, at least as a decision support tool. 3. Calculate the optimal factor exposure, assuming that the resulting portfolio has no idiosyncratic risk. This can be done by solving the optimization problem in Equation (11.6), assuming the factor model built in step 1 is described by Ra = RF + βf,a · Rf + εa and σa2 = β · F · β + σε2a , Rf being the factor returns f

forecasted in step 2 and σ the target volatility:  βf · Rf max βf

s.t.

f

β · F · β  σ 2

(11.6)

where βf is the factor exposure of the optimal portfolio. 4. In the last step, a portfolio of individual securities matching the optimal factor exposures βf and minimizing the specific risk is constructed. This can be done using the following algorithm:

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(a) Determine the number of securities in the target portfolio. This number is a compromise between maximizing the diversification effect, the size of the investment universe, transaction costs and the portfolio value. Reasonable sizes are between 20 and 100 securities. Denote this size by n. (b) Define the maximal weight of any single position in the portfolio. Usually a maximal position size m of 3/n or 2.5 %, whichever is larger, gives reasonable results. This restriction has no theoretical meaning. Its sole goal is to avoid small uncertainties in the forecasted parameters; that is Rf and F lead to large biased portfolio positions. Alternatively, techniques similar to those described in Chapter 10 can be applied. (c) Randomly select n securities out of the investment universe using a uniform random number generator. Alternatively, a probability proportional to the market value of the security can be used to choose assets randomly. (d) Calculate the weights wi for the n selected securities by solving the following optimization problem: min wi

s.t.

n 

wa2 · σε2a

a=1

0  wa  m n  wa · βf,a βf =

∀a ∈ {1, . . . , n} ∀f

(11.7)

a=1 n 

wa = 1

a=1

The portfolio holdings adequately represent the forecasted factor returns. This portfolio construction algorithm can be adjusted using variations of the mean-variance framework as described in Chapter 10. 11.3.2 Example Consider the CAPM. It is very often used in the context of single currency equity portfolio construction algorithms. Under this model Ra = RF + βa · (RM − RF ) + εa Using historical data, I estimate the exposure βa for each stock in my investment universe, the DAX index, to the market return RM represented by the return of the DAX index. The results obtained are shown in Table 11.3. Next I forecast the market return RM for the next time period as being RM = 1.1 %. I fix the target factor based volatility to σ = 10 %, as shown in Equation (11.6) and limit the number of securities to n = 10. Applying the dimension reducing factor model portfolio construction algorithm, I calculate the optimal portfolio holdings as shown in Table 11.3. The larger the sensitivity to the factor, the smaller the exposure to that security.

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Table 11.3 Results from applying the dimension reducing factor model based portfolio construction algorithm using the CAPM to the DAX universe Asset Adidas Allianz BASF Bayer BMW Commerzbank Continental Daimler Deutsch Bank Deutsche B¨orse Deutsche Lufthansa Deutsche Post Deutsche Postbank Deutsche Telekom E.ON

βa

σεa (%)

0.30 2.04 0.80 1.36 0.91 1.67 0.94 0.91 0.98 0.70 0.97 0.68 0.65 0.81 0.59

4.79 11.35 5.15 8.58 6.08 10.00 6.85 7.29 6.32 7.57 6.73 6.59 5.95 5.75 4.63

wa (%)

7.9

4.1 12.5

23.6

Asset Fresinus Henkel Hypo Real Estate Infineon Linde MAN Merck Metro M¨unchener R¨uck RWE SAP Siemens ThyssenKrupp TUI VW

βa

σεa (%)

0.03 0.45 1.01 0.98 0.90 0.99 0.47 0.91 2.14 0.66 1.00 1.01 1.32 1.50 0.84

5.30 5.27 7.28 9.65 6.52 8.23 8.08 6.60 12.57 6.13 7.41 6.61 8.47 10.24 9.80

wa (%)

30.0 1.8 1.7

18.4 0.0

0.0

Source: data sourced from Bloomberg Finance LP, author’s calculations.

11.3.3 Advantages and drawbacks Systematic risk based factor model type portfolio construction algorithms focus on forecasts associated with systematic risk. Any idiosyncratic risk is diversified away. The main advantages are that: • The complexity of the portfolio construction algorithm is determined by the number of factors rather than the dimension of the investment universe and its decomposition in asset classes or assets. • The sensitivities other than volatility can be taken into account. • The traditional mean-variance based techniques can be leveraged. On the downside, systematic risk based factor model portfolio construction approaches require forecasts of the factor returns as input that are unnatural and difficult to formulate in a qualitative framework. In addition, the quality of the portfolio constructed is sensitive to the explanatory power of the factor model and the quality with which its parameters are estimated.

11.4 FACTOR SENSITIVITY BASED PORTFOLIO CONSTRUCTION Rather than use a factor model to reduce the dimension of a mean-variance based portfolio construction approach, it can be used to construct a portfolio with a specific sensitivity to one or more factors. Consider, for example, the fixed income duration factor model. This

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model describes the sensitivity of the portfolio’s return to changes in the factor’s value, that is the interest rates. Let f be a set of factors. Then, a sensitivity based factor model describes the asset return Ra as Ra =



f·

f

∂W + εa ∂f

(11.8)

where W is the asset’s value, ∂W /∂f the sensitivity of the value of asset a to factor f , f the expected or forecasted change of the factor and εa the residual or unexplained return. In a duration based model, ∂W /∂f represents the asset’s duration or sensitivity of the asset return to one unit of change of the interest rates and f represents the forecasted change in interest rates. The relation in Equation (11.8) is valid for small changes in the factor’s value. Based on the factor model in Equation (11.8), I define the factor model sensitivity based portfolio construction algorithm as follows: 1. Develop a factor model based on the sensitivity of asset returns to changes in the factors, the factors being the exogenous variables. 2. Forecast the change in the factors. 3. Define the maximal sensitivities of the portfolio return to a unit of change in these factors, that is fmax . 4. Randomly select a subset of securities as described in steps 4.a to 4.c of the dimension reducing factor model based portfolio construction algorithm in Section 11.3.1. Solve the optimization problem in the following equation:  min wa2 · σε2a wa

a



∂W  fmax ∂f a 0  wa  m  wa = 1 wa · f ·

∀f

(11.9)

a

11.5 COMBINING SYSTEMATIC AND SPECIFIC RISK BASED PORTFOLIO CONSTRUCTION ALGORITHMS It is possible to combine the dimension reducing portfolio construction model with the Treynor and Black model. In this case, the securities selected in step 4.c of Section 11.3.1 are those that have the largest absolute weight in the Treynor and Black approach. The resulting algorithm is described as: 1. Define the ratio of the total risk attributed to – the specific risk, that is σε , and – the systematic risk, that is σS , such that σ 2 = σS2 + σε2 is the total portfolio risk. 2. Solve the Treynor and Black problem.

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3. Independently solve the dimension reducing factor model based portfolio construction problem with a target risk of σS , selecting the n securities as those with maximal absolute weight in step 2. 4. Re-scale the weights
wεa from step 2, that is the of the Treynor and Black
solution 2 2 · σ2 = σ2 − σ2 − 2 , taking into account the problem, such that a wε,a w · σ ε,a ε,a a S,a S residual specific risk from the dimension reducing factor model based portfolio construction portfolio. 5. Combine the portfolios from steps 2 and 3 into one portfolio.

12 Dynamic Portfolio Construction Dynamic portfolio construction models distinguish themselves from static, one-period models, as described in Chapters 10 and 11, by the fact that they dynamically adjust the portfolio structure over time. Rather than determine the portfolio structure only at the beginning of the investment horizon, it is adjusted continuously (in continuous time models) or at discrete moments in time (in discrete time models). There are two main reasons why a dynamic portfolio construction model should be used: • New information, on which the market forecasts depend and therefore the portfolio structure is based, becomes available over time. This ignores the fact that information almost certainly makes the portfolio structure suboptimal. In addition, parameters and other time series, like liabilities, cannot be summarized in static information. • Due to market movements, the portfolio structure may violate investment restrictions, like maximal value at risk or exposure to foreign assets. The portfolio holdings require regular adjustments to remain in line with the investor’s utility function, namely risk aversion. Consider the following sample dynamic portfolio construction algorithm: 1. Forecast the sign of the relative market returns between equities and bonds. 2. If equities are forecasted to outperform bonds, then allocate 60 % to equities and 40 % to bonds. If bonds are forecasted to outperform equities, then allocate 40 % to equities and 60 % to bonds. 3. If new information becomes available that changes the sign of the relative return forecast between equities and bonds, reformulate a new forecast by proceeding to step 1 of the algorithm. This step ensures that available information is transferred as soon as it becomes available in the portfolio structure. 4. If the weight of the asset forecasted to outperform exceeds 65 % or drops below 55 %, proceed to step 2 of the algorithm and rebalance or reallocate the portfolio weights to their original proportions. This step ensures that the risk of the portfolio is mitigated. It can be seen in this simple example that the portfolio structure is adjusted dynamically when specific events occur. New information is processed as soon as it becomes available and the portfolio structure is adjusted accordingly, if needed. The adjustment suggested by step 4 of the algorithm could also be executed at fixed intervals in time, for example every month, rather than being triggered by an external even. Dynamic portfolio construction algorithms have been the focus of a large amount of theoretical research. The work of Merton (1969, 1971) can be seen as the starting point of continuous-time portfolio theory.1 He applied stochastic control theory to the 1 Actually, the first known continuous-time model was introduced far earlier by Louis Ferdinand Bachelier in his dissertation called Th´eorie de la sp´eculation (Bachelier, 1900). However, his approach was so far ahead of his time, in addition to a minor flaw in using Brownian motion processes for asset prices that could become negative, that it did not get accepted.

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portfolio construction problem. Unfortunately, except for some special cases, finding optimal portfolio solutions to the portfolio construction problem reduces to solving the Hamilton–Jacobi–Bellman equation, that is solving a highly nonlinear set of partial differential equations, for which even numerical solutions prove elusive. With the advancement in stochastic calculus, the martingale approach, a more elegant approach, was developed by Karatzas et al. (1987), Karatzas (1989) and Cox and Huang (1989) to solve the dynamic portfolio construction problem. The main idea is to establish a separation of the portfolio problem in a static optimization problem, finding the optimal terminal wealth and consumption process, and a representation problem, that is computing the dynamic portfolio adjustment strategy. A third category of research has focused on solving dynamic portfolio construction problems using stochastic programming techniques. These techniques were developed by Cari˜no et al. (1994), Consigli and Dempster (1998), Klaassen (1998) and Mulvey and Ziemba (1998), to name the most important contributions. The idea is to obtain an approximate description of the true uncertainty of the objective function by randomly generating scenarios from the underlying asset distributions. Unfortunately, at least up to now, most dynamic portfolio construction models have found limited applicability in actively managed investment processes. The difficulties in using these models in alpha generating investment processes are: • They require the estimation of a large number of parameters, much larger than those required for single-period models. • The flexibility of modeling the evolution of asset prices over time makes the resulting dynamic portfolio construction models very sensitive to the model assumptions and prone to model error. • It proves very difficult to define the investor’s utility function over time, especially the evolution of risk aversion. • The developed dynamic portfolio construction models are highly complex and therefore very difficult and time consuming to solve. In nearly all cases numerical optimization algorithms must be applied to nonlinear problems with a large number of variables and constraints.2

12.1 DYNAMIC PORTFOLIO CONSTRUCTION MODELS Define the investment horizon of the investor by H and the start time of the investment by t0 . I call a portfolio construction model a myopic or single-period model if the portfolio structure determined at time t0 is kept unchanged until time H . Any actual or potential information becoming available after t0 is ignored. The Markowitz mean-variance portfolio optimization model is a typical myopic model, where the investor’s time horizon H equals the forecasting horizon. I distinguish between three types of dynamic portfolio construction model with respect to the time dimension: • Anticipative models. In an anticipative model, the investment manager takes all decisions at time t0 . The portfolio structure is adjusted periodically or when external events occur 2 The Russel–Yasuda Kasai model (Cari˜ no et al., 1994) used 17 asset classes, 10 periods and 2048 scenarios. It consisted of 637 variables and 324 constraints in its original form and 348 401 variables and 249 909 constraints in its extended version.

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or parameters change. The rules or algorithms, including their parameters, defining these changes over time are determined at time t0 and do not depend on future information becoming available. A prudent planning approach is taken as there are no opportunities to adjust how the portfolio structure is modified later on. • Adaptive models. In an adaptive model, the investment manager uses new information becoming available between time t0 and H as soon as possible to adjust the portfolio structure over time. The portfolio construction model takes place in a learning environment. A typical investment process for solving the asset and liability problem of a pension fund can be classified in this category. • Recourse models. A recourse dynamic portfolio construction model is a model that combines the anticipative and adaptive models into a single mathematical framework. It seeks a portfolio structure that not only anticipates future observations but also takes into account information becoming available over time to make adaptive or recourse decisions. A typical example of a recourse model is an investment manager considering both future movements of equity prices through a trend following investment process (adaptive) as well as a monthly portfolio rebalancing strategy (anticipation). Figure 12.1 illustrates the four different types of portfolio construction model including the myopic one. If between t0 and H no new information becomes available, then recourse models reduce to anticipative ones. There are two types of anticipative portfolio construction model of practical use, that is: • models that anticipate changes in the uncertainty or risk and • models that are based on expected future asset returns and their changes. The first category comprises so-called rebalancing algorithms whereas the second one is made of stochastic programming models. (i) myopic

t0

H

(ii) anticipative

t0

H

(iii) adaptive

t0

H

(iv) recourse

t0

decision

H portfolio structure change

FIGURE 12.1 Decision and portfolio structure adjustments in (i) myopic, (ii) anticipative, (iii) adaptive and (iv) recourse portfolio construction models

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It is easy to imagine that dynamic models are significantly more complex in development, for example modeling and assumptions formulation, as well as implementation. Therefore the foremost question to answer is when should dynamic models be used and when are myopic or one-period models most appropriate. There are four key reasons for advocating the use of dynamic models. These are: • Known or unknown cash flows occur over the investor’s time horizon. Any consumption based investor seeking to maximize the utility function exhibits such cash flows. Similarly, aiming to use an investment process to outperform a dynamically changing liability structure requires the use of a dynamic model. • Changes in return or risk expectations over the investor’s time horizon. This is especially the case when new information becomes available or when, for example, economic cycles are forecasted, like asset growth over the first five years followed by a recession. • Transaction costs that make investment decisions inter-time dependent. The decision is taken of whether an adjustment of the portfolio structure will have a larger or smaller performance impact than the cost to implement it. • Risk management considerations, requiring the portfolio risk to meet certain criteria over time. For example, a value-at-risk limit must be met at the end of any month. It can easily be seen that, based on these criteria, any actively managed investment process would require the use of a dynamic portfolio construction model. The question that then pops up is if and when is iteratively applying a myopic model an efficient dynamic portfolio construction model. The iterative myopic portfolio construction approach can be classified as a simplified adaptive model in which the decision is based on all relevant information available at the time the decision is taken, except for the previously taken decision and the resulting portfolio structure as well as the expected future decision that will be taken. Using consecutive myopic portfolio construction models is efficient if the return expectations over time are constant. Alexander and Francis (1986) show that applying a sequence of myopic portfolio construction models is optimal if: • the investor’s utility function is iso-elastic,3 • his marginal utility function with respect to the terminal wealth is positive, that is ∂U /∂WH > 0, and decreasing, that is ∂ 2 U /∂WH2 < 0, and • asset returns follow a serially independent multivariate normal distribution. Quintana and Putnam (1994) have shown that if the investor’s utility function is additive over time, then, the optimal portfolio construction model is a sequence of single-period myopic portfolio construction models.

12.2 DYNAMIC PORTFOLIO CONSTRUCTION ALGORITHMS 12.2.1 Optimal rebalancing Consider an investor with a constant relative risk aversion (CRRA), that is averse to a given percentage loss of wealth independent of the absolute level. Then the utility function can 3 An iso-elastic utility function characterizes those investors whose relative attitudes toward risk are independent of the level of wealth.

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187

be expressed as U (wT ) =

1 · wT1−a 1−a

where wT is the wealth at the investment horizon T and a a risk aversion parameter. Merton (1969) has show that, given a constant risk free rate, a single risky asset whose price evolves according to a geometric Brownian motion process and no transaction costs, the optimal allocation to the risky asset for a CRRA investor is constant for all time periods t such that µ − rF wt,R = 2 σ ·a where wt,R is the percentage to be invested in the risky asset, (µ, σ ) the parameters of the geometric Brownian motion process and rF the continuous compounded risk free rate. This result describes a simple anticipative portfolio construction algorithm. It can be extended to take into account transaction costs. Magill and Constantinides (1976), Taksar et al. (1988) and Davis and Norman (1990) showed that, for a single risky asset, the optimal rebalancing strategy is characterized by a no-trade zone around the target risk asset proportion. The no-trade region is a region in which it is optimal, after transaction costs, not to rebalance the portfolio. For proportional transaction costs, it is optimal to rebalance to the boundary of the no-trade zone when outside it. With fixed transaction costs, the rebalancing from outside the no-trade region should be to a state at the internal surface in the no-trade region, but never a full rebalance. Akian et al. (1996) analyzed the rebalancing problem for CRRA investors faced with proportional transaction costs. Leland (2000) as well as Sun et al. (2006a, 2006b) proposed a different approach based on divergence from a target tracking error rather than expected return losses. 12.2.2 Stochastic programming The stochastic programming approach is a common technique to solve recourse based portfolio construction problems. It has been well studied since Danzig’s paper in 1995. Its key advantages are that it allows many of the real world features to be captured, such as transaction costs, risk aversion, regulatory constraints, intermittent cash flows, short term and long term utility functions, and many other features. However, the resulting optimization problem very often turns out to be hard to solve due to its size (number of variables and number of constraints) or even impossible (due to a nondifferentiable or nonconvex objective function). Nevertheless, recent advancements in numerical and combinatorial optimization techniques have made stochastic programming problems more tractable. The idea of stochastic programming is to subdivide the investor’s investment horizon in a large number of planning periods. Then, at each stage, multiple scenarios representing possible outcomes of the markets, based on the previous period’s outcome, are generated. These scenarios can be represented by so-called scenario trees, as shown in Figure 12.2 for a single-equity index price. Typically Monte Carlo simulation techniques combined with stochastic process definitions underlying the considered asset classes are used. At the beginning of each planning period, an investment decision is taken and an optimal portfolio structure calculated. The decisions for each period are calculated such that the expected value of the utility function over all scenarios is maximized. In addition, constraints as

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Positive Alpha Generation: Designing Sound Investment Processes 123.45 ...

116.51

... 119.45 117.92

109.45

... 107.80

100.00

... 101.00 121.21 ...

112.91

... 97.34 121.20

103.45

... 107.99

... 101.01

t0

t1

t2

t3

FIGURE 12.2 Sample scenario tree representing possible index levels for an equity asset class

well as the relationship between the portfolio values of the different periods are taken into account. Consider the dynamic portfolio construction problem of determining the proportions to invest in different asset classes at discrete points in time. Denote the investment horizon by H and the different asset classes by a0 , a1 , . . . , aA , where a0 represents the risk free asset. The risk free asset is usually considered to be a fixed term deposit invested for the whole planning period at a fixed interest rate known at the beginning of the planning period. Let s ∈ S represent a single scenario. Assume that at the beginning of each period, the portfolio structure is readjusted. For each asset a, each planning time period t and each scenario s, I define the following parameters and decision variables: • R(a, t, s) is the return of asset a over the planning period t under scenario s. • TC(a) are the transaction costs for asset a as a percentage of the total value of the trade. • V  (a, t, s) is the value invested in asset a at the beginning of the time period t under scenario s before any trade has been executed. • V (a, t, s) is the value invested in asset a at the beginning of the time period t under scenario s after all trades have been executed. • V (t, s) is the value of the portfolio at the beginning of
the time period t under scenario s after all trades have been executed, that is V (t, s) = a V (a, t, s). • V (0) is the initial portfolio value.

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• T (a, t, s) is the amount traded in asset a at the beginning of the time period t under scenario s. The value is positive if the trade is a buy trade and negative if it is a sell trade. Based on these definitions, I define the general stochastic programming model implementing the dynamic portfolio construction algorithm as max

V (a,t,s)

s.t.




s V (H, s)

a V (a, 0, s) = V (0), ∀s a V (a, H, s) = V (H, s), ∀s V  (a, t, s) = (1 + R(a, t − 1, s)) ·V (a, t − 1, s), ∀a, t
1, s V (a, t, s) = V  (a, t, s) + T (a, t,
s), ∀a = cash, t, s V (cash, t, s) = V  (cash, t, s) − a=cash T (a, t, s)
− a=cash T C(a) · |T (a, t, s)|, ∀t, s V (a, t + 1, s) = V (a, t + 1, s  ), ∀a, t, s, s  , R(a, t, s) = R(a, t, s  )

utility function start portfolio value end portfolio value asset return trades cash management

(12.1)

nonanticipative constraint

Solving the optimization problem in Equation (12.1) provides a dynamic portfolio construction solution maximizing the given utility function. In addition to formulating the stochastic optimization problem, successful stochastic programming dynamic portfolio construction modeling depends on: • the definition of a sound multiperiod utility function and • the generation of sufficient realistic scenarios. Formally, the development of a stochastic programming algorithm for solving a dynamic portfolio construction problem can be subdivided into four steps: 1. Subdivide the investment horizon into multiple consecutive investment periods, called planning periods. The size of each planning period and its number should be chosen such that – the number of periods is large enough to reflect the dynamics of the markets, – the length of each planning period is consistent with an underlying economic interpretation, – the parameters used to model the assets’ prices and returns can be considered constant over the time horizon of a single planning period and – readjusting the portfolio at the beginning of each planning period is realistic both from a theoretical as well as a practical point of view. 2. Develop a scenario generator that generates possible future outcomes of the considered markets. It is important that – the number of scenarios generated is large enough to minimize the introduced sampling error, – the individual scenarios are realistic in their outcome and significant in their probability of realization and – the underlying mathematical model is sound.

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3. Define the investor’s utility function. The easiest utility function is to maximize the value of the portfolio at the end of the investment horizon, that is the terminal wealth. The utility function needs to model – the investor’s return expectations at the end of the investment horizon and, optionally, at the beginning of each planning period and – the investor’s risk tolerance or risk aversion at any moment in time. 4. The fourth and last step completes the stochastic programming model by adding constraints. There are two types of constraints to be modeled: – First, constraints ensuring the intertemporal consistency of the stochastic programming problem must be added. These constraints are called nonanticipative constraints. They ensure, for example, that the portfolio value at the beginning of the planning period t is equal to the portfolio value at the end of the planning period t − 1, adjusted for the transaction costs due to changes in the portfolio structure, if any. In addition, they ensure that decisions taken at time t do not use information from any time period later than t. – Second, investor specific restrictions are added to the optimization problem, for example a restriction requiring the equity exposure to be no larger than 30 % at the beginning of each planning period. A number of mathematical programming languages have been developed to make the specification of stochastic programming problem models easier. The most common ones are AMPL4 (Fourer et al., 1993), AIMMS (developed by Paragon Decision Technology), GAMS5 (general algebraic modeling system) and MPS (developed by IBM). It is important to note that none of these languages solves the described optimization problem. They allow an easy transformation of the mathematical specification in computer readable form. They also allow the data handing required by associating databases directly to the model to be tackled effectively. Finally, these languages allow interaction with numerous optimization software packages to solve the resulting stochastic programming problems. 12.2.3 Continuous time approaches From a theoretical point of view, continuous time portfolio construction approaches have received a lot of interest since the work of Merton (1969). Although the achieved results cannot be implemented directly in practice, they show important properties of dynamic portfolio construction algorithms. As the derivation of continuous time models is very technical, I will only focus on an interpretation of the results. The interested reader will find details in Brandt (2007) or Merton (1990), to name just a couple. The most important result states that, if the expected returns of the considered assets are not independent over time the optimal weights to be invested in risky assets wR is determined as wR = wM + wH (12.2) 4 The AMPL language was developed at the Bell Laboratories to describe high complexity problems for large scale mathematical computation (i.e. large scale optimization and scheduling type problems). One of its authors, Brian Kernighan, was heavily involved in the design and development of the UNIX operating system and is co-author of the first book on the C programming language. 5 Initial research and development of GAMS was funded by the International Bank for Reconstruction and Development.

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where wM is the so-called myopic demand weight and wH the hedged demand weight. The myopic demand weight is determined as a function of: • the risky asset’s first and second moments, that is its mean and variance, and • the investor’s relative risk aversion. The second term, the hedging demand weight, depends on the relationship between the state variables of the asset return generating model and the risky asset’s return distribution over time. Intuitively, the investor takes positions to hedge against undesirable effects of changes in the state variables. Possible state variables are, for example, inflation levels, GDP growth or jobless claims.

12.3 A PRACTICAL EXAMPLE Different variations of the dynamic stochastic programming portfolio construction models have been implemented in practice. The most common one is the Russel–Yasuda Kasai model (Carino et al., 1994). Other models are the Towers Perrin (Mulvey, 1996), the Mitsubishi Trust, the Swiss Bank Corporation, the Daido Life and the Banca Fideuram models (Mulvey and Ziemba, 1998). To illustrate the model construction process, I consider a pension fund with the following characteristics: • The investment horizon considered is H = 10 years, subdivided into 10 one-year planning periods. • The liabilities L(t) at the beginning of each planning period t are determined as a deterministic function of the short term interest rate, such that L(0) = $10 000 000 and L(t) = (1 + RF (t) + 1.2 %) · L(t − 1) for t > 0 where RF (t) is the risk free rate at the beginning of the planning period t. Define by A(t) the value of the asset portfolio at time t. The pension fund has defined its utility function as: • maximizing the coverage ratio at the end of the investment horizon H , that is A(H )/L(H ), • assuring a maximal value at risk at the 95 % confidence level at the beginning of each planning period of no more than 75 % of the surplus value, that is 0.75 % · (A(t) − L(t)), and • satisfying the regulatory requirements, that is investing no more than 40 % in equities and 30 % in foreign assets at any moment in time. In addition, the pension fund has defined the investment universe as well as its decomposition into asset classes, as shown in Table 12.1. Based on a quantitative model, the pension fund determined the forecasted or expected returns and volatilities for each asset class, as also shown in Table 12.1. Investment horizon and planning periods. The investment horizon and planning periods given by the pension fund are considered adequate. They represent a realistic horizon for rebalancing the fund’s overall asset allocation.

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Positive Alpha Generation: Designing Sound Investment Processes Table 12.1 Investment universe, asset classes, expected returns and volatilities considered by the pension fund Asset class Cash Domestic bonds Foreign bonds Domestic equities International equities Commodities Hedge funds

Forecasted/expected return

Volatility (%)

RF (t) RF (t) + 100 bp RF (t) + 150 bp RF (t) + 4.0 % t  5 RF (t) + 5.0 % t > 5 RF (t) + 5.0 % t  5 RF (t) + 5.5 % t > 5 RF (t) + 300 bp 4.0 %

0.0 2.5 6.5 14.0 17.0 8.5 5.5

Source: sample data

Scenario generation. I use an approach of modeling returns of different asset classes using stochastic processes. I model the risk free interest rates based on the Cox et al. (1985) model such that RF (t) = a · (b − RF (t)) · t + σRF ·



RF (t) · W

(12.3)

where the parameter a represents the speed of the mean reversion of interest rates to its long term average b, RF (t) the interest rate at time t, t a one-year time horizon, σRF the volatility of the interest rates and W a Brownian motion process. Then RF (t + 1) = RF (t) + RF (t) and the total return of the cash asset class is defined for the planning period t to be RF (t). To model the other asset classes, I assume for the sake of simplicity of the example that they follow a multivariate geometric Brownian motion process, that is

1 S = µ(t) − ·
· S · t +
· S · W 2

(12.4)

where S is a vector of asset prices, µ(t)a vector of expected returns based on Table 12.1 and the risk free rate generated using Equation (12.3),
the covariance matrix and W a Brownian motion process. Figure 12.3 shows a sample set of innovations of the asset prices resulting from the scenario generation approach. In practice more sophisticated models with more elaborated assumptions and parameter estimation approaches may be required. Utility function. I define the utility function as maximizing the expected proportion between the asset value and the liabilities at the end of the investment horizon, that is max

h(a,t,s)

  s

a

h(a, H, s) · S(a, H, s) − L(H, s)



where h(a, t, s) is the optimal number of units of asset a held at time t under scenario s, S(a, t, s) the asset’s price at time t under scenario s and L(t, s) the liabilities at time t under scenario s.

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160 150

Asset price

140 130 120 110 100 90 80 0

1

2

3

4

5

6

7

8

9

10

Planning period Source: sample data, author's illustration

FIGURE 12.3 Sample set of 20 asset price scenarios generated by the proposed scenario generator

In addition, I require that the value at risk at the 95 % confidence level at the beginning of each planning period of no more than 75 % by adding the following restriction to the problem: ∀s, t:1.65 ·

 (h(t, s) ∗ S(t, s)) ·
· (h(t, s) ∗ S(t, s))  75 % · (h(t, s) · S(t, s) − L(t, s))

where ∗ denotes the product component by component, rather than the scalar product. Constraints. Then I add the nonanticipative constraints to the problem formulated, that is ∀s, t < H : h(t + 1, s) · S(t + 1, s) = h(t, s) · S(t, s) ∀t, s, s  : S(t, s) = S(t, s  ) ⇒ h(t, s) = h(t, s  ) as well as, for consistency, the constraints ∀a, t, s : h(a, t, s)
0 Finally, I add the restrictions on maximal exposure to equities and foreign exposure, assuming that commodities as well as hedge funds are considered to be foreign investments. This can be done by adding the following restrictions: ∀t, s :



h(a, t, s) · S(a, t, s)  50 % ·

a = equity

∀t, s :



a = foreign



h(a, t, s) · S(a, t, s)

a

h(a, t, s) · S(a, t, s)  30 % ·



h(a, t, s) · S(a, t, s)

a

By solving this formulated problem using a quadratic optimizer, I obtain the optimal weights for each planning period, as shown in Figure 12.4.

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Positive Alpha Generation: Designing Sound Investment Processes 100% 80% 60% 40% 20% 0% 0

1

2

3

4

5

6

7

Cash

Domestic bonds

Foreign bonds

Foreign equities

Commodities

Hedge funds

8

9

10

Domestic equities

Source: sample data, author’s calculations

FIGURE 12.4 Optimal asset class weights at the beginning of each planning period

It can be seen that the asset allocation is rather stable. This is mainly due to assumed stable liabilities to be matched. In addition, the foreign fixed income proportion is reduced in favor of cash starting at year six. In fact, the increased expectations in equities allow the investment in risky foreign bonds to be reduced, which offers decent diversification but low returns per unit of risk.

Part V Portfolio Implementation

13 Transaction Costs, Liquidity and Trading Transaction costs as well as market liquidity play an important role in the implementing and trading value chain module. It is often the least understood and least appreciated module. The input to this module consists of a set of portfolio changes that ultimately result in trades. These changes may be specified at different levels of granularity. For example, buying 1 % of German equities modeled by the MSCI Germany index or selling 0.5 % exposure to the Ford Motor Company share and buying exposure to the General Motors Corporation share instead are such changes. The goal is to implement those changes in the portfolio: • with a minimal cost and performance impact and • treating all investors impacted by the trades fairly. Although some portfolio managers tend to disagree, the goal of the portfolio implementation and trading module is not to generate alpha through market timing. It is executing the changes quickly, without error and at a favorable price. When talking to portfolio managers, I have often heard the argument that a certain portfolio change cannot be implemented because the transaction costs are too high. However, when looking at the required trades, their liquidity and associated costs, it often became clear that the reason for refraining from implementing a given portfolio change was disbelief in the market forecast behind the change rather than the actual costs. I would even argue that in most markets, even in fixed income markets, the conviction needed to formulate a forecast is such that the expected alpha is larger than the transaction costs incurred. Consider a government bond market exhibiting transaction costs of 50 bp (basis point, equal to 0.1 %) per trade. Assume that my forecast is that the 10-year zero interest rate will rise. Then, the interest rates must rise by more than 5 bp, assuming a parallel shift of the yield curve to cover the transaction costs. Honestly, no investment strategists will express any conviction in a yield rise forecast of less than 5 bp. Nevertheless, when handled improperly, transaction costs and market impacts when trading large volumes can have a significant adverse performance impact. A sound understanding of transaction costs can be a powerful help in achieving investment performance success. It can even be a component of the key competitive advantage of an investment manager.

13.1 UNDERSTANDING TRANSACTION COSTS AND MARKET LIQUIDITY 13.1.1 The iceberg model Transaction costs can be subdivided into three components, that is: • explicit or visible costs, • implicit or invisible costs and • opportunity costs.

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This subdivision of transaction costs is often called the iceberg model as only a small fraction of these costs are actually visible, that is explicit. Explicit costs are the visible or charged costs of trading, namely ticket fees, brokerage costs, taxes and other duties, custodian commissions as well as soft dollar fees.1 These costs can be calculated explicitly. They tend to be rather small. Implicit costs, by contrast, are those costs that the market makers or counterparty charge for accepting to take the other side of a trade. They are mainly the bid–ask spread costs as well as market impact costs, that is changes in asset prices due to actually trading, usually large sizes. They reward the counterparties for taking the risk that the trade is an informed trade, that is that the portfolio manager executes the transaction because he or she is in possession of superior information or has processed information in a superior way to the counterparty. It is the cost charged for providing liquidity. A third category of transaction costs are opportunity costs. These are the indirect costs incurred from asset prices changing during the execution of a trade or delaying a trade actively or passively, for example due to the lack of liquidity. Opportunity costs also occur when buying an asset that does not exactly match the performance of the specified portfolio change. This may be the case, for example, when buying a liquid Euro STOXX 50 future, instead of buying the stocks in the broader Euro STOXX index, when the latter one was the required portfolio change. Similarly, the indivisibility of certain assets or their minimum trade sizes may have as a consequence that the trade executed does not exactly match the required portfolio change. This is, for example, the case when buying bonds, which are only traded in large fixed lots at reasonable implicit transaction costs. 13.1.2 Modeling transaction costs To take into account transaction costs in an investment process, it is necessary to formalize them. In general, explicit costs are modeled by an affine function ca (n) as shown in the following equation:  + + if n > 0  k +n·v 0 if n = 0 ca (n) = (13.1)  − k + n · v − if n < 0 where n represents the trade size in asset a, k + and k − are fixed costs and v + and v − variable costs for buying and selling asset a. I distinguish between buy and sell transactions, especially because taxes are usually not charged in the same way for buy and sell trades. For example, in the UK market a duty of 50 bp is charged when buying a stock but not when selling it. In addition, if the sell trade results in a net short position, the cost of collateral needs to be taken into account. The function ca (n) should be convex and differentiable, except at the origin, that is where n = 0, where it must evaluate to ca (0) = 0. Modeling indirect and opportunity costs are more complex as additional external variables attached with a degree of uncertainty come into play. Figure 13.1 illustrates the bid–ask spread costs as a function of the market liquidity, represented by the quote depth2 and the traded position size. 1 They also include collateral costs, time involved in acquiring knowledge, usually represented by the management fee, as well as record keeping, controlling and auditing costs. 2 The quote depth is the position size that can be traded without changing the price. On an exchange, this is the size of the current best bid, respectively offer price, made.

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asset price

quote depth

bid price

ask price

traded position size

FIGURE 13.1 Bid–ask spread as a function of the market liquidity and size of the position to be traded

13.1.3 The investor’s utility function with respect to transaction costs Knowing and understanding the transaction costs is the first step in designing a successful implementation and trading value chain module. Next, the change in the investor’s utility function due to the presence of transaction costs and market illiquidity should be considered. Let me first consider the CAPM framework. In this framework, I have E[Ra ] = RF +

cov(Ra , RM ) · (E[RM ] − RF ) var(RM )

(13.2)

Mayshar (1981) has shown that in the presence of current holdings and transaction costs, the pricing equation (13.2) needs to be rewritten as

cov(Ra , RM ) E[Ra ] = RF + C · δa · · (E[RM ] − RF ) + γa · var(Ra ) (13.3) var(RM ) where C is a measure of the marginal transaction cost and δa and γa two nonnegative parameters. In contrast with the CAPM, the variance of the asset itself significantly affects the expected return. This is consistent with the observation that, in times of uncertainty, that is in times of heightened volatility, bid–ask spreads tend to widen and therefore transaction costs tend to increase. A consequence of the expect utility maximization investor in the presence of transaction costs is that the investment manager will not always trade on information, that is forecasts that are expected to have an impact on the asset’s price. This leads nearly certainly to misalignments between the prices and the fundamental value of an asset. Such a misalignment will last until it is larger than the transaction costs, the moment at which it will be arbitraged away or reduced to a level smaller than the transaction costs. As different investors face different transaction costs, these opportunities may be exploited. Furthermore, deciding when to execute a trade will have an impact on the performance of the asset, independent of the forecasts, which are represented by its expected fundamental value. Note the subtle difference between the investment manager’s asset return expectation, the investment manager’s fair value for the asset and the market price. This difference is illustrated in Figure 13.2.

Positive Alpha Generation: Designing Sound Investment Processes

asset price

200

market price forecasted fair value current fair value

1) optimal but inefficient

current fair value

market price

market price

current fair value

2) optimal and efficient

3) sub-optimal but efficient

FIGURE 13.2 Relationship between market prices, current fair values and expected or forecasted fair value

In the first case, although the current fair value is below the forecasted one, the current market price is above and therefore buying the asset will not result in the expected return. The second case illustrates a situation where both the market price and the current fair value are below the forecasted fair value. Thus buying the asset at market price will, given the forecast being correct, result in a positive alpha. The third case illustrates a market environment in which executing the trade will result in a positive alpha, but less than expected. Transaction costs may void the realizable profits. The decision whether or not to execute a trade needs to take into account the forecasted fair value, the current fair value and the current market price. 13.1.4 Impact of liquidity on trading and costs The sole fact of wanting to buy or sell an asset introduces a liquidity risk. Every transaction is an exchange between two parties. In the context of trading, the complexity of finding one or more counterparties that are willing to take the opposite side of a trade for a given price and a given quantity is what defines market liquidity. Market liquidity can be decomposed into two categories, that is: • endogenous market liquidity and • exogenous market liquidity. Endogenous market liquidity is specific to a trade and as such to an investment manager. It is mainly driven by the size of a trade (the larger the size, the greater the endogenous liquidity risk) and the corresponding cost. A reason for reduced endogenous market liquidity is if the counterparty has a high conviction that the investment manager is executing an informed trade, that is has a competitive information advantage. Endogenous market liquidity is best managed by decomposing large trades into multiple consecutive smaller trades such that the bid–ask spread implied by the size is reasonable with respect to the volatility of the asset price. In addition, it is important to ensure that the information of the overall size of the trade is hidden from the market. If the counterparty or counterparties are able to find out or guess the overall size of the aggregated trades, they may and usually will use that information in pricing the asset. Successfully decomposing a large trade into smaller trades requires:

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• knowledge of the counterparties, that is market makers serving the market and providing the liquidity, • knowledge of how the asset price is expected to change purely on the fact that there is supply or demand of a given size on the market, that is the information that the trade initiating party may have competitive information about the asset and therefore its future price, and • knowledge of the asset’s volatility in order to be able to determine a time frame for executing the trade without suffering from adverse market movements. The drawback of decomposing trades is that the asset price may change between trades and negatively affect the performance. Therefore the investment manager has to find a compromise between the size of the trades and their scheduling over time. Exogenous market liquidity is the result of market characteristics. It is common to all market participants and unaffected by the action of any single one, although it can be affected by the joint action of a large number of individual market participants. The cost of exogenous market liquidity is characterized by the volatility of observed bid–ask spreads. It can be quantified, and therefore incorporated into the transaction cost function, by cost of liquitiy = γ · s + ξ · σs where s is the average spread, σs the spread volatility and γ and ξ two constants. 13.1.5 Structural impact of transaction costs on markets A key question often studied in the literature is whether or not lower transaction costs make markets more efficient (Gu and Hitt, 2001; Gurbaxani and Whang, 1991; Malone et al., 1987). Reductions of transaction costs can be directly linked to cost savings and a number of indirect benefits, such as: • the reduction of agency costs, • increased monitoring and coordination, and • the creation of new types of markets, especially in the derivative area. All these aspects are beneficial to all market participants, except perhaps for market makers who may see their commissions reduced. Reduced transaction costs also have as consequence that new participants enter the market. Due to this lower barrier to entry, the new participants may be poorly informed and as such increase the number of trades that are due to noise rather than actual information. This in consequence increases volatility and has a negative impact on indirect transaction costs. The consequence is that there is a nonzero optimal level of transaction costs that balances between the advantages and drawbacks of low costs. 13.1.6 Best execution The overall goal of the implementation and trading module is known in the investment management community under the name of best execution. In 2002, the CFA Institute published guidelines on the key components that are required to ensure best execution. These components are: • a process, that is formal guidelines and policies, that ensures that all trades are executed so as to maximize the value of the portfolio,

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• a set of disclosure guidelines regarding the trading venues, agents and techniques implemented, as well as actual or potential conflicts of interest, and • a record keeping process that documents the compliance with the defined process and disclosure guidelines. Consider a fixed income trading desk responsible for best execution of all government bond trades. First, the universe of possible counterparties is defined, including major electronic trading platforms. Then for each buy (sell) trade, the trading desk is required to obtain and document on paper at least three quotes from different counterparties for the required trade size within a timeframe of 10 minutes. In addition, at least one quote obtained through an electronic trading platform is required and documented. If this process succeeds, the trade is executed at the lowest (highest) price. The dealer documents and discloses the difference between the average of the received quotes and the actually executed price. If the process fails, that is insufficient quotes are received, the dealer returns the trade to the investment manager, including the received quotes, and asks for advice. The investment manager then decides whether he or she is willing to execute the trade with less than three competitive quotes available. If this is the case, the dealer documents this fact and documents the average quote from at least three dealers for smaller sizes. No trade is executed if less than three quotes are available for any size, unless approved by a senior manager of the investment management department. To assure independence, all trades are executed on a first come and first served basis. In addition, a Chinese wall between the trader and any advisory or investment banking personnel ensures that no conflict of interest occurs.

13.2 THE ACTION AND CONTEXT OF TRADING Before studying how to design an efficient as well as effective trading approach to be integrated into an investment process, it is important to understand the context in which trades are executed as well as the roles of the different market participants. 13.2.1 Role of market participants and their trading goals I distinguish between three categories of market participants, that is the market taker, the market maker and the intermediary. A market taker acts like a buyer in a nonfinancial industry. He or she looks at the assets and their respective prices that are offered and then decides to buy or not to buy. Depending on the type of market, a bargaining action precedes the transaction. In many investment processes the investment manager acts as a market taker. A market maker, on the other hand, acts like a seller in a nonfinancial industry. He or she offers to buy or sell a certain volume of assets at a given price. In an exchange traded environment, the offer is entered in the order book, as shown in Figure 13.3, and is made visible to all market participants. For example, market participants have entered commitments into an order book to buy 8000 shares at a price of 91.40. On the sell or bid side, an offer for 4000 shares at 91.45 exists. In off-exchange markets, whether over the counter or on so-called dark or hidden liquidity platforms, the offer is only made available to the requesting counterparty. Transparency is reduced and therefore the information about the potential existence of an informed trader is not divulged to the market. In most cases, investment banks act as market making participants. Market makers provide liquidity to the market. They are paid for that liquidity providing that they offer to buy at a lower price

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15 000 @91.60 5 000 @91.55 7 500 @91.50 ask side

bid side

8 000 @91.40

price

4 000 @91.45

5 500 @91.35 6 800 @91.30

25 000 @91.25

FIGURE 13.3 Structure of an order book

than they offer to sell. In addition, this bid–ask spread rewards them for their potential information disadvantage. The third category of market participants are the intermediaries, also called brokers. Rather than take either side in a transaction, they play the role of an intermediary. As such they offer a certain number of services such as finding an opposing side of a trade, supplying market information, as well as assuring discretion. For this service, a commission is paid. Using a broker has numerous advantages for an investment manager: • The investment manager does not have to screen the market for available offers, and therefore can focus more on core capabilities of forecasting markets. • The intermediary provides anonymity to those accessing the market. The intermediary acts as a trustee. Although the market may still find out the existence of a potential informed trader, his or her identity is hidden. In addition, as brokers usually represent multiple investors, large volumes do not necessarily mean well informed trades. • Finally, informed traders with opposing expectations with each other match orders of market takers. Such matches avoid both parties having to pay for the liquidity providing service, in addition to the commission paid to the broker. As advantageous as using an intermediary may sound, their services come at a price. The commissions may outweigh the saved hidden transaction costs. Potential conflicts of interest exist because the intermediary’s primary goal is to sell a service, rather than provide best execution. If deciding to use a broker service, it should be selected carefully and the provided service monitored thoroughly. In addition, using intermediaries is best suited for: • executing large sized trades,

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• accessing markets where the investment manager has insufficient knowledge of local practices,3 • when only a few, if any, market makers exist or • when markets are very illiquid, that is the average daily traded size is small compared to the order and the bid–ask spread published by market makers is large. 13.2.2 Order types There are many types of orders that an investment manager may use to execute trades. They distinguish themselves from each other on how they prioritize between speed of execution and quality of price. The most common and useful trade order types, from an investment manager’s perspective, are: • market order, that is instruction to execute a trade promptly at the prevailing market price – speed is preferred over price, • limit order, that is instruction to execute a trade at a specific price, accepting that the order execution may be delayed or even fail – price is preferred over speed, and • at-price order, that is agreement with a counterparty to trade an asset at a predefined price and quantity, usually at the opening or closing price. If the investment manager’s forecast is such that a large market impact of the forecasts is expected, market orders are to be preferred. On the other hand, if the forecasts are of a fair value or mean reversion type, a limit order may be preferred. The decision as to which kind of order type to use should also include the potential opportunity costs from not being able to execute the trade. At-price orders can usually be preferred if a given price must be matched, whatever that price is. This is, for example, the case when replicating an index, as described in Chapter 15.4 At-price orders are very often combined with basket trades.5 Such a trade could be to buy the stocks of the French CAC 40 index according to the weighting in the index at a price related to the index level at closing of the Paris Euronext exchange. The price paid for each individual security becomes irrelevant, as only the basket’s price counts. One of the skills an investment manger must have is to be able to select the order type that will maximize the investor’s utility function, including opportunity costs. 13.2.3 Market types Over time a large number of different types of markets have evolved. Each market type is characterized by serving one or more specific goals. Figure 13.4 classifies the most important market types from an investment manager’s perspective along two dimensions: • open or closed, that is whether or not the counterparty may select to trade or not solely based on whom it is trading with, and 3 Investment managers may have to ask themselves the question whether they are capable of forecasting markets where they have insufficient understanding of the working of trading in those markets. 4 It is an open question whether at-price orders can actually be classified as best execution, as the value of the portfolio is not maximized, but rather the defined utility function given by the investor to the investment manager maximized. 5 A basket trade is a trade of a set of assets at once at a single price of the basket. Basket trades allow, for example, all the shared parts in an index to be bought according to their weighting in the index with a single trade.

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public

Exchanges - quote driven - order driven - auction driven

private

open

Closed book markets - electronic crossing networks - liquidity platforms

closed

X Over the counter transactions

FIGURE 13.4 Classification of different market types

• public or private, that is whether the information about trading opportunities is publicly available or not. In an exchange based market, orders and offers are published, either by outcry or, more often, on an electronic platform. The price is formed through a price discovery mechanism, which equilibrates between supply and demand. Sometimes an auction mechanism is used as a price discovery mechanism. Open but private market types, like liquidity platforms or electronic crossing networks (ECN) (Hendershott and Mendelson, 2000), work in a similar way as exchanges, except that they do not provide a price discovery mechanism. An order is executed if a match is found, usually anonymously. In a closed market, called over-the-counter markets, the trade is agreed as a deal between two parties. No price discovery mechanism exists in the trading process by itself.6 The mechanism is based on mutual trust. In addition, a counterparty may offer different prices depending on the business relationship existing. As for selecting the most appropriate order type, the investment manager adds, or destroys, value by choosing the market to trade on and, in the case of closed markets, the counterparty to trade with.

13.3 IMPLEMENTATION AND TRADING AS A MODULE OF AN INVESTMENT PROCESS VALUE CHAIN Although there is not much to gain in terms of alpha generating capabilities from designing a sound implementation and trading module, there is much to lose if the job is done poorly. The implementation and trading module must be set up with a total quality management (Feigenbaum, 1991) approach in mind. This means that the following four steps must be followed: • The focus must be on continuous process improvement through making the process visible, repeatable and measurable. • The process must deliver the results it is supposed to deliver. • The way the user executes the trading process must be used to improve it. 6 The investment manager usually precedes the conclusion of an OTC trade by a customized price discovery mechanism, for example using a broker as an intermediary or requesting quotes from different possible OTC counterparties.

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• The trading process must be sound, in the sense that it follows a logic understandable and auditable by a third party. These four characteristics ensure that trades are executed correctly the first time. At the center of the design decision stands the question ‘How much information advantage must I have in order to recover the costs of trading?’ Assuming that sufficient forecasting capabilities have been captured, the goal of the implementation and trading module can be rephrased as implementing the portfolio change suggested by the portfolio construction module in terms of market exposure by executing a sequence of trades minimizing their performance impact after costs. Satisfying that goal can be achieved through an eight-step implementation and trading module development process: 1. Select the investment vehicles to use in implementing the portfolio changes. The key focus is on – matching risk exposure, – aligning the underlying costs with the investment horizon and – ensuring that sufficient liquidity is available. Multiple alternatives should be selected and analyzed to offer the greatest flexibility. 2. Understand and quantify the transaction costs of each possible investment vehicle. 3. Relate the identified transaction costs to the inherent imprecision of the forecasts and their time horizon. 4. Decide at what level or module of the value chain the transaction costs should be taken care of during the investment process. 5. Define the utility function to optimize during the sequencing of the execution of the trades. 6. Define the specific trading algorithms to be applied as well as which algorithm should be applied in what situation. 7. Simulate the developed algorithms and measure their impact. 8. Depending on the simulation results, adjust steps 1, 4, 5, 6 and 7. Contrary to common practice, I recommend defining the trading process and the associated algorithms to use beforehand, rather than develop them in an ad hoc fashion as markets evolve. The goal must be to act rather than to react. 13.3.1 Taking into account transaction costs There are different modules of the investment process value chain in which transaction costs can be taken into account: Market forecasting. The forecasts formulated are adjusted for explicit and implicit transaction costs. The main advantage of taking care of transaction costs at this stage is that the costs are considered exactly where the information advantage or forecast is generated. However, a major drawback is that investor or portfolio specific information is and should not be available at this stage. In addition the investor’s utility function in the presence of transaction costs is dependent on the volatility of current holdings, which again should be independent of the formulated forecasts. Risk management. As transaction costs can be related, at least in part, to an information uncertainty, for example through bid–ask spread volatility, the associated risk may be

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modeled as part of the market risk. The drawbacks of considering transaction costs at the market forecasting level are lifted. Unfortunately, the complexity of the risk management approach will increase significantly. The added value is very often eliminated by the inherent modeling imprecision, both at the model parameter side as well as the structure of the resulting model itself. Portfolio construction. Considering transaction costs at the portfolio construction module level allows the constructed portfolio to be conditioned on the cost impacts of any updates. The advantage of such an approach is that the decisions taken are in the context of the whole portfolio including both market forecasts, their inherent imprecision, the risk model, as well as the investor utility function and restrictions. Unfortunately, two major drawbacks exist. First, deciding whether or not to execute a trade exponentially increases the complexity of most portfolio construction algorithms, as the utility function is no longer continuous. Second, such an approach requires that the market forecasts are formulated at the level of precision not smaller than the transaction cost size. Implementation and trading. The decision on when and at what price to trade becomes a local decision. The trading decision is tightly related to the market and only loosely coupled with the whole portfolio structure. As the decision is a local one, the complexity of the overall investment process is only marginally affected. The trade execution may even be outsourced. The major drawback is that the overall portfolio view is lost. Indeed, the sum of locally optimal trading decisions may result in a globally suboptimal trading strategy. The selection of the most appropriate value chain module to take into account transaction costs depends on: • the precision and frequency of market forecasts, • the complexity of the resulting investment process and • the optimality of the result, that is maximization of the investor’s utility function. In practice, a combination of considering transaction costs at the portfolio construction and risk management as well as implementation and trading module levels has proven to be the most effective for a large class of investment processes. 13.3.2 Trading algorithms There are numerous trading algorithms. Most of them have been developed for specific utility functions or are based on specific properties of the traded asset’s time series characteristics. Any trading algorithm should implement the following three steps: 1. Determine the trade-off between urgency and size of the order. 2. Prioritize the trades according to the determined trade-offs and the requirement of executing buy and sell trades side by side to match the cash balance and risk. 3. Derive the most appropriate approach to the market, that is whether using direct access, a broker intermediary or OTC transactions, to name just a few. The most general trading algorithm is the so-called opportunistic participation strategy or the making and taking trading algorithm. The idea is to subdivide a large trade into a number of smaller trades. The market is entered through two dimensions, that is:

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let p = price at which the trader is willing to buy/sell let ε = expected/average premium for providing liquidity let m = maximal trade size that will not divulgue the possession of relevant information s ← number of units of an asset to buy/sell while s > 0 loop make a market, that is enter a limit order of size min(s, m) and price p − ε, respectively p + ε, for a buy, respectively a sell, trade if the limit order is executed, that is q units are traded, then s ←s−q adjust the existing limit order such that its size is again min(s, m), if necessary end if if a counterparty is willing to sell, respectively buy, a quantity q (via a limit order of their own) at a price between p − ε and p, respectively p and p + ε, then buy, respectively sell, that quantity s ←s−q end if end loop

FIGURE 13.5 Opportunistic participation trading strategy algorithm

• The small trades are offered to the market at a limit price, the limit price p being slightly less than the price p = p ± ε the investment manager is willing to pay for a buy order or slightly higher for a sell order, p being the price the trader is willing to pay. • As soon as another market maker offers the asset for a price between p ± ε and p, the investment manager buys, respectively sells, the asset until the prices move outside the range. Figure 13.5 describes the opportunistic participation strategy algorithm. A second important trading algorithm is that of smart order routing. A large order is subdivided into multiple smaller orders. They are then routed through different competing brokers, exchanges and other liquidity providers. Depending on the assets, their liquidity and the size of the expected trade, different approaches for trading need to be used. Matching buy trades with sell trades of equivalent sizes as well as having similar risk characteristics is important, eventually subdividing larger trades on one side to match smaller trades on the other side. Hard trades should be done first. The focus must be on maximizing after cost return rather than minimizing costs. Brokers should be selected based on their skills. They should become dependent on the investment manager rather than the opposite. They should be trusted, but controlled. Finally, it is very important to ensure that during trading the cash account against which trades are settled is always balanced. This means that buy and sell trades should be executed side by side.

13.4 EQUITY ASSET ALLOCATION TRADING APPROACH EXAMPLE Consider a region based equity asset allocation investment process. The input to the implementation and trading module is an optimal asset allocation, that is weights relative to a benchmark, as shown in Table 13.1, as well as confidence ranges per asset class. The confidence ranges express the inherent imprecision of the forecasting as well as the risk

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Table 13.1 Sample optimal asset allocation for a regional equity asset allocation investment process European Union (%)

United Kingdom (%)

United States of America (%)

25.0 23.5 −1.5 ±0.5 25.0 Trade

15.0 14.5 −0.5 ±0.2 11.0 Trade

40.0 42.5 2.5 ±1.6 43.0 Do not trade

Benchmark Optimal portfolio Relative weights Confidence range Current portfolio

Japan (%)

20.0 19.5 −0.5 ±0.2 21.0 Trade

Source: sample data, author’s calculations.

management modules of the investment process. For example, the optimal position in EMU equities is to be underweight by 1.5 %. However, due to the inherent imprecision of the portfolio construction algorithm, underweight between −1.0 and −2.0 % cannot be distinguished from each other from a modeling point of view. Therefore, if the current position is within that confidence range, as is the case for the current US equities portfolio holdings, that is 43.0 % ∈ {42.5 % ± 1.6 %}, no trade is necessary. Developing an implementation and trading approach means defining a process that translates the current portfolio weights, on an asset class exposure basis, as close as possible to the optimal portfolio weights, or at least within the confidence ranges. Let me illustrate the development process using the eight-step approach proposed. 13.4.1 Investment vehicle selection Depending on the trade size as well as the existing portfolio holdings and the time horizon of the forecasts, I use the following investment vehicles: • For large trade sizes, that is more than 1 000 000 USD exposure and short forecasting horizon, that is less than three months, I use exchange traded futures, like the S&P 500 or the FTSE 100 future, with a maturity consistent with the forecasting horizon. • For large trade sizes but long forecasting horizons, I rely on basket trades on the underlying asset classes. I decompose the trade size in subtrades of the maximal size for a basket trade that a single broker is willing to trade. • For small sized trades independent of the forecasting horizon, I rely on exchange traded funds (ETFs). 13.4.2 Transaction costs Next, I determine the expected transaction costs under normal market conditions. I distinguish between brokerage fees, bid–ask spread and maximal daily volume. 13.4.3 Forecasting imprecision and time horizon I decide to trade in general only in those asset classes that have a current position that is outside the optimal confidence range shown in Table 13.1. In my example, this means trade

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in EMU, UK and Japanese equities. For example, the current weight in United Kingdom equities, that is 11.0 % is outside the confidence range of 14.3 %–14.7 %. 13.4.4 Value chain module taking transaction costs into account As I relate the decision whether or not to execute a trade based on the optimal asset allocation, the confidence ranges and the current portfolio asset allocation, transaction costs are taken into account indirectly through the forecasting imprecision and the portfolio construction and risk management module level. The transaction costs determined are used to support the selection of the most appropriate investment vehicle, rather than decide whether or not to execute a transaction. I assume that an asset allocation that is significantly different from the current position is backed up by sufficient expected excess return from the forecast exceeding the transaction costs. 13.4.5 Utility function I model the trade-off between speed of implementation, that is minimizing the opportunity costs, and price, that is minimizing implicit costs, through the following three rules: • Trade at most 20 % of the average daily volume over the last 10 days, when trading futures. • Trade at most once per hour the size of the quoted size by any pure market maker7 when trading ETFs. • Execute basket trades for the largest size for which the counterparty agrees to the closing price minus a spread not larger than the value weighted average bid–ask spread of 90 % of the securities in the basket. 13.4.6 Trading algorithm I propose to implement the following trading algorithm: 1. Determine whether the trade is large, that is if its size exceeds USD 1 000 000 on a portfolio-by-portfolio level. 2. Calculate the aggregated amount to trade in large sized trades and in small sized trades. 3. Choose futures or basket trades on the basis of whether the forecast underlying the trade is over a time horizon shorter or larger than three months. 4. Subdivide the aggregated large sized and small sized trades into subtrades according to the defined utility function. 5. Execute pairwise buy and sell trades using for futures and ETFs the opportunistic participation strategy on the main exchange, that is the exchange providing the highest liquidity in the given future, and using basket trades, requesting quotes for the determined size from at least three brokers and choosing the best one. 7 A pure market maker is a market participant whose sole role is to provide liquidity, and as such has usually signed an agreement with the exchange to guarantee most of the time a minimal quote size with a maximal spread. For example, the Swiss exchange requires market makers for equity index based ETFs to comply with the following requirement during 90 % of the official trading hours of the Swiss exchange (on a monthly basis). The market maker is required to provide bid and ask prices for amounts of at least EUR 50 000 that do not deviate from the indicative NAV (net asset value) by more than 2 %, that is 1 % on either side.

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Table 13.2 Sample execution of the developed trading investment process

Buy UK – futures Sell Japanese – basket Sell EMU – futures

Day 1 (%)

Day 2 (%)

Day 3 (%)

Total (%)

+1.25 −0.75 −0.50

+1.25 −0.75 −0.50

+1.00 −0.50 −0.50

+3.50 −2.00 −1.50

Source: sample data, author’s calculations.

13.4.7 Simulation Rather than simulate the performance impact of the developed process, I monitor its quality through measuring the implementation shortfall costs. Table 13.2 illustrates the trades executed for the portfolio allocation shown in Table 13.1. Based on the daily volume analyses performed, I decide that trading more than 1.25 % in UK FTSE 100 futures on any day will have a nonnegligible market impact. Therefore, I decompose the long UK future trade into three trades, two at 1.25 % and one at 1.0 %. To have no active position at the end of any day, I sell positions against the trades in UK futures. I decide to do so proportionally in Japanese as well as EMU equities, resulting in selling −0.75 %, respectively −0.50 % Japanese equities, and taking short positions for 0.50 % of the portfolio value in EMU futures. Therefore, I am able to execute the required portfolio changes over a three-day time horizon.

14 Using Derivatives A derivative instrument is a contract between two parties to exchange cash, securities or other goods, called the valued item, at predefined dates in the future under a number of conditions. The value of a derivative instrument is linked to the price of an underlying asset, like the price of the IBM stock, economic information, like inflation or interest rates, or another derivative instrument, like an interest rate swap. The historical origins of forward derivative contracts are obscure. Organized futures markets began in 1848 when the Chicago Board of Trade opened (Kolb, 2000). Many different types of derivative instruments exist. Most of those that find active use in an investment process can be classified into one of the following three categories: • Futures and forwards. Two parties agree to exchange a valued item at a fixed date in the future for a price fixed today. • Options. One party agrees to sell to or buy from a second party a valued item for a price fixed today. The second party may decide whether or not to buy or sell the valued item. • Swaps. Two parties agree to exchange predefined cash flows or valued items over a given time period in the future. A key feature of derivative instruments as a portfolio construction or implementation tool is that they allow an investor to gain economic exposure to a given asset without actually having to own that asset. This de-coupling is one reason for an increased liquidity, and thus lower transaction costs of derivatives versus their cash equivalents. The asset may or may not exist physically.

14.1 DERIVATIVE INSTRUMENT CHARACTERISTICS Table 14.1 lists the most common derivatives and classifies them with respect to their counterparty and their underlying assets. The interested reader can find extensive details on derivatives, their valuation and use in finance in Fabozzi (1997), Hull (2005) or Kolb (2000). 14.1.1 Futures and forwards A futures contract is an agreement between a buyer (seller) and an established futures exchange or its clearinghouse, in which the buyer (seller) agrees to buy (sell) a specified amount of a valued item such as a stock or a bond, at a specified price at a defined date in the future. Futures contracts are standardized, exchange traded and marked to market daily. Most of them settle in cash rather than requiring physical delivery, commodity futures being the most prominent exception. In addition, the counterparty is always the exchange or its clearinghouse.

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Positive Alpha Generation: Designing Sound Investment Processes Table 14.1 List of some of the most common derivative instruments Asset class

Exchange traded

Over the counter

Fixed income

Interest rate futures Options on interest rate futures Bond futures Option on bond futures

Equities

Index futures Mini futures Options on stocks Options on indices Options on index futures Currency futures Options on currencies

Forward rate agreements Interest rate swaps Interest rate options Inflation swaps Repurchase agreements Credit default swaps Credit default options Total return swaps Warrants Options

Foreign exchange

Others

Commodity futures Commodity index futures Volatility futures

Currency forwards Currency forward options Cross currency swaps Weather swaps Volatility swaps Life expectancy swaps

Forward contracts provide the same economic exposure to the investor. However, they are traded over the counter. Forwards are usually not marked to market, exposing both parties to credit risk. Futures, as well as forwards, are simple and efficient instruments to gain or reduce exposure to specific assets or indices. Index futures allow a gain or reduction in exposure to all the stocks in an index through a single transaction. Bond futures allow interest rate risk exposure to be managed without having to trade in specific, less liquid government bonds. Asset allocation based investment processes are very efficiently implemented using futures, especially if the forecasting horizon is short and frequent changes are expected. In addition, future contracts allow gaining exposure to specific markets without entering into the associated currency exposure. Modeling futures and forward prices Futures, as well as forward contracts, are usually priced under the no-arbitrage assumption. This means that the price is determined in such a way that it is not possible to replicate the derivative payoff at a smaller (larger) price. The most common model used is the cost-of-carry model (Kolb, 2000). In this model, Price (asset) = Price (cash) + Price (future/foward) More formally, if the cash flows of the underlying asset as well as interest rates are nonstochastic, markets perfect and there are no taxes, then the price of a futures contract is given by Ft,T = At · er·(T −t) − Ct,T

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215

where Ft,T is the future/forward price at time t for a contract that matures at time T , At equals the spot price of the underlying asset at time t, Ct,T equals the value of the cash flows at time T paid by the underlying assets between time t and T , and r · (T − t) equals the continuous compounded risk free interest rate spanning the period between t and T . The difference between the futures price and the underlying cash asset, that is At − Ft,T , is called the basis. It represents the cost paid or return obtained for holding the cash asset. Futures and forward prices are equal up to a difference due to the daily resettlement feature of future contracts. The difference between the two prices is equal to the interest rates earned on the profits of the margin account. Empirical studies (Cornell and Reinganum, 1981) have shown that the pricing differences are insignificantly different from zero and can thus be ignored.

14.1.2 Options A call (put) option is a contract in which the seller of the option grants the buyer the right to purchase (sell) from the seller of the option a designated valued item at a specified price, the strike price, at a specified date (European option) or within a specific period (American option). In contrast with futures and forwards, an option has a value at the time it is issued. The buyer of the option pays this value to the seller at the beginning of the contract. Options may either be traded on an exchange (standardized) or over the counter (customized). In contrast with so-called symmetric derivatives, like futures or forward contracts, only the buyer faces a credit or counterparty risk. Options allow a gain in exposure to a given asset or asset class without bearing the downside risk. The use of options is most common in capital protection and preservation targeting investment processes. Options also allow actively managing the exposure to the volatility of the underlying asset’s price rather than the price itself. The use of options is limited in well structured investment processes because if an investment manager has an opinion or forecast about the future of an asset’s price, it is in many cases more efficient to gain exposure through instruments with symmetric payoffs, like futures, rather than options that require an initial premium to be paid. The premium represents the insurance cost for the forecast being wrong. As the underwriter of this insurance, that is the issuer of the option, has no knowledge of the investment manager’s forecast, that is an information asymmetry exists in favor of the investment manager, the option is in general expensively priced in the context of the market forecast of the investment manager. Valuing options The idea behind pricing options is identical to the one for pricing futures or forwards. A payout replicating portfolio strategy is designed and valued under the no-arbitrage assumption. An in-depth description can be found in Hull (2005). Black and Scholes (1973) have shown that, under a number of restrictions, the value or price of a European call-and-put option of a given asset can be expressed analytically. Then the price ct at time t of a European call option is defined by ct = At · N (d1 ) − K · e−r·(H −t) · N (d2 )

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Positive Alpha Generation: Designing Sound Investment Processes

where At is the underlying asset price at time t, K the strike or exercise price, r the continuous compounded interest rate, H the maturity and N (·) = cumulative normal distribution function

σ2 At + r+ · (H − t) ln K 2 d1 = √ σ · H −t √ d2 = d1 − σ · H − t The price pt of a European put option at time t is defined by pt = K · e−r·(H −t) · N (−d2 ) − At · N (−d1 ) In contrast with other pricing models, option prices depend on an unobservable parameter, the volatility of the underlying asset. This volatility must be estimated, which makes the pricing of the option depend on the used estimation algorithm. 14.1.3 Swaps A swap is a contract whereby two parties agree to exchange periodic cash flows or valued items based on predefined amounts as well as market data. The dollar amount is based on some predefined dollar principal called the notional amount of the swap. The most common swap derivative is the interest rate swap. In the context of an investment process, total return swaps are often very useful. In a total return swap one party pays the total return of an agreed upon index, for example the Lehman global aggregate bond index, at a fixed frequency and the other party pays the risk free interest rate, plus eventually a spread. Total return swaps allow the return of any asset or index to be replicated efficiently without needing to trade in the underlying asset. They may also, for example, be used to gain exposure to markets with sophisticated regulations, where the investment manager is unable to invest directly. Valuing swaps Each swap can be subdivided into two instruments, where the investor is long the first instrument and short the second. In the case of an interest rate swap, the two instruments are a fixed rate bond and a floating rate note. The price of a swap is determined in such a way that at the beginning of the transaction, the net position of the two instruments is equal to zero. Consider, for example, an interest rate swap. The floating rate note instrument has initially and at any interest rate reset date a value of 100. The coupon payment of the fixed rate instrument is then determined such that its value also equals 100. 14.1.4 Derivative instruments versus underlying assets A derivative instrument provides the same market or price exposure characteristics as the underlying asset. Other characteristics, like tax issues or liquidity constraints, are not

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217

replicated by the derivative instrument. More generally, derivative instruments differ from their underlying assets through the following features: • Derivative instruments do not require the underlying asset to be traded effectively in the same volume. Derivatives can thus provide higher liquidity. • Some derivatives allow the underlying assets to be selected from a predefined collection of assets at maturity. This option provides greater liquidity as the underlying cash asset can be the most liquid asset in the collection. Derivatives in this category can especially be found in the fixed income world, like, for example, bond futures or credit default swaps. • Derivative instruments may be treated differently from their associated underlying assets from an income as well as capital gain/loss tax perspective. • The physical handling of the underlying asset may be cost inefficient. • Derivatives can provide investors access to exposure that cannot be easily achieved through physical assets. On the other hand, derivatives include additional costs that are very often not accounted for: • Derivatives assume that both parties can borrow and lend cash at the risk free rate, that is the LIBOR rate. Investors generally do not have access to such facilities. • The cost of providing collateral though either haircuts or margin accounts is not included in the derivative’s pricing and thus the realized total return. • More sophisticated portfolio management system capabilities are required than for managing portfolios solely composed of cash instruments. 14.1.5 Handling collateral Counterparties of derivative instruments, whether exchanges or investors, require collateral to be deposited in order to mitigate the counterparty default risk. Exchange traded derivatives require the buyer (seller) to have a cash margin account. At the end of each trading day, the derivative is valued or marked-to-market and the profit or loss is credited to or debited from the margin account. If the cash amount on the margin account drops below a minimum margin requirement, the investor gets a margin call and has to deposit additional cash into the margin account. The minimal required amount is determined by the exchange such that the risk of loss in the case of a counterparty default is minimal. If the investor does not deposit the required cash in the margin account, his or her derivative contract is closed out. Table 14.2 lists some sample margins required from instruments traded on the Eurex futures exchange as of 31 January 2008. Over the counter derivatives also require collateral deposits either in the form of credit lines or other assets, like treasury bills. The amount of collateral required is called the haircut and is similar to the margin for exchange traded derivatives. Its size depends on the asset used as collateral, the volatility of the derivative’s underlying asset, as well as the time to maturity of the derivative. Haircuts are, in contrast to margins, significant in size as they have to be able to cover the profit and loss of the derivative over its whole existence, rather than just one day. Usually there is flexibility as to which securities can be used as collateral.

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Positive Alpha Generation: Designing Sound Investment Processes

Table 14.2 Margin requirements for some derivative instruments traded on the Eurex futures exchange as of 31 January 2008 Number of contracts 50 (long)

−35 (short) 105 (long)

10 (long) −40 (short)

Contract description

Maturity

Index point value

Price

Margin

Future on Dow Jones Euro STOXX Energy Future on 3 m EURIBOR Future on Swiss government bond 8–13 years, 6 % Future on VSTOXX volatility index Future on SMI stock index

March 08

¤50

397.2

100 000

June 08

¤2 500

96.065

16 686

June 08

Sfr. 1 000

119.240

88 579

May 08

¤1 000

26.60

122 000

March 08

Sfr. 10

7 640

159 636

Source: data sourced from Bloomberg Finance LP, Eurex margin calculator, author’s calculations.

From an investor’s perspective it is important to note that the assets used as collateral cannot be invested elsewhere, and thus have an impact on the overall performance of the portfolio. 14.1.6 Legal aspects In 1992 the International Swaps and Derivatives Association (ISDA) created a standardized contract for derivative transactions. This contract, also known under the name of the ISDA master agreement, defines the framework that two parties have to agree to before entering into derivative transactions with each other. It was revised in 2002. The ISDA master agreement is composed of two parts: • the pre-printed part, which cannot be amended, and • the schedule, which describes the specific agreement between the two parties, like, for example, the termination currency1 or modified sections of the master agreement. The master agreement does not by itself include details on any specific transaction. The details of each derivative transaction are documented in a separate document, called the confirmation agreement. Confirmation agreements are usually quite short. They specify the specific terms of a given transaction and refer to the master agreement for any generalities, like the calculation of fixed and floating amounts payable under an interest rate swap. An important characteristic of the ISDA agreement is that the master agreement and all confirmations form a single agreement from a legal point of view. This allows the parties to aggregate and net out any amounts owed by each other. The counterparty risk is therefore reduced to the net amount of all open transactions. This netting out provision is very 1 The termination currency is the currency in which all open derivative transactions are settled if an ISDA master agreement is terminated.

Using Derivatives Notional amount Trade date Effective date Termination date Fixed amounts Fixed rate payer Fixed rate payer payment date

Fixed rate Fixed rate day count fraction Fixed rate period end dates Floating amounts Floating rate payer Floating rate payer payment date

Floating rate option Floating rate designated maturity Initial floating rate setting Floating rate spread Floating rate reset date Floating rate day count fraction Floating rate period end dates Business days Calculation agent Governing law

219 EUR 150 000 000.00 30 July 2007 3 August 2007 1 July 2017 Bank Primary Semi-annually, on each 1 January and 1 July, commencing on 1 January 2008 and ending on the Termination Date, subject to adjustment in accordance with the ‘Following Business Day Convention’ 3.75% ‘30/360’ Not adjusted Best Asset Management, Inc. Semi-annually, on each 1 January and 1 July, commencing on 1 January 2008 and ending on the Termination Date, subject to adjustment in accordance with the ‘Following Business Day Convention’ ‘EUR LIBOR BBA’ 6 months 2.34% + 0.86% First day of each ‘Calculation Period’ ‘Actual/360’ Adjusted according to the ‘Following Business Day Convention’ London and Frankfurt Bank Primary Frankfurt, Germany

FIGURE 14.1 Sample excerpt from a confirmation agreement describing the terms of a transaction

important for financial institutions that have a large number of open derivative transactions since it allows them to allocate capital only against the net position rather than the gross amount. Figure 14.1 shows a sample excerpt from a confirmation agreement describing the terms of a interest rate swap transaction between Bank Primary and Best Asset Management.

14.2 USING DERIVATIVES TO IMPLEMENT AN INVESTMENT STRATEGY Derivative instruments seem to be very efficient and cost effective tools to implement active investment strategies. However, Fong et al. (2005) have shown that the use of derivatives only has a negligible impact on the analyzed portfolio return and risk characteristics. In addition, they found that options are mostly used to implement momentum strategies rather than engage in alpha generating transactions. Nevertheless, derivatives can have a significant

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Positive Alpha Generation: Designing Sound Investment Processes

positive impact on the portfolio construction process as they allow a seamless and efficient implementation of short term investment views if handled properly. There are four key areas2 used for derivative instruments rather than cash assets to generate positive alpha in an investment process: • Derivative instruments can be used to increase or decrease the economic exposure to specific assets or risk factors on a short term basis. The change in exposure is based on forecasting skills and is expected to generate positive alpha. Usually futures or forwards are used to achieve these changes in market exposure. • Derivative instruments can be used to construct nonlinear payoffs. For example, an investment manager may expect a given stock to generate a positive risk adjusted excess return over the risk free rate over the next three months, but forecasts a 10 % probability that the company will go out of business. In this case an option based investment strategy will allow the investor to participate in the upward stock movement while being protected against bankruptcy. • A third reason for using derivatives is to transform one payoff into another. An investment manager may, for example, have to match a liability cash flow stream of 5 % per year. In this case, the investment manager may transform the return of the portfolio into a fixed 5 % cash flow stream using swap based instruments. • Finally, derivatives can be used as cheap alternatives to cash assets. For example, entering into a Bund future contract is in general cheaper than buying an actual 10-year German government bond. An additional advantage of using derivative instruments as a mechanism to transfer market forecasts into portfolios is the explicit maturity associated with them. One of the key issues seen in the implementation of many investment processes is the handling of forecasting time horizons. Investment managers tend to keep positions resulting from incorrect market forecasts in the portfolio, fearing to realize losses. Many portfolio managers differentiate, at least implicitly, between realized and unrealized losses.3 The fact that derivative instruments have an explicit maturity, which should ideally be selected congruent with the time horizon of the forecast, avoids this behavior and requires that profits and losses resulting from market forecasts are realized. 14.2.1 Gaining market or factor exposure Let me consider an investment manager whose skills are forecasting regional absolute equity market returns over a one-week time horizon. The most efficient way to transfer these skills into a portfolio is to use equity index futures rather than trade underlying stocks or funds. The investment manager invests cash proportional to the target exposure in a one-week term deposit, as well as in the margin account required by the futures exchange. He or she then enters into long future contracts in the equity markets where there is a positive return 2 Many more reasons for using derivatives exist, but these are not directly linked to alpha generation. Such reasons may be tax treatments, capital requirement management, etc. 3 Differentiating between unrealized and realized profits or losses is useful or even necessary in the context of marking portfolios to their book value, as insurance companies do, or when there is a difference due to tax reasons. These cases are outside the scope of this book and the described mechanisms for designing sound investment processes.

Using Derivatives

221

expectation using highly liquid futures on major market indices, like S&P 500 for the US or FTSE 100 for the United Kingdom. As the futures generate profit and loss, those are credited to the margin account. In the event of a margin call, cash is withdrawn from the term deposit. If the interest rate on the margin account or the term deposit is less than the risk free rate, that is the LIBOR rate,4 the total return of the investment will be lower than a similar investment in cash assets, even before transaction costs. If, instead of entering into a futures contract, the investor uses an OTC forward transaction, cash proportional to the underlying exposure is invested in a risk free asset with a maturity congruent to that of the forward contract. This investment is usually accepted by the counterparty as sufficient collateral. Using swap derivatives is a third alternative to gain exposure to an equity market. In a swap transaction, the investor pays on a regular basis the risk free rate to the counterparty and receives in return the total return of the underlying asset. Swaps, in contrast to futures or forwards, usually have a long time to maturity. It is not necessary to re-enter into the same type of transaction over and over again. In most swap transactions, the underlying asset is not exchanged and thus the credit or counterparty risk is limited to the exchange of intermediate cash flows. Table 14.3 shows the advantages and drawbacks of futures, forwards and swaps to gain exposure to an asset or risk factor. Depending on the forecasting capabilities, their time horizon, the underlying asset and the current physical portfolio holding, different derivative instruments are most appropriate. A fourth way to gain exposure to equity markets is to buy call options on indices representing the selected equity markets and invest the remainder in a risk free asset. Table 14.3 Comparison of different symmetric derivatives to gain exposure to an underlying asset

Market coverage Availability Collateral requirements Counterparty risk Transaction costs Liquidity a Depends

Cash asset

Future

Forward

Swap

Any market

Major indices

Most marketsa

Most marketsa

Exchange Not applicable

Exchange Margin account

None

OTC Yes, for net exposure Medium, only net exposure at risk

High

None as the exchange is the counterparty Low

OTC Yes, for underlying assets High

Mediuma

Mediuma

Low to high

Low to high

Medium

Low

on the difficulty for the counterparty to hedge the resulting risk.

14.2.2 Reducing exposure or hedging The derivative instruments used to gain asset exposure can also be deployed to reduce or hedge asset or risk factor exposure. This can be done by taking short rather than long 4

The applicable LIBOR rate depends on the time to maturity of the futures contract entered into.

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positions in the corresponding derivative instruments. For example, an investor holds a portfolio of stocks replicating the S&P/ASX 505 index. Entering in S&P/ASX 50 short futures positions for the value of the held portfolio eliminates all risk exposure to the Australian stock market. The total return of the portfolio is then equal to R (portfolio) = R (S&P/ASX 50 index) − R (future on S&P/ASX 50 index) = R (cash). Such a construction is also called synthetic cash replication. Assume that the portfolio has a value of VP , the future a contract size of CF per index point and a price or index level of PF . Then, in order to replicate the market exposure of the portfolio VP number of contracts = C F · PF long contracts have to be entered into and VP held in cash. Although this implementation looks very similar to the associated long position, it bears a certain number of pitfalls that need to be handled correctly. Indeed, as with a long future position, the investor needs to have a margin account with a sufficient cash amount deposited to cover daily losses when the future increases in value. As, by construction of the portfolio, the investor does not hold any cash, he or she needs to sell stock, bearing the associated costs, to finance the margin account. If the market movements are small, this cash management issue is easy to handle. However, if the movements in the market are significant in the adverse direction, cash has to be generated to cover the margin calls on the margin account. The losses faced by the investor may be significant due to the fact that the costs of selling stocks may have to be borne to cover the losses from a short future position. Using forward transactions to hedge positions rather than futures allows this cash flow mismatch risk to be mitigated at the expense of potentially higher collateral requirement and initial transaction costs. Swap derivatives offer an intermediate solution with respect to matching cash flows and posting collateral. It is critical to manage the liquidity constraints from using derivatives efficiently in order to avoid large losses. Many anecdotal cases exist where the liquidity mismanagement has led to large losses or even bankruptcy.6 A typical example illustrating the liquidity management issue is the Metallgesellschaft Refining and Marketing (MGRM) case. In 1992, MGRM implemented the following investment strategy. First, it agreed to sell specified amounts of petroleum products on a monthly basis for up to 10 years, at fixed prices. In order to hedge the risk from fluctuating petroleum prices on its long term commitments, it shorted short term energy futures contracts. However, this hedging strategy failed to take into account that when oil prices dropped, the gains from the sale of oil are realized in the long term, but the losses from the short energy futures positions are realized immediately through marked-to-marking. When petroleum prices started to drop, MGRM faced a cash flow crisis. In December 1993, 5 Australian Securities Exchange (ASX) stock market index constructed and calculated by Standard & Poor’s. It is a market capitalization weighted index including the 50 largest publicly traded companies listed and traded on the ASX. 6 See, for example, http://www.erisk.com/Learning/CaseStudies.asp.

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223

the company cashed in positions at a loss that totaled more than $1 billion. Ultimately, federal officials, fearing investor panic, stepped in and shut the company down. KPMG, the liquidators, estimated losses at $1.3 billion. Minimum variance hedging If the asset to hedge or gain exposure is not identical to the asset underlying the considered future, the proportion of future contracts to be bought or sold needs to be adjusted. The most appropriate approach to adjust this proportion, called the hedge ratio, is to minimize the risk of the performance of the market being different from that of a cash plus future contract replication portfolio. This risk can be defined as shown in the following equation: σP2 = σA2 + HR2 · σF2 − 2 · HR · σF · σA · ρA,F

(14.1)

where σP is the volatility of the portfolio, σA the volatility of the asset, σF the volatility of the asset underlying the future, ρA,F the correlation between the asset and the future’s underlying asset and HR the required hedge ratio. Minimizing leads to the minimum variance hedge ratio HR: HR = ρA,F ·

σA σF

(14.2)

In practice, hedging a portfolio using the minimum variance hedging approach can be subdivided into three steps: (1) Estimate the variance of the portfolio, the asset underlying the future contract used, as well as the correlation between them using historical return data. (2) Calculate the minimum variance hedging ratio HR using Equation (14.2). (3) Compute the number of long or short futures contracts required, taking into account the value of a single future contract VF as well as the value or proportion VP of the portfolio to be hedged using number of contracts =

VP · HR VF

(14.3)

Consider an investor who has a physical position of 10 million CHF indexed to the SPI index. As liquid futures only exist on the narrower SMI index, the investment manager chooses a minimum variance approach to hedging his or her position. First, using six-month daily data, the volatility and correlation of the two indices are estimated, that is σSMI = 13.96 %, σSPI = 13.56 % and ρSMI,SPI = 0.9969. This leads to a hedge ratio of HR = 0.96. As the notional value of one SMI future contract is about 85 240 CHF by end 2007, the investment manager enters into 11 short positions to approximately hedge the 10 million exposure to the SPI index. 14.2.3 Payout replication In addition to adjusting exposure either upwards or downwards, derivatives, especially options, can be used to replicate nonlinear payouts. Consider an investor who forecasts that

224

Positive Alpha Generation: Designing Sound Investment Processes Table 14.4 Sample option based portfolio transferring a low stock market volatility forecast into a portfolio Position 1 −2 1

Instrument

Strike

Maturity

Price

Long Nikkei call option Short Nikkei call option Long Nikkei call option

15 500 16 500 17 500

Mar. 08 Mar. 08 Mar. 08

947.14 429.62 158.30

Source: data sourced from Bloomberg Finance LP.

Japanese stock market volatility will remain low over the next three months. To transfer this forecast into the portfolio he or she enters into the options trades shown in Table 14.4. The cost of this trade is JPY 246.2 per option portfolio. This constructed portfolio will have a positive value at maturity if the Nikkei 225 index closes between 15 500 and 17 500 on 13 March 2008, the maturity date of the options. The payout is maximal when the Nikkei 225 index equals 16 500. Another use of options to generate positive alpha is to sell call options slightly out of the money, if the market forecast is that the stock price will not exceed the sold option’s strike price. In that case, the option premium is cashed in. To avoid liquidity risk, the investor should hold the underlying stock on when the option is sold. Generally, the counterparty buying the option will require stocks as collateral.

14.3 EXAMPLE 14.3.1 Gaining exposure using futures Consider an EUR denominated total return oriented investment solution to be sold as a UCITS7 investment fund. The investment solution aims at generating positive alpha through taking long and short positions in the major developed equity markets. Figure 14.2 represents the investment manager’s current investment strategy. This strategy cannot be implemented or constructed in a mutual fund without the use of derivatives. Indeed, the regulator requires that all short positions are entered only through derivative instruments and not though lending of cash assets and selling those. In addition, the regulator requires that the total gross exposure of all derivatives is no larger than 100 % and the net exposure is not negative. In this example the net exposure equals 0.34 % and the gross exposure 27.87 %. Developing a portfolio construction process requires (i) selecting derivative instruments to gain exposure to the different equity markets and (ii) setting up a process to handle cash flows from profits and losses in the different equity markets. First, I decide to invest the assets in the fund in short term money market instruments with a maturity up to one month, keeping around 20 % of the assets on overnight deposits with different financial institutions. These 20 % will allow me to cover the daily profits and losses from the derivatives used to implement the investment strategy. Next, I decide to use exchange traded futures on the main indices in each market to represent the different markets. As exchange traded futures are marked-to-market, profit and losses become translated into physical cash on a daily 7 Undertaking for collective investments in transferable securities, the European Union regulatory framework for mutual funds.

Using Derivatives

225

8.0% 5.63 %

4.0%

2.46 %

2.0%

1.70 %

0.99 %

1.24 %

0.85 %

1.23 %

0.0% −2.0%

−1.46 %

−4.0%

−2.26 %

Australia

Hong Kong

Japan

Switzerland

The Netherlands

Italy

Spain

Germany

France

Canda

−4.53% United States

−6.0%

−2.21 %

−3.32 % United Kingdom

Market exposure

6.0%

Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 14.2 Sample long–short investment strategy

basis. This is advantageous as it will not require any sophisticated maturity matching. As long as the long–short investment strategy does not change, either from a forecast or risk side, I do not need to execute any derivative trades. I only have to ensure that my margin accounts have sufficient cash. Table 14.5 and Table 14.6 show the portfolio I am holding to implement the investment strategy illustrated in Figure 14.2. At the end of each work day, the profits and losses for all long as well as short futures positions are calculated and netted with the margin account. Once this is done, I adjust the cash amount in the margin account such that margin requirements are covered by moving Table 14.5 Derivative positions to implement the long–short investment strategy illustrated in Figure 14.2 Number of contracts −8 −6 −17 8 −5 5 2 −10 11 12 21 6

Contract name

S&P 500 future/Mar. 08 S&P/TSE composite future/Mar. 08 FTSE 100 future/Mar. 08 CAC 40 future/Mar. 08 DAX future/Mar. 08 IBEX 35 future/Mar. 08 MIB future/Mar. 08 ASX future/Mar. 08 SMI future/Mar. 08 Nikkei 225 future/Mar. 08 Hang Seng future/Mar. 08 S&P/ASX 200 future/Mar. 08

Trade price

1 466.3 790.5 6 320.0 5 555.5 7 948.0 15 290.0 38 443.0 507.7 8 402.0 15 060.0 27 087.0 6 200.0

Source: data sourced from Bloomberg Finance LP, author’s calculations.

Exposure in EUR

¤−2 038 013 ¤−656 870 ¤−1 492 079 ¤444 440 ¤−993 500 ¤764 500 ¤384 430 ¤−1 015 400 ¤556 190 ¤1 108 047 ¤2 533 911 ¤555 434

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Positive Alpha Generation: Designing Sound Investment Processes Table 14.6 Margin requirements and cash positions to implement the long–short investment strategy illustrated in Figure 14.2 Notional

Security

Price

Value in EUR

5 165 700 2 500 000 2 500 000 4 000 000 1 000 000 5 000 000 5 000 000 20 000 000

Margin cash account with A Overnight deposit with A Overnight deposit with B Overnight deposit with C Term deposit – 3 months Term deposit – 3 months Term deposit – 6 months West LB – FRN (EURIBOR + 5bp)

1.00 1.00 1.00 1.00 99.08 99.09 98.18 99.90

5 165 700 2 500 000 2 500 000 4 000 000 990 800 4 954 500 4 909 000 19 980 000

Source: data sourced from Bloomberg Finance LP, author’s calculations.

Futures trading

Cash management

repeat next day

Adjust margin account

check if exposures are still in line yes no

check if cash is efficiently invested no Reinvest cash efficiently yes

Trade futures

FIGURE 14.3 Portfolio construction part of an investment process using derivatives

cash from or to the different overnight deposits concluded. Then, I check if my market exposure is still in line with my investment strategy based on (i) absolute portfolio values and (ii) relative market movements. If this is no longer the case, I buy or sell futures contracts to bring the portfolio exposure again in line with the investment strategy. Finally, I check if some of the cash holdings may need investment or re-investment using higher yielding instruments than overnight deposits. Figure 14.3 illustrates the portfolio construction investment process part for handling derivatives. 14.3.2 Managing beta exposure in an equity portfolio Consider an investment manager whose skill set is to time equity markets, that is actively manage the beta exposure of a portfolio in a CAPM based framework. To illustrate how this investment manager could transfer his or her skill into portfolios using derivatives, I consider the EMU equities market, modeled by the Dow Jones Euro STOXX index8 as the CAPM market portfolio. The investment manager holds an indexed portfolio of actual securities tracking the Dow Jones Euro STOXX index. To implement the market timing strategy, the investment manager buys or sells futures contracts in order to adjust the portfolio beta 8

The index covers approximately 95 % of the free float equity market capitalization of the Eurozone.

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according to the forecasts. Unfortunately, there is not a future on the Dow Jones Euro STOXX index, but only on its smaller brother the Dow Jones Euro STOXX 50 index. I chose a minimum variance hedging strategy to apply beta adjustments to the underlying portfolio. I first estimate the variance of the underlying portfolio and the used future, as well as the correlation between both using 10-year monthly historical price return data over the time period December 1997 to December 2007. I use the standard equally weighted sample estimator, leading to an estimated portfolio variance of σP = 18.81 %, a future variance of σF = 19.45 % and a correlation of ρP ,F = 0.9909 between the portfolio and the used future. Then I calculate the minimum variance hedging ratio as HR =

18.81 % σP · 0.9909 = 0.9586 · ρP ,F = σF 19.45 %

Finally, in order to reduce the beta exposure of the portfolio by 10 %, assuming a portfolio value of VP = 17 450 000, a future contract value of VF = 43 070.00 and the contract size being EUR 10.00 per index point sell 17 450 000 VP · 0.9586 · 10 % ≈ 9 293 · HR · 10 % = VF 43 070.00 Dow Jones Euro STOXX 50 future contracts. As a practical matter, the risk minimizing hedge ratio can also be calculated using the regression shown in the following equation: RDow Jones Euro STOXX = α + HR · RDow Jones Euro STOXX50 + ε

(14.4)

where Rindex is the return of the indicated index, giving the same result. A similar approach can be used to adjust the duration of a fixed income portfolio. If more than one risk factor need to be adjusted, like hedging both the duration as well as the convexity of a fixed income portfolio, a multifactor regression approach similar to the regression shown in Equation (14.4) can be used. 14.3.3 Hedging currency risk using forwards Consider a Swedish fixed income investment manager managing a single currency fixed income portfolio denominated in SEK. Instead of holding SEK denominated bonds, the investment manager decides to hold EUR denominated bonds and hedge out the EUR/SEK currency risk. Two approaches to eliminate the currency risk from such a fixed income portfolio exist: • The investment manager can enter into currency swap transactions, swapping the euro cash flows from each individual security back into Swedish krona. • Instead of entering into numerous swap contracts, the investment manager may sell forward the foreign currency using forward contracts. Due to the complexity of setting up the ISDA contractual agreements, the fact that a swap transaction would be required for each bond, as well as the long term nature of over-the-counter swap transactions (costs, difficulty to unwind, etc.), the currency swap approach is rarely used.

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To hedge the fixed income portfolio, the Swedish investment manager uses short term currency forward contracts and agrees to exchange cash in euro against cash in Swedish krona at a specified point in the future at a fixed exchange rate based on Equation (14.5). The investment manager must thus decide: • how much euro cash to exchange at maturity, that is the notional of the forward contract, and • when to perform the cash exchange, that is determine the forward contract’s maturity, also called its rolling horizon. Ideally, the notional amount of the forward contract should be equal to the future value of the bonds in the portfolio at the future’s maturity date. This future value can be calculated either: • using the current portfolio holdings and market implied future interest rates or • using the current portfolio holdings and interest changes forecasted by the investment manager. In fixed income portfolios, the first approach is commonly used. This is reasonable if the forecasting horizon is significantly longer than the maturity horizon of the used forward transactions. Using the no-arbitrage approach, and the fact that a forward contract can be modeled by a long and a short position in a zero coupon bond with maturity equal to the maturity of the forward contract, it is possible to calculate the forward exchange rate of the contract as    H  1 + RD ·  360  ≈ S · 1 + (R − R ) · H  (14.5) F =S·   D  X 360  H  1 + RX · cost of carry 360 where F is the forward exchange rate, S the current sport exchange rate, RD the risk free rate of the domestic or bought currency (in my example, the SEK), RX the risk free rate of the foreign or sold currency (in my example, the EUR) and H the maturity of the contract in days. The risk free rates are determined for the horizon of the forward contract. Assuming that S = 9.0307, RD = RSEK = 3.2675 %, RX = REUR = 3.7748 % and H = 90, the applicable forward exchange rate would be F = 9.0194. As the goal is to hedge the currency risk over the whole existence of the fixed income portfolio and not only the time of a single forward contract, the investment manager has to re-enter into a new currency forward contract every T days. This process is called rolling a currency hedge. It implies realizing the profit and loss of the forward contract, which is netted with the profit and loss of the underlying fixed income portfolio expressed in the domestic currency, that is in SEK. The cash flows have to be managed carefully and eventually portions of the fixed income portfolio sold or bought. The maturity of the forward contracts is determined as a trade-off between: • transaction costs, suggesting long maturity forward contracts, that is infrequent rolls, and • the difference between the realigned and the predicted portfolio return as well as the interest rate risk inherent in any forward contract, suggesting short maturity forward contracts, that is frequent rolls.

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A time horizon of one to three months is considered a reasonable trade-off for investment grade fixed income portfolios. For more volatile portfolios, like high yield bond portfolios or even equity portfolios, currency forward contract maturities of one month or less are more adequate. If the goal of the investment manager is to outperform a hedged benchmark rather than fully eliminate currency risk, the hedging approach chosen with respect to hedging amount and forward contract maturity should be matched to that of the used benchmark calculation algorithm.

Part VI Investment Products and Solutions

15 Benchmark Oriented Solutions The most common investment solutions, at least with respect to institutional investors, are so-called benchmark oriented solutions. The investment manager is given by the investor a benchmark or index and is then assigned the task to either replicate (passive solutions) or outperform (active solutions) the performance of the benchmark. The definition of the benchmark and its performance is the responsibility of the investor. Usually, the benchmark is constructed in such a way that it exhibits a positive risk premium and maximizes the investor’s utility function with respect to return and risk trade-off. Sometimes it matches expected future liability cash flow streams. The investment manager of an active benchmark oriented solution is supposed to add value, that is alpha, through investment decisions leading to active portfolio positions relative to the benchmark. To do so, he or she is expected to take risk orthogonal to the risk of the benchmark. Benchmark oriented solutions started to flourish in the 1970s when investment managers lost the opportunity to manage the total risk or volatility of the portfolio (Ineichen, 2005). This change in approach was driven by unsatisfactory results as well as the agency problem between the investment manager and the investor. It is still an open debate whether better results are achieved when the total risk decision is in the hand of the investor (through the selection of a benchmark) or the investment manager (through the management of volatility or the use of a strategic asset allocation).

15.1 BENCHMARKS There are three key reasons from an investor’s perspective to prefer a benchmark oriented solution and define a benchmark: • First, and foremost, the benchmark can and should model the liability stream that needs to be matched by the investment portfolio’s return stream. For example, an insurance policy linked to the returns of an investment portfolio (a so-called unit linked policy) guarantees to pay a minimum return of 1.5 % per annum. The benchmark needs to be constructed to produce an annual total return of 1.5 % with minimal risk.1 • The second reason for opting for a benchmark oriented solution arises from an investor basing his or her investment strategy on the capital asset pricing model (CAPM) (Sharpe, 1964). The CAPM states that the utility maximizing portfolio for any investor is composed of a combination of a risk free asset, for example treasury bills, and a risky portfolio that represents the market portfolio. The proportions in both assets define the benchmark and are determined so as to match the investor’s preference for risk and return. • The third reason for relying on a benchmark oriented solution is to solve the agency problem, that is separate the responsibilities for the different components of the portfolio’s return.

1 Another approach to match the liability stream is to implement a capital protection investment process, as described in Chapter 17, or to use asset and liability management techniques, as presented in Chapter 19.

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15.1.1 Defining a benchmark A benchmark is defined as a universe of assets combined with a weighting of each of these assets. Three key decisions need to be taken in order to define a benchmark. The first decision that must be taken when constructing a benchmark is to define its expected return and risk characteristics. The risk characteristics can be specified explicitly, for example as a maximal drawdown or as a value at risk. However, they can also be defined implicitly through the selected benchmark weights. The return expectations are usually specified implicitly by assuming that the benchmark is constructed as an efficient portfolio with respect to its risk and return characteristics. The benchmark’s return expectations can also be defined through explicit forecasting return expectations for the individual assets composing the benchmark. Second, the investment universe, that is the opportunity set, must be defined. The benchmark universe is defined as a subset of the investment universe, ideally equal to it. It should be selected such that the individual assets can actually be traded, that is be investable. Each security in the benchmark universe is associated with a weight, the so-called benchmark weight, ideally chosen so as to remove specific or diversifiable risk from the assets in the benchmark. A reasonable benchmark weighting scheme for the 500 largest stocks in the United States would be to use their market capitalization as weight. The benchmark so constructed equals the S&P 500 index. The benchmark defines the utility function of the investor at a particular point in time. As the risk tolerance, expressed, for example, through the funding ratio or surplus at risk, as well as the expectations in asset returns, changes over time, a benchmark must be monitored and, if necessary, reviewed and adjusted. Some benchmarks include automatic adjustment rules. For example, a balanced benchmark composed of 50 % Citigroup world government bond index and 50 % FTSE world equities is reset to the 50 %/50 % weight at the end of each month. I exclude definitions of the form Index + α bp, like LIBOR + 300 bp, as benchmark definitions as it is not possible to invest in, that is replicate, these benchmarks. Indeed, such specifications can be subdivided into a benchmark, the index and an outperformance or alpha target α. If the benchmark is specified with respect to a number of characteristics, like, for example, the key rate duration exposure, I assume that there is a portfolio of traded securities that has exactly the same characteristics as the specified benchmark. 15.1.2 The case for an investment policy The investment policy determines the terms and conditions along which the investment manager is mandated to manage the investor’s portfolio (Leibowitz, 2004). It is made up of a benchmark called the investment policy portfolio and a set of restrictions. It represents: • the investor’s liability structure or required rate of return, • the risk tolerance and risk preference, and • the expected asset returns, either explicitly or implicitly. The investment policy is determined by the investor, that is the beneficiary or the agent or consultant. One of the reasons for having an investment policy is the separation of responsibility: • The investor is responsible for determining the investment policy.

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• The investment manager is responsible for implementing the investment policy and eventually, generating alpha on top of the investment policy’s return. The restrictions formulated as part of the investment policy represent the investor’s combination of objective and subjective expression of risk tolerance: (i) with respect to overall market movements and especially (ii) with respect to the skills and alpha capabilities of the investment manager. Restrictions refrain the investment manager from taking up risk in areas unwanted by the investor. Indeed, the risk taken in a portfolio can be subdivided as shown in Figure 15.1. Each investment decision, from the benchmark selection, through the tactical asset allocation, to the security selection, entails a certain amount of risk and an expected return in the form of alpha. The different investment decisions should be taken such that they are independent from each other; that is their risks should be orthogonal. This provides the best diversification and maximizes the expected alpha per unit of risk according to the fundamental law of active management. In Figure 15.1, the risk due to the tactical asset allocation component of the investment solution only slightly increases the total portfolio risk, whereas the security selection component actually decreases the total risk. The total portfolio risk can be decomposed into a component related to the benchmark or investment policy risk, called the total risk or volatility, and an ideally orthogonal component, the active risk or tracking error, expected to be rewarded by alpha. It is important to note that the investment policy is a decision taken at a particular point in time. As time moves on: • the liability structure changes, • the risk preference, expressed, for example, as the funding ratio or surplus-at-risk, adjusts and • the long term market expectations change. Therefore the investment policy must be reviewed and updated on a regular basis. Keep in mind that, as shown in Chapter 2, the selection of the benchmark, that is the investment

active risk

security selection risk total portfolio risk tactical asset allocation risk benchmark/investment policy risk

total risk

FIGURE 15.1 Sample combination of different sources of risk. Active risk components as tactical asset allocation or security selection may or may not decrease total risk while increasing active risk

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Table 15.1 Universe of asset classes in which the pension fund plan is investing, including return and volatility expectations and the resulting investment policy or benchmark portfolio Asset class

Cash GBP Long Gilts UK equities International equities Benchmark portfolio

Expected excess return (% p.a.)

Volatility (% p.a.)

Policy portfolio/ benchmark (%)

0.0 1.4 4.2 4.8 2.5

0.3 4.4 13.6 15.8 5.6

9.5 51.2 17.6 21.7 100.0

Source: data sourced from Bloomberg Finance LP, author’s calculations.

policy portfolio, is also an active investment decision, but a decision taken by the investor rather than the investment manager. Example Consider a pension fund plan that is required to achieve an excess rate of return of 2.5 % over the risk free rate per annum. This target return is determined based on the liability structure. The pension fund currently has a funding ratio of 110 %, that is it is overfunded. For the sake of simplicity, I consider that the pension fund can only invest in the asset classes shown in Table 15.1 above. I also assume that the pension fund plan’s return and volatility expectations are as indicated in the same table. Based on the funding ratio of 110 % and a subjective risk aversion of the pension fund plan, I require the value-at-risk in absolute terms at the 95 % confidence level over a one-month time horizon to be no more than 1.4 % of the fund’s value in excess of the liabilities, that is currently 2.5 %. From these assumptions, I derive the investment policy portfolio shown in Table 15.1 (right column). This portfolio has an expected excess rate of return of 2.5 % and a value-at-risk of 2.19 %, or an annualized volatility of 2.5 %. Ten years of historical data were used to estimate the volatility and correlations between the different asset classes. 15.1.3 Investment universe and indices The investment universe defines the opportunity set on which skills are applied. It is therefore mandatory that the investment universe be made of investable securities. There must be a market for the individual assets. It is common to define the investment universe through a single or a combination of market indices. A market index is: (i) a collection of assets, usually, but not necessarily, with similar characteristics, (ii) a weighting scheme of these assets and (iii) a rule set showing how the assets and weights change over time and how corporate actions, like cash flows, are handled. There are two categories of indices, so-called price indices and total return indices. In a price index, intermediate cash flows, like dividends or coupon payments, are discarded.

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Table 15.2 Sample five-asset equity index Security

BNP Paribas Soci´et´e G´en´erale AXA Cr´edit Agricole Dexia AFG

Shares outstanding

Price at date T

Price at date T + 1

930 467 477 461 028 716 2 090 061 928 1 497 322 301 1 148 871 204 191 428 937

79.9 125.4 28.07 31.36 20.37 112.2

80.3 124.7 26.95 31.47 21.09 114.34

Total return (%) 0.5 −0.56 −0.07 0.35 3.53 1.91

Market weight (%) 26.3 20.5 20.8 16.6 8.3 7.6

Source: data sourced from Bloomberg Finance LP, author’s calculations.

The index value is the weighted sum of the prices of the assets. In a total return index, cash flows are reinvested, either in the security that generated them or in all the securities of the index according to a predefined scheme. This reinvestment may occur at the date where the cash flow is price relevant, the so-called ex-date, or at a predefined date, usually the end of the month or quarter. Table 15.2 illustrates a simple five-asset equity index. Furthermore, total return indices are classified as gross or net indices, depending on whether the cash flows are considered before or after taxes. If the investor cannot receive the cash flows gross of taxes, net total return indices should be used. There are three common weighting schemes for indices: • Equally weighted or price weighted. The index value is the normalized sum of the prices of all securities in the index. • Market capitalization or amount outstanding weighted. The weight is determined by the number of securities outstanding multiplied by its price. • Free floating weighted. The weight is determined by the number of securities that are actually available for buying or selling multiplied by their price. In addition, especially in government fixed income indices, the weights are sometimes determined by the GDP or other macroeconomic variables of the issuing country. In smaller markets, like, for example, the European high yield market, the maximal weight per issue may be capped in order to avoid a concentration on a few large issuers. Whatever weighting scheme is used, it is important that the weighting can actually be replicated in a portfolio of reasonable size. Indices are usually constructed in such a way that they will change over time. Indices are reviewed on a monthly, quarterly or annual basis. It is common that these changes, especially if they are not rule based, are published some days before they come into effect. The constituents or universe of an index are either determined by a rule or a committee. In a rule based approach, a security enters or leaves the index if it satisfies a certain number of quantitative criteria. In a committee based approach, qualitative and judgmental factors are added to the inclusion and exclusion decision process. In a total return index, cash flows are reinvested. Any index, however thoroughly it is constructed, is biased. The most common bias is the survivorship bias. Well performing companies are added to the index while poorly performing ones are removed. In fixed income indices, highly indebted companies or countries are

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most represented in the indices. This leads very often to a bad credit bias in fixed income indices, especially in low investment grade indices.

15.2 PASSIVE BENCHMARK ORIENTED INVESTMENT SOLUTIONS At first it may look strange to have a section on passive benchmark oriented or indexed solutions in a book about designing investment processes to generate positive alpha. Techniques used in passive investment solutions are very useful for actively managed portfolios. For example, efficiently handling cash flows from corporate actions allows the transfer mechanism to be improved, that is implementing the third success factor of an investment process. Even further down the road, many investment strategies can be seen as a benchmark or index to be replicated in a portfolio. Compared with passive solutions, the key difference is that the strategy changes frequently. Finally, actual investor restrictions or other portfolio construction constraints, like size, may not allow the investment strategy, that is the formulated skills on the opportunity set, to be transferred fully into the investor portfolio. Techniques like stratified sampling or synthetic replication originating from the passive investment solutions world can be applied. 15.2.1 A case for passive investing Although I advocate throughout this book the concept of active management, situations exist where an indexed or passive solution is preferable. An investment manager that does not have skills in a given opportunity set or investment universe may resort to indexed solutions to gain exposure to the considered universe. An investment manager who bases an investment process on the capital asset pricing model and an underlying market equilibrium is required to replicate the market portfolio. The added value of the investment manager relates to translating the investor’s utility function into a relative exposure between the risk free asset and the market portfolio. In addition, designing an efficient transfer mechanism to implement the market portfolio is a second added value provided to the investor by the investment manager. Investment universes exist where the efficient market hypothesis applies better than in others. Such a universe is, for example, the opportunity set of large capitalized US stocks. If the investment manager wants to cover the world equity markets as an opportunity set, he or she may resort to an indexed solution for the US market and deploy skills and resources in other, potentially less efficient, markets, like Hong Kong or Singapore. Similarly, the investment manager may believe that his or her forecasting skills are average in a given market segment, that is on a given opportunity set. As Sharpe (1991) has shown, the market return is equal to the weighted average of the return of all market participants whether they are active or passive investors. Thus average skills cannot lead to above average returns, that is positive alpha. Therefore a passive investment strategy is the best possible approach in the absence of superior forecasting skills. An investment manager may have alpha generating skills, but the skills may be insufficient to cover the incurred costs for: (i) generating the skills, that is performing the forecasting and decision making process, and

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(ii) transferring the skills into portfolios, that is transaction costs, including market liquidity and opportunity costs. In such a situation a cost efficient passive implementation may be preferable. Some risk adverse investors prefer indexed products over actively managed investment solutions because they are more transparent and easier to understand. As the investor is ultimately responsible for the selected investment solution, he or she may be willing to give up alpha for the sake of understanding and transparency. 15.2.2 Investment processes for indexing The goal of any indexing or passive investment process (IIP) is to replicate as closely as possible the performance of the underlying index or benchmark. The simplest approach is full replication. A portfolio that exactly replicates the index, with respect to the securities and weights, is held. Any changes in the index, including corporate actions, are fully replicated in the portfolio. A variation of the full replication investment process is only to replicate the largest positions in an index. For example, selecting all securities such that their total weight equals 80 % of the total index weight and fully replicating that subset of securities is in many circumstances an efficient and effective approximation of the full replication investment process. If the index to be replicated satisfies any of the following properties, alternative techniques are more appropriate, that is: – the index contains a large number of securities, – the portfolio size to be invested is small, – the transaction costs, respectively the bid–ask spread, are large in absolute terms or relative to the invested asset size or – the market liquidity is thin. Alternative techniques are defined in a way such that the expected return and risk characteristics of the index are approximated as closely as possible. There are three categories of approximation techniques, that is (i) stratified sampling, (ii) optimized sampling and (iii) synthetic replication. Any such approximation technique calculates a portfolio structure minimizing a given utility function. This utility function is defined with respect to: – limiting the number of securities to be held, – the transaction costs, like bid–ask spread, associated with the securities held and – the liquidity of the securities, that is the average daily turnover. Other variations of this utility function are possible and adequate to replicate specific indices. Stratified sampling In statistics, stratifying is a technique of sampling from a population. It proceeds by grouping all securities of an index into relatively homogeneous and mutually exclusive cells. Next, one or more securities are selected from each cell in order to optimize a given utility

240

Positive Alpha Generation: Designing Sound Investment Processes 1. Define the criteria of the securities used to classify them into individual cells. Such criteria may be the industry sector or the home country. 2. Associate each security of the index to be replicated to one and only one cell according to the selected criteria. 3. Calculate for each cell its weight as the sum of the weights of its constituents. 4. Select from each cell at least one security in order to optimize the given utility function. The selection may be random or systematic. 5. Associate the securities from each cell with the weight of the cell divided by the number of securities selected from that cell. Other weighting schemes for securities selected from a given cell are possible.

FIGURE 15.2 Stratified sampling portfolio construction algorithm

function. Usually this utility function expresses the transaction costs or an upper bound on the number of securities to be held. The stratified sampling algorithm is described in detail in Figure 15.2. The main advantage of the stratified sampling investment process is its simplicity in implementing as well as focusing on the important characteristics of the securities in the index. Unfortunately, the key disadvantage is the subjective selection of the classification criteria, which may introduce a selection bias. Factor model techniques may be used to select the criteria and overcome some of the criteria selection bias issues. A homogeneity criterion may be valid at a given moment in time but may change throughout time. Consider the DAX German equity index shown in Table 15.3. I define the classification criterion as the ICB.2 I then select the security with the largest weight in each cell, except in the financial section. In the financial section, with nearly 25 % of weight in the index, I chose an insurance company as well as a bank to include in the stratified sample. This leads to a stratified portfolio consisting of 10 securities approximating the return of 30 securities in the DAX index. The chosen securities are indicated with their respective weights in the first column of the table. Optimized sampling In contrast to the stratified sampling approach, optimized sampling techniques rely on an underlying risk model. The most common risk model used is the Markowitz mean-variance framework. An optimized sampling portfolio of an index is calculated so as to minimize the active risk or tracking error between the sampled portfolio and the index, taking into account the utility function as a restriction. The optimized sampling algorithm searches for a subset of securities and calculates associated weights such that the tracking error between the optimized portfolio and the index is minimized. Formally, the optimized sampling algorithm is described in Figure 15.3. The utility function is usually defined as upper bounds on the number of trades and/or transaction costs. The risk model can be extended by adding penalties from certain types of transactions. To calculate the optimized sampling portfolio on the DAX index, I use three years of weekly history of return data of the 30 securities in the index as of 31 December 2007. The choice of a three-year history is based on some stocks not existing before 2005 and 2

Industry classification benchmark, a definitive classification scheme defined by Dow Jones and FTSE.

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Table 15.3 German DAX index constituents as of 31 December 2007, including their weights and their ICB industry classification Sector/security Basic materials Linde Bayer 13.64 % BASF Consumer goods Henkel Adidas BMW Continental Volkswagen 16.58 % Daimler Consumer services TUI Metro 2.26 % Lufthansa Health case Merck 1.57 % Fresenius

Weight (%) 13.64 1.45 6.00 6.19 16.58 0.86 1.29 1.72 1.79 2.81 8.11 2.26 0.47 0.74 1.04 1.57 0.72 0.85

Sector/security Financials Deutsche Postbank Hypo Real Estate Commerzbank Deutsche B¨orse M¨unchner R¨uck 9.89 % Deutsche Bank 14.60 % Allianz Industrials MAN ThyssenKrupp Deutsche Post 15.57 % Siemens Technology Inifineon 4.74 % SAP Telecommunications 5.32 % Deutsche Telekom Utilities RWE 15.83 % E.ON

Weight (%) 24.49 0.63 0.92 1.97 3.36 3.63 5.65 8.33 15.57 1.40 1.85 2.46 9.85 4.74 0.75 3.98 5.32 5.32 15.83 5.66 10.17

Source: data sourced from Bloomberg Finance LP, author’s calculations.

1. Define a risk model τ (.). 2. Select a subset P of securities from the index I and calculate the associated weights by solving the following optimization problem: min

τ (P , I )

s.t.

utility function restriction

P

FIGURE 15.3 Optimized sampling algorithm using a risk model

recent history being more appropriate to describe the security risk. Alternatively, I could have applied the expectation maximization missing data replacement algorithm described in Chapter 5 and used a larger history. I define the risk model as the equally weighted covariance matrix of the price returns. I define the utility function as the constraint that the optimal portfolio does not contain more than 10 securities and no short positions or leverage are allowed. Solving the optimization problem shown in Figure 15.3, I obtain the optimized sampling portfolio in Figure 15.4. The largest position is E.ON, with a weight of 16.3 % coinciding with E.ON also having the largest weight in the index, that is 10.2 %. Interestingly, the optimized sampling portfolio contains a holding in Fresenius, although its weight is only 0.9 % in the index. This can be

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Positive Alpha Generation: Designing Sound Investment Processes 12.9 %

Siemens 7.5 %

SAP 3.8 %

Fresenius

16.3 %

E.ON 5.1 %

Deutsche Telekom

4.1 %

Lufthansa

15.6 %

Deutsche Bank 12.1 %

Daimler

11.6 %

BASF

10.9 %

Alliaz 0%

2%

4%

Index weight

6%

8%

10 %

12 %

14 %

16 %

18 %

Optimized sampling weight Source: data sourced from Bloomberg Finance L P, author’s calculations

FIGURE 15.4 Portfolio weights of a 10 security portfolio replicating the DAX index calculated using the optimized sampling technique (only the securities included in the optimized sampling portfolio are shown and compared with their weights in the index)

explained by its low correlation with other stocks in the index, the correlation being less than 0.33 with all other stocks, averaging to a low 0.24 and therefore exhibiting characteristics not covered by any other stock. Synthetic replication The synthetic replication techniques build on the ideas of both the stratified and the optimized sampling approaches. First, a model of the characteristics of the securities is defined. In general a factor model is used. For example, in a single currency government bond index, the factors could be the duration as well as convexity of the index. Then the exposure of the index to these factors is calculated. Finally, securities or derivative contracts are selected so as to match the corresponding factor exposures and optimize a given utility function with respect to costs and holdings. Very often the utility function minimized is the idiosyncratic or residual risk of the portfolio. The instruments bought to match the factor exposure need not be identical to those in the index. This is one of the key differences between synthetic replication and other indexing investment processes. It allows cheaper and more liquid assets to be used rather than be bound to the securities in the index to be replicated. For example, to achieve a given duration exposure, an interest rate swap could be concluded. Consider the one-factor capital asset pricing model described in Chapter 4. Each stock in the DAX index is associated with a beta factor exposure. I calculate the factor exposure, that is the beta of each individual stock, by using ordinary least squares regression. I use the same data set described in the example on optimized sampling, that is three years of weekly historical price returns. Figure 15.5 shows the distribution of the beta values relative to the weights in the index. I choose 10 stocks and associated weights such that the beta factor exposure of the constructed portfolio matches that of the index as closely as possible.

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25 % DAX index Stratified sampling portfolio

Density

20 %

15 %

10 %

5%

0%

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

Beta Source: data sourced from Bloomberg Finance LP, author’s calculations

FIGURE 15.5 Beta density of the DAX index as well as the portfolio synthetically replicating the index based on the capital asset pricing model

Table 15.4 Stratified sampling portfolio replicating the DAX index based on the beta density distribution shown in Figure 15.5 Asset

Adidas Allianz BASF Bayer BMW Commerzbank Continental Daimler Deutsche Bank Deutsche B¨orse Lufthansa Deutsche Post Postbank Dt. Telekom E.ON

Index Beta Portfolio Asset weight (%) weight (%) 1.30 8.37 6.16 6.00 1.74 1.98 1.80 8.10 5.64 3.34 1.06 2.46 0.62 5.24 10.12

0.69 1.09 0.96 0.95 0.88 1.21 1.12 1.10 1.10 0.96 0.87 0.66 0.86 0.63 1.10

6.7 17.0

20.0

16.0 16.0

Fresenius Henkel Hypo Re. Est. Infineon Linde MAN Merck Metro Mnch. R¨uck RWE SAP Siemens ThyssenKrupp TUI VW

Index Beta Portfolio weight (%) weight (%) 0.86 0.86 0.90 0.78 1.45 1.43 0.72 0.74 3.64 5.67 3.95 9.95 1.86 0.48 2.76

0.45 0.76 1.13 1.02 0.87 1.35 0.63 0.66 0.98 0.90 0.75 1.29 1.47 0.84 0.87

3.3

4.0

7.0 7.0 3.0

Source: data sourced from Bloomberg Finance LP, author’s calculations.

The resulting portfolio, shown in Table 15.4, differs from the one calculated using the optimized sampling technique. Rather than invest in Daimler, Deutsch Bank and E.ON, who all have a beta exposure of around 1.1, the synthetic replication portfolio only invests in the E.ON security. In addition, high beta stocks like ThyssenKrupp and Man are included in the portfolio to cover the upper spectrum of the index’s beta distribution.

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Table 15.5 Comparison of different investment processes for replicating the performance of an index Full replication

Largest holdings

Stratified sampling

Optimized sampling

Synthetic replication

Replication precision

Best

Depends on homogeneity of selected criteria

Good under normal market conditions

Depends on significance of factors in determining security returns

Calculation complexity Implementation complexity

None

High if largest holdings represent large proportion of index Low

Very low

High

Very high

Easy

Easy

Easy

Easy to implement

Easy to implement

Model driven, risk based approach

Performance may deviate significantly from index if coverage is low

Identification of homogeneous selection criteria may be nontrivial

Calculation complexity, poor performance in extreme market situations

Complex, as securities with corresponding factor exposure must be selected Model driven, does not require securities in index to be bought Selection of adequate factors, calculation of factor exposure and selection of securities matching factors

Key advantages

Key drawbacks

Easy, number of trades equal to number of securities in index Easy to implement for small indices Inappropriate for large indices

The quality of the resulting portfolio depends on the adequateness of the selected factors to model the risks inherent in the index and the return components driving the performance of the index. Comparison of indexing investment processes Table 15.5 compares the five indexing investment processes presented with respect to their different characteristics. Depending on the characteristics of the index to be replicated, the underlying universe or market, as well as its characteristics with respect to liquidity and transaction costs, different techniques are most appropriate.

15.3 ACTIVE BENCHMARK ORIENTED INVESTMENT SOLUTIONS I define an active benchmark oriented investment solution as the combination of: • a benchmark or investment policy and • a set of, ideally uncorrelated, alpha sources.

Benchmark Oriented Solutions

245

The expected return due to the benchmark is generally large compared to the expected return from the individual alpha sources. An alpha source is defined by the investment manager holding relative over- and underweights versus the benchmark in the portfolio. For example, in a Swiss equity portfolio, having the SMI index as benchmark, the investment manager holds 8.26 % in the UBS stock versus a benchmark weight of 7.16 % and 5.13 % in the Credit Suisse stock, underweighting the latter one by 1.1 %. If UBS outperforms Credit Suisse Group, then the portfolio return will be larger than the benchmark return. This return difference defines the generated alpha and the relative weights, alpha source. No net short positions and no leverage are allowed.3 This means, for example, that a position in Syngenta with a weight of 3.49 % in the benchmark cannot be underweighted by more than 3.49 %. Based on the fundamental law of active management, multiple independent sources of alpha should be combined. In the Swiss equity example, additional sources of alpha are over- and underweighting large capitalized stocks versus small capitalized ones, as well as individual industry sectors. Formally, the design of an active benchmark oriented solution can be subdivided into three stages: • Select a benchmark or investment policy, that is a strategic risk/return profile, that best matches the investor’s needs and liabilities. • Select independent or quasi-independent sources of alpha. Each alpha source is constructed using the process approaches described in Chapters 4 through 12. • Combine the sources of alpha with the benchmark using, for example, a risk budgeting approach, taking into account solution specific restrictions. Alpha sources are defined by actively deviating from the benchmark. These deviations lead to active risk. Assume that the alpha source selected is independent of the benchmark. Then the active risk stemming from the alpha sources is orthogonal to the benchmark. It is useful to think about the total level of active risk in terms of information ratio (Winkelmann, 2000). Individual sources of alpha should therefore be selected based on their expected information ratio. Most investment managers primarily think in terms of excess return within the individual alpha sources. However, managing the amount of active risk taken over time is also a source of alpha. This alpha component is known as market timing. One of the most important steps in an investment process of an actively managed solution is the combination of the different sources of alpha with the benchmark. Suppose the objective of this step is to maximize the total portfolio’s expected information ratio. Alpha sources are only added when their expected outperformance is positive. However, the combination of different alpha sources is not straightforward. This is due to: – the correlations between the expected returns of the different alpha sources and – the portfolio restrictions that do not allow the skills to be transferred fully into the portfolio. Winkelmann (2000) suggests applying the following general principles, that is that the portfolio’s information ratio is maximized when the ratio of the marginal impact on the 3 More recently so-called 130/30 investment products that allow limited leverage and short positions to be taken have been introduced.

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Positive Alpha Generation: Designing Sound Investment Processes

total portfolio’s expected alpha to the marginal impact on the total portfolio’s active risk is the same for all alpha sources. Formally, this means that ∀a, b :

return contributiona return contributionb = risk contributiona risk contributionb

resulting in ∀a, b :

wa wb = RBa RBb

(15.1)

where RBa is the risk budget or percentage marginal contribution to risk allocated to asset a with weight wa such that ∂σ wa RBa = · ∂wa σ A combination of weights satisfying Equation (15.1) maximizes the portfolio’s information ratio. 15.3.1 Practical aspects When applying the presented investment process for managing active benchmark oriented solutions, there are two important practical considerations to be taken into account: • the transferability, that is the limitations on the alpha generation imposed by restrictions, and • risk management, that is rebalancing the portfolio to the initial risk characteristics. In an ideal world, there are no restrictions or constraints. In fact, any restriction imposed reduces the opportunity set available to generate positive alpha. I recommend avoiding range or weights based restrictions as much as possible, and recommend using both total active portfolio risk bounds as well as marginal contributions to active risk restrictions on individual alpha sources. Figure 15.6 illustrates such a risk based restriction approach, combined with active risk budgeting. Litterman et al. (2000) suggest reflecting risk based restrictions as the so-called risk policy in the investor’s investment policy and guidelines. 15.3.2 Portfolio rebalancing As markets evolve and alpha sources add value to the portfolio, the individual risk characteristics of the portfolio change. From time to time, the portfolio needs to be rebalanced to become again in line with the original or target risk profile. Litterman et al. (2000) propose implementing a risk monitoring process based on three risk states, that is green (risk is in line with expectations), yellow (risk is at unusual levels and needs attention) and red (risk indicates an extremely rare situation that needs review and eventually action). The final decision as to when to rebalance a portfolio needs to be taken in the light of transaction costs and investor restrictions. Changes in asset prices imply that the relative portfolio weights change over time. Consider an investment solution with a benchmark of 50 % bonds, modeled by the Citigroup

Benchmark Oriented Solutions

247 Benchmark asset allocation Target volatility 70 %–80 % of FTSE world volatility

Tactical asset allocation MCV ≈ 10 % TE ≈ 2.0 %

Diversification assets

Return enhancing assets

Cash MCV ≈ 1 % TE ≈ 0.1%

Real estate MCV ≈ 10 % TE ≈ 4.0 %

Emerging market equities MCV ≈ 8 % TE ≈ 2.2 %

Domestic bonds MCV ≈ 15% TE ≈ 0.7%

Hedge funds MCV ≈ 10 % TE ≈ 6.5 %

High yield bonds MCV ≈ 8 % TE ≈ 3.2 %

Foreign bonds MCV ≈ 10% TE ≈ 1.7%

Private equities MCV ≈ 8 % TE ≈ 4.1 %

Domestic equities MCV ≈ 15% TE ≈ 2.5% Foreign equities MCV ≈ 15% TE ≈ 2.8%

FIGURE 15.6 Risk based restriction approach combined with risk budgeting approach, with MCV representing the percentage marginal contribution to total volatility and TE the active risk or tracking error associated with the individual asset class or alpha source

world government bond index, and 50 % equities, modeled by the MSCI world index, of a USD based investor. Now assume that the bond markets have achieved a return of −4 % since the last rebalancing of the benchmark and portfolio and equities of +12 %. Then the portfolio weights, initially assumed to be 45 % bonds and 55 % equities, that is a 5 % overweight, have drifted to 9 % whereas the benchmark weights are 46.1 %/53.8 %, thus leading to an overweight of 4.9 % in equities versus bonds. The portfolio rebalancing investment process stage, part of the portfolio construction value chain module, tries to answer the following questions: • When should the portfolio be rebalanced, that is the weights readjusted to the target weights? • To what target weights should the weights be adjusted? To answer these questions, the investment manager needs to take into account: – – – – –

risk resulting from the deviation from the target weights, asset volatilities and correlations, overall portfolio risk or volatility, transaction costs and liquidities.

248

Positive Alpha Generation: Designing Sound Investment Processes Table 15.6 Trigger events initiating a portfolio rebalancing where wa is the target or optimal weight and wa the current portfolio’s deviation from it Event

Measurement

Absolute weight deviation

∃a : |wa |
cte

Relative weight deviation

∃a : |wa /wa |
cte √ w  ·
· w
cte √ √ w  ·
· w/ w  ·
· w
cte √ ∂ w ·
· w/∂wa
cte

Absolute risk deviation Relative risk deviation Marginal risk deviation

There are two types of portfolio rebalancing algorithms, that is the calendar based rebalancing and percentage of portfolio based rebalancing algorithms. In calendar based approaches, the portfolio weights are readjusted to their target weights at fixed dates, usually at the end of the month, end of quarter or end of year. It is a common practice for benchmark weights in asset allocation solutions to be rebalanced at the end of each month. The main advantages of this approach are its simplicity and the discipline with which it can be implemented. On the downside, its timing is independent of market movements. In addition, the resulting strategy is pro-cyclical; that is the weights of outperforming assets are reduced and those of underperforming assets increased. Profits and losses are realized. Percentages of portfolio based rebalancing strategies rely on trigger events, rather than time, to initiate portfolio rebalancing. The thresholds are defined in terms of absolute or relative weight deviations, active risk versus the target allocation or event in terms of marginal contribution to risk. Table 15.6 illustrates some of the most common trigger mechanisms used in practice. Percentages of portfolio based approaches are closely related to the theoretical concept of the no-trade region (Atkinson et al., 1997). They allow a tighter control and optimize the rebalancing frequency. On the downside, they require frequent monitoring, usually on a daily basis, of the portfolio weights. In addition, the definition of the trigger events is ad hoc. In 1997 McCalla proposed the equal probability rebalancing strategy. The triggering ranges are defined for each asset class as a common multiple of the volatility of that asset class. Each asset class is thus equally likely to trigger a rebalancing.4 The main advantage of any rebalancing algorithm is to reduce the expected losses from not tracking the optimal allocation or weights. In theory, the basic cost of not rebalancing is the present value of expected utility loss (Leland, 2000). Two approaches are used to implement the rebalancing strategy, that is: • use cash market instruments, that is bonds, equities and funds, or • rely on derivatives, like futures, swaps or forward transactions. Figure 15.7 presents the process I recommend implementing for portfolio rebalancing. It is built on the idea of using transaction cost efficient derivatives for small and frequent 4

The equally likely condition is subject to the assumption that asset class returns are normally distributed.

Monitoring

Benchmark Oriented Solutions

trigger event

249

no

margin call

Futures market rebalancing

Cash market rebalancing

yes

no

yes

close future positions

∆(portfolio + futures vs. target weights) too large

yes

net margin accounts

rebalance using cash markets

add cash to margin account

buy/sell futures

no

FIGURE 15.7 Description of the rebalancing investment process step

deviations from the target weights and cash market instruments for larger but more infrequent deviations. It should be executed on a daily basis.

15.4 CORE–SATELLITE SOLUTIONS The core–satellite investment solution approach blends both indexed and active investment strategies and portfolios in order to achieve a more consistent risk profile (Scherer, 2007; Singleton, 2005; Swensen, 2000). Figure 15.8 shows the structure of a core–satellite solution. The indexed portfolio is called the core subportfolio. The actively managed portfolio is called the satellite subportfolio. The satellite subportfolio is usually smaller in size or weight than the core subportfolio. In an ideal world, the core portfolio makes up 100 % of the assets and the satellites are made up of independent long–short overlays with zero market exposure and uncorrelated alpha sources. Core–satellite solutions have become more

Indexed core portfolio

Active Satellite C

FIGURE 15.8 Core–satellite investment solution structure

Active Satellite B

Active Satellite D

Active Satellite A

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Positive Alpha Generation: Designing Sound Investment Processes

Table 15.7 Sample core–satellite diversified growth type portfolio based on the market efficiency subdivision for a USD based investor Subportfolio

Asset class

Core Core Core Core Satellite Satellite Satellite Satellite

US equities European equities Japanese equities Equities Asia except Japan Emerging markets equities Commodities Private equities5 Emerging markets bonds

Weight (%)

Active risk (%)

Costs (bp)

30 25 20 10 7 8 10 5

≈ 0.0 ≈ 0.0 ≈ 0.0 ≈ 0.5 ≈8 ≈5 ≈ 15 ≈5

1 4 4 15 75 50 150 60

Source: sample data

Table 15.8 Possible core–satellite subdivision criteria

Core Satellite

Efficient market hypothesis

Investment skills

Costs

Risk

Liquidity

Efficient Inefficient

Low High

Cheap Expensive

Market risk Active risk

Liquid Illiquid

and more popular over recent years. Some core–satellite strategies are also known under the name of diversified growth strategies or new balanced solutions. There are different reasons why subdividing the portfolio into the two distinct subportfolios of core and satellite is sound. One argument is that the core represents the set of investments in efficient markets where generating alpha is very hard. The satellite subportfolio is composed of actively managed exposures to inefficient markets. These markets could be, for example, emerging markets bonds, private equity or hedge fund strategies. Table 15.7 shows a sample core–satellite portfolio based on this argument. Another reason for the decomposition would be to distinguish the asset classes where the investment manager has skills, the satellite subportfolio, from the other ones where an indexed portfolio is preferred. Similarly, a decomposition of the asset classes into those with mainly market exposure, the core, and those with high alpha expectations is possible. Some investment consultants have introduced core–satellite investment strategies in order to reduce costs and thus perform arbitrage. The core portfolio is invested using cheap indexed strategies. The satellite asset classes are managed with a high active risk and therefore are more expensive to manage. However, their cost per unit of active risk still remains cheap. This is due to the fact that the investment manager prices these products based on volume. Table 15.8 summarizes different possible criteria for decomposing the portfolio in a core and a satellite subportfolio. Core–satellite portfolios can be constructed using the portfolio construction techniques illustrated in Chapters 10, 11 and 12. Especially the risk budgeting based approaches, like 5 Private equities are managed against a pure cash benchmark and therefore the target active risk corresponds approximately to the target volatility.

Benchmark Oriented Solutions

251

the ones summarized at the beginning of this section, are adequate. Nevertheless, precautions with respect to the definition of risk have to be taken as: • some of the satellite asset classes exhibit nonnormal return patterns and • the others include significant illiquidity risk. Scherer (2007) as well as Till (2005) describe some risk management approaches that can be applied in the core–satellite context. The focus on risk management moves from volatility risk to default, devaluation and liquidity risk.

15.5 A SAMPLE BENCHMARK ORIENTED SOLUTION Consider a Dutch pension fund with a current surplus of 15 %. Furthermore, assume that the pension fund’s liabilities have a ten-year duration. I forecast that the equity risk premium of global equities will be 2 % in excess of the 10-year government bond yield over the next five years. Based on this information, I advise the investor to implement the investment policy or benchmark shown in Table 15.9. Table 15.9 Strategic asset allocation representing a pension fund’s investment universe and benchmark Asset class

Index

Cash in EUR

Citigroup 3-month money market index Citigroup 10+ government bond index

20

MSCI world (hedged)

50

MSCI emerging markets EPRA/NAREIT Europe

5 15

EUR denominated government bonds with a 10+ year maturity Global equities (developed markets, hedged) Emerging markets equities European property

Weight (%) 5

Source: sample data

This policy has an expected excess return of 0.4 % over a ten-year zero coupon government bond, forecasting a total return of 9.3 % for the property asset class. Furthermore, the value-at-risk at the 99 % confidence level over a one-year time horizon is 14.7 %, which is in line with the surplus at risk. As an investment manager, I recommend to the pension fund to implement an actively managed benchmark oriented solution, implementing the following alpha sources: • Use investment grade corporate bonds up to a rating of Aa2/AA depending on the forecasts on credit spread movements. • Implement a country/region based tactical asset allocation investment solution on developed and emerging equity markets. • Use a passive implementation of the individual developed equity market country/region asset classes because I assume that developed markets are efficient with respect to the efficient market hypothesis.

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Positive Alpha Generation: Designing Sound Investment Processes

• Implement an active stock selection investment solution for emerging market equities combined with a country over- and underweighting, as I believe that emerging markets are less efficient, in the sense of the efficient market hypothesis, as developed markets. • Buy a third party actively managed property fund which has ex-ante volatility no larger than that of the EPRA/NAREIT European property index. • Implement a cash–equity market timing overlay.

16 Absolute Positive Return Solutions Absolute positive return (APR) solutions, also called total return or target return solutions, represent probably the fasted growing category of investment solutions, but is also the least well understood.

16.1 WHAT ABSOLUTE POSITIVE RETURN CAN MEAN Absolute positive return solutions, as any other investment solutions, are characterized by the two dimensions return and risk. In addition, in the mind of the investor, the return of a market portfolio plays an important role in defining absolute positive return solutions. Based on these three dimensions, I define the five most common expectations of the investor in APR solutions: 1. The investor wants to participate in the return of the market portfolio if that return is positive and achieves a return equal to the risk free rate if the market portfolio’s return is negative. 2. The investor aims to achieve a positive return or a positive excess return over the risk free rate, in any short term period, usually months or quarters. The focus is on the frequency of positive returns, respectively excess returns, rather than their size. Infrequent negative returns, independent of their size, are tolerated, for example a loss in an equity market crash. 3. The investor wants to avoid negative returns, especially large negative returns. The focus is on the size of the negative returns, rather than their frequency. 4. The investor seeks returns that are not correlated with the returns of a market portfolio. The risk reduction or diversification effect is the primary characteristic sought after. 5. The investor wants to maximize the expected return under a given maximal statistical loss, for example defined as the value-at-risk at a 99 % confidence level over a one-month time horizon. It can be seen that the patterns of returns play an important role in the definition of the investor’s utility function or preference. Figure 16.1 illustrates three different return patterns with identical mean and standard deviations, aiming at three different types of investors. Investor A prefers steady positive returns. To achieve this steadiness, infrequent losses are accepted, like in May 2006 or November 2007. Investor B, on the other hand, prefers to avoid large losses, but is less resilient to more frequent small losses. Finally, investor C is mainly aiming at diversification, that is low correlation with equity market returns, in this example with a correlation of +0.1. All return time series shown exhibit the same mean return of 7.0 % per annum and a standard deviation of 5.0 % per annum. In the APR expectation type 1 and type 5, the primary focus is on return whereas in APR expectations 2 and 3, the aim is mainly set on risk management. The APR expectation 4 is most relevant in combination with other assets or investment solutions. However, some

254

Positive Alpha Generation: Designing Sound Investment Processes 8.0% 6.0% 4.0% 2.0% 0.0% −2.0% −4.0%

Investor A (type 2)

Investor B (type 3)

Investor C (type 4)

Dec.07

Oct.07

Nov.07

Sep.07

Jul.07

Aug.07

Jun.07

Apr.07

May 07

Mar.07

Jan.07

Feb.07

Dec.06

Oct.06

Nov.06

Sep.06

Jul.06

Aug.06

Jun.06

Apr.06

May 06

Mar.06

Jan.06

−8.0%

Feb.06

−6.0%

Market portfolio Source: author’s illustration

FIGURE 16.1 Different return patterns representing different investors’ preferences in absolute positive return solutions having an identical mean return of 7.0 % p.a. and a standard deviation or volatility of 5.0 % p.a.

investors have a preference for it solely explained by behavioral aspects of a good feeling of not losing money at the same time as others.

16.2 SATISFYING THE INVESTOR’S EXPECTATIONS Rather than question the investor’s definition of expectation in APR solutions, the investment manager should build an investment solution and an associated investment process that: • best approximates the investor’s utility function, that is their APR expectation, and • that they believe to be capable of delivering. A key success factor for APR solutions is to find an efficient mix between these two often contradicting goals. I distinguish between three types of capabilities that can be at the basis of building APR solutions: 1. engineering skills, 2. statistical arbitrage and diversification skills, and 3. market forecasting or alpha generating skills. 16.2.1 Applying engineering skills Engineering skill based APR solutions are constructed in order to ensure or guarantee certain portfolio return or payout characteristics. A key point of engineering skill based solutions is that they are deterministic or rule based and usually do not rely on statistical properties or market forecasting skills. Call-and-put options, as described in Chapter 17, used in implementing capital protected solutions, are the most common instruments used

Absolute Positive Return Solutions

255

within engineering skills. In fact, any capital protection or preservation solution can be classified as an engineering skill based APR solution. These approaches aim at investors with a type 1 APR expectation. Engineering skills can also be applied to construct type 4 APR solutions by hedging systematic or market risks, leaving only specific risks. For example, a US investor defining his or her market portfolio through the S&P 500 index could be interested in holding the following APR type 4 solution, that is: – a long position in a world equity portfolio, for example replicating the FTSE All World index, – a short position in the S&P 500 index through short futures on the S&P 500 with an exposure value equal to the long world equity position1 and – a margin cash account to handle the profit and loss from the future positions. This portfolio structure would be combined with a rebalancing strategy, for exposure matching and cash management, as described in Chapter 13. Its return exhibits a low correlation with the S&P 500 index and is therefore perceived to be independent of the market portfolio return. Another type of engineering skill based solution uses leverage to provide positive excess return based on scaling illiquidity premia on various assets. The investment manager invests in a well diversified portfolio of illiquid fixed income instruments. Then interest rate and credit risk are hedged by using interest rate futures and credit default swaps. As the remaining expected excess return tends to be rather small, the portfolio is leveraged up to the level where the portfolio’s expected return matches the investor’s target return. As liquidity premia under normal market conditions are rather stable, such a solution provides an APR solution for investors of types 2 and 4. The remaining risk of such a strategy is that, due to its illiquidity, a forced sale of assets from the portfolio will have a significant negative impact on the performance, which can wipe out the whole illiquidity premia return realized so far. Indeed, as could be observed during the US subprime crisis in 2007/2008, numerous collateralized investment vehicles, like CDOs or CLOs, lost value due to investors having to liquidate their positions and no buyers being available. 16.2.2 Using statistical arbitrage or diversification skills In contrast with engineering based approaches, statistical approaches rely on statistical properties of asset returns to satisfy the investor’s goal. I distinguish between two types of statistical approaches, that is: • the arbitrage approaches and • the diversification approaches. Statistical arbitrage techniques exploit statistical inefficiencies in the market. Most of these techniques are based on the following investment process: 1. Identify assets whose price or return is mean–reverting to its so-called fair value. 2. Determine the fair value, using no arbitrage pricing models or other statistical techniques. 1

Alternatively, the short position could be constructed so as to neutralize the beta risk of the world equity portfolio.

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Positive Alpha Generation: Designing Sound Investment Processes

3. If the current asset price is cheap relative to its fair value, take a long position in the asset; otherwise, if the asset is expensive, sell it. 4. Wait until the asset achieves its fair value and realize the profit. These types of investment processes are mainly used in hedge funds and are described in more detail in Chapter 18. The capabilities required to produce such APR solutions successfully are the execution of the four steps in the investment process. In practice, many statistical arbitrage strategies failed because they were unable to wait until the asset prices reverted to their fair value. In some cases the cost of waiting, through borrowing cash to take the long or short position in the mis-priced asset, exceeded the profit from waiting for the fair value to be attained and the profit realized. A second category of statistical techniques to produce APR solutions are diversification techniques. The idea behind these techniques is to construct portfolios consisting of a large number of assets that are not or only slightly correlated with each other. Figure 16.2 shows an example for such a portfolio. Through the effect of diversification, negative returns are reduced and the portfolio performance stabilized. The Yale endowment fund (Swensen, 2000) is probably the most prominent advocate of this approach. The success of statistical diversification relies on three key assumptions, that is: • In the long run, on average, assets generate positive excess returns over the risk free rate, or no investor would be willing to hold these assets. • The investment manager is capable of forecasting volatilities of and correlations between asset classes and adjusts the portfolio structure accordingly. • The investor is capable of waiting sufficiently long for the statistical properties to materialize. Cash; 5% Private equity; 9% Emerging market bonds; 8%

Domestic equities; 14%

High yield bonds; 6%

Global properties; 12%

Commodities; 7%

International equities; 31%

Emerging market equities; 8% Source: author’s illustration

FIGURE 16.2 A well diversified portfolio structure

Absolute Positive Return Solutions

257

Although over a short investment horizon, statistical diversification approaches do not necessarily satisfy any of the five APR expectations, over a sufficiently long time horizon, they are adequate solutions satisfying APR expectations 4 as well as 5. Another approach that can be classified in the statistical diversification skill set is to rely on asset classes that have, by construction, a low correlation with market returns. The most prominent example is the catastrophic bonds asset classes (Canter et al., 1997; Litzenberger et al., 1996), that is bonds whose coupon payment is contingent on an external event, like a hurricane or an earthquake. For example, the Swiss Re cat bonds total return index, as a proxy for catastrophic bonds, has a correlation of −0.0721 with global equities, modeled by the MSCI world total return index, and +0.0782 with global bonds, modeled by the Citigroup world government bond total return index, when estimated using weekly historical data between December 2004 and December 2007, the Swiss Re cat bond index being published for the first time at the end of 2004. 16.2.3 Forecasting skill based APR solutions Rather than relying on structural or statistical characteristics of the markets, in forecasting skill based APR solutions the added value is based on the investment manager’s forecasting capabilities. As Ineichen (2007) points out, the goal of a forecasting skill based absolute return solution is to have an idea generating or return forecasting process for the upside and a risk management or risk forecasting process for the downside. The separation between the upside and the downside should result in some form of call option-like return diagram. Forecasting based APR solutions aim primarily at investors with APR expectations of types 1, 2 and 3. As described in Chapter 2, their success depends on the investment manager having skills to formulate forecasts that are rewarded in the marketplace; that is the skills must be related to the investment universe and be transferable into a portfolio. As the way skills are rewarded in the marketplace changes over time, the investment manager must adapt his or her forecasting capabilities. In that sense, the forecasting process does not differentiate itself from the approach taken actively to manage benchmark oriented solutions. Depending on the investor’s expectations, additional focus should be put on the risk management side. A possible approach is to define a risk budget available for each time period. Then, the portfolio representing the formulated forecasts must be constructed such that it is expected to lose no more than the attributed risk budget over that time period. Depending on the investor, the risk budget may be adjusted over time based on the realized return. Consider an investor with an APR expectation of type 3, that is who is very sensitive to extreme losses. For such an investor, in addition to the skill based return generating investment process, it is useful to hedge extreme event risks. This can be realized by buying deep out-of-the-money put positions for each long position held and calls options for each short position. Such a portfolio is illustrated in Table 16.1, investing solely in cash and equities. The out-of-the-money options remove the tail or extreme events risk from the portfolio. It is important that the forecasting process generates many independent market views in order to exploit fully the fundamental law of active management. Unfortunately, for high return expectations from the investor’s side, not enough forecasts achieving them may exist. In such a situation, leverage is the most appropriate approach to increase the expected return rather than decrease the diversification effect of multiple independent market views.

258

Positive Alpha Generation: Designing Sound Investment Processes Table 16.1 Sample APR portfolio based on forecasting skills hedging extreme event risk

Holding +27 +160 −145 +67 +160 +145

Security S&P 500 future EuroSTOXX 50 future Nikkei 225 future Put @ 1 350 on S&P 500 Put @ 3 900 on EuroSTOXX 50 Call @ 17 500 on Nikkei 225 (cost of insurance)

Maturity

Price

Value (in USD)

June 08 June 08 June 08 21.06.2008 20.06.2008 13.06.2008

1 515.90 4 354.00 15 660.00 126.00 97.10 260.00

10 232 325 10 102 673 −19 908 380 884 200 225 303 330 535 1 400 038

Source: data sourced from Bloomberg Finance LP, author’s calculations.

16.3 THE RELATIONSHIP BETWEEN RISK AND RETURN Investors with expectations of types 1, 2 or 3 often target a given excess return, for example 200 bp in excess of the one-month EURIBOR2 interest rate level. The question is, what does that return target mean, in terms of risk and required skills? If the investment manager were capable of forecasting returns or excess returns of the individual asset classes, a portfolio with a given return target and minimizing risk could be constructed. Unfortunately, this is very often not the case. In this situation, the following rule of thumb helps: • Determine the information ratio (IR) the investment manager is expected to achieve, based on his or her realized performance. Setting IR = 0.5 is reasonable if insufficient historical experience is available. • Transform the target excess return, for example, 1 month EURIBOR + 150 bps, into a target risk by using the following equation, where α is the target excess return and τ the target risk: IR =

α α ⇔τ = τ IR

(16.1)

• Construct a portfolio using a risk budget of τ , for example 3 % = 150 bp/0.5. This rule of thumb is especially useful as risk is less variable than return and easier to manage in a portfolio context. It also shows that the better the skills of the investment manager, the less risk he needs to take in order to achieve a given return target. Figure 16.3 presents the results of the following simulation. Each month I randomly invest either in a long or a short position in an S&P 500 future such that the investment has a probability of 56 % in achieving a positive return, corresponding to an information ratio of IR ∼ = 0.5.3 The exposure taken in equities, long or short, is such that the risk τ at the beginning of the investment period is 2 %. The compounded realized α is shown over a five-year period, between December 1998 and December 2007, resulting in an annualized ex-post return of 1.5 % or IR = 0.86, slightly higher than predicted by Equation (16.1), that is α = IR · τ . 2 The Euribor (Euro Interbank Offered Rate) is a daily reference interest rate based on the averaged rates at which banks offer to lend unsecured funds to other banks in the Euro wholesale money market. It is fixed on a daily basis by the European Banking Federation (EBF) at about 11:00 am and is a filtered average of interbank deposit rates offered by a large panel of designated contributor banks. 3 Indeed, solving equation Pr{IR
0} = 56 % by assuming that IR follows a normal distribution results in E[IR] = 0.5.

Absolute Positive Return Solutions

259

160 140

50 %

Portfolio value

120 40 %

100 80

30 %

60

20 %

40 10 %

20

0%

D ec . Ju 97 n. D 98 ec . Ju 98 n. D 99 ec . Ju 99 n. D 00 ec . Ju 00 n. D 01 ec . Ju 01 n. D 02 ec . Ju 02 n. D 03 ec . Ju 03 n. D 04 ec . Ju 04 n. D 05 ec . Ju 05 n. D 06 ec . Ju 06 n. 07

0

Long-short position size

60 %

Alpha (indexed) [left scale]

Gross exposure to the S&P 500 index [right scale] Source: data sourced from Bloomberg Finance LP, author’s illustration

FIGURE 16.3 Illustration of the relationship between alpha and the gross exposure of the portfolio for a fixed risk budget of 2 %

16.4 LONG-ONLY FORECASTING BASED SOLUTIONS The basic idea behind long-only APR solutions is as follows. Consider the one-month EURIBOR rate as benchmark. Then actively position the portfolio against that benchmark taking τ units of active risk according to Equation (16.1) and the targeted alpha. Assuming the forecasted scores shown in Table 16.2, the target asset allocation for a 1 month EURIBOR + 150 bp would be as shown in the same table. It can be seen from the example in Table 16.2 that forecasting based long-only portfolios exhibit two key characteristics: • They only invest in assets with an expected excess return larger than the risk free rate. • The resulting portfolio is not a well diversified portfolio, but a portfolio investing in those asset classes that exhibit a high expected risk adjusted return. Table 16.2 Long-only APR investment solution for an EUR based investor Investment universe Asset class Cash Bonds in EUR Bonds in USD Equities EMU Equities UK Equities USA Equities Japan

Expected excess return (%)

Volatility (annualized) (%)

3.2 1.7 −2.5 6.4 4.2 7.3 2.5

1.5 4.5 7.8 14.5 13.5 18.2 23.5

Source: sample data, author’s calculations

Portfolio weight (benchmark weight) (%) 45 0 0 27 5 23 0

(100) (0) (0) (0) (0) (0) (0)

260

Positive Alpha Generation: Designing Sound Investment Processes 10 9 8 7 6 5 4 3 2

EURIBOR + 400bp

Sep.07

Jan.07

May 07

Sep.06

Jan.06

May 06

Sep.05

Jan.05

May 05

Sep.04

Jan.04

May 04

Sep.03

Jan.03

May 03

Sep.02

Jan.02

EURIBOR + 600bp

May 02

Sep.01

Jan.01

May 01

Sep.00

Jan.00

May 00

Sep.99

Jan.99

0

May 99

1

EURIBOR + 200bp

Source: data sourced from Bloomberg Finance L P, author’s calculations

FIGURE 16.4 Number of asset classes outperforming the one-month EURIBOR rate + 200bp, 400bp, respectively 600bp, over a one-month time horizon using the following asset classes, in euro: cash, bonds EUR, bonds USD, bonds JPY, equities EMU, equities UK, equities US, equities Japan, equities emerging markets, commodities

Indeed, the higher the target return, the less diversification the portfolio will exhibit. A question that comes up is whether it is possible in any situation to achieve the target return, assuming perfect hindsight. This boils down to the question of whether in any time period asset classes exist having an excess return higher than the return targeted. Figure 16.4 shows, for any given month over the last 10 years, the number of asset classes having a higher return than the target return. It can be seen that the higher the return target, the more difficult it gets to forecast the correct asset class. Whereas for a return target of EURIBOR + 200 bp an average of 47 % of the considered asset classes can be selected, for a return target of EURIBOR + 600 bp the average number of outperforming asset classes, since December 1998, using a monthly time horizon, drops to only 42 %. In exactly 10 different months between December 1998 and December 2007, none of the selected asset classes outperformed the EURIBOR + 200 bp benchmark. APR investment management is not only about forecasting but also about risk management. Three possible approaches to risk management exist, that is: • pro-cyclical, reducing the risk when the realized performance deteriorates and increasing it with improving performance, • anti-cyclical, increasing risk when the realized return is deteriorating and decreasing it otherwise, and • skill based, where the higher the confidence in the forecasts, the more risk is taken independent of past performance. The skill based risk management approach is consistent with the management of the uncertainty of the forecasts. It can be compared to the way a car driver adjusts driving speed. For example, when approach a crossing, the speed is reduced, and when entering the freeway, the speed is increased.

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The pro-cyclical risk management approach is consistent with a surplus-at-risk driven approach. The risk budget available is a function of the realized return serving as a cushion. This approach is most appropriate for investors with expectations of types 1 and 5, that is investors wanting to participate in the return of the market portfolio or investors seeking to maximize their expected return under a given maximal statistical loss. The anti-cyclical approach is only sound for investors who look at return in a binary way; that is either the target is achieved or not. This approach is also often described by the phrase if you are in trouble, then double. Unfortunately, in some situations, it may be the only option to have a nonzero probability of achieving the targeted return. Whichever approach for risk management is chosen, at the end is a function: • of the investment manager’s skills in forecasting risk, that is formulating an opinion on the uncertainty, and • of the investor’s needs and utility function. The chosen approach should be agreed upon by the investor and the investment manager.

16.5 THE PORTABLE ALPHA APPROACH A portable alpha approach as an APR solution can be constructed as follows. Take a benchmark oriented investment solution, for example a Japanese stock selection equity fund with the Nikkei 300 index as benchmark, that consistently outperforms. Then, in addition to holding a long position in the fund, a portable alpha solution adds a short position in the benchmark underlying the fund, that is, in my example, the Nikkei 300 index. This can be achieved, for example, through holding a short position in futures representing the index or highly correlated with the index. The resulting portfolio returns is the difference between the return of the investment solution, the Japanese stock fund, and its benchmark, the Nikkei 300 future. This return can be written as risk free rate + α where α is the excess return generated by the fund manager. This approach is called portable alpha, because the alpha is ported out of its original solution context. Figure 16.5 illustrates the idea. The risk free rate is generated by investing the proceeds from the short position in the benchmark asset in cash. If the short position is realized using futures, the return or profit and loss of the future is equal to the index return minus the risk free rate.4 16.5.1 Key success factors Successfully implementing a portable alpha strategy, whether as a standalone APR solution or as a building block of a diversified portfolio or a core–satellite solution, essentially depends on three factors, that is: • identifying skills, that is investment funds or solutions, that systematically generate outperformance or positive alpha, 4

This result can be easily derived using no arbitrage arguments.

262

Positive Alpha Generation: Designing Sound Investment Processes short benchmark position

fund

portable alpha

alpha excess return

benchmark return

+

= risk free rate cash

alpha excess return risk free rate cash

short benchmark return

FIGURE 16.5 Illustration of the decomposition and combination of fund returns in a portable alpha solution

• constructing a short portfolio whose return matches the return of the benchmark of the outperforming investment funds or solutions, and • implementing the long and short positions and handling the cash flows both in size and maturity between the two investments. In order to exploit the fundamental law of active management fully, a large investment universe with numerous different and independent skill sets should be chosen. The benchmark underlying the alpha generating fund as well as the associated restrictions should be analyzed for their impact, usually negative, on the alpha generating capabilities. Indeed, any restriction reduces the size of the opportunity set and therefore the possibilities of generating positive excess return. For example, if the fund manager is not allowed to hold net short positions, he or she can underweight a position at most up to its weight in the benchmark, to generate alpha. A second important aspect is the actual portability of the excess return. This means that it must be possible to take a short position in the benchmark of the underlying portfolio or, at least, in assets highly correlated with the benchmark. Techniques used in passive benchmark oriented solutions and described in Chapter 15 may be used. The task becomes more difficult if the benchmark of the fund is highly customized or contains a large number of constituents, like, for example, the Lehman global aggregate corporate bond index, or includes illiquid securities, like high yield bond benchmarks. In these cases, hedging the benchmark exposure can become tricky and sometimes even void the alpha generated. Finally, as the alpha is extracted from the fund, using derivatives or security borrowing techniques, sufficient credit lines and cash are needed to be able to avoid any liquidity issues that may arise. 16.5.2 A portable alpha investment process Although the conceptual idea of portable alpha solutions is simple, assuming capabilities to identify significant alpha generating sources, the difficulty resides in the details of their

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implementation. The associated investment process can be decomposed in the following three steps: 1. Start by identifying funds or investment managers with alpha generating capabilities. Although identifying a single manager is sufficient, I recommend relying on multiple sources of alpha to diversify the information or alpha generating risk of any single source of skills. In this selection process, it is important to identify the benchmark or market exposure that is not part of the alpha generated through skills. This benchmark must be, in its definition, stable over time. The exposure to the different alpha sources should be managed through a risk budgeting based approach, using, for example, an approach similar to Lee’s (2000) optimal risk budgeting. 2. Focus on identifying and implementing the derivatives overlay to remove the benchmark return from the portfolio. There are two types of instruments that can be used efficiently, that is futures or swaps:5 – The main advantage of futures is their generally high liquidity, their low transaction costs and their limited counterparty risk.6 On the downside, futures require the holding of a margin account, that is allocating assets that are no longer available to the investment manager for generating alpha. In addition, the daily mark-to-market introduces a horizon mismatch between the derivative short position and the long alpha generating fund position. – On the other side, swap transactions, especially total return swaps as described in Chapter 13, are highly customizable and therefore good candidates for taking short positions in benchmarks. They are especially useful for removing the exposure of complex benchmarks. However, they are over-the-counter transactions and therefore there is no competitive market. Pricing tends to reduce the advantage of customization.7 3. The third and probably the most practical issue to address is the handling of profits and losses both from the alpha generating fund and the short position in the fund’s benchmark. In theory, the profit and loss of the return due to the benchmark in the fund and the short position should match and therefore be netted on a daily basis. In practice, however, using a limit based strategy, similar to rebalancing strategies described in Chapter 12, is advisable. A given amount of cash, depending on the volatility, that is the risk of the benchmark, needs to be set aside on the short position’s margin account. Alternatively the fund may be deposited as collateral and the margin requirements financed through borrowing and a credit line. Each time the value of the margin account has exceeded a certain level, for example increased by 50 % or decreased to 20 % of its initial value, a rebalancing between the fund and the margin account should be performed. This process is illustrated in Figure 16.6, where a dot indicates when cash was moved from, respectively to, the margin account. The margin account cushion available must also cover for potential periods in which the fund manager underperforms its benchmark, that is produces negative alpha. 5 An alternative to a short future position or a swap transaction is to borrow the security or securities from an investment bank and sell them on the market. Conceptually such a transaction can be identified as a swap transaction. 6 The counterparty of any future transaction is an exchange, rather than an investor or broker. The marked-to-market approach with daily netting reduces the credit risk to the overnight risk of the exchange as the counterparty. 7 The counterparty providing the swap transaction will charge for the complexity of and the risk in hedging the swap transaction on the market. The issues of using a futures based hedging approach now resides with the swap counterparty.

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Positive Alpha Generation: Designing Sound Investment Processes

16 000 14 000 12 000 10 000 8 000 6 000 4 000

Margin account value

Lower bound

Upper bound

Dec.06

Jun.06

Dec.05

Jun.05

Dec.04

Jun.04

Dec.03

Jun.03

Dec.02

Jun.02

Dec.01

Jun.01

Dec.00

Jun.00

Dec.99

Jun.99

Dec.98

0

Jun.98

2 000 Dec.97

Margin account value (in '000 $)

18 000

Rebalancing

Source: author’s illustration

FIGURE 16.6 Illustration of a possible rebalancing strategy between the alpha generating fund and the margin account of the short benchmark position

16.5.3 Example Consider a USD based investor aiming at achieving an APR of a one-month LIBOR + 400 bp. They have an APR expectation of type 2. I start by searching for an investment manager having skills systematically to produce positive alpha. Based on the concepts described in this book, I ideally should be that investment manager myself. Assume that I have found three investment managers managing the funds shown in Table 16.3. For each fund I select the most appropriate strategy to remove the benchmark exposure or return. In the case of the US large capitalized equity fund with MSCI USA as the benchmark I use the S&P 500 future and calculate the correlation between those two indices. Table 16.3 Sample set of benchmark oriented funds exhibiting consistently a positive alpha and the associated derivative overlay instruments used to take a short position in the benchmarks of these funds Holding (%) 38 −38 28 −28 32 −32 −32 2

Fund

Benchmark

Expected alpha (bp)

Information ratio

US large capitalized equities S&P 500 future (with a correlation of 0.998 with the MSCI USA index) US small capitalized equities CME Russell 2000 index Japanese equities TSE TOPIX future Sell JPY/USD forward Margin cash account

MSCI USA

400

0.50

Russell 2000

300

0.75

200

0.40

0

n/a

Source: sample data

TOPIX

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265

A correlation of 0.998 indicates that no adjustment for potential mismatches is necessary. I use the Russell 2000 future traded on the Chicago mercantile exchange to gain a short position in the US small capitalized equities fund. Finally, I sell TOPIX futures on the Tokyo stock exchange against the Japanese equity fund. In addition, I hedge the JPY currency risk stemming from the benchmark by selling JPY against USD for the position’s notional value using forward contracts. I use three-month futures and forward transactions as a compromise between matching performance, liquidity management and transaction costs. Based on a value-at-risk at the 99 % level over a one-day horizon for the different benchmarks, I set aside 2 % in cash to handle any intermittent profit and loss occurring during the three months between the different forward and futures contracts. If, during any three-month period the margin account value drops below its minimum or exceeds its initial value by 50 %, I rebalance the holdings in the fund and the future. As can be seen from Table 16.3, the constructed portfolio has an expected return of only a one-month LIBOR + 300 bp, rather than 400 bp. I could increase the return of the portfolio by decreasing the diversification and invest only in the US large capitalized equities funds. Another approach to achieve the return targeted by the investor would be to ask the individual investment managers to increase their active risk to a level where the return scales to a one-month LIBOR + 400 bp based on Equation (16.1). Unfortunately, this is very often not possible, due to underling benchmark and other potential restrictions. I therefore decide to use leverage, that is borrow 33 % additional cash, using the existing positions as collateral, and invest it in the same portfolio strategy. Assuming that I can borrow at the one-month LIBOR rate, I can construct a portfolio with an expected return of a one-month LIBOR + 400 bp. 16.5.4 Advantages and drawbacks The main advantage of the portable alpha APR approach is that I can take advantage of existing investment management skills. Portfolio engineering techniques allow the extraction of the pure skill based return from the portfolio. The assumption that the skill set is not correlated with market return, a reasonable assumption when consistent skills exist,8 allows me to generate positive return in excess of the risk free rate that is uncorrelated with the market return. In addition, I can use leverage to achieve nearly any reasonable return target. On the downside, I need to implement a thorough cash management process, which may be costly. In addition, the fact that the alpha generation portfolio is managed against a benchmark introduces unnecessary restrictions. This drawback could be lifted by managing a pure long–short portfolio rather than taking the detour through a benchmark oriented fund and a short position in a benchmark. Such investment solutions are described in Chapter 18 in the context of hedge funds.

16.6 COMBINING ABSOLUTE POSITIVE RETURN AND BENCHMARK ORIENTED SOLUTIONS Similarly to the concept of core–satellite portfolios described in Chapter 15, APR solutions can be combined with passive benchmark oriented solutions. Such an approach is sound 8 This zero correlation of returns due to skills with market returns is consistent with market returns behaving like a normally distributed random variable and returns due to skill exhibit persistence.

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Absolute return risk budget

Synthetic benchmark exposure

+

Physical alpha portfolio

= Benchmark risk budget

Physical benchmark exposure

Total benchmark return

Generated alpha

FIGURE 16.7 Combining APR solutions with passive benchmark oriented solutions to outperform a given benchmark

if I can match my liabilities, as described in Chapter 19, by a benchmark portfolio. The construction of a combined solution proceeds in two steps: 1. First, decompose the total risk budget into – market or benchmark risk and – active or APR risk. 2. Next, deploy the asset base such that the APR manager is fully funded for the risk allocated and a 100 % benchmark exposure is achieved. The construction is illustrated in Figure 16.7. The realized return is equal to the benchmark return plus the alpha generated by the APR manager. It is important to note the decoupling of the benchmark from the alpha. Therefore the APR manager does not need to have skills in the investment universe of the benchmark.

17 Capital Protection and Preservation Approaches A third category of investors put capital preservation ahead of return expectations. These investors look for portfolio insurance strategies that ensure that the value of their portfolio, at some specified date in the future, will not fall below some specified level. Leland (1980) has shown that there are two classes of investors for whom portfolio insurance maximizes their expected utility. These are: (1) Investors who have average market return expectations, but whose risk aversion increases with wealth more rapidly than average, so-called safety first investors. (2) Investors who have average risk aversion, but whose expectations of market returns are more optimistic than average, that is well diversified investment managers who expect on average to generate positive alpha. These investors have a hyperbolic absolute risk aversion (HARA) utility function, that is have a preference for consumption over savings and are risk averse. Two classes of portfolio insurance investment processes or strategies exist. These are: • static strategies characterized by a buy-and-hold approach and • dynamic strategies characterized by requiring continuous portfolio adjustments. In contrast with most other investment solutions presented in this book, portfolio insurance investment processes or strategies do not require any kind of forecasting or alpha skills. Portfolio insurance investment processes solely rely on: • selection of parameters describing the investor’s utility function, for example the minimal required wealth level and time horizon, and • thoroughly executing the selected investment process. Nevertheless, they may be used in conjunction with an alpha generating investment solution. A portfolio investment process can be developed to protect the investor from incorrect market forecasts by the investment manager rather than falling markets.

17.1 THE INVESTOR’S UTILITY FUNCTION The importance of a portfolio insurance investment process (PIIP) results from the investor having a risk preference that is asymmetric around the mean, as shown in Figure 17.1. In addition, the investor needs to specify the following parameters of the utility function: • the maturity, that is the date at which the investor wants to recoup his or her assets, • the floor or protection level, that is the minimum portfolio value at maturity,

Positive Alpha Generation: Designing Sound Investment Processes

Payout value

268

Value of the performance asset

FIGURE 17.1 Expected payoff diagram at maturity from a typical investor with a HARA utility function

• the performance asset, that is the selection of the market in which the investor expects to participate, and • the return expectation or risk aversion, that is the expected participation in the performance asset’s movements. The investor may want to specify intermediate cash flows to finance consumption. They can be either handled as separate portfolios with maturities matching the expected cash flow dates or they may be integrated into a single PIIP or even made contingent on the performance of the investment process. As defining a specific date as a maturity date is an abstract concept for many investors, the floor may be specified for a fixed set of dates, for example every 31 December or continuously, as, for example, the maximum of yesterday’s floor and the portfolio value a year ago. The floor required by the investor may be specified as a function of the portfolio value at a given point in time, instead of being fixed. For example, the floor could be defined as 95 % of the highest value of the portfolio at any time in the past. The performance asset may be a single security, like the IBM share, an individual market index, like the Russell 2000 index, or even an actively managed portfolio, like a convertible arbitrage hedge fund.

17.2 PORTFOLIO INSURANCE INVESTMENT PROCESSES The goal of any portfolio insurance investment process is to guarantee a floor value F (T ) at maturity T and participate in the performance of a given market, called the performance asset. I denote by V (t) the value of a portfolio managed according to a PIIP at time t. S(t) denotes the value of the underlying performance asset at time t, r the continuous compounded risk free rate and T the maturity date. It is common to subdivide the assets of an insured portfolio into two categories: • the risk free asset, that is the asset that ensures or contributes to the protection of the floor at maturity, and • the risky asset, that is the investment that allows the investor to participate in the performance asset.

Path dependent

269

Stop-loss processes

Constant proportion portfolio insurance (CPPI) Variable proportion portfolio insurance (VPPI)

Path independent

Capital Protection and Preservation Approaches

Cash plus call option Equity plus put option

Dynamic call replication Dynamic put replication

Static

Dynamic

FIGURE 17.2 Classification of the portfolio insurance investment processes along the two dimensions static versus dynamic processes and performance asset path dependent versus independent

Figure 17.2 classifies some of the most common PIIPs, with respect to whether they are static or dynamic and whether the portfolio value at maturity depends on the path of the price of the performance asset. 17.2.1 Simple strategies The simplest PIIP is the so-called stop-loss strategy. The investor starts with investing 100 % of the portfolio in the performance asset. If, at time t  T the value of the portfolio drops below the discounted floor value, that is V (t) = F (T ) · e−r·(T −t) , then the performance asset is sold and 100 % of the portfolio is invested in the risk free asset. Executed correctly, this strategy ensures a payoff, as shown in Figure 17.1. Unfortunately, this simple strategy has two major flaws. First, as prices of the performance asset are not always continuous, it may not be possible to sell the performance asset at price F (T ) · e−r·(T −t) , thus not guaranteeing the floor value at maturity. In addition, once the performance asset’s price has crossed the barrier of the floor, the portfolio will no longer participate in the performance asset’s return. Brennan and Solanki (1981) have shown that such a strategy is, under most circumstances, inconsistent with expected utility maximization. A second simple PIIP is to buy for F (T ) · e−r·T a zero coupon bond, that is a risk free asset, and for the remaining amount the performance asset. This strategy will guarantee the floor value F (T ) at maturity, unless the zero coupon bond issuer defaults. In addition, the investor will participate to a small degree in the performance asset’s return. At maturity, the portfolio value equals V (T ) = F (T ) +

 S(T )  · 1 − e−r·T S(0)

Unfortunately, this strategy is nearly always suboptimal with respect to expected utility maximization. 17.2.2 Option based portfolio insurance The option based portfolio insurance (OBPI) investment process was introduced in 1976 by Leland and Rubinstein (1988). It is based on the seminal work of Black and Scholes (1973)

270

Positive Alpha Generation: Designing Sound Investment Processes

on option pricing. The investor buys a zero coupon bond with a face value of the floor F (T ) at maturity. The remaining proceeds are then used to buy at the money European call options on the performance asset. The option portfolio represents the risky asset. The value of an OBPI based portfolio at any moment in time is given by V (t) = F (T ) · e−r·(T −t) + n · C(t, F (T ), S(t), T , σS (t), r) where C(t, K, S(t), T , σS (t), r) is the price of a European call option on the performance asset at time t with maturity T , strike price K, price S(t), implied volatility σS (t), and r the risk free rate. In addition, n=

V (0) − F (T ) · e−r·T C(0, F (T ), S(0), T , σS (0), r)

defines the number of call options that can be bought with the proceeds not invested in the zero coupon bond. A similar protection strategy can be engineered by holding 100 % of the performance asset and a put with a strike price of F (T ) on the performance asset. The equivalence between the bond plus call and equity plus put PIIP is given by the put–call parity theorem. OBPI investment processes have two key advantages. They are static structures and therefore require only trading when the PIIP is set up, and they are independent of the path of the performance asset’s value. On the other hand, the investor in an OBPI investment process faces various disadvantages. The performance of such an investment process depends on the number n of call options that can be bought with the proceeds not invested in the zero coupon bond. As the price of an option depends on the volatility of the underlying performance asset as well as the risk free rate, the possible performance participation rate depends on the value of these two parameters at setup. High volatilities and/or low levels of interest rates adversely affect the expected participation rate in the performance asset return. In addition, as the volatility is not observable, liquidity of the option market plays an important role. The more liquid the market is, the higher the probability that the volatility value used for pricing the option is actually closed to the fair value. If the option is an over-the-counter option, then the investor also faces counterparty risk, which may become significant as the option moves deep in the money. It may not be possible to buy options on the performance asset with the required strike price and/or maturity. In this case, the investor may dynamically replicate the option’s payoff. In complete markets, this can be done using a self-financing and duplicating strategy. Similarly, no zero coupon bond may exist for the specified maturity. Shorter maturity zero bonds or coupon bonds need to be used instead and the reinvestment risk needs to be hedged, for example by using swaptions.1 It is important to note that OBPI strategies guarantee the floor value only at maturity. The portfolio value may drop below the floor value before maturity due to raising interest rates, declining performance asset prices and/or decreases in volatility of the performance asset. Later Figure 17.5, Figure 17.6 and Figure 17.7 illustrate the evolution of the OBPI based portfolio value in different market environments. 1

A swaption is an option on an interest rate fixed for a floating swap agreement.

Capital Protection and Preservation Approaches

271

17.2.3 Constant proportion portfolio insurance The constant proportion portfolio insurance (CPPI) investment process is a dynamic strategy. It was introduced by Perold (1986) for fixed income instruments and by Black and Jones (1988) for equity instruments. It is based on the idea to subdivide the portfolio into a portion invested in a performance asset and the remaining portion in a risk free asset. The percentage invested in the risky asset is determined by an exogenous parameter, called the multiple m, and adjusted continuously. Both the floor and the multiple represent the investor’s risk preference or risk tolerance. At any given moment in time the cushion C(t) is defined as C(t) = max{V (t) − F (T ) · e−r·(T −t) , 0} The cushion represents the portion of the portfolio that can be lost without adverse effects on the capital protection at maturity. The CPPI investment process then defines that m · C(t) is invested at any given time in the performance asset and V (t) − m · C(t) in the risk free asset. Figure 17.3 illustrates the CPPI investment process. Depending on the rate of return of the performance asset, the amount to be invested in the risk free asset increases or decreases. In fact, the more the value of the performance asset drops, the less risky asset and the more risk free asset are held in the portfolio to ensure that at maturity the portfolio’s value is at least equal to the floor value F (T ). Depending on the multiple and the recent performance asset performance, the proportion invested in the risky asset may exceed 100 %. The higher the multiple, the more the investor will participate in a sustained increase in the performance asset. Nevertheless, the higher the multiple, the faster the portfolio will approach

Transaction costs

Cushion C (t) Portion of portfolio that can be lost

Risky asset

CPPI Portfolio Net portfolio value V (t)

Cushion C(t+1) Portion of portfolio that can be lost

Transaction costs

Risky asset

Risky asset m. C (t +1)

m. C (t )

Net portfolio value

Floor Discounted value of minimal portfolio value at maturity

Date t

CPPI Portfolio

V (t +1) Risk free asset V(t ) – m C (t )

Risk free asset

Market performance

FIGURE 17.3 Illustration of the CPPI investment process

Risk free asset V (t +1) – m. C (t +1)

Date t + 1

Floor Discounted value of minimal portfolio value at maturity

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Positive Alpha Generation: Designing Sound Investment Processes

the floor when there is a sustained decrease in the performance asset. Black and Perold (1992) showed that, in a complete market, the CPPI strategy can be characterized as expected utility maximizing, when the utility function is piecewise HARA. In continuous time and continuous price environments, the CPPI strategy provides capital protection for all nonnegative multiples. Unfortunately, neither time, nor prices of the performance asset are usually continuous. Therefore, in a discrete world, 1/m represents the maximal percentage drop of the performance asset’s price possible between two executed adjustment trades without having an adverse effect on the floor protected by the CPPI investment process. The CPPI investment process provides three key advantages over other PIIPs. CPPI portfolios are simple two asset portfolios that can be implemented for any type of performance asset. CPPI portfolios allow the investor to specify risk tolerance and targeted market participation through the specified multiple. If allowed, CPPI portfolios take advantage of leveraging the effect of good performance of the performance asset. It is easy to add specific features, like contingent intermediate cash flows or adjustable floors based on the portfolio’s value to the investment process. For example, the floor may be defined as F (t) = 90 % · V (t) instead of F (t) = e−r·(T −t) , making it independent of time and interest rate levels. Nevertheless, the investor also faces certain important disadvantages. In contrast with OBPI investment processes, the performance of CPPI based investment processes depends on the path, that is the evolution of the value of the underlying performance asset. As the strategy requires dynamic rebalancing of the target portfolio, transaction costs and market liquidity may adversely affect the performance of the CPPI portfolio or even fail to ensure the floor value at maturity. If the floor value at maturity is defined as a function of interest rates, it may not be possible to invest in the risk free asset at the prevailing rate when rebalancing the portfolio. This disadvantage may be mitigated, for example, by buying a swaption, allowing the risk free asset to be invested at any given time in the future at the required level of interest rates. Practical considerations implementing a CPPI based investment process In practice it is not feasible to continuously rebalance the target portfolio such that m · C(t) is invested in the performance asset. Transaction costs, discrete price movements, as well as restrictions on leverage, require implementing an approximation strategy for rebalancing the target portfolio. Key parameters are the minimal trade size when either buying or selling the performance asset, minimal cushion size, as well as minimal changes in the performance asset’s price before trading. Figure 17.4 shows a possible algorithm handling a CPPI based investment process, where t represents the date/time just before the portfolio readjustment. The gap or shortfall risk is defined as the risk that the investment process will fail to guarantee the floor value at maturity. If the volatility of the performance asset is known, or at least an upper bound is known, then it is possible to specify the CPPI parameters, that is the multiple and the floor such that the shortfall probability or the gap risk is bound by a constant.

Capital Protection and Preservation Approaches

273

Each day at 10 am, 2 pm and 4 pm execute the following instructions: Calculate NAV2 V (t) of the target portfolio adjusted for management and other fees. Calculate the floor value F (T ) · e−r·(T −t) using current risk free interest rate levels. Calculate the cushion value C(t) = V (t) − F (T ) · e−r·(T −t) . Calculate the theoretical investment in the performance asset P T (t) as m · C(t). Let P C(t) be the current investment in the performance asset. Let T C denote the incurred transaction costs, in percent, of the performance assets sold, usually 5 to 10 bp. if P C(t)  P T (t) then if P C(t) = 0 then if C(t)/V (t)
ρ1 then buy P T (t) · (1 − T C) of the performance asset else do not trade P T (t) − P C(t) else if
ρ2 then buy (P T (t) − P C(t)) · (1 − T C) of the performance V (t) asset else do nothing else if P C(t) > P T (t) then if P T (t)/V (t)  ρ3 then sell P C(t) · (1 + T C), that is all holding in the performance asset P C(t) − P T (t)
ρ4 then sell (P C(t) − P T (t)) · (1 + T C) of the performance asset else if AV (t) else do not trade end if Charge transaction costs • • • •

ρ1 ρ2 ρ3 ρ4

is the minimal cushion size before investing in the performance asset, is the percentage change in risky asset holding before buying more risky asset, is the threshold at which the risk asset is completely sold and is the percentage change in risky asset holding before selling the risky asset.

These parameters determine the trading frequency and thus the gap risk as well as the incurred transaction costs.

FIGURE 17.4 Sample algorithm for implementing a CPPI based investment process

17.2.4 Variable proportion insurance The variable proportion portfolio insurance (VPPI) investment process is an extension of the CPPI techniques. Instead of keeping the multiple m constant, it is adjusted dynamically. The risk preference of an investor in a CPPI investment process is specified by the combination of the multiple and the performance asset. There are two reasons for using a VPPI investment process instead of a CPPI one: • The investor’s risk preference changes over time; for example, the investor becomes more risk averse because he or she may be creating a family or retiring. 2

NAV, the net asset value, that is the portfolio value V (t) net of all costs.

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• The characteristics of the performance asset change over time. For example, the volatility, that is the riskiness of the performance asset, may increase or decrease, thus requiring an adjustment of the multiple to keep the overall PIIP in line with the investor’s risk preference. The investment manager may have risk forecasting capabilities. The performance asset, instead of being a given market index, can, for example, be an actively managed total return portfolio. In this portfolio, the ex-ante volatility would be adjusted according to the confidence levels in the forecasts, thus decreasing volatility in uncertain environments. Depending on the ex-ante volatility, the multiple of the VPPI would be adjusted using 1 day 1 day a formula similar to m = 1/VaR99.9 % , where VaR99.9 % is the one-day value at risk of the performance asset portfolio with a 99.9 % confidence level, leaving only a minimal gap risk.

17.3 COMPARING DIFFERENT PORTFOLIO INSURANCE INVESTMENT PROCESSES Both CPPI and the OBPI based portfolio insurance strategies have their advantages and disadvantages. The absence of arbitrage implies that neither of the two payoffs is greater than the other for all terminal values of the performance asset. The two payoff functions intersect each other. For any parameterization of the financial markets, that is risk free rate and performance asset, there is at least one value for the CPPI multiple m such that the OBPI strategy dominates/is dominated by the CPPI one. Figure 17.5, Figure 17.6 and Figure 17.7 illustrate the performance of OBPI as well as CPPI based portfolio insurance investment processes under different market conditions and compare them to the performance asset’s return. I model the performance asset through a Brownian motion stochastic process with an expected 220 200 180 160 140 120

Participating asset (S&P 500) OBPI portfolio value

Dec.98

Oct.98

Aug.98

Jun.98

Apr.98

Feb.98

Dec.97

Oct.97

Aug.97

Jun.97

Apr.97

Feb.97

Dec.96

Oct.96

Aug.96

Jun.96

Apr.96

Feb.96

80

Dec.95

100

CPPI portfolio value Floor Source: data sourced from Bloomberg Finance L P, author’s calculations

FIGURE 17.5 Performance simulation of OBPI and CPPI investment processes in an upwards trending market represented by the S&P 500 index between 1996 and 1998

Capital Protection and Preservation Approaches

275

120 110 100 90 80 70

Participating asset (S&P 500) OBPI portfolio value

Oct.03

Aug.03

Jun.03

Apr.03

Feb.03

Dec.02

Oct.02

Aug.02

Jun.02

Apr.02

Feb.02

Dec.01

Oct.01

Aug.01

Jun.01

Apr.01

Feb.01

50

Dec.00

60

CPPI portfolio value Floor

Source: data sourced from Bloomberg Finance L P, author’s calculations

FIGURE 17.6 Performance simulation of OBPI and CPPI investment processes in a downwards trending market represented by the S&P 500 index between 2001 and 2003 120

110

100

Participating asset (S&P 500) OBPI portfolio value

Dec.78

Oct.78

Aug.78

Jun.78

Apr.78

Feb.78

Dec.77

Oct.77

Aug.77

Jun.77

Apr.77

Feb.77

Dec.76

Oct.76

Aug.76

Jun.76

Apr.76

Feb.76

80

Dec.75

90

CPPI portfolio value Floor Source: data sourced from Bloomberg Finance L P, author’s calculations

FIGURE 17.7 Performance simulation of OBPI and CPPI investment processes in a sideways moving market represented by the S&P 500 index between 1976 and 1978

return µ = 12 % and volatility σ = 18 %, and set the risk free rate to r = 4 %. I use a three-year maturity horizon and 0.1 % two-way transaction costs. No management fee is charged. I set the parameters such that ρ1 = 5 %, ρ2 = 1 %, ρ3 = 1.0 % and ρ4 = 2 %. Black and Rouhani (1989) have shown that an OBPI investment process performs better if the performance asset’s value increases moderately. CPPI strategies perform better if markets drop or increase by small or large amounts. Bertrand and Prigent (2002) and Prigent (2007)

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proved that an OBPI investment process is a generalized form of a CPPI one, where the multiple is allowed to vary, that is a special case of a VPPI investment process with  

S(t) σ2 · (H − t)  + r+  ln F (H ) · e−r·H 2   S(t) · N  √    σ · H −t m(t, S(t)) =

C(t, S(t), F (H ), H, r)

where N (.) is the cumulative distribution function of a standard normally distributed random variable and C(·, ·, ·, ·, .) the price of a European call option. The multiple that makes the CPPI investment process equal to the OBPI one is the amount to be invested in the performance asset to replicate the European call option diminished by the OBPI cushion, which is the call option’s value. A major drawback of any insurance strategy, whether OBPI, CPPI, or even VPPI, is that the underlying strategy is pro-cyclical; that is when the risky asset loses in value, it is sold to the market, increasing the price pressure and resulting in a downturn price spiral. This effect has, for example, been significant during the 1986 ‘Black Monday’ market crash.

17.4 MANAGING RISK There are two categories of risks that an investment manager implementing a CPPI or dynamically replicating an OBPI investment process must address. These are so-called: • external risks, due to potential external events, like market crashes or liquidity crunches, disrupting the normal investment process, and • internal risks, due to failures in the investment process, like valuation or model errors. All risks finally cumulate in the so-called gap risk. The gap risk is the risk that, at maturity, the portfolio value is below the floor. On the external risk side, the most common risks translating into gap risk are: • liquidity risk, that is the risk that the investment manager cannot sell the performance asset at the current market price in sufficient volume, • discrete price risk, that is the risk that the investment manager is not able to sell at any given moment in time and at any given price, and • extreme event risk, that is prices of the performance asset drop to a value such that the portfolio value falls below the discounted floor value. External risks can be managed by selecting model parameters, like multiple values, such that the probability of an external risk having a material negative impact on the portfolio’s value is minimal. The difficulty in managing external risks is that they are not covered by traditional market theory. Extreme event theories (Embrechts et al., 2002) combined with Monte Carlo simulations are usually required to get a reasonable approximation of external risks. External risks can be covered by the investment manager by allocating a portion of his or her own capital to it or it can be sold to an insurance company, an investment bank or packaged and sold to the market.

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277

The most important internal risks to be mitigated when implementing a dynamic PIIP are: • model risk, that is the risk that the model underlying the investment process is incorrect under certain market conditions, for example that the model does not correctly handle large changes in the risk free interest rate, • valuation risk, that is the risk that the input parameters, like volatility, used for valuation purposes are poor approximations, and • counterparty risk, that is the risk that a counterparty offering a protective feature in the investment process, like a zero coupon bond or a derivative structure, defaults and therefore does not deliver on the contractual promises. Techniques similar to those illustrated in Chapter 5 for mitigating forecasting model risk can and should be applied to manage both model as well as valuation risk. Using multiple independent sources for parameters used in valuation methods allow the impact that adverse parameters provide to be reduced. As described in Chapter 14, counterparty risks can be managed efficiently by using marked-to-market techniques or by requiring collateral, or a combination of both approaches.

17.5 DESIGNING A CLIENT SPECIFIC CAPITAL PROTECTION SOLUTION Consider BestCorp, a corporation that wants to finance a ¤100 mio project, where ¤20 mio are due in one year and the remaining ¤80 mio in four years’ time. Instead of holding the funds in fixed income securities, BestCorp wants to participate in the European stock market in such a way that the financing structure is guaranteed. Today is 31 December 2007. I represent the European stock market through the Dow Jones Euro STOXX 50 index. I propose to implement an OBPI investment process for the first cash flow, that is the ¤20 mio due in one year’s time and a CPPI investment process for the remaining ¤80 mio. I choose an OBPI strategy for the short maturity cash flows because there are sufficient exchange traded and liquid European call options on the selected equity index. I select an option with a maturity date of December 20083 and a strike price of 4 400 that is closest to the current level of 4 404.64 of the index. Such an option trades at 352.90 with a contract size of 10. To implement the protective bond floor, I buy a one-year commercial paper from a corporate having the highest possible short term rating, that is P-1/A-1+ from Moody’s, respectively Standard & Poor’s. Such an issue is available at 12-month LIBOR −10 bp, that is at 95.55. This investment will guarantee the cash flow of ¤20 mio at 31 December 2008. As there are no exchange traded options with a four-year maturity on the Euro STOXX 50 index and as I want to avoid customized OTC options due to counterparty risk as well as pricing issues, I propose to implement a CPPI based investment process to protect the ¤80 mio cash flow due in December 2011. I use futures on the Euro STOXX 50 index to represent the performance asset. I implement the risk free asset by buying a portfolio of zero coupon bonds stripped from government bond issues by European Union member countries having a notional value of ¤80 mio and a maturity closes, but not later than 31 December 3

Options as well as futures at the Eurex expire on the third Friday of the respective month.

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Notional

Security

Option based portfolio 2 520 · 10 European call options on Euro STOXX 50, strike price at 4400, maturity December 2008 ¤20 mio Commercial paper @ LIBOR – 10 bp Constant proportion portfolio insurance portfolio 299 · 10 Long futures on Euro STOXX 50 expiring March 2008 ¤30 mio 0 Germany 07/04/2011, principal strip ¤30 mio 0 Spain 10/31/2011, principal strip ¤30 mio 0 The Netherlands 07/15/2011, principal strip ¤1.26 mio Cash account @ LIBOR – 25 bp ¤1.03 mio Futures margin account

Price

352.90 4.65 % 4 435 86.86 85.48 86.71 4.25 %

Value (mio)

¤0.89 ¤19.11 ¤13.264 ¤26.06 ¤25.64 ¤26.01 ¤1.26 ¤1.03

Source: data sourced from Bloomberg Finance LP, Eurex margin calculator, author’s calculations.

2011. The resulting portfolio is shown in Table 17.1. I invest the remainder of the proceeds, that is ¤1.257 mio, in a bank cash deposit and ensure that sufficient cash is available on the margin account required by the futures held, currently ¤3 443 per future contract. As BestCorp is characterized as very risk averse, I use a multiple of m = 3. This leads me to the initial portfolio for the CPPI investment strategy shown in Table 17.1, initially holding a future exposure of ¤13.56 mio, that is 299 long future contracts. As an additional feature, I invest the return from the call option in the OBPI part of the investment process at the end of the first year into the CPPI structure to increase the available cushion and enhance the expected participation in the European equity market.

4

Futures exposure.

18 Hedge Funds Although hedge funds have become popular among investors only during the last decade, they find their roots in 1949. Indeed, the term hedge fund dates back to a fund founded by Alfred Winslow Jones in 1949 (Johnson, 2007). His strategy was to sell short some stocks while buying others. In that way he hedged part of the market risk. In 1952 he converted the fund into a limited partnership and started charging a 20 % incentive or performance fee. By 1969, there were some 140 funds that could be characterized as hedge funds. The industry was relatively quiet until the early 1990s when an increasing number of high profile investment managers entered the hedge fund world, seeking fortune and fame. The success stories then took a hit in the late 1990s, early 2000 with the failure of prominent hedge funds, including the LTCM1 fund (Lowenstein, 2000). It is only through an increased move toward regulation, stemming from the regulator as well as the industry, that hedge funds have again become popular among investors. What makes a hedge fund unique and different from benchmark oriented and especially absolute oriented solutions, as described in Chapters 15 and 16? Hedge funds aim to generate an asymmetric return payout by participating in the upside price movements of assets and avoiding the downside (Ineichen, 2007). In that sense they are similar to capital protection solutions, as described in Chapter 17, at least from the perspective of their goals. However, rather than relying on an engineered structure, hedge funds use their investment skills to produce returns that have a low or no correlation to market returns. To do so, they rely heavily on the use of derivatives, leverage and illiquidity. In addition, the skill in hedge funds is often expressed in sophisticated quantitative models, as shown in Chapter 6, rather than relying on qualitative approaches, as described in Chapter 5. From an investor’s perspective, there are also differences between hedge funds and traditional investment solutions. Hedge funds are generally not transparent to the investor. This absence of transparency is often explained by the argument that providing information about the fund’s structure and strategy would diminish its competitive advantage and, therefore, its alpha generating capabilities. Hedge funds are also illiquid investments. The investor can only redeem parts after a given notice period, usually three to 12 months, depending on the hedge fund strategy. In addition, it is common practice for hedge funds to charge a high management fee of 2 % and a performance incentive fee of 20 % of all returns in excess of the agreed upon target return, usually the risk free rate plus a spread. Finally, in many countries hedge funds are only available to qualified investors, that is institutional investors. The retail or private banking investor can often only participate in hedge funds through structured products or fund-of-fund solutions.

1 Long Term Capital Management was a hedge fund founded in 1994 by John Meriwether, which had the Nobel prize winners Myron Scholes and Robert C. Merton on its board of directors. It was implementing highly leveraged fixed income arbitrage strategies.

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18.1 SUCCESS FACTORS OF HEDGE FUNDS The classification of success factors underlying hedge funds are the same as those for traditional investment processes,2 that is: • investment opportunities, • forecasting skills and • a transfer mechanism. One difference comes from how the investment universe is decomposed, respectively the investment opportunities classified. In a hedge fund investment process, the investment universe is selected and decomposed based on the skill set of the investment manager. For example, the universe in a merger arbitrage based hedge fund is composed of stocks from companies that are expected to participate in a merger or have announced a merger. Companies not open for participating in a merger and acquisition transaction are not considered part of the investment universe. In a traditional investment process, the investment universe is defined by skill independent characteristics, like countries or sectors. Let me compare the three success factors in the context of a hedge fund to those of a traditional investment process. Understanding the commonalities and the differences between the two is key in developing a successful hedge fund investment process. 18.1.1 Investment universe In the context of the fundamental law of active management, the universe should be chosen as broad as possible, maximizing the value of breadth, as shown in Chapter 2. Hedge funds follow this approach by defining the investment universe as all assets on which the available skills can potentially be applied. Table 18.1 compares the characteristics defining the investment universe of hedge funds and traditional investment processes. The definition of the investment universe is driven by the investment manager rather than the investor. Possible investment universes are, for example, distressed securities, convertible bonds and associated equities, futures, credit default swaps and corporate bonds. Table 18.1 Comparing the investment universe characteristics of hedge funds and traditional investment processes Characteristics

Hedge funds

Traditional

Subdivision of universe Universe selection Broadness of universe

Skill based Investment manager Focused

Asset based Investor Diversified

2 I denote by traditional investment processes, benchmark oriented or absolute positive return oriented alpha generating investment processes, as described in Chapters 15 and 16.

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18.1.2 Forecasting skills The performance of a hedge fund is, at least in the ideal world, attributable to the investment manager’s skills and not a function of randomness. This makes the portfolio returns rather independent of market returns and somewhat predictable. The forecasting skills enter into the fundamental law of active management through the information coefficient. As in traditional investment processes, the added value or generated alpha depends on the quality of the skill set. However, in contrast to common investment processes for traditional investment solutions, the forecasts in hedge funds are usually risk factor driven rather than based on exogenous dimensions. The advantage of the approach chosen by many hedge funds is that there is a close relationship between alpha and skill as, in general, the market rewards investors for taking risk, assuming the right risks are taken. Table 18.2 summarizes the differences between hedge fund and traditional forecasting skills. The focus is much higher and quality is preferred over quantity. For example, a global macro hedge fund may focus on forecasting inflation and GDP growth, as inflation pressure and real growth are two risk factors, which, if correctly forecasted, are rewarded. Other hedge fund managers focus on determining the impact of unexpected changes in risk factors on asset prices. Another skill set often used in hedge funds is that of pricing complex security structures, like convertibles or distressed bonds, and arbitrage away resulting pricing inefficiencies. In theory, the skill set used by hedge fund investment managers could also be applied to traditional investment processes. In practice, this is rarely the case because: • the hedge fund skills are hard to communicate transparently to investors3 and • efficient instruments required to transfer the skills into the portfolio are not allowed or are very complex to use in the traditional long-only, unleveraged world, for example from a regulatory perspective.

Table 18.2 Comparing the forecasting skills of hedge funds and traditional investment processes Characteristics

Hedge funds

Traditional

Key focus Number of forecasting skills

Risk factor driven Focused of a few skills with high information coefficient Primary focus on information coefficient Quantitative modeling, pricing based

Theme driven Large number of skills with reasonable information coefficients Primary focus on breadth

Fundamental law of active management Most common approaches

3

Investors think in asset classes and securities rather than abstract risk factors.

Qualitative, judgmental

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Assume as a skill set the skill of identifying mispriced convertibles. The misprices are small and can only be reasonably exploited, net of both explicit and implicit transaction costs, with large sized trades and by applying leverage. 18.1.3 Transfer mechanism One of the biggest differences between traditional investment processes and hedge funds exists in the area of the transfer mechanisms, that is how the forecasting skills are transferred into the investor portfolios. Hedge funds use mainly risk driven transfer mechanisms. They translate the forecasts into risk adjusted exposures to the specific factors. Hedge funds rely on sophisticated instruments to gain specific risk exposures. They are not limited to cash assets and mainly depend on derivative instruments in the implementation and trading value chain module. Whereas the investor may be willing to allow the use of hedge fund like investment instruments and is usually not opposed to specific alpha generating skills, portfolio construction restrictions remain. In many cases, these restrictions are imposed by the regulator or the investor because of his or her education.4 Table 18.3 summarizes the key differences in investment instruments used by hedge funds managers and traditional portfolio managers. Table 18.3 Comparing the portfolio construction instruments used by hedge funds and traditional investment managers Characteristics

Hedge funds

Traditional

Net short positions Leverage Use of derivatives

Yes Yes Yes

Investment restrictions Benchmark Transfer function framework

Few Zero, cash return Defined by skill and investment universe Tailored to investment universe, focused on extreme events

No No Partially, only covered positions5 Many Index, composition of indices Mean-variance based, factor model based Volatility, value-at-risk, shortfall risk

Risk management

18.2 EXPLOITABLE ALPHA GENERATING SOURCES A commonality between benchmark oriented solutions, absolute positive return investment products and hedge funds is that they are based on the investment manager’s skill set. It is therefore important to understand the characteristics of the different possible types of skills to identify those where an investment manager believes to have a competitive advantage. In 4 Institutional investors are responsible for the decisions they delegate to the investment manager. It is that responsibility that makes them reluctant to allow the investment manager to use investment instruments that they do not fully understand. 5 Derivative positions, for example a short DAX future position, are often allowed when the investment manager also holds a long position in equivalent cash instruments, for example a long position in a DAX exchange traded fund.

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hedge funds, alpha generating skills define the investment style. Hedge funds can be seen as a special case of absolute positive return investment solutions, described in Chapter 16. I therefore use the same taxonomy to identify and classify skills. 18.2.1 Engineering based skills The main idea behind engineering based skills is to construct a portfolio that has specific characteristics. These characteristics are usually such that the expected return, respectively risk adjusted return, is larger than the risk free rate. I distinguish between two types of engineering skills, that is: • identifying mispriced securities whose value will converge over time to a known value, its fair value, and • identify specific risk premiums and extract them. An example of a set of mispriced securities is a set of two bonds from the same issuer, with similar maturities, but issued in different currencies. By holding a long position in one bond and a short position in the other, combined with a currency swap eliminating the currency exposure risk, a positive return can be generated quasi risk free. The difficulty of such approaches is that the surrounding environment, like, for example, the costs of borrowing to enter in a short position or required collateral, must be taken into account. Consider buying an illiquid bond, for example a private placement, and finance it with a more liquid bond from the same issuer. The illiquidity premium, assuming I can hold the portfolio through maturity, can be obtained quasi risk free. The difficulties with engineering based skills are: • finding a reasonable number of investment opportunities and • correctly constructing the portfolio such that the expected return is positive both in theory and, after transaction costs, in practice. Table 18.4 summarizes various hedge fund investment strategies classified according to their required skill set and investment universe. 18.2.2 Statistical arbitrage skills The second category of skills that are very common in hedge fund investment processes are so-called statistical arbitrage skills. The idea is to take advantage of pricing asymmetries. In contrast with engineering skills, statistical arbitrage skills are based on statistical properties of asset or asset class returns. For example, bond A is valued expensive when compared to bond B, based on the investment manager’s proprietary valuation model. In addition, statistical analysis shows that, on average, such valuation differences persist for at most six months. The prices revert thereafter to their fair value. In this case, the investment manager takes a short position in bond A and a long position in bond B in his hedge fund. Other kinds of statistical arbitrage skills can be found in the area of mergers and acquisitions. The investment manager forecasts that a given announced or expected merger will happen and, as a consequence, the prices of the involved companies will converge. In contrast with engineering skills, the investment manager has to formulate a forecast on an event in the future, which may or may not materialize. In general, the forecast is

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Table 18.4 Classification of the most common hedge fund investment strategies according to their underlying main skills and investment universe Category

Engineering

Statistical arbitrage

Market forecasting

Directional

Systematic trading

Market timing

Short seller, distressed securities, emerging markets, global macro

Relative value

Fixed income arbitrage, convertible arbitrage Long–short credit

Long–short equity, equity market neutral

Long–short selection Risk premium

Liquidity premium, private placements

Event driven

Risk arbitrage Merger arbitrage, regulator arbitrage

Forecasting events

based on statistical analysis. In addition to the risk of incorrect forecasting, the difficulties found in applying engineering skills are also relevant. Many of the hedge fund investment processes rely on statistical arbitrage in one form or another. 18.2.3 Forecasting skills The third category of skills is based on explicitly forecasting events related to markets. In contrast with traditional investment processes, in hedge funds the focus is on high quality forecasts, that is with high hit ratios, rather than forecasts that can be easily transferred into portfolios. Usually the structure of the forecasts adjusts over time as the reward by the market for correct forecasts evolves. An example of such forecasting skills is formulating expectations for inflation relative to the market implied forecast. This forecast is transferred into the hedge fund portfolio using long, respectively short, positions in inflation linked and nominal yield bonds. If no inflation linked bonds exist, proxies from other markets can be used or inflation-linked swaps entered into. It is common that the forecasts are nondirectional, that is do not relate directly to the direction of the performance of equity or bond markets. Forecasting skill based investment processes are very similar to traditional absolute positive return investment processes, except for the additional flexibility in implementing, which is transferring the forecasts into the investor’s portfolio. As can be seen from Table 18.4, there are many hedge fund styles that are directional and based on forecasting. Therefore, a number of hedge funds are quite similar to traditional absolute positive return solutions. I believe that a truly successful investment manager should provide pure alpha generating solutions based on the hedge fund’s success factors combined with the traditional solution transparency and pricing approach.

18.3 ISSUES SPECIFIC TO HEDGE FUNDS When developing an investment process for managing a hedge fund, there are some specificities that need to be taken care of (Anderson, 2006; Nelken, 2005). These specificities, although not limited to hedge funds, play a major role in them.

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18.3.1 Pricing and valuation Many hedge fund investment processes use OTC derivatives and illiquid assets as part of their portfolio construction and trading process. These instruments are difficult to value as no regulated market exists on which prices are quoted. This, combined with the fact that they are usually acquired using leverage, that is third party financing, makes asset pricing a key issue in hedge funds. It is therefore important that a valuation setup is defined before a specific security is bought and not afterwards. The valuation setup must be able to answer the following questions at any given time, whether under normal market conditions or in market turmoil environments: • At what price and average volume is the asset traded under normal market conditions? • What is the fair value of the asset and how is it determined? By how much does the market price deviate from the fair value under normal market conditions? • What is the liquidation price of the whole position expected to be held in the asset under both normal market conditions and extreme event scenarios? Answering these questions allows the assets of a hedge fund to be valued for different purposes. In addition, they provide risk management information to whether it is sound to buy or hold a given position size at a given moment in time. Multiple external pricing sources should be used in the valuation process. It is even recommended that a binding agreement should be obtained for executable sizes of illiquid or OTC assets with specific counterparties. For example, when entering into a 10-year total return swap on the FTSE/Xinhua 25 index,6 the contract may contain a clause that the seller of the swap, usually an investment bank, agrees to buy back the contract at swap fair value − 0.5 % · t −

1 2

· closest S&P 500 future bid–ask spread

where t is the remaining time to maturity of the swap in years. The fair value calculation would be delegated to an independent valuation service provider. 18.3.2 Risk management In addition to managing market and credit risks, it is important that a hedge fund investment process contains provisions for managing the following risks, especially in turmoil market environments: • Liquidity risk, that is the risk that an asset can only be sold at a significant discount to its fair value or even that no counterparty is willing to buy the asset, • leverage based risk, that is the risk of a margin or haircut call requiring assets to be sold in order to raise cash in a downward moving market with poor liquidity, • counterparty risk, that is the risk that a given counterparty of an OTC transaction fails on its commitment, especially when the transaction is used for hedging, that is risk management, and 6 The FTSE/Xinhua 25 index includes the 25 largest Chinese companies comprising H shares and red chip shares, ranked by total market capitalization and listed on the Hong Kong stock exchange.

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• systemic risk, that is the risk that the action of another market participant forces the hedge fund to sell assets or close derivative positions at prices significantly below fair value. Taking exposures to these risks is in general part of a hedge fund’s investment process. Therefore they should be mitigated through: • diversification and • contractual agreements with the investors and counterparties about forced closures.

18.3.3 Management fees In the hedge fund industry it is common that the investment manager is rewarded for his or her skills through a performance or incentive fee. The idea is that the investor pays the investment manager a percentage, usually 20 %, of the realized performance or outperformance versus an agreed target return. The investment manager only receives the performance fee when the value of the fund exceeds the highest net asset value (NAV), the hurdle rate, previously achieved. For example, consider a hedge fund launched at an NAV of ¤1 000. Assume that it rises to ¤1 300 in the first year. Then a performance fee of 20 % on the 30 % return, that is ¤ 60, is due. If during the second year the NAV drops to ¤1 200, no performance fee is payable. If, at the end of the third year, the NAV is ¤1 430, a performance fee is due only on the 10 % return from the highest NAV, that is ¤1 300, rather than the full return from ¤1 200 to ¤1 430. This algorithm for calculating the performance fee is called the high watermark algorithm. Performance fees usually focus only on return and not on risk adjusted return. A performance fee can be interpreted as a call option written by the investor. The investment manager is long the call. However, in contrast with options on market indices, the investment manager defines both the return and the volatility of the underlying of the performance fee option. The upside potential is infinite, whereas the downside is limited to not receiving the performance fee, or losing the investor as an investor (Goetzmann et al., 2003). Hedge fund managers are thus engaged in some form of arbitrage in expectations. This situation creates a moral hazard issue that any investor should be aware of. An investment manager should therefore propose a performance fee on a risk adjusted return basis and provide sufficient transparency for the investor to exercise a control function.

18.4 DEVELOPING A HEDGE FUND INVESTMENT PROCESS Let me describe the key steps in developing a successful hedge fund investment process. I focus on the alpha generating as well as risk management aspects. Black (2004) provides a detailed overview of the other relevant aspects, like business strategies, operations or regulatory issues: 1. Determine the skill set at the basis of the hedge fund. 2. Derive the investment universe, that is the opportunity set, and the investment style. 3. Develop a risk management approach focusing on – market and credit risk, – extreme event risk and

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– hedge fund predominant risks, as described in Section 18.3.2. 4. Implement a transfer mechanism focusing on – portfolio construction, – liquidity and margin–call management and – collateral handling. The competitive advantage and potential uniqueness of a hedge fund depends on the selected skill set. The focus is usually on a few high performing skills. The selection process should start, similarly to the development of generic forecasting process, as shown in Chapter 5, with: (i) defining the expected outcomes, (ii) formulating the underlying market or economic theory and (iii) specifying the key assumptions made. Once the skill set, and therefore also the investment style, has been defined, the investment universe and the available assets and derivative instruments to be used should be selected. Very often, at this stage a prime broker7 is selected. As traditional transfer mechanisms, described in Chapters 10, 11 and 12, do not allow incorporating the needed sophisticated risk models, a dedicated risk measurement and management process should be set up. This risk management process must be very stringent and should cover the following areas: – market and credit risk management under normal conditions, – extreme events risk handling through scenario analysis, historical backtesting and Monte Carlo simulations, – liquidity and counterparty risk measurement and management, including collateral used, and – systemic risk contingency planning for risk that does not originate with the hedge fund investment process itself. A traffic light system approach has proven to be sound. Depending on the outcome of the risk measurement process, the light remains green, switches to orange or turns red. If the light is green, no action needs to be taken. When the light, as calculated by the risk measurement process, turns to orange, the portfolio positions must be reviewed and, if deemed necessary, changes implemented. In the case where the light turns to red, the initiating risk positions must be closed, realizing the associated profit or loss in order to avoid an uncontrollable state of the portfolio. In the fourth and final stage, a transfer mechanism must be defined. As the skills are usually only loosely coupled to specific assets or asset classes, this stage is highly financial engineering based. Techniques similar to those described by Mason et al. (1995) prove relevant. In addition, collateral management techniques as well as cash management approaches, such as those shown in Chapter 14, are important. 7 A prime broker, usually an investment bank, provides the core services needed by a hedge fund, that is custodian services, security lending, financing facilities, customized trading technology, as well as operational support by acting as the hedge fund’s primary operations contact with all other broker dealers.

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In contrast with other investment solutions described in this book, no investor needs or utility function definitions are considered. On the one hand, investor needs are not relevant: • because the goal is to produce a systematic absolute positive return, usually of the form LIBOR + α, and • because the investment process is not transparent. However, on the other hand, it is this absence of investor needs that make hedge fund solutions inappropriate for many investors, including those with specific regulatory restrictions. It may nevertheless be argued that the return uncorrelated with market returns delivered by hedge funds is an investor need and therefore hedge funds are relevant as portfolio diversifiers. 18.4.1 A sample investment process for a commodity hedge fund The underlying idea of the proposed hedge fund investment process is to enter into long–short trades on pairs of commodities. The pair trades are chosen such that they have low or even negative correlations, allowing the construction of a low risk portfolio expecting positive absolute return. They are transferred into the portfolio using an optimal risk budgeting based static portfolio construction algorithm. The implementation and trading is executed solely relying on exchange traded futures on the individual commodities. I assume that I am capable of forecasting the sign of the excess returns between pairs of commodities, for example grain will outperform wheat or natural gas will outperform crude oil over the next three months. In addition, I assume that I am capable of predicting the probability of these forecasts being correct. Assuming that the relative returns between pairs of commodities follow a normal distribution, I calculate, for a given confidence level, the expected return per unit of risk solving equation confidence = Pr{IR
0}

(18.1)

of the considered pair of commodities. The investment universe is defined as the set of commodity futures contracts traded on the CME.8 As my forecasting horizon is three months, I use futures maturing in three months. The margin cash as well as additional collateral is invested, whenever possible, in overnight deposits and three-month US treasury bills. I rely on a traditional volatility based risk model under normal market conditions. I enter into future contracts for no more than 20 % of the average weekly traded positions over the last three months. In addition, I limit the leverage to 400 % and as such always keep cash or treasury bills for at least 25 % of the portfolio exposure. I limit any single position to 10 % of the total exposure of the portfolio. I introduce an automatic position closing algorithm, if any position exceeds by more than 2.5 % the maximal allowed exposure. The risk arbitrage transfer mechanism works as follows: 1. I start by constructing, for each pair of assets for which I have formulated a forecast, a long–short portfolio with a fixed volatility, for example 1.0 %. 8 Chicago Mercantile Exchange, also called The Merc. It was founded in 1898 as the Chicago Butter and Egg Board.

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Table 18.5 Sample commodity hedge fund risk management and portfolio construction outcome

Confidence Coffee Corn Crude oil Gold Natural gas Silver Sugar Wheat Risk Alpha Weight

Sugar vs coffee

Natural gas vs crude oil

Silver vs gold

Wheat vs corn

Portfolio

53 % −3.3 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 3.3 % 0.0 % 1.0 % 0.2607 0.7034

52 % 0.0 % 0.0 % −3.6 % 0.0 % 3.6 % 0.0 % 0.0 % 0.0 % 1.0 % 0.1737 0.6760

52 % 0.0 % 0.0 % 0.0 % −6.6 % 0.0 % 6.6 % 0.0 % 0.0 % 1.0 % 0.1737 0.5748

54 % 0.0 % −3.3 % 0.0 % 0.0 % 0.0 % 0.0 % 0.0 % 3.3 % 1.0 % 0.3479 0.8077

−2.32 % −2.63 % −2.47 % −3.79 % 2.47 % 3.79 % 2.32 % 2.63 % 1.0 % 0.6817

Source: data sourced from Bloomberg Finance LP, sample forecasts, author’s calculations

Table 18.6 Historical correlations between forecasted commodity pairs

Sugar vs coffee Natural gas vs crude oil Silver vs gold Wheat vs corn

Sugar vs coffee

Natural gas vs crude oil

Silver vs gold

Wheat vs corn

1.0000 −0.3367 −0.1782 0.0119

−0.3367 1.0000 0.0141 −0.2374

−0.1782 0.0141 1.0000 −0.2575

0.0119 −0.2374 −0.2575 1.0000

Source: data sourced from Bloomberg Finance LP, author’s calculations

2. Then I select all pairs for which the correlations among them are less than 0.25. If any pair has a correlation larger than 0.25, it is eliminated from the set of considered pairs of commodities. Table 18.5 shows a sample of four forecasted pairs retained, their correlations being shown in Table 18.6. 3. Finally, I calculate the weights attributed to each pair maximizing the expected alpha of the portfolio based on the forecasted confidence levels and the associated information ratios, or alphas, derived using Equation (18.1). The resulting weights of the sample portfolio are shown in Table 18.5 assuming a total portfolio risk of at most 1.0 %. 4. Three days before the futures settlement date, in order to avoid physical delivery, I close all positions and realize the profit and losses.

18.5 HEDGE FUNDS AS AN ASSET CLASS A question that very often comes up is whether or not hedge funds are an asset class. The short answer is ‘no’, because hedge funds are an investment strategy implemented using assets or asset classes. However, that is only part of the truth. Let me recall the properties of an asset class. The securities forming the asset class must be homogeneous. Assuming a

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Excess return over LIBOR (ann.) (%)

Correlation with USD cash

10.8 8.6 −0.5 10.7 9.5 11.4 6.2 13.4 12.0 6.9 4.3 6.0 9.4

7.5 4.6 16.8 15.6 2.8 5.5 3.6 10.4 9.8 11.9 0.5 4.5 14.1

6.4 4.2 −4.8 6.4 5.2 7.0 1.8 9.1 7.7 2.6 0.0 1.6 5.0

0.11 0.21 0.06 −0.08 0.27 0.04 0.06 0.09 0.12 −0.04 1.00 0.13 0.10

0.08 0.00 0.15 −0.12 0.07 −0.14 0.05 0.19 0.00 0.24 0.13 1.00 −0.11

Correlation with US equities

Standard deviation (ann.) (%)

Index Convertible arbitrage Dedicated short bias Emerging markets Equity market neutral Event driven Fixed income arbitrage Global macro Long/short equity Managed futures USD cash USD government bonds US equities

Correlation with USD government bonds

Hedge funds strategy

Average return (ann.) (%)

Table 18.7 Characteristics of different hedge fund asset classes as well as their correlations to traditional asset classes based on monthly data from the CS/Tremont hedge fund indices between December 1993 and December 2007

0.49 0.15 −0.76 0.48 0.36 0.55 0.06 0.24 0.59 −0.10 0.10 −0.11 1.00

Source: data sourced from Bloomberg Finance LP, author’s calculations

narrow set of skills, this may or may not be the case. The asset class must be distinguishable, that is exhibit times of significant relative performance. Table 18.7 shows the characteristics of the most common hedge fund strategies as well as their correlation with cash, government bond and equity asset classes. The property of mutually exclusive is somewhat harder to verify. Figure 18.1 shows a graphical representation of the correlations between the different hedge fund styles and traditional asset classes. As can be seen, the correlation between the hedge fund strategies is low, confirming the ability to distinguish criteria. Depending on the strategy of the hedge fund, it is satisfied. For example, long/short equity hedge funds are not mutually exclusive with the equity asset class as they have a correlation at 0.59. However, when comparing fixed income arbitrage to equity market neutral strategies, the mutually exclusive condition is satisfied. Therefore, based on these properties, hedge funds can be classified as asset classes. However, the main difficulty with considering hedge funds as an asset class is the required forecasting capabilities. It is very difficult, if not impossible, to forecast the return of a hedge fund. This would even be the case when hedge funds are transparent. Indeed, forecasting hedge fund returns is closely related to forecasting whether or not the hedge fund investment manager’s forecasts are correct. This is actually equivalent to forecasting asset or asset class returns, making the hedge fund investment manager redundant. This argument holds as long as the hedge fund manager is producing alpha based purely on skills. As any excess return over the risk free rate can be related to some sort of skill, as shown in Chapter 2, forecasting hedge fund performance is therefore at least as difficult as forecasting markets.

Hedge Funds

291 Cash Bonds Equities Hedge funds Convertible arbitage Dedicated short 0.50–1.00

Emerging markets

0.00–0.50

Equity markets neutral

−0.50–0.00

Event driven

−1.00–−0.50

Fixed income arbitage Global macro Long–short

C as Bo h n E ds C on He qui ve dg tie rti e s b f D le und Em edic arb s Eq a it ui erg ted age ty in m g m sho ar ke ark rt t e Fi xe E s ne ts di ve u nc n tra om t d l e riv G arb en lo ba itag e l L M o ma an ng cr ag –s o ed ho fu rt tu re s

Managed futures

Source: data sourced from Bloomberg Finance L P, author’s calculations

FIGURE 18.1 Correlation between different hedge fund strategies and traditional asset classes based on monthly data from the CS/Tremont hedge fund indices between December 1993 and December 2007 Table 18.8 Percentage of best and worst performing asset classes on a monthly basis between 1993 and 2007 based on monthly data from the CS/Tremont overall hedge fund index, the Citigroup USD one-month money market index (cash), the Citigroup US government bond index (bonds) and the MSCI USA equity index (equities) Asset class Cash Bonds Equities Hedge funds

% best

% worst

14 21 40 24

26 27 35 13

Source: data sourced from Bloomberg Finance LP, author’s calculations

Another question that comes up very often is whether hedge funds are a required asset class, assuming forecasting skills. Tables 18.8 to 18.10 show how often a specific asset class was the best performing, respectively worst performing, asset class and compare hedge funds to cash, bonds and equities. If only allowed to take a single long position, in 76 % of cases hedge funds would not be part of the constructed portfolio. This is even the case for 80 %, respectively 85 %, of the monthly period when considering the specific strategies market neutral or fixed income arbitrage. On the other hand, hedge funds have been the worst asset

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Positive Alpha Generation: Designing Sound Investment Processes Table 18.9 Percentage of best and worst performing asset classes on a monthly basis between 1993 and 2007 based on monthly data from the CS/Tennant market neutral hedge fund index, the Citygroup USD one-month money market index (cash), the Citigroup US government bond index (bonds) and the MSCI USA equity index (equities) Asset class Cash Bonds Equities Market neutral hedge funds

% best

% worst

7 24 50 20

23 27 38 13

Source: data sourced from Bloomberg Finance LP, author’s calculations

Table 18.10 Percentage of best and worst performing asset classes on a monthly basis between 1993 and 2007 based on monthly data from the CS/Tremont fixed income arbitrage hedge fund index, the Citigroup USD one-month money market index (cash), the Citigroup US government bond index (bonds) and the MSCI USA equity index (equities) Asset class Cash Bonds Equities Fixed income arbitrage hedge funds

% best

% worst

6 26 53 15

24 27 38 12

Source: data sourced from Bloomberg Finance LP, author’s calculations

class only in about 13 % of the considered monthly periods. This shows that hedge funds, as measured by the used indices, provide a better protection against negative returns than traditional asset classes.

19 Liability Driven Investing In recent years, liability driven investing (LDI) has become one of the most prominent approaches used by pension and endowment funds1 to determine their investment strategy. LDI has emerged as a solution approach to the asset and liability management (ALM) problem. The basic idea is simple. First, determine the liabilities and their characteristics, such as their interest rate sensitivities. Then, develop an investment solution that matches or at least approximates the liability characteristics. The investment strategy is determined conditioned on the liabilities. Rather than generating a positive return or outperform a market portfolio, the liabilities are considered the benchmark to match or outperform. The portfolio of a pension fund should not be evaluated in terms of market risk and return, but rather in terms of liabilities or underfunding measures (Boender et al., 1998; MacBeth et al., 1994).

19.1 THE CONCEPT OF LIABILITY DRIVEN INVESTING As the LDI approach is the investment manager’s view of the ALM problem (Martenelli, 2006), the liabilities are considered given: • explicitly, for example as future cash flow streams or as their net present value, • through a parametric specification, for example as a target return of CPI + 50 bp or as a function of contributions, GDP growth, interest rates and mortality rates, or • as a simulation model, that is an algorithm that calculates the expected liabilities based on a given set of scenarios. 19.1.1 The goal of a pension fund The goals of a pension fund, or more specifically the structure of its liabilities, can be classified along the four agents involved in a pension plan, as shown in Figure 19.1. Plan sponsors, whether corporations or plan participants, are responsible for defining the overall aim of the pension plan. They determine the structure of the future cash flows, as a function of contributions and claims. They focus on minimizing the risk of failing to meet the defined cash flows, that is an underfunding situation. Employees, or active plan participants, aim to achieve at least a predefined high return on their contributions. The retirees, as the recipients of pensions, aim to ensure that the payments received do not diminish their purchasing power. Their goals are usually formulated in terms of inflation protection. Finally, the regulator often imposes additional restrictions to ensure the solvency of the pension plan. 1 I use the term ‘pension’ or ‘endowment fund’ as a proxy for any investor who performs an investment to meet a set of future liabilities. An individual taking up a mortgage and wanting to amortize it or parents saving to finance their newly born’s education fall in the same category as pension funds, investment trusts or family offices, to name just a few.

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Positive Alpha Generation: Designing Sound Investment Processes Plan sponsor

Employees

Pension plan

Regulator

Retirees

FIGURE 19.1 Agents involved in determining the goals and therefore the liability structure of a pension fund

19.1.2 Analyzing the liabilities At the beginning of the development of an LDI solution stands the analysis of the liability structure and its characteristics. The liability structure is a consequence of the different goals that a pension plan is pursuing. The liability characteristics can be analyzed along three key dimensions: • Expected future cash flows. Contributions and pensions make up the most important cash flows of a pension fund. • Promised accruals. Especially defined contribution pension plans often guarantee, explicitly or implicitly, a minimal rate of return on the assets matching the liabilities; that is the liabilities increase at a predefined rate. This rate may either be fixed, for example at 4 %, or linked to economic figures, like inflation, long term interest rates or economic growth. • Factor sensitivity. In the context of LDI, the key characteristics of a stream of liabilities are their sensitivity to factors, like interest rates, inflation and life expectations, to name the most important ones. If these factor exposures can be matched by equivalent factor exposures on the asset side, their risk can be removed from the pension plan. The analysis of liabilities can be made as a snapshot in time through the calculation of a net present value of the future cash flows or considered over time as a sequence of expected cash flows. Figure 19.2 illustrates a liability structure of a pension fund over time. In the short to medium term, the liabilities increase up to the year 2029. This is usually the case when the number of employees and salaries increase. Starting in 2030, the expected liabilities begin to decrease. More cash is paid out to retirees than is contributed by active employees. Although showing the liabilities over time, Figure 19.2 represents a snapshot based on current assumptions about: • the evolution of the number of employees and retirees, • the salary structure over time and • the mortality rate, to name the most important parameters. It is often useful to use stochastic modeling techniques, as described in Chapter 12, to represent the uncertainty related to the parameters, also called risk factors, that impact the liabilities. Relying on an LDI strategy is most relevant if a significant relationship of the risk factors underlying the liabilities with risk exposures in the asset market can be shown. On the

Liability Driven Investing

295

35 30

(in mio.)

25 20 15 10 5

20 08 20 11 20 14 20 17 20 20 20 23 20 26 20 29 20 32 20 35 20 38 20 41 20 44 20 47 20 50 20 53 20 56 20 59 20 62

0

Source: sample data, author’s calculations

FIGURE 19.2 Sample liability structure of a pension fund

other hand, if the liabilities have no direct relationship with asset markets, for example if the liabilities are required to accrue with a fixed rate of 2.5 %, or if the liabilities are not evaluated at market value, LDI strategies will not be suitable. In this case either benchmark oriented, absolute return or capital presentation based solutions are appropriate.

19.2 PORTFOLIO CONSTRUCTION IN A LIABILITY DRIVEN INVESTMENT CONTEXT The techniques used in LDI are often call immunization, as the goal is to immunize the risk from the liabilities with the same risk on the asset side. From a practical standpoint, these techniques can be classified into several categories. The first category, called cash flow matching, constructs a portfolio that ensures a perfect match between the cash flows from the portfolio of assets and the liabilities. This construction process can be best performed using financial engineering techniques. The most appropriate instruments are fixed income securities, fixed rate, floating rate and especially zero coupon bonds, combined with customized interest rate or total return swaps. The fixed income securities ensure that the generated future cash flows and the swap transactions match these cash flows to the actual liability cash flows. Cash flow matching techniques require the future cash flows to be either known explicitly with certainty or expressed as a function of current market levels, especially interest rates and inflation. In addition, the cash flows should be outflows rather than inflows. Inflows require the conclusion of sophisticated forward agreement transactions on future cash flows. For example, buying today a five-year zero coupon bond at a value of ¤10 mio with a settlement date in one year requires an expected cash flow of ¤10 mio to be invested in one year at the current prevailing market rates. Such transactions are nonstandard and will incur a nonnegligible cost. The major advantage of the cash flow matching approach is that is allows a perfect risk management. If the cash flows are perfectly matched, there is not any risk, except for a default risk of

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the involved counterparties2 in the transactions. Cash flow matching techniques are most appropriate in satisfying the needs of the retiree, the regulators and partially the sponsors in the four agents pension fund model. If using cash flow matching is not possible because of unknown cash flows, the complexity of the resulting transactions or the unavailability of securities or counterparties to engineer the matching portfolio, risk factor matching techniques can be applied. First, the liabilities are analyzed and key risk factors identified. Usually these risk factors are interest rate risk and inflation risk. The risk factor matching portfolio is constructed in order to match the determined risk sensitivities of the liabilities (Leibowitz and Weinberger, 1981, 1982, 1983). For example, if the interest rate risk of the liabilities can be characterized by an 11-year duration, the risk factor matching portfolio can be constructed as a fixed income portfolio having an 11-year duration. The interest rate risk due to parallel shifts of the yield curve is immunized. Depending on the structure of the liabilities and their changes over time as well as the changes in the asset markets, the risk factor matching portfolio requires dynamic adjustments. Dynamic adjustments are always required if the match between the assets and liabilities is imperfect. Exactly when and how often to adjust the portfolio structure depends on the risk implied by the mismatch, that is risk, opportunity costs and transaction costs. Rebalancing techniques, similar to those described in Chapter 15 for benchmark oriented solutions, can be applied. The risk factor matching techniques are most appropriate if the changes of the liabilities are driven by common risk factors that can be matched by risk factors found in the asset market. In contrast with cash flow matching techniques, risk factor matching approaches are much more flexible and easy to implement. Their major drawback is that they focus solely on removing identified risk from the relationship between assets and liabilities. No excess returns, as aimed at by the employees, one of the four agents involved, are generated. Rather than matching specific risk factors of the liabilities with assets, risk exposure management techniques focus on the overall uncertainty of assets not matching liabilities. Consider liabilities with a net present value of L(t) at time t and an asset portfolio with a value of A(t). A solution to the ALM problem requires that A(t)
L(t) at any time t. The risk exposure management approach consists in constructing an asset portfolio such that the probability of A(t) < L(t) is minimized. At a given point in time, this probability can be, for example, measured as the value-at-risk. The asset portfolio is constructed such that A(t) − L(t)
−VaRα (A(t))

(19.1)

for a given confidence level α, for example set at 84 % (McCarthy, 2000), and a given time horizon, for example one year. The key advantages of the risk exposure management approach are that they: • allow consideration of portfolios with assets other than fixed income securities and • do not focus solely on matching or immunizing risks, but also allow generation of excess return. Risk exposure management approaches require dynamic readjustments of the portfolio structure over time as A(t) − L(t) changes. The resulting portfolios are adjusted pro-cyclically, 2

Bond issuers and swap counterparties.

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297

similar to CPPI based investment strategies, the floor being defined as a function of L(t) and VaRα (A(t)) rather than a fixed value. Unfortunately, in some situations, a solution does not exist to the problem in Equation (19.1). In addition, if the time horizon between two observations of A(t) − L(t) is short, for example one month, opportunities may be missed.3 The fourth portfolio construction approach focuses on the expected returns rather than on risk. In the return matching approach, the rate of change of the liabilities is first derived. For example, the most common Swiss pension fund plan’s liabilities grow at a rate around 5.5 %. Then, in a second step, long term equilibrium rates of return for a large set of asset classes are determined. Finally, a portfolio is constructed such that the expected rate of return of the portfolio matches the rate of change of the liabilities, and the risk, measured in terms of volatility or downside risk, is minimized. The techniques for defining a benchmark, described in Chapter 15, should be applied to construct that portfolio. Alternatively, different sources of alpha can be combined to achieve the target return. The methodologies for generating portable alpha solutions, as described in Chapter 16, are applicable. The choice of whether to use a solution based on the benchmark definition or portable alpha, or a combination of both, depends on the time horizon over which the return should be generated. Short time horizons require absolute positive return like approaches. If the time horizon is large, that is more than three to five years, benchmark oriented solutions are preferred. Indeed, the diversification effect of the benchmark materializes. In addition, the fact that returns increase linearly over time whereas risk only with the square root of time increase the expected risk adjusted return with time. In contrast with the other three approaches, priority is given to the performance aspect, rather than the risk management or immunization goal. The main drawback of the return matching approach is that the focus is no longer on matching risk between assets and liabilities. If the return generating approach fails, the mismatch between assets and liabilities may significantly increase. Table 19.1 summarizes the different approaches for constructing LDI portfolios and characterizes their specifications.

19.3 LIABILITY DRIVEN INVESTMENT SOLUTIONS Defining the portfolio construction approach provides the toolbox needed to construct LDI solutions. I distinguish between three types of LDI solution offerings that an investment manager can provide, that is: • building block based approaches, • cash flow transforming approaches or • risk characteristic based approaches. In all the solutions described, the focus is put on the engineering aspect, rather than the alpha generating skills. Nevertheless, both alpha generating capabilities as well as risk management skills can and should form a significant part of an investment manager’s LDI value proposition. 3 Indeed, as the risk increases by the square root of time and return linearly with time, a larger risk budget in terms of value-at-risk is available if considering a longer time horizon. This effect represents what I call the diversification over time effect.

298

Positive Alpha Generation: Designing Sound Investment Processes Table 19.1 Comparison of LDI portfolio construction approaches

Engineering complexity Asset classes used Actor preference Skills required

Cash flow matching

Risk factor matching

Risk exposure matching

Return target matching

High

Medium

Medium

Low

Fixed income Fixed income Sponsor, regulator, Sponsor, regulator retiree Engineering Engineering, risk management

Forecasting dimension Risk of approach

None

Characteristics

Immunization

Change of liabilities

All All Sponsor, employee, Employee regulator Risk management, Long/short term long term alpha alpha, risk management Risk factors Total risk Asset returns, total risk Change of Invalid risk Insufficient liabilities, selection measure forecasting of risk factors skills Immunization Portfolio return, Portfolio return risk management

19.3.1 Building block based approaches The liability structure of a pension fund can be subdivided into three different time-frames, as shown in Figure 19.3. The first five years cash flows are matched using the cash flow matching approach. For the next five to 20 years, the cash flow matching approach is no longer realistic, because of the lack of available instruments and the uncertainty inherent in the liability structure. Risk factor exposure methods are most appropriate for matching risk over this medium term time horizon. Finally, for cash flows extending beyond 20 years,4 the risk exposure method is most appropriate. For the second and third time horizons, individual building blocks, for example collective investment vehicles or mutual funds, can be constructed and provided to the investor. The risk factor exposure time horizon could be matched by a combination of investment products with constant durations, as shown in Table 19.2. The constructed portfolio matches a 6.6 years interest rate sensitivity risk. The investor or the investor’s investment consultant can then combine the building blocks to match the specific duration risk of the investor’s liabilities. In addition, using more than two blocks allows key rate durations of the liabilities to be matched rather than only the duration. To match the liabilities with a long term horizon, a well diversified portfolio with an equity like risk premium is adequate. Table 19.3 illustrates such a diversified portfolio. The diversification effect allows, when correctly constructed, an equity like risk premium to be achieved with an expected lower volatility. Indeed, the constructed portfolio has a volatility of about 80 % of that of a large capitalized equity market index, hedged in GBP. In addition, the correlation with domestic equities is 0.82. Both the fixed duration as well as the growth portfolio can be managed passively or actively. An active component can add, assuming investment management skills, additional 4 The best time horizon for switching from risk factor exposure based methods to risk characteristics based models depends on the specificities of the pension fund’s liabilities and their uncertainty as well as their exposure to risk factors that cannot be matched by exposures on the asset market.

Liability Driven Investing

299

35 30 25

(in mio)

20 15 10 5 cash flow matching

risk factor exposure (duration) matching

risk exposure (surplus value-at-risk) matching

20 08 20 11 20 14 20 17 20 20 20 23 20 26 20 29 20 32 20 35 20 38 20 41 20 44 20 47 20 50 20 53 20 56 20 59 20 62

0

Source: sample data, author's calculations

FIGURE 19.3 Typical structure of the liabilities of a pension fund, when the expected future liabilities are a function of interest rates, inflation and economic growth Table 19.2 Sample mix of fixed income investment products with constant duration for a UK based investor combined to achieve a specific interest rate risk exposure and a target duration of 6.6 years Weight (%)

Maturity bucket (years)

40 30 20 10

3 to 5 5 to 7 7 to 10 10 to 20

Investment product description

Government Government Government Government

bond portfolio bond portfolio and corporate bond portfolio and corporate bond portfolio

Target duration 4 6 8.5 15

Source: sample data.

return or alpha without increasing the total risk of the individual components and the total portfolio. However, the active risk taken must be chosen carefully so as to be orthogonal to the risk of the building blocks. The advantage of the building block approach is its scalability. The investor can combine the individual components to match specific liabilities. Especially consultants prefer this approach as they can efficiently use the building blocks to advise their investors on customized solutions. The drawback lies in the fact that the full power of diversification and risk management is not used as the different components of the portfolio are constructed independently. Indeed, the potential diversification effect between the risk factor exposure and the risk exposure building blocks is not actively managed. 19.3.2 Cash flow transforming based approaches The idea of cash flow transforming based approaches is to provide, as an overlay, a portfolio that transforms a sequence of cash flows independent of the liabilities into a sequence of cash flows matching the liabilities or at least their risk characteristics. Rather than decompose the

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Table 19.3 Sample diversified portfolio for a UK based investor with an expected risk premium of 3.7 %, about 2.5 % higher than large capitalized equity markets, about 80 % of the volatility of equity markets and a correlation of 0.82 with UK equities Weight Asset classa (%)

5 10 5 10 5 15 5 15 10 20

UK equities, indexed to FTSE 100 Small capitalized UK equities International developed equities (hedged in GPB) Emerging markets equities Global private equity Convertible bonds High yield bonds Emerging markets UK property International property Portfolio

Expected risk Volatility Correlation to premium (annualized) UK equity (%) (%) market 1.0 2.8 1.3 5.5 5.1 2.6 0.1 2.2 4.4 5.5 3.7

13.4 17.9 12.8 24.6 16.1 11.7 11.1 15.0 17.2 14.9 10.8

1.00 0.73 0.91 0.69 0.74 0.71 0.49 0.52 0.47 0.58 0.82

Source: data sourced from Bloomberg Finance LP, author’s forecasts, author’s calculations a Unless specifically mentioned, all asset classes are actively managed with a target tracking error of 1 % to 2 %. The additional alpha generated through active management is not included in the shown expected risk premium. Assuming an information ratio of 0.5 for each active manager of the individual asset classes, an additional 50 bp to 1 % of return will be expected.

ALM problem in time, as is done in the building block approach, the cash flow transforming approach decomposes the problem in space. First, the investment manager manages, actively or passively, a portfolio such that it generates a set of cash flows, usually of the form LIBOR + x, every six to 12 months. This portfolio may be either a pure fixed income portfolio or a portable alpha portfolio, where the expected alpha matches the required rate of return x. Depending on the time horizon, as well as the current surplus, that is the difference between the current portfolio’s value and the present value of the liabilities, more or less risk may be taken to generate LIBOR + x. Then, in a second step, a swap transaction, usually concluded with an investment bank, is entered into which transforms the LIBOR + x cash flows into cash flows that match the liability stream. For example, assume that the pension fund guarantees a rate of return of CPI + 50 bp on the paid contributions. The swap transaction would exchange a cash flow of LIBOR + x fixed over the whole duration of the swap against a cash flow of CPI + 50 bp. Usually such swap transactions are concluded for time horizons of five to 20 years. At the end of the time horizon, a new swap transaction with a different LIBOR + x paying cash flow is set up. Figure 19.4 illustrates the cash flows of a cash flow transforming based portfolio, including the value of the cash flow generating portfolio. Ideally the latter one would behave like a floating rate bond. However, due to the additional risk required to generate LIBOR + x, additional volatility in its value is necessary. 19.3.3 Risk characteristic based portfolios In risk characteristic based approaches, controlling and managing the risk of the portfolio relative to the liabilities stand in the center. Risk may be measured at the total portfolio level or at the individual risk factor level.

Liability Driven Investing

301

30%

4.00

25%

3.50

20%

3.00

15% 10%

2.50

5%

2.00

0%

1.50

−5%

1.00

−10%

0.50

−20%

0.00 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037

−15%

Paying cash flow

Receiving cash flow

Portfolio value (right axis, in mio. £) Source: sample data, author's calculations

FIGURE 19.4 Sample cash flow transforming portfolio values and exchanged cash flows assuming no cash inflows or outflows of the portfolio

The risk at the total portfolio level is, for example, measured in terms of the surplus at risk. To construct a total risk managed portfolio, I start by formulating market return expectations. The time horizon for these forecasts should be chosen as a compromise between: • maximizing the confidence in the forecasts, • minimizing the time horizon of the forecasts and • matching the time horizon of the liabilities. Then I construct a portfolio, using a mean-variance based approach, as shown in Chapter 10. It is common practice to call the constructed portfolio the strategic asset allocation or Table 19.4 Sample combination of portable alpha strategies to achieve a LIBOR + 200 bp target return set by the liabilities of a UK pension fund assuming independence of the different alpha sources Weight (%) 15 25 30 10 20

Benchmark oriented investment strategy –portable alpha generation

Risk (%)

Information ratio expected return

Global tactical asset allocation –short benchmark position using futures S&P 500 based stock selection –short S&P 500 index future GBP fixed income duration management –short–long Gilt future Global alpha currency portfolio –no short position needed Emerging markets stock selection portfolio –short ETF on emerging markets portfolio (assuming independence of the alpha sources)

2.75

IR = 0.75 LIBOR + 2.06 % IR = 0.40 LIBOR + 1.96 % IR = 0.60 LIBOR + 1.50 % IR = 0.83 LIBOR + 2.99 % IR = 0.35 LIBOR + 2.28 % IR = 1.00 LIBOR + 2.00 %

Source: sample data

4.90 2.50 3.60 6.50 2.01

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Positive Alpha Generation: Designing Sound Investment Processes

benchmark, if the time horizon is long, usually three to five years at least. On the other hand, if the time horizon is short, the portfolio is called an alpha portfolio, and approaches described in Chapter 16 can be applied. A different solution to the same problem is to combine different sources of alpha into a portfolio, rather than different assets. Table 19.4 shows a possible portfolio constructed based on portable alpha strategies targeting a return of LIBOR + 200 bp. The success of this portfolio relies solely on the skills of the individual managers and the independence of their skills. Rather than manage risk at the total portfolio level, the risk of the liabilities can be decomposed into individual risk factors and matched by an asset portfolio. The techniques used are similar to those used in the building block solutions. However, in addition, the correlations between the individual risk factors are taken into account. This allows the expected return for a given level of risk to be increased.

19.4 A PROCESS FOR DETERMINING A LIABILITY DRIVEN INVESTMENT SOLUTION In addition to generating alpha as a component of an LDI solution, the added value provided by the investment manager lies in constructing a solution, in collaboration with the investor and/or the investor’s investment consultant, that best: • matches the liabilities and • satisfies the involved agents’ utility functions. The LDI solution construction process can be summarized by the following six steps: 1. Determine the liability structure5 and understand the involved agents’ potentially conflicting utility functions. 2. Analyze and decompose the liabilities in cash flows, risk factor exposures, risk exposures and their relationships with the existing asset base. 3. Based on the structure of the liabilities, select an LDI portfolio construction technique. 4. Based on the selected portfolio construction technique, the available investment manager’s skills in alpha generation and the agents’ utility functions, determine the most appropriate LDI solution approach. 5. Construct the portfolio and verify its characteristics with respect to the liabilities using, for example, statistical tests or scenario analysis. 6. Monitor and, if necessary, revise the portfolio constructed by re-executing steps 1 to 5. Each investment manager should adapt the proposed process to formulate a personal unique value proposition.

5

Determining the liability structure is usually done by an actuary rather than the investment manager.

Part VII Quality Management

20 Investment Performance Measurement Measuring and understanding the performance, that is the alpha generated by an investment process, is a necessary quality assurance step. The quality must be measured and controlled at each stage of the value chain, both at the module as well as at the interface between the module levels. Every investment process should be wrapped into a formal performance measurement framework whose goals are: • ensuring the quality of information gathering and processing and the resulting forecasts and investment decisions, • monitoring the quality of the risk management process step and • guaranteeing the quality of the transfer mechanism, that is the setup that ensures that the forecasts are transformed in a risk managed way into actual portfolio returns or alpha. However, performance measurement should not only be seen as a controlling tool. It should be an integral part of any investment process and, as such, contribute to the alpha generating process. This implies that the performance measurement must relate the realized performance to actual investment decisions taken. Consider an equity investment process based on a three-factor model – style, size and momentum. The performance measurement framework must measure and relate the performance to these three factors. In addition, the relationship between the investment decisions about the three factors and their contribution must be explicated. Ideally, the information about the forecasting capabilities on the individual factors should be used to budget risk to the individual factor exposures. Consider, for example, an investment manager who has much better forecasting capabilities in the growth versus value forecasts, that is the style factor, than in the large versus small capitalization or the momentum factor forecasts. In this case, a growth and small capitalization forecast should result in a larger overweight in growth stocks than in small capitalization securities purely based on the available skills. This approach requires that the past forecasting skills are a good proxy for future forecasting skills. This is a requirement in order for an investment process to generate persistent positive alpha. Implementing a sound performance measurement framework is therefore a key component of a successful investment process, but it is only as good as the usage that is made of its results. Only when systematic actions are associated with the potential measurement outcomes will the performance measurement contribute to the overall success. This last feedback loop step, combined with precise action points, is unfortunately only rarely found in investment processes.

20.1 PERFORMANCE MEASUREMENT DIMENSIONS The performance of a portfolio can be analyzed along the two dimensions return and risk. Along each dimension, both absolute and relative approaches are possible. The possible approaches of measuring the portfolio performance are illustrated in Figure 20.1. Along the return dimension, the performance may be measured in absolute terms, that is relative to

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Positive Alpha Generation: Designing Sound Investment Processes

risk dimension

investment decisions exogenous factors benchmark return risk free rate zero return return dimension

FIGURE 20.1 Dimensions along which the performance of a portfolio can be measured

a zero or risk free return, a benchmark, exogenous factors or investment decisions. In the context of performance measurement as a decision support tool, the portfolio return should always be related to the investment decisions taken. Measurement of the performance purely along the risk dimension is very often omitted. This is probably due to the fact that risk is not directly observable. Nevertheless, I highly recommend measuring the relationship between the ex-ante or forecasted risk and the ex-post or realized return. Over a sufficiently long investment horizon, these two measures should be consistent in statistical terms. Techniques similar to those described in Chapter 8 for backtesting a risk model can be applied. Finally, the return and risk dimensions should be related to each other in order to determine the risk adjusted realized return. On the one hand, risk adjusted return measures allow measuring if the risk budgeted to individual investment decisions has actually produced the desired return. On the other hand, risk adjusted return measures control whether the return achieved is actually due to investment decisions rather than purely due to a higher risk exposure. Consider a portfolio P1 with a return of 5.0 % and an ex-post volatility of 7.5 %. Consider a second portfolio P2 with a total return of 7.5 %, but with an ex-post volatility of 19.5 % over the same time horizon. In a purely return based performance measurement framework, portfolio P2 is ranked better than P1 . However, in a simple risk adjusted framework, measuring the realized return per unit of ex-post risk would assign a value of 5.0 %/7.5 % = 0.67 to portfolio P1 and 7.5 %/19.5 % = 0.38 to portfolio P2 , thus preferring P1 over P2 . Sometimes it is useful to relate the ex-post return to the ex-ante risk, especially when measuring the impact of risk budgets on realized returns. 20.1.1 Decomposing performance I define performance attribution as the process of decomposing return and risk along the investment manager’s investment decisions in order to measure the value added or destroyed by each individual decision. I talk about return attribution if I only consider decomposing return along the investment decisions. Consider again the three-factor equity investment process, that is forecasting the return of the style, the size and the momentum factors. An example of a return attribution framework is to decompose and relate the portfolio return to the decisions associated with the three factors, that is growth versus value, large versus small capitalization and momentum versus mean reversion. This means, for example, answering the question of which part of the portfolio return is due to the decision that growth stocks

Investment Performance Measurement

307

will outperform value stocks. Some authors use the term of performance attribution to denote the decomposition of excess returns, independent of the decomposition dimension. I define performance contribution as the process of decomposing the portfolio return, either absolute or relative, and associate it with exogenous factors, like interest rates or subcomponents (such as asset classes) of the portfolio. In the literature the term performance contribution is also very often used to denote the decomposition of absolute returns.

20.2 SETTING UP A PERFORMANCE MEASUREMENT FRAMEWORK The development of a sound integrated performance measurement framework (PMF) can be seen as a development project in its own right. This is especially the case as it touches all modules of the investment process value chain and potentially impacts them. Special care must be taken to ensure consistency between the investment process and the performance measurement framework. This unfortunately means that many off-the-shelf PMF software systems are of limited use within an investment process. They are relegated to an external controlling function. However, they can be very useful tools to uncover hidden flaws in an investment process, for example in identifying unintended risks. Consider a region based asset allocation investment process. A PMF not related to the investment process, that is the region based decomposition of the investment universe, may identify structural issues in the sense that the resulting asset allocation is extremely exposed to certain sectors, a dimension that is not actively managed. The development of a PMF can be subdivided into five major steps: 1. Identify the goal of each module of the investment process and clarify the decisions taken. 2. Identify the management of the uncertainty behind each decision taken in each module. 3. Design a performance measurement approach that associates the output of the module to the decisions taken, filtering out the impact of the input received from preceding modules. 4. Relate the decisions of each module to the final outcome, that is the portfolio return, alpha or risk adjusted alpha. 5. Define potential cases for action for each measured performance figure. Figure 20.2 illustrates the decomposition scheme of the PMF and its relation to the individual modules of the investment process. The selection and design of a specific measurement approach in step 3 requires the following questions to be answered: • Is the portfolio managed against a benchmark or an absolute return target? In the case of an absolute target, is the reference return zero or the risk free rate?1 • How do the decisions taken relate to the portfolio structure? Is the approach top-down or bottom-up? How many decision layers are there and how are they interlinked? Is the decision process based on asset classes or exogenous factors? 1 In the latter case a benchmark oriented PMF can be applied setting the benchmark to be a portfolio of 100 % cash.

Positive Alpha Generation: Designing Sound Investment Processes Portfolio construction & risk management

Decision 9

Decision 7

Implementation and trading

Decision 6

Decision 5

Decision 3

Decision 2

Decision 1

Market forecasting

Decision 8

308

Portfolio return /portfolio excess return

FIGURE 20.2 Relation between the decomposed portfolio return and the modules of the value chain of an investment process

• What is the time horizon of the investments? When and how often is the portfolio structure adjusted? How do decisions taken relate over time? • What performance measurement output is needed? Is considering only return sufficient or should risk be taken into account? What is the level of detail required? • What data are available to perform the performance calculation?

20.3 BASICS OF PERFORMANCE MEASUREMENT 20.3.1 Return measurement Under the assumption that no cash flow occurs between time 0 and T , the discrete total return of a portfolio is calculated as R=

VT −1 V0

(20.1)

where Vt is the portfolio value at time t. If a cash flow Ct−1 occurs at time t − 1, then the time weighted total return is calculated as R0−T =

T * t=1

Vt Vt−1 + Ct−1

−1

This requires valuing the portfolio each time an external cash flow occurs. The modified Dietz method allows the portfolio return to be calculated over a period of time, assuming a constant rate of return during the period using VT − V0 − R0−T = V0 +

T
t=0



T


Ct

t=0

T · Ct T −t

(20.2)

Investment Performance Measurement

309

This approximation avoids having to re-evaluate the portfolio at each cash flow. The modified Dietz method assumes that the intermediate cash flows are invested at the end of the day they occur. The money weighted return or internal rate of return of a portfolio is the discount rate I0−T , which equates the discounted ending portfolio value with its corresponding beginning value, that is T  VT Ct V0 = + t (1 + I0−T ) (1 + I0−T )T t=0

The money weighted total return formula should not be used to calculate the portfolio return in the context of an investment process. The time weighted approach should be used. If no cash flows occur between time t − 1 and t, the time weighted and money weighted formals give the same return. If a portfolio is composed of A assets, each with a weight wa,t and a return of Ra,t , then the return of the portfolio can be calculated as Rt =

A 

wa,t · Ra,t

a=1

When the return Ra,t is expressed as continuous compounded, that is ra,t = ln(1 + Ra,t ), then    rt = ln 1 + wa,t ·era,t assuming that


a

wa,t = 1, that is the portfolio is fully invested.

Trade date and value date handling When buying or selling a security, it is common practice to distinguish between the trade date and the value date. The trade date is the date at which the trade is executed or the derivative contract is signed. The economic exposure to the assets starts at the trade date. The value date is the date at which cash is exchanged against the bought or sold asset. The value date is usually later than the trade date. It is also sometimes called the settlement date. For example, if I buy a corporate bond today that has a value date in two weeks, my economic exposure to the issue of the bonds starts as of today. If the bond defaults between today and the value date in two weeks, I will receive on the value date a defaulted bond, rather than the possibility to void the transaction. From a performance measurement point of view, I am interested in the economic exposure of my portfolio rather than the actual positions. Therefore a PMF should rely on trade date valuation. Coping with derivatives Derivates, in contrast with cash assets,2 only provide an economic exposure to an asset or asset class. For example, one DAX future contract expiring in June 2007 provides on 29 2 A cash asset denotes a physical asset, like a stock or a bond, but not a contractual agreement, like a future or forward. The market on which cash assets are traded is called the cash market. The cash market has its name because the securities traded are traded against an exchange of cash.

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Positive Alpha Generation: Designing Sound Investment Processes

Table 20.1 Illustration of the actual basis and the cash equivalent basis method for valuing portfolios containing derivatives and calculating their performance Asset – cash equivalent

Value at time t − 1

Value at time t

Profit and loss

Return (%)

S&P March 2008 future

1 493.1 $373 275 114.81 ¤114 810 592.40 SFr. 5 924 95.830 $239 575

1 477.2 $369 300 113.11 ¤113 110 282.20 SFr. 2 822 95.765 $239 412

$−3 975

−1.6

¤−1 700

−1.48

SFr. −3 102

−52.36

$−162.2

−0.07

Bund March 2008 future European call on SMI (March 2008 at 8 500) 30 days EUR/USD future (March 2008)

Source: data sourced from Bloomberg Finance LP, author’s calculations.

May 2007 an economic exposure of ¤194 2253 to the DAX index, that is the 30 underlying securities in the DAX index. However, its value is equal to the unrealized profit and loss, that is ¤350, assuming a closing of 7 755 index points of the DAX on 28 May 2007. In this sense, derivatives can provide leverage or net short positions. Two different methods exist for calculating the performance of derivative instruments, that is the actual basis and the cash equivalent basis methods. In the actual basis method, the constituents of a portfolio are valued at their market price and the derivatives are valued at their profit and loss. Table 20.1 illustrates this method. The cash equivalent method attempts to eliminate the effect of the leverage by restating the derivative positions as an equivalent position in the cash market. The S&P future, for example, is replaced by a portfolio of stocks replicating the S&P index for an equivalent of $373 275, as illustrated in Table 20.1. The Global Investment Performance Presentation Standards (GIPS) require that, in the presence of leverage, both the actual as well as the cash equivalent performance figures are stated. The cash equivalent performance figures show the investment manager’s capabilities of generating alpha through forecasting markets, whereas the difference between the performance calculated using actual and the cash equivalent approach expresses the efficiency of using leverage. In most traditional portfolios, where leverage is not allowed and the portfolio manager is required to hold the underlying security of any derivative, for example $373 275 in cash for holding a long S&P 500 future position or a portfolio of stocks indexed to the S&P for a value of $3 732 750 when holding short 10 March 2008 S&P 500 future contracts, the actual method should be used. Handling currency returns I distinguish between the local currencies, that is the currencies to which each asset is exposed economically, and the base or reference currency, that is the currency in which the performance is reported. Let RL denote the local currency return of an asset, RB the return 3

One index point of the DAX index equals

¤25, that is an index level of 7 769 equals ¤194 225.

Investment Performance Measurement

311

of the same asset expressed in the base currency and RC the return of the local currency versus the base currency. Then 1 + RB = (1 + RL ) · (1 + RC )

(20.3)

A na¨ıve interpretation of this return decomposition would be to state that RL is due to the market exposure and RC to the currency exposure, and RL · RC the residual or unexplained return. Karnosky and Singer (1994) have shown that this interpretation is flawed. Indeed, it is not possible to gain exposure to a currency outright and receive the return RC . A currency exposure independent of a market can only be gained through the forward market. Let FL denote the risk free rate of the local currency and FB the risk free rate of the base currency. Then the return RF of a currency forward transaction is given by 1 + RF = (1 + RC ) ·

(1 + FL ) (1 + FB )

Now reconsider Equation (20.3): 1 + RB = (1 + RL ) · (1 + RC ) (1 + FB ) (1 + FL ) · (1 + RC ) · = (1 + RL ) · (1 + FL ) (1 + FB )       market return = (1 + RH )

currency return = (1 + RF )

where RH denotes the hedged asset return. Therefore a correct attribution of the return RB is to identify RH as the market return and RF as the currency return. The residual or cross product RH · RF can be either expressed as such or allocated proportionally to the asset and currency return. A possible simple decomposition would be RB = RH + 12 · RH · RF + RF + 12 · RH · RF       market return

currency return

Other more advanced schemes are possible. Consider a portfolio with dedicated currency exposures not associated with individual assets. Denote by wa the weight invested in asset a in the local currency la and
by wc the 4 In a fully invested portfolio exposure to currency c independent of the assets. wa = 1
and wc = 0. Then the return of the portfolio can be written as RB =

 a

=



wa · Ra,B +



wc · Rc,F

c

wa · ((1 + Ra,H ) · (1 + Rl(a),H ) − 1) +

a



wc · Rc,F

c

where the currency return of the base currency is RB,F = 0. 4 An exposure of w to currency c is modeled by an opposing exposure in the portfolio’s based currency, that is c wB = −wc .

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Positive Alpha Generation: Designing Sound Investment Processes

In order to calculate the weights of this portfolio at the end of the period, it is necessary to introduce an additional synthetic asset class to which the profit and loss of the currency forwards is associated. If the portfolio has a cash or risk free asset class in the base currency, this asset can be used to attribute the currency forward profit and loss:  wa,t+1 = wa,t · (1 + Ra,L ) · (1 + Ra,C )  wc,t+1 = wc,t · (1 + Rc,F ) ,  wa/c,t+1  wa/c,t+1 =
 b=a/c wb,t+1 
 wP &L,t+1 = wP &L,t+1 + c wc,t+1  
  wB,t+1 = wB,t+1 − c wc,t+1    wa/c,t+1 = wa/c,t+1

+ step 1 step 2

step 3

It is important to note that hedged return + currency forward return = unhedged return Indeed, hedging an asset with a value of $100 means hedging the currency exposure of $100. If the asset is worth $110 at the end of the time period, only the currency risk versus $100 is hedged, but the $10 return obtained is not hedged against currency risk. Aggregating return over time Let Rt denote the return of a portfolio over consecutive time periods. The aggregated return between time t = 1 and t = T is calculated as the geometric sum, that is R0−T =

T *

(1 + Rt ) − 1

t=1


If the return is continuously compounded, then r0−T = Tt=1 rt . The portfolio return, expressed as an annualized return, is calculated as Rannualized = (1 + R1−T )365/T −1 .5 If only return at the portfolio level is considered, aggregation over time does not pose any problem. However, as soon as the return components of a portfolio need to be decomposed, the process becomes tricky. Consider a portfolio with the following returns over a two time period time horizon:  R1 = wa,1 · Ra,1 a  R2 = wa,2 · Ra,2 a

Then the return over the two time periods is equal to R1−2 = (1 + R1 ) · (1 + R2 ) − 1 5 Depending on the applicable day count conventions, respectively the number of days per year, the annualization factor 365 may be replaced by 360 or even 366.

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313

     = 1+ wa,1 · Ra,1 · 1 + wa,2 · Ra,2 − 1 a a     wa,1 · Ra,1 + wa,2 · Ra,2 + wa,1 · wb,2 · Ra,1 · Rb,2 = 

a

=



a

a

(wa,1 · Ra,1 + wa,2 · Ra,2 ) + a  

  a

(20.4)

b b

return due to asset a

(wa,1 · wb,2 · Ra,1 · Rb,2 )    return due to the interaction between assets a and b

The main issue with the decomposition shown in Equation (20.4) is the attribution of the return due to the interaction of different assets at different time periods, the so-called cross product return. Denote by f (wa,1 , wb,2 , Ra,1 , Rb,2 ) a function such that wa,1 · wb,2 · Ra,1 · Rb,2 = wa,1 · (f (wa,1 , wb,2 , Ra,1 , Rb,2 ) · wb,2 · Ra,1 · Rb,2 ) + wb,2 · ((1 − f (wa,1 , wb,2 , Ra,1 , Rb,2 )) · wa,1 · Ra,1 · Rb,2 ) Then Equation (20.4) can be written as R1−2 =

 0

   wa,1 · Ra,1 + f (wa,1 , wb,2 , Ra,1 , Rb,2 ) · wb,2 · Ra,1 · Rb,2 a b    + wa,2 · Ra,2 + (1 − f (wa,1 , wb,2 , Ra,1 , Rb,2 )) · wb,1 · Ra,1 · Rb,2 b

Therefore the cross product return can be attributed to either of the involved asset classes. A simple definition for the function f isf (·) = 0.5, attributing half of the cross product to the first and half to the second asset involved. Menchero (2000) proposed an optimized approach for calculating the function f (·) linking coefficients. Davies and Laker (2001) proposed a commutative approach, that is an approach where changing order of the time periods does not change the attribution of the cross product return. Recursive models for linking multiperiod returns have been proposed by different authors (Bonafede et al., 2002; Campisi, 2004; Cari˜no, 1999; Frongello, 2002). The proposed algorithms all have their unique advantages and drawbacks, with respect to commutativity, metric preservation and objectivity. The most important aspect for choosing a particular chaining approach in a PMF is to understand its limitations and ascertain its impact on the calculated performance decomposition and its interpretation. My experience has shown that whatever technique is chosen, its impact on decisions taken based on the calculated return components is small. Significance of realized returns The success of an investment process is closely related to the skill set of the investment manager, as indicated in Chapter 2. It is therefore important to find out whether the realized portfolio return is due to skills of the investment manager or solely due to luck. The first test to be performed is to find out whether or not the generated positive alpha is significant. To do so, I formulate the following hypothesis H0 : α > 0 Ha : α  0

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Then classical statistical hypothesis testing can be applied. The main difficulty unfortunately does not lie with the statistical test itself, but with the size of the sample data set required to obtain significant results. The standard error of the t-statistic of the realized alpha can be written as α 1 t-statistic = √ and standard error ≈ √ N N where N is the sample size. 20.3.2 Multidimensional performance figures A performance measure should allow portfolios to be compared and ranked, both against each other and against their benchmark or the risk free rate. Such a measure should be maximized by the investment manager, so that, if that maximum is exceeded, special skills can be inferred. Satchell and Pedersen (2002) have shown that any performance measure should satisfy the maximum principle, that is: • the performance measure is based on a maximized expected utility function of the investor and • the performance measure is defined as a ratio between the excess return and risk. Considering only the realized return does not satisfy the maximum principle. Since the mid 1960s, different risk adjusted performance ratios have been proposed: • Sharpe ratio. The Sharpe ratio SR (Sharpe, 1966) expresses the portfolio’s excess return over the risk free rate in relation to the portfolio volatility: SR =

RP − RF σP

The Sharpe ratio can be interpreted as the steepness of the capital market line in the capital asset pricing model. It normalizes the excess return to a unit of risk, risk being expressed as the volatility of the portfolio returns. • Information ratio. The information ratio IR is the counterparty of the Sharpe ratio in the context of an actively managed portfolio. The risk free rate is replaced by the benchmark return, a return that is considered risk-less from the investment manager’s perspective. The volatility is replaced by the tracking error or active risk taken by the investment manager: RP − RB IR = τP ,B • Treynor ratio. The Treynor ratio TR (Treynor, 1965) assumes the one-factor CAPM, the factor exposure being denoted by β. The excess return portfolio is related to the factor exposure, that is RP − RF TR = β

Investment Performance Measurement

315

Under the assumption that the one-factor model is the correct market model, the portfolio risk is completely described by that factor. • Modigliani–Modigliani ratio. The Modigliani–Modigliani ratio MM (Modigliani and Modigliani, 1997), also denoted by M 2 , is a combination of the Sharpe ratio in the presence of a benchmark and the information ratio. The excess return is related to the risk differential between the benchmark and the portfolio: MM =

σB · (RP − RB ) σP

• Sortino ratio. The Sortino ratio SP (Sortino and Price, 1994) is a modification of the Sharpe ratio considering only returns below a given threshold H . The threshold may be fixed, observable or random. It is often defined as the rate of return on long-dated bonds: SP =

RP − H θP ,H

where θP ,H denotes the downside risk or semi-variance relative to the threshold H . • Calmar ratio. The Calmar ratio CR is a performance measurement ratio often used in the hedge fund industry. Rather than relying on a statistical measure of risk, the largest drawdown is used: |RP | CR = δP where δP denotes the maximal observed drawdown. It is a common practice to use historical data to calculate the ex-post volatility or risk. Sometimes the realized return is also related to the ex-ante or forecasted risk measure. This is especially useful if the investment manager actively changes the risk taken in a portfolio. The presented performance ratios are typically expressed in an annualized form, but the annualized ratios will be different depending on how the return and risk figures are annualized. Goodwin (1998) has studied various annualization approaches. He concludes that the method of annualizing surprisingly matters little in the interpretation of the ratios, as long as the annualization method is used in a consistent way. I recommend using the continuously compounded excess return approach. Continuously compounding makes the return data additive and the calculations simpler. I have T H  · (ln(1 + RP ,t ) − ln(1 + RB,t )) T t=1 

2 t  1 √ T

σ = H· · ·e (ln(1 + RP ,t ) − ln(1 + RB,t )) − T −1 H

e=

t=1

where H is the number of time periods per year, that is H = 12, if using monthly return data.

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Positive Alpha Generation: Designing Sound Investment Processes

20.4 PERFORMANCE ATTRIBUTION The goal of a performance attribution PMF is to decompose the portfolio performance and attribute or relate it to the different investment decisions taken. There are two categories of performance attribution models, that is: • the asset grouping models and • the factor models. 20.4.1 Asset grouping model methods The asset grouping performance attribution model can be applied to any investment process in which the investment decisions translate in allocating a specific weight, usually relative to a benchmark, to a subset of assets, so-called asset classes. An example of such an investment process is a sector allocation investment process. The investment manager forecasts the relative performance of different industry sections and then overweights the industry sectors that are forecast to outperform and underweights those that are expected to underperform. The most used performance attribution model in the asset grouping category is the Brinson approach (Brinson and Fachler, 1985; Brinson et al., 1986; Brinson et al., 1991). Although the model is known under the name of the Brinson model, an earlier description was published by a working group of the Society for Investment Analysis in London, United Kingdom, as early as 1972. Let wB,a denote the weight of asset class a in the benchmark B, wP ,a the portfolio weight of asset class a, RB,a the benchmark return of asset class a, RP ,a the portfolio return of asset class a, RB the overall benchmark
return, that is RB =
a wB,a · RB,a , and RP the overall portfolio return, that is RP = a wP ,a · RP ,a . The Brinson approach subdivides the portfolio return into four compartments, that is: • the benchmark return RB

RB =



wB,a · RB,a

a

• the allocation return, sometimes also called the market timing return  RA = (wP ,a − wB,a ) · RB,a

(20.5)

a

• the selection return, also known under the name of the stock picking return  RS = wB,a · (RP ,a − RB,a ) and

(20.6)

a

• the interaction return RI =



(wP ,a − wB,a ) · (RP ,a − RB,a )

a

such that

RP = RB + RA + RS + RI

(20.7)

Figure 20.3 illustrates the decomposition of the portfolio return along the two dimensions, asset class weights and asset class returns.

Investment Performance Measurement

317

Ra,P Selection return

Interaction return

Rs = ∑ wa,B . (Ra,P − Ra,B)

RI =

a

∑ (wa,P − wa,B).(Ra,P − Ra,B) a

Ra,B Benchmark return

Allocation return

RB = ∑ (wa,B − Ra,B)

RA = ∑ (wa,P − wa,B).Ra,B

a

a

wa,B

wa,P

FIGURE 20.3 Brinson et al. (1986) return attribution model

The allocation return RA expresses the return due to selecting different weights from those of the benchmark for the different asset classes. The selection return RS describes the difference of return in a given asset class independent of its relative weights. The interaction return RI is the return component due to the interaction between the allocation and the selection return. It can be understood as the impact of the allocation on the return generated by the selection. Sometimes
the selection and interaction return are combined, replacing Equation (20.6) by RS = a wP ,a · (RP ,a − RB,a ), such that RP = RB + RA + RS Alternatively, rather than measure the allocation effect as the return due to relative weights, that is (wP ,a − wB,a ) · RB,a , the allocation effect is defined as the excess return over the benchmark return, that is  RA = (wP ,a − wB,a ) · (RB,a − RB ) (20.8) a

The allocation performance attribution RA,a = (wP ,a − wB,a ) · (RB,a − RB ) for asset a is positive when the overweighted asset class outperforms the benchmark rather than has a positive return. This is in line with the assumption that the risk free return of an active investment manager is the benchmark return. Therefore a positive allocation contribution should only be shown if the allocation effect has a contribution larger than the benchmark return. If only the sum of the allocation contributions over all asset classes is considered, the results using Equations (20.5) and (20.8) are equal. Table 20.2 illustrates both approaches for a regional equity asset allocation investment process. The Brinson approach assumes a single-period buy-and-hold investment strategy. In order to measure the performance attribution over a long time period, with different changes in the relative weights in the portfolio, the model must be applied to each time period individually and the results aggregated over time using an approach described in Section 20.3.1, ‘Aggregating return over time’. A time period is characterized by the time horizon over which no trades are executed and no cash flows happen.

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Table 20.2 Illustration of the different variants of the Brinson performance attribution model for a single period Asset class

wB,a %

wP ,a %

Switzerland UK Germany France Italy Spain Total

9.5 24.8 17.7 15.6 18.5 13.9 100.0

11.5 21.8 22.2 12.1 17.0 15.4 100.0

RB,a % 2.60 3.20 1.70 4.10 3.80 2.20 2.99

RP ,a % 2.35 3.45 1.75 4.40 3.80 1.90 2.88

RA,a (bp) 5 −10 8 −14 −6 3 −14

RS,a (bp) −2 6 1 5 0 −4 5

RI,a (bp) −1 −1 0 −1 0 0 −3

 RA,a (bp)

 RS,a (bp)

−1 −1 −6 −4 −1 −1 −14

−3 5 1 4 0 −5 3

Source: sample data, author’s calculations.

20.4.2 Factor model methods The idea behind the factor model performance attribution is to define a set of factors that correspond to the investment decisions taken. The portfolio return is then attributed to these factors. The steps involved in calculating the performance attribution according to a factor model are: 1. Define an explanatory or conditioned factor model as described in Chapter 5 such that the factor exposures coincide with the investment decisions taken or the translation of the investment decisions in a portfolio structure. 2. Estimate the return of each factor, usually using the benchmark data at the security level. Alternatively, the whole investment universe can be used. As the goal is to explain the portfolio return rather than forecast it, using in-sample data to estimate the factor returns is valid. 3. Calculate the exposure of the benchmark and portfolio to the individual factors. 4. The difference between the realized return and the sum of the residual factor return is called the residual or unexplained return. Table 20.3 illustrates a PMF on a three-factor interest rate based investment process. The investment decisions are translated into portfolio holdings, exposures to one year, five years and 10 years duration, that is a position on three points of the yield curve. All available noncallable government bonds with a maturity of no less than one year as the universe are used to estimate the factor return, equally weighting each bond. The so-called butterfly structure implemented in the sample portfolio contributed −0.10 % to the overall portfolio return. The negative excess return is mainly due to the underweight in the five-year factor not being compensated by the overweight in the one-year factor exposure. For asset grouping performance attribution models, the key issue is to relate the measured performance to the actual investment decisions taken. Jensen’s alpha In 1968 Jensen introduced a model to decompose the realized return of a portfolio in order to show the value added by the investment manager. He considered the single-factor CAPM.

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Table 20.3 Factor model based performance attribution model for a government bond fixed income investment process forecasting return at the duration bucket level Factor

1 year 5 years 10 years Total

Benchmark exposure (%)

Portfolio exposure (%)

Factor return (%)

Relative return contribution (%)

25.0 35.0 40.0

30.0 27.0 43.0

0.70 3.50 4.80

0.0350 −0.2800 0.1440 −0.1010

Source: sample data, author’s calculations.

Ex-post, the portfolio return in any time period can be expressed as RP − RF = α + β · (RM − RF ) + ε

(20.9)

where ε ∼ N (0, σ ) and α as well as β are two constants estimated using the OLS regression estimator; α is called Jensen’s alpha and represents the value added by the investment manager. If an investment manager has forecasting skills, then Jensen’s alpha will be significantly positive.6 The model can also be applied to a benchmark oriented portfolio, replacing the market portfolio return by the benchmark portfolio return. Jensen’s model is a special case of a factor model based PMF in which the factor return, that is the market portfolio return, is given and the exposure to the factor is estimated rather than the inverse. Fama’s performance components model Fama (1972), building on the work by Sharpe (1966), Treynor (1965) and Jensen (1968, 1969), proposed a performance decomposition model that allows attributing the portfolio performance to four effects, that is: • • • •

net security selection effect, effect of diversification and foregone diversification, investment manager’s market risk and investor’s expected market risk.

Fama assumes that the systematic risk of any portfolio can be characterized by the beta parameter of the CAPM. Let RA (β) denote the return of portfolio A with risk β. In addition, he assumes that any efficient portfolio can be decomposed in a portion invested in the risk free asset and a portfolio invested in the market portfolio. Let RE (β) denote the return on the combination of the risk free asset and the market portfolio such that the resulting portfolio has a risk of β. The return can be written as RE (β) = ω · RF + (1 − ω) · RM and the portfolio excess return as RP − RF    portfolio excess return

=

(R − R (β ))  P E P  return attributed to selection

− (RE (βP ) − RF )    return attributed to risk

6 The significance of alpha being positive can be tested using a standard t-test with the null hypothesis being √ H0 : α > 0 and the alternative hypothesis being Ha : α  0. The test statistic is calculated as tn−1 = α/(s · n), where n is the number of observations of α, α the sample mean and s the sample standard deviation.

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The return attributed to selection measures the return due to systematic risk taken by the investment manager. It is equivalent to Jensen’s alpha. Let T denote the portfolio composed of the risk free asset and the market portfolio such that the risk of the portfolio is equal to that of the actual portfolio in terms of volatility rather than beta. Then the return attributed to the net selectivity is defined as return attributed to net selectivity =

(R − R (β ))  P E P  return attributed to selection

−

(R (β ) − R (β ))  E P  E T  return attributed to diversification

If the return attributed to net selectivity is negative, then the investor bears a risk that was not compensated, that is that could have been diversified away. It is also possible to decompose the return attributed to risk in a similar way. Let βU denote the target risk or risk objective of the investor, which may be smaller or larger than the risk of the actual portfolio. Then R (β ) − RF =  E P  return attributed to risk

(RE (βP ) − RE (βU ))    return attributed to risk taken by the investment manager

−

(RE (βU ) − RF )   

return attributed to the investor s risk

The return attributed to risk taken by the investment manager is the return component that is due to the investment manager deviating from the target risk level defined by the investor. The decomposition approach proposed by Fama can also be applied using a benchmark portfolio rather than the market portfolio as factor. With minor modifications it can also be applied using a multifactor model or even a different definition of risk.

20.5 PERFORMANCE CONTRIBUTION Performance contribution approaches decompose the portfolio return and associate its components to the components of the portfolio. In its simplest form, the return contribution of an asset class is calculated as
wa · Ra . Performance contribution figures are usually additive cross-sectional, that is R = a wa · Ra . Table 20.4 illustrates a performance contribution for a global balanced portfolio. The currency return is separated from the market return and the cross product attributed to the currency return. In addition, the return of each asset class is decomposed in a hedged market return and a currency forward return using the approach described in Section 20.1.3, ‘Handling currency returns’.

20.6 THE PROCESS BEHIND THE PROCESS A PMF is only as good as the usage of its output. I distinguish between two types of approaches for using the calculated performance figures: • The color zone approach defines a process that indicates to the investment manager when a possible issue arises where action may be required. The actual action is not

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Table 20.4 Performance contribution of a global balanced portfolio for an EUR based investor during the month of November 2007 (weights in percent, return contributions in basis points)

Currency

Local return

Cash

Weight Local return Market contribution Currency contribution Total contribution Weight Local return Market contribution Currency contribution Total contribution Weight

Bonds

Equities

Total

Local return Market contribution Currency contribution Total contribution Weight Market return Currency return Total return

Europe

UK

Switzerland

Unitex States

Japan

EUR 0

GBP −321

CHF −16

USD 29

JPY −43

Total

n/a

5.00 % 0.40 % 2.0

5.0 % n/a 2.0

0.0

0.0

2.0

2.0

20.0 % −0.11% −2.2

2.5 % 1.32 % 3.0

0.0 % −0.72% 0.0

7.5 % 0.08 % 0.3

5.0 % 0.19 % 2.5

35.0 % n/a 3.6

0.0

−7.8

0.0

2.5

−3.7

−9.1

−2.2

−4.8

0.0

2.8

−1.2

−5.5

20.0 %

15.0 %

3.0 %

15.0 %

7.0 %

60.0 %

−0.45% −9.0

0.51 % 6.0

−3.64% −10.4

−0.57% −9.1

−3.27% −20.7

n/a −43.2

0.0

−46.7

−1.0

4.9

−5.1

−47.9

−9.0

−40.7

−11.4

−4.2

−25.8

−91.1

45.0 % −9.2 0.0

17.5 % 9.0 −54.5

3.0 % −10.4 −1.0

22.50 % −8.7 7.3

12.0 % −18.1 −8.9

100.0 % −37.5 −57.0

−9.2

−45.5

−11.4

−1.4

−27.0

−94.6

Source: data sourced from Bloomberg Finance LP, author’s calculations.

defined. It is left to human analysis and judgmental outcome. Color zone approaches aim at identifying: – performance outliers, that is extreme realized returns not in line with the taken risk, and – persistent, although generally small, negative performance contributions. They are usually applied at the portfolio level or at a high level of aggregation. • Invasive approaches define rules and algorithms that adjust the investment process behavior. They can be seen as integrated components of the decision process. They target mitigating false investment decisions. Such approaches are usually applied at the investment decision or asset level, that is at a low level of aggregation.

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20.6.1 Color zones approaches Let me illustrate a possible color zone approach. I assume that the implemented PMF calculates, for all benchmark oriented solutions I manage, on a monthly basis: (i) the ex-post or realized excess return and (ii) the ex-ante or target tracking error. Furthermore, I define a monthly excess return as an outlier if it exceeds in absolute terms the value of 1.96 τ , where τ is the ex-ante tracking error. I differentiate between four stages in my color zone approach: • The healthy or green zone. No action is needed. • The sick or red zone. Immediate action is required. • The observation or orange zone. The measured portfolio excess returns are not in line with the target, but no immediate action is needed. • The healing or blue zone. Actions have been applied to the portfolio and their effects on the excess return are observed. A portfolio is in the sick stage if either the current month’s excess return has been an outlier or if it has been in the observation stage for at least three months. It is classified as in observation as soon as the monthly return is negative but no less than −1.96 τ and moves out of observation after three consecutive months with positive return. If it stays for four consecutive months in the observation stage, it is classified as sick. A portfolio that no longer satisfies the condition for being sick moves to the healing stage, where it is observed until four consecutive months of positive excess return have been observed. This process is summarized in Figure 20.4. Litterman et al. (2000) present a similar approach called the three-zone approach. Bailey et al. (2007) propose a similar color zone approach. −1.96 t ≤ a ≤ 0

−1.96 t ≤ a ≤ 0

0 < a < 1.96 t

0 < a < 1.96 t In observation

Healthy a ≤−1.96 t or a ≥1.96 t

a>0

Healing

a≤0

0 < a < 1.96 t

a ≤ −1.96 t or a ≥ −1.96 t

a ≤ −1.96 t or a ≥1.96 t

Sick

a ≤ 0 or a ≥1.96 t

FIGURE 20.4 Sample color zone approach to monitor the excess return of a portfolio in relation to the ex-ante risk taken

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if not forecast history exists then ct ← 56 % let ct−1 denote the confidence in the most recent forecast if the most recent available forecast was correct then if ct−1 < 68 % then ct ← ct−1 + 1 % if the most recent forecast available was incorrect then if ct−1 > 51 % then ct ← ct−1 − 1 % else the forecast is not considered but kept for future measurement

FIGURE 20.5 Algorithm to determine the confidence in a forecast based on past performance, expressed as the hit ratio of that forecast, the confidence level thresholds being chosen based on experience

The advantages of the color zone approaches are that they are easy to develop and implement. They can be tailored to the human thought process. Their main disadvantages are that they are judgmental, and may therefore miss quality issues, and they usually trigger an action but do not define the associated action. 20.6.2 Invasive approaches Consider an investment process based on the optimal risk budgeting framework presented in Chapter 10. In this framework, each forecast is associated with a confidence level. Rather than have the investment professionals specify the confidence level, I determine the confidence level based on the past forecasting capabilities of the investment manager. I define the confidence at time t in a forecast according to the algorithm shown in Figure 20.5. This algorithmic approach makes the confidence level formulation process objective and purely based on past performance of the investment manager. Such an approach is consistent with the goal to produce systematic positive alpha. The main advantages of invasive approaches are that they define the action to be taken based on the realized performance. They avoid situations in which incorrect forecasts are kept over a long time period. On the negative side, as a rule based approach, they do not cope with extreme situations that are not anomalies. For example, an invasive approach may require significant portfolio changes after an event such as 11 September 2001, although from a judgmental point of view, this may not be necessary. They also assume that past performance is a good indicator for future performance, consistent with the assumption of an alpha generating investment process being persistent. I recommend the implementation of an invasive approach only with a veto mechanism by the investment manager.

20.7 PRACTICAL CONSIDERATIONS IN PERFORMANCE MEASUREMENT There are many practical pitfalls to avoid when implementing a PMF and the associated software system performing the calculations, called the performance measurement system (PMS). First and foremost, it is important that the PMS uses consistent data. This means that if the portfolio is compared to a benchmark, the portfolio must be valued using the same approach and asset prices as the benchmark. Table 20.5 shows a small sample of a euro

324

Positive Alpha Generation: Designing Sound Investment Processes Table 20.5 Valuation of a small high yield portfolio using different price sources

Issuer Central Eur. Dist. Corp. El Paso GMAC Hertz ISS Global Ineos Group TUI Nordic Telecom Wind Acquisition Finance

Maturity

Coupon

Mid quote

Last paid

Benchmark

25.07.2012 06.05.2009 14.09.2009 01.01.2014 18.09.2010 15.02.2016 10.12.2012 01.05.2016 01.12.2015

8.000 7.125 4.750 7.875 4.750 7.875 5.125 8.250 9.750

99.8633 101.8200 88.3651 96.2500 94.2390 84.6667 88.4741 101.2825 106.7435

99.7950 101.8292 89.4957 96.2500 94.4183 84.6667 88.3761 101.1667 106.7435

100.2500 101.5000 88.9264 95.0000 94.1650 83.5000 88.2500 101.0000 107.0000

Source: data sourced from Bloomberg Finance LP as of 31 December 2007

high yield bonds portfolio valued using different pricing sources. None of the shown pricing sources is unrealistic; nevertheless the resulting price differences are around 0.25 %, with a maximum difference of 1.40 %. It is best practice to value the portfolio positions according to the algorithm and prices applied to calculate the benchmark performance. In addition, currencies should be priced at a fixed point in time. The quasi-standard exchange rates used in performance measurement as well as in most benchmark calculation algorithms are the Reuters/WM exchange rates.7 Derivatives are best valued at their opening price, as the opening price is usually determined by an auction mechanism and is used as the final price at settlement date. For other securities, it is common practice to use market closing prices from the main exchange where the security trades. Alternatively, the last bid price quote before the end of day auction may be used, especially when the market is less liquid. If a portfolio contains collective investment vehicles (CIV) that are valued according to their published net asset value (NAV), rather than a market price, it is important to associate the NAV used with the date for which the CIV is priced rather than the date on which the NAV is published. It is common practice to value a CIV, that is calculate its NAV, with closing prices of the previous day. For example, an actively managed US equity portfolio is valued on 24 April 2007. The NAV of 25 April is based on closing prices as of 24 April. Therefore in the performance measurement, the NAV of 25 April must be associated with the date of 24 rather than 25 April. A completely different issue when introducing a PMS is the possibility provided to the investment manager for gaming the system. Gaming the system means managing the portfolio with a target to maximize the performance attribution output rather than the portfolio alpha. For example, if only return is measured, the investment manager may be attracted to take large risky positions or, if the performance is not decomposed in market and currency returns, the portfolio manager may focus on currency management rather than the underlying markets. It is therefore important to identify possible gaming situations and add management restrictions to avoid a situation where the portfolio managers diverge from 7 The Reuters/WM exchange rates are calculated according to the following algorithm. For actively traded currency exchange rates, bid, ask, as well as traded prices, are observed on a second-by-second basis over a one-minute time period symmetrically around 4 pm UK time. Median bid and ask rates are calculated and the mid rates determined. In addition, the actually traded rates are included to determine the exchange rate.

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their management style. This, of course, is only an issue if the performance measurement system is used as an independent controlling instrument rather than as a decision support tool, which I advocate it should be in the context of designing a sound investment process.

20.8 EXAMPLES OF PERFORMANCE MEASUREMENT FRAMEWORKS Let me illustrate the design process of a PMF on the following two examples: • The first example is based on a single currency fixed income investment process for managing a portfolio of corporate bonds. • The second example is based on a global asset allocation investment combined with selecting actively managed funds for each asset class. 20.8.1 Fixed income investment process performance measurement Consider the following three-stage single currency corporate bond investment process. The investment manager first forecasts interest rate movements at the one-year, five-year and 10-year horizon. Then the total return for each of the three buckets is calculated and an absolute allocation is derived for the three buckets, taking into account both the forecasted return as well as the associated risk, measured in terms of modified duration. The weight per bucket is considered the investment decision whose performance I want to measure. Any bond in the portfolio is mapped to one or two of the buckets, based on its modified duration. For example, a six-year duration (with actual duration 5.25) bond is mapped two-thirds to the five-year bucket (with duration 4.25) and one-third to the 10-year bucket (with duration 7.25). In a second step, the investment manager determines the credit allocation of the portfolio and considers an absolute allocation among the four sectors, government bonds, financials, industrials and mortgage backed securities. The investment decision to measure is defined as the weight per sector based on a forecast of credit spread changes. Finally, the investment manager selects individual securities that match the selected interest rate and credit spread forecasts by minimizing the bond specific default risk, diversifying the portfolio holdings and minimizing transaction costs. This selection decision is the third investment decision to be measured. Table 20.6 summarizes the developed investment process. Note that it is not possible to observe the performance contribution of each investment decision directly. For example, holding a seven-year UBS bond with a yield spread of 30 bp contributes to the interest rate decision due to its seven-year time to maturity and to the sector decision being a financial security and an issuer decision, the investor having selected a UBS bond rather than a Credit Suisse bond, as an example. The performance of the proposed investment process is best measured using a factor model based performance attribution approach. The overall factor model can be described by Ra,t = R0,t +  Sa,t =

3 

da,i,t · Ri,t + Sa,t

(20.10)

i=1

Ea,t Rs,t + Ea,t

if a is a government bond if a belongs to sector s

(20.11)

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Positive Alpha Generation: Designing Sound Investment Processes Table 20.6 Illustration of a single currency corporate bond investment process Step 1

Step 2

Step 3

Risk factor Decomposition of risk factor Forecasts

Interest rates Maturity/duration buckets Interest rates and resulting total returns

Credit spreads Sectors

Issuer default risk Issues

Credit spreads and resulting excess returns

Transfer mechanism

Risk based weighting of buckets

Weighting of sectors

Default risk, diversification effect and transaction costs Buy, hold and sell decisions

Stage 1 – – – – –

Select a universe of government bonds and calculate the total return for each bond. Calculate the exposure of each government bond to the three maturity/duration buckets. Estimate the factor returns R0,t and Ri∈{1,2,3},t of Equation (20.10). For each bond in the portfolio, calculate its exposure da,i,t to the maturity/duration buckets. Calculate the performance attribution from each bond to the interest rate decisions as wa,t · da,i,t · Ri,t , where wa,t is the weight of bond a in the portfolio.
– If the total return attributed to a single interest rate decision, that is a wa,t · da,i,t · Ri,t , is negative and larger in absolute terms than 50 % of the absolute sum of the total return attributed to interest rate decisions, then limit the maximal exposure to that interest rate decision for the next time period to two-thirds of its current weight.

Stage 2 – Select a universe of corporate bonds and calculate the total return of each bond (the universe should be a super-set of the bonds in the portfolio, that is include all bonds in the portfolio). – For each bond, calculate Sa,t using Equation (20.10) and the factor returns estimated in stage 1. – Estimate the sector return Rs,t of Equation (20.11) using an ordinary least squares estimator. – Calculate the performance attribution from each bond to the sector allocation decision as wa,t · Rs,t
and their sum for each sector a wa,t · Rs,t .
– If the total return attributed to a single credit spread decision, that is a wa,t · Rs,t , is negative and larger in absolute terms than 50 % of the absolute sum of the total return attributed to credit spread decisions, then limit the maximal exposure to that sector for the next time period to two-thirds of its current weight. Stage 3 – Calculate the residual return for each bond in the portfolio as  Ea,t = Ra,t − R0,t +

3 

 da,i,t · Ri,t

− Rs,t

i=1

– If the residual return Ea,t is negative and three times as large as the second largest residual return, reduce the position in that bond by half.

FIGURE 20.6 Factor model performance measurement framework algorithm

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Table 20.7 Illustration of the factor model based performance measurement framework algorithm described in Figure 20.6 Weight (%)

Bond description

7.9 14.0 6.6 3.6 6.1 3.7 5.2 14.7 14.1 14.5 7.6 2.0 100.0

ABN AMRO Bank Abertis Infrastr. AIB Mfg Bank Air Liquide Air Prod. & Chem. Belgelec Carrefour Credit Suisse Deutsche Bahn EDP Henkel Hypo Real Estate Total

Duration

3.806 7.047 4.561 2.240 3.675 2.238 3.042 7.380 7.633 9.026 4.611 1.094 5.977

Sector

Rate contrib. (%)

Sector contrib. (%)

Specific Return (%)

Total return (%)

Bank Industrial Bank Industrial Industrial Financial Industrial Bank Transport. Utility Industrial Bank

−0.27 −0.84 −0.41 −0.13 −0.25 −0.13 −0.13 −0.82 −0.81 −0.73 −0.42 −0.13 −0.57

−0.25 −0.31 −0.25 −0.31 −0.31 −0.30 −0.31 −0.25 −1.55 −0.27 −0.31 −0.25 −0.46

−0.16 −0.09 0.37 −0.33 −0.16 0.08 −0.03 0.25 0.21 −0.06 1.09 0.67 0.13

−0.69 −1.24 −0.28 −0.77 −0.71 −0.35 −0.47 −0.82 −2.15 −1.06 0.36 0.30 −0.90

Source: bond data sourced from Bloomberg Finance LP, sample weights and return data, author’s calculations.

where Ra,t is the realized return of bond a between time t − 1 and t, R0,t a constant, Ri∈{1,2,3},t the return due to the interest rate bucket i, da,i,t the exposure of bond a to the interest rate bucket i, Rs,t the factor return associated with sector s and Ea,t the residual or bond specific return. Figure 20.6 describes the three consecutive steps of the algorithm implementing the proposed performance measurement framework. It proceeds by first extracting the performance due to the interest rate forecasting decisions using Equation (20.10). Then, I calculate the sector returns as the weighted excess return using Equation (20.11). Finally, the residual return is attributed to the bond specific return. Table 20.7 shows an example performance attribution for a hypothetical portfolio of corporate bonds. The total portfolio performance over the measured period is −0.90 %, of which −0.57 % is attributed to adverse interest rate movements, −0.46 % to spread widening and a positive contribution of 0.13 % due to selection of the individual bonds. Note also the actions associated with the performance measurement outcomes at each stage of the decision process. I assume, for the sake of simplicity, a fixed risk budget associated with each of the three investment decision steps. In a real investment process, the relative risk budget between the three decision categories would also be related to the performance attribution outcome. 20.8.2 Global asset allocation investment process Consider the following simple equity asset allocation investment process. The investment manager forecasts the total return of each of the five equity regions, Switzerland, European Union, United Kingdom, Unites States and Japan, modeled by their main stock market

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Positive Alpha Generation: Designing Sound Investment Processes

for each time period (a time period is defined as the period between two trades) in the performance measurement period loop calculate the benchmark weights wB,a,t at the beginning of the period, calculate the portfolio weights wP ,a,t at the beginning of the period, calculate the benchmark total return RB,a,t over the time period for each asset class, calculate the portfolio total return RP ,a,t over the time period for each asset class using the net asset values of the funds implementing the asset classes,
– calculate the benchmark contribution as RB,t = a wB,a,t · RB,a,t , – calculate the performance attributable to the asset allocation or market forecasting as RA,t =
a (wP ,a,t − wB,a,t ) · RB,a,t ,  = – calculate the performance attributable to the fund selection capabilities as RS,t
w · (R − R ). P ,a B,a a P ,a

– – – –

end loop aggregate the return attribution for the individual time periods using the aggregation over time performance figure framework using – the decomposition RP = RB + RA + RS for each time period and – a residual attribution function such that f (·) = 0.5.

FIGURE 20.7 Algorithm implementing an asset grouping performance attribution framework based on the Brinson approach Table 20.8 Illustration of the asset grouping based performance measurement framework algorithm described in Figure 20.7 Asset class Switzerland European Union United Kingdom Unites States Japan Total

wB,a

wB,a

3.70 % 17.90 % 12.70 % 44.20 % 21.50 %

0.95 % 20.30 % 14.20 % 40.65 % 23.90 %

RB,a

RP ,a

RA,a

 RS,a

−3.65 % −0.50 % 1.25 % 4.65 % −2.60 % 1.43 %

−3.94 % 0.75 % 1.15 % 3.97 % −1.74 % 1.48 %

0.10 % −0.01 % 0.02 % −0.17 % −0.06 % −0.12 %

0.00 % 0.25 % −0.01 % −0.28 % 0.21 % 0.17 %

Source: sample data, author’s calculations.

indices. Then an asset allocation is constructed relative to a benchmark using the traditional Markowitz mean-variance approach using a fixed tracking error of 2 %. The relative weights versus the benchmark represent the investment decisions whose performance I want to measure. They represent the forecasts transferred in a unique way into portfolio weights. In the second step of the investment process, the investment manager selects actively managed funds for each of the five asset classes. Here, the investment decisions to measure are whether or not the investment manager has selected funds that outperform their corresponding benchmark. For the sake of simplicity I do not take into account the actual active risk taken by each fund manager and assume that the benchmark index of the selected funds is the same as the one used to model the different asset classes.

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Using an asset grouping approach is the most adequate way to attribute the performance of the portfolio to (a) the benchmark selected, (b) the asset allocation or market return forecasting decisions and (c) the fund selection decisions. I implement a Brinson based approach as shown in Figure 20.7. In order to simplify the reporting, I attribute the residual or cross product return to the allocation decision. Table 20.8 shows a sample use of the proposed performance measurement framework algorithm. The portfolio outperforms its benchmark by 5 bp, of which −12 bp is due to the asset allocation effect and +17 bp from security selection. The investment manager showed security selection skills over the measured period, but failed to add alpha through asset allocation skills.

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Index A absolute positive return investment solutions (APR) alpha 254, 257–8, 261–6, 283 arbitrage/diversification skills 254, 255–7 benchmark oriented investment solutions 265–6 concepts 4–5, 253–66, 283, 305–7, 325–7 definition 253–4 engineering skills 254–5, 265 forecasting skills 254, 257–8 hedge funds 283 long-only forecasting based solutions 259–61 portable alpha approach 261–6, 301–2 risk management 260–1 risk–return framework 258–9, 305–7 skills needs 254–8 absolute risk, concepts 96–9, 305–29 absolute total returns, concepts 48–9, 305–29 active investment management 1–7, 9–34, 233–8, 244–52 see also investment management benchmark oriented investment solutions 233–8, 244–52, 265–6, 301–2 core-satellite investment solutions 249–52, 261–2 fundamental law 19–22, 104–5, 159–60, 176, 245–6, 280–2 success factors 1, 21–2, 261–6 active risk see also asset risk concepts 96–9, 250–2, 266, 328–9 definition 17–18 actively managed assets (AM) see also hedge funds concepts 43–4 adaptive dynamic multiperiod portfolio construction models, concepts 185–6

agency costs benchmark oriented investment solutions 233–4 transaction costs 201 aggregated returns over time, performance measurement 312–13, 317–18 agile and scalable characteristic, value net approach 30–1 AIC see Akaike’s information criterion Akaike’s information criterion (AIC) 71–2 allocation returns, concepts 316–18 ALM see asset and liability management alpha see also investment manager skills; risk free rates; risk–return. . .; total returns. . . absolute positive return investment solutions 254, 257–8, 261–6, 283 benchmark oriented investment solutions 235, 238–52 concepts 1–7, 15–22, 23–4, 26–7, 37–9, 75–9, 93, 106–7, 149–52, 158–61, 235, 238–52, 254, 257–8, 261–6, 280–4, 301–2, 305–29 definition 1, 15 derivatives 219–29 exploited investment opportunities 19–21, 172, 282–4 formula 15 hedge funds 282–4, 290–2 idiosyncratic (diversifiable) risk 172–81 Jensen’s alpha 318–20 key generating success factors 4–5, 11–22, 75–9, 172, 238–52, 261–6, 280–4 performance measurement 4–5, 297, 305–29 portable alpha APR approach 261–6, 301–2

risk budgeting approaches 149–59, 245–52, 323–7 risk management perspectives 95–6, 149–52 systematic risk 172, 176–81 uncertainty risks 95–6 alpha returns, definition 15–16 alpha thesis 45 alternative risk measures 129–45 American options 215–16 AMPL 190 anarchy, decision making 87–8 ANN see artificial neural networks anticipative dynamic multiperiod portfolio construction models, concepts 184–6 approximations, quantitative forecasting processes 57–60 APR see absolute positive return. . . APT see arbitrage pricing theory arbitrage opportunities hedge funds 283–4 market inefficiencies 38–9, 42–4, 62–8, 254, 255–7, 274, 279, 283–4 arbitrage pricing theory (APT) 42–4, 52, 62–8 architects, value chains 25–8 ARMA see autoregressive moving average models artificial neural networks (ANN) 62, 65–8 asset allocation investment opportunities approach concepts 12–15, 25, 28–9, 84–5, 99–104, 129–33, 150–2, 208–11, 233, 325, 327–9 examples 14–15, 208–11 asset classes concepts 11–22, 46–8, 58–60, 99–104, 162–70, 192–4, 213–29, 236–52, 289–92, 316–29 definition 11, 289–90

342 asset classes (Continued) examples 14–15, 192–4 hedge funds 289–92 rankings 14–15, 192–4 asset grouping models, performance attribution 316–18, 327–9 asset and liability management (ALM) 4–5, 293–302 asset re-pricing techniques, nonparametric risk models 140, 143 asset risk see also active risk concepts 17, 96–9, 129–45, 328–9 definition 17, 96 asset types, risk premiums 43–4, 48–55, 251–2 assets definition 11 types 43–4 asymmetrical distributions 133–8 see also return distributions at-price orders 204 auctions 205 autocorrelation concepts 71–2, 104, 122–4 handling methods 122–4 automatic order matching tool 23–4 autoregressive moving average models (ARMA) 62–8

B backtesting phase covariance matrix estimates 125–8 quantitative forecasting processes 60, 71–9, 125–8 backward-looking information, forward-looking information 46–8 barriers to entry 30 basket trades 209–11 Basle Committee on Banking Supervision 124 Bayesian models 62–8, 82–9, 105, 115, 166–70 benchmark oriented investment solutions see also active. . .; passive. . . absolute positive return investment solutions 265–6 concepts 4–5, 233–52, 265–6, 301–2 decision making 234–52 definitions 233–4 examples 251–2, 301–2 indices 236–44 investor’s perspectives 233–8 reasons 233 benchmark oriented mean-variance model, concepts 158

Index benchmark portfolios see also market. . . concepts 11, 13, 15–22, 96–9, 157–8, 208–11, 228–9, 233–52 definition 11, 16 information ratios 18–19, 258–60, 301–2, 314–15 selection issues 16, 18–19, 235–6 benchmark returns, concepts 316–29 benchmark risk, definition 17–18 best execution concepts 201–4 best practice basic principles, investment processes 21–2 beta exposure, derivatives 226–9, 242–3 bibliography 331–40 bid–ask spread costs see also transaction costs concepts 198–211 BL. . . see Box–Ljung test statistic ‘Black Monday’ crash of 1986 276 Black and Scholes option pricing model 118–19, 215–16, 269–70 Black–Litterman model 17, 49, 64, 166–7, 176 algorithms 166–7, 176 concepts 166–7, 176 block maximal EVT models 135–8 blue zone, color zone PMF approaches 322–3 bonds 11–12, 19, 25, 40–3, 52–8, 65, 75–9, 104–7, 120–2, 131–2, 139, 150–2, 183–4, 192–4, 197, 214–29, 236, 246–8, 250–2, 256–66, 270, 284, 290–2, 298–9, 309, 321–7 see also fixed-income securities equities 40, 42–3, 65, 75–9, 104–7, 183–4, 192–4, 250–2 ratings 251–2 yield curves 42–3, 47–8, 65, 75–9, 139–40, 197 zero coupon bonds 228–9, 270 bootstrapping techniques, nonparametric risk models 140, 142 bottom-up approaches 25, 31–4, 41, 44–55, 154, 307–8 boundary and extreme events forecasting model tests 71–3, 77–9, 127–8, 286–8 Box–Ljung test statistic (BL. . .) 124 breadth of investment opportunities concepts 20–2, 280–2 definition 20 Brinson performance attribution model 316–18, 327–9 brokers (intermediaries), concepts 203–5, 209–11, 287 Brownian motion models 62–8, 73, 183–4, 187, 192–4, 274–8

budgeting risk see risk budgeting approaches building block based investment solutions, liability driven investing 297–302

C calendar based portfolio rebalancing algorithm 248–9 call options 118–19, 215–16, 221, 224, 254–5, 270, 276–8, 310 Calmar ratio 315 capital asset pricing model (CAPM) see also conditioned explanatory quantitative forecasting models concepts 2, 41–4, 57, 60–1, 62–8, 98–9, 113–16, 157, 166–7, 171, 173–6, 178–9, 199–200, 226–7, 233, 238–9, 242–3, 314–15, 318–19 definition 41 capital assets (CA) see also bonds; equities; property. . .; stocks concepts 43–4 capital protection and preservation investment solution approaches see also portfolio insurance. . . client specific designs 277–8 concepts 4–5, 267–78 dynamic strategies 267, 269, 271–8 static strategies 267, 269 CAPM see capital asset pricing model CART framework 65 see also hierarchical decision trees cash 43–4, 236, 251–2, 290–2, 321 see also risk free assets cash flow matching, liability driven investing 295–7, 298, 299–304 cash liquidity risk, definition 94 CBOT see Chicago Board of Trade certainty equivalent framework, concepts 153–4 change management issues investment processes 2 value chains 28–31 checklist benefits, structured design methods 3–4 Chicago Board of Trade (CBOT) 75, 213 CIV see collective investment vehicles classical decision theory 82–4 classifications, information 46–8 closed market types 204–5 coherence concepts 45–6, 50–5, 98–104 definition 50 coherent risk measures, subadditivity 17–18, 98–9 collaborative and systemic characteristic, value net approach 30–1

Index collateral, derivatives 217–18, 220–2, 225–9, 263–5, 277 collective investment vehicles (CIV) 324–5 color zone approaches, performance measurement 320–3 commercial paper 277–8 commissions, brokers 203–4, 209–11 commodities 12–15, 43–4, 192–4, 250, 256–7, 288–9 see also consumption and valuing storing assets communications decision making processes 82, 83–8, 89 initiators 83–8 mediums 83 competencies, investment processes 21–2 competitive advantages see also investment decision skills concepts 3, 23–4, 31–4 value chains 23–4, 31–4 competitive pricing, investment products 23, 33–4 concave shape, risk function 18, 41–2 Conditional Value at Risk (CVaR) 131, 165–6 conditioned explanatory quantitative forecasting models 57, 61–8, 113–14 see also capital asset pricing model; forecasts confidence levels 9, 49–50, 72–3, 88–9, 95–6, 105–6, 126–8, 209–11, 288–9 see also hit ratio confirmation agreements, derivatives 218–19 consistency concepts 1–7, 45–6, 50–5, 69–70, 323–9 definition 2, 50 examples 55 constant proportion portfolio insurance (CPPI) concepts 271–8, 297 practical implementation considerations 272–3 constant relative risk aversion (CRRA) 186–7 constants, notation uses 6–7 consumption and valuing storing assets (CV) see also commodities concepts 43–4 continuous time models, concepts 183–4, 190–4 continuously compounded excess returns 315–16 contractual agreements, hedge funds 286 convertibles 281–4, 290–1

343 core capabilities see also investment decision skills concepts 3, 254–8 core-satellite investment solutions 249–52, 261–2 corner portfolios, concepts 153–4 corporate bonds see bonds correlations 7, 13, 58–79, 105–6, 109, 118–19 notation uses 7 cost of capital see also discounted cash flow concepts 41, 84 cost-of-carry model 214–15 counterparty risk see credit risk covariance 7, 20–1, 40–1, 70, 99–100, 109–28, 152–61 see also volatility. . . matrix 99–100, 104, 110–28, 152–70, 192–4 notation uses 7 tests 125–8 Cox, Ingersoll and Ross short term interest rate forecasting model 63, 142–3 CPPI see constant proportion portfolio insurance credit default swaps 255 credit risk concepts 18, 93–4, 144–5, 215–29, 285–7, 325–7 definition 18, 93–4, 144–5, 285–6 models 144–5 CreditMetrics approach 145 CreditRisk+ model 145 cross product returns 313, 329 cross-time sample covariance matrix, definition 122–3 CRRA see constant relative risk aversion currencies 11, 45, 51, 58, 67–70, 134, 150–2, 158–9, 214–29, 301–2, 310–12, 320–4 derivatives 227–9, 311–12 performance measurement 310–12, 320–4 currency swaps 227–9 current ratios 47 CVaR see Conditional Value at Risk

D data-handling parameter estimations 68–71, 77–9, 100–7, 110–19 dates, value/trade-date contrasts 309 DCF see discounted cash flow DDM see dividend discount model debt assets see also bonds concepts 43–4 decision making see also Bayesian models; investment decision skills algorithms 81, 86–8

benchmark oriented investment solutions 234–52, 265–6 building processes 83–9 characteristics 81 classical school 82–4 communications 82, 83–8, 89 concepts 1–7, 11–12, 81–9, 186, 238 critique 86–8 democratic decisions 85–9 examples 88–9 forecasts 81–9 formulaic decisions 86–8 goals 81–2, 86–7 information 18, 81–9, 94–6, 104–7 mechanisms 85–8 monarchic decisions 86–8 organizational structures 86–7 ‘out of the box’ thinking 87–9 performance measurement 305–29 steps 81–2 theory 82–3 utility functions 81–9, 267–8 weightings 88–9, 234–52 decision trees 65 decision-based information 46–8 decomposition approaches see also value chains investment opportunities 12–13, 15–17 investment processes 21–2, 23–34, 149–70 market risk categories 98–9, 250–1 modeling risk 98–104, 150–2 performance measurement 306–8 portfolio construction 149–70 returns 15–17, 42–4 roles/responsibilities delegation strategy 23–34 volatility 119–22 default risk, credit risk 18, 93–4, 144–5, 325–7 degree of information, forecasting communication approaches 83–4 democratic decisions 85–9 derivatives 4–5, 11, 75–9, 118–19, 201, 213–29, 242–3, 248–9, 263–6, 279–92, 300, 309–10 see also forward. . .; futures; options; swaps alpha 219–29 beta exposure 226–9, 242–3 characteristics 213–19 collateral 217–18, 220–2, 225–9, 263–5, 277 concepts 4–5, 11, 118–19, 213–29, 242–3, 248–9, 279–92, 309–10 confirmation agreements 218–19 currency risk hedging 227–9, 311–12 definition 213 examples 224–9

344 gaining exposure 220–1, 224–7 haircuts 217–18 hedge funds 279–92 hedging 221–3, 227–9, 279–92 historical background 213 investment strategy implementation 219–29 ISDA master agreement 218–19, 227–8 legal aspects 218–19, 227–8 long–short investment strategies 220–9, 284 margin requirements 217–18, 220–1, 225–9, 263–5 marked-to-market valuations 217–18, 277 minimum variance hedge ratio 223 payout replication 220, 223–4 performance measurement 309–10 portfolio construction 219–29 prices 118–19, 214–16, 324 reducing exposure 220–3, 226–7 types 213–14 underlying assets 216–17 uses 219–29, 242–3, 248–9, 254–5, 263–6, 269–70, 276–8, 279–92, 300 deterministic information 82–3 Dietz method 308–9 digital characteristic, value net approach 30–1 disclosure guidelines, best execution 202 discounted cash flow (DCF) concepts 31–4, 41–4, 58–9, 84 definition 41 discovery mechanism, prices 205 discrete time models, dynamic multiperiod portfolio construction models 183–94 discrete total returns, concepts 308–14 distinguishable property, investment opportunities 13, 14–15 distressed bonds 281–4 distributions see return distributions diversifiable risk see idiosyncratic. . . diversified portfolios 40, 98–9, 120–2, 154–61, 176–7, 254, 255–7, 286, 298–300, 319–20 diversifying risk component, volatility decomposition approaches 120–2 dividend discount model (DDM), definition 41 dividend yield 47 downside risk model see mean semi-variance model duration model 98, 171, 179–80, 298–9, 318–19, 325–7 Durbin-Watson test 71–2 see also autocorrelation dynamic multiperiod portfolio construction models

Index concepts 4–5, 183–94, 235–6, 246–9 continuous time models 183–4, 190–4 critique 183–4, 186 definition 183–6 discrete time models 183–94 drawbacks 184 examples 191–4 myopic models 184–6 optimal rebalancing 186–7, 246–9, 255 sample 183–4 stochastic programming techniques 184, 187–94 theoretical background 183–4 types 184–6, 191 uses 183–4, 186

E earnings dimension, exogenous information classification 47–55 ECB see European Central Bank econometric theories 57–79 see also quantitative forecasting processes economic cycles 47–55, 65, 186 economic factor model 114 economic underlying theories, quantitative forecasting processes 60–8, 75–9 efficient frontiers 152–70 efficient market hypothesis see also market efficiencies concepts 37–8, 45, 61–2 electronic crossing networks (ECN) 205 electronic platforms, prices 205 elliptic distributions 131–4 see also return distributions EM see expectation maximization algorithm emerging markets 2, 88–9, 250–2, 256–7, 290–2, 300–2 employees, pension funds 293–4 endogenous information classification, concepts 46–55, 101–2, 200–1 endogenous market liquidity, concepts 200–1 energy assets 14–15 engineering and project management skills 1, 21–2, 254–5, 265, 283–4 see also project management. . . absolute positive return investment solutions 254–5, 265 hedge funds 283–4 environmental risk model component 99–103 equilibrium portfolio, definition 173–4 equilibrium quantitative forecasting models 62–8, 166–7, 172–81

equities 25, 32–4, 40–55, 58, 65, 75–9, 104–7, 120–2, 129–33, 150–2, 162–70, 174–6, 183–4, 192–4, 208–11, 214–29, 236, 246–8, 250–2, 256–66, 290–2, 300, 321, 327–9 see also stocks bonds 40, 42–3, 65, 75–9, 104–7, 183–4, 192–4, 250–2 risk premiums 43, 48, 65, 251–2, 283–4, 298–300 equity premium puzzle, definition 40 error maximizations, concepts 113 estimable model structures, forecasts 61–2, 76–9 estimates mean 112–13 parameters 63–8, 77–9, 99–100, 102–7, 110–28, 153–4 quantitative forecasting processes 58–79, 99–100, 104–7, 109–28 ETFs see exchange-traded funds European Central Bank (ECB) 50 European options 118–19, 215–16, 270, 276–8, 310 event risk see also idiosyncratic. . .; systematic. . . concepts 98–9 EVT see extreme value theory ex-ante information ratio 19, 151–2 ex-post information ratio 18, 151–2 excess returns see also information ratio; residual risk; risk premiums CAPM 2, 41–4, 314–15, 318–19 color zone PMF approaches 320–3 concepts 18–19, 40–4, 261–6, 307–29 definition 18 performance measurement 314–29 exchange-traded derivatives see also derivatives concepts 213–29 exchange-traded funds (ETFs) 209–11, 301–2 exchanges see public market types exogenous information classification, concepts 46–55, 101–2, 200–1 exogenous market liquidity, concepts 201 expectation maximization algorithm (EM), missing data 70–1 expected hit ratio see hit ratio expected results, quantitative forecasting processes 58–60, 75–9 expected returns see also utility functions concepts 13–15, 17–18, 41–55, 97–107, 191–4, 297, 305–29 forecasts 13–15, 41–55, 191–4, 305–29

Index risk definitions 17–18 expected shortfall risk see Conditional Value at Risk expected values, notation uses 6–7 explicit (visible) transaction costs, concepts 197–211 explicitness and transparency benefits, structured design methods 3–4 exponential smoothing 62–8 exposure based risk models concepts 101–2, 138–40, 220–3 fixed-income securities 139–40 external risks, definition 276–7 extreme event distributions 135–8, 286–8 see also return distributions extreme value theory (EVT) 135–8

F factor based quantitative forecasting models 61–70, 75–9, 113–14, 122–4 factor exposure risk decomposition approaches 101–2, 220–2 factor model based portfolio construction 12–13, 61–2, 70, 75–9, 103–4, 113–14, 171–81, 325–7 complexity benefits 177–8 concepts 171–81 critique 171, 175–6, 179 drawbacks 171, 175–6, 179 risk 171–2 sensitivity based uses 179–80 simplicity benefits 177–8 Treynor and Black portfolio construction model 171, 172–6, 180–1 factor model methods, performance attribution 316, 318–20, 325–7 factor model sensitivity based portfolio construction algorithm, definition 180 fair values, prices 199–200 Fama, E.F. 64, 113–14, 171, 173–4, 319–20 Fama and French model 171, 173–4 Fama’s performance components model 319–20 fast flow characteristic, value net approach 30–1 fat tails 133–8 see also return distributions feedback loops, investment solutions 22, 24 fees brokers 203–4, 209–11 hedge funds 279, 286 five-asset equity index example 237–8 fixed-income securities 25, 43, 98, 104–7, 164, 171, 179–80,

345 214–29, 236, 290–2, 298–9, 318–19, 325–7 see also bonds duration model 98, 171, 179–80, 298–9, 318–19, 325–7 exposure based risk models 139–40 performance measurement example 325–7 time horizons 228–9 flexibility tradeoffs, investment processes 2 floating rate notes 216, 300 forecasts see also investment decision skills; qualitative. . .; quantitative. . . approaches 13–14, 44–55, 57–79, 206–7 basic principles 14 bottom-up approaches 44–55, 307–8 categories 48–9, 58–60 coherence concepts 45–6, 50–5 combined information 4, 13–14, 81–9 communications 82, 83–9 concepts 1–7, 11–12, 13–22, 25–34, 35–89, 206–7, 254, 257–8, 280, 281–4, 305–29 consistency concepts 45–6, 50–5, 69–70, 323–9 decision making 81–9 definition 13 estimable model structures 61–2, 76–9 examples 14–15, 51–5, 74–9 expected returns 13–15, 41–55, 191–4, 305–29 fundamental law of active investment management 20–1, 104–5, 159–60, 176, 245–6, 281–2 hedge funds 280, 281–2, 284, 290–2 information 4, 13–14, 45–55, 57–79, 94, 183–94, 305–29 market efficiencies 37–9 market expectations 45–55 model risk 18, 74–9 optimal aggressiveness factor 167–70 performance measurement 305–29 risk premiums 43–4, 48–55, 65, 251–2 simplicity benefits 13, 48–9 skills needs 26–7, 58–60, 104–7, 238–9, 254, 257–8, 280–2 stages 44–55, 57–60 tests 58–60, 71–9 top-down approaches 44–55, 307–8 transaction costs 206–11 translation processes 45–55, 149–70

value chains 25–34, 206–11 volatility 109–28 formal model, quantitative forecasting processes 57–79 formulaic decisions 86–8 forward rate agreements (FRAs) 139–40, 213–14 forward-looking information, backward-looking information 46–8 forwards 11, 45, 139–40, 213–29, 248–9 see also derivatives concepts 213–15, 220–9, 248–9 definition 213–14 modeling prices 214–15 prices 214–15 uses 220–9, 248–9 Fr´echet distribution 135–8 fund-of-fund based investment products 12–13, 151–2, 279 fundamental law of active investment management 19–22, 104–5, 159–60, 176, 245–6, 280–2 fundamentals, exogenous information classification 47–55 futures 75–9, 198, 209–11, 213–29, 248–9, 263–6, 277–8, 309–10 see also derivatives concepts 213–29, 248–9, 263–6, 277–8, 309–10 definition 213–14 margin requirements 217–18, 225–9, 263–5 modeling prices 214–15 performance measurement 309–10 prices 214–15 uses 220–9, 248–9, 263–6, 277–8

G GARCH processes 57, 103–4, 116–19, 124 see also unconditioned pure quantitative forecasting models GDP 42, 47–55, 70, 101, 191, 237, 281–2, 293 see also economic. . . generalized extreme value distributions (GEV) 135–8 generalized least squares (GLS) 64–8 GEV see generalized extreme value distributions global asset allocation investment process performance measurement example 325, 327–9 Global Investment Performance Standards (GIPS) 310 global macro hedge funds 281–4, 290–2 GLS see generalized least squares goals, investment processes 21–2, 23, 86–7

346 government bonds 19, 52–5, 197, 220, 236, 250–2, 290–2, 298–9, 318–19 see also bonds GPD see Pareto distributions green zone, color zone PMF approaches 322–3 growth stocks, style based investment opportunities approach 12–13, 305 Gumbell distribution 135–8

H haircuts, derivatives 217–18 Hamilton–Jacobi–Bellman equation 184 HARA see hyperbolic absolute risk aversion head and shoulders formations 66–7 hedge funds 4–5, 19, 38, 109, 121–2, 192–4, 247, 279–92 see also actively managed assets alpha 282–4, 290–2 arbitrage opportunities 283–4 asset classes 289–92 commodities 288–9 concepts 4–5, 279–92 contractual agreements 286 fees 279, 286 forecasting skills 280, 281–2, 290–2 forecasts 280, 281–2, 284, 290–2 historical background 279 investment processes 286–92 investment solutions 4–5, 279–92 investment strategies 283–4 investment universe 280–4, 286–7 portfolio construction 282 pricing issues 285–6 risk management 285–8 skills needs 280–4 specific issues 284–6 success factors 280–2 traditional investment process contrasts 280–2, 291–2 transfer mechanisms 280, 282, 287–8 transparency concerns 279, 288 valuations 285–6 hedge ratio, definition 223 hedging, derivatives 221–3, 227–9, 279–92 heteroskedasticity 104 heuristic search techniques 60 hierarchical decision trees 65 see also CART framework historical simulations, nonparametric risk models 140–2 hit ratio (HR) see also alpha; information ratio concepts 9, 49–50, 72–3, 88–9, 105–6

Index definition 19 homogeneity property, investment opportunities 13, 14–15 HR see hit ratio hyperbolic absolute risk aversion (HARA), concepts 267–8, 272

I IC see information coefficient iceberg model, transaction costs 197–8 idiosyncratic (diversifiable) risk alpha 172–81 concepts 40, 98–9, 172–81 definition 40, 98–9, 172 portfolio construction 172–81 i.i.d assumption 122 immunization techniques, liability driven investing 295–6 impact of information risk, concepts 104–7 implementation methods see also portfolio implementation performance measurement 305, 307–8 value chains 25–6, 27–34 implicit (invisible) transaction costs, concepts 197–211 implied return multiples, concepts 157 implied volatilities 45–55, 118–19 indices concepts 236–44 definition 236–7 investment processes 239–44 investment universe 236–8 optimized sampling algorithm 239, 240–2, 244 passive (indexed) investment management 39, 233, 238–44, 249–52 stratified sampling 239–40, 244 synthetic replication 222–3, 239, 242–4, 266 types 236–7 weightings 237–44 indirect costs see also opportunity costs; transaction costs concepts 198–211 industrial metals 14–15 industry classification factor models 113–14 inflation rates 45–8, 52–5, 57, 76–9, 191, 281–4 information see also transfer mechanisms classifications 46–8 decision making 18, 81–9, 94–6, 104–7

dynamic multiperiod portfolio construction models 4–5, 183–94, 235–6 endogenous/exogenous information classifications 46–55, 101–2, 200–1 forecasts 4, 13–14, 45–55, 57–79, 94, 183–94, 305–29 gathering risk 104–7 interfaces 24–34, 84–5 losses 2 market efficiencies 37–55 objective/subjective contrasts 46–8 performance measurement 305–29 processing risk 104–7 structure 46–8 information coefficient (IC) see also investment manager skills concepts 20–2, 104–7 definition 20 information ratio (IR) see also excess returns; investment manager skills concepts 18–22, 258–60, 301–2, 314–15 definition 18, 20 information risk concepts 18, 94–6, 104–7, 166–70 definition 18, 94, 95–6, 104 levels of detail 104–5 mean-variance based portfolio construction models 166–70 risk management 18, 94–6, 104–7, 166–70 innovations 23–34 see also investment products institutional investors 28–9, 233 interaction returns, concepts 316–18 interest rate risk, definition 139 interest rate swaps 139–40, 213–14, 216–29, 242–3 interest rates 15, 45–6, 57, 63–4, 75–9, 105, 106, 139–40, 142–4, 213–14, 216–29, 234, 242–3, 265, 277–8, 288, 300–2 interfaces, investment processes 22, 23–34, 84–5 intermediaries see brokers internal rate of return, concepts 309–14 internal risks, definition 276–7 International Swaps and Derivatives Association (ISDA) 218–19, 227–8 introduction 1–7 invasive approaches, performance measurement 321, 323 inverse optimization, concepts 157 investment decision skills see also competitive advantages; core capabilities; forecasts

Index concepts 1–7, 11–12, 14–22, 23–34, 37–9, 81–9, 104–7, 150–2, 238–9, 254–8, 280–4, 313–29 definition 1, 2 development methods 3 hedge funds 280–4 performance measurement 4–5, 297, 313–29 investment management 1–7, 9–34, 93–107, 170, 203–4, 233–52, 297, 305–29 see also active. . .; organizational structures; value chains brokers 203–4 market efficiencies 37–9 organizational structures 21–2, 24–34 performance measurement 4–5, 297, 305–29 risk management perspectives 93–6 success factors 1 types 28–9 investment manager skills see also alpha; hit ratio; information ratio alpha formula 15–16, 238–9 concepts 1–7, 15–22, 23–34, 37–9, 43–4, 93–107, 150–2, 219–29, 238–9, 254–8, 313–29 derivatives 219–29 performance measurement 4–5, 297, 313–29 significance of realized returns 313–14 value chains 23–34 investment opportunities see also breadth. . . concepts 11–22, 172, 280–2 decomposition approaches 12–13, 15–17 definition 12–13 distinguishable property 13, 14–15 examples 14–15 fundamental law of active investment management 20–1, 104–5, 159–60, 176, 245–6, 281–2 homogeneity property 13, 14–15 investment universe 12–13, 14–15, 32–4, 51–2, 99–100, 150–2, 172–6, 191–4, 234–52, 280–4, 286–7 mutually exclusive property 13, 14–15 investment philosophy 45 investment policies, concepts 234–6 investment process risk, definition 94 investment processes see also alpha; consistency; investment decision skills; investor needs; transfer mechanisms

347 absolute positive return investment solutions 4–5, 253–66, 283, 305–7, 325–7 benchmark oriented investment solutions 4–5, 233–52, 265–6, 301–2 best practice basic principles 21–2 capital protection and preservation investment solution approaches 4–5, 267–78 challenges 2–3 change management issues 2 competencies 21–2 concepts 1–7, 20–2, 31–4, 100–4, 205–11, 238–52, 280–92 decomposition modular approach 21–2, 23–34, 149–70, 305–29 definition 1 design issues 25–8 development processes 21–2 engineering and project management skills 1, 21–2, 254–5, 265, 283–4 environmental risk model component 99–103 examples 31–4 flexibility tradeoffs 2 fundamental law of active investment management 19–22, 104–5, 159–60, 176, 245–6, 280–2 goals 21–2, 23, 86–7 hedge funds 286–92 indices 239–44 interfaces 22, 23–34, 84–5 key challenges 2–3 modeling risk 99–107 organizational structures 21–2 overview 1, 4–5 performance measurement 4–5, 297, 305–29 PIIP 267–78 portable alpha APR approach 262–6, 301–2 portfolio implementation 205–11 portfolio rebalancing 186–7, 246–9, 255 risk measure selections 129–33 sophistication levels 2 success characteristics 1–2, 4–5, 21–2, 261–6, 280–2 tests 22 top-down approach steps 22, 25, 32–4, 307–8 value chains 4–5, 22, 23–34, 205–11 volatility decomposition 121–2 investment products 23, 28–31, 33–4, 231–302 see also innovations; investment solutions competitive pricing 23, 33–4 organizational structures 28–31

quality issues 23 investment solutions 1–7, 22, 25–34, 231–302, 325–7 see also investment processes; investor needs absolute positive return investment solutions 4–5, 253–66, 283, 305–7, 325–7 benchmark oriented investment solutions 4–5, 233–52, 265–6, 301–2 capital protection and preservation investment solution approaches 4–5, 267–78 concepts 1–7, 22, 25–34, 231–302 core-satellite investment solutions 249–52, 261–2 definition 1 feedback loops 22, 24 hedge funds 279–92 liability driven investing 4–5, 233–4, 297–302 overview 4–5 PIIP 267–78 portable alpha APR approach 261–6, 301–2 value chains 25–34 investment strategies derivatives 219–29 hedge funds 283–4 investment universe 12–13, 14–15, 32–4, 51–2, 99–100, 150–2, 172–6, 191–4, 234–52, 280–4, 286–7 hedge funds 280–4, 286–7 indices 236–8 investor needs see also investment solutions absolute positive return investment solutions 4–5, 253–66, 325–7 benchmark oriented investment solutions 4–5, 233–52, 265–6, 301–2 capital protection and preservation investment solution approaches 4–5, 267–78 concepts 1–7, 26–34, 149, 233–52, 253–66, 267–78 definition 1 liability driven investing 4–5, 233–4, 293–302 preferences 131–2, 267–78 risk attitudes 26, 40, 95, 131–2, 186–7, 233–52, 253–66, 267–78 utility functions 1–7, 14, 38–9, 41–2, 81–9, 95–6, 152–61, 166–7, 171–2, 186–94, 199–202, 205–11, 233–52, 254–66, 267–78, 314–15 investor reporting skills needs 27–8

348 value chains 25–6, 27–34 investor-aligned characteristic, value net approach 30–1 investor-types organizational structures 30–1 investor’s perspectives absolute positive return investment solutions 253–4 benchmark oriented investment solutions 233–8 risk management 93–6 IR see information ratio ISDA see International Swaps and Derivatives Association

J Jensen’s alpha 318–20 Jones, Alfred Winslow 279 judgmental approaches to marketing see qualitative forecasting processes

K Kalman filter models 62, 67–8 KMV model 145 knowledge 23–34 Kupiec test 126–7

L LDI see liability driven investing Lee’s optimal risk budgeting model 167–70, 176 legal aspects of derivatives 218–19, 227–8 legal risk, definition 18, 94–5 leverage hedge funds 279–92 mean-variance based portfolio construction models 160, 179 liability driven investing (LDI) analysis processes 294–5 building block based investment solutions 297–302 cash flow matching 295–7, 298, 299–304 concepts 4–5, 233–4, 293–302 definition 293 immunization techniques 295–6 investment solutions 4–5, 233–4, 297–302 portfolio construction 295–7, 302 risk exposure management techniques 296–7, 300–2 risk factor matching 296–8, 300–2 limit orders 204 linear factor quantitative forecasting models 62–8 liquidity issues concepts 4–5, 14, 27, 197–211, 239, 251, 279–92 definitions 200–1

Index examples 208–11 hedge funds 279–92 impacts 200–2 trading 200–2 transaction costs 200–2 liquidity risk, definition 18, 94, 285 listwise deletion techniques, missing data 70–1 long-only APR forecasting based solutions 259–61 long-only (traditional) investment solutions 4–5, 259–61 long–short investment strategies, derivatives 220–9, 284 long–short mean-variance model, concepts 158, 168–70 long–short portfolios 168–70, 220–9, 284, 290–3 LTCM 279 luck 37

M macroeconomic forecasts 32–4, 42, 47–55, 82, 114 making and taking trading algorithm 207–8 manager selection investment opportunities approach 12–13, 48 margin requirements, derivatives 217–18, 220–1, 225–9, 263–5 marginal contribution to risk (MCR), concepts 101–3, 120–2, 150–2 marked-to-market valuations, derivatives 217–18, 277 market assumptions 46–8, 52, 57–79, 100–4, 151–2 market efficiencies 37–55, 61–8, 153–61, 254, 255–7, 274, 279, 283–4 alpha 37–8 arbitrage opportunities 38–9, 42–4, 62–8, 254, 255–7, 274, 279, 283–4 concepts 37–9, 45, 61–2, 153–61 empirical studies 37 forms 37–8 market equilibrium theory 17, 166–7, 172–3 market expectations 45–55 market inefficiencies, concepts 38–9 market liquidity risk see also liquidity. . . definition 94 market makers 202–5 market orders 204 market participants 202–5 market portfolios see also benchmark. . .; portfolio. . . concepts 11, 13, 17 definition 11, 17 market risk see also modeling. . .; systematic. . .

concepts 17–18, 93–107, 170, 172, 250–2, 286–8 decomposition categories 98–9, 250–1 definition 17–18, 93 market takers 202–5 market timing return coefficient, definition 16 market types, trading 204–5 Markov chain Monte Carlo models (MCMC models) 62, 68 Markowitz mean-variance model 50, 95, 97, 154–61, 164, 184, 240–2, 328–9 see also mean-variance. . . concepts 154–61, 164, 184, 240–2, 328–9 risk free assets 154–61 martingale approach 184 mathematical risk model specification component 99–100, 103–4 matrices, notation uses 6–7 maximal drawdown see peak to bottom risk measure maximum likelihood estimators (ML) 60, 64–8, 100, 103–4 maximum principle, performance measurement 314–15 MCMC models see Markov chain Monte Carlo models MCR see marginal contribution to risk mean 4–5, 18–20, 47–55, 58–9, 60–1, 72–3, 78–9, 95, 97, 112–13, 131–2, 149–70, 171, 179, 301–2, 328–9 mean absolute error (MAE) 72–3, 78–9 mean absolute percentage error (MAPE) 72–3, 78–9 mean belief of investors, prices 38 mean error (ME) 72–3, 78–9 mean percentage squared error (MPSE) 72–3, 78–9 mean reversion 47–55, 58–9, 103–4, 204, 306–7 mean semi-variance model, concepts 164–6 mean squared error (MSE) 72–3, 78–9 mean-variance based portfolio construction models see also portfolio. . . alternative models 161–70 Black–Litterman model 17, 49, 64, 166–7, 176 concepts 4–5, 18–20, 95, 97, 149–70, 171, 179, 301–2, 328–9 critique 157–70, 171, 179 currency restrictions 158–9 different product types 157–8 drawbacks 160–70, 171 information risk 166–70 investor restrictions 159–60, 170

Index Lee’s optimal risk budgeting model 167–70, 176 leverage restrictions 160, 179 Markowitz mean-variance model 50, 95, 97, 154–61, 164, 240–2, 328–9 min–max efficient model 161–4, 170 parameter drawbacks 160–70, 171 re-sampled efficient frontier approach 161–2 restriction-handling advantages 157–60 risk free assets 154–61 Meriwether, John 279 Merton, Robert C. 183–4, 190–1, 279 Merton’s continuous time algorithm 183–4, 190–1 min–max efficient portfolio model 161–4, 170 minimum-variance hedge ratio (HR) 223 minimum-variance portfolios 152–61, 223 mispricing opportunities 172–6 missing data, quantitative forecasting model parameters 70–1, 104, 110–11 ML see maximum likelihood. . . model risk concepts 18, 74–9, 94–5 definition 74, 94 sources 74, 94 modeling risk see also volatility risk model axiomatic approach 17–18, 97–8 components 99–104 concepts 4–5, 93–107, 129–33, 140–4, 240–2 credit risk models 144–5 decomposition approaches 98–104, 150–2 development processes 99–104 environmental components 99–103 mathematical model specification component 99–100, 103–4 monotonicity axiom property 97–9, 154 multiple models 129–33 nonparametric risk models 140–4 parameter estimation component 99–100, 102–4 positive homogeneity axiom property 97–9 subadditivity axiom property 17–18, 98–9 tests 125–8 theory 96–9 time horizons 102–3, 110–11, 124, 154 translation invariance axiom property 97–9

349 modeling transaction costs, concepts 198–206 models prices 2, 39–44 returns 39–44 modern portfolio theory 109, 154–61 Modigliani–Modigliani ratio (MM) 315 module designers see also decomposition approaches value chains 25–8, 197, 205–11, 305–29 momentum based information 48–55 monarchic decisions 86–8 money weighted returns, concepts 309–14 monotonicity axiom property, modeling risk 97–9, 154 Monte Carlo simulations 60, 68, 71–3, 118–19, 140, 142–5, 187–90, 276–7, 287–8 mortgage backed securities 325–7 moving averages 57, 62–8 see also GARCH processes multi-factor model 171 multidimensional performance figures 314–15 multilayer neural network models 60 multiperiod portfolio construction models 4–5, 183–94 see also dynamic. . . multiple risk models 129–33 multivariate normal distributions 133–4, 165 mutually exclusive property, investment opportunities 13, 14–15 myopic models see also dynamic. . .; Markowitz mean-variance model concepts 184–6

N NAV see net asset value negative returns 253–4 net asset value (NAV) 324–5 no trade theorem 38 noise traders 38–9 nonlinear models ANN 62, 65–8 concepts 62, 64–8, 144 hierarchical decision trees 65 Kalman filter models 62, 67–8 MCMC models 62, 68 pattern matching models 62, 66–8 nonlinear payoffs, risk measures 144 nonparametric risk models, concepts 140–4 norm, notation uses 6–7 normal distributions 60–1, 99–100, 103–7, 116–19, 129, 131–3, 216 see also return distributions normal mixture distributions 133–4

notation uses 6–7, 96–7 null hypothesis tests 77–9, 126–7

O OAFs see optimal aggressiveness factors objective/subjective information contrasts 46–8 OBPI see option based portfolio insurance observation equations, Kalman filter models 67–8 observations, quantitative forecasting processes 57–79 OLS see ordinary least squares opaque/transparent model contrasts, quantitative forecasting processes 58–9 open market types 204–5 operational excellence 27 operational risk see also settlement risk definition 18, 94 opportunistic participation strategy 207–8 opportunity costs see also indirect costs; transaction costs concepts 197–211, 239 optimal aggressiveness factors (OAFs) 167–70 optimal rebalancing, dynamic multiperiod portfolio construction models 186–7, 246–9, 255 optimized sampling algorithm, concepts 239, 240–2, 244 option based portfolio insurance (OBPI) 269–78 options see also derivatives Black and Scholes option pricing model 118–19, 215–16, 269–70 concepts 118–19, 213, 215–29, 254–5, 269–78, 310 definition 213, 215 performance measurement 310 PIIP 269–70, 276–8 prices 118–19, 215–16 types 215–16 uses 219–29, 254–5, 269–70, 276–8 orange zone, color zone PMF approaches 322–3 order books 202–4 order types 204–5 ordinary least squares (OLS) 60, 64–8, 78, 242–3, 319 organizational structures critique 30–1 decision making processes 86–7 examples 25, 29, 31–4 investment process structure 21–2 investment products 28–31

350 types 30–1 value chains 24–34 OTC see over the counter ‘out of the box’ thinking, decision making 87–9 over the counter (OTC) 205, 207–8, 213–29, 285 see also derivatives overview of the book 4–7

P pairs, return forecast concepts 49, 88–9 parameters consistency needs 69–70, 104 data-handling estimations 68–71, 77–9, 100–7, 110–19 estimation techniques 63–8, 77–9, 99–100, 102–4, 110–28, 153–4 mathematical risk model specification component 99–100, 104 mean-variance based model drawbacks 160–70, 171 missing data 70–1, 104 modeling risk component 99–100, 102–4 PIIP 267–8 quantitative forecasting processes 57–79, 104, 110–28 reliability needs 69–70, 104 volatility risk model 110–19 Pareto distributions 73, 129, 135–8 partial derivatives 138–40 partial differential equations 184 passive (indexed) investment management concepts 39, 233–44, 249–52 core-satellite investment solutions 249–52, 261–2 critique 238–9 pattern matching models 62, 66–8 payout replication, derivatives 220, 223–4 peak to bottom risk measure 129–30 pension funds 191–4, 236, 251–2, 293–302 see also liability driven investing people competencies 21–2 percentage of portfolio based rebalancing algorithm 248–9 perceptrons, ANN 65–8 performance attribution, concepts 306–7, 316–29 performance contribution, concepts 307–8, 320–9 performance measurement basics 308–15 Brinson performance attribution model 316–18, 327–9 Calmar ratio 315 color zone approaches 320–3 concepts 4–5, 297, 305–29

Index consistency needs 323–4 currencies 310–12, 320–4 decomposition approaches 306–8 derivatives 309–10 dimensions 305–7 examples 325–9 Fama’s performance components model 319–20 fixed income investment process example 325–7 framework implementation 305, 307–29 global asset allocation investment process example 325, 327–9 information ratio 18–22, 258–60, 301–2, 314–15 invasive approaches 321, 323 investment manager skills 4–5, 297, 313–29 Jensen’s alpha 318–20 maximum principle 314–15 Modigliani–Modigliani ratio 315 multidimensional figures 314–15 output uses 320–3 pitfalls 323–5 practical considerations 323–5 ratios 314–15 return measurement basics 308–14 Sharpe ratio 314–15 Sortino ratio 315 performance measurement framework (PMF) 307–29 PIIP see portfolio insurance investment process plan sponsors, pension funds 293–4 PMF see performance measurement framework political dimensions, exogenous information classification 47–55, 82 portable alpha APR approach 261–6, 301–2 portfolio construction 4–5, 14–19, 25–6, 27–34, 50, 94, 147–94, 207–11, 282, 295–7 see also dynamic. . .; factor model. . .; mean-variance. . . concepts 4–5, 14–19, 25–6, 27–34, 50, 94, 147–94, 207–11, 282 definition 149 derivatives 219–29 hedge funds 282 idiosyncratic (diversifiable) risk 172–81 liability driven investing 295–7, 302 model risk 94 modular approaches 149–70, 305–29 performance measurement 305–29 principles 14 quantitative forecasting processes 149–70

risk budgeting approaches 149–59, 167–70, 245–52, 258–60, 266, 323–7 selection of approach 170 single period mean-variance based portfolio construction 149–70 systematic risk 172, 176–81 transaction costs 207–11 translation processes 45–55, 149–70 Treynor and Black portfolio construction model 171, 172–6, 180–1 value chains 25–6, 27–34, 207–11 portfolio implementation see also derivatives; liquidity issues; trading; transaction costs concepts 4–5, 25–6, 27–34, 195–229, 272–3, 307–29 investment process value chains 205–11 performance measurement 307–29 skills needs 27 value chains 25–6, 27–34, 205–11 portfolio insurance investment process (PIIP) asset categories 268–9 concepts 267–78 CPPI 271–8, 297 dynamic strategies 267, 269, 271–8 OBPI 269–78 options 269–70 parameters 267–8 risk management 276–7 simple strategies 269 static strategies 267, 269 portfolio management see also dynamic multiperiod. . .; mean-variance. . .; static factor. . .; transfer mechanisms concepts 1–7, 11–22, 43–4, 149–70, 195–229 performance measurement 305–29 portfolio optimization 50, 95, 97, 109–28, 154–61 see also Markowitz. . . portfolio rebalancing 186–7, 246–9, 255 portfolio risk see also market risk; risk. . . algorithms 97, 129–33 concepts 96–107, 109, 129–45, 150–2 decomposition categories 98–9, 150–2 definitions 129–33 portfolios, definition 11 positive alpha see alpha positive homogeneity axiom property, modeling risk 97–9 precious metals 12, 14–15 premiums options 215–16

Index risk premiums 18–19, 40–4, 48–55, 65, 251–2, 283–4, 298–300 price indices see also indices definition 236–7 price to book ratios 47 price/earnings ratio 47 prices see also transaction costs arbitrage opportunities 38–9, 42–4, 62–8 auctions 205 CAPM 2, 41–4, 57, 60–1, 62–8, 98–9, 113–16, 157, 166–7, 171, 173–6, 178–9, 199–200, 226–7, 233, 238–9, 242–3, 314–15, 318–19 competitive pricing 23, 33–4 derivatives 118–19, 214–16, 324 discovery mechanism 205 fair values 199–200 forwards 214–15 futures 214–15 hedge funds 285–6 market efficiencies 37–55 mean belief of investors 38 models 2, 39–44 no trade theorem 38 options 118–19, 215–16 risk premiums 18–19, 40–4, 48–55, 65, 251–2, 283–4, 298–300 SDF 39–44, 52–5 swaps 216 pricing kernel 39–44 see also stochastic discount factor principal component techniques 60–1, 63–8, 114 private information 46–51 private market types 205 probabilities, Bayesian models 62–8, 82–9, 105, 115, 166–70 process based quantitative forecasting models 62–8, 116–18 project goals see also investment processes concepts 21–2, 23 project management skills see also engineering. . . best practice rules 21–2, 254–5 concepts 1, 21–2, 283–4 property assets 11, 43–4, 251–2, 300 public information 46–51 public market types 205 pull-based information interface exchanges 24, 33–4, 84–5, 89 pure factor model exposure based portfolio construction 176–81 push-based information interface exchanges 24, 33–4, 84–5, 89 put options 215–16, 254–5, 270

351

Q qualitative forecasting processes see also forecasts concepts 4–5, 37–55, 62–8, 81–9, 105–7 quality controls, interfaces 24, 33–4 quality management 4, 22, 23, 303–29 see also performance. . . investment products 23 quantitative forecasting processes see also forecasts approximations 57–60 backtesting phase 60, 71–9, 125–8 building methods 57–60 concepts 4–5, 14, 48, 57–79, 81–9, 110–19, 149–70, 191–4, 279, 281–2 conditioned explanatory forecasting models 57, 61–8, 113–14 data-handling parameter estimations 68–71, 77–9, 100–7, 110–19 examples 74–9 expected results 58–60, 75–9 formal model 57–79 hedge funds 279, 281–2 market assumptions 57–79, 100–4, 151–2 model risk 74–9 observations 57–79 parameters 57–79, 104, 110–28 portfolio construction processes 149–70 structural definition 57–79 tests 58–60, 71–9, 125 theoretical model 58–68, 76–9 transparent/opaque model contrasts 58–9 types 57, 60–8, 110–19 unconditioned pure forecasting models 57, 61–8, 74–9 underlying theory 58–68, 75–9 quote depth, transaction costs 198–211

R R square statistic 77–9 random variables, notation uses 6–7 rankings, return forecast concepts 49, 55, 192–4, 314–15 ratings, bonds 251–2 rational investors 95 see also utility functions ratios concepts 47–55, 314–15 performance measurement 314–15 re-sampled efficient frontier approach, concepts 161–2 real world models 76–9, 99–104 record-keeping processes, best execution 202 recourse dynamic multiperiod portfolio construction models, concepts 185–6

red zone, color zone PMF approaches 322–3 regulators, pension funds 293–4 regulatory constraints, dynamic multiperiod portfolio construction models 187–90 relative risk, concepts 96–9, 183–4, 209–11, 305–29 relative total returns, concepts 49, 65, 305–29 reliability needs, parameters 69–70, 104 reputation risk, definition 94–5 residual risk see also excess returns definition 18 responsibilities see roles/responsibilities. . . retirees, pension funds 293–4 return attribution, concepts 306–29 return distributions see also normal. . . alternatives 132–8 elliptic distributions 131–4 extreme event distributions 135–8, 286–8 Pareto distributions 73, 129, 135–8 return matching approach, liability driven investing 297 return ranges, concepts 49 returns see also excess. . .; expected. . . absolute positive return investment solutions 4–5, 253–66, 325–7 aggregated returns over time 312–13, 317–18 benchmark oriented investment solutions 233–52, 265–6, 301–2 bottom-up forecasting returns 44–55, 307–8 Brinson performance attribution model 316–18, 327–9 CAPM 2, 41–4, 57, 60–1, 62–8, 98–9, 113–16, 157, 166–7, 171, 173–6, 178–9, 199–200, 226–7, 233, 238–9, 242–3, 314–15, 318–19 categories of forecasts 48–9, 58–60 color zone approaches 320–3 decomposition approaches 15–17, 42–4 determinants 42 discrete total returns 308–14 Fama’s performance components model 319–20 forecasts 13–22, 44–55, 206–7 internal rate of return 309–14 Jensen’s alpha 318–20 mean semi-variance model 164–6 mean-variance based portfolio construction models 4–5, 18–20,

352 returns (Continued) 95, 97, 149–70, 171, 179, 301–2, 328–9 models 39–44 money weighted returns 309–14 notation uses 6–7 performance attribution 306–7, 316–29 performance contribution 307–8, 320–9 performance measurement 305–29 risk–return framework 4–5, 14–22, 152–70, 172–6, 258–9, 305–29 SDF 39–44, 52–5 selection returns 16, 316–29 time weighted total returns 308–14 top-down forecasting returns 44–55, 307–8 risk see also modeling. . .; volatility absolute positive return investment solutions 4–5, 253–66, 305–6 attitudes 26, 40, 95, 131–2, 186–7, 233–52, 253–66, 267–78 benchmark oriented investment solutions 233–52, 265–6, 301–2 color zone approaches 320–3 definitions 17, 74, 93–6, 109, 129–33 performance measurement 305–29 types 17–18, 32, 40, 74, 93–6, 172, 235–6, 276–7, 285–6 risk adjusted realized alpha 1–2 risk aversion 4–5, 26, 40, 95, 131–2, 186–7, 253–4, 267–78 capital protection and preservation investment solution approaches 4–5, 267–78 concepts 186–7, 267–78 CRRA 186–7 HARA 267–8, 272 risk budgeting approaches, concepts 96, 98–9, 149–59, 167–70, 245–52, 258–60, 266, 323–7 risk exposure management techniques, liability driven investing 296–7, 300–2 risk factor matching, liability driven investing 296–8, 300–2 risk free assets (RF) see also cash concepts 43–4, 96–9, 150–2, 154–61, 220–9, 233–4, 268–78 mean-variance model 154–61 risk free rates alpha formula 15 concepts 6–7, 15–16, 18, 40–4, 57, 75–9, 96–9, 112–13, 118–19, 150–2, 154–61, 192–4, 220–9, 261–6, 314–29 notation uses 6–7, 96–7 risk function, concave shape 18, 41–2

Index risk management 4–5, 18, 25–6, 27–34, 91–145, 149–70, 206–11, 250–2, 260–1, 276–7, 285–8, 296–302, 305–29 see also modeling risk; tracking errors; volatility absolute positive return investment solutions 260–1 alpha 95–6, 149–52 aspects 27, 93–107, 260–2 concepts 18, 27, 93–107, 206–11, 250–2, 260–1, 276–7, 305–29 CPPI 276–7, 297 definition 18, 27, 93 hedge funds 285–8 information risk 18, 94–6, 104–7, 166–70 investment management perspectives 93–6 investor’s perspectives 93–6 OBPI 276–7 performance measurement 305–29 perspectives 93–6 PIIP 276–7 skills needs 27 transaction costs 206–11 value chains 25–6, 27–34, 206–11 risk measures 17–18, 91–145 see also volatility. . . alternative measures 129–45 concepts 109–28, 129–45 definitions 129–33 exposure based risk measure 138–40 nonlinear payoffs 144 nonparametric risk models 140–4 risk models see modeling risk risk premiums see also excess returns; systematic risk asset types 43–4, 48–55, 251–2 concepts 18–19, 40–4, 48–55, 65, 251–2, 283–4, 298–300 definition 40 forecasts 43–4, 48–55, 65, 251–2 risk reward returns see also alpha definition 15–16 risk–return framework 4–5, 14–22, 152–70, 172–6, 258–9, 305–29 absolute positive return investment solutions 258–9 alpha formula 15 RiskMetrics 112, 116–19, 139–40 risky assets, PIIP 268–78 robustness and assumption forecasting model tests 71–3, 77–9, 127–8, 161–4 roles/responsibilities delegation strategy, decomposition approaches 23–34 Russel–Yasuda Kasai model 191

S sample forecasting models 72–3, 77–9, 103–4, 110–12, 122–4 satellite portfolios, concepts 249–52, 261–2 scaling risk, time horizons 124 scenario analysis 60, 129–33, 162–70, 187–94 Scholes, Myron 279 scores, return forecast concepts 49 SDF see stochastic discount factor seamless executions 2 see also consistency securities lending 263 security selection investment opportunities approach 12–13, 25 selection issues, benchmark portfolios 16, 18–19, 235–6 selection returns concepts 16, 316–29 definition 16, 316 self-documenting benefits, structured design methods 3–4 semi-strong form of efficient markets, concepts 37–9 semi-variance 95–6, 164–6 sensitivity based uses, factor model based portfolio construction 179–80 sentiment dimension, exogenous information classification 47–55 set of restrictions, investment policies 234–5 settlement risk 94 see also operational risk Sharpe ratio 314–15 short positions, hedge funds 279–92 shrinkage based models, concepts 114–15, 122–4, 166–7 simplicity benefits factor model based portfolio construction 177–8 forecasts 13, 48–9 single period factor model based portfolio construction see also factor model. . . concepts 171–81 single period mean-variance based portfolio construction see also mean-variance. . . concepts 152–70 six sigma 27 skills needs see investment manager skills soft commodities 14–15 Sortino ratio 315 specific risk see idiosyncratic (diversifiable) risk square root of time rule (SROT) 124 SROT see square root of time rule

Index standard deviation 95–107, 131–2, 153–4 see also volatility. . . standard error 78–9, 314 star business models 30 state equations, Kalman filter models 67–8 static factor based portfolio construction models, concepts 4–5, 61–2, 171–81 static models see single period. . . statistical arbitrage/diversification techniques absolute positive return investment solutions 254, 255–7 hedge funds 283–4 statistical forecasting model tests 71–3, 77–9, 125–8 statistical risk, concepts 95–6 statistical underlying theories, quantitative forecasting processes 60–8, 75–9 stochastic control theory, concepts 183–4 stochastic discount factor (SDF) concepts 39–44, 52–5, 62–8 two-factor model 42, 52 stochastic programming techniques, concepts 184, 187–94 stochastic volatility models (SV) 116–19 stocks 11 see also equities stop-loss PIIP strategies 269 strategic asset allocation, concepts 301–2 strategic metals 11–12 strategy, definition 23, 26 stratified sampling, concepts 239–40, 244 stress testing techniques, nonparametric risk models 140, 143–4 strong form of efficient markets, concepts 37–8 structured design methods, concepts 3–4 structured products 279 style based investment opportunities approach 12–13, 305 subadditivity axiom property, modeling risk 17–18, 98–9 subjective/objective information contrasts 46–8 subprime crisis of 2007/2008 255 success characteristics active investment management 1, 21–2, 261–6 alpha 4–5, 11–22, 75–9, 172, 238–52, 261–6 formula 2 hedge funds 280–2

353 investment processes 1–2, 4–5, 21–2, 261–6, 280–2 SV see stochastic volatility models swaps 11, 139–40, 213, 216–29, 242–3, 248–9, 255, 263–6, 285, 300 see also derivatives concepts 213, 216–29, 242–3, 248–9, 255, 263–6, 285, 300 definition 213, 216 prices 216 types 214 uses 220–9, 242–3, 248–9, 255, 263–6, 285, 300 synthetic replication, concepts 222–3, 239, 242–4, 266, 311–12 systematic risk see also market risk alpha 172, 176–81 concepts 40–4, 98–9, 172, 176–81, 286 portfolio construction 172, 176–81

T t statistics 77–9, 103–4, 133–4, 314 tangency portfolios, concepts 156–7 target return solutions see absolute positive return investment solutions TC see transfer coefficient teams, democratic decisions 85–9 technical analysis, pattern matching models 62, 66–8 tests backtests 60, 71–9, 125–8 boundary and extreme events forecasting model tests 71–3, 77–9, 127–8 categories 71 investment processes 22 modeling risk 125–8 quantitative forecasting processes 58–60, 71–9, 125 robustness and assumption forecasting model tests 71–3, 77–9, 127–8, 161–4 volatility risk model 125–8 three-factor model 171 threshold exceeding EVT models 135–8 time horizons 1, 14–15, 16, 48–55, 61–2, 110–11, 124, 141, 154, 184–94, 209–11, 308–14 dynamic multiperiod portfolio construction models 4–5, 183–94 fixed-income securities 228–9 forecast principles 14–15, 48–55, 61–2, 209–11 investment solutions 1 market timing return coefficient 16 modeling risk 102–3, 110–11, 124, 154 scaling risk 124

time series analysis 47–55, 57–79, 140–4 consistency needs 69–70 reliability needs 69–70 time weighted total returns, concepts 308–14 TIPS 52–5 top-down approaches forecasts 44–55, 307–8 investment processes 22, 25, 32–4, 307–8 total return indices see also indices definition 236–7 total return swaps 11, 216, 285 total returns of an investment 1–7, 15–16, 236–7, 253–66, 305–29 see also absolute positive return. . .; alpha; return. . . total risk of components, risk decomposition approaches 102, 120–2, 150–2, 233–6 tracking errors see also risk. . . concepts 4–5, 7, 26, 96–9, 328–9 notation uses 7 trade date, value date contrasts 309 trading algorithms 207–8, 210–11 best execution 201–4 concepts 4–5, 197–211, 307–29 context 202–5 examples 208–11 investment process value chains 205–11 liquidity impacts 200–2 market participants 202–5 market types 204–5 order types 204–5 performance measurement 307–29 transaction costs 207–11 value chains 4–5, 22, 23–34, 197–211 training issues, ANN 66 transaction costs see also prices components 197–8 concepts 4–5, 160, 186–94, 197–211, 221–2, 239, 325–7 examples 208–11 forecasts 206–11 liquidity impacts 200–2 modeling 198–206 portfolio construction 207–11 risk management 206–11 structural impacts on markets 201 utility functions 199–202, 210–11 value chains 206–11 transfer coefficient (TC) concepts 20–2 definition 20 transfer mechanisms

354 transfer mechanisms (Continued) see also portfolio implementation; portfolio management algorithms 149–70 concepts 1–7, 11–22, 149–70, 195–229, 238–9, 280–2, 287–8, 305–29 definition 1, 2, 14, 149 examples 14–15 fundamental law of active investment management 20–1, 104–5, 159–60, 176, 245–6, 282 hedge funds 280, 282, 287–8 needs 149 performance measurement 305–29 weightings 14–15, 16–17 translation invariance axiom property, modeling risk 97–9 translation processes 45–55, 149–70 transparency concerns, hedge funds 279, 288 transparent/opaque model contrasts, quantitative forecasting processes 58–9 transpose, notation uses 6–7 Treynor and Black portfolio construction model 171, 172–6, 180–1 Treynor ratio 314–15 two-factor SDF model 42, 52 two-fund separation theorem, concepts 153–4

U UCITS 224 unconditioned pure quantitative forecasting models see also forecasts; GARCH processes concepts 57, 61–8, 74–9 examples 74–9 underlying assets, derivatives 213–29 underlying theory, quantitative forecasting processes 58–68, 75–9 undiversifiable risk see systematic risk unique selling proposition see also competitive advantages concepts 31–2 unit risk portfolios, concepts 168–70 unit-linked insurance policies 233 utility functions see also expected returns; rational investors concepts 1–7, 14, 38–9, 41–2, 81–9, 95–6, 152–61, 166–7, 171–2, 186–94, 199–202, 205–11, 233–52, 254–66, 267–78, 314–15 decision making 81–9, 267–8 definition 95, 152, 192

Index formula 95, 152, 192 PIIP 267–8 transaction costs 199–202, 210–11

V valuation dimensions, exogenous information classification 47–55 valuations see also prices consistency needs 323–5 hedge funds 285–6 value added see alpha Value at Risk (VaR) 95–6, 130–2, 186, 251–2, 298–9 value chains 4–5, 22, 23–34, 197–211, 305–29 see also decomposition approach architects 25–8 change management issues 28–31 competitive advantages 23–4, 31–4 components 23–5 concepts 4–5, 22, 23–34, 197 critique 29–30, 31 definition 23, 25–6 design issues 25–8, 197, 205–11 examples 25–6, 31–4 forecasts 25–34, 206–11 implementation methods 25–6, 27–34 inhibiting factors 29–30 interfaces 23–34 investment processes 4–5, 22, 23–34, 205–11 investment solutions 25–34 investor reporting 25–6, 27–34 module designers 25–8, 197, 205–11, 305–29 organizational structure 24–34 performance measurement 305–29 portfolio construction 25–6, 27–34, 207–11 portfolio implementation 25–6, 27–34, 205–11 risk management 25–6, 27–34, 206–11 roles/responsibilities delegation strategy 23–34 skills needs 23–34 trading 4–5, 22, 23–34, 197–211 transaction costs 206–11 value net approach 30–1 value date, trade date contrasts 309 value net approach 30–1 value stocks, style based investment opportunities approach 12–13, 305 valuing storing assets 43–4 variable proportion portfolio insurance (VPPI), concepts 273–4

variance 4–5, 7, 18–21, 40–1, 58, 60–1, 71–9, 95–6, 109–28, 149–70, 180–1 see also mean-variance. . .; volatility. . . variance gamma distributions 133–4 vectors, notation uses 6–7 visualization communication mediums 83–4 volatility 4–7, 17–18, 20–1, 45–55, 57–79, 95, 97–107, 109–28, 172–6, 191–4, 217–18, 235–52, 298–300, 319–20 see also risk. . . autocorrelation issues 122–4 decomposition approaches 119–22 definitions 109–10, 130–1, 172 forecasts 109–28 implied volatilities 45–55, 118–19 investment process decomposition approach 121–2 marginal risk decomposition approach 120–2, 150–2 notation uses 6–7 total risk decomposition approach 120–2, 150–2 uses 109 volatility risk model see also modeling risk concepts 109–28, 129–33 definitions 109–10 mean estimates 112–13 parameters 110–19 tests 125–8 theory 109–10 VPPI see variable proportion portfolio insurance

W weak form of efficient markets, concepts 37–8 Weibull distribution 135–8 weightings benchmarks 234–52 decision making 88–9, 234–52 indices 237–44 Lee’s optimal risk budgeting model 167–70, 176 transfer mechanisms 14–15, 16–17 white noise 57–8 Wiener processes 63

Y yield curves 42–3, 47–8, 65, 75–9, 139–40, 197 yield spreads 75–9, 325–7

Z zero coupon bonds 228–9, 270 zero sum games 38–9