Research in Finance, Volume 26

RESEARCH IN FINANCE RESEARCH IN FINANCE Series Editor: John W. Kensinger Volumes 6–25: Edited by Andrew H. Chen RE...

Author: John F. Kensinger

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RESEARCH IN FINANCE

RESEARCH IN FINANCE Series Editor: John W. Kensinger Volumes 6–25:

Edited by Andrew H. Chen

RESEARCH IN FINANCE VOLUME 26

RESEARCH IN FINANCE EDITED BY

JOHN W. KENSINGER University of North Texas, Denton, TX, USA

United Kingdom – North America – Japan India – Malaysia – China

Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2010 Copyright r 2010 Emerald Group Publishing Limited Reprints and permission service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements. The opinions expressed in these chapters are not necessarily those of the Editor or the publisher. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-84950-726-4 ISSN: 0196-3821 (Series)

Awarded in recognition of Emerald’s production department’s adherence to quality systems and processes when preparing scholarly journals for print

CONTENTS LIST OF CONTRIBUTORS

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INTRODUCTION

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REAL OPTIONS ‘‘IN’’ ECONOMIC SYSTEMS: EXPLORING SYSTEMIC DISTURBANCE CAUSES AND CURES S. B. von Helfenstein MANAGING REAL OPTIONS IN NOT-FOR-PROFIT ORGANIZATIONS: THE CASE OF SHELL SPACE John W. Kensinger and Stanley T. Crawford O-SCORE FINANCIAL DISTRESS RISK ASSET PRICING Syou-Ching Lai, Hung-Chih Li, James A. Conover and Frederick Wu THE LONG-TERM RELATIONS UNDER CLIMATE CHANGE BETWEEN ECONOMIC ACTIVITY AND METAL UTILIZATIONS USING THE FORGETTING FACTOR Andrew H. Chen, Jack Penm and R. D. Terrell

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IMPROVED DIVERSIFICATION THROUGH A MIX OF OIL AND EQUITIES Helen Xu

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CHANGES IN TRADING VOLUME AND RETURN VOLATILITY ASSOCIATED WITH S&P 500 INDEX ADDITIONS AND DELETIONS Eric C. Lin

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MODELLING THE US SWAP SPREAD Hon-Lun Chung, Wai-Sum Chan and Jonathan A. Batten PRICING AND RISK MANAGEMENT OF VARIABLE ANNUITIES AND EQUITY-INDEXED ANNUITIES Guanghua Cao, Andrew H. Chen and Zhangxin Chen

CONTENTS

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183

LIST OF CONTRIBUTORS Jonathan A. Batten

Hong Kong University of Science & Technology, Hong Kong, China

Guanghua Cao

Applied Mathematics, Southern Methodist University, Dallas, TX, USA

Wai-Sum Chan

The Chinese University of Hong Kong, Hong Kong, China

Andrew H. Chen

Southern Methodist University, Dallas, TX, USA

Zhangxin Chen

University of Calgary, Calgary, Canada

Hon-Lun Chung

Hong Kong Polytechnic University, Hong Kong, China

James A. Conover

University of North Texas, Denton, TX, USA

Stanley T. Crawford

Richardson Independent School District, Richardson, TX, USA

John W. Kensinger

University of North Texas, Denton, TX, USA

Syou-Ching Lai

Graduate Institute of Finance and Banking, National Cheng Kung University, Tainan, Taiwan

Hung-Chih Li

Graduate Institute of Finance and Banking, National Cheng Kung University, Tainan, Taiwan

Eric C. Lin

California State University, Sacramento, CA, USA

Jack Penm

Australian National University, Canberra, Australia vii

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LIST OF CONTRIBUTORS

R. D. Terrell

Australian National University, Canberra, Australia

S. B. von Helfenstein

Braver P.C., Boston, MA, USA

Frederick Wu

University of North Texas, Denton, TX, USA

Helen Xu

Holy Names University, Oakland, CA, USA

INTRODUCTION The current volume in the Research in Finance series features an international set of contributors. The overall theme of the volume is a timely topic capturing one of the leading issues of the year: coping with ‘‘systemic’’ risk. Sarah von Helfenstein (of Braver PC, a professional services ﬁrm near Boston, Massachusetts) addresses real options within economic systems. This chapter offers very interesting insights into the so-called systemic risks within global ﬁnancial markets. Instead of tying these risks into accounting issues, this chapter argues that the systemic risks arise from rational exercise of the various real options that are inherently present within a global system of inter-related ﬁnancial players (including state-sponsored and private players). Certain patterns of option exercise could manifest symptoms consistent with so-called systemic failure. The incoming series editor, John Kensinger (University of North Texas, USA), who has been involved in developing real options tools for several decades, offers the case of decisions about creation and utilization of shell space as a means of gaining insight into how non-proﬁt organizations use the real options approach. His coauthor, Stanley Crawford (an administrator in the Richardson, Texas Independent School System who is working toward his doctorate in education with a minor in ﬁnance) offers valuable insights into the different criteria that inﬂuence decisions in secondary education organizations (and other non-proﬁt entities), compared with similar decisions in business organizations. Shell space is created during new construction when spaces are enclosed from the elements but with interiors left unﬁnished, often with plumbing and heating, ventilation, and air conditioning systems not fully operational. Such shell space offers an option for later expansion of useable space at lower cost than new construction would require. Acquiring this option to expand is a classic case of buying a real option with clearly identiﬁable underlying asset (the potential for additional ﬁnished space) and readily measureable exercise price (the cost of completion). Options to defer choices until more information becomes available, plus options to abandon with partial recovery of capital may also be present. In the case of a business, the decisions about acquisition and utilization of such options would be based on market value maximization. In the case of a ix

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INTRODUCTION

school system, hospital, or other non-proﬁt organization, a major concern is holding the cost of adding new space from increasing strongly. Management motivation may also be different, with concern about reducing the difﬁculty of getting approval for expansion in future years. Therefore, compared with business executives, non-proﬁt administrators have incentives to spend more on the acquisition of expansion options, while exercising expansion options earlier, and exercising abandonment options later (or not at all). In Chapter 3, a team of accounting professors and a ﬁnance professor offer new evidence that traditional accounting ‘‘indicators’’ of impending ﬁnancial distress may not work well at predicting difﬁculties. This team consists of Syou-Ching Lai, Hung-Chih Li, (accounting professors at the Graduate Institute of Finance and Banking, National Cheng Kung University, Tainan, Taiwan), James A. Conover (a ﬁnance professor and former department chair at the University of North Texas, Denton, Texas, USA), and Frederick Wu (an accounting professor and former chair of the Accounting Department at the University of North Texas). They examine statistical tools used to foresee ﬁnancial distress, ﬁnding that the established tools have difﬁculty reliably predicting ﬁnancial distress. They offer insight into the difﬁculties analysts have experienced recently in their efforts to anticipate the ﬁnancial crisis. The difﬁculties they reveal in the efforts to anticipate ﬁnancial distress show the ﬂaws present in any hope of gaining early warning from accounting data to avert ‘‘systemic’’ shortfalls. Then, the outgoing series editor, Andrew H. Chen (Southern Methodist University, USA), with coauthors Jack Penm and R.D. Terrell (College of Business and Economics, The Australian National University), offer new insight into recent economic upheavals. They examine the time series interactions of gross domestic product and industrial production, with the utilization and consumption of important metals such as copper and steel. They ﬁnd very worthwhile insights not just from data about major industrial economies but also several emerging economies. Helen Xu (Holy Names University, Oakland, California) offers new evidence that crude oil prices consistently display negative correlation with common stock. Thus, crude oil offers improved diversiﬁcation when it is included in a portfolio with stock. She lays out a simple procedure using futures contracts so that ordinary investors could garner the various beneﬁts that are available for bulk dealers in physical oil. These ﬁndings offer signiﬁcant insight for investors trying to decide how to achieve better diversiﬁcation during periods of economic turmoil. The last three chapters have derivatives and stock market indices in common. Eric Lin (California State University, Sacramento) investigates

Introduction

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what happens to a stock when it is brought into a major index such as the S&P 500 or removed from the index. When a stock is in the index, it is subjected to all of the inﬂuences that result when arbitrage traders buy the index and sell the individual stocks in the correct proportions to offset the risk inherent in holding the equities (or conversely, sell the index and buy the individual shares). Eric develops clear evidence for a volume effect and a volatility effect. When stocks are included in the index, trading volume increases and variability of returns shifts slightly (but the reverse does not occur when stocks are removed from the index). Although there is a small and statistically signiﬁcant shift in beta (the measure of market risk) when stocks enter the index, this beta shift is so small that it is not economically signiﬁcant. So it can be said that market-adjusted volatility does not increase. These results should be reassuring for investors who might be fearful about the rationality of equity markets in the presence of derivatives. Jonathan Batten (of the Hong Kong University of Science & Technology, and Macquarie Graduate School of Management, Sydney, Australia) offers new insights into the yield difference between a US Treasury bond and a swap of equivalent maturity. He and coauthors Han-Lun Chung (Hong Kong Polytechnic University) and Wai-Sum Chan (Chinese University of Hong Kong) analyze how the swap spread is affected by macroeconomic sentiment – such as inﬂation expectations or business cycle effects. During periods of economic downturn, spreads typically widen due to portfolio rebalancing into Treasury bonds and away from riskier instruments. This chapter could be very interesting for readers seeking better insight into current economic difﬁculties. In the closing chapter, we hear more about stock index products from Guanghua Cao (recent graduate of the doctoral program in applied mathematics at Southern Methodist University, Dallas, Texas), Andrew Chen (Distinguished Professor of Finance at Southern Methodist University, Dallas, Texas), and Zhangxin Chen (Schulich School of Engineering, University of Calgary, Canada). They focus on variable annuities and equity-indexed annuities, which have embedded put and call options. Analyzing these embedded options from an engineering perspective, they ﬁnd an optimum product mixture of those contracts for insurance companies (optimal in the sense of facilitating the deployment of capital in the most efﬁcient manner). John W. Kensinger Series Editor

REAL OPTIONS ‘‘IN’’ ECONOMIC SYSTEMS: EXPLORING SYSTEMIC DISTURBANCE CAUSES AND CURES$ S. B. von Helfenstein ABSTRACT As global economic systems become increasingly more complex and dynamic and the universal language of historical accounting is being profoundly altered, the theory and tools we use in neo-classical economics, traditional ﬁnance, and valuation are beginning to prove inadequate to the tasks being required of them. Hence, there is a need to consider new avenues of thought and new tools. In this conceptual chapter, I explore the use of real options ‘‘in’’ engineering systems design as a means to achieve more rigorous and insightful results in the design and valuation of economic systems, particularly that of the ﬁrm. In the process, I gain further insight into the causes and cures for systemic disturbances generated by the presence and selection of real options in economic systems.

$

This work is a tribute to Dr. Richard de Neufville, whose seminal contributions to his and our ﬁelds provide new avenues for theoretical exploration and critical problem-solving.

Research in Finance, Volume 26, 1–32 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026004

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S. B. VON HELFENSTEIN Globalization of ﬁnancial markets is a trend that is often discussed but rarely quantiﬁed. The growth of cross-border capital ﬂows—or the value of purchases and sales of ﬁnancials assets by investors from different countries—is one vivid illustration of ﬁnancial globalization at work y The growth of capital ﬂows is having a profound impact on ﬁnancial markets and economies around the world. Much attention has focused on the risks that cross-border investments create – such as the ‘hot’ money that can surge into a country suddenly and rush out just as fast, leaving ﬁnancial ruin in its wake. More recently, concern has grown as increasingly powerful state investors from Asia and oil-exporting nations buy ever-larger investments in US, European, and other markets y Since 1990, global capital ﬂows have grown faster than the value of world trade, world GDP, or the world’s ﬁnancial assets y The growth of large, sophisticated institutional investors and other new ﬁnancial intermediaries has y contributed to the rise in cross-border capital ﬂows. These investors – more than individuals – seek investment opportunities globally. In 2006, McKinsey Global Institute (MGI) estimates that pension funds, mutual funds, and insurance companies around the world had $59.4 trillion in assets under management, nearly triple their size in 1995. Over the past ﬁve years, hedge funds and private-equity ﬁrms have also tripled in size, reaching $2.2 trillion in assets under management by the end of 2006. Farrell et al. (2008, pp. 43–44)

1. INTRODUCTION The prior passage from the McKinsey Global Institute fourth annual report on global capital markets is but a single example of the ever-increasing complexity, interdependency, and riskiness of early 21st century economic systems. Changes in the environment continue at warp speed, outpacing the theoretical and practical developments necessary to describe, discuss, manage, and harvest them effectively. Various stakeholders are applying vast amounts of intellectual and ﬁnancial capital to the tasks at hand. However, they have inherited a static, linear, and deterministic design space from neo-classical economics and traditional ﬁnance that is no longer adequate to meet the challenges facing it. Oddly enough, ﬁnancial accounting, that 500-year-old ‘‘linguistic’’ platform critical to all economic endeavors, has been targeted as the source of the problem, while the real source has been generally ignored. This conceptual chapter discusses a number of the challenges generated by the increasing scope and pace of change. First, it suggests more realistic deﬁnitions of economic systems and the ﬁrm as an economic system. It then provides several examples of the effects of the existence and exercise of real options within economic systems. Several approaches to ﬁrm organizational design and valuation are described, with a focus on using applications of real options ‘‘in’’ engineering systems as the possible solution to the difﬁculties inherent in these processes. It concludes by discussing the

Real Options ‘‘in’’ Economic Systems

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relevance of real options ‘‘in’’ economic systems to systemic disturbances and suggests avenues for future research.

2. PROBLEM STATEMENT Modern economic systems are experiencing exponential growth in complexity, interdependence, and risk, as well as equally diminishing transparency. The events creating such dramatic shifts in the environment have been broadly discussed and analyzed by a range of stakeholders. But effective solutions do not yet exist because a paradigm shift is required to identify and develop them. Instead of concentrating on the problems created by the current linear, deterministic paradigm promoted by neo-classical economics and traditional ﬁnance, the ﬁnancial regulators have recently added fuel to a burning platform by moving ﬁnancial accounting away from its historical transaction basis (i.e., cost and cash) toward a prospective valuation basis (i.e., estimates and opinion). This imposes upon economic systems, at all levels, a potential Tower of Babel effect in which agents within these systems no longer speak a common language, while appearing to use a common unit of measure. Assuming that the direction and pace of change cannot be altered, economics, ﬁnance, and valuation will have to develop better theory and tools to accomplish the mandates set before them. Real options analysis may offer a solution. A detailed discussion of this problem statement follows.

2.1. Financial Accounting is on Trial Basic ﬁnancial accounting is structured on a set of generally useful rules by which all agents within economic systems can describe themselves, their net resources, and their activities in a common language. This common language enables transactions to take place at all levels in the markets on a reasonably level playing ﬁeld. It provides for the exchange of more or less complete information between parties. It provides the base on which strategic planning and competitive games can be designed and implemented. While the taxing and other regulatory authorities have introduced high levels of complexity into basic accounting systems, it is accounting systems that allow them to set their ‘‘handicaps’’ (game restrictions) and fund their activities. Current ﬁnancial accounting rules are complicated but workable because, in the end, every agent resource and activity can be traced back

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to cash, the fundamental unit and common denominator of real-world exchange. Although ﬁnancial accounting serves a critical purpose that cannot be eliminated or replaced, dynamic shifts at the meta-system, system, and sub-system level have created communication issues regarding the effects of such changes on both systems and agents. As ﬁnancial accounting is the common language for business resources and activities, it has got routinely criticized for not measuring up to the challenges in the environment, for not being able to represent the intangible and the esoteric in a universally accepted way. While no one has asked if any other discipline could do better, accounting regulators worldwide have made the decision to opt for ‘‘relevance’’ over ‘‘reliability’’ and invented a new discipline, Fair Value Accounting, codiﬁed in November 2007 in Statement of Financial Accounting Standards No. 157 (SFAS 157), ‘‘Fair Value Measurement.’’ To those not interested or embroiled in accounting or valuation, it seems an innocuous move. Many in the ﬁnance community have welcomed it as an improvement over the ‘‘dinosaur’’ of historical cost basis accounting. Whether we know or care about SFAS 157, it is about to change all worlds. In effect, it has altered the foundational language of economic systems radically, in favor of constructed markets and hypothetical events built on valuation principals, rather than real-world ones based in actual transactions. While for years valuation professionals and in-house corporate ﬁnance teams have grappled with how to assign values to speciﬁc ﬁrm resources and projects without ignoring the other resources and projects that contribute to such values, this problem was primarily conﬁned to the context of business acquisitions and certain types of business entities. The new Fair Value standard subjects the entire balance sheet (and income statement) of all ﬁrms to such treatment on an ongoing basis, effectively disaggregating the ﬁrm. As the contents of ﬁnancial reports ﬂow back into the markets as information on which real-world decisions are made at every level, Fair Value accounting and reporting will create signiﬁcant and unforeseen effects on the state of the system. New approaches to collecting and analyzing market and ﬁrm data will be required to manage the introduction of increased levels of subjectivity into these data. A painful example of this is the ‘‘mark-to-market’’ accounting rule that replaces historical asset costs with current valuations on the balance sheet. ‘‘Mark-to-market’’ has been credited with greatly exacerbating the ﬁnancial collapse of 2008, on both the upward and the downward paths of the real estate bubble. The following is an abbreviated list of system shifts that have caused so much frustration to be directed at ﬁnancial accounting.

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2.1.1. The ‘‘Discovery’’ and Inﬂuence of Intangible Assets on Organizational Structure, Growth, Complexity, and Value As market attention has shifted toward intangible assets and away from tangible ones as the source of ﬁrm value, ﬁnancial accounting faces a dilemma. It has to devise methods of measuring and recording the inﬂuence of such assets on the organization and its beneﬁt streams (i.e., outputs). Yet, most of these assets are neither separable nor transferable, two attributes necessary for resource measurement, management, and exchange. 2.1.2. The Explosion of Operating and Financial Complexity in Economic Systems Worldwide Industry and cross-industry consolidation, globalization, and new and exotic markets, industries, and products require an increasingly broad range of organizational and transaction structures, many of which are highly complex. Financial accounting has been asked to address such complexity in meaningful and accurate ways. 2.1.3. The Exponential Increase in Market and Transaction Complexity Increasingly sophisticated investment vehicles, enhanced computer-based trading and desktop trading, 24 7 markets, global currency ﬂows, Internet collaboration, consolidating exchanges, and exchanges that operate as public companies all feed on and generate ﬁnancial information on a real-time basis. It has become increasingly difﬁcult to trace cash and value from the inception of a transaction to its ﬁnal reﬂection in ﬁnancial statements. Financial accounting is being asked to incorporate high levels of complexity and speed into procedures and processes that were designed for lower levels of both. 2.1.4. The Increased Presence of Governmental and Regulatory Inﬂuence over Every Area of Life Organizations are expending massive amounts of time and resources to adapt to and mitigate the requirements of government and regulatory bodies. Tax rules continue to burgeon, constantly threatening organizational viability. Financial accounting must keep pace with the constant turbulence. 2.2. Financial Valuation, Yesterday’s Red-Headed Stepchild, has Become Today’s Darling as We Attempt to Quantify and Resolve Such Issues. But is She Ready? Financial valuation, an esoteric hybrid of all business disciplines, is currently practiced in a linear, deterministic, but prospective, manner with

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varying degrees of rigor. There are many theoretical and practical issues on which no two valuation analysts fully agree. For this ﬁeld, governed by informed professional judgment, beauty is truly in the eyes of the beholder. SFAS 157 institutes ﬁnancial valuation as the arbiter of value for the net resources, and related beneﬁt streams, of the ﬁrm for ﬁnancial reporting purposes. Yet, ﬁnancial valuation has its own set of disabilities that have not been resolved. Two such disabilities follow. 2.2.1. Traditional Valuation Approaches no Longer Sufﬁce to Capture the Complex Realities of Dynamic Economic Systems These same approaches will also be inadequate to meet the demands of fair value accounting. If we have no means by which to measure resources and activities in dynamic systems, we may not be able to manage them either. This will introduce more randomness and higher risk into the system. 2.2.2. Non-Linear Valuation Approaches, Such As Real Options Analysis, Contain Complexities, and Challenges that are Beyond the Average Practitioner Some of these challenges are: difﬁculties in developing fundamental inputs for the models; difﬁculties of creating consistent structure and repeatability of real options problems and solutions; a low level of computational transparency for average users; and no standardized methods for checking projected results against actual ones. What makes the post-SFAS 157 world different is that these same challenges will now inhere in all aspects of accounting, a non-ﬁnance discipline. This, in turn, will ensure that a high degree of subjectivity and complexity will become embedded in both ﬁnancial reporting and market prices, leading to decreased capacity for rational decision-making by system agents.

2.3. There is Increased Potential of Unforeseen and Unforeseeable Disturbances and Random Acts of Violence that Disrupt Economic Systems on a Global Scale Dr. Nassim Taleb calls these random events ‘‘black swans’’ and suggests that there is ‘‘an ingrained tendency in humans to underestimate outliers y Left to our own devices, we tend to think that what happens every decade in fact only happens once every century, and, furthermore, that we know what’s going on’’ (Taleb, 2007, p. 141).

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2.4. Increased Complexity and Turbulence Create Increased, and Different, Sources of Risk, as Well as New Sets of Questions Can modern economic systems afford to view risk and complexity in the traditional way? Should ﬁnancial reporting more explicitly reﬂect the complexity and risk inherent in organizational resources and activities? Do we need better methods of describing both the state of the system and the economic systems functioning within it? Do modern economic systems actually function in the ways that are commonly described in the traditional literature and ﬁnancial reporting? If they do not, how should we describe them? These are just a few of the issues that arise from the scale and scope of change within the global economic environment.

3. REVIEW OF THE RELEVANT LITERATURE The fundamental research for this chapter takes an eclectic approach, drawing on academic research, academic books, practitioner manuals, and papers; Financial Accounting Standards Board pronouncements; the popular press; and insider tips from ‘‘The Street.’’ Readings cross various disciplines such as economics, valuation, real options analysis and risk management, physics and complexity science. I sought to (a) explore the manner in which various agents within global economic systems describe, structure, leverage, and create/harvest value from increasingly complex and risky systems; (b) identify what appear to be the most powerful and genuinely useful approaches to solving the stated problems. As the scope of the research was substantial, only those readings that are directly applicable to the focus of this chapter have been used herein.

3.1. Overview of Findings Research ﬁndings are as widely varied as the readings themselves. Overall, we discover that economies, markets, and ﬁrms are systems that undergo signiﬁcant disturbances generated by endogenous optionality that is either not recognized (causing inadequate decision-making), not understood (causing short-sighted or disingenuous decision-making), or not measured from the correct paradigmatic base (causing model and system failure).

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At the level of national/international economic systems and markets, we ﬁnd that the Austrian School of Economics provides a theoretical structure more amenable to reality than that offered by neo-classical economics. At the levels of the ﬁrm and of ﬁeld practice, we ﬁnd that quantitative approaches to measuring and managing ﬁrm and market economic systems seem to pursue three general directions: (1) build complex models to describe and manage increasing complexity; (2) build simpliﬁed models to describe and manage increasing complexity; and (3) build layered models that reduce complexity as they progress through a sequence. All of these practical approaches involve varying degrees of computational complexity and potential/actual ‘‘black boxes,’’ that is, lack of transparency. The remainder of this chapter further explores these ﬁndings.

4. ECONOMIC SYSTEMS 4.1. General Description Simplistically, economic systems come in many sizes and forms. They can be as large as the global marketplace, a national economy, an industry, or a ﬁrm. They can be as small as a family unit, although no smaller because any system requires more than one agent to exist. Similar to other systems, economic systems are governed by explicit and implicit rules that affect all agents within the system in varying ways. The kinds and combinations of rules governing a particular economic system classify it under monikers such as ‘‘capitalism,’’ ‘‘market,’’ ‘‘oligopoly,’’ or ‘‘start-up.’’ Discussion and management of economic systems requires consideration of ‘‘social stochasticity’’ (Wang, 2005, p. 38), that is, the social consequences and uncertainties surrounding agent decisions and initiatives. Unlike the static, closed, equilibrium models propounded by neo-classical economics and traditional ﬁnance, real-world economic systems are open, complex, and adaptive – living. If they are not, they may be kept on life support, but eventually they will die (as in the demise of the Iron Curtain) or rupture open in an unmanageable chaos of birthing (as with the hypercapitalism being born out of that closed system, the People’s Republic of China). Why? Because economic systems are created by and built upon open, complex, adaptive, living biological systems, called human beings, and life must beget life. The very life-bearing properties of these systems, however, have created a dilemma. Until fairly recently, only limited means have been available to

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describe them quantitatively without the use of highly simpliﬁed, linear, deterministic models. While much of this is due to a general absence of powerful yet accessible technological tools, most of it is due to the fact that it is just plain more direct and less time-consuming to think about and use deterministic models. A broad-brush approach has been considered sufﬁcient for most decisions and transactions. In addition, until fairly recently, those guardians of national economic systems, the regulators, agreed. It appears that systemic disturbances leading to increasingly draconian oversight by these regulators may push theoreticians and practitioners in the ﬁnancial disciplines to ﬁnally embrace approaches and methodologies that properly reﬂect the true nature of the economic systems they describe, measure, and manage. Real options analysis is one such approach.

4.2. The Firm as an Economic System There are almost as many theories of the ﬁrm as there are researchers to build them. Such theories have been amply explored elsewhere and do not need re-examination. The goal here is to establish a solid foundation for the use of real options ‘‘in’’ the ﬁrm as an economic system. To accomplish this, we need to demonstrate that the inner life of the ﬁrm is open, dynamic, complex, and adaptive, that it resembles the economic system of a market in which risk and reward are critical determinants of value. 4.2.1. The Inner Life of the Firm Resembles a Market System I suggest that the ﬁrm is a complex adaptive system, rather than just a portfolio of resources, options, rights, knowledge assets, or projects. The following description draws the parallel between the ﬁrm and what we know about market systems. Firm core processes are market participants, or agents, and are dynamically combining and recombining into portfolios of capabilities, while also competing for scarce resources (capital, knowledge assets, and infrastructure). Each core process is made up of a portfolio of sub processes (e.g., ‘‘supply chain’’) that enables it to carry out activities and produce outputs. Each core process is affected by the risk and uncertainty, the availability of resources, the property rights, the transaction costs, and the real investment options available to the ﬁrm and to other core processes within the ﬁrm. The ﬁrm is more than the sum of its parts (processes), since these parts are constantly overlapping, competing, and shifting dynamically internally and externally as they interact with stakeholders and competitors. In addition, the ﬁrm and its agents

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S. B. VON HELFENSTEIN (processes) are characterized by change, uncertainty, and risk. Differences between exogenous markets and endogenous markets with regards to change, uncertainty and risk are a matter of degree not kind. It is arguable as to which market experiences a higher degree of change, uncertainty and risk. The [ﬁrm] is created and maintained by various kinds of individual, institutional, and organizational investors and stakeholders y Top management functions much like an investment fund or portfolio manager, deciding where to allocate scarce resources and add or divest investments that allow the ﬁrm to continue to function with varying degrees of health. Each core process has its own management team that also functions much like a portfolio manager. (Nelson, 2005, p. 39)

These characteristics of the endogenous market of the ﬁrm are virtually identical to those of the exogenous market. Thus, we may assume that the ﬁrm is an economic system with market attributes. 4.2.2. As for All Market Systems, Risk and Reward are Key Determinants of Firm Value There are many commonly discussed and important sources of risk and reward to the ﬁrm. Some are exogenous and some endogenous to it. All must be considered when discussing corporate strategy, considering corporate projects or transactions, attempting to identify the key value drivers of a ﬁrm or value the ﬁrm itself, invest in it or close it down. The challenging and everchanging nature of risk and its relationship to reward is perhaps the biggest source of discussion and debate in the boardroom, the halls of academia, the ofﬁces of practitioners, and on ‘‘The Street.’’ Measurement of risk and reward is a key component of assessing value. Yet, as the economic system of the ﬁrm becomes increasingly complex and sophisticated and path dependencies between sources of risk and reward increase, traditional measurement approaches and techniques have less and less appeal. One possible solution comes from the ﬁelds of computer and complexity science and quantum physics. It has been named sub-corporate ﬁnance. While attempting to resolve an intractable problem in process reengineering, Drs. Valery Kanevsky (mathematics) and Tom Housel (computer science), proposed that it is possible to describe all organizational outputs in terms of common units of change that can be linked to the knowledge assets of the ﬁrm (Housel & Kanevsky, 1995; Kanevsky & Housel, 1998). This nugget of theory became the foundation for the application of principles of ﬁnance to the ﬁrm at a sub-corporate level in Nelson (2005). Kanevsky and Housel suggest that Businesses are open systems – systems that exchange substance and energy with their environments. As such, businesses have the capability, through their processes, to change

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the structure of raw material inputs (i.e., substance, energy, information) into ﬁnal products/services. In the language of thermodynamics, this change in structure can be measured in terms of the corresponding change in entropy, when the input state a is transformed into output state b by process P (i.e., b ¼ P(a)). Assume that this change can further be represented as a set of ‘elementary’ changes that are minute enough to become identical in terms of the corresponding amount of entropy they cause. This assumption about the equivalence of elementary changes can be expanded across any ﬁnite number of processes with predetermined outputs. This allows the comparison in terms of entropy among any set of processes by means of elementary changes y This concept can be applied to calculating the value added by business processes by calculating the entropy of K-complexity caused by the process to transform an input to its process output. To accomplish this, we will employ the parallelism between business processes and computations y (Kanevsky & Housel, 1998, pp. 278–280)

They also suggest that Since a unit of K-complexity represents a unit of change and is equivalent to a unit of Shannon information, all process outputs can be standardized by describing them in terms of the number of units of Shannon information (i.e., bits) required to produce them, given the state of the technology used in the process. All outputs can also be described in terms of the time required by an ‘average’ learner to learn how to produce them. ‘Learning Time’ can be considered a surrogate for the amount of organizational knowledge required to produce the outputs y [allowing us to] describe outputs in terms of learning time [and to assign] a common unit, the Knowledge Unit (Km), y to represent the amount of organizational knowledge required to produce the outputs. (paraphrase in Nelson, 2005, p. 7)

This proportionality between units of change, units of Kolmogorov Complexity, units of information, and units of organizational knowledge has profound implications for the concept of the ﬁrm as an economic system and the quantitative measurement of risk and reward. Sub-corporate ﬁnance proposes that, based on the concept of common units of change that can be counted and monetized as ﬁrm or system processes transform ‘‘inputs’’ of various kinds into ‘‘outputs’’ of other kinds, risk becomes a descriptor for the expected change in uncertainty that will occur during such process state transformations. Reward (ROI, value added) thus becomes a descriptor for the actual change in uncertainty that occurs during state transformations. As a market transaction can be considered a form of process in which one asset undergoes a state transformation into another, risk and reward, as redeﬁned by sub-corporate ﬁnance take on new and useful attributes. Taking this concept further, If we apply the proportionalities we described [earlier] in which DE (change in entropy, uncertainty) E K(y|x) (conditional Kolmogorov complexity) E bits E Km, and we

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S. B. VON HELFENSTEIN agree that risk is a change in uncertainty (DF), then risk and Km are also proportionate and represent the same common unit of measure. As a result, we suggest that measuring the change in entropy embodied in process outputs of the organization in common units, Km, is equivalent to measuring risk. This in turn ties risk measurement directly to the knowledge assets of the organization and only indirectly to the movements of ‘the market’ and competitors. (Nelson, 2005, p. 31)

This provides a radical but useful link between risk and knowledge assets, a link already implied in the literature. Thus, while sub-corporate ﬁnance has not yet been operationalized, it could provide valuable new insights. In addition, the data required to populate its models is abundant and comes from deep within the ﬁrm, close to sources and drivers of value. As such, not only does it support the proposition that the ﬁrm is a true economic system, complete with its own endogenous, quantiﬁable sources of risk and reward, but it could eventually provide new data sources for the application of real options ‘‘in’’ the economic system of the ﬁrm.

5. OPTIONALITY WITHIN ECONOMIC SYSTEMS As with all open systems, economic systems are pregnant with optionality (i.e., the right, but not the obligation, to pursue different pathways to navigate, manage, and harvest system opportunities and challenges). The dynamics of innovation, ‘‘creative destruction,’’ evolution, revolution, booms and busts, change, complexity, and risk are byproducts of optionality. Expectations, those elements from which choice is made, drive the scope and scale of optionality. According to Austrian School economists, imagination generates expectations. ‘‘[E]xpectations do not refer to entities of the world as it is, nor yet to predeﬁned entities of the world as it might be, but to entities existing in, and created by, the imagination of each decision maker. In other words, the raw material for expectations is provided not by the world directly, but by imagination at work on the world. For this reason, a man may have expectations about future events and actions that have not occurred to anyone else’’ (Rizzo, 1979, p. 36). Optionality has two key attributes, uncertainty and ﬂexibility, both with the capacity for creating or destroying systems value. Neo-classical economics and traditional ﬁnance recognize and attempt to quantify uncertainty (generally equating it with risk), but ﬁnd a variety of theoretical and practical rationale for ignoring ﬂexibility at other than the ‘‘product’’

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level (e.g., ﬁnancial options and other derivatives; market trading algorithms; goods and services available for consumption). As the following examples will demonstrate, this results in sub-optimal decision processes and system-wide disturbances.

5.1. Optionality within the Global Economic System Much has already been written about the causes and effects of the Great Collapse of 2008. The Brookings Institution issued a cluster of papers on this topic, titled the ‘‘Fixing Finance Series.’’ One of these, ‘‘The Origins of the Financial Crisis,’’ provides a concise and thoughtful analysis that we use as a baseline for diagrammatic representations of ignored, misrepresented, and mis-measured optionality in the global economic system (Baily, Litan, & Johnson, 2008). Fig. 1 describes system participants and roles. Fig. 2 describes system optionality as exercised by various participants. It is worth noting that down to the level of ‘‘Specialists and Brokers,’’ economic system participants relied on linear, deterministic models and theory based on neo-classical economics and traditional ﬁnance as the basis from which to make decisions regarding what they knew was an asset bubble. They pursued the optionality within the system by ignoring the full range of ﬂexibility available to them and mis-measuring the uncertainty involved. Specialists, brokers, and home buyers were culpable participants to the extent that they falsiﬁed or stretched the facts to pursue self-interest within increasingly loose mortgage lending standards. However, they operated under information asymmetry with regard to broader systems issues. This, along with dishonesty and a desire to ‘‘work the system,’’ caused them to misunderstand their own option set and make sub-optimal decisions.

5.2. Optionality within the Economic System of the Firm One of the most crisp examples of optionality within the economic system of the ﬁrm is the decision process during mergers and acquisitions. In an M&A scenario, both buyer and seller have a separate set of options to consider, some but not all of them conjoining. Some options are created by tax rules; some by the organizational and operational structures and portfolios of assets and liabilities of the two companies; some by the types and motivations of investors and intermediaries; some by the personal concerns

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U.S. Congress – Established affordable housing goals and pushed/forced lenders to meet them. Federal Reserve – A private banking system given “federal” authority for monetary policy. Kept interest rates at 1% for a long period to encourage strong economic growth. Foreign Institutions – Directly funded U.S. companies and mortgage debt instruments (a very large capital inflow) on very favorable terms, fuelling housing boom. U.S Regulatory Structure – Federal Deposit Insurance Corporation, Office of Comptroller and Currency, Office of Thrift Supervision, Financial Accounting Standards Board, Securities Exchange Commission, State regulatory systems – Overlapping authority, silo mentality, high degree of complexity. Government Sponsored Enterprises (GSEs; i.e., Fannie Mae and Freddie Mac) – Bought mortgage loans from banks to facilitate lending and lower interest rates. Issued first mortgage-backed securities. Guaranteed investors against default and pre-payment losses on underlying assets. Their lower cost of borrowing and lower capital requirements led to overleveraging. By 2008, over $5.4 trillion in mortgage debt, of which $1 trillion was sub-prime and Alt-A. Institutional Investors & Rating Agencies – Bought mortgages. Created, bought, sold, and rated mortgage-back investment vehicles and related instruments. Specialists and Brokers – Sold mortgages but did not fund them. Home-Buyers – Took advantage of the “system.” Overleveraged due to “something for nothing” syndrome and no significant penalties for failure to perform.

Fig. 1.

Financial Crisis System Participants and Roles.

and goals of current owners and top management. Even the regulators have a say in the decision process if either the acquirer or the target is publicly held. Legal, accounting, and valuation advisors also put their imprimatur on deals. This creates a highly dynamic, complex, non-linear environment, and decision process. However, this process is developed and guided almost entirely by static, linear, deterministic discounted cash ﬂow and valuation models, neoclassical economic thinking, and a tunnel focus on uncertainty/risk mitigation while ignoring system ﬂexibility completely. The consequences of this approach have been well-documented as disastrous. A high proportion of mergers and acquisitions fail. Human capital and

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U.S. Congress – Exercised social engineering options without consideration of consequences. Federal Reserve – Misunderstood consequences of monetary policy option exercised, leading to favoring near-term growth over amelioration of asset pricing bubble with long-term effects. Foreign Institutions – Exercised options with consequences for U.S. stability and sovereignity not yet fully understood. U.S Regulatory Structure – Option exercise included: Softening or eliminating consequences of failure to perform by various nongovernment system participants; imposing high-risk, valuation based accounting rules (“mark-to-market”) while ignoring possibility of ‘fat tails’ in distribution curves; other similar, linear choices. Government Sponsored Enterprises (GSEs; i.e., Fannie Mae and Freddie Mac) – Exercised option to take advantage of quasi-governmental status and guarantees to ensure organizational control of mortgage lending markets and related profitmaking. Ignored, misunderstood and mis-measured actual optionality and uncertainty available. Institutional Investors & Rating Agencies – Exercised option to use tranched Mortgage-Backed Securities and Collateralized Debt Obligations to bundle and repackage high risk debt and spread risk widely throughout system. Rated the new securities based on riskspreading features rather than underlying risk. Used high-risk Credit Default Swaps to “insure” general investors against default risk of underlying assets, while ignoring overall riskiness of those assets and misunderstanding the riskiness of the new securities.

Fig. 2.

Financial Crisis Exercise of Optionality by Participants.

infrastructure systems never integrate properly and are often discarded, wasting untold monetary and organizational value. Technologies and products never realize the potential for which they were purchased. High quality projects are abandoned and low quality projects pursued based on turf wars, mistaken priorities, and other never-discussed or mis-measured rationale. The picture is not pretty. Throughout the system, practical solutions to such problems are being sought. I will ﬁrst examine two solutions offered from within the current paradigm. I will then discuss two solutions from within the real options paradigm – one in current use and one that I suggest is the most fruitful avenue for future exploration.

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6. CURRENT PARADIGM – BUILD COMPLEX MODELS TO DESCRIBE AND MANAGE COMPLEX ECONOMIC SYSTEMS The following are examples of approaches based on building complex models to describe and manage complex systems. They are both taken from the ﬁeld of valuation and are written by practitioners, and thus are not part of a body of academic literature. 6.1. Model A: Valuation of Complex Capital Structures Accounting rules such as Mark-to-Market and the move to Fair Value Accounting create major challenges for the valuation of complex capital structures for mergers and acquisitions, ﬁnancing, and other purposes. Th[e] growing trend toward fair value presents signiﬁcant challenges in valuing privatelyheld companies with complex capital structures. Because it is necessary to ﬁrst value those securities with superior claims to common equity, many valuation specialists, auditors, and ﬁnancial executives now ﬁnd themselves forced to enter a jungle of complex capital valuation. Depending upon the provisions associated with the components of a complex capital structure, accurate valuation of common stock in this environment may require sophisticated simulation models. Until recently, however, there was very little guidance – much less convergence of thought y within the appraisal community. Even where such guidance exists, it is unnecessarily conﬂicting, and more important, incapable of handling such commonplace features as cash distributions prior to liquidity events and performance-based vesting. (Chamberlain, Hill, Kamma, & Karam, 2007, p. 1)

To address such challenges, a team of valuation professionals and academics propose a methodology, based in simulation techniques, to integrate two extant valuation methods: the Options Pricing Method (OPM) and the Probability Weighted Expected Return Method (PWERM). These two methods are commonly used in the ﬁeld, having been propounded in a 2004 AICPA Practice Aid, ‘‘Valuation of Privately-Held-Company Equity Securities Issued as Compensation.’’ ‘‘OPM takes as a starting point the current enterprise value and, using a volatility estimate that captures the market risk of the underlying business, models the stochastic evolution of this value over time. The various equity classes are then viewed as [European] option-like claims on this underlying value y’’ (Chamberlain et al., 2007, p. 4). The OPM is performed using the closed-form Black–Scholes option pricing method that does not allow for performance-based vesting or other path-dependent events.

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‘‘PWERM explicitly takes into account the random nature and timing of potential future liquidity events–y–[C]urrent enterprise value is the probability-weighted sum of the discounted future liquidity outcomes’’ (Chamberlain et al., 2007. pp. 4–5). The discount rate utilized is that of the underlying asset (i.e., the company), thus preventing the method from capturing the changes in risk over time and over equity classes. The integrative method incorporates the following steps: (1) determine the risk-neutral distribution of underlying asset values; (2) simulate future asset values using this distribution and the selected end-points that represent various liquidity events; (3) infer beneﬁt stream paths (EBITDA, cash ﬂow) from these asset value paths; (4) use the beneﬁt stream paths to determine the cash distributions resulting from path-dependent events (such as performance-based vesting) and the effect of such distributions on endpoint liquidation values; (5) using traditional priority rules, allocate the enterprise values determined in steps 1–4 to the various equity classes; (6) discount the resulting pay-off values by the risk-free rate (because the underlying asset has been simulated under risk-neutral conditions); and (7) repeat these steps and take an average to conclude a ﬁnal value for each equity class (Chamberlain et al., 2007, p. 8). This model allows the analyst to set ‘‘conditions believed to resemble the ones that prevail in reality, and [launch] a collection of simulations around possible events,’’ where there are no constraints on the number of input variables that can be used, and the analyst can ‘‘generate thousands, perhaps millions, of random sample paths, and look at the prevalent characteristics of some of their features’’ (Taleb, 2004, p. 46). Yet, the model is complex and involves a high degree of informed professional judgment throughout. For the simulation alone, the analyst must select those few variables that demonstrate signiﬁcant inﬂuence over the resulting outputs, check for correlation among these, and ignore the rest. The analyst must also select the probability distribution and parameters for that distribution that represent a ‘‘best ﬁt’’ for input variability. ‘‘[B]ut, picking the right distribution and the parameters for the distribution remains difﬁcult for two reasons. The ﬁrst is that few inputs that we see in practice meet the stringent requirements that statistical distributions demand y The second is that the parameters still need to be estimated after the distribution is picked y [yet the available data for this purpose is regularly insufﬁcient or unreliable]’’ (Damodaran, 2007, pp. 165–167). The act of performing simulations may provide a mistaken sense of having rigorously investigated all aspects of a matter, when the adage ‘‘garbage in, garbage out,’’ still prevails.

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While this proposed methodology supplies a real option like attempt to resolve what appear to be conﬂicting issues in more traditional methodologies and solve an important problem, it increases model risk by increasing model complexity and the need for subjective inputs without the true rigor of a real options approach.

6.2. Model B: Valuation of Complex Tax Issues Related to Organizational Form A debate has raged in the valuation community for years regarding the effect on value of the tax attributes belonging to Sub-Chapter S-corporations. Should S-corporations be assigned higher values than C-corporations based on their tax attributes? After all, (1) investors in S-corporations avoid paying the dividend tax at the individual level, but C-corporation investors cannot avoid this tax (Note that both investor classes pay taxes on income earned at the corporate level); and (2) S-corporation shareholders can increase the tax basis in their stock through retained earnings, while C-corporation shareholders cannot. Even the Federal Tax Court has entered the debate, issuing a number of decisions since 1999 that have created further confusion and discussion. To address this issue and more precisely quantify any additional value that S-corporation status might bring to its shareholders, ﬁve valuation experts have developed models. Four of these, each named for its progenitor, are complex enough to be virtually proprietary, although they have all made their way into practical use to one degree or another. The ﬁfth method, the ‘‘simpliﬁed model,’’ explains and compares the other four and offers a streamlined approach to modeling the same issues. The fundamental components of the ‘‘simpliﬁed model’’ are: ‘‘(1) a traditional discounted cash ﬂow (which can be expanded for any holding period or contracted to a single-period capitalization); (2) recognition of the beneﬁt of the avoided dividend tax; and (3) recognition of the capital gains beneﬁt of the ability to build up basis’’ (Fannon, 2008, p. 4–1). The model requires the analyst to consider, select, and quantify the following assumptions: (1) annual distribution percentages and amounts; (2) the probability that the likely hypothetical buyer, under a fair market value standard, will qualify to maintain the S-corporation status; (3) the level of risk associated with shareholder ability, or lack thereof, to realize basis build-up; (4) the estimated holding period before the hypothetical buyer will ‘‘ﬂip’’ his investment in the company; and (5) the federal income tax rates to

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be used for the company and the shareholders (Fannon, 2008, pp. 4–1 through 4–2). Each assumption requires the analyst to apply varying degrees of informed professional judgment based on varying sets of facts and analyst perspectives. Model complexity and risk are increased due to the number of factors to be considered and path dependencies that cannot be addressed. More importantly, these models may suggest that if a company’s tax attributes have a substantial inﬂuence on its value, we should consider modeling the complete range of tax attributes of every company investigated during valuation analysis since companies have widely differing tax attributes based on their economic system design. This would require an approach to modeling the economic system of the ﬁrm that is not currently available within traditional valuation practice.

7. THE REAL OPTIONS PARADIGM – A SIMPLIFIED MODEL TO DESCRIBE AND MANAGE COMPLEX ECONOMIC SYSTEMS In Strategic Investment: Real Options and Games, Drs. Hans Smit and Lenos Trigeorgis synthesize corporate ﬁnance, industrial organization, corporate strategy (strategic planning), and value into a simpliﬁed model that describes and manages the complexity of the effect of ﬁrm optionalities and strategic games on value creation. Their basic premise is as follows: In the past decade, the strategic management ﬁeld has seen the development of two main views. One view is that ﬂexibility is valuable. As the competitive environment of most ﬁrms changes quite frequently, ﬂexibility in investments should allow ﬁrms to optimize their investments and value creation. The other view is that commitment is valuable because it can inﬂuence the strategic actions of competitors. This creates the opportunity to realize better payoffs (and shareholder value). Both views are supported by theoretical arguments and a large body of research. The ﬂexibility view partly draws on the resource-based view of the ﬁrm and core-competency arguments: a ﬁrm should invest in resources and competencies that give it a distinctive chance to pursue a set of market opportunities y The commitment view is ﬁrmly anchored in industrial organization and game theory, which during the nineties were increasingly adopted in the strategy ﬁeld. Since both views have a theoretical justiﬁcation, a key question is under what circumstances each can inform strategic decisions. (Smit & Trigeorgis, 2004, p. 35)

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Their response to their own question is the following model: Expanded ðstrategicÞ NPV ¼ ðpassiveÞ NPV þ flexibility ðoptionÞ value þ strategic ðgame theoreticÞ value Conceptually, they consider the ﬁrm as a portfolio of options, or ‘‘‘bundle of opportunities’ [requiring] a balance between exploiting current cash-generating advantages and generating new options.’’ The correlations and interactions (‘‘interproject synergies’’) and the ‘‘intertemporal (compound option) effects’’ among the ﬁrm’s strategic and operational projects (options) as well as the risk attributes of various stages in these projects create the value of the ﬁrm (i.e., the portfolio of options). (Smit & Trigeorgis, 2004, pp. 80–81) The value created by and within this portfolio can be quantiﬁed using a binomial option pricing model. This approach to valuing the ﬁrm’s portfolio of options provides a richer and more realistic assessment of value than traditional, linear discounted cash ﬂow models. In addition, ﬁrms experience a ‘‘strategic impact’’ from the investment decisions they make in contexts in which they are aware of the actions and interactions of rivals that will affect project value. In such contexts, ‘‘[g]ame theory can be helpful in analyzing strategic investment decisions y [F]ollowing the rules of game theory can help reduce a complex strategic problem into a simple analytical structure consisting of four dimensions [(1) identiﬁcation of the players, (2) the timing or order in which the players make their decisions, (3) the available actions and information set, and (4) the payoff structure attached to each possible outcome]y . [G]ame theory is also a helpful valuation tool for strategic decisions because it encompasses a solution concept that can help in understanding or predicting how competitors will behave, and it also provides an equilibrium strategy and values for the strategic decisions’’ (Smit & Trigeorgis, 2004, pp. 171–172). Smit and Trigeorgis present an integrated model by which to discuss game theory in terms of real options analysis. This holistic model is simplistically described in the following. When a ﬁrm engages in multistage (sequential) games under uncertainty and wishes to analyze its strategic choices using game theory analysis, management will build a strategic decision tree by which to lay out available choices and moves. By moving backward through this decision tree structure, much like the certainty-equivalent binomial tree used in real options analysis, the option value at each node of the tree can be calculated using risk-neutral probabilities and OPM. ‘‘This new approach makes it

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possible to value complete strategies in a competitive context in a fashion that is consistent with both modern economics and ﬁnance theory’’ (Smit & Trigeorgis, 2004, p. 181). It also provides a simpliﬁed, but powerful, framework by which to investigate the effects of market competition and strategic planning on ﬁrm value. There are no traditional valuation tools that can address this important issue.

8. THE PROPOSED SOLUTION – LAYERED REAL OPTIONS MODELS THAT REDUCE COMPLEXITY DURING USE The proposed solution has its origins in the seminal work of Dr. Richard de Neufville, Massachusetts Institute of Technology, in ﬂexibility in engineering systems design. While there are still aspects of this solution that have not been operationalized, the underlying research and accompanying theory is extensive. Sections 8.1 and 8.2 will use the research of two of Dr. de Neufville’s engineering students as a platform for the further exploration of the application of real options ‘‘in’’ economic systems. Both suggest thought-provoking methods for scanning the optionality within the system and reducing design space complexity during use.

8.1. The Distinction between Real Options ‘‘on’’ and Real Options ‘‘in’’ Projects Before we proceed further, we must identify the difference between real options ‘‘on’’ and ‘‘in’’ projects. Real options ‘‘on’’ projects is the most common application of real options analysis. It is utilized to value projects for which future management decisions regarding project direction (i.e., switch, put on hold, continue to the next stage, shut down, and so forth) have been explored. While is it a fruitful and important means by which to develop more realistic project values, it offers no consideration of or insight into the inner workings of the projects being valued. As these inner workings often have substantial inﬂuence over eventual project direction or success, this may be considered a sizeable oversight. Real options ‘‘in’’ projects, coming out of the ﬁeld of large-scale engineering systems design, considers the inner workings of projects. Its concern is to identify and provide ﬂexibility (i.e., options) from the inside

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out that, while not necessarily focused on optimization, will ensure a high quality/most desirable project design, considering resource and other constraints and the state of the system.

8.2. Using Common Analytic Tools and Informed Professional Judgment to Simplify System Design Space In his masters thesis, ‘‘Facing Reality: Design and Management of Flexible Engineering Systems,’’ Michel-Alexandre Cardin, PhD candidate, MIT, provides a useful way of thinking about the kinds of problems that inhere in both engineering and economic systems design. His statement of the problem is as follows (Cardin, 2007, p. 15): Designers of engineering systems always seek for better approaches to improve the value and performance of a system. They seek the best combinations of design elements and management decision rules before selecting a particular design. In doing so, they assume one particular evolution of the uncertain variable(s) affecting their system over its intended useful life y One problem with this approach is that the future is uncertain. The uncertain variables affecting the value and performance of the system may turn out completely different than originally assumed. Therefore, it is possible that designers choose a design conﬁguration that performs extremely well under the scenario originally assumed, if it occurs, but very poorly if reality turns out otherwise. If designers consider several scenarios of the uncertain variables before committing to a particular design, another problem emerges. In addition to considering several possible combinations of design and management decision rules under a particular scenario, they need to ﬁnd the best combination for each possible scenario y The number of possible combinations y can become intractable very rapidly. If ﬂexibility is considered as a way to adapt the system to take even more advantage of unexpected upside opportunities, or to reduce losses in case of downside events, the problem becomes even larger and harder to tackle.

This could equally well describe the problem related to valuation and performance measurement of a complex economic system containing optionality using the tools and models of traditional ﬁnance and economics. Cardin proposes a ﬁve-stage layered decision model to explore and parse the design space of the system and identify the most promising design solutions, given the state of the system.

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8.2.1. Critical Deﬁnitions Cardin provides a number of deﬁnitions that are foundational to his methodology. They are paraphrased and quoted in the following (Cardin, 2007, pp. 19, 21, 22, 24, 27–29). Design elements are the building blocks of an engineering system. Management decision rules codify selected management and operating behaviors for consideration in design models. Uncertain variables are ‘‘variables outside of designers’ and program managers’ control that can affect the value and performance of the systems y Such variables can take on different behaviors over the course of a project’s useful life.’’ An operating plan is ‘‘a way to manage and operate a system that combines a particular set of design elements and management decision rules under a particular uncertain variable scenario.’’ The combinatorial space represents ‘‘the spectrum of all possible combinations of design elements, management decision rules, and uncertain variable scenarios that designers can investigate to ﬁnd the best design under all possible manifestations of uncertainty.’’ As the combinatorial space is too vast to be fully explored, Cardin cites (de Neufville, 2006) as suggesting the choice of a limited set of uncertain variable scenarios, followed by the selection of a limited set of combinations of design elements and management decision rules (i.e., operating plans) that are best suited for these scenarios. ‘‘This limited set of operating plans forms a catalog of operating plans, where each operating plan is suited to a particular scenario of the uncertain variables.’’ Optimization is not a desired consideration in forming a catalog. Flexibility is. Flexibility in an engineering system is the system’s capacity to adapt to uncertain and unexpected conditions in a relatively efﬁcient manner that also enhances system value and performance. de Neufville (2005) identiﬁes two primary sources of ﬂexibility in engineering systems: ‘‘in’’ and ‘‘on’’ the system. Flexibility ‘‘in’’ a system is created by properly exploiting technical design elements. Flexibility ‘‘on’’ a system is created by an appropriate and insightful set of management decisions on behalf of the system as a whole, without necessarily affecting elements of design. 8.2.2. Methodology The goal of Cardin’s methodology is to explore the combinatorial space, using tools and concepts familiar to ﬁrm, industry, or government managers, to identify, think about, and adopt the catalog of operating plans that will most improve (not necessarily optimize) system value and

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performance. He suggests the following ﬁve layered stages, quoted and paraphrased in the following (Cardin, 2007, pp. 35–52). 8.2.2.1. Build an Initial Deterministic Model of the Engineering System. The main design elements, management decision rules, performance and valuation metrics, and sources of uncertainty (uncertain variables) for the system are identiﬁed and selected. 8.2.2.2. ‘‘For Each Source of Uncertainty, Propose a Limited Set of Uncertain Variable Scenarios and Review the Initial Model’’ in Light of These. Cardin suggests that simulations be performed on several of the most likely scenarios, using the deterministic projections developed for the abovementioned step ‘‘Build an Initial Deterministic Model of the Engineering System.’’ Once an adequate number of scenarios have been simulated, they can be analyzed to discover which scenarios provide the most compelling general categories for future consideration. The analysis is conducted using brainstorming and informed professional judgment. 8.2.2.3. Using Tools such as Engineering System Matrices Brainstorming and Informed Professional Judgment. Identify the critical sources of ﬂexibility in the system, and position these in the uncertain variable scenario models. 8.2.2.4. Using a Concept from Statistical Experiment Design Called Factorial Analysis Scan the Combinatorial Space and Create a Catalog of Operating Plans. Cardin suggests that the requirements of ‘‘full factorial analysis’’ are excessive in light of typical managerial resource and time constraints. Instead, he presents two methods for reducing the number of experiments required to conduct factorial analysis. One of these is fractional factorial analysis, in which ‘‘only a subset of [all possible] combinations [of factor levels] is selected to perform experiments and measure the system’s response.’’ The second, favored by Cardin, is a search algorithm called adaptive OFAT, for which there are two models, one for discrete factor levels and one for continuous factor levels. The discrete model is more suitable for this search problem. 8.2.2.5. Assess the Expected Value and Performance Added by the Catalog of Operating Plans versus Those Exhibited in the Original, Static, Inﬂexible Operating Plan. Monte Carlo simulations are run to develop probability distributions of expected net present values for uncertain variable scenarios

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within a catalog. These probability distributions are then recast as histograms and also as cumulative distribution functions using Value at Risk and Gain curves, allowing managers to visualize the exact beneﬁts of ﬂexibility in the proposed system design. 8.2.3. Conclusion While this methodology leaves certain computational issues unresolved, it provides a valuable and practical way to think about and simplify engineering system design space and identify ﬂexibility within the system that enhances value and performance. I suggest it can be applied to economic systems equally effectively.

8.3. Using a Layered Computational Model to Simplify System Design Space ‘‘Real Options ‘in’ Projects and Systems Design – Identiﬁcation of Options and Solutions for Path Dependency,’’ the Ph.D. dissertation (MIT) by now Dr. Tao Wang, also discusses the identiﬁcation and use of real options ‘‘in’’ engineering systems design, but from a more purely computational perspective. This section will provide a simpliﬁed overview of Wang’s proposed real options model. 8.3.1. Another Look at Challenges in Engineering Systems Design As also stated by Cardin, the design of physical systems, such as large-scale engineering systems, involves identifying and incorporating a vast array of technical constraints (real options, or ﬂexibility, ‘‘in’’ the system) that are highly interdependent and path dependent, conditions that, traditionally, are not and cannot be considered in calculating the value of such projects. These kinds of projects require designs that can be adapted over time due to their longevity, the high degree of uncertainty associated with future system requirements, their sheer size and scope, and the magnitude of the investment. In addition, engineering system scale, scope, and complexity are being profoundly affected by ‘‘globalization, new technological capabilities, rising consumer expectations, and increasing social requirements’’ (Wang, 2005, p. 29). Much like standard corporate project planning, traditional deterministic engineering systems design makes use of projected expected values of uncertain variables, passive recognition of such uncertainties, and a focus on economies of scale. Flexible engineering systems design of the sort proposed

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by de Neufville, Cardin, and Wang considers ‘‘sequences of probability functions at multiple points in time,’’ proactive management of uncertainties, and attention given to strategies other than economies of scale (Wang, 2005, pp. 22–23). The need to consider ‘‘social stochasticity,’’ that is, the economic and social consequences and uncertainties surrounding large-scale engineering projects, has become an increasingly critical design element because ‘‘[a]ny technical systems are to serve human’s needs’’ (Wang, 2005, p. 38). Thus, engineering systems design and implementation require the consideration of high degrees of complexity, uncertainty, and ﬂexibility (i.e., optionality), none of which is properly captured using traditional linear deterministic methodologies. Wang suggests a layered real options computational model that can address these challenges effectively. His model is described in more detail in the following (Wang, 2005, pp. 138–152, paraphrased and quoted). 8.3.2. Layer One – The Screening Model The top layer of Wang’s approach is a screening model ‘‘established to screen out the most important variables and interesting real options (ﬂexibility). The screening model is a simpliﬁed, conceptual, low-ﬁdelity model for the system. Without losing the most important issues, it can be easily run many times to explore an issue, while the full, complete highﬁdelity model is hard to establish and costly to run many times. From another perspective y we can think of it as the ﬁrst step of a process to reduce the design space of the system.’’ This model requires the use of simplifying assumptions such removing sequential build-out timing considerations and uncertain variable stochasticity. It uses non-linear programming to perform sensitivity analysis on key system variables to identify optimal designs for each set of variables. Once optimal designs have been selected, they are compared with each other to ﬁnd the real options (ﬂexibilities) that are common to and beneﬁcial for all sets. Like Cardin’s model, optimal value creation, but not necessarily optimization, is the goal. 8.3.3. Layer Two – The High-Fidelity Simulation Model The next layer is a high-ﬁdelity simulation model, by which selected design sets are analyzed using technical and economic uncertainties that were not considered in Layer One. This model adds back uncertain variable stochasticity but does not consider timing issues. It tests for the robustness, reliability, and beneﬁts of the design sets and provides information by which they can be reﬁned.

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8.3.4. Layer Three – The Option Valuation Model Using the results of Layer Two, an option valuation model is run on the ﬁnal design contenders. This model now includes timing and strategic considerations in the analysis by recasting [a standard binomial lattice] in the form of a stochastic mixed-integer programming model [in which the binomial tree is maximized] subject to constraints consisting of 0-1 integer variables representing the exercise of the options ( ¼ 0 if not exercised, ¼ 1 if exercised y [S]uch reformulation empowers analysis of complex path-dependent real options ‘in’ projects for engineering systems y Technical constraints in the screening model are modiﬁed in the real options timing model. Since the screening and simulation models have identiﬁed the conﬁguration of design parameters, these are no longer treated as design variables. On the other hand, the timing model relaxes the assumption of the screening model that the projects are built together all at once. It decides the possible sequence of the construction of each project in the most satisfactory designs for the actual evolution of the uncertain future.

To assist those readers who do not have an expertise in the kinds of programming used in his analysis, Wang provides the following descriptions: Mathematical programming studies the mathematical properties of maximizing or minimizing problems, formulates real world problems using mathematical terms, develops and implements algorithms to solve the problems. Sometimes mathematical programming is mentioned as optimization or operations research y Stochastic programming is the method for modeling optimization problems that involve uncertainty y In stochastic programming, some data are random, whereas various parts of the problem can be modeled as linear, non-linear, or dynamic programming y A mixed integer programming problem is the same as the linear or non-linear problem except that some of the variables are restricted to take integer values while other variables are continuous. (Wang, 2005, pp. 39–42)

Wang notes a deﬁciency of stochastic mixed-integer programming – it is difﬁcult to tell if the result is a global or a local optimum. However, he maintains that even if the optimums provided by his methodology are local rather than a global, the overall results are superior to those available through traditional methodologies or human intuition. 8.3.5. Conclusion Wang’s methodology can be executed easily and rapidly on an ordinary laptop computer once a problem has been programmed. I suggest that, in spite of its computational complexity, its application to economic systems should be further explored.

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9. APPLICATIONS OF REAL OPTIONS ‘‘IN’’ ECONOMIC SYSTEMS Just as engineering systems design, implementation, and valuation can no longer be properly captured using traditional linear deterministic methodologies, so we face the same barriers for the design, implementation, and valuation of the economic system of the ﬁrm. Both the Cardin and Wang models may provide a means by which to address these challenges effectively. To support this concept, I will consider the two examples of complex valuation problems discussed in Section 6 and one additional example from the work of Joseph Schumpeter that could be analyzed and resolved using the proposed models.

9.1. Designing and Valuing Complex Capital Structures As stated in Sections 5.2 and 6.1, complex capital structures arise from mergers and acquisitions, deal ﬁnancing, and other equally dynamic scenarios. These scenarios are mismatched with the valuation theory and models currently in use. The real options models discussed in Section 8 are ideally suited for transaction design (pre-acquisition) and for postacquisition valuation purposes because they are able to actually explore the range of real options/ﬂexibility available ‘‘in’’ possible or actual deal and capital structures. If deal structures and post-acquisition valuations included non-linearities and path dependencies as part of the strategic option set, not only would management and investors gain a clear picture of the uncertainty and ﬂexibility available in the system and be able to negotiate better deals, but the future success of the combined entity might be less at risk.

9.2. Designing and Valuing Complex Tax Attributes The complexity and range of the tax attributes of ﬁrms is enormous as ﬁrms continuously seek to minimize the effect of taxes on corporate and investor wealth. As stated in Section 6.1, once we begin to take tax attributes into account for one design element of the ﬁrm (here, its legal structure), would it not be useful to also consider the effect of taxation on other value-creating or value-destroying design elements? For example, the valuation of intangible assets under SFAS 141(R) includes quantiﬁcation of the future tax beneﬁts attributable to the

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amortization of such assets. Yet, using common linear models, we cannot discuss and quantify a realistic picture of the expected future beneﬁts of these assets. Thus, we may be overvaluing them. In another example, when valuing privately held companies by using public company benchmarks, we are forced to ignore the tax attributes of the benchmarks because they vary so widely and, in fact, may not be available to ‘‘outsiders.’’ Yet, these tax attributes may be signiﬁcantly different from those of the subject privately held company and from the default attributes we use in our cash ﬂow models. Again, this has the potential to skew the valuation of the subject company. Both proposed engineering systems design models could be used to explore taxation real options ‘‘in’’ the ﬁrm and provide assistance to ﬁnancial managers and others who perform various aspects of transaction and organizational design and to accountants and valuation analysts who need to quantify tax effects and put them into ﬁnancial statements. While considering the effects of taxation on every aspect of the ﬁrm seems both unnecessary and burdensome, the proposed design models could narrow the taxation design space for a particular ﬁrm and reduce the number of options for consideration. Once the most beneﬁcial conﬁguration of tax model design was established for a particular ﬁrm, it could be revisited and updated over time to take into consideration changing conditions. This process would enable the ﬁrm to design and pursue a rigorous, consistent tax policy and track the effects of its taxation decisions on proﬁtability and ﬁrm health.

9.3. Capturing Creative Destruction and Other Attributes of Economic System Life Cycles Of the capitalist economic system and its sub-systems, Joseph Schumpeter wrote, Capitalism y is by nature a form or method of economic change and not only never is but never can be stationary. And this evolutionary character of the capitalist process is not merely due to the fact that economic life goes on in an social and natural environment which changes and by its change alters the data of economic action; this fact is important and these changes (wars, revolutions and so on) often condition industrial change, but the are not its prime movers. Nor is this evolutionary character due to a quasi-automatic increase in population and capital or to the vagaries of monetary systems of which exactly the same thing holds true. The fundamental impulse that sets and keeps the capitalist engine in motion comes from the new consumer goods, the new methods of production or transportation, the new markets, the new forms of industrial organization that capitalist enterprise creates y [This is a] process of

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S. B. VON HELFENSTEIN industrial mutation – if I may use that biological term – that incessantly revolutionizes the economic structure from within, incessantly destroying the old one, incessantly creating a new one. This process of Creative Destruction is the essential fact about capitalism. (Schumpeter, 1942, pp. 82–83)

The Austrian School economists also recognize such dynamism and disequilibrium in system life cycles by including explicit consideration of time as a critical component of economic system structure. Time manifests itself at least four different ways in the structure of economics. A world of disappointment and surprise is one of disequilibrium and discoordination. The plans of market participants are continually being discoordinated by the imperfect fulﬁllment of expectations. Disequilibrium implies that opportunities for mutually advantageous exchange exist, and that those who possess superior information will reap a kind of arbitrage proﬁt by seizing these opportunities y [T]he contrast between [the Austrians generally] and the neoclassical theory of value y cannot be stressed too much. It is essentially the distinction between a world in which time plays a vital role, and one in which the passing of time may be ignored. (Rizzo, 1979, pp. 1–2, 38)

Without solutions such as the proposed ‘‘real options ‘in’ systems’’ methods, we cannot map the process of creative destruction in the life cycles of ﬁrms or economies. We have limited, impoverished ways by which to discuss and quantify the effects of time. I suggest the proposed models might allow us to create richer and more realistic design space, generate new insights about ﬁrm value creation/destruction and long-term sustainability, and open new avenues for investigation and innovation in ﬁnance and economics.

9.4. The Obstacle of Computational Complexity While a somewhat lesser concern in Cardin’s methodology, computational complexity is a very real obstacle to the further exploration and utilization of Wang’s model for economic system design and valuation. The question we must ask ourselves at this juncture is whether it is preferable to make increasingly complex, sequential, and subjective adjustments to various traditional model variables to attempt to capture ﬁrm complexity or to explore and build complex computational models that can be run on an average laptop computer and embed the ability to investigate the effects of all variables simultaneously. The former eventually leads us so deeply into a maze of informed professional judgment that we lose all sense of reality concerning the speciﬁc ‘‘ﬁrm as economic system’’ we are valuing. The latter, while requiring the use of informed professional

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judgment in structuring the rules by which the programming models are built, might allow us to explore large design/valuation spaces while minimizing the role of subjectivity and speculation. Simple, deterministic calculations built on convoluted estimates and opinions, or complex, dynamic computations built on simple rules and judgment calls? That may be the choice facing us in this matter.

10. AVENUES FOR FUTURE RESEARCH AND WHY IT MATTERS Clearly, further research must be performed to understand if the ‘‘real options ‘in’ systems’’ models can be utilized in the manner proposed herein, within the time and resource constraints of normal ﬁrm and consulting environments. Then the challenges of operationalizing the models must be addressed. If these challenges could be successfully addressed and the resulting models made readily available to the ﬁnance and valuation communities, at reasonable cost, only our imaginations would be the limit to the further applications of these concepts. Why does this matter? We have offered many examples already. Accounting regulators want accounting to function like ﬁnance, leading to the corruption of ﬁnancial statements and market data and the effective disaggregation of the ﬁrm into a collection of resources and claims against them. But traditional ﬁnance and valuation do not currently have the tools or concepts to solve this problem. Both disciplines are being required to do what neither can. Economic system participants at all levels make choices within a dynamic, high-risk environment while using static, linear, deterministic models that cause them to ignore, misunderstand, or mis-measure the effects of uncertainty and ﬂexibility on the system. Firms make material decisions on a daily basis, using tools and concepts that fail to identify, consider, or quantify critical optionalities that could change ﬁrm destiny. Can we afford to continue in this manner? We believe that the real options ‘‘in’’ economic systems might provide an answer.

REFERENCES Baily, M., Litan, R., & Johnson, M. (2008, November). The origins of the ﬁnancial crisis, The Initiative on Business and Public Policy. Fixing Finance Series-Paper 3. Brookings Institution.

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Cardin, M.-A. (2007, June) Facing reality: Design and management of ﬂexible engineering systems. Masters thesis, Masssachusetts Institute of Technology. Chamberlain, T., Hill, J., Kamma, S., & Karam, Y. (2007). Navigating the jungle of valuing complex capital structures in privately held companies: An integrative simulation approach. Journal of Business Valuation and Economic Loss Analysis, 2(2), Article 5. Damodaran, A. (2007). Strategic risk taking. New Jersey: Wharton School Publishing. de Neufville, R. (2005). ‘‘Lecture notes,’’ ESD. 71, Engineering Systems Division, Massachusetts Institute of Technology, Cambridge, MA. de Neufville, R. (2006). Analysis methodology for the design of complex systems in uncertain environment: Application to mining industry. Unpublished working document. Fannon, N. J. (2008). Guide to the valuation of subchapter S corporations. Portland, OR: Business Valuation Resources. Farrell, D., Lund, S., Fo¨lster, C., Bick, R., Pierce, M., & Atkins, C. (2008). Mapping global capital markets: Fourth annual report. San Francisco, CA: McKinsey Global Institute. Housel, T., & Kanevsky, V. (1995). Reengineering business processes: A complexity theory approach to value added. Information Systems and Operational Research (INFOR), 33(4), 251. Kanevsky, V., & Housel, T. (1998). The learning-knowledge-value cycle. In: G. von Krogh, J. Roos & D. Kleine (Eds), Knowing in ﬁrms: Understanding, managing and measuring knowledge (pp. 269–284). London: Sage. Nelson, S. (2005, June). Sub-corporate ﬁnance: New sources of data for real options analysis. Paper presented at 9th Annual International Real Options Conference. Rizzo, M. J. (Ed.) (1979). Time, uncertainty, and disequilibrium. Lexington, MA: D.C. Heath and Company. Schumpeter, J. A. (1942). Capitalism, socialism and democracy. New York: Harper & Row Publishers, Inc. Smit, H. T. J., & Trigeorgis, L. (2004). Strategic investment: Real options and games. New Jersey: Princeton University Press. Taleb, N. N. (2004). Fooled by randomness. New York: The Random House, Inc. Taleb, N. N. (2007). The black swan. New York: The Random House, Inc. Wang, T. (2005). Real options ‘‘in’’ projects and systems design – identiﬁcation of options and solution for path dependency. Ph.D. dissertation, Massachusetts Institute of Technology.

MANAGING REAL OPTIONS IN NOT-FOR-PROFIT ORGANIZATIONS: THE CASE OF SHELL SPACE John W. Kensinger and Stanley T. Crawford ABSTRACT The authors are a ﬁnance professor and an administrator in a major suburban independent school district who minored in ﬁnance while working toward his doctorate in education. We have used the case of shell space to discover the different incentives non-proﬁt administrators have in the acquisition, recognition, and rational exercise of real options by their organizations (compared with managers of for-proﬁt businesses). Shell space is space within a new building that has been enclosed against the elements, but not yet ﬁnished for its intended future use. The shell space can be viewed as a set of complex options (along the lines of the Stulz– Johnson options to choose among a group of several possible ﬁnished outcomes with different costs of exercise). A business executive could be expected to make the acquisition decision based on the value drivers know to impact such options. In the not-for-proﬁt arena, though, decisions about the acquisition and use of options are driven by incentives that arise

Research in Finance, Volume 26, 33–50 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026005

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from within the organization or emanate from the politically elected (or appointed) board of trustees.

INTRODUCTION Sarah Van Helfenstein (previous chapter) describes insightfully how we can better understand what is now being labeled ‘‘systemic risk’’ as just the result of rational exercise of the wide variety of real options that exist within economic systems.1 There is a growing literature on the issues arising from managing real options within business organizations, but there has yet been little discussion of the often very different incentives for creating, capturing, nurturing, or exercising real options in the not-for-proﬁt organizations that make up a large portion of the overall economic/political landscape. Increasingly, the focus is on optimal exercise of real options as the critical executive function, as opposed to accurate calculation of the options’ respective values.2 In this work, we explore the incentives for non-proﬁt decision-makers at various stages of real options acquisition, recognition, and exercise, beginning with the case of shell space and then extending to other real options situations. These incentives give worthwhile insight into the rational exercise of real options within a major portion of our economic system, and hence improved insight into recent ﬂares of systemic risk. In their essence, options provide value by making it possible to participate in the opportunity represented by owning an asset in hopes of seeing its value increase, while keeping potential losses limited (the value of the option comes from the losses avoided by holding it). Hans Stoll (1969) demonstrates how a simple call option – which gives the holder the privilege, without the obligation, of buying the speciﬁed underlying asset at a speciﬁed price (called the exercise price or the strike price) within a speciﬁed period (the expiration) – can be replicated. A perfect replication of the call option can be created by purchasing the underlying asset, assisted by a loan taken against the promised payment of the exercise price on the expiration date, and protected by a put option that provides its holder the privilege (without the obligation) to sell the speciﬁed underlying asset at the same exercise price, within the same expiration limit. The holder of this package gains full participation in the potential for increasing value of the asset, with reduced investment due to the power of borrowing, and protection from losing money if the value of the asset falls below the exercise price (if necessary, the put could be exercised to sell the asset for the amount needed to settle the debt), and therefore, the loss

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is limited to no more than the initial net investment. Therefore, having a call option is like buying the underlying asset with the help of ﬁnancial leverage, while keeping the insurance provided by the put. In developing the literature on the real options approach for capital investment decisions in business organizations, the underlying premise has been that when shareholders (and other investors) perceive the real options and learn details about them, they will include these option values in the process of evaluating the securities issued by the ﬁrm. Real options are thought to resemble ﬁnancial options such as exchange-traded calls and puts, except that the underlying assets are real assets (as opposed to ﬁnancial assets such as stocks) and that the rules governing exercise come from the real world rather than from a legal contract. Then, relationships that drive the value of ﬁnancial options are extended to analyze decisions about acquiring and exercising real options, with decision-makers thought to be seeking maximum market value at each decision. A decision-maker of a for-proﬁt business would face strong difﬁculties in estimating the value added to the organization by having the real option and assessing the change in value if the option were exercised, but the focus would be on market value or a reasonable proxy of it. When real options are created or acquired by not-for-proﬁt organizations, though, there is not a strong connection with evaluation processes in ﬁnancial markets. A nonproﬁt administrator would face measures of value that may be much less clear. In the case of a school administrator, the broad objective is to provide quality education within the bounds of acceptable cost. Quality is difﬁcult to capture in quantitative measures, except perhaps through proxies such as student/teacher ratios, average class size, scores on standardized tests, graduation rates, or percentage of graduates who proceed to college. Cost may be reduced in the political mix to simpliﬁed proxies such as annual expenditures per pupil. The harried administrator may have incentives to focus on these admittedly imprecise measures of beneﬁt versus cost. Also, administrators may be driven by more organization-speciﬁc incentives that we will now explore in the context of shell space. It might charitably be asserted that school administrators, for example, want to achieve educational objectives in the most cost effective manner. Yet, as we will see upon examination of the incentives for decision-makers, the rationale for acquisition and exercise of real options can be quite different for decision-makers in not-for-proﬁt organizations than those in for-proﬁt business organizations. We will start with the case of shell space and then extend the analysis to other real options3 that are prevalent in the not-for-proﬁt arena.

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THE CASE OF SHELL SPACE Shell space in new buildings refers to areas that have been fully enclosed from the environment with an exterior shell but left unﬁnished pending future commitment of resources to the completion. While in this fully enclosed but unﬁnished state, the shell space provides real options for which the underlying asset is ﬁnished interior space such as classrooms in a school or operating rooms in a hospital. The exercise price is the cost of completing the construction. There is no ﬁxed expiration date for such real options, but the remaining useful life of the initial complex imposes limitations on how long it is possible to delay full completion (if not with the expensive ﬁnishing of, say, an operating room in a hospital; then at least with something less expensive but still useful, such as conference space).4 Shell space is less expensive than the alternative of ﬁnishing the space during the initial construction, but it can be signiﬁcantly more expensive than simply preparing the new building for later expansion by strengthening the framework to allow for later addition of higher stories, or even framing out (but not enclosing) an upper story. Indeed, when such a steel framework, crowning another wing of the building, was clearly visible through the classroom window, one of the authors regularly used it as an example when explaining the concept of real options to ﬁnance students.

DECISION TO ACQUIRE SHELL SPACE The cost of adding shell space seems at ﬁrst to be a straightforward function of cost per square foot. Still, the whole picture deserves a closer look. Enclosing a space with roof and exterior walls involves opportunities for economies of scale (the area of enclosed space expands exponentially relative to additions upon the building’s length, width, or height). Also, building cost per square foot may be lower for a large-scale project on a single site, compared with the same total space scattered over smaller scale projects on multiple sites (or on adjacent sites at different times). Therefore, total construction costs may be reduced (while still remaining within budget constraints for the initial period) by means of building a large enclosure with some shell space reserved for the future. Also, the aesthetics of the ﬁnished building may be enhanced by fully enclosing the desired exterior while holding down initial period costs by keeping some shell space in reserve.

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Incentives for Those Making Decisions to Acquire Shell Space For the decision-maker in a not-for-proﬁt organization, shell space offers a way to maximize the total enclosed space that can be acquired within the conﬁnes of a limited budget for the initial period. Once the expansion option is in hand, it gives the school district superintendent or hospital administrator leverage with the politically elected (or appointed) board of trustees when the time comes to ﬁnish the space. Then, the argument might be that the original investment would be wasted without the authorization for further resources to ﬁnish the interior. Faced with a given initial limitation, therefore, the administrator has incentives to enclose shell space to gain more persuasion in future budget negotiations. Of course, this is counter to the purely business description of an option as conveying the privilege, but not the obligation, to exercise. Unlike a decision-maker in a for-proﬁt business, who seeks to acquire real options when the value of the option exceeds the cost of acquisition, the administrator of a not-for-proﬁt organization has no incentive to focus on the value of the options acquired. Instead, this administrator has incentives to focus on maximizing the beneﬁts to the organization that can be acquired within a given limitation upon expenditures in the initial time period (and any beneﬁts to the organization are not reﬂected in market transactions). Choosing the location of the shell space within the building may also be different in a not-for-proﬁt organization. To gain the greatest leverage for future funding, a school or hospital administrator has incentives to pick visible locations for the shell space – thus increasing the public relations impact of the constant visible reminder that future funding is needed. With a for-proﬁt business, the incentives are more in line with keeping the shell space strategically located, or even discretely out-of-sight (so as not to call attention from competitors).

DECISIONS TO EXERCISE SHELL SPACE REAL OPTIONS The options involved in shell space are complex. Besides having loosely deﬁned expiration and variable exercise prices, there may be multiple alternative uses for a given space that involve different ﬁnishing costs. In a hospital, for example, a given space might be ﬁnished for use as an operating suite (with high ﬁnishing cost due to the specialized plumbing and wiring

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requirements), or alternatively into a space for medical imaging equipment, or a conference space (with much lower ﬁnishing costs than other alternatives). In a school, the space might be used for classrooms, administration, cafeteria, or sports facilities – all with different ﬁnishing costs. Furthermore, any decision to exercise the option cancels the possibility of exercising in another way at a future time when more information has become available.

Luehrman’s Intuition about When to Exercise Real Options The challenge of making the best choices when exercising real options has become a central issue in the real options literature. Tim Luehrman (1998) offers an intuitive guide to optimal exercise of real options through his now classic ‘‘tomato garden’’ analogy (see Fig. 1). Panel A shows the direction of increasing value for the option. As we move from left to right along the horizontal ‘‘value-to-cost’’ scale, we follow a continuum starting at low value of the underlying asset relative to the cost of exercise (where the option is ‘‘out of the money’’ and would not be exercised). Moving to the right along the continuum, we pass the point where the value of the asset equals the exercise price (value-to-cost ratio is 1.0) and then move into the territory where the option might be proﬁtably exercised (here the option is ‘‘in the money’’). As we move from top to bottom along the vertical ‘‘volatility’’ scale, we start where there is low potential for the value of the asset to change very much during the time remaining. Moving down, we follow a continuum toward ever more and more potential for the value of the asset to move (either up or down). Given the limitation on liability that is inherent in options, increased potential for movement translates into higher value for the option (because of full participation in upward moves, with limited exposure to downward moves). The dashed line moving diagonally from upper left toward lower right represents the combined effect of the twin vectors. Luehrman divides this option space into six regions to develop the intuition about deciding when to exercise. He asks the reader to think of a gardener cultivating tomatoes and deciding when to pick them (i.e., when to exercise the real option). The gardener’s competition is a ﬂock of birds that appear unannounced from time to time and would love to eat the tomatoes before the gardener picks them. Region 1 in the option space (high on the value-to-cost scale and low on the volatility scale) contains options that are ready to exercise. Exercising

Managing Real Options in Not-for-Proﬁt Organizations

Fig. 1.

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When to Exercise? Source: Luehrman (1998).

the option is proﬁtable now, and there is low likelihood of further movement in the underlying asset’s value. Indeed, the greatest concern here is the possibility that the competition might arrive (in the tomato garden analogy, the fruit is ripe and should be harvested before the birds get it). Region 2 generates the most difﬁculty for the decision-maker. The crop is ready to harvest, yet still has potential to improve with a bit more time on the vine. In this situation, the greatest concern would be to gather information about the competitors. If they are nearby and active, it would be advisable to harvest. If there are no signs of activity among the competitors, it might be worthwhile to let the fruit continue to improve before harvesting. In region 3, the options are ‘‘near the money’’ and so offer little or no proﬁt if exercised. There is still high potential for improvement in value, and therefore, unless there are immediate threats from the competition, the fruit

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should be left to ripen on the vine. Only if the competitors present an imminent threat would it be advisable to harvest early and hope the fruit would ripen satisfactorily on a sunny windowsill. In region 4, the difﬁculty for the decision-maker has become much reduced. The options are ‘‘out-of-the-money’’ with substantial potential for improvement if the fruit were left on the vine to ripen. Therefore, the advice would normally be to delay the harvest. In region 5, we have ‘‘out-of-the-money’’ options with smaller likelihood of improvement. Now would not be a good time to harvest, and this assessment would probably not change. Finally, in region 6, we have ‘‘outof-the-money’’ options with no potential for the value of the underlying asset to improve. These we should write off and take our losses.

Options to Expand When management decisions are subject to market discipline, the incentive is to maximize the combined value of the initial-phase facility plus the value of expansion options. The primary variables that drive the value of expansion options are time remaining until expiration and volatility of value for the underlying assets. More time and volatility translate into higher option values. Perhaps the biggest difference in the way decisions are made about exercising options to expand existing ﬁnished space for non-proﬁts lies in the strong disincentive to simply allow the option to expire without being exercised. With a for-proﬁt business, options would normally only be exercised if the added value of the ﬁnished space exceeds the cost of exercise. For the non-proﬁt administrator, though, letting the option expire (leaving the shell space unﬁnished) could make future funding initiatives more difﬁcult. Thus as the building ages (so that ﬁnishing space within it seems less sensible), the incentive would be to choose one of the alternative uses that involves relatively low ﬁnishing cost and remove the inconvenient reminder of past decisions (even if the added value of the ﬁnished space might not be judged sufﬁcient to offset the cost, if the matter were subject to market discipline). Thus a non-proﬁt administrator might be tempted to exercise options that lie within region 5 (or even region 6) of Fig. 1. When beneﬁts to the organization are not reﬂected in market transactions, moreover, any decisions about exercising shell space options may be driven in very different ways than would be the case in a business organization. Rather than seeking the opportune time for value maximization, plus the optimal use that offers the largest difference between ﬁnishing costs and market value

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added, non-proﬁt administrators seek to maximize internal organizational beneﬁts. For a purely business decision, choosing the opportune time for exercise would involve consideration of the option value lost by committing to a particular use for the space, when substantial uncertainty remains concerning the value of alternative uses relative to their associated costs, and sufﬁcient time remains to await further information. Administrators of non-proﬁt organizations, though, have incentives to get the shell space ﬁnished as soon as funding becomes available. It makes them appear more effective than would be the case if they waited for more information even though funding were available immediately. Therefore, options that lie within region 3 of Fig. 1, or even region 4, might be exercised immediately. Moreover, the most valuable use of the space, in the eyes of a non-proﬁt administrator, could be the use that makes the others within the organization most happy, yet still pleases the politically chosen board of trustees. For a school administrator, classroom space in hand today could trump ofﬁce space or sport facilities, because new classroom space makes teachers happy and helps public relations. For a hospital administrator, more operating room space in hand might trump laboratory space or space for high-technology medical imaging systems that may be very useful but offer less beneﬁt for staff physicians or lower public relations impact.

Options to Defer It may be that at the time of initial construction, the decision-maker has difﬁculty anticipating the exact nature of the organization’s future needs. Then, shell space offers the opportunity to gain the beneﬁts of scale economies during the initial construction, thus maximizing the space than can be enclosed with the funding available in the initial period. With options to defer, the variables with greatest impact on option value are time and volatility (more of either translates into higher value). For a business organization, the delay would provide the added value of being able to await future arrival of information needed to clarify the best use of the space. When the organization does not face market discipline, there is not an incentive to maximize market value added from a decision. For a non-proﬁt organization, therefore, the array of goals would be (1) maximize the funding that is available immediately through the political process and (2) use the leverage to gain additional funding as soon as possible thereafter. Awaiting future resolution of uncertainty would be a lesser consideration for a non-proﬁt administrator than for a business executive.

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FURTHER BENEFITS OF SHELL SPACE AND INCENTIVES INVOLVED WITH OTHER REAL OPTIONS Earlier Completion of Future Projects For school or hospital administrators (and other non-proﬁts as well), new funding must wait for existing capacity to be exhausted, or for there to be a widespread perception that it soon will be. Then the time required to start new construction and complete the work leads to an extended period of excess demand for limited available capacity (schools must resort to temporary portable buildings and hospitals must delay procedures or send patients to alternative locations, resulting in foregone revenues). Having shell space ready in advance of the capacity pressures makes it possible to respond more quickly and restore the organization to normal proﬁciency. This is a kind of switching option (an option that allows switching from one state to another). For a non-proﬁt administrator, the incentives for possessing such options are focused on reduced administrative sacriﬁces required to stabilize an unbalanced situation (so, less disruption). Less Disruption When capacity shortfalls loom immediately, one common approach to remediation is to shufﬂe several departments into compressed space to accommodate a function that has been dislocated by the onset of new construction. If shell space had been created in advance, however, some of it might be placed in temporary use while other shell space undergoes ﬁnish work. This is a kind of ﬂexibility option.5 Again, for a non-proﬁt administrator, the incentives for possessing such options are focused on reduced administrative sacriﬁces required to stabilize an unbalanced situation.

Better Facility Layout By planning ahead, placing shell space strategically so that adjacent existing functions can be expanded into it, long-term functional relationships can be established (such as classrooms clustered around cafeteria space in schools, or operating suites near necessary laboratory facilities in hospitals). This is another sort of ﬂexibility option, for which the above incentives apply.

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Temporary Use of Shell Space Shell space might be sealed until future development, or it could be placed in temporary use as storage space, or even (with climate control equipment enabled) as conference space. In a school, shell space originally designated for cafeteria or auditorium use might even be partitioned into temporary classroom space as an alternative to portable buildings. Again, this is another sort of ﬂexibility option for which the above incentives again apply. Enhanced Objectivity for Future Decisions about Space Utilization When functional departments feel that their future needs are well represented in resource allocation negotiations, their representatives might be more candid. If they doubt that their needs are fairly assessed, they may provide biased inputs into the resource allocation process. Effective deployment of shell space by non-proﬁt administrators may therefore result in more candid inputs into future resource allocation negotiations. This would offer greater value the more uncertainty there is about future needs.

EXTENSION TO OTHER REAL OPTIONS THAT OCCUR AMONG NOT-FOR-PROFIT ORGANIZATIONS School districts allocate money for everything from capital expenditures, such as new schools, additions to existing schools and other facilities, to the purchase of textbooks and maintenance supplies. Expenditures occur at both the district and the building levels. Here also, administrators try to make cost-effective ﬁnancial decisions and still achieve desired educational objectives. In these decisions, other real options can be found. Portable Buildings versus Permanent Structures School administrators often must wait until existing facilities are utilized at full capacity (or forecast to soon reach capacity) before they can seek funding for new facilities. This calls for short-term measures to deal with the facility shortages, because average time for completion of new school

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construction is three years.6 Often, the solution is to acquire portable buildings. These portable buildings convey real options, because when they are no longer needed in one place they can be moved to other locations, mothballed for later use, or sold to other districts. The real options reduce the risk of acquiring portable buildings; and in a business environment, these options would be analyzed as abandonment options. For a business, such options would be acquired if their impact on market value, net of cost, were positive. For a non-proﬁt organization, the rationale for acquisition would focus on reduction of stress on the organization during the time needed for constructing permanent facilities. For a business organization, exercising the abandonment option would be triggered when the value of the underlying asset exceeds the cost of exercise. Please remember here that the cost of exercise includes the value foregone when an option is extinguished before it has ﬁnally expired. For non-proﬁt organizations, though, market value is not a pressing issue. The non-proﬁt administrator might be more concerned about the appearance of wasting resources if the portable units were mothballed and stored. Likewise, the non-proﬁt administrator might be more concerned about the appearance of failure (or desperation) if portable units were sold – before the end of their useful lives – to another district. The portable units would then continue to be deployed within the system, even if not fully utilized. Therefore, non-proﬁt administrators have disincentives against exercising abandonment options. Such options may be allowed to languish until they ﬁnally expire unused. One way to ease the stress on school administrators while at the same time returning many of the future decisions to the market value arena would be for the school system to lease the temporary units from a business enterprise. The added value of such arrangements could be shared with the school system through lease rates that are more attractive than the alternative of buying.

Land for Future Development Acquiring land in a key location offers the opportunity, at some time in the future, of exchanging the vacant space for any of several alternative uses (with no set expiration date by which the choice must be consummated). Such options are similar to the options analyzed by Rene Stulz (1982) and Herb Johnson (1987).7 One thing that can clearly be said about the valuedrivers for such options is that the options are more valuable the lower the

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correlation among the values of the different underlying assets (i.e., the alternative uses). As exercising the options extinguishes any remaining value generated by the possibility for choosing an alternative use in the future, there would be an incentive in the for-proﬁt environment to wait for resolution of uncertainty about which of the alternative uses might ﬁnally emerge as the most valuable. In a school system (or other non-proﬁt environment), the pressure is toward early exercise, to avoid the appearance of indecision or the perception that the original acquisition was hasty or even wasteful. The land would likely be used for the ﬁrst alternative that becomes ripe for development (i.e., in terms of Fig. 1, options that are in regions 3, 4, or even 5 might be exercised immediately). When there is no reference to the impact of management decisions on any market estimation of the value of the organization, option values often will not be maximized.

Contraction of Services Now let us turn to an option to reduce or eliminate service or capacity. In this particular scenario, a portion of a school system is operating signiﬁcantly below the capacity available from its facilities, with a low ratio of students to teachers (with personnel costs being the primary driver of costs for the school system, having fewer students per teacher adds substantially to the cost per pupil). Furthermore, enrollment in this sector is expected to continue shrinking. As the administrators consider this scenario, they realize that there is an option to reduce service/production (an option to shrink is a less extreme version of an abandonment option). In the case of a for-proﬁt business, the focus for decisions would be on taking the actions that maximize market value of the organization. For the administrators of a non-proﬁt organization, however, the focus would be on reducing the stresses within the organization. Rather than cleanly reducing staff through termination or layoffs, the incentives for the administrators would be to allow gradual staff reduction by attrition, with gradual reassignment of remaining staff. Rather than closing facilities and selling unused resources, the incentives would be toward slower expansion in growing areas, with redistricting to divert students from other areas into the facilities that are experiencing declining enrollments. While such moves would give wide distribution to low levels of stress, administrators would avoid high levels of stress within concentrated areas. From a value-based perspective, though, the actions could be value diminishing.

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Too Big to Fail State governments (and even the federal government) sometime provide put options to large urban school systems that are considered too big (or too politically important) to be allowed to fail. Therefore, when such school systems extend themselves to the point of potential insolvency, they can rely on the prospect of a bailout from higher levels of government. The existence of such put options is well known to create ‘‘moral hazard’’ incentives for taking ill-advised risks. Whereas in the previous examples, we have seen multiple instances of early extinguishment of options when further risk is well advised from the perspective of market value maximization, here we encounter the prospect of non-proﬁt organizations potentially incurring risks that are ill advised from a market value perspective, because of the put options that are given to them from higher levels of government. Here, there is a large element of what has been called systemic risk.

CONCLUDING REMARKS The real options approach allows decision-makers to more accurately picture what a given expenditure decision may entail. Plus, real options allow for management intervention at varying points throughout the project’s life. We have used the case of shell space to discover the different incentives non-proﬁt administrators have in the acquisition, recognition, and rational exercise of real options by their organizations (compared with managers of for-proﬁt businesses). While business executives can be expected to base such decisions on the anticipated impact on market value of the securities issued by the ﬁrm, non-proﬁt administrators have no connection to a securities market valuation process. In the not-for-proﬁt arena, decisions about the acquisition and use of options are driven by incentives that arise from within the organization or emanate from the politically elected (or appointed) board of trustees. Shell space is space within a new building that has been enclosed against the elements, but not yet ﬁnished for its intended future use. Often, it is isolated from the climate control systems of the building. The shell space can be viewed as a set of complex options (along the lines of the Stulz–Johnson options to choose among a group of several possible ﬁnished outcomes with different costs of exercise). A business executive could be expected to make

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the acquisition decision based on the estimated values of the expansion options and deferral options the shell space represents. In turn, the exercise decision could be expected to reﬂect the impact on the combined value of the existing facility and the remaining options, thus considering the potential loss of value from the remaining options that are extinguished upon exercise. Also, a business decision-maker would not ﬂinch at allowing the options to expire unused, if the added value from exercise does not offset the cost of ﬁnishing. When deciding whether to acquire shell space as part of the initial phase of construction, non-proﬁt administrators are concerned about gaining the most organizational value from the initial funding. The economies of scale in construction allow a larger space to be enclosed with the funds available initially, if some of the spaced is shelled in reserve for future expansion. Administrators would stringently avoid the prospect of the options being allowed to expire unused, because of the appearance of waste. The incentives would be to exercise the options at the earliest opportunity, and perhaps even base the choice among the alternative uses, at least in part, upon the amount of available funding. In short, the incentives are to use the creation of shell space as a means of maximizing the space for the organization that can be gained from the initial funding and then use the unﬁnished space as leverage to gain more funding in the future. In sum, non-proﬁt administrators have incentives that lead them sometimes to acquire expansion options at higher prices than would forproﬁt executives. Non-proﬁt administrators also have incentives that lead them to exercise expansion options sooner, while exercising abandonment options later (or not at all). Furthermore, large urban school districts may be on the receiving end of put options provided by state (or even federal) levels of government because these districts are considered too big (or too politically important) to be allowed to fail. Such districts can rely on the prospects of a bailout. These put options create perverse incentives for taking illadvised risks. We have seen several examples, therefore, of situations in which nonproﬁt administrators have incentives to extinguish expansion options early, when further risk is well advised from the perspective of market value maximization. Then, we have encountered the prospect of non-proﬁt organizations that are ‘‘too big to fail’’ potentially incurring risks that are ill advised from a market value perspective.

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NOTES 1. See also Van Helfenstein (2008). 2. See Majd and Pindyck (1987) for an early discussion of the decision about when to exercise real options. 3. For a thorough listing of the variety of real options, see Trigeorgis (1995, 1996) and Trigeorgis and Mason (1987). 4. For an excellent discussion of shell space in new hospital construction, see Roark, Brooks, and Kilgore (1993). 5. See Trigeorgis and Mason (1987) or Triantis and Hodder (1990) for further discussion of managerial ﬂexibility. 6. See the 2008 Annual School Construction Report, a supplement to School Planning and Management, February 2008 (CR1-CR16). 7. Estimation of values for such options requires an iterative computer procedure implemented by the algorithm created by Mark Schervish (1985), or the algorithm created by Boyle and Tse (1990).

REFERENCES Boyle, P. P., & Tse, Y. K. (1990). An algorithm for computing values of options on the maximum or minimum of several assets. Journal of Financial and Quantitative Analysis, 25, 215–228. Johnson, H. (1987). Options on the maximum or the minimum of several assets. Journal of Financial and Quantitative Analysis, 22, 277–283. Luehrman, T. A. (1998). Strategy as a portfolio of real options. Harvard Business Review, 76(5), 89–101. Majd, S., & Pindyck, R. S. (1987). Time to build, option value, and investment decisions. Journal of Financial Economics, 18, 7–27. Roark, K., Brooks, B., & Kilgore, K. (1993). Costs and beneﬁts of shell space construction – model for cost/beneﬁt analysis of shell space construction. Healthcare Financial Management, 47(10), 46–52. Schervish, M. J. (1985). Algorithm AS195: Multivariate normal probabilities with error bound. Applied Statistics, 34(1), 81–87. Stoll, H. (1969). The relationship between put and call option prices. Journal of Finance, 24(5), 801–824. Stulz, R. (1982). Options on the minimum or the maximum of two risky assets. Journal of Financial Economics, 10, 161–185. Triantis, A., & Hodder, J. (1990). Valuing ﬂexibility as a complex option. Journal of Finance, 55, 549–565. Trigeorgis, L. (Ed.) (1995). Real options in capital investment: Models, strategies, and applications. Westfort, CT: Praeger Publishers. Trigeorgis, L. (Ed.) (1996). Real options: Managerial ﬂexibility and strategy in an uncertain world. Cambridge: MIT Press. Trigeorgis, L., & Mason, S. (1987). Valuing managerial operating ﬂexibility. Midland Corporate Finance Journal, 5(1), 14–21.

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Van Helfenstein, S. B. (2008). Acquisitions, creative destruction, and taxes: Applications of real options in economic systems. Presented at the 12th Annual International Conference on Real Options, Rio de Janiero, Brazil, July 11.

FURTHER READING Amram, M. (2002). Value sweep: Mapping corporate growth opportunities. Boston: Harvard Business School Press. Amram, M., & Kulatilaka, N. (1999a). Real options: Managing strategic investment in an uncertain world. Boston: Harvard Business School Press. Amram, M., & Kulatilaka, N. (1999b). Disciplined decisions: Aligning strategy with the ﬁnancial markets. Harvard Business Review, 77(1), 95–104. Amram, M., & Kulatilaka, N. (2000). Strategy and shareholder value creation: The real options frontier. Journal of Applied Corporate Finance, 13(2), 15–28. Bonini, C. (1977). Capital investment under uncertainty with abandonment options. Journal of Financial and Quantitative Analysis, 12, 39–54. Brennan, M., & Schwarz, E. (1985). Evaluating natural resource investments. The Journal of Business, 58, 135–158. Chen, A. H., Conover, J. A., & Kensinger, J. W. (1998). Valuing ﬂexible manufacturing facilities as options to exchange assets. Quarterly Journal of Economics and Finance, 38, 651–674 (special issue on the role of competition and strategy). Chen, A. H., Conover, J. A., & Kensinger, J. W. (2006). Extending the real options approach by including virtual options. 10th Annual International Conference on Real Options, Columbia University, New York City, 14–17 June. Dixit, A. K., & Pindyk, R. S. (1994). Investment under uncertainty. Princeton, NJ: Princeton University Press. Dixit, A. K., & Pindyk, R. S. (1995). The options approach to capital investment. Harvard Business Review, 73(3), 105–115. Howe, K. M., & McCabe, G. M. (1983a). On optimal asset abandonment and replacement. Journal of Financial and Quantitative Analysis, 18, 295–305. Howe, K. M., & McCabe, G. M. (1983b). On optimal asset abandonment and replacement. Journal of Financial and Quantitative Analysis, 18, 295–305. Kensinger, J. (1987). Adding the value of active management into the capital budgeting equation. Midland Corporate Finance Journal, 5(1), 31–42. Margrabe, W. (1978). The value of an option to exchange one asset for another. Journal of Finance, 33, 177–198. Margrabe, W. (1982). A theory of the price of a claim contingent on N asset prices. Working Paper 8210, September. School of Government and Business Administration, George Washington University. McDonald, R., & Siegel, D. (1985). Investment and the valuation of ﬁrms when there is an option to shut down. International Economic Review, 26, 331–349. Myers, S. C. (1977). The determinants of corporate borrowing. Journal of Financial Economics, 5, 147–175. Myers, S. C. (1984). Finance theory and ﬁnancial strategy. Interfaces (January–February), 126–137.

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Myers, S. C., & Majd, S. (1990). Abandonment value and project life. Advances in Futures and Options Research, 4(1), 1–21. Paddock, J., Siegel, D., & Smith, J. (1988). Option valuation of claims on real assets: The case of offshore petroleum leases. Quarterly Journal of Economics, 103(3), 479–508. Porter, M. E. (1985). Competitive advantage. New York: The Free Press. Robichek, A. A., & VanHorne, J. C. (1967). Abandonment value and capital budgeting. Journal of Finance, 22, 577–589. Robichek, A. A., & VanHorne, J. C. (1969). Abandonment value and capital budgeting: Reply. Journal of Finance, 24, 96–97. Shapiro, A. C. (1985). Corporate strategy and the capital budgeting decision. Midland Corporate Finance Journal, 3(1), 22–36. Siegel, D., Smith, J., & Paddock, J. (1987). Valuing offshore oil properties with option pricing models. Midland Corporate Finance Journal, 5(1), 22–30. Stulz, R., & Johnson, H. (1985). An analysis of secured debt. Journal of Financial Economics, 14, 501–522.

O-SCORE FINANCIAL DISTRESS RISK ASSET PRICING Syou-Ching Lai, Hung-Chih Li, James A. Conover and Frederick Wu ABSTRACT We examine explicitly priced ﬁnancial distress risk in post-1990 equity markets. We add a ﬁnancial distress risk factor to Fama and French’s (1993) three-factor model, based on Grifﬁn and Lemmon’s (2002) ﬁndings that ﬁnancial distress is not fully captured by the book-to-market factor. We test three-factor and four-factor capital asset pricing models using both annual buy-and-hold analysis and monthly time series analysis across portfolios adjusted for common book-to-market, size, and ﬁnancial distress factors. We ﬁnd empirical support for an Ohlson (1980) O-score-based ﬁnancial distress risk four-factor asset pricing model in the U.S. and Japanese markets.

Fama and French (1993) propose a three-factor – a market factor, a ﬁrm size factor, and a book-to-market equity (BE/ME) factor – capital asset pricing model to explain monthly stock returns for portfolios of stocks noted in Fama and French (1992). Later literature extends the three-factor model to incorporate additional pricing factors, such as Carhart (1997) adding momentum as a fourth factor. Several authors investigate ﬁnancial distress as a systematic risk affecting asset returns Research in Finance, Volume 26, 51–94 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026006

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(Campbell, Hilscher, & Szilagyi, 2008; Daniel & Titman, 1997; Dichev, 1998; Ferguson & Shockley, 2003; Grifﬁn & Lemmon, 2002). We add country-speciﬁc O-score ﬁnancial distress as an explicit fourth factor to the three-factor Fama and French model and test the model in two national equity markets. Our ﬁnancial distress O-score four-factor model outperforms the three-factor model and the Carhart four-factor model in pricing the U.S. and Japanese equity market returns. In our view, a large strand of the Fama and French three-factor model literature seeks to explain cross sectional stock market factors in studies using natural groupings such as skilled or unskilled mutual fund managers (Carhart, 1997), risk of bankruptcy distress (Dichev, 1998), analyst coverage (Grifﬁn & Lemmon, 2002), and return reversals around earnings announcements (Grifﬁn and Lemmon). A second strand seeks to investigate the underlying factors explaining value and growth stock returns or to test the three-factor model in different samples such as international markets (Arshanapalli, Coggin, & Doukas, 1998a; Arshanapalli, Coggin, Doukas, & David, 1998b; Campbell et al., 2008; Chen & Zhang, 1998; Daniel & Titman, 1997; Fama & French, 1998; Grifﬁn & Lemmon, 2002; Lakonishok, Shleifer, & Vishny, 1994; Lau, Lee, & McInish, 2002). A third strand of research extends the capital asset pricing model beyond Fama and French’s (1993) three-factor model (Carhart, 1997; Chen & Zhang, 1998, 2008; Vassalou & Xing, 2004; Von Kalckreuth, 2005). We incorporate ﬁnancial distress risk into the three-factor model, following this third strand of research, after noting that authors have not explicitly incorporated ﬁnancial distress risk as a systematic risk factor. We develop a four-factor model that builds on Fama and French’s threefactor model to incorporate Grifﬁn and Lemmon’s ﬁndings that Ohlson’s (1980) O-score measure captures ﬁnancial distress risk beyond Fama and French’s three factors in the 1965–1996 time period for U.S. data. Our fourfactor capital asset pricing model adds Ohlson’s ﬁnancial distress O-score as an economy-wide factor to Fama and French’s three factors – a ﬁnancial distress risk four-factor model with country-speciﬁc parameter estimates for the United States and Japan. We test the ﬁt of the return generating process for our four-factor model against Fama and French’s (1993) threefactor model and Carhart’s (1997) momentum-based four-factor model in portfolios of value stocks and growth stocks in the U.S. and the Japanese equity markets. In our 1991–2006 U.S. and Japan Datastream samples, our ﬁnancial-distress-risk-based four-factor model has stronger explanatory power than either the Fama and French three-factor model or the Carhart (1997) momentum-based four-factor model. In addition, the

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country-speciﬁc four-factor models explain the returns of both value and growth stocks in Japan and the United States when partitioned into large and small stocks.

1. LITERATURE REVIEW 1.1. Fama and French Three-Factor Stock Return Model Theory and International Tests Fama and French (1993, 1996a, 1996b, 2008) propose the following threefactor asset pricing model to explain asset returns in periodt: ri;t ¼ ai;t þ mi;t MTBt þ si;t SMBt þ hi;t HMLt þ i;t

(1)

where ri,t ¼ asset or portfolio i’s return minus the risk-free rate ðri;t rf ;t Þ; ai;t ¼ intercept; MTBt ¼ market excess return ðrm;t rf ;t Þ; SMBt ¼ the difference between the returns on portfolios of small and big stocks (Small Minus Big); HMLt ¼ the difference between the returns on portfolios of high and low book value to market value stocks (High Minus Low); i;t ¼ an error term; and mi ; si ; hi ¼ asset or portfolio i’s regression coefﬁcients for MTBt , SMBt , and HMLt . As pointed out by Fama and French (2004), one important implication of the asset pricing test is that the intercept, ai , in the time series regression is zero, following the logic of Jensen’s alpha for portfolio returns. They indicate that, using this criterion, the Fama and French model captures much of the variation in average returns for portfolios formed on size, BE/ME and other price ratios that cause problems for the capital asset pricing model (CAPM), in U.S. stock markets. Fama and French (1998) ﬁnd empirical support for their model compared to an international CAPM. Arshanapalli et al. (1998a, 1998b), Chen and Zhang (1998), Halliwell, Heaney, and Sawicki (1999) and Gaunt (2004) use intercept analysis and adjusted R-square analysis to examine the ﬁt of the Fama and French threefactor model in many international stock markets ﬁnding strong empirical support for the three-factor model over domestic single-factor CAPM models.

1.2. Momentum Four-Factor Model Theory and Tests Jegadeesh and Titman (1993) ﬁnd that abnormal returns derived from momentum strategies are not fully priced by the three-factor Fama and

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French model. However, part of the abnormal returns generated in the ﬁrst year after portfolio formation dissipates during the following two years. Carhart (1997) builds on Jegadeesh and Titman (1993) to study persistence in mutual fund performance during 1962–1993. Carhart adds a one year momentum factor – calculated as the difference between portfolio returns for the highest 30% and lowest 30% momentum stocks – to the Fama and French three-factor model and forms a momentum-based four-factor model to explain mutual fund returns. ri;t ¼ ai;t þ mi;t MTBt þ si;t SMBt þ hi;t HMLt þ pi;t PR1YRt1 þ i;t

(2)

where PR1YRt1 is the difference between the returns on portfolios of highest 30% and lowest 30% momentum stocks for the prior year and pi;t ¼ asset or portfolio i’s regression coefﬁcients for PR1YRt1 and the rest of the variables are the same as the three-factor model in Eq. (1). Carhart sorts ﬁrms into deciles based on high to low portfolio returns to create a return-based factor. Carhart compares the performance of the CAPM to both the Fama and French three-factor model and the momentum four-factor model. Carhart (1997) ﬁnds that incorporating momentum as a fourth factor is important for explaining equity mutual funds average returns and risk-adjusted returns: ‘‘In tests not reported, I ﬁnd that the four-factor model substantially improves on the average pricing errors of the CAPM and the three-factor model,’’ (Carhart, 1997, p. 62). Carhart ﬁnds that his momentum four-factor model provides additional explanatory power for up to one year after portfolio formation.

1.3. Financial Distress Theory and Tests Dichev (1998, p. 1146) notes that several authors (Chan, Chen, & Hsieh, 1985; Fama & French, 1992) suggest that the size and book to market effects might be proxying for a ﬁrm distress risk factor.1 Consequently, Dichev explicitly studies bankruptcy risk in NYSE, AMEX, and NASDAQ stocks from 1981 to 1995, comparing Altman’s (1968) Z-score and Ohlson’s (1980) O-score measures. Dichev (1998, p. 2317) ﬁnds ‘‘bankruptcy risk is not rewarded by higher returns. Thus a distress factor is unlikely to account for the size and book-to-market effects.’’ Dichev ﬁnds that ﬁrms with high bankruptcy risk earn substantially lower than average returns since 1980 with either measure and that Ohlson’s model displays a stronger negative association between bankruptcy risk and subsequent returns.

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Grifﬁn and Lemmon (2002) test portfolio pricing of ﬁnancial distress with a three-factor Fama and French model using the same samples and time period as Fama and French (1998). Grifﬁn and Lemmon examine the U.S. stock market based on ﬁve quintiles of ﬁnancial distress risk using Ohlson’s (1980) measure of distress risk, termed O-score. Using buy-and-hold returns, Grifﬁn and Lemmon ﬁnd that value portfolios outperform growth portfolios and that high O-scores are positively related to stock returns. The difference in stock returns for ﬁrms with the highest risk of distress is twice as large for high BE/ME securities relative to low BE/ME securities compared to other groups (Grifﬁn and Lemmon, 2002, p. 2334). In summary, Grifﬁn and Lemmon (2002) ﬁnd that the Fama and French three-factor model explains the returns more completely if the ﬁnancial distress of the ﬁrms is further classiﬁed.2 Grifﬁn and Lemmon (2002, p. 2317) ﬁnd that ‘‘Among ﬁrms with the highest distress risk as proxied by Ohlson’s (1980) O-score, the difference in returns between high and low book-to-market securities is more than twice as large as that in other ﬁrms. This large return differential cannot be explained by the three-factor model or by differences in economic fundamentals.’’ These ﬁndings suggest that the market does not fully impound available ﬁnancial distress information into market prices.3 Campbell et al. (2008) examine ﬁnancial distress risk in the U.S. stock market as a predictor of asset prices. They construct an empirical monthly index for each company with accounting and market-pricing variables. Alphas for three-factor Fama and French and four-factor Carhart regressions indicate that distressed stocks have very low returns, particularly after correcting for risk using the Fama and French three-factor model. They investigate many explanations for apparent underperformance of distressed stocks or ‘‘the distress anomaly’’ (p. 2923).

2. RESEARCH METHODOLOGY We investigate using Grifﬁn and Lemmon’s (2002) ﬁnancial distress results by proposing a fourth empirical systematic risk factor in the Fama and French (1993) model in place of the prior-one-year momentum factor suggested by Carhart (1997) who incorporated the Jegadeesh and Titman (1993) momentum ﬁndings into the Fama and French (1993) empirical asset pricing model. Grifﬁn and Lemmon (2002) suggest that the ﬁnancial distress risk of a ﬁrm has a signiﬁcant impact on the rate of return so we add a fourth aggregate factor, the ﬁnancial distress risk factor, OLMH, to the

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Fama and French three-factor model to test the structure of factors inﬂuencing small and large value and growth stock returns in Japan and the United States. Our model is ri;t ¼ ai;t þ mi;t MTBt þ si;t SMBt þ hi;t HMLt þ oi;t OLMH t þ i;t

(3)

where all variables are the same as the Fama and French variables presented in Eq. (1) plus an additional variable. We add a ﬁnancial distress risk premium, OLMH t which equals the value-weighted average rate of return difference between the portfolio containing the lowest 20% O-score ﬁrms and the portfolio containing the highest 20% O-score ﬁrms. The coefﬁcient oi;t is the coefﬁcient for asset i in period t. 2.1. Data and Period of Study To get independent results from Grifﬁn and Lemmon (2002), we include only recent (1991–2006) data for the two largest developed country stock markets: the United States and Japan. Our data overlaps Grifﬁn and Lemmon’s 1965–1995 U.S. Compustat and Center for Research in Securities Prices (CRSP) data and includes a separate nation. Financial distress risk pricing in another developed nation with similar laws, accounting data and reliance on capitalism but with different patterns of trade and industry should reveal the robustness of the ﬁnancial distress risk factor asset pricing model. Both countries have similar attitudes in using stock markets and debt markets as sources of capital and also toward using capitalism as a mechanism for allocating resources in their economies. Both countries have substantial differences as well. Their national cultures, social structures, political structures and corporate cultures should be reﬂected in the factors underlying stock market returns. In particular, corporate management philosophies, workers’ attitudes toward their companies and national tax structures differ enough that size, book-to-market, market risk and ﬁnancial distress factors should be different between the United States and Japan. Our two nation tests of competing three-factor and four-factor models should have enough similarities and differences in the common risk factors to reveal differences in asset pricing for these two markets. 2.1.1. Stock Returns Our daily returns sample is drawn from non-ﬁnancial public Datastream ﬁrms in the United States and Japan for both currently trading and defunct

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57

securities.4 We follow Fama and French (1996a, 1996b) and Liew and Vassalou (2000) and exclude ﬁnancial stocks, insurance stocks, holding companies for securities such as unit trusts and depository receipts, preferred shares, cross listings, warrants, and duplicates. We use July 1991 to June 2006 for the United States Russell 1000 Datastream ﬁrms and May 1995 to April 2005 for the Japan Datastream ﬁrms. Japanese ﬁrms are those listed on the Tokyo or Osaka Stock Exchanges or the Over-the-Counter Securities Exchange for Japan listed in Thomson Financial Datastream (JASDAQ). Closing quotations and the value of each stock are from the Thomson Financial Datastream database. Daily and monthly returns are calculated from the closing quotation adjusted for cash dividends and bonus shares. Additional details are in Table 1. 2.1.2. Accounting Variables On the basis of Grifﬁn and Lemmon’s (2002) tests, we use the Ohlson (1980) O-score as a proxy for the risk of ﬁnancial distress. Because O-score is calculated using accounting variables from annual ﬁnancial statements, we allow up to a six-month lag for ﬁrms to publish the statements, following the approach taken in Fama and French (1992) and subsequent literature. We gather Thomson Financial accounting statement data from July 1991 to June 2006 for the United States and from May 1995 to April 2005 for Japan, respectively. We eliminate ﬁrms with a negative BE/ME when we use BE/ME as a factor to classify stocks.

2.2. Financial Distress Factor We classify stocks into value stocks and growth stocks assuming that BE/ME is an adequate proxy, following Chan and Chen (1991), Chan, Hamao, and Lakonishok (1991), Chan, Jegadeesh, and Lakonishok (1995), Lakonishok et al. (1994) and Chan and Lakonishok (2004) who suggest that the explanatory ability of BE/ME for the rate of return is very robust. Following Grifﬁn and Lemmon (2002), we sort stocks into three BE/ME groups: the lowest 30% (growth stocks), the middle 40% (blended), and the highest 30% (value stocks). If the ratio of the book value of equity to the market value of equity is low, the ﬁrm’s future is favorable relative to its historical cost and so is deﬁned as a ‘‘growth stock.’’5 On the contrary, if the ratio is high, it is a ‘‘value stock.’’

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Table 1.

Portfolio Classiﬁcations and Variable Deﬁnitions.

Portfolio Classiﬁcations Annually, all ﬁrms are grouped into 15 portfolios in the United States and 15 portfolios in Japan. The top row represents three groupings of individual ﬁrm book-to-market-equity (BE/ME) ratios. Group L is the lowest 30% and represents the growth ﬁrms. Group M is the middle 40% and represents blended ﬁrms (mixture of growth and value). Group H is the highest 30% and represents value ﬁrms. The columns partition the probability of ﬁnancial distress factor, O-score, into ﬁve quintile groupings. O-score group LO (low O-score) is the portfolio grouping that contains ﬁrms with the lowest 20% of O-score values. O-score group LO represents the ﬁrms with the lowest ﬁnancial distress risk. O-score groups 2, 3, and 4 are portfolios that contain the ﬁrms with the second lowest, third lowest, and fourth lowest O-score values. O-score group HO (high O-score) is the portfolio grouping that contains ﬁrms with the highest quintile O-score values for that year, or the group with the highest ﬁnancial distress risk. To read the table, the O-score quintiles are broken down into the HML grouping that is appropriate. For example, the cell labeled LLO is a portfolio that has ﬁrms with the lowest 30% of BE/ME ratios and lowest O-score quintile for that year. Likewise, the cell labeled L2 represents a portfolio containing the ﬁrms with the lowest 30% of BE/ME ratios for the year and the second lowest O-score quintile. The cell labeled M4 represents a portfolio containing the ﬁrms with the middle 40% BE/ME ratio and the fourth lowest O-score quintile grouping. Finally, the cell labeled HHO represents a portfolio containing the ﬁrms with the highest BE/ME ratio and the highest O-scale quintile. O-Score

O-score portfolios

BE/ME

LO 2 3 4 HO

HML Portfolios L

M

H

LLO L2 L3 L4 LHO

MLO M2 M3 M4 MHO

HLO H2 H3 H4 HHO

Notes: To create size groupings, we sort the ﬁrms within each of the ﬁfteen 15 U.S. portfolios and within the ﬁfteen 15 Japanese portfolios by size. We create 30 portfolios for each country by grouping the ﬁrms that are greater than the median size (large ﬁrms) within a portfolio. The remaining ﬁrms are the small ﬁrms in each portfolio, respectively. We measure each variable as follows: 1. Rate of return. Stock rights and dividends are adjusted on ex-rights days and ex-dividend days to create daily multipliers and daily returns, but we do not reinvest cash dividends. We calculate the monthly and annual rates of return with each respective accumulated multiplier from the daily rates of return. 2. Risk-free rate. For the U.S. risk-free rate, we use the treasury bill rate. For the Japanese riskfree rate, we use the commercial paper rate. 3. Market portfolio return. We follow Fama and French’s (1992, 1993) methodology of excluding ﬁnancial and security companies in market portfolio return calculations. We use either the rate of return for ﬁrms included in the United States Russell 1000 index (United States sample) or the rate of return for all ﬁrms traded in Japan (Japan sample). 4. Book-to-market equity (BE/ME). Book equity of common stock equals total shareholder’s equity of the ﬁrm minus the amount of preferred stock. Market value of the common stock equals the closing quotation multiplied by the number of outstanding common shares. BE/ME is calculated by dividing the book equity by the market value. 5. Size. Size, or the market value of the stock, is calculated by multiplying the closing quotation of the stock on the last transaction day of the year by the number of outstanding shares.

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2.2.1. Financial Distress in the United States Ohlson (1980) uses a logit model to construct a ﬁnancial alarm model that we use as our proxy for the likelihood of ﬁnancial distress. To develop the model, Ohlson chooses 105 bankrupt company stocks and 2,058 nonbankrupt stocks from both NYSE/AMEX and OTC ﬁrms. All sample ﬁrms are in the manufacturing industry and are selected from 1970 to 1976. He applies nine ﬁnancial variables to estimate a logit model to predict the probability of ﬁnancial distress. Ohlson’s logistic regression model is shown as follows (Ohlson, 1980, pp. 118, 121): 8 > < 1:32 0:407SIZE t þ 6:03TLT At 1:43WCTAt O scoret ¼ þ0:0757CLCAt 2:37NITAt 1:83FUT Lt > : þ0:285INTWO 1:72OENEG 0:521CHIN t

t

(4)

t

where SIZE t ¼ ln (total assets/GNP price-level index); TLTAt ¼ total liabilities/total assets; WCTAt ¼ working capital/total assets; CLCAt ¼ current liabilities/current assets; NITAt ¼ net income/total assets; FUT Lt ¼ funds provided by operations/total liabilities; INTW Ot ¼ 1, if net income is negative for the last two years, zero if otherwise; OENEGt ¼ 1, if total liabilities exceeds total assets, zero if otherwise; and CHIN t ¼ ðNI t NI t1 Þ=ðjNI t j þ jNI t1 jÞ, where NIt is net income for the most recent period. For the U.S. stock markets, we calculate annual O-score values using Ohlson’s original parameter estimates.6 A higher probability of bankruptcy represents a lower quality ﬁrm, implying that the probability of bankruptcy for stocks with high ﬁnancial distress risk (high O-score) is higher. For stocks with low ﬁnancial distress risk (low O-score), the probability of bankruptcy is low. 2.2.2. Financial Distress in Japan To obtain the economy-wide estimates for the probability of ﬁnancial distress risk in Japan, we estimate Japanese parameters for the logistic model with the same explanatory variables and approach taken by Ohlson (1980), but with our 1995 to 2005 Japanese accounting data. We use annual data for all ﬁnancially distressed Japanese ﬁrms and for all normal Japanese ﬁrms listed in the Datastream database to estimate the Japanese equivalent logistic regression parameters.

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Our estimates for the Ohlson logistic model’s parameters for Japan are: 8 > < 0:022 0:053SIZE t 2:653TLTAt 1:707WCTAt O scoreðJapanÞt ¼ 2:738CLCAt þ 4:369NITAt þ 2:102FUT Lt > : þ0:065INTWO 0:211OENEG þ 0:712CHI N t t t (5) where the accounting variables have the same deﬁnitions as those in Eq. (4). Our O-score (Japan) ﬁnancial distress model’s estimated parameters are different from the U.S. O-score model parameters, demonstrating the differences in the U.S. economy and Japanese economy. However, the Japanese model’s parameter estimates are similar in size to U.S. parameters, demonstrating the similarities in pricing ﬁnancial distress with accounting variables in the two economies. When we constrain the Japanese O-score logistic model estimates to be the same as the U.S. O-score estimates (results not presented), we reject the null hypothesis of similarity at the 0.001 level. We use the O-score (Japan) model’s estimates to calculate an annual Japanese O-score for each Japanese ﬁrm in our sample for each year. The yearly Japanese O-score ﬁnancial distress probability uses each Japanese ﬁrm’s accounting data, similar to the procedure used for the U.S. ﬁrms. For all the Japanese tests conducted in the rest of the chapter, we use the ﬁrm-speciﬁc Japanese O-scores calculated using our estimated O-score (Japan) parameters.7

2.3. O-Score and HML Portfolio Groupings We investigate the ﬁt of three competing stock market return generating processes for both the U.S. sample and the Japanese sample using a similar ﬁrm portfolios approach used in Fama and French (1993, 1998) and Grifﬁn and Lemmon (2002). We form 15 portfolios of ﬁrms (in each country) that are similar in (a) their magnitude of ﬁnancial distress and (b) BE/ME. First, we sort all non-ﬁnancial ﬁrms by their previous year’s O-scores into ﬁve quintile groups from the smallest to the largest in each of our samples.8 The ﬁve quintile O-score groupings are labeled LO (low O-score), 2, 3, 4, and HO (high O-score). The high O-score (HO) group has the high probability for a ﬁrm to suffer ﬁnancial distress compared to any of the other groups and the low O-score (LO) group indicates the low probability for a ﬁrm to suffer ﬁnancial distress relative to the other groups.

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61

Second, we further partition each of the ﬁve O-score quintile’s sample according to their previous year’s BE/ME ratio. We assign the ﬁrms into one of three HML portfolios: the highest 30% of values are assigned to a portfolio termed H (value), the middle 40% of values to a portfolio labeled M (blended), and the lowest 30% of values to a portfolio labeled L (growth). Combining our annual O-score groupings with our annual HML (BE/ME) groupings, we classify all ﬁrms into one of ﬁfteen portfolios as listed in Table 1 for each of the two countries for each year. For our initial buy-and-hold analysis, we buy and hold each of the 15 portfolios for a year and calculate the annual return on investment for each equally weighted portfolio. We resort and reclassify portfolios annually. We compare the rates of return between high and low HML groups for each O-score group and between high and low O-score groups for each HML group. Because of the unknown characteristics of the middle HML group (usually termed blended stocks, denoted by M in our tables), we do not overemphasize this group but report all test results for completeness of the analysis.

2.4. Regression Model Analysis of Fama and French-Type Three- and Four-Factor Models In each country, we employ ordinary least squares (OLS) monthly timeseries regressions to test the explanatory power of the three asset pricing models. First, we use the parameter estimates for the Fama and French model (Eq. 1) variables – MTB, SMB, and HML – to estimate the goodness of ﬁt. Second, we use Carhart’s (1997) momentum four-factor model (Eq. 2) that adds momentum, in the form of a factor labeled PR1YR, to the threefactor model. We compare its explanatory power relative to the Fama and French three-factor model. Third, we use our ﬁnancial distress four-factor model (Eq. 3) that adds O-score, in the form of a factor labeled OLMH, to the three-factor model. We compare its explanatory power beyond the Fama and French three-factor model to explain monthly stock returns.

3. EMPIRICAL RESULTS AND ANALYSIS 3.1. O-Score, Book-to-Market Equity, and Returns To separately examine the relationship between BE/ME and O-scores, we classify portfolios into three groups based on BE/ME, in each of the ﬁve

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O-score quintiles as described earlier and summarized in Table 1. Tables 2 and 3 present summary statistics for the stocks in each portfolio in the United States and Japan, respectively.9 Within the quintiles of O-score in the United States and in Japan, the probabilities of bankruptcy for the portfolios with low, medium and high BE/ME exhibit a similar pattern from low to high as expected. For the highest quintile of O-score in the United States, however, low BE/ME ﬁrms have the highest probability of bankruptcy at 0.51, whereas high BE/ME ﬁrms have the lowest probabilities of bankruptcy at 0.49, which are statistically identical. Similarly, for the highest quintile of O-score in Japan, low BE/ME ﬁrms have the highest probability of bankruptcy at 0.68 and high BE/ME ﬁrms have lowest probabilities of bankruptcy at 0.62, but again the difference is not statistically signiﬁcant. The ﬁnding of low BE/ME ratios in ﬁrms in the highest O-score group in both the United States and Japan is puzzling. Table 2 shows that when U.S. ﬁrms in the highest O-score quintile are partitioned into low, middle, and high BE/ME values, the market value of the lowest BE/ME grouping is much larger than the market value of the ﬁrms in the highest BE/ME grouping ($31.5 million market value compared to $6.5 million market value, respectively). However, the puzzle of low BE/ME ﬁrms in the highest quintile is explained when further analyzed. For all O-score quintiles, Table 2 reveals a ﬁnding of a pattern of much higher market values for all low BE/ME ﬁrms relative to high BE/ME ﬁrms. In addition, within the low, middle, and high BE/ME groupings, the average market value of ﬁrms increases with the level of ﬁnancial distress indicated by its O-score quintile. Table 3 reveals very similar patterns for Japan. Among all low BE/ME stocks in the United States, the second to highest O-score ﬁrms have smaller 12-month prior returns and the smallest 36-month prior returns. For both the United States and Japan, high BE/ME ﬁrms have monotonically decreasing prior 12-month average returns across increasing O-score quintile, but the prior 36-month average returns share this pattern only for the lowest four O-score quintiles. Comparing the lowest O-score quintile to the highest O-score quintile for each country reveals higher 12-month and 36-month returns for LO ﬁrms than for HO ﬁrms for all three BE/ME categories. Table 2 reveals that return on assets is negative for the highest O-score quintile across BE/ME categories in the United States, resulting in a decline in retained earnings growth for that quintile. This is consistent with high ﬁnancial distress as proxied by the O-score quintile. Table 3 reveals

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Table 2. Summary Statistics of Firm Characteristics for United States Portfolios: Sorted by BE/ME and the Probability of Financial Distress. O-Score Portfolios

Book-to-Market-Equity Portfolios: L (Growth); M (Blended); H (Value) O-score average (probability) BE/ME average (ratio) Number of ﬁrms per year

LO 2 3 4 HO

L

M

H

L

M

H

L

M

H

0.16 0.20 0.23 0.27 0.51

0.16 0.21 0.23 0.28 0.47

0.16 0.21 0.25 0.31 0.49

0.1531 0.1677 0.1715 0.1736 0.1487

0.3637 0.3610 0.3606 0.3567 0.3562

0.8033 0.8881 0.8827 1.1193 0.8543

49 49 49 50 52

51 51 51 51 51

50 50 50 50 50

Prior 12-month average return (%)

LO 2 3 4 HO

Prior 36-month average return (%)

L

M

H

66.00 31.00 24.00 15.00 30.00

37.00 22.00 21.00 15.00 17.00

30.00 21.00 13.00 9.00 5.00

Market value average ($ millions) L LO 2 3 4 HO

M

H

6,092.54 3,380.51 2,106.22 7,601.46 3,391.64 2,352.19 13,887.48 5,778.58 3,758.85 40,472.71 12,480.44 6,595.52 31,507.00 18,665.96 6,517.33

L

M

H

Retained earnings growth (%) L

M

H

159.00 198.00 177.00 0.3649 0.0271 0.2367 130.00 106.00 118.00 0.3541 1.6463 0.4335 73.00 89.00 83.00 0.2614 0.1626 0.1203 59.00 71.00 69.00 0.1879 0.0756 0.0446 112.00 103.00 94.00 0.0884 1.2142 0.9035 Market leverage average (ratio)

Return on assets average (%)

L

M

H

L

M

H

1.6295 1.4383 1.9428 2.0414 8.1212

1.5323 1.2232 1.3033 1.6772 1.9450

1.8075 1.5408 1.7447 1.9849 2.0328

11.18 11.85 11.65 10.34 6.40

7.13 7.91 7.60 5.50 1.60

5.09 5.50 4.57 2.53 5.14

Notes: Firms included in the Russell 1000 index from July 1991 to June 2006 are ranked independently based on their values of ﬁnancial distress (O-score) calculated using Ohlson’s (1980) model and book-to-market-equity (BE/ME) as described in Table 1. Prior 12-month stock returns are percentages of equal-weighted buy-and-hold returns from July to June in the year prior to ranking. The 36-month prior stock returns are equal-weighted buy-and-hold returns from July three years ago to June in the year of ranking. Growth in retained earnings is the percentage change in retained earnings on the balance sheet over the year before ranking. Market capitalization is the market value of ﬁrm. Leverage is the ratio of total book assets less book equity to-market equity. Return on assets is the ratio of income before extraordinary items to total book assets.

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Table 3. Summary Statistics of Firm Characteristics for Japan Portfolios: Sorted by BE/ME and the Probability of Financial Distress. O-Score Portfolios

Book-to-Market-Equity Portfolios: L (Growth); M (Blended); H (Value) O-score average (probability)

LO 2 3 4 HO

BE/ME average (ratio)

L

M

H

L

M

H

L

M

H

0.26 0.37 0.45 0.53 0.68

0.28 0.36 0.41 0.47 0.59

0.29 0.37 0.43 0.50 0.62

0.237 0.233 0.169 0.144 0.064

0.848 0.878 0.877 0.908 0.904

3.532 3.153 8.298 28.044 35.665

218 223 223 223 224

292 298 298 298 298

221 223 223 223 223

Prior 12-month return average (%)

LO 2 3 4 HO

Prior 36-month average (%)

Retained earnings growth (%)

L

M

H

L

M

H

L

M

H

24.0 18.5 15.0 16.0 5.2

11.4 07.7 14.6 9.8 5.9

12.0 7.8 6.9 5.8 2.8

31.2 8.0 6.9 11.2 12.5

25.2 13.7 21.3 22.5 14.4

23.6 19.0 31.8 30.6 15.4

0.51 0.20 0.11 0.12 0.48

0.15 0.00 0.07 0.39 0.28

0.10 0.84 0.09 0.67 0.36

Market value average ($ millions)

LO 2 3 4 HO

Number of ﬁrms per year

Market leverage average ratio

Return on assets average %

L

M

H

L

M

H

L

M

H

13,687.00 32,820.00 46,021.00 65,963.00 65,471.00

225.00 224.00 149.00 111.00 132.00

21.00 32.00 31.00 32.00 30.00

0.17 0.43 0.57 1.78 1.34

0.75 0.84 1.42 2.34 4.83

1.68 2.19 10.71 37.46 30.50

1.4 2.5 2.6 0.2 6.6

1.7 2.4 2.3 1.3 0.2

00.0 0.5 0.3 0.6 2.7

Notes: Firms in Japan from May 1995 to April 2005 are ranked independently based on their values of ﬁnancial distress (O-score) calculated using our Japanese parameter estimates for Ohlson’s (1980) model and book-to-market-equity (BE/ME) as discussed in Table 1. Prior 12-month stock returns are percentages of equal-weighted buy-and-hold returns from May to April in the year prior to ranking. The 36-month prior stock returns are equal-weighted buyand-hold returns from May three years ago to April in the year of ranking. Growth in retained earnings is the percentage change in retained earnings on the balance sheet over the year prior to ranking. Market capitalization is the market value of ﬁrm. Leverage is the ratio of total book assets less book equity to market equity. Return on assets is the ratio of income before extraordinary items to total book assets.

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65

a different pattern in Japan, showing that all high BE/ME ﬁrms have low return on assets, with three quintiles showing negative return on assets. Interestingly, the average percentage growth in retained earnings patterns does not match the average percentage return on assets patterns across O-score quintiles and BE/ME portfolio types for Japan, indicating potential differences in accounting practices in the two countries. Within the low BE/ME quintile in the United States, retained earnings growth in the year before ranking is the smallest for the highest O-score ﬁrms. For the United States, the prior 12-month return and the return on assets are consistent indicators of ﬁrm ﬁnancial performance across O-score quintiles in high BE/ME ﬁrms. The higher a ﬁrm’s ﬁnancial distress quintiles, the lower are the return on assets, growth in retained earnings, and prior 12-month return. Interestingly, the prior 36-month stock market return average displays a similar pattern. Taken together, these ﬁndings indicate that the negative shocks to book equity may explain the low BE/ME ratios of these ﬁrms in the United States. To further examine whether O-score and BE/ME are both related to distress risk, we report summary market leverage statistics in Tables 2 and 3. Market leverage, measured as the ratio of the book value of liabilities to the market value of equity, is positively related to both O-score and BE/ME for Japan but is positively related to O-score only for the United States. Both O-score and BE/ME are negatively related to return on assets, and positively related to leverage, which is consistent with the view that both O-score and BE/ME are related to distress risk. When we compute the Spearman rank correlations between O-score and BE/ME, we ﬁnd a value of 0.054 and 0.0046 in the United States and in the Japanese stock markets, respectively. These results suggest that O-score contains information related to distress risk that the BE/ME ratio does not capture. If the BE/ME ratio and O-score both capture unique information related to the ﬁnancial distress risk factor, both O-score and BE/ME should be considered in pricing stock returns.

3.2. Summary Statistics for Sales and Investment Ratios for Portfolios Tables 4 and 5 show the percentage of growth in sales of the low BE/ME portfolios is larger than that of high BE/ME portfolios in the United States and Japan, respectively. The sales-to-book assets ratio of the low BE/ME portfolios is higher than that of the high BE/ME portfolio. The median market value of equity to sales ratio of the low BE/ME portfolios is higher

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Table 4. Sales and Investments Ratio Medians For United States Portfolios: Sorted by BE/ME and the Probability of Financial Distress. O-Score Portfolios

Book-to-Market-Equity Portfolios: L (Growth); M (Blended); H (Value) Median growth in sales (percentage)

LO 2 3 4 HO

L

M

H

L

M

H

39.40 25.00 16.86 12.00 61.19

24.89 17.78 11.70 9.90 92.16

22.97 13.40 12.14 6.94 29.34

1.2800 1.3543 1.2016 1.1037 0.8030

1.1718 1.3208 1.1983 1.0890 0.8183

1.2069 1.1480 1.0460 0.7960 0.7830

Median capital expenditure/book assets (ratio)

LO 2 3 4 HO

Median sales/book assets (ratio)

Median market value/sales (ratio)

L

M

H

L

M

H

0.0358 0.0399 0.0312 0.0276 0.0289

1.6020 1.4245 1.5315 1.8406 5.3841

0.7497 0.8290 0.8449 1.0950 1.1530

6.5890 4.2200 3.9680 4.0880 7.0014

1.6020 1.4245 1.5315 1.8406 5.3841

0.7497 0.8290 0.8449 1.0950 1.1530

Notes: Firms included in the Russell 1000 index from July 1991 to June 2006 are ranked independently based on their values of ﬁnancial distress (O-score) calculated using Ohlson’s (1980) model and book-to-market-equity (BE/ME) as described in Table 1. We report the median of growth in sales, sales to book asset, market value of equity to sales, and capital expenditures to book assets.

than that of high BE/ME portfolios in the United States and Japan, which is consistent with Grifﬁn and Lemmon’s (2002) evidence that investors favor the ﬁrms with high market equity to sales ratios than those with lower market equity to sales levels. The capital expenditure-to-asset ratio in the portfolio with low BE/ME (growth ﬁrms) is lower than portfolio with higher BE/ME (either blended or value ﬁrms). We conclude that these data show low BE/ME portfolios perform well relative to high BE/ME portfolios. Grifﬁn and Lemmon (2002) point out that investors may overreact to the information about the future growth potential of ﬁrms with low BE/ME. Their ﬁndings suggest that investors are anticipating improving sales growth and proﬁtability of those future sales for ﬁrms with low BE/ME relative to ﬁrms with high BE/ME.

67

O-Score Financial Distress Risk Asset Pricing

Table 5. Sales and Investments Ratio Medians For Japan Portfolios: Sorted by BE/ME and the Probability of Financial Distress. O-Score Portfolios

Book-to-Market-Equity Portfolios: L (Growth); M (Blended); H (Value) Median growth in sales (percentage)

LO 2 3 4 HO

L

M

H

L

M

H

25.0 52.6 5.9 3.6 9.4

4.2 4.8 8.9 7.7 2.1

1.0 1.2 0.1 1.1 4.3

0.830 0.868 0.850 0.759 0.584

0.907 1.053 1.125 1.135 1.160

0.900 1.057 1.102 1.138 1.903

Median capital expenditure/book assets (ratio)

LO 2 3 4 HO

Median sales/book assets (ratio)

Median market value/sales (ratio)

L

M

H

L

M

H

0.001 0.002 0.002 0.011 0.020

0.005 0.002 0.004 0.004 0.007

0.007 0.001 0.003 0.002 0.003

1.069 2.198 4.976 5.857 8.379

0.837 0.710 0.621 0.480 0.379

0.316 0.266 0.225 0.177 0.145

Notes: Japanese ﬁrms from 1995 to 2006 are ranked independently based on their values of the probability of ﬁnancial distress (O-score) calculated using Ohlson’s (1980) model and book-tomarket-equity (BE/ME). We report the median of growth in sales, sales to book assets, market value of equity to sales and capital expenditures to book assets.

3.3. Buy-and-Hold Returns for the Portfolios Sorted on BE/ME and O-Score To investigate whether return differences captured by partitioning ﬁrms into the three BE/ME portfolios and the ﬁve O-score quintiles are signiﬁcant, Tables 6 and 7 display the annual buy-and-hold returns and the return differences for all ﬁrms in the United States and Japan, respectively. Furthermore, we report the return differences captured by high versus low BE/ME portfolios and high versus low O-score deciles for small and large ﬁrms, respectively10. We partition stocks in the United States and Japan into 15 portfolios. These portfolios are the three sizes of BE/ME (lowest 30%, middle 40%, and highest 30%) within each of the ﬁve O-score quintiles calculated for the respective countries. Furthermore, we partition each of the 15 portfolios of all ﬁrms into two sub-portfolios based on their market

68

SYOU-CHING LAI ET AL.

Table 6. Average Annual Buy-and-Hold Returns for Size Portfolios in the United States: Sorted by BE/ME and the Probability of Financial Distress. O-Score Portfolios

L

M

H

Ret(H)(L)

p-value

28.59% 18.21% 15.66% 14.59% 23.06%

26.88% 21.48% 20.75% 17.69% 34.25%

0.12% 2.91% 5.19% 2.33% 12.07%

0.213 0.112 0.032 0.143 0.035

5.53% 0.0410

7.36% 0.0920

26.92% 15.78% 14.31% 15.16% 28.22%

22.54% 28.38% 21.88% 21.14% 27.80%

4.64% 3.07% 8.42% 4.29% 9.65%

0.34 0.08 0.02 0.07 0.02

9.03% 0.02

1.30% 0.15

5.26% 0.09

Large-ﬁrm BE/ME portfolios LO 25.79% 2 15.87% 3 16.65% 4 13.31% HO 21.07%

30.11% 19.69% 16.36% 13.45% 15.12%

31.88% 18.62% 18.44% 14.80% 35.68%

6.09% 2.75% 1.79% 1.49% 14.61%

0.06 0.11 0.14 0.12 0.06

14.99% 0.00

3.81% 0.13

All ﬁrms BE/ME portfolios LO 26.76% 2 18.57% 3 15.56% 4 15.36% HO 22.18% Ret(HO)(LO) p-value

4.59% 0.0010

Small-ﬁrm BE/ME portfolios LO 27.19% 2 25.31% 3 13.45% 4 16.85% HO 18.15% Ret(HO)(LO) p-value

Ret(HO)(LO) p-value

4.72% 0.00

Notes: Percentage value-weighted annual buy-and-hold returns for ﬁrms in the United States from July 1991 to June 2006 are displayed for portfolios formed by ranking with probabilities of ﬁnancial distress (O-score) calculated using Ohlson’s (1980) model and book-to-market-equity (BE/ME). Stocks are ranked into three groups by size (all stocks, small stocks, and large stocks). Size-adjusted groupings are a simple average of the large and small time series. The tests for statistical differences between groups are based on the time series of monthly returns from July 1991 to June 2006. The high minus low BE/ME portfolio differences are calculated within the same distress groups by forming a portfolio that is long in the high BE/ME portfolio and short in the low BE/ME portfolio. Differences in ﬁnancial distress portfolio returns are calculated from high distress portfolios minus low distress portfolios within each BE/ME grouping. Statistical signiﬁcance at the 0.05 level separately.

69

O-Score Financial Distress Risk Asset Pricing

Table 7. Average Annual Buy-and-Hold Returns for Size Portfolios in Japan: Sorted by BE/ME and the Probability of Financial Distress. O-Score Portfolios

L

H

Ret(H)(L)

p-value

6.62% 0.91% 1.13% 6.07% 3.82%

12.42% 14.68% 26.49% 14.60% 12.32%

13.14% 16.87% 27.93% 13.81% 24.60%

0.084 0.072 0.067 0.044 0.004

11.57% 0.067

2.80% 0.311

0.10% 0.494

1.41% 6.90% 5.81% 8.92% 9.41%

3.34% 1.58% 3.51% 9.16% 3.54%

15.84% 19.51% 45.27% 15.03% 12.54%

14.43% 26.40% 39.47% 6.11% 21.95%

0.090 0.043 0.103 0.231 0.098

10.82% 0.069

0.19% 0.488

3.30% 0.285

2.84% 2.51% 8.69% 7.35% 15.16%

9.89% 3.39% 1.25% 2.98% 4.09%

9.00% 9.85% 7.71% 14.17% 12.10%

11.85% 7.33% 16.40% 21.52% 27.25%

0.259 0.249 0.031 0.020 0.003

12.31% 0.157

5.79% 0.269

3.09% 0.395

All ﬁrms BE/ME portfolios LO 0.72% 2 2.19% 3 1.44% 4 0.78% HO 12.28% Ret(HO)(LO) p-value Small ﬁrms LO 2 3 4 HO Ret(HO)(LO) p-value Large ﬁrms LO 2 3 4 HO Ret(HO)(LO) p-value

M

Notes: Percentage value-weighted annual buy-and-hold returns for ﬁrms in Japan from May 1995 to April 2005 are displayed for portfolios formed by ranking with probabilities of ﬁnancial distress (O-score) calculated using Ohlson’s (1980) model and book-to-market-equity (BE/ME). Stocks are ranked into three groups by size (all stocks, small stocks, and large stocks). The sizeadjusted groupings are from a simple average of the large and small time series. The tests for statistical differences between groups are based on the time series of monthly returns from May 1995 to April 2005. The high minus low BE/ME portfolio differences are calculated within the same distress groups by forming a portfolio that is long in the high BE/ME portfolio and short in the low BE/ME portfolio. Similarly, differences in ﬁnancial distress portfolio returns are calculated using returns from high-distress portfolios minus low-distress portfolios within each BE/ME grouping. Statistical signiﬁcance at the 0.05 level separately.

70

SYOU-CHING LAI ET AL.

capitalization (small and large) given the importance of the size factor in the Fama and French three-factor model and in our results in Tables 2 and 3. We ﬁnd that a book-to-market effect exists for buy-and-hold portfolios across O-score quintiles in the U.S. and Japanese stock markets. Table 6 shows differences at the 5% signiﬁcance level in annual buy-andhold returns for high BE/ME portfolios versus low BE/ME portfolios and within two of the ﬁve O-score quintiles for all the U.S. ﬁrms. Table 7 shows differences within two of the ﬁve O-score quintiles for all Japanese ﬁrms. Table 6 displays the average annual size-adjusted percentage returns differentials for all ﬁrms between the U.S. portfolios with high BE/ME ratios versus ﬁrms with low BE/ME ratios. The returns of the high BE/ME minus the low BE/ME portfolios are signiﬁcantly different for O-score quintile portfolio 3, at 5.19%, and the O-score quintile HO at 12.07%, but not for the other three quintiles. The all-ﬁrms portfolio tests are further partitioned into the small and larger ﬁrms within each of the 15 portfolios. For the United States, the returns differences are clearly due to the small ﬁrms, which have an 8.42% difference for O-score quintile 3 and a 9.65% difference for O-score quintile HO. All other small-ﬁrm U.S. O-score quintile differences between high BE/ME portfolios and low BE/ME portfolios are not statistically signiﬁcantly different from zero. No large-ﬁrm O-score quintile differences are statistically signiﬁcantly different from zero for the U.S. ﬁrms. Table 7 displays that the returns differential due to the book-to-market effect is signiﬁcant in the all-Japanese ﬁrms’ portfolios in O-score quintile 4 at 13.81% and in O-score quintile HO at 24.60%. When partitioned into small and large Japanese ﬁrms, we ﬁnd that the small-ﬁrm O-score quintile 2 high BE/ME versus low BE/ME return difference is statistically signiﬁcantly at 26.4% and that the large-ﬁrm O-score quintiles have high BE/ME return differences of 16.4%, 21.52% and 27.25% in O-score quintiles 3, 4, and HO, respectively. An O-score effect, or ﬁnancial distress effect, is noticeable in the United States buy and hold evidence but not in the Japanese evidence. Table 6 shows the returns of the high O-score quintile portfolio, HO, minus the returns of the low O-score quintile portfolio, LO, for the all-ﬁrms portfolio grouping shows a statistically signiﬁcant difference of 4.59% and 5.53% in the low BE/ME portfolios (growth) and the middle BE/ME portfolios (blended ﬁrms), but no statistical difference in the high BE/ME (value) portfolios, for the United States. For U.S. small ﬁrms, the high ﬁnancial distress O-score quintile, HO, earns 9.03% less for the low BE/ME portfolio but the high-minus-low O-score differences are not signiﬁcant for the

O-Score Financial Distress Risk Asset Pricing

71

blended or the value portfolios. For U.S. large ﬁrms, the high ﬁnancial distress O-score quintile, HO, earns 4.72% less for the low BE/ME portfolios (growth) and 14.99% less for the middle (blended) portfolio but is not signiﬁcant for the value portfolio. For Japan, Table 7 shows that although we observe the same patterns of lower returns for high ﬁnancial distress portfolios, HO, compared to low ﬁnancial distress portfolios, LO, especially for low BE/ME portfolios, the differences are not statistically signiﬁcant at the 5% level. Our ﬁndings for the U.S. market are consistent with Dichev’s (1998) evidence that the ﬁrms with low distress risk can earn higher average stock returns than ﬁrms with higher ﬁnancial distress risk. However, the high stock returns occur for the portfolio with high BE/ME (value stocks) in the highest O-score quintile (high ﬁnancial distress, HO) for large ﬁrms, consistent with the ﬁrms having higher ﬁnancial distress risk. Similarly, for Japan, higher stock returns are observed for the Japanese portfolios with high BE/ME ratios (value stocks) compared to portfolios with low BE/ME ratios (growth stocks) across O-score quintiles for all ﬁrms, small ﬁrms, and large ﬁrms.

3.4. Fama-and-French (1993) Three-Factor Model – Monthly Data Tests Although the buy-and-hold return data for average portfolio returns is interesting, an important question investigated in the literature that explores Fama and French’s (1993) asset pricing model is whether individual stock returns depend on common factors. We wish to test the goodness of ﬁt of the three alternative asset pricing models in Eqs. (1)–(3). For each country, we would like to test the ﬁt of the three models inside each of the 30 portfolios formed based on a ﬁrm’s ﬁnancial distress O-score, relative BE/ME, and relative size. As discussed previously, each U.S. ﬁrm is classiﬁed into one of ﬁfteen portfolios: ﬁve relative O-score quintiles, ranging from low ﬁnancial distress to high ﬁnancial distress, and three relative BE/ME classiﬁcations, ranging from low BE/ME (growth) to middle BE/ME (blended) to high BE/ME (value). These 15 portfolios are examined in Tables 2 and 4. Next, we partition each of the 15 all-ﬁrms portfolios into small ﬁrms (smallest half ) and large ﬁrms (largest half) for a total of 30 portfolios. These are the same 30 sub-portfolios used in Table 6 and discussed in the previous section. Similarly for Japanese ﬁrms, we partition them into 15 similar portfolios: ﬁve relative O-score quintiles and three relative BE/ME classiﬁcations, using the same methods discussed earlier. These 15 portfolios are examined in

72

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Tables 3 and 5. We then partition each of the 15 Japanese all-ﬁrms portfolios into small ﬁrms and large ﬁrms for a total of 30 portfolios. These are the same 30 sub-portfolios used in Table 7, discussed earlier. First, we investigate how the three-factor model of Fama and French (1993) performs in its ability to explain the 30 portfolio returns in the United States and the 30 portfolio returns in Japan. In Table 8 through Table 11, we exhibit estimated regression coefﬁcients and regression statistics for the portfolios in each BE/ME, O-score, and size group in the United States and Japan.11 For the United States, Tables 8 and 9 show that the coefﬁcients of the market factor, MTB, are positive for all 15 small-ﬁrm portfolios and all 15 large-ﬁrm portfolios. The coefﬁcients of the small-minus-big factor, SMB, for twelve of the small-ﬁrm portfolios and nine of the large-ﬁrm portfolios are signiﬁcantly different from zero at a 5% signiﬁcance level. The coefﬁcients of the book-to-market factor, HML, for 12 of the small-ﬁrm portfolios and eight of the large-ﬁrm portfolios are signiﬁcantly different from zero. One of the small-ﬁrm portfolios and four of the largeﬁrm portfolios have signiﬁcantly negative HML coefﬁcients. Furthermore, we ﬁnd most coefﬁcients of the book-to-market factor, HML, increase monotonically as ﬁrms move from low to high book-to-market ratio, with the exception of high O-score small ﬁrms, which is reversed. In Tables 8 and 9, the adjusted R-square in the United States is higher than 40% for most of the portfolios and greater than 50% for 14 of the large-ﬁrm portfolios. For 10 of the 15 small-ﬁrm portfolios and 11 of the 15 largeﬁrm portfolios, the intercepts are signiﬁcantly different from zero, indicating that the Fama and French three-factor model does not price stock returns completely in the United States. The Gibbons, Ross, and Shanken (GRS) (1989) test of the alpha are both signiﬁcant, indicating rejection of the null hypothesis that all of the alpha coefﬁcients are equal to zero for U.S. small ﬁrms (Table 8) and U.S. large ﬁrms (Table 9), allowing some room for improvement. Tables 10 and 11 show that 16 of the 30 coefﬁcients of the market-to-book factor (MTB), 15 of the 30 coefﬁcients of the size factor (SMB), and 8 of the 30 coefﬁcients of the BE/ME factor (HML) are signiﬁcantly different from zero for small and large ﬁrms in the Japanese stock market. In addition, for Japanese ﬁrms, 11 of the 15 small-ﬁrm portfolio intercepts and 10 of the 15 large-ﬁrm portfolio intercepts are not signiﬁcantly different from zero indicating a very good ﬁt for the Fama and French three-factor model. Likewise, the GRS test is not signiﬁcant for both groups of portfolios (failing to reject the Fama and French model’s ﬁt). In addition,

73

O-Score Financial Distress Risk Asset Pricing

Table 8.

Three-Factor Monthly Regressions: U.S. Small-Firm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

a^ LO 2 3 4 HO

0.024 0.168 0.002 0.009 0.024

0.022 0.015 0.002 0.008 0.020

1.247 0.852 2.222 1.021 0.324

0.229 0.387 0.863 0.250 0.053

0.113 0.023 0.019 0.175 0.013

3.550 2.760 0.320 1.680 3.750

0.528 0.002 0.076 0.017 0.091 0.269 0.059 0.509 0.430 0.309

0.437 0.229 0.063 0.778 0.276

13.900 14.200 13.000 13.880 15.330

0.434 0.144 0.012 0.010 0.343

0.218 0.024 0.135 0.341 0.185

4.120 3.270 1.960 1.510 4.150

14.740 14.480 13.740 14.700 14.930

0.583 0.421 0.539 0.568 0.911

2.350 0.380 0.420 0.330 2.190

4.010 6.120 8.980 6.070 6.970

1.900 4.580 3.660 3.970 2.400

0.001 0.006 0.751 0.095 0.000

2.850 1.060 0.080 0.080 2.580

0.180 0.240 1.460 3.040 1.730

H

0.000 0.001 0.052 0.132 0.000

0.059 0.000 0.000 0.000 0.017

^ p-value (m) 14.550 15.270 15.070 16.060 14.680

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

p-value (^s) 7.000 8.540 8.240 8.930 5.340

0.020 0.705 0.678 0.746 0.030

^ tðhÞ 0.366 0.260 0.330 0.371 0.061

M p-value ð^aÞ

t(^s)

h^ LO 2 3 4 HO

L

^ tðmÞ

s^ LO 2 3 4 HO

H

tð^aÞ

m^ LO 2 3 4 HO

M

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

^ p-value (h) 2.780 2.320 2.930 3.790 5.250

0.005 0.289 0.936 0.934 0.011

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

GRS F(^a) 7.0079

P(F) 0.0002

Adjusted R2 (%) LO 2 3 4 HO

43.63 30.90 30.99 20.74 42.39

49.00 45.10 49.07 60.70 50.05

43.42 50.19 50.46 42.84 55.80

Notes: U.S. small-ﬁrm estimates of the Fama and French 3-factor model Eq. (1) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

74

SYOU-CHING LAI ET AL.

Table 9.

Three-Factor Monthly Regressions: U.S. Large-Firm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.021 0.008 0.016 0.019 0.009

0.455 0.241 0.264 0.607 0.478

0.034 0.025 0.017 0.024 0.007 0.017 0.014 0.003 0.021 0.004

1.064 0.290 0.032 0.388 0.514

3.390 1.300 3.870 4.010 1.990

0.797 0.512 0.565 0.860 0.382

0.436 0.309 0.002 0.232 0.553

16.610 18.700 23.870 15.990 21.990

0.549 0.354 0.394 0.616 0.266

3.350 3.490 3.600 0.690 0.240

0.001 0.196 0.000 0.000 0.048

15.020 16.550 21.380 23.220 23.600

16.650 23.490 23.600 19.310 19.760

0.000 0.001 0.169 0.002 0.000

0.001 0.001 0.000 0.489 0.809

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

p-value (^s)

1.064 0.118 3.890 0.144 2.130 0.341 0.414 2.550 0.490 1.820 0.199 0.022 4.100 5.610 4.700 0.095 0.498 1.210 2.650 3.280 0.318 7.202 0.690 1.660 12.170

0.000 0.012 0.000 0.229 0.494

^ tðhÞ

0.767 0.047 3.950 0.290 0.206 0.265 2.610 0.230 0.132 0.024 4.220 1.190 0.088 0.343 5.940 2.720 0.231 4.888 0.940 1.780

H

^ p-value (m)

tð^sÞ

h^ LO 2 3 4 HO

4.980 3.450 1.380 3.220 3.600

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.889 0.628 0.000 0.009 0.098

0.035 0.071 0.000 0.001 0.000

^ p-value (h) 5.000 1.720 1.900 3.340 12.190

0.000 0.010 0.000 0.000 0.347

0.775 0.818 0.237 0.007 0.076

0.000 0.088 0.059 0.001 0.000

Adjusted R2 (%) LO 2 3 4 HO

66.73 62.13 67.92 55.34 52.82

50.95 50.94 73.15 72.24 52.63

28.63 50.24 54.35 58.59 96.44

GRS F(^a) P(F) 5.5598 0.0011

Notes: U.S. large-ﬁrm estimates of the Fama and French three-factor model Eq. (1) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

75

O-Score Financial Distress Risk Asset Pricing

Table 10.

Three-Factor Monthly Regressions: Japan Small-Firm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

a^ LO 2 3 4 HO

0.021 0.079 0.033 0.050 0.014

0.051 0.046 0.028 0.007 0.029

1.481 1.089 1.541 1.710 1.836

1.012 0.691 0.498 0.754 0.601

0.034 2.066 0.078 0.633 0.211 1.790 0.022 0.472 0.100 2.358

1.326 1.194 0.071 0.586 0.610

2.744 1.989 2.026 2.935 2.655

1.456 1.737 0.342 0.078 1.668

1.980 2.655 1.463 1.914 1.869

0.609 0.776 2.860 0.289 2.254

0.081 0.150 0.339 0.325 0.330

0.041 0.528 0.076 0.638 0.020

3.373 2.468 0.048 1.796 2.224

0.007 0.049 0.000 0.004 0.009

2.098 1.737 1.938 2.798 2.420

0.148 0.085 0.733 0.938 0.098

0.544 0.439 0.005 0.773 0.026

0.000 0.009 0.146 0.058 0.064

0.001 0.015 0.962 0.075 0.028

p-value (^s) 0.092 0.011 0.004 0.072 0.013

^ tðhÞ 0.081 2.444 0.212 2.583 0.566 1.963 0.364 1.742 0.092 1.663

H

^ p-value (m)

0.153 0.065 0.497 1.699 2.399 1.467 0.464 0.331 0.806 2.583 1.809 1.778 0.081 0.157 1.102 2.935 2.098 2.183 0.876 0.822 0.098 1.822 1.486 2.399 0.083 0.368 0.243 2.521 1.704 1.815

0.058 0.338 0.717 0.229 0.001

M p-value (^a)

tð^sÞ

h^ LO 2 3 4 HO

L

^ tðmÞ

s^ LO 2 3 4 HO

H

tð^aÞ

m^ LO 2 3 4 HO

M

0.018 0.073 0.038 0.140 0.091

0.145 0.078 0.031 0.018 0.072

^ p-value (h) 1.754 1.726 1.688 1.748 1.835

0.016 0.011 0.052 0.084 0.099

0.038 0.085 0.055 0.006 0.017

0.082 0.087 0.094 0.083 0.069

Adjusted R2 (%) LO 2 3 4 HO

76.0 58.3 90.8 70.2 70.6

94.1 61.7 18.3 71.0 57.5

85.9 50.9 5.00 53.1 45.7

GRS F(^a) P(F) 0.2338 0.8688

Notes: Japan small-ﬁrm estimates of the Fama and French three-factor model Eq. (1) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

76

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Table 11.

Three-Factor Monthly Regressions: Japan Large-Firm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

a^ LO 2 3 4 HO

0.055 0.073 0.014 0.092 0.057

0.060 0.042 0.027 0.091 0.099

0.883 0.838 2.312 0.390 0.751

0.479 0.739 1.111 0.363 0.551

0.059 2.224 0.021 1.309 0.011 1.726 0.049 1.579 0.273 3.373

1.291 0.329 1.107 2.875 0.980

1.178 2.116 3.715 1.130 2.772

1.946 0.658 0.540 0.868 2.551

0.751 1.679 2.623 0.564 2.374

0.411 2.399 0.113 0.845 2.744

0.028 0.193 0.087 0.117 0.001

0.477 0.443 0.238 0.633 0.300

1.328 0.889 1.577 3.932 1.877

0.241 0.036 0.000 0.261 0.006

1.855 1.678 1.855 1.971 1.778

0.054 0.512 0.590 0.387 0.012

0.682 0.018 0.910 0.400 0.007

0.454 0.096 0.010 0.574 0.019

0.187 0.376 0.118 0.000 0.063

p-value (^s)

^ tðhÞ 0.830 1.778 0.358 1.737 0.289 2.399 0.486 1.835 0.866 1.683

H

^ p-value (m)

0.748 0.577 0.078 1.673 1.971 2.617 0.097 0.941 0.629 0.012 2.744 1.835 1.855 0.007 0.381 0.345 0.041 1.906 1.772 2.157 0.059 0.030 0.811 1.033 2.935 2.183 2.744 0.004 0.510 0.222 0.100 2.583 2.045 1.828 0.011

0.844 0.524 0.816 0.079 0.297

M p-value (^a)

tð^sÞ

h^ LO 2 3 4 HO

L

^ tðmÞ

s^ LO 2 3 4 HO

H

tð^aÞ

m^ LO 2 3 4 HO

M

0.051 0.069 0.079 0.031 0.043

0.010 0.067 0.033 0.007 0.0700

^ p-value (h) 1.742 1.989 1.766 1.537 1.998

0.078 0.085 0.018 0.069 0.095

0.06 0.096 0.066 0.051 0.078

0.0840 0.049 0.0800 0.1270 0.048

Adjusted R2 (%) LO 2 3 4 HO

31.6 60.6 87.2 0.60 67.9

10.9 60.7 74.6 22.4 59.8

32.6 15.1 14.1 89.4 63.9

GRS F(^a) P(F) 0.6602 0.6182

Notes: Japan large-ﬁrm estimates of the Fama and French three-factor model Eq. (1) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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according to the adjusted R-squares for the 30 Japanese stock market regressions, the explanatory power of the Fama and French three-factor model is greater than 50% for 20 of the 30 portfolios. This supports the contention that the three-factor model of Fama and French prices stock returns well in the Japanese stock market but there is room for improvement.

3.5. Carhart (1997) Momentum Four-Factor Model – Monthly Data Tests Jegadeesh and Titman (1993) ﬁnd that past winners realize consistently higher returns than past losers. Carhart (1997) adopts the prior-one-year momentum anomaly proposed by Jegadeesh and Titman to build a fourfactor model which consists of a market factor (MTB), a ﬁrm size factor (SMB), a BE/ME factor (HML), and the prior-one-year return momentum factor (PR1YR). This model, labeled Eq. (2) earlier, is the Fama and French three-factor model plus a momentum factor. We use Carhart’s (1997) four-factor model to examine the factors underlying stock returns for our 30 U.S. stock portfolios and our 30 Japanese stock portfolios. When adding an additional factor to an existing well-ﬁtting model (the Fama and French three-factor model), we need to consider the problem of multicollinearity. For each of the overall stock market portfolios, we calculate the variance inﬂationary factor (VIF) of the PR1YR factor. We obtain a VIF for PR1YR in Carhart’s four-factor model of 2.196 for the United States and 1.070 for the Japanese stock market. Because these two VIFs are much smaller than 10, we conclude that the problem of multicollinearity for Carhart’s four-factor model is relatively unimportant (Lee, 1993). Tables 12 and 13 present the estimated regression coefﬁcients and regression statistics for each of the 30 U.S. portfolios based on Carhart’s four-factor model in the United States. Similarly, Tables 14 and 15 present the estimated regression coefﬁcients and regression statistics for each of the 30 Japanese stock market portfolios. For the United States, Table 12 shows that although the estimated coefﬁcients of intercept terms are not signiﬁcantly different from zero in 14 of the 15 for small-ﬁrm portfolios and the GRS test is insigniﬁcant, the adjusted R-square coefﬁcients are very low, with two being negative. In addition, for small ﬁrms 14 of the 15 market factor coefﬁcients and all 15 PR1YR factor coefﬁcients are not signiﬁcantly differently from zero, indicating Carhart’s four-factor model is not suitable for pricing small-ﬁrm

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Four-Factor Carhart Monthly Regressions: U.S. Small-Firm Estimates.

Table 12. O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.009 0.009 0.003 0.003 0.016

0.012 0.004 0.001 0.004 0.007

0.923 0.645 -2.255 0.892 0.471

0.006 0.166 0.709 0.005 0.227

0.000 0.019 0.005 0.010 0.002

0.750 0.840 0.280 0.330 1.460

0.201 0.338 0.345 0.938 0.051

0.800 0.620 1.990 0.950 0.460

0.564 0.095 0.091 0.045 0.443

0.026 0.008 0.252 0.480 0.277

0.454 0.154 0.011 0.003 0.350

0.035 0.037 0.126 0.325 0.167

0.559 0.416 0.502 0.551 0.887

2.500 0.470 0.222 0.250 2.230

0.164 0.082 0.011 0.064 0.070

0.112 0.113 0.077 0.130 0.143

0.455 0.400 0.781 0.741 0.145

0.353 0.259 0.310 0.362 0.594

2.970 1.130 0.070 0.020 2.610

0.200 0.400 0.400 1.260 0.060

0.426 0.533 0.048 0.342 0.643

1.450 0.820 0.100 0.700 0.710

0.226 0.595 0.895 0.639 0.429

0.968 0.043 0.578 0.202 0.818

0.995 0.825 0.314 0.996 0.779

0.841 0.692 0.687 0.210 0.954

p-value (^s) 2.850 2.500 3.000 3.780 5.170

0.150 0.060 1.840 2.880 1.750

0.014 0.640 0.682 0.806 0.027

0.883 0.955 0.0680 0.005 0.0820

0.005 0.013 0.003 0.000 0.000

^ p-value (h) 2.670 2.300 2.740 3.670 5.120

0.290 0.380 1.350 2.880 1.560

0.003 0.260 0.941 0.982 0.010

^ tðpÞ 0.117 0.045 0.150 0.079 0.111

H

^ p-value (m)

^ tðhÞ

p^ LO 2 3 4 HO

0.040 2.040 0.560 1.280 0.230

tð^sÞ

h^ LO 2 3 4 HO

1.220 0.530 0.130 0.470 0.790

0.010 0.220 1.010 0.010 0.280

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.770 0.707 0.177 0.004 0.120

0.008 0.023 0.007 0.000 0.000

^ p-value (p)

1.270 1.540 1.130 1.570 1.820

1.200 0.550 1.800 1.090 1.300

0.148 0.415 0.921 0.483 0.479

0.206 0.126 0.261 0.119 0.071

0.233 0.586 0.073 0.278 0.195

GRS F(^a) 0.3320

P(F) 0.8023

Adjusted R2 (%) LO 2 3 4 HO

4.18 0.72 0.45 1.05 2.06

1.19 0.24 2.22 4.50 1.84

3.70 1.67 5.25 7.06 13.27

Notes: U.S. small-ﬁrm estimates of the Carhart four-factor model Eq. (2) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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Four-Factor Carhart Monthly Regressions: U.S. Large-Firm Estimates.

Table 13. O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.020 0.006 0.016 0.019 0.007

0.035 0.016 0.007 0.014 0.019

0.403 0.151 0.265 0.655 0.565

0.621 0.261 0.048 0.387 0.607

0.024 0.024 0.017 0.002 0.005

3.120 1.020 3.700 4.010 1.610

0.794 0.507 0.563 0.860 0.378

1.063 0.339 0.200 0.093 0.314

0.370 0.342 0.011 0.154 5.526

0.380 0.150 0.380 0.840 0.770

0.546 0.349 0.393 0.617 0.262

0.766 0.204 0.133 0.087 0.227

0.115 0.412 0.023 0.502 7.198

3.860 2.530 4.070 5.590 2.620

0.006 0.010 0.001 0.006 0.010

0.004 0.003 0.002 0.001 0.011

0.002 0.309 0.000 0.000 0.108

0.540 0.320 0.060 0.550 0.620

4.670 2.110 1.210 0.670 1.640

0.046 0.264 0.025 0.347 4.891

3.910 2.570 4.190 5.920 2.680

4.970 1.870 1.190 0.930 1.750

0.300 0.290 0.010 0.200 1.830

0.701 0.883 0.707 0.404 0.442

0.380 0.710 0.070 0.570 0.970

0.270 0.220 0.140 0.100 0.810

0.000 0.002 0.204 0.002 0.002

0.002 0.001 0.001 0.681 0.809

0.592 0.750 0.954 0.584 0.534

0.766 0.770 0.989 0.842 0.070

p-value (^s) 0.470 1.800 0.150 3.300 12.090

0.000 0.013 0.0001 0.0001 0.010

0.0001 0.037 0.229 0.503 0.103

0.636 0.074 0.883 0.001 0.0001

^ p-value (h) 0.280 1.700 0.240 3.370 12.130

0.000 0.011 0.0001 0.0001 0.008

^ tðpÞ 0.011 0.004 0.001 0.009 0.001

H

^ p-value (m)

^ tðhÞ

p^ LO 2 3 4 HO

3.100 3.270 3.500 0.410 0.240

t(^s)

h^ LO 2 3 4 HO

4.830 3.220 1.270 3.090 3.230

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.0001 0.063 0.237 0.356 0.082

0.782 0.092 0.810 0.001 0.0001

^ p-value (p) 0.610 0.220 0.090 0.820 0.030

0.706 0.479 0.943 0.572 0.333

0.791 0.825 0.888 0.919 0.420

0.540 0.823 0.927 0.414 0.979

Adjusted R2 (%) LO 2 3 4 HO

6.22 1.80 7.35 15.00 28.00

10.43 0.36 1.43 1.35 0.12

0.33 0.33 2.23 4.48 16.41

GRS F(^a) 5.0046

P(F) 0.0024

Notes: U.S. large-ﬁrm estimates of the Carhart four-factor model Eq. (2) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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Table 14.

Four-Factor Carhart Monthly Regressions: Japan Small-Firm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.247 0.458 0.277 0.033 0.033

0.141 0.113 0.458 0.768 0.324

1.750 2.002 1.072 0.570 0.710

0.875 0.931 1.233 0.475 1.134

0.422 0.863 1.611 0.350 1.023

1.321 1.430 1.447 0.105 0.139

0.529 0.867 0.045 0.100 0.250

0.076 0.349 0.213 0.782 0.328

0.812 0.333 2.185 1.273 0.311

5.368 3.587 3.207 1.045 1.694

0.071 0.193 0.706 0.365 0.095

0.078 0.144 0.321 0.258 0.342

0.204 0.572 0.941 0.783 0.033

1.975 1.890 0.164 0.222 0.726

0.398 0.759 0.440 0.016 0.094

0.128 0.225 0.690 1.159 0.500

0.244 0.212 0.208 0.920 0.895

1.241 0.725 0.812 2.179 0.581

0.003 0.016 0.024

4.000 2.710 2.733 1.296 2.517

0.043 0.305 0.516 0.337 0.036

0.282 0.444 2.718 0.860 0.290

0.344 0.151

0.380 1.516 0.426 1.630 0.075

0.105 0.117 0.876 0.833 0.500

1.553 1.729 1.672 0.036 0.285

0.744 0.833 1.946 4.020 1.411

0.311 0.022 0.344 0.344 0.021

0.010 0.042 0.041 0.251 0.053

0.270 0.501 0.453 0.081 0.587

0.691 0.271 0.590 0.049 0.417

0.719 0.190 0.688 0.164 0.943

^ p-value (h) 0.084 0.853 0.247 0.741 0.087

0.789 0.675 0.042 0.429 0.783

^ tðpÞ 0.628 1.335 1.984 0.487 1.431

0.312 0.591 0.137 0.015 0.266

p-value (^s)

0.421 1.237 0.575 2.595 0.885

0.458 0.540 0.917 0.904 0.977

H

^ p-value (m)

^ tðhÞ

p^ LO 2 3 4 HO

1.126 3.274 1.045 1.046 3.332

tð^sÞ

h^ LO 2 3 4 HO

1.125 0.574 1.772 3.653 1.253

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.666 0.612 0.401 0.407 0.373

0.937 0.433 0.815 0.492 0.934

^ p-value (p) 1.221 3.695 0.938 1.059 3.399

0.181 0.144 0.155 0.972 0.787

0.490 0.443 0.109 0.010 0.217

0.276 0.014 0.391 0.338 0.019

GRS F(^a) 5.6315

P(F) 0.0949

Adjusted R2 (%) LO 2 3 4 HO

88.6 63.1 92.9 43.7 35.9

93.6 59.6 44.2 91.6 63.6

77.8 86.6 28.8 71.6 89.4

Notes: Japan small-ﬁrm estimates of the Carhart four-factor model Eq. (2) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

81

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Table 15.

Four-Factor Carhart Monthly Regressions: Japan LargeFirm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.409 0.140 0.119 0.054 0.054

0.145 0.123 0.042 0.276 0.118

0.128 0.981 2.536 0.472 0.515

0.660 0.389 1.142 0.758 0.593

0.528 0.043 0.419 0.195 0.462

1.174 0.799 0.698 0.302 0.595

0.554 0.904 0.324 0.009 0.571

0.624 0.539 0.353 0.912 0.212

2.546 0.281 0.237 3.188 0.576

0.131 2.002 5.314 0.947 2.027

0.643 0.486 0.876 0.057 0.360

0.525 0.350 0.246 0.739 0.289

0.400 0.024 0.183 0.952 0.203

1.023 3.327 1.223 0.033 4.053

0.482 0.091 0.143 0.052 0.151

0.116 0.224 0.020 0.252 0.026

0.293 0.460 0.517 0.775 0.578

0.698 0.700 2.405 0.804 2.153

1.189 1.748 1.341 1.744 1.385

0.495 0.371 0.057 0.403 0.974

1.110 1.673 3.094 0.193 2.392

0.936 1.062 0.874 1.321 1.771

2.231 0.513 0.296 6.373 0.903

0.901 0.102 0.003 0.387 0.098

1.067 0.401 0.646 0.225 1.282

0.264 0.869 0.091 0.577 0.206

0.686 0.564 0.815 0.450 0.283

0.252 0.833 0.204 0.325 0.099

0.516 0.515 0.061 0.458 0.084

0.076 0.630 0.779 0.001 0.408

p-value (^s) 0.631 0.080 0.410 3.431 0.575

0.353 0.021 0.276 0.975 0.010

0.288 0.141 0.238 0.142 0.225

0.556 0.940 0.698 0.019 0.590

^ p-value (h) 0.732 1.143 0.119 1.358 2.573

0.317 0.155 0.027 0.855 0.062

^ tðpÞ 0.801 0.031 0.555 0.200 0.258

H

^ p-value (m)

^ tðhÞ

p^ LO 2 3 4 HO

1.296 0.223 1.461 1.092 2.025

tð^sÞ

h^ LO 2 3 4 HO

0.429 0.617 0.247 0.819 1.203

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.392 0.337 0.422 0.244 0.137

0.497 0.305 0.910 0.233 0.050

^ p-value (p) 1.515 0.122 1.494 0.862 0.871

0.335 0.705 0.547 0.831 0.256

0.802 0.425 0.931 0.589 0.845

0.190 0.908 0.196 0.428 0.423

GRS F(^a) 0.1926

P(F) 0.8953

Adjusted R2 (%) LO 2 3 4 HO

33.2 54.2 85.8 27.8 71.0

5.4 59.0 69.5 12.6 52.1

44.6 1.6 28.7 88.9 62.4

Notes: Japan large-ﬁrm estimates of the Carhart four-factor model Eq. (2) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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stock returns in the U.S. stock market. For U.S. large ﬁrms, in Table 13, ﬁve of the 15 intercepts are not signiﬁcantly different from zero and the GRS test is signiﬁcant indicating that the intercept coefﬁcients are not jointly zero. But adjusted R-square coefﬁcients are much lower than with the threefactor model, with ﬁve being negative and none being greater than 28%. Matching the U.S. small-ﬁrm results, all 15 U.S. large-ﬁrm market factor, MTB, coefﬁcients and all 15 PR1YR factor coefﬁcients are not signiﬁcant, indicating that Carhart’s four-factor model does not price U.S. large-ﬁrm or small-ﬁrm returns very well in our post-1990 sample. Compared to the Fama and French three-factor model, although the Carhart four-factor model has higher explanatory power of portfolio returns in the U.S. stock market for small ﬁrms, if measured solely by examining the lowered number of signiﬁcant intercept coefﬁcients, all coefﬁcients of the prior-one-year return factor, PR1YR, are not signiﬁcantly different from zero for both small and large portfolios of U.S. ﬁrms that are sorted into similar ﬁnancial distress and BE/ME levels, indicating a limitation of the Carhart four-factor model to explain asset pricing in U.S. portfolios containing assets with similar levels of ﬁnancial distress and BE/ME levels. For the Japanese stock market, Table 14 shows that only one coefﬁcient of the size factor (SMB) and one coefﬁcient of the book-to-market-equity factor (HML) for small ﬁrms are signiﬁcantly different from zero. For large ﬁrms, Table 15 shows three of the Japanese size factor coefﬁcients are signiﬁcantly negative and two of the 15 book-to-market-equity factor coefﬁcients are different from zero, one positive and one negative. For Japanese small ﬁrms, Table 14 shows that seven of the 15 market factor (MTB) coefﬁcients are signiﬁcantly different from zero and are positive. For Japanese large ﬁrms, Table 15 illustrates that two of the 15 market factor coefﬁcients are signiﬁcant and are positive. For large ﬁrms in Japan, none of the coefﬁcients of the momentum factor, PR1YR, are signiﬁcantly different from zero. In contrast to the large ﬁrms, three of the coefﬁcients of the momentum factor, PR1YR, are signiﬁcantly different from zero for the small ﬁrms. Like the U.S. evidence, this indicates a limitation of the Carhart four-factor model to explain asset pricing in Japanese portfolios containing assets with similar levels of ﬁnancial distress and BE/ME levels. For the Carhart model, only three of the Japanese small-ﬁrm portfolios have signiﬁcant intercept coefﬁcients compared to four in the three-factor model, indicating a better ﬁt. None of the Japanese larger ﬁrm portfolios have signiﬁcant intercept coefﬁcients compared to ﬁve signiﬁcant intercepts

O-Score Financial Distress Risk Asset Pricing

83

in the three-factor model, also indicating a better ﬁt. Further evidence that both models ﬁt according to the GRS test is that none of the GRS tests are signiﬁcant for either the three-factor model or the four-factor Carhart model. Finally, the adjusted R-square statistic increases for 10 of the 15 small Japanese ﬁrms but is lower for 12 of the 15 large Japanese ﬁrm portfolios in the Carhart model compared to the three-factor model. Overall, although Carhart’s four-factor model has higher explanatory power of portfolio returns compared to the three-factor model in the Japanese stock market, as measured by the lowered number of signiﬁcant intercept terms, the PR1YR factor is not signiﬁcantly different from zero in 27 of 30 regressions, indicating that Carhart’s four-factor momentum model is not a suitable model for pricing stock returns in the Japanese stock market when stocks have similar levels of ﬁnancial distress and BE/ME.

3.6. Financial Distress Four-Factor Model – Monthly Data Tests Some previous studies, such as Dichev (1998), suggest that O-score can be a very good proxy for ﬁnancial distress. Grifﬁn and Lemmon (2002) argue that O-score captures some relevant information that the BE/ME factor (HML) does not capture about stock returns. Vassalou and Xing (2004) ﬁnd that the size and BE/ME factors do not signiﬁcantly capture default risk of ﬁrms. We add an O-score-based ﬁnancial distress factor into Fama and French’s three-factor model as the fourth factor in a four-factor ﬁnancial asset pricing model. This O-score ﬁnancial distress four-factor model is labeled Eq. (3). Our model consists of a market factor (MTB), a ﬁrm size factor (SMB), a BE/ME factor (HML), and the ﬁnancial distress proxy factor, denoted by OLMH, which is the difference between portfolio average returns of the top ﬁnancial distress quintile and the bottom ﬁnancial distress quintile. To consider the problem of multicollinearity caused by the introduction of an additional factor into a previously well-speciﬁed model, we calculate the VIF of the O-score factor in our U.S. sample and in our Japanese sample. We ﬁnd an O-score factor VIF of 1.906 in the U.S. stock market and 1.375 in the Japanese stock market, respectively. Because the two VIF factors are much smaller than 10, we conclude that the problem of multicollinearity for the regressions run for our O-score ﬁnancial distress four-factor model is relatively unimportant (Lee, 1993).

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In Tables 16–19, we show the estimated regression coefﬁcients and regression statistics for the portfolios in each of our 30 United States and 30 Japanese BE/ME, O-score, and size groups for the four-factor model. Tables 16 and 17 show that most estimated coefﬁcients, for the market-tobook factor (MTB), the size factor (SMB), the BE/ME factor (HML) and the O-score factor (OLMH) for large and small ﬁrms in the United States, are signiﬁcantly different from zero. Speciﬁcally, 44 of the 60 U.S. small ﬁrms’ factor coefﬁcients, and 44 of the 60 U.S. large ﬁrms’ factor coefﬁcients are statistically signiﬁcantly different from zero at the 5% signiﬁcance level. Furthermore, we ﬁnd that for both large and small high-BE/ME ﬁrms (value stocks) and for small low-BE/ME stocks (small, growth stocks), the coefﬁcient of the O-score factor (OLMH) decreases with the O-score from low to high in the United States, conﬁrming the distress risk puzzle. This evidence is consistent with the U.S. buy-and-hold returns evidence presented in Table 6. Compared to the Fama and French three-factor model’s results (Table 8 and Table 9), the adjusted R-square statistics increase for small and large ﬁrms, especially for small ﬁrms (all adjusted R-squares rise for small ﬁrms), after the O-score factor is added in the U.S. stock market. The improvement in adjusted R-squares is noticeable for all ﬁve O-score quintile portfolios for value stocks (high BE/ME), improving up to 29% for small ﬁrms (all value stocks had increased adjusted R-squares) and by up to 33% for large ﬁrms (two portfolios had slight decreases in adjusted R-squares). Although the O-score factor coefﬁcients are signiﬁcantly different from zero for 20 of 30 U.S. portfolios, 20 of the 30 U.S. intercept coefﬁcients are still signiﬁcantly different from zero, which is the same number of signiﬁcant intercept coefﬁcients as the Fama and French three-factor model. Furthermore, the GRS tests for both small ﬁrms and large ﬁrms continue to be signiﬁcant for the O-score four-factor model formulation compared to the threefactor model results, indicating rejection of the null hypothesis of the joint test that each intercept for the 15 portfolios of small ﬁrms equals zero. This implies that the ﬁnancial distress four-factor risk model can be further improved to price stock returns in the U.S. stock market, in future research. Overall, we conclude that our O-score ﬁnancial distress four-factor model has higher explanatory power than the Fama and French three-factor model in the U.S. stock market to explain stock market portfolio returns. Taken together, we ﬁnd that our O-score ﬁnancial distress four-factor model might be more adequate than the three-factor model in explaining the crosssectional structure of U.S. stock returns.

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Four-Factor Financial Distress Monthly Regressions: U.S. Small-Firm Estimates.

Table 16. O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.020 0.016 0.000 0.009 0.024

0.018 0.014 0.007 0.005 0.020

1.323 0.821 2.270 1.020 0.363

0.311 0.403 0.882 0.300 0.055

0.005 0.020 0.016 0.016 0.010

3.000 2.580 0.040 1.660 4.020

0.568 0.160 0.010 0.015 0.324

10.810 11.190 9.940 10.890 12.370

0.889 0.149 0.133 0.067 0.246

0.387 0.057 0.177 0.276 0.302

0.021 0.060 0.268 0.001 0.552

0.419 0.061 0.240 0.607 0.195

0.035 0.096 0.293 0.736 0.683

0.450 0.650 0.540 0.330 1.110

0.400 0.081 0.248 0.009 0.203

0.427 0.082 0.102 0.258 0.009

11.360 12.200 12.010 13.020 11.620

0.000 0.000 0.000 0.000 0.000

1.073 0.632 0.367 0.601 0.867

0.110 0.330 1.340 0.000 3.070

3.020 0.670 1.890 0.080 1.720

0.001 0.000 0.092 0.310 0.000

0.389 0.000 0.002 0.001 0.056

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

p-value (^s)

4.980 4.660 5.150 5.490 5.680

4.830 4.530 5.570 6.270 7.570

0.692 0.514 0.592 0.745 0.270

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

^ p-value (h)

4.710 4.450 5.920 4.050 4.340

6.660 4.300 4.070 4.590 5.600

0.917 0.745 0.182 0.998 0.003

6.480 3.740 2.770 2.590 2.490

0.003

^ tðoÞ 0.684 0.360 0.273 0.222 0.252

H

^ p-value (m)

^ tðhÞ

o^ LO 2 3 4 HO

0.003 0.011 0.968 0.099 0.000

tð^sÞ

h^ LO 2 3 4 HO

0.860 3.960 3.140 3.480 1.930

3.450 3.040 1.690 1.020 4.040

11.630 11.450 10.720 11.640 11.930

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

^ p-value (o)

4.210 0.930 1.240 2.620 0.100

0.502 0.060 0.932 0.087

0.000 0.087 0.215 0.010 0.923

0.000 0.000 0.006 0.010 0.014

GRS F(^a) 3.0882

P(F) 0.0286

Adjusted R2 (%) LO 2 3 4 HO

47.88 40.62 70.24 31.31 53.47

57.30 61.08 62.37 66.82 50.02

72.17 68.67 70.76 59.84 61.53

Notes: U.S. small-ﬁrm estimates of the O-score four-factor model Eq. (3) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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Table 17.

Four-Factor Financial Distress Monthly Regressions: U.S. Large-Firm Estimates.

O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.016 0.003 0.015 0.018 0.007

0.028 0.013 0.003 0.012 0.021

0.556 0.346 0.248 0.611 0.436

0.721 0.373 0.041 0.353 0.515

0.012 0.020 0.014 0.000 0.018

2.660 0.450 3.620 3.890 1.510

0.682 0.217 0.064 0.288 5.097

14.430 10.640 11.360 14.210 15.610

1.277 1.008 0.640 0.879 0.578

1.662 0.733 0.148 0.260 0.312

0.000 0.214 0.309 0.594 0.041

0.083 0.242 0.529 0.101 0.237

1.046 0.846 0.332 0.232 5.144

5.770 4.690 4.080 5.030 3.580

0.531 0.550 0.083 0.021 0.217

0.662 0.433 0.384 0.183 0.006

14.260 12.200 14.910 9.610 13.060

0.000 0.000 0.000 0.000 0.000

6.910 4.260 0.820 1.670 1.430

1.378 0.229 0.330 0.647 2.535

0.000 1.230 2.430 4.200 0.320

4.530 4.830 1.000 0.230 2.540

5.050 3.370 1.960 1.380 8.680

0.030 0.005 0.004 0.935 0.278

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.411 0.096 0.154

0.000 0.001 0.051 0.169 0.000

^ p-value (h) 8.230 1.130 2.420 4.770 5.300

1.000 0.219 0.016 0.000 0.752

11.740 3.600 3.840 3.310 7.260

0.000 0.000

0.420 1.740 3.640 0.800 1.350

5.200 4.760 4.030 2.220 0.050

0.000 0.008 0.511 0.006 0.001

p-value (^s)

^ tðoÞ 1.288 0.479 0.343 0.294 2.277

H

^ p-value (m)

^ tðhÞ

o^ LO 2 3 4 HO

0.009 0.654 0.000 0.000 0.134

tð^sÞ

h^ LO 2 3 4 HO

2.190 2.870 2.950 0.080 1.090

4.270 2.700 0.660 2.780 3.540

11.330 14.510 14.950 13.510 15.530

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.672 0.084 0.000 0.424 0.180

0.000 0.260 0.017 0.000 0.000

^ p-value (o)

0.320 0.818 0.012

0.000 0.000 0.000 0.028 0.957

0.000 0.000 0.000 0.001 0.000

GRS F(^a) 6.2963

P(F) 0.0004

Adjusted R2 (%) LO 2 3 4 HO

56.04 73.17 47.92 54.88 60.57

52.43 59.80 60.72 51.45 50.22

42.98 45.90 47.72 92.10 96.90

Notes: U.S. large-ﬁrm estimates of the O-score four-factor model Eq. (3) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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Four-Factor Financial Distress Monthly Regressions: Japan Small-Firm Estimates.

Table 18. O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.013 0.063 0.041 0.040 0.032

0.064 0.057 0.033 0.043 0.055

1.499 1.128 1.519 1.735 1.883

1.045 0.720 0.510 0.880 0.668

0.063 0.082 0.449 0.044 0.134

0.112 0.445 0.615 0.327 0.210

0.238 0.641 0.179 0.762 0.132

0.216 0.204 0.101 1.392 0.672

1.399 1.204 0.526 0.641 0.695

2.458 1.829 3.322 2.655 2.417

0.143 0.516 0.619 0.114 0.215

0.233 0.022 0.283 0.097 0.635

0.166 0.759 1.611 0.347 0.145

1.906 1.693 1.784 1.971 1.683

0.094 0.197 0.109 0.127 0.240

0.168 0.142 0.062 0.635 0.339

0.911 0.657 0.539 0.744 0.834

3.663 2.219 0.459 1.805 2.729

0.015

3.816 2.475 1.327 2.395 1.954

2.444 1.742 1.828 2.239 2.583

0.251 0.165 3.289 0.114 0.481

1.989 1.862 1.796 1.772 2.286

0.070 0.001 0.009 0.017

2.016 2.183 1.742 1.683 3.030

1.693 2.254 2.098 2.358 2.196

0.223 0.484 0.190 0.599 0.028

0.000 0.015 0.187 0.018 0.053

0.000 0.028 0.647 0.074 0.007

0.059 0.093 0.077 0.051 0.095

0.016 0.084 0.070 0.027 0.011

0.093 0.026 0.038 0.020 0.030

^ p-value (h) 2.494 2.444 2.133 2.303 2.098

0.049 0.065 0.075 0.079 0.024

1.663 2.583 1.963 0.814 1.914

0.046 0.031

1.828 1.971 1.482 1.688 2.358

2.286 2.045 1.809 1.802 2.087

0.087 0.341 0.731 0.566 0.480

p-value (^s)

^ tðoÞ 0.369 0.051 3.024 0.278 0.432

H

^ p-value (m)

^ tðhÞ

o^ LO 2 3 4 HO

1.226 0.703 1.318 0.527 2.222

tð^sÞ

h^ LO 2 3 4 HO

1.728 0.956 0.344 0.576 0.709

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.070 0.051 0.141 0.094 0.020

0.014 0.016 0.035 0.023 0.038

^ p-value (o)

0.084 0.095 0.003

0.024 0.043 0.073 0.074 0.000

0.099 0.011 0.052 0.417 0.058

GRS F(^a) 0.2894

P(F) 0.8322

Adjusted R2 (%) LO 2 3 4 HO

71.5 51.3 89.4 64.8 65.9

94.9 57.7 2.50 82.8 59.8

90.7 41.2 33.6 64.3 70.4

Notes: Japan small-ﬁrm estimates of the O-score four-factor model Eq. (3) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

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Four-Factor Financial Distress Monthly Regressions: Japan Large-Firm Estimates.

Table 19. O-Score Portfolios

BE/ME Portfolios L

M

H

L

M

a^ LO 2 3 4 HO

0.024 0.065 0.013 0.091 0.051

0.097 0.070 0.029 0.097 0.102

0.963 0.830 2.285 0.391 0.745

0.515 0.767 1.113 0.369 0.554

0.157 0.005 0.050 0.026 0.245

0.367 1.032 0.290 1.353 1.330

0.303 0.835 0.019 0.040 0.428

0.093 0.263 0.317 0.731 0.261

1.193 0.345 1.068 2.853 1.008

1.982 1.923 3.705 1.024 2.537

0.062 0.433 1.128 0.088 0.226

0.060 0.128 0.214 0.564 0.334

1.360 0.224 0.551 0.737 0.467

1.790 1.862 2.016 2.444 1.699

0.939 0.094 0.323 0.009 0.073

0.032 0.327 0.025 0.071 0.735

0.714 0.304 0.773 0.179 0.186

0.811 1.788 2.374 0.519 2.163

1.731 1.668 1.663 2.007 2.007

0.275 0.541 0.729 0.231 1.183

1.778 1.855 2.697 2.468 2.583

1.792 0.891 1.532 3.720 1.904

0.057 0.000 0.308 0.012

2.551 1.737 2.468 1.842 1.709

1.899 1.802 2.551 1.754 2.157

0.142 0.941 0.633 0.650 0.017

0.419 0.076 0.019 0.605 0.033

0.076 0.374 0.128 0.000 0.059

0.076 0.065 0.046 0.016 0.092

0.086 0.000 0.099 0.047 0.047

0.060 0.074 0.012 0.082 0.033

^ p-value (h)

2.270 2.157 1.954 1.989 1.460

2.655 2.860 2.551 1.790 1.709

0.373 0.285 0.619 0.429 0.024

p-value (^s)

2.098 1.294 1.760 2.444 2.087

0.078 0.066 0.008 0.015 0.011

2.321 1.989 1.731 1.877 1.772

0.012

^ tðoÞ 1.145 0.190 0.456 0.264 0.328

H

^ p-value (m) 0.050

^ tðhÞ

o^ LO 2 3 4 HO

1.478 0.074 0.478 0.455 2.412

tð^sÞ

h^ LO 2 3 4 HO

0.894 1.073 0.499 0.794 2.284

M p-value (^a)

^ tðmÞ

s^ LO 2 3 4 HO

L

tð^aÞ

m^ LO 2 3 4 HO

H

0.025 0.033 0.053 0.049 0.147

0.038 0.198 0.081 0.016 0.039

^ p-value (o)

0.085 0.015 0.068 0.090

0.009 0.005 0.012 0.076 0.090

0.022 0.049 0.086 0.063 0.079

GRS F(^a) 0.4528

P(F) 0.7340

Adjusted R2 (%) LO 2 3 4 HO

79.3 54.5 91.1 1.30 63.9

14.9 66.9 69.5 67.3 52.5

74.5 10.2 19.8 90.3 66.6

Notes: Japan large-ﬁrm estimates of the O-score four-factor model Eq. (3) for portfolios stratiﬁed by ﬁnancial distress level (O-score) and book-to-market-equity (BE/ME). Portfolios and variables are described in Table 1. Statistical signiﬁcance at the 0.05 level separately.

O-Score Financial Distress Risk Asset Pricing

89

In Japan, Tables 18 and 19 also tell us that the O-score factor plays a key role in explaining the returns of each of the 30 portfolios in the Japanese stock market since the O-score coefﬁcients for 14 of 30 portfolios are signiﬁcantly different from zero. Relative to the Fama and French threefactor model, the adjusted R-square statistics results are mixed, with increases for seven of the 15 small ﬁrms and nine of the 15 large ﬁrms. The increase effect is noticeable for value stocks, increasing for eight of the 10 value portfolios by up to 42% (decreases are 4.9% for large ﬁrms and 9.7% for small ﬁrms, both in the second lowest O-score quintile). Three of 30 intercept coefﬁcients for the O-score ﬁnancial distress four-factor model are signiﬁcantly different from zero and neither GRS test is signiﬁcant, indicating a good ﬁt of the four-factor model for both small and large Japanese ﬁrms. This compares to 9 signiﬁcant intercept terms of the 30 intercept terms in the Fama and French three-factor model which also had GRS tests that are not signiﬁcant. Taken together, the increased R-squares (weak evidence), the signiﬁcant O-score coefﬁcients (stronger evidence), and the decreased number of signiﬁcant intercept terms (stronger evidence) lead us to believe that the O-score ﬁnancial distress four-factor model adds value to the Fama and French three-factor model in pricing stock returns in the Japanese stock market.

4. CONCLUSIONS We use a direct proxy of the likelihood of ﬁnancial distress, developed by Ohlson (1980) and denoted by ‘‘O-score’’ to examine the relationships among BE/ME, distress risk, and stock returns in the U.S. and Japanese stock markets using 1991–2006 Datastream data. We build a four-factor pricing model using the Fama and French (1993) three-factor model plus a ﬁnancial distress factor (O-score). We compare the ﬁt of our O-score ﬁnancial distress four-factor model with the Fama and French three-factor model and with Carhart’s (1997) momentum-based four-factor model with monthly data. Buy-and-hold empirical results show that, in the United States, stocks with high BE/ME (value stocks) have higher returns than the stocks with low BE/ME (growth stocks) within the same O-score quintile for all ﬁrms. For the United States, we also ﬁnd stocks in the lowest O-score quintile have higher returns than the stocks in the highest O-score quintile for both growth stocks and blended stocks, which is consistent with the ﬁndings of

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Dichev (1998). This ﬁnding is consistent with Grifﬁn and Lemmon (2002) and Vassalou and Xing (2004). Because the Japanese corporate ﬁnance market has many similarities and differences compared to the U.S. market and prior research has not applied the Ohlson (1980) ﬁnancial distress risk measure to Japanese ﬁrms, our analysis of Japanese buy-and-hold returns for ﬁnancial distress quintiles presents new evidence. For Japan, our buy-and-hold results show higher returns for value stocks than for growth stocks. When we compare our Japanese ﬁndings to our U.S. ﬁndings, we ﬁnd that value stocks outperform growth stocks for several quintiles in both countries. A major difference is that high ﬁnancial distress ﬁrm quintile in Japan has a large negative return, whereas the high ﬁnancial distress quintile for the United States has a large positive return. Another major difference is that stock returns for U.S. ﬁrms are all higher than Japanese ﬁrms, regardless of whether it is a value or growth ﬁrm or high or low ﬁnancial distress. A third major difference is that for the value stocks, most portfolios of Japanese ﬁrms exhibit negative average returns while U.S. ﬁrms exhibit positive average returns. According to our long–short buy-and-hold ﬁndings in the United States and Japan, we conclude that value stocks could bring investors a higher return than growth stocks and, furthermore, that going long in value stocks and going short in growth stocks, within the highest O-score quintile, would make the highest returns. Additionally, those return differentials increase even more with small ﬁrms in the United States (Table 6) and large ﬁrms in Japan (Table 7). To investigate further, we examine monthly time series regressions for the Fama and French three-factor model and two competing four-factor models. We ﬁnd that the Carhart momentum factor model does not ﬁt well when compared to the Fama and French three-factor model in our 30 portfolios that hold ﬁnancial distress risk and BE/ME constant, in either the U.S. stock market or the Japanese stock market. We ﬁnd that our O-score ﬁnancial distress four-factor model explains portfolio stock returns more completely than Fama and French’s (1993) three-factor model in both the U.S. and Japanese stock markets in our 30 portfolios that hold constant ﬁnancial distress risk and BE/ME. We ﬁnd (1) fewer intercept coefﬁcients that are signiﬁcantly different from zero, (2) the adjusted R-square becomes higher, and (3) signiﬁcant O-score factor coefﬁcients. These three ﬁndings imply that a ﬁrm’s O-score captures relevant pricing information that the three original Fama and French factors do not incorporate. We attribute the differences in our results across the two nations to differences in U.S. and Japanese corporate governance and ﬁnancial

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91

practices, political and cultural differences and economic performance. Higher Japanese ﬁnancial leverage for value ﬁrms and lower Japanese ﬁnancial leverage for growth ﬁrms shown by comparing Table 3 (Japan) to Table 2 (U.S.) has an impact on the Japanese-speciﬁc parameter estimates of the Ohlson (1980) ﬁnancial distress metric that are constructed from ﬁnancial accounting data. U.S. ﬁrms also have higher average returns on assets than Japanese ﬁrms except in the highest ﬁnancial distress quintile. We believe that the country-speciﬁc parameter estimates for the Ohlson (1980) ﬁnancial distress measure captures most cultural, social, and economic ﬁrm pricing differences for these two countries, allowing a better set of estimates of the factors generating stock prices beyond existing models.

NOTES 1. Daniel and Titman (1997) model the return generating process with three models arguing for a characteristic-based model. 2. Von Kalckreuth (2005) presents a potential explanation for Grifﬁn and Lemmon (2002)’s ﬁndings as a ‘‘wreckers theory’’ of ﬁnancial distress. By contrast, Chen and Zhang (2008) take a different approach than the literature reviewed here. They offer an explanation for three-factor models and both momentum and ﬁnancial distress anomalies with a neoclassical approach. This alternative approach explains stock returns as outcomes of economic processes rather than explaining the stock return generation process as being the result of the market, book to market, and size factors found in the Fama and French (1993) literature strand. Ferguson and Shockley (2003) examine ﬁnancial distress risk’s role in pricing equity securities by creating portfolios based on relative leverage and relative ﬁnancial distress, testing them in the U.S. equity market. They use debt to equity to measure relative ﬁnancial leverage and Altman’s (1968) Z to measure relative ﬁnancial distress. They ﬁnd that debt to equity and Altman’s Z are important time series variables when added to the Fama and French factors and the single-factor CAPM model. 3. Grifﬁn and Lemmon (2002) examine high BE/ME constructed portfolios (value) versus low BE/ME constructed portfolios (growth) by using Fama and French’s three-factor model within sorted deciles of Ohlson’s (1980) bankruptcy risk proxy for ﬁrm distress (O-score), building on Dichev’s (1998) work contrasting the ability of Ohlson’s (1980) measure and Altman’s (1968) measure to predict ﬁnancial distress. 4. Firms that cease trading or are merged are not excluded. If a ﬁrm ceases trading, its stock price on the date that it resumes trading on the same exchange or another exchange or over the counter will be used. But if a ﬁrm is permanently delisted or removed from the counter, the ﬁnal stock price will be taken as zero. A merged company will be excluded if the merger’s details are not veriﬁable. We follow the same rules for our Datastream data as Fama and French (1993) and Grifﬁn and Lemmon (2002) who use CRSP data from 1965 to 1996. Of course, Japanese data are not extensive for years before our data sample.

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5. Theoretically, the book value of equity of a ﬁrm is primarily based on historical costs and thus does not reﬂect the future value of the ﬁrm. In contrast, the market value of equity does reﬂect the ﬁrm’s future. 6. Grifﬁn and Lemmon (2002) do not modify the O-score parameter estimates, even though they use July 1965 to June 1996 data which overlaps Ohlson’s 1970–1976 data period. 7. Where appropriate in our later tests, we compare our test results for O-scores calculated for Japanese ﬁrms using the U.S. parameters to O-scores calculated using the logistic model parameters estimated in this section for Japan. In our tests, factor models for Japanese data conducted with O-scores calculated with U.S. O-score parameters do not ﬁt as well as factor models for Japanese data conducted with the O-score (Japan) parameters. 8. Grifﬁn and Lemmon (2002) follow the approach taken in Fama and French (1993, 1998). Because Grifﬁn and Lemmon’s data sample covers U.S. data from 1965 to 1996 and our sample of 1991 to 2006 includes 10 years of data after that time period, we present our U.S. information that is similar to Grifﬁn and Lemmon’s Table II and Table III. 9. Similar to Grifﬁn and Lemmon’s Table I. 10. Similar to Grifﬁn and Lemmon’s Table II. 11. Similar to Grifﬁn and Lemmon’s Table III.

ACKNOWLEDGMENTS The authors thank Chunchi Wu, Dave Davidson and session participants at the 2009 American Accounting Association National Meeting for comments and suggestions. Jeng-Luen Tsay and Rih-tai Jian provided valuable assistance with data and data analysis. Jessica Conover provided valuable editing assistance. All errors that remain are due to the authors.

REFERENCES Altman, E. I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance, 23, 589–609. Arshanapalli, B., Coggin, T. D., & Doukas, J. (1998a). Multifactor asset pricing analysis of international investment strategies. Journal of Portfolio Management (Summer), 10–23. Arshanapalli, B., Coggin, T. D., Doukas, J., & David, H. S. (1998b). The dimension of international equity style. Journal of Investing, 15–30. Campbell, J. Y., Hilscher, J., & Szilagyi, J. (2008). In search of distress risk. Journal of Finance, 63, 2899–2939. Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance, 52, 57–82.

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Chan, K. C., & Chen, N. F. (1991a). Structural and return characteristics of small and large ﬁrms. Journal of Finance, 46, 1467–1485. Chan, K. C., Chen, N.-F., & Hsieh, D. (1985). An exploratory investigation of the ﬁrm size effect. Journal of Financial Economics, 14, 451–471. Chan, L. K. C., Hamao, Y., & Lakonishok, J. (1991b). Fundamentals and stock returns in Japan. Journal of Finance, 46, 1739–1764. Chan, L. K. C., Jegadeesh, N., & Lakonishok, J. (1995). Evaluating the performance of value versus glamour stocks: The impact of selection bias. Journal of Financial Economics, 38, 269–296. Chan, L. K. C., & Lakonishok, J. (2004). Value and growth investing: Review and update. Financial Analysts Journal, 60, 71–86. Chen, L., & Zhang, L. (2008). Neoclassical factors. Working Paper. Department of Finance, University of Michigan. Chen, N.-F., & Zhang, F. (1998). Risk and return of value stocks. Journal of Business, 71, 501–555. Daniel, K., & Titman, S. (1997). Evidence on the characteristics of cross sectional variation in stock returns. Journal of Finance, 52, 1–33. Dichev, I. D. (1998). Is the risk of bankruptcy a systematic risk? Journal of Finance, 53, 1131–1147. Fama, E. F., & French, K. R. (1992). The cross-section of expected stock returns. Journal of Finance, 47, 427–465. Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33, 3–56. Fama, E. F., & French, K. R. (1996a). Multifactor explanations of asset pricing anomalies. Journal of Finance, 51, 55–84. Fama, E. F., & French, K. R. (1996b). The CAPM is wanted, dead or alive. Journal of Finance, 51, 1947–1958. Fama, E. F., & French, K. R. (1998). Value versus growth: The international evidence. Journal of Finance, 53, 1975–1999. Fama, E. F., & French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of Economic Perspectives, 18, 25–47. Fama, E. F., & French, K. R. (2008). Average returns, B/M, and share issues. Journal of Finance, 63, 2971–2995. Ferguson, M. F., & Shockley, R. L. (2003). Equilibrium ‘‘anomalies’’. Journal of Finance, 58, 2549–2580. Gaunt, C. (2004). Size and book to market effects and the Fama-French three factor asset pricing model: Evidence from the Australian stock market. Accounting and Finance, 44, 27–44. Gibbons, M. R., Ross, S. A., & Shanken, J. (1989). A test of the efﬁciency of a given portfolio. Econometrica, 57, 1121–1152. Grifﬁn, J. M., & Lemmon, M. L. (2002). Book-to-market equity, distress risk, and stock returns. Journal of Finance, 57, 2317–2336. Halliwell, J., Heaney, J., & Sawicki, J. (1999). Size and book-to-market effects in Australian share markets: A time series analysis. Accounting Research Journal, 12, 122–137. Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efﬁciency. Journal of Finance, 48, 65–91.

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Lakonishok, J., Shleifer, A., & Vishny, R. W. (1994). Contrarian investment, extrapolation, and risk. Journal of Finance, 49, 1541–1578. Lau, S. T., Lee, C. T., & McInish, T. H. (2002). Stock returns and beta, ﬁrms size, E/P, CF/P, book-to-market, and sales growth: evidence from Singapore and Malaysia. Journal of Multinational Financial Management, 12, 207–222. Lee, C. F. (1993). Statistics for business and ﬁnancial economics. Lexington, MA/Toronto: DC. Heath and Company. Liew, J., & Vassalou, M. (2000). Can book-to-market size and momentum be risk factors? Journal of Financial Economics, 57, 221–245. Ohlson, J. A. (1980). Financial ratios and the probabilistic prediction of bankruptcy. Journal of Accounting Research, 18, 109–131. Vassalou, M., & Xing, Y. (2004). Default risk in equity returns. Journal of Finance, 59, 831–867. Von Kalckreuth, U. (2005). A ‘‘wreckers theory’’ of ﬁnancial distress. Discussion Paper. Series 1: Economic Studies No 40/2005, Deutsche Bundesbank, Frankfurt, Germany.

THE LONG-TERM RELATIONS UNDER CLIMATE CHANGE BETWEEN ECONOMIC ACTIVITY AND METAL UTILIZATIONS USING THE FORGETTING FACTOR Andrew H. Chen, Jack Penm and R. D. Terrell ABSTRACT In this chapter, we apply an efﬁcient subset of vector error correction model (VECM) using the forgetting factor to examine the cointegration under climate change of the time series of the gross domestic product (GDP) and the industrial production and that of the utilization and consumption of important metals such as copper and steel in some important OECD countries as well as some selected newly industrialized Asian and Latin American countries. Both the long-term and the shortterm dynamic relations among these variables are examined and the implications are discussed.

1. INTRODUCTION Recently, there has been renewed interest among economists in the utilization and consumption of important metals such as copper and steel Research in Finance, Volume 26, 95–111 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026007

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in major world economies (see Penm & Terrell, 2003). It has been well recognized that the consumptions of copper and steel have been closely associated with the general economic activity in a nation. This stems from the belief that a growth in a nation’s economic activity stimulates an increase in the demand for consumptions of these important metals. Furthermore, the level of utilization of these metals has often been considered as an indicator of a nation’s stages of industrialization and its development of services and technology management. In recent years, the stability of the relationship between the levels of the metal utilization and that of the general economic activity has received considerable attention. This interest has been primarily in response to a relative slowdown in the growth of the level of utilization of these metals in some consuming countries, despite of their continuing growth in the general economic activity. Several factors may have contributed to a possible change in the relationship between general economic activity and utilization of these important metals. These include the substitution of materials in manufacturing, a more efﬁcient use of metals, and changing consumer preferences. This chapter tests two hypotheses. The ﬁrst hypothesis is whether there are stable long-term relationships between the levels of economic activity, and copper and steel consumption, in selected consuming countries all of which are experiencing climate change. The absence of stable long-term relationships would indicate that, over a long period, the levels of copper and steel consumption are less dependent on the level of general economic activity. If stable long-term relationships exist among general economic activity and copper and steel consumption, then, after temporary deviations in the short term, consumption of copper and steel would revert to their traditional long-term relationships with general economic activity. The second hypothesis is whether the levels of economic activity under climate change lead to a change in the same direction in copper and steel consumption, if comovements in the same direction exist in cointegrating relationships. Such identiﬁed ‘‘uni-directional’’ comovements indicate increased/decreased copper and steel consumption from stronger/weaker economic growth. Identiﬁcation of such long-term relationships and unidirectional comovements, if they exist, would be essential in forecasting consumption of steel and copper. Powerful computing equipment has had a dramatic impact on mineral resources modeling and simulations and has motivated development of innovative computing-intensive time series approaches to ﬁnancial services and to trade assessment. Signiﬁcant advances in powerful computing equipment provide faster computational speed, larger amounts of memory,

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and more accurate numerical results, than traditional computing. New time series methodology has speciﬁed models in a more sophisticated manner, used the data in highly adaptive ways, and facilitated major innovations in development of management in mineral resources and technology management. Against that background, increasingly sophisticated resource industryoriented approaches to development of resource services and technology management are now of central importance, driven by applications of increasing electronic scale and complexity. As a result, development of these approaches becomes essential, providing resource ﬁnance managers with a window of opportunity to make a signiﬁcant contribution to the frontier of industry-oriented and academic research. In this chapter, we undertake research in ﬁnancial resource modeling and data analysis using new and important time series approaches. While trade and environmental investment decisions still operate in an uncertain market, and human judgment can never be fully replaced, quantitative time series analysis has an important role to play in guiding effective decisions and setting trade strategy. We construct and utilize subset vector error correction model (VECM; see Penm & Terrell, 2003) as a basis for the cointegration test. The unbiased VECM estimation approach is asymptotically equivalent to the maximum likelihood estimation. We also include a forgetting factor in the estimation of the patterned VECMs. The forgetting factor technique is a data weighting process that allows the estimation to place greater weight on more recent observations and less weight on earlier data. In such estimation, the effects on the underlying relationships of slow evolution generated by the causal linkage process will be accounted for. As to why a subset time series model is used, subset modeling includes full-order models, and researchers use this approach whenever measurements exhibit some periodicity. If the underlying true time series process has a subset structure, the suboptimal full-order speciﬁcation can give rise to inefﬁcient estimates and inferior projections. Our forgetting factor approach improves the estimated parameter proﬁle, model structure and performance reliability for assessing complex relationships involving slowly evolving long-term effects, such as climate change. These qualities are not found in conventional time series approaches involving only full-order models. Subset modeling is superior to full-order modeling for discovering complex relationships, as has been clearly indicated in Penm and Terrell (2003), Chen, Penm and Terrell (2006), and Penm (2007). After adopting the hypothesis of long-term cointegration, the VECM shows the short-term dynamic relationships among those variables involved

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in selected important OECD and selected newly industrialized Asian and Latin American, countries. For the OECD region, a cointegration test is undertaken for the United States, Japan, and the European Union (EU) initial 15-member countries. Argentina, Mexico, and Brazil are involved in the test for the Latin American region for both copper and steel. For the Asian region South Korea, Taiwan and India are included in the test for both copper and steel, and Indonesia is added for the test on steel. The remainder of this chapter is constructed as follows. In Section 2, we outline warnings about climate change identiﬁed in the Stern Review. In Section 3, we discuss Australia’s copper, iron ore, and steel exports. In Section 4, we outline the construction of patterned VECM, which demonstrates the ‘‘presence and absence’’ restrictions on the coefﬁcients of subset time series systems, including full-order systems. Also, brief descriptions of the forgetting factor techniques used for estimation are given. In Section 5, we brieﬂy describe data sources for testing purposes. We then detail the methodology of cointegration and present the estimation results in Section 6. In Section 7, a summary is given.

2. WARNINGS FROM THE STERN REVIEW The Stern Review (Stern, 2006) indicates that the potential impact of climate change could create risks of major disruption to economic and social activity and suggests the resulting climate change will produce about US$7 trillion economic and environmental loss. Severe climate change will make world climate conditions harsher and render drought, storms, cyclones, heat waves, ﬂoods, and tsunamis more likely to occur in numerous areas of the globe. The direct economic loss from natural disasters is doubling every 10 years. The insurance and reinsurance sector has an inherent exposure to the direct effects of climate change. As suggested in the International Panel on Climate Change (IPCC) 2000 Special Report on Emission Scenarios, weather-related events of all magnitudes resulted in about US$710 billion in insured and uninsured economic losses between 1985 and 1999. Climate change–related risks are increasingly considered for speciﬁc ‘‘susceptible’’ sectors, such as hydroelectric and mineral projects, and irrigation, agriculture, metal, and tourism sectors. Also, the lost hydropower production would be US$2.75 billion per annum by 2060. Most recently, scientists have been able to construct evidence of climate change from collected information on temperature, rainfall, and other weather variables from measuring stations all over the world, including the

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most important metal trade countries of Australia including United States, Japan, the initial 15-member countries of the EU, Korea, Taiwan, India, Brazil, Mexico, Indonesia, and Argentina. Furthermore, earth orbiting satellites and other technological advances have enabled scientists to examine the big picture, collecting many different types of information about the above countries and their climate on a sophisticated scale. The obvious major reported evidence for climate change is that (i) sea level rose about 17.5 centimeters in the last century, though in the past decade the rate of rise nearly doubled; (ii) levels of carbon dioxide have recently been higher than in the past decade; (iii) global surface air temperatures rose about three-quarters of a degree Celsius in the past century; (iv) the top 650 m of oceans has recently shown warming of about 0.181F; and (v) many species of plants and animals are already responding to global warming, moving to higher elevations. Scientists have predicted climate change impacts in the long run, which include a general rise in surface temperature; changes in seasonal temperature variation and rainfall patterns; variations in soil moisture and water resources; alteration of agricultural climate zones and crop growth periods; and increases in the incidence of severe weather events such as ﬂoods and droughts. Additionally, crop productivity, growth distribution of vegetation, forestry growth patterns, sea levels, and marine production operations are also expected to be negatively impacted, and thus directly threaten one-sixth of the world’s population. Globally, in the absence of policy interventions, the long-run adverse relationship between gross domestic product (GDP) and greenhouse gas (GHG) emissions per head is likely to persist. The Stern Review suggests that global warming could eventually shrink the global economy by 20 percent, although taking immediate action would cost just one percent of global GDP. Utilizing renewable energy resources and reducing GHG emissions have become an internationally popular and discernible trend. All industrialized and developing countries, including the most important metal trade countries of Australia listed above, have endeavored to save oil, gas, and coal consumption as an approach to improve competition and meet environmental objectives in the long run. The Kyoto Protocol is an amendment to the United Nations Framework Convention on Climate Change, which was negotiated in Kyoto, Japan, in December 1997. Concern over global warming had prompted the world’s governments to negotiate the Kyoto Climate Change Treaty, which requires the world’s largest economies to cut their overall emissions of GHG. The protocol assigns mandatory emission limitations for the reduction of GHG emissions to the signatory nations.

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Australia is required to cut emissions below 108 percent of 1990 levels by 2008–2012, with further reductions to be negotiated in future. Furthermore, the protocol includes ‘‘ﬂexible mechanisms’’ that allow Australia to meet its GHG emission limitation by purchasing GHG emission reductions from elsewhere. The United States is the world’s largest emitter of GHGs. The USA’s plans to curb emissions of GHGs must be placed in the context of science and politics. The American Clean Energy and Security Act of 2009 envisages the following cuts in GHG emissions in the energy sector in the United States: Calendar year/Emission allowances (in millions), 2012/4,770; 2015/4,942; 2020/4,873; 2030/3,533; 2035/2,908; 2040/2,284; 2050; and each year thereafter 1,035. Japan is the world’s ﬁfth biggest emitter of GHGs (after the United States, China, India, and Russia), and is the only one of the ﬁve that is under pressure to meet a GHG emissions limit. For its size, Taiwan is a major emitter of GHGs. The EU accounts for around 10 percent of global emissions. The EU has decided to work as a unit to meet its emissions targets as suggested by Kyoto. South Korea is the world’s 10th highest emitter of GHGs and plays a signiﬁcant role in the global climate change arena. GHG emissions and climate change have become important issues of national discussion with a view to reducing GHG emissions. Taiwan produces more carbon dioxide than most developing nations. Having the natural environment of a subtropical island, Taiwan is very vulnerable to the impacts of climate change and is yet to meet its own GHG reduction targets. India is the world’s fourth largest economy and ﬁfth largest GHG emitter. India has a number of policies that contribute to climate mitigation by reducing GHG emissions. In some respects, India’s emissions are low compared to those of other major economies. Indonesia is the third largest global emitter of GHGs. The challenge for Indonesia is to create appropriate and effective adaptation and mitigation strategies to meet GHG emission reduction and avoidance targets. Mexico ranks as the 14th largest emitter of GHGs in the world. Mexico has agreed with the World Bank to lead the Latin America and the Caribbean world toward development of lower carbon emissions. Brazil is the world’s eighth largest emitter of GHGs. Yet, it has an unusual emissions proﬁle, with 75 percent emissions resulting from unsustainable land use and intensive deforestation, as the Amazon in Brazil comprises one of the world’s largest forests and ecosystems. Although Argentina makes only a small contribution to world GHG emissions, it is already experiencing the effects of global warming. More research is being undertaken to understand the precise nature of the impact of global warming on Argentina.

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3. AUSTRALIA’S COPPER, IRON ORE, AND STEEL EXPORTS The substantial risks in insurance markets and their increasing complexity clearly require a better understanding of impacts of climate change on energy, mineral and metal resources, and energy commodity price movements, such that the resultant strategy for adaptation to climate change by the insurance industry can be managed more effectively. Speciﬁcally, one of the biggest challenges to a modern natural disaster insurance manager is that of setting out an array of reactive and proactive options across time for reducing overall losses to the community. Not only does our approach have the potential to reduce insurers’ exposure, and in so doing limit the level of claims to encourage managers to adopt disaster mitigation measures, but it also could enable a more selective risk exposure of insurance items, so that good risks are rewarded and poor risks are penalized. Mineral and metal energy resources play an important role in Australia’s economy. The Australian Bureau of Agricultural and Resource Economics (ABARE) 2008 annual report indicates that since 2000, mineral and metal energy resources directly contribute about ﬁve percent of GDP annually in Australia, representing about AU$50 billion in 2008. Around two-thirds of Australian mineral and energy production has been exported since 2000. Each of these energy commodities, in particular copper, iron ore, and steel, recorded substantial increases in both export prices and volumes in that period. Many experts believe that the most cost-effective way to reduce GHG emissions is through increased energy efﬁciency. Copper could play a signiﬁcant role in making the nations of the world more energy efﬁcient. Among the engineering metals, copper is the best conductor of heat and electricity. By using copper instead of less energy efﬁcient materials, more of the electricity generated can be used to reap beneﬁts from the products we use. The increased electrical efﬁciency reduces electrical demand, which in turn reduces the consumption of fossil fuels. Reduced fossil fuel consumption means reduced emissions of GHGs, which in turn reduces society’s impact on climate change. About 70 percent of all copper consumption is used to beneﬁt from copper’s enhanced thermal and electrical energy efﬁciency properties. The price of copper is also an important factor for most copper supply and export countries, including Australia. Before the end of 2003, the world market price of copper was relatively stable. Since 2004, the price began to move upwards considerably. In 2000, Australia ranked as one of the top ﬁve largest producers in the world of mined copper. In 2007, ABARE indicated that Australia’s copper exports reached the level of nearly AU$7 billion.

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The value of world exports of iron and steel increased by 203 percent over the period 1985 through 2002, reaching nearly US$144 billion, and the share in world commodities exports rose by about half a percent, reaching almost 11 percent in 2002. Australia has become one of the world’s largest export countries for iron ore and steel. In 2007, ABARE indicated that Australia’s iron ore and steel exports reached nearly AU$11 billion. The climate change challenge has induced changes in the production, distribution, and consumption patterns of metal energy resources, in particular steel. During the steelmaking process that is based on processing iron ore or scrap, GHG emissions occur at many stages. However, through major advances in technology in the steel industry in North America, Western Europe and Japan have reduced energy consumption per unit of production by about 50 percent in the past decade. The most promising approach to limit GHG emissions is recycling. This approach exhibits the lowest CO2 substitution cost and avoids drastic revisions of steel-making practices. Other promising approaches include smelting reduction and increased use of natural gas, plant biomass, electricity, or hydrogen vectors in more innovative ways to achieve a reduction in CO2 emissions. In the OECD countries overall, direct processrelated emissions from iron and steel production account for about 2.4 percent of total GHG emissions. Furthermore, iron ore and steel exports that form one of Australia’s largest mineral earners are expected to increase. This is because Australia’s iron ore industry has undergone a renewed investment growth following the emergence of new ore types, rising production and generally improved exchange rates of the Australian dollar against the US dollar. To improve understanding of trade complexity in the energy metal resource industries in conditions of climate change, it is crucial to utilize sophisticated time series decision support approaches to identify energy metal consumption patterns, which focus on the short-term dynamics and the long-term relationship between the consuming sectors and the economic activities. The research outcomes will provide a better understanding of the nature of such complexity problems in the energy metal trade and have direct application in solving those practical problems that are of both regional and global concern.

4. METHODOLOGY 4.1. The Forgetting Factor The use of forgetting factor in time series analysis has attracted considerable interest in recent years. For example, Penm and Terrell (2003) utilize a

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forgetting factor in subset autoregressive modeling of the spot aluminum and nickel prices on the London Metal Exchange. The use of the forgetting factor technique to estimation and simulation of ﬁnancial market variables has been reported by Brailsford, Penm, and Terrell (2002). Consider a vector autoregressive (VAR) model of the following form: zðtÞ þ

q X

At zðt tÞ ¼ eðtÞ

(1)

t¼1

where z(t) is a k 1 vector of wide-sense stationary series. e(t) is a k 1 vector of independent and identically distributed random process with E{e(t)} ¼ 0 and EfeðtÞe0 ðt tÞg ¼ O if t ¼ 0 and ¼ 0 if t40: At , t ¼ 1; . . . ; q are k k matrices of coefﬁcients. The observations zðtÞ½t ¼ 1; . . . ; Tg are available. Let kðtÞ ¼ l1 ðtÞ . . . . . . ln ðtÞ denotes a 1 k vector associated with time t. Following O’Neill, Penm, and Penm (2007), a strategy for determining the value of the forgetting factor kðtÞ is as follows. ki ðtÞ ¼ kZtþ1 if 1 t Z and ¼ 1 if Zot T for i ¼ 1; . . . ; n

(2)

Eq. (2) means that ‘‘forgetting’’ of the past occurs from time Z. No forgetting is involved from time Zþ1 to time T. If k ¼ 1 for every t, then we obtain the ordinary least squares solution. If 0oko1, the past is weighted down geometrically from time Z. In theory, the value of k could be different between ki ðtÞ (a so-called variable forgetting factor). For simplicity, we only consider the ﬁxed forgetting factor case in which the value of k is constant for ki ðtÞ. This means that the coefﬁcients in Eq. (1) are estimated to minimize, " #" #0 q q T X X X kðtÞ zðtÞ At yðt tÞ zðtÞ At zðt tÞ (3) t¼1

t¼1

t¼1

One important issue relating to the use of the forgetting factor in estimation is how to determine the value of k in applications. The conventional method is based on arbitrary or personal choices. Penm and Terrell (2003) propose to determine the value of k using the bootstrap. In this study, their recommended method is adopted for the determination of the value of k. While Brailsford et al. (2002) also propose a procedure to determine the value of dynamic forgetting factor for nonstationary systems, we have focused on the use of a ﬁxed forgetting factor in this study, because applications of a ﬁxed forgetting factor to forex market movements is likely to be more predictable.

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4.2. VECM for an I(1) System In constructing VECM for an I(1) system, from Eq. (1) we have Aq ðLÞ ¼ I þ

q X

At Lt

t¼1

where L denotes the lag operator, and LzðtÞ ¼ zðt 1Þ. It is assumed that the roots of jAq ðLÞj ¼ 0 lie outside or on the unit circle to ensure that z(t) can contain I(1) variables. Of note, z(t) is integrated of order d, I(d), if it contains at least one element that must be differenced d times before it becomes I(0). Furthermore, z(t) is cointegrated with the cointegrating vector, b, of order g, if buz(t) is integrated of order (d–g), where z(t) has to contain at least two I(d) variables. Following Penm and Terrell (2003), the equivalent VECM for Eq. (1) can then be expressed as follows: Aq ð1Þzðt 1Þ þ Aq1 ðLÞDzðtÞ ¼ eðtÞ

(4)

where z(t) contains variables of the types I(0) and I(1). Note that ‘D’ represents the difference, DzðtÞ ¼ zðtÞ zðt 1Þ and eðtÞ is stationary. Eq. (4) can be rewritten as follows: A zðt 1Þ þ Aq1 ðLÞDzðtÞ ¼ eðtÞ

(5)

where A ¼ Aq ð1Þ and A zðt 1Þ is stationary, and the ﬁrst term in Eq. (5) is the error correction term. The term Aq1 ðLÞDzðtÞ is the VAR part of the VECM. Because y(t) is cointegrated of order 1, the long-term impact matrix, A , must be singular. As a result, A ¼ ab0 and b0 zðt 1Þ is stationary, where the rank of A is r (0o r os), and a and b0 are matrices of dimensions s r and r 2s, respectively. The columns of b are the cointegrating vectors and the rows of a are the loading vectors. Penm and Terrell (2003) demonstrate that a system that involves both cointegrated and stationary series can be characterized by sparse patterned VECM that includes full-order models. This patterned VECM can be used for estimation of such a cointegrated system. The development course of the climate change is a long-term slowly evolving underlying process, and the effects of climate change will be exhibited in the detected long-term cointegrating relations.

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Our search algorithm originally proposed by Penm and Terrell (2003) to select the optimal sparse VECM and the associated patterned a and b is brieﬂy described below. 1. To begin this algorithm, we ﬁrst identify the optimal sparse patterned VECM using model selection criteria. 2. After the optimal sparse patterned VECM is identiﬁed, the rank of the long-term impact matrix is then computed using the singular value decomposition method so that the number of cointegrating vectors in the system will be known. 3. A tree-pruning algorithm that avoids evaluating all candidates is then implemented for the search of all acceptable sparse patterns of the loading and cointegrating vectors. 4. The identiﬁed candidates of the sparse patterned cointegrating vectors are estimated by the method based on a triangular VECM representation proposed in Penm and Terrell (2003). 5. The estimation of the associated candidates for the sparse patterned loading vectors is carried out by the Yule–Walker estimation method with linear restrictions. 6. The optimal sparse patterned a and b are ﬁnally selected by model selection criteria. Furthermore, for copper consumption, the VECM of Eq. (5) can be rewritten as follows: "

PðLÞ

GðLÞ

#"

DC t

#

FðLÞ YðLÞ DE t " # " 1# t C t1 ¼ 2 t E t1

" þ

# a1 b1 a2 b1

1

b2 b1

where " yðtÞ ¼ " a¼

Ct Et a1 a2

#

" ; eðtÞ ¼

#

" ; b¼

b1 b2

1t 2t #

"

# ; A

q1

ðLÞ ¼

PðLÞ

GðLÞ

FðLÞ

YðLÞ

# (6)

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Eq. (6) can then be described as follows: " # " # 8 Ct1 dC t > > > ½PðLÞ GðLÞ þ d1 ¼ 1t > > mE t1 dE t < " # " # > C t1 dC t > > > ½FðLÞ YðLÞ þ d2 ¼ 2t > : mE t1 dE t

ð7Þ

ð8Þ

where d1 ¼ a1 b1 ; d2 ¼ a2 b1 and m ¼ b2 =b1 . For steel consumption, Eq. (6) becomes " # " # 8 S t1 dS t > > > ½PðLÞ GðLÞ þ d1 ¼ 1t > > mE t1 dE t < " # " # > St1 dSt > > ½FðLÞ YðLÞ > þ d2 ¼ 2t > : mE t1 dE t

ð9Þ

ð10Þ

where C and S denote copper and steel consumptions, respectively, and E denotes GDP. Pð0Þ ¼ 0; Gð0Þ ¼ 0;Fð0Þ ¼ I; Yð0Þ ¼ I.

5. DATA SOURCES In this chapter, reﬁned copper consumption was used to approximate copper consumption in each country investigated. Apparent consumption of ﬁnished steel was used in the estimation for OECD countries, while apparent consumption of crude steel was used in the estimation for developing countries. Both GDP and industrial production were used to approximate the level of general economic activity. Data for reﬁned copper consumption were obtained from the World Bureau of Metal Statistics. Data for apparent consumption of crude steel are published by the International Iron and Steel Institute (IISI), while those for ﬁnished steel consumption came from Datastream. The macroeconomic data, GDP and industrial production, were obtained from the International Monetary Fund. The sample periods used, subject to data availability, in the estimation are presented in Table 1.

Long-term Relations between Economic Activity and Metal Utilizations

Table 1. Country The United States Japan The EU countries South Korea Taiwan India Indonesia Mexico

Argentina Brazil a

107

Data Samples Subject to Data Availability. Series

Sample Perioda

Copper Steel Copper Steel Copper Steel Copper Steelb Copperb Steelb Copperb Steelb Copper (a) GDP (b) Industrial production Steelb Copperb Steelb Copperb Steelb

1982(1)–2005(4) 1986(2)–2005(4) 1982(1)–2005(4) 1986(2)–2005(4) 1982 (1)–2005(4) 1986(2)–2005(4) 1982(1)–2005(4) 1967–2005 1967–2005 1959–2005 1967–2005 1972–2005 1981(1)–2005(4) 1982(1)–2005(4) 1967–2005 1969–2005 1969–2005 1969–2005 1969–2005

Quarterly GDP data are seasonally adjusted except for South Korea and Mexico. Annual data.

b

6. MODELING RESULTS In this chapter, cointegration theory was utilized to test for the existence of long-term relationships between general economic activity and copper and steel consumption in major consuming countries. Following Engle and Granger (1987), a set of variables is said to be cointegrated, if the individual variables are nonstationary, such as copper and steel consumption and the level of general economic activity, but a linear combination of these variables becomes stationary. In other words, the test for cointegration shows whether random shocks to the relationships between these variables tend to dissipate over time or whether the shocks have a permanent effect on their relationships whereby copper and steel consumption and the level of general economic activity tend to drift apart from each other in an independent manner. Evidence that cointegration exists among a set of variables provides strong support for the presence of long-term relationships among them.

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ANDREW H. CHEN ET AL.

VECM Indicating Relationships between Copper Consumption and General Economic Activity.

Table 2.

Eq. (7) for copper consumption: PðLÞDCt þ GðLÞDE t þ d1 Ct1 þ mE t1 ¼ 1t . A ﬁxed forgetting factor with the value 0.99 applies to all observations. Variables are in logarithms. C denotes copper consumption, E gross domestic product, g deseasonalized E, and P industrial production. General economic activity is indicated by either E or P as proxies.. Di, i ¼ 1, 2, 3 are seasonal dummies, d is ﬁrst difference, and jt statisticsj are shown in brackets. For simplicity, 1t is not displayed below. The United States DC t ¼ 2:1558 0:1135DCt2 þ 0:3665DCt4 þ 3:4135DE t1 0:5772ðCt1 0:5563 E t1 Þ ð4:73Þ

ð1:36Þ

ð4:47Þ

ð2:82Þ

ð6:31Þ

ð2:15Þ

DC t ¼ 2:2561 þ 0:3261DCt4 þ 2:14721DPt þ 1:3735Pt3 0:6786ðCt1 0:6443Pt1 Þ ð5:07Þ

ð4:12Þ

ð2:78Þ

ð1:79Þ

ð7:11Þ

ð4:44Þ

where Pt denotes industrial production, and replaces Et in Eq. (7). Japan DC t ¼ 2:993 þ 0:157DCt2 þ 0:257DCt4 þ 2:055DE t 0:742ðCt1 0:401 E t1 Þ ð5:89Þ

ð1:81Þ

ð2:94Þ

ð1:62Þ

ð4:48Þ

ð6:84Þ

DC t ¼ 1:597 0:577DCt1 þ 2:111DPt 0:415ðCt1 0:471Pt1 Þ ð3:36Þ

ð5:68Þ

ð2:88Þ

ð2:51Þ

ð3:58Þ

The European Union countries DC t ¼ 1:968 0:543DCt1 0:578DCt2 0:493Ct3 þ 4:391DE t1 0:472 ðC t1 0:537E t1 Þ ð2:06Þ

ð3:07Þ

ð4:31Þ

ð5:17Þ

ð2:73Þ

ð2:28Þ

ð2:22Þ

DC t ¼ 1:419 0:585DCt1 0:636DCt2 0:533DC t3 þ 1:877DPt1 þ 1:787DPt2 0:499ðCt1 0:831Pt1 Þ ð1:62Þ

ð2:98Þ

ð4:33Þ

ð5:53Þ

ð2:02Þ

ð1:93Þ

South Korea gt ¼ E t 10:503 þ 0:414D1 þ 0:283D2 þ 0:236D3 ð94:35Þ

ð2:61Þ

ð1:77Þ

ð1:47Þ

DC t ¼ 1:958 0:553DCt1 0:391DCt2 0:266DC t3 0:499 ðCt1 1:374gt1 Þ ð3:41Þ

ð3:53Þ

ð2:82Þ

ð2:07Þ

ð3:23Þ

ð2:77Þ

DC t ¼ 0:739 0:368DCt1 0:636DC t2 0:672DCt3 0:766ðCt1 1:1791 Pt1 Þ ð1:91Þ

ð1:92Þ

ð1:95Þ

ð1:83Þ

ð4:15Þ

ð3:61Þ

Taiwan DC t ¼ 28:553 1:152 ðCt1 1:697E t1 Þ ð5:18Þ

ð5:32Þ

ð5:12Þ

India DC t ¼ 1:199 þ 0:497DCt2 þ 3:018DE t1 0:596ðCt1 0:753E t1 Þ ð1:86Þ

ð3:88Þ

ð3:15Þ

ð3:78Þ

ð4:88Þ

DC t ¼ 1:151 þ 0:336DCt2 þ 2:915DPt 0:647 ðCt1 0:635Pt1 Þ ð2:78Þ

ð2:71Þ

ð2:95Þ

ð4:93Þ

ð4:22Þ

Argentina DC t ¼ 11:325 þ 0:324DCt1 þ 2:186DE t 1:989DE t1 0:749 ðCt1 1:696 E t1 Þ ð3:68Þ

ð1:87Þ

ð2:61Þ

ð1:88Þ

Mexico gt ¼ E t 6:437 þ 0:047D1 þ 0:018D2 þ 0:063D3 ð29:87Þ

ð1:77Þ

ð1:58Þ

ð2:35Þ

DC t ¼ 1:761 0:531 ðCt1 1:659gt1 Þ ð4:47Þ

ð4:53Þ

ð2:05Þ

DC t ¼ 1:833 þ 1:542DPt 0:615 ðCt1 1:388Pt1 Þ ð1:70Þ

ð1:99Þ

ð4:92Þ

ð2:53Þ

ð3:93Þ

ð3:91Þ

ð2:25Þ

ð2:11Þ

Long-term Relations between Economic Activity and Metal Utilizations

Table 3.

109

VECM Indicates Relationships between Steel Consumption and General Economic Activity.

Eq. (9) for steel consumption: PðLÞDSt þ GðLÞDE t þ d1 St1 þ mE t1 ¼ 1t . A ﬁxed forgetting factor with the value 0.99 applies to all observations. Variables are in logarithms. S denotes steel consumption. E gross domestic product, g deseasonalised E, and P industrial production. General economic activity is indicated by either E or P as proxies.. Di, i ¼ 1, 2, 3 are seasonal dummies, d is ﬁrst difference, and jt statisticsj are shown in brackets. For simplicity, 1t is not displayed below. The United States DS t ¼ 1:133 0:241DSt2 þ 3:315DE t þ 3:032DE t1 0:185ðSt1 0:782E t1 Þ ð1:93Þ

ð2:21Þ

ð3:56Þ

ð3:15Þ

ð2:31Þ

ð2:15Þ

Japan DS t ¼ 1:818 þ 0:221DSt2 0:203DSt3 0:267DSt4 þ 1:791DE t þ 1:720DE t2 0:295 ðSt1 0:773E t1 Þ ð3:31Þ

ð2:05Þ

ð1:72Þ

ð2:28Þ

ð2:18Þ

ð3:51Þ

ð3:97Þ

ð3:93Þ

DS t ¼ 0:717 0:315DSt1 0:231DSt3 0:257DSt4 þ 0:950DE t þ 1:982DE t1 0:177 ðSt1 1:138Pt1 Þ ð1:71Þ

ð2:82Þ

ð2:52Þ

ð2:38Þ

ð2:80Þ

ð3:12Þ

ð3:20Þ

ð3:07Þ

The European Union countries DS t ¼ 2:187 0:201DSt2 0:171DSt3 þ 0:468DSt4 þ 4:062DE t þ 2:722DE t1 0:411 ðSt1 1:028E t1 Þ ð3:83Þ

ð2:12Þ

ð2:15Þ

ð5:08Þ

ð2:31Þ

ð1:52Þ

ð4:85Þ

DS t ¼ 1:337 0:191DSt2 0:185DSt3 þ 0:447DSt4 þ 2:3521DPt 0:443 ðSt1 1:558Pt1 Þ ð2:51Þ

ð2:28Þ

ð2:51Þ

ð5:03Þ

ð2:37Þ

ð4:41Þ

ð3:57Þ

South Korea DS t ¼ 1:291 0:315DSt2 þ 2:327DE t 0:208 ðS t1 1:332E t1 Þ ð1:48Þ

ð3:31Þ

ð3:82Þ

ð2:71Þ

ð2:22Þ

DS t ¼ 2:237 0:335DSt2 þ 1:048DPt 0:391 ðSt1 0:947Pt1 Þ ð3:31Þ

ð3:38Þ

ð3:37Þ

ð3:07Þ

ð2:73Þ

Taiwan DS t ¼ 3:238 þ 2:065DE t 0:323 ðSt1 1:258E t1 Þ ð2:77Þ

ð1:78Þ

ð2:63Þ

ð2:68Þ

India DS t ¼ 0:277 þ 0:383DSt3 þ 0:968DE t1 0:573 ðSt1 1:097E t1 Þ ð2:88Þ

ð2:77Þ

ð1:98Þ

ð4:13Þ

ð4:08Þ

DS t ¼ 4:642 þ 0:201DSt1 þ 0:293DSt2 þ 0:418DSt3 þ 1:347DPt 0:793 ðSt1 0:878Pt1 Þ ð4:17Þ

ð1:29Þ

ð1:83Þ

ð2:77Þ

ð2:78Þ

Indonesia DS t ¼ 2:712 þ 3:535DE t 0:487 ðSt1 1:118E t1 Þ ð2:07Þ

ð2:08Þ

ð2:33Þ

ð2:38Þ

Mexico DS t ¼ 0:357 þ 3:012DE t 0:447 ðSt1 1:255E t1 Þ ð1:78Þ

ð6:75Þ

ð4:21Þ

ð4:42Þ

DS t ¼ 1:633 þ 2:327DPt 0:427 ðSt1 1:128Pt1 Þ ð3:55Þ

ð6:91Þ

ð3:93Þ

ð4:08Þ

Brazil DS t ¼ 1:572 þ 1:691DE t þ 1:739DE t1 0:422 ðSt1 1:193E t1 Þ ð2:58Þ

ð3:22Þ

ð3:71Þ

ð3:68Þ

ð3:73Þ

ð4:23Þ

ð4:52Þ

ð3:32Þ

110

ANDREW H. CHEN ET AL.

In the course of applying VECM to test the cointegrating relationships, all variables are log transformed. Unit root tests indicate that all transformed series are I(1). We apply a ﬁxed forgetting factor with the value 0.99 to the stochastic system involved. We then conduct the search procedures proposed indicated in Section 4 to obtain the optimal sparse patterned VECM. The optimal VECM is utilized to test for the existence of cointegrating relationships between copper and steel consumption and general economic activity in major consuming countries. The estimated sparse patterned VECM are presented in Tables 2 and 3. It is noteworthy that, in addition to general economic activity, many other factors, such as own price and prices of substitutes and complements, could also signiﬁcantly affect copper and steel consumption, especially in the short term. These factors should also be incorporated when forecasting.

7. SUMMARY The results of this study generally support the view that long-term cointegrating relationships and uni-directional comovements exist between copper and steel consumption and general economic activity in selected consuming countries. As the development course of climate change is a long-term slowly evolving but underlying process, the effects of weather shocks caused by climate change are exhibited in the detected long-term cointegrating relations. The linking sign between metal consumption and economic activities detected in all cointegrating relationships is consistent with the hypothesis that ‘‘uni-directional’’ comovements exist in major consuming countries experiencing conditions of climate change.

REFERENCES Brailsford, T., Penm, J., & Terrell, R. D. (2002). Selecting the forgetting factor in subset autoregressive modelling. Journal of Time Series Analysis, 23, 629–650. Chen, A., Penm, J., & Terrell, R. D. (2006). An evolutionary recursive algorithm in selecting statistical subset neural network/VDL ﬁltering. Journal of Applied Mathematics and Decision Sciences, 2, 1–12. Article ID 46592. Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica, 55, 251–276. O’Neill, T., Penm, J., & Penm, J. S. (2007). A subset polynomial neural networks approach for breast cancer diagnosis. International Journal of Electronic Healthcare, 3(3), 293–302.

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Penm, J. (2007). The recursive ﬁtting of multivariate complex subset ARX models. Applied Mathematical Sciences, 1(23), 1129–1143. Penm, J., & Terrell, R. D. (Eds). (2003). Collaborative research in quantitative ﬁnance and economics. Australia: Evergreen Publishing. Stern, N. (2006). Stern review executive summary (700p.). New Economics Foundation.

IMPROVED DIVERSIFICATION THROUGH A MIX OF OIL AND EQUITIES Helen Xu ABSTRACT This study presents evidence of a statistically signiﬁcant negative correlation between crude oil and equities over the past 20 years. Including proper proportions of negatively correlated assets in a diversiﬁed portfolio can improve the ratio of reward relative to risk, and therefore, adding crude oil with equities into a diversiﬁed portfolio can provide superior portfolio performance, compared with equities alone. Because crude oil prices held stable for nearly a century before the oil crisis of 1973, and oil derivatives did not begin trading actively on public markets until the 1980s, the diversiﬁcation value of oil is a relatively new phenomenon. Also contributing to the phenomenon, the majority of oil reserves and the majority of crude oil production capacity worldwide are held by entities that are not traded in public equity markets, and therefore, the diversiﬁcation beneﬁts of oil cannot be fully realized by holding a portion of the global market portfolio of equities.

Research in Finance, Volume 26, 113–126 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026008

113

114

HELEN XU

1. INTRODUCTION This study presents evidence of statistically signiﬁcant negative correlation between crude oil and equities from 1987 through 2005. The methodology developed for the Goldman Sachs Commodity Index (GSCI) is applied to measure the risk and return proﬁle of crude oil (Goldman Sachs Commodity Index Manual, 2005 edition). Although the GSCI method has been used to construct the returns of a basket of commodities, it has never been applied on individual commodities in previous investigations of the relationship between commodity returns and equity returns. Froot (1995) applied GSCI methodology to construct the total returns for individual commodities, but when he studied the relationship between individual commodities and other assets, he used the percentage changes in the spot price of individual commodities because of data limitation. Rzepczynski, Belentepe, Feng, and Lipsky (2004) used cumulative returns method to ﬁgure out the crude oil returns. Georgiev (2004) calculated the spot returns, roll yield, and total returns of energy commodities, but he generated futures price data series by using a weighted average of the prices of the two contracts closest to expiration, which is different from GSCI method. Using the GSCI methodology allows the use of readily available futures prices to extract information about the returns that can be earned by a wholesale dealer who buys and holds crude oil – including the potential for lending crude oil and receiving interest paid in kind.1 By following the trading strategy involved in the GSCI methodology, an ordinary investor can capture much of the beneﬁt available to wholesale dealers who buy and hold crude oil. The results show that the return for buying and holding crude oil is signiﬁcantly negatively correlated with the S&P 500 Index. Including proper proportions of negatively correlated assets in a diversiﬁed portfolio can improve the ratio of reward relative to risk, and therefore, adding oil into a diversiﬁed portfolio of equities can provide superior portfolio performance, compared with equities alone. Because oil prices held steady for nearly a century before the oil crisis of 1973 and oil derivatives did not begin trading actively on public markets until the 1980s, the diversiﬁcation value of oil is a relatively new phenomenon. Although oil derivatives began trading on the public exchanges in the early 1980s, it was not until 1987 that volume reached mature levels. Our study begins with data from this point and continues through the end of 2005. Also contributing to the phenomenon, the majority of oil reserves and the majority of crude oil production capacity worldwide are held by entities that

Improved Diversiﬁcation through a Mix of Oil and Equities

115

are not traded in public equity markets. Therefore, the diversiﬁcation beneﬁts of oil cannot be fully realized by holding a portion of the global market portfolio of equities.

2. DATA Crude oil futures contract trading volumes before 1987 were not very signiﬁcant. As a result, the price information of crude oil futures contracts may not be very efﬁcient before 1987. Therefore, daily crude oil futures contract closing prices are collected over the period from 1987 to 2005. The S&P 500 Composite Index and the CRSP Value-Weighted Market Index are used to represent investment in stocks. The monthly returns of both indexes are collected from the CRSP database. This study also collects the monthly 91-day Treasury bill returns data from the CRSP database and the daily 91-day Treasury bill yield data from the Federal Reserve Bank database.

3. METHODOLOGY This study directly estimates the return for buying and holding commodities based on the methodology developed to construct the GSCI. According to the GSCI Manual (2005), the quantity of each commodity in the index is determined by the average quantity of production during the past ﬁve years. The GSCI Total Return Index measures the return of a fully collateralized commodity investment that is rolled forward from the ﬁfth to the ninth business day of each month. Using GSCI total return calculation methodology from the GSCI Manual, this study constructs a portfolio consisting of a certain value of the light crude oil futures contract and the equivalent value of 91-day Treasury bills and then calculates the return of the constructed portfolio from 1987 to 2005 monthly. The monthly return of the constructed portfolio consists of the monthly spot return of crude oil, return from the rolling process, and monthly return from holding Treasury bills. On the rolling day, which is set to be the ﬁfth business day of the rolling month, the roll yield is equal to the ratio of the nearest futures contract closing price to the next nearest futures contract closing price minus one. Energy futures contracts mature every month; thus, the rolling process occurs on the ﬁfth business day each month. Then, the following formula is used to calculate monthly total return of crude oil. Total return is

116

HELEN XU

the crude oil investment return. Total return ¼ spot return þ roll yield þ Treasury bill return After deriving the buy and hold return estimation, we consider the average monthly total return, standard deviation of total returns, and Pearson correlation coefﬁcients between monthly total returns of crude oil and corresponding returns on stocks. For crude oil, the calculation period is a monthly observation rolling 3 years. Based on the estimation of the risk and return proﬁle of crude oil, this study decides whether crude oil provides diversiﬁcation beneﬁts for shareholders’ portfolios. To illustrate the diversiﬁcation beneﬁt of crude oil with stocks, the author calculates the risk and returns proﬁles for portfolios with different combinations of crude oil and S&P 500 Index. Then, this study follows Georgiev’s (2001) method to measure the diversiﬁcation beneﬁt of crude oil by the change in the Sharpe ratio. The Sharpe ratio change is equal to the Sharpe ratio of the portfolio consisting of 60% S&P 500 Index investment and 40% crude oil investment minus the Sharpe ratio of the portfolio consisting of 100% S&P500 Index investment. The more the Sharpe ratio increases, the more diversiﬁcation beneﬁt crude oil provides. As the return data of crude oil is monthly, the Sharpe ratio calculation period for crude oil is monthly observations rolling 3 years.

4. RESULTS 4.1. Negative Correlation between Crude Oil and Equity Table 1 reports the correlation coefﬁcient between the monthly rates of return on crude oil, and the monthly rates of return on S&P 500 market index is 0.136 and signiﬁcant at the 5% level. The correlation coefﬁcient between the Table 1. Risk and Return Proﬁle of Crude Oil.

Crude oil S&P500 Index CRSP Value-Weighted Index

Mean (%)

Standard Deviation (%)

Correlation Coefﬁcients with Crude Oil

12.24 12.36 12

17.08 15.21 15.42

1 0.136 0.106

Note: Mean return and standard deviations are annualized. Statistically signiﬁcant at 5% level (two-tailed test).

117

Improved Diversiﬁcation through a Mix of Oil and Equities

monthly rates of return on crude oil and the monthly rates of return on the CRSP Value-Weighted Market Index is 0.106 but not statistically signiﬁcant. The reason that we ﬁnd two correlation coefﬁcients with different signiﬁcance levels may be because S&P 500 Index represents the ‘‘old economy’’ and crude oil investment is more closely related to the ‘‘old economy.’’ Overall, it is safe for us to say that crude oil has negative or zero correlation with the stock market. Table 1 also reveals that crude oil has approximately the same average return and standard deviation as the overall stock market. Usually the commodities market is thought to be more volatile than the stock market. Investment in commodities incurs very high risk. Surprisingly, this study ﬁnds that crude oil has almost the same standard deviation as stocks. Crude oil has almost the same risk and return proﬁle as stocks have, but is negatively correlated with stocks. Therefore, if crude oil is added into a portfolio consisting of equities, crude oil will be able to reduce portfolio risk and at same time keep the portfolio average rate of return. Crude oil can provide diversiﬁcation beneﬁt for equities. Fig. 1 shows that during most of the period crude oil is negatively correlated with stocks. When the stock market crashed on Monday, October 19, 1987, however, the crude oil market was not affected signiﬁcantly. In the 1990 Gulf War, there was a spike in crude oil market; yet, the stock market was sluggish. 0.3 crude oil total return S&P 500 return

0.2

-0.2

-0.3

Fig. 1. Crude Oil Investment Returns versus S&P 500 Returns.

1/30/2005

1/30/2004

1/30/2003

1/30/2002

1/30/2001

1/30/2000

1/30/1999

1/30/1998

1/30/1997

1/30/1996

1/30/1995

1/30/1994

1/30/1993

1/30/1992

1/30/1991

1/30/1990

1/30/1989

-0.1

1/30/1988

0 1/30/1987

rate of return

0.1

118

HELEN XU

Table 2.

Correlation Coefﬁcients between Crude Oil and S&P 500 Index. Correlation Coefﬁcients

01/1987–12/1989 02/1987–01/1990 y 01/2003–12/2005 Mean Standard deviation t-Test

0.1 0.16 y 0.18 0.13 0.22 8.05

Statistically signiﬁcant at 1% level (two-tailed test).

Before the Asian ﬁnancial crisis, OPEC signiﬁcantly increased the production of crude oil due to the economic developments of East Asian countries; yet nonetheless, because of the Asian ﬁnancial crisis, the demand for crude oil by East Asian countries suddenly declined in 1998. OPEC adjusted the crude oil production level, and thus crude oil prices rose in 1999. At the same time, stock prices dropped signiﬁcantly when the Asian ﬁnancial crisis occurred at the end of 1997 and recovered at the end of recession. The Dotcom Bubble in 2000 and 2001 moved investors’ money from the stock market into the commodity market. During the 9/11 events, the crude oil market also dropped signiﬁcantly. Consequently, there was a spike in crude oil return at the end of 2001. Concurrent with the Iraq War in 2003, crude oil returns rose when the stock market dropped. Overall when crude oil returns rise, S&P 500 Index returns decrease and vice versa. If they are combined together into one portfolio, the risk of the resulting portfolio will be signiﬁcantly smaller than the risk of each individual component. To further examine the correlation relationship between crude oil returns and stock returns, this investigation calculates the rolling correlation coefﬁcients from 1987 to 2005 with monthly observations and a three-year rolling period (see Table 2). The average three-year correlation coefﬁcient is 0.13 and signiﬁcantly different from zero at the 1% level. Assuming investors apply a three-year correlation coefﬁcient as the ex ante expectation of the relationship between crude oil returns and stock returns, this result implies that the average expected correlation between these two investments is signiﬁcantly negative. Fig. 2 illustrates correlation coefﬁcients for all of the rolling time periods. For most of the time, correlation coefﬁcients fall below zero and the minimum is 0.6, and even when correlation coefﬁcients are positive, the

119

Improved Diversiﬁcation through a Mix of Oil and Equities

correlation coefficients

0.4 correlation coefficients

0.2

01/2003-

09/2001-

05/2000-

01/1999-

09/1997-

05/1996-

01/1995-

09/1993-

05/1992-

01/1991-

09/1989-

-0.4

05/1988-

-0.2

01/1987-

0

-0.6 -0.8 rolling time periods

Fig. 2.

Correlation between Crude Oil Returns and S&P 500 Index Returns.

Table 3.

Diversiﬁcation Beneﬁt of Crude Oil with Equities.

Portfolio Composition 100% of crude oil 90% of crude oil and 10% 80% of crude oil and 20% 70% of crude oil and 30% 60% of crude oil and 40% 50% of crude oil and 50% 40% of crude oil and 60% 30% of crude oil and 70% 20% of crude oil and 80% 10% of crude oil and 90% 100% of S&P 500 Index

of of of of of of of of of

S&P S&P S&P S&P S&P S&P S&P S&P S&P

500 500 500 500 500 500 500 500 500

Index Index Index Index Index Index Index Index Index

Rate of Return

Standard Deviation

0.0102 0.01021 0.01022 0.01023 0.01024 0.01025 0.01026 0.01027 0.01028 0.01029 0.0103

0.0493 0.0437 0.0390 0.0350 0.0320 0.0305 0.0304 0.0320 0.0349 0.0389 0.0439

maximum is only 0.2. This evidence shows that crude oil and stocks are negatively correlated most of the time.

4.2. Diversiﬁcation Beneﬁts Both Table 3 and Fig. 3 exemplify the diversiﬁcation beneﬁt of crude oil with equities. The portfolio consisting of 40% crude oil investment and 60% S&P 500 Index has the minimum variance and also is the optimal. The Sharpe ratio is also applied on both portfolio 1 (100% S&P 500 index) and portfolio 2 (60% S&P 500 index and 40% crude oil) to statistically measure the diversiﬁcation beneﬁt of crude oil with stocks.

portfolio rate of return

120

HELEN XU 0.01032 0.0103 0.01028 0.01026 0.01024 0.01022 0.0102 0.01018 0

0.01

0.02

0.03

0.04

0.05

0.06

standard deviation

Fig. 3.

Table 4.

Diversiﬁcation Beneﬁt of Crude Oil.

Sharpe Ratios for Portfolio 1 and for Portfolio 2.

01/1987–12/1989 02/1987–01/1990 y 01/2003–12/2005 Mean Standard deviation Wilcoxon Signed-Rank Test Two-Sample t-test

Sharpe Ratio for Portfolio 1

Sharpe Ratio for Portfolio 2

0.16 0.06 y 0.39 0.18 0.21

0.21 0.13 y 0.58 0.23 0.20

5.1 5.71

Statistically signiﬁcant at 1% level (two-tailed test).

Table 4 summarizes the calculated Sharpe Ratios for portfolio 1 and for portfolio 2 from year 1987 to year 2005. We use monthly observations and three-year rolling period to calculate the rolling Sharpe ratios. The average three-year Sharpe ratio for a portfolio consisting of 100% investment in the S&P 500 index is 0.18. If we replace 40% of investment in the S&P 500 index with light crude oil investment, then the average three-year Sharpe ratio of the new portfolio increases to 0.23. Both the Wilcoxon signed-rank test and the two-sample t-test show that the average Sharpe ratio of portfolio 1 is signiﬁcantly less than the average Sharpe ratio of portfolio 2. Therefore, adding crude oil into equities signiﬁcantly increases the diversiﬁcation proﬁle of the equity portfolio. Fig. 4 demonstrates the Sharpe ratio changes across time. The portfolio including 60% of investment in the S&P 500 index and 40% of investment in crude oil has dominated the portfolio consisting of 100% investment in the S&P 500 index during 2/3 of rolling periods from 1987 to 2005.

0

0.2

0.4

0.6

Sharpe ratio of portfolio 1 Sharpe ratio of portfolio 2

-0.4

Fig. 4.

Sharpe Ratios Comparison between Portfolio 1 and Portfolio 2.

89 90 90 91 91 92 92 92 93 93 94 94 94 95 95 96 96 97 97 97 98 98 99 99 99 00 00 01 01 02 02 02 03 03 04 04 04 05 05 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20 20 20 20 20 20 20 20 20 20 20 20 20 2/ 05/ 10/ 03/ 08/ 01/ 06/ 11/ 04/ 09/ 02/ 07/ 12/ 05/ 10/ 03/ 08/ 01/ 06/ 11/ 04/ 09/ 02/ 07/ 12/ 05/ 10/ 03/ 08/ 01/ 06/ 11/ 04/ 09/ 02/ 07/ 12/ 05/ 10/ 1 7- 7- 7- 8- 8- 9- 9- 9- 0- 0- 1- 1- 2- 2- 2- 3- 3- 4- 4- 4- 5- 5- 6- 6- 7- 7- 7- 8- 8- 9- 9- 9- 0- 0- 1- 1- 2- 2- 2-0.2 98 98 98 98 98 98 98 98 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 00 00 00 00 00 00 00 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /2 /2 /2 /2 /2 /2 /2 01 06 11 04 09 02 07 12 05 10 03 08 01 06 11 04 09 02 07 12 05 10 03 08 01 06 11 04 09 02 07 12 05 10 03 08 01 06 11

sharpe ratio

0.8

Improved Diversiﬁcation through a Mix of Oil and Equities 121

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4.2.1. Why Does Crude Oil Have Almost the Same Average Rate of Return as Stocks? This study tries to explain this phenomenon from the following two aspects. First, demand for energy products is increasing but the supply is limited. World economy development increases the demand for energy commodities. The resources for energy commodity are limited; hence, energy commodity prices keep going up. Second, the energy commodity market is usually a market in backwardation, and thus investors obtain positive returns from rolling commodity futures contracts forward. In terms of cost of carry model, the futures contract price is equal to spot price plus cost of storage plus interest expense minus dividend yield obtained from the underlying asset. Because of crude oil loans, crude oil generates the crude oil loan rate of return, which is same as the dividend yield. When crude oil loan interest rates are greater than costs of storage and interest expenses, the futures price falls below the spot price. As futures contracts go near to maturity, futures price go up to approximate the spot price. If investors keep taking long positions in crude oil futures contracts, they are able to capture a positive return from this approximation process.

4.2.2. Why is There a Negative Correlation between Crude Oil and Equities? First, industrial transportation costs, material costs, and other operation costs are very closely related to crude oil price. When the price of oil goes up, companies’ operation costs increase. Therefore, crude oil price increases have negative effects on the future outlook of the whole economy, and stock price drop correspondently. Crude oil price increases also cause the cost of living to increase. Investors will reduce their investment on stocks, and thus stock price decreases. Second, commodities offer a natural hedge for inﬂation. Becker and Finnerty (2000) ﬁnd that equity and debt typically lose value during periods of unexpected inﬂation. Commodity investments rise with inﬂation, so thus, commodity investment is negatively correlated with equity. Third, Gorton and Rouwenhorst (2005) contend that commodities and equities show different behaviors in business cycles. In the beginning of recession, stock prices usually drop but commodities prices do not drop signiﬁcantly. At the end of recession, stock prices go up but commodities prices may decrease. The crude oil is a real asset and its prices are determined by the supply and demand of crude oil. Stocks are ﬁnancial assets. These two kinds of assets can possibly show different patterns of price behavior.

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5. RELATED RESEARCH Dusak (1973) explains that the futures price data is more accessible than spot price data and using futures prices avoids the need to estimate storage costs directly. Therefore, futures prices near maturity might be used as a proxy for spot price. Cost of carry theory also shows that futures prices tend to move together with underlying spot prices and that at maturity futures prices equal spot prices, or else there would be risk-free arbitrage. As a result, most studies examine the systematic risk and return of spot commodities through commodity futures markets. Black (1976) and Dusak (1973) state that futures contracts do not have value by themselves because net cash ﬂow is zero when futures transactions occur. The returns of futures contracts only derive from the returns of the underlying commodities. Dusak (1973) contends that futures transactions are actually leveraged transactions of the underlying spot commodity. The systematic risk in a futures market comes entirely from the systematic risk of the underlying spot market. Following Dusak’s argument, extensive studies examine the systematic risk and return of commodity futures contracts to infer the systematic risk and return of spot commodities. The methodologies used to calculate commodity returns are typically classiﬁed into four categories: the percentage change in the futures prices, the percentage change in the futures prices plus Treasury bill, a rollover strategy to measure futures return, and the GSCI style investment return. No matter what kind of methodology is used, most previous studies show that some commodities have zero or negative correlations with stocks and bonds. Dusak (1973) applies the futures price near maturity to approximate underlying spot commodity price and the percentage change in the futures prices to approximate underlying spot commodity risk premium and shows that systematic risk and mean returns of wheat, corn, and soybean were near zero over the period 1952–1967. Fama and French (1987) also use futures prices of maturing contracts to measure spot prices. For the simple monthly returns, 5 out of 21 commodities are found to have signiﬁcant and positive return; for the monthly logarithmic returns, 19 out of 21 commodities have zero return. Kolb (1992) uses the same methodology to derive the daily mean returns of 29 commodities and shows that currencies, ﬁnancials, and precious metals rarely have positive returns. Kolb (1996) examines 4,735 futures contracts on 45 commodities. He also ﬁnds that there is no positive relation between systematic risk and realized return for futures contracts. Lee, Leuthold, and Cordier (1985) examine the relationship between daily returns of the commodity futures market index (CFI) and the stock market

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index (S&P 500) over the period from 1972 to 1981 and ﬁnd that they are independent of each other. Including commodity futures contracts in equity portfolios may reduce risk and improve portfolio performance. Bodie and Rosansky (1980) ﬁnd that an equally weighted portfolio consisting of 23 commodities had almost the same return and standard deviation as an equally weighted common stock portfolio over the period between 1950 and 1976. They ﬁnd that 15 out of 23 commodities had negative correlations with the stock market. They also ﬁnd that the systematic risk exposures of commodities are inversely correlated with their mean returns. They apply two methodologies to measure commodity returns. One follows Dusak’s (1973) method, while the other calculates commodity futures return as the simple rate of return on commodities futures plus the risk-free rate because commodity exchanges permit the posting of Treasury bills as margin and investors can earn interest on Treasury bills. Fortenbery and Hauser (1990) apply the same methodology to show there are very small correlation coefﬁcients between agricultural commodity returns and stock returns. De Roon, Nijman, and Veld (2000) apply a rollover strategy to measure futures return. The rollover strategy is to roll the nearest contract to the next nearest contract on expiration month. The percentage change of futures contract prices is taken as futures return. They show that most futures contracts outside the ﬁnancial groups are zero correlated with the stock market. Financial futures are positively correlated to the stock market, while gold and silver futures are negatively correlated to the stock market. The fourth method is to calculate commodity investment return – GSCI style investment return. The GSCI Manual (2005) states that the GSCI represents the returns that would be earned by holding only passive long positions in commodity futures with the long positions fully collateralized with Treasury bills. The GSCI return represents a fully collateralized return, and thus is comparable to the returns of stocks and bonds. The GSCI investment return comes from three sources: spot return from price changes in the underlying commodities; roll yield from rolling the nearest futures contracts to the next nearest futures contracts each month; and Treasury bill yield. Donohue, Froot, and Light (1992) show that by adding 5% of GSCI into a 60/40 stock/bond portfolio, the average returns over the 1970–1990 period would have increased from 9.6% to 9.8%, and the standard deviation of returns would have decreased from 12.1% to 11.5%. Ankrim and Hensel (1993) show that the monthly correlation of returns between the GSCI and the S&P 500 is 0.06, the monthly correlation between GSCI and the Ibbotson Intermediate Government Bonds Index is 0.11, and the correlation between the S&P 500 and the Ibbotson Intermediate Government

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125

Bonds Index is 0.25. Greer (2000) shows that total return from the GSCI index is comparable in magnitude and volatility to equity returns but is negatively correlated with stocks and bonds. Georgiev (2001) ﬁnds that over the period 1990–2001, GSCI returns had a correlation of 0.04 with the S&P 500, a correlation of 0.02 with the Lehman Government/Corporate Bond Index, a correlation of 0.03 with the MSCI World Index, and a correlation of 0.05 with the Lehman Global Bond Index. Jensen, Mercer, and Johnson (2002) examine the diversiﬁcation beneﬁts of commodity futures for a traditional portfolio that consists of U.S. stocks, international stocks, corporate bonds, and Treasury bills over the period 1973–1999. Consistent with previous results, commodity futures can enhance portfolio performance very signiﬁcantly. Metals and agricultural commodities offer the most diversiﬁcation beneﬁts. Gorton and Rouwenhorst (2005) construct an equally weighted commodity futures index and examine its monthly returns over the period from July 1959 to December 2004. The risk premium on the commodity index is shown to be as same as the risk premium on equities, but commodity returns are negatively correlated with equity and bond returns.

6. CONCLUSIONS The results reported here show that crude oil is negatively correlated with stocks but have almost the same rate of return as stocks.2 If crude oil is mixed with equities, it can improve the diversiﬁcation proﬁle of the portfolio. The changes mean that gaining exposure to crude oil make the market more complete.

NOTES 1. Typically, borrowers repay the loan of crude oil by returning the amount borrowed plus additional oil as payment of interest on the loan. This yield accounts for the backwardation typically displayed by oil futures. 2. We also have examined the risk and return proﬁle of gold. Gold is shown to have zero correlation with equities and has a rate of return almost equal to zero. Therefore, gold may not be a good investment and gold does not provide diversiﬁcation beneﬁts when included with equities in a portfolio. We also notice that the standard deviation of bi-monthly rates of return on gold is less than the standard deviation of bi-monthly rates of return on stock investments. Gold investment is less risky than stock and crude oil.

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REFERENCES Ankrim, E., & Hensel, C. (1993). Commodities in asset allocation: A real-asset alternative to real estate. Financial Analysts Journal (May–June), 20–29. Becker, K., & Finnerty, J. (2000). Indexed commodity futures and the risk and return of institutional portfolios. OFOR Working Paper. Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1–2), 167–179. Bodie, Z., & Rosansky, V. (1980). Risk and return in commodity futures. Financial Analysts Journal, 36(May–June), 3–14. De Roon, F. A., Nijman, T. E., & Veld, C. (2000). Hedging pressure effects in futures markets. The Journal of Finance, 55(3), 1437–1456. Donohue, N., Froot, K. A., & Light, J. O. (1992). Investment linked to commodity futures. Harvard Business School Case 293-017. Dusak, K. (1973). Futures trading and investor returns: An investigation of commodity market risk premiums. Journal of Political Economy, 81(6), 1387. Fama, E., & French, K. (1987). Commodity future prices: Some evidence of forecast power, premiums, and the theory of storage. The Journal of Business, 60(1), 55–73. Fortenbery, T. R., & Hauser, R. J. (1990). Investment potential of agricultural futures contracts. American Journal of Agricultural Economics, 72(3), 721. Froot, K. A. (1995). Hedging portfolios with real assets. Journal of Portfolio Management, Summer, 6–77. Georgiev, G. (2001). Beneﬁts of commodity investment. Journal of Alternative Investment, 4(1), 40–48. Georgiev, G. (2004). Active long-only investment in energy futures. Journal of Alternative Investment, 7(2), 32–43. Goldman Sachs Commodity Index Manual. (2005 edition). Gorton, G., & Rouwenhorst, G. (2005). Facts and fantasies about commodity futures. Yale ICF Working Paper no. 04–20. Greer, R. J. (2000). The nature of commodity index returns. The Journal of Alternative Investments (Summer), 45–52. Jensen, G. R., Mercer, J. M., & Johnson, R. R. (2002). Tactical asset allocation and commodity futures. The Journal of Portfolio Management, 28(4), 100–111. Kolb, R. (1992). Is normal backwardation normal? Journal of Futures Markets, 12, 75–92. Kolb, R. (1996). The systematic risk of futures contracts. Journal of Futures Markets, 16, 631–654. Lee, C. F., Leuthold, R. M., & Cordier, J. E. (1985). The stock market and the commodity futures market: Diversiﬁcation and arbitrage potential. Financial Analyst Journal, 41(4), 53–60. Rzepczynski, M., Belentepe, C. Y., Feng, W., & Lipsky, P. (2004). Black gold - trading crude oil for greater portfolio efﬁciency: A comparison with commodity indices. Journal of Alternative Investment, 7(2), 44–50.

CHANGES IN TRADING VOLUME AND RETURN VOLATILITY ASSOCIATED WITH S&P 500 INDEX ADDITIONS AND DELETIONS Eric C. Lin ABSTRACT When a stock is added into the S&P 500 Index, it in effect becomes crosslisted in the Index derivative markets. When index-based trading strategies such as index arbitrage are executed, the component stocks are directly affected by such trading. We ﬁnd increased volatility of daily returns, plus increased trading volume for the underlying stocks. Utilizing a list of S&P 500 Index composition changes over the period September 1976 to December 2005, we study the market-adjusted volume turnover and return variance of the stocks added to and deleted from the Index. The results indicate that after the introduction of the S&P 500 Index futures and options contracts, stocks added to the S&P 500 experience statistically signiﬁcant increase in both trading volume and return volatility. Both daily and monthly return variances increase following index inclusion. When stocks are removed from the index, though, neither volatility of returns nor trading volume experiences any signiﬁcant

Research in Finance, Volume 26, 127–154 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026009

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change. So, we have new evidence showing that Index inclusion changes a ﬁrm’s return volatility, and supporting the destabilization hypothesis. An old Wall Street adage says ‘‘It takes volume to make prices move.’’

INTRODUCTION Exchange trading in S&P 500 futures and options contracts began in 1982 and 1983, respectively. The popularity of these contracts soared soon after their introductions (see Vijh, 1994). The implied dollar trading volume in these contracts soon exceeded that in the cash securities. Harris (1989) reported, ‘‘By 1987, the average daily dollar volume in the S&P 500 futures contracts alone exceeded the dollar volume of cash S&P 500 trade by a factor of about two, while the dollar value of the daily net change in total open interest is about 8% of S&P 500 stock dollar volume.’’1 When a stock is added into an index such as the S&P 500, it in effect becomes cross-listed in the index derivative markets, thus becoming subject to substantially increased arbitrage pressure. Prices of index futures and options are co-integrated with the spot market, linked to the prices of underlying securities by index arbitrage. The existence of index derivatives contracts creates additional routes for arbitrageurs to trade. Moreover, index-based trading strategies create additional order ﬂows that must be absorbed by the market. In general, there is a positive relationship between trading volume and the magnitude of price changes in the ﬁnancial markets (see, for example, Karpoff, 1987). Stoll and Whaley (1987) point out the cash settlement feature of index futures contracts, which require index arbitrageurs to unwind positions in the spot index securities. The ‘‘unwinding’’ of index arbitrage positions, instead of the traditional delivery settlement method, tends to induce price pressures that temporarily cause price movements in the component shares. Short-term price changes, resulted from program trading transactions that buy or sell a large portfolio of component stocks (block trades), are inevitable in the presence of index-based trading programs.2 The results of this study include four key ﬁndings. First, we learn that stocks being added to the S&P 500 Index experience signiﬁcantly higher trading volume and return volatility (in both daily and monthly stock return series) following the effective date of inclusion (not the announcement date, but the date the inclusion becomes effective). This increase is evident only

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129

during the period after index derivatives trading had become well established (during the second sub-period of the sample, 1986–2005). This ﬁnding suggests that the increase in volatility may be related to heavy trading in the derivatives contracts. Second, for index deletions, there is no signiﬁcant change in either trading activity or return volatility, either before or after trading in index futures became well established. Third, the increase in variance became signiﬁcant starting in 1986 and remained signiﬁcant in most of the following years after. These ﬁndings are compatible with the result of Harris (1989), documenting a positive difference between S&P 500 stock and non-S&P 500 stock return volatilities, beginning in the year 1985. Fourth, we ﬁnd strong evidence that turnover changes are positively related to changes in volatility. Finally, we ﬁnd weak evidence that a small beta increase is associated with the added ﬁrms during the second subperiod. The small shift in beta of 0.04 is statistically, but not economically, signiﬁcant. Furthermore, our results are independent of the methodologies we employ in estimating return volatility. We offer the following explanations for the empirical results. Trade in index derivative contracts has a fundamental effect (not a temporary effect) on the stock return distribution of a security being included in the S&P 500 Index. Firms removed from the index experience no signiﬁcant change in trading volume and return variance because the market capitalization of these stocks generally becomes extremely small as they exit the index. As a result, they are not (or perhaps minimally) affected by index trading. This study contributes to two groups of literature. First, it documents an increase in return volatility associated with index addition. This result is particularly useful to option traders and risk management programs. Additionally, this new evidence supports Shleifer’s (1986) imperfect substitute hypothesis. Second, this study adds additional support that derivatives trading may ‘‘fundamentally’’ destabilize the underlying cash securities.

RESULTS The empirical results are divided into three parts. We begin with abnormal trading volume. Then we consider the volatility effect associated with Index additions and deletions. Finally, we ﬁnd that there is no signiﬁcant alteration of the stock’s systematic (market) risk. Complete information about data and methodology are available in the Appendix.

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TRADING VOLUME EFFECT We use the abnormal turnover ratio methodology similar to Harris and Gurel (1986) and Chen, Noronha, and Singal (2004). In the turnover ratio approach, volume turnover is simply individual ﬁrm trading volume divided by total shares outstanding. The ratio is then divided by overall market volume measured by the total trading volume of the New York Stock Exchange (NYSE).3 The market-adjusted turnover ratio tests whether postinclusion (deletion) volume is different from pre-inclusion (deletion) volume. If there is abnormal trading around Index changes, the mean turnover ratio (MTR) will deviate signiﬁcantly from unity. Table 2 provides results of our trading volume analysis. For stocks added to the Index, the turnover ratios in post-1986 periods are signiﬁcantly different from those prior to the year 1986. We use the year 1986 as the cutoff year as Vijh (1994) shows that the total (implied) dollar volume related to S&P 500 Index-based trading strategies for the year 1986 is almost twice as much as that for the year 1985 (Table 1). As a result, we compare the turnover ratios of two distinct periods: (1) September 1976 to December 1985, when Index-based trading was less important, and (2) January 1986 to December 2005, after trading in index derivatives had become well established. For the ﬁrst period, the mean turnover around the effective day of inclusion is not signiﬁcantly different from the ‘‘normal’’ turnover, which is measure using trading volume before the actual event day. In fact, our result indicates that during this period when Index component stocks are less likely to be affected by trading in the derivative markets, there is no abnormal trading volume associated with the company being included in the S&P 500. In other words, an entry to the Index portfolio did not change the trading volume of the addition during this period. In the period 1986–2005, though, the mean turnover around the effective day is 1.093. The p-value of the t-test is less than 0.001, indicating that the post-inclusion volume is signiﬁcantly higher than the volume during normal trading days, by almost 10%. This provides evidence that index additions are likely to experience an increase in trading volume following their entry into the Index, coinciding with the underlying cash securities becoming directly linked to the trading in the Index derivative products. We obtain similar abnormal volume results when we extend our ‘‘event’’ period longer, up to 150 trading days after the effective. The result suggests that there is permanent change in volume for 1986–2005 portion of the data. Our results

Index options

Futures options

2,935 8,069 12,364 15,056 19,505 19,045 11,354 10,560

161 678 947 1,444 2,446 2,895 1,553 1,679

– 14 12 8 1,683 6,205 4,817 6,274

– 1 1 1 42 187 132 199

– 281 673 1,090 1,886 1,877 735 1,162

– 24 52 105 237 285 101 185

Contracts Dollars Contracts Dollars Contracts Dollars

Index futures

S&P 500 Trading Volume

161 703 1,000 1,550 2,725 3,367 1,786 2,063

Total dollars

– 10,595 64,288 90,805 113,151 101,827 57,433 58,371

Contracts – 177 977 1,686 2,684 3,044 1,503 1,721

Dollars

Index Options

S&P 100 Trading Volume

S&P 500 Index Trading Strategies and Trading Volume.

Source: Reproduced from Vijh (1994, RFS).

1982 1983 1984 1985 1986 1987 1988 1989

Year

Table 1.

16,670 21,845 23,309 27,774 36,010 48,143 41,118 42,022

Round lots

Stocks

495 775 773 981 1,389 1,889 1,366 1,556

Dollars

NYSE Trading Volume

Changes in Trading Volume and Return Volatility 131

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are generally in line with those reported in previous studies (a review of the literature follows later in this work). Next, we examine the trading volume around the time when an S&P 500 component stock is removed from the Index. We expect that there should be no abnormal volume around the deletion event, even during the period of heavy index-based trading. This is because the deletions typically represent an extremely small fraction of the Index at the time of removal. Most of the deletions that survived the sample screening procedure are removed for lack of representation. These stocks are usually the smallest ﬁrms in the S&P 500, in terms of market capitalization. As a result, derivatives trading would have little or no impact on the volume of these ‘‘beaten-down’’ shares. Hence, there should be no abnormal volume surrounding the removal day. In Table 2, the MTRs during both periods are not signiﬁcantly different from one, which means that subsequent to the removal, the trading volume of the deleted stocks are close to the normal volume (pre-deletion volume). In the period 1976–1985, the MTR is 0.935% or 93.5% of the normal volume. Although the volume is lower following the deletion, the decrease in volume is not statistically or economically different than the trading volume in the pre-removal period of (31, 91). Similarly, for the period covering 1986–2005, the MTR of 0.974 is close to unity and is not statistically different from unity. Our results remain unchanged if we extend the event period up to 150 trading days after the effective day. This indicates that there is no abnormal trading volume around the time a company is taken Table 2.

Index Changes and Volume Effects.

S&P 500 Index Changes: Volume Effect Additions

197609-198512

198601-200512

Initial sample Final sample Turnover ratio

96 0.994

247 1.093

p-value

0.901

o.001

197609-198512

198601-200512

Initial sample Final sample Turnover ratio

18 0.935

60 0.974

p-value

0.533

0.714

Deletions

Signiﬁcance at the 1% level.

Changes in Trading Volume and Return Volatility

133

out of the S&P 500 Index in the full sample period September 1975 to December 2005 or in the subperiods – September 1975 to December 1985 as well as January 1986 through December 2005. In sum, we ﬁnd signiﬁcant increase in trading volume for stocks added to the S&P 500, but only during the period when dollar volume in S&P 500 Index-based derivatives (e.g., index futures and index options) is considered important. The excess volume is close to 10% of the normal trading volume in days before the actual inclusion. In the period from 1976 to 1985, index additions are not associated with trading volume change. As for index deletions, we ﬁnd results supporting our hypotheses that there is no abnormal trading volume when a company is removed from the Index. Upon further investigation, we document similar results when extending the event period from 120 to 150 trading days following the effective day. Thus, it can be argued that the volume effect associated with index membership changes is permanent.

VOLATILITY EFFECT We employ various measures of stock return volatility and ﬁnd that our results are independent of the methods used for estimating return variance. We investigate whether volatilities change as ﬁrms enter or leave the S&P 500 Index, in periods before and after 1986 when the dollar volume in S&P Index derivative products became signiﬁcant. First, simple stock return variances are computed using daily return series. The post-change (prechange) return variance is estimated using 60 trading days in the interval from day þ61 (day 31) to day þ120 (day 90). These time intervals correspond to the intervals in the volume effect analysis. Second, we look at a measure of idiosyncratic volatility – residual return variance (see Elliott, Van Ness, Walker, & Warr, 2006). The residual variance measures a ‘‘stock’s idiosyncratic risk and is the variance of the difference between the return on the ﬁrm’s stock and the return on the market portfolio.’’ The CRSP AMEX-NYSE-NASDAQ equally weighted index is used as a proxy of the market index.4 The ﬁnal measure of volatility is based on Nelson’s (1991) exponential GARCH model. We estimate the conditional variances for each additions and deletions using stock returns from day 150 to day þ150. After the return variances are calculated, we compare the distribution of variances before and after the actual S&P 500 Index changes. The Wilcoxon Signed Ranks test and paired t-test are used to determine whether there is a change

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Table 3.

Volatility Effect – Return Variance Measure. S&P 500 Index Changes: Volatility Effect Return variance

Additions sample September 1976 to December 1985 % positive ¼ 44.12% January 1986 to December 2005 % positive ¼ 63.75% Deletions sample September 1976 to December 1985 % positive ¼ 28.57% September 1976 to December 1985 % positive ¼ 47.27%

Pre-inclusion

Postinclusion

Paired t

0.000514

0.000459

0.000875

0.00114

4.41

0.000611

0.00090

0.584

0.00135

0.00132

0.196

1.09

Wilcoxon signed ranks (Z)

1.527 4.873

1.527

0.31

Signiﬁcance at the 1% level.

in return variance around the time a company is included in or removed from the Index. Tables 3–5 assess the volatilities of additions and deletions surrounding the effective day. We obtain the same results regardless of the methods we use to estimate volatilities. The daily return variances for added ﬁrms average 0.00088 and 0.0011 before and after the S&P 500 changes for the period 1986–2005, and average 0.00051 and 0.00046 before and after the effective day during 1976–1985. Both Wilcoxon Signed test and paired t-test indicate that added companies experience signiﬁcant increase in return variance [Wilcoxon Z-statistic (p-value) 4.87 (o0.01); paired t-statistic (p-value) 4.41 (o0.01)], but only over the period when index-based trading achieves record volume in 1986. Similar results are obtained in the analysis of residual return variance and EGARCH conditional variance. Both residual return variance and EGARCH conditional variance are signiﬁcantly higher after a stock is added to the S&P 500 over the period 1986 to 2005. Tables 3–5 also report the percentage of stocks that experience higher volatility. The percentage ranges from over 60% to about two-thirds of the additions. Thus, our results do not appear to be driven by a few outliers.

135

Changes in Trading Volume and Return Volatility

Table 4.

Volatility Effect – EGARCH Conditional Variance.

S&P 500 Index Changes: Volatility Effect – EGARCH Conditional Variance EGARCH conditional variance

Additions sample September 1976 to December 1985 % positive ¼ 46.60% January 1986 to December 2005 % positive ¼ 66.53% Deletions sample September 1976 to December 1985 % positive ¼ 42.86.57% January 1986 to December 2005 % positive ¼ 44.07%

Pre-inclusion

Post-inclusion

Paired t

0.000490

0.000465

0.000914

0.00111

4.70

0.00055

0.00080

0.776

0.19

0.00148

0.00195

1.39

0.249

1.12

Wilcoxon signed ranks (Z)

0.21 6.23

Signiﬁcance at the 10% level. Signiﬁcance at the 1% level.

Table 5.

Volatility Effect – Residual Return Variance.

S&P 500 Index Changes: Volatility Effect – Idiosyncratic Risk Residual return variance

Additions sample September 1976 to December 1985 % positive ¼ 43.14% January 1986 to December 2005 % positive ¼ 60.58% Deletions sample September 1976 to December 1985 % positive ¼ 38.10% January 1986 to December 2005 % positive ¼ 46.67%

Pre-inclusion

Post-inclusion

0.00044

0.00041

0.000766

0.000985

4.24

0.000505

0.000883

0.809

0.678

0.00121

0.00153

1.15

0.375

Signiﬁcance at the 1% level.

Paired t

0.62

Wilcoxon signed ranks (Z)

0.986 4.450

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In the period from September 1976 to December 1985, both Wilcoxon Signed test and paired t-test fail to reject the null hypothesis of no change in volatility for added ﬁrms. In addition, it is shown that volatility actually decreases for more than half of the added ﬁrms in this period. In general, the results indicate that for added ﬁrms, volatility does not change in this period before 1986. The results lend additional support to the notion that trading in the index derivative markets may lead to an increase in the volatility of the underlying shares as we ﬁnd dramatically different results in the two subperiods. Figs. 1 and 2 further show that post-inclusion volatility is higher following the introduction of index futures and option contracts. These show mean squared daily returns (MSDRs) from 150 days before index inclusion until 150 days after, for each of the stocks in the samples. Fig. 1 gives a clear picture that during the time prior to index derivatives trading becoming well established, there was no shift in volatility of returns from the time before the time after inclusion (the inclusion event is day 0). Fig. 2 gives a clear picture of an upward shift in volatility upon inclusion in the index, during the period after index derivatives trading becoming well established. To further understand how return volatility is inﬂuenced by trading volume in the derivative markets, we examine the observed volatility

8E-4

MSDR

6E-4

4E-4

2E-4

0E0 -150

-100

-50

0 Day

50

100

150

Fig. 1. S&P 500 Index Additions (September 1976–1985). This Chart Shows the Median Squared Daily Returns, Surrounding the Effective Date, for Stocks Added to the S&P 500 Index During the Period September 1976 to December 1985.

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Changes in Trading Volume and Return Volatility

5E-4

MSDR

4E-4

3E-4

2E-4

1E-4

-150

-100

-50

0

50

100

150

Day

Fig. 2. Index Additions (January 1986–December 2005). This Chart Shows the Median Squared Daily Returns, Surrounding the Effective Date, for Stocks Added to the S&P 500 Index During the Period January 1986 to December 2005.

dynamics around index additions by breaking down the two periods further. Table 6 illustrates shows the detailed year-to-year volatility comparisons for index additions, beginning with the year 1981, which is the full year before the introduction of S&P 500 Index futures contracts. Return volatility is not signiﬁcantly different before and after index additions, generally from 1981 to 1985. Tests of volatility change for these ﬁve years indicate that there is no change in return variance for additions. Except the year 1981, the remaining four years in this period are associated with more ﬁrms that have higher post-inclusion volatilities. In fact, more than 55% of added ﬁrms have higher post-inclusion volatilities, comparing with 34% of added ﬁrms in the period from 1976 to 1981. Immediately Around the Crash of 1987 Note that, in Table 6, the year 1986 is the ﬁrst year in the sample period that we ﬁnd higher return volatility (signiﬁcant at the 5% level) in the postinclusion period.5 There were a total of 14 additions in the ﬁnal sample and more than 70% of the added ﬁrms (10 of 14) experience higher post-change volatility. The EGARCH conditional (daily) variance for included stocks average 0.00048 and 0.0006 before and after the S&P500 changes for the year.

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Table 6. Year (Positive/ Negative) 1976–1980 (14/22) 1981 (3/11) 1982 (10/8) 1983 (3/2) 1984 (9/9) 1985 (9/6) 1986 (10/4) 1987 (3/1) 1988 (4/5) 1989 (14/4) 1990 (6/3) 1986–1990 (37/17) 1991–1995 (17/15) 1996–2005 (107/53)

Volatility Changes around 1982 and 1983 (EGARCH Volatility).

Pre-Inclusion Volatility

Post-Inclusion Volatility

Change in Volatility

Wilcoxon Signed Ranks

0.000491 0.000446 0.000509 0.000580 0.000540 0.000411 0.000483 0.000560 0.000267 0.000320 0.000478 0.000398 0.000490 0.001150

0.000407 0.000401 0.000600 0.000612 0.000500 0.000405 0.000600 0.000850 0.000234 0.000362 0.000662 0.000488 0.000571 0.001400

0.000084 0.000045 0.000091 0.000032 0.000040 0.000006 0.000117 0.000290 0.000033 0.000042 0.000184 0.000090 0.000081 0.000250

1.82 1.60 1.07 1.21 0.54 0.63 1.92 1.10 1.00 1.98 1.84 3.00 1.57 5.13

Signiﬁcance at the 10% level. Signiﬁcance at the 5% level. Signiﬁcance at the 1% level.

To examine years immediately following 1986, we must deal with the crash of 1987. Consistent with previous research, we removed ﬁrms with effective day that is 120 trading days around the crash. There are 14 index additions excluded from our sample for this reason. For the year 1987, there were only four added ﬁrms available for statistical analysis. We still report Wilcoxon signed test results, but we must interpret the results carefully. In 1987, three of the four newly included stocks show higher post-change volatility. The average pre- and post-inclusion variance are 0.00056 and 0.00085, however, the change in volatility is not statistically signiﬁcant probably due to small sample size. We did not ﬁnd volatility change in 1988, but ﬁnd signiﬁcant increases in return variance in 1989 and 1990. For the period 1986–1990, the post-inclusion volatility of 0.000488 is signiﬁcantly higher that the pre-inclusion volatility of 0.000398 at the 1% signiﬁcance level. The result supports our hypothesis that post-inclusion volatility is higher for index addition in this period when index trading volume reached record highs. It is interesting to note that in the period preceding 1982, the postinclusion volatility is actually lower than the pre-inclusion volatility. The

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Wilcoxon test shows that the negative change in volatility is statistically signiﬁcant. Table 6 also indicates signiﬁcantly higher post-change volatility for added ﬁrms in periods following 1990. Both periods 1991–1995 and 1996–2005 show increases in conditional variances with the later period providing more signiﬁcant increase. Despite the activation of circuit breakers and other forms of exchange trading curbs following the crash of 1987, we continue to ﬁnd higher post-inclusion volatility in our sample of S&P 500 additions. For companies deleted from the Index, we ﬁnd evidence in support of our hypotheses that no change in volatility is associated with deletions. As we expect, in a market value-weighted index like the S&P 500, ﬁrms being excluded from the Index typically represent an extremely small fraction of the Index, as a result volatility should not change around removal days regardless of whether or not there is signiﬁcant trading in the index derivative markets. We ﬁnd no signiﬁcant change in volatility of the ﬁrms that were removed from the S&P 500. Less than 50% (ranging from 28.57% in the period 1976– 1985 to 47.27% in the 1986–2005 period) of the deleted ﬁrms experience an increase in return volatility although the average post-deletion volatilities are generally higher than pre-deletion volatilities. Figs. 3 and 4 gives clear

4E-4

MSDR

3E-4

2E-4

1E-4

0E0 -150

-100

-50

0

50

100

150

Day

Fig. 3. Index Deletions (1976–1985). This Chart Shows the Median Squared Daily Returns, Surrounding the Effective Date, for Stocks Deleted from the S&P 500 Index During the Period January 1976 to December 1985.

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8E-4

MSDR

6E-4

4E-4

2E-4

0E0 -150

-100

-50

0

50

100

150

Day

Fig. 4. Index Deletions (1986–2005). This Chart Shows the Median Squared Daily Returns, Surrounding the Effective Date, for Stocks Deleted from the S&P 500 Index During the Period January 1986 to December 2005.

pictures that return volatility does not change after a ﬁrm is removed from the S&P 500 Index. Additionally, we examine the relationship between turnover (percentage) change and change in volatility. We calculate percentage turnover change by subtracting pre-inclusion turnover from post-inclusion turnover and divide this ratio by the pre-inclusion turnover. The ‘‘most active’’ group consists of the top decile of ﬁrms experiencing highest turnover change. The second group includes ﬁrms that experience no change in turnover, and the third group includes ﬁrms that experience strongest turnover decline. Table 7 reports the results, indicating that there is a positive relationship between turnover and return variance. The higher the percentage turnover, the greater the volatility increase.

MARKET RISK MEASURE The results concerning total variability of returns raise the question of whether there is any change in systematic (market) risk surrounding Index composition changes. We estimate stock betas around the effective day using the methodology speciﬁed in Scholes and Williams (1977), to cope

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Table 7. Linking Turnover and Volatility. S&P 500 Index Changes Portfolio group

Mean % turnover change

Pre-inclusion volatility

Post-inclusion volatility

Paired tstatistic

114.60

0.001047

0.001953

4.04

3.53

0.000688

0.000794

1.13

0.00116

0.00124

0.64

Group 1 turnover increase Group 2 turnover no change Group 3 turnover decrease

–25.84

Signiﬁcantly different from zero at the 1% level. Signiﬁcantly different from zero at the 5% level.

Table 8.

Market Risk.

S&P 500 Index Changes Stock Beta

Additions 1976–1985 1986–2005 Deletions 1976–1985 1986–2005

Pre-inclusion beta

Post-inclusion beta

Change in beta

Wilcoxon signed ranks

1.28 1.25

1.23 1.29

0.05 0.04

0.705 1.35

0.72 1.05

0.65 0.89

0.07 0.16

0.487 0.715

Signiﬁcance at the 10% level.

with the issue of non-synchronous trading. The Scholes and Williams beta has been shown to outperform ordinary least squares (OLS) beta in a number of empirical studies. For additions, we estimate pre- and postchange betas using daily returns in the [31, 150] and [þ31, þ150] windows around the effective day. But for deletions, we begin with day 16 and day þ16 and estimate betas using 120 trading day returns, due to data limitations of the deleted companies. The Wilcoxon Signed test is used to determine whether there is beta shift around the actual Index inclusion or deletion. Table 8 presents the results regarding beta. The mean pre-inclusion beta is not signiﬁcantly different from the post-inclusion beta for the sample of Index additions in the period covering 1976–1985. But, over the period

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1986–2005 we ﬁnd a small increase in beta, statistically (not economically signiﬁcant) signiﬁcant at the 10% level.6 This ﬁnding is similar to results documented in Vijh (1994) and Barberis, Shleifer, and Wurgler (2005). The analysis of beta adds to our understanding of risk change associated with index additions. On the contrary, we ﬁnd no change in beta around the event of index removals.

EXCHANGE TRADING COMPARED WITH OVER-THE-COUNTER Furthermore, Table 9 shows the volatility tests for the NYSE and Nasdaq stocks being added to the S&P 500 Index. The results show that both groups of stocks experience signiﬁcant increase in return volatility following inclusion. The results indicate that Nasdaq-based ﬁrms experience even higher increase in volatility than NYSE-listed companies.7 Table 10 shows the analysis of monthly return variance. The monthly return volatility of an added ﬁrm increases from 0.0177 to 0.021 [Wilcoxon Z-statistic (p-value) 3.31 (o0.01)]. The results indicate that index inclusions experience signiﬁcant increase in long-interval return volatility measure. This ﬁnding is consistent with the hypothesis that derivative trading fundamentally destabilizes the underlying securities (see Harris, 1989).8

RELATED RESEARCH Harris (1989) discusses two paradigms describing the impact derivative markets have on the volatility of the spot markets. First, large transactions Table 9.

NYSE vs NASDAQ. Index Additions

Exchange or market

No. of Firms

Pre-inclusion Volatility

Postinclusion Volatility

Paired t-statistic

Wilcoxon Z

NYSE % positive ¼ 63.40%

153

0.00058

0.00071

3.63

3.79

Nasdaq % positive ¼ 77.03%

74

0.0014

0.00194

5.53

5.26

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Changes in Trading Volume and Return Volatility

Monthly Return Variance.

Table 10.

Index Additions Monthly return variance Period

197609|198512 198601|200512

Pre-inclusion volatility

Post-inclusion volatility

Paired t-statistic

Wilcoxon Z

0.0111 0.0177

0.0106 0.0210

0.54 2.47

0.29 3.31

Signiﬁcantly different from zero at the 1% level.

in the derivatives markets may result in transaction spillover to the underlying spot markets, inducing liquidity pressure. In other words, trade in the derivative contracts may cause related transactions in the cash markets that are often too large to be absorbed by the market (i.e., order imbalances). Such transactions, according to Harris (1989) and Vijh (1994) may be associated with mechanical arbitrage activities, portfolio insurance operations, and program trading. The notion of price pressure suggests that price changes are transitory and may be attributed to temporary trading imbalances, induced by index-based trading programs. This argument implies that return volatility measured over short intervals (such as daily) will be greater for the added stocks subsequent to the effective day, but that return volatility estimated over longer intervals (i.e., weekly and monthly) will be the same. This prediction is consistent with the price pressure hypothesis in that stock prices revert close to pre-announcement levels (see Harris & Gurel, 1986). The second paradigm asserts that trading in futures and options markets fundamentally destabilizes the value formation process in cash markets. Under this framework, both short- and long-interval measures of return volatility should be larger after a stock is ofﬁcially included in the index portfolio. In other words, large ongoing transactions resulting from arbitrage, program trading, and portfolio insurance operations cause permanent changes in prices of the underlying securities. The change in long-interval volatility measures may be associated with long-run demand shift of the component stocks.9 There are two main lines of reasoning to account for the change in volume and volatility, resulting from index derivatives transactions. One interpretation is that stock return variability is positively related to the information arrivals accompanied by trading volume. This argument is based on how

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information is incorporated into security prices. As the market digests new information, prices are adjusted to reﬂect a new set of available information. The other is based on how market makers respond to large block trades caused by arbitrage. This is a market microstructure perspective, looking at the volatility of price changes as market makers adjust prices based on their portfolio risk and inventory risk. Prices may also change in response to liquidity demand requiring market makers to provide immediate transactions when large transactions come to the marketplace. Both interpretation suggest also a positive relationship between trading volume and return volatility.

SPILLOVER Stoll and Whaley (1987) look at market-wide trading activities and stock price changes around derivative expiration days. They ﬁnd that trading volume and volatility of the S&P 500 Index increase signiﬁcantly around expiration days. These volume and price effects, though, are not associated with non-S&P stocks. French and Roll (1986) have investigated how stock return volatility varies in response to different levels of trading. They document higher stock volatilities when the stock market is open for trading, with non-market session hours are linked to lower volatility. Their ﬁndings are consistent with the positive volume–volatility relationship. In the information framework, an increase in trading is typically accompanied by additional information that is being priced in the marketplace. Ross (1989) suggests that the volatility of prices is directly related to the rate of information ﬂowing into the market. Similarly, Cox (1976), Copeland (1976), Epps and Epps (1976), Tauchen and Pitts (1983), and Jennings and Barry (1983) provide insights as to whether price changes are linked to information arrivals. These models provide insights as to how information production is related to price volatility. Derivative markets offer additional channels for information to be disseminated, implying that information is more likely to be discovered and transmitted between the markets. Security prices are adjusted to reﬂect new information, and thus, price movements may directly correspond to information arrivals. Cox (1976) investigates the information effect of futures trading and whether there is a relationship between information production and the prices of the spot assets. Cox demonstrates that futures trading activities are associated with an increase in information production of the underlying securities and prices of the spot assets respond quickly to the updated information set. Vijh (1994) points out that the large trading in S&P 500

Changes in Trading Volume and Return Volatility

145

products may affect prices because, ‘‘Simply by chance the buy orders will dominate sell orders on certain days while the sell orders will dominate buy orders on other days.’’ In addition, Dufﬁe, Kupiec, and White (1990) argue that index arbitrage may cause price changes as large transactions are executed in the spot markets, resulting in reduced liquidity. Stoll and Whaley (1987) and French and Roll (1986) show that stock variance is strongly related to trading activities. Moreover, derivative trading is also subject to margin calls that at times of order imbalance may trigger additional price pressure. Santoni (1987) documents an inverse relation between S&P 500 Index futures trading volume and volatility of the S&P 500 market index, suggesting that an increase in futures trading activities leads to a reduction in spot market volatility. Moreover, Bessembinder and Seguin (1992) provide evidence that stock market volatility is negatively correlated to (total) trading volume in the cash markets. Trades in the futures markets are directly related to the trading volume in the underlying spot securities. However, when the authors decompose trading activities, they ﬁnd that only ‘‘unexpected’’ trading volume in the spot securities is positively correlated with volatility. Expected changes in volume do not affect volatility. Moreover, Edwards (1988a, 1988b) ﬁnds that the introduction of futures contracts is not related to volatility changes in the underlying cash markets.

DESTABILIZATION Stein (1987) contends that fewer informed traders may be attracted to derivative markets. The increase in the number of noise traders may reduce the information content of the market prices, resulting in price destabilization.10 Index derivative transactions are likely to increase information production and the rate of information transmitted to the market. As a result, trading in the derivative markets may be related to volatility changes in the spot assets. Pruitt and Wei (1989) provide further evidence supporting the short-term price effect (price pressure). Their study shows that institutional ownership increases following a ﬁrm’s inclusion in the S&P 500 Index. As institutional investors are associated with larger trading transactions, it is more likely to cause temporary order imbalances, which in turn lead to higher price changes. Jones, Kaul, and Lipson (1994) decomposes daily trading volume into number of trades and average trade size and examines their impact on the volatility of stocks traded in the NASDAQ national market. They ﬁnd that

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number of transactions is the most important measure of trading activity that explains volatility changes although size of trade is also an inﬂuencing factor. Ho and Macris (1984) suggest that the market makers adjust bid-ask spreads when they face large order ﬂows. Market makers, in response to liquidity constraints, often carry additional inventory to cope with possible order imbalances, resulting in suboptimal inventory holdings. This inventory cost is then reﬂected in security prices resulting in short-term price changes. Several other studies have also argue that large transactions tend to increase costs associated with market making services and these costs are associated with stock prices being deviated from their intrinsic (fundamental) values.

THIS STUDY’S RESULTS IN PERSPECTIVE Previous studies investigating the effects of index composition changes suggest that the volatility of the added or deleted ﬁrm does not change around the announcement or effective date, and thus, the observed price response to the event of index changes cannot be attributed to change in risk.11 Dhillon and Johnson (1991) study the prices of options for companies added to the S&P 500 Index, during the period 1984–1988. The results indicate that, around the announcement date, call prices increase but put prices decline, leading to inconclusive evidence as to whether return variances for the added ﬁrms change. Studies in index composition changes, following Dhillon and Johnson (1991), have generally regarded index change announcements as non-volatility induced events. Our study is closely related to a body of literature investigating the impact of index trading strategies on the volatility of the underlying securities. Traditional ﬁnance theory suggests that derivative markets are linked to the underlying spot market by mechanical arbitrage trading (see Grossman, 1988). When cash securities are overpriced (underpriced) relative to the derivative markets, arbitrageurs could sell (buy) the cash assets and take long (short) positions in the derivatives. These arbitrage transactions continue to take place until both markets converge to equilibrium. Arbitrage transactions tend to create additional large order ﬂows in the underlying market as the arbitrage mechanism works to correct prices. Previous empirical studies have examined the relationship between trading volume and volatility. Karpoff (1987) and Gallant, Rossi, and Tauchen (1992) have shown a positive relation between volume and the absolute value of price changes. Thus, it can be argued that arbitrage transactions may result in abnormal trading, which in turn causes price movements.

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In this study, we build on the work of Harris (1989) and Vijh (1994) and directly examine trading volume and security return volatility for ﬁrms that are added to and deleted from the S&P 500 Index from September 1976 to December 2005. We are particularly interested in the trading volume and volatility of index additions and deletions around effective date, the ﬁrst day when the actual change is reﬂected in the index composition. To investigate the impact of index derivatives, the full sample period is partitioned into two subperiods, covering the period September 1976–1985 and 1986–2005. This ﬁrst subperiod is related to a period of relative lower index derivative dollar volume since the S&P index futures (options) were not available until 1982 (1983). Subsequent to the ﬁrst subperiod, the dollar volume on the index derivatives contracts reached record highs. Thus, the second subperiod focuses on the effects of transactions (such as index arbitrage, portfolio insurance, and program trading). Our main goal is to determine whether the index trading volume affects the turnover and return volatility of the underlying stocks. Unlike earlier studies, we employ a list of index additions and deletions to study the impact of index trading strategies.12

CONCLUSION This study investigates the trading volume and volatility of companies added to and deleted from the S&P 500 Index, in the period following the introduction of S&P 500 Index futures and options (1986–2005). Following the empirical framework of Vijh (1994), we ﬁnd signiﬁcant increase in both trading volume and return volatility after a ﬁrm is included in the index. This result is not found during the period prior to the introduction of index derivative securities. To our best knowledge, we are the ﬁrst to document an increase in return volatility associated with index addition. Upon further investigation, we ﬁnd that both daily and monthly return variances increase for the added ﬁrms, indicating that the price effect due to index changes is not solely due to short-term price pressure. The empirical evidence supports a long-term downward sloping demand curve for stocks. We ascribe the change in risk to index arbitrage transactions although we cannot rule out other factors (such as portfolio insurance operations and program trading) inﬂuencing the volatility of the added ﬁrms. Furthermore, we document a positive relationship between turnover change and volatility change – the greater the change in turnover, the higher the change in return volatility following inclusion. This provides evidence

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that volatility of the added ﬁrms is affected by trading volume resulting from index trading strategies. Our result is consistent with Karpoff (1987). For the deleted stocks, no signiﬁcant changes in trading volume and return volatility are found for deleted ﬁrms. We argue that the market value of these ﬁrms relative to the market value of the index become extremely small at the time they are removed. Since the S&P 500 Index is a market value-weighted portfolio, the deleted ﬁrms are not signiﬁcantly affected by index trading as their index weights become trivial.

NOTES 1. The quote is from Harris (1989, p. 1155). 2. The introduction of the S&P 500 ETF in 1993 allows index arbitrageurs to trade the index portfolio more easily. Before 1993, most index arbitrage transactions are carried out using program trading. 3. We use the NYSE market volume, consistent with earlier studies. 4. We obtain similar results using the CRSP AMEX-NYSE-NASDAQ valueweighted index. 5. We were able to conﬁrm Dhillon and Johnson’s (1991) results, using index inclusion data from 1984–1988. Volatility of the added ﬁrms does not change during this period. 6. We obtained similar results with regard to systematic risk, using the conventional OLS approach. 7. Elliott and Warr (2003) document that Nasdaq-based ﬁrms experience much higher excess returns than NYSE-based companies upon their inclusion into the S&P 500. They argue that the result may be related to the unique specialist program of the NYSE. 8. Harris (1989) supports the price pressure hypothesis as he ﬁnds that daily (not weekly) volatility measures are higher for S&P 500 stocks. 9. See Shleifer (1986). In general, previous studies investigating the S&P Effect support the long-term downward sloping demand curves for stock hypothesis. 10. The question whether trade in index derivatives destabilizes the underlying has been debated in the literature. Previous studies have also found that derivatives trading decreases the return volatility of the spot securities. Leading examples include Edwards (1988a, 1988b), Conrad (1989), and Bessembinder and Seguin (1992). 11. In the current literature, change in risk is not found to be associated with index additions and deletions. There are several competing hypotheses explaining the market reactions to the announcement (effective) of index compositions. They include price pressure, imperfect substitutes, liquidity, information content, and investor recognition. A recent analytical review of the related studies can be found in Elliott et al. (2006). 12. Harris (1989) compares S&P 500 stock return volatilities to the volatilities of a matched set of stocks, after controlling for cross-sectional differences in ﬁrm attributes such as size, beta, and liquidity.

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Harris, L. (1989). S&P 500 cash stock price volatilities. Journal of Finance, 44, 1155–1175. Harris, L., & Gurel, E. (1986). Price and volume effects associated with changes in the S&P 500 list: New evidence for the existence of price pressures. Journal of Finance, 41(4), 815–829. Ho, T., & Macris, R. (1984). Dealer bid–ask quotes and transaction prices: An empirical study of some AMEX options. Journal of Finance, 39, 23–45. Jennings, R. H., & Barry, C. (1983). Information dissemination and portfolio choice. Journal of Financial and Quantitative Analysis, 18, 1–19. Jones, C. M., Kaul, G., & Lipson, M. L. (1994). Transactions, volume, and volatility. Review of Financial Studies, 7, 631–651. Karpoff, J. M. (1987). The relation between price changes and trading volume: A survey. Journal of Financial and Quantitative Analysis, 22, 109–126. Nelson, D. (1991). Conditional heteroscedasticity in asset returns: A new approach. Econometrica, 59, 347–370. Pruitt, S. W., & Wei, K. C. J. (1989). Institutional ownership and changes in the S&P 500. Journal of Finance, 44(2), 509–513. Ross, S. A. (1989). Information and volatility: The no-arbitrage martingale approach to timing and resolution irrelevancy. Journal of Finance, 44, 1–17. Santoni, G. J. (1987). Has programmed trading made stock prices more volatile? Federal Reserve Bank of St. Louis. Review, 69, 18–29. Scholes, M., & Williams, J. (1977). Estimating beat from non-synchronous data. Journal of Financial Economics, 5, 309–327. Shleifer, A. (1986). Do demand curves for stocks slope down. Journal of Finance, 41, 579–590. Stein, J. C. (1987). Information externalities and welfare-reducing speculation. Journal of Political Economy, 95, 1123–1145. Stoll, H. R., & Whaley, R. E. (1987). Expiration day effects of index options and futures. Financial Analysts Journal, 43, 16–28. Tauchen, G. E., & Pitts, M. (1983). The price variability-volume relationship on speculative markets. Econometrica, 51, 485–505. Vijh, A. (1994). S&P 500 trading strategies and stock betas. Review of Financial Studies, 7, 215–251. Wurgler, J., & Zhuravskaya, E. (2002). Does arbitrage ﬂatten demand curves for stocks? Journal of Business, 75(4), 583–608.

APPENDIX. SAMPLE AND DATA The initial sample consists of all additions and deletions occurring between September 1976 and December 2005. We gathered information about these changes from two sources. First, we obtained index changes for the period September 1976 through December 2000 from Jeffrey Wurgler. This dataset was used in two earlier S&P Index studies – Wurgler and Zhuravskaya (2002) and Barberis et al. (2005). The remaining data on index changes were collected from the Standard and Poor’s company website. The sample period begins in September 1976 because before that time, S&P did not publicly announce index changes.

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The initial sample consists of 181 additions for the September 1976 to December 1985 period and 515 additions for the January 1986 to December 2005 period. For the entire sample period September 1976 to December 2005), there are 696 additions and 696 deletions. There are, on average, between 20 and 25 stocks added to (deleted from) the S&P 500 Index each year. Since S&P index additions and deletions are often associated with other contemporaneous corporate events (e.g., spin-offs, merger and acquisition, and restructuring), we use the following set of criteria to screen out ﬁrms that are not pure cases of inclusion or deletion. First, we exclude index changes resulted from merger, acquisition, or restructuring. Second, we remove index additions involving merger/acquisition transactions that do not actually include a new company to the index portfolio. For instance, when a non-index company acquires an S&P 500 ﬁrm and is subsequently added to the index, we exclude such addition from our sample. To make certain that we have a clean sample in the analysis of trading volume and return volatility, we search the LexisNexis Academic database for confounding events (such as earnings, dividend, split, ﬁnancing/ investment announcements during the period from three days before the announcement date to seven days subsequent to the effective date (see Denis, McConnell, Ovtchinnikov, & Yu, 2003). In addition, we require that the there must be sufﬁcient stock returns, trading volume, and shares outstanding data around the effective day. For the trading volume analysis, the post-change (event) period covers the interval [þ61, þ120]. We also extend the post-inclusion turnover ratio up to 150 trading days after the effective. No index additions in our sample survive less than 150 days. For our volatility and market risk tests, the required daily returns span 300 trading day surround the effective day. The ﬁnal (clean) sample includes 364 additions and 90 deletions. The Center for Research in Security Prices (CRSP) database is used to obtain daily returns, daily trading volume, and shares outstanding for the ﬁrms used in the analysis. We obtain NYSE trading volume from the historical data archive library on its website.

Methodology Abnormal Volume Measurement We analyze trading volume around the effective date of S&P 500 Index changes using procedure similar to those in Harris and Gurel (1986), Elliott and Warr (2003), and Chen et al. (2004). Our purpose is to determine

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whether excess turnover is associated with index changes, before and after the introduction of S&P 500 Index-based trading strategies. Following Chen et al. (2004, p. 1907), we use turnover (trading volume divided by shares outstanding) instead of trading volume, so that unusually high volume in a few large stocks does not disproportionately affect the market volume. The volume turnover is calculated by Eq. (A2). The denominator is the market-adjusted volume during the ‘‘estimation’’ period. The estimation period covers the interval, [61, 120]. The market-adjusted turnover is the ratio of individual stock volume divided by market volume. The numerator is the ‘‘event’’ period turnover adjusted by total market volume during the post-change interval of [þ61, þ120]. In Eq. (A2), Tit is the volume turnover for stock i at time t, the subscript m refers to the market index. Consistent with previous studies, we use the NYSE trading volume as a proxy for market level volume. The pre- (post-change) turnover ratio is the 60-day average trading turnover (with a minimum of 30 days) beginning 61 trading days before (after) the effective date. Thus, trading before (after) the effective date must last for at least 90 days. We calculate the pre- and postchange turnover ratio for each index change in our sample and test whether the MTR across all index changes is signiﬁcantly different from unity. T it ðTurnoverÞ ¼

V it , Sit

(A1)

EDþ120 P

T it T mt t¼61 TRi ðTurnover RatioÞ ¼ ED120 , P T it t¼61 T mt MTR ¼

N X

TRi

(A2)

(A3)

i¼1

Volatility Measurement We investigate four measures of stock return volatility surrounding the event – variance of daily stock returns, residual standard deviation, and EGARCH conditional variance. For each index change, we calculate variance of stock returns from the period prior to (subsequent) the effective day. We use Elliott et al. (2006) idiosyncratic expression to measure residual variance: ‘‘the residual standard deviation measures the stock’s idiosyncratic risk and is the standard deviation of the difference between the return on the

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ﬁrm’s stock and the return on the CRSP Equally-weighted portfolio.’’ For the pre-change (normal) period we measure this difference over the [61, 120] window, and for the post-change period we use the period [þ61, þ120]. We then compare each pair of pre- and post-change return variances in our sample. The autoregressive conditional heteroscedasticity (ARCH) was ﬁrst developed by Engle (1982). Later, the generalized (GARCH) form of ARCH, proposed by Bollerslev (1986), allows for ‘‘lagged variances and the further lagging of the error term.’’ Nelson (1991) further extends the GARCH form to incorporate ‘‘volatility clustering’’ and the ‘‘leverage effect’’ that exists in ﬁnancial data. The speciﬁcation proposed, known as exponential GARCH (EGARCH), allows for an asymmetric response to positive and negative price changes. The general EGARCH model begins with a simple univariate framework where no other variables (except past values of returns) can be used in predicting mean returns. The mean return process can generally be expressed as rt ¼ m þ fðLÞrt1 þ t ;

t ¼ r þ 1; . . . ; T

(A4)

where f(L) is a polynomial in the lag operator L, that is, f(L) ¼ f1þf2LþyþfpLp1. The error term et describes the unpredictable component of the returns. A common assumption about its behavior is that it follows a GARCH-type process, namely that t jI t1 Nð0; s2t Þ where It1 is the information available at time t1 and s2t follows a process s2t ¼ a0 þ at 2t1 þ b1 s2t1 in the GARCH (1, 1) representation (Bollerslev, 1986), and rﬃﬃﬃ 2 2 2 t1 t1 log st ¼ o þ b log st1 þ a þg st1 st1 p

(A5)

in the Nelson (1991) exponential GARCH [EGARCH(1, 1)] representation. The variance equation shows that the model is basically a ‘‘weighted moving average’’ of past volatility (one-period lag) and residuals from the mean regression estimations. ‘‘A typical characteristic of asset returns is volatility clustering where one period of high volatility is followed by more of the same and then successive periods of low volatility ensue’’ (Bollerslev, Chou, & Kroner, 1992). The EGARCH model offers several advantages over other ARCH models.

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First, the EGARCH model can deal with volatility clustering and the leverage effect. Second, unlike GARCH, the EGARCH model ‘‘imposes no positive constraints on estimated parameters and explicitly accounts for asymmetry in asset return volatility, thereby avoiding possible misspeciﬁcation in the volatility process’’ (Glosten, Jaganathan, & Runkle, 1993). Furthermore, Nelson (1991) points out that the ‘‘EGARCH model also allows for a general probability density function (i.e., generalized error distribution, GED), which allows for distributions involving non-normality.’’ This approach makes fewer assumptions about the distribution of the measured volatility series. As Bollerslev et al. (1992) and several others suggest ‘‘imposing the normality assumption could bias the estimates.’’ We use both the Berndt-Hall-Hall-Hausman (hereafter, BHHH) and Marquardt optimization algorithms in the iteration process. The BHHH method outperforms the Marquardt approach, in terms of the percentage of processes that were successfully converged. In our experiments, all of our EGARCH conditional variances converged using the BHHH approach, however, less than 75% successfully converged under the Marquardt algorithm. However, our signiﬁcance level for the documented increase in variance does not change using either procedure. We thus report only the results of the BHHH optimization algorithms. Market Risk Following Scholes and Williams (1977), we estimate stock betas by adjusting for nonsynchronous trading (infrequent trading). This methodology has been shown to outperform the conventional OLS technique. Scholes and Williams (1977) propose a model to incorporate nonsynchronous trading. Infrequent trading may cause a bias in beta estimation procedure. Lo and MacKinlay (1990) contend that ‘‘thin trading induces a negative autocorrelation in stock returns, an overstatement of the return variance, and a downward bias in the market risk.’’ To deal with the problems, Scholes and Williams (1977) derive a consistent estimate for beta: ^ ^þ ^ ^b ¼ bi þ bi þ bi i 1 þ 2r^ m

(A6)

þ where b^i , b^i , and b^i , respectively, are the OLS estimates of the slopes of regression of asset i’s returns on one-period lag, concurrent, and oneperiod ahead of the market index; r^ m is the ﬁrst-order autocorrelation of the index return.

MODELLING THE US SWAP SPREAD Hon-Lun Chung, Wai-Sum Chan and Jonathan A. Batten ABSTRACT The dynamics between ﬁve-year US Treasury bonds and interest rate swaps are examined using bivariate threshold autoregressive (BTAR) models to determine the drivers of spread changes and the nature of the lead–lag relation between the two instruments. This model is able to identify the economic – or threshold – value that market participants consider signiﬁcant before realigning their portfolios. Speciﬁcally, three different regimes are identiﬁed: when the swap spread in the previous week is either high or low, the Treasury bond market leads the swap market. However, when the swap spread is low, none of the markets leads each other. Thus, yield movements are shown to be governed by the direction and magnitude of the change in the swap spread, which in turn provides an economic insight into the rebalancing between swap and bond portfolios.

1. INTRODUCTION Trading in Treasury bonds and interest rate swaps comprise two key activities in the global ﬁnancial marketplace. Treasury bonds are regarded as risk-free securities and carry the highest ratings in local markets by rating Research in Finance, Volume 26, 155–181 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026010

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agencies,1 whereas interest rate swaps are derivative contracts traded in over-the-counter markets. Importantly, counterparties through the use of collateral support and netting arrangements can eliminate almost, if not entirely, the credit risk associated with these instruments. In fact recent academic research (Collin-Dufresne & Solnik, 2001; Feldhutter & Lando, 2008 among others) assumes that the swap contract is free of default risk. The swap comprises counterparties with two offsetting sets of underlying cash-ﬂows that generally contain a ﬁxed and ﬂoating rate component.2 Financial market participants can use the swap to hedge existing interest rate exposure, or for speculative interest rate risk taking. For example, if interest rates are expected to decline, investors ‘buy’ or invest in the ﬁxed rate, whereas if they expect interest rates to increase, they buy or invest in the ﬂoating rate. The reverse is true of those who wish to borrow. Nonetheless, the swap spread, representing the yield difference between a bond and a swap of equivalent maturity, is affected by macroeconomic sentiment – such as inﬂation expectations, or business cycle effects (Cortes, 2006; Ito, 2007). This effect is most apparent during periods of economic downturn when spreads typically widen due to portfolio rebalancing into Treasury bonds and away from riskier instruments; a result consistent with an increase in risk-aversion. During boom periods, when the probability of default in corporate bond markets declines, sentiment concerning interest rate direction becomes the primary concern, although the effect is known to vary based on swap maturity (Huang, Chen, & Camacho, 2008) and underlying interest rate volatility (Malhotra, Bhargava, & Chaudhry, 2005). Overall, the interplay between default and interest rate expectations results in time-varying spreads, which have critical impacts for the ﬁnancial decision making by corporations, traders and portfolio managers The objective of this chapter is to determine the exact nature of the relation between swaps and risk-free bonds through the application of nonlinear threshold models, which previously have been widely used for investigating the dynamics within currency and stock markets (Chappell, Mistry, & Ellis, 1996; Tsay, 1998). These techniques are speciﬁcally applied to an investigation of the lead–lag dynamics between the US Treasury bond and US$-denominated swap markets where the change in the swap spread is the dependent variable. US Treasury bonds comprise the largest government bond market in the world, whereas US-denominated interest rate swaps comprise daily turnover in excess of US$81.3 trillion (BIS, 2008). We build upon earlier investigations in interest rate and swap markets (e.g. Malhotra et al., 2005; Ito, 2007) by utilising a new class of bivariate threshold autoregressive (BTAR) models (Chan & Cheung, 2005) to capture

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the regime-switching, lead–lag dynamics that exist between the US bond and swap markets. The BTAR model is chosen for a number of reasons. First, this model provides an exact measure of the economic incentive for a portfolio investor to shift funds, in this case between two ﬁnancial instruments – the swap and a ﬁxed rate bond of equivalent maturity. This measure, termed a ‘threshold’ or ‘critical’ value in the BTAR model, may also be interpreted as the hidden cost necessary for ﬁnancial market participants to shift between these two asset classes. Second, if indeed these thresholds values can be identiﬁed, then they can be used to anticipate the change in the yield curve dynamics. This will allow traders to be more cautious in managing risk and help policymakers and central banks ﬁne tune monetary policies. Third, the threshold value can be expressed in terms of interest rate percentages or ‘basis points’; a number that can be easily understood and interpreted by ﬁnancial markets. This is quite different from the information provided by other models, such as Markov Switching Models (Hamilton, 1996). The study uses weekly ﬁve-year US Treasury bond yield and interest rate swap rates from January 1995 to December 2004. This period is a representative period for both markets with the swap market having fully matured since its inception in the early 1980s. This period also provides a novel setting for investigation of the impact of regime change since it includes the longest period of economic expansion in the United States, the Russian bond default, the near failure of long-term capital management (LTCM) and historically low Fed Funds and Treasury bond yields. The chapter is set out as follows: in Section 2, a brief review of the recent literature on spread trading and modelling is provided; then in Section 3, the data used in the study and the six possible scenarios of bond and swap price movements are explained. Section 4 provides details on the lead–lag modelling techniques utilised as well as the BTAR model. The results are presented in Section 5, which also allows for some concluding remarks.

2. LITERATURE REVIEW A number of studies have examined the lead–lag relationship between different markets, and the majority use the intraday price data from stock indices and stock index futures. For example, Kawaller, Koch, and Koch (1987) examine the intraday price relationship between the Standard & Poor’s (S&P) 500 futures and the S&P 500 Index, whereas Harris (1989) studies the ﬁve-minute changes in the S&P 500 Index and futures contracts

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over the 10-day period surrounding the October 1987 stock market crash. Stoll and Whaley (1990) explore the time-series properties of the 5-min intraday returns of the stock index and stock index futures contracts, and Chan (1992) examines the asymmetric lead–lag relationship between futures and component stocks. Thus, the current study provides valuable new insights into the interest rate and swap markets, which are generally overlooked by academic investigators. A common question raised by researchers’ concerns is whether the lead– lag relationship between different markets changes over time as a result of changing exogenous or endogenous factors. That is, the way in which different instruments interact may be a regime-dependent phenomenon that varies if the internal or external environment changes. To help address this question, Tsay (1998) studies the relationship between three-month and three-year interest rates and uses the difference between the logarithms of the two interest rates as the threshold variable. He identiﬁes three different regimes that represent economic expansion, a stable economy and economic slowdown. Although Ito (2007) investigates interest rate spreads on Japanese-denominated swaps and Malhotra et al. (2005) does so in the US setting, neither consider regime dependent threshold values or utilise BTAR modelling techniques. More recently, Huang et al. (2008) investigate the determinants of variations in the yield spreads between Japanese yen interest rate swaps and Japan government bonds for a similar period to this study (from 1997 to 2005), although they use a smooth transition vector autoregressive model to analyse the impact of various economic shocks on swap spreads. They ﬁnd that GARCH volatility is useful for identifying regime change. More speciﬁcally they identify the end of the Japanese banking crisis as a signiﬁcant control variable and that the impact of economic shocks on swap spreads varies across maturity and regimes. This ﬁnding is also consistent with Ito (2007) who ﬁnds the effect of Treasury interest rates and the term structure yield difference between long and short rates (the slope of the yield curve) on swap spreads also varies by spread maturity. However, the economic implications of these results are moderated by the failure to include threshold values. The class of threshold autoregressive (TAR) models (Tong, 1978, 1983) has now been widely employed in the literature to explain the various empirical nonlinear phenomena that are observed in many ﬁnancial and economic time series. Yadav, Pope, and Paudyal (1994) suggest that TAR models are potentially of interest whenever ﬁnancial decisions are triggered by the threshold values of a control variable, such as arbitrage in the presence of

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transaction costs and market interventions by regulators. Brooks and Garrett (2002) use self-exciting threshold autoregressive models (SETAR) to explain the daily dynamics of the FTSE 100 index basis. To date this is the ﬁrst study to apply these techniques to understand the dynamics of the relation between interest rate swaps and the underlying ﬁxed rate bond.

3. DATA The weekly closing rates of US Treasury bond yields and US$ interest rate swap rates from January 1995 to December 2004 are used in this study. In line with the market practice for end of week portfolio realignment, we use weekly data. Doing so, also overcomes stickiness that is otherwise evident in daily data. Thus, signiﬁcant changes in the swap spread is clearer in weekly data, whereas monthly observations lose information, and the number of observations is fewer. The data were downloaded from the Bloomberg Fixed Income Database, on which the Treasury bond yields and swap rates are monitored closely by thousands of traders worldwide. A sample start date in the mid-1990s is more appropriate due to the tremendous growth in the interest rate derivative markets from its commencement in the early 1980s and the more recent structural change in the pricing and trading of swaps, such as the introduction of master agreements from the International Swaps and Derivatives Association, Inc. (ISDA) and the netting agreements for credit risk reduction that were developed in the early 1990s. As noted earlier, the price data from 1995 to 2004 covers a diverse range of economic experiences, including the later period covering the longest episode of economic expansion in US history, the Asian ﬁnancial crisis and the near-failure of LTCM. Depending on the expected direction of interest rate movement, market participants will have a preference for using different instruments when they expect the spread between the two markets (the swap spread) to narrow or widen. There are basically six different scenarios. In the ﬁrst four of the scenarios, the interest rates in the Treasury and swap markets are moving in the same direction, which means changes in the two markets are positively correlated. In the last two scenarios, the interest rates in the Treasury market and the swaps market are moving in different directions, which means changes in the two markets are negatively correlated. These different scenarios are detailed below: Scenario I – Bond and swap rates rise (positive correlation) with a widened swap spread. In this scenario, market participants prefer to pay ﬁxed in a swap than to short sell government bonds, assuming that the

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transaction cost and liquidity of the two markets are similar. This scenario usually occurs when the market expects interest rates to increase and corporations pay ﬁxed to hedge their interest rate exposure, which is known as ‘macro hedging’. Scenario II – Bond and swap rates decline (positive correlation) with a widened swap spread. In this scenario, market participants prefer to purchase government bonds than receive ﬁxed in a swap, all other things being equal. This usually occurs when there is uncertainty in the ﬁnancial markets, such as resulted from the Russian default in 1998. The rush to high-quality and liquid assets is termed the ‘ﬂight to quality’ and the ‘ﬂight to liquidity’ by Longstaff (2004) and occurred once again during the subprime crisis of 2007/2008. Scenario III – Bond and swap rates increasing (positive correlation) with a reducing swap spread. In this scenario, market participants prefer to short sell government bonds than pay ﬁxed in a swap, all other things being equal. This usually happens after a ﬁnancial crisis, when investors switch from quality debt securities to risky debt securities. Scenario IV – Bond and swap rates declining (positive correlation) with a reducing swap spread. In this scenario, market participants prefer to receive ﬁxed in a swap than purchase government bonds, all other things being equal. This occurs when investors use interest rate swaps as a hedge for their ﬂoating rate assets and mortgage backed securities. Interest rate swaps can be used to lock in the interest rate that is received for ﬂoating rate securities and hedge mortgage backed securities that possess a negative convexity. The use of receive ﬁxed in interest rate swaps to hedge the capital loss of mortgage backed securities became very common after the US Treasury cut down the volume of long-dated US government bonds that it issued. This is commonly known as ‘mortgage hedging’ in the ﬁnancial markets. Scenario V – Narrowing swap spread with swap and bond rates converging (negative correlation). In this scenario, market participants sell bonds and receive ﬁxed in the swaps market simultaneously, which causes a narrowing of the swap spread. Scenario VI – Widening swap spread with swap and bond rates converging (negative correlation). In this scenario, market participants buy bonds and pay ﬁxed in the swaps market simultaneously, which causes a widening of the swap spread. For scenarios V and VI, the interest rates in the Treasury bond and swaps markets are moving in opposite directions. Given that the expected direction of movement of the swap spread may have an impact on the preference for the usage of an

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instrument, we use the change in the swap spread as the threshold variable to test whether it can cause regime switching. The summary statistics for the yields and swap spreads are recorded in Table 1. Note that over the sample period the average yield for the ﬁve-year Treasury bond is 5.01%, and the average ﬁve-year swap rate is 5.54%, which gives an average swap spread of 52 basis points over the period. The volatilities are similar, at 1.31% and 1.32%, respectively. Both the bond yield and the swap rate exhibit negative skewness, which means that yields tend to stay at the high end of the range. This is conﬁrmed by the median bond yield and the median swap rate of 5.39% and 5.86%, respectively, which are higher than the average over the sample period. Both the bond yield and the swap rate have a bimodal distribution, which means that they tend to remain at either the high end or at the low end of the range. This was in line with the monetary policy of the Federal Reserve over the period, during which the Fed Fund target rate was usually lifted or lowered in consecutive Federal Open Market Committee (FOMC) meetings so that interest rates would either be kept below 3.0% to avoid a liquidity crunch, or maintained above 5.0% to curtail inﬂationary pressures in the broader economy. The swap spread has positive skewness, and the median swap spread is 45 basis points, which is 8 basis points less than the average. The swap rate is Table 1. Summary Statistics of the Government Bond Yield, Swap Rate and Swap Spread. Government Bond Yield (GOVt)

Interest Rate Swap Rate (IRSt)

Swap Spread (SSt)

Mean Standard deviation Skewness Kurtosis Maximum Median Minimum ADF

5.013 1.313 0.338 1.959 7.866 5.390 2.030 1.812

5.537 1.318 0.464 2.120 8.210 5.858 2.371 1.710

0.524 0.231 0.536 2.174 1.055 0.452 0.016 1.567

Correlation GOVt IRSt SSt

1.000 0.985 0.066

1.000 0.109

1.000

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8

May 00

7

Percent

6 Dec 04 5 4 3

Oct 98 USGG5YR USSW5

2

Jun 03

1 95

96

97

98

99

00

01

02

03

04

USGG5YR – 5-year Treasury Bond Yield USW5 – 5-year Swap Rate

Fig. 1.

US Five-Year Treasury Bond Yield and Swap Rate.

highly correlated with the Treasury bond yield (correlation coefﬁcient of 0.985), but the swap spread is not correlated with either the bond yield or the swap rate. All three of the yield time-series failed to pass the Augmented Dickey–Fuller (ADF) test, and thus cannot be considered to be weakly stationary. The trend in ﬁve-year US Treasury and swap yields is clear in Fig. 1, which shows that the Treasury bond yield was at its high of 7.87% in January 1995 and declined to a low of 4.94% in October 1998. This was due to the ‘Flight to Quality’ phenomenon that was triggered by the Russian default in August 1998. The Federal Reserve tightened interest rates after Y2K, and the Treasury bond yield rose to a high of 6.77% in May 2000. Following the technology stocks crash, the Federal Reserve eased the Fed Fund target rate to an all-time low of 1.0%, and the Treasury bond yield declined to 2.03% in June 2003. With concerns over a ‘bubble’ developing in the property market in the United States, the Fed Fund rate was raised again in May 2004, and in December 2004 the Treasury bond yield was 3.61%.

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Modelling the US Swap Spread 1.2 May 00 1.0

Oct 98

Percent

0.8 0.6 Jan 95 0.4 0.2

Jun 03

Dec 04

0.0 USSS5 -0.2 95

96

97

98

99

00

01

02

03

04

USSS5 – 5-year swap spread

Fig. 2.

US Five-Year Swap Spread.

Fig. 2 plots the US ﬁve-year spread over the sample period and highlights the volatility that existed in the swap spread. Before 1998, the swap spread seldom moved above 40 basis points, but it increased sharply following the Russian default and the near failure of LTCM. By October 1998, the ﬁveyear swap spread had increased to 90 basis points (from its mean of 52.4 basis points), which contributed to the record loss of US$1.6 billion for the hedge fund. After the crisis triggered by the collapse of LTCM, the swap spread also broke through the 90 basis point level twice, in 1999 and 2000, before declining to 34 basis points in June 2003, then rising slightly to 42 basis points in December 2004. It is interesting to note that the direction of interest rates and the swap spread in terms of the previously discussed in Scenarios (I to VI). First, note the two different settings for the two periods 1995–2000 and 2000–2004. From January 1995 to May 2000, interest rates and the swap spread overall moved in opposite directions, that is, interest rates tended to decline, whereas the swap spread increased (the negative correlation in Scenario VI). However, from June 2000, the swap spread began to move in the same direction as interest rates: initially, the swap spread narrowed from June 2000 to May 2003 with the decline in interest rates (the positive correlation in Scenario II),

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Table 2. Analysis of the Change in the Swap Spread from 1994 to 2004. Period

– 1 2 3 4 5 6 7 8 9 10 11 Period

Date

IRSt

GOVt

SSt

Change in IRSt

Change in GOVt

Change in SSt

06/01/1995 02/01/1998 16/10/1998 05/02/1999 06/08/1999 03/12/1999 09/06/2000 17/05/2002 02/08/2002 16/05/2003 07/05/2004 31/12/2004

7.99 6.03 4.95 5.54 6.84 6.69 7.40 5.03 3.92 2.68 4.48 4.03

7.87 5.61 4.04 4.96 5.91 6.07 6.36 4.59 3.21 2.38 3.94 3.61

0.34 0.42 0.90 0.58 0.92 0.61 1.04 0.43 0.71 0.29 0.54 0.42

– 1.97 1.08 0.60 1.30 0.16 0.72 2.38 1.11 1.24 1.81 0.46

– 2.26 1.56 0.92 0.95 0.16 0.29 1.77 1.39 0.82 1.56 0.34

– 0.08 0.48 0.33 0.35 0.31 0.43 0.61 0.28 0.42 0.25 0.12

Date

Swap Spread

Scenario

Duration (months)

– 1 2 3 4

06/01/1995 02/01/1998 16/10/1998 05/02/1999 06/08/1999

– Widen Widen Narrowed Widen

– Scenario Scenario Scenario Scenario

5 6

03/12/1999 09/06/2000

Narrowed Widen

Scenario V Scenario I

4 6

7

17/05/2002

Narrowed

Scenario IV

23

8 9

02/08/2002 16/05/2003

Widen Narrowed

Scenario II Scenario IV

3 9

10

07/05/2004

Widen

Scenario I

11

31/12/2004

Narrowed

Scenario IV

II II III I

– 36 9 4 6

12 8

Observation

– Buy bond Buy bond Sell bond Pay ﬁxed in swap Curve depart Pay ﬁxed in swap Received ﬁxed in swap Buy bond Received ﬁxed in swap Pay ﬁxed in swap Received ﬁxed in swap

Explanation

Flight to quality Flight to quality Unwind position Macro hedging Unknown Macro hedging Mortgage hedging Flight to quality Mortgage hedging Macro hedging Mortgage hedging

and both increased after June 2003 (the positive correlation in Scenario I). These observations highlight the importance of monitoring regime-switching effects when modelling the time-varying properties of the spreads. Table 2 provides more detail on these general trends by dividing the sample period into 11 sub-periods with the ﬁrst panel showing the levels in the swap (IRS) and bond (GOV) and spread (SS) rates and their respective

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Modelling the US Swap Spread

changes. The bottom panel in Table 2 explains whether the swap spread narrowed or widened, its duration and the Scenario (from I to VI). The explanation for the observed behaviour is explained in the last column. Thus, if we observe the highs and lows in the swap spread and compare the relative movement of the change in the Treasury bond yield relative to the change in the interest rate swap rate, one can see that the widening of the swap spread from January 1998 to October 1998 and from May 2002 to August 2002 was mainly due to a ﬂight to quality (Scenario II). However, the more recent widening of the swap spread from 2003 to 2004 was due to macro hedging (Scenario I). As interest rates declined sharply from June 2000, institutions tended to receive ﬁxed in swaps to hedge their investment in mortgage backed securities that possess negative convexity (Mortgage Hedging Scenario IV). The narrowing of the swap spread in the ﬁrst part of 2000 was therefore closely related to the sharp decline in the swap rate. Given that the interest rate and swap spread levels are not weakly stationary, the ﬁrst difference in the three time series is used in the subsequent analysis. The summary statistics, for the ﬁrst difference (or change) in the Government bond and interest rate swap yields, are provided in Table 3. Descriptive statistics for the averages of the weekly change in the Treasury bond yield and the swap rate are both negative, at 0.8 basis points, whereas the standard deviations are quite high, at 14.9 basis

Table 3. Summary Statistics of the Change in the Government Bond Yield, Swap Rate and Swap Spread. Change in Government Bond Yield (z1t)

Change in Interest Rate Swap Rate (z2t)

Threshold Variable Change in Swap Spread (yt)

Mean Standard deviation Skewness Kurtosis Maximum Median Minimum ADF

0.008 0.149 0.323 3.804 0.584 0.009 0.488 9.802

0.008 0.153 0.419 4.291 0.655 0.010 0.464 9.552

0.000 0.052 0.795 15.906 0.267 0.001 0.424 11.825

Correlation z1t z2t yt

1.000 0.099 0.941

1.000 0.244

1.000

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HON-LUN CHUNG ET AL.

points and 15.3 basis points, respectively. The average of the change in the swap spread is zero, and the standard deviation is 5.2 basis points. Both the change in the Treasury bond yield and the change in the swap rate are positively skewed, whereas the change in the swap spread is negatively skewed. The three time series are proved to be weakly stationary after the ADF test. It is interesting to note that the correlations between the yields and spread (bottom panel of Table 3) shows that the change in the swap spread is highly correlated with the change in the Treasury bond yield (correlation coefﬁcient of 0.941), but only weakly correlated with the change in the swap rate (correlation coefﬁcient of 0.244). The very weak correlation between the change in the Treasury bond yield and the change in the swap rate (correlation coefﬁcient of 0.099) is bad news for institutions that use Treasury bonds to hedge their swap positions.

4. METHODOLOGY 4.1. Modelling Techniques On the basis of Chan’s (1992) study of stock indices and index futures, the lead–lag behaviour of the change (D) in the US Treasury bond (GOVt) and the interest rate swap (IRSt) at time (t) can be examined using the following regression, Eq. (1), where e is a random variable: DIRSt ¼ a þ

4 X

bk DGOVtþk þ t

(1)

k¼4

Given that the change in the Treasury bond yield and the change in the swap rate are stationary time series, the Granger causality test (up to lag L) can be performed using the following sets of regressions (Eq. (2)): DIRSt ¼ c þ

L P

d i DIRSti þ

i¼1

DGOVt ¼ f þ

L P i¼1

L P

ej DGOVtj þ t

j¼4

gi DGOVti þ

L P

(2) hj DIRStj þ t

j¼1

BTAR modelling techniques are then applied to the examination of the dynamic relationship between the change in the ﬁve-year US Treasury yield (DGOVt) and the change in the ﬁve-year interest rate swap rate (DIRSt),

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where Zt ¼ (z1t, z2t)u with z1t ¼ DGOVt ¼ GOVtGOVt1 and z2t ¼ DIRSt ¼ IRStIRSt1. The series under study is the weekly closing prices over the period January 1995 to December 2004, which gives 522 observations. For the threshold variable, deﬁned as yt ¼ DSSt ¼ SStSSt1, the weekly change in the swap spread (DSSt) is used. The threshold variable series is plotted in Fig. 3, which displays yt from January 1995 to December 2004. The volatility spikes in the series are clear, especially during 1996 and 2003. Tiao and Box (1981) suggest summarising the cross-correlation relationship of the data series Zt ¼ (z1t, z2t)u, using indicator matrices where the indicator symbols (þ), () and ( ), where (þ) denotes a value that is greater than twice the estimated standard error, () denotes a value that is less than twice the estimated standard error and ( ) denotes an insigniﬁcant value that is based on the aforementioned criteria. The (i, j) element of the indicator matrix at lag l summarises the signiﬁcance of the lag-l crosscorrelation when the component series zjt leads the component series zit. Furthermore, the diagonal elements summarise the signiﬁcance of the sample autocorrelations for each series. Analogous to the Tsay (1989) procedure for univariate TAR modelling, Tsay (1998) extends the univariate 0.3 0.2 0.1

Percent

0.0 -0.1 -0.2 -0.3 -0.4

FDUSSS5

-0.5 95

96

97

98

99

00

01

02

FDUSSS5 – First Difference of 5-year Swap Spread

Fig. 3. Change in Five-Year Swap Spread.

03

04

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HON-LUN CHUNG ET AL.

method to the multivariate situation. In this section, we consider a bivariate time series Zt ¼ (z1t, z2t)u. A k-regime BTAR (d; p1,y, pk) model deﬁned as 8 p1 P ð1Þ > > wð1Þ Uð1Þ if ytd r > j Ztj þ at ; 0 þ > > j¼1 > > > > p2 > P > ð2Þ ð2Þ > Uð2Þ if r1 oytd r2 < w0 þ j Ztj þ at ; j¼1 (3) Zt ¼ > > . . . . > . .. .. .. > > . > > p > k P > ðkÞ ðkÞ ðkÞ > > > : w0 þ j¼1 Uj Ztj þ at ; if rk1 oytd where k is the number of regimes in the model, d is the delay parameter, pi is the autoregressive order in the ith regime of the model, wðiÞ 0 are (2 1)are (2 2)-dimensional matrix dimensional constant vectors, and UðiÞ j parameters for i ¼ 1, y, k. The threshold parameters satisfy the constraint The innovational vectors in the ith N ¼ r0or1or2oyor P1=2k1ork ¼ N. P1=2 ¼ e , where are symmetric positive deﬁnite regime satisfy aðiÞ t t i i matrices, and {et} is a sequence of serially uncorrelated normal random vectors with a mean of 0 and a covariance matrix I, the (2 2)-dimensional identity matrix. The threshold variable ytd is assumed to be stationary and depends on the observable past history of Ztd. For example, we can set ytd ¼ g0 Ztd

(4)

where gu is a pre-speciﬁed (2 1) dimensional vector. When g ¼ (1, 0)u, the threshold variable is simply ytd ¼ z1,td. When g ¼ (1/2, 1/2)u, the threshold variable is the average of the two elements in Ztd.

4.2. Nonlinearity Testing Given p ¼ max{p1, y, pk} and d r p, we can observe the bivariate vector time series {Z1, y, Zn}. It should be noted that the threshold variable ytd in Eq. (1) can only assume values in Y ¼ {ypþ1d ,y, ynd}. Let (i) be the time index of the ith smallest observation in Y. Tsay (1998) considers the multivariate generalisation of the ordered regression arrangement. Rolling

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Modelling the US Swap Spread

ordered bivariate autoregressions in 0 0 1 0 1 Z0ð1Þþd1 Zð1Þþd 0 0 BZ C B B ð2Þþd C B 1 Zð2Þþd1 B C B B .. C ¼ B .. .. .. B . C B. . . @ A @ 0 0 1 ZðjÞþd1 . . . ZðjÞþd

the form 1 0 0 1 0 0 1 að1Þþd Z0ð1Þþdp o0 B 0C B 0 C Z0ð2Þþdp C C B U1 C B að2Þþd C C B C B C C B .. C þ B .. C (5) .. C B C B C . A @. A @ . A U0p a0ðjÞþd Z0ðjÞþdp

can be arranged successively, where j ¼ m, mþ1, y, np, and m is the number of start-up observations in the pordered autoregression. ﬃﬃﬃ pﬃﬃﬃ Tsay (1998) suggests a range of m (between 3 n and 5 n). Different values of m can be used to investigate the sensitivity of the modelling results with respect to the choice. It should be noted that the ordered autoregressions are sorted by the variable ytd, which is the regime indicator in the BTAR model. Let ^ðmþ1Þþd denote the one-step-ahead standardised predictive residual from the least-squares ﬁtted multivariate regression for j ¼ m. Tsay (1998) provides the direct computational formula for ^ðmþ1Þþd , but it can easily be obtained from many commonly used statistical software packages (Timm & Mieczkowski, 1997). Analogous to the univariate case, if the underlying model is a linear autoregressive process, then the predictive residuals are white noise and are uncorrelated with the regressor X0t ¼ f1; Z0t1 ; Z0t2 ; . . . ; Z0tp g. However, if Zt follows a threshold process, then the predictive residuals are correlated with the regressor. Tsay (1998) utilises this property and considers the multivariate regression ^ 0ðlÞþd ¼ X0ðlÞþd B þ w0ðlÞþd

(6)

for l ¼ mþ1, y, np, where B is the matrix regression parameter, and wu(l)þd is the matrix of the residuals. The problem of testing nonlinearity is then transformed into the testing of the hypothesis H0: B ¼ 0 in this regression. Tsay (1998) employs the test statistic CðdÞ ¼ ðn p m kp 1Þ fln jS0 j ln jS1 jg

(7)

where |S| denotes the determinant of the matrix S, and 1 S 0 ¼ npm 1 S 1 ¼ npm

np P

^ðlÞþd ^0ðlÞþd

l¼m1 np P

w^ ðlÞþd w^ 0ðlÞþd

l¼m1

(8)

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HON-LUN CHUNG ET AL.

where w^ is the least-squares residual of regression (6). Under the null hypothesis that Zt is linear, Tsay (1998) shows that C(d) is asymptotically a chi-squared random variable with (pk2þk) degrees of freedom.

4.3. Model Speciﬁcation, Estimation and Diagnostic Checking To perform the C(d) test for nonlinearity in Eq. (7), the values of both p and d must be given. In practice, we can select p from the partial autoregression matrix (PAM) of Zt. Tiao and Box (1981) deﬁne the PAM at lag l, which is denoted by P(l), to be the last matrix coefﬁcient when the data are ﬁtted to a vector autoregressive process of order l. This is a direct extension of the deﬁnition of Box and Jenkins (1976) of the partial autocorrelation function for a univariate time series. The PAM P(l) of a linear vector AR(p) process are zero for l W p. This ‘cut-off’ property provides useful information for the identiﬁcation of the order p. Once p is selected, d is chosen, such that it provides the most signiﬁcant C(d) statistic. In univariate TAR modelling, various scatterplots are used to specify the number of regimes k and the threshold parameters (i.e. the r values). Unfortunately, these plots are not applicable to high-dimensional multivariate TAR analysis. Following Tong (1983), Akaike’s information (AIC) is used to search for these parameters. Given p, d, k and Rk ¼ {r1, y, rk1}, the full-length ordered bivariate autoregression can be divided into different regimes. For the jth regime of the data, allow a general model of the form Zj ¼ AjU(j)aj, where Zj ¼ ðZ0ðpj1 þ1Þþd ; Z0ðpj1 þ2Þþd ; . . . ; Z0ðpj Þþd Þ0 0 ðjÞ 0 UðjÞ ¼ ðo00 ; U0 ðjÞ 1 ;...;U p Þ

aj ¼ ða0ðpj1 þ1Þþd ; a0ðpj1 þ2Þþd ; . . . ; a0ðpj Þþd Þ0 0 1 1 Z0ðpj1 þ1Þþd1 Z0ðpj1 þ1Þþdp B C B 1 Z0ðpj1 þ2Þþd1 Z0ðpj1 þ2Þþdp C B C C Aj ¼ B .. .. .. B .. C B. C . . . @ A 0 0 ... Zðpj Þþdp 1 Zðpj Þþd1

(9)

and pj is the largest value of (j) such that {rj1oz(j)rrj} for j ¼ 1, y, k1. We deﬁne p0 ¼ 0 and pk ¼ np. The number of observations in the jth regime is nj ¼ pjpj1. The least-squares estimate of Uj can be obtained by

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Modelling the US Swap Spread

the ordinary multivariate least-squares method: ^ ðjÞ ¼ ðA0 Aj Þ1 ðA0 Zj Þ F j j

(10)

The residual variance–covariance matrix of the jth regime can then be obtained by X ^ j

n

j 1X ¼ fa^ðpj1 þtÞþd a^0ðpj1 þtÞþd g nj t¼1

(11)

Finally, the AIC of the bivariate ﬁtted TAR model in Eq. (1) is deﬁned as AICð p; d; k; Rk Þ ¼

k X j¼1

X ^ fnj ln j þ 2kðkp þ 1Þg

(12)

Given p and d, we can search for the parameters k and Rk by minimising the AIC. Owing to the computational complexity and possible interpretations of the ﬁnal model, k is usually restricted to be a small number, such as 2 or 3. For the threshold parameters Rk, the data may be divided into subgroups according to the empirical percentiles of ytd and use the AIC to select the r values. Finally, the AIC is used to reﬁne the AR order (pkrp) in each regime. To guard against incorrect speciﬁcation of the model, a detailed diagnostic analysis of the residuals is required. This includes an examination of the plots of the standardised residuals and the sample cross-correlation (SCC) matrices of the residuals (Tiao & Box, 1981).

5. RESULTS 5.1. Lead–Lag and Causality Testing Table 4 reports the results if the simple lead–lag analysis (Eq. (1)). In this analysis the only coefﬁcient that is statistically signiﬁcant is b0. Neither the lead variables (b1 to b4) nor the lag variable (b1 to b4) are statistically signiﬁcant. The results indicate that there is no lead–lag relationship between the change in the swap rate with respect to the change in the Treasury bond yield for the lead variables, or the lag variable, over the entire sample period. The next analysis involves pairwise testing for Granger causality (Eqs. (2) and (3)). The results for this testing are reported in Tables 5–7, where

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HON-LUN CHUNG ET AL.

Table 4.

Results of the Simple Lead–Lag Study. Coefﬁcient

t-Statistics

b4 b3 b2 b1 b0 b1 b2 b3 b4

0.017 0.002 0.010 0.011 0.964 0.020 0.000 0.020 0.004

1.098 0.110 0.626 0.699 61.444 1.300 0.018 1.248 0.263

R2 F-Statistics

0.885 431.782

Signiﬁcance at the 5% level.

Table 5 reports the results for the complete sample and Tables 6 and 7 for two sub-periods. For the entire sample period (January 1995 to December 2004), the null hypothesis that the change in the Treasury bond yield does not Granger-cause a change in the swap rate is rejected (F-statistic ¼ 5.029) at least at the 95% level. The null hypothesis is also rejected for periods from one lag up to ﬁve lags (F-statistic ¼ 3.697 to 2.677) at least at the 95% level. However, the null hypothesis that a change in the swap rate does not Granger-cause a change in the Treasury bond yield cannot be rejected, which shows that the change in the Treasury bond yield has only a unilateral causality on the change in the swap rate. If the sample period is divided into two sub-periods, the ﬁrst from January 1995 to May 2000 (Table 6), then the result is similar to that for the whole sample period (reported in Table 5): there is unilateral causality with the change in the Treasury bond yield Granger-causing the change in the swap rate. However, for the ﬁrst sub-period, the F-statistics (3.853 to 2.352) are only signiﬁcant (at least at the 95% level) from three lags to ﬁve lags. For the second sub-period, from June 2000 to December 2004, the results are reported in Table 7. These ﬁndings are also consistent with the ﬁrst subperiod and the overall sample. However, the F-statistics (7.031 and 3.349, respectively) are only signiﬁcant (at least the 95% level) for the one lag period and two lag period only. An interesting result is that the change in the swap rate is marginally shown to Granger-cause a change in the Treasury bond yield for one lag period (at the 10% level with F-statistic ¼ 3.855).

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Modelling the US Swap Spread

Table 5.

Results of the Granger Causality Tests (January 1995 to December 2004). Pairwise Granger Causality Tests

Null Hypothesis Lag ¼ 1 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 2 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 3 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 4 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 5 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt

Observations

F-Statistics

p-value

520

5.029

0.025

520

0.243

0.622

519

3.697

0.025

519

0.487

0.615

518

3.852

0.010

518

0.387

0.763

517

2.850

0.023

517

0.559

0.692

516

2.677

0.021

516

0.708

0.617

Signiﬁcance at the 5% level.

5.2. Bivariate Threshold Autoregressive Models We ﬁrst examine the SCC matrices using indicator symbols and conclude that there are no moving average elements in the BTAR models. Then the PAM of the observed bivariate vector time series are observed, with the PAM matrices summarised using indicator symbols in Table 8. The likelihood ratio statistic M(l) can be used to test the null hypothesis that the PAM are zero matrices. Originally, Bartlett (1938) shows that the

174

HON-LUN CHUNG ET AL.

Results of the Granger Causality Tests (January 1995 to May 2000).

Table 6.

Pairwise Granger Causality Tests (January 1995 to May 2000) Null Hypothesis Lag ¼ 1 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 2 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 3 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 4 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 5 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt

Observations

F-Statistics

p-value

278

0.792

0.374

278

0.312

0.577

277

2.485

0.085

277

0.963

0.383

276

3.852

0.010

276

0.940

0.422

275

2.692

0.032

275

0.962

0.429

516

2.352

0.041

516

0.726

0.604

Signiﬁcance at the 5% level.

M(l) statistic is asymptotically w2 distributed with four degrees of freedom if the null hypothesis is true. From Table 8, one can observe that the M(l) statistics drop signiﬁcantly after l ¼ 3 (from 17.81 at lag 3 to 3.76 at lag 4). Therefore, it is possible to tentatively specify p ¼ 3 for the C(d) test for nonlinearity. This allows the C(d) test with p ¼ 3, drp and m ¼ 150. These results for nonlinearity are then reported in Table 9. Since the critical value for this test is 23.68. The results clearly reject the linear hypothesis, which indicates that BTAR-type nonlinearity is detected in the data. The test statistics also suggest using the delay parameter d ¼ 1.

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Modelling the US Swap Spread

Table 7.

Results of the Granger Causality Tests (June 2000 to December 2004).

Pairwise Granger Causality Tests (June 2000 to December 2004) Null Hypothesis Lag ¼ 1 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 2 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 3 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 4 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt Lag ¼ 5 Change in GOVt does not Granger-cause a change in IRSt Change in IRSt does not Granger-cause a change in GOVt

Observations

F-Statistics

p-value

241

7.031

0.009

241

3.855

0.048

240

3.349

0.037

240

2.169

0.117

239

2.401

0.069

239

1.583

0.194

238

1.662

0.160

238

1.183

0.319

237

1.468

0.201

237

1.262

0.281

Signiﬁcance at the 5% level.

With 522 observations, it is possible to consider the possibilities of BTAR models with two or three regimes, that is, k ¼ 2 or 3. Given p, d and k, a grid search method can be used to select the thresholds by minimising the AIC values that are deﬁned in Eq. (12). Let Pa ðytd Þ be the empirical a-th percentile of ytd. For the two-regime models assume r 2 ½P10 ðytd Þ; P90 ðytd Þ. For the three-regime models, assume that r1 2 ½P10 ðytd Þ; P45 ðytd Þ and r2 2 ½P55 ðytd Þ; P90 ðytd Þ. Table 10 shows the selected threshold values under different combinations of (k, p, d). It is indicated that the overall AIC is 5,182.38 when k ¼ 3, p ¼ 3, d ¼ 1, r^1 ¼ 0:033 and

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HON-LUN CHUNG ET AL.

Indicator Matrices for the Partial Autoregression Matrices.

Table 8. Lag (l)

1

þ

50.47

M(l) Lag (l)

3

3.77

7

M(l)

2

8.85

6.41

þ

17.81

8

4

Note: The critical value for the M(l) test is

Table 9.

1.37

1.14

6.11

5.83

11

w20:95;4

6

3.13

10

12

1.72

¼ 9:49.

Tests for Nonlinearity.

1 28.63

d C(d)

3.76

9

5

2 17.56

3 19.99

Note: The critical value for the C(d) test is w20:95;14 ¼ 23:68.

Table 10.

Selection of k, p, d and the Threshold Values..

k

p

d

r^1

r^2

AIC

2 3

3 3

1 1

0.049 0.033

0.017

5141.69 5182.38

r^2 ¼ 0:017: One can further reﬁne the model by allowing different autoregressive orders for different regimes. The AIC selects (p1, p2, p3) ¼ (2, 3, 1) with the least squares estimation results of the speciﬁed model provided in Table 11. In this table the results for the ﬁrst (k ¼ 1, p ¼ 1), second (k ¼ 2, p ¼ 0) and third regime (k ¼ 3, p ¼ 3) are provided. The indicator matrices and the residual PAM are also examined and do not show any model inadequacy [M(l) for lags 1–6 take values 2.84 to 1.70ocritical value of 9.49].

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Modelling the US Swap Spread

Table 11.

Model Estimation Results.

^ ðkÞ The estimated coefﬁcients: F j (A)

The ﬁrst regime (k ¼ 1, p1 ¼ 1, n1 ¼ 83) Lag( j) 0 1

0 0:74 0:78 0

(B)

0:19

0:22

The second regime (k ¼ 2, p2 ¼ 0, n2 ¼ 294) Lag( j) 0

0 0

(C)

The third regime (k ¼ 3, p3 ¼ 3, n3 ¼ 142) Lag( j) 0 1

0 0:78 0:70 0

0:29 0:19

2 0:51 0:62 0:44

0:37

3 0:39 0:60 0:17

0:40

The element in the coefﬁcient matrix is statistically signiﬁcant at the 5% level.

Using the BTAR modelling framework, three regimes can now be constructed for the dynamics of the Treasury and the interest rate swaps markets. The ﬁrst regime occurs when the weekly change in the swap spread is negative and more than 3.3 basis points (yt1rr1 ¼ 0.033). The second regime occurs when (r1 ¼ 0.033ryt1rr2 ¼ 0.017). The third regime exists when the weekly change in the swap spread is positive and more than 1.7 basis points (yt1Wr2 ¼ 0.017). The weekly lead–lag relationship between the Treasury and interest rate swaps markets in the k-th regime ^ ðkÞ reported in Table 11. can be examined using the off-diagonal elements F j

If any of the upper off-diagonal elements of the estimated matrices are signiﬁcant, then a change in the Treasury bond yield has a lead effect on the change in the swap rate. If any of the lower off-diagonal elements of the estimated matrices are signiﬁcant, then a change in the swap rate has a lead effect on the change in the Treasury bond yield. The results are now summarised in Table 12, which shows a with statistically signiﬁcant lead–lag relationship (with the Government bond leading the swap market) in the ﬁrst regime at lag 1, no relationship in the second regime, and a statistically signiﬁcant relationship at lags 2 and 3 in the third regime. The swap market does not lead the bond market in any of the three regimes.

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HON-LUN CHUNG ET AL.

Table 12.

Lag

Analysis of the Weekly Lead–Lag Relationship.

Government Bond

Interest Rate Swaps Market

Leads

Leads

Interest Rate Swaps Market

Government Bond

Size

(A) The ﬁrst regime 1 0.78

t-ratio

Lag

Size

t-ratio

1 2 3

0.29 0.37 0.17

0.83 1.54 0.65

2.60

(B) The second regime No signiﬁcant lead–lag effects found (C) The third regime 1 0.70 2 0.62 3 0.60

1.94 2.38 2.07

Signiﬁcance at the 5% level.

5. CONCLUSIONS This chapter examines the dynamics between US Treasury bond and swap markets using BTAR models. This approach, which may be applied to other interest rate products in other markets, tests whether the lead–lag relationship between the bond market and the swap market is a nonlinear dynamic process, and second whether this relationship is governed by the change in the interest rate differential, or spread, between these two markets. The ﬁndings from the BTAR models may be summarised in Fig. 4 where three regimes are identiﬁed. In the ﬁrst regime, the Treasury market leads the swaps market when the change in the swap spread is negative and more than 3.3 basis points on a weekly basis. In the second regime, there is no signiﬁcant lead–lag relationship between the two markets when the change in the swap spread falls within a narrow range of negative 3.3 basis points and positive 1.7 basis points. In the third regime, the Treasury bond market leads the swaps market when the change in the swap spread is positive and more than 1.7 basis points. Simple lead–lag studies do not reveal any particular information about the dynamics between the two markets. However, Granger causality tests are useful for revealing the overall movement of the two markets, because it shows that the change in the Treasury bond yield can Granger-cause a

179

Modelling the US Swap Spread __________________________________________________________________________ Regime I – Change in Swap Spread (ΔSSt-1) is more than –3.3 basis point per week ΔGOVt leads ΔIRSt (the coefficient is positive) Number of observations = 83 (15.99% of the total sample) ΔSSt-1 < -3.3 b.p. Only 1 lag is significant ______________________________________________ Regime II – Change in Swap Spread (ΔSSt-1) is between –3.3 basis point and + 1.7 basis point per week No lead-lag -3.3 b.p. < ΔSSt-1< 1.7b.p. Number of observation = 294 (56.65% of the total sample) ______________________________________________ Regime III – Change in Swap Spread (ΔSSt-1) is more than 1.7 basis point per week Mixed lead-lag relation (two conditions) Number of observations = 142 (27.36% of the total sample) First condition ΔSSt-1 > 1.7 b.p. ΔGOVt leads ΔIRSt (the coefficient is positive) Longer lags (1, 2, 3) are significant Second condition ΔIRSt lead ΔGOVt (the coefficient is negative) Lags 2 marginally significant __________________________________________________________________________

Fig. 4.

Three Regimes.

change in the swap rate. Subjective judgment can enhance the performance of the Granger causality test. For example, one can identify that lag 3 to lag 5 are signiﬁcant for the ﬁrst sub-period from January 1995 to May 2000, whereas lag 1 and lag 2 are signiﬁcant in the second sub-period from June 2000 to December 2004. By dividing the data into two sub-periods, we also ﬁnd that for the second sub-period, the change in the swap rate can marginally Granger-cause a change in the Treasury bond yield for one lag period. However, this information cannot be revealed if a subjective decision is not made. The BTAR model, however, can reveal more information without the use of subjective decision making. The regime switch process can occur in individual observations because the switching is determined by the state of the threshold variable, namely, the magnitude and direction of the change in the swap spread. The dominance of Treasury bonds in leading the swaps market occurs when either a widening or a narrowing of the swap spread occurs. This seems to fall in line with the ‘Flight to Quality’ and ‘Flight to Liquidity’ phenomena and is probably due to the nature of Treasury securities. Usually, no credit limit is required for transactions to be carried out for ﬁnancial institutions, and Treasury securities have a minimal impact on the balance sheets of various institutions.

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Overall, the BTAR model provides a deeper insight than simple lead–lag studies and Granger causality tests into the dynamics between the US Treasury bonds and interest rate swaps markets. Importantly, this approach can be applied to other interest rate product in other markets. By identifying the regimes and the conditions for change in the regimes, market participants and regulators can become more informed about the probable changes that will occur in the Treasury bond and interest rate swaps markets. Similar to the ﬁndings of Chappell et al. (1996), who identify the bounds within which the French Franc/Deutschmark exchange rate kept to before the launch of the Euro, the movements of the Treasury bond market and the interest rate swaps market are governed by the direction and magnitude of the change in the swap spread. The BTAR model is able to identify the threshold value of the change in the swap spread that bond and swap market participants considered to be signiﬁcant. Further research can be conducted to explain the existence of the threshold values.

NOTES 1. The main rating agencies are Standard & Poors, Moody’s and Fitch Investor Services. For convenience we use the Standard & Poors’ notation. The ﬁxed rate side of the swap by convention in ﬁnancial markets is expressed as a spread over the riskfree ﬁxed rate bond. 2. Bank of International Settlements (BIS, 2008) statistics for June 2007 show that interest rate swaps account for 52.6% of outstandings (US$ 516.4 trillion) in overthe-counter (OTC) derivatives, with plain vanilla or simple ﬁxed-ﬂoating swaps accounting for most swap turnover. Of this swap total, about 30% (US$ 81.28 trillion) are denominated in US$.

REFERENCES Bank for International Settlements (BIS). (2008). Semi annual OTC derivatives turnover at end June 2007, Basle, Switzerland. Available at http://www.bis.org/statistics/derstats.htm. Retrieved on April 21, 2008. Bartlett, M. S. (1938). Further aspects of the theory of multiple regression. Proceedings of the Cambridge Philosophical Society, 34, 33–40. Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control (2nd ed.). San Francisco: Holden-Day. Brooks, C., & Garrett, I. (2002). Can we explain the dynamics of the UK FTSE 100 stock and stock index futures markets? Applied Financial Economics, 12, 25–31. Chan, K. (1992). A further analysis of the 1eadlag relationship between the cash market and stock index market. Review of Financial Studies, 5, 123–152.

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Chan, W. S., & Cheung, S. H. (2005). A bivariate threshold time series model for analyzing Australian interest rates. Mathematics and Computers in Simulation 68, 68, 429–437. Chappell, P. J., Mistry, P., & Ellis, C. (1996). A threshold model for the French Franc/ Deutschmark exchange rate. Journal of Forecasting, 15, 155–164. Collin-Dufresne, P., & Solnik, B. (2001). On the term structure of default premia in the swap and LIBOR markets. Journal of Finance, 56, 1095–1114. Cortes, F. (2006). Understanding the term structure of swap spreads. Bank of England Quarterly Bulletin 46, 1, 45–56. Feldhutter, P., & Lando, D. (2008). Decomposing swap spreads. Journal of Financial Economics, 88, 375–405. Hamilton, J. D. (1996). Speciﬁcation testing in Markov-switching time-series models. Journal of Econometrics, 70, 127–157. Harris, L. (1989). The October 1987 S&P 500 stock-future basis. Journal of Finance, 44(1), 77–99. Huang, Y., Chen, C. R., & Camacho, M. (2008). Determinants of Japanese Yen interest rate swap spreads: Evidence from a smooth transition vector autoregressive model. Journal of Futures Markets, 28(1), 82–107. Ito, T. (2007). The analysis of interest rate swap spreads in Japan. Applied Financial Economics Letters, 3(1-3), 1–4. Kawaller, I. G., Koch, P. D., & Koch, T. W. (1987). The temporal price relationship between S&P 500 futures and the S&P 500 index. Journal of Finance, 42, 1309–1330. Longstaff, F. A. (2004). The ﬂight-to-liquidity premium in US Treasury bond prices. Journal of Business, 77(3), 511–526. Malhotra, D. K., Bhargava, V., & Chaudhry, M. (2005). Determinants of treasury-LIBOR swap spreads. Review of Paciﬁc Basin Financial Markets and Policies, 8(4), 687–705. Stoll, H. R., & Whaley, R. E. (1990). The dynamics of stock index and stock index futures returns. Journal of Financial and Quantitative Analysis, 25, 444–468. Tiao, G. C., & Box, G. E. P. (1981). Modelling multiple time series with applications. Journal of the American Statistical Association, 76, 802–816. Timm, N. H., & Mieczkowski, T. A. (1997). Univariate and multivariate general linear models: Theory and applications using SAS software. North Carolina: SAS Publishing. Tong, H. (1978). On a threshold model. In: C. H. Chen (Ed.), Pattern recognition and signal processing. Amsterdam: Sijhoff and Noordoff. Tong, H. (1983). Threshold models in non-linear time series analysis. New York: SpringerVerlag. Tsay, R. S. (1989). Testing and modelling threshold autoregressive processes. Journal of the American Statistical Association, 84, 231–240. Tsay, R. S. (1998). Testing and modelling multivariate threshold models. Journal of the American Statistical Association, 93, 1188–1202. Yadav, P. K., Pope, P. F., & Paudyal, K. (1994). Threshold autoregressive modelling in ﬁnance: The price difference of equivalent assets. Mathematical Finance, 4, 205–221.

PRICING AND RISK MANAGEMENT OF VARIABLE ANNUITIES AND EQUITY-INDEXED ANNUITIES Guanghua Cao, Andrew H. Chen and Zhangxin Chen ABSTRACT A variety of equity-linked insurance contracts such as variable annuities (VA) and equity-indexed annuities (EIA) have gained their attractiveness in the past decade because of the bullish equity market and low interest rates. Due to the complexity of their inherent nature, pricing and risk management of these products are quantitatively challenging and therefore have become sources of concern to many insurance companies. From a ﬁnancial engineer’s perspective, the options in VA and those embedded in EIA can be modeled as puts and calls, respectively, and enable the use of numerical option pricing techniques. Additionally, values of VA and EIA move in opposite directions in response to changes in the underlying equity value. Therefore, for insurers who offer both businesses, there are natural offsets or diversiﬁcation beneﬁts in terms of economic capital (EC) usage. In this chapter, we consider two speciﬁc products: the guaranteed minimal account beneﬁt (GMAB) and the point-to-point (PTP) EIA contract, which belong to the VA and EIA classes respectively. Taking into account mortality risk and suboptimal dynamic lapse behavior, we build a framework that quantiﬁes the value of each Research in Finance, Volume 26, 183–212 Copyright r 2010 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0196-3821/doi:10.1108/S0196-3821(2010)0000026011

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product and the natural hedging beneﬁts based on risk-neutral option pricing theory. With Monte Carlo simulation and ﬁnite difference methods being implemented, an optimum product mixture of those two contracts is achieved that deploys capital the most efﬁciently.

1. INTRODUCTION The market for equity-linked insurance such as variable annuities (VA) and equity-indexed annuities (EIA) has grown tremendously over recent years and has become a signiﬁcant segment of our capital markets. This has been evidenced by the growing sales that have reached $113 billion for VA and $13 billion for EIA in 2003.1 This is partly thanks to the bullish US equity market along with relatively low interest rates over the past decade, which have led policyholders to be more aware of investment opportunities outside the traditional insurance sector so that they can enjoy the beneﬁts from ﬁnancial markets in conjunction with investment guarantees and tax advantages. Different from traditional insurance products, these equitylinked insurance contracts provide policyholders mortality or maturity protection as well as the beneﬁcial return based on the equity market’s performance. The pricing and risk management of these products are quantitatively challenging and therefore have become sources of concern to both the regulator and the many insurance companies. For instance, pricing these annuity contracts is complicated with mortality risk and dynamic lapse2 behavior involved; also, the limited capital of a life insurance company constrains the volume of its VA and EIA business; thus, how to deploy the economic capital (EC)3 more efﬁciently turns out to be an urgent problem to frame. It is important to stress that from an option pricing perspective, the options in VA and those embedded in EIA can be modeled as puts and calls, respectively, which will be shown in detail later. However, with mortality and dynamic lapse risk involved, pricing these contracts becomes numerically challenging and needs special techniques for its complicated features such as path dependency. The values of these embedded options move in opposite directions in response to underlying equity price changes. Suppose both products share the same underlying equity process, then these two types of options have payoffs that can partially offset each other, thus natural diversiﬁcation beneﬁts exist in a portfolio that contains both VA and EIA products, and

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therefore, the EC requirements for that annuity writers can be reduced. From the insurance company’s (risk management) point of view, it will be very useful to quantify these diversiﬁcation beneﬁts and derive an optimal business mix based on the most efﬁcient way to deploy the capital. The framework of this chapter, which differs from previous literatures, is based on this purpose. Perhaps, the best way to illustrate this intuition is through a simple numerical example. Table 1 provides the Value at Risk (VaR) and standard deviation of a European put, a European call, and a 50/50 mixture of these two options (i.e., a straddle) at time horizons of both 1 and 2 years. This example assumes both options are at-the-money, have maturity of 4 years, and are based on the same underlying equity price that follows a geometric Brownian motion with drift m ¼ 8%, non-dividend-paying, volatility s ¼ 0.2, risk-free rate r ¼ 2%, and initial price S0 ¼ 1. It is shown in Table 1 that the straddle portfolio has a much lower VaR and standard deviation than the average of these two options, which can be explained by Fig. 1. The correlation between the prices of a put and a call is negative: when one option is in-the-money (implies a higher price), the other one is likely to be out-of-the-money (implies a lower price). This natural diversiﬁcation lowers down both the VaR and the standard deviation of that straddle portfolio (red line in Fig. 1). And it will be shown later that similar diversiﬁcation effect also exists in portfolio that contains both VA and EIA. There have been some previous literature in this area. For research on VA, Brennan and Schwartz (1976), Boyle and Schwartz (1977), and Brennan and Schwartz (1979) ﬁrst introduced the famous Black–Scholes–Merton (Merton, 1973) framework into this ﬁeld. They assumed complete markets for both ﬁnancial and mortality risk and derived risk-neutral price formulae. More recent work on equity-linked life insurance was done by Bacinello and Ortu (1993a, 1993b, 1996), Aase and Persson (1994), and Nielsen and Table 1. Tenor (Years)

1 2

Diversiﬁcation of a Put and Call.

99% Value at Risk

Standard Deviation

Put

Call

50/50 Mix

Put

Call

50/50 Mix

0.30 0.38

0.76 1.22

0.38 0.61

0.07 0.09

0.16 0.27

0.06 0.11

Note: The bold/italic numbers are used to show the effect of risk reduction (in terms of 99% VaR and Std. Dev.) from diversiﬁcation.

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GUANGHUA CAO ET AL. Diversification example (1 year time horizon) 1.4 1.2 •

Option price

1

Put Call 50/50 Mixture

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

0

1.2

1.4

1.6

1.8

2

Stock price

Fig. 1.

Diversiﬁcation of a Put and a Call. (1-year time horizon).

Sandmann (1995). These authors allowed the risk-free interest rate to be stochastic. Follmer and Sonderman (1986) assumed an incomplete mortality market and introduced the concept of risk-minimizing strategies, which was extended by Moller (1998). Hardy (2003) offered risk-neutral pricing and dynamic hedging analyses on VA. Milevsky applied an optimal control technique to analyze VA with mortality and lapse risk (Milevsky & Salisbury, 2002) as a best stopping time problem and concluded that in today’s market, the guaranteed minimum death beneﬁt (GMDB) products were overpriced (Milevsky & Posner, 2001), and in contrast, the guaranteed minimum withdrawal beneﬁt (GMWB) products were underpriced (Milevsky & Salisbury, 2004). In the ﬁeld of EIA research, Tiong (2000) used Esscher transforms and derived closed form pricing formulae for several types of EIA products: point-to-point (PTP), cliquet, and lookback, which were also covered by Hardy (2003). Lin and Tan (2003) extended the model to include stochastic interest rates. This chapter applied the Black–Scholes–Merton option pricing framework along with a complete mortality and lapse market. Oppose to an

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187

optimal control approach, lapse behavior is modeled as a function of time and underlying equity performance that can be economically irrational and suboptimal. Based on this framework, we developed analytic formulas and ﬁnite difference schemes to price both VA and EIA, which enable the EC calculation and optimization. The rest of this chapter is organized as follows. We present the framework in Section 2. Analytical formulas including risk-neutral pricing and EC calculating are implemented on two speciﬁc products: guaranteed minimum account beneﬁt (GMAB) in Section 2.1, and the PTP EIA contract in Section 2.2, which belong to the VA and EIA classes respectively. In Section 2.3, we introduced a ﬁnite difference approach to price GMAB and PTP. In Section 2.4, we analyzed the EC of a GMAB/PTP mixture portfolio based on a Monte Carlo simulation and ﬁnite difference hybrid algorithm. An optimal combination of these two products is achieved which employs EC the most efﬁciently. We conclude in Section 2.5 with closing remarks and summary.

2. FORMULATION 2.1. GMAB Contract, Valuation, and Economic Capital 2.1.1. Product Description VA are tax-deferred, complex structured equity, and interest rate investment vehicles. They provide money-back guarantees on a separate mutual fund account, and these guarantees can be viewed as put options with an increasing strike price. Different from usual ﬁnancial products that are paid up-front, premiums of these products are paid by installments, with a proportional beneﬁt charge that is deduced from the underlying mutual fund account on a periodic basis. The simplest VA product is the GMAB, which provides the beneﬁciary a minimal guarantee in the event that the policyholder dies or contract matures, whichever one comes ﬁrst. In this chapter, we focus on a GMAB account. An example of a GMAB contract is as follows: at initiation, t ¼ 0, the policyholder enters into a contract by paying the insurance company an initial amount P. The insurance company immediately invests the amount P into a mutual fund (such as an S&P 500 index fund) and there is no further payment from the policyholder. The insurance company guarantees a rate of return rg up to the end of contract, when the beneﬁciary will receive the

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greater of either the current mutual fund account value or the guaranteed amount. In exchange, the insurance company charges a certain percent of account amount as the contract fees. The guaranteed payment can be triggered by mortality or maturity, but not by lapse behavior: If the policyholder decides to lapse the VA contract before maturity, he/she can get his/her mutual fund account value back after some penalty fees charged, but the guarantee is not redeemable. 2.1.2. GMAB without Mortality and Lapse Consider a GMAB contract with $1 initial account value and maturity time N (in years). Ignoring any mortality and lapse risk, the embedded option in GMAB turns to be a plain vanilla European put. For the rest of this chapter, the underlying equity price is assumed to satisfy a geometric Brownian motion, the interest rate is assumed to be constant, and continuous compounding will be used for simplicity. This framework is similar to Hardy (2003). Given time horizon n prior to maturity, let Gn be the guaranteed amount, Gn ¼ erg n 1; 0 n N As we discussed before, Gn is going to be the strike price for its embedded option. Let m be management fee rate that was charged to policyholder’s account and fF n g be the account value process that satisﬁes, F n ¼ emn

Sn ; 0nN S0

At any time t ¼ n prior to N, suppose the underlying stock price is Sn. The embedded put option value in GMAB can then be calculated as follows: rðNnÞ HN Hn ¼ EQ n ½e

where

þ rg N mN S N H N ¼ ðGN F N Þ ¼ e e S0 mN þ e ¼ eðmþrg ÞN S0 S N S0 þ

In the formula above, E Q n ½d is expectation under risk-neutral measure Q. The HN term, which is the ﬁnal cash ﬂow of the GMAB contract that happens at maturity N, is equivalent to the payoff of a vanilla European

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put option. If we take notation V put ðS 0 ; K; r; d; s; tÞ as the price of a vanilla European put, then under the Black–Scholes–Merton framework (Black & Scholes, 1973), the closed form of such an option value can be written as follows: emN V put ðSn ; eðmþrg ÞN S 0 ; r; d; s; N nÞ S0 emN ¼ erg NrðNnÞ Fðd 2 Þ Sn edðNnÞ Fðd 1 Þ S0

Hn ¼

where 2

logðS n =S 0 Þ ðm þ rg ÞN þ ðr d þ s2 ÞðN nÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s Nn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s N n

d1 ¼

For a GMAB contract, the net value of adding the guarantee to the VA product at time n, noted by NVn ðS n Þ, can be formulated as the difference between two parts: the embedded option (guarantee) value from time n to maturity N, and the present value of the beneﬁt charge (noted as fn), as a portion of the total management fees charged to the policyholder’s account. NVn ðSn Þ has the following form: NVn ðS n Þ ¼ H n f n and Z

N

fn ¼ n

erðtnÞ E Q n ½F t mdt ¼

1 Sn S0

Z

N

emt mdt ¼

n

1 Sn ½emn emN S0

The corresponding EC of GMAB is deﬁned as the percentile risk measure of NVðS n Þ: P½NVn ðSn Þ NV0 ðS 0 Þ ECGMAB o1 b where b is the conﬁdence level. As NVn ðS n Þ is monotonic,4 its analytical EC (or equivalent, VaR) can be directly calculated (Fong & Lin, 1999) in the following way: Var½f ðSÞ ¼ f ðVar½SÞ

if f ðSÞ is monotonic

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Supposing that a 99% conﬁdence level (notice this is under realistic measure) is applied, the EC under current framework is as follows: ECGMAB ¼ NVn;99% NV0 ¼ H n;99% f n;99% NV0 emN Vput ðS n;99% ; eðmþrg ÞN S 0 ; r; d; s; N nÞ f n;99% NV0 S0 pﬃﬃ s2 ¼ erg N Fðd 2 Þ emNdðNnÞ eðmd 2 Þn2:33s n Fðd 1 Þ pﬃﬃ s2 f n ðS 0 eðmd 2 Þn2:33s n Þ NV0 ¼

where pﬃﬃﬃ 2 2 ðm d s2 Þn 2:33s n ðm þ rg ÞN þ ðr d þ s2 ÞðN nÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d1 ¼ s Nn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s N n NV0 ¼ H 0 f 0

2.1.3. GMAB with Mortality and Lapse In the previous section, mortality and lapse risk were totally ignored. In the real world, it is the involvement of mortality and lapse that distinguish GMAB from the normal ﬁnancial instruments. Mortality leads to stochastic contract maturity time, and the lapse feature gives the policyholder an opportunity to abandon the contract. (Lapse happens when policyholders terminate payments without having paid the full value of contract, usually at the cost of penalty.) Let CðtÞ be the percentage of policyholders that survive and do not lapse before time t, q(t) and lðS t ; tÞ be the simultaneous mortality and lapse intensities (or hazard rates), respectively. Independence between lapse risk and mortality risk is also assumed. Under a continuous time model, CðtÞ has the following form: Rt ½lðS ;uÞþqðuÞdu CðtÞ ¼ e 0 u Standard actuarial practice treats mortality risk as diversiﬁable or nonsystematic, which means the mortality risk can be eliminated by issuing a large enough number of equivalent contracts.5 In this chapter, we adhere to R this assumption. Then, the beneﬁts of a life insurance contract turn to be CðtÞqðtÞPðtÞ dt, where P(t) represents the payoff at time t.

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However, because equity market performance has huge impact on the policyholder’s lapse behavior (Shumrak, Greenbaum, Darley, & Axtell, 1999; Milevsky & Salisbury, 2002), lapse risk is not fully diversiﬁable and therefore cannot be hedged by simply issuing a large number of contracts. Lapse rate l has form of lðS t ; tÞ, and survival probability CðtÞ depends on the whole underlying equity price path fS t g prior to n. A number of researchers model the lapse behavior as a policyholder’s rational decision and treat VA as an American-typed option with best stopping time always approachable. In this chapter, we suggest that the lapse behavior of both VA and EIA policyholders can be irrational and suboptimal just like other life insurance products and build the model in a different way.6 We introduce the dynamic lapse multiplier to model dynamic lapse. At any time n, the instantaneous lapse rate can be modeled as follows: lðS n ; tÞ ¼ f ðR; tÞ l B where R¼

Fn 1 ¼ Sn eðmþrg Þn Gn S 0

The actual lapse rate l is the product of the base lapse rate lB7 and the dynamic lapse multiplier f ðR; tÞ. f ðR; tÞ depends on the ratio of guaranteed value to market value (GV/MV). The dynamic lapse multiplier is a nondecreasing function in variable Sn, which means a GMAB policyholder is more likely to lapse when the embedded option is more out-of-the-money (i.e., when the ratio of account value and guarantee is high). Taking survival probability into account, the risk-neutral price of the embedded option is as follows: Z N þ þ rðtnÞ rðNnÞ e CðtÞqðtÞðG F Þ dt þ e CðNÞ ðG F Þ Hn ¼ EQ t t N N n n

(1) and the PV of the fees f n ¼ EQ n

Z

N

CðtÞ erðtnÞ F t mdt

n

Let NVn ðS n Þ be the net value of adding the guarantee to the VA product, which is NVn ðS n Þ ¼ H n f n

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Taking into account the mortality and lapse risk, EC of GMAB is deﬁned in the same way as previously deﬁned, P½ðNVn ðS n Þ NV0 ðS 0 ÞÞ ECGMAB o1 b Because of the path dependency of NV n ðS n Þ, an analytic form of ECGMAB is difﬁcult to achieve. In later sections of this chapter, a Monte Carlo simulation/ﬁnite difference hybrid algorithm is implemented to calculate ECGMAB . Fig. 2 illustrates the impact of mortality and lapse risk on the VA embedded option value Hn and fee amount fn as function of underlying equity price S, shown as red and blue lines respectively. The solid lines are the case when no mortality and lapse risk is taken into account; the dashed lines in the left plot represent a 5% constant mortality rate model, and the dashed lines in the right plot denote a 5% base lapse rate model. As observed, mortality risk reduces fee amount fn (as less people pay premium payments) and has opposite effect on Hn upon underlying equity price. VA: Impact of Mortality Risk

VA: Impact of Lapse Risk 0.8

0.8 VA VA VA VA

0.7

0.6

0.6

0.5

0.5 Value

Value

0.7

option without mortality fee without mortality option with mortality fee with mortality

0.4

0.3

0.2

0.2

0.1

0.1

0

0.2 0.4 0.6 0.8 1 Equity price S

Fig. 2.

1.2 1.4 1.6 1.8 2

option without lapse fee without lapse option with lapse fee with lapse

0.4

0.3

0

VA VA VA VA

0

0 0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8 2

Equity price S

Impact of Mortality and Lapse Risk on VA.

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Pricing and Risk Management of VA and EIA

Mortality triggers early exercise; thus, when equity price is low, mortality causes in-the-money put option exercise and the contract becomes less proﬁtable to the insurer; when equity price is high, early exercise is suboptimal to policyholders as the put option time value is lost entirely, turns the contract to be more in insurer’s favor. In contrast, lapse behavior always reduces fee amount fn and Hn, as less people keep paying premium and more people surrender their embedded options. In a simpler case, if lapse risks are assumed to be independent from the market (implying l(t) does not depend on Sn), a more simpliﬁed form of the GMAB would be accessible. Let BSPðn; tÞ be, at any time n, the value of the put option embedded in GMAB that matures at t, without taking lapse and mortality into account. From the previous section we know that emt V put ðS n ; eðmþrg Þt S 0 ; r; d; s; t nÞ S0 emt ¼ erg trðtnÞ Fðd 2 Þ Sn edðtnÞ Fðd 1 Þ S0

BSPðn; tÞ ¼

where 2

logðSn =S0 Þ ðm þ rg Þt þ ðr d þ s2 Þðt nÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s tn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s t n d1 ¼

Unlike Eq. (1), CðtÞ is no longer path-dependent and therefore can be factored out from the risk-neutral expectation. The embedded put option value in GMAB can be written as follows: Z N CðtÞqðtÞBSPðn; tÞdt þ CðNÞ BSPðn; NÞ Hn ¼ n

The PV of the fees is Z

N

fn ¼ n

¼

mS n S0

e Z

Rt 0

½lðuÞþqðuÞdu

N

e

Rt 0

erðtnÞ E Q ½F t mdt

½lðuÞþqðuÞdu

emðtnÞ dt

n

Proposition 1. In the case where both mortality and lapse risk are independent from the underlying equity prices, function NVn ðSn Þ is monotonically decreasing.

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Proof. See the appendix.

As NVn ðSn Þ is monotonic, its analytical EC (or equivalent, VaR) can be directly calculated in the same way as in the previous section (Fong & Lin, 1999): ECGMAB ¼ NVn;99% NV0 ¼ H n;99% f n;99% NV0 where Z

N

CðtÞqðtÞBSP99% ðn; tÞdt þ CðNÞBSP99% ðn; NÞ

H n;99% ¼ n

Z pﬃﬃ s2 f n;99% ¼ eðmd 2 Þn2:33s n m

N

e

Rt 0

½lðuÞþqðuÞdu

emðtnÞ dt

n pﬃﬃ s2 emt V put ðS 0 eðmd 2 Þn2:33s n ; eðmþrg Þt S 0 ; r; d; s; t nÞ S0 pﬃﬃ s2 ¼ erg trðtnÞ Fðd 2 Þ emt eðmd 2 Þn2:33s n edðtnÞ Fðd 1 Þ

BSP99% ðn; tÞ ¼

pﬃﬃﬃ 2 2 ðm d s2 Þn 2:33s n ðm þ rg Þt þ ðr d þ s2 Þðt nÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d1 ¼ s tn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s t n NV0 ¼ H 0 f 0

2.2. Valuation and Economic Capital of PTP 2.2.1. Product Description Unlike VA, EIA are general account8 assets. EIA contracts vary between insurance companies and the simplest EIA product is called PTP. This provides the beneﬁciary return on an index, but with a minimal guarantee (which is call-like) at the contract’s maturity (usually death protection is included). An example of a PTP contract is as follows: at the initiation, t ¼ 0, the policyholder enters into a contract by paying the insurance company an initial amount P. The insurance company invests the amount P into the bond market, and there is no further payment from the policyholder.

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The insurance company guarantees a ﬁxed rate of return rg (with a prespeciﬁed guaranteed proportion) up to the end of the contract (guaranteed payment can be caused by mortality, maturity, or lapse decided by the policyholder), when the beneﬁciary will receive the greater of either the return on an index (with a pre-speciﬁed participation rate) or the guaranteed amount. If the policyholder lapses the EIA contract before maturity, he/she can get the guaranteed amount back after some penalty fees are charged, but the return on that index is not redeemable. 2.2.2. PTP without Mortality and Lapse Consider a simple PTP contract with $1 initial account value and maturity time N (in years) with ﬁxed-interest rate rg and guaranteed proportion Z (95% or 100% is common). Also, assume the underlying equity index price follows geometric Brownian motion with constant risk-free rate and volatility. Let Gn ¼ Z erg n ; 0 n N be the amount of account value that is guaranteed. Similar to a GMAB contract, Gn is going to be the strike price for its embedded option. Let Sn represent the value at n of the equity index used. Given a participation rate a, the beneﬁciary of embedded call option payoff at maturity will be

þ

SN 1þa 1 Z erg N H N ¼ ðF N G n Þþ ¼ S0 þ a S 0 rg N ¼ S N ðZe ð1 aÞÞ a S0 with FN ¼

SN 1þa 1 S0

where FN is the available amount for participation. At any time noN, the embedded call value on this contract can be formulated through risk-neutral pricing theory. rðNnÞ HN Hn ¼ EQ n ½e

Let notation V call ðS 0 ; K; r; d; s; tÞ represent the price of a standard European call. Under the Black–Scholes–Merton framework (Black & Scholes, 1973), the closed form of the embedded option value Hn can be

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written as follows:

a S 0 rg N Hn ¼ V call Sn ; ðZe ð1 aÞÞ; r; d; s; N n a S0 aSn Fðd 1 Þ ðZerg N ð1 aÞÞerðNnÞ Fðd 2 Þ ¼ edðNnÞ S0 where logðaS n =½S 0 ðZe d1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s N n

rg N

s2 ðN nÞ ð1 aÞÞÞ þ r d þ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s Nn

Similar to a GMAB contract, the net value of adding the guarantee to the PTP product at time n, noted by NVn ðS n Þ, can be formulated as the difference between two parts: the ﬁrst part is the embedded option (guarantee) value, from time n to maturity N; the second part is the present value of the fee that is going to be charged in the future (noted as fn). NVn ðSn Þ has the following form: NVn ðS n Þ ¼ H n f n where Z

N

erðtnÞ ðr rg ÞZdt ¼

fn ¼ n

r rg Z½1 erðNnÞ r

The corresponding EC of the PTP is deﬁned as the percentile risk measure of NVðSn Þ: P½NVn ðSn Þ NV0 ðS0 Þ ECPTP o1 b where b is the conﬁdence level. As NVn ðSn Þ is again monotonic,9 its analytical EC (or equivalent, VaR) is accessible (Fong & Lin, 1999). Supposing that a 99% conﬁdence level (notice this is under realistic measure) is applied, the EC under the current framework is as follows: ECPTP ¼ NVn;99% NV0 ¼ H n;99% f n NV0 pﬃﬃ s2 ¼ edðNnÞ aeðmd 2 Þnþ2:33s n Fðd 1 Þ ðZerg N ð1 aÞÞerðNnÞ Fðd 2 Þ f n NV0

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with

pﬃﬃﬃ s2 s2 n þ 2:33s n þ logða=½ðZerg N ð1 aÞÞÞ þ r d þ ðN nÞ md 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d1 ¼ s N n pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s N n NV0 ¼ H 0 f 0

2.2.3. PTP with Mortality and Lapse The effect of taking mortality into consideration in a PTP contract is similar to the GMAB case. By using the same terminology, let CðtÞ be the percentage of policyholders that survive and do not lapse before t, qðtÞ and lðSt ; tÞ be the mortality and lapse intensities (or equivalently, hazard rates), respectively. Independence between lapse risk and mortality risk is also assumed. Then we see that Rt ½lðS ;uÞþqðuÞdu CðtÞ ¼ e 0 u Similar to GMAB, lapse risk is not fully diversiﬁable and CðtÞ depends on the whole underlying equity price path fSn g prior to t. At any time n, the instantaneous lapse rate can be modeled as follows: lðS n ; tÞ ¼ f ðR; tÞ l B with R¼

Gn S 0 ¼ Z erg n F n Sn

The actual lapse rate l is the product of the base lapse rate lB and the dynamic lapse multiplier f ðR; tÞ. f ðR; tÞ depends on the ratio of market value to guaranteed value (MV/GV, which is different from GMAB). The dynamic lapse multiplier is again a non-decreasing function in variable Sn, which means a PTP policyholder tends to lapse more likely when the embedded option is more out-of-the-money (i.e., when the ratio of account value and guarantee is high). Taking survival probability into account, the risk-neutral price of the embedded option at time n is as follows: Z N þ þ Q rðtnÞ rðNnÞ e CðtÞqðtÞðF t Gt Þ dt þ e CðNÞ ðF N GN Þ Hn ¼ En n

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And the PV of the fees is Z

N

CðtÞ erðtnÞ ðr rg ÞZdt n Z N CðtÞ erðtnÞ dt ¼ ðr rg ÞZ

fn ¼

n

Let NVn ðS n Þ be the net value of adding the guarantee to the PTP product, which is NVn ðS n Þ ¼ H n f n Taking into account the mortality and lapse risk, the EC of PTP is deﬁned as follows: P½NVn ðSn Þ NV0 ðS0 Þ ECPTP o1 b An analytical form of ECPTP is difﬁcult to achieve. In this chapter, a Monte Carlo simulation/ﬁnite difference hybrid algorithm is implemented to calculate ECPTP. Fig. 3 illustrates the impact of mortality and lapse risk on the EIA embedded option value Hn and fee amount fn as function of underlying equity price S, shown as red and blue lines respectively. The solid lines are the case when no mortality and lapse risk is taken into account; the dashed lines in the left plot represent a 5% constant mortality rate model, and the dashed lines in the right plot denote a 5% base lapse rate model. Similar to Fig. 2, mortality risk reduces fee amount fn (as less people pay premium payments) and has opposite effect on Hn upon underlying equity price. Mortality triggers early exercise; thus, when equity price is low, the out-the-money call option is exercised unproﬁtably and the contract becomes more proﬁtable to the insurer; when equity price is high, call option is in-the-money and early exercise is in policyholder’s favor. In contrast, lapse behavior always reduces fee amount fn and Hn, as less people keep paying premium and more people surrender their embedded options. In a simpler case, if lapse risks are assumed to be independent from the market (which means l(t) does not depend on Sn), a clearer form of the PTP would be accessible. Let BSCðn; tÞ be, at any time n, the value of the call option embedded in PTP that matures at t, without taking lapse and

199

Pricing and Risk Management of VA and EIA EIA: Impact of Mortality Risk

EIA: Impact of Lapse Risk

0.7

0.7 EIA option without mortality EIA fee without mortality EIA option with mortality EIA fee with mortality

0.6

EIA option without lapse EIA fee without lapse EIA option with lapse EIA fee with lapse

0.6

0.5

0.4

0.4 Value

Value

0.5

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

2

0

0

0.2 0.4 0.6 0.8

Equity price S

Fig. 3.

1

1.2 1.4 1.6 1.8

2

Equity price S

Impact of Mortality and Lapse Risk on EIA.

mortality into account. From the previous section, we know that

a S 0 rg t V call S n ; ðZe ð1 aÞÞ; r; d; s; t n BSCðn; tÞ ¼ S0 a dðtnÞ aS n Fðd 1 Þ ðZerg t ð1 aÞÞerðtnÞ Fðd 2 Þ ¼e S0 with logðaS n =½S0 ðZe d1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s t n

rg t

s2 ðt nÞ ð1 aÞÞÞ þ r d þ 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s tn

Here CðtÞ is no longer path-dependent and therefore can be factored out from the risk-neutral expectation. The embedded call option value in PTP

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can be written as follows: Z N Hn ¼ CðtÞqðtÞBSCðn; tÞdt þ CðNÞ BSCðn; NÞ n

The PV of the fees is Z f n ¼ ðr rg ÞZ

N

CðtÞ erðtnÞ dt

n

where NVn ðSn Þ is again monotonic through similar steps to those in the proof of Proposition 1. The EC of PTP can be calculated through the same way as in the last section (Fong & Lin, 1999): ECPTP ¼ NVn;99% NV0 ¼ H n;99% f n NV0 with Z

N

CðtÞqðtÞBSC99% ðn; tÞdt þ CðNÞ BSC99% ðn; NÞ

Hn ¼ n

pﬃﬃ S s2 a 0 V call ðS 0 eðmd 2 Þnþ2:33s n ; ðZerg t ð1 aÞÞ; r; d; s; t nÞ a S0 pﬃﬃ s2 ¼ edðtnÞ aeðmd 2 Þnþ2:33s n Fðd 1 Þ ðZerg t ð1 aÞÞerðtnÞ Fðd 2 Þ

BSC99% ðn; tÞ ¼

pﬃﬃﬃ s2 s2 rg t md n þ 2:33s n þ logða=½Ze ð1 aÞÞ þ r d þ ðt nÞ 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d1 ¼ s tn pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 ¼ d1 s t n NV0 ¼ H 0 f 0

2.3. Pricing VA/EIA: A Finite Difference Approach 2.3.1. Methodology Oppose to the Monte Carlo simulation, ﬁnite difference is a fast and highly efﬁcient method to compute the irregular and path-dependent VA/EIA option value.10 The ﬁrst step we take is to remove path dependency. Let H n ¼ H n ðCðtÞ; S; tÞ be the option value of VA or EIA and CðtÞ be the survival

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probability at time t. Intuitively, we have H n ðCðtÞ; S; tÞ ¼ CðtÞ H n ð1; S; tÞ This implies that option value Hn is proportional to survival probability C(t). For example, the price of an option with half policyholders left should be exactly half of the price when 100% of policyholders stay in the contract, given other conditions unchanged. Therefore, we let notation H n ðS; tÞ stand for H n ð1; S; tÞ for simplicity. Similarly, we have f n ðCðtÞ; S; tÞ ¼ CðtÞ f n ð1; S; tÞ In addition, we use f n ðS; tÞ for f n ð1; S; tÞ. We call H n ðS; tÞ and f n ðS; tÞ as all-survival prices. Both H n ðS; tÞ and f n ðS; tÞ are Markovians and can be solved through a partial differential equation (PDE) approach. In addition, we introduce the discrete mortality and lapse model (DMLM): Assuming mortality, lapse behavior and fee charging only happen discretely at nodes t0 ; t1 ; . . . ; tN , as shown in the following graph:

Black-Scholes PDE holds in any period (ti, ti+1)

t ti

ti+1

Mortality and lapse event only happens discretely at nodes t0, t1,..., tn

The Black–Scholes PDE holds for both H n ðS; tÞ and f n ðS; tÞ in between every open interval ðti ; tiþ1 Þ, because no mortality, lapse, or fee charge event happens between nodes: @V 1 2 2 @2 V @V rS sS þ rV ¼ 0 2 @t 2 @S @S

on ðti ; tiþ1 Þ

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In this PDE, t stands for time to maturity. Let x ¼ lnðSÞ to get constant coefﬁcients.

@V 1 2 @2 V 1 2 @V rd s (2) s þ rV ¼ 0 on ðti ; tiþ1 Þ @t 2 @x2 2 @x Under the DMLM framework, the ﬁnite difference method will be applicable. 2.3.2. Crank–Nicholson Scheme þ For any time node ti, we split it into two nodes t i and ti , which are inﬁnitely 11 close. − Black-Scholes PDE holds in any period (ti+,ti+1 )

S

t − + t i−1 t i−1

ti−

ti+

− + ti+1 ti+1

Mortality and lapse event only happens discretely at nodes t0, t1,...,tN

We use the Crank–Nicholson scheme here for its second-order accuracy and non-conditional convergence. Let V nk ¼ V nk ðSk ; tþ n Þ and nþ1 ¼ V ðS ; t Þ; the discretization form of Eq. (2) is V nþ1 k nþ1 k k

V nþ1 V nk 1 1 2 V nkþ1 2V nk þ V nk1 1 2 V nkþ1 V nk1 n k s s þ r d ¼ rV k 2 2 2 Dt Dx2 2Dx " #

nþ1 2V nþ1 þ V nþ1 1 1 2 V nþ1 1 2 V nþ1 k k1 kþ1 V k1 nþ1 þ s kþ1 s þ r d rV k 2 2 2 Dx2 2Dx

This can be simpliﬁed in matrix form:

1 1 I DtA V nþ1 ¼ I þ DtA V n 2 2 where I is identity matrix and A is triangular with constant coefﬁcients. Eq. (2) has accuracy of OðDx2 þ Dt2 Þ and can be solved quickly by Thomas’ algorithm with FLOP counts OðNÞ. Special attention pays to all time nodes t n where mortality, lapse, and fee charge occur. According to DMLM, at t n , there is qDt percentage of policyholders die (implying that portion of the total option is exercised) and

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203

lDt percentage of policyholders lapse (meaning that portion of option is abandoned). This can be incorporated into schemes as follows: VA : H n ðS; tþ Þ ¼ qDt maxðK S; 0Þ þ ð1 qDt lDtÞH n ðS; t Þ EIA : H n ðS; tþ Þ ¼ qDt maxðS K; 0Þ þ ð1 qDt lDtÞH n ðS; t Þ Both q and l depend on (S, t) and are computed at each grid ðS k ; tn Þ. For fee charge fn, it satisﬁes PDE (Eq. (2)) inside each interval ðtþ i ; tiþ1 Þ. At time node tn , according to DMLM, qDt þ lDt units of policyholders exit their contract; meanwhile, there are mSDt (for VA) or ðr rg ÞZDt (for EIA) amount of extra fee charged. This can be modeled as follows: VA : f n ðS; tþ Þ ¼ ð1 qDt lDtÞ ðf n ðS; t Þ þ SmDtÞ EIA : f n ðS; tþ Þ ¼ ð1 qDt lDtÞ ðf n ðS; t Þ þ ðr rg ÞZDtÞ The initial condition of VA and EIA option value H n ðS; tÞ is contract’s payoff function at maturity t ¼ 0. The fee charge fn has zero initial value for both VA and EIA: VA : H n ðS; 0Þ ¼ maxðK S; 0Þ EIA : H n ðS; 0Þ ¼ maxðS K; 0Þ f n ðS; 0Þ ¼ 0

2.4. Economic Capital and Conditional VaR for VA and EIA Mixture As introduced at the beginning of this chapter, natural diversiﬁcation effects exist for a portfolio that includes both VA (which is put-like) and EIA (which is call-like) products. Suppose both products share the same underlying equity process. Then, such a portfolio can be modeled as a straddle (or strangle), that is, whenever either product is in-the-money, the other one is likely to be out-of-the-money. More speciﬁcally, when stock price is low and VA is in-the-money, the option value embedded in EIA drops and draws the portfolio value to remain regular; when the stock price is high and EIA is in-the-money, not only is the option value embedded in VA drop, but also the policyholder’s account was charged by the insurance company with higher management fees; both lower the total loss of the whole portfolio. Therefore, the risk to the insurer that provides these products is reduced. In this chapter, we pick both VaR and Conditional VaR12 (CVaR) as risk measures.

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The following ﬁgures illustrate the diversiﬁcation effect between VA and EIA. The ﬁrst ﬁgure is the price distribution histogram of VA, EIA, and a mixture that contains 50% VA and 50% EIA. Compared to VA or EIA, the mixture has a very concentrated distribution range around zero. The second ﬁgure is the price of VA, EIA, and mixture as a function of underlying stock price. Compared to VA or EIA, the mixture curve is ﬂatter and less sensitive to stock moves. These all imply a smaller VaR.

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Pricing and Risk Management of VA and EIA

In the next section, we will introduce the Monte Carlo simulation/ﬁnite difference hybrid framework to calculate EC. 2.4.1. Economic Capital and Conditional VaR Calculation In this chapter, a Monte Carlo simulation/ﬁnite difference hybrid framework is used to valuate EC of both products and the diversiﬁcation beneﬁts. The simulation algorithm consists of the following steps: 1. Divide time t and space x ¼ lnðSÞ up into M by N discrete intervals: 0 ¼ t1 ot2 o . . . otM ¼ T X Min ¼ x1 ox2 o . . . oxN ¼ X Max

2.

3.

4.

5.

Let fS k ¼ expðxk Þjk ¼ 1; . . . ; Mg be the stock space nodes. Set up the ﬁnite difference grid. Given any time horizon n, we solve PDE backwards till time t ¼ n. We can get option and fee all-survival prices at all space nodes at t ¼ n: fH kn jk ¼ 1; . . . ; Ng and ff kn jk ¼ 1; . . . ; Ng. Simulate the equity price paths from time 0 to time t ¼ n. Let NSM be the total number of simulations runs. The equity price at t ¼ n is fS n;i ji ¼ 1; . . . ; NSMg. We also simulate survival probability CðtÞ along these equity paths till t ¼ n (taking mortality qðtÞ and lapse rate lðS; tÞ into account): fCn;i ji ¼ 1; . . . ; NSMg. This is called the outer simulation paths. At time t ¼ n for each simulation path i, we have equity price S n;i . Supposing Sn;i is located between two adjoining stock space nodes, ~ ~ ~ S k oSn;i oSkþ1 oSkþ2 , we interpolate a quadratic polynomial13 between k k ~ k~ þ 1; k~ þ 2g to compute all-survival option value H~ n;i and fH n ; f n jk ¼ k; fee f~n;i , respectively. The actual option, fee, and VA/EIA net value are calculated as follows: H n;i ¼ Cn;i H~ n;i f ¼ Cn;i f~ n;i

n;i

NVn;i ¼ H n;i f n;i 6. Repeat steps 4 and 5 until we get VA and EIA net value NVn;i for all simulation paths i ¼ 1; . . . ; NSM as shown in the ﬁgure below. We can compute EC and Conditional VaR at 99% level of NVn;i .

For a portfolio P that includes both VA and EIA products, let w be the weight of VA. We can optimize w to minimize the portfolio’s EC14 at any time horizon t ¼ n: EIA min ðVaR½wNVVA n þ ð1 wÞNVn Þ

0owo1

w can be optimized through usual iteration algorithms such as Newton’s method or gradient descent. The following graph is an example of the portfolio EC as the function of w: Economic Capital at t=1

0.25

EC

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4 0.5 0.6 W eight of VA

0.7

0.8

0.9

1

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Pricing and Risk Management of VA and EIA

Table 2 provides the EC requirements and conditional VaR for VA, EIA, and the optimal VA/EIA mixture based on different time horizons.15 The optimal weight column is the percentage weight of VA in the optimal portfolio. Graphical results are listed in Fig. 4. In this example, the natural hedging effect is signiﬁcant. In the ﬁrst 5 years, the optimal mixtures have an average of 43% smaller EC requirements than VA, and these are 78% smaller compared to EIA. These optimal mixtures also have Conditional VaR that are superior to any single product. Economic Capital Requirements and CVaR for VA, EIA, and Mixture.

Table 2. Tenor (Years)

1 2 3 4 5

Economic Capital (VaR 99%)

Optimal Weight (VA, %)

VA

EIA

Optimal portfolio

0.19 0.26 0.32 0.37 0.42

0.29 0.57 0.89 1.26 1.70

0.07 0.13 0.19 0.25 0.31

50 57 64 70 75

Conditional VaR

VA

EIA

Optimal portfolio

0.22 0.31 0.37 0.43 0.48

0.35 0.70 1.11 1.57 2.13

0.08 0.16 0.23 0.30 0.36

Note: The bold/italic numbers are used to show the effect of risk reduction (in terms of Eco Capital and Cond. VaR) from an optimized VA/EIA portfolio.

Economic Capital (EC) Comparison 2.00

VA EIA Optimal Portfolio

1.60

EC

1.20 0.80 0.40 0.00 1

2

3

4

5

Tenor (Years)

Fig. 4.

Economic Capital Requirement for VA, EIA, and the Optimal Mixture.

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It is also observed in Table 2 that the EC of EIA goes up tremendously. This is because for shortening a call, there is not an upper bound for the future loss. While in the mixture portfolio, the loss from EIA is balanced out by the moneyness of the option embedded in VA and the fees charged from policyholder’s account, as shown in the optimal portfolio column. In Table 2, the weights of VA in the optimal mixture portfolio grow gradually. This is because the capital demand from EIA increases quickly and thus requires more weight in the VA to offset.

2.5. Conclusion This chapter contributes to the literature in the area of equity-linked insurance contract pricing and analyzing natural diversiﬁcation beneﬁts between VA and EIA products. These beneﬁts result from the reason that the values of VA and EIA move in opposite directions in response to a change in the underlying equity value. The author modeled VA and EIA in the risk-neutral option pricing framework and implemented ﬁnite difference pricing scheme. Numerical examples show that natural hedging is feasible and the beneﬁts are signiﬁcant, which enables insurance companies’ capital to be deployed more efﬁciently.

NOTES 1. Source: National Association for Variable Annuities (NAVA). 2. The term ‘‘lapse’’ means a policyholder unwinds his insurance contract, liquidates his account, and exits. Usually a certain penalty fee is necessary for the cost of breaking an existing contract. 3. Economic capital (EC) in this chapter is deﬁned as the difference between 99% Value at Risk (VaR) of product’s net value and initial value: ECðtÞ ¼ VaR99% ðVðtÞÞ Vð0Þ. 4. Monotonicity of function NVn ðSn Þ is implied by the negativeness of its ﬁrst derivative with respect to Sn. 5. For incomplete mortality market analysis, please refer to Follmer and Sonderman (1986). 6. This is because life insurance policyholders are neither ﬁnancial professionals nor institutional investors, and lapse does happen for reasons unrelated to the equity performance. Liquidity problems and defaults can be examples. 7. Base lapse rate can be inﬂuenced by macro-economic factors such as domestic economy and federal rates. 8. Differs from VA, the owners of general accounts could lose part or all of their investments if the insurer defaults.

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9. Monotonicity of function NVn ðS n Þ is implied by the positiveness of Hn’s ﬁrst derivative with respect to Sn. Here fn is not a function of Sn and therefore has no contribution to dðNVn ðS n ÞÞ=dS n . 10. Monte Carlo simulation results can be greatly improved if we take the no-mortality/lapse option value as a control variate. However, such approach would still be a lot slower than using the ﬁnite difference algorithm described in this chapter. 11. The split technique here is for implementation purpose only. 12. Conditional VaR is deﬁned as the conditional expectation of random variable that exceeds its VaR: CVaRa ðXÞ ¼ EðXjX4VaRa ðXÞÞ. CVaR is coherent and therefore is usually considered as a better alternative risk measure to VaR. CVaR is also called Expected Shortfall or Expected Tail Loss in Finance. Please refer to Pﬂug (2000) for more detail. 13. We use quadratic interpolation here to be consistent with the second-order accuracy of Crank–Nicholson scheme. 14. Alternatively we can run optimization targeting CVaR of the portfolio, which is not covered in the scope of this chapter. 15. Valuation parameters are as follows: VA: Maturity N ¼ 10 years, guaranteed interest rate rg ¼ 2%, premium charge 2%. EIA: Maturity N ¼ 10 years, guaranteed interest rate rg ¼ 2%, guaranteed amount Z ¼ 100%, participation rate a ¼ 70% Mortality rate: 1%, base lapse rate: 2%. Equity drift m ¼ 12%, volatility s ¼ 0.2, dividend yield d ¼ 2%. 16. Here, the authors intentionally skipped rigorous mathematical proof of the interchange of derivative and integral. Precisely, this formula is valid only when the following technical conditions hold: (1) Both CðtÞqðtÞBSPðn; tÞ and dðCðtÞqðtÞBSPðn; tÞÞ=dS n are continuous; (2) Both CðtÞqðtÞBSPðn; tÞ and dðCðtÞqðtÞBSPðn; tÞÞ=dSn are bounded by a L1 function. See Cheney (2001) for example.

REFERENCES Aase, K., & Persson, S. (1994). Pricing of unit-linked life insurance policies. Scandinavian Actuarial Journal, 1, 26–52. Bacinello, A. R., & Ortu, F. (1993a). Pricing equity-linked life insurance with endogenous minimum guarantees. Insurance: Mathematics and Economics, 12, 245–257. Bacinello, A. R., & Ortu, F. (1993b). Pricing guaranteed securities-linked life insurance under interest-rate risk. Actuarial Approach for Financial Risks, 1, 35–55. Bacinello, A. R., & Ortu, F. (1996). Fixed income linked life insurance policies with minimum guarantees: Pricing models and numerical results. European Journal of Operational Research, 91, 235–249. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(May–June), 637–659. Boyle, P. P., & Schwartz, E. S. (1977). Equilibrium prices of guarantees under equity-linked contracts. The Journal of Risk and Insurance, 44, 639–680.

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Brennan, M. J., & Schwartz, E. S. (1976). The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financial Economics, 3, 195–213. Brennan, M. J., & Schwartz, E. S. (1979). Alternative investment strategies for the issuers of equity linked life insurance policies with an asset value guarantee. Journal of Business, 52, 63–93. Cheney, W. E. (2001). Analysis for applied mathematics. New York: Springer. Follmer, H., & Sonderman, D. (1986). Hedging of non-redundant contingent claims. In: W. Hildenbrand & A. Mas-Colell (Eds), Contributions to mathematical economics (pp. 205–223). Amsterdam: Elsevier Science. Fong, H. G., & Lin, K. (1999). A new analytical approach to value at risk. Journal of Portfolio Management, 25(5), 88–98. Hardy, M. R. (2003). Investment guarantees: Modeling and risk management for equity-linked life insurance. New Jersey: Wiley. Lin, X. S., & Tan, K. S. (2003). Valuation of equity-indexed annuities under stochastic interest rate. North American Actuarial Journal, 7(4), 72–91. Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(Spring), 141–183. Milevsky, M. A., & Posner, S. (2001). The titanic option: Valuation of the guaranteed minimum death beneﬁt in variable annuities and mutual funds. The Journal of Risk and Insurance, 68(1), 55–79. Milevsky, M. A., & Salisbury, T. S. (2002). The real option to lapse and the valuation of deathprotected investments. Working paper. Milevsky, M. A., & Salisbury, T. S. (2004). Static and dynamic valuation of guaranteed minimum withdrawal beneﬁts. Working paper. Moller, T. (1998). Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bulletin, 28(1), 17–47. Nielsen, J. A., & Sandmann, K. (1995). Equity-linked life insurance: A model with stochastic interest rates. Insurance: Mathematics and Economics, 16, 225–253. Pﬂug, G. Ch. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In: S. Uryasev (Ed.), Probabilistic constrained optimization: Methodology and applications. Norwell, MA: Kluwer Academic Publishers. Shumrak, M., Greenbaum, M., Darley, V., & Axtell, R. (1999). Modeling annuity policyholder behavior using behavioral economics and complexity science. Canadian Institute of Actuaries Annual Meeting. Tiong, S. (2000). Valuing equity-indexed annuities. North American Actuarial Journal, 4, 149–170.

APPENDIX Notations Gn rg N Fn Hn

9 9 9 9 9

guaranteed level guaranteed interest rate maturity of product account value at time n value of embedded option

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Pricing and Risk Management of VA and EIA S0 m Z a V put ðS 0 ; K; r; d; s; tÞ CðtÞ lðS t ; tÞ q(t) NVn fn b EC CVaR BSCðn; tÞ BSPðn; tÞ

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

underlying equity price at time 0 management fee of VA charged each year guaranteed amount of EIA participation rate price of a vanilla European put survival probability instantaneous lapse rate instantaneous mortality rate net value of guarantee value of beneﬁt charge conﬁdence level economic capital Conditional Value at Risk European call price at time n and maturities at t European put price at time n and maturities at t

Proof of Proposition 1 Proposition 1. In the case where both mortality and lapse risk are independent from the underlying equity price, the function NVn ðS n Þ is monotonically decreasing. Proof. If both risks are independent from the underlying equity price Sn, NVn ðS n Þ has the following form: NVn ðS n Þ ¼ H n f n with Z

N

CðtÞqðtÞBSPðn; tÞdt þ CðNÞ BSPðn; NÞ

Hn ¼ n

The PV of the fees is S n fn ¼ S0

Z

N

e

Rt 0

½lðuÞþqðuÞdu

emðtnÞ dt

n

Now, take the ﬁrst derivative of both Hn and fn with respect to Sn:16 Z N dH n dðBSPðn; tÞÞ dðBSPðn; NÞÞ ¼ CðtÞqðtÞ dt þ CðNÞ dS n dS dSn n n

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As dðBSPðn; tÞÞ emt dðtnÞ ¼ e Fðd 1 Þo0 S0 dSn we know that Hn is monotonically decreasing. For fn, Z N Rt df n ½lðuÞþqðuÞdu ¼ e 0 emðtnÞ dt 0 dS n S0 n which implies that fn is monotonically increasing. Therefore, NVn ðS n Þ is monotonically decreasing.