The Credit Market Handbook: Advanced Modeling Issues (Wiley Finance)

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The Credit Market Handbook Advanced Modeling Issues H. GIFFORD FONG, EDITOR Journal of Investment Management (JOIM)

WILEY John Wiley & Sons, Inc.

The Credit Market Handbook

Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States. With ofﬁces in North America, Europe, Australia and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding. The Wiley Finance series contains books written speciﬁcally for ﬁnance and investment professionals as well as sophisticated individual investors and their ﬁnancial advisors. Book topics range from portfolio management to e-commerce, risk management, ﬁnancial engineering, valuation and ﬁnancial instrument analysis, as well as much more. For a list of available titles, visit our Web site at www.Wiley Finance.com.

The Credit Market Handbook Advanced Modeling Issues H. GIFFORD FONG, EDITOR Journal of Investment Management (JOIM)

WILEY John Wiley & Sons, Inc.

Copyright © 2006 by the Journal of Investment Management. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data The credit market handbook : advanced modeling issues / H. Gifford Fong, Editor. p. cm. – (Wiley ﬁnance series) Includes index. ISBN-13: 978-0-471-77862-2 (cloth) ISBN-10: 0-471-77862-1 (cloth) 1. Credit–Mathematical models. 2. Risk management–Mathematical models. 3. Default (Finance)–Mathematical models. I. Fong, H. Gifford. II. Series. HC3701.C74 2006 332.701–dc22

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Contents Introduction Executive Chapter Summaries

ix xi

CHAPTER 1 Estimating Default Probabilities Implicit in Equity Prices

1

Tibor Janosi, Robert Jarrow, and Yildiray Yildirim Introduction The Model Structure Description of the Data Estimation of the State Variable Process Parameters Equity Return Estimation Analysis of the Time Series Properties of the Parameters Analysis of Fama–French Four-Factor Model with No Default Analysis of a Bubble Component (P/E ratio) in Stock Prices Analysis of the Default Intensity Relative Performance of the Equity Return Models Comparison of Default Intensities Based on Debt versus Equity Conclusions Notes References Appendix

1 3 7 9 12 22

30 32 33 33 35

CHAPTER 2 Predictions of Default Probabilities in Structural Models of Debt

39

26 26 27 29

Hayne E. Leland Introduction Recent Empirical Studies Structural Models and Default Risk

40 42 43

vi

CONTENTS

The Default Boundary in Exogenous and Endogenous Cases The Default Probability with Constant Default Barrier Calibration of Models: The Base Case Matching Empirical Default Frequencies with the L-T Model Matching Empirical DPS with the L-S Model The Moody’s–KMV Approach Some Preliminary Thoughts on the Relationship Between the KMV Approach and L-S/L-T Conclusions Acknowledgments Postscript Appendix Notes References

CHAPTER 3 Survey of the Recent Literature: Recovery Risk

45 46 46 47 50 54 55 57 57 58 58 58 62

65

Sanjiv R. Das Introduction Empirical Attributes Recovery Conventions Recovery in Structural Models Recovery in Reduced-Form Models Measure Transformations Summary and Speculation References

CHAPTER 4 Non-Parametric Analysis of Rating Transition and Default Data

65 67 69 70 71 73 75 75

77

Peter Fledelius, David Lando, and Jens Perch Nielsen Introduction Data and Outline of Methodology Estimating Transition Intensities in Two Dimensions One-Dimensional Hazards and Marginal Integration Conﬁdence Intervals Transitions: Dependence on Previous Move and Duration Multiplicative Intensities Concluding Remarks Acknowledgments

77 82 83 88 90 91 94 98 98

Contents

Notes References

CHAPTER 5 Valuing High-Yield Bonds: A Business Modeling Approach

vii 99 99

101

Thomas S. Y. Ho and Sang Bin Lee Introduction Speciﬁcation of the Model A Numerical Illustration Empirical Evidence Implications of the Model Conclusions Acknowledgments Appendix Notes References

CHAPTER 6 Structural versus Reduced-Form Models: A New Information-Based Perspective

102 104 108 113 115 115 116 116 117 117

118

Robert A. Jarrow and Philip Protter Introduction The Setup Structural Models Reduced-Form Models A Mathematical Overview Observable Information Sets Conclusion Acknowledgment Notes References

CHAPTER 7 Reduced-Form versus Structural Models of Credit Risk: A Case Study of Three Models

118 120 120 123 125 128 129 129 129 130

132

Navneet Arora, Jeffrey R. Bohn, and Fanlin Zhu Introduction Merton, Vasicek–Kealhofer, and Hull–White Models Data and Empirical Methodology Results

133 137 141 144

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CONTENTS

Conclusion Acknowledgments Notes References

CHAPTER 8 Implications of Correlated Default for Portfolio Allocation to Corporate Bonds

160 161 161 162

165

Mark B. Wise and Vineer Bhansali Introduction A Model for Default The Portfolio Problem Sample Portfolios with Zero Recovery Fractions Sample Portfolios with Nonzero Recovery Fractions Concluding Remarks Acknowledgments Notes References

165 168 171 176 181 182 184 184 184

CHAPTER 9 Correlated Default Processes: A Criterion-Based Copula Approach

186

Sanjiv R. Das and Gary Geng Introduction Description of the Data Copulas and Features of the Data Determining the Joint Default Process Simulating Correlated Defaults and Model Comparisons Discussion Acknowledgments Appendix: The Skewed Double Exponential Distribution Notes References

Index

187 190 191 196 204 210 212 212 215 216 219

Introduction Credit analysis is undergoing a dramatic transformation owing to rigorous quantitative treatment. Probability theory, statistical modeling, and the modern theories of ﬁnance are being applied to help establish the critical function of how to determine credit risk. This book showcases a series of papers drawn from the published works in the archives of the Journal Of Investment Management (JOIM). Our intent is to bring together noted authors and their work to provide a rich framework of research in the credit analysis area. Our ﬁrst chapter, “Estimating Default Probabilities Implicit in Equity Prices,” by Janosi, Jarrow, and Yildirim, focuses on the use of equity prices as input in the probability of default measure. Leland, in “Predictions of Default Probabilities in Structural Models of Debt,” evaluates alternative structural model methodologies. Das, in “Recovery Risk,” reviews the literature for estimating recovery (loss given default). “Non-Parametric Analysis of Rating Transition and Default Data,” by Fledelius, Lando, and Nielsen, illustrates the use of non-parametric and smoothing methods for analyzing rating transitions. Ho and Lee of term structure fame provide a model for high-yield bond valuation. Jarrow and Protter provide new insights into the comparison of structural versus reduced-form models in “Structural versus Reduced-Form Models: A New Information-Based Perspective.” This topic is further explored by Arova and Bohn in “Reduced-Form versus Structural Models of Credit Risk.” The last two chapters focus on the issue of correlated default. Das and Geng, in “Correlated Default Processes: A Criterion-Based Copula Approach,” describe the use of copulas in modeling correlated default, while Wise and Bhansali, in “Implications of Correlated Default for Portfolio Allocation to Corporate Bonds,” consider the optimization problem dealing with correlated default. Our thanks to the authors for their contributions to this book and to the general literature. Thanks also to Christine Proctor of Gifford Fong Associates and John Wiley and Sons, Inc. for their tireless efforts on behalf of this book.

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Executive Chapter Summaries

CHAPTER 1 ESTIMATING DEFAULT PROBABILITIES IMPLICIT IN EQUITY PRICES Tibor Janosi, Robert Jarrow, and Yildiray Yildirim This chapter uses a reduced-form credit risk model to estimate default probabilities implicit in equity prices. The equity return model developed includes the possibility of default, market risk premiums, and price bubbles. For a cross section of ﬁrms, this equity return model is estimated using monthly returns in a time-series regression. Three conclusions are obtained. First, the analysis supports the feasibility of estimating default probabilities implicit in equity returns. This new estimation procedure provides a third alternative to using either structural models with equity prices or reduced-form credit risk models with debt prices for estimating default probabilities. Second, we ﬁnd that equity returns contain a bubble component not captured by the traditional Fama–French four-factor model for equity’s risk premium. This bubble component, proxied by the ﬁrm’s price-earnings ratio, is signiﬁcant for many of the ﬁrms in our sample. Third, due to noise introduced in equity returns by price bubbles and market risk premium, the estimated default probabilities may confound these quantities, giving biased estimates with large standard errors. Indeed, the default probabilities obtained herein are larger than those previously obtained using either logit models based on historical data or those obtained implicitly from debt prices. By extrapolation, these results cast additional doubt on the precision of default probabilities obtained using structural models with equity prices.

CHAPTER 2 PREDICTIONS OF DEFAULT PROBABILITIES IN STRUCTURAL MODELS OF DEBT Hayne E. Leland This chapter examines default probabilities predicted by alternative “structural” models of risky coporate debt. We focus on default probabilities rather

xi

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EXECUTIVE CHAPTER SUMMARIES

than credit spreads because (1) they are not affected by additional market factors such as liquidity and tax differences, and (2) prediction of the relative likelihood of default is often stated as the objective of bond ratings. We have three objectives:

1. To distinguish “exogenous default” from “endogenous default” models. 2. To compare these models’ predictions of default probabilities, given common inputs. 3. To examine how well these models capture actual average default frequencies, as reﬂected in Moody’s Investors Service (2001) corporate bond default data 1970–2000.

We ﬁnd the endogenous and exogenous default boundary models ﬁt observed default Frequencies very well for horizons 5 years and longer, for both investment-grade and non-investment-grade ratings. Shorter-term default frequencies tend to be underestimated. This suggests that a jump component should be included in asset value dynamics. Both types of structural models ﬁt available default data equally well. But the models make different predictions about how default probabilities and recovery rates change with changes in debt maturity or asset volatility. Further data and testing will be needed to test these differences. Finally, we compare and contrast these structural models’ default predictions with a simpliﬁed version of the widely used Moody’s–KMV “distance to default” model described in Crosbie and Bohn (2002).

CHAPTER 3 SURVEY OF THE RECENT LITERATURE: RECOVERY RISK Sanjiv R. Das This chapter surveys a selection of recent working papers on recovery rates, providing a framework for extant research. Simpler versions of models are also presented with a view to aid accessibility and pedagogical presentation. Despite the obvious empirical difﬁculties encountered with recovery rate data, modeling advances are making possible better quantiﬁcation and measurement of recovery, and will result in innovative contracts to span this risk.

Executive Chapter Summaries

xiii

CHAPTER 4 NON-PARAMETRIC ANALYSIS OF RATING TRANSITION AND DEFAULT DATA Peter Fledelius, David Lando, and Jens Perch Nielsen Is the current rating of a bond issuer sufﬁcient to determine the probability of default or the probability of a rating change? The empirical evidence says no. Previous history matters, and probabilities depend on whether the ﬁrm entered into its current rating class through a downgrade or an upgrade, and on the amount of time spent in the current rating class. Non-parametric analysis of rating transition intensities is a powerful way of visualizing such effects and is therefore useful both for quickly understanding the behavior of a rating system and for exploring data before setting up a full statistical model. In this chapter we illustrate the use of non-parametric and smoothing methods for analyzing rating transitions by showing how the time spent in a particular rating class and the direction of the move into this class inﬂuence transition intensities away from the class.

CHAPTER 5 VALUING HIGH-YIELD BONDS: A BUSINESS MODELING APPROACH Thomas S. Y. Ho and Sang Bin Lee This chapter provides a model for valuing high-yield bonds. The model asserts that a corporate bond is a contingent claim on the ﬁrm, which in turn is a contingent claim on the business risks. Therefore, the bond model takes the debt structure, capital structure, operating leverage, and other aspects of a business model into account. The model is empirically testable and the chapter shows that the model can better explain some empirical results. A corporate bond valuation model that takes a business modeling approach has many applications. In corporate ﬁnance, we can identify the impact of the operating leverage and the ﬁnancial leverage on the risk transfer to the stakeholders of the ﬁrm. The model can be used for the design of the debt structure in relation to the business model of the ﬁrm. In equity research, the model solves for the appropriate cost of equity of the business risk. As a result, the model enables us to compare the stock value of ﬁrms in a similar business but with different capital structure, debt structure, and operating leverage. In bond research, we can value the senior and junior corporate bonds within the debt package, taking the business model into

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EXECUTIVE CHAPTER SUMMARIES

account. The model can be used for investment in stocks or bonds and in corporate ﬁnance in capital structure decisions.

CHAPTER 6 STRUCTURAL VERSUS REDUCED-FORM MODELS: A NEW INFORMATION-BASED PERSPECTIVE Robert A. Jarrow and Philip Protter For modeling credit risk, two classes of models exist: structural and reducedform. Structural models often assume that default occurs when the ﬁrm’s asset value hits a barrier, whereas reduced-form models use a hazard rate framework to model default. These models are viewed as disjoint and competing, and there is heated debate as to which class of models is best for predicting default and/or pricing credit-risky instruments. This chapter shows that structural and reduced-form models are not disjoint model types, but rather the same model containing different informational assumptions. Structural models assume knowledge held by the ﬁrm’s managers: continuous observations of the ﬁrm’s asset values and liabilities. In contrast, reduced-form models assume the knowledge observed by the market: the information generated by a set of state variables, and the ﬁrm’s default time and recovery rate process. It is shown that structural models can be transformed into reduced-form models as the information becomes less reﬁned—from that observed by the ﬁrm’s management, to that which is observed by the market. Given this insight, the current debate in the credit risk literature about these two model types needs to be redirected. The debate should be focused on whether or not the model should be based on the information set observed by the market. For pricing and hedging credit risk, the information set observed by the market is the relevant one. This is the information set used in the economy to determine prices. For internal risk management purposes with respect to its own credit risk, however, a structural model may be preferred.

CHAPTER 7 REDUCED FORM VERSUS STRUCTURAL MODELS OF CREDIT RISK: A CASE STUDY OF THREE MODELS Navneet Arora, Jeffrey R. Bohn, and Fanlin Zhu In this chapter we compare two structural models [basic Merton and Vasicek–Kealhofer (VK) as implemented by Moody’s–KMV] and one

Executive Chapter Summaries

xv

reduced-form model [Hull–White (HW)] of credit risk. In order to make the model testing relevant to practical implementation of these models, our evaluation criteria must address concerns faced by practitioners in practice (as opposed to theoretical concerns.) We propose that two useful purposes for credit models are default discrimination and relative value analysis. Default discrimination is relevant for quantitative risk and portfolio management. Relative value analysis is relevant for mark-to-market exercises for portfolio management and decision support in a credit trading environment. We test the ability of the Merton and VK models to discriminate defaulters from nondefaulters based on default probabilities generated from information in the equity market. We test the ability of the HW model to discriminate defaulters from nondefaulters based on default probabilities generated from information in the bond market. We ﬁnd the VK and the HW models exhibit comparable accuracy ratios and substantially outperform the simple Merton model. We also test the ability of each model to predict spreads in the credit default swap (CDS) market as an indication of each model’s strength as a relative value analysis tool. We ﬁnd the VK model tends to do the best across the full sample and relative subsamples except for cases where an issuer has many bonds in the market. In this case, the HW model tends to do the best. The empirical evidence will assist market participants in determining which model is most useful based on their “purpose in hand.” Note that these tests end up producing evaluations of not only the modeling approaches, but also the different data sources. Models relying on poor or scarce data are not that useful regardless of how good they seem in theory. On the structural side, a basic Merton model is not good enough; appropriate modiﬁcations to the framework make a difference. On the reduced-form side, the quality and quantity of data make a difference; many traded issuers will not be well modeled in this way unless they issue more traded debt. In addition, bond spreads at shorter tenors (less than two years) tend to be less correlated with CDS spreads. This makes accurate calibration of the term structure of credit risk difﬁcult when relying on bond data. The widespread availability and reliability of equity data tend to produce better results for the more sophisticated structural model evaluated in these tests. In cases where an issuer has many outstanding bonds and data for these bonds are widely available, the reduced-form model produces better results. For practitioners looking across the broad cross section of traded credit instruments, the data requirements for robust reduced-form modeling and the availability of robust equity-based measures should inform discussions on which modeling approach to use. Moreover, users of reduced-form models looking to price other credit-risky securities such as CDSs should bear in mind the potential impact on bond spreads of other risks resulting from interest rate movements and changes in liquidity. These effects can differ across size and spread levels, thereby distorting the performance of these models. A model such as VK is relatively more stable in its performance

xvi

EXECUTIVE CHAPTER SUMMARIES

across various categories by size and spreads. This model’s strength partially owes to its structural framework that uses equity data—which are less contaminated by other risks—and partially owes to its more sophisticated implementation. The overall results emphasize the importance of empirical evaluation when assessing the strengths and weaknesses of different types of credit risk models.

CHAPTER 8 IMPLICATIONS OF CORRELATED DEFAULT FOR PORTFOLIO ALLOCATION TO CORPORATE BONDS Mark B. Wise and Vineer Bhansali The possibility of correlated defaults makes the problem of optimal allocation to corporate bonds interesting for both plain vanilla corporate bond investors and investors in structured credit products. Because the loss distribution of these portfolios does not have to be normal, the applicability of classic mean–variance allocation machinery is called into question. In this chapter we show that under very general assumptions for the utility function of investors and a large range of parameters, the optimal allocation decision of an investor can still be well approximated by the ﬁrst three moments of the loss distribution. The intuition developed by our numerical results is valuable for investors who are looking to invest in corporate bonds or structured products such as collateralized debt obligations (CDOs) or their tranches. This chapter provides a road map for setting these fees. It draws on a marketing research technique, conjoint analysis, and straightforward optimization procedure to guide fund managers toward making better decisions. These decisions have potential beneﬁts for both mutual fund companies in terms of increased proﬁtability and for investors in terms of increased utility. To demonstrate our approach we conducted a conjoint analysis with 50 mutual fund investors. We ﬁnd that, although most mutual fund investors strongly prefer low front-end loads, signiﬁcant opportunities for price discrimination exist in this marketplace. Our model uncovers the potential for multiple efﬁcient fee structures for different classes of fund shares, and links this potential to observable customer characteristics. Investment management companies can use these observed linkages to better target their pricing and overall marketing efforts.

Executive Chapter Summaries

xvii

CHAPTER 9 CORRELATED DEFAULT PROCESSES: A CRITERION-BASED COPULA APPROACH Sanjiv R. Das and Gary Geng We are witnessing an explosive growth in structured products based on baskets comprising a large number of defaultable bonds. These take the form of collateralized debt obligations (CDOs) or basket default swaps, and many others. Modeling a large system of correlated default in order to price and manage the risk of these new products is thus a fast-growing area of research. The joint movement of default probabilities is not usually governed by a multivariate normal distribution, which is often assumed for tractability. Moving beyond the normal distribution requires more sophisticated modeling. Copulas are a general and facile technical device for the modeling of correlated default processes with varying assumptions on the joint distribution. Although there is a growing literature on the technical modeling of default using copulas, the practicalities involved in ﬁtting copulas to a default system have not received empirical attention. This chapter seeks to ﬁll this gap. We develop a methodology to calibrate the joint default process of hundreds of issuers. To determine the appropriate choice of the joint default process, we propose a new metric. This metric accounts for three different aspects of default correlation: (1) level, (2) asymmetry, and (3) tail-dependence and extreme behavior. Given a choice of copula, the joint distribution of default probabilities may be estimated, and the system calibrated for simulation purposes. This chapter goes a step further and provides a methodology for comparison among copulas as well. The approach is based on a new metric that is practically motivated, and may be used by practitioners in deciding which copula to employ in their models. The designers of CDOs will be able to improve their pricing of CDO tranches. Rating agencies will be able to sharpen their assessments of securities based on correlated default. Traders in CDOs who are engaged in capital arbitrage will have a range of models to choose from. Insurance companies and depository institutions will be able to ﬁt copulas to both the asset and liability sides of their balance sheets, and assess the joint default risk of the ﬁrm in entirety. In addition, regulators may use copulas to model systemic risk by aggregating correctly across entities. This approach thus has practical value for many market participants.

CHAPTER

1

Estimating Default Probabilities Implicit in Equity Prices Tibor Janosi,a Robert Jarrow,b,∗ and Yildiray Yildirimc

This chapter uses a reduced-form credit risk model to estimate default probabilities implicit in equity prices. For a cross section of ﬁrms, a time-series regression of monthly equity returns is estimated. We show that it is feasible to infer the ﬁrm’s probability of default implicit in equity returns. However, the existence of price bubbles and the difﬁculty in modeling equity price risk premium confound the estimation of these default probabilities, generating potentially biased estimates with large standard errors. Comparing these default intensities with those obtained from historical data or implicitly from debt prices conﬁrms this result.

1. INTRODUCTION Given the recent exponential growth in the credit derivatives market (see Risk Magazine, 2000), credit risk modeling and estimation have become topics of interest. The theoretical literature is quite extensive (see Bielecki and Rutkowski, 2000, for a review). The empirical literature estimating reducedform credit risk models has concentrated on using debt prices (see Dufﬁe, 1999; Dufﬁe and Singleton, 1997; Dufﬁe et al., 2000; Janosi et al., 2002; Madan and Unal, 1998), credit derivative prices (see Hull and White, 2000, a Computer Science Department, Cornell University, Ithaca, NY, USA. b Johnson Graduate School of Management, Cornell University, Ithaca, NY, and

Kamakura Corporation, USA. c School of Management, Syracuse University, Syracuse, NY, USA. ∗ Corresponding author. Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853, USA. Tel.: +1 607 255 4729; e-mail: [email protected].

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THE CREDIT MARKET HANDBOOK

2001), or bankruptcy histories (see Altman, 1968; Chava and Jarrow, 2002; Shumway, 2001; Zmijewski, 1984). Equity prices have only been used to estimate default parameters for structural models (see Delianedis and Geske, 1998). The purpose of this chapter is to use equity prices in conjunction with a reduced form credit risk modeling approach to estimate default probabilities. The approach utilized is a slight generalization of the model contained in Jarrow (2001). The data used for this investigation are equity prices from the Center for Research in Security Prices (CRSP) and debt prices from the University of Houston’s Fixed Income Database over the time period May 1991–March 1997. The observation interval is 1 month. Debt prices consist of bids taken from Lehman Brothers trading sheets on the last calendar day in each month, see Warga (1999) for additional details. Fifteen different ﬁrms are included in this study, where the ﬁrms are chosen to stratify various industry groupings: ﬁnancial, food and beverages, petroleum, airlines, utilities, department stores, and technology. The same 15 ﬁrms as in Janosi et al. (2002) are included so that a comparison of the different estimation procedures can be performed. Eight different models for equity returns are investigated herein, the simplest models containing no default. Using a rolling estimation procedure for each month during this observation period, the equity model’s parameters (including the bankruptcy parameters) are estimated using a time-series regression on monthly equity returns. In this procedure, only information available to the market at the time the equations are estimated is utilized. First, the analysis supports the feasibility of estimating default probabilities implicit in equity returns. In a relative comparison of the eight models, in-sample root mean squared error goodness-of-ﬁt tests and outof-sample generalized cross-validation statistics support the necessity of including default parameters into the equity return model. The best performing default intensity depends on the spot rate of interest but not on an equity market index, conﬁrming similar results that Janosi et al. (2002) obtained when using debt prices. Second, we ﬁnd that equity returns contain a bubble component not captured by the Fama and French (1993, 1996) four-factor model for equity’s risk premium. This bubble component, proxied by the ﬁrm’s P/E ratio, is signiﬁcant for many of the ﬁrms in our sample. Third, due to the possible existence of equity price bubbles and the difﬁculty in modeling equity risk premium, the default intensity estimates obtained appear to confound these quantities. Indeed, a comparison of the default intensity estimates obtained herein with those obtained using either historical data or implicitly from debt prices indicates that the equity-based default intensities are signiﬁcantly larger. By extrapolation,

Estimating Default Probabilities Implicit in Equity Prices

3

this possible model misspeciﬁcation also casts doubt on the reliability of the equity-based default probability estimates obtained using structural models as in Delianedis and Geske (1998), conﬁrming the previous conclusions of Jarrow and van Deventer (1998, 1999) and Jarrow et al. (2002) in this regard. This is also consistent with the inability of structural models, using equity price information, to explain credit spreads in corporate debt; see Collin-Dufresne et al. (2001), Eom et al. (2002), and Huang and Huang (2002). The previous literature estimating reduced-form credit risk models using debt prices includes Dufﬁe (1999), Dufﬁe and Singleton (1997), Dufﬁe et al. (2000), Janosi et al. (2002), and Madan and Unal (1998). Dufﬁe and Singleton (1997) estimate swap spreads, Madan and Unal (1998) estimate yields on thrift institution certiﬁcates of deposit, and Dufﬁe et al. (2000) estimate credit and liquidity spreads for Russian debt. Both Dufﬁe (1999) and Janosi et al. (2002) estimate default intensities using US corporate debt. As mentioned earlier, we will compare our estimated default intensities with those from Janosi et al. (2002). Bankruptcy prediction models using historical bankruptcy data include Altman (1968), Chava and Jarrow (2002), Shumway (2001), and Zmijewski (1984), among others. A structural model for estimating default intensities is Delianedis and Geske (1998). An outline of this chapter is as follows. Section 2 presents the model structure. Section 3 provides a description of the data, and Section 4 estimates the state variable process parameters. The equity model parameter estimation is discussed in Sections 5–10. Section 11 compares the default parameter intensities estimated using the equity model to those obtained using debt prices by Janosi et al. (2002). A conclusion is provided in Section 12.

2. THE MODEL STRUCTURE This section introduces the notation and provides a generalization of the reduced-form credit risk model for equity returns contained in Jarrow (2001). Default-free zero-coupon bonds of all maturities and a ﬁrm’s common stock are traded. Markets are assumed to be frictionless with no arbitrage opportunities, but equity prices may contain bubbles. Let p(t, T) represent the time t price of a default-free dollar paid at time T where 0 ≤ t ≤ T. The instantaneous forward rate at time t for date T is deﬁned f (t, T) = −∂ log p(t, T)/∂T, with the spot rate of interest r(t) = f (t, t).

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THE CREDIT MARKET HANDBOOK

Consider a ﬁrm issuing equity. This ﬁrm may default. Let τ be the random variable representing the ﬁrst time of default and let N(t) =

1 if τ ≤ t 0 otherwise

(1.1)

denote its point process. We assume that this point process has an intensity, denoted by λ(t). λ(t) gives the approximate probability of the ﬁrm’s default over the time interval [t, t + ].1 Equity pays dividends and has a liquidating payoff at time TL . The time t value of all of these payments equals the value of the equity (per share), denoted by ξ(t). These promised dividends and liquidating payoff are made, unless the ﬁrm defaults. If default occurs, the equity holders lose everything.2 We need to develop some notation for these promised payments to equity. The regular dividends Dt are paid at times t = 1, 2, . . . , TD . We assume that these dividends are deterministic quantities, placed in an escrow account and paid for sure.3 The requirement that these dividends are deterministic implicitly determines the date TD . For many equities, TD will be a month or less. The liquidating dividend L(TL ) is paid at time TL unless default occurs prior to this date. The liquidating dividend consists of the time TL (future) value of all unannounced and random dividends paid over (TD , TL ), plus the remaining resale value of the ﬁrm at time TL . Let S(t) represent the time t present value of the liquidating dividend, conditional upon no default prior to time t. There is some evidence for example, the recent price growth of Internet stocks4 , that stock prices contain a “bubble” or “monetary value” component; see Jarrow and Madan (2000). For simplicity, we model the bubble component as a random process that is proportional to the present value of the liquidating dividend: t S(t) e 0 µθ (u) du − 1

(1.2)

where µθ (u) ≥ 0 is the continuous return in the stock price due to the bubble component. Given this set-up, it is easy to see5 that the per share equity value at time t is given by t  µθ (u) du + TD D p(t, j)  S(t)e 0 j≥t j ξ(t) = if t < τ   0 if t = τ

(1.3)

5

Estimating Default Probabilities Implicit in Equity Prices

The share price consists of the present value of the liquidating dividend (S(t)) compounded by the bubble (µθ (t)), and the (announced) deterministic dividends (Dj for j = t, . . . , TD ). To obtain an empirical formulation of the above model, more structure needs to be imposed on the stochastic nature of the economy. Exactly following Jarrow (2001),6 we consider an economy that is Markov in three state variables: the spot rate of interest, the cumulative excess return on an equity market index, and the liquidating dividend process. For the spot rate of interest, we use a single-factor model with deterministic volatilities, sometimes called the extended Vasicek model. This model has two parameters: a mean reversion parameter (a) and the spot rate’s volatility (σr ). The second state variable Z(t) is the cumulative excess return on an equity market index. The equity market index follows a geometric Brownian motion with volatility (σm ). The correlation coefﬁcient between the return on the market index and changes in the spot rate of interest is (ϕrm ). The third state variable is the liquidation value of the ﬁrm’s equity, denoted by L(t). This liquidation value is assumed to follow a geometric Brownian motion with volatility (σL ) and with ϕmL and ϕrL representing the correlation of the liquidation value with the market index and changes in the spot rate of interest, respectively. For analytic tractability, the default intensity process is assumed to be linear in the spot rate of interest and the cumulative excess return on the equity market index; that is, λ(t) = λ0 + λ1 r(t) + λ2 Z(t)

(1.4)

where λ0 , λ1 , λ2 are constants. Under this structure, it is shown in Jarrow (2001) that the present value of the liquidating dividend can be rewritten as S(t) =

L(t) −λ1 σ 2 (t,TL )−λ1 σL ϕr L tTL b(u,T ∗ ) du 1 L e p(t, TL ) 2 /2

× e−λ2 ϕrm η(t,TL )−λ2 σL ϕmL (TL −t)

ν(t, TL )

where ν(t, T) = p(t, T)e−λ0 (T−t)−λ1 µ1 (t,T) 2

2

× e+(2λ1 +λ1 )σ1 (t,T)/2−λ2 Z(t)(T−t) 3 λ2 /6 2

× e+(1+λ1 )λ2 ϕrm η(t,T)+[T−t]

(1.5)

6

THE CREDIT MARKET HANDBOOK

µ1 (t, T) =

σ12 (t, T) =

T t T

f (t, u) du +

T

b(u, T)2 du/2

t

b(u, T)2 du

t

b(u, t) = σr (1 − e−a(t−u) )/a and η(t, T) = −(σr /a3 )[1 − e−a(T−t) ] + (σr /a2 )e−a(T−t) (T − t) + (σr /2a)[T − t]2 Substitution of equation 1.5 into the stock price equation 1.3 yields the ﬁnal valuation formula. Unfortunately, observing only a single value for the stock price at each date leaves this system underdetermined as there are more unknowns (L(t), λ0 , λ1 , λ2 ) than there are observables (ξ(t)).7 To overcome this situation, the stochastic process for L(t) is used to transform equation 1.3 into a time-series model for the ﬁrm’s equity returns. Unfortunately, this transformation introduces the equity price’s risk premium into the estimation procedure. In this regard, it is shown in the appendix that8

 T ξ(t) − j≥tD Dj p(t, j)  − r(t − ) ≈ log  T D ξ(t − ) − j≥t− Dj p(t − , j) p(t, TL ) b(t − , TL )2 + λ0 + λ 1 + log 2 p(t − , TL 

− λ2 [Z(t)(TL − t) − Z(t − )(TL − t + )] + λ1 λ2 ϕrm b(t, TL )(TL − t) + [σL L (t − ) + µθ (t − ) − (1/2)σξ2 ] + ε(t − )

(1.6)

where ε(t − ) ≡ σL (wL (t) − wL (t − )) and L (t) is the liquidation value’s risk premium. Equation 1.6 gives a time-series expression for the stock’s return over the time period [t − , t]. This is a generalization of the typical asset-pricing

Estimating Default Probabilities Implicit in Equity Prices

7

model to include a ﬁrm’s default parameters. This expression forms the basis for our empirical estimation in the subsequent sections. One can think of this model for equity returns in three different but related ways. The ﬁrst interpretation of equation 1.6 is that it is equivalent to a reduced-form credit risk model for the ﬁrm’s equity. This interpretation follows the method of derivation. The second interpretation of equation 1.6 is that it is a type of structural model for the ﬁrm’s equity where the ﬁrm’s liquidation value (assets less liabilities) is exogenously given and default occurs according to a default intensity process correlated with the randomness inherent in the ﬁrm’s liquidation value. Finally, the third interpretation of equation 1.6 is that it is a generalized asset-pricing model with bankruptcy explicitly parameterized within the equity’s return process. Given this perspective, equation 1.6 makes explicit the bond market factor discussed in Fama and French (1993).

3. DESCRIPTION OF THE DATA The time period covered in this study is May 1991–March 1997. The interval for computing equity returns is 1 month. For each ﬁrm, and for each month in the observation period, we will be ﬁtting a time-series regression of equity returns going back in time 4 years (48 months). Thus, we lose the ﬁrst 4 years of our observation interval, giving time-series regressions for each ﬁrm and for each month from May 1995–March 1997. All individual ﬁrm equity data (including earnings, dividends, etc.) are obtained from CRSP. For the equity market index, we used the S&P 500 index. For estimating an equity risk premium, we will employ the Fama– French benchmark portfolios (book-to-market factor (HML), small ﬁrm factor (SMB), and a momentum factor (UMD). These monthly portfolio returns were obtained from Ken French’s webpage.9 The US Treasury prices used for this investigation were obtained from the University of Houston’s Fixed Income Database. The data consist of monthly bid prices of all outstanding bills, notes, and bonds taken from Lehman Brothers’ trading sheets on the last calendar day in each month; see Warga (1999) for additional details. Being such a large database (containing over two million entries), the potential for data errors is quite large. Indeed, a careful examination of the data conﬁrmed this suspicion. Hence, we ﬁltered the data to remove obvious data errors. We excluded Treasury bonds with inconsistent or suspicious issue/dated/maturity dates and matrix prices. Lastly, using a median yield ﬁlter of 2.5%, we also removed US Treasury debt listings whose yields exceeded the median yield by this percent. After ﬁltering, there are approximately 29,100 US Treasury prices left in the sample set.

8

THE CREDIT MARKET HANDBOOK

The same 20 ﬁrms as in Janosi et al. (2002) were initially selected for analysis. These ﬁrms were selected to stratify various industry groupings: ﬁnancial, food and beverages, petroleum, airlines, utilities, department stores, and technology. Due to unavailability of balance sheet data or stock prices, ﬁve of these companies were eliminated. The remaining 15 ﬁrms included in this study and the industry represented by each ﬁrm are provided in Table 1.1. The Moody’s and S&P ratings for each company’s debt issues at the start of our sample period (May 24, 1991) are also included. As mentioned previously, the interval for equity returns is 1 month. The monthly return interval was chosen for two reasons. First, the default parameter estimation using debt prices in Janosi et al. (2002) was based on monthly data, so monthly equity returns will provide an equivalent comparison. Second, and more importantly, it is believed that the use of monthly data for equities eliminates market-microstructure noise more prevalent in smaller return intervals (daily or weekly) (see Dimson, 1979; Schwartz and Whitcomb, 1977a,b; Smith, 1978).

Table 1.1

Details of the Firms Included in the Empirical Investigation

Financials Bankers Trust NY Merrill Lynch & Co Food & Beverages Pepsico Inc Coca-Cola Ent. Inc Airlines AMR Corporation Southwest Airlines Utilities Carolina Power Light Texas Utilities Ele Co Petroleum Mobil Corp Department Stores Sears Roebuck + Co Wal-Mart Stores, Inc Technology Eastman Kodak Company Xerox Corp Texas Instruments Intl Bus Machines

Ticker Symbol

SIC Code

First Date Used in the Estimation

Last Date Used in the Estimation

Number of Bonds

Moody’s

S&P

bt mer

6022 6211

01/31/1994 12/31/1991

04/30/1994 03/31/1997

3 14

Al A2

AA A

pep cce

2086 2086

12/31/1991 12/31/1991

03/31/1997 06/30/1994

8 3

Al A2

A AA−

amr luv

4512 4512

02/29/1992 05/31/1992

08/31/1994 03/31/1997

2 3

Baal Baal

BBB+ A−

cpl txu

4911 4911

08/31/1992 04/30/1994

01/31/1993 03/31/1997

3 4

A2 Baa2

A BBB

mob

2911

12/31/1991

02/29/1996

3

Aa2

AA

s wmt

5311 5331

12/31/1991 12/31/1991

08/31/1996 03/31/1997

7 3

A2 Aa3

A AA

ek

3861

01/31/1992

09/30/1994

3

A2

A−

xrx txn ibm

3861 3674 3570

12/31/1991 10/31/1992 01/31/1994

03/31/1997 03/31/1997 03/31/1997

4 3 3

A2 A3 Al

A A AA−

Ticker Symbol is the ﬁrm’s ticker symbol. SIC is the standard industry code. Number of Bonds corresponds to the number of distinct bond issues used in the estimation. Moody’s refers to Moody’s debt rating for the company’s senior debt on the ﬁrst date used in the estimation. S&P refers to S&P’s debt rating for the company’s debt on the ﬁrst date used in the estimation.

Estimating Default Probabilities Implicit in Equity Prices

9

4. ESTIMATION OF THE STATE VARIABLE PROCESS PARAMETERS To implement the estimation of the equity return process, we use a two-step procedure. In step one, we ﬁrst estimate the parameters for the state variable processes. Step two uses these estimates as constants in the equity return estimation. Step two is discussed in Section 5.

4.1 Spot Rate Process Parameter Estimation The inputs to the spot rate process evolution are the forward rate curves over an extended observation period ( f (t, T) for all months t ∈ January 1975–March 1997) and the spot rate parameters (a, σr ). For the estimation of the forward rate curves, a two-step procedure is also utilized. First, for a given time t, the discount bond prices (p(t, T) for various T) are estimated by solving the following minimization problem:   choose

(p(t, T) for all relevant T ≤ max{Ti : i ∈ It })   bid 2 to minimize i∈It [Bi (t, Ti ) − Bi (t, Ti ) ]

(1.7)

where

Bi (t, Ti ) =

ni

Ct j p(t, tj )

j=I

is a US Treasury security with coupons of Cj dollars at times tj for j = 1, . . . , ni , where tni = Ti is the maturity date; It is an index set containing the various US Treasury bonds, notes, and bills available at time t; and Bi (t, Ti )bid is the market bid price for the ith bond with maturity Ti . The discount bond prices’ maturity dates T coincide with the maturities of the Treasury bills, and the coupon payment and principal repayment dates for the Treasury notes and bonds. Step 2 is to ﬁt a continuous forward rate curve to the estimated zerocoupon bond prices (p(t, T) for all T ≤ max{Ti : i ∈ I}). We use the maximum smoothness forward rate curve as developed by Adams and van Deventer (1994) and reﬁned by Janosi and Jarrow (2002). Brieﬂy, we choose the unique piecewise, fourth-degree polynomial with the left and right

10

THE CREDIT MARKET HANDBOOK

end points left “dangling” that minimizes

max{Ti : i∈It } t

|∂ 2 f (t, s)/∂s2 |ds

For the estimation of the spot rate parameters (a, σr ), the procedure follows that used in Janosi et al. (2002). A rolling estimation of the parameters using only information available at the time of the estimation is performed, making the parameter estimates (at , rt ) dependent on time t as well. The procedure is based on an explicit formula for the variance of the default-free zero-coupon bond prices under the extended Vasicek model (see Heath et al., 1992). For = 1/12 (a month), the expression is vart [log(P(t + , T)/P(t, T)) − r(t)] = σrt2 (e−at (T−t) − 1)2 /a2t

(1.8)

First, we ﬁx a time to maturity T − t ∈ {3 months, 6 months, 1 year, 5 years, 10 years, the longest time to maturity of an outstanding Treasury bond closest to 30 years}. Then, we ﬁx a current month t ∈ {May 1991–March 1997}. Going backward in time 60 months (5 years), we compute the sample variance, denoted νt T, using the smoothed forward rate curves previously generated. Note that the sample variance depends on both the date of estimation and the bond’s maturity. Then, for each month t ∈ {May 1991–March 1997}, to estimate the parameters (σrt , at ) we run a nonlinear regression νtT = σrt2 (e−at (T−t) − 1)2 /a2t + etT

(1.9)

across the bond time to maturities T − t ∈ {1/4, 1/2, 1, 5, 10, longest time to maturity closest to 30} where etT is the error term. The parameter estimates are

art σrt

Min

Mean

Max

Std. dev.

0.0109 0.0100

0.0282 0.0109

0.0428 0.0117

0.0101 0.0004

11

Estimating Default Probabilities Implicit in Equity Prices

The R2 for each of these monthly nonlinear regressions (not reported) exceeded 0.99 in all cases. The spot rate volatility (σrt ) is nearly constant over this period. In contrast, the mean reverting parameter (art ) appears to be more volatile. To test for the time-series stability of these parameter estimates, a unit root test was performed.10 For the volatility σrt , the test rejects a unit root, implying the time series is stationary. In contrast, one cannot reject a unit root for the mean reverting parameter at .

4.2 Market Index Parameter Estimation Although the equity returns are monthly, for estimating the parameters of the market index we use daily data. This is done because daily data are available for the market index, and the higher frequency data will provide less noisy estimates because market microstructure considerations are less important for an index (than they are for individual ﬁrms). Daily observations of the market return and the 3-month T-bill yield are available from CRSP. Using the daily S&P 500 index price data and the daily 3-month T-bill spot rate data, we estimate the parameters of the market index process (σm , ϕrm ). As previously mentioned, this estimation is based on daily data ( = 1/365). As before, the procedure involves a rolling estimation of the parameters using only information available at the time of the estimation. For a given day t ∈ {May 24, 1990–March 31, 1997}, we go back in time 365 business days and estimate the time-dependent sample variance and correlation coefﬁcients (σmt , ϕrmt ) using the sample moments, that is, 2 σmt

= vart

M(t) − M(t − ) M(t − )

1

and ϕnnt = corrt

M(t) − M(t − ) , r(t) − r(t − ) M(t − )

The parameter estimates are

σmt ϕt

Min

Mean

0.0982 −0.2706

0.1261 −0.0990

Max

Std. dev.

0.1897 0.1262

0.0270 0.1142

(1.10)

12

THE CREDIT MARKET HANDBOOK

The market volatility is relatively constant between 0.1 and 0.2 over this observation period. The correlation coefﬁcient appears to be more variable. As before, to test for the stability of the parameters a unit root test was performed. The results show that a unit root can be rejected at the 90% conﬁdence level for the market volatility but not for the correlation coefﬁcient.11 With the parameter estimates for the market volatility (σmt ) and the daily 3-month Treasury bill yields, the excess cumulative return on the market process Z(t) is computed,12 starting the time series on May 24, 1991.

5. EQUITY RETURN ESTIMATION Given the state variable parameters as estimated in the previous sections, this section presents the estimated equity return model. The basis for this estimation is equation 1.6. To empirically implement equation 1.6, we need to specify models for both the risk premium and the equity price bubble. Following Fama and French (1993, 1996), we use a four-factor assetpricing model with the factors being the excess return on a market portfolio, SMB(t), HML(t), and UMD(t), that is, M(t) − M(t − ) σL L (t − , X(t − )) = β0 − r(t − ) M(t − ) + β1 [SMB(t)] + β2 [HML(t)] + β3 [UMD(t)] (1.11) where SMB(t) is the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks at time t, HML(t) is the difference between the return on a portfolio of high-book-to-market stocks and the return on a portfolio of low-book-to-market stocks at time t, and UMD(t) is a momentum factor. The equity price bubble is proxied by the price-earnings ratio and possibly the stock’s own variance, that is, Price − (1/2)σξ2 (t) + µθ (t − , X(t − )) = β4 [σξ2 (t)] + β5 (t) Earnings (1.12) where σξ2 (t) ≡ var

ξ(t) − ξ(t − ) ξ(t − )

1

13

Estimating Default Probabilities Implicit in Equity Prices

The stock’s own variance is included with an arbitrary coefﬁcient to see if it differs from its theoretical value of −(1/2) in equation 1.6. There is also a concern that the SMB(t), HML(t), and UMD(t) factors may already include an adjustment for bubbles. For this reason, the subsequent regressions are run both with and without the P/E ratio included. For the estimation, we ﬁx (TL − t) = 20 years and we set TD = t. The ﬁrst restriction makes the ﬁrm’s valuation horizon 20 years, making equity comparable with long-term debt. The second restriction implies that all future dividends are viewed as random. Consequently, we only need to make an adjustment for dividends in the payout month. Substitution of the above into equation 1.6 yields:  ξ(t)   − r(t − )    ξ(t − )    if no dividend over [t − , t]  ξ(t)  log − r(t − )    ξ(t − ) − Dx p(t − , x)   if dividend at x ∈ [t − , t]

log

≈ 0 + 1 log +

p(t, TL ) p(t − , TL )

b(t − , TL )2 2

+ 2 [Z(t)(TL − t)

− Z(t − )(TL − (t − ))] M(t) − M(t − ) − r(t − ) + β0 M(t − ) + β1 [SMB(t)] + β2 [HML(t)] + β3 [UMD(t)] Price + β4 [σξ2 (t)] + β5 (t) Earnings where 0 = +λ0 + λ1 λ2 ϕrm b(t, TL )(TL − t) 1 = λ1 2 = −λ2

(1.13)

14

THE CREDIT MARKET HANDBOOK

To ensure that the intensity process is non-negative when both λ1 and λ2 are zeros, we impose the constraint that 0 ≥ 0 in the estimation. The ﬁnal computations for the default parameters are λ0 = (0 /) − 1 2 ϕrm b(t, TL )(TL − t) λ1 = 1 λ2 = −2 The time period covered is May 1991–March 1997. For each ﬁrm, a timeseries regression is run using 48 months of historical data. Thus, the ﬁrst regression estimation occurs 4 years into the data set on May 31, 1995. For each subsequent month, until March 1997, the regression is re-estimated and parameter estimates obtained. This generates 23 regressions for each ﬁrm’s returns. As before, only information available to the market at the time of the estimation is utilized. This rolling estimation procedure gives a monthly time series of parameter estimates (λ0t , λ1t , λ2t , β0t , β1t , β2t , β3t , β4t , β5t ) based on 46 (48 – 2) months of overlapping data. The choice of a 4-year estimation period was based on trading off the stability of the estimates versus larger standard errors. Although longer estimation periods are likely to make the standard errors smaller, they also imply that structural shifts are more likely to occur, making the estimated parameters less stable. Eight different models for equity returns are estimated. The models differ with respect to the number of independent variables included in the regression. Models 1 and 2 have no default (λ0 ≡ λ1 ≡ λ2 ≡ 0). They differ only in the inclusion of a P/E ratio (β5 ≡ 0 or not). Models 3–8 include default, and they differ with respect to the default intensity investigated and the inclusion of the P/E ratio or not. In particular, models 3 and 4 have only λ0 nonzero. Models 5 and 6 have both λ0 and λ1 nonzero. Models 7 and 8 have all default parameters nonzero. These eight models are nested and a relative comparison of model performance is subsequently provided. To summarize the monthly time series estimates across all models and across all times, Table 1.2 provides the average values for the point estimates of the parameters and their t-statistics.13 The average adjusted R2 is also included. The values in Table 1.2 are averages over the number of months in the observation period (May 1995–March 1997) for which the linear regression estimates of the parameters are computed.14 Summary statistics for various F-tests are also provided. The ﬁrst F-test is for the null hypothesis (β0t = β1t = β2t = β3t = β4t = 0). Given are the average P-scores of the F-tests (across the number of regressions). The F-tests for models 3, 5, and 7 test for the joint hypothesis that all default parameters are zero, that is, λ0t = 0, λ0t = λ1t = 0, and λ0t = λ1t = λ2t = 0, respectively. The F-tests

λ0

0.1851 1.3881 0.3244 1.2759 0.1859 1.3817 0.3446 1.3526 0.1976 1.4670 0.3526 1.4129

0.2643** 2.0564 0.3161 1.0685 0.2584** 1.9930 0.2970 0.9655 0.2468** 1.8878 0.2649 0.8587

Model 2

Food & Beverages 3—Pepsico lnc (pep) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

2—Merrill Lynch & Co (mer) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

0.0420 0.1695 0.0317 0.1165 0.0163 0.0662 0.0109 0.0387

0.1406 0.6617 0.1710 0.7739 0.1214 0.5707 0.1513 0.6847

λ1

−0.0098 0.8235 −0.0100 0.8120

−0.0168 1.1576 −0.0167 1.1432

λ2

1.1323∗∗ 3.7001 1.1177∗∗ 3.5745

2.0601∗∗ 5.6185 1.9509∗∗ 5.3521 1.8714∗∗ 5.0857 1.8729∗∗ 4.9728 1.8468∗∗ 4.2686 1.8555∗∗ 4.2352 0.1971 −0.0362 0.1827 −0.0310

1.7055∗∗ 5.5380 1.7051∗∗ 5.5066 1.6377∗∗ 5.2676 1.6041∗∗ 5.0361 1.5122∗∗ 4.1266 1.4390∗∗ 3.7553 −1.4618 −0.3581 −1.5191 −0.3763

β0

−0.0058∗ −1.4687 −0.0057∗ −1.4432

0.0046 1.0160 0.0033 0.7323 0.0033 0.7421 0.0033 0.7342 0.0035 0.7653 0.0034 0.7466 0.0031 0.6715 0.0030 0.6487

0.0071∗∗ 1.8227 0.0068∗∗ 1.7420 0.0061∗ 1.5567 0.0057∗ 1.4421 0.0065∗ 1.6377 0.0062∗ 1.5408 0.0060∗ 1.4694 0.0057 1.3782

β1

−0.0046 −1.2377 −0.0046 −1.2189

0.0055 1.2179 0.0035 0.7375 0.0034 0.7097 0.0035 0.7279 0.0035 0.6451 0.0036 0.6597 0.0032 0.5695 0.0033 0.5670

0.0102∗∗ 2.7373 0.0097∗∗ 2.5609 0.0088∗∗ 2.2794 0.0084∗∗ 2.1534 0.0079∗∗ 1.8964 0.0073∗∗ 1.6815 0.0072∗∗ 1.7106 0.0066∗ 1.5082

β2

Averages of the Parameter Estimates and t -scores from the Equity Model Regression

Financials 1—Bankers Trust NY (bt) Model 1

Table 1.2

0.0053∗ 1.5369 0.0049 1.3207

0.0063∗ 1.5668 0.0041 0.9959 0.0026 0.6240 0.0024 0.5489 0.0023 0.5291 0.0022 0.4903 0.0030 0.6549 0.0028 0.6191

−0.0018 −0.4994 −0.0024 −0.6517 −0.0029 −0.8168 −0.0028 −0.7874 −0.0036 −0.9790 −0.0036 −0.9708 −0.0032 −0.8617 −0.0032 −0.8541

β3

−0.8301∗∗ −4.3056 −0.9515∗∗ −4.3600

−1.7785∗∗ −8.1829 −1.9064∗∗ −8.2971 −1.9523∗∗ −8.4074 −1.9552∗∗ −8.3476 −1.9474∗∗ −8.2949 −1.9505∗∗ −8.2079 −1.9320∗∗ −8.1654 −1.9348∗. −8.0735

−2.3971∗∗ −2.3400 −2.8753∗∗ −2.3452 −3.4759∗∗ −2.6669 −3.5133∗∗ −2.6798 −3.4779∗∗ −2.6482 −3.5364∗∗ −2.6819 −3.5981∗∗ −2.7542 −3.6556∗∗ −2.7857

β4

0.0053 0.2765

−0.0008 −0.0851

−0.0017 −0.1569

−0.0023 −0.2172

0.0089∗∗ 1.7342

−0.0060 −0.7129

−0.0061 −0.7244

−0.0054 −0.6020

0.0034 0.7640

β5

0.5889

0.5877

0.8486

0.8477

0.8441

0.8432

0.8426

0.8412

0.8379

0.8292

0.5284

0.5208

0.5105

0.5025

0.5028

0.4967

0.4901

0.4873

R2

0.7647

0.0000

0.7547

0.6244

0.7553

0.5592

0.7275

0.3857

0.1008

0.0000

0.4726

0.4325

0.4674

0.4394

0.5216

0.3083

0.4676

0.0000

F-Test

0.0841 0.6271 0.3186 0.9245 0.0845 0.6180 0.3328 0.9341 0.0872 0.6355 0.3723 1.0168

λ0

(Continued)

0.2255 0.8963 0.2226 0.8748 0.2467 0.9980 0.2431 0.9711 0.2506 1.0267 0.2503 1.0149

Model 4

Model 3

Model 2

0.0000 0.0000 0.0000 0.0000

Airlines 5—AMR Corporation (amr) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

4—Coca-Cola (cce) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Table 1.2

−0.4447 −1.4881 −0.4390 −1.4567 −0.4291∗ −1.4390 −0.4226 −1.4060

0.0094 0.0484 0.0421 0.1931 −0.0024 −0.0015 0.0316 0.1483

λ1

0.0249 −1.1105 0.0255 −1.1225

−0.0035 0.3777 −0.0061 0.5364

λ2

1.7603∗∗ 4.3989 1.9060∗∗ 4.8285 1.7603∗∗ 4.2846 1.9060∗∗ 4.7145

0.9856∗∗ 2.2356 0.9706∗∗ 2.1831 0.9211∗∗ 2.0779 0.9103∗∗ 2.0344 1.3239∗∗ 2.5788 1.3083∗∗ 2.5229 5.8008∗ 1.6003 5.8969∗ 1.6021

1.0959∗∗ 3.4971 1.0721∗∗ 3.3861 1.0859∗∗ 2.9236 1.0313∗∗ 2.7222 0.5229 0.1955 −0.0044 −0.0011

β0

0.0065 1.3297 0.0060 1.2410 0.0065 1.2610 0.0060 1.1764

−0.0028 −0.5606 −0.0025 −0.4840 −0.0041 −0.7770 −0.0038 −0.6907 −0.0057 −1.0530 −0.0054 −0.9616 −0.0054 −0.9962 −0.0050 −0.8961

−0.0059∗ −1.4903 −0.0065∗ −1.6000 −0.0059∗ −1.4430 −0.0064∗ −1.5436 −0.0063∗ −1.5102 −0.0070∗∗ −1.6455

β1

0.0008 0.1303 0.0012 0.2359 0.0008 0.1285 0.0012 0.2328

0.0001 −0.0089 −0.0002 −0.0607 −0.0003 −0.0693 −0.0005 −0.1140 0.0027 0.4505 0.0024 0.3948 0.0035 0.5922 0.0032 0.5275

−0.0049 −1.2944 −0.0058∗ −1.4473 −0.0050 −1.2179 −0.0061 −1.3965 −0.0052 −1.2625 −0.0066∗ −1.4841

β2

−0.0028 −0.5812 −0.0027 −0.5693 −0.0028 −0.5741 −0.0027 −0.5622

0.0009 0.1999 0.0007 0.1582 0.0003 0.0822 0.0002 0.0561 0.0027 0.5240 0.0025 0.4874 0.0024 0.4650 0.0023 0.4317

0.0046 1.2792 0.0053 1.4093 0.0046 1.1967 0.0051 1.3045 0.0049 1.2656 0.0056 1.4027

β3

−0.2679 0.2237 −0.8753 −0.3361 −0.2679 0.1048 −0.8753 −0.1961

0.5367 0.9575 0.5427 0.9561 −0.8996 −0.2253 −0.8706 −0.2095 −1.1775 −0.3747 −1.1430 −0.3542 −1.2668 −0.4320 −1.2459 −0.4194

−1.3522∗. −4.6109 −1.2471∗∗ −4.4420 −1.3592∗∗ −4.5569 −1.2688∗∗ −4.3840 −1.4123∗∗ −4.5377 −1.3114∗∗ −4.3540

β4

−0.0100∗∗ −1.9470

−0.0100∗∗ −1.9794

0.0221 0.5758

0.0205 0.5049

0.0215 0.5302

0.0243 0.5613

−0.0523 −0.8496

−0.0457 −0.7608

−0.0436 −0.7454

β5

0.4100

0.3543

0.4100

0.3543

0.2455

0.2368

0.2115

0.2043

0.1667

0.1581

0.1764

0.1675

0.6023

0.5956

0.5944

0.5894

0.5936

0.5887

R2 F-Test

0.0685

0.0756

0.0724

0.0000

0.5735

0.3130

0.6180

0.2999

0.5924

0.4780

0.5772

0.0000

0.4356

0.9442

0.4781

0.8507

0.4869

0.6246

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.2486 1.1639 0.4029 0.9320 0.2167 1.0517 0.2009 0.5453 0.2153 1.0131 0.1939 0.5259

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

0.0015 0.0195 0.0204 0.1347 0.0015 0.0202 0.0247 0.1632 0.0014 0.0186 0.0243 0.1628

Utilities 7—Carolina Power Light (cpl) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

6—Southwest Airlines Co (luv) Model 1

Model 8

Model 7

Model 6

Model 5

0.2135 1.2838 0.2162 1.2784 0.1985 ", 1.1775 0.2028 1.1816

−0.7461∗∗ −1.7237 −0.7405∗ −1.6337 −0.7722∗∗ −1.7537 −0.7694∗∗ −1.6703

−0.2262 −0.8031 −0.3201 −1.1553 −0.2615 −0.9433 −0.3658 −1.3552

−0.0099 0.8539 −0.0096 0.8276

−0.0073 0.3367 −0.0077 0.3580

−0.0272 1.3987 −0.0295∗ 1.6313

0.8464∗∗ 3.4567 0.8692∗∗ 3.4627 0.8452∗∗ 3.3190 0.8641∗∗ 3.3663 0.6614∗∗ 2.2047 0.6790∗∗ 2.2413 −1.0943 −0.4656 −1.0226 −0.4373

2.0166∗∗ 3.1244 1.9524∗∗ 2.9775 1.9001∗∗ 2.9004 1.8841∗∗ 2.8407 2.5738∗∗ 3.4433 2.5646∗∗ 3.3338 1.3422 0.2600 1.2768 0.2153

1.9712∗∗ 4.0794 2.2250M 4.6410 −2.8612 −0.6147 −2.9848 −0.7042

−0.0031 −1.0833 −0.0029 −0.9636 −0.0031 −1.0666 −0.0028 −0.9208 −0.0025 −0.8351 −0.0020 −0.6682 −0.0030 −0.9768 −0.0025 −0.8090

0.0034 0.4052 0.0032 0.3842 0.0028 0.3169 0.0026 0.2807 0.0005 0.0318 0.0003 0.0041 −0.0002 −0.0490 −0.0004 −0.0826

0.0058 1.0795 0.0049 0.9422 0.0048 0.8927 0.0038 0.7245

0.0081∗∗ 2.7034 0.0084∗∗ 2.7031 0.0081∗∗ 2.6154 0.0085∗∗ 2.6785 0.0069∗∗ 2.1013 0.0074∗∗ 2.2060 0.0064∗∗ 1.9236 0.0068∗∗ 2.0200

0.0113∗ 1.5801 0.0109∗ 1.5162 0.0100 1.3567 0.0098 1.2747 0.0149∗∗ 1.8983 0.0148∗∗ 1.8072 0.0145∗∗ 1.8144 0.0144∗∗ 1.7187

0.0024 0.4104 0.0036 0.6597 0.0012 0.2143 0.0024 0.4523

0.0060∗∗ 2.2368 0.0064∗∗ 2.2443 0.0060∗∗ 2.0881 0.0065∗∗ 2.2081 0.0048∗∗ 1.6044 0.0053∗∗ 1.7529 0.0051∗∗ 1.6896 0.0056∗∗ 1.8106

0.0014 0.2680 0.0001 0.0848 −0.0007 −0.0306 −0.0007 −0.0320 0.0030 0.4334 0.0028 0.4096 0.0033 0.4647 0.0032 0.4464

−0.0015 −0.3012 −0.0009 −0.1805 −0.0010 −0.2007 −0.0002 −0.0397

−1.8684∗∗ −10.7690 −1.6545∗. −10.4473 −1.8686∗∗ −10.5615 −1.6537∗. −10.3224 −1.8507∗∗ −10.6269 −1.6434∗∗ −10.3838 −1.8352∗∗ −10.4632 −1.6729∗∗ −10.2392

−1.6570∗∗ −4.0185 −1.7540∗∗ −4.0080 −1.8198∗∗ −4.1869 −1.8199∗∗ −4.1351 −1.7444∗∗ −4.0043 −1.7507∗∗ −3.9784 −1.7367∗∗ −3.7165 −1.7418∗∗ −3.6858

−0.3713 0.0564 −1.0727 −0.2787 −0.3585 0.0650 −1.0852 −0.2785

−0.0061 −0.4042

−0.0068 −0.4392

−0.0058 −0.3607

−0.0039 −0.4180

0.0059 −0.0446

0.0047 −0.0436

−0.0424 −0.4210

0.0410 0.7684

−0.0116∗∗ −2.2907

−0.0111∗∗ −2.1406

0.8416

0.8406

0.8367

0.8348

0.8301

0.8276

0.8294

0.8275

0.5370

0.5338

0.5329

0.5301

0.4947

0.4909

0.4796

0.4663

0.4797

0.4082

0.4331

0.3669

0.5987

0.5845

0.5864

0.5397

0.6597

0.6866

0.6732

0.0000

0.6808

0.2532

0.7065

0.1702

0.6616

0.4773

0.4857

0.0000

0.0294

0.5419

0.0416

0.3243

λ0

(Continued)

0.0208 0.1722 0.1189 0.6085 0.0148 0.1187 0.0990 0.4662 0.0100 0.0758 0.0903 0.4118

0.3137** 2.0807 0.3038** 2.0012 0.2923** 2.0143 0.2853** 1.9478 0.3179** 2.1802 0.3108** 2.1040

Model 2

Department Stores 10—Sears Roebuck + Co (s) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

Petroleum 9—Mobil Corp (mob) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

8—Texas Utilities Ele Co (txu) Model 1

Table 1.2

0.1288 0.9309 0.1062 0.7204 0.1215 0.8823 0.1013 0.6927

0.3495∗∗ 2.2115 0.3400∗∗ 2.1258 0.3524∗∗ 2.2318 0.3449∗∗ 2.1517

λ1

−0.0093 0.9046 −0.0095 0.8999

0.0031 −0.4410 0.0031 −0.4214

λ2

1.7989∗∗ 4.9673 1.8000∗∗ 4.9109

0.6879∗∗ 3.5411 0.6894∗∗ 3.4993 0.6915∗∗ 3.4903 0.6977∗∗ 3.5211 0.5683∗∗ 2.4188 0.5939∗∗ 2.4838 −1.0949 −0.4177 −1.1141 −0.3860

0.2394 0.9406 0.2289 0.8906 0.1219 0.4848 0.1159 0.4577 −0.1838 −0.6802 −0.1828 −0.6678 0.3425 0.2814 0.3446 0.2679

β0

0.0076∗∗ 1.7257 0.0077∗∗ 1.7029

−0.0040∗∗ −1.7330 −0.0040∗∗ −1.7207 −0.0040∗∗ −1.7137 −0.0039∗∗ −1.6896 −0.0036∗ −1.5405 −0.0037∗ −1.5411 −0.0040∗∗ −1.6474 −0.0039∗ −1.6236

−0.0040 −1.3261 −0.0041 −1.3279 −0.0055∗∗ −1.8441 −0.0056∗∗ −1.8153 −0.0043∗ −1.4823 −0.0045∗ −1.4826 −0.0048∗ −1.5949 −0.0049∗ −1.5812

β1

0.0091∗∗ 2.2308 0.0091∗∗ 2.1942

0.0031 1.3104 0.0029 1.2137 0.0031 1.3185 0.0029 1.1782 0.0022 0.8667 0.0021 0.8082 0.0018 0.7026 0.0016 0.6470

0.0032 1.0673 0.0033 1.0838 0.0016 0.4914 0.0017 0.5256 −0.0004 −0.1872 −0.0003 −0.1400 −0.0006 −0.2461 −0.0006 −0.2036

β2

0.0002 0.0558 0.0002 0.0370

0.0031 1.4253 0.0031 1.4223 0.0029 1.3094 0.0025 1.0502 0.0023 0.9591 0.0020 0.8308 0.0025 1.0606 0.0023 0.9280

0.0035 1.2746 0.0034 1.2506 0.0015 0.5680 0.0016 0.5676 −0.0002 −0.0642 −0.0002 −0.0486 −0.0004 −0.1406 −0.0004 −0.1258

β3

−1.6245∗∗ −6.8985 −1.6207∗∗ −6.8048

0.8424 1.1439 1.5378 0.9900 0.4696 0.4455 0.9164 0.6263 0.7714 0.5468 1.1824 0.7100 0.9026 0.6053 1.1509 0.6972

−3.0857∗∗ −1.4274 −3.0423∗∗ −1.3262 −9.3617∗∗ −2.5419 −9.1432∗∗ −2.4208 −8.7331∗∗ −2.4598 −8.6330∗∗ −2.3671 −9.2299∗∗ −2.6065 −9.1404∗∗ −2.5058

β4

0.0004 0.0896

−0.0067 −0.5050

−0.0077 −0.5981

−0.0087 −0.7237

−0.0028 −0.3968

0.0005 −0.0277

0.0004 −0.0639

−0.0001 −0.1703

−0.0003 −0.2301

β5

0.7251

0.7250

0.4420

0.4294

0.4165

0.3991

0.4104

0.3918

0.4077

0.3943

0.3835

0.3783

0.3594

0.3513

0.2847

0.2715

0.2278

0.2101

R2

0.8993

0.0000

0.5932

0.6201

0.5572

0.6784

0.5255

0.7177

0.6808

0.0000

0.5869

0.0084

0.4957

0.0299

0.4086

0.0868

0.3541

0.0000

F-Test

0.2045** 1.7729 0.2060** 1.7510 0.2003** 1.7218 0.2022** 1.7029 0.2000** 1.7069 0.2011** 1.6815

0.1722* 1.5028 0.0849 0.1929 0.1703 1.4745 0.0857 0.1913 0.1744* 1.5007 0.0849 0.1871

Model 5

Model 4

Model 3

Model 2

0.4802** 3.0286 0.4450** 2.3317 0.4895** 3.0747

Technology 12—Eastman Kodak Company ek) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

11—Wal-Mart Stores, Inc (wmt) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

0.2574 0.9315

0.1400 0.5867 0.1227 0.5196 0.1579 0.6539 0.1409 0.5888

−0.0592 −0.2150 −0.0650 −0.2319 −0.0826 −0.3101 −0.0868 −0.3192

0.0027 −0.3628 0.0024 −0.3459

−0.0120 0.8340 −0.0118 0.8139

0.8666∗∗ 1.9389 0.8786∗∗ 2.0149 0.7649∗∗ 1.8573 0.7882∗∗ 1.8946 0.5459 1.1273

1.1142∗∗ 3.1272 0.9708∗∗ 2.7181 1.0112∗∗ 2.8215 0.9761∗∗ 2.6957 0.8920∗∗ 2.0960 0.8732∗∗ 2.0412 1.2975 0.7260 1.2250 0.6983

1.6604∗∗ 4.5993 1.6583∗∗ 4.5331 1.7253∗∗ 3.9934 1.7283∗∗ 3.9507 −0.3629 −0.0926 −0.3231 −0.0818

0.0011 0.2073 0.0024 0.4374 0.0009 0.1667 0.0016 0.2895 0.0017 0.3327

0.0061 1.2819 0.0048 0.9742 0.0054 1.1369 0.0046 0.9270 0.0058 1.2140 0.0050 0.9973 0.0061 1.2471 0.0053 1.0424

0.0072∗∗ 1.6622 0.0071∗ 1.6097 0.0070∗ 1.6030 0.0069∗ 1.5384 0.0066∗ 1.4739 0.0065 1.4218

0.0058 1.0854 0.0058 1.0978 0.0059 1.1717 0.0059 1.1580 0.0046 0.8331

−0.0023 −0.6456 −0.0042 −1.1116 −0.0032 −0.8660 −0.0042 −1.0791 −0.0039 −1.0052 −0.0047 −1.1817 −0.0038 −0.9672 −0.0046 −1.1357

0.0076∗∗ 1.8643 0.0076∗∗ 1.8399 0.0083∗∗ 1.8681 0.0083∗∗ 1.8465 0.0079∗∗ 1.7580 0.0079∗∗ 1.7356

0.0041 0.8540 0.0048 1.0228 0.0007 0.1475 0.0013 0.2631 −0.0007 −0.1585

−0.0104∗∗ −2.5727 −0.0125∗∗ −3.0620 −0.0122∗∗ −2.9302 −0.0127∗∗ −3.0526 −0.0129∗∗ −2.9736 −0.0133∗∗ −3.0635 −0.0135∗∗ −3.0628 −0.0139∗∗ −3.1403

−0.0017 −0.4050 −0.0016 −0.3818 −0.0014 −0.3365 −0.0013 −0.3033 −0.0010 −0.2289 −0.0009 −0.2066

−2.5096∗∗ −2.1621 −3.5253∗∗ −2.9134 −4.6018∗∗ −3.7628 −4.7336∗∗ −3.8075 −4.6967∗∗ −3.8302

−1.6027∗∗ −6.4881 −1.6049∗∗ −6.8081 −1.7160∗∗ −6.7938 −1.7035∗∗ −6.5209 −1.7159∗∗ −6.7080 −1.7067∗∗ −6.4437 −1.7124∗∗ −6.6328 −1.7034∗∗ −6.3458

−1.7049∗∗ −7.3206 −1.7028∗∗ −7.2227 −1.7140∗∗ −7.1450 −1.7128∗∗ −7.0403 −1.7185∗∗ −7.1536 −1.7160∗∗ −7.0422

0.0088 0.5366

0.0305∗∗ 1.9362

0.0288 0.4718

0.0282 0.4672

0.0292 0.4848

0.0481∗∗ 1.8077

−0.0006 −0.1108

−0.0008 −0.1585

−0.0006 −0.1255

0.3502

0.3436

0.3342

0.3036

0.2610

0.7851

0.7748

0.7824

0.7713

0.7805

0.7688

0.7770

0.7309

0.7500

0.7497

0.7449

0.7445

0.7417

0.7414

0.0855

0.4925

0.0583

0.0839

0.0000

0.3215

0.9794

0.3074

0.6799

0.2938

0.3282

0.1709

0.0000

0.8588

0.2301

0.8263

0.2084

0.8567

0.1193

0.4484** 2.3463 0.4781** 2.9428 0.4335** 2.2397

λ0

(Continued)

0.2090** 1.7526 0.2316** 1.8683 0.2190** 1.7784 0.2488** 1.9319 0.2291** 1.8413 0.2580** 1.9806

Model 4

Model 3

Model 2

0.4585** 2.3821 0.6052** 2.2246

14-Texas Instruments (txn) Model 1

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

13—Xerox Corp (xrx) Model 1

Model 8

Model 7

Model 6

Table 1.2

−0.1717 −0.8382 −0.2085 −1.0008 −0.1694 −0.8127 −0.2034 −0.9612

0.2738 0.9810 0.2547 0.9105 0.2714 0.9630

λ1

−0.0055 0.1351 −0.0047 0.0851

0.0108 −0.4136 0.0119 −0.4436

λ2

1.8865∗∗ 3.2322 1.7710∗∗ 3.0068 1.6216∗∗ 2.9045 1.6467∗∗ 2.9321

1.2132∗∗ 3.9747 1.2573∗∗ 3.9863 1.0930∗∗ 3.6500 1.1510∗∗ 3.7360 1.2496∗∗ 3.5418 1.3552∗∗ 3.6874 0.2122 0.6103 0.4475 0.7016

0.5574 1.1447 2.5346 0.5916 2.7523 0.6245

β0

0.0121∗∗ 1.6837 0.0128∗∗ 1.7743 0.0120∗∗ 1.7538 0.0115∗∗ 1.6593

0.0103∗∗ 2.6915 0.0105∗∗ 2.7132 0.0092∗∗ 2.5085 0.0095∗∗ 2.5493 0.0087∗∗ 2.3048 0.0089∗∗ 2.3474 0.0085∗∗ 2.2263 0.0087∗∗ 2.2654

0.0025 0.4718 0.0016 0.3110 0.0025 0.4614

β1

0.0043 0.7490 0.0051 0.8591 0.0027 0.5159 0.0019 0.4037

0.0018 0.5044 0.0024 0.6278 0.0007 0.2410 0.0014 0.4143 0.0018 0.5039 0.0028 0.7512 0.0015 0.4439 0.0025 0.6818

0.0044 0.8047 0.0047 0.8613 0.0046 0.8355

β2

0.0057 0.8613 0.0044 0.6572 0.0007 0.0978 0.0007 0.0916

−0.0045 −1.3459 −0.0041 −1.2157 −0.0062∗∗ −1.7953 −0.0059∗∗ −1.6946 −0.0054∗ −1.4742 −0.0048 −1.3116 −0.0055∗ −1.4723 −0.0050 −1.3210

−0.0001 −0.0424 −0.0006 −0.1344 0.0001 −0.0017

β3

−1.2969∗∗ −4.3193 −1.3995∗∗ −4.3960 −1.5654∗∗ −5.2857 −1.5630∗∗ −5.0617

−0.0673∗∗ −4.4713 −0.1420∗∗ −4.4791 −0.8459∗∗ −5.8850 −1.1548∗∗ −5.9673 −1.0034∗∗ −5.9317 −1.4161∗∗ −6.0628 −1.0472∗∗ −5.9300 −1.4584∗∗ −6.0472

−4.8483∗∗ −3.8896 −4.6179.∗ −3.6679 −4.7782∗∗ −3.7394

β4

−0.0083 −0.3976

0.0111 1.0504

−0.0028 −0.9213

−0.0030 −0.9883

−0.0025 −0.8275

−0.0019 −0.6178

0.0109 0.6520

0.0101 0.6072

β5

0.6350

0.6291

0.6083

0.5997

0.6372

0.6248

0.6336

0.6210

0.6224

0.6130

0.6125

0.6048

0.3721

0.3589

0.3613

R2 F-Test

0.5676

0.1243

0.3454

0.0000

0.4024

0.4066

0.3652

0.2815

0.4468

0.3091

0.5569

0.0000

0.4240

0.0976

0.4650

0.4606** 2.3408 0.6180** 2.2450 0.4560** 2.3349 0.5710** 2.1218

−0.2366 −0.6163 −0.2677 −0.6848 −0.1984 −0.5150 −0.2234 −0.5680

0.3041 1.3755 0.2940 1.3131 0.3426* 1.5996 0.3332* 1.5322 0.3324* 1.5415 0.3227* 1.4740

−0.6866∗∗ −2.1204 −0.6800∗∗ −2.0763 −0.7040∗∗ −2.1559 −0.6970∗∗ −2.1109

0.0085 −0.0678 0.0087 −0.0718

0.0317 −1.1841 0.0296 −1.1086 0.9832∗∗ 2.0200 1.0009∗∗ 2.0352 0.8861∗∗ 1.8288 0.9050∗∗ 1.8431 1.5084∗ 2.7341 1.5194∗∗ 2.7191 3.1610 0.6100 3.2185 0.6092

1.8318∗∗ 2.7818 1.8893∗∗ 2.8358 7.4758∗∗ 1.7555 7.1323∗∗ 1.6770 −0.0092∗ −1.4656 −0.0092∗ −1.4598 −0.0090∗ −1.4502 −0.0090∗ −1.4371 −0.0110∗∗ −1.8386 −0.0110∗∗ −1.8160 −0.0112∗∗ −1.8344 −0.0112∗∗ −1.8090

0.0111∗ 1.5817 0.0104∗ 1.4626 0.0125∗∗ 1.7373 0.0119∗ 1.6216 −0.0076 −1.2198 −0.0075 −1.1911 −0.0080 −1.3135 −0.0079 −1.2775 −0.0032 −0.5187 −0.0031 −0.5019 −0.0030 −0.4854 −0.0029 −0.4655

0.0040 0.6849 0.0034 0.5904 0.0054 0.8434 0.0049 0.7665 −0.0113∗∗ −2.0725 −0.0103∗∗ −1.7640 −0.0126∗∗ −2.3075 −0.0117∗∗ −1.9893 −0.0091∗∗ −1.6619 −0.0084∗ −1.4399 −0.0087∗ −1.5317 −0.0079 −1.3107

0.0020 0.2898 0.0021 0.3036 0.0015 0.2132 0.0016 0.2175 0.4104 1.0783 0.1364 0.7597 −1.0142 −0.3027 −1.2047 −0.3822 −1.3997 −0.6027 −1.5464 −0.6467 −1.3941 −0.5959 −1.5573 −0.6512

−1.5919∗∗ −5.1557 −1.5895∗∗ −4.9065 −1.6093∗∗ −5.2705 −1.6197∗∗ −5.0215

−0.0012 −0.4741

−0.0011 −0.4409

−0.0014 −0.5111

−0.0016 −0.5997

−0.0058 −0.2559

−0.0090 −0.4403

0.6331

0.3996

0.3956

0.3877

0.3841

0.3173

0.3125

0.2867

0.2812

0.6528

0.6487

0.6397

0.6456

0.0928

0.6688

0.0716

0.6189

0.2402

0.5534

0.0000

0.5866

0.1904

0.5253

0.1731

In each cell under the columns (λ0 , λ1 , λ2 , β0 , β1 , β2 , β3 , β4 , β5 ), the ﬁrst number is average parameter estimates across the months in the observation period from the equity model regressions. They are presented for each company and for each model type, separated by industries. The second entry is the average t-score for the corresponding average parameter estimate. This t-score is adjusted for the fact that the regressions contain overlapping time intervals. All t-scores test the null hypothesis that the coefﬁcient is zero, except for β4 . For β4 , the null hypothesis is −1/2. Models 1 and 2 have no default. Models 3 and 4 have a constant default intensity. Models 5 and 6 have the default intensity dependent on the spot rate of interest. Models 7 and 8 have the default intensity dependent on the spot rate of interest and a market index. The number of observations per regression is 48. The number of regressions in the average is 23. The average R2 is given. The F-Test column contains the average P-score where the P-scores are obtained from the F-tests of the individual regressions. The P-score from an individual F-test corresponds to the probability of rejecting the null hypothesis when it is true. The ﬁrst row corresponds to the null hypothesis (β0 = β1 = β2 = β3 = β4 = 0). The F-tests for models 3, 5, and 7 test for the joint hypothesis that all default parameters are zero, i.e., λ0 = 0, λ0 = λ1 = 0, and λ0 = λ1 = λ2 = 0, respectively. The F-tests from models 2, 4, 6, and 8 test the hypothesis that β5 = 0. **Signiﬁcant at 10% level. *Signiﬁcant at 15% level.

Model 8

Model 7

Model 6

Model 5

Model 4

Model 3

Model 2

15—International Business Machines (ibm) Model 1

Model 8

Model 7

Model 6

Model 5

22

THE CREDIT MARKET HANDBOOK Eastman Kodak Company: l(t) = l0 + l1 r(t) + l2 z(t)

0.75 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

0.7 0.65 0.55 0.25 0.5 0.45 0.4 0.35 0.3

M

ar

93

92 ec

92 D

ct

92

2

92 O

Au g

Ju n

92

91

r9 Ap

Fe b

ec D

ct

91

91 O

Au g

M

ay

91

0.25

FIGURE 1.1 Time series estimates of Eastman Kodak Company’s intensity function.

from models 2, 4, 6, and 8 test the hypothesis that β5t = 0. The subsequent sections discuss these statistics and various tests for the relative performance of the different equity models. A typical time series graph of the default intensities for Eastman Kodak (ek) using models 3–8 is shown in Figure 1.1. As depicted, all six models exhibit similar patterns in the default intensities. The magnitude of the default intensity appears to be quite large, exceeding 0.3 for all dates and models. Table 1.3 provides some summary statistics (both in- and out-of-sample) regarding the quality of the models ﬁt to the data. Included are the average root mean squared error, the average generalized cross-validation statistic, the average default intensity, the average standard error of the default intensity, and the average 1-year default probability.

6. ANALYSIS OF THE TIME SERIES PROPERTIES OF THE PARAMETERS Under the assumed model structure, the default and equity risk premium parameters λ0t , λ1t , λ2t , β0t , β1t , β2t , β3t ) should be constant across time.

23

Estimating Default Probabilities Implicit in Equity Prices Table 1.3

Summary Statistics for Model Performance Avg GCV

Avg RMSE

λ

se(λ)

Financials 1—Bankers Trust NY (bt) Model 1 0.0027 Model 2 0.0027 Model 3 0.0027 Model 4 0.0028 Model 5 0.0028 Model 6 0.0029 Model 7 0.0028 Model 8 0.0029

0.0488 0.0490 0.0482 0.0485 0.0485 0.0487 0.0482 0.0484

0.1851 0.3244 0.1929 0.3531 0.1400 0.2968

0.0178 0.0736 0.0183 0.0775 0.0217 0.0811

2—Merrill Lynch & Co (mer) Model 1 0.0038 Model 2 0.0037 Model 3 0.0036 Model 4 0.0038 Model 5 0.0038 Model 6 0.0039 Model 7 0.0039 Model 8 0.0040

0.0585 0.0572 0.0564 0.0568 0.0567 0.0573 0.0566 0.0572

0.2643 0.3161 0.2605 0.2985 0.2148 0.2322

Food & Beverages 3—Pepsico Inc (pep) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

0.0026 0.0027 0.0027 0.0028 0.0028 0.0029 0.0029 0.0030

0.0483 0.0487 0.0486 0.0488 0.0491 0.0493 0.0494 0.0495

4—Coca-Cola (cce) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

0.0054 0.0056 0.0054 0.0057 0.0054 0.0056 0.0055 0.0057

0.0693 0.0697 0.0689 0.0694 0.0679 0.0684 0.0674 0.0678

Airlines 5—AMR Corporation (amr) Model 1 0.0044 Model 2 0.0042 Model 3 0.0046 Model 4 0.0044 Model 5 0.0047 Model 6 0.0045 Model 7 0.0047 Model 8 0.0043

0.0627 0.0607 0.0635 0.0614 0.0636 0.0610 0.0624 0.0592

6—Southwest Airlines Co (luv) Model 1 0.0113 Model 2 0.0116 Model 3 0.0114 Model 4 0.0119 Model 5 0.0111 Model 6 0.0116 Model 7 0.0116 Model 8 0.0121

0.1005 0.1007 0.0998 0.1005 0.0972 0.0981 0.0980 0.0989

0.2486 0.4029 0.1798 0.1642 0.1468 0.1246

Utilities 7—Carolina Power and Light (cpl) Model 1 0.0017 Model 2 0.0017 Model 3 0.0017 Model 4 0.0018 Model 5 0.0017 Model 6 0.0018 Model 7 0.0018 Model 8 0.0019

0.0386 0.0389 0.0390 0.0393 0.0386 0.0388 0.0385 0.0388

0.0015 0.0204 0.0121 0.0354 −0.0289 −0.0040

1-ydf

Avg Y Values

R2

0.1686 0.2683 0.1758 0.2920 0.1316 0.2519

0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048 0.0048

0.4873 0.4901 0.4967 0.5028 0.5025 0.5105 0.5208 0.5284

0.0166 0.0929 0.0171 0.1001 0.0217 0.1062

0.2319 0.2604 0.2290 0.2496 0.1925 0.1963

0.0027 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027 0.0027

0.8292 0.8379 0.8412 0.8426 0.8432 0.8441 0.8477 0.8486

0.0841 0.3186 0.0850 0.3349 0.0742 0.3506

0.0240 0.1220 0.0251 0.1309 0.0288 0.1404

0.0799 0.2696 0.0805 0.2812 0.0696 0.2930

0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018

0.5877 0.5889 0.5887 0.5936 0.5894 0.5944 0.5956 0.6023

0.2255 0.2226 0.2246 0.2213 0.3234 0.3252

0.0640 0.0660 0.0631 0.0651 0.0697 0.0719

0.1851 0.1838 0.1828 0.1811 0.2530 0.2547

0.0156 0.0156 0.0156 0.0156 0.0156 0.0156 0.0156 0.0156

0.1675 0.1764 0.1581 0.1667 0.2043 0.2115 0.2368 0.2455

0.0000 0.0000 −0.0131 −0.0186 −0.1407 −0.1595

0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015

0.3543 0.4100 0.3543 0.4100 0.3669 0.4331 0.4082 0.4797

0.0513 0.2547 0.0494 0.2591 0.0671 0.2835

0.2160 0.3219 0.1524 0.1244 0.1176 0.0817

−0.0069 −0.0069 −0.0069 −0.0069 −0.0069 −0.0069 −0.0069 −0.0069

0.4663 0.4796 0.4909 0.4947 0.5301 0.5329 0.5338 0.5370

0.0078 0.0233 0.0078 0.0231 0.0100 0.0254

0.0015 0.0197 0.0133 0.0356 −0.0286 −0.0024

−0.0055 −0.0055 −0.0055 −0.0055 −0.0055 −0.0055 −0.0055 −0.0055

0.8275 0.8294 0.8276 0.8301 0.8348 0.8367 0.8406 0.8416

0.0000 0.0000 −0.0112 −0.0159 −0.1212 −0.1345

0.0631 0.0597 0.0654 0.0610 0.0717 0.0657

(Continued)

24 Table 1.3

THE CREDIT MARKET HANDBOOK (Continued) Avg GCV

Avg RMSE

λ

se(λ)

8—Texas Utilities Ele Co (txu) Model 1 0.0017 Model 2 0.0018 Model 3 0.0016 Model 4 0.0017 Model 5 0.0015 Model 6 0.0016 Model 7 0.0015 Model 8 0.0016

0.0393 0.0393 0.0378 0.0379 0.0361 0.0363 0.0358 0.0361

0.3137 0.3038 0.3096 0.3021 0.3406 0.3336

0.0227 0.0231 0.0211 0.0215 0.0232 0.0238

Petroleum 9—Mobil Corp (mob) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

0.0010 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011 0.0011

0.0302 0.0305 0.0305 0.0305 0.0305 0.0306 0.0303 0.0304

0.0208 0.1189 0.0212 0.1043 −0.0208 0.0577

Department Stores 10—Sears Roebuck + Co(s) Model 1 0.0036 Model 2 0.0038 Model 3 0.0035 Model 4 0.0037 Model 5 0.0037 Model 6 0.0038 Model 7 0.0038 Model 8 0.0040

0.0571 0.0577 0.0556 0.0562 0.0559 0.0565 0.0560 0.0567

11—Wal-Mart Stores, Inc (wmt) Model 1 0.0036 Model 2 0.0034 Model 3 0.0035 Model 4 0.0036 Model 5 0.0037 Model 6 0.0037 Model 7 0.0038 Model 8 0.0038

0.0564 0.0546 0.0555 0.0551 0.0558 0.0555 0.0560 0.0557

Technology 12—Eastman Kodak Company (ek) Model 1 0.0052 Model 2 0.0050 Model 3 0.0045 Model 4 0.0047 Model 5 0.0046 Model 6 0.0048 Model 7 0.0048 Model 8 0.0049

0.0683 0.0662 0.0627 0.0630 0.0626 0.0629 0.0630 0.0631

13—Xerox Corp (xrx) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

0.0027 0.0028 0.0025 0.0026 0.0026 0.0026 0.0027 0.0027

0.0486 0.0489 0.0467 0.0468 0.0469 0.0468 0.0471 0.0471

14—Texas Instruments (txn) Model 1 0.0096 Model 2 0.0098 Model 3 0.0088 Model 4 0.0090 Model 5 0.0091 Model 6 0.0094 Model 7 0.0091 Model 8 0.0094

0.0926 0.0923 0.0875 0.0877 0.0881 0.0882 0.0869 0.0874

1-ydf

Avg Y Values

R2

0.2688 0.2615 0.2673 0.2618 0.2888 0.2837

0.0024 0.0024 0.0024 0.0024 0.0024 0.0024 0.0024 0.0024

0.2101 0.2278 0.2715 0.2847 0.3513 0.3594 0.3783 0.3835

0.0135 0.0370 0.0143 0.0442 0.0158 0.0462

0.0201 0.1028 0.0216 0.0910 −0.0212 0.0457

0.0115 0.0115 0.0115 0.0115 0.0115 0.0115 0.0115 0.0115

0.3943 0.4077 0.3918 0.4104 0.3991 0.4165 0.4294 0.4420

0.2045 0.2060 0.1973 0.1989 0.1511 0.1528

0.0132 0.0137 0.0136 0.0141 0.0183 0.0190

0.1840 0.1851 0.1775 0.1787 0.1381 0.1394

0.0007 0.0007 0.0007 0.0007 0.0007 0.0007 0.0007 0.0007

0.7250 0.7251 0.7414 0.7417 0.7445 0.7449 0.7497 0.7500

0.1722 0.0849 0.1772 0.0918 0.1878 0.0969

0.0167 0.1055 0.0174 0.1082 0.0223 0.1158

0.1577 0.0668 0.1625 0.0740 0.1700 0.0793

−0.0180 −0.0180 −0.0180 −0.0180 −0.0180 −0.0180 −0.0180 −0.0180

0.7309 0.7770 0.7688 0.7805 0.7713 0.7824 0.7748 0.7851

0.3799 0.3557 0.3945 0.3675 0.4112 0.3851

0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099 0.0099

0.2610 0.3036 0.3342 0.3436 0.3502 0.3613 0.3589 0.3721

0.1836 0.2021 0.1848 0.2072 0.1725 0.1965

0.0042 0.0042 0.0042 0.0042 0.0042 0.0042 0.0042 0.0042

0.6048 0.6125 0.6130 0.6224 0.6210 0.6336 0.6248 0.6372

0.3635 0.4322 0.3579 0.4332 0.4300 0.4727

0.0089 0.0089 0.0089 0.0089 0.0089 0.0089 0.0089 0.0089

0.5997 0.6083 0.6291 0.6350 0.6331 0.6397 0.6487 0.6528

0.4802 0.4450 0.5023 0.4620 0.5324 0.4931

0.2090 0.2316 0.2106 0.2385 0.1953 0.2259

0.4585 0.6052 0.4489 0.6048 0.5750 0.6790

0.0262 0.0365 0.0265 0.0367 0.0328 0.0431

0.0199 0.0214 0.0205 0.0221 0.0244 0.0261

0.0437 0.0836 0.0462 0.0866 0.0570 0.0993

(Continued)

25

Estimating Default Probabilities Implicit in Equity Prices Table 1.3

(Continued) Avg GCV

Avg RMSE

15—International Business Machines (ibm) Model 1 0.0066 0.0769 Model 2 0.0069 0.0775 Model 3 0.0066 0.0757 Model 4 0.0069 0.0764 Model 5 0.0062 0.0725 Model 6 0.0065 0.0732 Model 7 0.0064 0.0727 Model 8 0.0067 0.0734 Average Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

0.0044 0.0045 0.0043 0.0044 0.0043 0.0044 0.0044 0.0045

0.0597 0.0595 0.0584 0.0586 0.0580 0.0581 0.0579 0.0580

λ

0.3041 0.2940 0.3086 0.2996 0.3349 0.3268

0.2115 0.2596 0.2080 0.2462 0.2030 0.2371

se(λ)

1-ydf

0.0505 0.0522 0.0468 0.0484 0.0551 0.0570

0.0301 0.0710 0.0302 0.0732 0.0360 0.0803

R2

Avg Y Values

0.2548 0.2464 0.2543 0.2467 0.2775 0.2715

0.0059 0.0059 0.0059 0.0059 0.0059 0.0059 0.0059 0.0059

0.2812 0.2867 0.3125 0.3173 0.3841 0.3877 0.3956 0.3996

0.1797 0.2118 0.1761 0.2004 0.1641 0.1860

0.0026 0.0026 0.0026 0.0026 0.0026 0.0026 0.0026 0.0026

0.5018 0.5174 0.5213 0.5317 0.5417 0.5526 0.5562 0.5670

Given are the average generalized cross-validation statistics (GCV), the average root mean squared error (RMSE), the average default intensity (λ), the average standard error of the default intensity (se(λ)), the average 1-year default probability (1-ydf), the average value for the dependent variable in the equity model regression, and the average R2 . The number of observations per regression is 48. The averages are taken across all months from the equity model regressions. The number of regressions in the average is 23.

Table 1.4

Unit Root Test Performance across All Companies λ0

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8

3/15 2/15 2/15 3/15 1/15 3/15

λ1

0/15 2/15 0/15 1/15

λ2

β0

β1

β2

β3

β4

2/15 2/15

2/15 3/15 2/15 3/15 4/15 1/15 3/15 2/15

0/15 1/15 1/15 1/15 1/15 3/15 4/15 4/15

1/15 1/15 0/15 0/15 1/15 1/15 1/15 1/15

2/15 2/15 1/15 0/15 3/15 2/15 1/15 2/15

4/15 3/15 4/15 3/15 2/15 4/15 2/15 2/15

β5 2/15 2/15 3/15 3/15

The entries under the columns correspond to the number of companies for the relevant coefﬁcient where the null hypothesis of a unit root is rejected at the 90% level. There are 15 total companies—tests for a unit root.

The 4-year estimation period was selected to better ensure this hypothesis, by minimizing the structural shifts in the economy that would more likely occur using a longer horizon interval. Given measurement error in the input data (equity prices and the state variable parameters) and its effect on the parameter estimates, we test the hypothesis that the time series variation in these parameters is solely due to random (white) noise. Alternatively stated, we test to see if the parameter estimates follow a random walk around a given mean. A unit root test is used in this regard. Table 1.4 contains a summary of the unit root rejections across model types. As seen, we accept a unit root (non-stationarity) for almost all the parameters in all the models. This includes both the default parameters and the risk premium coefﬁcients. Acceptance of the unit root implies that the model parameters may be non-stationary. This nonstationarity could be due to a confounding of the default and risk premium parameter estimates. Given that the underlying variables related to both

26

THE CREDIT MARKET HANDBOOK

quantities are correlated, multicollinearity in the linear regression may be a problem.

7. ANALYSIS OF FAMA–FRENCH FOUR-FACTOR MODEL WITH NO DEFAULT Before analyzing the default parameters, it is important to document the performance of the simple Fama–French four-factor model with no default. Table 1.2 contains the estimates for the coefﬁcients of the Fama–French fourfactor model with and without a P/E ratio (models 1 and 2, respectively) and their t-scores. The F-test for model 1 in Table 1.2 tests the hypothesis that the model is signiﬁcant (i.e., β0t = β1t = β2t = β3t = β4t = 0). For every ﬁrm, the average p-value for this F-test is 0.0000, strongly rejecting the hypothesis of no signiﬁcance. This test conﬁrms the need to include the Fama–French four-Factor model to explain stock risk premiums. The average R2 for model 1 is 0.5018.

8. ANALYSIS OF A BUBBLE COMPONENT (P/E RATIO) IN STOCK PRICES This section tests for the signiﬁcance of a bubble component in equity returns by testing the null hypothesis that the P/E ratio is insigniﬁcant, that is, β5t = 0. The F-test for model 2 in Table 1.2 also tests this hypothesis. For models 1 and 2 (not including default), the averages p-values for three ﬁrms (mer, amr, and ek) are signiﬁcantly different from zero. The individual t-scores for β5t show signiﬁcance for four ﬁrms (mer, amr, wmt, and ek), conﬁrming this conclusion. This represents 20% (=3/15) to 26% (=4/15) of our ﬁrms. For models 3–8 (including default), the average F-test gives signiﬁcance for only one of these three ﬁrms (amr). The average t-scores for β5t conﬁrm this reduced signiﬁcance. For amr alone (among the three: mer, amr, and ek), the estimated coefﬁcient for the constant in the regression (λ0 ) is zero. It appears that for models with no default (models 1 and 2), the P/E ratio proxies for a bubble component in stock prices not contained in the four factors of Fama–French. But, for the models with default (models 3– 8), the P/E ratio becomes insigniﬁcant. The inclusion of a constant in the regression model (λ0 ) appears to confound the bubble component in stock prices.15 An additional test for a possible model misspeciﬁcation with respect to the bubble component is provided by the t-score for the stock’s own variance (β4 ). This t-score tests for the null hypothesis that β4 = −1/2,

Estimating Default Probabilities Implicit in Equity Prices

27

the theoretical value as given in equation 1.6. As indicated, for all models and for all but four ﬁrms (cce, amr, mob, and ibm), one can reject the null hypothesis that β4 = −1/2. This rejection is strong evidence consistent with the stock’s own variance proxying for stock price bubbles. In summary, this section provides evidence consistent with the existence of an equity price bubble not captured in the Fama–French fourfactor model. Both the P/E ratio and the stock’s own variance appear to be signiﬁcant explanatory variables in the equity return regression model.

9. ANALYSIS OF THE DEFAULT INTENSITY As mentioned earlier, the average default intensity parameters and t-scores are contained in Table 1.2. The ﬁrms’ estimates are presented in industry groupings for easy comparison. First to be noticed is that the ﬁt of the linear regressions is quite good. The average R2 varies between 0.1581 (for cce, model 3) to 0.8486 (for mer, model 8). From Table 1.3, the average R2 across all models varies from 0.5018 for model 1 to 0.5670 for model 8. R2 uniformly increases across ﬁrms with increasing model complexity up to model 8. This is to be expected because R2 is a measure of the in-sample ﬁt, and as we progress from model 1 to model 8, more independent variables are added to the linear regression. Second, it is interesting to examine the signs of the coefﬁcients for the default intensity parameters. The signs of λ1 and λ2 indicate the sensitivity of the ﬁrm’s default likelihood to changes in the spot rate and the cumulative excess return on the equity market index, respectively. For example, for wmt (Wal-Mart Stores) the sign of λ1 is positive indicating that as interest rates rise, the likelihood of default increases. Continuing, the sign of λ2 is negative, indicating that as the market index rises, the likelihood of default decreases. The signs of these coefﬁcients differ across ﬁrms and between ﬁrms within an industry. An example of different signs within an industry is for the department stores grouping, where Sears Roebuck and Company (s) and Wal-Mart Stores, Inc. (wmt) have contrasting signs for both the interest rate and market index variables. Next, we discuss the statistical signiﬁcance of these point estimates. For λ0 , the point estimate is signiﬁcantly different from zero in model 3 for seven ﬁrms (mer, txu, s, wmt, ek, xrx, txn, and ibm). The F-test for model 3 also tests the hypothesis that λ0t = 0. This test conﬁrms the average t-score results because the average p-scores are low (below 15% for ﬁve ﬁrms). Including the P/E ratio in model 4 eliminates the signiﬁcance of the default parameter λ0 for two of the seven ﬁrms (mer and wmt), indicating a possible confounding

28

THE CREDIT MARKET HANDBOOK

of the default parameter estimate with equity price bubbles. This suspicion is conﬁrmed for more complex models 5–8. The pattern with respect to the signiﬁcance of the default parameter λ0 is similar to that previously discussed for models 3 and 4. With respect to the spot rate coefﬁcient, λ1 , the signiﬁcance of its t-scores varies across ﬁrms and model types. For four out of the 15 ﬁrms, the average t-score is signiﬁcantly different from zero for at least one of models 5–8. This observation is also supported by the F-test for model 5 (the joint hypothesis λ0t = λ1t = 0). The average p-scores for this F-test are less than 20% for ﬁve ﬁrms (luv, txu, ek, txn, and ibm). This evidence strongly supports the inclusion of the interest rate variable in the default intensity model. Finally, with respect to the market index coefﬁcient, λ2 , the average t-scores indicate that it is insigniﬁcant from zero for all ﬁrms except one (amr). Unfortunately, the introduction of the independent variable (Z(t)(TL − t) − Z(t − )(TL − (t − ))) into this regression causes a severe multicollinearity problem with another independent variable in the equity risk premium, the excess return on the market portfolio. As seen in models 7 and 8, for all ﬁrms the introduction of (Z(t)(TL − t) − Z(t − )(TL − (t − ))) causes the point estimates of β0 to change dramatically from their values in models 1–6 and β0 becomes insigniﬁcantly different from zero. The correlation matrix in Table 1.5 conﬁrms this multicollinearity problem. The correlation between these two variables is 0.9934. This implies that the estimates of the coefﬁcients λ2 and β0 cannot be separated using this regression model. This evidence is consistent with a confounding of the default probability estimates with those of the equity’s risk premium. As seen from Table 1.3, the impact of these different default intensity models on the point estimates of the default intensities can be dramatic. For example, for mer the average default intensity varies from −0.0569 in model 7 to 0.3534 for model 6. Negative default intensities should be interpreted as being a point estimate of zero. Similar patterns can also be observed for the other ﬁrms in our sample. This sensitivity is consistent with a confounding of the default probability model with the equity risk premium and bubble component. Table 1.5 Correlation Matrix for Selected Independent Variables in the Equity Model Regression

λ1 λ2 β0 β1 β2 β3

λ1

λ2

β0

β1

β2

β3

1.0000 0.4685 0.4580 −0.4072 0.2607 0.2896

0.4685 1.0000 0.9934 −0.1277 −0.3539 0.1141

0.4580 0.9934 1.0000 −0.1449 −0.3765 0.1060

−0.4072 −0.1277 −0.1449 1.0000 −0.5153 0.0850

0.2607 −0.3539 −0.3765 −0.5153 1.0000 −0.0267

0.2896 0.1141 0.1060 0.0850 −0.0267 1.0000

Estimating Default Probabilities Implicit in Equity Prices

29

Also documented in Table 1.3 are the magnitudes of these default probability estimates. As indicated, they are quite high relative to historical default frequencies. Indeed, the average default intensity across all ﬁrms and all models exceeds 0.20, whereas the magnitude of the average historical default intensities observed from bankruptcy data is less than 0.01 (see Chava and Jarrow, 2002). Although this difference could be due to the fact that these estimates obtained are the risk-neutral probabilities, as opposed to the statistical probabilities, more likely the difference is due to a confounding of the default intensity model with both the equity risk premium and price bubble. As previously highlighted, many of the preceding test results are consistent with this second interpretation. Also to be noticed at this juncture are the magnitudes of the RMSE of the regression model in comparison to the average magnitude of the dependent variable. The RMSE is an estimate of the standard error of the unpredictable component of equity returns. The average magnitude of the dependent variable is the average equity return over this period. As indicated in Table 1.3, the standard error of the unpredictable component of the equity’s return is over 10 times the magnitude of its average value. This is true for all ﬁrms in our sample. This illustrates the magnitude of the “noise” in equity prices relative to our predictive ability using the Fama–French four-factor model. This noise inhibits our ability to estimate the default intensities with a great deal of precision. In summary, an analysis of the default intensity parameters in the equity return regression model shows that although it is feasible to estimate the likelihood of default, the estimated intensity process parameters are confounded by the equity risk premium and price bubble component. Both default risk and the equity’s risk premium appear to be positively correlated. Including both variables in a regression yields an upward bias in the estimated default probability (relative to historical default frequencies).

10. RELATIVE PERFORMANCE OF THE EQUITY RETURN MODELS This section studies the relative performance of the eight equity return models. For each ﬁrm and for each model’s regression, both a root mean squared error statistic (RMSE) and a generalized cross-validation statistic (GCV) are computed. The RMSE statistic measures the “average” pricing error between the model and the market price. It is an in-sample goodness-of-ﬁt measure. As with all in-sample goodness of ﬁt measures, a potential problem with RMSE is that it may provide a biased picture of the quality of model performance due to a model overﬁtting the noise in the data. The second GCV

30

THE CREDIT MARKET HANDBOOK

test statistic is designed to partially overcome this problem, as an out-ofsample goodness-of-ﬁt measure that is predictive in nature.16 The lower the GCV statistic, the better the out-of-sample model ﬁt. The average RMSE and GCV statistics for each ﬁrm and model are contained in Table 1.6. For 13 of the 15 ﬁrms, the RMSE statistic is smallest (or within 0.0001 of the smallest value) for models 3–6. For all 15 ﬁrms, the GCV statistic is smallest (or within 0.0001 of the smallest value) for models 3–6. This is strong evidence consistent with the importance of including the default parameters into the equity return model and the feasibility of using equity returns to infer default intensity estimates. Surprisingly, for no ﬁrms except one (txu) do models 7 and 8 have the smallest GCV statistics. Despite the multi-collinearity problem present when including Z(t) into the default intensity process, this is strong evidence consistent with the insigniﬁcance of the λ2 coefﬁcient. This relative performance analysis conﬁrms the insigniﬁcance of the λ2 coefﬁcient documented in Janosi et al. (2002) for the same ﬁrms, but using debt prices. In summary, in terms of the GCV statistic, the best-ﬁtting models are 3–6. Given the previous evidence with respect to the signiﬁcance of the interest rate variable in the default intensity process, the preferred equity return models are probably 5 and 6.

11. COMPARISON OF DEFAULT INTENSITIES BASED ON DEBT VERSUS EQUITY This section investigates the equality between the default intensities estimated using the equity returns with the default intensities as estimated in Janosi et al. (2002). Using the identical structure, the identical data, and the same time period as employed above, Janosi et al. (2002) estimate the expected loss per unit time λ(t)(1 − δ) implicit in debt prices, using a reduced-form credit risk model. Here, 0 ≤ δ ≤ 1 corresponds to the recovery rate on defaulted debt. They selected 20 different ﬁrms, 15 of which overlap with this study (see Table 1.1). Janosi et al. (2002) estimated ﬁve different liquidity premium models. We use only the best-ﬁtting model, the constant liquidity premium. To match the best-ﬁtting equity pricing model (model 6), we use the intensity function from Janosi et al. (2002) without the market index included. For this credit risk model, we have the monthly time series of the estimated expected loss per unit time λ(t)(1 − δ) for each of the 15 companies from Janosi et al. (2002) and their standard errors. Unfortunately, due to nonoverlapping periods of observations in the two studies, only 10 ﬁrms are included in this comparison. The ﬁve ﬁrms omitted are: amr, bt, cce, cpl, and ek.

31

Estimating Default Probabilities Implicit in Equity Prices Table 1.6 Test for the Equivalence between the Default Intensities Based on Debt Prices versus Equity Prices

amr bt cce cpl ek ibm luv mer mob pep s txn txu wmt xrx

η0

η1

Recovery Range

Avg(λd (t))

Avg(λe (t))

N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A −1.3253 −1.8008 −4.3092 −5.0623 −2.5844 −2.0105 −1.7100 −1.8682 0.5209 0.8286 0.4483 0.6984 3.0442 1.7638 0.5267 0.9931 0.5775 0.5621 1.4597 1.1763

N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 30.6004 2.1955 84.5898 5.2471 35.6537 1.4645 29.2491 1.7266 −8.6911 −0.7300 −7.0506 −0.5852 −50.7298 −1.5520 −7.1674 −0.7136 −9.3437 −0.4802 −27.3956 −1.1658

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

None

0.0103

0.2996

None

0.0104

0.1642

[0.99, 1.00]

0.0100

0.2985

[0.99, 1.00]

0.0036

0.2311

[0.95, 1.00]

0.0082

0.3349

[0.88, 1.00]

0.0086

0.2058

[0.94, 1.00]

0.0083

0.6048

[0.93, 1.00]

0.0061

0.3021

[0.00, 1.00]

0.0036

0.0918

[0.96, 1.00]

0.0056

0.2385

In the fourth and ﬁfth columns, the average default intensities are provided. The debt estimates are for a zero recovery rate. The equity estimates are for the midpoint of the recovery rate from column 3. The equity estimation is from May 31, 1995 to March 31, 1997 for all the companies. The debt estimation time period is contained in Table 1.1. This table represents the intersecting dates from both experiments. For the ﬁrst three columns, we (t) = ηi + ηi r(t) + εt . We report the range of recoveries that makes H0 : ηi = 0 and (t) − λi estimate λi 0 1 0 equity debt η1 = 0 insigniﬁcant at the 95% level. The ﬁrst number is the point estimate, the second is the t-statistic. Only ibm i and luv are rejected for all possible δ.

Available are estimated intensities λie (t) from the equity returns for ﬁrm i in month t, and estimated expected losses (per unit time) aid (t) = λid (t)(1−δ i ) from Janosi et al. (2002) for ﬁrm i in month t. The null hypothesis to be tested is (debt) aid (t)/(1 − δ i ) = λie (t) (equity) for all ﬁrms i and all months t. Given an assumed value for the recovery rate δ i , we can do a pairwise t-test (using the standard errors of the estimates) of the difference [aid (t)/(1 − δ i )] − λie (t) for a given ﬁrm i for each month t. Under the null hypothesis this difference is zero. Unfortunately, this is a joint test of the null hypothesis and the assumed value of δ i . To eliminate this joint hypothesis,

32

THE CREDIT MARKET HANDBOOK

we test for equality of these differences over a range of different values for δ i from 0 to 1. If there is a value for δ i where the null hypothesis [˜aid (t)/(1 − δ i )] − λie (t) = 0 is not rejected, then this value for δ i is an estimate of the recovery rate. The relevant summary statistics for these tests are contained in Table 1.6. For all ﬁrms but two (ibm and luv), there is some recovery rate such that the two estimates can be viewed as equivalent. This is strong evidence consistent with equality of the implicit estimation procedures across both equity and debt prices. However, the recovery rate needed to obtain equality of the two estimates is in excess of 88%, for all ﬁrms except one (wmt). This recovery rate is higher than the average recovery rate of 67% contained in Moody’s (1992) for senior secured debt over the time period 1974–1991. This overestimate of the recovery rate suggests an upward bias in the estimated default rates obtained from equity returns, conﬁrming the conclusions from the previous analyses.

12. CONCLUSIONS This chapter uses a reduced-form model to estimate default probabilities implicit in equity returns. The model implemented is a generalization of the model contained in Jarrow (2001). The time period covered is May 1991–March 1997. Monthly equity returns on 15 different ﬁrms are studied. The ﬁrms are chosen to provide a stratiﬁed sample across various industry groupings. Three general conclusions can be drawn from this investigation. First, equity returns can be used to infer a ﬁrm’s default intensities. This is a feasibility result. Second, equity returns appear to contain a bubble component, as proxied by the ﬁrm’s P/E ratio. Bubbles in equity returns are not completely captured by the four-factor model of Fama and French (1993, 1996). Third, due to this imprecision in modeling equity risk premiums, the point estimates of the default intensities confound with the equity risk premium. Estimated default probabilities using equity returns are larger than those obtained based on either historical bankruptcy data or implicitly using debt prices. This conclusion casts doubts upon the reliability of the default probability estimates obtained from equity prices using structural models as in Delianedis and Geske (1998) (conﬁrming the previous conclusions of Jarrow and van Deventer, 1998, 1999, and Jarrow et al., 2002, in this regard). This is also consistent with the inability of structural models using equity price information to explain credit spreads in corporate debt (see Collin-Dufresne et al., 2001; Huang and Huang, 2002; Eom et al., 2002).

Estimating Default Probabilities Implicit in Equity Prices

33

NOTES 1. The intensity process is deﬁned under the risk neutral probability. 2. This assumption is easily relaxed; see Jarrow (2001). 3. One could assume that these dividends could be defaulted on as well; see Jarrow (2001). 4. See Money Magazine April 1999, p. 169 for Yahoo’s P/E ratio of 1176.6. 5. This is a simple no arbitrage restriction that the present value of the sum of multiple cash ﬂows equals the sum of the present values of the cash ﬂows. 6. For the explicit equations, see Jarrow (2001). 7. This is in contrast to the typical situation where there are multiple debt issues outstanding that can be utilized to implicitly estimate default intensities when using debt prices. 8. The appendix contains a minor correction to the formula contained in Jarrow (2001). 9. The address is: http://web.mit.edu/kfrenc/www/data_library.html. 10. We perform a Dickey–Fuller (DF) test. The DF test statistic is the t-statistic for the ρ coefﬁcient in the regression yt − yt−1 = µ + ρyt−1 + εt where µ, ρ are constants and εt is an error term. The null hypothesis of a unit root for yt is ρ = 0. A rejection of the null hypothesis implies that there is no unit root. The unit root test statistics are σr (−2.6348) and ar (−1.1632). 11. The unit root test statistics are σm (−3.9407) and ϕ (−1.3479). 12. The exact formula for this computation is in Jarrow (2001). 13. The t-score is adjusted to reﬂect the fact that the regressions contain overlapping time intervals; see Janosi et al. (2002) for more details on the adjustment. 14. This is not to be confused with the number of observations used in the time t regression for a particular ﬁrm. At the time t regression, we use 48 months of data. 15. This should not be surprising. In the standard implementation of the CAPM, the constant term in the regression equation is called the “alpha” and it is used to represent abnormal returns. If the bubble component is not adequately modeled, its time series variation would appear in the estimate of this coefﬁcient. 16. Roughly speaking, the GCV statistics measure the average predictive error obtained by systematically eliminating each data point from the time series regression, predicting that point’s value with the regression, and then measuring the “average” predictive error that results, after adjusting for degrees of freedom (see Wahba, 1985).

REFERENCES Adams, K. and van Deventer, D. (1994). “Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness.” Journal of Fixed Income June, 52–62. Altman, E. I. (1968). “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy.” Journal of Finance 23, 589–609.

34

THE CREDIT MARKET HANDBOOK

Bielecki, T. and Rutkowski, M. (2000). Credit Risk: Modelling, Valuation and Hedging. New York: Springer-Verlag (in press). Chava, S. and Jarrow, R. (2002). “Bankruptcy Prediction with Industry Effects.” Working paper, Cornell University. Collin-Dufresne, P., Goldstein, R. and Martin, J. (2001). “The Determinants of Credit Spread Changes.” Journal of Finance 54, 2177–2207. Delianedis, G. and Geske, R. (1998). “Credit Risk and Risk Neutral Default Probabilities: Information about Rating Migrations and Defaults.” Working paper, UCLA. Dimson, E. (1979), “Risk Measurement when Shares are Subject to Infrequent Trading.” Journal of Financial Economics 7, 197–226. Duffee, G. (1999). “Estimating the Price of Default Risk.” The Review of Financial Studies 12, 197–226. Duffie, D. and Singleton, K. (1997). “An Econometric Model of the Term Structure of Interest Rate Swap Yields.” Journal of Finance 52, 1287–1321. Dufﬁe, D. and Singleton, K. (1999). “Modeling Term Structures of Defaultable Bonds.” Review of Financial Studies 12, 197–226. Dufﬁe, D., Pedersen, L. and Singleton, K. (2000). “Modeling Sovereign Yield Spreads: A Case Study of Russian Debt.” Working paper, Stanford University. Eom, Y., Helwege, J. and Huang, J. (2002). “Structural Models of Corporate Bond Pricing: An Empirical Analysis.” Working paper, Penn State University. Fama, E. and French, K. (1993). “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33, 3–56. Fama, E. and French, K. (1996). “Multifactor Explanations of Asset Pricing Anomalies.” Journal of Finance 60, 55–84. Heath, D., Jarrow, R. and Morton, A. (1992). “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation.” Econometrica 60, 77–105. Huang, J. and Huang, M. (2002). “How Much of the Corporate Treasury Yield Spread Is Due to Credit Risk.” Working paper, Penn State University. Hull, J. and White, A. (2000). “Valuing Credit Default Swaps I: No Counterparty Default Risk.” Journal of Derivatives 8, 29–40. Hull, J. and White, A. (2001). “Valuing Credit Default Swaps II: Modeling Default Correlations.” Journal of Derivatives 8, 12–23. Janosi, T. and Jarrow, R. (2002). “Maximum Smoothness Forward Rate Curves.” Working paper, Cornell University. Janosi, T., Jarrow, R. and Yildirim, Y. (2002). “Estimating Expected Losses and Liquidity Discounts Implicit in Debt Prices.” Journal of Risk 5, 1–39. Jarrow, R. (2001). “Default Parameter Estimation Using Market Prices.” Financial Analysts Journal 57, 75–92. Jarrow, R. and van Deventer, D. (1998). “Integrating Interest Rate Risk and Credit Risk in Asset and Liability Management.” Asset and Liability Management: The Synthesis of New Methodologies. Risk Publications. Jarrow, R. and van Deventer, D. (1999). “Practical Usage of Credit Risk Models in Loan Portfolio and Counterparty Exposure Management.” Credit Risk Models and Management. Risk Publications.

Estimating Default Probabilities Implicit in Equity Prices

35

Jarrow, R. and Madan, D. (2000). “Arbitrage, Martingales and Private Monetary Value.” Journal of Risk 3, 73-90. Madan, D. and Unal, H. (1998). “Pricing the Risks of Default.” Review of Derivatives Research 2, 121–160. Moody’s Special Report (1992). Corporate Bond Defaults and Default Rates. New York: Moody’s Investors Service. Risk Magazine (2000). “Credit Risk: A Risk Special Report.” (March). Schwartz, R. and Whitcomb, D. (1977a). “The Time-Variance Relationship: Evidence on Autocorrelation in Common Stock Returns.” Journal of Finance 32, 41–55. Schwartz, R. and Whitcomb, D. (1977b). “Evidence on the Presence and Causes of Serial Correlation in Market Model Residuals.” Journal of Financial and Quantitative Analysis June, 291–313. Shumway, T. (2001). “Forecasting Bankruptcy More Accurately: A Simple Hazard Model.” Journal of Business (in press). Smith, K. (1978). “The Effect of Intervaling on Estimating Parameters of the Capital Asset Pricing Model.” Journal of Financial and Quantitative Analysis June, 313–332. Warga, A. (1999). Fixed Income Data Base. University of Houston, College of Business Administration (www.uh.edu/∼awarga/lb.html). Wahba, G. (1985). “A Comparison of GCV and GLM for Choosing the Smoothing Parameter in the Generalized Spline Smoothing Problem.” Annals of Statistics 13, 1378–1402. Zmijewski, M.E. (1984). “Methodological Issues Related to the Estimation of Financial Distress Prediction Models.” Journal of Accounting Research 22, 59–82.

APPENDIX From the appendix in Jarrow (2001), we have: 

 TD ξt − j≥t Dj ν(t, j: e)  log  TD∗ ξt− − j≥t− Dj ν(t − , j: e) t Lt µθ (u) du + λ0 + = log Lt− t− p(t, TL ) 2 − λ1 − log − b(t − , TL ) /2 p(t − , TL ) − λ2 [Z(t)(TL − t) − Z(t − )(TL − t + )] − λ21 b(t − , TL )2 /2 − λ22 (TL − t)2 /2 + λ1 σL ϕrL b(t − , TL ) + λ2 σL ϕmL (TL − t)

36

THE CREDIT MARKET HANDBOOK

In the previous expression, the following quantities are unobservable: ϕrL , ϕmL . To eliminate these quantities from this expression, we compute the variance of the preceding expression.

 TD ξ(t) − j≥t Dj ν(t, j:e)  σξ2 (t) ≡ vart log  TD Dj ν(t − , j: e) ξ(t − ) − j≥t− Lt p(t, TL ) = vart log − λ1 − log Lt− p(t − , TL ) −b(t − , TL )2 /2 



−λ2 [Z(t)(TL − t) − Z(t − )(TL − t + )]

But we have:

b(t − , TL )2 − 2

p(t, TL ) − log p(t − , TL )

= µ1 (t, TL ) − µ1 (t − , TL ) + DET = [b(t, TL )r(t) − b(t − , TL )r(t − )]/σr + DET = b(t, TL )[r(t) − r(t − )]/σr + DET where DET indicates nonrandom terms. Also note that

r(t) − r(t − ) = σr

t t−

e−a(ν−t) dW(ν) + DET

Substitution gives

b(t − , TL )2 p(t, TL ) − − log 2 p(t − , TL )

t = b(t, TL ) e−a(ν−t) dW(ν) + DET t−

37

Estimating Default Probabilities Implicit in Equity Prices

Combined, we get: σξ2 (t)

= vart σL [wL (t) − wL (t − )] − λ1 b(t, TL )

t

e

−a(ν−t)

dW(ν)

t−

−λ2 (TL − t)[Z(t) − Z(t − )] + DET] Computing this variance yields: σξ2 (t) ≈ λ21 b(t, TL )2 + λ22 (TL − t)2 + σL2 −2λ1 b(t, TL )σL ϕrL − 2λ2 σL ϕmL (TL − t) + 2λ1 λ2 ϕrm b(t, TL )(TL − t) where we have used the facts:

t e−a(ν−t) dW(ν) = vart t−

t t−

e−2a(ν−t) dν = (1 − e2a )/2a ≈ ∇

and

t

t−

e−a(ν−t) dν = (1 − ea )/a ≈

Rearranging the terms gives: − σξ2 (t)/2 + σL2 /2 + λ1 λ2 ϕrm b(t, TL )(TL − t) ≈ −λ21 b(t, TL )2 /2 − λ22 (TL − t)2 /2 + λ1 b(t, TL )σL ϕrL + λ2 σL ϕmL (TL − t) Substitution gives the result:  TD ξt − j≥t Dj ν(t, j: e)  log  TD ξt− − j≥t− Dj ν(t − , j: e) t Lt = log µθ (u) du + λ0 + Lt− t− p(t, TL ) + b(t − , TL )2 /2 + λ1 log p(t − , TL ) 

− λ2 [Z(t)(TL − t) − Z(t − )(TL − t + )] − σξ2 (t)/2 + σL2 /2 + λ1 λ2 ϕrm b(t, TL )(TL − t)

38

THE CREDIT MARKET HANDBOOK

Using Girsanov’s theorem, we have the original Brownian motion: wL (t) = t ˆ L (t) is a Brownian motion under the statistical w ˆ L (t) + 0 L (u) du where w measure and L (u) is the liquidation value’s risk premium. Hence, log(Lt /Lt− ) = r(t − ) + σL L (t − ) − (1/2)σL2 + σL [wL (t) − wL (t − )] Thus, we have the ﬁnal result: 

 TD∗ ξt − j≥t Dj ν(t, j: e)  − r(t − ) log  TD∗ ξt− − j≥t− Dj ν(t − , j: e) = σL L (t − ) + µθ (t − ) − σξ2 (t)/2 p(t, TL ) + λ0 + λ1 log + b(t − , TL )2 /2 p(t − , TL ) − λ2 [Z(t)(TL − t) − Z(t − )(TL − t + )] + λ1 λ2 ϕrm b(t, TL )(TL − t) + σL [wL (t) − wL (t − )]

CHAPTER

2

Predictions of Default Probabilities in Structural Models of Debt Hayne E. Lelanda

This chapter examines default probabilities predicted by alternative “structural” models of risky coporate debt. We focus on default probabilities rather then credit spreads because (1) they are not affected by additional market factors such as liquidity and tax differences, and (2) prediction of the relative likelihood of default is often stated as the objective of bond ratings. We have three objectives: 1. To distinguish “exogenous default” from “endogenous default” models. 2. To compare these models’ predictions of default probabilities, given common inputs. 3. To examine how well these models capture actual average default frequencies, as reﬂected in Moody’s (2001) corporate bond default data 1970–2000. We ﬁnd the endogenous and exogenous default boundary models ﬁt observed default frequencies very well for horizons and longer, for both investment grade and non-investment grade ratings. Shorter-term default frequencies tend to be underestimated. This suggests that a jump component should be included in asset value dynamics. Both types of structural models ﬁt available default data equally well. But the models make different predictions about how default probabilities and recovery rates change with changes in debt maturity or asset volatility. Further data and testing will be needed to test these differences. Finally, we compare and contrast these structural models’ default predictions with a simpliﬁed version of the widely used Moody’s-KMV “distance to default” model described in Crosbie and Bohn (2002).

a Haas School of Business, University of California, Berkeley, USA.

39

40

THE CREDIT MARKET HANDBOOK

1. INTRODUCTION This chapter examines the default probabilities (DPs) that are generated by alternative “structural” models of risky corporate bonds.1 We have three objectives: 1. To distinguish “exogenous default” from “endogenous default” models. 2. To compare these models’ predictions of default probabilities, given common inputs. 3. To examine how well these models capture actual average default frequencies, as reﬂected in Moody’s (2001) corporate bond default data 1970–2000. Our analysis is limited to structural models of debt and default.2 These models assume that the value of the ﬁrm’s activities (“asset value”) moves randomly through time with a given expected return and volatility. Bonds have a senior claim on the ﬁrm’s cash ﬂow and assets. Default occurs when the ﬁrm fails to make the promised debt service payments. We focus on two sets of structural models that have been quite widely used in academic and/or practical applications: those with an “exogenous default boundary” that reﬂects only the principal value of debt, and those with an “endogenous default boundary,” where default is chosen by management to maximize equity value.3 Models with a default boundary that depends only on the principal value of debt include the pioneering work of Black and Scholes (1973) and Merton (1974). To exploit the analogy with options, these authors restricted themselves to zero-coupon debt. With zero-coupon bonds the default boundary is zero until the bond matures, and therefore default never occurs prior to the bond’s maturity. At maturity, the ﬁrm defaults if asset value is less than the bond’s principal value. Subsequent work with both endogenous and exogenous default boundaries has focused on the default risk of coupon-paying bonds. In these models, default can occur prior to bond maturity. Typical of recent work using an exogenous default barrier is the Longstaff and Schwartz (1995) model (hereafter “L-S”).4 The L-S model considers coupon-paying bonds with an exogenous default boundary that remains constant through time. While the mathematics allows for any constant default boundary, L-S consider a default boundary that equals the principal value of debt. The ﬁrm defaults when asset value ﬁrst falls beneath debt principal value (“negative net worth”). However, Huang and Huang (2002) (hereafter “H-H”) and others have pointed out that ﬁrms often continue to operate with negative net worth. They also note that the implied default costs must be extremely high to explain the relatively low recovery rates on corporate

Predictions of Default Probabilities in Structural Models of Debt

41

bonds. H-H argue that it is more reasonable to specify a default barrier that is some fraction β (less than or equal to one) of debt principal.5 We shall use the L-S model, with arbitrary β, as representative of exogenous default models. The key aspect of exogenous models is that the default barrier depends only on the level of debt principal. An endogenous default boundary was introduced by Black and Cox (1976), and subsequently extended by Leland (1994), Leland and Toft (1996), and Acharya and Carpenter (2002). These models assume that the decision to default is made by managers who act to maximize the value of equity. At each moment, equity holders face the question: Is it worth meeting promised debt service payments? The answer depends on whether the value of keeping equity “alive” is worth the expense of meeting the current debt service payment.6 If the asset value exceeds the default boundary, the ﬁrm will continue to meet debt service payments—even if asset value is less than debt principal value, and even if cash ﬂow is insufﬁcient for debt service (requiring additional equity contributions). If the asset value lies below the default boundary, it is not worth “throwing good money after bad”: The ﬁrm does not meet the required debt service, and defaults.7,8 In the endogenous default model, the default boundary is that which maximizes equity value. It is determined not only by debt principal, but also by the riskiness of the ﬁrm’s activities (as reﬂected in value process), the maturity of debt issued, payout levels, default costs, and corporate tax rates. Black and Cox (1976) and Leland (1994) derive the optimal endogenous default value for the case of perpetual (inﬁnite maturity) debt. Leland and Toft (1996) (“L-T”) extend the analysis to consider ﬁrms issuing debt of arbitrary maturity. Since they show that debt maturity affects the default boundary and, therefore, default probabilities, their approach is required to calibrate the endogenous model to empirical default data. We use the L-T model in subsequent analysis. To capture the idea of a long-term or “permanent” capital structure, L-T presume that debt is continuously rolled over.9 Total outstanding principal, debt service payments, and average debt maturity remain constant through time, even though each individual bond has a ﬁnite maturity.10 This stationary capital structure requires constant debt service payments, and the value-maximizing default boundary is also constant through time. The structure of the chapter is as follows. Section 2 discusses some recent related empirical work. Section 3 considers the determinants of default risk in structural models, and how they relate to more traditional approaches to default risk. Section 4 examines how the default boundary is determined in the exogenous and endogenous default models. Section 5 shows how cumulative default probabilities (DPs) at different horizons are related to default

42

THE CREDIT MARKET HANDBOOK

boundaries and asset value dynamics. Section 6 introduces “base case” parameters for subsequent prediction of default. Sections 7 and 8 examine how well the L-T and L-S models match the default frequencies observed by Moody’s (2001) over the period 1970–2000 for A-rated, Baa-rated, and B-rated debt. Section 8 also shows how changes in default probabilities (and potential bond ratings) predicted by the two approaches differ, given changes in a ﬁrm’s leverage and debt maturity. This section provides testable hypotheses that distinguish the models. Section 9 provides a brief description of the KMV approach. Section 10 offers some thoughts on similarities and differences between KMV, L-S, and L-T models. Section 11 concludes.

2. RECENT EMPIRICAL STUDIES Three recent papers examine related questions. Huang and Huang (2002) (H-H) use several structural models, including L-S and L-T, to predict yield spreads. They calibrate inputs, including asset volatility, for each model so that certain target variables are matched. The targets include leverage, equity premium, recovery rate, and cumulative default probability at a single time horizon. That horizon is 10 years in their base case, although they separately consider a 4-year horizon. Thus, inputs such as asset volatility will differ across the models H-H consider, although they are trying to match observed default frequencies over a common time period. H-H show that, given their calibration, the models make quite similar predictions on yield spreads.11 Noting that empirically observed yield spreads relative to Treasury bonds are considerably greater than these predicted spreads (particularly for highly rated debt), H-H conclude that additional factors such as illiquidity and taxes must be important in explaining market yield spreads. In contrast with H-H, we focus on predicting default probabilities rather than yield spreads. Rather than choosing different inputs to allow the different models to match observed default frequencies, we instead choose common inputs (including volatility) across models, to observe how well they match observed default statistics. In a recent paper using Merton’s (1974) approach, Cooper and Davydenko (2004) (“C-D”) have recently proposed a procedure somewhat the inverse of H-H. Rather than calibrate on past default probabilities to predict (the default portion of) yield spreads, C-D propose to predict expected default losses on any corporate bond based on its current yield spread, given information on leverage, equity volatility, and equity risk premia. To derive realistic numbers, C-D conclude that the correct measure of a bond’s yield spread for calibrating asset volatility is not the yield spread between that

Predictions of Default Probabilities in Structural Models of Debt

43

bond and a Treasury bond of similar maturity, but rather the spread between that bond and an otherwise-similar AAA-rated bond. In contrast with C-D, we focus on structural models’ ability to predict the probability of default (not the expected losses from default).12 But the C-D approach raises several interesting questions for future research, including whether the recovery rate can potentially be untangled from the probability of default, whether the C-D approach can be extended beyond the Merton model to consider coupon-paying bonds, and how well the structural models that match historical default probabilities can predict the yield spread relative to AAA-rated bonds. Eom et al. (2004) examine yield spread predictions across several structural models. While of considerable interest for bond pricing, the work does not directly address default probabilities.13

3. STRUCTURAL MODELS AND DEFAULT RISK Prediction of corporate bond default rates has been an objective of ﬁnancial analysis for decades. Moody’s, and Standard & Poor’s have provided bond ratings for almost a century. Key variables in the rating process include: ■ ■ ■ ■ ■ ■

Asset–liability ratios. Coverage ratios (cash ﬂows or EBIT relative to debt service payments). Business prospects (growth of cash ﬂows or return on assets). Dividends and other payouts. Business risks (volatility of cash ﬂows or value of assets). Asset liquidity and recovery ratios in default.

Exactly how these and perhaps other variables are balanced with one another to determine a rating remains the proprietary information of the rating agencies. Structural models are an attempt to create a rigorous formula to predict cumulative default probabilities at different horizons. Their inputs are similar in nature to those used by ratings agencies: 1. Asset value (value of the ﬁrm without leverage) at time t, V(t). Asset value typically evolves as a “geometric random walk”: dV(t) = (µ − δ)dt + σ dZ(t), V(t)

V(0) = V0

(2.1)

44

THE CREDIT MARKET HANDBOOK

where µ is the expected (objective) total rate of return on assets V, equal to r + λ, the risk-free rate plus the equity premium; λ the equity risk premium, δ the fractional payout rate on assets (to both debt and equity), σ the volatility of the rate of return on total assets, and {Z} a Wiener process. V0 is the initial asset value at t = 0, normalized to V0 = 100. All parameters are assumed constant through time. This modeling was introduced by Black and Scholes (1973) and Merton (1974), and has been used in many subsequent studies.14 Note that the total expected rate of return on assets µ comprises an “expected growth of asset value” rate µ − δ, plus a “payout” rate δ on asset value. 2. Debt is speciﬁed by three parameters: P the debt’s principal value, C the debt’s annual coupon ﬂow, and T the debt’s maturity.15 The coupon C is determined such that at time 0, the market value of debt D = P. Note that initial leverage can be represented by P/V(0). While it is possible to extend both the L-S and L-T models to consider multiple classes of debt, we consider here the case where the ﬁrm issues a single class of debt. Note that annual debt service payments are C +P/T. 3. Default boundary, VB . The default boundary is an asset value VB < V0 at which, when ﬁrst reached, the ﬁrm ceases to meet its debt service payments.16 Different structural models make different assumptions about how VB is determined. We discuss these models in Section 4 below. 4. Default costs, α. α is the fraction of asset value VB lost when default occurs. The fraction α lost in default is assumed to be a constant across default boundary levels.17 Costs of default not only include the direct costs of restructuring and/or bankruptcy, but also the loss of value resulting from employees leaving, customers directing business to other ﬁrms, trade credit being curtailed, and the possible loss of growth options. These “indirect” costs are likely to be several times greater than the direct costs. Note that the recovery rate, the fraction of original principal value received by bondholders in the event of default, is given by R=

(1 − α)VB P

(2.2)

The parameter α will be chosen later such that the predicted recovery rate (equation 2.2) will match empirical estimates. Loss given default (LGD) is the proportion 1 − R. 5. Default-free interest rate, r. We assume r, the continuously-compounded default-free interest rate, is a constant through time.18

Predictions of Default Probabilities in Structural Models of Debt

45

4. THE DEFAULT BOUNDARY IN EXOGENOUS AND ENDOGENOUS CASES 4.1 Exogenous Default Boundary: The L-S Model Recall from Section 1 that the exogenous model (L-S) postulates a default barrier VB-LS = β P

β≤1

(2.3)

Note that default boundary equation 2.3 depends only on debt principal P, and therefore is not affected by debt maturity T, coupon rate C, ﬁrm risk σ , payout rate δ, the riskless rate r, or default costs α. Also, note from equation 2.2 that equation 2.3 implies that the recovery rate R = (1 − α)β

(2.4)

and, therefore, the recovery rate in the L-S model, is invariant to leverage as well as debt maturity, coupon rate, ﬁrm risk, payout rate, default costs. In contrast, endogenous models typically have recovery rates that vary with these parameters, even though α is a constant.19

4.2 Endogenous Default Boundary: The L-T Model The L-T model assumes that the decision to default (and therefore the default boundary) is chosen optimally by a manager maximizing equity value. To derive a constant default barrier, L-T assume a “stationary” capital structure. Debt of a constant maturity T is continuously rolled over (unless default occurs). As debt matures, it is replaced by new debt with equal principal value and maturity. Total debt principal will be constant at a level P, with coupon ﬂow C. Although the ﬁrm has an aggregate debt structure that is presumed stationary, any particular bond can be valued as in L-S, but with the default barrier VB-LT used for valuation rather than the default barrier VB-LS . The optimal default boundary in L-T is given by the (complex) formula VB-LT =

(C/r)(A/rT) − B) − AP/(rT) − τ Cx/r 1 + αx − (1 − α)B

(2.5)

where the variables A and B are deﬁned in the Appendix. C is the coupon that results in the bond selling initially at par, and τ is the corporate income

46

THE CREDIT MARKET HANDBOOK

tax rate. Note that, like bond prices, VB-LT depends only on the risk-neutral drift r and not on the actual drift µ. The default boundary VB-LT decreases with maturity T, volatility σ , and the risk-free rate r, and increases with default costs α and increases more than proportionately with principal P.

5. THE DEFAULT PROBABILITY WITH CONSTANT DEFAULT BARRIER With a constant default barrier, the mathematics of European-style barrier options are applicable. The ﬁrst-passage time probability density function for reaching a barrier VB at time t from a starting value V0 > VB is given by f (t; b, µ, δ, σ ) =

σ

√

b 2t 3

× e−(b+(µ−δ−0.5σ

2 )t)/(2σ 2 t)

(2.6)

where b = ln(V0 /VB ). The ﬁrst-passage time cumulative probability function determines the cumulative DP at time t: √ DP(t; b, µ, δ, σ ) = N[(−b − (µ − δ − 0.5σ 2 )t)/σ t] + e−2b(µ−δ−0.5σ

2 )/σ 2

√ × N[(−b + (µ − δ − 0.5σ 2 )t)/σ t].

(2.7)

In the Merton (1974) model, default never occurs before the zero-coupon bond matures at T. At T, VB = P and the probability of default is √ N[(−b − (µ − δ − 0.5σ 2 )T)/σ T]20

(2.8)

It is obvious that equation 2.8 is always less than equation 2.7 at t = T. Since the default boundary in the Merton model is zero for all t until T, default is always less likely in the Merton model than in the constant default barrier model with VB = P.

6. CALIBRATION OF MODELS: THE BASE CASE We follow Huang and Huang (2002) in many of our parameter choices, focusing on debt with a Baa rating. Base case parameters are given in Table 2.1.

Predictions of Default Probabilities in Structural Models of Debt

47

Table 2.1 Parameter

Symbol

Default-free interest rate

r

Asset risk premium

λ

Expected asset return Asset volatility

µ σ

Payout rate Corporate tax rate

δ τ

Debt principal

P

Debt coupon

C

Maturity

T

Fractional default costs

α

Value Assumed and Rationale 8%, as in H-H, about the average 10-year Treasury rate, 1970–2000. 4%, the asset risk premium. Consistent with an equity premium of about 6% when the average ﬁrm has about 35% leverage. 12%, the sum of r + λ. 23%. Individual stocks average about 35% volatility with a correlation of 0.20, consistent with a diversiﬁed equity portfolio risk of about 20% (the long-term S&P 500 index volatility). If the average S&P 500 ﬁrm has leverage of about 35%, average asset volatility will be about 23% per annum.21 6%, as assumed by H-H.22 15%, representing the corporate tax rate offset by the personal tax rate advantage of equity returns.23 43.3 in the base case, implying a leverage ratio on asset value of 43.3%, the average leverage ratio of Baa-rated ﬁrms as reported in Standard & Poor’s (1999). We also consider leverage ratios of 32.2% (for single-A-rated ﬁrms) and 65.7% (for single-B-rated ﬁrms). Coupons that allow the bonds to sell at par P. These are determined numerically. Yield spread is given by C/P − r. 10 years in the base case. We also consider bond maturities of 5 and 20 years.24 30%, which will imply a 51% recovery rate in the base case, the recovery rate assumed by H-H.25

7. MATCHING EMPIRICAL DEFAULT FREQUENCIES WITH THE L-T MODEL For the base case in the previous section, with 43.3% leverage, the predicted bankruptcy barrier VB-LT is 31.7 (recall V0 = 100). The recovery rate is 51.2%, which matches H-H’s recovery rate. The solid line in Figure 2.1 plots the default probability (DP) as a function of time horizon, as predicted by the L-T model for the base case with Baa-rated bonds. The dotted line plots the average cumulative default frequencies for Baa-rated debt as a function of horizon, as given by Moody’s (1998, Exhibit 28) for the period 1970–1997. Recall that Baa-rated debt is consistent with an average leverage of 43.3% and a 10th-year maturity for newly issued debt (implying a 5th-year average maturity for total ﬁrm debt in the L-T model). The DPs predicted by the L-T model seem quite reasonable for Baa-rated debt, at least for horizons beyond 5 years. But default probabilities at shorter time horizons are systematically and substantially underestimated.26 When we consider riskier (single-B-rated) debt, with the ﬁrm having leverage of 65.7%, the L-T model does not look good in the base case, as can be seen in Figure 2.2. The actual default probabilities are considerably higher for all default horizons than predicted by the L-T approach. But there is no reason to

48

THE CREDIT MARKET HANDBOOK

DP : 1970–2000

0.1 0.08 0.06 0.04 0.02 0 0

5

10

15

20

Default horizon in years

FIGURE 2.1 Using the L-T model for Baa-rated debt (43.3% leverage and 10-year maturity), predicted (solid line) and actual (dotted line) default probabilities plotted as functions of time horizon.

DP : 1970–2000

0.5 0.4 0.3 0.2 0.1 0 0

5

10 15 Default horizon in years

20

FIGURE 2.2 Using the L-T model for single-B-rated debt (65.7% leverage and 10-year maturity), predicted (solid line) and actual (dotted line) default probabilities plotted as functions of time horizon.

think that the base case—which was predicated on the volatility of a typical ﬁrm—is relevant to ﬁrms issuing B-rated debt. While Stohs and Mauer (1996) show that B-rated debt has about the same average maturity as Baa-rated debt, there is reason to believe that the ﬁrms with B-rated debt have higher asset volatility. Unfortunately, publicly available data that link ﬁrm volatility to initial debt rating are unavailable. Using the L-T approach and base case parameters (other than asset risk), an average asset volatility can be estimated by ﬁnding the volatility that best

49

Predictions of Default Probabilities in Structural Models of Debt

DP : 1970–2000

0.5 0.4 0.3 0.2 0.1 0

0

5

10 15 Default horizon in years

20

FIGURE 2.3 Using the L-T model for B-rated debt (65.7% leverage and 10-year maturity; asset volatility of 32.0%), predicted (solid line) and actual (dotted line) default probabilities plotted as functions of time horizon.

approximates actual DPs. Figure 2.3 depicts the DPs predicted by the L-T model when asset volatility of ﬁrms with B-rated debt is σ = 32.0%.27 At the assumed level of volatility, the ﬁt is again quite good.28 Recovery is predicted to be slightly lower (50.6%) and the yield spread on a 10-year bond is estimated at 414 basis points (versus an historical average of 408 basis points for B-rated bonds in the Lehman Bond Index, over the period 1973–1993). Unfortunately, we have insufﬁcient data to determine whether an asset volatility of 32.0% is empirically reasonable for ﬁrms with bonds having a B rating. For A-rated debt, the ﬁt of predicted and actual DPs is illustrated in Figure 2.4 for an asset volatility of 22.3% (slightly lower than the base case). The predicted recovery rate of 51.6% is marginally higher than the base case. It is clear that the L-T approach is capable of reproducing the general shapes of default probabilities for bonds with different initial ratings. It matches default rates quite accurately over longer horizons, given the choice of asset volatility. However, it underestimates default probabilities for shorter horizons, and the problem is relatively more severe for investmentgrade debt. We return to this point in Section 11. The goodness of ﬁt is quite sensitive to the volatility of ﬁrm assets. If, for example, the volatility of ﬁrm assets in the base case were increased to 25%, the predicted default frequency over 20-years would rise from 11 to 16%. Lowering the default cost coefﬁcient from 30% to 15% lowers the 20 year default probability from 11% to 10% (and it would also increase the recovery rate from 51.2% to 59.4%).

50

THE CREDIT MARKET HANDBOOK

DP : 1970–2000

0.04 0.03 0.02 0.01 0 0

5

10 15 Default horizon in years

20

FIGURE 2.4 Using the L-T model for A-rated debt (32% leverage and 10-year maturity; asset volatility = 22.3%), predicted (solid line) and actual (dotted line) default probabilities plotted as functions of time horizon.

8. MATCHING EMPIRICAL DPS WITH THE L-S MODEL The L-S model does not offer guidance on selecting the variable β in the default barrier equation VB-LS = βP H-H suggest choices of 1.00 and 0.60 for β. For the base case, those coefﬁcients yield the default probabilities shown in Figures 2.5 and 2.6. Obviously, β equal to neither 1.00 nor 0.60 provides a good ﬁt to the base case parameters. An intermediate value seems best—but what value? A possible answer comes from the following observations. Observation 1 For any endogenously determined boundary VB-LT , there exists a β = β ∗ such that VB-LS = VB-LT Proof : Choose β∗ =

VB-LT P

(2.9)

51

Predictions of Default Probabilities in Structural Models of Debt

EDF for beta 1.00

0.2

0.15

0.1

0.05

0 0

5

10 15 Default horizon in years

20

FIGURE 2.5 Using the L-S model for Baa-rated debt (43.3% leverage and 10-year maturity; β = 1.00), predicted (dashed line) and actual (dotted line) empirical default frequencies plotted as functions of time horizon.

EDF for beta 0.60

0.1 0.08 0.06 0.04 0.02 0 0

5

10

15

20

Default horizon in years

FIGURE 2.6 Using the L-S model for Baa-rated debt (43.3% leverage and 10-year maturity; β = 0.60), predicted (dashed line) and actual (dotted line) empirical default frequencies plotted as functions of time horizon.

Recalling from equation 2.3 that VB-LS = βP, and substituting equation 2.9, gives the desired result. Thus, for any given set of parameter values in Section 4 (and the resulting VB-LT in the L-T model) we can always choose β = VB-LT /P for the L-S model, in which case the default boundary of the L-S model coincides with the endogenous default boundary of the L-T model.29

52

THE CREDIT MARKET HANDBOOK

Observation 2 At β = β ∗ , the recovery rate will be the same in both L-T and L-S models. Proof : From equations 2.2, 2.4, and 2.9, it follows that RL-T = (1 − α)VB-LT /P = (1 − α)β ∗ P/P = RL-S . Note that if R∗ is a target recovery rate that has been used to determine the default cost parameter α in the L-T model, it follows immediately from equation 2.4 that β ∗ = R∗ /(1 − α) for the L-S model. Thus, the choice of β for the L-S model can also be motivated as that which gives a recovery ratio equal to the target ratio R∗ , given α. Observation 3 For any given set of the parameters in Section 4, the cumulative default probabilities at all t will be identical for the L-T and L-S models, when β = β ∗ . Proof : From equations 2.6 and 2.7, when all parameters in Section 4, plus VB , are the same for both the L-T and L-S models. To match the VB-LT = 31.7 in the base case of Baa-rated debt, with P = 43 (implying 43% leverage), it is necessary from equation 2.9 to choose β = 31.7/43 = 0.731 By construction of this β, both models have the same default boundary and recovery ratios in the base case for Baa-rated bonds. Thus, the DP graph for the L-S model is identical to that of the L-T model in Figure 2.1. We use β = 0.731 in comparing the default probability predictions of the exogenous (L-S) and endogenous (L-T) default models for B-rated and A-rated bonds. While the L-S model can always replicate the DP curves produced by the L-T model, the opposite is not true. The L-S model potentially has more ﬂexibility to ﬁt the data than the L-T model. For example, any combination of (1 − α)β for the L-S model generates a 51.2% recovery rate and thus will ﬁt the recovery data equally well.30 The combination (α = 14.7%, β = 0.60) generates the same recovery rate, using a default cost number that some studies may suggest is more reasonable than α = 0.30. But such a combination yields very poor predictions of default probabilities: See Figure 2.6, which examines exactly this case. Indeed, the best ﬁt of default probabilities for the L-S model (given the parameters excluding α) is when α = 0.30 and β = 0.731. With σ = 23%, the L-S model (like the L-T model) underestimates the DPs for B-rated debt, where leverage is 65.7%. Raising asset volatility to 32% produces an almost identical ﬁt when β = 0.731. The L-S default curve is very slightly higher than the L-T default curve in Figure 2.3, but it is not reproduced separately since the curves are practically indistinguishable. The DP curves are similar because the endogenous recovery ratio in the

53

Predictions of Default Probabilities in Structural Models of Debt

L-T model remains virtually unchanged from the base case (50.7% versus 51.2%), despite the increase in volatility and greater leverage of B-rated ﬁrms. The equilibrating VB-LS would require β = R/(1−α) = 0.507/0.70 = 0.724, which is very close to 0.731. It is impossible to claim one of these is superior to the other, since the default probability curves are virtually identical, and the empirical cumulative default frequencies have considerable dispersion about their average.31 A similar situation arises for A-rated debt, where leverage averages 32%. At an asset volatility of 22.3%, fractionally lower than the base case, both the L-S model with β = 0.731 and the L-T model again predict almost the same default probabilities. Matching the L-S model’s endogenously determined recovery rate of 51.6% for A-rated debt would require β = R/(1 − α) = 0.516/0.70 = 0.737. Retaining β = 0.731 results in insigniﬁcant differences in default probabilities between the exogenous and endogenous models. Other differences in the models’ predictions, however, are potentially testable. For example, the endogenous (L-T) default boundary falls with asset volatility or with debt maturity, ceteris paribus, and thus the model predicts that recovery ratios decline with higher ﬁrm risk or with greater debt maturity. The exogenous (L-S) model predicts no change in the recovery rate or default frequency as these variables change. We consider the case where a ﬁrm (with Baa-rated debt in the base case) wishes to issue 20-year debt rather than 10-year debt. Figure 2.7 shows the cumulative default probabilities predicted by the two models. The L-S model predicts no change in the default probability curve (the higher dashed line),

DP : 1970–2000

0.1 0.08 0.06 0.04 0.02 0 0

5

10 15 Default horizon in years

20

FIGURE 2.7 For Baa-rated debt (43.3% leverage and 20-year maturity), predicted (L-S model: dashed line; L-T model: solid line) and actual (dotted line) default probabilities plotted as functions of time horizon.

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THE CREDIT MARKET HANDBOOK

since the exogenous default boundary is invariant to maturity.32 This is the upper curve in Figure 2.7. The lower curve is DP predicted by L-T for the ﬁrm issuing 20-year debt. It is lower than the DP since VB-LT is lower than with 10-year debt.33

9. THE MOODY’S–KMV APPROACH As outlined in Crosbie and Bohn (2002) and Crouhy et al. (2000), the KMV (now Moody’s–KMV) approach consists of four steps: 1. Estimate asset value and volatility. Current asset values V0 and estimated future asset volatility σ are “backed out” from observed equity value, volatility, and leverage, using a proprietary variant of the Black and Scholes/Merton option pricing model. 2. Calculate a “Default Boundary”VB-KMV . KMV calculate the boundary as VB-KMV = PST +

1 PLT 2

where PST is the principal of short-term liabilities and PLT the principal of long-term liabilities. It appears that KMV considers a liability to be “short term” if it is due within the horizon t over which the default probability is computed. Longer horizons thus have a higher fraction of “short-term” liabilities, with a consequent higher default barrier.34 A special case of interest is a “rolled over” capital structure similar to L-T, with constant total debt P and equal amounts P/T of each maturity debt, 1, . . . , T. If the default horizon is t years, then 1 Min(t, T) Min(t, T) P+ 1− P T 2 T 1 1 Min(t, T) + = P 2 2 T

VB-KMV (t, T) =

(2.10)

The KMV default boundary depends on the debt maturity T. But unlike either L-S or L-T, the KMV boundary also depends on time t. Therefore equation 2.7 does not apply.

Predictions of Default Probabilities in Structural Models of Debt

55

3. Calculate the distance to default (DD). DD(t) is measured by the number of standard deviations between the (log of the) ratio of the expected future asset level V(t) (given V0 ) and the KMV default boundary at t, using the current asset level V0 and asset return volatility σ estimated in step (i). This is given by DD(t) =

ln(V0 /(VB-KMV (t))) − (µ − δ − 0.5σ 2 )t √ σ t

(2.11)

where we have suppressed the argument T in VB-KMV (t, T). 4. Map DD (t) into DP (t). KMV has an extensive proprietary database relating ﬁrms’ empirical cumulative default frequencies t periods in the future to estimated DD(t) and t. It thus creates a mapping f such that the default probability (termed EDF, expected default frequency) is EDF(t) = DP(t) = f (DD(t), t)

(2.12)

This DP(t) can then be compared with the actual average default frequencies of bonds with different ratings, to derive an “implicit” bond rating. In their publicly available papers, KMV show that if asset values followed the diffusion process equation 2.1, then their DP at horizon t will be given by DP(t) = N[−DD(t)]

(2.13)

where N(·) is the standard normal cumulative distribution function.35 Note that equation 2.13 is simply the DP for a zero-coupon bond with maturity t and principal VB-KMV (t, T) that is given in equation 2.10. This links KMV’s approach with Merton’s (1974).

10. SOME PRELIMINARY THOUGHTS ON THE RELATIONSHIP BETWEEN THE KMV APPROACH AND L-S/L-T Given the VB-KMV (t, T) function as in equation 2.10, and that asset values follow the diffusion process in equation 2.1, we can compute KMV’s DP(t) from equation 2.12.

56 Merton KMV default probability

THE CREDIT MARKET HANDBOOK

0.1 0.08 0.06 0.04 0.02 0 0

5

10 15 Time to default in years

20

FIGURE 2.8 Default probability plotted as a function of horizon t for the base case.

Figure 2.8 plots the DP as a function of horizon t for the base case.36 For the ﬁrst 12 years, the plot is not too dissimilar to either L-T or L-S with β = 0.731. When the horizon t reaches the term of debt maturity T = 10, the KMV default boundary no longer increases with t. This explains the kink at t = 10 and the ﬂattening of the KMV DP (or EDF).37 As with L-S, (but not L-T), the KMV DP in Figure 2.8 will not shift upward as default costs α change, nor will recovery rates R be affected by debt maturity or ﬁrm risk. As with L-T (but not L-S), the KMV DP shifts downward with increased debt maturity. In common with both L-S and L-T, DP increases with leverage and volatility.38 Of course, the sensitivities of DPs to changes in parameters are different across the different models. At a more conceptual level, having the distance-to-default DD(t) as the sole argument in KMV’s DP (whether through equation 2.13 or a more general relationship in equation 2.12) is a strong restriction. Observe that equation 2.13 gives the probability of being beneath VB-KMV (t) at the horizon t only, and not the probability of falling beneath the barrier VB-KMV (τ ) at any time τ within the horizon [0, t]. This is in contrast with L-S and L-T, where the argument for DP is given by equation 2.7. That equation has DD as the ﬁrst term in its argument, but has a second term as well. This implies that DD is not a sufﬁcient statistic for DP in equation 2.7, and that ordinal rankings of the two criteria can differ. For example, at longer horizons, the DD(t) as computed in equation 2.13 can decline with t; that is, the estimated default frequency can actually decrease with a longer horizon, even though this should not be possible for a cumulative default probability function. For the base case, the KMV DP is decreasing with the time horizon when t exceeds 25 years.39

Predictions of Default Probabilities in Structural Models of Debt

57

11. CONCLUSIONS We have compared structural models’ abilities to predict observed default rates on corporate bonds. The endogenous L-T model derives a constant default boundary that optimizes equity value. The exogenous L-S model assumes a constant default boundary through time, at a level equal to an exogenously speciﬁed fraction β of debt principal value. β = 1 (a positive net worth condition) is a logical choice for this parameter, but gives poor predictions of default probabilities and recovery rates. Thus, we consider β a free parameter for the exogenous model. When β can be chosen freely, the exogenous model can always replicate the endogenous model’s default predictions, but not vice versa. However, we found that the β which equates the models’ defaults predictions is in fact the best choice to explain observed default rates for the period 1970–2000. Both models predict the general shape and level of default probabilities for A-, Baa-, and B-rated debt quite well for horizons exceeding 5 years. Observed longer-term default frequencies across the various ratings can be predicted in both models by changing asset volatilities only—all other parameters remained at their base case levels when leverage is adjusted to reﬂect the difference across credit ratings. The models do make different predictions about default probabilities when maturity, asset volatility, or default costs are changed, ceteris paribus. But additional data will be required to determine whether the models perform well in these dimensions. Both the endogenous and exogenous models examined here have underpredicted default probabilities at shorter time horizons. A similar problem exists when structural models are used to predict yield spreads: spreads approach zero as debt maturity becomes short. Some authors, including H-H and C-D, have suggested that the models’ underprediction of observed yield spreads (relative to equivalent Treasury bonds) reﬂects a liquidity spread in addition to the credit spread that the structural models (accurately) measure. But while liquidity differences might explain the observed yield spread underestimates, they cannot explain underestimates of predicted default frequencies. Including jumps in the asset value process (equation 2.1) may solve the underestimation of both default probabilities and yield spreads.40 This remains an important future research objective.

ACKNOWLEDGMENTS The author thanks Dirk Hackbarth for his comments and invaluable assistance with the calculations and ﬁgures. Ronald Anderson and Christopher Hennessy also provided insights and comments. An earlier version of this

58

THE CREDIT MARKET HANDBOOK

chapter was presented at the GRETA conference, “Assessing the Risk of Corporate Default,” Venice, Italy, 2002.

POSTSCRIPT Since this chapter was completed, a paper by Schaefer and Strebulaev (1974) has estimated the average underlying asset volatility σ for bonds with different ratings. Their Table VIII estimates asset volatilities of 22% for A-rated bonds, 22% for Baa-rated bonds, and 31% for B-rated bonds, for the period December 1996–December 2001. These estimates are remarkably close to the asset volatilities of 22.3% for A-rated bonds, 23% for Baa-rated bonds, and 32% for B-rated bonds that we determined would provide the best default probability ﬁts for the period 1970–2000 in Figures 2.3, 2.1, and 2.4, respectively.

APPENDIX The coefﬁcients A and B in equation 2.5 are given by √ √ A = 2a e−rT N[aσ T] − 2zN[zσ T] −

√ 2 √ n[zσ T] σ T

√ 2e−rT √ n[aσ T] + (z − a) σ T √ √ 2 2 1 N[zσ T] − √ n[zσ T] + (z − a) + 2 B = − 2z + 2 zσ T zσ T σ T +

(r − δ − (σ 2 /2)) σ2 2 2 ((aσ ) + 2rσ 2 )1/2 z= , σ2

a=

x=a+z

where n(·) and N(·) denote the standard normal density and cumulative distribution functions, respectively.

NOTES 1. Default probability is similar to the “expected default frequency” (EDFTM ) used by Moody’s–KMV.

Predictions of Default Probabilities in Structural Models of Debt

59

2. Thus, we do not make comparisons with reduced-form models as developed (among others) by Jarrow and Turnbull (1995), Jarrow et al. (1997), and Dufﬁe and Singleton (1999). Uhrig-Homburg (2002) provides an excellent survey of both reduced-form and structural models. 3. A “default boundary” is a level of asset value, perhaps time dependent, such that the ﬁrm will default on its debt if its asset value falls beneath this level. For models whose state variable is cash ﬂow, the default boundary is a sufﬁciently low level of cash ﬂow that the ﬁrm defaults on its debt. 4. Other models with exogenous default include Geske (1997), Brennan and Schwartz (1980), Kim et al. (1993), Nielsen et al. (1993), Briys and de Varenne (1997), and Collin-Dufresne and Goldstein (2001). 5. For example, the Moody’s–KMV approach (which differs in some other details) postulates an exogenous default barrier that is equal to the sum of shortterm debt principal plus one-half of long-term debt principal (Crosbie and Bohn, 2002). See Section 9 in this chapter. 6. A fuller description of endogenous default is provided in Leland and Toft [1996, Section IIB]. 7. By assumption, the ﬁrm is not allowed to liquidate operating assets to meet debt service payments in most structural models. Bond indentures often include covenants restricting asset sales for this purpose. 8. We presume that default occurs whenever the debt service offered is less than the amount promised. Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1996) introduce “strategic debt service,” where bargaining between equity and debtholders can lead to less debt service than promised, but without formal default. We do not pursue this approach here, in part because these papers assume debt with inﬁnite maturity, whereas we focus on debt of differing maturities. 9. To be more speciﬁc, debt is continuously offered at a maturity T, and continuously retired at par value at maturity. In the stationary case, the total principal of outstanding debt P is constant, as are the coupons C paid and the average maturity (that equals T/2). 10. As with the L-S approach, any bond issued by the ﬁrm can be readily valued in such an environment as long as the overall capital structure is constant through time (implying the default boundary is constant through time). It is assumed that all debt issues default when the boundary is reached, consistent with the cross-default provisions that are typically included in bond covenants. 11. This is not entirely surprising. If the probability of default is similar by construction, the credit spread (involving a transformation from the actual probability process to the risk-neutral process) is likely to be similar as well. 12. We focus on default probability rather than expected losses not because we necessarily believe it is a more important statistic, but rather because (1) statistics are readily available on default frequencies for different bond rating classes, and (2) it is what Standard & Poor’s claim their ratings predict (at least on a relative basis). While Moody’s empirical studies focus on their ratings’ ability to predict default frequencies, they claim that their ratings seek to explain both default frequency and loss given default.

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13. Some results in the Eom et al. paper are puzzling. The average leverage, volatility, payout, and other parameters in their data set give rise to an underestimate of average yield spreads by the L-T model. This is consistent with Huang and Huang’s (2003) results. Yet Eom et al. conclude that the L-T model substantially overestimates spreads. 14. Zhou (2001) and Delianedis and Geske (2001) consider an asset value process with both a diffusion and a jump component. Keeping volatility constant, jumps tend to increase the default probabilities over short periods, while default probabilities over longer periods are less affected. Dufﬁe and Lando (2001) consider the case where asset value is observed only with noise, giving results similar to models with jumps. Goldstein et al. (2001) postulate current cash ﬂow or “EBIT,” rather than asset value, to be the state variable. However, because asset value is typically a ﬁxed multiple of cash ﬂow in these alternative models, the resulting analytics are virtually identical. Following Merton (1974) and others, we focus on models where asset value is the stated variable. 15. We do not consider callable or convertible debt, although that can be considered within the context of dynamic models (e.g., Fischer et al., 1989; Leland, 1998; Goldstein et al., 2001; Collin-Dufresne and Goldstein, 2001). 16. As noted previously, we ignore the possibility of “strategic debt service” as in Anderson and Sundaresan (1996) and Mella-Barral and Perraudin (1996). 17. Seniority is not a determinant of α here, since there is only one class of debt. 18. Kim et al. (1993), Nielsen et al. (1993), Briys and de Varenne (1997), and Longstaff and Schwartz (1995) allow for stochastic default-free interest rates. A negative correlation between asset values and interest rates, as seems to accord with empirical evidence, reduces the estimated yield spreads, although the effect is quite small. Acharya and Carpenter (2002) introduce stochastic interest rates in an endogenous bankruptcy model. 19. When VB is determined endogenously, as in Leland and Toft (1996), the ratio R = (1 − α)VB /P tends to fall with P, implying R falls with leverage. There is some evidence suggesting a negative correlation between default rates (positively related to leverage) and recovery ratios; see Altman et al. (2002) and Bakshi et al. (2001). However, this may also reﬂect seniority differences. 20. Note that the argument of equation 2.8 is the same as (negative) “d2” in the Black–Scholes option pricing formula with proportional dividends δ, but with the actual drift µ replacing the risk-neutral drift r. 21. A recent study by KMV (2002) provides an example where the asset volatility of a low-risk ﬁrm (Anheuser–Busch) was estimated to be 21%, and the asset volatility of a very high-risk ﬁrm (Compaq) was estimated at 39%. 22. More highly leveraged ﬁrms might be expected have higher payout rates, as interest expense will be greater (although dividends will be less). We do not attempt to estimate this, however. 23. If the corporate tax rate is 35%, the personal tax on bond income is 40%, and the tax rate on stock returns is 20%, then the effective tax advantage of debt is 1 − (1 − 0.35)(1 − 0.20)/(1 − 0.40) = 0.133, or a bit less than 15%. 24. In L-T, T is the maturity of the bonds that are continuously being offered. Because of the stationary capital structure, bond maturities are uniformly distributed between 0 and T. Thus, average maturity of the debt structure will be

Predictions of Default Probabilities in Structural Models of Debt

25.

26.

27. 28.

29.

30. 31.

32.

33.

34.

61

T/2. Stohs and Mauer (1996) ﬁnd an average debt maturity of 4.92 years for B-rated ﬁrms (consistent with T = 9.84 years), 4.60 years for Baa-rated ﬁrms (T = 9.20), and 4.52 years for single-A-rated ﬁrms (T = 9.04). Costs of default include not only the direct costs of restructuring and/or bankruptcy, but also the loss of value resulting from workers resigning, customers directing business to other potential sellers, trade credit being curtailed, and possible loss of growth options. These “indirect” costs are likely to be several times greater than the direct costs. While Andrade and Kaplan (1998) ﬁnd default costs to be 10–20% for ﬁrms that had undergone highly leveraged buyouts previously, high leverage will typically be desirable for ﬁrms with relatively low default costs. Thus, their sample is likely to have downwardly biased default costs relative to the average ﬁrm. Short-term yield spreads also tend to be underestimated by structural models. There is some possibility that the data are the problem at short horizons, not the model. Default probabilities (and yield spreads) are highly convex in leverage. Thus, the average short-term default rate on a 48%-leveraged ﬁrm and a 38%leveraged ﬁrm will be higher than the default rate of a 43%-leveraged ﬁrm rated Baa. The data reﬂect average default rates across leverages in the Baa category; but the model estimates default rates of the average leverage. However, even with 48% leverage, the predicted default rate at very short maturities seems too low. Jump models may be required to explain higher short-term default rates; see Section 11. Best ﬁt here is the volatility (to the nearest 0.1%) that best matches (informally) the empirical PD. Note that the graph of actual defaults is ﬂat between 17 and 20 years, indicating no defaults over this interval. This seems a curiosity of the time period examined, and reﬂects the fact that default frequencies can vary considerably through time. The opposite does not hold: For a given set of parameter values in Section 4, VB-LT is uniquely determined. Thus, except for the β satisfying equation 2.9, the models will not have equal default boundaries nor generate equal DP. But recall that any β other than equation 2.9 will not generate a default boundary that maximizes the value of equity. Recall α = 0.30 was required in the L-T model to generate the observed recovery rate of 51.2%. Recall the average cumulative default frequency (over 1970–2000) after t years is the average of the cumulative default frequencies of different cohorts of bonds of a given rating, t years after issuance. The default histories of different cohorts have considerable variability; see Moody’s (2001). If it is argued that β should change with bond maturity, it is necessary to specify how. The L-S model does not provide an answer, although the L-T model implies one. The L-T model estimates that the credit spread on the longer-term debt actually rises slightly, by eight basis points, despite the fall in the default boundary and default probabilities. This reﬂects the fact that the recovery rate has fallen. Recall that both the L-S and L-T models have a constant default barrier, although dynamic variants may have VB upwardly-ratcheted (on average) with time. See, for example, Leland (1998) and Goldstein et al. (2001).

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35. We stress that KMV do not necessarily believe that asset values follow the diffusion process in equation 2.1, although they use it for illustrative purposes. We, nonetheless, use it to compare the differences in default probability estimates that the various approaches would make under the same set of assumptions about asset value movements. 36. Recall that a maturity T = 10 implies an average debt maturity of T/2 = 5 years when the capital structure is stationary. 37. KMV forecasts default probabilities only to a horizon of 5 years, so what happens much beyond this may be of academic interest only. 38. The assertions in this paragraph are not dependent on the speciﬁc asset volatility form equation 2.1 chosen, but simply reﬂect properties of DD in equation 2.10 and VB-KMV (t, T) in equation 2.9. 39. Because KMV limits forecasts of EDFs to 5 years, this is a theoretical but not a practical difﬁculty. Furthermore, the fact that f in Eq. (2.11) can depend separately on t can offset the decline in DD(t). 40. Zhou (2001) shows that asset value jumps can explain why yield spreads remain strictly positive even as maturity approaches zero.

REFERENCES Acharya, V. and Carpenter, J. (2002). “Corporate Bond Valuation and Hedging with Stochastic Interest Rates and Endogenous Bankruptcy.” Review of Financial Studies, 15, 1355–1383. Altman, E., Brady, B., Resti, A. and Sironi, A. (2002). “The Link between Default and Recovery Rates: Implications for Credit Risk Models and Procyclicality.” Working Paper, Stern NYU. Anderson, R. and Sundaresan, S. (1996). “Design and Valuation of Debt Contracts.” Review of Financial Studies, 9, 37–68. Anderson, R. and Sundaresan, S. (2000). “A Comparative Study of Structural Models of Corporate Bond Yields: An Exploratory Investigation.” Journal of Banking and Finance, 24, 255–269. Andrade, G. and Kaplan, S. (1998). “How Costly is Financial (not economic) Distress? Evidence from Highly Levered Transactions that became Distressed.” Journal of Finance, 53, 1443–1493. Bakshi, G., Madan, D. and Zhang, F. (2001). “Recovery in Default Modeling: Theoretical Foundations and Empirical Applications.” Working Paper, University of Maryland. Black, F. and Cox, J. (1976). “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” Journal of Finance, 31, 351–367. Black, F. and Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81, 637–659. Brys, E. and de Varenne, F. (1997). “Valuing Risky Fixed Rate Debt: An Extension.” Journal of Financial and Quantitative Analysis, 32, 239–248.

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63

Collin-Dufresne, P. and Goldstein, R. (2001). “Do Credit Spreads Reﬂect Stationary Leverage Ratios?” Journal of Finance, 56, 129–157. Crosbie, P. and Bohn, J. (2002). “Modeling Default Risk.” KMV. Delianedis, G. and Geske, R. (2001). “The Components of Corporate Credit Spreads: Default, Recovery, Tax, Jumps, Liquidity, and Market Factors.” Working Paper 22-01, Andersen School, University of California, Los Angeles. Dufﬁe, G. (1999). “Estimating the Price of Default Risk.” Review of Financial Studies, 12, 197–226. Dufﬁe, D. and Singleton, J. (1999). “Modeling the Term Structures of Defaultable Bonds.” Review of Financial Studies, 12, 687–720. Dufﬁe, D. and Lando, D. (2001). “Term Structures of Credit Spreads with Incomplete Accounting Information.” Econometrica, 69, 633–664. Fischer, E., Heinkel, R. and Zechner, J. (1989). “Optimal Dynamic Capital Structure Choice: Theory and Tests.” Journal of Finance, 44, 19–40. Goldstein, R., Ju, N. and Leland, H. (2001). “An EBIT-based Model of Optimal Capital Structure.” Journal of Business, 47, 483–512. Huang, J.-Z. and Huang, M. (2002) (revised May 2003), “How Much of the Corporate-Treasury Yield Spread is Due to Credit Risk? Results from a New Calibration Approach.” Working Paper, GSB, Stanford University. Jarrow, R. and Turnbull, S. (1995). “Pricing Derivatives on Financial Securities Subject to Credit Risk.” Journal of Finance, 50, 53–86. Jarrow, R., Lando, D. and Turnbull, S. (1997). “A Markov Model for the Term Structure of Credit Risk Spreads.” Review of Financial Studies, 10, 481–523. Leland, H. (1994). “Corporate Debt Value, Bond Covenants and Optimal Capital Structure.” Journal of Finance, 49, 1213–252. Leland, H. and Toft, K. (1996). “Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads.” Journal of Finance, 51, 987–1019. Leland, H. (1998). “Agency Costs, Risk Management and Capital Structure.” Journal of Finance, 53, 1213–1243. Longstaff, F. and Schwartz, E. (1995). “Valuing Risky Debt: A New Approach.” Journal of Finance, 50, 789–820. Mella-Barral, P. and Perraudin, W. (1997). “Strategic Debt Service.” Journal of Finance, 52, 531–566. Merton, R. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance, 29, 449–470. Moody’s Investors Services (2001). “Historical Default Rates of Corporate Bond Issuers, 1920–2000,” Special Comment. Moody’s Investors Services (1999). “The Evolving Meaning of Moody’s Bond Ratings.” Special Comment. Nielsen, L., Saà-Requejo, J. and Santa-Clara, P. (1993). “Default Risk and Interest Rate Risk: The Term Structure of Default Spreads.” Working Paper, INSEAD. Schaefer, S. and Strebulaev, I. (1974). “Structural Models of Credit Risk are Useful: Evidence From Hedge Ratios of Corporate Bonds” (London Business School working paper, May 2004).

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Stohs, M. and Mauer, D. (1996). “The Determinants of Corporate Debt Maturity Structure.” Journal of Business, 69, 279–312. Uhrig-Homburg, M. (2002). “Valuation of Defaultable Claims: A Survey.” Schmalenbach Business Review, 54, 24–57. Zhou, C. (2001). “The Term Structure of Credit Spreads with Jump Risk.” Journal of Banking & Finance, 25, 2015–204.

Keywords: Default frequencies; structural models; credit risk

CHAPTER

3

Survey of the Recent Literature: Recovery Risk Sanjiv R. Dasa

This chapter surveys selections of recent working papers on recovery rates, providing a framework for extant research. Simpler versions of models are also presented with a view to aid accessibility and pedagogical presentation. Despite the obvious empirical difﬁculties encountered with recovery rate data, modeling advances are making possible better quantiﬁcation and measurement of recovery and will result in innovative contracts to span this risk.

1. INTRODUCTION Default risk has two main components: the risk of default occurrence and the risk of recovery. The extant literature has focused widely on the former, and much too little attention has been given to recovery risk. This is changing, and many working papers on recovery risk have been written recently. This chapter reviews what we know about recoveries and highlights some of the new thinking in this area. In discrete time, we may write the one-period spread as determined by the probability of default, which we denote as λ, and the loss rate given default (LGD) L = 1 − φ, where φ is the recovery rate. We restrict ourselves to an exposition in discrete time. Hence, we consider intervals ending at times {t1 , t2 , . . . , tN }, where there are N periods and corresponding time points. More generally, we may think of the credit spread for a given maturity as being a function of a vector of forward probabilities of default and a Santa Clara University, Santa Clara, CA 95053, USA. E-mail: [email protected]

65

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recovery rates, as well as some ﬁrm-speciﬁc constants. We may write the spread (S) as St = f (λ, φ)

(3.1)

where λ = {λt1 , λt2 , . . . , λtn }. Likewise, we have recovery rates φ = {φt1 , φt2 , . . . , φtN }. It is natural to expect that the relationship between spreads and default probabilities is a positive one, and that between spreads and recovery rates is negative. Hence, ∂S > 0, ∂λ

∂S <0 ∂φ

(3.2)

This also suggests, but does not necessitate, a negative relationship between default rates and recovery rates, that is, Corr[λ, φ] < 0. In some cases, this negative relationship is derived endogenously in the model (as in variants of the Merton (1974) model. In other cases, it is imposed exogenously, as in the paper by Bakshi et al. (2001). Before proceeding to models and empirical reality, it is worth summarizing the statistics available on recovery rates. These are all available in the work of Altman and Fanjul (2004) and other sources, many provided by Altman. The broad statistics on recovery seem to remain unchanged (mostly) year after year with uncanny stability. Consider Table 3.1, drawn from Altman and Fanjul. As can be seen, the overall median recovery rate is 40% and this has been very much the value for many years. Seniority determines most of the cross-sectional variation in recoveries. Recovery rates are highly variable within each category, for the standard deviations are relatively large in comparison to the means. This suggests that accurate estimates of recovery may be difﬁcult to ﬁnd.

Table 3.1

Recovery Rates for Publicly Traded Corporate Bonds Median (%)

Mean (%)

Senior secured Senior unsecured Senior subordinated Subordinated Discount bonds

54 42 32 32 18

53 35 30 29 21

23 27 25 23 18

All

40

34

25

Seniority

Std. Dev.

These rates are drawn from the results presented in Altman and Fanjul (2004). Recovery rates are computed based on prices within 30 days of default, and are percentages of the value of the bond prior to default. The sample covers the period 1974–2003.

Survey of the Recent Literature: Recovery Risk

67

2. EMPIRICAL ATTRIBUTES Apart from the raw statistics delineated above, there are many empirical ﬁndings that have recently been uncovered in the growing literature in this area. These are as follows: 1. There is a strong negative relationship between default rates and recovery rates on default. (Note that these are realized default rates, and are different from probabilities of default (PDs), which are forward-looking metrics.) Altman et al. (ABRS) (2005) ﬁnd that this negative relationship is strong for various linear and nonlinear speciﬁcations. The R-squares are in the range of 65% and sometimes as high as 80%. The relationship is deﬁnitely nonlinear, as linear speciﬁcations result in much lower R2 s of about 50%. In the ABRS paper, recovery rates appear to be negatively convex in the default rate; that is, as default rates increase, recovery rates fall, but tend to fall less with increasing default. There is a tendency toward an asymptotic lower bound for recovery rates; it suggests that the distributions of recovery rates are likely to be positively skewed. The negative correlation of default and recovery is natural in models where both are driven by a common systematic factor. Estimates of the correlation vary widely. Hu and Perraudin (2002) ﬁnd a negative correlation (≈−0.2) in their sample of US ﬁrms, whereas Carey and Gordy (2003) ﬁnd only a very low and weak negative relationship in a 30-year sample of US defaults. Bakshi et al. (2001) report, in their study of lesser quality investment-grade debt, that a 4% increase in the probability of default results in a 1% decrease in recovery rates. Frye (2000) reports that in times of economic distress, recovery rates can drop by as much as 25%. 2. Recovery rates are very volatile. We can see this from the standard deviation of recovery rates in Table 3.1. A look at recovery rates over time, as presented in ABRS, shows that aggregate recovery rates went to a high of 62% in 1987, and reached a low of about 23% in 1990. This volatility, coupled with the negative correlation of default and recovery, implies that risk management measures such as credit valueat-risk (cVaR) that ignore these aspects of recovery empirics, must surely understate their risks. Simulation exercises in ABRS attempt to quantify the extent of this impact. 3. The supply of defaultable debt is also an important determinant of recovery rates. This thesis is developed and tested in the work of ABRS. They ﬁnd that the total amount of high-yield bonds outstanding is inversely related to recovery rates in a statistically signiﬁcant manner.

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4. Seniority matters. From Table 3.1 we can see that this is clearly the case. Acharya et al. (2003) ﬁnd this effect in a study of defaulted bonds over the last two decades, after controlling for many other causes. 5. Recovery rates depend on industry sector. This ﬁnding in Acharya et al. (2003) is robust to many controls. The authors imply that industry effects may be driving the inﬂuence of defaultable bond supply on recovery rates. An important implication of this work is that models of recovery rates should incorporate industry factors, and from the point of view of diversiﬁcation, it makes sense to choose bonds across sectors, rather than within sectors. Sector recovery risk may also be hedged using sector-speciﬁc basket default contracts. 6. Regime effects impact recovery rates. As discussed, Frye (2000) shows that recoveries in economic downturns are substantially (≈25%) lower than what we see in normal times. In a recent paper, Hu (2004) provides an analysis of recovery rate distributions by seniority, rating class, and regime, and ﬁnds very different outcomes by regime. She ﬁts regimedependent beta distributions for recovery and ﬁnds starkly different shapes of the distributions. The implications for Monte Carlo simulation of recoveries is that regime-shifting is important to include in the model, especially for long-dated portfolios. 7. The beta distribution is proving to be the ﬁtting model of choice. The probability density function for recovery rates is usually modeled as follows: f (φ; a, b) =

φ a−1 (1 − φ)b−1 β(a, b)

(3.3)

where a and b are the parameters of the distribution, and β(a, b) is . Hu (2004) the beta function. The beta function is β(a, b) = (a) (b)

(a+b) shows that recovery rates over the last two decades ﬁt this distribution well, and that there are two primary determinants of the ﬁt: (1) the economic regime in which the recovery rate resides, and (2) the seniority of defaulted debt. She ﬁts recovery rate data by setting the parameters of the beta distribution as functions of economic regimes and seniority. The differences across regime are striking, which implies that Monte Carlo simulation of recovery rates should also be based on regimes in order to capture scenarios in a comprehensive manner. Overall, there is a clear relationship of recovery rates to defaults, industry factors, seniority, and economic regimes. All this evidence is also consistent with a high element of systematic risk driving default and recovery.

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69

3. RECOVERY CONVENTIONS Recovery rates are usually expressed in percentage terms, although under different metrics. Computing recovery depends on both the speciﬁcation of the dollar amount recovered (the numerator) and the base against which comparison is made (the denominator). While there are many ways in which the recovered amount may be assessed, two approaches seem to be common. 1. Recovery is measured as the value of bonds within the month after default. An observed price is sufﬁcient to determine the value for this purpose. 2. Recovery is measured at the time of the ﬁnal resale of the bonds, or the value in the ﬁnal restructuring under a court agreement. This usually occurs long after the default date itself. Which of these conventions is used often depends on the purpose for which recovery rates are required. In derivative pricing models, such as those required for valuing default swaps, the former measure is the more appropriate one. However, in empirical analyses, in which the end-game value of a Chapter 7 or 11 outcome is being assessed, quite clearly we are interested in the latter speciﬁcation. The second part of the recovery rate speciﬁcation lies in expressing the recovered amount as a percentage of a baseline value. We encounter three popular conventions here: 1. Recovery of par (RP): Here the recovered amount is expressed as a fraction of the par value of the security. 2. Recovery of Treasury (RT): Recovery is a fraction of the value of a similar risk-free bond. 3. Recovery of market value (RMV), where the recovery amount is taken as a fraction of the price of the security if it had not defaulted. This is sometimes similar to expressing recovery as a percentage of the price of the bond just prior to default, provided default comes as a complete surprise. In a recent model, Bakshi et al. (2001) ﬁt models to BBB bond data and assessed which of these conventions resulted in a better empirical ﬁt. They found that the best approach was the Recovery of Treasury (RT) assumption. They also discussed how to take their ﬁtted model and move between the physical and risk-neutral measures for recovery rates. We will consider this later.

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4. RECOVERY IN STRUCTURAL MODELS The line of structural models beginning with Merton (1974) are based on modeling the underlying ﬁrm value, and then determining the values of riskneutral default and recovery as functions of ﬁrm value. Thus, both default and recovery are endogenous in this class of models. We undertake a simple exposition here. Let the initial value of the ﬁrm be V0 . At the maturity (T) of the debt in the ﬁrm, the value of the ﬁrm VT must be below the face value of debt D for the ﬁrm to be in default. It is easy to see that this directly provides the recovery rate, that is, Recovery rate = φ =

VT < 1, D

if default occurs

(3.4)

The expected recovery is simply VT < D V T D

(3.5)

1 E[VT |VT < D] D

(3.6)

E[φ] = E = Note that

E(VT ) = E[VT |VT ≥ D] + E[VT |VT < D] The previous two equations may be used to write the equation for recovery as follows: 1 {E[VT ] − E[VT |VT ≥ D]} D       rT 1 −rT V0 − e =e E[VT |VT ≥ D]

 D  

E[φ] =

(3.7)

(3.8)

Asset or nothing call

1 rT e {V0 − V0 N(d1 )} D V0 rT = e {1 − N(d1 )} D ln(V0 /D) + r + 12 σV2 T d1 = √ σ T =

(3.9) (3.10)

(3.11)

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Survey of the Recent Literature: Recovery Risk

This is a very simple expression for recovery rates in the Merton model. It is easily checked that this implies an endogenous negative correlation between default and recovery. This analytic equation may also be easily extended to models where a default barrier is used. There are structural models in which the recovery rate is exogenously imposed and is not derived within the model. This is the approach taken in the model developed by CreditMetrics (Gupton et al. 1997). In fairness, this idea was originally employed in a model by Nielsen et al. (1993). This model posits the form originally used in Black and Cox (1976), where default occurs not just at maturity, as in the Merton model, but also at times prior to maturity, if the value of the ﬁrm accesses a default boundary. The innovation adopted in the CreditMetrics model is to make the default barrier stochastic, which plays the important role of making spreads larger and more volatile.

5. RECOVERY IN REDUCED-FORM MODELS In reduced-form models, the recovery rate is not endogenously available as in structural models. It needs to be exogenously supplied. Hence, it is an additional input to the modeling process. As noted in the beginning of this chapter, spreads are a function of the probability of default and the loss rate (1 − φ). Hence, it is not possible to disentangle recovery rates from spreads without making an assumption about the probability of default, or vice versa. This has been pointed out in many articles (see, e.g., Dufﬁe and Singleton, 1999) and various suggestions have been made to work around this problem. A recent paper by Chan-Lau (2003) develops a way of extracting recovery rates from the default swap markets. These markets are fast becoming the calibration anvil for reduced-form models. The innovation in this approach lies in exploiting the relationship of default probabilities and recovery rates to spreads in a manner that achieves bounds on recovery rates, rather than point estimates. We brieﬂy review this approach here. Consider a discrete-time model with N periods, indexed by i = 1, 2, . . . , N. Hence, we have times 0, t1 , . . . , ti , . . . , tN . For simplicity, we let the intervals h for each period be constant, that is, h = tN /N. The forward rate between ti−1 and ti is denoted as f (ti ), which is the one-period forward rate ending at time ti . Within each period, default probabilities are assumed to remain constant, and are denoted as λi ≡ λ(ti ). Default probabilities are stated in annualized form. Therefore, the survival probability from time 0 to time ti is   i s(ti ) = exp − (3.12) λ(tj )h j=1

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Suppose the N-period default swap premium is cN % per year. Then, the premium paid per period on a notional of $100 is equal to cN × h. We may express the expected discounted value of the premium payment in period i as follows:   i pi = cN h s(ti ) exp− (3.13) f (tj )h j=1

The total expected premium on the entire default swap is P=

N

Pk

(3.14)

k=1

Assume that the recovery rate is constant and is, as before, denoted by φ. Then, the expected discounted payoff from the default payment on the swap in period i is  Di = [1 − e−λi h ]s(ti−1 ) exp−

i

 f (tj )h

(3.15)

j=1

The total expected discounted payoffs on default are D=

N

Dk

(3.16)

k=1

In a fair-value default swap we must have the premiums equalling the payoffs in discounted expectation (under the martingale measure): P=D

(3.17)

Given a set of swap premiums, we may bootstrap the values of s(ti ) for all maturities. We begin with N = t1 , and assuming a ﬁxed φ, we solve for s(t1 ), given a known premium c1 . Next, we use premium c2 , and the previously computed s(t1 ) to obtain s(t2 ). And so on up the maturity curve until we have all the survival probabilities. Chan-Lau (2003) suggests that we may think of the survival probabilities as functions of the chosen recovery rate φ. Setting the recovery rate too high would imply negative survival probabilities in order to be consistent with

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spreads. Hence, there must be a “maximal” recovery rate that admits a set of acceptable survival probabilities across all maturities. ¯ is extracted in Chan-Lau’s workThis maximal recovery rate, labeled φ, ing paper for various emerging market debts, cross-sectionally, on a daily basis. The time series of maximal recovery rates is then compared across countries to determine the features of cross-border correlations. Interestingly, he also ﬁnds that φ¯ drops sharply in advance of Argentina’s recent default crisis.

6. MEASURE TRANSFORMATIONS In both of the preceding sections, recovery rates were extracted from models via calibration to the prices of traded securities. Therefore, the recovery rates are obtained under the pricing (risk-neutral) probability measure. For the purpose of risk management, these need to be converted into recovery rates under the statistical (physical) measure. This matter is ably addressed in the paper by Bakshi et al. (2001). We present their model (in simpliﬁed form) here. In the world under the physical measure, we deﬁne the probability of default to be λ¯ , a constant. The recovery rate is random and will be high or low as follows: φ=

φH φL

with probability p¯ with probability 1 − p¯

(3.18)

Naturally, conversion to and from the risk-neutral and physical measures requires the risk premium for recovery risk. This premium depends on the speciﬁc utility function chosen. We work here without necessarily specifying the form of the utility function. Assume a representative investor with initial wealth W0 , which includes a defaultable security that promises to pay F at maturity. Utility is designated to be a function of terminal wealth (W) in this, a one-period model, that is, U(W). The investor can buy a default protection contract on the defaultable security now at cost C0 . In the event of default, the investor’s payoff from the protection contract will be CH or CL , with CH > CL , dependent on the recovery rate realized with the probabilities above. Interest rates are zero in the model. We set the number of units of the hedge contract to be x. The outcomes of the model are threefold: (1) No default occurs, (2) Default occurs and recovery is φH , or (3) Default occurs and recovery

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is φL . The three respective terminal wealth values are as follows:   W0 − xC0 W = W0 + x[CH − C0 ] − (1 − φH )F   W0 + x[CL − C0 ] − (1 − φL )F

¯ w/prob (1 − λ) w/prob λ¯ p¯ ¯ − p) ¯ w/prob λ(1

(3.19)

We may write the expected utility of these outcomes as ¯ ¯ pU[W ¯ ¯ E[U(W)] = (1 − λ)U[W 0 − xC0 ] + λ 0 + x(CH − C0 ) − (1 − φH )F] ¯ − p)U[W ¯ + λ(1 0 + x(CL − C0 ) − (1 − φL )F]

(3.20)

It is easy now to solve for x, by differentiating the expected utility with respect to x and setting the result equal to zero: Take

¯ dE[U(W)] = 0, dx

Solve for x

(3.21)

¯ Substituting x in E[U(W)] gives E¯ ∗ [U(W)], that is, the optimal utility function. This preceding analysis was carried out under the physical measure. Were the same optimization done with a risk-neutral investor, the expected utility would be E[U(W)] = E(W) = (1 − λ)W0 + λ p[W0 − (1 − φH )F] + λ(1 − p)[W0 − (1 − φL )F]

(3.22) (3.23)

¯ ¯ and p¯ as where E(·), λ, and p are written differently compared to E(·), λ, before, to signify that the analysis is under the risk-neutral measure. Also, note that there is no purchase of a default protection contract, as the investor is neutral to credit risk. ¯ ¯ p} ¯ to Equating E[U(W)] = E(W), we get a single equation mapping {λ, {λ, p}. In the special case that φH = φL = 1, there is no recovery risk. In this case we may solve for λ in terms of λ¯ . Once we have this, then we may use the equation above to solve for p. The risk premium for recovery would be ¯ related to the Radon–Nikodym derivative (p/p). Bakshi et al. (2001) calibrate a full-blown version of this model to BBB bonds, and in out-of-sample forecasts, ﬁnd the best empirical support for the RT assumption. In addition, by offering a way in which physical measure

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75

recovery assumptions may be taken and translated into risk-neutral ones for pricing, there is much practical value in this model. In a follow-up paper to the abovementioned work, Zhang (2003) develops a multifactor model of default swap pricing, and examines the implied recovery forecasts from the model of Argentina’s sovereign debt default. The model separates recovery rates from default probabilities. In the period much before default, the model calibrates very well, but performs less well closer to default, no doubt on account of additional factors affecting the market not picked up in the parsimonious model that was used. Short maturity contracts are particularly hard to price. This model is a tour de force. It provides closed-form solutions, and as expected, ﬁnds that risk-neutral default probabilities are higher than those under the statistical measure. The default-risk premium is very volatile over the sample. The model’s factors align well with the slope of the US term structure, the level of long-term rates, and the J. P. Morgan emerging market bond index (EMBI).

7. SUMMARY AND SPECULATION Recovery models are truly proving that creative theory helps makes good sense of possibly weak data. We know a lot now about the determinants of recovery rates and their relationship to default probabilities. Improvements in our models are resulting in tighter bounds on the range of recovery rates. These models are tractable and simple to calibrate. Models help in transforming general uncertainty into quantiﬁable risk. It is at the cusp of this metamorphosis that new markets are born. We appear to be close to seeing rapid growth in markets for recovery risk.

REFERENCES Acharya, V., Bharath, S. and Srinivasan, A. (2003). “Understanding the Recovery Rates on Defaulted Securities.” Working Paper, London Business School. Altman, E. and Fanjul, G. (2004). “Defaults and Returns in the High-Yield Bond Market: Analysis through 2003.” Working Paper, NYU Salomon Center. Altman, E., Resti, A. and Sironi, A. (2003). “Default Recovery rates in Credit Risk Modeling: A Review of the Literature and Empirical Evidence.” Working Paper, New York University. Altman, E., Brady, B., Resti, A. and Sironi, A. (2005). “The Link between Default and Recovery Rates: Theory, Empirical Evidence and Implications.” Journal of Business 79(1), forthcoming.

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Bakshi, G., Madan, D. and Zhang, F. (2001). “Recovery in Default Risk Modeling.” Working Paper, University of Maryland. Black, F. and Cox, J. (1976). “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” Journal of Finance 31(2), 351–367. Carey, M. and Gordy, M. (2003). “Systematic Risk in Recoveries on Defaulted Debt.” Working Paper, Federal Reserve Board, Washington. Chan-Lau, J. A. (2003). “Anticipating Credit Events Using Credit Default Swaps, with an Application to Sovereign Debt Crisis.” IMF Working Paper. Dufﬁe, D. J. and Singleton, K. J. (1999). “Modeling Term Structures of Defaultable Bonds.” Review of Financial Studies 12, 687–720. Frye, J. (2000). “Collateral Damage Detected.” Working Paper, Federal Reserve Bank of Chicago, Emerging Issues Series, October, 1–14. Gupton, G., Finger, C. and Bhatia, M. (1997). “CreditMetrics Technical Document.” J. P. Morgan & Co., New York. Hu, W. (2004). “Applying MLE Analysis to Recovery Rate Modeling of U.S. Corporate Bonds.” MS Thesis, UC Berkeley, Haas School of Business. Hu, Y.-T. and Perraudin, W. (2002). “The Dependence of Recovery Rates and Default.” Working Paper, Birkbeck College. Jarrow, R. A. (2001). “Default Parameter Estimation Using Market Prices.” Financial Analysts Journal 57(5), 75–92. Jokivuolle, E. and Peura, S. (2003). “Incorporating Collateral Value Uncertainty in Loss-Given-Default Estimates and Loan-to-Value Ratios.” European Financial Management 9(3), 299–314. Merton, R. C. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” The Journal of Finance 29, 449–470. Nielsen, L., Saa-Requejo, J. and Santa-Clara, P. (1993). “Default Risk and Interest Rate Risk: The Term Structure of Credit Spreads.” Working Paper, INSEAD & UCLA. Zhang, F. X. (2003). “What Did the Credit Market Expect of Argentina Default? Evidence from Default Swap Data.” Working Paper, Federal Reserve Board. Zhou, C. (2001). “The Term Structure of Credit Spreads with Jump Risk.” The Journal of Banking and Finance 25, 2015–2040.

Keywords: Recovery; loss rate; risk-neutral; measure

CHAPTER

4

Non-Parametric Analysis of Rating Transition and Default Data Peter Fledelius,a David Lando,b,∗ and Jens Perch Nielsena

We demonstrate the use of non-parametric intensity estimation—including construction of pointwise conﬁdence sets—for analyzing rating transition data. We ﬁnd that transition intensities away from the class studied here for illustration strongly depend on the direction of the previous move but that this dependence vanishes after 2–3 years.

1. INTRODUCTION The key purpose of rating systems is to provide a simple classiﬁcation of default risk of bond issuers, counterparties, borrowers, and so on. A desirable feature of a rating system is, of course, that it is successful in ordering ﬁrms so that default rates are higher for lower-rated ﬁrms. However, this ordering of credit risk is not sufﬁcient for the role that ratings are bound to play in the future. A rating system will be put to use for risk management purposes, and the transition and default probabilities associated with different ratings will have concrete implications for internal capital allocation decisions and for solvency requirements put forth by regulators. The accuracy of these a Royal & Sun Alliance. b Department of Finance at the Copenhagen Business School, Frederiksberg,

Denmark. ∗ Corresponding author. Copenhagen Business School. E-mail: dl.ﬁ@cbs.dk.

77

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decisions and requirements depends critically on a solid understanding of the statistical properties of the rating systems employed. It is widely documented that the evolution of ratings displays different types of non-Markovian behavior. Not only do there seem to be cyclical components but there is also evidence that the time spent in a given state and the direction from which the current rating was reached affects the distribution of the next rating move. Even if this is consistent with stated objectives of the rating agencies (as discussed below), it is still of interest to quantify these effects since they improve our understanding of the rating process and of the forecasts one may wish to associate with ratings. In this chapter, we use non-parametric techniques to document the dependence of transition intensities on duration and previous state. This exercise serves two key purposes. First, we show that these effects can be more clearly demonstrated in a non-parametric setting where the speciﬁc modeling assumptions are few. For example, we ﬁnd that the effect of whether the previous move was a downgrade or an upgrade vanishes after about 30 months since the last move but that it is signiﬁcant up to that point in time. We also consider stratiﬁcation of ﬁrms in a particular rating class according to the way in which the current rating class was reached, reinforcing the results reached, for example, in Lando and Skødeberg (2002). Again, we are able to quantify how long the effect persists. To do this requires a notion of signiﬁcance and we base this on the calculation of pointwise conﬁdence intervals by a bootstrap method, which we explain in this chapter. The second purpose is to point more generally to the non-parametric techniques as a fast way of revealing whether there is any hope of ﬁnding a certain property of default rates or transition probabilities in the data, how to possibly parameterize this property in a parametric statistical model, and even to test whether certain effects found in the data are signiﬁcant using a non-parametric conﬁdence set procedure. We illustrate this from beginning to end, keeping our focus on a single rating class and jointly modeling the effects of the previous rating move, duration in the current rating, and calendar time. While the technique is used to examine only a few central issues in the rating literature, it is clear that the methods can be extended to other areas of focus. For example, the question of whether transitions depend on economic conditions, key accounting variables for the individual ﬁrms, and other covariates may also be addressed. Whatever the purpose, the nonparametric techniques used in this chapter combined with the smoothing have two main advantages. First, whenever we formulate hypotheses related to several covariates, there is a serious thinning of the data material. Consider, for example, what happens if we study downward transitions from a rating class as a function of, say, calendar time t and duration since last

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79

transition x. In this case, we have not only limited our attention to a particular rating class, but for every transition that occurs, we also have a set of covariates in two dimensions (calendar time and time in state) and we wish to say something about the dependence of the transition intensities on this pair of covariates. This inevitably leads to a “thinness” of data. The smoothing techniques help transform what may seem as very erratic behavior into a recognizable systematic behavior in the data. In this way, even if we have been ambitious in separating out the data into many categories and to a dependence of more than one covariate, we are still able to detect patterns in the data—or to conclude that certain patterns are probably not in the data. The pictures we obtain can be much more informative than a parametric test. Second, if we detect patterns in the data, we may often be interested in building parametric models. With more dimensions in the data, tractability and our ability to interpret the estimators require some sort of additive or a multiplicative structure to be present in these parametric models. The methods employed in this chapter allow one not only to propose more suitable parametric families for the one-dimensional hazards, but they also help detect additive or multiplicative structures in the data using a technique known as marginal integration. In essence, marginal integration gives us non-parametric estimators for the marginal (one-dimensional) hazard functions based on a joint multivariate hazard estimation. The idea is to ﬁrst estimate the joint hazard and then marginally integrate. If an additive or multiplicative structure is present, this integration gives the marginal intensities and—very importantly—these estimators of one-dimensional hazards are not subject to problems with confounding factors. To explain what we mean by this, consider a case where there are two business cycle regimes, one “bad” with a high downgrade intensity from a particular state and one “good” with a low intensity. Assume that we are primarily interested in measuring the effect on the downgrade intensity of duration, that is, time spent in a state. If the sample of ﬁrms with short duration consists mainly of ﬁrms observed in the bad period and the sample of ﬁrms with long duration consists mainly of ﬁrms observed in the good period, then a one-dimensional analysis of the downgrade intensity as a function of duration may show lower intensities for long duration even if this effect is not present in the data. The effect shows up simply as a consequence of the composition of the sample and is really due to a business cycle effect. In our analysis, this problem is avoided by modeling the intensity as a function of calendar time and duration and then ﬁnding the contribution of duration to the intensity through marginal integration. The main technical aspect of this chapter is the smoothing technique itself. For the reader interested in pursuing the statistical methods in more

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depth, we comment brieﬂy on the related literature from statistics. Throughout this study we use the so-called “local constant” two-dimensional intensity estimator developed by Fusaro et al. (1993) and Nielsen and Linton (1995). This local constant estimator can be viewed as the natural analogue to the traditional Nadaraya–Watson estimator known from nonlinear regression as a local constant estimator; see Fan and Gijbels (1996). We could have decided to use the so-called “local linear” two-dimensional intensity estimator as deﬁned in Nielsen (1998). This would have paralleled the development in regression, where local linear regression is widely used, primarily due to its convenient properties around boundaries; see Fan and Gijbels (1996). However, our primary focus is an introduction of these novel non-parametric techniques to credit rating data. We have, therefore, decided to stick to the most intuitive procedures and avoid any unnecessary complications. Rating systems have become increasingly important for credit risk management in ﬁnancial institutions. They serve not only as a tool for internal risk management but are also bound to play an important role in the Basel II proposals for formulating capital adequacy requirements for banks. Partly as a consequence of this, there is a growing literature on the analysis of rating transitions. To our knowledge, the ﬁrst literature to analyze non-Markovian behavior and “rating drift” are Altman and Kao (1992a–c) and Lucas and Lonski (1992). Carty (1997) also examines various measures of drift. The deﬁnitions of drift vary, but typically involve looking at proportions of downgrades to upgrades either within a class or across classes. It is important to be precise about the deviations from Markov assumptions. It is perfectly consistent with Markovian behavior, of course, to have a larger probability of downgrade than upgrade from a particular class. In this chapter we are not concerned with this notion of “drift.” Rather, we are interested in measuring whether the direction of a previous rating move inﬂuences the current transition intensity. We are also interested in measuring the effect of time spent in a state on transition intensities. Note that variations in the intensity as a function of time spent in a state could still be consistent with time-nonhomogeneous Markov chains, but the marginal integration technique allows us to ﬁlter out such effects related to calendar time (or business cycles) and show non-Markovian behavior. The non-parametric modeling of calendar time is similar to that of Lando and Skødeberg (2002), where a semiparametric multiplicative intensity model, with a time-varying “baseline intensity,” is used to analyze duration effects and effects of previous rating moves. There the analysis ﬁnds strong effects, particularly for downgrade intensities. This chapter can be seen as providing data analysis which, ideally, should precede a semi-parametric or parametric modeling effort. The graphs displayed in this chapter allow one to visually inspect

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81

which functional forms for intensities as functions of covariates are reasonable and whether a multiplicative structure is justiﬁed. Another example of hazard regressions is given in Kavvathas (2000), which also examines, among a host of other issues, non-Markovian behavior. The extent to which ratings depend on business cycle variables is analyzed in, for example, Kavvathas (2000), Nickell et al. (2000), and Bangia et al. (2002). Nickell et al. (2000) use an ordered probit analysis of rating transitions to investigate sector and business cycle effects. The same technique was used in Blume et al. (1998) to investigate whether rating practices had changed during the 1990s. Their analysis indicates that ratings have become more “conservative” in the sense of being inclined to assign a lower rating. In this chapter, we do not model these phenomena using a parametric speciﬁcation, but changes due to business cycle effects and policy changes are captured through the intensity component depending on calendar time. The non-parametric techniques shown here allow us to get a more detailed view of some of the mechanisms that may underlie the nonMarkovian behavior. One simple explanation is, of course, that the state space of ratings is really too small, and that if we add information about rating outlooks and even watchlists, this brings the system much closer to being Markovian; see Hamilton and Cantor (2004). In Löfﬂer (2003, 2004) various stylized facts or stated objectives about the behavior of ratings are examined through models for rating behavior. There are two effects that we look more closely at in this chapter. First, in Cantor (2001) the attempt of Moody’s to rate “through the cycle” is supported by the taking of a rating action “only when it is unlikely to be reversed within a relatively short period of time.” We will refer to this as “reversal aversion.” Second, avoidance of downgrades by multiple notches could lead to a policy by which a ﬁrm having experienced a rapid decline is assigned a rapid sequence of one-notch downgrades. Our results for the rating class Baa1 examined here (and for three other classes as well, not shown here) support the reversal aversion, whereas there is some support for the sequential downgrading. We will return to this point below. A type of occurrence exposure analysis similar to ours is performed in Sobehart and Stein (2000) but our methods differ in two important respects. First, in Sobehart and Stein (2000) the covariates are ordered into quantiles. However, for default prediction models and for the use in credit risk pricing models, we need the intensity function speciﬁed directly in terms of the levels of the covariates. Obviously, the changing environment of the economy changes the composition of ﬁrms and hence a company can change the level of a ﬁnancial ratio without changing the quantile or vice versa. Furthermore, there is a big difference between smoothing over the levels themselves and

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over the quantiles. When smoothing is based on quantiles, we have little control over the bandwidth in the sense that members of neighboring quantiles may be very distant in terms of levels of ﬁnancial ratios. Second, the intensity smoothing techniques we used are formulated directly within a counting process framework for the analysis of survival or event history data. Hence, the statistical properties of our estimators are much more well understood than those of a “classical” regression smoother applied to the intensity graphs. As in Sobehart and Stein (2000), we also consider the use of marginal integration in the search for adequate ways of modeling joint dependence of several covariates, that is, whether several covariates have additive or multiplicative effects. An outline of our chapter is as follows: Section 2 describes our data, which are based on Moody’s ratings of US corporate bond issuers over a 15-year period. Section 3 describes our basic model setup with special focus on the smoothed intensities. We describe the basic technique in which we smooth occurrence and exposure counts separately before forming their ratio to obtain intensity estimates. Section 4 describes the procedure of marginal integration by which we obtain one-dimensional hazard rate estimators from our two-dimensional estimators. Section 5 explains the process for building conﬁdence sets. Section 6 discusses our main empirical ﬁndings and Section 7 discusses the choice of additive and multiplicative intensity models and why we should prefer a multiplicative model to an additive model. Section 8 concludes.

2. DATA AND OUTLINE OF METHODOLOGY Our data are complete rating histories from Moody’s for US corporate issuers and we base our results in this study on data from the period after the introduction in April 1982 of reﬁned rating classes. The data contain the exact dates of transitions, but for the purpose of our smoothing techniques we discretize them into 30-day periods. There is little loss of important information using this grid instead of the exact dates, since the bandwidths we will use for smoothing cover a much wider interval. We study the transition intensity as a function of chronological time and duration in current class. In order to establish a duration, we allow for a run-in time of 50 periods starting April 26, 1982, so the study actually starts on June 4, 1986. At this date we can assign each issuer a duration in the current state, which is between 1 and 50 periods. The observation period of transitions covers the time from June 4, 1986, to January 9, 2002, which gives us 190 periods of 30 days. As described below, we will be splitting the

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83

data further according to whether the previous rating change was an upgrade or a downgrade, or there has been no previous rating change recorded. The fundamental quantity we model in this chapter is an intensity α of a particular event that depends on time and some other covariate x. The event may be a single type of event (such as “default”) or it may be an aggregation of several types of events (such as an “upgrade”). If we denote the next occurrence of the event as τ , the intensity provides a local expression for the probability of the event in the sense that the conditional probability Px,t exists, given that an event can happen at time t (if we consider upgrades, then the particular ﬁrm under consideration is actually observed at time t) and that the covariate is around x. Heuristically, we have Px,t (τ ∈ (t, t + t)) = α(t, x)t + o(t) A more formal deﬁnition of α is based on counting process theory and the concept of stochastic intensities; see Andersen et al. (1993) and Nielsen and Linton (1995). For the purpose of this chapter the deﬁnition above is sufﬁcient. We are interested in considering duration effects, and while it is possible to see some effect on the bivariate graphs, we will study mainly the effects in one dimension. The procedure we use is the following: ■

■

■

Obtain a non-parametric estimator for the bivariate intensity by smoothing exposures and occurrences separately. Construct univariate intensity estimators through “marginal integration”—a technique described below. Obtain pointwise conﬁdence sets for the univariate intensities by bootstrapping the bivariate estimator and for each simulation integrating marginally.

We will now describe each procedure in turn and at each step illustrate the technique graphically on a particular rating class.

3. ESTIMATING TRANSITION INTENSITIES IN TWO DIMENSIONS Our ﬁrst goal is to estimate the bivariate function α. We can obtain a preliminary estimator for the intensity α(t, x) for a given time t and duration x

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as follows: Assume that the event we are interested in is “downgrade.” First, count the total number of ﬁrms E(t, x) whose covariates were in a small interval around x and t. Among these ﬁrms, count the number O(t, x) of ﬁrms that were downgraded. A natural estimator for the intensity is then the occurrence/exposure ratio

α(t, ˜ x) =

O(t, x) E(t, x)

This analysis is performed on a “grid” of values of x and t. Even if our original data set is large, we are left with many “cells” in our grid with no occurrences, or, even worse, with no exposures. This is because we have a two-dimensional set of covariates, calendar time and duration, and because we further stratify the data according to the direction of the previous rating move. Whereas most previous studies such as Blume et al. (1998) and Nickell et al. (2000) analyze the old system with eight rating classes, we look at the rating classes with modiﬁers as well. This decreases exposure, but actually increases occurrences since many moves take place within the ﬁner categories. All in all, this leaves us with many cells in our grid with no information. The purpose of smoothing is to compensate for this thinness by using information from “neighbor” cells in the grid. Smoothing implicitly assumes that the underlying intensity varies smoothly with the covariates, but makes no other assumptions on the functional form. While non-parametric intensity techniques have been widely accepted in actuarial science and biostatistics for years—see Nielsen (1998) and Andersen et al. (1993) for further literature—these methods are still not used much in ﬁnance. We can compute the raw occurrence exposure ratios as above for every considered combination of x and t in our grid and subsequently smooth the intensity function using a two-dimensional smoothing procedure. This would correspond to the internal type of regression estimators; see Jones et al. (1994). However, the external intensity estimator used in this chapter applies smoothing to occurrence and exposure separately. The main reason is that there is evidence from the one-dimensional case that the “external” type of estimator is more robust to volatile exposure patterns than the “internal” estimator; see Nielsen and Tanggaard (2001), who compare the “internal” estimator of Ramlau-Hansen (1983) with the external type similar to the one used in this chapter. Since we do experience volatile exposure patterns, the external smoothing approach described above, therefore seems appropriate for our study.1

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The exact smoothing procedure adapted to our data is deﬁned as follows: We have a grid of N = 50 points in the x-direction and T = 190 points in the t-direction. We compute the smoothed two-dimensional intensity estimator as the ratio α(t, ˆ x) =

¯ t,x O E¯ t,x

(4.1)

where the numerator and denominator, the smoothed versions of the “raw” occurrences and exposures, are deﬁned as

¯ t,x = O

N T

Kb1 (x − x1 ) × Kb2 (t − t1 )Ot,x

x1 =1 t1 =1

E¯ t,x =

T N

Kb1 (x − x1 ) × Kb2 (t − t1 )Et,x

x1 =1 t1 =1

where we have used the so-called Epanechnikov kernel functions

|x − x1 |2 Kb1 (x − x1 ) = 0.75 × I{|x−x1 |

The Epanechnikov kernel function adds more weight to observations close to the point of interest than, for example, a uniform kernel, and less or no weight to points far from the point of interest. The bandwidths essentially determine how far from a grid point (t, x) the occurrences and exposures in other grid points should affect the intensity estimates at (t, x). In this chapter, we choose the grid size and select the bandwidths b1 , b2 by visual inspection, trying to increase the bandwidth if the graphs appeared too ragged, and to decrease it if there were signs of oversmoothing. We used a (common) bandwidth of b1 = b2 = 25. This allowed us to smooth out most local changes and capture the long-term trends in data.2 It is illustrative to go through an example of a kernel smoothing step by step. Consider throughout this chapter the class Baa1. In Figure 4.1 we see

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Raw Exposure

Smoothed Exposure

FIGURE 4.1 A graphical illustration of the “exposure” matrix E(t, x), that is, the number of ﬁrms “exposed” to making a transition as a function of time and duration, and its smoothed version. The rating class is Baa1. The exposures are divided into “buckets” covering 30-day periods in both duration and calendar time. Hence, an exposure of n at a given grid point (t, x) tells us that at the beginning of a 30-day period starting at date t there were n ﬁrms that had been in the state between 30(x − 1) and 30x days. Firms that leave the class are distributed between their old exposure class and their new exposure class.

a graphical representation of the total number of exposures of ﬁrms in our data set as a function of time and duration. We show both the raw counts and the smoothed version of these counts. Figure 4.2 shows downgrade activity among the same ﬁrms and for the same grid deﬁnition as for the exposures. Again, both the raw counts and the smoothed versions are displayed. We see very erratic patterns for the raw counts that are hard to interpret. Naturally,

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87

Raw Downgrades

Smoothed Downgrades

FIGURE 4.2 A graphical illustration of the raw occurrence matrix and its smoothed version for the event downgrade from Baa1 as a function of time and duration. The deﬁnitions are the same as in Figure 4.1.

for the downgrade activity, there are many zeros, since for a small handful of ﬁrms that typically occupy a particular grid point, a transition is rare. From the smoothed exposure matrix and the smoothed occurrence matrix we are in a position to obtain a bivariate intensity estimate for downgrades. The result is shown in Figure 4.3. Note that this technique maintains the impression of continuous data. In most other studies with sparse data, one has to group data to obtain reasonable statistical results. We do not have to subgroup data in a ﬁxed number of groups when we use the smoothing technique described above.

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THE CREDIT MARKET HANDBOOK Downgrade Intensity

FIGURE 4.3 The smoothed downgrade intensity as a function of time and duration obtained by dividing the smoothed downgrade matrix by the smoothed exposure matrix. The rating category is Baa1.

4. ONE-DIMENSIONAL HAZARDS AND MARGINAL INTEGRATION The two-dimensional intensity estimator is a good starting point for our analysis. However, we will often prefer a simpler structure where we can interpret the marginal effect of each explanatory variable. We are primarily interested in the marginal effect of duration (time since last transition), but to make sure we can trust our conclusions we also take into account the effect of different calendar times. Let α(t, x) be the two-dimensional true intensity. Consider the following models:

α(t, x) = α1 (t) × α2 (x) × c−1 α(t, x) = α1 (t) + α2 (x) − c

The ﬁrst model is a multiplicative model. The second model is additive. In both models we can estimate α1 (t) and α2 (x) by marginal integration. (See Linton and Nielsen (1995) for the simple regression analogue and Linton et al. (2003) for a mathematical analysis of the more complicated intensity estimators considered in this paper.) For our data set, the estimators can be

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0.0

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10

20 30 Duration (months)

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0

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FIGURE 4.4 The marginally integrated upgrade and downgrade intensities as a function of duration in current state (top graph) and calendar time (bottom graph). The rating class is Baa1 and we consider here only the ﬁrms that are downgraded into this class. written as αˆ 1 (t) =

N 1 α(t, ˆ x) N x=1

αˆ 2 (x) =

T 1 α(t, ˆ x) T t=1

cˆ =

1 α(t, ˆ x) T × N x,t

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Note that estimators are the same in the multiplicative and the additive model. So we add more structure to the model by using marginal integration, and we obtain more directly interpretable estimators. Figure 4.4 shows the marginally integrated downgrade intensity with respect to duration and calendar time, respectively, obtained from the smoothed downgrade intensity presented in Figure 4.3. It is interesting to note the strong cyclical behavior of the downgrade intensity component depending on calendar time, whereas the duration effect is very stable, at least for our base case Baa1. The fact that the cyclical behavior is so pronounced is a strong reason to specify a two-dimensional mode before studying the effects of duration. However, as we will see below, we need to be careful in interpreting the graph, since we will see that the Baa1-rated ﬁrms actually display a signiﬁcant heterogeneity with respect to the previous rating move, which needs to be accounted for.

5. CONFIDENCE INTERVALS Our particular implementation of the bootstrap method is based on the following four steps: ■

■

From the observed occurrence Ot,x and exposure Et,x we calculate the estimator α(t, ˆ x) of α(t, x) using the techniques from Section 3. We then simulate n new sets of occurrences. Each observation point in the occurrence matrix is simulated as follows: k,∗

Ot,x = bin(Et,x , α(t, ˆ x)) − Et,x × α(t, ˆ x) + Ot,x

■

where the ﬁrst two terms give us the difference between the simulated and the expected number of transitions, and the last term corrects the overall level of the simulated occurrence matrix. k,∗ Third, for each new simulated occurrence matrix Ot,x a new estimator αˆ k,∗ (t, x) is calculated based on this new occurrence matrix and the original exposure Et,x . We store the αˆ k,∗ (t, x), k = 1, . . . , 1000. For a given duration x and year t, all αˆ k,∗ (t, x) are ordered as αˆ [1],∗ (t, x), . . . , αˆ [1000],∗ (t, x)

■

The upper 97.5% and lower 2.5% pointwise conﬁdence interval can now be calculated as αˆ [975],∗ (t, x) and αˆ [25],∗ (t, x).

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It is possible to construct conﬁdence intervals for both the twodimensional estimator itself and for smooth functionals of it. Fledelius et al. (2004) provide a general framework for bootstrapping pointwise conﬁdence intervals and give a heuristic argument for why bootstrapping works. In this chapter, we only construct conﬁdence intervals for our marginally integrated intensity estimator for which the asymptotic theory was established in Linton et al. (2003).

6. TRANSITIONS: DEPENDENCE ON PREVIOUS MOVE AND DURATION We now use the procedure outlined above to investigate some key hypotheses on rating behavior. Our main concern is the inﬂuence of previous rating moves on the downgrade and upgrade intensity from the current state. While the stratiﬁcation according to the previous move is clearly important in itself, the duration effect adds an interesting perspective because it allows us to study for how long the stratiﬁcation remains important. We also show the marginally integrated upgrade and downgrade intensities over calendar time and ﬁnally show that the multiplicative model lying behind this marginal integration seems to describe data well. While it is possible in principle to investigate the issues for all rating classes, we have decided to focus on one class, Baa1, for the graphs presented here. This class has a fairly large number of observations. When too few observations are present, we must use very large bandwidths, and we therefore obtain “ﬂat” intensity estimates. The other classes we investigated were A1, Baa2, and Ba1, but showing graphs for all of these would be excessive. We began by stratifying the exposures according to the direction of the previous rating move. This means that the exposure matrix E(t, x) deﬁned above for each rating category is divided into three groups: Those whose previous move was an upgrade, those whose previous move was a downgrade, and those for which there is no information, typically because the current rating class is the ﬁrst recorded. In Figure 4.5 we see the result for the rating category Baa1. The pattern displayed here is typical of the classes we investigated. The potentially signiﬁcant effect we are searching for is in the classiﬁcation between previous upgrade and previous downgrade. When the event is upgrade, a previous downgrade and no information on the previous move have similar intensities, and when the event is downgrade, a previous upgrade and no information are similar. We found this to be true for the other categories as well. We therefore focus on the difference in the estimated intensities depending on whether the previous move was an upgrade or a downgrade. Our

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FIGURE 4.5 Downgrade and upgrade intensities from Baa1 stratiﬁed according to direction of the previous move with no information on the previous move as a separate category. goal is to check whether and for which durations there are signiﬁcant differences. To analyze this “rating drift” issue, we present intensity estimates as a function of duration in a state with conﬁdence bounds. Figure 4.6 shows the downgrade and upgrade from intensities from Baa1, respectively, as a function of how long the company has been in class Baa1 and stratiﬁed according to the previous move. The intensities are shown with bootstrapped pointwise conﬁdence intervals.

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0.05

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FIGURE 4.6 Downgrade and upgrade intensities with conﬁdence bounds from Baa1 as a function of duration, stratiﬁed by the previous move.

Clearly, the most pronounced effect is seen for downgrades from Baa1, where for most durations there is a signiﬁcantly higher intensity of downgrades for the ﬁrms that were downgraded into Baa1 than for those that were upgraded into Baa1. The downgrade intensity for companies that were downgraded into Baa1 is fairly constant for the ﬁrst 25–30 periods; after that we see a decrease in the level. The downgrade intensity for companies upgraded into Baa1 starts out at low values, and we see an increase

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in intensity with length of stay. If a company is upgraded into Baa1, we see very little downgrade activity in the ﬁrst 10–20 periods. The gap between the intensities disappears after 25–30 periods. In summary, the previous rating change direction gives extra information for companies that have stayed in Baa1 for less than 30 periods of 30 days, that is, somewhere between 2 and 3 years. The upgrade intensities display a pattern similar to that for downgrades in the sense that the data for the previous transition contains important information for around 25 to 30 periods. This is evident from the low upgrade intensity in the ﬁrst 25 periods for companies downgraded into Baa1. Both cases seem to support the observation that rating reversals are rare in Moody’s rating history. They also show that the memory is approximately 25 to 30 periods, equivalent to 2–2.5 years. We also investigated other classes and the conclusions were similar. It is clear that the largest effect of stratiﬁcation is for the cases where the event considered is a downgrade. Here, the intensity is signiﬁcantly lower when the previous move was an upgrade. In most cases, but not for Ba1, the downgrade intensity is relatively constant as a function of duration in the state when the previous move was a downgrade. This is evidence that the difference in intensity is not so much due to ﬁrms that are given a sequence of downgrades “one at a time” but rather a “reversal aversion effect” that a ﬁrm is unlikely to be upgraded following a downgrade. The effect seems to vanish after between 20–30 periods in a state. We also ﬁnd that the upgrade intensities seem to be higher when the previous move was an upgrade than when it was a downgrade. The effect is not as pronounced as for downgrades, but it is still signiﬁcant. Again, the history seems more consistent with rating reversal aversion than with rating momentum in that the intensity for upgrade on the condition that the previous move was an upgrade is relatively constant across duration.

7. MULTIPLICATIVE INTENSITIES In our study, we have noticed a clear intensity variation with calendar time, and we therefore cannot investigate the effect of duration through a one-dimensional analysis. We have to consider the two-dimensional intensity case. However, the one-dimensional estimates that we obtain through marginal integration from the smoothed two-dimensional estimates can be interpreted properly only if the intensity model can reasonably be thought of as multiplicative or additive, as explained in Section 4. We therefore implement a visual inspection of the multiplicative and the additive intensity models and compute a measure for squared error to see which one ﬁts the

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Downgrade Intensity

FIGURE 4.7 The bivariate intensity estimator for the class Baa1 when the previous move was a downgrade.

data best. The visual inspection consists of comparing the two-dimensional estimator obtained by adding or multiplying the two marginally integrated intensities with the smoothed but unrestricted two-dimensional estimator. In Figure 4.7 we show the smoothed but unrestricted bivariate downgrade intensity for ﬁrms that were downgraded into Baa1. If we perform a marginal integration as explained in Section 4 to obtain the downgrade intensity as a function of both duration and calendar time, we obtain the results shown in Figure 4.8 (where we also show the marginally integrated upgrade intensities for comparison). From the marginally integrated intensities we can now show the bivariate intensity, assuming a multiplicative and an additive structure, respectively. The resulting bivariate intensities are shown in Figure 4.9. A visual inspection conﬁrms that the multiplicative intensity is closer to the unrestricted intensity estimate. Certainly, the difference between these graphs and the unrestricted estimators in Figure 4.7 does not lead us to reject the assumption of a multiplicative structure, even if the multiplicative structure is smoother and does not capture all of the features of the unrestricted estimator. We would also prefer the multiplicative structure to the additive structure based on a computation of squared errors and for reasons of interpretation. The multiplicative model is ˆˆ x) = αˆ 1 (t) × αˆ 2 (x) × cˆ −1 α(t,

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0.01 0.02 0.03 0.04 0.05

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0.01 0.02 0.03 0.04 0.05

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1995 Year

2000

FIGURE 4.8 The marginally integrated upgrade and downgrade intensities as functions of duration in current state (top graph) and calendar time (bottom graph). The rating class is Baa1 and we consider here only the ﬁrms that are downgraded into this class. ˜˜ x) denote the corresponding intensity estimator in the additive Let α(t, ˆˆ x) and α(t, ˜˜ x) model. We calculated the sums of squared errors (SSEs) for α(t, deﬁned as 2 ˆˆ x) α(t, ˆ x) − α(t, x,t

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Multiplicative Downgrade Intensity

Additive Downgrade Intensity

FIGURE 4.9 The bivariate intensity estimator for ﬁrms downgraded into class Baa1 assuming a multiplicative intensity structure (top graph) and an additive intensity structure (bottom graph). The multiplicative (additive) structure deﬁnes the intensity as the product (sum) of the marginally integrated intensities. and 2 ˜˜ x) α(t, ˆ x) − α(t, x,t

The event considered was a downgrade from Baa1, and to avoid the problems with heterogeneity within this class, we chose only the ﬁrms that were downgraded into Baa1. We found an SSE in the multiplicative model of 0.048

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and an SSE in the additive model of 0.103. The SSE calculation, therefore, supports selecting the multiplicative model for downgrades.

8. CONCLUDING REMARKS This chapter applies non-parametric smoothing techniques to the study of rating transitions in Moody’s corporate default database. The techniques give a powerful way of visualizing data. We illustrate their use with a detailed study of both duration dependence of transitions and the direction of a previous rating move. The patterns we see for Baa1 (and for the other classes we investigated but do not display here) are consistent with a policy of “rating reversal aversion” in which the ratings through a period of 2–3 years show a reduced tendency of moving in a direction opposite to the direction in which they were moved into the current state. It is also consistent with Moody’s stated objectives, as formulated in Cantor (2001), of changing ratings only when a reversal is unlikely to take place in the foreseeable future. This pattern is signiﬁcant in both directions; that is, downgrades are less likely for ﬁrms that were upgraded into the state than for ﬁrms downgraded into the state, and the pattern is reversed when studying upgrades. The effect of stratiﬁcation is, however, most pronounced for downgrade activity. One may also attribute this observed pattern to an effect caused by multinotch downgrades being carried out one notch at a time for some ﬁrms despite the fact that they have in reality experienced a credit quality decline. If this were the pattern, we should see signiﬁcant duration effects on downgrade activity as a function of duration but we found a clear pattern of this only for the rating class Baa2. We show clear calendar time effects in our data set, as demonstrated, in Figure 4.8, and this is consistent with, for example, Nickell et al. (2000). We condition out those effects through the marginal integration procedure. This procedure works well under either an additive or a multiplicative intensity structure and we ﬁnd support for the treatment of the model as a multiplicative model, consistent with the semiparametric Cox regression studied, for example, in Lando and Skødeberg (2002). About the time of completion of this work, the study of Hamilton and Cantor (2004) emerged, which as earlier mentioned, studies the role of outlooks and watchlists. This also lends support to the non-Markovian pattern found here but the precise role of the outlooks in our setting awaits further analysis.

ACKNOWLEDGMENTS We are grateful to Moody’s for providing the data and in particular to Richard Cantor and Roger Stein for their assistance. Helpful comments from

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an anonymous referee are also gratefully acknowledged. All errors and opinions are, of course, our own. Lando acknowledges partial support by the Danish Natural and Social Science Research Councils.

NOTES 1. To our knowledge, the asymptotic theory is also only fully understood in the external case. 2. We could have chosen to work with quantitative criteria for bandwidth selection. There is extensive literature on bandwidth selection with the simple kernel density estimator as the primary object of interest. In our intensity case, the analogue to the most widely used bandwidth selection procedure, cross-validation, was introduced in Nielsen and Linton (1995) and Fledelius et al. (2004). Automatic bandwidth selection turns out to be extremely complicated in practice and no one method has yet obtained universal acceptance; see Wand and Jones (1995). For example, the simplest and most widely used method, cross-validation, is well known to be a bad bandwidth selector for many data sets; see Wand and Jones (1995).

REFERENCES Altman, E. and Kao, D. (1992a). Corporate Bond Rating Drift: An Examination of Credit Quality Rating Changes over Time. IFCA, Charlottesville: The Research Foundation of the Institute of Chartered Financial Analysts. Altman, E. and Kao, D. (1992b). “The Implications of Corporate Bond Rating Drift.” Financial Analysts Journal 3, 64–75. Altman, E. and Kao, D. (1992c). Rating drift in high yield bonds. The Journal of Fixed Income 1, 15–20. Andersen, P., Borgan, Ø., Gill, R. and Keiding, N. (1993). Statistical Models Based on Counting Processes. New York: Springer. Bangia, A., Diebold, F., Kronimus, A., Schagen, C. and Schuermann, T. (2002). “Ratings Migration and the Business Cycle, with Applications to Credit Portfolio Stress Testing.” Journal of Banking and Finance 26(2–3), 445–474. Blume, M., Lim, F. and MacKinlay, A. (1998). “The Declining Credit Quality of US Corporate Debt: Myth or Reality.” Journal of Finance 53(4), 1389–1413. Cantor, R. (2001). “Moody’s Investors Service Response to the Consultative Paper Issued by the Basel Committee on Bank Supervision a New Capital Adequacy Framework.” Journal of Banking and Finance 25, 171–185. Carty, L. (1997). Moody’s Rating Migration and Credit Quality Correlation, 1920– 1996. Special Comment, Moody’s Investors Service, New York. Fan, J. and Gijbels, I. (1996). Local Polynomial and Its Applications. London: Chapman and Hall.

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Fledelius, P., Guillen, M., Nielsen, J. and Vogelius, M. (2004). “Two-dimensional Hazard Estimation for Longevity Analyses.” Scandinavian Actuarial Journal 2, 133–156. Fusaro, R., Nielsen, J. P. and Scheike, T. (1993). “Marker Dependent Hazard Estimation. An Application to Aids.” Statistics in Medicine 12, 843–865. Hamilton, D. and Cantor, R. (2004). Rating Transitions and Defaults Conditional on Watchlist. Outlook and Rating History. Special Comment, Moody’s Investors Service, New York. Jones, M. C., Davies, D. and Park, B. U. (1994). “Versions of Kernel-type Regression Estimators.” Journal of the American Statistical Association 89, 825–832. Kavvathas, D. (2000). “Estimating Credit Rating Transition Probabilities for Corporate Bonds.” Working Paper, Department of Economics, University of Chicago. Lando, D. and Skødeberg, T. (2002). “Analyzing Rating Transitions and Rating Drift with Continuous Observations.” The Journal of Banking and Finance 26, 423–444. Linton, O. and Nielsen, J. P. (1995). “A Kernel Method of Estimating Structure Nonparametric Regression Based on Marginal Integration.” Biometrika 82, 93–100. Linton, O., Nielsen, J. and de Geer, S. V. (2003). “Estimating Multiplicative and Additive Hazard Functions by Kernel Methods.” Annals of Statistics 31, 464–492. Löfﬂer, G. (2003). “Avoiding the Rating Bounce: Why Rating Agencies Are Slow to React to New Information.” Working Paper, Goethe-Universität, Frankfurt. Löfﬂer, G. (2004). “An Anatomy of Rating Through the Cycle.” Journal of Banking and Finance 28, 695–720. Lucas, D. and Lonski, J. (1992). “Changes in Corporate Credit Quality 1970–1990.” The Journal of Fixed Income 1, 7–14. Nickell, P., Perraudin, W. and Varotto, S. (2000). “Stability of Ratings Transitions.” Journal of Banking and Finance 24(1–2), 203–227. Nielsen, J. P. (1998). “Marker Dependent Kernel Hazard Estimation from Local Linear Estimation.” Scandinavian Actuarial Journal 2, 113–124. Nielsen, J. and Linton, O. (1995). “Kernel Estimation in a Nonparametric Marker Dependent Hazard Model.” Annals of Statistics 23, 1735–1748. Nielsen, J. P. and Tanggaard, C. (2001). “Boundary and Bias Correction in Kernel Hazard Estimation.” Scandinavian Journal of Statistics 28, 675–698. Ramlau-Hansen, H. (1983). “Smoothing Counting Processes by Means of Kernel Functions.” Annals of Statistics 11, 453–466. Sobehart, J. and Stein, R. (2000). “Moody’s Public Firm Risk Model: A Hybrid Approach to Modeling Short Term Default Risk.” Moody’s Investors Service. Global Credit Research. Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. London: Chapman and Hall.

Keywords: Credit ratings; transition probabilities; non-Markov effects; non-parametric analysis

CHAPTER

5

Valuing High-Yield Bonds: A Business Modeling Approach Thomas S. Y. Hoa,∗ and Sang Bin Leeb

This chapter proposes a valuation model of a bond with default risk. Extending from the Brennan and Schwartz real option model of a ﬁrm, the chapter treats the ﬁrm as a contingent claim on the business risk. This chapter introduces the “primitive ﬁrm,” which enables us to value ﬁrms with operating leverage relative to a ﬁrm without operating leverage. This chapter emphasizes the business model of the ﬁrm, relating the business risk to the ﬁrm’s uncertain cash ﬂow and its assets and liabilities. In so doing, the model can relate the ﬁnancial statements to the risk and value of the ﬁrm. The chapter then uses Merton’s structural model approach to determine the bond value. This model considers the ﬁxed operating costs as payments of a “perpetual debt,” and the ﬁnancial debt obligations are junior to the operating costs. Using the structural model framework, we relative-value the bond to the observed ﬁrm’s market capitalization and provide a model that is empirically testable. We also show that this approach can better explain some of the high-yield bond behavior. In sum, this model extends the valuation of high-yield bonds to incorporate the business models of the ﬁrms and endogenizes the ﬁrm value stochastic process, which, in practice, is a key element in high-yield valuation. We have shown that in relating the ﬁrm’s business model to the ﬁrm value, the resulting ﬁrm value stochastic process affects the bond value signiﬁcantly.

a Thomas Ho Company, Ltd., New York, NY, USA. b School of Business Administration, Hanyang University, Seoul, 133-791, Korea. ∗ Corresponding author. Thomas Ho Company, Ltd., 55 Liberty Street, 4B, New

York, NY, 10005-1003, USA. Tel.: 1-212-571-0121; e-mail: tom.ho@thomasho. com.

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1. INTRODUCTION There has been much research on the valuation of corporate bonds with credit risks in the past few years. The impetus of the research may be driven by a number of factors. Recently, there has been a surge of bonds facing signiﬁcant credit risks, as a result of the downturn of the economy after the burst of the new economy bubble. For example, in the telecommunication sector, a number of ﬁrms have declared default because of the excess supply of telecommunication infrastructure and ﬁnancial obligations. Another reason is the impending change in regulations in risk management. Increasingly, regulators are demanding more disclosure of risks from the ﬁnancial institutions and the measures of credit risks in the ﬁrm’s investment portfolio. The ﬁnancial disclosure would lead to the examination of the adequacy of capital for the ﬁrms. Finally, the credit risk model is important for the use of a number of recent ﬁnancial innovations. These innovations include collateralized debt obligations, credit default swaps, and other credit derivatives that have demonstrated signiﬁcant growths in the past few years. The credit risk model is important in determining these securities’ values and managing their risks. Valuing bonds with credit risk must necessarily be a complex task. A high-yield bond tends to have the business risk of the bond’s issuer. And, therefore, the valuing of a high-yield bond may be as involved as valuing the equity of the issuer. Indeed, both the bonds and the equity of a ﬁrm are contingent claims on the ﬁrm value. One approach to valuing a high-yield bond is that of Merton (1974). The model views the ﬁrm’s equity as a call option on the ﬁrm value and applies the Black–Scholes model to value a corporate bond. This approach does not require the investors to know the proﬁtability of the ﬁrm and the market expected rate of return of the ﬁrm. The model needs to know only the prevailing ﬁrm value and its stochastic process. In essence, according to the Merton model, a defaultable bond is a default-free debt embedded with a short position of a put option on the ﬁrm value, with the strike price equaling the face value of the debt and the time to expiration equaling the maturity of the bond. More generally, models that view a high-yield bond as a bond with an embedded put option are called structural models. There are many extensions of the Merton model. One general extension is the use of a trigger default barrier that speciﬁes the condition for default. For example, the Longstaff and Schwartz1 (1995) model allows the ﬁrm to default at any time whenever the ﬁrm value falls below a barrier. This approach views that a bond has a barrier option embedded in a defaultfree bond. This model has been extended by Saa-Requejo and Santa-Clara (1999), allowing for the stochastic strike price; Briys and de Varenne (1997)

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allow the barrier to be related to the market value of debt. Such extensions assume the stochastic ﬁrm value captures all of the business risk of the ﬁrm. They do not model the business of the ﬁrm and they, in particular, ignore the importance of the negative cash ﬂows of a ﬁrm in triggering the event of default. To avoid such shortcomings, the Kim et al. (1993) model assumes that the bondholders get a portion of the face value of the bond at default, which is based on the lack of cash ﬂow to meet obligations. They deﬁne the default trigger point as a net cash ﬂow at the boundary, when the ﬁrms cannot pay for the interests and dividends. Brennan and Schwartz use the real option approach to determine the ﬁrm value as a contingent claim on the business risk. Using this approach, they model the value of a mining company. The real option valuation approach extends the Merton model to specify the business model of a ﬁrm and, therefore, the approach values the corporate bonds as compound options on the business risks. This chapter takes this real option approach to value the high-yield bonds. Speciﬁcally, we model the business of the ﬁrm and its operating cash ﬂows contingent on the business risks. Using the structural model’s compound option concept, we determine the default conditions of a ﬁrm, given its capital structure and the business model. In essence, our approach endogenizes the trigger default barrier of the ﬁrm using the ﬁrm’s business model and the capital structure. Speciﬁcally, we propose that a ﬁrm’s ﬁxed operating costs play a signiﬁcant role in triggering default of the bond’s debt. When the ﬁrm has a negative operating income that cannot be ﬁnanced internally, the ﬁrm must necessarily seek funding in the capital markets. However, if the ﬁrm value is low in relation to all of its future ﬁnancial obligations, then the ﬁrm may not be able to fund the negative operating income, leading to default. Indeed, some bonds are considered risky because of the ﬁrm’s high operating leverage, even though the ﬁnancial leverage may be low. In comparison with the structural models in the research literature, our model suggests that the ﬁrm value stochastic process is not a simple lognormal process. Instead, the ﬁrm value follows an “option” price process. And the debt is not a risk-free bond embedded with a put option. It is embedded with a compound option and it is a “junior debt” to the ﬁxed costs. This approach has broad implications to debt valuation. Our model suggests that the pricing of defaultable bonds must include more ﬁnancial information of a ﬁrm: in particular, the ﬁnancial and operating leverage of the ﬁrm. The model allows for the ﬁrm to default before the bond matures by allowing the negative cash ﬂow to trigger a default. Since the model does not require an exogenously speciﬁed trigger default function, but solves for the default condition using the option pricing approach, we can use the model

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to price the bonds using the ﬁrm’s ﬁnancial statements, which are widely available. Therefore, the model can be tested empirically. In this chapter, we provide some empirical evidence to support the validity of the model. While this chapter provides a simple model, we show that the approach is very general. Extensions of the model will be left for future research. The chapter proceeds as follows. We describe the model in Section 2, presenting the assumptions made in the model. For the clarity of the exposition, in Section 3, we provide a numerical example, showing how the model can be used to market available data. Section 4 presents some empirical evidence on the validity of the model. Section 5 discusses some of the implications of the model and, ﬁnally, Section 6 provides the conclusions.

2. SPECIFICATION OF THE MODEL This section presents the assumptions of the model. Similar to the Merton model, we assume that the market is perfect, with no transaction costs. We assume that there are corporate and personal taxes such that the assumptions are consistent with the Miller model. The corporate tax rate of the ﬁrm is assumed to be τc . In this world, the capital structure does not affect the value of the ﬁrm. We use a binomial lattice framework to construct the risk processes. We assume that the yield curve is ﬂat and is constant over time at an annual compounding rate of rf . The bond valuation model is based on a real option model. Speciﬁcally, we begin with the description of the business risk of the ﬁrm by depicting the primitive ﬁrm latticeV p (n, i). We then build the ﬁrm value latticeV(n, i), which includes some considerations of a valuation of a ﬁrm: ﬁxed costs and taxes. Finally, we use the ﬁrm value lattice to analyze all of the claims on the ﬁrm value, based on the Miller and Modigliani (MM) framework.

2.1 Primitive Firm The ﬁrm, the equity, the debt, and all other claims on the ﬁrms are treated as the contingent claims to the primitive ﬁrm. The primitive ﬁrm is the underlying “security” to all these claims and it captures the business risk of the ﬁrm. We begin with the modeling of the business risk. We assume that the ﬁrm has a ﬁxed capital asset and the capital asset generates uncertain revenues. The gross return on investment (GRI) is deﬁned as the revenue generated per $1 of the capital asset. GRI is a capital asset turnover ratio. In this simpliﬁed

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model, we consider a ﬁrm is endowed with a capital asset that does not depreciate and can generate perpetual revenues. Speciﬁcally, we assume that GRI follows a binomial lattice process that is lognormal (or multiplicative) with no drift, a martingale process, where the expected GRI value at any node point equals the realized GRI at that node point (n, i), where n is the number of time steps and i denotes the state of the world. Speciﬁcally, GRI(n + 1, i + 1) = GRI(n, i) exp(σ ) with probability q, where 0 < q < 1 and GRI(n + 1, i) = GRI(n, i) exp(−σ ) with probability 1 − q. The market probability q is chosen so that the expected value of the risk over one period is the observed risk at the beginning of each step. That is, the risk follows a martingale process stated below: GRI(n, i) = q × GRI(n + 1, i + 1) + (1 − q) × GRI(n + 1, i)

(5.1)

where q=

1 − e−σ eσ − e−σ

σ being the volatility of the risk driver. We assume that the MM theory can be extended to the multi-period dynamic model described above. In this extension, we assume that all of the individuals make their investment decisions and trading at each node on the lattice. These activities include the arbitrage trades described in the MM theory. The results of the theory apply to each node. Therefore, there is a cost of capital ρ at each node dependent on the business risk and not on the risk of the ﬁrm’s free cash ﬂows, which depend on the operating leverage. The cost of capital ρ of a sector is the required rate of return for that sector business risk, assuming that the ﬁrm has no ﬁxed costs. By way of contrast, the cost of capital of MM theory assumes a particular ﬁrm, taking the operating leverage as given. The cost of capital of a ﬁrm according to the MM theory reﬂects the risk of the free cash ﬂows of the ﬁrm. Here, we extend this concept of MM theory to a business sector and use a ﬁrm with no operating leverage as the benchmark for comparison.

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Since the production process is the same at each node on the lattice, the ﬁrm risk is the same at each node. For this reason, we will assume that the cost of capital is constant in all states and time. According to the MM theory, the ﬁrm value is the present value of all of the ﬁrm’s free cash ﬂow along all of the paths on the lattice. In particular, the lattice of the primitive ﬁrm value is given as V p (n, i) =

CA × GRI(n, i) × m ρ

(5.2)

where m is the gross proﬁt margin. By the deﬁnition of the binomial process of the gross return on investment, we have V p (n + 1, i + 1) = V p (n, i)eσ

(5.3)

Further, since the cost of capital of the ﬁrm is ρ, the ﬁrm pays a cash dividend of Cu = V p (n, i) × ρ × eσ p

Therefore, the total value of the ﬁrmVu , an instant before the dividend payment in the up state, is Vu = V p × (1 + ρ) × eσ p

(5.4)

p

Similarly, the total value of the ﬁrmVd , an instant before the dividend payment in the down state, is Vd = V p × (1 + ρ) × e−σ p

(5.5)

Then, the risk neutral probability p is deﬁned as the probability that ensures the expected total return is the risk-free return: p

p

p × Vu + (1 − p) × Vd = (1 + rf ) × V p p

(5.6)

p

Substituting V p , Vu , Vd into the equation above and solving for p, we have p=

A − e−σ eσ − e−σ

(5.7)

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107

where A=

1 + rf 1+ρ

Note that as long as the volatility and the cost of capital are independent of the time n and state i, the risk-neutral probability is also independent of the state and time, and is the same at each node point on the binomial lattice. We have now changed the measure from market probability to the risk-neutral probability. We will use this risk-neutral probability to determine the values of the contingent claims.

2.2 The Firm Value We assume that the ﬁrm pays out all of the free cash ﬂows. Let the ﬁxed cost be FC, which is constant over time and state. In the case of negative cash ﬂow, we assume that the ﬁrm gets tax credits, and the ﬁrm raises the funds from equity. This assumption is quite reasonable since tax credits can carry forward for over 20 years; the government in essence participates in the business risks of the ﬁrm and it should not affect the basic insight of the model. The cash ﬂow is the revenue net of the operating costs, ﬁxed costs, and taxes: CF(n, i) = (CA × GRI(n, i) × m − FC) × (1 − τ )

(5.8)

This model assumes that the ﬁrm has no growth over this time horizon. This assumption is quite reasonable because the high-yield companies often cannot implement growth strategies. Further, the model can be extended in a straightforward manner to incorporate growth for a ﬁrm when growth is important to its bond pricing. Ho and Lee (2004) provide an extension of the model with growth, allowing for optimal investment decisions. The terminal value at each state in the binomial lattice at the horizon date has four components: the present value of the gross proﬁt; the present value of the ﬁxed costs, which takes the possibility of future default into account; the present value of the tax that is approximated as a portion of the pretax ﬁrm value; and, ﬁnally, the cash ﬂows of the ﬁrm at each node point. Following the Merton (1974) model, we assume that the ﬁrm pays no dividends after the planning period and the primitive ﬁrm follows a price dynamic described below: dV p = ρV p dt + σp V p dZ where dZ is the Wiener process.

(5.9)

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The present value of the ﬁxed costs is determined as a hypergeometric function, according to Merton (1974), since we assume that the ﬁrm can go default in the future and the ﬁxed costs are not paid in full. The lattice of the ﬁrm value is determined by rolling back the ﬁrm values, taking the cash ﬂows into account. The ﬁrm value at the terminal period at each node is CA · GRI(n, i) · m CA · GRI(n, i) · m − V(n, i) = Max ρ−g ρ−g + (CA × GRI(n, i) × m − FC) (1 − τc ), 0 (5.10) where ( g) is the present value of the perpetual risky ﬁxed cost, and ( g) is the valuation formula of the perpetual debt given by Merton (1973) presented in the Appendix. In the intermediate periods, the ﬁrm value is determined by backward substitution: p × V(n + 1, i + 1) − (1 − p) × V(n + 1, i) V(n, i) = Max (1 + rf ) +(CA × GRI(n, i) × m − FC) × (1 − τc ), 0

(5.11)

2.3 Debt Valuation and Market Capitalization We assume that the ﬁrm value is independent of the debt level. The value of the bond is determined by the backward substitution approach. The stock lattice is the ﬁrm lattice net of the bond lattice. We ﬁrst consider the terminal conditions for the bond to be Min(debt obligation at T, ﬁrm value at T). We then conduct the backward substitutions such that we apply the valuation rule at each node point: Min(backward substitution bond value + bond cash ﬂow, ﬁrm value). Following the standard methodology, the rolling back procedure leads to the value of the debt at the initial value. The market capitalization of the ﬁrm is the ﬁrm value net of the debt value.

3. A NUMERICAL ILLUSTRATION We assume that the yield curve is ﬂat and is constant over time at a 4.5% annual compounding rate. The market premium is deﬁned as the market

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109

expected return net of the risk-free rate, which is assumed to be 5%. The tax rate of the ﬁrm is 30%, which is assumed to be the marginal tax rate. The model is based on a ﬁve-step binomial lattice. It is a one-factor model, with only the business risk. The model is arbitrage free relative to the underlying values of a ﬁrm that bears all the business risks of the revenues. We use one ﬁrm, Hilton Hotels, in the sector of consumer and lodging, as an example to illustrate the implementation of the model. On the evaluation date October 31, 2002, the market capitalization is reported to be $4,944 million, the stock volatility estimated to be the 1-year historical volatility of Hilton’s stock, is 51.9%, and the stock beta is 1.255. Using the capital asset pricing model, we estimate that the expected rate of return of the stock r = 4.5% + 1.255 × 5% = 10.775% The ﬁnancial data are given from the ﬁnancial statements as follows. The revenue is $2834 million, with the operating cost of $1,542 million and ﬁxed cost of $726 million. The capital asset is $7,714 million with the long-term debt of $5,823 million and interest costs of $357 million. Using these data, we can calculate GRI = Proﬁt margin (m) =

Revenue = 0.367 Capital asset Revenue − operating cost = 45.58% Revenue

3.1 The GRI Lattice To generate the GRI lattice, we have to assume the sector cost of capital and the volatility initially to begin the iterative process. Let us assume the cost of capital spread to be 1.57% and the sector volatility 11.54%. The assumed data are an initial input for the nonlinear optimization procedure. Using the assumed data, we can calculate the expected returns of the sector (ρ): ρ = 0.045 + 0.0157 = 0.0607 We can now determine the primitive ﬁrm lattice based on the model: V p (n, i) =

CA × GRI(n, i) × m ρ

We can also calculate the cash ﬂow of the primitive ﬁrm at each node point. The lattice of cash ﬂow at each state, based on the capital asset of $7,714

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THE CREDIT MARKET HANDBOOK

million, is CFp (n, i) = CA × GRI(n, i) × m

(5.12)

The risk neutral probability is p(n, i) =

[(1 + rf )/(1 + ρ)] − e−σ eσ − e−σ

= 0.414223

3.2 The Firm Value The ﬁxed costs of the ﬁrm stay the same. The gross margin also remains constant. The ﬁrm defaults when the ﬁrm cannot ﬁnance the ﬁxed costs. All excess cash ﬂows are paid out. A lattice of cash ﬂow at each state is developed. At each node, the cash ﬂow is the revenue net of the operating costs, ﬁxed costs, and taxes; CF(n, i) = (CA × GRI(n, i) × m − FC) × (1 − τ ) The terminal value at each state is given below. By assuming that the longterm growth rate g is 1.57%, the ﬁrm value at the terminal period at each node is

CA · GRI(n, i) · m CA · GRI(n, i) · m − ρ−g ρ−g + (CA × GRI(n, i) × m − FC)

V(n, i) = Max

×(1 − τ ), 0 where ( g) is given in the Appendix. At the terminal date, the time horizon where n = 5, the ﬁrm value for each node is State (i) Firm value ($mn)

0 2,663.545

1 5,867.177

2 10,635.13

3 17,099.26

4 25,472.04

5 36,115.54

Now, we roll back the ﬁrm value from the terminal value, taking the cash ﬂows into account. In the intermediate periods, the ﬁrm value is determined

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Valuing High-Yield Bonds: A Business Modeling Approach

36115.54 29829.59 24355.28 19603.22 15500.84

25472.04 20523.04

16248.04

11986.82

12585.19 9480.37

17099.26 13267.72 10635.13

10019.47 7305.15

7784.25 5867.18

5447.84 3901.21

2663.55 Today

1 yr

2 yr

3 yr

4 yr

5 yr

FIGURE 5.1 Lattice of the ﬁrm value. by backward substitution: V(n, i) = Max

p × V(n + 1, i + 1) − (1 − p) × V(n + 1, i) (1 + rf )

+ (CA × GRI(n, i) × m − FC) × (1 − τ ), 0 The resulting lattice is shown in Figure 5.1.

3.3 Debt Valuation and Market Capitalization The debt structure of Hilton Hotels is somewhat complicated, with eight bonds, and is shown in Table 5.1. Using the debt structure given above, we can calculate the promised cash ﬂow of the bonds, which are $118.377 thousand at year 0, $380.053 thousand for the period from year 1 to year 3, and $867.553 and $7,133.518 thousand at years 4 and 5, respectively. Time Debt cash ﬂow ($mn)

0 118.377

1 380.053

2 380.053

3 380.053

4 867.553

5 7,133.518

We use the lattice of the ﬁrm model, the lattice of the debt cash ﬂow, and the standard backward substitution procedure to determine the debt value. The resulting lattice of the debt is shown in Figure 5.2.

112 Table 5.1

THE CREDIT MARKET HANDBOOK Debt Package of Hilton Hotels

Observed Date

Maturity

Coupon Rate

Principal

Price

# of Outstanding

20030228 20030228 20030228 20030228 20030228 20030228 20030228 20030228

20130228 20060515 20091215 20171215 20080515 20121201 20070415 20110215

0.061 0.05 0.072 0.075 0.076 0.076 0.08 0.083

1000 1000 1000 1000 1000 1000 1000 1000

100 96 99 92 102 99 104 105

3,473,000 500,000 200,000 200,000 400,000 375,000 375,000 300,000

7133.518 867.553 380.053 380.053 380.053 118.377

7133.518 867.553

380.053 380.053

380.053

7133.518 867.553 7133.518

380.053 380.053

867.553 7133.518

380.053 867.553

7133.518

Today

1 yr

2 yr

3 yr

4 yr

5 yr

FIGURE 5.2 Debt cash ﬂows.

The market capitalization value is the ﬁrm value net of the debt value. Finally, given that the bond price is $5,818 million, the internal rate of return is 9.26%, and the credit risk spread, which is the internal rate of return net of the risk-free rate, is 476 basis points. From the two nodes of the ﬁrst period of the stock lattice and the sector lattice, the lattice of the primitive ﬁrm, we can calculate the stock volatility and the sector volatility.

3.4 Calibration Recall that this procedure thus far has assumed the following input data: long-term growth rate of the ﬁrm g, sector volatility σp , and sector expected excess return ρ under market probability. There are in essence three unknowns. These data are not observed directly. They are just initial data to enable us to proceed with the algorithm in determining the bond value.

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Valuing High-Yield Bonds: A Business Modeling Approach

However, the model derives the following values: market capitalization and the stock volatility, which we can directly observe. Further, note that the excess return of the stock can be determined by the capital asset pricing model and by the equation below, which is based on the market price of risk of a contingent claim. This provides the third constraint to the nonlinear search with three control variables.

Excess return (stock) = Excess return (sector) ×

stock volatility sector volatility

(5.13)

Therefore, we can use a nonlinear estimation procedure in perturbing the assumed data, the long term growth rate and the sector volatility, such that the model-derived value equals the observed values. Market capitalization and stock volatility = observed market capitalization and stock volatility.

Market capitalization Calibrated stock value

4944.6 4944.6

Given stock volatility Estimated stock volatility

0.51898 0.51925

4. EMPIRICAL EVIDENCE We use an example of McLeodUSA to demonstrate empirically the relationship between the market capitalization and the bond package value. We show that as the market capitalization of the ﬁrm falls, the bond package value would also fall. However, the bond value falls more precipitously than the market capitalization as the ﬁrm approaches bankruptcy. We also show that the Merton model of the corporate bond can explain much of this relationship between the market capitalization and the bond value. But the model tends to understate the acceleration of the fall in the bond prices when the market capitalization falls below a certain value. We use our model to explain this relationship between the market capitalization and the bond package value. Speciﬁcally, we assume that the gross return on investment declines, leading the fall in the market capitalization and the bond value. It is quite reasonable to make this assumption. A source

114

THE CREDIT MARKET HANDBOOK

of the ﬁnancial problem of McLeodUSA was that the expectation of the demand for communication had been falling along with an excess supply of communication networks. As a result, the market revised its expectations of the returns of the McLeodUSA assets, and hence the fall of GRI. We use the calibrated business model of McLeodUSA. The sector volatility and the implied long-term growth rate are estimated to be 0.0935 and 0.0332, respectively. Further, we add a spread to the Treasury curve in discounting the bond such that the bond price equals the observed price initially. The use of a spread is reasonable because we have discussed that bonds tend to have a liquidity spread and a risk premium (a market price of risk for the model risks not captured by the model) in predicting defaults. Referring to Figure 5.1, the results show that the proposed model provides relatively better explanatory power of the observations than the Merton model. In particular, using the proposed model, we predict the precipitous fall of the bond value better than when we use the Merton model. The empirical evidence can be explained intuitively. McLeodUSA is a communication company that has signiﬁcant operating costs. Note that the debts outstanding do not mature in the next 3 years and there is no immediate maturity of crisis for the ﬁrm in the short run. However, because of the signiﬁcant ﬁxed cost resulting in losses of the operation, the ﬁrm’s operation has to be supported by external ﬁnancing. When the market revises downward on the returns of the assets, the market capitalization falls, leading to the situation where McLeodUSA fails to have access to the capital market to fund its negative operating costs. In essence, the ﬁrm defaults on its operating costs, something that the Merton model would not have considered (Figure 5.3).

FIGURE 5.3 Stock value versus bond value of McLeodUSA.

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115

5. IMPLICATIONS OF THE MODEL The proposed model is an extension of Merton’s structural model, where we have used the relative valuation approach to solve for valuation of the market value of debt. Therefore, we have not used any historical estimation of the survival rate of the bonds, which require the model to hypothesize the appropriate market discount rate for the expected cash ﬂow. Also, this approach does not require any assumption of the recovery ratio, which is endogenous in the structural model. When compared with the Merton model, this model predicts that the ﬁrm can default without the crisis of maturity. The default event can be triggered by the ﬁxed costs. This model can explain the market observation that some ﬁrm has a low credit rating and trades with a high-yield spread, even though the market debt-to-equity ratio is low. This model of the bond price is sensitive to the stock price, as with the Merton model. The main advantage of the proposed model is that it allows us to use more detailed information about the ﬁrm, and therefore the bond valuation model is more realistic. Finally, the model is empirically testable. The ﬁnancial data, bond yield spreads, and stock prices are relatively accessible.

6. CONCLUSIONS This chapter proposes a valuation of bonds with credit risks. The main contribution of the chapter is to introduce the concept of a primitive ﬁrm. The primitive ﬁrm has neither operating leverage nor ﬁnancial leverage. Given the risk class of this primitive ﬁrm, we can determine the value of a ﬁrm with operating and ﬁnancial leverage as a contingent claim to the primitive ﬁrm. This approach enables us to introduce the business model of a ﬁrm in the high-yield bond valuation. Relating the high-yield bond pricing to the ﬁrm’s business is standard in practice, and this approach provides a more robust analytical framework for practitioners in the high-yield area. The model provides intuitive insights into the high-yield bond behavior. For example, the model shows that the ﬁxed costs of the ﬁrm can be viewed as the “perpetual debt” of the ﬁrm, senior to the ﬁnancial debt obligations. Such a treatment of the ﬁxed costs signiﬁcantly affects the probability distribution in the event of default of a ﬁrm and the valuation of the debt. Furthermore, it shows that the bond price is more sensitive to the market capitalization. As the market capitalization falls, the ﬁrm with signiﬁcant ﬁxed costs would fall precipitously with the market capitalization, before the event of default becomes imminent. For this reason, the model shows that the relationship

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THE CREDIT MARKET HANDBOOK

of the probability of default and the bond price must also depend on the ﬁxed cost level. This may explain the observation that the rating of the bond tends to lag behind the market pricing of the bonds. Since the bond rating measures the probability of default, which is different from the bond pricing as this model shows, the extent of the lag must depend on the ﬁrm’s operating leverage. This also suggests that models that use bond value or market capitalization to predict the probability of default can also be erroneous. We show that the bond value depends on both the recovery ratio and the probability of default. Since the recovery ratio depends on the ﬁrm’s operating leverage, the ﬁrm default probability cannot be related simply to the market capitalization or the bond price without taking the ﬁrm’s business model into account. Finally, the proposed model shows that the use of a primitive ﬁrm can have broad implications for future research. The study of the primitive ﬁrms for different market sectors will enable us to better understand the high-yield bond valuation.

ACKNOWLEDGMENTS We would like to thank Yoonseok Choi, Hanki Seong, Yuan Su, and Blessing Mudavanhu for their assistance in developing the models and in our research. We also would like to thank Owen Graduate School of Management at Vanderbilt University, seminar participants of Management for valuable comments. Any errors are ours.

APPENDIX The valuation formula of the perpetual debt was given by Merton (1973): 2 FC (2FC/σ 2 V)2rf /σ (V, ∞) = 1− rf (2 + (2rf /σ 2 )) 2rf 2rf −2FC ×M , 2 + , σ2 σ 2 σ 2V

(A.1)

where V is the primitive ﬁrm value, FC the ﬁxed cost per year, rf the riskfree rate, the gamma function (deﬁned below), σ the standard deviation

Valuing High-Yield Bonds: A Business Modeling Approach

117

of GRI, and M(·) the conﬂuent hypergeometric function (deﬁned below). a 1 −b/V b 2FC = −(1 + a)bFC e M a, 2 + a, − 2 brf V σ V + eb/V FC aV (2 + a) b + (1 + a)(b − aV) 1 + a, V where a=

2rf σ

and

∞ 2FC , (x) = t x−1 e−t dt, σ2 0

∞ t a−1 e−t dt (a, x) =

, 2

b=

x

NOTES 1. See p. 792, Assumption 4, Longstaff and Schwartz (1995).

REFERENCES Briys, E. and de Varenne, F. (1997). “Valuing Risky Fixed Rate Debt: An Extension.” Journal of Financial and Quantitative Analysis 32(2), 239–248. Ericsson, J. and Reneby, J. (1998). “A Framework for Valuing Corporate Securities.” Applied Mathematical Finance 5(3), 143–163. Ho, T. S. Y. and Lee, S. B. (2004). The Oxford Guide to Financial Modeling. New York: Oxford University Press. Longstaff, F. A. and Schwartz, E. M. (1995). “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt.” Journal of Finance 50(3), 789–819. Saá-Requejo, J. and Santa-Clara, P. (1999). “Bond Pricing with Default Risk.” Working Paper, University of California, Los Angeles. Shimko, D., Tejima, N. and van Deventer, D. (1993). “The Pricing of Risky Debt when Interest Rates are Stochastic.” Journal of Fixed Income 3(2). Vasicek, O. (1977). “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics 5, 177–188.

Keywords: High yield bonds; business model; structural model; contingent claim theory; operating leverage; default

CHAPTER

6

Structural versus Reduced-Form Models: A New Information-Based Perspective Robert A. Jarrowa,∗ and Philip Protterb

This chapter compares structural versus reduced-form credit risk models from an information-based perspective. We show that the difference between these two model types can be characterized in terms of the information assumed known by the modeler. Structural models assume that the modeler has the same information set as the ﬁrm’s manager—complete knowledge of all the ﬁrm’s assets and liabilities. In most situations, this knowledge leads to a predictable default time. In contrast, reduced-form models assume that the modeler has the same information set as the market—incomplete knowledge of the ﬁrm’s condition. In most cases, this imperfect knowledge leads to an inaccessible default time. And so we argue that the key distinction between structural and reduced-form models is not whether the default time is predictable or inaccessible, but whether or not the information set is observed by the market. Consequently, for pricing and hedging, reduced-form models are the preferred methodology.

1. INTRODUCTION For modeling credit risk, two classes of models exist: structural and reducedform. Structural models originated with Black and Scholes (1973) and Merton (1974), and reduced-form models originated with Jarrow and a Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853,

USA. b School of Operations Research, Cornell University, Ithaca, NY 14853-3801, USA. ∗ Corresponding author. Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853, USA. E-mail: [email protected].

118

Structural versus Reduced-Form Models

119

Turnbull (1992), and were subsequently studied by Jarrow and Turnbull (1995), Dufﬁe and Singleton (1999), and others. These models are viewed as competing (see Bielecki and Rutkowski, 2002; Rogers, 1999; Lando, 2003; Dufﬁe, 2003), and there is a heated debate in the professional and academic literature as to which class of models is best (see Jarrow et al., 2003, and references therein). This debate usually revolves around default prediction and/or hedging performance. The purpose of this chapter is to show that these models are not disconnected and disjoint model types, as is commonly supposed, but rather are really the same model containing different informational assumptions. Structural models assume complete knowledge of a very detailed information set, akin to that held by the ﬁrm’s managers. In most cases, this informational assumption implies that a ﬁrm’s default time is predictable. But this is not necessarily the case.1 In contrast, reduced-form models assume knowledge of a less detailed information set, akin to that observed by the market. In most cases, this informational assumption implies that the ﬁrm’s default time is inaccessible. Given this insight, one sees that the key distinction between structural and reduced-form models is not in the characteristic of the default time (predictable versus inaccessible), but in the information set available to the modeler. Indeed, structural models can be transformed into reducedform models as the information set changes and becomes less reﬁned from that observable by the ﬁrm’s management to that which is observed by the market. An immediate consequence of this observation is that the current debate in the credit risk literature about these two model types is misdirected. Rather than debating which model type is best in terms of forecasting performance, the debate should be focused on whether the model should be based on the information set observed by the market or not. For pricing and hedging credit risk, we believe that the information set observed by the market is the relevant one. This is the information set used by the market, in equilibrium, to determine prices. Given this belief, a reduced-form model should be employed. As a corollary of this information structure, the characteristic of the ﬁrm’s default time is determined—whether it is a predictable or totally inaccessible stopping time. Surprisingly, at this stage in the credit risk literature, there appears to be no disagreement that the asset value process is unobservable by the market [see, especially, Duan (1994) and Ericsson and Reneby (2002, 2003) in this regard]. Although not well understood in terms of its implication, this consensus supports the use of reduced-form models. In addition, without the continuously observed ﬁrm’s asset value, the available information set often implies that the ﬁrm’s default time is inaccessible, and that a hazard rate model should be applied.

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An outline of this chapter is as follows: Section 2 sets up the common terminology and notation. Section 3 reviews structural models, and Section 4 reviews reduced-form models. Section 5 links the two model types in a common mathematical structure. Section 6 provides the market information-based perspective, and Section 7 concludes the chapter.

2. THE SETUP Credit risk investigates an entity (corporation, bank, individual) that borrows funds, promises to return these funds under a prespeciﬁed contractual agreement, and may default before the funds (in their entirety) are repaid. ¯ Given Let us consider a continuous time model with time period [0, T]. ¯ on this time interval is a ﬁltered probability space {(, G, P), (Gt : t ∈ [0, T])} satisfying the usual conditions, with P the statistical probability measure. ¯ will be key to our subsequent arguThe information set (Gt : t ∈ [0, T]) ments. We take the perspective of a modeler evaluating the credit risk of a ﬁrm. The information set is that information held by the modeler. Often, in the subsequent analysis, we will be given a stochastic process Yt that may be a vector valued process, and we will want to study the information set generated by its evolution. We denote this information set by σ (Ys : s ≤ t). Let us start with a generic ﬁrm that borrows funds in the form of a zero-coupon bond promising to pay a dollar (the face value) at its maturity, ¯ Its price at time t ≤ T is denoted by v(t, T). Let this time T ∈ [0, T]. be the only liability of the ﬁrm. Also traded are default-free zero-coupon bonds of all maturities, with the default-free spot rate of interest denoted by rt . Markets for the ﬁrm’s bond and the default-free bonds are assumed to be arbitrage-free; hence, there exists an equivalent probability measure Q such that all discounted bond prices are martingales with respect to the t ¯ The discount factor is e− 0 rs ds . Markets information set {Gt : t ∈ [0, T]}. need not be complete, so that the probability Q may not be unique.

3. STRUCTURAL MODELS This section reviews the structural models in an abstract setting. As mentioned in the introduction, structural models were introduced by Black and Scholes (1993) and Merton (1974). The simplest structural model is used to illustrate this approach. The key postulate emphasized in our presentation is ¯ that the modeler observes contains that the information set {Gt : t ∈ [0, T]} the ﬁltration generated by the ﬁrm’s asset value. Let the ﬁrm’s asset value be denoted by At . Then, Ft = σ (As : s ≤ t) ⊂ Gt .

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Let the ﬁrm’s asset value follow a diffusion process that remains nonnegative: dAt = At α(t, At )dt + At σ (t, At )dWt

(6.1)

where αt and σt are suitably chosen so that equation 6.1 is well deﬁned, and Wt is a standard Brownian motion. See, for example, Theorem 71 on page 345 of Protter (2004), where such equations are treated. Given the liability structure of the ﬁrm, a single zero-coupon bond with maturity T, and face value 1, default can happen only at time T. And default happens only if AT ≤ 1. Thus, the probability of default on this ﬁrm at time T is given by P(AT ≤ 1) The time 0 value of the ﬁrm’s debt is T v(0, T) = EQ [min(AT , 1)]e− 0 rs ds

(6.2)

Assuming that interest rates rt are constant, and that the diffusion coefﬁcient σ (t, x) = σ is constant (that is, the ﬁrm’s asset value’s volatility is constant), this expression can be evaluated in closed form. The expression is v(0, T) = e−rT N(d2 ) + At N(−d1 )

(6.3)

where N(·) is the cumulative function, d1 = √ standard normal distribution √ [log(At ) + (r + σ 2 /2)T]/σ T, and d2 = d1 − σ T. This is the original risky-debt model of Black and Scholes (1973) and Merton (1974), where the ﬁrm’s equity is viewed as a European call option on the ﬁrm’s assets with maturity T and a strike price equal to the face value of the debt. To see this, note that the time T value of the ﬁrm’s equity is AT − min(AT , 1) = max(AT − 1, 0). The right-hand side of this expression corresponds to the payoff of the previously mentioned European call option on the ﬁrm’s time T asset value. Because the Black–Scholes and Merton models have default occurring only on one date, the models have since been generalized to allow default prior to time T if the asset’s value hits some prespeciﬁed default barrier, Lt . The economic interpretation is that the default barrier represents some debt covenant violation. In this formulation, the barrier itself could be a stochastic process. Then, the information set must be augmented to include

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this process as well, that is, Ft = σ (As , Ls : s ≤ t). We assume that in the event of default, the debt holders receive the value of the barrier at time T. Other formulations are possible.2 In this generalization, the default time becomes a random variable and it corresponds to the ﬁrst hitting time of the barrier τ = inf {t > 0 : At ≤ Lt }

(6.4)

Here, the default time is a predictable stopping time.3 Formally, a stopping time τ is a non-negative random variable such that ¯ (see Protter, 2004). A stopping the event {τ ≤ t} ∈ Ft for every t ∈ [0, T] time τ is predictable if there exists a sequence of stopping times (τn )n≥1 such that τn is increasing, τn ≤ τ on {τ > 0} for all n, and limn→∞ τn = τ a.s. Intuitively, a predictable stopping time is “known” to occur “just before” it happens, since it is “announced” by an increasing sequence of stopping times. This is certainly the situation for the structural model constructed above. In essence, although default is an uncertain event and thus technically a surprise, it is not a “true surprise” to the modeler, since it can be anticipated with almost certainty by watching the path of the asset’s value process. The key characteristic of a structural model that we emphasize in this chapter is the observability of the information set Ft = σ (As , Ls : s ≤ t), and not the fact that the default time is predictable. Given the default time in equation 6.4, the value of the ﬁrm’s debt is given by T v(0, T) = EQ [1{τ ≤T} Lτ + 1{τ >T} 1]e− 0 rs ds

(6.5)

If interest rates rt are constant, the barrier is a constant L, and the asset’s volatility σt is constant; then, equation 6.5 can be evaluated explicitly, and its value is v(0, T) = Le−rT Q(τ ≤ T) + e−rT [1 − Q(τ ≤ T)] 2

(6.6)

(1−2r/σ ) × N(h (T)), h (T) = where Q(τ ≤ T) = N(h 2 1 √ 1 (T)) + A0 e √ 2 [− log A0 − (r − σ /2)T]/σ T, and h2 (T) = [− log A0 + (r − σ 2 /2)T]/σ T (see Bielecki and Rutkowski, 2002 p. 67). Returning again to the more general risky-debt model contained in equation 6.5, note that interest rates are stochastic. Using the simple zerocoupon bond liability structure as expressed above, this more general

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formulation includes the models of Shimko et al. (1993), Nielsen et al. (1993), Longstaff and Schwartz (1995), and Hui et al. (2003). Generalizations of this formulation to more complex liability structures include the papers by Black and Cox (1976), Jones et al. (1984), and Tauren (1999). From the perspective of this chapter, the key assumption of the structural approach is that the modeler has the information set generated by continuous observations of both the ﬁrm’s asset value and the default barrier. This is equivalent to the statement that the modeler has continuous and detailed information about all of the ﬁrm’s assets and liabilities. This is the same information set that is held by the ﬁrm’s managers (and regulators in the case of commercial banks). This information set often implies that the default time is predictable, but this is not necessarily the case.

4. REDUCED-FORM MODELS This section reviews reduced-form models in an abstract setting. Reducedform models were originally introduced by Jarrow and Turnbull (1992) and subsequently studied by Jarrow and Turnbull (1995), and Dufﬁe and Singleton (1999), among others. As before, the simplest structure is utilized to illustrate the approach. The key postulate emphasized here is that the modeler observes the ﬁltration generated by the default time τ and a vector of state variables Xt , where the default time is a stopping time generated by a Cox process Nt = 1{t≤t} with an intensity process λt depending on the vector of state variables Xt (often assumed to follow a diffusion process), that is, Ft = σ (τ , Xs : s ≤ t) ⊂ Gt . Intuitively, a Cox process is a point process where, conditional on the information set generated by the state variables ¯ the conditioned process is Poisover the entire time interval σ (Xs : s ≤ T), son with intensity λt (Xt ); see Lando (1998). In reduced-form models, the processes are normally speciﬁed under the martingale measure Q. In this formulation, the stopping time is totally inaccessible. Formally, a stopping time τ is a totally inaccessible stopping time if, for every predictable stopping time S, Q{ω : τ (ω) = S(ω) < ∞} = 0.4 Intuitively, a totally inaccessible stopping time is not predictable; that is, it is a “true surprise” to the modeler. The difference between predictable and totally inaccessible stopping times is not the key distinction between structural and reduced-form models that we emphasize herein. To complete this formulation, we also give the payoff to the ﬁrm’s debt in the event of default, called the recovery rate. This is usually given by a stochastic process δt , also assumed to be part of the information set available

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to the modeler, that is, Ft := σ (τ , Xs , δs : s ≤ t). This recovery rate process can take many forms [see Bakshi et al. (2001) in this regard]. We assume, to be consistent with the structural model in the previous section, that the recovery rate δτ is paid at time T. We emphasize that the reduced-form information set requires less detailed knowledge on the part of the modeler about the ﬁrm’s assets and liabilities than does the structural approach. In fact, reduced-form models were originally constructed to be consistent with the information that is available to the market. In the formulation of the reduced-form model presented, the probability of default prior to time T is given by ¯ Q(τ ≤ T) = EQ EQ (N(T) = 1 | σ (Xs : s ≤ T)) T = 1 − EQ e− 0 λs ds

(6.7)

The value of the ﬁrm’s debt is given by T v(0, T) = EQ [1{τ ≤T} δτ + 1{τ >T} 1]e− 0 rs ds

(6.8)

Note that a small, but crucial, distinction between the pricing equation 6.5 in the structural model and the pricing equation 6.8 in the reduced-form model is that the recovery rate process is prespeciﬁed by a knowledge of the liability structure in the structural approach, whereas here it is exogenously supplied. This distinction is characteristic of a reduced-form model to the extent that the liability structure of the ﬁrm is usually not continuously observable, whereas the resulting recovery rate process is. This formulation includes the model of Jarrow and Turnbull (1995), Jarrow et al. (1997), Lando (1998), Duffie and Singleton (1999), Madan and Unal (1998), and others. For example, if the recovery rate and intensity processes are constants (δ, λ), then this expression can be evaluated explicitly, generating the model in Jarrow and Turnbull (1995) where the debt’s value is given by v(0, T) = p(0, T) δ + (1 − δ)e−λT

(6.9)

T where p(0, T) = EQ e− 0 rs ds . Other extensions of this model include the inclusion of counterparty risk; see Jarrow and Yu (2001).

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5. A MATHEMATICAL OVERVIEW This section relates structural models to reduced-form models by concentrating on the information sets held by the modeler. We claim that if one changes the information set held by the modeler from more to less information, then a structural model with default being a predictable stopping time can be transformed into a hazard-rate model with default being an inaccessible stopping time. To see this, we imbed the different approaches into one unifying mathematical framework. Let us begin with a ﬁltered complete probability space (, G, P, G), where G = (Gt )t≥0 . The information set G is the information set available to the modeler. On this space, we assume a given Markov process A = (At )t≥0 , where At represents the ﬁrm’s value at time t. Often one assumes that A is a diffusion and that it solves a stochastic differential equation driven by Brownian motion and the Lebesgue measure. For simplicity, let us assume exactly that; that is, At = 1 +

t 0

σ (s, As )dWs +

t 0

µ(s, As )ds

In a structural model, one assumes that the modeler observes the ﬁltration generated by the ﬁrm’s asset value F = (σ (As : 0 ≤ s ≤ t))t≥0 from which one can divine the coefﬁcients σ and µ. Consequently, F ⊂ G. Default occurs when the value of the ﬁrm crosses below some threshold level process L = (Lt )t≥0 , where it is assumed that once default occurs, the ﬁrm cannot recover. While this default barrier can be a stochastic process, for the moment we assume that it is a constant. We then let τ = inf{t > 0 : At ≤ L} Default occurs at the time τ . The stopping time τ will then be G predictable. As an alternative to the simple model just described, it is reasonable to assume that the modeler does not continuously observe the ﬁrm’s asset value. Indeed, there appears to be no disagreement in the literature that the asset value process is unobservable [see, especially, Duan (1994) and Ericsson and Reneby (2002, 2003) in this regard]. Then, given the modeler has partial information, the question then becomes how to model this partial information? A method proposed by Dufﬁe and Lando (2001) is to obscure the process A by observing it only at discrete time intervals (not continuously) and by adding independent noise. One then obtains a discrete time process Zt = At + Yt , where Yt is the added noise process, and which is observed at

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times ti for i = 1, . . . , ∞. Since one sees Z and not A, one has a different ﬁltration of observable events: H = (Ht )t≥0 ⊂ G, where Ht = σ (Zti : 0 ≤ ti ≤ t), up to null sets. Here, τ need not be a stopping time for the ﬁltration H. One is now interested in the quantity P(τ > u | Ht ) However, for this example, Dufﬁe and Lando (2001) assume a structure such that they are able to show that t τ is still a stopping time, but one that is totally inaccessible with 1{t≥τ } − 0 λ(s, ω)ds a local martingale. Colloquially, one says that the stopping time τ has the intensity process λ = (λt )t≥0 .5 The reason for this transformation of the default time τ from a predictable stopping time to an inaccessible stopping time is that, between the time observations of the asset value, we do not know how the asset value has evolved. Consequently, prior to our next observation, default could occur unexpectedly (as a complete surprise). Under this coarser information set, the price of the bond is given by equation 6.8 with the recovery rate L. Here, the structural model, due to both information obscuring and reduction, is transformed into an intensity-based hazard-rate model. Kusuoka (1999) has proposed a more abstract version of the Dufﬁe– Lando model where the relevant processes are observed continuously and not discretely. In this approach, Kusuoka does not specify where the default time τ comes from, but rather he begins with the modeler’s ﬁltration H ⊂ G and a positive random variable τ . He then expands the ﬁltration “progressively” so that in the expanded ﬁltration J ⊂ G the random variable τ is a stopping time. [The interested reader can consult Chapter VI of Protter (2004) for an introduction to the theory of the expansion of ﬁltrations.] This ﬁltration expansion is analogous to adding noise to the asset value process as in Dufﬁe and Lando (2001). Unfortunately, Kusuoka (1999) makes the rather restrictive hypothesis that all H martingales remain martingales in the larger ﬁltration J, thereby limiting the application of his model. Another variant of introducing noise to the system is that proposed by Giesecke and Goldberg (2003), where the default barrier itself is taken to be a random curve. (This is usually taken to be a horizontal line of the form y = L, but where the level L itself is unknown, and modelled by making it random.) The modeler cannot see this random curve, which is constructed to be independent of the underlying structural model. Since the default time depends on a curve that cannot be observed by the modeler, the default time τ is rendered totally inaccessible. Nonetheless, Giesecke and Goldberg (2003) still assume that the ﬁrm’s asset value process is observed continuously, making their structure a kind of hybrid model, including both

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reduced-form elements and strong assumptions related to structural models, namely, that one can observe the asset’s value. Çetin et al. (2004) take an alternative approach to those papers just cited. Instead of adding noise to obscure information as in Dufﬁe–Lando, or beginning with the investor’s ﬁltration and expanding it as in Kusuoka (1999), Çetin et al. begin with a structural model as in Dufﬁe–Lando, but where the modeler’s ﬁltration G is taken to be a strict subﬁltration of that available to the ﬁrm’s managers. Starting with the structural model similar to that described above, Çetin et al. redeﬁne the asset value to be the ﬁrm’s cash ﬂows. The relevant barrier is now Lt = 0, all t ≥ 0. Here, the modeler observes only whether the cash ﬂows are positive, zero, or negative. They assume that the default time is the ﬁrst time, after the cash ﬂows are below zero, when the cash ﬂow both remains below zero for a certain length of time and then doubles in absolute magnitude. Under this circumstance, the default time is totally inaccessible and the point process has an intensity, yielding an intensity-based hazard-rate model. The approach of Çetin et al. (2004) involves what can be seen as the obverse of ﬁltration expansion: namely, ﬁltration shrinkage. However, the two theories are almost the same: after one shrinks the ﬁltration, in principle, one can expand it to recover the original one. Nevertheless, the mathematics is quite different, and it is perhaps more natural for the default time to be a stopping time in both ﬁltrations, and to understand how a predictable default time becomes a totally inaccessible time when one shrinks the ﬁltration. A common feature of these approaches to credit risk is that for the modeler’s ﬁltration H, the default time τ is usually a stopping time, and it changes in nature from a predictable stopping time to a totally inaccessible stopping time. One can now consider the increasing process 1{t≥τ } , which may or may not (depending on the model) be adapted to H. In any event, a common thread is that 1{t≥τ } is, or at least its projection onto H is, a submartingale [see, e.g., Protter (2004) for the fact that its projection is a submartingale], and the increasing process of its Doob–Meyer decomposition can be interpreted as its compensator in H. Since τ is totally inaccessible, will be continuous, and usually one can verify that is of the form t t = 0 λs ds, where the process λ then plays the role of its arrival intensity under the ﬁltration H. Thus, the overall structure is one of ﬁltrations, often with containment relationships, and how stopping times behave in the two ﬁltrations. The structural models play a role in the determination of the structure that begets the default time; but as the information available to the modeler is reduced or obscured, one needs to project onto a smaller ﬁltration, and then the default time (whether changed, obscured, or remaining the same) becomes totally inaccessible, and the compensator of the one jump point

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process 1{t≥τ } becomes the object of interest. If can be written in the form t t = 0 λs ds, then the process λ can be interpreted as the instantaneous rate of default, given the modeler’s information set.

6. OBSERVABLE INFORMATION SETS In the mathematical overview, we observed that a structural model’s default time is modiﬁed when one changes the information set used by the modeler. Indeed, it can be transformed from a predictable stopping time to a totally inaccessible stopping time. From our perspective, the difference between structural and reduced-form models does not depend on whether the default time is predictable or inaccessible, but rather on whether the default time is based on a ﬁltration that is observed by the market or not. Structural models assume that the modeler’s information set G is that observed by the ﬁrm’s managers—usually continuous observations of the ﬁrm’s asset value At and liabilities Lt processes. In contrast, reducedform models assume that the modeler’s information set G is that observed by the market, usually the ﬁltration generated by a stopping time τ and continuous observations of a set of state variables Xt . As shown in the previous section, one can start with the larger information set G, modify it to a reduced information set H, and transform a structural model to a reduced-form model. In the review of the literature contained in the previous section, using this classiﬁcation scheme, we see that the model of Giesecke and Goldberg (2003) is still a structural model, even though it has a totally inaccessible stopping time, while the models of Dufﬁe and Lando (2001), Kusuoka (1999), and Çetin et al. (2004) are reducedform models. Giesecke and Goldberg’s (2003) is still a structural model because it requires the continuous observation of the ﬁrm’s asset value, which is not available to the market, whereas in the models of Dufﬁe and Lando (2001), Kusuoka (1999), and Çetin et al. (2004), the information sets assumed are those available to the market. Which model is preferred—structural or reduced-form—depends on the purpose for which the model is being used. If one is using the model for risk management purposes—pricing and hedging—then the reduced-form perspective is the correct one to take. Prices are determined by the market, and the market equilibrates based on the information that it has available to make its decisions. In marking-to-market, or judging market risk, reduced-form models are the preferred modeling methodology. Instead, if one represents the management within a ﬁrm, judging its own ﬁrm’s default risk for capital considerations, then a structural model may be preferred. However, this is

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not the approach one wants to take for pricing a ﬁrm’s risky-debt or related credit derivatives.

7. CONCLUSION The informational perspective of our chapter implies that to distinguish which credit risk model is applicable—structural or reduced-form—one needs to understand what information set is available to the modeler. Structural models assume that the information available is that held by the ﬁrm’s managers, while reduced-form models assume that it is the information observable to the market. Given this perspective, the deﬁning characteristic of these models is not the property of the default time—predictable or inaccessible—but rather the information structure of the model itself. If one is interested in pricing a ﬁrm’s risky debt or related credit derivatives, then reduced-form models are the preferred approach. Indeed, there is consensus in the credit risk literature that the market does not observe the ﬁrm’s asset value continuously in time. This implies, then, that the simple form of structural models illustrated in Section 3 above does not apply. In contrast, reduced-form models have been constructed, purposefully, to be based on the information available to the market.

ACKNOWLEDGMENT This author is supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092.

NOTES 1. An example would be where the ﬁrm’s asset value follows a continuous time jump diffusion process. 2. For example, this could be easily modiﬁed to be the value of the barrier paid at the default time τ . This simple recovery rate process is selected to simplify equation 6.6. 3. If the asset price admits a jump, then the default time usually would not be predictable. From the perspective of this chapter, however, this would still be called a structural model. 4. Recall that the formulation of most reduced-form models takes place under the martingale probability measure Q. 5. It is not always the case that the compensator of a totally inaccessible stopping time has an intensity. Having an intensity is equivalent to requiring that the

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REFERENCES Bakshi, F., Madan, D. and Zhang, F. (2001). “Recovery in Default Risk Modeling: Theoretical Foundations and Empirical Applications.” Working Paper, University of Maryland. Bielecki, T. and Rutkowski, M. (2002). Credit Risk: Modeling, Valuation and Hedging. New York: Springer-Verlag. Black, F. and Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81, 637–654. Black, F. and Cox, J. C. (1976). “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” Journal of Finance 31, 351–367. Çetin, U., Jarrow, R., Protter, P. and Yildirim, Y. (2004). “Modeling Credit Risk with Partial Information.” The Annals of Applied Probability, 14(3), 1167–1178. Duan, J. C. (1994). “Maximum Likelihood Estimation using Price Data of the Derivative Contract.” Mathematical Finance 4(2), 155–167. Dufﬁe, D. (2003). Dynamic Asset Pricing Theory, 3rd ed. Princeton U. Press. Dufﬁe, J. D. and Lando, D. (2001). “Term Structure of Credit Spreads with Incomplete Accounting Information.” Econometrica 69, 633–664. Dufﬁe, D. and Singleton, K. (1999). “Modeling Term Structures of Defaultable Bonds.” Review of Financial Studies 12(4), 197–226. Ericsson, J. and Reneby, J. (2002). “Estimating Structural Bond Pricing Models.” Working Paper, Stockholm School of Economics. Ericsson, J. and Reneby, J. (2003). “The Valuation of Corporate Liabilities: Theory and Tests.” Working Paper, SSE/EFI No. 445. Giesecke, K. “Default and Information.” Working Paper, Cornell University. Giesecke, K. and Goldberg, L. (2003). “Forecasting Default in the Face of Uncertainty.” Working Paper, Cornell University. Hui, C. H., Lo, C. F. and Tsang, S. W. (2003). “Pricing Corporate Bonds with Dynamic Default Barriers.” Journal of Risk 5(3), 17–39. Jarrow, R., Lando, D. and Turnbull, S. (1997). “A Markov Model for the Term Structure of Credit Risk Spreads.” Review of Financial Studies 10, 481–523. Jarrow, R. and Turnbull, S. (1992). “Credit Risk: Drawing the Analogy.” Risk Magazine 5(9). Jarrow, R. and Turnbull, S. (1995). “Pricing Derivatives on Financial Securities Subject to Credit Risk.” Journal of Finance 50(1), 53–85. Jarrow, R. and Yu, F. (2001). “Counterparty Risk and the Pricing of Defaultable Securities.” Journal of Finance 56, 1765–1799. Jarrow, R., van Deventer, D. and Wang, X. (2003). “A Robust Test of Merton’s Structural Model for Credit Risk.” Journal of Risk 6(1), 39–58.

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Jones, E., Mason S. and Rosenfeld, E. (1984). “Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation.” Journal of Finance 39, 611–627. Kusuoka, S. (1999). “A Remark on Default Risk Models.” Advances in Mathematical Economics 1, 69–82. Lando, D. (1998). “On Cox Processes and Credit Risky Securities.” The Review of Derivatives Research 2, 99–120. Lando, D. (2003). Credit Risk Modeling: Theory and Applications, unpublished manuscript. Longstaff, F. A. and Schwartz, E. S. (1995). “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt.” Journal of Finance 50(3), 789–819. Madan, D. and Unal, H. (1998). “Pricing the Risks of Default.” Review of Derivatives Research 2, 121–160. Merton, C. R. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29, 449–470. Nielsen, T. L., Saa-Requejo, J. and Santa-Clara, P. (1993). “Default Risk and Interest Rate Risk: The Term Structure of Default Spreads.” Working Paper, INSEAD. Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd edn. New York: Springer-Verlag. Rogers, L. C. G. (1999). “Modelling Credit Risk.” Working Paper, University of Bath. Shimko, D., Tejima, H. and van Deventer, D. (1993). “The Pricing of Risky Debt when Interest Rates are Stochastic.” Journal of Fixed Income 3, 58–66. Tauren, M. (1999). “A Model of Corporate Bond Prices with Dynamic Capital Structure.” Working Paper, Indiana University. Zeng, Y. (2004). Working Paper, Cornell University.

Keywords: Credit risk; structural models; reduced from models; hazard rate models; risk debt; credit derivatives

CHAPTER

7

Reduced-Form versus Structural Models of Credit Risk: A Case Study of Three Models Navneet Arora,a Jeffrey R. Bohn,a,∗ and Fanlin Zhua

In this chapter, we empirically compare two structural models [basic Merton and Vasicek–Kealhofer (VK)] and one reduced-form model [Hull– White (HW)] of credit risk. We propose here that two useful purposes for credit models are default discrimination and relative value analysis. We test the ability of the Merton and VK models to discriminate defaulters from nondefaulters based on default probabilities generated from information in the equity market. We test the ability of the HW model to discriminate defaulters from nondefaulters based on default probabilities generated from information in the bond market. We ﬁnd the VK and HW models exhibit comparable accuracy ratios and substantially outperform the simple Merton model. We also test the ability of each model to predict spreads in the credit default swap (CDS) market as an indication of each model’s strength as a relative value analysis tool. We ﬁnd the VK model tends to do the best across the full sample and relative subsamples except for cases where an issuer has many bonds in the market. In this case, the HW model tends to do the best. The empirical evidence will assist market participants in determining which model is most useful based on their “purpose in hand.” On the structural side, a basic Merton model is not good enough; appropriate modiﬁcations to the framework make a difference. On the reduced-form side, the quality and quantity of data make a difference; many traded issuers will not be well modeled in this way unless they issue more traded debt. In addition, bond spreads at shorter tenors (less than 2 years) tend to be less correlated with CDS spreads. This makes accurate calibration of the term structure of credit risk difﬁcult from bond data. a Research Group, Moody’s KMV, 1620 Montgomery Street, San Francisco, CA 94111, USA. ∗ Corresponding author. E-mail: [email protected].

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1. INTRODUCTION Complete “realism” is clearly unattainable, and the question whether a theory is realistic “enough” can be settled only by seeing whether it yields predictions that are good enough for the purpose in hand or that are better than predictions from alternative theories (Friedman, 1953). This insight presented decades ago applies to the current debate regarding structural and reduced-form models. While much of the debate rages about assumptions and theory, relatively little is written about the empirical application of these models. The research results highlighted in this chapter improve our understanding of the empirical performance of several widely known credit pricing models. In this way, we can better evaluate whether particular models are good enough for our collective purpose(s) in hand. Let us begin this discussion by considering some of the key theoretical frameworks developed for modeling credit risk. Credit pricing models changed forever with the insights of Black and Scholes (1973) and Merton (1974). Jones et al. (1984) punctured the promise of these “structural” models of default by showing how these types of models systematically underestimated observed spreads. Their research reﬂected a sample of ﬁrms with simple capital structures observed during the period 1977–1981. Ogden (1987) conﬁrmed this result, ﬁnding that the Merton model underpredicted spreads over US Treasuries by an average of 104 basis points. KMV (now Moody’s–KMV or MKMV) revived the practical applicability of structural models by implementing a modiﬁed structural model called the Vasicek–Kealhofer (VK) model (see Vasicek, 1984; Crosbie and Bohn, 2003; Kealhofer, 2003a,b). This VK model is combined with an empirical distribution of distance-to-default to generate the commercially available Expected Default Frequency™ or EDF™ credit measure. The VK model builds on insights gleaned from modiﬁcations to the classical structural model suggested by other researchers. Black and Cox (1976) model the default point as an absorbing barrier. Geske (1977) treats the liability claims as compound options. In this framework, Geske assumes the ﬁrm has the option to issue new equity to service debt. Longstaff and Schwartz (1995) introduce stochastic interest rates into the structural model framework to create a two-factor speciﬁcation. Leland and Toft (1996) consider the impact of bankruptcy costs and taxes on the structural model output. In their framework, they assume the ﬁrm issues a constant amount of debt continuously with ﬁxed maturity and continuous coupon payments. CollinDufresne and Goldstein (2001) extend the Longstaff and Schwartz model by introducing a stationary leverage ratio, allowing ﬁrms to deviate from their target leverage ratio in the short run only.

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While empirical evidence is still scant, a few empirical researchers have begun to test these model extensions. Lyden and Saraniti (2000) compare the Merton and the Longstaff–Schwartz models and ﬁnd that both models underpredict spreads; the assumption of stochastic interest rates does not seem to change the qualitative nature of the ﬁnding. Eom et al. (2003) ﬁnd evidence contradicting conventional wisdom on the bias of structural model spreads. They ﬁnd structural models that depart from the Merton framework tend to overpredict spreads for the debt of ﬁrms with high volatility or high leverage. For safer bonds, these models, with the exception of Leland–Toft, underpredict spreads. On the commercial side, MKMV offers a version of the VK model applied to valuing corporate securities that is built on a speciﬁcation of the default-risk-free rate, the market risk premium, liquidity premium, and expected recovery in the context of a structural model. The VK model framework is used to produce default probabilities deﬁned as EDF credit measures and is then extended to produce a full characterization of the value of a creditrisky security. This model appears to produce unbiased robust predictions of corporate bond credit spreads (see Bohn, 2000a,b; Agrawal et al., 2004, for more details). Some important modiﬁcations to the typical structural framework include estimation of an implicit corporate-risk-free reference curve instead of using the US Treasury curve. Some of the underprediction found in the standard testing of the Merton model likely results from choosing the wrong benchmark curve in the sense that the spread over US Treasuries includes more than compensation for just corporate credit risk. The assumption here is that the appropriate corporate default-risk-free curve is closer to the US swap curve (typical estimates are 10–20 basis points less than the US swap curve). The MKMV implementation of the VK model allows for a timevarying market risk premium, which materially improves the performance of the model. Other important modiﬁcations to the framework include the speciﬁcation of a liquidity premium that may be associated with the ﬁrm’s access to capital markets and the assumption of a time-varying expected recovery amount. All of these modiﬁcations contribute to producing a more usable structural model. The structural model is particularly useful for practitioners in the credit portfolio and credit risk management ﬁelds. The intuitive economic interpretation of the model facilitates consistent discussion regarding a variety of credit risk exposures. Corporate transaction analysis is also possible with the structural model. If an analyst wants to understand the impact on credit quality of increased borrowing, share repurchases, or the acquisition of another ﬁrm, the structural model naturally lends itself to understanding the transaction’s implications. In general, the ability to diagnose the inputs and outputs of the structural model in terms of understandable economic variables

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(e.g., asset volatility as a proxy for business risk and the market’s assessment of an enterprise’s value) facilitates better communication among loan originators, credit analysts, and credit portfolio managers. The other major thread of credit-risk modeling research focuses on “reduced-form” models of default. This approach assumes a ﬁrm’s default time is inaccessible or unpredictable and driven by a default intensity that is a function of latent state variables. Jarrow and Turnbull (1995), Dufﬁe and Singleton (1999), Hull and White (2000), and Jarrow (2001) present detailed explanations of several well-known reduced-form modeling approaches. Many practitioners in the credit trading arena have tended to gravitate toward this modeling approach, given its mathematical tractability. Jarrow and Protter (2004) argue further that reduced-form models are more appropriate in an information theoretic context, given that we are unlikely to have complete information about the default point and expected recovery. Strictly speaking, most structural models assume complete information.1 Jarrow and Protter’s claim rests on the premise that a modeler has only as much information as the market, making the reduced-form approach more realistic. In practice, however, the complete information assumption in structural models is an approximation designed to facilitate a simpler way of capturing the various economic nuances of how a ﬁrm operates. The strength or weakness of a model should be evaluated on its usefulness in real-world applications. A reduced-form model, while not compromising on the theoretical issue of complete information, suffers from other weaknesses including lack of clear economic rationale for deﬁning the nature of the default process. An alternative way of characterizing the differences between the two models is that structural models are closer to models that use fundamentals for pricing, whereas reduced-form models are closer to models that rely on relative pricing. Reduced-form models are characterized by ﬂexibility in their functional form. This ﬂexibility is both a strength and a weakness. Given the ﬂexible structure in the functional form for reduced-form models, ﬁtting a narrow collection of credit spreads is straightforward. However, this ﬂexibility in functional form may result in a model with strong in-sample ﬁtting properties, but poor out-of-sample predictive ability. Since this type of model reﬂects a framework not directly rooted in an explanation of why a ﬁrm defaults (i.e., less grounded in the economics driving default than in mathematical tractability), diagnosing how to improve performance of these models can be challenging. In addition, difﬁculties in interpretation of results can be acute when modeling large cross sections of debt instruments— particularly when there is a high degree of heterogeneity in terms of credit quality. Without empirically testing the costs and beneﬁts of any particular modeling approach, it is premature to draw conclusions based on purely theoretical arguments.

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The empirical testing of reduced-form models is still nascent. The reason relates back to the lack of theoretical guidance on characterizing the default intensity process. Dufﬁe (1999) found that the parameter estimates using a square-root process of intensity can be fairly unstable. Another reason relates to the bond data on which these models are usually calibrated. These data are typically indicative, creating data problems like a slow “leakage” of information into the price, thereby producing misleading results.2 Transaction bond price sources like the Trade Reporting and Compliance Engine (TRACE) may alleviate data problems, but these sources are new and do not provide detailed time series of data. Other sources of bond data continue to be plagued by missing and inaccurate data. A ﬁnal reason involves the difﬁculty in empirically separating the merits of the modeling framework and the quality of the underlying data, given that bond data are typically used to ﬁt the model as well as test the model. Structural models are primarily based on equity price data and will not suffer from this difﬁculty when they are then tested on bond data. The recent availability of credit default swap data provides a new opportunity to understand the power of both the structural and reduced-form modeling frameworks.3 The crucial question for academicians and practitioners alike is which modeling approach is better in terms of discriminating defaulters from nondefaulters and identifying relative value? The objective of this chapter is to shed empirical light on this question. We test the performance of a classic Merton model, the VK structural model as implemented by MKMV, and a reduced-form model based on Hull and White (2000) (HW) in separating defaulting and nondefaulting ﬁrms in the sample (also known as power-curve testing). We also look at the three models’ performance in explaining the levels and cross-sectional variance of credit default swap data. These three models are chosen because they represent three key stages of the development of the literature in credit risk modeling. The Merton model was the original quantitative structural approach for credit risk modeling. The VK model represents a more realistic and meaningful model for practitioners. (This model was the ﬁrst commercially marketed structural model.) The HW model is a reduced-form approach that was developed to address parameter stability problems associated with existing approaches as described in Dufﬁe (1999). The choice of credit default swap data for testing ensures a neutral ground on which the success of the different models can be evaluated. None of the models is calibrated on the data used for testing. This testing strategy enables us to avoid the pitfalls of testing models on data similar to the data used to ﬁt the models. In this way, we conduct a fair out-of-sample test.4 The chapter is arranged as follows: Section 2 describes the basic methodologies of the Merton, VK, and HW models. Section 3 discusses the data

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and the empirical methodology used in the tests. Section 4 presents the results. We also elaborate on some of the robustness checks we conducted on our results in this section. Section 5 concludes.

2. MERTON, VASICEK–KEALHOFER, AND HULL–WHITE MODELS 2.1 Merton Model Merton (1974) introduced the original model that led to the outpouring of research on structural models. Merton modeled a ﬁrm’s asset value as a lognormal process and assumed that the ﬁrm would default if the asset value, A, fell below a certain default boundary X. The default was allowed at only one point in time, T. The equity, E, of the ﬁrm was modeled as a call option on the underlying assets. The value of the equity was given as E = A[d1 ] − X exp[−rT][d2 ]

(7.1)

where log[A/X] + µ + 12 σ 2 T d1 = √ σ T √ d2 = d1 − σ T and represents the cumulative normal distribution function. The debt value, D, is then given by D=A−E

(7.2)

The spread can be computed as s=−

1 A log [d2 ] + exp[rT][−d1 ] T X

(7.3)

where A is the initial asset value of the ﬁrm, X is the default barrier for the ﬁrm (i.e., if the ﬁrm’s asset value A is below X at the terminal date T, then the ﬁrm is in default), µ is the drift of the asset return, and σ is the volatility of the asset returns. We include this model in our analysis to start with a simple framework as an initial benchmark. A comparison of the performance of a Merton model

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with the MKMV implementation of the VK model (which reﬂects substantial modiﬁcation to the basic Merton framework) will illuminate the impact of relaxing many of the constraining assumptions in the Merton framework.

2.2 VK Model MKMV provides a term structure of physical default risk probabilities using the VK model. This model treats equity as a perpetual down-and-out option on the underlying assets of a ﬁrm. This model accommodates ﬁve different types of liabilities: short-term liabilities, long-term liabilities, convertible debt, preferred equity, and common equity. MKMV uses the option-pricing equations derived in the VK framework to obtain the market value of a ﬁrm’s assets and its associated asset volatility. The default point term structure (i.e., the default barrier at different points in time in the future) is determined empirically. MKMV combines market asset value, asset volatility, and the default point term structure to calculate a distance-to-default (DD) term structure. This term structure is translated into a physical default probability, better known as an EDF credit measure, using an empirical mapping between DD and historical default data log[A/XT ] + µ − 12 σ 2 T DDT = √ σ T

(7.4)

XT in the VK model has a slightly different interpretation than in the Merton model. In the context of the VK model framework, if the ﬁrm’s market asset value A falls below XT at any point in time, then the ﬁrm is considered to be in default. In the DD-to-EDF empirical mapping step, MKMV uses the VK model to estimate a term structure of this default barrier to generate a DD term structure that can be mapped to a default-probability term structure—hence the subscript T for the default barrier X. The basic methodology is discussed in Crosbie and Bohn (2003), Kealhofer (2003a), and Vasicek (1984). This model departs from the traditional structural model in many ways. First, it treats the ﬁrm as a perpetual entity that is continuously borrowing and retiring debt. Second, by explicitly handling different classes of liabilities, it is able to capture richer nuances of the capital structure. Third, it calculates its interim asset volatility by generating asset returns through a de-levering of equity returns. This calculation is different from more common approaches that compute equity volatility and then de-lever it to compute asset volatility. Fourth, MKMV generates the ﬁnal asset volatility by blending the interim empirical asset volatility as computed in the manner explained above together with a modeled volatility estimated from

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comparable ﬁrms. This step helps ﬁlter out noise generated in equity data series. The default probability generated by the MKMV implementation of the VK model is called an expected default frequency or EDF credit measure. These modiﬁcations address many of the concerns raised by Eom et al. (2003) regarding the tendency of Merton models to overestimate spreads for riskier bonds and underestimate spreads for safer bonds. This estimation process also results in term structures of default probabilities that are downward-sloping for riskier ﬁrms and upward-sloping for safer ﬁrms. This pattern is consistent with the empirical credit-migration patterns found in the data.5 Once the EDF term structure is obtained, a related cumulative EDF term structure can be calculated up to any term T referred to as CEDFT . This is then converted to a risk-neutral cumulative default probability CQDFT using the following equation: CQDFT = N[N −1 [CEDFT ] + λ · sqrt(R2 ) · sqrt(T)]

(7.5)

where R2 is the square of correlation between the underlying asset returns and the market index returns, and λ is the market Sharpe ratio.6 The spread of a zero-coupon bond is obtained as s=−

1 log[1 − LGD · CQDFT ] T

(7.6)

where LGD stands for the loss given default in a risk-neutral framework. We make two different sets of assumptions for the LGD and market Sharpe ratio in our implementation of the VK model for the purposes of valuation: ■

■

A ﬁxed value of 0.55 for LGD and a value of 0.4 for the market Sharpe ratio. These choices are consistent with the overall historical estimates of the market Sharpe ratio at MKMV (see Agrawal et al., 2004). The level is also generally consistent with estimates from the equity markets. The LGD estimate is consistent with typical recoveries indicated in defaulted bond data. A sector- and seniority-wise constant LGD, calibrated from the aggregated cross-sectional bonds on any given day. To calibrate, one has to ﬁrst have a typical value of the LGD for the entire sample. A value of 0.55 for LGD is assumed to do this calibration. Using this assumed value of 0.55, one can calibrate the market Sharpe ratio from bond data. After substituting the value of the Sharpe ratio, one can calibrate the sector- and seniority-speciﬁc LGD. This approach is described in detail in Agrawal et al. (2004).

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Our ﬁndings are robust to both assumptions.7 In this chapter, we report only the results based on using the MKMV-supplied EIS (EDF-implied spreads), which assumes the second approach. Note that the ﬂoating leg of a simple CDS (i.e., a single payment of LGD paid out at the end of the contract with a probability of CQDFT ) can also be approximated with this relationship.

2.3 HW Model Hull and White (2000) provide a methodology for valuing credit default swaps when the payoff is contingent on default by a single reference entity and there is no counterparty default risk. Instead of using a hazard rate for the default probability, this model incorporates a default density concept, which is the unconditional cumulative default probability within a period regardless of what happens in other periods. By assuming an expected recovery rate, the model generates default densities recursively based on a set of zerocoupon corporate bond prices and a set of zero-coupon Treasury bond prices. Then the default density term structure is used to calculate the premium of a credit default swap contract. The two sets of zero-coupon bond prices can be bootstrapped from corporate coupon bond prices and Treasury coupon bond prices. They show the credit default swap (CDS) spread s to be T ˆ + A(t))]q(t)v(t) dt [1 − R(1 s = 0T 0 q(t)[u(t) + e(t)] dt + π u(t)

(7.7)

where T is the life of the CDS contract, q(t) is the risk-neutral default probability density at time t, A(t) is the accrued interest on the reference obligation at time t as a percentage of face value, π is the risk-neutral probability of no credit event over the life of the CDS contract, w is the total payments per year made by the protection buyer, e(t) is the present value of the accrued payment from previous payment date to current date, u(t) is the present value of the payments at time t at the rate of $1 on the payment dates, and ˆ is the expected recovery rate on the reference obligation in a risk-neutral R world. The risk-neutral default probability density is obtained from the bond data using the relationship qj =

j−1 Gj − Bj − i=1 qi αij αjj

(7.8)

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where αij is the present value of the loss on a defaultable bond j relative to an equivalent default-free bond at time ti . αij can be described as αij = v(ti )[Fj (ti ) − Rj (ti )Cj (ti )]

(7.9)

Cj is the claim made on the jth bond in the event of default at time ti , while Rj is the recovery rate on that claim, Fj is the risk-free value of an equivalent default-free bond at time ti , while v(ti ) is the present value of a sure payment of $1 at time ti . In this framework, one can infer a risk-neutral default risk density from a cross section of bonds with various maturities. As long as the bonds measure the inherent credit risk and have the same recovery as used in the CDS, one should be able to recover a fair price for the CDS based on the prices of the obligor’s traded bonds.

3. DATA AND EMPIRICAL METHODOLOGY 3.1 Data The data set consists of bond data for each model’s implementation, as described in the preceding section, combined with data on actual CDS spreads used to test each model’s predicted prices. Corporate bond data were provided by EJV, a subsidiary of Reuters. The data include prices quoted daily from 10/02/2000 to 6/30/2004. We selected US dollar-denominated corporate bonds only. There are 706 ﬁrms that have at least two bonds that can be used for bootstrapping. CDS data are from CreditTrade and GFInet, two active CDS brokers. Of the 706 ﬁrms, 542 ﬁrms have CDS data. Therefore, we restricted our analysis to these 542 ﬁrms. The ﬁnal sample tested represented a reasonable cross section of ﬁrms with traded debt and equity mitigating concerns about biases arising from sample selection. While the EJV data are indicative in nature and subject to the price “staleness” concerns described above, the CDS data are likely to be closer to actual transacted prices. (CDSs tend to trade more often than bonds.) The lagging nature of the bond data will be somewhat of a handicap for the HW reduced-form model calibrated on those data. Part of the objective of this study is to present model estimations as they would be done in practice so the nature of the available data is as relevant as the estimation approach. The Merton default probabilities and implied spreads are generated using time series of equity data and ﬁnancial statements on the sample of ﬁrms from COMPUSTAT. The VK daily default probabilities are MKMV EDF credit measures.

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The period of study is particularly interesting as it covers two years of recession in many industrialized countries and includes several ﬁrms with substantial credit deterioration (e.g., Ahold, Fiat, Ford, Nortel, and Sprint). This time period also includes ﬁrms embroiled in major accounting and corporate governance scandals, such as Enron and WorldCom. We required each ﬁrm to have a minimum of two bonds to be included in our data set. The highest number of bonds tracked for any particular issuer in this data set is 24.

3.2 Methodology For the Merton model, we used the default point as 80% of the overall liabilities of the ﬁrm. We chose this number because it performed the best in terms of default predictive power. The concept of default predictive power is discussed in the next section. Regardless, we tested other default point speciﬁcations and found that the qualitative nature of the results are not sensitive to how we specify the default point. For the reduced-form model, we ﬁrst derived the corporate zero rates8 by bootstrapping. This procedure facilitates calculating zero-coupon yield curves from market data. For each ﬁrm, all senior-unsecured straight bonds (i.e., bonds without embedded options, such as those derived from convertibility and callability) are ranked by their time to maturity from 0 to 6 years. For each 3-month interval, one bond at most is selected. To estimate the zero rate, the ﬁrm must have at least two bonds outstanding with suitable price data. The bootstrapping procedure is done using a MATLAB ﬁnancial Toolbox function that takes bond prices, coupon rates, maturities, and coupon frequencies as inputs. The procedure is described in detail in Hull (1999). Using this procedure, we obtain zero rates corresponding to relevant bond maturities. Since many ﬁrms have two to ﬁve bonds and the zero rates from bootstrapping are very noisy, a linear interpolation of these zero rates may lead to unrealistic forward rates. To mitigate this issue, we use a two-degree polynomial function to approximate the zero curve. We then use the ﬁtted function to generate the zero rates every 3 months. Corporate zero-coupon bond prices are calculated using the 3-month-interval zero rates. We obtain the Treasury zero rates from 1 month to 30 years from Bloomberg. Treasury zero-coupon bond prices and forward prices are obtained from the risk-free zero curve. Table 7.1(a) through (c) show the descriptive statistics of the bond and CDS data. In Table 7.1(a), we see that more than 28% of the sample have only two bonds usable for bootstrapping. About 4% of the ﬁrms in the

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Table 7.1(a) Cumulative Percentage Distribution of Issuers by Number of Bonds Used in Bootstrapping for Zero-Coupon Yield Curve Number of Bonds =2 ≤3 ≤4 ≤5 ≤6 ≤7 ≤8 ≤9 ≤10 ≤11 ≤12 ≤13 ≤14 ≤15 ≤16 ≤17 ≤18 ≤19 ≤20 ≤21 ≤22 ≤23 ≤24

Percentage of Sample 28.84 45.83 57.32 65.35 70.96 75.74 79.38 83.31 85.89 87.99 89.68 91.25 92.47 93.59 94.48 95.32 96.09 96.78 97.34 98.01 98.56 99.22 100.00

Table 7.1(b) Cumulative Percentage Distribution of Time to Maturity of Bonds Used in Bootstrapping for Zero-Coupon Yield Curve Time to Maturity ≤0.25 ≤0.50 ≤0.75 ≤1.00 ≤1.25 ≤1.50 ≤1.75 ≤2.00 ≤2.25 ≤2.50 ≤2.75 ≤3.00 ≤3.25 ≤3.50 ≤3.75 ≤4.00 ≤4.25 ≤4.50 ≤4.75 ≤5.00 ≤5.25 ≤5.50 ≤5.75 ≤6.00

Percentage of Sample 5.99 10.50 14.93 19.47 24.03 28.54 33.10 37.71 42.16 46.69 51.25 55.69 59.94 64.19 68.43 72.69 76.87 81.09 85.30 89.20 92.35 95.37 98.33 100.00

sample have more than 20 bonds that were used for bootstrapping. This limited number of bonds per issuer poses difﬁculties given that the accuracy of the implied default probability density depends on the number of bonds available. Table 7.1(b) shows that the time to maturity of the bonds is fairly uniformly distributed between 0 and 6 years, with the density dropping off at the extremes.

144 Table 7.1(c) Spreads

THE CREDIT MARKET HANDBOOK Cumulative Percentage Distribution of CDS by Range of

CDS Spread ≤25 ≤50 ≤75 ≤100 ≤125 ≤150 ≤175 ≤200 ≤225 ≤250 ≤275 ≤300 ≤325 ≤350 ≤375

Percentage of Sample 26.77 48.47 64.02 73.07 78.42 82.86 86.21 88.33 90.17 91.49 92.82 93.81 94.69 95.50 96.16

Table 7.1(c) shows that the underlying issuers span a wide cross section of ﬁnancial health, as measured by CDS spreads.

4. RESULTS 4.1 Default Predictive Power of Models We ﬁrst test the ability of the three models to predict defaults. We do this by rank ordering the ﬁrms in our sample in their probability of default from highest to lowest. We then eliminate x percent of the riskiest ﬁrms from our sample and compute the number of actual defaults that were avoided using this simple rule. This number is expressed as a percentage of the total number of defaults, y%. We vary x from 0 to 100 and ﬁnd a corresponding y for each x. Ideally, for a sample of size N with D defaults in it, when x = D/N, then y should be 100%. This would imply that the default predictive model is perfect, that is, each ﬁrm eliminated would, in fact, be part of the group that actually defaults. The larger the area under the curve of y against x, the better the model’s power to minimize both Type I error (holding a position in a ﬁrm that later defaults) and Type II error (avoiding a position in a ﬁrm that does not default). This area is deﬁned as the model’s accuracy ratio (AR) (see Stein, 2002, 2003, for a more detailed discussion of default model performance evaluation). For a random default-risk measure without any predictive power, the x–y graph should be a 45◦ straight line. The more area between the 45◦ line and the power curve, the more accurate the measure. There are two caveats in interpreting results from this test. First, it is a limited sample: Data on bonds and CDSs are restricted to fewer ﬁrms than the

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data available for equities. Of course, a ﬁrm that does not issue tradeable bonds may not be as interesting to practitioners; however, the increasing interest in trading bank loans and devising new synthetic credit instruments creates demand for analytics to evaluate instruments for which traded bonds and CDSs are not available. Given the large potential for trading credit risk beyond bonds, the applicability of an equity-based structural model to a much more extensive data set is critical to expanding the coverage of ﬁrms and developing market liquidity. The second caveat is that reduced-form models are designed to provide risk-neutral probabilities of default. The order of these may not be the same as that of physical probabilities. For example, a ﬁrm with a low physical probability of default but high systematic risk in its asset process may have a higher risk-neutral probability of default compared to a ﬁrm with a relatively higher physical probability of default but with no systematic risk in its asset process. As credit investors move toward building portfolios with more optimal return–risk proﬁles, distinguishing physical from risk-neutral default probabilities becomes critical. In this test, we make the strong assumption that the order stays the same. Figure 7.1(a) shows the results of our default prediction test. In this ﬁgure, there are three default-risk measures: default probabilities from a simple Merton structural model, default probabilities from the VK model, and

100 90

Percent of defaults excluded

80 70 60 50 40 30 20 HW Power Merton Power VK Power

10 0 0

10

20 30 40 50 60 70 80 Percent of population excluded

90

100

FIGURE 7.1(a) A comparison of predictive power across models as given by the accuracy ratio. The accuracy ratios for the VK model, HW model, and Merton models are 0.801, 0.785, and 0.652, respectively.

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Merton Model

VK Model

1.2 1

EDF

0.8 0.6 0.4 0.2 –40

–30

–20

0 –10 0 Months

10

20

30

FIGURE 7.1(b) Behavior of risk-neutral default probabilities before default for different models. Date 0 represents the date of default.

default probabilities from the HW reduced-form model. All three measures are cumulative 1-year risk-neutral default probabilities. As we can see, the VK structural model ranks the highest in its ability to predict default with an AR of 0.801. The HW model approach is not too far behind with an accuracy ratio of 0.785. The basic Merton model, however, is far behind with 0.652. This demonstrates that with proper calibration of default models, both equities and bonds can be effective sources for information about impending defaults. Figure 7.1(b) shows the median default probabilities from the three models for every defaulted ﬁrm in the data set before and after default. Month 0 reﬂects the month in which each company defaulted. Negative numbers reﬂect the number of months before default and positive numbers reﬂect the number of months after default. As we can see from the graph, for the limited sample where the defaulted companies did, in fact, have traded bonds, both the VK model and the reduced-form model predict defaults reasonably well. The classic Merton model trails the other two models in terms of signalling distress closer to the actual event of default.

4.2 Levels and Cross-Sectional Variation in CDS Spreads We next test the ability of the three models to predict the CDS spread levels and explain the cross-sectional variation in CDS spreads. As discussed above,

Reduced-Form versus Structural Models of Credit Risk

147

the conventional wisdom is that structural models, in general, underpredict actual credit spreads. If this were indeed the case, then one would expect that the CDS spreads predicted by the Merton model would generally underestimate the observed level of CDS spreads. For a modiﬁed structural approach, such as the VK structural model, previous empirical evidence (see Eom et al., 2003) suggests that one should expect the model implied CDS spreads to be underpredicted for safer ﬁrms and overpredicted for riskier ﬁrms. Some of this tendency can be explained by the functional form used for transforming physical default probabilities into risk-neutral default probabilities; the function in the VK framework may be increasing the risk-neutral probabilities too much for high-risk ﬁrms. That said, the model structure does not imply a particular bias in either direction. There is no established pattern in the empirical literature with respect to the accuracy or bias of the spreads predicted by reduced-form models. If bond markets are a fair reﬂection of the inherent risk of a ﬁrm and the corporate bond spreads do, in fact, reﬂect primarily default risk, then the reduced-form model-implied CDS spreads should be an unbiased predictor of CDS spreads. To the extent the reduced-form model is a biased predictor of CDS spreads, the likely cause is factors other than default risk (e.g., liquidity) driving spreads in the corporate bond market. Figures 7.2(a) through (c) show the performance of the three models in their ability to predict the CDS spreads. The graphs show Error = Market CDS − Model CDS as histograms for each model. Consistent with previously published research, we see that the Merton model substantially underpredicts the actual CDS spread, as demonstrated by the right skew of the frequency chart. Both the reduced-form model and the VK structural model seem to be skewed toward the left side, indicating that these models overpredict CDS spreads. However, the skew is much larger for the reduced-form model, with more than 10% of the sample reﬂecting overestimation of CDS spreads by more than 200 bps. This compares to 3.5% of the sample where the VK structural model overestimates CDS spreads by more than 200 bps. The median error in the case of the VK structural model is −33.28 bps, which seems to be consistent with the claim of Eom et al. (2003) that more sophisticated structural models overestimate credit risk. This bias is smaller than the median error of −72.71 bps found for reduced-form models. Consistent with the existing literature, the median error in the case of the Merton model is a positive 53.99 bps, implying that the Merton model does underestimate credit risk, even when measured by CDS spreads. Note also that the VK model generated the smallest median absolute error (Table 7.2).

148

THE CREDIT MARKET HANDBOOK 12

10

Percent

8

6

4

2

0 –200 –165 –130 –95 –60 –25 10 45 HW_error

80 115 150 185

FIGURE 7.2(a) Histogram of difference between the market and the model CDS prices (as given by Market CDS – Model CDS) for HW reduced-form model.

Table 7.2(a) Difference between Market CDS and Model CDS Prices (as given by Market CDS–Model CDS) for Firms in Different CDS Buckets Percentile Difference

Model

Firms with CDS ≤100 bps

Firms with CDS >100 bps

All Firms

p25

HW VK Merton

−111.70 −83.62 21.98

−135.16 −68.60 107.30

−118.75 −80.18 27.34

p50

HW VK Merton

−70.73 −39.68 37.17

−77.96 −7.57 146.11

−72.71 −33.28 53.99

p75

HW VK Merton

−39.02 −10.13 58.24

−33.34 62.51 238.45

−37.63 2.01 102.33

149

Reduced-Form versus Structural Models of Credit Risk 12

10

Percent

8

6

4

2

0 –200 –165 –130 –95 –60 –25 10 45 VK_error

80 115 150 185

FIGURE 7.2(b) Histogram of difference between the market and the model CDS prices (as given by Market CDS–Model CDS) for the VK structural model.

Table 7.2(b) Difference between Market CDS and Model CDS Prices (as Given by Market CDS – Model CDS) for Firms with Different Numbers of Bonds Available Percentile Difference

Model

Firms with 2–4 Bonds

Firms with 5–9 Bonds

Firms with 10–15 Bonds

Firms with 16–24 Bonds

p25

HW VK Merton

−146.62 −88.81 24.50

−120.08 −68.43 34.89

−102.10 −69.12 29.10

−92.29 −73.79 28.32

p50

HW VK Merton

−80.63 −37.62 51.64

−73.30 −27.21 60.40

−66.99 −26.63 55.14

−63.47 −34.86 48.74

p75

HW VK Merton

−32.92 1.80 107.77

−41.15 6.68 98.08

−37.05 3.47 106.46

−39.79 −6.68 88.01

150

THE CREDIT MARKET HANDBOOK 12

10

Percent

8

6

4

2

0 –200 –165 –130 –95 –60 –25 10 45 Merton_error

80 115 150 185

FIGURE 7.2(c) Histogram of difference between the market and the model CDS prices (as given by Market CDS–Model CDS) for Merton model.

Finally, Figure 7.3 shows the time series of the correlation of market CDS spreads with modeled CDS spreads. This correlation reﬂects the ability of a model to explain the cross-sectional variation of market CDS spreads. If the modeled CDS spreads are the same as the market CDS spreads, then the correlation will be exactly 1. In general, a better correspondence between the levels of modeled and realized market spreads will lead to a higher correlation. As evident from this ﬁgure, the VK model performs the best in this regard. Surprisingly, the HW reduced-form model performs the worst. Table 7.3(a) shows the median of the slope coefﬁcients that result from regressing the market CDS on the modeled CDS cross-sectionally on a daily

151

Reduced-Form versus Structural Models of Credit Risk Times series of correlation of market CDS versus model CDS Reduced-form

VK

Merton

1

Correlation

0.8 0.6 0.4 0.2 0 –0.2 –0.4 12/06/99

04/19/01

09/01/02 Date

01/14/04

05/28/05

FIGURE 7.3 Time series of correlation between market CDS spreads and model CDS spreads. For each day, the correlation was computed based on cross-sectional data of market and model CDS prices.

Table 7.3(a) Slope Coefﬁcients Resulting from Regression of Market CDSs on Model CDSs by Firms in Different CDS Spread Buckets. The slope coefﬁcients were computed daily for a 3-year period. The different quartiles of the time series distribution of the slope coefﬁcients are reported here Percentile Slope Coefﬁcient

Model

Firms with CDS ≤100 bps

Firms with CDS >100 bps

All Firms

p25

HW VK Merton

0.05 0.46 0.43

0.00 0.14 0.25

0.01 0.58 0.78

p50

HW VK Merton

0.33 0.68 0.82

0.00 0.17 0.35

0.07 0.76 1.26

p75

HW VK Merton

0.50 0.95 1.47

0.01 0.21 0.50

0.19 0.94 1.95

basis. In general, if the model is an unbiased estimator of the realized spread, then this slope should be 1. While both the VK and Merton models deviate from 1, with median slopes of 0.76 and 1.26, respectively, the HW model’s median slope is at 0.07, which seems unusually low. The median R-squareds of these regressions (which should also be the squares of the correlation between the market and modeled spreads) of the HW model, the VK model, and the Merton model stand at 0.09, 0.48, and 0.26, respectively, once again demonstrating the strength of the VK structural model in explaining the cross-sectional variation of CDS spreads.9 These results are particularly striking, given that VK-model-based EDF credit measures rely mostly on input from equity markets.

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THE CREDIT MARKET HANDBOOK

Table 7.3(b) Slope Coefﬁcients Resulting from Regression of Market CDSs on Model CDSs by Firms with Numbers of Bonds Available. The slope coefﬁcients were computed daily for a 3-year period. The different quartiles of the time series distribution of the slope coefﬁcients are reported here Percentile Slope Coefﬁcient

Model

Firms with 2–4 Bonds

Firms with 5–9 Bonds

Firms with 10–15 Bonds

Firms with 16–24 Bonds

p25

HW VK Merton

0.01 0.57 0.77

0.20 0.55 0.79

0.39 0.34 0.56

0.66 0.16 0.16

p50

HW VK Merton

0.05 0.73 1.26

0.36 0.73 1.57

0.57 0.56 1.29

0.82 0.56 0.63

p75

HW VK Merton

0.13 0.92 2.04

0.53 1.05 2.89

0.84 0.92 3.36

0.96 0.95 2.90

4.3 Further Diagnosis of Results In an attempt to understand the overall results, we drilled deeper into the nature of the relationship between CDS and bond data. Some market participants and academicians10 claim that while CDS and equity markets are relatively liquid, the bond markets are more impacted by liquidity problems. If liquidity across these markets differs in this way, one would expect that time to maturity would impact the correlation between bond and CDS prices. Two explanations are possible for this impact. First, bonds that are closer to maturity usually have a low probability of defaulting, and hence a low spread as implied by structural models. Therefore, any spread due to noncredit risk represents a larger fraction of overall bond spread. Since this “noncredit risk component” does not have to necessarily correlate across different markets, bonds with less time to maturity are likely to have lower correlation with their corresponding CDS spreads. Second, bonds with less time to maturity are more likely to be “off-the-run” securities (i.e., the time to maturity of these types of bonds makes them less attractive to market participants focused on matching particular durations) compared to bonds with greater time to maturity. Assuming that “on-the-run” securities are more liquid, bonds with more time to maturity are less likely to be impacted by liquidity problems. In Figure 7.4, we further examine the correlation between the bond and CDS markets by dividing the bonds into three different-tenor buckets: ■ ■ ■

0–2 years 2–4 years 4–6 years

These correlations are cross-sectional. We also plot the time series behavior of these correlations. Since the daily correlations are too noisy, the plot

153

Reduced-Form versus Structural Models of Credit Risk Times-series analysis of correlation between OAS and CDS 0∼2 years

2∼4 years

4∼6 years

1

Correlation

0.8

0.6

0.4

0.2

0 /00

/15

03

/05 /04 /02 /02 /04 /01 /01 /00 /03 /17 /01 /10 /24 /14 /19 /05 /01 /28 02 08 12 05 01 04 11 10 06

Date

FIGURE 7.4 Time series of correlation between market CDS spreads and bond spreads across different-tenor buckets. For each day, the correlation was computed based on cross-sectional data of market CDS and bond spreads.

shows the 5-day moving average. As expected, the correlation is fairly low for the low-maturity bonds. Note that even after calculating a 5-day moving average, the correlations are extremely noisy. Equations 7.8 and 7.9 demonstrate that the zero rates can be highly sensitive to any noise in the data. If one zero rate is adversely affected by noise, the entire term structure of zero rates can be impacted. Low correlations may be a consequence of the noise in the data and not necessarily reﬂective of each model’s performance. An alternative way of testing the reduced-form framework would be to use less noisy data for calibration, that is, CDS data. Unfortunately, at present, most of the CDS contracts are liquid only at the 5-year tenor. Using these data, we would observe a zero rate at only one point on the term structure. Any testing that follows would have to assume a ﬂat term structure for zero rates. Nonetheless, this test would be useful given that the zero-rate term structure would be less sensitive to noise in the data. Therefore, one could alternatively calibrate the data on CDS data and test it on bond data. This test, while interesting, would limit the ability of reduced-form models to capture nonﬂat term structures of credit risk. Second, this test would only validate the usefulness of pricing bonds with CDS data and not vice versa. The model created by Hull and White (2000) is intended for pricing CDS contracts using bond spreads, not necessarily the other way around. In addition, we acknowledge that it may be possible to achieve superior pricing of CDS contracts using bond data ﬁltered by tenor and other liquidity

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THE CREDIT MARKET HANDBOOK

indicators. Unfortunately, few guidelines exist for creating these ﬁlters. We leave construction of these ﬁlters and further investigation of these issues to future research.11

4.4 Robustness Tests In this section, we subject our results to more scrutiny by analyzing various subsets of the data to determine whether our results are attributable to the presence of outliers. One possible explanation for our results may be that healthier ﬁrms behave differently than riskier ﬁrms. We again conduct our tests on these two subsets of ﬁrms separately. Our proxy for a relatively healthy ﬁrm is one where the CDS spread is less than 100 bps. As seen in Table 7.1(c), about 73% of the sample falls in the healthier category. This percentage is in line with our expectations; most of the CDS data are concentrated among larger and less risky ﬁrms. Table 7.2(a) shows that the negative bias of the VK model is typically evident for ﬁrms with CDS spreads below 100 bps. For riskier ﬁrms, the modeled spreads are largely unbiased, with the median error around −7.6 bps. This result is inconsistent with the claim of Eom et al. (2003) that more sophisticated structural models underpredict credit risk at lower levels of risk, and overpredict credit risk at higher levels of credit risk. In comparison, the Merton model consistently underpredicts credit risk, as can be seen by the positive errors for both classes of CDS. The HW reducedform model overpredicts credit risk for both classes. This overprediction is signiﬁcantly high and even the 75th percentiles of errors are fairly negative for both classes of CDS. Table 7.4(a) highlights the ability of the models to explain the crosssectional variation of CDS spreads. We ﬁnd that although on aggregate, the HW model is outperformed by the Merton and VK models, on the subset of data with CDS < 100 bps, the HW model outperforms the Merton model. The VK model outperforms the other two models in all other subsets of the data except the 75th percentile of ﬁrms with CDS < 100 bps. One possible explanation for the HW model’s difﬁculties may involve the fact that most issuers have few outstanding bonds. A larger number of bond issues will increase the efﬁciency of estimating the default density. To test for this possibility, we divided our sample into ﬁrms with two bonds, ﬁrms with three to four bonds, ﬁrms with ﬁve to nine bonds, ﬁrms with 10 to 15 bonds, and ﬁrms with 16 to 24 bonds.12 Table 7.2(b) shows that the negative bias of the VK and HW models is present in all subsamples of the data. This result indicates that the bias is not being caused by ﬁrms with few outstanding bonds. The negative bias of the HW model is larger compared to that of the VK model for all data subsets.

155

Reduced-Form versus Structural Models of Credit Risk

Table 7.4(a) Ability of a Model to Explain Cross-Sectional Variation of Market CDS Spreads, Given by R-Squared of Regression of Market CDS on Model CDS by Firms in Different CDS Buckets. The R-squareds were computed daily for a 3-year period. The different quartiles of the time series distribution of R-squareds are reported here Percentile R-Squared

Model

Firms with CDS ≤100 bps

Firms with CDS >100 bps

All Firms

p25

HW VK Merton

0.08 0.25 0.07

0.00 0.20 0.04

0.02 0.40 0.16

p50

HW VK Merton

0.36 0.40 0.18

0.01 0.25 0.08

0.09 0.48 0.26

p75

HW VK Merton

0.62 0.54 0.32

0.04 0.30 0.13

0.21 0.56 0.36

Table 7.4(b) Ability of Model to Explain Cross-Sectional Variation of Market CDS Spreads, Given by R-Squared of Regression of Market CDS on Model CDS by Firms with Different Numbers of Bonds Available. The R-squared was computed daily for a 3-year period. The different quartiles of the time series distribution of R-squareds are reported here Percentile R-Squared

Model

Firms with 2–4 Bonds

Firms with 5–9 Bonds

Firms with 10–15 Bonds

Firms with 16–24 Bonds

p25

HW VK Merton

0.02 0.42 0.17

0.17 0.32 0.09

0.45 0.27 0.09

0.64 0.04 0.01

p50

HW VK Merton

0.07 0.50 0.28

0.38 0.46 0.24

0.72 0.44 0.29

0.77 0.29 0.04

p75

HW VK Merton

0.18 0.60 0.43

0.61 0.63 0.43

0.85 0.63 0.50

0.87 0.47 0.08

The Merton model, on the other hand, has a consistently positive bias in all categories, indicating that the model’s underestimation of credit risk is fairly consistent across various subsamples of the data. Table 7.4(b) examines the impact of the availability of information, as measured by the number of bonds by an issuer, on the ability of the HW model to explain the cross-sectional variation of the CDS spreads. Interestingly, we ﬁnd that the HW model outperforms the other two models when there are more than 10 bonds available to calibrate the default probability density. This improvement most likely results from the greater amount of cross-sectional information, in terms of the number of bonds, available to calibrate the default probability density. Similarly, in Table 7.3(b), we see that the sensitivity of the realized CDS spread to the modeled spread, as measured by the median slope of cross-sectional regression of market spreads on model spreads, increases with the number of bonds available. The ability of both the VK model and the Merton model to explain the cross-sectional variation declines among ﬁrms with more than 15 bonds issued, as observed by the lower R-squareds. This result is surprising, given

156

THE CREDIT MARKET HANDBOOK

that these models do not use bond information in their calibration. The result is most likely driven by the fact that ﬁrms that issue such large numbers of bonds have other variables impacting their spreads (e.g., interest-rate risk). These ﬁrms account for only 5% of the data, as seen in Table 7.1(a). That said, in terms of debt outstanding, these ﬁrms constitute a larger percentage (but not a majority) of the dollar amount of corporate debt outstanding. For example, on June 2004, ﬁrms that had more than 10 bonds used in bootstrapping (after applying all the ﬁlters) represented 40% of the total amount outstanding in our sample. Similarly, ﬁrms that had more than 15 bonds available for bootstrapping represented about 28% of the amount outstanding. Regardless of the measure, the majority of the ﬁrms tested in this research exercise did not have a large number of bonds outstanding. We have seen that, all else equal, the performance of the model improves with the number of bonds available for bootstrapping. Further characterization of the sample of proliﬁc issuers helps us interpret the results on other drivers of model performance. Large ﬁrms tend to be the ones that issue more bonds.13 Table 7.5(a) shows that large ﬁrms account for relatively more of the proliﬁc issuers than small ﬁrms. Large issuers also tend to be less risky. Table 7.5(c) demonstrates this fact by showing the distribution of CDS spreads for the entire sample compared to the distribution of CDS spreads for the largest ﬁrms (deﬁned by log(size) > 11). The large-ﬁrm sample does, in fact, have a higher percentage of ﬁrms with lower CDS spreads. We also ﬁnd that large issuers tend to issue debt of duration similar to smaller ﬁrms. This can be seen from the comparison of the distribution of time to maturity across bonds of large ﬁrms and bonds of the overall sample in Table 7.5(b). Therefore, as a percentage contribution to the overall spread, interest rate risk may dwarf default risk for these types of larger and safer issuers. Corporate bonds issued by small ﬁrms tend not to be transacted as much, making liquidity risk relatively more important in determining their spreads. These circumstances create countervailing inﬂuences on the performance of the HW model using bond data to ﬁt CDS spreads. To the extent that interest rate risk or liquidity risk overpowers default risk as the primary driver of CDS spreads, an HW model calibrated on bond data will not perform as well. Given these circumstances, we cannot predict ex ante how ﬁrm size will impact model performance. We test the impact of size on model performance by further dividing the subsamples based on the number of bonds available for bootstrapping. This procedure is designed to isolate the effect of size from the effect of number of bonds available for bootstrapping (since the two are somewhat positively correlated). For each subsample, we compute the median size of the ﬁrms. We then divide each subsample into two groups: one group where ﬁrm size exceeds the subsample median and the other group where ﬁrm size

Reduced-Form versus Structural Models of Credit Risk

157

Table 7.5(a) Cumulative Percentage Distribution of Issuers by Number of Bonds Used in Bootstrapping for Zero-Coupon Yield Curve Number of Bonds =2 ≤3 ≤4 ≤5 ≤6 ≤7 ≤8 ≤9 ≤10 ≤11 ≤12 ≤13 ≤14 ≤15 ≤16 ≤17 ≤18 ≤19 ≤20 ≤21 ≤22 ≤23 ≤24

Percentage of Sample 28.84 45.83 57.32 65.35 70.96 75.74 79.38 83.31 85.89 87.99 89.68 91.25 92.47 93.59 94.48 95.32 96.09 96.78 97.34 98.01 98.56 99.22 100.00

Percentage of Sample with log (Size) > 11 18.37 30.38 38.33 44.66 49.12 52.83 56.11 58.43 61.17 64.45 67.72 70.73 73.24 75.97 78.56 81.32 84.06 86.32 88.09 90.29 93.14 96.49 100.00

Table 7.5(b) Cumulative Percentage Distribution of Time to Maturity of Bonds Used in Bootstrapping for Zero-Coupon Yield Curve Time to Maturity ≤0.25 ≤0.50 ≤0.75 ≤1.00 ≤1.25 ≤1.50 ≤1.75 ≤2.00 ≤2.25 ≤2.50 ≤2.75 ≤3.00 ≤3.25 ≤3.50 ≤3.75 ≤4.00 ≤4.25 ≤4.50 ≤4.75 ≤5.00 ≤5.25 ≤5.50 ≤5.75 ≤6.00

Percentage of Sample 5.99 10.50 14.93 19.47 24.03 28.54 33.10 37.71 42.16 46.69 51.25 55.69 59.94 64.19 68.43 72.69 76.87 81.09 85.30 89.20 92.35 95.37 98.33 100.00

Percentage of Sample with log (Size) > 11 6.26 11.38 16.39 21.46 26.35 31.14 35.91 40.73 45.26 49.77 54.33 58.81 62.99 67.10 71.16 75.23 79.18 83.16 87.14 91.12 93.74 96.23 98.59 100.00

is less than the subsample median. Table 7.6(a) reports the performance of each model in explaining the cross-sectional variation in CDS spreads (as measured by the R-squareds of the regression of CDS spreads on modeled spreads) across the two size categories for each subsample.

158 Table 7.5(c) Spreads

THE CREDIT MARKET HANDBOOK Cumulative Percentage Distribution of CDSs by Range of

CDS Spread

Percentage of Sample

≤25 ≤50 ≤75 ≤100 ≤125 ≤150 ≤175 ≤200 ≤225 ≤250 ≤275 ≤300 ≤325 ≤350 ≤375

26.77 48.47 64.02 73.07 78.42 82.86 86.21 88.33 90.17 91.49 92.82 93.81 94.69 95.50 96.16

Percentage of Sample with log (Size) > 11 47.72 69.87 81.53 87.46 90.37 93.02 94.68 95.72 96.43 96.85 97.41 97.97 98.40 98.68 98.86

Table 7.6(a) Comparison of Different Models’ Ability to Explain Cross-Sectional Variation (Measured by R-squared of Cross-Sectional Regression of Market CDS on Model CDS Prices) across Large and Small Firms, Controlling for Number of Bonds Available for Bootstrapping. The R-squareds were computed daily for a 3-year period. The different quartiles of the time series distribution of R-squareds are reported here. The asterisk indicates that the cross-sectional correlation between market and model CDS prices is actually negative; there are a large number because R-squared is the square of the correlation here Below Median Size

Above Median Size

Number of Bonds

Model

p25

p50

p75

p25

p50

p75

2–4 bonds

HW VK Merton

0.09 0.34 0.11

0.37 0.46 0.20

0.57 0.57 0.31

0.01 0.37 0.24

0.04 0.48 0.46

0.11 0.62 0.69

5–9 bonds

HW VK Merton

0.27 0.32 0.10

0.50 0.52 0.32

0.76 0.71 0.63

0.05 0.27 0.08

0.35 0.42 0.18

0.67 0.62 0.42

10–15 bonds

HW VK Merton

0.27 0.31 0.09

0.64 0.59 0.46

0.85 0.80 0.76

0.40 0.16 0.07

0.73 0.48 0.31

0.89 0.68 0.65

16–24 bonds

HW VK Merton

0.46 0.19 0.02

0.72 0.50 0.22

0.88 0.70 0.47

0.64 0.01 0.10*

0.85 0.16 0.00

0.92 0.35 0.01

The VK model performs fairly consistently across the different subsamples. The HW model performance is a little more varied. As the number of bonds outstanding increases, the HW model performs progressively better. The classic Merton model consistently underperforms the other models in almost every subsample. One interesting pattern is the HW outperformance for large proﬁlic issuers. We suspect these results reﬂect the extent to which default risk is not a primary determinant of spreads for large, low-risk, proliﬁc issuers of long-term debt. The performance of the structural models relies on the sensitivity of CDS spreads to changes in the equity-based default probability measures. The standard Merton model reﬂects primarily equity price movement, which may not be a primary determinant of CDS spreads

159

Reduced-Form versus Structural Models of Credit Risk

Table 7.6(b) Summary Statistics of CDS Spreads across Firms with Different Numbers of Bond Issues. All statistics were based on daily observations in the period 2000/10 to 2004/06, and the numbers reported here are the medians across the daily time series of summary statistics Percentile Std. deviation |p75 − p25| Median Mean Std. deviation/mean NOBS

Firms with 2–4 Bonds

Firms with 5–9 Bonds

Firms with 10–15 Bonds

Firms with 16–24 Bonds

111.29 99.77 68.20 107.74 1.03 88

99.83 62.14 70.00 102.69 0.97 35

81.20 80.00 71.39 95.13 0.85 17

62.63 48.71 65.20 83.22 0.75 21

for large ﬁrms. The VK model, on the other hand, includes modiﬁcations to the speciﬁcation of the default point, the estimation of asset volatility, and the interaction of ﬁrm asset value and the default model in such a way as to capture more of the determinants of CDS spreads for all ﬁrms, regardless of size. As a result, its performance is consistent across subsamples. For the most proliﬁc issuers (i.e., outstanding bonds greater than 16), we should interpret the results with care given the small number in this subsample. Table 7.6(b) reports some of the characteristics of the cross-sectional distribution of the CDS spreads in this sample, categorized by ﬁrms with different numbers of bond issues available. As we move to ﬁrms with a larger number of bonds, the CDS spreads are less diverse, as indicated by the standard deviation and the interquartile range (|p75 − p25|) of the spreads. There is also a substantial reduction in the number of CDS observations available on a daily basis (as seen by the median number of observations each day). R-squared is not a reliable statistic when the variation in the dependent variable (in this case, the CDS spreads) is low, or the number of observations is low. Our proliﬁc-issuer subsamples suffer from both of these inadequacies. For example, upon breaking the sample of ﬁrms with more than 15 bonds into two halves according to size (as in Table 7.6a), the median daily number of observations in each half is about nine. This, along with the lowest coefﬁcient of variation (standard deviation/mean) leads to an unreliable R-squared measure. The subsamples for ﬁrms with fewer numbers of bonds outstanding do not suffer from this particular difﬁculty. All results should be considered with these characteristics in mind. In summary, the testing demonstrates that for the vast majority of ﬁrms, a structural model such as the VK model developed at MKMV, calibrated on a time series of equity data, works better in measuring credit risk relative to an HW reduced-form model calibrated on a cross section of corporate bonds. The VK model substantially outperforms a simple implementation of the standard Merton model. For practitioners looking across the broad cross section of traded credit instruments, the data requirements for robust reduced-form modeling and the availability of robust equity-based measures should inform discussions of which modeling approach to use. Moreover,

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THE CREDIT MARKET HANDBOOK

users of reduced-form models looking to price other credit-risky securities such as CDSs should bear in mind the potential impact on bond spreads of other risks resulting from interest rate movements and changes in liquidity. These effects can differ across size and spread levels, thereby distorting the performance of these models. A model such as VK is relatively more stable in its performance across various categories by size and spreads. The model’s strength is partially due to its structural framework that uses equity data, which are less contaminated by other risks, and partially due to its more sophisticated implementation.

5. CONCLUSION In this chapter, we empirically test the success of three models in their ability to measure credit risk. These models are the Merton model, the Vasicek–Kealhofer model, and the Hull–White model. These three models were chosen because they represent three key stages in the development of the theoretical literature in credit risk. These models also represent two main approaches for credit-risk modeling: the structural approach and the reduced-form approach. This research is the ﬁrst attempt at comparing these types of models in their ability to discriminate defaulted ﬁrms from nondefaulted ﬁrms, and to predict the levels and explain the cross-sectional variation of CDS spreads. The advantage of these data is that they are not used in the calibration of any of the models, thereby facilitating a true out-of-sample test. The VK model marginally outperformed the HW model in terms of its default predictive power. Both of these models consistently outperformed the Merton model in their default predictive power. Despite the advantages stated by proponents of reduced-form models, the HW model largely underperforms a sophisticated structural model (as implemented by MKMV) in its ability to predict levels and explain cross-sectional variation of CDS spreads. Interestingly enough, the HW model outperforms the simple Merton model when a given ﬁrm issues a large number of bonds. In these cases of ﬁrms issuing more than 10 bonds at any given time, the HW model can also outperform (in terms of explaining the cross-sectional variation of CDS spreads) the more sophisticated VK structural model for some subsets of low-risk corporate issuers. At this time, the number of such ﬁrms is small. In our sample, the HW model was more effective in its ability to explain the cross-sectional variation of the CDS spreads only for the largest 5% of the ﬁrms, in terms of the number of issues outstanding on which data were available. Even for this sample, the error in terms of the difference between actual and predicted levels of spreads was much larger for the HW model when compared

Reduced-Form versus Structural Models of Credit Risk

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to the VK model. The performance of the VK model is consistent across large and small ﬁrms. Surprisingly, the performance of the HW model worsens as it is applied to larger ﬁrms that issue few bonds. The performance of the HW model could have been impacted due to the sensitivity of the model calibration to the underlying bond data at various tenors coupled with the propensity of bond data to be “noisier” at shorter terms. In order to improve a reduced-form model’s performance using bond data, better ﬁlters will need to be devised to eliminate noisy data, thus facilitating more efﬁcient calibration. We leave this type of investigation to future research. The overall results emphasize the importance of empirical evaluation when assessing the strengths and weaknesses of different types of credit risk models.

ACKNOWLEDGMENTS The authors would like to thank Deepak Agrawal, Ittiphan Jearkjirm, Stephen Kealhofer, Hans Mikkelsen, Martha Sellers, Roger Stein, Shisheng Qu, Bin Zeng, and seminar participants at MKMV and the Prague Fixed Income Workshop (2004) for invaluable suggestions and advice. We also thank an anonymous referee whose suggestions helped improve the chapter substantially. We are grateful to Jong Park for providing the code involving the implementation of the Merton model. All errors are ours.

NOTES 1. Giesecke and Goldberg (2004) show that it is possible to develop a structural model in which the modeler also has incomplete information about the default point, making the time to default inaccessible even in a structural model. Dufﬁe and Lando (2001) propose a “hybrid” model that assumes accounting information is noisy, thereby making the default time inaccessible in the context of a structural model. 2. Because indicative prices do not always reﬂect all the information in the market, a researcher may mistakenly conclude that a particular model predicts price changes when, in fact, an actual trade would have reﬂected an immediate price change, eliminating any predictive power of the model. See Dufﬁe (1999) for more discussion on this problem of stale bond prices. 3. See, for example, Ericsson and Reneby (2004). 4. The three models take different sources of data as inputs. While the Merton and VK models mostly rely on equity data, the HW model is calibrated with bond data. Therefore, any result in this test is a reﬂection of the framework as well as the quality of data available to the models as input. One of the strengths of a

162

5.

6.

7. 8. 9. 10. 11. 12.

13.

THE CREDIT MARKET HANDBOOK model, especially from a practitioner’s point of view, is that it should yield accurate results based on data that are easily available and accessible. A conceptually powerful framework, while intellectually stimulating, can be fairly meaningless if any application of it relies on data that are either unavailable or of poor quality. Note that Helwege and Turner (1999) ﬁnd evidence of upward-sloping term structures of spreads for risky corporate bonds. When we look at the behavior of distance-to-default (DD) for ﬁrms, the downward-sloping term structures of default probabilities for high-risk issuers appear to be more typical. In other research completed by Agrawal and Bohn (2005), these contradictory ﬁndings are reconciled by investigating the differences in par spreads and zero-coupon spreads as well as the differences in spreads calculated from new bond issues and bond issues traded in the secondary market. While Helwege and Turner see upward-sloping term structures in par spreads for new bond issues, the more recent research ﬁnds that zero-coupon spreads from the secondary market for high-yield bonds reﬂect the downward-sloping term structures predicted in the theory and reﬂected in the DD data. Agrawal and Bohn (2005) also show that under certain conditions zero-coupon spreads can be downward sloping when par spreads are not. The normal and abnormal inverse functions act as translators of the default probability estimate without requiring that the default probability estimate, itself, be generated from a normal distribution. In this case, the CEDFT represents the physical cumulative default probability and was calculated from the empirical distribution estimated at MKMV. A detailed derivation of this expression can be found in Agrawal et al. (2004). This step helps improve performance of modeled bond spreads against market bond spreads. A zero rate is the implicit interest on a zero-coupon bond of a given maturity. Median R-squareds of 0.09, 0.48, and 0.26 correspond to median correlations of approximately 30%, 70%, and 51%. See Longstaff et al. (2004), for example. We thank an anonymous referee for suggesting most of the ideas in this subsection. The number of bonds represents the number available for bootstrapping after using all the ﬁlters described above—not the actual number of bonds outstanding. We measure the size of a ﬁrm by its book asset value. A large ﬁrm is considered to have book assets in excess of about $60 billion [log (60,000) is approximately 11; our data are reported in millions of dollars].

REFERENCES Agrawal, D., Arora, N. and Bohn, J. (2004). “Parsimony in Practice: An EDF-Based Model of Credit Spreads.” White Paper, Moody’s KMV.

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Agrawal, D. and Bohn, J. (2005). “Humpbacks in Credit Spreads.” White Paper, Moody’s KMV. Black, F. and Cox, J. (1976). “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” Journal of Finance 31, 351–367. Black, F. and Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81, 637–659. Bohn, J. (2000a). “A Survey of Contingent-Claims Approaches to Risky Debt Valuation.” Journal of Risk Finance 1(3), 53–78. Bohn, J. (2000b). “An Empirical Assessment of a Simple Contingent-Claims Model for the Valuation of Risky Debt.” Journal of Risk Finance 1(4), 55–77. Collin-Dufresne, P. and Goldstein, R. (2001). “Do Credit Spreads Reﬂect Stationary Leverage Ratios?” Journal of Finance 56, 1929–1957. Collin-Dufresne, P., Goldstein, R. and Martin, S. (2001). “The Determinants of Credit Spread Changes.” Journal of Finance 56, 2177–2208. Crosbie, P. and Bohn, J. (2003). “Modeling Default Risk.” White Paper, Moody’s KMV. Dufﬁe, G. (1999). “Estimating the Price of Default Risk.” Review of Financial Studies 12(1), 1997–2026. Dufﬁe, D. and Lando, D. (2001). “Term Structures of Credit Spreads with Incomplete Accounting Information.” Econometrica 69, 633–664. Dufﬁe, D. and Singleton, K. (1999). “Modeling the Term Structure of Defaultable Bonds.” Review of Financial Studies 12, 687–720. Eom, Y., Helwege, J. and Huang, J. (2003). “Structural Models of Corporate Bond Pricing: An Empirical Analysis.” Review of Financial Studies. Ericsson, J. and Reneby, J. (2004). “An Empirical Study of Structural Credit Risk Models Using Stock and Bond Prices.” Institutional Investors Inc., pp. 38–49. Friedman, M. (1953). The Methodology of Positive Economics. Essay in Essays on Positive Economics, University of Chicago Press. Geske, R. (1977). “The Valuation of Corporate Liabilities as Compound Options.” Journal of Financial and Quantitative Analysis pp. 541–552. Giesecke, K. and Goldberg, L. (2004). “Forecasting Default in the Face of Uncertainty.” Journal of Derivatives 12(1), 11–25. Helwege, J. and Turner, C. (1999). “The Slope of the Credit Yield Curve for Speculative Grade Issuers.” Journal of Finance 54(5), 1869–1884. Hull, J. and White, A. (2000). “Valuing Credit Default Swaps: No Counterparty Default Risk.” Working Paper, University of Toronto. Hull, J. (1999). Options, Futures and Other Derivatives, 4th edn. Englewood Cliffs, NJ: Prentice Hall Publications. Jarrow, R. (2001). “Default Parameter Estimation Using Market Prices.” Financial Analysts Journal 57, 75–92. Jarrow, R. and Protter, P. (2004). “Structural versus Reduced Form Models: A New Information-Based Perspective.” Working Paper, Cornell University. Jarrow, R. and Turnbull, S. (1995). “Pricing Derivatives on Financial Securities Subject to Default Risk.” Journal of Finance 50, 53–86.

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Kealhofer, S. (2003a). “Quantifying Credit Risk I: Default Prediction.” Financial Analysts Journal, 59(1), 30–44. Kealhofer, S. (2003b). “Quantifying Credit Risk II: Debt Valuation.” Financial Analysts Journal, 59(3), 78–92. Kim, J., Ramaswamy, K. and Sunderasan, S. (1993). “Does Default Risk in Coupons Affect the Valuation of Corporate Bonds? A Contingent Claims Model.” Financial Management 22, 117–131. Leland, H. and Toft, K. (1996). “Optimal Capital Structure, Endogeneous Bankruptcy, and the Term Structure of Credit Spreads.” Journal of Finance 51, 987–1019. Longstaff, F. and Schwartz, E. (1995). “Valuing Risky Debt: A New Approach.” Journal of Finance 50, 789–820. Longstaff, F., Mithal, S. and Neis, E. (2004). “Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit-Default Swap Market.” Working Paper, Anderson School of Management, UCLA. Lyden, S. and Saraniti, D. (2000). “An Empirical Examination of the Classical Theory of Corporate Security Valuation.” Research Paper, Barclays Global Investors. Merton, R. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29, 449–470. Ogden, J. (1987). “Determinants of the Ratings and Yields on Corporate Bonds: Tests of the Contingent Claim Model.” The Journal of Financial Research 10, 329–339. Stein, R. (2002). “Benchmarking Default Prediction Models: Pitfalls and Remedies in Model Validation.” White Paper, Moody’s KMV. Stein, R. (2003). “Are the Probabilities Right? A First Approximation to the Lower Bound on the Number of Observations Required to Test for Default Rate Accuracy.” White Paper, Moody’s KMV. Vasicek, O. (1984). “Credit Valuation.” White Paper, Moody’s KMV.

Keywords: Credit risk; default risk; default probability; structural mode; reduced-form model; Merton model; equity; corporate bonds; credit default swaps (CDS); power curves; credit spreads

CHAPTER

8

Implications of Correlated Default for Portfolio Allocation to Corporate Bonds Mark B. Wisea,∗ and Vineer Bhansalib

This chapter deals with the problem of optimal allocation of capital to corporate bonds in ﬁxed income portfolios when there is the possibility of correlated defaults. Using a multivariate normal copula function for the joint default probabilities, we show that retaining the ﬁrst few moments of the portfolio default loss distribution gives an extremely good approximation to the full solution of the asset allocation problem. We provide detailed results on the convergence of the moment expansion and explore how the optimal portfolio allocation depends on recovery fractions, level of diversiﬁcation, and investment time horizon. Numerous numerical illustrations exhibit the results for simple portfolios and utility functions.

1. INTRODUCTION Investors routinely look to the corporate bond market in particular, and to spread markets in general, to enhance the performance of their portfolios. However, for every source of excess return over the risk-free rate, there is a source of excess risk. When sources of risk are correlated, the allocation decision to the risky sectors, as well as allocation to particular securities in that sector, can be substantially different from the uncorrelated case. Since a California Institute of Technology, Pasadena CA 91125, USA. b PIMCO, 840 Newport Center Drive, Suite 300, Newport Beach, CA 92660, USA. ∗ Corresponding author. California Institute of Technology, Pasadena CA 91125,

USA. E-mail: [email protected].

165

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the joint probability distribution of returns of a set of defaultable bonds varies with the joint probabilities of default, recovery fraction for each bond, and number of defaultable bonds, a direct approach to the allocation problem that incorporates all these factors completely can only be attempted numerically within the context of a default model. This approach can have the shortcoming of hiding the intuition behind the asset allocation process in practice, which leans very heavily on the quantiﬁcation of the ﬁrst few moments, such as the mean, variance, and skewness. In this chapter, we take the practical approach of characterizing the portfolio default loss distribution in terms of its moment expansion. Focusing on allocation to corporate bonds (the analysis in this chapter can be generalized to any risky sector which has securities with discrete payoffs) we will answer the following questions: ■

■

In the presence of correlated defaults, how well does retaining the ﬁrst few moments of the portfolio default loss distribution do, for the portfolio allocation problem, as compared to the more intensive full numerical solution? For different choices of correlations, probabilities of default, number of bonds in the portfolio, and investment time horizon, how does the optimal allocation to the risky bonds vary?

In this chapter we consider portfolios consisting of risk-free assets at a return y and corporate zero-coupon bonds with return ci for ﬁrm i, and study how correlations between defaults affect the optimal allocation to corporate assets in the portfolio over some time horizon T. We assume that if company i defaults at time t, a fraction Ri (the recovery fraction) of the value of its bonds is recovered and reinvested at the risk-free rate. We quantify the impact of risk associated with losses from corporate defaults1 on portfolio allocation. To compensate the investor for default risk, the corporate bond’s return ci is greater than that of the risk-free assets in the portfolio. The excess return, or spread of investment in bonds, of ﬁrm i can be decomposed into two parts. The ﬁrst part of the spread, which we call λi , arises as the actuarially fair value of assuming the risk of default. The second part, which we call µi , is the excess risk premium that compensates the corporate bond investor above and beyond the probabilistically fair value. In practice, µi can arise due to a number of features not directly related to defaults. For example, traders will partition µi into two pieces, one for liquidity and one for event risk, µi = li +ei . Most low-grade bonds may have a full percentage point arising simply from liquidity premiums, and µi can ﬂuctuate to large values in periods of credit stress. Liquidity li is systematic and one would expect it to be roughly equal for a similar class of bonds. Event risk ei contributes to µi due to the

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167

ﬁrm’s speciﬁc vulnerability to factors that affect it (e.g., negative press). In periods of stress, the default probabilities li and ei all increase simultaneously, and the recovery rate expectations Ri fall, leading to a spike in the overall spread. Since these variables can be highly volatile, the reason behind the portfolio approach to managing credit is to minimize the impact of nonsystematic event risk in the portfolio.2 The excess risk premium itself is not very stable over time. Empirical research shows that the excess risk premium may vary from tens of basis points to hundreds of basis points. For instance, in the BB asset class, if we assume a recovery rate of 50% and default probability of 2%, the actuarially fair value of the spread is 100 basis points (bp). However, it is not uncommon to ﬁnd actual spreads of the BB class to be 300 bp over Treasuries (Altman, 1989). The excess 200 bp of risk premium can be decomposed in any combination of liquidity premium and event risk premium, and is best left to the judgment of the market. When the liquidity premium component is small compared to event risk premium, we would expect the portfolio diversiﬁcation and methods of this chapter to be valuable. One other factor must be kept in mind when comparing historical spreads to current levels. In the late 80s and early 90s, the spread was routinely quoted in terms of a Treasury benchmark curve. However, the market itself has developed to a point where spreads are quoted over both the LIBOR swap rate and the Treasury rate, and the swap rate has gradually substituted the Treasury rate as the risk-free benchmark curve. This has two impacts. First, since the swap spread (swap rate minus Treasury rate for a given maturity) in the US is signiﬁcant (of the order of 50 bp as of this writing), the excess spread must be computed as a difference to the swap yield curve. Second, the swap spread itself has been very volatile during the last few years, which leads to an added source of non-default-related risk in the spread of corporate bonds when computed against the Treasury curve. Thus, the 200 bp of residual spread is effectively 150 bp over the swap rate in the BB example, of which, for lack of better knowledge, equal amounts may be assumed to arise from liquidity and event risk premium over the long term. The allocation decision to risky bonds strongly depends on the level of risk aversion in the investor’s utility function, and the required spread for a given allocation will go up nonlinearly as risk aversion increases. The value of the optimal fraction of the portfolio in corporate bonds, here called αopt , cannot be determined without knowing the excess risk premium part of the corporate returns. They provide the incentive for a risk-averse investor to choose corporate securities over risk-free assets. In this chapter we explore the convergence of the moment expansion for αopt , using utility functions with constant relative risk aversion. Our work indicates that (for αopt less than unity) αopt is usually determined by the mean, variance, and skewness of the portfolio default loss probability distribution.

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The sensitivity to higher moments increases as αopt does. Some measures of default risk, for example, a VAR analysis,3 may be more sensitive to the tail of the default loss distribution. We also examine how the optimal portfolio allocation scales with the number of ﬁrms, time horizon, and recovery fractions. Historical evidence suggests that on average, default correlations increase with the time horizon.4 For example, Lucas (1995) estimates that over 1-year, 2-year, and 5-year time horizons, default correlations between Ba-rated ﬁrms are 2%, 6%, and 15%, respectively. However, the errors in extracting default correlations from historical data are likely to be large since defaults are rare. Also, these historical analyses neglect ﬁrm-speciﬁc effects that may be very important for portfolios weighted toward a particular economic sector. Furthermore, in periods of market stress, default probabilities and their correlations increase dramatically (Das et al., 2001). There are other sources of risk associated with corporate securities. For example, the market’s perception of ﬁrm i’s probability of default could increase over the time horizon T, resulting in a reduction in the value of its bonds. For a recent discussion on portfolio risk due to downgrade ﬂuctuations, see Dynkin et al., 2002. Here we do not address the issue of risk associated with ﬂuctuations in the credit spread, but rather focus on the risk associated with losses from actual defaults. In the next section a simple model for default is introduced. The model assumes a multivariate normal copula function for the joint default probabilities. In Section 3 we set up the portfolio problem. Moments of the fractional corporate default loss probability distribution are expressed in terms of joint default probabilities and it is shown how these can be used to determine αopt . In Section 4 the impact of correlations on the portfolio allocation problem is studied using sample portfolios where all the ﬁrms have the same probabilities of default and the correlations between ﬁrms are all the same. The recovery fractions are assumed to be zero for the portfolios in Section 4. The impact of nonzero recovery fractions on the convergence of the moment expansion is studied in Section 5. Concluding remarks are given in Section 6. This work is based on Wise and Bhansali (2002). It extends the results presented in that chapter to arbitrary time horizons (allowing default to occur at any time) and makes more realistic assumptions for the consequences of default.

2. A MODEL FOR DEFAULT It is convenient for discussions of default risk to introduce the random variables nˆ i (ti ). nˆ i (ti ) takes the value 1 if ﬁrm i defaults in the time horizon ti and

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Implications of Correlated Default for Portfolio Allocation

zero otherwise. The joint default probabilities are expectations of products of these random variables: Pi1 ,...,im (ti1 , . . . , tim ) = E[nˆ i1 (ti1 ) · · · nˆ im (tim )]

(8.1)

when i1 = i2 · · · = im . Pi (ti ) is the probability that ﬁrm i defaults in the time period ti and Pi1 ,...,im (ti1 , . . . , tim ) is the joint probability that the m ﬁrms i1 , . . . , im default in the time periods ti1 , . . . , tim . We assume that the joint default probabilities are given by a multivariate normal copula function (see, for example, Li, 2000). Explicitly, P1,...,n (t1 , . . . , tn ) =

1

−χ1 (t1 )

dx1 · · ·

(2π )n/2 det ξ −∞   1 (−1) × exp − xi ξij xj  2

−χn (tn ) −∞

dxn

(8.2)

ij

(−1)

where the sum goes over i, j = 1, . . . , n, and ξij is the inverse of the n × n correlation matrix ξij . For n not too large the integrals in equation 8.2 can be done numerically or, since defaults are rare, analytic results can be obtained using the leading terms in an asymptotic expansion of the integrals. The choice of a multivariate normal copula function is common but somewhat arbitrary. In principle, the copula function should be chosen based on a comparison with data. For recent work along these lines, see Das and Geng (2002). In equation 8.2, the n × n correlation matrix ξij is usually taken to be the asset correlation matrix. This has the advantage of allowing a connection to stock prices (Merton, 1974). However, this assumption is not necessary. For example, the assets, aˆ i could be functions of normal random variables, aˆ i = gi (ˆzi ). Suppose default occurs if the assets aˆ i cross the thresholds Ti . Then, the condition for default on the normal production factor variables zˆ i is that (−1) they cross the thresholds gi (Ti ). In such a model it is natural to interpret ξij as the correlation matrix for the normal production factor variables zˆ i . If the functions gi are linear, then the assets are also normal and their correlation matrix is also given by ξij . However, if the functions gi are not linear, the joint probability distribution for the assets is not multivariate normal and can have fat tails. Unless the gi are speciﬁed, it is not possible to connect stock prices to the correlation matrix ξij and the default thresholds χi . However, even when the functions gi are not known, the default thresholds or “equivalent distances to default” χi (ti ) and the correlation matrix ξij are determined by

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the default probabilities Pi (ti ) and the default correlations dij (t), and so have a direct connection to measures that investors use in quantifying security risk. In our work we assume that the correlation matrix is time-independent. Equation 8.2 in the case n = 1 gives Pi (ti ) =

1 (2π )1/2

−χi (ti ) −∞

1 dxi exp − x2i 2

(8.3)

Hence, for an explicit choice for the time dependence of the default probabilities, the default thresholds are known. A simple (and frequently used) choice for the default probabilities is Pi (ti ) = 1 − exp(−hi ti )

(8.4)

where the hazard rates hi are independent of time. Since nˆ i (ti )2 = nˆ i (ti ) it follows that the correlation of defaults between two different ﬁrms (which we choose to label as 1 and 2) is d12 (t) =

E[nˆ 1 (t)nˆ 2 (t)] − E[nˆ 1 (t)]E[nˆ 2 (t)] (E[nˆ 1 (t)2 ] − E[nˆ 1 (t)]2 )(E[nˆ 2 (t)2 ] − E[n2 (t)]2 )

P12 (t, t) − P1 (t)P2 (t) =√ P1 (t)(1 − P1 (t))P2 (t)(1 − P2 (t))

(8.5)

The default model we are adopting is not as well motivated as a ﬁrst passagetime default model (Black and Cox, 1976; Longstaff and Schwartz, 1995; Leland and Toft, 1996, etc.) where the assets undergo a random walk, and default is associated with the ﬁrst time the assets fall below the liabilities. However, it is very convenient to work with. We will evaluate the integrals in equation 8.2 by numerical integration. For this we need the inverse and determinant of the correlation matrix. It is very important that the correlation matrix ξij be positive semi-deﬁnite. If it has negative eigenvalues the integrals in equation 8.2 are not well deﬁned. Typically, a correlation matrix that is forecast using qualitative methods will not be mathematically consistent and will have some negative eigenvalues. A practical method for constructing the consistent correlation matrix that is closest to a forecast one is given in Rebonato and Jäckel (2000). For the portfolios discussed in Sections 4 and 5, the n × n correlation matrix ξij is taken to have all of its off-diagonal elements the same, that is, ξij = ξ for i = j. Such a correlation matrix has one eigenvalue equal to 1 + (n − 1)ξ and

Implications of Correlated Default for Portfolio Allocation

171

the others equal to 1 − ξ . Consequently, its determinant is det[ξij ] = (1 − ξ )n−1 [1 + (n − 1)ξ ]

(8.6)

Its inverse has diagonal elements, (−1)

ξij

=

1 + (n − 2)ξ 1 + (n − 2)ξ − (n − 1)ξ 2

(8.7)

and off-diagonal elements (i = j), (−1)

ξij

=−

ξ 1 + (n − 2)ξ − (n − 1)ξ 2

(8.8)

In the next section we consider the problem of portfolio allocation for portfolios consisting of corporate bonds subject to default risk and risk-free assets. The implications of default risk are addressed using the model discussed in this section. Other sources of risk—for example, systematic risk associated with the liquidity part of the excess risk premium—are neglected.

3. THE PORTFOLIO PROBLEM Assume a zero-coupon bond from company i grows in value (if it does not default) at the (short) corporate rate ci (t) and that if the company defaults, a fraction Ri of the value of that bond at the time of default is reinvested at the risk-free (short) rate y(t). Then, the random variable for the value of this bond at some time T in the future is

T T dnˆ i (s) Vˆ i (T) = exp dτ ci (τ ) (1 − nˆ i (T)) + Ri ds Vi (0) ds 0 0

× exp

s

0

dτ ci (τ )dτ +

T

0

dτ y(τ )

(8.9)

where Vi (0) is the initial value of the zero-coupon bond. In equation 8.9 we have continuously compounded the returns and we have assumed that T is less than or equal to the maturity date of the bond.5 Note that the random variable dnˆ i (t)/dt is equal to the Dirac delta function, δ(t − ti ), if company i defaults at the time ti and is zero if it does not default. The ﬁrst term in equation 8.9 gives the value if the company does not default before time T

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and the second term (proportional to Ri ) gives the value if the company defaults before time T. The corresponding formula for a risk-free asset is Vrf (T) Vrf (0)

= exp

T 0

dτ y(τ )

(8.10)

Note that we are not treating the (short) risk-free rate y(τ ) as a random variable, although it is certainly possible to generalize this formalism to do that. The (short) corporate rate is decomposed as ci (t) = y(t) + λi (t) + µi (t)

(8.11)

where µi is the excess risk premium and λi is the part of the corporate short rate that compensates the investor in an actuarially fair way for the fact that the value of the investment can be reduced through corporate default. In other words,

Vˆ i (T) E Vi (0)

=

µi =0

Vrf (T)

(8.12)

Vrf (0)

Taking the expected value, equation 8.12 implies that

1 = (1 − Pi (T)) exp

T 0

dτ λi (τ ) + Ri

T 0

dPi (s) exp ds ds

0

s

dτ λi (τ ) (8.13)

Differentiation with respect to T gives λi (T) =

dPi (T) 1 − Ri dT 1 − Pi (T)

(8.14)

Using equation 8.4 for the time dependence of the probabilities of default, equation 8.14 implies the familiar relation λi (T) = hi (1 − Ri )

(8.15)

The above results imply that the total wealth, after time T, in a portfolio consisting of risk-free assets and zero-coupon corporate bonds that are subject to default risk is (assuming the bonds mature at dates greater than or

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Implications of Correlated Default for Portfolio Allocation

equal to T) ˆ W(T) = W0 exp +α

T 0

dτ y(τ )

fi exp

T 0

i

(1 − α)

dτ (λi (τ ) + µi (τ )) − ˆl(T)

(8.16)

where W0 is the initial wealth, α the fraction of corporate assets in the portfolio, ˆl(T) the random variable for the fractional default loss over the time period T, Ri the recovery fraction, and fi denotes the initial fraction of corporate assets in the portfolio that are in ﬁrm i ( i fi = 1). In equation 8.16 the sum over i goes over all N ﬁrms in the portfolio and the default loss random variable is given by ˆl(T) =

fi × exp

i

− Ri

T 0

T 0

dτ (λi (τ ) + µi (τ )) nˆ i (T)

dnˆ i (s) × exp ds ds

dτ (λi (τ ) + µi (τ ))

s

0

(8.17)

Integrating by parts, ˆl(T) =

fi ×

T

exp 0

i

dτ (λi (τ ) + µi (τ ))

× nˆ i (T)(1 − Ri ) + Ri × exp

s

0

T

ds nˆ i (s)(λi (s) + µi (s))

0

dτ (λi (τ ) + µi (τ ))

(8.18)

The expected value of ˆl(T) is E[ˆl(T)] =

fi exp

i

− Ri

T

ds 0

T 0

dτ (λi (τ ) + µi (τ )) Pi (T)

dPi (s) × exp ds

0

s

dτ (λi (τ ) + µi (τ ))

(8.19)

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Fluctuations of the fractional loss ˆl(T) about its average value are described by the variable δ ˆl(T) = ˆl(T) − E[ˆl(T)]

(8.20)

The mean of δ ˆl is zero and the probability distribution for δ ˆl determines the default risk of the portfolio associated with ﬂuctuations of the random variables nˆ i (t). The moments of this probability distribution are v(m) (T) = E[(δ ˆl(T))m ]

(8.21)

Using equation 8.1 and the property nˆ i (t1 )nˆ i (t2 ) = nˆ i (min[t1 , t2 ]), the moments v(m) (T) can be expressed in terms of the joint default probabilities. The random variable for the portfolio wealth at time T is ˆ W(T) = W0 exp

T 0

dτ y(τ ) [1 + αx(T) − αδ ˆl(T)]

(8.22)

where x(T) = −1 +

fi × exp

T 0

dτ (λi (τ ) + µi (τ ))

− E[ˆl(T)]

(8.23)

Using equations 8.13 and 8.19 we ﬁnd that x(T) =

fi exp

i

× exp

0

T 0

dτ µi (τ ) − 1 + Ri

T

ds 0

dPi (s) ds

s dτ (λi (τ ) + µi (τ )) × 1 − exp

T

s

dτ µi (τ ) (8.24)

Note that x(T) vanishes if the excess risk premiums µi do. To ﬁnd out what value of α is optimal for some time horizon T, a utility function is introduced that characterizes the investor’s level of risk aversion. Here, we use utility functions of the type Uγ (W) = W γ /γ that have constant relative risk aversion,6 1 − γ . The optimal fraction of corporates,

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Implications of Correlated Default for Portfolio Allocation

ˆ αopt , maximizes the expected utility of wealth E[Uγ (W(T))]. Expanding the ˆ utility of wealth in a power series in δ l and taking the expected value gives ˆ E[Uγ (W(T))] =

γ (W0 /γ ) exp

×

1+

∞ m=2

γ

T

0

d τ y(τ ) (1 + αx(T))γ

(m − γ ) × (−γ )(m + 1)

α 1 + αx(T)

m v

(m)

(T)

(8.25) where is the Euler Gamma function. The approximate optimal value of α obtained from truncating the sum in equation 8.25 at the mth moment is denoted by αm . The focus of this chapter is on portfolios that are not leveraged and have αopt less than unity. We will later see that typically for such portfolios, the αm converge very quickly to αopt so that for practical purposes only a few of the moments v(m) need to be calculated. Note that the value of αopt is independent of the risk-free rate y(t). The expressions for ˆl(T) and x(T) are complicated but they do simplify for short times T or if all the recovery fractions are zero. For example, if T is small ˆl(T)

fi (1 − Ri )nˆ i (T)

(8.26)

i

and x(T) fi i

T 0

ds µi (s)

(8.27)

On the other hand, if all the recovery fractions vanish (i.e., Ri = 0), then, ˆl(T) =

fi × exp

i

T 0

dτ (λi (τ ) + µi (τ )) nˆ i (T)

(8.28)

and x(T) =

i

fi exp

T 0

dτ µi (τ ) − 1

(8.29)

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If x(T) is zero, then αopt is also zero. It is x(T) that contains the dependence on the excess risk premiums that provide the incentive for a risk-adverse investor to choose corporate bonds over risk-free assets. An approximate formula for αopt can be derived by expanding it in x(T):

αopt =

∞ sn (T) n=1

n!

x(T)n

(8.30)

The coefﬁcients si (T) can be expressed in terms of the moments, v(m) (T). Explicitly, for the ﬁrst two coeffcients, s1 (T) =

1 (1 − γ )v(2) (T)

(8.31)

and s2 (T) = −

(2 − γ )v(3) (T) (1 − γ )2 (v(2) (T))3

(8.32)

For γ < 1, including the third moment reduces the optimal fraction of corporates when v(3) (T) is positive.

4. SAMPLE PORTFOLIOS WITH ZERO RECOVERY FRACTIONS Here, we consider very simple sample portfolios where the correlations, default thresholds, and excess risk premiums are the same for all N ﬁrms in the portfolio, that is, ξij = ξ , χi (t) = χ (t) and µi (t) = µ.7 Then, all the probabilities of default are the same, Pi (t) = P(t), and the joint default probabilities are also independent of which ﬁrms are being considered, Pi1 ,...,im (t1 , . . . , tm ) = P12,...,m (t1 , . . . , tm ). The time dependence of the default probabilities is taken to be given by equation 8.4. Given our assumptions, the hazard rates are then also independent of ﬁrm, hi = h. We also take the portfolios to contain equal assets in the ﬁrms so that fi = 1/N, for all i, and assume all the recovery fractions are zero. Since the portfolio allocation problem does not depend on the risk-free rate y or the initial wealth, we set y = 0 and W0 = 1.

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Implications of Correlated Default for Portfolio Allocation

With these assumptions the expressions for ˆl(T) and x(T) become ˆl(T) = 1 exp(hT + µT) nˆ i (T) N

(8.33)

x(T) = exp(µT) − 1

(8.34)

i

and

For random defaults, the probability of a fractional loss of ˆl = exp(hT + µT)n/N in the time horizon T is (1 − P(T))N−n P(T)n N!/(N − n)!n! and so the mth moment of the default loss distribution is

v

(m)

(T) = exp(mhT + mµT)

N n n=0

× (1 − P(T))N−n P(T)n

N

m

− P(T)

N! (N − n)!n!

(8.35)

The expected utility of wealth is

ˆ = E[Uγ (W(T))]

N N! 1 (1 − P(T))N−n P(T)n γ (N − n)!n! n=0 n − P(T) × 1 + αx(T) − α N γ × exp(hT + µT)

(8.36)

Having the explicit expression for the utility of wealth in equation 8.36, let us compare results of the moment expansion for the optimal fraction of corporates αm with the all-orders result, αopt . These are presented in Table 8.1 in the case h = 0.02, µ = 100 bp, and γ = −4. Results for different values of the number of ﬁrms N and different time horizons T are shown in Table 8.1. We also give in columns three and four of Table 8.1 the volatility vol =

v(2) (T)

(8.37)

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Table 8.1 Optimal Fraction of Corporates for Sample Portfolios with Random (ξ = 0) Defaults. Other Parameters Used Are h = 0.02, γ = −4, and µ = 100 bp T (yr)

N

Vol

Skew

α2

α3

α4

α5

αopt

1 1 1 5 5 5

1 5 10 1 5 10

0.14 0.064 0.045 0.34 0.15 0.11

6.9 3.1 2.2 2.8 1.2 0.87

0.098 0.50 1.0 0.090 0.49 1.1

0.079 0.40 0.81 0.074 0.39 0.85

0.077 0.39 0.78 0.072 0.37 0.75

0.077 0.38 0.77 0.072 0.36 0.73

0.077 0.38 0.76 0.072 0.36 0.71

and skewness v(3) (T) skew = 3 v(2) (T)

(8.38)

of the portfolio default loss probability distribution. Increasing N gives a larger value for the optimal fraction of corporates because diversiﬁcation reduces risk. This occurs very rapidly with N. For N = 1, µ = 100 bp, T = 1 yr, and γ = −4 the optimal fraction of corporates is only 7.7%. By N = 10, increasing the number of ﬁrms has reduced the portfolio default risk so much that the 100 bp excess risk premium causes a portfolio that is 76% corporates to be preferred. For all the entries in Table 8.1 the moment expansion converges very rapidly, although for low N it is the small value of αopt that is driving the convergence. Since in equation 8.25 the term proportional to v(m) has a factor of α m accompanying it, we expect good convergence of the moment expansion at small α. The focus of this chapter is on portfolios that are not leveraged and have αopt < 1. But a value αopt > 1 is not forbidden when ﬁnding the maximum of the expected utility of wealth. This occurs at lowest order in the moment expansion in the last row of Table 8.1. Next we consider the more typical case where there are correlations, ξ = 0. Table 8.2 gives values of α2 –α5 for T = 1 yr, µ = 100 bp, h = 0.02, and γ = −4. Portfolios with N = 10, 50, and 100 are considered and values for ξ between 0.5 and 0.25 are used. Again, the convergence of the moment expansion is quite good. Although just including the variance (i.e., α2 ) can be off by a factor of almost two, α3 is usually a reasonable approximation to the true optimal fraction of corporates. The convergence is worse, the larger the value of αopt ; for values of αopt less than 40% we ﬁnd that α3 is within about 10% of αopt . In the last row of Table 8.2 the value of α5 is equal to unity. However, we know that the true value of αopt must be less than unity. For α = 1 there is some ﬁnite (but tiny) chance of investors losing all their wealth and for γ < 0, the utility of zero wealth is −∞.

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Implications of Correlated Default for Portfolio Allocation

Table 8.2 Moment Expansion for Optimal Portfolio with Correlated Defaults for T = 1 Yr, µ = 100 bp, h = 0.02, and γ = −4 T (yr) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

N

µ (bp)

ξ

dij

Vol

Skew

α2

α3

α4

α5

10 10 10 10 10 10 50 50 50 50 50 50 100 100 100 100 100 100

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

0.50 0.45 0.40 0.35 0.30 0.25 0.50 0.45 0.40 0.35 0.30 0.25 0.50 0.45 0.40 0.35 0.30 0.25

0.152 0.126 0.102 0.082 0.064 0.049 0.152 0.126 0.102 0.082 0.064 0.049 0.152 0.126 0.102 0.082 0.064 0.049

0.070 0.066 0.063 0.060 0.057 0.054 0.059 0.054 0.050 0.045 0.041 0.037 0.057 0.053 0.048 0.043 0.039 0.035

5.2 4.9 4.5 4.2 3.8 3.5 5.5 5.2 4.8 4.4 4.0 3.6 5.6 5.3 5.0 4.6 4.2 3.8

0.42 0.47 0.52 0.57 0.63 0.70 0.59 0.70 0.84 1.0 1.2 1.5 0.62 0.75 0.91 1.1 1.4 1.8

0.31 0.35 0.39 0.43 0.48 0.53 0.42 0.49 0.58 0.70 0.85 1.1 0.44 0.52 0.62 0.76 0.94 1.2

0.29 0.33 0.36 0.40 0.45 0.50 0.38 0.45 0.53 0.63 0.77 0.94 0.40 0.47 0.56 0.68 0.84 1.1

0.29 0.32 0.36 0.40 0.44 0.49 0.37 0.43 0.51 0.61 0.74 0.90 0.39 0.45 0.54 0.65 0.80 1.0

Correlations dramatically affect the dependence of the optimal portfolio allocation on the total number of ﬁrms. For the ξ = 0.5 entries in Table 8.2, the optimal fraction of corporates is 0.29, 0.37, and 0.39 for N = 10, 50, and 100, respectively. For N = 10,000 we ﬁnd that α5 = 0.40. Increasing the number of ﬁrms beyond 100 only results in a small increase in the optimal fraction of corporates. When defaults are random the moments of the portfolio default loss distribution go to zero as N → ∞. For example, with ξ = 0, the variance of the default loss distribution is v(2) = exp(2hT + 2µT)

P(1 − P) N

(8.39)

and the skewness of the default loss distribution is v(3) (v(2) )3/2

=√

1 (1 − 2P) NP(1 − P)

(8.40)

For random defaults, as N → ∞, the distribution for δ ˆl approaches the trivial one where δ ˆl = 0 occurs with unit probability. However, for ξ > 0 the moments v(m) go to nonzero values in the limit N → ∞ and the default loss distribution remains nontrivial and non-normal. In Table 8.3 the time horizon is changed to T = 5 yr but the other parameters are the same as in Table 8.2. The value of αopt is always smaller than in Table 8.2, indicating that the compounding of the excess risk premium does not completely compensate for the added risk associated with the greater probability of default. The convergence of the moment expansion and the

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THE CREDIT MARKET HANDBOOK

Table 8.3 Moment Expansion for Optimal Portfolio with Correlated Defaults for T = 5 Yr, µ = 100 bp, h = 0.02, and γ = −4 T (yr) 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

N

µ (bp)

ξ

dij

Vol

Skew

α2

α3

α4

α5

10 10 10 10 10 10 50 50 50 50 50 50 100 100 100 100 100 100

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

0.50 0.45 0.40 0.35 0.30 0.25 0.50 0.45 0.40 0.35 0.30 0.25 0.50 0.45 0.40 0.35 0.30 0.25

0.246 0.213 0.182 0.153 0.127 0.102 0.246 0.213 0.182 0.153 0.127 0.102 0.246 0.213 0.182 0.153 0.127 0.102

0.193 0.184 0.175 0.166 0.158 0.149 0.174 0.163 0.152 0.141 0.129 0.118 0.172 0.160 0.149 0.137 0.125 0.113

2.3 2.2 2.1 2.0 1.9 1.8 2.4 2.3 2.2 2.1 2.0 1.8 2.4 2.3 2.2 2.1 2.0 1.8

0.29 0.32 0.36 0.40 0.45 0.51 0.36 0.42 0.49 0.58 0.71 0.89 0.38 0.44 0.52 0.62 0.76 0.98

0.22 0.24 0.27 0.30 0.34 0.38 0.27 0.31 0.35 0.41 0.49 0.60 0.28 0.32 0.37 0.43 0.52 0.65

0.21 0.23 0.25 0.28 0.31 0.35 0.25 0.28 0.32 0.38 0.44 0.54 0.26 0.29 0.33 0.39 0.47 0.57

0.20 0.22 0.25 0.27 0.31 0.34 0.24 0.28 0.31 0.36 0.43 0.51 0.25 0.28 0.33 0.38 0.45 0.54

Table 8.4 Moment Expansion for Optimal Portfolio with Correlated Defaults for T = 1 Yr, N = 100, µ = 60 bp, h = 0.02, and γ = −2 T (yr) 1 1 1 1 1 1

N 100 100 100 100 100 100

µ (bp) 60 60 60 60 60 60

ξ

dij

Vol

0.50 0.45 0.40 0.35 0.30 0.25

0.152 0.126 0.102 0.082 0.064 0.049

0.057 0.053 0.048 0.043 0.039 0.035

Skew 5.6 5.3 5.0 4.6 4.2 3.8

α2

α3

α4

α5

0.62 0.74 0.90 1.1 1.4 1.7

0.47 0.56 0.68 0.82 1.0 1.3

0.45 0.53 0.63 0.76 0.94 1.2

0.44 0.52 0.62 0.74 0.92 1.1

dependence of the optimal fraction of corporates on the number of bonds is similar to that in Table 8.2. Table 8.3 indicates that the volatility of the portfolio default loss probability distribution increases as the time horizon increases, and the skewness decreases as the time horizon increases. The same behavior occurs for random defaults and this can be seen from equations 8.39 and 8.40 with the small probability of default P increasing with time. Next we consider the effect of changing the utility function to one with a lower level of risk aversion. In Table 8.4, γ = −2 is used. Equation 8.31 suggests that an excess risk premium of 60 bp would yield roughly the same portfolio allocation as the excess risk premium of 100 bp used in Tables 8.2 and 8.3. In Table 8.4 we consider portfolios with 100 ﬁrms and take the same parameters as in Table 8.2 apart from the lower level of risk aversion, γ = −2, and the lower excess risk premium, µ = 60 bp. It is not surprising that the values of α2 in Table 8.4 are close to those in Table 8.2. However, α5 is typically about 10% larger than in Table 8.2. The convergence of the moment expansion is better in Table 8.4 than in Table 8.2.

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Implications of Correlated Default for Portfolio Allocation

Table 8.5 Probability of Default and Convergence of the Moment Expansion for T = 1 Yr, ξ = 0.50, N = 100, and γ = −4 T (yr) 1 1 1 1 1

N 0.02 0.04 0.06 0.08 0.10

µ (bp) 100 200 300 500 600

ξ

dij

Vol

0.50 0.50 0.50 0.50 0.50

0.152 0.189 0.214 0.231 0.246

0.057 0.092 0.121 0.148 0.173

Skew 5.6 4.0 3.2 2.7 2.4

α2

α3

α4

α5

0.62 0.50 0.44 0.52 0.46

0.44 0.36 0.32 0.35 0.32

0.40 0.33 0.29 0.32 0.29

0.39 0.32 0.29 0.31 0.28

The examples in this section have all used an annualized hazard rate of 2%; however, the convergence of the moment expansion is similar for signiﬁcantly larger default probabilities. In Table 8.5 the convergence of the moment expansion is examined for portfolios with 100 corporate bonds and hazard rates of 2, 4, 6, 8, and 10%. The value of ξ is ﬁxed at 0.50 and the time horizon is 1 year. The convergence of the moment expansion is good for all the values of h used in Table 8.5. For longer time horizons the increased default risk is not compensated for by the compounding of the excess risk premium. For example, if we take the parameters of the last row of Table 8.5 but increase the time horizon to T = 5 years, then α2 = 0.25, α3 = 0.21, α4 = 0.18, and α5 = 0.17.

5. SAMPLE PORTFOLIOS WITH NONZERO RECOVERY FRACTIONS In this section we consider portfolios like those in Section 4 but allow the recovery fractions Ri = R to be nonzero. Here, we focus on short time horizons using the approximate formulas, ˆl(T) = (1 − R)nˆ i (T)

(8.41)

x(T) = µT

(8.42)

and

appropriate to that regime. Recall that if R is not zero the actuarially fair corporate spread is λ = h(1 − R). Table 8.6 shows the impact of nonzero recovery on αopt . The parameters used are γ = −4, T = 1 year, N = 10, R = 0.5, h = 0.04, and µ = 50 and 100 bp. Doubling the hazard rate to 0.04 gives an actuarially fair corporate spread of 200 bp, which is the same as in Tables 8.2 and 8.3. Note that even though the actuarially fair spread in Table 8.6 is the same as in Table 8.2, nonzero recovery reduces the default risk and so the excess risk premium must be reduced by a factor of 2 to get values of αopt close to those in Table 8.2.

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Table 8.6 Recovery and Moment Expansion for Optimal Portfolio with Correlated Defaults Using T = 1 Yr, N = 10, h = 0.04, R = 0.5, and γ = −4 T (yr)

N

µ (bp)

ξ

dij

Vol

Skew

α2

α3

α4

α5

1 1 1 1 1 1 1 1 1 1 1 1

10 10 10 10 10 10 10 10 10 10 10 10

50 50 50 50 50 50 100 100 100 100 100 100

0.50 0.45 0.40 0.35 0.30 0.25 0.50 0.45 0.40 0.35 0.30 0.25

0.152 0.126 0.102 0.082 0.064 0.049 0.152 0.126 0.102 0.082 0.064 0.049

0.050 0.048 0.046 0.043 0.041 0.039 0.050 0.048 0.046 0.043 0.041 0.039

4.6 4.5 4.3 4.2 4.0 3.8 4.6 4.5 4.3 4.2 4.0 3.8

0.40 0.44 0.49 0.54 0.60 0.67 0.81 0.90 1.0 1.1 1.2 1.4

0.32 0.36 0.39 0.44 0.48 0.54 0.57 0.63 0.70 0.78 0.86 0.96

0.31 0.34 0.38 0.42 0.47 0.52 0.53 0.59 0.65 0.72 0.80 0.88

0.31 0.34 0.38 0.42 0.46 0.51 0.52 0.57 0.63 0.70 0.78 0.86

For random defaults one can easily see why recovery reduces portfolio default risk. Taking the hazard rate to be small, the time horizon to be short, and the defaults to be random, λT N

(8.43)

1−R NλT

(8.44)

v(2) (T) (1 − R) and v(3) (T)

(v(2) (T))3/2

where λ is the actuarially fair spread. Hence, with λ held ﬁxed, the variance and skewness of the portfolio default loss probability distribution decrease as R increases. Since we are using the approximate formulas in equations 8.41 and 8.42, we would not get exact agreement with Table 8.2 even if R = 0. For example, with T = 1 year, N = 10, γ = −4, µ = 100 bp, h = 0.02, ξ = 0.5, and R = 0, using equations 8.41 and 8.42 yields α2 = 0.44, α3 = 0.33, α4 = 0.31, and α5 = 0.30. On the other hand, the ﬁrst row of Table 8.2 is α2 = 0.42, α3 = 0.31, α4 = 0.29, and α5 = 0.29.

6. CONCLUDING REMARKS Default correlations have an important impact on portfolio default risk. Given a default model, the tail of the default loss probability distribution is difﬁcult to compute, often involving numerical simulation8 of rare events. The ﬁrst few moments of the default loss distribution are easier to calculate, and in the default model we used, this computation involved some simple numerical integration. More signiﬁcantly, the ﬁrst few terms in the moment

Implications of Correlated Default for Portfolio Allocation

183

expansion have a familiar meaning. Investors are used to working with the classic mean, variance, and skewness measures and have developed intuition for them and conﬁdence in them. In this chapter, we studied the utility of the ﬁrst few moments of the default loss probability distribution for the portfolio allocation problem. The default model we use assumes a multivariate normal copula function. However, the corporate assets themselves are not necessarily normal or lognormal and their probability distribution can have fat tails. When there are correlations, the portfolio default loss distribution does not approach a trivial9 or normal probability distribution as the number of ﬁrms in the portfolio goes to inﬁnity. Correlations dramatically decrease the effectiveness of increasing the number of ﬁrms, N, in the portfolio to reduce portfolio default risk. We explored the convergence of the moment expansion for the optimal fraction of corporate assets αopt . Our work indicates that, for αopt less than unity, the convergence of the moment expansion is quite good. The convergence of the moment expansion gets poorer as αopt gets larger. We also examined how αopt depends on the level of risk aversion, the recovery fraction, and investment time horizon. The value of αopt depends on the utility function used. In this chapter, we used utility functions with constant relative risk aversion, 1 − γ . It is possible to make other choices and whereas the quantitative results will be different for most practical purposes, we expect that the general qualitative results should continue to hold. Our main conclusions from the examples in Sections 4 and 5 are as follows:

■

■

In the presence of correlated defaults, the default loss distribution is not normal. Nevertheless, we ﬁnd that retaining just the ﬁrst few moments of the default loss probability distribution yields a good estimate of the optimal fraction of corporates. As αopt gets smaller, the convergence of the moment expansion improves. For the examples in Sections 4 and 5 with αopt near 40%, retaining just the mean, variance, and skewness of the default loss probability distribution determined αopt with a precision better than about 15%. For small numbers of corporate bonds, increasing the number of ﬁrms decreases portfolio default risk. However, correlations cause this effect to saturate as the number of bonds increases. For the examples in Sections 4 and 5 (with production factor or asset correlations 0.50 ≥ ξ ≥ 0.25), the optimal fraction of corporates does not increase signiﬁcantly if N is increased beyond 100 ﬁrms.

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■

Increasing the recovery fractions decreases portfolio risk and this is true even when the probabilities of default increase so that the actuarially fair spreads remain constant. With probabilities of default given by Pi (T) = 1 − exp(−hi T) and the hazard rates hi constant, continuous compounding of a constant excess risk premium does not exactly compensate for the increase in the default probability with time. The optimal fraction of corporates decreased in going from a T = 1-year to a T = 5-year investment horizon.

■

ACKNOWLEDGMENTS The authors would like to thank their colleagues at PIMCO for enlightening discussions. Mark B. Wise was supported in part by the Department of Energy under contract DE-FG03-92-ER40701.

NOTES 1. In this chapter, we use annualized units, so quoted default probabilities, hazard rates, and corporate rates are annualized ones. 2. The authors would like to thank David Hinman of PIMCO for enlightening discussions on this topic. 3. See, for example, Jorion (2001). 4. Zhou (2001) derives an analytic formula for default correlations in a ﬁrst passage-time default model and ﬁnds a similar increase with time horizon. 5. Assuming that after maturity the value of a zero-coupon corporate bond is reinvested at the risk-free rate. It is straightforward to generalize the analysis of this chapter to portfolios containing zero-coupon bonds with different maturities, some of which are less than the investment horizon. Similarly, default risk for portfolios containing coupon-paying bonds can be studied since each coupon payment can be viewed as a zero-coupon bond. 6. See, for example, Ingersoll (1987). 7. We also assume that µ is independent of time. 8. See, for example, Dufﬁe and Singleton (1999) and Das et al. (2001). 9. The trivial distribution has δ ˆl = 0 occuring with unit probability.

REFERENCES Altman, E. I. (1989). “Measuring Corporate Bond Mortality and Performance.” Journal of Finance 44(4), 909–921.

Implications of Correlated Default for Portfolio Allocation

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Black, F. and Cox, J. (1976). “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” Journal of Finance 31, 351–367. Das, S., Fong, G. and Geng, G. (2001). “The Impact of Correlated Default Risk on Credit Portfolios.” Working Paper, Department of Finance, Santa Clara University and Gifford Fong Associates. Das, S., Freed, L., Geng, G. and Kapadia, N. (2001). “Correlated Default Risk.” Working Paper, Department of Finance, Santa Clara University and Gifford Fong Associates. Das, S. and Geng, G. (2002). “Modeling the Process of Correlated Default.” Working Paper, Department of Finance, Santa Clara University and Gifford Fong Associates. Dufﬁe, D. and Singleton, K. (1999). “Simulating Correlated Defaults.” Working Paper, Stanford University Graduate School of Business. Dynkin, L., Hyman, J. and Konstantinovsky, V. (2002). “Sufﬁcient Diversiﬁcation in Credit Portfolios.” Lehman Brothers Fixed Income Research (working paper). Ingersoll, J. (1987). Theory of Financial Decision Making. Lanham, MD: Rowman and Littleﬁeld Publishers Inc. Jorion, P. (2001). Value at Risk, 2nd edn. New York: McGraw-Hill Inc. Li, D. (2000). “On Default Correlation: A Copula Function Approach.” Working Paper 99-07, The Risk Metrics Group. Leland, H. and Toft, K. (1996). “Optimal Capital Structure, Endogenous Bankruptcy and the Term Structure of Credit Spreads.” Journal of Finance 51, 987–1019. Longstaff, F. and Schwartz, E. (1995). “A Simple Approach to Valuing Risky Floating Rate Debt.” Journal of Finance 50, 789–819. Lucas, D. (1995). “Default Correlation and Credit Analysis.” Journal of Fixed Income, 4(4), 76–87. Merton, R. (1974). “On Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29, 449–470. Rebonato, R. and Jäckel, P. (2000). “The Most General Methodology for Creating a Valid Correlation Matrix for Risk Management and Option Pricing Purposes.” The Journal of Risk 2(2), 17–26. Wise, M. and Bhansali, V. (2002). “Portfolio Allocation to Corporate Bonds with Correlated Defaults.” Journal of Risk 5(1), Fall. Zhou, C. (2001). “An Analysis of Default Correlations and Multiple Defaults.” The Review of Financial Studies 12(2), 555–576.

Keywords: Asset correlations; default; corporate bonds; portfolio allocation

CHAPTER

9

Correlated Default Processes: A Criterion-Based Copula Approach Sanjiv R. Dasa,∗ and Gary Gengb

In this chapter, we develop a methodology to model, simulate, and assess the joint default process of hundreds of issuers. Our study is based on a data set of default probabilities supplied by Moody’s Risk Management Services. We undertake an empirical examination of the joint stochastic process of default risk over the period 1987 to 2000 using copula functions. To determine the appropriate choice of the joint default process we propose a new metric. This metric accounts for different aspects of default correlation, namely, (1) level, (2) asymmetry, and (3) tail dependence and extreme behavior. Our model, based on estimating a joint system of over 600 issuers, is designed to replicate the empirical joint distribution of defaults. A comparison of a jump model and a regime-switching model shows that the latter provides a better representation of the properties of correlated default. We also ﬁnd that the skewed double exponential distribution is the best choice for the marginal distribution of each issuer’s hazard rate process, and combines well with the normal, Gumbel, Clayton, and Student’s t copulas in the joint dependence relationship among issuers. As a complement to the methodological innovation, we show that (1) Appropriate choices of marginal distributions and copulas are essential in modeling correlated default; (2) Accounting for regimes is an important aspect of joint speciﬁcations of default risk; and (3) Misspeciﬁcation of credit portfolio risk can occur easily if joint distributions are inappropriately chosen. The empirical evidence suggests that improvements over the standard Gaussian copula used in practice are indeed possible.

a Santa Clara University, Santa Clara, CA 95053. b Amaranth Group Inc., Greenwich, CT 06831. ∗ Corresponding author. Santa Clara University, Santa Clara, CA 95053, USA.

E-mail: [email protected].

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187

1. INTRODUCTION Default risk at the level of an individual security has been extensively modeled using both structural and reduced-form models.1 This chapter examines default risk at the portfolio level. Default dependences among issuers in a large portfolio play an important role in the quantiﬁcation of a portfolio’s credit risk exposure for many reasons. Growing linkages in the ﬁnancial markets have led to a greater degree of joint default. While the actual portfolio loss due to the default of an individual obligor may be small unless the risk exposure is extremely large, the effects of simultaneous defaults of several issuers in a well-diversiﬁed portfolio could be catastrophic. In order to efﬁciently manage and hedge the risk exposure to joint defaults, a model for default dependencies is called for. Further, innovations in the credit market have been growing at an unprecedented pace in recent years and will likely persist in the near future. Many newly developed ﬁnancial securities, such as collateralized debt obligations (CDOs), have payoffs depending on the joint default behavior of the underlying securities.2 In order to accurately measure and price the risk exposure of these securities, an understanding and accurate measurement of the default dependencies among the underlying securities is essential.3 Accurate speciﬁcation of default correlations is required in the pricing of basket default contracts. Correlation speciﬁcations are critical, as the baskets are not large enough to ensure diversiﬁcation. Finally, the Basle Committee on Banking Supervision has identiﬁed poor portfolio risk management as an important source of risk for banking institutions. As a result, banks and other ﬁnancial institutions have been required to measure their overall risk exposures on a routine basis. Default correlation is again an important ingredient to integrate multiple credit risk exposures. In this chapter, we use Moody’s default database to develop a parsimonious numerical method of modeling and simulating correlated default processes for hundreds of issuers. Using copula functions, we capture the stylized facts of individual default probabilities as well as the dynamics of default correlations documented in Das et al. (2001a) (DFGK, 2001a). Our methodology is criterion-based; that is, it uses a metric that compares alternative speciﬁcations of the joint default distribution using three criteria: (1) the level of default risk, (2) the asymmetry in default correlations, and (3) the tail dependence of joint defaults. We elaborate on these in what follows. The base unit of joint credit risk is the probability of default (PD) of each issuer. Many popular approaches exist for computing PDs in the market place, developed by ﬁrms such as KMV Corporation, Moody’s Risk Management Systems (MRMS), RiskMetrics, and so forth. In the spirit of reduced-form models, the default probability of issuer i at time t is usually

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expressed as a stochastic hazard rate, denoted as λi (t), i = 1, . . . , N. This chapter empirically examines the joint stochastic process for hazard rates, λi (t), for N issuers. We ﬁt a joint dynamic system to the intensities (or hazard rates) of roughly 600 issuers classiﬁed into six rating categories. Our approach involves capturing the correlation of PD levels across the six rating classes, and the correlation of ﬁrms’ PDs within each rating class. We ﬁnd (1) the appropriate copula function (for the joint distribution), (2) the stochastic process for rating-level default probabilities, and (3) the best set of marginal distributions for individual issuer default probabilities. At the rating level, we choose from two classes of stochastic processes for the average hazard rate of the class: a jump-diffusion model and a regime-switching one. At the issuer level, we choose from three distributional assumptions: a normal distribution, a Student’s t distribution, and a skewed double exponential distribution. We compare these distributions for each issuer using four goodness-of-ﬁt criteria. At the copula level, we choose from four types: normal, Student’s t, Gumbel, and Clayton. These inject varying amounts of correlation emanating from the joint occurrence of extreme observations.4 The various combinations of choices at the rating, issuer, and copula levels result in 56 different systems of joint variation. To compare across these, we develop a metric to determine how well different multivariate distributions ﬁt the observed covariation in PDs. The metric accounts for the level of correlations, asymmetry of correlations, and the tail dependence in the data. An important issue we look at here is tail dependence. Tail dependence is the feature of the joint distribution that determines how much of the correlation between default intensity processes comes from extreme observations and how much from central observations. For example, imagine two different bivariate distributions with the same standard normal marginal distributions. One distribution has a normal copula and the other a Student’s t copula with a low degree of freedom. The former has lower tail dependence than the latter, which generates more joint tail observations. Hence, from a risk point of view, given the marginal distributions, the second joint distribution would be riskier for a large class of risk-averse investors. Our study sheds light on the tail dependence for default risk in a representative data set of US ﬁrms. Our study of joint default risk for US corporates using copula techniques ﬁnds that the best choice of copula depends on the marginal distributions and stochastic processes for rating level PDs. The Student’s t copula is best when the jump-diffusion model is chosen for the default intensity of each rating class. However, the normal or Clayton copula is more appropriate when

189

Correlated Default Processes: A Criterion-Based Copula Approach 0.032 0.03 0.028

probability of default

0.026 0.024 0.022 0.02 0.018 0.016 0.014 0.012 Jan87

May88

Sep89

Feb91

Jun92

Nov93

Mar95

Aug96

Dec97

Apr99

Sep00

FIGURE 9.1 Time series of average PDs. This ﬁgure depicts the average level of default probabilities in the data set. The data show the presence of two regimes, one in which PDs were high, as in the early and later periods of the data, and the other in which PDs were much lower, less than half of those seen in the high-PD regime. Complementary analysis of correlated default by subperiod on the same data set in the same time frame is presented in DFGK (2001a). the regime-switching model is chosen for rating level intensities. In general, our metric prefers a regime-switching model to a jump-diffusion one (for rating-level intensities), irrespective of the choice of marginal distributions. A perusal of the time-series plot of the data (see Figure 9.1) provides intuitive support for this result. While the regime-switching model for rating-level intensities is best, no clear winner emerges among the competing copulas. The interdependence between copula and marginal distributions justiﬁes our wide-ranging search of over 56 different speciﬁcations. We also assess the impact of different copulas on risk-management measures. An examination of the tail loss distributions shows that substantial differences are possible among the 56 econometric speciﬁcations. Arbitrarily assigning high kurtosis distributions to a copula with tail dependence may result in the unnecessary overstatement of joint default risk. Since different copulas inject varied levels of tail dependence, the metric developed in this

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chapter allows ﬁne-tuning of the speciﬁcation, which enhances the accuracy of credit VaR calculations. The rest of the chapter proceeds as follows. In Section 2 we brieﬂy describe the data on intensities. Section 3 provides the ﬁnance reader with a brief introduction to copulas, and Section 4 contains the estimation procedure and results. Section 5 presents the simulation model and the metric used to compare different copulas. Section 6 concludes.

2. DESCRIPTION OF THE DATA Our data set comprises issuers tracked by Moody’s from 1987 to 2000. For each issuer, we have PDs based on the econometric models for every month the issuer was tracked by Moody’s. Moody’s calibrates the PDs to match the level of realized defaults in the economy. Issuers are divided into seven rating classes. Rating classes 1 through 6 reﬂect progressively declining credit quality. Many more issuers fall into rating class 7, which comprises unrated issuers; the PDs within this class range from high to low, resulting in an average PD that is close to the median PD of all of the other rating classes. We do not consider PDs from rating class 7. We considered ﬁrms that had a continuous sample over the period. We obtained data on a total of 620 issuers classiﬁed into six rating classes. Our data set does not contain data from ﬁrms in the ﬁnancial sector. These 1-year default probabilities were converted into intensities using an assumption of constant hazard rates, that is, λit = − ln(1 − PDit ). Table 9.1 provides a summary of Moody’s rating categories for bonds. The time series of average default probabilities is presented in Figure 9.1. Table 9.2 presents the descriptive statistics of our data from rating classes 1 through 6. The mean increases from rating class 1 to 6, as does the standard deviation. PD changes tend to be higher for lower-grade debt. Table 9.3 presents the dependence between rating classes measured by Kendall’s τ

Table 9.1 Moody’s Rating Categories for Bonds in Descending Order of Credit Quality 1. 2. 3. 4. 5. 6. 7.

Aaa/Aa bonds carry the smallest degree of investment risk and are generally known as high-grade bonds. A bonds possess many favorable investment attributes and are considered upper-medium-grade obligations. Baa bonds are considered medium-grade obligations. Such bonds lack outstanding investment characteristics and in fact have speculative characteristics as well. Ba bonds are judged to have speculative elements. Often the protection of interest and principal payments may be very moderate and thereby not well safeguarded during both good and bad times over the future. B bonds generally lack characteristics of the desirable investment. Assurance of interest and principal payments over any long period of time may be small. Caa/Ca/C bonds have a high degree of speculation. Such issues may either be in default or have elements of danger present with respect to principal or interest payments. Not rated.

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Table 9.2 Summary of Time Series of Average PDs for Each Rating Class. Each time series represents a diversiﬁed portfolio of issuers within each rating classa PD in Levels (%)

PD in Changes (%)

Rating Category

Mean

Std Dev.

Mean

Std Dev.

1 2 3 4 5 6

0.0728 0.1701 0.6113 1.4855 3.8494 6.5389

0.1051 0.4884 1.6808 2.6415 4.6401 4.7731

0.0012 0.0036 0.0044 0.0066 0.0138 0.0165

0.0441 0.1468 0.7317 1.2694 2.3576 3.2075

a Each time series represents a diversiﬁed portfolio of issuers within each rating class, equally weighted.

Table 9.3

Kendall’s τ for Probabilities of Default for Rating Classesa Correlations

Rating Category

1

2

3

4

5

6

1 2 3 4 5 6

1.0000 0.4254 0.1849 0.1447 0.0378 0.0362

0.6508 1.0000 0.2312 0.1393 0.0877 −0.0429

0.2724 0.4671 1.0000 0.0543 0.0352 0.0238

0.2868 0.4569 0.6416 1.0000 0.1245 −0.0276

0.2987 0.4052 0.4483 0.5382 1.0000 0.1032

0.2762 0.1794 0.1080 0.1708 0.2911 1.0000

a The upper right triangle (values located above 1.0000) reports the measure for PD levels. The lower left triangle (values located below 1.0000) reports the correlation for PD changes.

statistic, a means of determining the dependence between any two time series. Higher-grade debt evidences more rank correlation in PDs than low-grade debt. This highlights the fact that high-grade debt is more systematically related, and low-grade debt experiences greater idiosyncratic risk.

3. COPULAS AND FEATURES OF THE DATA 3.1 Definition We are interested in modeling an n-variate distribution. A random draw from this distribution comprises a vector X ∈ Rn = {X1 , . . . , Xn }. Each of the n variates has its own marginal distribution, Fi (Xi ), i = 1, . . . , n. The joint distribution is denoted as F(X). The copula associated with F(X) is a multivariate distribution function deﬁned on the unit cube [0, 1]n . When all the marginals, Fi (Xi ), i = 1, . . . , n, are continuous, the associated copula is unique and can be written as C(u1 , . . . , un ) = F(F1−1 (u1 ), . . . , Fn−1 (un ))

(9.1)

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Every joint distribution may be written as a copula. This is Sklar’s Theorem (Sklar, 1959, 1973). For an extensive discussion of copulas, see Nelsen (1999).5 Copulas allow the modeling of the marginal distributions separately from their dependence structure. This greatly simpliﬁes the estimation problem of a joint stochastic process for a portfolio with many issuers. Instead of estimating all of the distributional parameters simultaneously, we can estimate the marginal distributions separately from the joint distribution. Given the estimated marginal distribution for each issuer, we then use appropriate copulas to construct the joint distribution with a desired correlation structure. The best copula is determined by examining the statistical ﬁt of different copulas to the data.6 Copula techniques lend themselves to two types of credit risk analysis. First, given a copula, we can choose different marginal distributions for each individual issuer. By changing the types of marginal distributions and their parameters, we can examine how individual default affects the joint default behavior of many issuers in a credit portfolio. Second, given marginal distributions, we can vary the correlation structures by choosing different copulas, or the same copula with different parameter values.

3.2 Copulas Used in the Chapter In this chapter, in order to capture the observed properties of the joint default process, we consider the following four types of copulas: Normal copula: The normal copula of the n-variate normal (Gaussian) distribution with correlation matrix ρ, is deﬁned as Cρ (u1 , . . . , un ) = ρn (F −1 (u1 ), . . . , F −1 (un )) where (u) is the normal cumulative distribution function (CDF), and F −1 (u) is the inverse of the CDF. Student’s t copula: Let Tρ,ν be the standardized multivariate Student’s t distribution with ν degrees of freedom and correlation matrix ρ. The multivariate Student’s t copula is then deﬁned as follows: C(u1 , . . . , un ; ρ, ν) = Tρ,ν (tν−1 (u1 ), . . . , tν−1 (un )) where tν−1 is the inverse of the cumulative distribution function of a univariate Student’s t distribution with ν degree of freedom.

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Correlated Default Processes: A Criterion-Based Copula Approach

Gumbel copula: This copula was ﬁrst introduced by Gumbel (1960) and can be expressed as follows:  1/α  n  (− ln ui )α C(u1 , . . . , un ) = exp−

(9.2)

i=1

where α is the parameter determining the tail dependence of the distribution. Clayton copula: This copula, introduced in Clayton (1978), is as follows: C(u1 , . . . , un ) =

n

−1/α u−α i

−n+1

(9.3)

i=1

Again, α > 1 is a tail dependence parameter. The procedure for simulating an n-dimensional random vector from the above copulas can be found in Wang (2000), Bouye et al. (2000), and Embrechts et al. (2001).

3.3 Tail Dependence An important feature of the use of copulas is that it permits varying degrees of tail dependence. Tail dependence refers to the extent to which the dependence (or correlation) between random variables arises from extreme observations. Suppose (X1 , X2 ) is a continuous random vector with marginal distributions F1 and F2 . The coefﬁcient of upper tail dependence is: λU = lim Pr[X2 > F2−1 (z) | X1 > F1−1 (z)] z→1

(9.4)

If λU > 0, then upper tail dependence exists. Intuitively, upper tail dependence is present when there is a positive probability of positive extreme observations occurring jointly.7 For example, the Gumbel copula has upper tail dependence with λU = 2 − 21/α . The Clayton copula has lower tail dependence with λL = 2−1/α . The Student’s

√ √ t has equal upper and lower tail dependence with λU = v+1 1−ρ √ 2t¯v+1 . 1+ρ

Table 9.4 provides a snapshot of the tail dependence between rating classes. The results in the table present the correlations between rating

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Table 9.4 Correlations Between Hazard Rates among Rating Classes as Measures of Tail Dependence among observations in the bottom 10th, 20th, and 30th percentiles, and the correlations in the top 10th, 20th, and 30th percentiles; that is, the cutoffs are the 70th, 80th, and 90th percentiles. The values in the table are the correlations when observations from the second rating in the pair of rating classes lie in the designated portion of the tail of the distribution Rating Pair Rating

Rating

Lower Tail Dependence

Upper Tail Dependence

10%

20%

30%

70%

80%

Panel A: Tail Dependence in Levels 1 2 0.4853 1 3 0.0147 1 4 −0.1471 1 5 −0.0882 1 6 −0.2206 2 3 −0.1912 2 4 −0.2353 2 5 −0.0147 2 6 −0.1912 3 4 0.2500 3 5 −0.0294 3 6 −0.1765 4 5 0.0588 4 6 0.1471 5 6 0.2206

90%

0.2879 −0.3371 0.0379 −0.1439 −0.1705 −0.2197 −0.1326 −0.2045 −0.3447 0.2424 −0.1402 −0.1591 −0.1023 −0.0833 0.0455

0.2212 −0.1673 −0.0008 −0.1314 −0.0727 0.1004 −0.0857 −0.2065 −0.1559 0.2539 0.0694 −0.0955 0.0155 −0.0318 0.1020

0.7518 0.5837 0.6376 0.6865 0.5298 0.5967 0.6000 0.6718 0.4171 0.5559 0.5576 0.3535 0.7257 0.4922 0.4563

0.7386 0.5795 0.4545 0.6061 0.4583 0.5985 0.4015 0.5189 0.3826 0.3902 0.4205 0.4583 0.6439 0.6326 0.5152

0.7206 0.5441 0.1471 0.2059 −0.0735 0.5588 0.1324 0.0882 −0.1765 0.1765 0.2647 0.0147 0.4412 0.4853 0.2059

Panel B: Tail Dependence in Changes 1 2 0.5000 1 3 0.0441 1 4 0.1029 1 5 0.1029 1 6 0.0588 2 3 0.1618 2 4 −0.2647 2 5 −0.0147 2 6 0.1618 3 4 0.2941 3 5 −0.0441 3 6 0.0882 4 5 −0.0588 4 6 −0.0735 5 6 −0.0294

0.4394 0.0189 0.1439 −0.0455 −0.0455 0.0606 −0.0076 0.0189 0.0455 0.1364 −0.1212 0.0076 −0.0227 0.1515 0.1553

0.4106 0.0367 0.0188 −0.0269 −0.0580 0.0873 −0.0580 0.0645 0.0482 −0.0433 −0.0367 0.1216 0.1265 0.0971 0.0171

0.4253 0.0776 0.1429 0.1069 −0.2033 0.0204 0.1984 0.0302 −0.1559 0.0122 0.2082 −0.0694 0.2555 0.0041 0.0547

0.4015 0.0038 0.0644 0.1439 −0.1856 −0.0606 0.0606 0.2045 −0.1667 −0.0720 −0.0152 −0.1136 0.3864 0.1098 0.0341

0.3971 0.2206 0.1618 0.2794 0.0735 0.1324 0.2647 0.3382 −0.2500 −0.0441 0.0294 −0.3382 0.3824 −0.1324 −0.0147

pairs over observations in the tails of the bivariate distribution. Results are provided both in levels and in changes. There is strong positive correlation in the upper tail, evidencing upper tail dependence. Correlations in the lower tail are often low and negative, and hence there is not much evidence of lower tail dependence.

3.4 Empirical Features of Dependence in the Joint Distribution In order to compare the statistical ﬁtting of joint distributions associated with different copulas, we develop a metric that captures three features of the dependence relationship in the joint distribution: ■

Correlation levels: We wish to ensure that our copula permits the empirically observed correlation levels in conjunction with the other moments. This depends on the copula and the attributes of the data.

195

Correlated Default Processes: A Criterion-Based Copula Approach 0.35

R1 R2 R3 R4 R5 R6

0.3

Conditional correlation

0.25

0.2

0.15

0.1

0.05

0 –1.5

–1

–0.5

0

0.5

1

1.5

Exceedance Level

FIGURE 9.2 Asymmetric exceedance correlations of PDs for the different rating classes. The plots are consistent with the presence of asymmetric correlation. Complementary analysis of correlated default by subperiod in this same time frame is presented in DFGK (2001a) where different analyses are conducted on the same data set. ■

■

Correlation asymmetry: Hazard rate correlations are level-dependent, and are higher when PD levels are high—correlations increase when PD levels jump up, and decrease when PDs jump down. Tail dependence: It is important that we capture the correct degree of tail dependence in the data, that is, the extent to which extreme values drive correlations.

Following Ang and Chen (2002) and Longin and Solnik (2001), we present the correlations for different rating classes in a correlation diagram (see Figure 9.2). This plot is created as follows. We compute the total hazard rate (THR) across issuers (indexed by i) at each point in time (t), N that is, THRt = i=1 λi (t), ∀t. We normalize the THR by subtracting the mean from each observation and dividing by the standard deviation. We then segment our data set based on exceedance levels (ξ ) determined by the normalized THR. Our exceedance levels are drawn from the set

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ξ = {−1.5, −1.0, −0.5, 0, 0, 0.5, 1.0, 1.5}. (Note that zero appears twice because we account for both left and right exceedance.) Ang and Chen (2002) developed a version of this for bivariate processes. Our procedure here is a modiﬁcation for the multivariate case. For example, the exceedance correlation at level −ξ is determined by extracting the time series of PDs for which the normalized THR is less than −ξ , and computing the correlation matrix therefrom. We then ﬁnd the average value of all entries in this correlation matrix to obtain a summary number for the correlation in the rating class at the given exceedance level. These are plotted in Figure 9.2. There is one line for each rating class. The line extends to both the left and the right sides of the plot. To summarize, the exceedance graph shows (for each rating category) the correlation of default probabilities when the overall level of default risk exceeds different thresholds. As threshold levels become very small or very large, correlations decline, and the rate at which they decline indicates how much tail dependence there is. The three features of the dependence relationship are evident from the graph. First, the height of the exceedance line indicates the level of correlation, and we can see that high-grade debt has greater correlation than lower-grade debt (see Xiao (2003) for a similar result with credit spreads). Second, there is clear evidence of correlation asymmetry, as correlation levels are much higher on the right side of the graph; that is, when hazard rate changes are positive. Third, the amount of tail dependence is inferred from the slope of the correlation line as the exceedance level increases. The ﬂatter the line, the greater the amount of tail dependence. As absolute exceedance levels increase, we can see that correlation levels drop. However, they fall less slowly if there is greater tail dependence. We can see that lower-grade debt appears to have more tail dependence than higher-grade debt. Overall, the exceedance plot shows that bonds within high-quality ratings have greater default correlation than bonds within low-quality ratings. However, tail dependence is higher for lower-rated bonds. Both results are intuitive—high-grade debt tends to be issued by large ﬁrms that experience greater amounts of systematic risk, and low-grade ﬁrms evidence more idiosyncratic risk; however, when economy-wide default risk escalates, lowgrade bonds are more likely to experience contagion, leading to greater tail dependence.

4. DETERMINING THE JOINT DEFAULT PROCESS 4.1 Overall Method Our data comprise ﬁrms categorized into six rating classes. We average across ﬁrms within rating class to obtain a time series of the average

Correlated Default Processes: A Criterion-Based Copula Approach

197

intensity (λk ) for each rating class k. We assume that the stochastic processes for the six average λk s are drawn from a joint distribution characterized by a copula, which establishes the joint dependence between rating classes. The correlation matrix for the mean intensities is calculated directly from the data. The copula is set to one of four possible choices: normal, Student’s t, Gumbel, or Clayton. The form of the stochastic process for each average hazard rate is taken to be either a jump-diffusion or a regime-switching one. This setup provides for the correlation between rating classes. Next, the correlations within a rating class are obtained from the linkage of all ﬁrms in a rating class to the mean PD process of the rating category. We model individual ﬁrm PDs as following a non-negative stochastic process with reversion to the stochastic mean of each rating. To conduct a Monte Carlo simulation of the system, we propagate the average rating PDs using either a jump-diffusion or a regime-switching model, with correlations across ratings given by a copula. Then we propogate the individual issuer intensities using stochastic processes that oscillate around the rating means. The entire system consists of a choice of (1) stochastic process for average intensities (jump-diffusion versus regimeswitching), (2) copula (normal, Student’s t, Gumbel, Clayton), and (3) marginal distribution for issuer-level stochastic process for intensities (one of seven different options). All told, therefore, we have 56 alternate correlated default system structures to choose from.

4.2 The Estimation Phase Estimation is undertaken in a two-stage manner: (1) the estimation of the stochastic process for the mean intensity of each rating class, and (2) the estimation of the best candidate distribution for the stochastic process of each individual issuer’s intensity. We choose two processes for step (1). Both choices are made to inject excess kurtosis into the intensity distribution. First, we use a normal jump model. Second, we use a regime-switching model. Each is described in turn in the next two subsections.

4.2.1 Estimation of the Mean of Each Rating Class Using a Jump Model For our six rating classes (indexed by k) and all issuers (indexed by j), we compute intensities from the probabilities of default (Pkj , j = 1, . . . , Nk , k = 1, . . . , 6) in our data set. The intensities are computed as λkj = − ln(1−Pkj ) ≥ 0. We denote M as the total number of rating classes and Nk as the total number of issuers within the rating class for which data are available. Let λk (t) be the average intensity across issuers within rating class k. Nk Therefore, λk (t) = 1/Nk j=1 λkj (t), ∀t. We assume that λk (t) follows the

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stochastic process:

λk (t) = κk [θk − λk (t)] t + xk (t) λk (t) t

(9.6)

xk (t) = k (t) + Jk (t)Lk (qk , t) k (t) ∼ N[0, σk2 ] Jk (t) ∼ U[ak , bk ] 1 w/prob Lk (qk , t) = 0 w/prob

qk 1 − qk

We assume that the jump size Jk follows a uniform distribution over [ak , bk ].8 This regression accommodates a mean-reverting version of the stochastic process, and κk calibrates the persistence of the process. This process is estimated for each rating class k, and the parameters are used for subsequent simulation. Copulas are used to obtain the joint distribution of correlated default. The correlation matrix used in the copula comes from the residuals xk (t) computed in the regression above. We use maximum likelihood estimation (MLE) to obtain the parameters of this process. Since there is a mixture of a normal and a jump component, we can decompose the unconditional density function for the residual term xk (t) into the following: f [xk (t)] = qk f [xk (t) | Lk = 1] + (1 − qk )f [xk (t) | Lk = 0]

(9.7)

The latter density, f [xk (t) | Lk = 0], is conditionally normal. The marginal distribution of xk (t) conditional on Lk = 1 can be derived as follows: f [xk (t) | Lk = 1] =

bk

ak

1

× exp − (bk − ak ) 2π σk2

(xk (t) − Jk )2

2σk2

dJk (9.8)

Let z = ( Jk − xk (t))/σk . Then, 1 z2 1 exp − f [xk (t) | Lk = 1] = a −x (t) √ dz k k bk − a k 2 2π σ b − xk (t) a − xk (t) 1 k − k = σk σk bk − a k

bk −xk (t) σ

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Table 9.5 Results of Stochastic Process for Estimating Mean of the Intensities of Each Rating Class. Numbers below the estimates are the t -statistics from the estimation, which is undertaken by maximum likelihood Rating Class Parameter

5

6

θk

0.0524 7.73

0.1336 16.74

0.3499 2.75

1.1550 4.01

2.3812 1.99

5.4848 7.12

κk

0.4813 2.84

0.0018 0.01

0.6680 1.88

0.4811 1.87

0.4305 2.01

2.8350 4.19

σk

0.0487 14.17

0.1050 13.39

0.2518 12.48

0.2633 11.44

0.2872 8.51

0.5743 5.85

bk

0.0839 12.39

0.1316 5.64

0.1241 1.94

0.2070 1.58

0.2684 1.52

1.1231 5.50

ak

−0.0707 −6.33

−0.0644 −3.46

0.1241 1.96

0.2069 1.61

0.2684 2.37

0.0771 0.21

qk

0.1016 3.87

0.0777 3.18

0.1514 2.94

0.0950 2.07

0.2461 2.08

0.4374 2.07

Log-likelihood

1

−622.16

2

−462.95

3

−205.48

4

−133.56

−32.82

−35.03

In the case when ak = bk , the process has a constant jump size with probability 1, and the marginal distribution of xk (t) degenerates to a normal distribution with mean ak and standard deviation σk . This can be seen by letting bk = ak + c. We then have

ak − xk (t) ak + c − xk (t) 1 f [xk (t) | Lk = 1] = lim − c→0 c σk σk

ak + c − xk (t) 1 = lim φ c→0 σk σk

xk (t) − ak 1 = φ σk σk where φ(·) and (·) are the standard normal density and distribution functions, respectively. Because jumps are permitted to be of any sign, we may expect that bk ≥ 0 and ak ≤ 0. Also, if |bk | > |ak |, it implies a greater probability of positive jumps. When ak is close to bk , or when they have the same sign, the process has a constant or a one-direction jump, respectively. The estimation results are presented in Table 9.5. The mean of the intensity process (θk ) increases with rating class k, as expected. The variance (σk ) also increases with declining credit quality. Generally speaking, the probability of a jump (qk ) in the hazard rate is higher for lower quality issuers. We can see that rating classes 3 to 5 have a constant jump and rating 6 has only a positive jump. While the mean jump for all rating classes appears to be close to zero, that for the poorest rating class is much higher than zero. Hazard rates jump drastically when a ﬁrm approaches default.

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After obtaining the parameters, we compute residuals for each rating class. The residuals include randomness from both the normal and the jump terms. The covariance matrix of these residuals is stored for later use in Monte Carlo simulations. 4.2.2 Estimating the Mean Processes in a Regime-Switching Environment From Figure 9.1, we note that there are periods in which PDs are low, interjected by smaller sporadic regimes of spikes in the hazard rates. A natural approach to capture this behavior is to use a regime-switching model.9 In order to determine regimes, we ﬁrst compute the average intensity (hazard rate, approximately), λ¯ = N i=1 λi , across all issuers in our database. Within each regime the intensity is assumed to follow a square-root volatility model represented in discrete time as follows:

λ¯ r (t) = κr [θr − λ¯ r (t)] t + σr λ¯ r (t) t(t) r = {HI, LO}

(9.9)

The two regimes are indexed by r, which is either HI or LO. κr is the rate of mean reversion. The mean value within the regime is θr , and σr is the volatility parameter. The probability of switching between regimes comes from a logit model based on a transition matrix:

pLO 1 − pHI

1 − pLO pHI

where pr = exp(αr )/(1 + exp(αr )), r ∈ {LO, HI}. Estimation is undertaken using maximum likelihood. We ﬁx the values of θr based on the historical PDs, with a high–low regime cutoff of 2%. θLO was found to be 1.68% and θHI to be 2.40%.10 We then estimate the rest of the parameters in the above model. The estimation results are presented in Table 9.6. In the high-PD regime, the higher level of hazard rates is matched by a higher level of volatility. Our next step is to use the determined regimes to estimate the parameters of the mean process for each rating class, that is, λk , k = 1, . . . , 6. We ﬁt regime-shifting models to each of the rating classes, using a stochastic process similar to that in equation 9.9:

λ¯ k,r (t) = κk,r [θk,r − λ¯ k,r (t)] t + σk,r λ¯ k,r (t) t (k, t) r = {HI, LO}, k = 1, . . . , 6

(9.10)

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Table 9.6 Estimation Results of Regime-Switching Model for Regimes Estimated on Average Intensity ¯ Process λ¯ = N i=1 λi across All Issuers in Data Set. We designated periods in which λ ≤ 2% as the “low-PD” regime, and periods in which λ¯ > 2% as the “high-PD” regime. The two regimes are indexed by r, which is either HI or LO. κr is the rate of mean reversion. The mean value within the regime is θr , and σr is the volatility parameter. of switching between regimes comes from a logit model based The probability pLO 1 − pLO where pr = exp(αr )/(1 + exp(αr )), r ∈ {LO, HI}. Estimation is on a transition matrix: 1 − pHI pHI undertaken using maximum likelihood. Parameter

Low-PD Regime

θ κ t-stat σ t-stat α t-stat Log likelihood

1.68% 1.5043 2.4053 0.1681 14.2093 4.1286 4.6649 −195.5948

High-PD Regime 2.40% 1.7746 2.0558 0.2397 7.9849 3.3644 2.5428

Table 9.7 Estimation Parameters of Regime-Switching Model for Mean of Each Rating Class: This table presents parameters for the regime-switching model applied to the mean process for each rating class, i.e., λ¯ k , k = 1, . . . , 6 Rating Class Parameter

1

2

3

4

5

6

θLO t-stat

0.0559 0.40

0.1625 7.06

0.5678 11.90

1.3588 14.22

3.6367 45.24

6.3138 74.32

θHI t-stat

0.0991 0.36

0.3757 6.55

0.9818 4.91

2.2358 19.29

5.1216 18.74

7.8295 8.07

κLO t-stat

1.3535 0.23

0.5094 1.31

1.5855 2.86

1.1317 2.25

2.2365 3.49

6.2456 6.45

κHI t-stat

0.6152 0.35

0.1775 1.74

1.1717 0.95

2.8567 2.52

2.0833 1.67

1.8862 1.81

σLO t-stat

0.0568 17.63

0.1224 17.27

0.3214 16.19

0.2986 16.57

0.3338 16.92

0.6798 16.02

σHI t-stat

0.1103 8.41

0.2214 7.83

0.3275 7.84

0.3305 7.93

0.3681 8.15

0.8336 8.24

The results of the estimation are presented in Table 9.7. The parameters are as expected. The mean of the processes is higher in the high regime, as is the volatility. We now proceed to look at the choice of individual issuer marginal distributions. 4.2.3 Estimation of the Stochastic Process for Each Individual Issuer Assume that the intensity for each individual issuer follows a square-root process:

λkj (t) = κkj {θkj + γkj λk − λkj (t)} + σkj λkj (t) εkj (t) We normalize the residuals by dividing both sides by

(9.11)

λkj (t)dt. We then use

ordinary least squares to estimate parameters, which are stored for use in simulations. The resulting residuals are used to form the covariance matrix for each class.

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The long-term mean is deﬁned as the sum of a constant number θkj and the group mean λk (t) multiplied by a constant, γkj . This indexes the mean hazard rate for each issuer to the current value of the mean for the rating class. Consequently, there are four parameters for each regression, κkj , θkj , γkj , and σkj . Note that κkj should be positive; θkj and γkj can be positive or negative as long as they are not both negative at the same time. None of the issuers has negative θkj and γkj at the same time, although 20 issuers have negative κkj . We, therefore, deleted the 20 issuers with negative κ from our dataset for further analysis.11 4.2.4 Estimation of the Marginal Distributions We ﬁt the residuals from the previous section to the normal, Student’s t, and skewed double exponential distributions. We use prepackaged functions in Matlab for maximum-likelihood estimation (MLE) for the normal and Student’s t distributions. The likelihood function for the skewed double exponential distribution is as follows. Assume that the residual εkj has a normal distribution with mean γ V and variance V. Further, the variance V is assumed to have an exponential distribution with the following density function:

1 V pdf(V) = exp − V0 V0

(9.12)

Then, it can be shown that εkj has a skewed double exponential distribution with the following density function:

|εkj | λ − γ εkj pdf(ε) = exp − V0 λ

(9.13)

where λ=

V0 2 + γ 2 V0

See the Appendix for a derivation of these results. The log-likelihood function L is given as follows: n n 2 + γ 2 V0 n 2 |εi | +γ i L = − {log(V0 ) + log(2 + γ V0 )} − V0 2 i=1

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Correlated Default Processes: A Criterion-Based Copula Approach

203

4.2.5 Goodness of Fit of Marginal Distributions Since we choose one of three distributions for the residuals of each issuer, we use four criteria to decide the best one. These are as follows: ■

■

The Kolmogorov distance: This is deﬁned as the supremum over the absolute differences between two cumulative density functions, the empirical one, Femp (x), and the estimated ﬁtted one, Fest (x). The Anderson and Darling Statistic: This is given by |Femp (x) − Fest (x)| AD = max √ x∈R Fest (x)[1 − Fest (x)]

■

■

The AD statistic puts more weight on the tails compared to the Kolmogorov statistic. The L1 distance: This is equal to the average of the absolute differences between the empirical and statistical distributions. The L2 distance: This is the root-mean-squared difference between the two distributions.

We use these criteria to choose between the normal, Student’s t, and skewed double exponential distributions for each issuer. In each of our 56 simulated systems, the issuer level distributions are chosen from the following seven cases: 1. 2. 3. 4.

All marginal distributions were chosen to be normal. All marginal distributions are Student’s t. All marginals are from the skewed double exponential family. For each issuer, the best distribution is chosen based on the Kolmogorov criterion. 5. For each issuer, the best distribution is chosen based on the Anderson and Darling statistic. 6. Each marginal is chosen based on the best L1 distance. 7. Each marginal is chosen based on the best L2 distance. For each criterion, we count the number of times each distribution provides the best ﬁt. The results are reported in Table 9.8, which provides interesting features. Based on the Kolmogorov, Anderson–Darling, and L2 statistics, the skewed double exponential distribution is the most likely to ﬁt the marginals best. However, the L1 statistic ﬁnds that the normal distribution ﬁts most issuers better. The better ﬁt of the skewed double exponential as a marginal distribution comes from its ability to match the excess kurtosis of the distribution.

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Table 9.8 In this table we report the Estimation Results of the Fit of Individual Intensity Residuals to Various Distributions. The three distributions chosen were: double exponential, normal, and the Student’s t . We used four distance metrics to compare the empirical residuals to standardized distributions: the Kolmogorov distance, the Anderson–Darling statistic, and distances in the L1 and L2 norms. The table below presents the number of individual issuers that best ﬁt each of the distributions under the different metrics. A total of 619 issuers was classiﬁed in this way. Statistic Kolmogorov Anderson–Darling L2 L1

Double exponential

Normal

Student’s t

522 409 529 118

55 7 53 500

42 203 37 1

4.3 Calibration Our calibration procedure is as follows. First, using MLE, we determine the parameters of the various stochastic processes (rating-level averages and individual issuer PDs). Second, we choose one of the 56 possible correlation systems, as described above, and simulate the entire system 100 times to obtain an average simulated exceedance graph. Third, we compare the simulated exceedance graph with the empirical one (pointwise) and determine the mean-squared difference between the two graphs. This distance metric is used to determine a ranking of the 56 possible systems we work with.

5. SIMULATING CORRELATED DEFAULTS AND MODEL COMPARISONS 5.1 Overview In this section, we discuss the simulation procedure. This step uses the estimated parameters from the previous section to generate correlated samples of hazard rates. The goal is for the simulation approach to deliver a model with the three properties described in the previous section. Therefore, the asymmetric correlation plot from simulated data should be similar to that in Figure 9.2. We implement the simulation model with an additional constraint, whereby we ensure that the intensities (λk (t)) are monotonically increasing in rating level k. This prevents the average PD for a rating class from being lower than the average PD for the next better rating category; that is, λk (t) should be such that for all time periods t, it must be such that if i < j, then λi (t) < λj (t). This check is instituted during the simulation as follows. During the Monte Carlo step, if λi (t) > λj (t) when i < j, then we set λj (t) = λi (t). As an example, see Figure 9.3 where we plot the times series of λk , k = 1, . . . , 6 for a random simulation of the sample path of hazard rates.

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Correlated Default Processes: A Criterion-Based Copula Approach 15

Hazard rate * 100

10

5

0 0

20

40

60

80

100

120

140

Months

FIGURE 9.3 Simulated series of average PDs by rating class. The hazard rates are (from top to bottom of the graph) from rating class 6 to rating class 1. The simulation ensures that λi (t) < λj (t) if i < j.

5.2 Illustrative Monte Carlo Experiment As a simple illustration that our approach arrives at a correlation plot fairly similar to that seen in the data, we ran a naive Monte Carlo experiment. In this exercise, we assumed that all error terms were normal, except under some conditions, when we assumed the Student’s t distribution. We ran the Monte Carlo model 25 times and computed the average exceedance correlations for all rating classes. To obtain asymmetry in the correlations, we need to have higher correlations when hazard rates are high. To achieve this, the simulation uses the Student’s t distribution with six degrees of freedom when the average level of hazard rates in the previous period in the simulation is above the empirical average of hazard rates. These features provided the results in Figure 9.4. The similarity between the empirical exceedance graph (Figure 9.2) and the simulated one (Figure 9.4) shows that the two-stage

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0.35 R1 R2 R3 R4 R5 R6

0.3

Conditional correlation

0.25

0.2

0.15

0.1

0.05

0 –1.5

–1

–0.5

0 Exceedance Level

0.5

1

1.5

FIGURE 9.4 Asymmetric exceedance correlations of simulated PDs for the different rating classes. Again the plots are consistent with the presence of asymmetric correlation, and the degree of asymmetry is higher for the better rating categories. Monte Carlo model is able to achieve the three correlation properties of interest.

5.3 Determining Goodness of Fit The dependence structure among the intensities is depicted in the asymmetric correlation plots. We develop a measure to assess different speciﬁcations of the joint distribution. This metric is the average squared point-wise difference between the empirical exceedance correlation plot and the simulated one. The points in each plot that are used are for the combination of rating class (from 1 to 6) and exceedance levels (from −1.5 to +1.5). Hence, there is a total of 48 points in each plot that are used for computing the metric. Deﬁne the points in the empirical plot as hk,x , where k indexes the rating class and x indexes the exceedance levels. The corresponding points in the simulated plot are

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Correlated Default Processes: A Criterion-Based Copula Approach

denoted by hk,x . The metric q (a root-mean-squared estimation statistic) is as follows: 1.5 6 1 q= (hk,x − hk,x )2 48

(9.14)

k=1 x=−1.5

A smaller value of q implies a better ﬁt of the joint dependence relationship. In the following section, we use the simulation approach and this metric to compare various models of correlated default.

5.4 Calibration Results and Metric It is important to note that the joint dependence across all issuer intensities depends on three aspects: (1) the marginal distributions used for individual issuers, (2) the stochastic process used for each rating class, and (3) the copula used to implement the correlations between rating classes. Optimizing the choice of individual issuer marginal distributions will not necessarily achieve the best dependence relationship, because the copula chosen must also be compatible. In this chapter, we focus on four copulas (normal, Student’s t, Gumbel, and Clayton), and allow the rating class intensity stochastic process to be either a jump-diffusion model or a regime-shifting model. Tables 9.9 and 9.10 provide a comparison of the jump and the regimeswitching models across copulas. Twenty-eight systems use a jump-diffusion

Table 9.9 Comparison of Metric q Values for Jump-Diffusion and Regime-Shifting Models across the Normal (Gaussian) and Gumbel Copulas. For each model we generate the asymmetric correlation plot and then compute the distance metric. The asymmetric correlations (the metric q) are computed for the following seven cases: normal distribution, Student’s t distribution, skewed double exponential distribution, the combination of the best distributions based on Kolmogorov criterion, the combination of the best distributions based on the Anderson and Darling statistic, and the combination based on the L1 and L2 norms Gaussian Copula Case Normal Student’s t Double exponential Kolmogorov Anderson– Darling L2 L1

Gumbel Copula

Jump-Diffusion

Regime-Shifting

Jump-Diffusion

Regime-Shifting

0.0508 0.0375 0.0752

0.0276 0.0327 0.0355

0.0486 0.0416 0.0700

0.0462 0.0515 0.0381

0.0802 0.0447

0.0387 0.0315

0.0743 0.0452

0.0364 0.0450

0.0764 0.0667

0.0367 0.0318

0.0710 0.0625

0.0376 0.0401

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Table 9.10 Comparison of Metric q Values for the Jump-Diffusion and Regime-Shifting Models across the Clayton and Student’s t Copulas Metric for best correlated default model (Clayton and Student’s t copulas): this table presents the summary statistic for the asymmetric correlation metric to determine the best simulation model. For each model we generate the asymmetric correlation plot and then compute the distance metric. The asymmetric correlations (the metric q) are computed for the following seven cases: normal distribution, Student’s t distribution, skewed double-exponential distribution, the combination of the best distributions based on Kolmogorov criterion, the combination of the best distributions based on the Anderson and Darling statistic, the combination based on the L1 and L2 norms. Our metric comprises the average squared point-wise difference between the empirical exceedance correlation plot and the simulated one. This a natural distance metric. The points in each plot that are used are for the combination of rating class (from 1 to 6) and exceedance levels (from −1.5 to +1.5). Hence there are a total of 48 points in each plot which are used for computing the metric. Deﬁne the points in the empirical plot as hk,x , where k indexes the rating class and x indexes the exceedance levels. The corresponding points in the simulated plot are denoted hk,x . The metric q (a RMSE statistic) is as follows: 1.5 1 6 [h − hk,x ]2 . We report the results for both models, the jump-diffusion set up q = 48 k=1 x=−1.5 k,x and the regime-switching one, and two copulas, the Clayton and Student’s t copulas Clayton Copula Case Normal Student’s t Double exponential Kolmogorov Anderson– Darling L2 L1

Student’s t Copula

Jump-Diffusion

Regime-Shifting

Jump-Diffusion

Regime-Shifting

0.0484 0.0415 0.0694

0.0395 0.0473 0.0304

0.0446 0.0370 0.0672

0.0506 0.0564 0.0433

0.0738 0.0445

0.0293 0.0397

0.0714 0.0418

0.0404 0.0514

0.0703 0.0622

0.0296 0.0324

0.0681 0.0596

0.0423 0.0452

model for the average PDs in each rating, and 28 use a regime-switching model. We see that the values of the q metric are consistently smaller for the regime-switching model using all test statistics. This implies that the latter model provides a better representation of the stochastic properties among the issuer hazard rates. In particular, it ﬁts the asymmetric correlations better.12 We conclude that, for rating-level PDs, a regime-switching model performs better than a jump-diffusion model, irrespective of the choice of issuer process and copula.13 Tables 9.9 and 9.10 allow us to compare the ﬁt provided by the four copulas. We ﬁnd that when the jump-diffusion model is used, the Student’s t copula performs best, no matter which marginal distribution is used. However, when the regime-switching model is used, sometimes the normal copula works best and at other times the Clayton copula is better. The normal copula works best when the marginals are all normal or Student’s t, and when the marginals are chosen using the Anderson–Darling criterion. The Clayton copula is best when the marginals are all skewed double exponential or when the marginal criteria are Kolmogorov or L2 . The Student’s t copula is best when the L1 criterion is chosen.

Correlated Default Processes: A Criterion-Based Copula Approach

209

These results bear some explanation. A comparison of the correlations from the skewed double exponential distribution with those from the raw data reveals that the model greatly overestimates the up-tail dependences. The Clayton copula decreases the up-tail dependences and increases the lowtail dependences. As a result, it corrects some of the overestimation from the double exponential marginals, resulting in better ﬁtting. Hence, the Clayton copula seems to combine well with skewed double exponential marginals. On the other hand, the q metric favors the normal copula if the criterion chooses normal marginals (as with the normal, Student’s t, and L1 criteria), since less balancing of tail dependences is required.

5.5 Empirical Implications of Different Copulas The degree of tail dependence varies with the choice of copula. It is ultimately an empirical question as to whether the parameterized joint distribution does result in differing tail risk in credit portfolios. To explore this question, we simulated defaults using a portfolio comprising all the issuers in this study under the regime-switching model. Remember that we ﬁtted four copulas and seven different choices of marginal distributions, resulting in 28 different models, each with its attendant parameter set. To compare copulas, we ﬁx the marginal distribution and then vary the copula. As an illustration, we present Figure 9.5. This plots the tail loss distributions from the four copulas when all marginals are assumed to be normal. The line most to the right (the Gumbel copula) has the most tail dependence. There is roughly a 90% chance that the number of defaults will be less than 75; that is, a 10% probability that the number of defaults will exceed 75. For the same level of 75 defaults, the leftmost line (from the normal copula), there is only a 7% probability of the number of losses being greater than 75. By examining all four panels of the ﬁgure, we see that the ranking of copulas by tail dependence is unaffected by the choice of marginal distribution; that is, the normal copula is the leftmost plot, followed by the Student’s t, Clayton, and Gumbel copulas. In Figure 9.6 we plot the tail loss distributions for two models, the best ﬁtting one and the worst. The best ﬁt model combines the Clayton copula and marginal distributions based on the Kolmogorov criterion. The worst ﬁt copula combines the Student’s t copula with Student’s t marginals. A comparison of the two models shows that the worst ﬁtted copula in fact grossly overstates the extent of tail loss. Therefore, while there is tail dependence in the data, careful choice of copula and marginals is needed to avoid either under- or overestimation of tail dependence.

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102

100

Cumulative probability

98

96

94

92

90

88 70

Normal Gumbel Clayton StudentT 75

80 85 Number of defaults

90

95

FIGURE 9.5 Comparison of copula tail loss distributions. This ﬁgure presents plots of the tail loss distributions for four copulas when the marginal distribution is normal. The x-axis shows the number of losses out of more than 600 issuers, and the y-axis depicts the percentiles of the loss distribution. The simulation runs over a horizon of 5 years and accounts for regime shifts as well. The copulas used are: normal, Gumbel, Clayton, Student’s t.

6. DISCUSSION This chapter develops a criterion-based methodology to assess alternative speciﬁcations of the joint distribution of default risk across hundreds of issuers in the US corporate market. The study is based on a data set of default probabilities supplied by Moody’s Risk Management Services. We undertake an empirical examination of the joint stochastic process of default risk over the period 1987 to 2000. Using copula functions, we separate the estimation of the marginal distributions from the estimation of the joint distribution. Using a two-step Monte Carlo model, we determine the appropriate choice of multivariate distribution based on a new metric for the assessment of joint distributions.

211

Correlated Default Processes: A Criterion-Based Copula Approach 100 Clayton Copula, Kolmogorov Student Copula, Student T 99

Cumulative Probability (%)

98 97 96 95 94 93 92 91 90 65

70

75

80 Number of Defaults

85

90

95

FIGURE 9.6 Comparison of copula tail loss distributions: in this ﬁgure we plot the tail loss distributions for two models, the best ﬁtting one and the worst. The best ﬁt model combines the Clayton copula and marginal distributions based on the Kolmogorov criterion and the worst ﬁt copula combines the Student’s t copula with Student’s t marginals. The simulation runs over a horizon of 5 years and accounts for regime shifts as well.

We explore 56 different speciﬁcations for the joint distribution of default intensities. Our methodology uses two alternative speciﬁcations (jumps and regimes) for the means of default rates in rating classes. We consider three marginal distributions for individual issuer hazard rates, combined using four different copulas. An important extension to this model structure is the inclusion of rating changes. In our analysis, we centered ﬁrms as lying within the same rating for the period of the simulation, based on their most prevalent rating. The two-step simulation model would need to be enhanced to a three-step one, with an additional step for changes in ratings.14 Other than the myriad speciﬁcations, there are many useful features of the analysis for modelers of portfolio credit risk. First, we develop a simple metric to measure best ﬁt of the joint default process. This metric accounts

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for different aspects of default correlation—namely, level, asymmetry, and tail dependence or extreme behavior. Second, the simulation model, based on estimating the joint system of over 600 issuers, is able to replicate the empirical joint distribution of default. Third, a comparison of the jump model and the regime-switching model shows that the latter provides a better representation of the properties of correlated default. Fourth, the skewed double exponential distribution is a suitable choice for the marginal distribution of each issuer hazard rate process, and combines well with the Clayton copula in the joint dependence relationship among issuers. Our simulation approach is fast and robust, allowing for rapid generation of scenarios to assess risk in credit portfolios. Finally, the results show that it is important to correctly capture the interdependence of marginal distributions and copula to achieve the best joint distribution depicting correlated default. Thus, this chapter delivers the empirical counterpart to the body of theoretical papers advocating the usage of copulas in modeling correlated default.

ACKNOWLEDGMENTS We are extremely thankful for many constructive suggestions and illuminating discussions with Darrell Dufﬁe, Gifford Fong, Nikunj Kapadia, John Knight, and Ken Singleton, and participants at the Q-conference (Arizona 2001), Risk Conference (Boston 2002), and Barclay’s Global Investors Seminar (San Francisco 2002). We are also very grateful to two excellent (anonymous) prior referees for superb comments and guidance that resulted in a much-transformed chapter. The ﬁrst author gratefully acknowledges support from the Dean Witter Foundation and a Breetwor Fellowship. We are also grateful to Gifford Fong Associates and Moody’s Investors Services for data and research support for this chapter. The second author is supported by the Natural Sciences and Engineering Research Council of Canada.

APPENDIX: THE SKEWED DOUBLE EXPONENTIAL DISTRIBUTION Assume that a random variable X has a normal distribution with mean µ + γ V and variance V, where V has an exponential distribution with the

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following density function: pdf(V) =

1 V exp − V0 V0

(9.15)

Then X has a skewed double exponential distribution.

A.1 Density Function The density function is derived as follows:

1 V (x − µ − γ V)2 × exp − pdf(x) = dV exp − √ 2V V0 V0 2πV 0 ∞ 1 1 exp{γ (x − µ)} =√ √ 2πV0 V 0 γ 2 V0 + 2 (x − µ)2 − V dV × exp − 2V 2V0    1 2 + γ 2 V0  2πV0 =√ × exp −|x − µ| exp{γ (x − µ)}   V0 2 + γ 2 V0 2πV0     1 V0 2 + γ 2 V0 = × exp −|x − µ| + γ (x − µ)   V0 V 0 2 + γ 2 V0 λ |x − µ| = + γ (x − µ) exp − V0 λ

∞

1

A.2 Maximizing the Log-Likelihood Function This subsection deals with the technical details of the maximization of log-likelihoods for the skewed double exponential model. The likelihood function is n n L = − {log(V0 ) + log(2 + γ 2 V0 )} − |xi − µ| 2 + γ 2 V0 /V0 2 i=1

+γ

n i=1

(xi − µ)

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where m is the number of observations in which xi is greater than µ, l is the number of observations otherwise, and n = m + l. The ﬁrst-order condition can be derived as follows:   n n dL d|xi − µ|  2 + γ 2 V0  +γ = (−1) V0 dµ dµ i=1

i=1

2 + γ 2 V0 d|xi − µ| =0 × V0 dµ i=1

n n n 2γ V0 γ + |xi − µ| (xi − µ) =− − 2 2 + γ 2 V0 (2 + γ 2 V0 )/V0 i=1 i=1 n n nγ V0 V0 =− + (xi − µ) = 0 − γ |x − µ| i 2 + γ 2 V0 2 + γ 2 V0 i=1 i=1 n (γ 2 V0 − 2 − γ 2 V0 )/V02 n 1 γ2 − =− + |xi − µ| ×

2 V0 2 + γ 2 V0 2 (2 + γ 2 V0 )/V0 n

= −nγ −

dL dγ

dL dV0

i=1

=−

n(1 + γ 2 V

0) + V0 (2 + γ 2 V0 )

n i=1

|xi − µ|

V0 V0 (2 + γ 2 V0 )

=0

Solving the above ﬁrst-order conditions gives

µ= V0 = γ =

m2

2ml

+ l2 nml

x∈A1 xi

n 1 n

i=1 xi

x∈A2 xi

2

−µ

(l − m)2 (l − m)2 2ml n1 ni=1 xi − µ

where A1 = {x : xi − µ ≥ 0} and A2 = {x : xi − µ < 0}. Given a dataset, we may not be able to ﬁnd a value of µ to satisfy the above equations. However, by choosing µ to be as close as given by the ﬁrstorder conditions, we can ﬁt the data well with the skewed double exponential distribution.

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215

NOTES 1. See the structural models of Merton (1974), Geske (1977), Leland (1994), and Longstaff and Schwartz (1995); and the reduced-form models of Dufﬁe and Singleton (1999), Jarrow and Turnbull (1995), Das and Tufano (1996), Jarrow et al. (1997), Madan and Unal (1999), and Das and Sundaram (2000). 2. A CDO securitization comprises a pool of bonds (the “collateral”) against which tranches of debt are issued with varying cash ﬂow priority. Tranches vary in credit quality from AAA to B, depending on subordination level. Seller’s interest is typically maintained as a ﬁnal equity tranche, carrying the highest risk. CDO collateral usually comprises from a hundred to over a thousand bond issues. Tranche cash ﬂows critically depend on credit events during the life of the CDO, requiring Monte Carlo simulation [see, for example, Dufﬁe and Singleton (1999)] of the joint default process for all issuers in the collateral. An excellent discussion of the motivation for CDOs and the analysis of CDO value is provided in Dufﬁe and Garleanu (2001). For a parsimonious model of bond portfolio allocations with default risk, see Wise and Bhansali (2001). 3. As demonstrated by Das et al. (2001b) and by Gersbach and Lipponer (2000), both the individual default probability and the default correlations have a signiﬁcant impact on the value of a credit portfolio. 4. Extreme value distributions allow for the fact that processes tend to evidence higher correlations when tail values are experienced. This leads to a choice of fatter-tailed distributions with a reasonable degree of “tail dependence.” The support of the distribution should also be such that the PD intensity λi (t) lies in the range [0, ∞). 5. There is a growing literature on copulas. The following references contain many useful papers on the topic: Embrechts et al. (1997), Embrechts et al. (1999a,b), Frees and Valdez (1998), Frey and McNeil (2001a,b), Li (1999), Nelsen (1999), Lindskog (2000), Schonbucher and Schubert (2001), and Wang (2000). 6. Durrleman et al. (2000) discuss several parametric and non-parametric copula estimation methods. 7. Lower tail dependence is symmetrically deﬁned. The coeffcient of lower tail dependence is λL = lim Pr[X2 < F2−1 (z)|X1 < F1−1 (z)] z→0

(9.5)

If λL > 0, then lower tail dependence exists. 8. This speciﬁcation does permit the hazard rate to populate negative values. While this is not an issue during the estimation phase, the simulation phase is adjusted to truncate the shock if the negative support is accessed. However, we remark that this occurs in very rare cases, since λk (t) is the average across all issuers within the rating class, and the averaging drives the probability of negative hazard rates to minuscule levels. 9. Ang and Chen (2002) found this type of model to be good at capturing the three stated features of asymmetric correlation in equity markets.

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10. We found that the estimations of θr were very sensitive to the initial values used in the optimization. By ﬁxing these two values, the estimation turned out to be more stable. 11. The estimation of equation 9.11 relies on the assumption that the mean around which the individual hazard rate oscillates depends on the initial rating class of the issuer, even though this may change over time. Extending the model to map default probabilities to rating classes is a nontrivial problem, and would complicate the estimation exercise here beyond the scope of this chapter. Indeed, this problem in isolation from other estimation issues is complicated enough to warrant separate treatment and has been addressed in a chapter by Das et al. (2002). 12. This result is consistent with that of Ang and Chen (2002), who undertake a similar exercise with equity returns. 13. The fact that the Student’s t marginal distribution works best for the jump models may indicate that the jump model does not capture the dependences as well as the regime-switching model does, since the Student’s t distribution is good at injecting excess kurtosis into the conditional distribution of the marginals. 14. We are grateful to Darrell Dufﬁe for this suggestion.

REFERENCES Abramowitz, M. and Stegun, I. A. (1972). The Handbook of Mathematical Functions. New York: Dover. Ang, A. and Chen, J. (2002). “Asymmetric Correlation of Equity Portfolios.” Journal of Financial Economics 63(3), 443–494. Bouye, E., Durrleman, V., Nikeghbali, A., Riboulet, G. and Roncalli, T. (2000). “Copulas for Finance: A Reading Guide and Some Applications.” Working Paper, Credit Lyonnais, Paris. Clayton, D. G. (1978). “A Model for Association in Bivariate Life Tables and Its Application in Epidemiological Studies of Familial Tendency in Chronic Disease Incidence.” Biometrika 65, 141–151. Crouhy, M., Galai, D. and Mark, R. (2000). “A Comparative Analysis of Current Credit Risk Models.” Journal of Banking and Finance 24, 59–117. Das, S., Fan, R. and Geng, G. (2002). “Bayesian Migration in Credit Ratings Based on Probabilities of Default.” Journal of Fixed Income 12(3), 17–23. Das, S., Freed, L., Geng, G. and Kapadia, N. (2001a). “Correlated Default Risk.” Working Paper, Santa Clara University. Das, S., Fong, G. and Geng, G. (2001b). “The Impact of Correlated Default Risk on Credit Portfolios.” Journal of Fixed Income 11(3), 9–19. Das, S. and Sundaram, R. (2000). “A Discrete-Time Approach to Arbitrage-Free Pricing of Credit Derivatives.” Management Science 46(1), 46–62. Das, S. and Tufano, P. (1996). “Pricing Credit Sensitive Debt When Interest Rates, Credit Ratings and Credit Spreads Are Stochastic.” The Journal of Financial Engineering 5(2), 161–198.

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Das, S. and Uppal, R. (2000). “Systemic Risk and International Portfolio Choice.” Working Paper, London Business School. Davis, M. and Violet, L. (1999a). “Infectious Default.” Working Paper, Imperial College, London. Davis, M. and Violet, L. (1999b). “Modeling Default Correlation in Bond Portfolios.” In C. Alexander (ed.) ICBI Report on Credit Risk. Dowd, K. (1999). “The Extreme Value Approach to VaR: An Introduction.” Financial Engineering News 3(11). Dufﬁe, D. J. and Garleanu, N. (2001). “Risk and Valuation of Collateralized Debt Obligations.” Financial Analysts Journal 57(1), 41–59. Dufﬁe, D. J. and Singleton, K. J. (1999). “Simulating Correlated Defaults.” Working Paper, Stanford University, Graduate School of Business. Durrleman, V., Nikeghbali, A. and Roncalli, T. (2000). “Which Copula Is the Right One?” Working Paper, Groupe de Recherche Operationnelle, Credit Lyonnais, France. Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997). Modeling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag. Embrechts, P., McNeil, A. and Straumann, D. (1999a). “Correlation and Dependence in Risk Management: Properties and Pitfalls.” Working Paper, University of Zurich. Embrechts, P., McNeil, A. and Straumann, D. (1999b). “Correlation: Pitfalls and Alternatives.” Working Paper, Department Mathematik, ETH Zentrum, Zurich. Embrechts, P., Lindskog, F. and McNeil, A. (2001). “Modeling Dependence with Copulas and Applications to Risk Management.” Working Paper CH-8092, Dept. of Mathematics, ETZH Zurich. Frees, E. W. and Valdez, E. A. (1998). “Understanding Relationships Using Copulas.” North American Actuarial Journal 2(1), 1–25. Frey, R. and McNeil, A. J. (2001a). “Modeling Dependent Defaults.” Working Paper, University of Zurich. Frey, R. and McNeil, A. J. (2001b). “Modeling Dependent Defaults: Asset Correlation Is Not Enough.” Working Paper, University of Zurich. Genest, C. (1987). “Frank’s Family of Bivariate Distributions.” Biometrika 74, 549– 555. Genest, C. and Rivest, L. (1993). “Statistical Inference Procedures for Bivariate Archimedean Copulas.” Journal of the American Statistical Association 88, 1034–1043. Gersbach, H. and Lipponer, A. (2000). “The Correlation Effect.” Working Paper, University of Heidelberg. Geske, R. (1977). “The Valuation of Corporate Liabilities as Compound Options.” Journal of Financial and Quantitative Analysis 12(4), 541–552. Gumbel, E. J. (1960). “Distributions des valeurs extremes en plusiers dimensions.” Publ. Inst. Statist. Univ. Paris 9, 171–173. Jarrow, R. A. and Turnbull, S. M. (1995). “Pricing Derivatives on Financial Securities Subject to Credit Risk.” Journal of Finance 50(1), 53–85. Jarrow, R. A., Lando, D. and Turnbull, S. M. (1997). “A Markov Model for the Term Structure of Credit Spreads.” Review of Financial Studies 10, 481–523.

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Jarrow, R. A. and Yu, F. (2000). “Counterparty Risk and the Pricing of Defaultable Securities.” Working Paper, Johnson GSM, Cornell University. Kijima, M. (2000). “Credit Events and the Valuation of Credit Derivatives of Basket Type.” Review of Derivatives Research 4, 55–79. Lee, A. J. (1993). “Generating Random Binary Deviates Having Fixed Marginal Distributions and Speciﬁed Degree of Association.” The American Statistician 47, 209–215. Leland, H. E. (1994). “Corporate Debt Value, Bond Covenants and Optimal Capital Structure.” Journal of Finance 49(4), 1213–1252. Li, D. X. (1999). “On Default Correlation: A Copula Function Approach.” Working Paper 99-07, The RiskMetrics Group, New York. Lindskog, F. (2000). “Modeling Dependence with Copulas and Applications to Risk Management.” Working Paper, Risklab, ETH Zurich. Longin, F. and Solnik, B. (2001). “Extreme Correlation of International Equity Markets.” Journal of Finance 56, 649–676. Longstaff, F. A. and Schwartz, E. S. (1995). “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt.” Journal of Finance 50(3), 789–819. Madan, D. and Unal, H. (1999). “Pricing the Risks of Default.” Review of Derivatives Research 2(2/3), 121–160. Merton, R. (1974). “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29, 449–470. Nelsen, R. B. (1999). An Introduction to Copulas. New York: Springer-Verlag. Sklar, A. (1959). “Functions de repartition a n dimensions et leurs marges.” Publ. Inst. Statist. Univ. Paris 8, 229–231. Sklar, A. (1973). “Random Variables, Joint Distributions, and Copulas.” Kybernetica 9, 449–460. Schonbucher, P. and Schubert, D. (2001). “Copula-Dependent Default Risk in Intensity Models.” Working Paper, Dept. of Statistics, Bonn University. Wang, S. S. (2000). “Aggregation of Correlated Risk Portfolios: Models and Algorithms.” Working Paper, CAS. Wise, M. and Bhansali, V. (2001). “Portfolio Allocation to Corporate Bonds with Correlated Defaults.” Working Paper No. CALT-68-2365, California Institute of Technology. Xiao, J. (2003). “Constructing the Credit Curves: Theoretical and Practical Considerations.” Working Paper, RiskMetrics. Zhou, C. (2001). “An Analysis of Default Correlations and Multiple Default.” Review of Financial Studies 14, 555–576.

Keywords: Correlated default; copulas; tail dependence

Index A Abramowitz, M., 216 ABRS, 67 Accuracy ratio, 145. See also Hull-White model; Merton model; Vasicek-Kealhofer model Acharya, V., 62, 75 Actuarial science, 84 Adams, K, 34 Additive model, 88, 90 Aggregate debt structure, 45 Agrawal, D., 163 All-orders result, 177 Altman, E., 34, 62, 75, 99, 184 Andersen, P., 99 Anderson, R., 62 Anderson and Darling Statistic, 203 Anderson-Darling criterion, 208 Andrade, G., 62 Ang, A., 216 A-rated debt, default probabilities (L-T model usage), 50 Arora, Navneet, 163 Asset liquidity, 43 Asset process, systematic risk, 145 Asset value, 40, 43 process, 122, 125 Asset volatility, 138 calibration, 42–43 estimation, 48 generation, 138 increase, 52 Asset-liability ratios, 43 Asset-pricing model, 6–7 Asymmetric correlation, presence, 206 Asymmetric exceedance correlations. See PDs

B Baa-rated debt, default probabilities, 53 L-S model, usage, 51 L-T model, usage, 48 Backward substitutions, 108, 111

Bakshi, F., 130 Bakshi, G., 62, 76 Balance sheet data, unavailability, 8 Bangia, A., 99 Bankruptcy costs/taxes, impact, 133 histories, 2 Base case default probability/horizon contrast, 56 parameters, 42, 46–47 good ﬁt, 50 table, 47 Baseline intensity. See Time-varying baseline intensity Basle Committee on Banking Supervision, 187 Beta distribution, usefulness, 68 Bhansali, V., 185, 218 Bharath, S., 75 Bhatia, M., 76 Bielecki, T., 34, 130 Biostatistics, 84 Bivariate intensity estimator, 95, 97 non-parametric estimator, 83 Black, F., 62, 76, 130, 163, 185 Black option pricing model, 54 Black-Scholes models, default occurrence, 121 Blume, M., 99 Bohn, Jeffrey R., 63, 163 Bond value determination, 112 random variable, 171 usage, 116 Bonds. See Coupon-paying bonds credit quality order, Moody’s rating categories, 190 data, descriptive statistics, 142–143 embedded options, exclusion, 142 markets, liquidity problems, 152 maturity date, 171

219

220 Bonds (contd.) time to maturity, cumulative percentage distribution, 157 valuation, business modeling approach. See High-yield bond valuation Book-to-market factor, 7 Book-to-market stocks, 12 Bootstrapping, 156. See also Pointwise conﬁdence bond availability, 157 procedure, 142 usage. See Zero-coupon yield curve Borgan, Ø., 99 Bouye, E., 216 Brady, B., 62, 75 B-rated debt, default probabilities L-S model, usage, 51 L-T model, usage, 49 Briys, E., 117 Brownian motion, 38 Brys, E., 62 Bubble component analysis. See Stock prices Buckets, division, 86 Business cycles effects, 79, 81 regimes, existence, 79 Business model, usage, 103 Business modeling approach. See High-yield bond valuation Business prospects, 43 Business risks, 43, 104

C Calendar time, 89, 96 Calibration. See Debt Cantor, R., 99, 100 Capital allocation decisions. See Internal capital allocation decisions Capital asset turnover ratio, 104 Capital cost, 105. See also Firms Capital structure, 41. See also Long-term capital structure; Rolled over capital structure; Stationary capital structure usage, 103 Carey, M., 76 Carpenter, J., 62 Cash ﬂows. See Debt calculation, 109–110 growth, 43 Catty, L., 99 CDS. See Credit default swap

INDEX CEDF. See Cumulative EDF Center for Research in Security Prices (CRSP), 2 data, 7 T-bill yield, availability, 11 Çetin, U., 130 Chan-Lau, J.A., 76 Chava, S., 34 Chen, J., 216 Clayton, D.G., 216 Clayton copula, 193, 211 usage, 208 Collin-Dufresne, P., 34, 63, 163 COMPUSTAT, 141 Conﬁdence bounds, 92 Conﬁdence intervals, 90 Constant default barrier derivation, 45 inclusion. See Default probabilities Copulas, 191–196. See also Clayton copula; Gaussian copulas; Gumbel copulas; Normal copula; Student’s t copula; Worst ﬁt copula approach. See Criterion-based copula approach deﬁnition, 191–192 empirical implications, 209–210 function. See Multivariate normal copula function selection, 169 tail loss distributions, comparison, 210, 211 usage, 192–193 Corporate assets, fraction, 173 Corporate bonds credit spreads, predictions, 134 cross-section, 159–160 Merton model, 113 optimal fraction, 178 Corporate bonds, portfolio allocation (correlated default implications) introduction, 165–168 model, 168–171 notes, 184 portfolio problem, 171–176 references, 184–185 Corporate bonds, recovery rates. See Publicly traded corporate bonds Corporate rate, decomposition, 172 Corporate transactions analysis, 134 Corporate zero-coupon bonds, 166 Corporate-risk-free reference curve, 134

Index Correlated default processes, 186 data description, 190–191 features, 191–196 discussion, 210–212 introduction, 187–190 notes, 215–216 references, 216–217 Correlated defaults complementary analysis, 195 implications. See Corporate bonds overview, 204–205 presence, 166 simulation, 204–210 Correlation asymmetry, 195 choice, 166 levels, 194 matrix, 28, 169. See also Equity model regression Counterparty risk, inclusion, 124 Counting process theory, 83 Coupon-paying bonds, 40 Covariates, two-dimensional sets, 84 Coverage ratios, 43 Cox, J., 62, 76, 163, 185 Cox process, generation, 123 Credit default swap (CDS) contract life, 140 premium, calculation, 140 cumulative percentage distribution (spread range), 144, 158 data calibration, 153 choice, 136 prices, differences (histogram). See Hull-White reduced-form model; Merton model; Vasicek-Kealhofer structural model regression. See Market CDS regression; Model CDS spread, 140 trading frequency, 141 Credit default swap (CDS) spreads bond spreads correlation, time series. See Market CDS buckets, difference, 151 correlation, time series. See Market/model CDS spread correlation cross-sectional variation, 146–152, 155, 158

221 determinants, 159 levels, 146–152 prediction ability, 147 summary statistics, 159 Credit derivatives market, growth, 1 Credit deterioration, 142 Credit pricing models, 133 Credit quality, decline, 190 Credit risk analysis, types, 192 management, rating systems importance (increase), 80 measure, 102 overprediction, 154 Credit risk, reduced-form models/structural models (contrast), 132 data, 141–142 default predictive power, 144–146 empirical methodology, 141, 142–144 introduction, 133–137 notes, 161–162 references, 163–164 results, 144–160 diagnosis, 152–154 Credit risk models case study, 132 importance, 102 structure. See Reduced-form credit risk models CreditMetrics, 71 Criterion-based copula approach, 186 data description, 190–191 features, 191–196 discussion, 210–212 introduction, 187–190 notes, 215–216 references, 216–217 Crosbie, P., 63, 163 Cross-border correlations, features, 73 Cross-sectional variation. See CDS spreads Cross-validation statistic, 20. See also Generalized cross-validation Crouhy, M., 216 CRSP. See Center for Research in Security Prices Cumulative default frequencies, 53 Cumulative distribution functions, 58 Cumulative DPs, 41–42 determination, 46

222 Cumulative EDF (CEDF), 139 Cumulative normal distribution function, 137

D Das, Sanjiv R., 65, 185, 216, 217 Data description. See Correlated default processes; Criterion-based copula approach; Equity returns features. See Correlated default processes; Criterion-based copula approach problems, alleviation, 136 quality/quantity, difference, 132 thinness, 79 time-series plot, 189 Davies, D., 100 Davis, M., 217 DD. See Distance to default de Geer, S.V., 100 de Varenne, F., 62, 117 Debt. See also Zero-coupon debt borrowing/retiring, 138 cash ﬂows, 112 lattice, 111 default probabilities, L-T model (usage). See Baa-rated debt equity (contrast), impact. See Default intensities holders, barrier value, 122 maturity, 45 increase, 53 package. See Hilton Hotels principal, level, 41 speciﬁcation, 44 structure. See Aggregate debt structure valuation, 115 market capitalization, relationship, 108. See also High-yield bond valuation value, ﬁrm value, 112 Debt, structural models appendix, 58 calibration, 46–47 default probabilities, predictions, 39 empirical studies, 42–43 introduction, 40–42 notes, 58–62 references, 62–64 Debt-to-equity ratio. See Market Default. See Correlated defaults correlation, 170

INDEX instantaneous rate, 128 likelihood, increase, 27 thresholds, 169 Default barrier, 138 increase, 54 stochastic process, 125 Default boundaries, 44. See also Endogenous default; Exogenous default decrease, 46 Default costs, 44 Default data, non-parametric analysis introduction, 77–82 marginal integration, 88–90 methodology, outline, 82–83 notes, 99 one-dimensional hazards, 88–90 references, 99–100 transitions, move/duration dependence, 91–94 Default frequencies. See Shorter-term default frequencies matching. See L-T model Default implications, correlation. See Corporate bonds Default intensities analysis, 27–29 comparison, debt/equity contrast (impact), 30–32 equivalence tests, debt prices/equity prices (usage), 31 process, analytic tractability, 5 standard error average, 22 time series graph, 22 Default losses, prediction, 42 Default model, usage, 168–171 Default parameters estimation, 8 time series properties, analysis, 22–26 Default predictive power. See Credit risk Default probabilities (DPs), 170, 176 behavior. See Risk-neutral default probabilities constant default barrier, inclusion, 46 difference, 67 estimation. See Equity prices horizon, contrast. See Base case Kendall’s τ , 191 L-S model, usage. See Baa-rated debt; B-rated debt L-T model, usage. See A-rated debt; Baa-rated debt; B-rated debt; Single-B-rated debt

223

Index plotting, 47 predictions, 43. See also Debt time series, 190 underprediction, 57 Default process. See Joint default process Default protection contract, purchase, 73–74 Default rates, recovery rates (negative relationship), 67 Default risk, 166 implications, 171 literature, survey empirical attributes, 67–68 introduction, 65–66 measure transformations, 73–75 recovery conventions, 69 references, 75–76 speculation, 75 relationship. See Structural models Default swap. See Fair-value default swap total expected premium, 72 Default time, transformation, 126 Defaultable debt, supply, 67 Default-free bond, risk-free value, 141 Default-free interest rate, 44–45 Default-free zero-coupon bonds, 3 Default-risk measures, 145 Default-risk-free rate, 134 Delianedis, G., 34, 63 Density function, 202, 213 Dependence. See Tail dependence empirical features. See Joint distribution relationship, 196 DET indication, 37 Deventer, D., 117 Diebold, F., 99 Dimson, E., 34 Discount bond prices, 9 Discrete-time model, consideration, 71 Distance to default (DD) calculation, 55 DD-to-EDF empirical mapping step, 138 empirical distribution, 133 term structure, calculation, 138 Dividend payment, 4, 106 Doob-Meyer decomposition, 127 Dowd, K., 217 Downgrade activity, 86–87 Downgrade intensities, 92, 93 DPs. See Default probabilities Duan, J.D., 130 Duffee, G., 34, 63, 163 Dufﬁe, D., 34, 76, 130, 185, 217

Dufﬁe-Lando model, 126 Duration dependence. See Default data; Rating transition data effects, 90 Durrleman, V., 216, 217 Dynkin, L., 185

E Earnings per share (EPS), matching. See L-S model Eastman Kodak Company, intensity function (time series estimates), 22 EBIT, 43 Economy-wide default risk, increase, 196 EDF. See Expected default frequency Eigenvalues, 170 EIS. See Expected default frequency EJV (Reuters), 141 EMBI. See Emerging Market Bond Index Embrechts, P., 217 Emerging Market Bond Index (EMBI) (J.P. Morgan), 75 Empirical default frequencies, matching. See L-T model Empirical EPS, matching. See L-S model Endogenous default, 39 boundaries, L-T model, 45–46 contrast. See Exogenous default Eom, Y., 34, 163 Epanechnikov kernel functions, usage, 85 Equity issuance, consideration, 4 Equity model regression independent variables, correlation matrix, 28 parameter estimatics, averages, 15–19 t-scores, averages, 15–19 Equity prices, bubbles, 27 existence, 2 Equity prices, default probabilities (estimation), 1 appendix, 36–38 ﬁrms, involvement, 8 introduction, 1–3 model performance, summary statistics, 23–25 notes, 33–34 references, 34–35 Equity returns bubble component, 2, 32 computation, data description, 7–9

224 Equity returns (contd.) estimation, 12–20 models, 14 relative performance, 29–30 time-series model, 6 Equity risk parameters, time series properties (analysis), 22–25 premium, 29 Equity value, 137 Equivalence tests, debt prices/equity prices (usage). See Default intensities Ericsson, J., 117, 130, 163 Estimation parameters. See Regime-switching model phase. See Joint default process results. See Regime-switching model European call option, 121 Event downgrade, 87 Event risk, 166–167 premium, 167 Exceedance correlation plot, 206 Exceedance plot, 196 Excess risk premium, 167, 176 Exogenous default, 39 boundaries, L-S model, 45 endogenous default, contrast, 40 Expected default frequency (EDF), 55, 133 EDF-implied spreads (EIS), 140 Expected utility, differentiation, 74 Exposure matrix, graphical illustration, 86 External intensity estimator, 84

F Face value, receiving, 103 Fair-value default swap, 72 Fama, E., 34 Fama-French benchmark portfolios, 7 Fama-French four-factor model, analysis, 26 Fan, J., 99 Fan, R., 216 Fanjul, G., 75 FCFs. See Free cash ﬂows Filtration expansion, 126 generation, 123 reduction, 127 stopping time, 126 Finger, C., 76 Firms. See Primitive ﬁrm asset value, 119, 137

INDEX capital cost, 106 debt, payoff, 123 ﬁxed costs, 110 operating leverage, 116 perpetual debt, 115 value. See High-yield bond valuation lattice, determination, 108 volatility, 48 First-order condition, derivation, 214 First-passage time cumulative probability function, 46 Fischer, E., 63 Fixed cost, present value, 108 Fledelius, Peter, 77, 100 Fong, G., 185 Forward rates, 71 curves, estimation, 9 Four-factor asset-pricing model, 12 Fourth-degree polynomial, 9 Fractional default loss, 173 Fractional loss ﬂuctuations, 174 probability, 177 Free cash ﬂows (FCFs), 107 Freed, L., 216 Frees, E.W., 217 French, K., 34 Friedman, M., 163 Frye, J., 76 F-tests, usage, 14, 26 Fusaro, R., 100

G Galai, D., 216 Gamma function, 116–117 Gaussian copulas, 207 Generalized cross-validation (GCV) statistics, 29–30 Genest, C., 217 Geng, G., 185, 216 Geometric random walk, 43–44 Gersbach, H., 217 Geske, R., 34, 63, 163, 217 Giesecke, K., 130, 163 Gijbels, I., 99 Gill, R., 99 Girsanov’s theorem, usage, 38 Goldberg, L., 163 Goldstein, R., 34, 63, 163

225

Index Goodness of ﬁt. See Marginal distribution determination, 206–207 measure, 30 Gordy, M., 76 Gross return on investment (GRI), 104–108 binomial process, 106 decrease, 114 lattice generation, 109 usage. See High-yield bond valuation standard deviation, 116–117 Guillen, M., 100 Gumbel, E.J., 217 Gumbel copulas, 193, 207, 209 Gupton, G., 76

H Hamilton, D., 100 Hazard rates annualization, 181 change, 199–200 correlations, 194 graphical illustration, 205 independence, 176 joint dynamic system, 188 model, 125. See also Intensity-based hazard-rate model application, 119 samples, 204 Heath, D., 34 Heinkel, R., 63 Helwege, J., 34, 163 High-grade debt, issuance, 196 High-low regime cutoff, 200 High-yield bond valuation approach, 102 business modeling approach, 101 appendix, 116–117 introduction, 102–104 notes, 117 references, 117 empirical evidence, 113–114 model ﬁrm, value, 107–108 implications, 115 primitive form, 104–107 speciﬁcation, 104–108 numerical illustration, 108–113 calibration, 112–113 debt valuation, market capitalization (relationship), 111–112

ﬁrm, value, 110–111 GRI lattice, usage, 109–110 High-yield companies, 107 Hilton Hotels debt package, 112 debt structure, 111 example, 109 HML, 7, 12 factors, 13 Ho, Thomas S.Y., 101, 117 Horizon interval, 20 Hu, W., 76 Huang, J., 34, 35, 63, 163 Huang, M., 35 Hui, C.H., 130 Hull, J., 35, 163 Hull-White model (HW model), 132, 137, 140–141 accuracy ratio, 145 negative bias, 155 Hull-White reduced-form model, market/model CDS price differences (histogram), 148 HW model. See Hull-White model Hyman, J., 185

I Idiosyncratic risk, 196 Implied default probability density, 143 Independent variables, correlation matrix. See Equity model regression Information sets generation, 123 knowledge assumption, reduced-set models, 119 observation, 119, 128–129 Information-based perspective. See Structural models Ingersoll, J., 185 In-sample root mean squared error goodness-of-ﬁt tests, 2 Intensities. See Downgrade intensities; Multiplicative intensities; Upgrade intensities analysis. See Default intensities estimator. See Bivariate intensity estimator; Local constant two-dimensional intensity estimator; Local linear two-dimensional intensity estimator

226 Intensities (contd.) graphs, 82 issuer-level stochastic process, marginal distribution, 197 joint dynamic system, 188 mean estimation, stochastic process (results), 199 processes constants, 124 mean, 199 residuals, ﬁt (estimation results), 204 two-dimensional estimation. See Transition intensities Intensity-based hazard-rate model, 127 Interest, spot rate, 5 Interest rates. See Default-free interest rate constancy, 121 risk, 156 Internal capital allocation decisions, 77 Investment time horizon, 166 Issuers, cumulative percentage distribution, 143, 157

J Jäckel, P., 185 Janosi, Tibor, 1, 35 Jarrow, Robert, 1, 34, 35, 63, 76, 118, 130, 163, 164, 217, 218 Joint default probabilities, 174 Joint default process calibration, 204 determination, 196–204 estimation phase, 197 method, 196–197 Joint dependence calibration results, 207–209 metric, 207–209 Joint distribution, dependence (empirical features), 194–196 Joint probability distribution. See Returns Joint stochastic process, estimation problem, 192 Jokivuolle, E., 76 Jones, E., 131 Jones, M.C., 100 Jorion, P., 185 Jump model, usage. See Rating class Jump point process, 127–128 Jump size, uniform distribution, 198

INDEX Jump-diffusion model metric q values, 207, 208 usage, 188, 207–208 Junior debt, 103

K Kao, D., 99 Kapadia, N., 216 Kaplan, S., 62 Kavvathas, D., 100 Kealhofer, S., 164 Keiding, N., 99 Kernel smoothing, example, 85–86 Kijima, M., 218 Kim, J., 164 Kluppelberg, C., 217 KMV approach relationship. See L-S model; L-T model steps. See Moody’s-KMV approach KMV default boundary, 54 KMV DP, decrease, 56 Kolmogorov criterion, 209, 211 Kolmogorov distance, 203 Konstantinovsky, V., 185 Kronimus, A., 99 Kurtosis distributions, 189 injection, 197 Kusuoka, S., 131

L

L1 distance, 201 L2 distance, 203 Lando, David, 63, 77, 130, 131, 163, 217 Lebesgue measure, 125 Lee, A.J., 218 Lee, Sang Bin, 101, 117 Leland, H., 63, 164, 185 Leland, Hayne E., 39, 218 LGD. See Loss given default Li, D., 185, 218 LIBOR swap rate, 167 Likelihood function, 213–214 Lim, F., 99 Lindskog, F., 217, 218 Linear regression, multicollinearity, 25 Linton, O., 100 Lipponer A., 217 Liquidating dividend, present value, 5–6 Liquidating payoff, 4

Index Liquidation value, risk premium, 38 Liquidity premium, 30, 134 spread, 114 Lo, C.F., 130 Local constant two-dimensional intensity estimator, 80 Local linear two-dimensional intensity estimator, 80 Löfﬂer, G., 100 Log-likelihood function, 202 maximization, 213–214 Longin, F., 218 Longstaff, F., 63, 117, 130, 164, 185, 218 Long-term capital structure, 41 Long-term growth rate, 110 Lonski, J., 100 Loss given default (LGD), 44, 65, 139. See also Sector-speciﬁc LGD; Seniority-speciﬁc LGD Low-grade bonds, 166 Low-maturity bonds, 153 L-S model. See Exogenous default empirical EPS, matching, 50–54 KMV approach, relationship, 55–56 usage. See Baa-rated debt; B-rated debt L-T model. See Endogenous default empirical default frequencies, matching, 47–50 KMV approach, relationship, 55–56 usage. See A-rated debt; Baa-rated debt; B-rated debt; Single-B-rated debt Lucas, D., 100, 185 Lyden, S., 164

M MacKinlay, A., 99 Madan, D., 35, 62, 76, 130, 131, 218 Marginal distribution choices, 209 estimation, 202 goodness of ﬁt, 203–204 selection, 203 Marginal downgrade intensity, 89, 92, 93 Marginal effect, explanation, 88 Marginal hazard functions, non-parametric estimators (usage), 79 Marginal integration. See Default data; Rating transition data impact, 79 Marginally integrated intensities, 97

227 Marginally integrated upgrade/downgrade intensities, 89, 91, 96 Mark, R., 216 Market capitalization, 113 relationship. See Debt value, 112 debt-to-equity ratio, 115 index coefﬁcient, 28 parameter estimation, 11–12 probability, 105 risk premium, 134 Sharpe ratio, 139 value. See Recovery of market value volatility, constancy, 12 Market CDS regression, 151, 152 R-squared, 155, 158 Market CDS spreads bond spreads correlation, time series, 153 correlation, time series, 150 cross-sectional variation, explanation, 155 Market portfolio, return, 12 Market-microstructure noise, elimination, 8 Market/model CDS price differences, histogram, 148, 149. See also Hull-White reduced-form model; Merton model; Vasicek-Kealhofer structural model Market/model CDS spread correlation, time series, 151 Markov assumptions, deviation, 80–81 Martin, J., 34 Martingale measure, 123 Martingale process, 105 Matlab (MATLAB), 142, 202 Mauer, D., 64 Maximum likelihood estimation (MLE), 198 McLeodUSA, stock value/bond value (contrast), 113–114 McNeil, A., 217 Mean estimation, jump model usage. See Rating class Mean process estimation. See Regime-switching environment Mean reversion rate, 200 Measure transformations. See Default risk; Recovery; Recovery risk Mella-Barral, P., 63 Merton, R.C., 63, 76, 131, 164, 185, 218 Merton model, 137–138 accuracy ratio, 145

228 Merton model (contd.) comparison, 115 default occurrence, 121 extensions, 102–103 market/model CDS price differences, histogram, 150 Mikosch, T., 217 Miller and Modigliani model (MM model), 104–105 Miller model, 104 Mithal, S., 164 MKMV. See Moody’s-KMV MLE. See Maximum likelihood estimation MM model. See Miller and Modigliani model Model CDS, regression, 151, 152 Model-derived value, 113 Moment expansion. See Optimal portfolio convergence, 178 default/convergence probability, 181 Momentum factor, 7, 12 Monetary value component, 4 Monte Carlo experiment, illustration, 205–206 Monte Carlo simulation, 200 usage, 197 Moody’s Investors Services, 63 cycle, rating through, 81 default database, usage, 187 rating categories. See Bonds Risk Management Services, 210 Moody’s Special Report, 35 Moody’s-KMV (MKMV) approach, 133 implementation, 134, 138 steps, 54–55 Morton, A., 34 Move dependence. See Default data; Rating transition data Multiplicative intensities, 94–98 structure, 97 Multiplicative model equation, 95 usage, 88–89 Multiplicative structure, additive structure (preference), 95 Multivariate normal copula function, 168–169

N n-dimensional random vector, simulation procedure, 193 Negative net worth, 40

INDEX Neis, E., 164 Nelsen, R.B., 218 Nickell, P., 100 Nielsen, Jens Perch, 77, 100 Nielsen, L., 63, 76, 100, 131 Nikeghbali, A., 216, 217 Noise process, addition, 125–126 Noncredit risk component, 152 Non-default-related risk, 167 Non-linear optimization procedure, 109 Non-Markovian behavior, 78 existence, 80 Non-parametric analysis. See Default data; Rating transition data Non-parametric estimators, usage. See Marginal hazard functions Non-parametric techniques, usage, 78, 81 Nonzero recovery, 181 fractions, inclusion. See Portfolios Normal copula, 192 N-period default swap premium, 72 Null hypothesis, testing, 31–32 n-variate distribution, modeling, 191

O Occurrence exposures analysis, 81–82 stratiﬁcation, 91 Occurrence matrix, simulation, 90 Occurrence/exposure ratio, usage, 84 Off-diagonal elements, 171 Off-the-run securities, 152 Ogden, J., 164 One-dimensional hazards. See Default data; Rating transition data On-the-run securities, liquidity, 152 Operating leverage. See Firms Optimal portfolio moment expansion, 179, 180, 182 recovery, 182 Option price process, 103 Out-of-sample forecasts, 74–75 Out-of-sample generalized cross-validation statistics, 2 Out-of-sample model ﬁt, 30 Oversmoothing, signs, 85

P Parameters. See Base case estimates. See Risk premium

229

Index estimatics, averages. See Equity model regression estimation. See Market index; Spot rate process parameter estimation rolling estimation, 10 values set, 51 Park, B.U., 100 Payout rate, 44 PD. See Probability of default P/E. See Price-to-earnings ratio Pedersen, K., 34 Perpetual debt. See Firms valuation formula, 108, 116 Perraudin, W., 63, 76, 100 Peura, S., 76 Pointwise conﬁdence interval, 90 bootstrapping, 91 sets. See Univariate intensities Portfolios allocation correlated default implications. See Corporate bonds dependence, 179 default loss probability distribution, 178 variance/skewness, 182 moment expansion. See Optimal portfolio problems, 171–176. See also Corporate bonds samples nonzero recovery fractions, inclusion, 181–182 zero recovery fractions, inclusion, 176–181 volatility, 180 Power-curve testing, 136 Predictive power, comparison, 145 Price earnings ratio, proxy, 12 Price staleness, 141 Price-to-earnings ratio (P/E), analysis, 26–27 Primitive ﬁrm, 101, 104. See also High-yield bond valuation lattice, 104 determination, 109 value, 116 Probability of default (PD) asymmetric exceedance correlations, 195. See also Simulated PDs averages, simulated series, 205 levels, measure, 191 time series, 189 time series, summary, 191

Protection buyer, 140 Protter, Phillip, 130, 164 Publicly traded corporate bonds, recovery rates, 66

R Radom-Nikodym derivative, 74 Ramaswamy, K., 164 Ramlau-Hansen, H., 100 Ramlau-Hansen estimator, 84 Rating class downward transitions, 78 mean estimation, jump model (usage), 197–200 stochastic process, 207 Rating drift, 80 analysis, 92 Rating systems, importance (increase). See Credit risk management Rating transition data, nonparametric analysis introduction, 77–82 marginal integration, 88–90 methodology, outline, 82–83 notes, 99 one-dimensional hazards, 88–90 references, 99–100 transitions, move/duration dependence, 91–94 Ratings displays, evolution, 78 Raw occurrence exposure ratios, computation, 84 Raw occurrence matrix, graphical illustration, 87 Rebonato, R., 185 Recovery. See Reduced-form models; Structural models conventions. See Default risk; Recovery risk equations, 70 expectation, 70 fractions, 166 disappearance, 175 zero value, assumption, 176 ratios, 43 statistics, 66 Recovery of market value (RMV), 69 Recovery of par (RP), 69 Recovery of Treasury (RT), 69 assumption, 74–75

230 Recovery rates. See Publicly traded corporate bonds; Target recovery rate constants, 124 decrease, 53 dependence, 68 expression, 69 maximum, 73 measure transformations, 73–75 need, 32 regime effects, impact, 68 volatility, 67 Recovery risk, literature survey conventions, 69 empirical attributes, 67–68 introduction, 65–66 measure transformations, 73–75 references, 75–76 speculation, 75 Reduced-form credit risk models equivalence, 7 estimation, 3 structure, 3–7 Reduced-form framework, testing, 153 Reduced-form information set, 124 Reduced-form model-implied CDS spreads, 147 Reduced-form modeling frameworks, 136 Reduced-form models, 123–125 contrast, information-based perspective. See Structural models empirical testing, 136 knowledge assumption. See Information sets market/model CDS price differences, histogram. See Hull-White reduced-form model origination, 118 preference, 128 recovery, 71–73 structural models, contrast. See Credit risk Reference obligation, expected recovery rate, 140 Regime determination, 200 Regime effects, impact. See Recovery rates Regime-shifting models, metric q values, 207, 208 Regime-switching environment, mean process estimation, 200–201 Regime-switching model, 189 estimation parameters, 201 estimation results, 201

INDEX Reneby, J., 117, 130, 163 Residuals, normalization, 201 Resti, A., 62, 75 Returns, joint probability distribution, 166 Reversal aversion effect, 94 Riboulet, G., 216 Risk aversion, 174–175 level, decrease, 180 driver, volatility, 105 management, purpose, 73 Risk (magazine), 35 Risk premium. See Excess risk premium; Liquidation value; Market parameter estimates, 25 requirement, 73 Risk-free assets, 167, 172 Risk-free rate, 116 excess return, source, 165 Risk-free zero curve, 142 Risk-neutral cumulative default probability, 139 Risk-neutral default probabilities, 146 behavior, 145 density, 140–141 Risk-neutral probability, 106, 107 Rivest, L., 217 RMSE. See Root mean squared statistic RMV. See Recovery of market value Robustness tests, 154–160 Rogers, L.C.G., 131 Rolled over capital structure, 54 Rolling estimation procedure, 14 Roncalli, T., 216, 217 Root mean squared statistic (RMSE), magnitudes, 29–30 Rosenfeld, E., 131 RP. See Recovery of par R-squared statistic, reliability, 159 time series distribution quartiles, 155, 158 RT. See Recovery of Treasury Rutkowski, M., 34, 130

S Saà-Requejo, J., 63, 76, 117, 131 Santa-Clara, P., 63, 76, 117, 131 Saraniti, D., 164 Schaefer, S., 63 Schagen, C., 99

Index Scheike, T., 100 Scholes, M., 62, 130 Scholes/Merton option pricing model, 54 Schonbucher, P., 218 Schubert, D., 218 Schuermann, T., 99 Schwartz, E., 63, 164, 185 Schwartz, E.M., 117 Schwartz, E.S., 131, 218 Schwartz, R., 35 Sector volatility, 112 Sector-speciﬁc LGD, 139 Seniority, importance, 68 Seniority-speciﬁc LGD, 139 Sharpe ratio. See Market Shimko, D., 117, 131 Shorter-term default frequencies, 39 Short-term liabilities, 54 Shumway, T., 35 Simulated PDs, asymmetric exceedance correlations, 206 Simulation models comparisons, 204–210 overview, 204–205 Single-B-rated debt, default probabilities (L-T model usage), 48 Singleton, K., 34, 76, 130, 163, 185, 217 Sironi, A., 62, 75 Skewed double exponential distribution, 202, 207–208, 212–214 Skewed double exponential model, 213 Sklar, A., 218 Skødeberg, T., 100 Slope coefﬁcients, 151, 152 Small ﬁrm factor, 7 SMB, 7, 12 factors, 13 Smith, K., 35 Smoothed downgrade intensity, 88 Smoothed downgrade matrix, 88 Smoothed exposure matrix, 88 Smoothed two-dimensional intensity estimator, computation, 85 Smoothing procedure, deﬁnition, 85 quantiles, impact, 82 techniques, usefulness, 79 Sobehart, J., 100 Solnik, B., 218 Solvency requirements, 77 Spot rate process parameter estimation, 9–11

231 Spreads determinant, 158 underprediction, 134 writing, 66 Srinivasan, A., 75 SSEs. See Sums of squared errors State variables, 126 process parameters, estimation, 9–12 Stationary capital structure, 45 Statistical probability measure, 120 Stegun, I.A., 216 Stein, R., 100, 164 Stochastic process, 120, 123. See also Default barrier estimation, 201–202 Stock prices bubble component analysis, 26–27 existence, 4 data, unavailability, 8 Stock volatility, 113 Stohs, M., 64 Stopping time, 127, 128 inaccessibility, 123 Straumann, D., 217 Strebulaev, I., 63 Strike price, 121 Structural modeling frameworks, 136 Structural models, 120–123 contrast. See Credit risk default probabilities predictions. See Debt default risk, relationship, 43–44 preference, 128–129 recovery, 70–71 reduced-form models (contrast), information-based perspective introduction, 118–120 mathematical overview, 125–128 notes, 129–130 references, 130–131 setup, 120 usefulness, 134 Student’s t copula, 192 Student’s t distribution, 202 Sums of squared errors (SSEs), calculation, 96–98 Sundaram, R., 216 Sundaresan, S., 62, 164 Survival probabilities, 72–73 Swap yield curve, 167 Synthetic credit instruments, 145 Systematic risk, 196

232

T Tail dependence, 193–195 measures, 194 Tail loss distribution, 210. See also Copulas Tanggaard, C., 100 Target recovery rate, 52 Tauren, M., 131 Tejima, N., 117, 131 Term structure, estimation, 138 Terminal value, 107 Terminal wealth values, 74 THR. See Total hazard rate Time horizons, 47, 110. See also Investment time horizon change, 179–180 increase, 181 Time series distribution quartiles. See R-squared Time series estimates, summarization, 14–15 Time series properties, analysis. See Default parameters; Equity risk Time-dependent sample variance/correlation coefﬁcients, 11 Time-series model. See Equity returns Time-varying baseline intensity, 80 Toft, K., 63, 164, 185 Total expected discounted payoffs, 72 Total expected premium. See Default swap Total hazard rate (THR), 195–196 Trade Reporting and Compliance Engine (TRACE), 136 Transition intensities, two-dimensional estimation, 83–88 Transitions exact dates, 82 move/duration dependence. See Default data; Rating transition data Treasury coupon bond prices, 140 Trigger default function, 103 Tsang, S.W., 130 t-scores, averages. See Equity model regression t-statistics, 14 Tufano, P., 216 Turnbull, S., 63, 130, 164, 217 Turner, C., 163 Two-dimensional estimation. See Transition intensities Two-dimensional intensities estimator, 88 computation. See Smoothed two-dimensional intensity estimator

INDEX example, 94–95 Two-stage Monte Carlo model, 205–206 Type I error, 144 Type II error, 144

U Uhrig-Homburg, M., 64 UMD, 7 factors, 13 Unal, H., 35, 131, 218 Unit root rejections, summary, 25 Unit root test performance, results, 25 Univariate intensities estimators, 83 pointwise conﬁdence sets, 83 Unrestricted two-dimensional estimator, 95 Upgrade intensities, 92, 93 Uppal, R., 217 U.S. Treasury prices, usage, 7 U.S. Treasury securities, coupons, 9 Utility function, change, 180

V Valdez, E.A., 217 van Deventer, D., 34, 35, 130, 131 VAR analysis, 168 Varotto, S., 100 Vasicek, O., 117, 164 Vasicek-Kealhofer model (VK model), 132, 137, 138–140 accuracy ratio, 145 implementation, 139 MKMV implementation, 134 negative bias, 155 performance ability, 154 Vasicek-Kealhofer structural model, market/model CDS price differences (histogram), 149 Violet, L., 217 VK model. See Vasicek-Kealhofer model Vogelius, M., 100 Volatility. See Portfolios; Risk constancy. See Market estimation. See Asset volatility level, 49, 200

W Wahba, G., 35 Wand, M.P., 100 Wang, S.S., 218

233

Index Wang, X., 130 Warga, A., 35 Wealth, utility (expansion), 175 Whitcomb, D., 35 White, A., 35, 163 Wiener process, 107 Wise, M., 185, 218 Worst ﬁt copula, 211

X Xiao, J., 218

Y Yield curve, ﬂatness (assumption), 108 Yield spreads, prediction, 42 Yildirim, Yildiray, 1, 35, 130 Yu, F., 130, 218

Z Zechner, J., 63 Zeng, Y., 131 Zero recovery fractions, inclusion. See Portfolios Zero-coupon bond, 55 dollar payment, 120 liability structure, 122–123 maturity, 121 spread, 139 Zero-coupon corporate bonds, 172 Zero-coupon debt, 40 Zero-coupon yield curve, bootstrapping (usage), 143, 157 Zhang, F., 62, 76, 130 Zhou, C., 64, 76, 185, 218 Zhu, Fanlin, 132 Zmijewski, M.E., 35

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