Professional Perspectives on Fixed Income Portfolio Management, Vol. 4 (Frank J. Fabozzi Series)

Frontmatter-Prof Persp Page iii Thursday, July 24, 2003 10:09 AM Professional Perspectives on Fixed Income Portfolio M...

Author: Frank J. Fabozzi

355 downloads 1457 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Frontmatter-Prof Persp Page iii Thursday, July 24, 2003 10:09 AM

Professional Perspectives on

Fixed Income Portfolio Management Volume 4

FRANK J. FABOZZI EDITOR

John Wiley & Sons, Inc.

Frontmatter-Prof Persp Page i Thursday, July 24, 2003 10:09 AM



Frontmatter-Prof Persp Page ii Thursday, July 24, 2003 10:09 AM

THE FRANK J. FABOZZI SERIES Fixed Income Securities, Second Edition by Frank J. Fabozzi Focus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. Abate Handbook of Global Fixed Income Calculations by Dragomir Krgin Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi Real Options and Option-Embedded Securities by William T. Moore Capital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. Fabozzi The Exchange-Traded Funds Manual by Gary L. Gastineau Professional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi Investing in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia Pilarinu Handbook of Alternative Assets by Mark J. P. Anson The Exchange-Traded Funds Manual by Gary L. Gastineau The Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry The Handbook of Financial Instruments edited by Frank J. Fabozzi Collateralized Debt Obligations: Structures and Analysis by Laurie S. Goodman and Frank J. Fabozzi Interest Rate, Term Structure, and Valuation Modeling edited by Frank J. Fabozzi Investment Performance Measurement by Bruce J. Feibel The Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. Fabozzi The Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. Markowitz Foundations of Economic Value Added: Second Edition by James L. Grant Financial Management and Analysis: Second Edition by Frank J. Fabozzi and Pamela P. Peterson Measuring and Controlling Interest Rate and Credit Risk: Second Edition by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

Frontmatter-Prof Persp Page iii Thursday, July 24, 2003 10:09 AM



FRANK J. FABOZZI EDITOR

John Wiley & Sons, Inc.

Frontmatter-Prof Persp Page iv Thursday, July 24, 2003 10:09 AM

Copyright © 2003 by Frank J. Fabozzi. 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, 201748-6011, fax 201-748-6008, e-mail: [email protected]. 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 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, visit our web site at www.wiley.com.

ISBN: 0-471-26805-4

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Frontmatter-Prof Persp Page v Thursday, July 24, 2003 10:09 AM

Contents

Preface Contributing Authors

vii xiv

FIXED INCOME ANALYSIS AND STRATEGIES Risk/Return Trade-Offs on Fixed Income Asset Classes Laurent Gauthier and Laurie Goodman

1

Fixed Income Risk Modeling for Portfolio Managers Ludovic Breger

17

Tracking Error William Lloyd, Bharath Manium, and Mats Gustavsson

45

Consistency of Carry Strategies in Europe Antti Ilmanen and Roberto Fumagalli

77

The Euro Benchmark Yield Curve: Principal Component Analysis of Yield Curve Dynamics Lionel Martellini, Philippe Priaulet, and Stéphane Priaulet Dollar Rolling—Does It Pay? Jeffrey Ho and Laurie Goodman

103

131

CREDIT RISK AND CREDIT DERIVATIVES Valuing Corporate Credit: Quantitative Approaches versus Fundamental Analysis Sivan Mahadevan, Young-Sup Lee, David Schwartz, Stephen Dulake, and Viktor Hjort Maturity, Capital Structure, and Credit Risk: Important Relationships for Portfolio Managers Steven I. Dym

141

183

v

Frontmatter-Prof Persp Page vi Thursday, July 24, 2003 10:09 AM

vi

Contents

A Unified Approach to Interest Rate Risk and Credit Risk of Cash and Derivative Instruments Steven I. Dym

197

Implications of Merton Models for Corporate Bond Investors Wesley Phoa

211

Some Issues in the Asset Swap Pricing of Credit Default Swaps Moorad Choudhry

229

Exploring the Default Swap Basis Viktor Hjort

239

The Valuation of Credit Default Swaps Ren-Raw Chen, Frank J. Fabozzi, and Dominic O’Kane

255

STRUCTURED PRODUCTS An Introduction to Residential ABS John N. McElravey

281

Nonagency Prepayments and the Valuation of Nonagency Securities Steve Bergantino

303

The Role and Performance of Deep Mortgage Insurance in Subprime ABS Markets Anand K. Bhattacharya and Jonathan Lieber

325

Some Investment Characteristics of GNMA Project Loan Securities Arthur Q. Frank and James M. Manzi

339

A Framework for Secondary Market CDO Valuation Sivan Mahadevan and David Schwartz

365

Understanding Commercial Real Estate CDOs Brian P. Lancaster

395

Aircraft Valuation-Based Modeling of Pooled Aircraft ABS Mark A. Heberle

431

Index

439

Frontmatter-Prof Persp Page vii Thursday, July 24, 2003 10:09 AM

Preface

he articles in volume 4 of Professional Perspectives on Fixed Income Portfolio Management are grouped into three areas: Fixed Income Analysis and Strategies, Credit Risk and Credit Derivatives, and Structured Products.

T

FIXED INCOME ANALYSIS AND STRATEGIES In the lead article in this volume, “Risk/Return Trade-Offs on Fixed Income Asset Classes,” Laurent Gauthier and Laurie Goodman look at the risk/ return characteristics of major fixed-income asset classes over time in order to see if one asset class consistently outperforms another on a risk-adjusted basis. They first look at the Sharpe ratios for each asset class, and compare those to the duration-adjusted excess returns. The authors then use principal components analysis to identify the factors that are important in determining excess returns and duration-adjusted excess returns. Finally, Gauthier and Goodman examine the performance by asset classes after hedging out the market factors identified through the principal components analysis. The conclusions are quite robust: Overweighting spread product pays over time. Within spread products, mortgages and asset-backed securities tend to have a very favorable risk/return profile over time. The next four articles focus on the European fixed-income market and European asset managers and traders. In “Fixed Income Risk Modeling for Portfolio Managers,” Ludovic Breger discusses the important sources of risk in European fixed-income securities and how to build a reasonable risk model. The author addresses challenges such as accommodating different benchmarks and securities, or providing a wide coverage without compromising accuracy. The risk characteristics of a typical euro investment-grade corporate index are roughly halfway between the conservative and speculative ends of the risk spectrum. Although European fixed-income instruments are on average less risky than their U.S. dollar equivalent, this by no means implies that a sound risk management is less relevant.

vii

Frontmatter-Prof Persp Page viii Thursday, July 24, 2003 10:09 AM

viii

Preface

The growth in the popularity of total return management in the European fixed-income market has led portfolio managers, consultants, and pension funds to increasingly focus on ex ante tracking error to measure the risk in their portfolios relative to a market index. In “Tracking Error,” William Lloyd reviews three different methodologies for calculating tracking error and the assumptions associated with them. While very convenient and conceptually straightforward, he concludes that tracking error is not the best way to evaluate the relative risk in a fixed-income portfolio. Instead, Lloyd advocates the use of scenario analysis as a better method of determining the risk exposures in a fixedincome portfolio. Yield-seeking investment strategies are popular ways of trying to add value in active portfolio management. Most carry strategies—overweighting high-yielding assets and underweighting low-yielding assets— are profitable in the long run, but some strategies appear more risky than others. Antti Ilmanen and Robert Fumagalli in their article “Consistency of Carry Strategies in Europe” show that carry strategies are especially consistently profitable at short maturities. Among various structural tilts that real-money investors can make in their portfolios, replacing short-dated government debt with safe credits seems to offer the best reward for risk. They find similar patterns in all markets they examine, presenting empirical results from European and U.S. swapgovernment spread markets and credit markets. However, they find the results are more compelling for real-money investors than for leveraged investors because the latter need to factor in funding spreads. Moreover, as Ilmanen and Fumagalli note, the consistency of outperformance found is not as robust when investors go further down the credit curve than when they only shift from governments to highest-grade credits. The term structure of interest rates can take at any point in time various shapes and the key question from a risk management perspective is to understand how the term structure of interest rates evolves over time. There have been several studies of the term structure for the U.S. market. In “The Euro Benchmark Yield Curve: Principal Component Analysis of Yield Curve Dynamics” Lionel Martellini, Philippe Priaulet, and Stéphane Priaulet present an empirical analysis of the term structure dynamics in the euro-zone. They study both the zero-coupon euro interbank yield curve, and zero-coupon Treasury yield curves from five individual countries (France, Germany, Italy, Spain, and the Netherlands). Using principal components analysis, they find that three main factors typically explain more than 90% of the changes in the yield curve, whatever the country and the period under consideration. These factors can be interpreted as changes in the level, the slope, and the curvature of the term structure. Martellini, Priaulet, and Priaulet also find

Frontmatter-Prof Persp Page ix Thursday, July 24, 2003 10:09 AM

Preface

ix

strong evidence of homogeneity in the dynamics of the yield curve for different countries in the euro-zone, signaling an increasing financial integration. In “Dollar Rolling: Does It Pay?” Jeffrey Ho and Laurie Goodman look at the historical performance of a mortgage portfolio in which an investor holds a limited number of securities and dollar rolls these securities. This strategy is compared to the historical performance of a mortgage index. The authors show that on average, since 1992, rolling a small portfolio of TBA (“To be Announced”) securities outperformed a mortgage market index by 50 to 60 basis points. Even so, there are times when dollar rolling just does not pay. Generally, they find that dollar rolling is the most profitable during prepayment waves, it is less profitable during periods of limited supply.

CREDIT RISK AND CREDIT DERIVATIVES Several major events in the credit markets have put a new focus on valuing corporate credit. What methodologies can be used to value corporate credit? There are many potential answers to this question. Quantitative approaches have gained popularity recently, particularly structural models based on equity market inputs. The traditional fundamental approach, used for decades by most credit analysts, requires company and industry knowledge. In “Valuing Corporate Credit: Quantitative Approaches Versus Fundamental Analysis” Sivan Mahadevan, Young-Sup Lee, David Schwartz, Stephen Dulake, and Viktor Hjort compare fundamental approaches to valuing corporate credit with quantitative approaches, commenting on their relative merits and predictive powers. On the quantitative front, they review structural models, such as KMV and CreditGrades™. These models utilize information from the equity markets and corporate balance sheets to determine default probabilities or fair market spreads. Then they describe reduced form models. These models use information from the fixed-income markets to directly model default probabilities. Finally, the authors review simple statistical techniques such as factor models. These models are helpful in determining relative value. With respect to fundamental approaches, they provide an in depth examination of rating agency and credit analyst methodologies. Typical corporate bond pricing models simply add a risk premium to the riskless government bond yield. This fails to capture the diversity of bond structures and attendant risk differentials. The approach presented by Steven Dym in “Maturity, Capital Structure, and Credit Risk: Important Relationships for Portfolio Managers” recognizes the distinct risk

Frontmatter-Prof Persp Page x Thursday, July 24, 2003 10:09 AM

x

Preface

profiles of senior and subordinated debt. Dym shows how to relate changes in return on the firm’s physical assets to the prices, hence yields, of these instruments, and explains how maturity differences interact with seniority levels to produce important, albeit counterintuitive, price effects. Bond portfolio managers are today faced with an almost bewildering array of instruments and associated risk profiles. In his article “Unified Approach to Interest Rate Risk and Credit Risk of Cash and Derivative Instruments,” Steven Dym presents a unique, yet straightforward, way to think about the main variables in these instruments—pure interest rate risk and credit risk. The intuitive approach applies to fixed coupon bonds as well as to floating-rate notes, derivatives, and cash-derivative combinations. In the process, Dym throws some light on a number of counterintuitive relationships in the fixed-income marketplace. In the past few years, corporate bond investors have often observed an inverse correlation between a company’s stock price and the spread on its bonds. The so-called “Merton approach” to credit risk, which analyzes a firm’s capital structure using contingent claims theory, provides a theoretical explanation for this correlation. Merton models have become increasingly popular in the banking industry, and are most often used to predict default probabilities. In his article “Implications of Merton Models for Corporate Bond Investors,” Wesley Phoa describes how equity-based credit risk models can be interpreted by corporate bond investors focused on mark-to-market returns rather than default rates. Credit default swaps provide an efficient means of pricing pure credit, and by definition are a measure of the credit risk of a specific reference entity or reference asset. Asset swaps are well-established in the market and are used both to transform the cash flow structure of a corporate bond and to hedge against interest rate risk of a holding in such a bond. As asset swaps are priced at a spread over LIBOR, with LIBOR representing interbank risk, the asset swap spread represents in theory the credit risk of the asset swap name. By the same token, using the noarbitrage principle it can be shown that the price of a credit default swap for a specific reference name should equate the asset swap spread for the same name. However a number of factors, both structural and operational, combine to make credit default swaps trade at a different level to asset swaps. These factors are investigated by Moorad Choudhry in his article “Some Issues in the Asset-Swap Pricing of Credit Default Swaps.” He finds that the difference in spread, known as the default swap basis, can be either positive (the credit default swap trading above the asset swap level) or negative (trading below the asset swap). Further discussion of the default swap basis is provided by Viktor Hjort in “Exploring the Default Swap Basis.” He presents an overview of the factors driving default swaps and analyzes the relationship

Frontmatter-Prof Persp Page xi Thursday, July 24, 2003 10:09 AM

Preface

xi

between the cash and derivatives markets at the market, sector, and individual credit level. The default swap market is often perceived as driven primarily by technical factors particular to this market only. Hjort finds little evidence to support this view. Instead, the nature of the markets argues for a close correlation and for the default swap market effectively being positively correlated with, but more volatile than, a version of the underlying cash market—what the author defines as “high beta.” In the author’s view, the investment implications are that (1) investors should aim to get exposure to credit in whichever market is cheaper, and (2) investors should use the high-beta character of the default swap market to position themselves for major rallies or sell-offs. Trading the basis can allow investors to accomplish the first objective by picking up significant spread without changing the view on the credit. Hjort finds that being long the market that rallies the most can be as important as having the right call on the direction of the market itself so that investors can achieve the second objective. There are two approaches to pricing credit default swaps: static replication and modeling. Static replication is based on the assumption that if one can replicate the cash flows of a credit default swaps using a portfolio of tradable financial instruments, then the price of a credit default swap should equal the value of the replicating portfolio. In situations where either the credit default swap cannot be replicated or one does not have access to prices for the financial instruments in the replicating portfolio, it may become necessary to use a modeling approach. RenRaw Chen, Frank J. Fabozzi, and Dominic O’Kane focus on the modeling approach. In “The Valuation of Credit Default Swaps,” they explain how to determine the premium or spread for a single-name credit default swap, what factors affect its pricing, and how to mark-to-market credit default swaps. The authors show that this requires a model and set out the standard model that is used by the market.

STRUCTURED PRODUCTS The largest sector of the U.S. investment-grade market is the MBS/ABS sector. The MBS market, which includes both residential and commercial MBS, continues to grow. Agency MBS (which includes Ginnie Mae MBS and conventional MBS issuance by Fannie Mae and Freddie Mac) represents between 35% to 38% of most U.S. investment-grade broad-based bond market indexes. Add to this nonagency MBS and residential ABS, one realizes the importance of understanding these structured products in order to effectively manage a bond portfolio. While a much smaller

Frontmatter-Prof Persp Page xii Thursday, July 24, 2003 10:09 AM

xii

Preface

sector compared to the mortgage sector, the has been the substantial growth in ABS and CDOs. The list of products that have been securitized and the collateral used for CDOs continues to grow. The articles in this section discuss structured products. The maturation of securitization combined with a dramatic growth in consumer credit and a secular decline in interest rates fueled the development of nonconforming mortgage products such as home equity loans. These nonconforming mortgage products supply the collateral backing the residential ABS market. John McElravey describes the major features of the residential, or home equity loan, ABS market in “Introduction to Residential ABS.” The intent of the article is to provide the reader with a foundation for understanding and analyzing residential ABS collateral and structures as well as their investment attributes. An overview of nonagency prepayments and an introduction to the valuation of nonagency securities is provided in Steve Bergantino’s article “Nonagency Prepayments and the Valuation of Nonagency Securities.” The model, developed by Lehman Brothers, covers 15-and 30-year fixedrate jumbos, jumbo alt-As, conforming balance alt-As, and jumbo relos, explicitly incorporating the effects on prepayments of loan size, borrower credit quality, prepayment penalties, and geographic distribution. While the usage of mortgage insurance (MI) at the loan level to insure high loan-to-value mortgage loans against losses is fairly common, it is only recently that a variant of this technology, referred to as “deep MI,” has been used in subprime structured transactions. Anand Bhattacharya and Jonathan Lieber in “The Role and Performance of Deep Mortgage Insurance in Subprime ABS Markets” explain how the incorporation of deep MI into structured deals allows an issuer to obtain lower aggregate credit enhancement than other structured alternatives, such as subordination of cash flows. However, as with other options, the continued usage of this technology in the structured markets will be heavily determined by the cost of deep MI, which is a function of the ability and willingness of insurance providers to continue to underwrite this risk. Bhattacharya and Lieber point out that although the use of deep MI in the subprime ABS arena is relatively recent, the performance of deep MI as a credit enhancement tool so far appears to be quite promising. The GNMA multifamily mortgage market, also known as the project loan market, has been growing in both size and number of institutional investors involved. Research support for this market sector is still developing. Art Frank in “Some Investment Characteristics of GNMA Project Loan Securities” helps to close this research gap with analysis of both recent and long-term default and prepayment trends for GNMA project loans.

Frontmatter-Prof Persp Page xiii Thursday, July 24, 2003 10:09 AM

Preface

xiii

With the increased trading of collateralized debt obligations (CDOs) in recent years, the topic of CDO pricing has become increasingly important. In “A Framework for Secondary Market CDO Valuation,” Sivan Mahadevan and David Schwartz describe three fundamental approaches for valuing CDO tranches: the rerating methodology, the market value methodology, and the cash flow methodology. The approaches vary considerably in terms of computational complexity and required market savvy, but each can be useful for investors trying to evaluate opportunities in the market. In “Understanding Commercial Real Estate CDOs,” Brian Lancaster chronicles the rapid growth of the $13 billion commercial real estate (CRE) CDO market, the factors driving such growth, the market’s performance, issuer motivations in sponsoring CRE CDOs, and key factors for investors to consider in the purchase of CRE CDOs. He also analyzes the relative value of CRE CDOs versus other fixed-income instruments, arguing that they benefit from the overly conservative nature of the rating agencies methodologies. Finally, Lancaster stresses different types of CRE CDOs in light of the historic performance of the CRE markets and in so doing provides the investor with a methodology to discriminate among CRE CDOs. The market for aircraft ABS remains under severe stress due to the combination of a weak U.S. economy, the bankruptcy of several major U.S. carriers, the Iraq war of 2003, and SARS. Pooled aircraft ABS securities are suffering from a combination of lower cash flows and aircraft valuations. In “Aircraft Valuation-Based Modeling of Pooled Aircraft ABS,” Mark Heberle introduces a valuation-based model to provide a more robust means of analyzing pooled aircraft securitizations. This methodology uses assumptions about an aircraft’s future value prospects to drive a forward-looking portfolio valuation and related lease cash flows. The methodology presented by the author should help investors in this asset class to develop a more complete understanding of the correlation between aircraft values, lease revenue, and deal structure.

Frontmatter-Prof Persp Page xiv Thursday, July 24, 2003 10:09 AM

Contributing Authors

Steve Bergantino Anand K. Bhattacharya Ludovic Breger Ren-Raw Chen Moorad Choudhry Stephen Dulake Steven I. Dym Frank J. Fabozzi Arthur Q. Frank Roberto Fumagalli Laurent Gauthier Laurie Goodman Mats Gustavsson Mark A. Heberle Viktor Hjort Jeffrey Ho Antti Ilmanen Brian P. Lancaster Young-Sup Lee Jonathan Lieber William Lloyd Sivan Mahadevan Bharath Manium James M. Manzi Lionel Martellini

John N. McElravey Dominic O’Kane Wesley Phoa Philippe Priaulet Stéphane Priaulet David Schwartz

xiv

Lehman Brothers Countrywide Securities Corporation Barra, Inc. Rutgers University Centre for Mathematical Trading and Finance, CASS Business School, London Morgan Stanley Brocha Asset Management Yale University Nomura Securities International, Inc. Citigroup UBS Warburg UBS Warburg Barclays Capital Wachovia Securities, Inc. Morgan Stanley UBS Warburg Citigroup Wachovia Securities Morgan Stanley Countrywide Securities Corporation Barclays Capital Morgan Stanley Barclays Capital Nomura Securities International, Inc. University of Southern California and EDHEC Risk and Asset Management Research Center Banc One Capital Markets, Inc. Lehman Brothers, Inc. The Capital Group Companies HSBC-CCF and University of Evry Val d’Essonne AXA Investment Managers Morgan Stanley

1-Gauthier/Goodman Page 1 Thursday, July 24, 2003 10:44 AM

Risk/Return Trade-Offs on Fixed Income Asset Classes Laurent Gauthier, Ph.D. Director UBS Warburg Laurie Goodman, Ph.D. Managing Director UBS Warburg

n fixed-income markets, investors often pay inadequate attention to the historical risk/return characteristics of different asset classes. Thus, for example, if one asset class consistently outperforms another on a risk adjusted basis, then total rate-of-return money managers (whose performance is measure against an aggregate fixed income index) should consistently overweight that particular asset class. In this chapter, we look at the risk/return characteristics of major fixed-income asset classes over time in order to see if such opportunities exist. We will delve into Treasuries, noncallable Agency debentures, callable Agency debentures, mortgage-backed securities, asset-backed securities, and corporates (also referred to as “credit”). For robustness, we use several risk/return measures, each valuable for different purposes. Our plan of attack is as follows. We first focus on the Sharpe ratios for each asset class, then compare those to the duration-adjusted excess returns (which are returns over the relevant benchmark Treasury securities). In the second section, we run a principal components analysis to

I

1


2

PROFESSIONAL PERSPECTIVES ON FIXED INCOME PORTFOLIO MANAGEMENT

identify the common factors in the performance of fixed income asset classes. In the final section, we review a regression analysis of the returns over the risk-free rate. Our conclusion is that overweighting spread products over time pays. Within spread products, mortgages and asset-backed securities tend to have a very favorable risk/return profile over time.

THE DATA For this study, we used the total rate-of-return for the components of the SSB (Salomon Smith Barney) Broad Investment Grade (BIG) Index.1 Monthly return data on the major asset classes of Treasuries, mortgages, Agency debentures, and corporates is available going back well into the 1980s. However data quality on the callable Agency series looked suspect in its early years, and data for asset-backed securities were not available prior to January 1992. As a result, we only used data as far back as January 1992, and ran it up through March 2003, which is the most recent available when we were writing this article. SSB also calculates a duration-adjusted excess return series for each asset class in their index, which is available back to January 1995. That particular return series is calculated by subtracting out the weighted returns on each of the benchmark Treasuries that characterizes each index, with weightings determined by the partial effective durations.

SHARPE RATIOS We began our analysis by calculating the risk/return trade-off (the Sharpe ratio) for each of the major assets classes. This Sharpe ratio is given by the following equation: Sharpe Ratio = Average excess return ⁄ Standard deviation of return ra – rf = ---------------------σ ( ra – rf ) where ra is the return on the asset class, and rf is the risk free rate. We used 1-month LIBOR as the risk-free rate for our analysis. Exhibit 1 shows our findings. As can be seen, the average return (and 1

UBS, our employer, has licensed the SSB Yield Book and attendant data.


3

Risk/Return Trade-Offs on Fixed Income Asset Classes

EXHIBIT 1

Historical Returns Agency Agency NonCallable callable

MBS

ABS

Credit Credit NonTreasury Callable callable

Nominal Monthly Returns (1/1992–3/2003) Average Standard dev.

0.552 0.765

0.654 1.419

0.588 0.610 0.838 0.834

0.606 1.261

0.653 1.390

0.652 1.315

0.221 1.258 0.176

0.268 1.387 0.193

0.267 1.313 0.203

0.022 0.074 0.302

0.000 0.968 0.000

0.042 0.774 0.054

Excess Monthly Returns (= Nominal return minus 1-month LIBOR, 1/1992–3/2003) Average Standard dev. Ratio

0.167 0.756 0.221

0.269 1.414 0.190

0.202 0.225 0.828 0.834 0.244 0.269

Duration-Adjusted Returns (1/1995–3/2003) Average Standard dev. Ratio

0.030 0.225 0.135

0.055 0.273 0.201

0.068 0.074 0.306 0.251 0.221 0.297

average return over LIBOR) for noncallable Agencies and corporates is higher than that for the other asset classes (callable Agencies, MBS, ABS, and Treasuries). However the standard deviation of the return for both the credit and noncallable Agency categories is so much higher than that on other asset classes, that their Sharpe ratios end up lower. Meanwhile, the ABS, MBS, and callable Agency categories have much lower standard deviations than do the other asset classes. Thus, they end up with higher Sharpe ratios (0.27 on ABS, 0.24 on MBS, and 0.22 on callable Agencies.) In fact, the standard deviation of returns is strongly related to the duration of a security. That is, securities with higher durations will end up having higher returns when interest rates drop, and lower returns when interest rates rise compared to their shorter duration counterparts. Longer duration securities will have a higher standard deviation of excess returns, due to the historical volatility of interest rates. However, the problem with using Sharpe ratios as a guide to performance is that it assumes investors can leverage without limit, and that money can be freely borrowed ad infinitum at the risk-free rate. Thus along those theoretical lines investors should lever up shorter instruments rather than holding the longer duration instruments that constitute a chunk of SSB’s BIG Index. But in reality most total rate-of-return money managers


4


do have leverage constraints and therefore cannot leverage without limit. Thus, while Sharpe ratios are certainly one good measure of risk/return, that should not be the only measure; as portfolios containing only the asset classes with the highest Sharpe ratios would require more leverage than most portfolio managers are permitted. Besides, most fixed income portfolio managers are unwilling to put on a huge curve bet, which would be implicit in buying leveraged short paper versus non-leveraged longer paper.

DURATION-ADJUSTED EXCESS RETURNS A duration-adjusted excess return removes both implicit leverage and the curve bet. It essentially looks at the return on each asset class versus what a duration-equivalent portfolio of on-the-run Treasuries would have provided. The results of such an analysis are shown in the bottom section of Exhibit 1 (with returns also on a monthly basis). For example, Exhibit 1’s Agency NC return of 0.0555 means that Agencies have, on average, provided a duration-adjusted excess return of 5.5 basis points/month. Just as with the Sharpe ratio analysis the ABS and MBS categories provided the highest excess returns, while the noncallable credit series provided returns similar to Agency debentures. One interesting point about this analysis is that callable Agencies look worse than noncallable Agencies, which is the opposite of results from using Sharpe ratios. Also, the differential between MBS and Agency noncallables is much less pronounced than under Sharpe ratios. The reason for this point of interest is that OAS-based models are used in determining the partial durations implicit in duration-adjusted excess return calculations. To the extent that the market does not behave according to how the models work—there will be a bias in duration-adjusted excess returns. Let’s now attempt to quantify the effect of this bias that throws awry the effective duration. Exhibit 2 shows the average effective duration of each of the indices over our 11 plus-year period, as well as the latest duration. Obviously, in the current low rate environment, durations for the callable indices (Agency callables and mortgages) are considerably shorter than historical averages. Agency bullets are also much shorter than historically as the GSEs have altered their debt mix over the last decade, and are now issuing more at the front end of the curve (where they fund more favorably relative to LIBOR). The third row of Exhibit 2 is the empirical duration of each of the indices over the period we looked at. It is calculated as minus the coefficient of the regression of


5


EXHIBIT 2

Duration and Duration Directionality Agency Credit Agency NonNonCallable callable MBS ABS Treasury callable

SSB Avg Duration (1/1992–3/2003) SSB Latest Duration Empirical Index Duration (1/1992–3/2003) Measure of Index Duration Directionalitya

3.1

6.0

3.1

3.0

5.3

5.5

2.1 2.3

4.7 4.6

1.6 2.3

3.0 2.5

5.8 4.1

5.8 3.9

14%

0%

12% 7%

4%

25%

a Slope of the ratio of empirical to effective duration versus average 10-year Treasury yield (2-year rolling window).

monthly returns over changes in 10-year Treasury yields. Basically this measure shows the sensitivity of returns to interest rate levels. Now we could compare this empirical duration (Exhibit 2’s third row) to the exhibit’s first row (average effective duration). But since the interest rate environment has changed a great deal over the time period covered, and there have been changes in the indices’ composition, such a juxtaposition would not be very telling. To pinpoint directionality more accurately via a single numerical reading, we first constructed a specific measure for the discrepancy between empirical and effective durations. We used a 2-year rolling window (12 months of data before the observation + 12 months of data after the observation) to obtain the empirical duration of returns, which was expressed as a percentage of the average effective duration over the same period. To get a specific measure of directionality of durations, we then regressed the duration discrepancy over the 2-year average of 10year Treasury yields, with our measure of directionality taken from the slope of that regression. Our results are shown in the bottom row of Exhibit 2, and we have a handy intuitive interpretation that aids in understanding the results. For example, the coefficient for duration directionality on MBS is 12%, which suggests that a 100-basis-point rally would shorten the duration by 12% more than would be suggested by option-adjusted spread (OAS) models. Note that duration directionality is extremely low for both Agency bullets and for Treasuries, as would be expected. It is also higher for MBS and callable Agencies than for ABS. The only surprise may be the result listed for corporate bonds. However, realize that peri-


6


ods of low rates tend to be correlated with times of crises, during which corporates typically underperform. Thus, corporates should behave as if they have a shorter duration during time of low yields. This produces a bias in the average duration adjusted returns. Since the market has rallied over the period under consideration, the SSB average duration adjusted excess returns on the sections with high duration directionality are biased downward. This helps explain the weaker performance of ABS, MBS, and callable agencies on the duration adjusted excess return measures versus those using Sharpe ratios. The row just before the end of Exhibit 1 captures the standard deviation of excess returns. The conclusions are somewhat obvious: Treasuries have a very low standard deviation of excess returns (as we are simply capturing the on-the-run versus off-the-run basis), while the credit series has a very high standard deviation of excess returns (as duration alone is inadequate, since it only explains part of the return variability). The standard deviations for MBS, ABS, and callable and noncallable Agency series lie between those two extremes. The last line of Exhibit 1 shows (duration-adjusted excess returns)/ (standard deviation of these returns). We do not regard this number as particularly useful, as it overstates the standard deviation of sectors with high duration directionality, and hence understates the attractiveness of these sectors. Even so, some market participants do look at this measure.

FIXED INCOME RETURNS, BY ASSET CLASS To try to figure out what factors are important in determining excess returns and duration-adjusted excess returns, we ran a principal components analysis. The factors, or “components,” emerging from that process can then be matched to market factors to “explain” performance. Exhibit 3 shows the results of our principal component analysis. 1. Let’s look first at the top part of the exhibit, which “explains” nominal returns. Note that the first component explains 92.7% of the variation and looks exactly like the exposure to interest rates (duration). Note also that the order of magnitude of the coefficients on each of the indices looks very much like the average duration given in Exhibit 2. Exhibit 4 confirms this, showing a scatter plot of the return on Factor 1 versus the change in the 10-year Treasury yield. Factor 1 has a very clear linear relationship to changes in interest rates. Identification is provided in Exhibit 5, which looks at the correlation of each factor to various market measures (such as the slope of the 2–10 spread; 5-year


7


EXHIBIT 3

Principal Component Analysis Component 1

2

3

4

5

6

0.28 0.54 0.30 0.48 0.31 0.47

0.00 0.24 0.00 –0.82 0.15 0.49

0.41 –0.20 0.75 –0.28 0.19 –0.34

0.16 0.60 0.00 –0.11 –0.73 –0.25

0.00 0.46 –0.34 0.00 0.56 –0.60

0.85 –0.22 –0.46 0.00 0.00 0.00

3.1 95.8

2.3 98.1

0.9 99

0.5 99.5

— 1

0.28 0.52 0.65 –0.40 0.26 0.00

0.76 0.17 –0.23 0.00 –0.58 0.00

–0.10 0.67 –0.66

–0.53 0.45 0.21 0.12 –0.64 –0.22

–0.12

12.1 92.4

2.9 95.3

2.5 97.8

1.8 99.6

— 1.0

Nominal Returns Agy. Callable Agy. NC MBS Credit ABS Treasury Factor contribution (%) Cumulative Importance (%)

92.7 92.7

Duration-Adjusted Returns Agy. Callable Agy. NC MBS Credit ABS Treasury Factor contribution (%) Cumulative importance (%)

EXHIBIT 4

0.18 0.21 0.23 0.91 0.23 0.00 80.3 80.3

0.32

Relationship—Rates versus Nominal Return Factor #1

–0.18 0.97


8


EXHIBIT 5

Correlations—PCA Factors and Explanatory Variables

Nominal Return

Duration-Adjusted Return

EXHIBIT 6

Factor

Slope (2–10s)

10-yr Trsy

10-yr Swap Spd

5-yr Cap Vol

S&P 500

1 2 3 1 2 3

4% 15% 42% –6% 21% 16%

–89% –20% 13% 36% 1% 27%

–15% 38% –28% –63% –40% 4%

40% 27% –27% –43% –11% –15%

–9% –50% 0% 41% –24% –3%

Relationship—S&P 500 versus Nominal Return Factor #2

cap volatility, etc.). Looking across the row labeled “Factor 1,” we see that the 10-year yield has a correlation of –89% to the first factor of nominal returns. 2. The second most important factor in “explaining returns” by asset class is the credit specific factor. This alone explains another 3.1% of the nominal returns, which brings the cumulative total part “explained” up to 95.8%. Our identification of this factor was relatively easy—a high negative weighting on the credit index combined with a high positive weighting on Treasuries. Exhibit 6 confirms this identification, showing a strong relationship between Factor 2 and the S&P 500; and our correlation analysis in Exhibit 5 confirms this intuition as well. Factor 2 has a correlation of –50% to the S&P 500. Note: The weight on the credit index is –0.82, indicating that the



EXHIBIT 7

9

Relationship—Cap Volatility versus Nominal Return Factor #3

lower the S&P 500, the lower corporate bond returns will be, and vice versa. 3. The third aspect explaining returns by asset class is very clearly an optionality factor. Note that the coefficient on the assets classes that have some optionality (callable Agencies, MBS, and ABS) is positive, while the coefficient on the noncallable series (Treasuries, noncallable Agencies, and credit) is negative. Optionality actually involves several market factors, such as the shape of the curve and volatility. Exhibit 5 shows that the optionality factor has a very positive correlation with curve slope, but a negative relationship with 5-year cap volatility. This suggests that the steeper the curve (the slope), the better a callable series should do (as the options that have been implicitly written are now more out-of-the-money). The higher the volatility, the lower the return on the callable series. Exhibit 7 confirms the negative relationship between Factor 3 and volatility. Because the shape of the curve is also quite important, the relationship between volatility and Factor 3 is slightly less clear than it was between the first two factors. But the significant point is that the three factors together—Treasury yields, credit, and volatility—explain 98.1% of the variation in nominal returns of aggregate fixed-income indices. We now turn to explaining the duration-adjusted excess returns. These are actually much harder to “explain,” as we have already eliminated changes in interest rates (which we just showed to be the most important factor, accounting for 92.7% of return variation).


10 EXHIBIT 8


Relationship—Swap Spreads versus Duration-Adjusted Return Factor #1

1. Look first at the coefficients on Factor 1 in the bottom section of Exhibit 3. It is very clear from these that the most important factor is one governing all spread product. Swap spreads are certainly a proxy for this factor. Exhibit 5 shows that the 10-year swap spread has a – 63% correlation to Factor 1, which is far higher than that on any other market variables. Exhibit 8 confirms the strong relationship between swap spreads and the duration-adjusted return Factor 1. Note that this factor explains 80.3% of the variation in this series. 2. The second factor is a corporate-specific factor. Corporates have a negative factor coefficient, while all other asset classes have a positive factor coefficient. Exhibit 5 shows that this second factor has a clear negative relationship to the S&P 500. The relationship between Factor 2 and the S&P 500 is shown in Exhibit 9; it is quite a strong one. However Exhibit 5 also shows a negative relationship between 10-year swap spreads and the second factor, indicating that the factor identification is not as clean as it otherwise could be. Intuitively, the negative correlation between Factor 2 and swap spreads mitigates some of the effect of the first factor. This credit-specific factor (Factor 2) explains another 12.1% of the returns, bringing the total explanatory power to 92.4%. (Additional remaining factors are not easily identifiable.)

CAPTURING EXCESS RETURNS Now that we have figured out the factors which fundamentally matter in examining returns by fixed income asset class, we can look at using



EXHIBIT 9

11

Relationship—S&P 500 versus Duration-Adjusted Return Factor #2

these factors to capture excess returns by asset class. That is, if a given asset class still outperforms (after hedging out market factors), it suggests that over time, the asset class is a superior provider of excess returns. We will now apply the specific market factors we have identified in the prior section of our analysis. We first set up a series of regressions on nominal excess returns. These regressions use the excess returns each month as the dependent variable, with independent variables being the fundamental factors which we’ve discovered above that should matter— the level of Treasury rates, the shape of the Treasury curve, 10-year swap spreads, 5-year cap volatility and the S&P 500. Exhibit 10 displays the regression results. First look at Treasuries—for which the level of rates is the overwhelming factor powering the sector. The S&P 500 has a low coefficient, but it is significant and has the expected sign. We had expected the shape of the curve to be important—but it was not. Arguably, since the duration of the Treasury index is closer to the 10-year Treasury than to anything else, the curve effect was muted. Additionally, as explained below, we have multicollinearity problems with this analysis. For noncallable Agencies—the level of rates and swap spreads are significant. The S&P also enters significantly, with the expected sign, but is clearly less important than the level of rates or swap spreads. The shape of the curve and volatility are insignificant. There are no surprises here. Now let’s move on to the callable instruments:


12


EXHIBIT 10

Regression Results—Excess Returns Agency Agency NonCallable callable

MBS

ABS

Credit Credit NonTreasury Callable callable

Coefficients Intercept Slope 10-yr Treasury 10-yr Swap Spread 5-yr cap

0.080 0.690 –2.427 –0.020 –0.049

0.095 –0.116 –4.506 –0.036 0.006

0.118 0.462 –2.485 –0.029 –0.052

0.123 0.940 –2.503 –0.023 0.012

0.058 0.014 –3.973 –0.012 0.038

0.080 –1.071 –3.766 –0.051 0.033

0.094 –0.448 –3.986 –0.054 –0.013

S&P 500

–0.221

–2.555

–0.794

–1.381

–2.457

4.357

2.325

0.055 0.369 0.222 0.008 0.037 1.239

0.042 0.286 0.172 0.006 0.028 0.960

0.034 0.230 0.139 0.005 0.023 0.771

0.047 0.317 0.191 0.007 0.032 1.066

0.074 0.497 0.300 0.011 0.049 1.668

0.059 0.399 0.241 0.009 0.040 1.340

Standard dev. of coefficients Intercept Slope 10-yr Treasury 10-yr Swap Spread 5-yr cap S&P 500

0.034 0.230 0.139 0.005 0.023 0.774

T-statistics Intercept Slope 10-yr Treasury 10-yr Swap Spread 5-yr cap S&P 500 Resid. St. Dev. Original St. Dev. % Explained Alpha/Residual St. Dev. Sharpe ratio

2.3 3.0 –17.5 –3.9 –2.1 –0.3

1.7 –0.3 –20.3 –4.4 0.2 –2.1

2.8 1.6 –14.4 –4.6 –1.8 –0.8

3.6 4.1 –18.1 –4.6 0.5 –1.8

1.2 0.0 –20.8 –1.7 1.2 –2.3

1.1 –2.2 –12.6 –4.7 0.7 2.6

1.6 –1.1 –16.6 –6.2 –0.3 1.7

0.376 0.756 50%

0.602 1.414 43%

0.466 0.828 56%

0.375 0.834 45%

0.518 1.258 41%

0.811 1.387 58%

0.652 1.313 50%

0.21

0.16

0.25

0.33

0.11

0.10

0.14

0.22

0.19

0.24

0.27

0.18

0.19

0.20

■ For callable Agencies—all of the variables we used were significant,

excepting the S&P 500 (as expected). ■ MBS and ABS look a lot like callable Agencies—except that with MBS,

the slope of the curve is much less important; and in ABS, cap volatility is not significant, as the option component of this index is small.


13


EXHIBIT 11

Correlation Matrix

Slope 10-yr Treasury 10-yr Swap Spread 5-yr cap S&P 500

Slope

10-yr Trsy

10-yr Swap Spread

5-yr Cap

S&P 500

100% 6% –36% 11% –20%

6% 100% –4% –48% 11%

–36% –4% 100% 7% –12%

11% –48% 7% 100% –23%

–20% 11% –12% –23% 100%

■ For noncallable corporates—the level of rates, swap spreads and the

S&P 500 matter. (For the S&P 500, the coefficient is quite high, but the significance is less than we had hoped). Curve slope and volatility are insignificant. ■ For callable corporates—all factors except volatility are important. Note that callable corporate bonds tend to be much less callable than their Agency counterparts. That is they have long lock-outs before the call, and many calls are at a premium. Moreover, it is the volatility of the individual corporate/credit that matters more than implied interest rate volatility. Given all the factors that go into pricing a callable corporate, it is not surprising that 5-year cap volatility came in insignificant. It is important to realize that the coefficients on these regressions should not be regarded as gospel. There is a fairly high correlation between the independent variables, as shown in Exhibit 11. As a result, the coefficients will be less meaningful.2 Now we will focus on two aspects of our results. First, the intercept term on the regression should measure the hedged excess return. Note that the intercept is highest and most significant for MBS and ABS. For MBS, the intercept is 0.12, with a t-statistic of 2.8. For ABS the intercept is 0.12, with a t-statistic of 3.6. The intercepts for Agency and corporate paper are similar to each other, but clearly lower than MBS and ABS. Treasury paper has the lowest intercept. This indicates that after hedging, all spread product outperforms Treasuries. Second, the residual standard deviation (as shown in Exhibit 10) gives us some idea as to how much of the standard deviation of returns can be explained by market factors we have discussed. Note that for all asset classes, we “explained” from 41% to 58% (roughly, about one-half) of the variation in excess returns. 2

In regression-speak, we have a multicolinearity problem. We can solve that by orthogonalizing the variables, but that leaves us with numbers that are more difficult to interpret. So we just acknowledge the issue and live with it.


14


We can use these results to measure the risk/return trade-off of “hedged excess returns” for various asset classes. If we look at the intercept divided by the residual standard deviation, then ordinal results look roughly similar. MBS and ABS look better than all other asset classes. Agencies look better than corporates or Treasuries, while corporates outshine Treasuries. These results are shown in the bottom section of Exhibit 10. Note that this particular ranking of returns by asset class is very similar to the Sharpe ratios we obtained in the first section and repeated for convenience in the bottom row of Exhibit 10.

CONCLUSION In this article we looked at the risk/return trade-offs of the various fixed income asset classes. We found consistent outperformance on the MBS and ABS series. Here’s a quick review of the evidence: ■ The Sharpe ratios and duration-adjusted excess returns both indicated

the superior performance of the ABS and MBS sectors. ■ In addition, callable Agencies have done better (on a Sharpe ratio basis)

than noncallable Agencies. ■ Credit asset classes fared much more poorly than either structured

products or callable agencies on a Sharpe ratio basis, but better than Treasuries. ■ Looking at the average duration-adjusted excess returns, the noncallable credit has done better than the callable Agencies, but less well than noncallable Agencies. ■ Treasuries again fared the most poorly on a duration-adjusted basis. We then used a principal component analysis to examine the market factors that mattered most for excess returns and duration-adjusted excess returns. We identified the usual suspects: the level of rates, the shape of the curve, volatility, swap spreads and the S&P 500. We took that one step further, and used regression analysis to determine how much excess return and residual risk there was within each asset class after hedging out the market factors we identified. Again, ABS and MBS remain the best performing asset classes, whether measured by the alpha or the alpha divided by residual standard deviation. Looked at in this manner, the corporate series looks approximately as appealing as that for Agency debentures. Treasuries again were the poorest performer. Putting these results together, it appears that being overweighted in spread product is a strategy that historically pays on a risk/return basis



15

(as Treasuries are consistently the poorest performer). Moreover, ABS and MBS are consistently the best performing asset classes, regardless of the risk/return measures used, suggesting total return managers should have a consistent overweight to these sectors.


2-Breger Page 17 Wednesday, July 23, 2003 10:21 AM

Fixed Income Risk Modeling for Portfolio Managers Ludovic Breger, Ph. D. Manager Fixed Income Research Barra, Inc.

he European credit market, consisting mainly of euro and sterling denominated debt, is second only to the U.S. domestic market in terms of size, influence, and liquidity. Not surprisingly, European securities are becoming common in global portfolios. The recent turmoil in credit markets has shown once again that understanding risk is or should be a critical aspect of portfolio management. However, as the European credit market is a mosaic of widely different instruments, issuers, and currencies, identifying and forecasting the risk of European fixed income securities is not a simple task. This article will take the reader through the process of building a European risk model and discuss the important sources of risk in generic fixed income portfolios. Our intention is not to cover the whole spectrum of securities but to address some typical modeling challenges such as accommodating different benchmarks and securities, and providing a wide coverage without compromising accuracy. With a general framework in place, the model can be easily extended to cover more markets or bond types.

T

The author thanks Jean-Martin Aussant, Oren Cheyette, and Darren Stovel for insightful comments and suggestions on how to improve this article.

17


18


A FRAMEWORK FOR UNDERSTANDING AND MODELING RISK This discussion covers the main factors affecting bond returns in the European fixed income market, namely, the random fluctuations of interest rates and bond yield spreads, the risk of an obligor defaulting on its debt, or issuer-specific risk, and currency risk. There are also other, more subtle sources of risk. Some bonds such as mortgage-backed and asset-backed securities are exposed to prepayment risk but such instruments still represent a small fraction of the total outstanding European debt. Bonds with embedded options are exposed to volatility risk. However, it is not apparent that this risk is significant outside derivatives markets. A detailed understanding of correlations between asset returns is required to accurately estimate the risk of a portfolio. Unfortunately, estimating correlations directly is in practice impossible as unknowns severely outnumber observations even in relatively small portfolios. The standard solution is to decompose the portfolio’s vector of asset returns using market-wide common factors:1 r excess = X ⋅ f + r specific

(1)

where X = the matrix of asset exposures f = the vector of factor returns rspecific = the vector of asset residual returns not explained by factors or specific returns idiosyncratic to individual assets Decomposing returns is a key step in identifying, understanding, and modeling the sources of risk that are at work in the market. It is also crucial in understanding risk exposures. We begin our analysis by writing the excess returns of assets in a portfolio as r excess = ( r IR + r curr + r spread factor + r specific )

(2)

where rIR 1

= the vector of returns due to changes in interest rates

For more information on factor models, see for instance Richard C. Grinold and Ronald N. Kahn, “Multiple Factor Models for Portfolio Risk,” John W. Peavey III (ed.) A Practitioner’s Guide to Factor Models (Charlottesville, VA: AIMR, 1994).


Fixed Income Risk Modeling for Portfolio Managers

rcurr rspread factor rspecific

19

= the vector of returns due to changes in currency exchange rates = the vector of returns due to sector-wide changes in yields or credit spreads = the vector of specific returns not explained by common factors

Note that the decomposition implicitly ignores the predictable component of return that is irrelevant for risk modeling purposes.2 The return common horizon will be one month in most cases. Although daily or even weekly returns would provide a much larger data set, they are also on average much more sensitive to noise in bond data.3 We will also see in what follows that it is sometimes possible to use returns over a shorter time horizon. If the return factor model adequately accounts for common factors, then the specific returns are uncorrelated and we can write portfolio risk as T

2

σ = h⋅Σ⋅h

(3)

with T

Σ = X⋅Φ⋅X+∆

(4)

where h Σ Φ ∆

= = = =

the the the the

vector of portfolio holdings covariance matrix of asset returns covariance matrix of factor returns diagonal matrix of specific variances

Equation (4) will yield active risk forecasts when h is a vector of active holdings. The data that can go into computing factor returns will of course depend on what the factors are. It can include bond and index level data as well as currency exchange rates. Assume that we have the factor return series. To construct covariances, we could postulate that the underlying random processes are time stationary and compute covariances using equally weighted factor returns. We actually know that mar2

Some market idiosyncrasies such as settlement conventions are an important part of a valuation model but irrelevant to a risk model. 3 For instance, short-horizons spread returns observed for high-grade corporate bonds are small and are typically very noisy.


20


kets change over time and that recent data are more representative of current market conditions than are older data. A simple method for accommodating this fact consists in exponentially weighting factor returns to calculate the covariance matrix. The relative weight of returns from time τ in the past relative to the most recent returns is e–t/τ, where τ is a time-decay constant.4 The optimal time constant τ can be obtained empirically using, for instance, a maximum-likelihood estimator. However, series that are particularly volatile may require a different treatment (see for instance Currency Returns section). Much of the art of constructing a model goes in choosing relevant factors. Note that factors are descriptive and not explanatory. In other words, they allow for forecasting risk without necessarily being linked to the forces that really drive interest rates or returns. Let’s now proceed with a discussion of several classes of factors.

INTEREST RATE RISK Interest rate or term structure risk stems from movements in the benchmark interest rate curve. Excluding exchange rate risk, it is the main source of risk for most investment-grade bonds. Any reasonable model will include markets that are stable and actively traded. A typical coverage, taken from JP Morgan GBI Broad Index, is shown in Exhibit 1. Note the presence of two emerging markets. Building a term structure risk model for the European market involves choosing several benchmarks—at least one for each currency. A recent complication is that domestic government yields are no longer the universal choice. The LIBOR/swap curve has recently emerged as the euro zone preferred benchmark due to the absence of a natural sovereign yield curve and the growing liquidity and transparency of swap EXHIBIT 1

European Markets in JP Morgan GBI Broad Index as of January 1,

2003 Austria Belgium Czech Republic Denmark Finland France Germany 4

Greece Hungary Ireland Italy Netherlands Norway Poland

The half-life is τ ln 2.

Portugal Spain Sweden Switzerland United Kingdom



EXHIBIT 2

21

Examples of Sovereign Term Structures within the Euro Zone on July

31, 2002

curves. However, many markets continue to trade primarily with respect to the government benchmark. In some emerging markets, the absence of a liquid market for sovereign debt makes the LIBOR/swap curve the only available benchmark. A simple approach used at Barra and which permits alternative views is to use the sovereign term structure as local benchmark whenever possible and include a swap spread “intermediate” factor that can be added to the sovereign-based interest rate factors to allow interest rate to be expressed with respect with the swap curve. This swap spread factor will be described in more detail in the next section. In markets where the benchmark is already the LIBOR/swap curve, there is obviously no need for a swap factor. The existence of a euro zone born from the union of several legacy markets introduces an additional modeling challenge. More than one domestic government is issuing euro-denominated debt, and although yields have converged, some differences clearly remain that suggest building a set of factors for each legacy market. (See Exhibit 2 for some examples of sovereign term structures within the euro zone.) Some bonds also need to be analyzed almost completely independently of other assets. This is the case for Inflation Protected Bonds (IPBs) denominated in euro or sterling, which offer investors a “real” inflationadjusted yield. Such securities are weakly correlated with other asset classes and are exposed to a set of IPB-specific interest risk factors simi-


22


EXHIBIT 3

Examples of Shift, Twist and Butterfly Interest Risk Factors Shapes (Top) and Return Volatilities (Bottom) on July 31, 2002

lar in nature to the conventional interest rate factors but derived from IPB data and real yields. What should the interest rate factors be? Key rate durations, which are rate changes at the term structure vertices, seem a natural and somewhat appealing choice. However, because rates for different maturities are highly correlated, using so many factors is unnecessary, and causes difficulty with spurious correlations. Anywhere from 90% to 98% of term structure risk can in fact be modeled using only three principal components commonly referred to as shift, twist, and butterfly. The principal component analysis is now a fairly standard approach that we describe in more details in the Appendix to this article. Exhibit 3 shows examples of factor shapes. Note how principal components derived from Portuguese sovereign euro-denominated debt are very different from the German shapes. Such large differences within the euro zone confirm the need for a different set of factors in each legacy market. Shift, twist, and butterfly volatilities are shown in Exhibit 4. Quasiparallel shifts in the term structure are the dominant source of risk in all cases with volatilities ranging from 35 to 200 basis points per year. In spite of these large differences, term structure risk is relatively homogeneous across most markets and in particular within the euro zone. Note,



EXHIBIT 4

23

Interest Rate Factor Volatilities on July 31, 2002

again, that the differences in factor volatilities are sufficiently large to justify building separate legacy factors.5 As expected, the largest volatilities are observed for emerging market benchmarks, Czech Republic being the riskiest market. And not surprisingly, real yields appear to be more stable than their noninflation protected sovereign counterparts. 5

An alternative but less accurate approach would be to build a unique set of EMU interest rate factors and capture each legacy market idiosyncrasies with a spread factor.


24 EXHIBIT 5


Examples of Interest Rate Risk Breakdown Exposure

Risk (bp/yr)

Shift Twist Butterfly Shift Twist Butterfly Total Federal Republic of Germany 8.5% 07/16/07 Federal Republic of Germany 4.75% 07/04/28 Czeck Republic 6.95% 01/26/16

4.9 –0.3

–2.9

310

8

35

312

10.8 18.7

10.6

680

490

130

850

10.2 15.0

10.3

1,900 2,700

500

3300

The interest rate risk of any given bond will depend on the bond’s exposures to the factors and on correlations between factors. Exhibit 5 gives detailed risk decompositions for three sovereign bonds. The typical annualized risk of a straight bond issued by the Federal Republic of Germany varies from about 200 to 300 basis points to over 800 basis points, depending on its duration. At the other end of the spectrum, the interest rate risk of a bond issued by the Czech Republic can reach as much as 3,000 basis points,6 which exceeds the risk of most speculative corporate issues in developed markets. Clearly, such extreme cases will require special attention when controlling risk.

SPREAD RISK Until fairly recently, outside the U.S., U.K., and Japan, there were relatively few tradable nongovernment bonds. The recent explosion of the global corporate credit market now provides asset managers with new opportunities for higher returns and diversification. Unlike domestic government debt, however, corporate debt is exposed to spread risk, which arises from unexpected yield spread changes. For modeling purposes, such changes can again be decomposed into a systematic component that describes, for instance, a market-wide jump in the spread of Arated utility debt and can be captured by common spread factors, and an issuer or bond-specific component. This section discusses model market-wide spread risk, while the next section will address issuer specific spread risk and default risk. Data considerations are crucial in choosing factors. The choice of factors will be somewhat limited in markets with little corporate debt. Spread factors should increase the investor’s insight and be easy to interpret. Meaningful factors will in practice be somewhat connected to the 6

Note here how twistlike movements of the Czeck benchmark account for more risk than the shift distortions themselves. A simpler, duration-based model would severely under-forecast risk.



25

portfolio assets and construction process and allow a detailed analysis of market risk without threatening parsimony.

Swap Spread Factors First, as mentioned earlier, there is usually no universal benchmark in a given market. Again, a possible approach, used in Barra’s models, is to introduce a swap spread factor that describes the average spread between sovereign and swap rates and can conveniently allow spread risk to be expressed with respect to the LIBOR/Swap curve when interest rate risk factors are originally based on the sovereign yield curve. This same factor can also be used to compute spread risk in markets where there is not enough data to build a detailed credit block. It can also be used in markets where more detailed credit factors are available, but when there is not enough information to expose a bond to the appropriate credit factor. As we will see in what follows, this will be the case when a euro or sterling-denominated corporate bond is not rated. Based on the observation that bonds with larger spreads are on average more risky, Barra’s model assumes the following exposure to the swap factor: x = D eff + ( α – 1 ) ⋅ D spr OAS γ with α = max 1,  ------------ S

(5)

where Deff Dspread OAS S γ

= = = = =

the bond effective duration the bond spread duration the bond spread the swap spread a scaling exponent determined empirically and equal to 0.6

At the time of this writing, corporate bonds denominated in currencies others than euro and sterling are only exposed to the local interest factors and if it exists, the swap factor. This swap factor is roughly equivalent to a Financial AA spread factor, as the bulk of organizations that engage in swaps are AA-rated financial institutions. The swap model is coarser than the two local credit models discussed in the next section, but it performs adequately because spread changes are highly correlated within markets. Swap spread volatilities for several currencies are shown in Exhibit 6, with values that vary from about 15 bp/yr to 40 bp/yr. Also shown are the resulting spread risks in the euro and sterling markets for several rating categories. We will see further below that the swap model pre-


26


EXHIBIT 6

Swap Spreads (a) Swap Spread Annualized Volatility for Markets Covered in the Model

(b) Comparison between Typical Euro and Sterling Spread Volatilities Computed Using the Swap Factor for Different Rating Categories

dicts reasonably accurately both the absolute magnitude of the spread risk in each market and their relative values.

Credit Spread Factors The euro and sterling markets are broad and liquid markets. Accurately modeling spread risk in these two markets requires market-dependent, “credit blocks.”



EXHIBIT 7

27

Example of Sector and Rating Breakdown in the Euro Marketa Euro

Sectors

Ratings

Agency Financial Foreign sovereign Energy Industrial Pfandbrief Supranational Telecom Utility

AAA AA A BBB

BB B CCC a A nondomestic sovereign bond is exposed to the factor corresponding to its sector and rating. Due to the limited number of high-yield bonds outstanding, noninvestment grade factors are only broken down by ratings.

Various considerations drive the choice of spread factors. Factors built on little data can end up capturing a large amount of idiosyncratic risk and be representative of a few issuers rather than the market. A corollary is that it is often wiser to avoid building separate factors for thin industries. Spread factors should be meaningful for the investor, and somehow be related to the process of constructing a portfolio. An obvious and natural approach is to capture fluctuations in the average spread of bonds with the same sector and rating. As the size of the high-yield European bond market is still modest, there is unfortunately not enough data to construct sector-by-rating factors for speculative ratings. The simplest alternative is then to construct rating-based factors. A typical sector and rating breakdown for the euro market is given in Exhibit 7. Note that using market-adjusted ratings as opposed to conventional agency ratings can increase the explanatory power of sector-by-rating spread factors. The idea is to adjust the rating of bonds with a spread that is not too different from the average spread observed within their rating category. For instance, a AA-rated euro-denominated bond with a spread equal to 200 basis point would be reclassified into a BBB-rated bond.7 Credit spreads are computed with respect to the local swap curve to accommodate for the swap spread factor. 7

See Ludovic Breger, Lisa Goldberg, and Oren Cheyette, “OAS Implied Ratings,” Barra Research Insight, 2002.


28

EXHIBIT 8


Euro and Sterling Spread Factor Volatilities as of November 30, 2002

Note that arbitrage considerations imply that the spread risk of issues from the same obligor should be independent of the market. Why then do we need two sets of credit factors? After all, a model with only one set would be more parsimonious. Empirical evidence simply shows that spread risk is indeed currency-dependent, at least for higher credit quality issuers.8 Volatilities for selected factors are displayed in Exhibit 8. Spread risk in the euro and sterling markets is, on average, comparable. Looking now in more details, sterling factors tend to be more volatile than 8

Alec Kercheval, Lisa Goldberg, and Ludovic Breger, “Modeling Credit Risk: Currency Dependence in Global Credit Markets,” Journal of Portfolio Management, Winter 2003, pp. 90–100.



29

euro factors for AAA, AA, and A ratings, and less volatile for lower ratings. This is a trend already seen in Exhibit 6 that confirms that the swap factor would be a simpler but meaningful alternative. Significant differences exist for individual factors that illustrate the need for currency-dependent factors (see for instance the Telecom A and BBB factors). Also note how the high volatilities of the Energy, Utility, and especially Telecom factors reflect the recent problems in these industries. Each corporate bond will only be exposed to one of these factors, with an exposure that will typically increase with the bond’s maturity. A rule of thumb is that it will be comparable to the bond’s exposure to the shift factor. The spread risk of almost all AAA, AA, and A rated bonds will be less than their interest rate risk, and it is only for BBB rated bonds and in some very specific market sectors such as Energy and Telecoms that spread risk starts exceeding benchmark risk. Spread risk is by far the dominant source of systematic risk for high-yield instruments.

Emerging Markets Spread Factors Emerging debt can be issued either in the local currency or in any other external currencies (i.e., Mexico issuing in euro or sterling). These two types of debt do not carry the same risk,9 and need to be modeled independently. “Internal” risk was discussed in the interest risk section and we will now address external risk. A rather natural approach is to expose emerging market bonds to a spread factor. The sovereign spread factor turns out to be a poor candidate as the risk of emerging market debt strongly depends on the country of issue. Exhibit 9 shows average Argentinean monthly spread returns from June 30, 1999 to June 30, 2002 for U.S. dollar-denominated debt. The collapse of the peso, the illiquidity of the financial system, and the brutal decline in the economic activity are all reflected in Argentinean returns. Chilean spreads remained virtually unaffected despite a strong economic link between the two countries. As a result, any accurate model will need at least one factor per country of issue. The amount of data available for building emerging market spread factors is unfortunately scarce. First, there are often at best only a few bonds issued by sovereign issuers in emerging markets. The second problem is that there are mostly U.S. dollar-denominated. Even when some bonds denominated in, say, euro are available, there is generally little returns history. In some cases, we will seek risk forecasts for an 9

External debt is more risky than internal debt. If needed, sovereign governments can generally raise taxes or print money to service their internal debt. A shortage of external currencies can be more dramatic. This can be seen in the credit ratings delivered by agencies.


30

EXHIBIT 9


Examples of Spread Returns for Two Emerging Markets

issuer with no history of issuance in a specific currency. Since the risk of an emerging market bond is directly related to the credit worthiness of the sovereign issuer, which is independent of the currency of denomination, we can actually borrow from the history of U.S. dollar-denominated emerging market returns to forecast spread volatilities in other currencies. Spread return data can be obtained from an index such as JP Morgan Emerging Markets Bond Index Global (EMBIG). Strictly speaking, these factors are applicable only to sovereign and sovereign agency issuers, based on the inclusion criteria for, say, EMBIG if we happen to use this particular index to estimate emerging markets spread factors. However, many issuers of external debt domiciled in these markets carry a risk that is comparable to the corresponding sovereign issuers, so that it is reasonable to use the sovereign factor as a proxy for corporate issuers. Emerging market spread volatilities are shown in Exhibit 10. The spread risk of Latin American obligors tend to be above average, currently largest for Argentinean and Brazilian markets that have a spread risk comparable to a B to CCC-rated euro corporate. The risk of Asian issuers is on the other hand below average and comparable to the interest rate observed in developed markets. We clearly observe a wide spectrum of risk characteristics that confirms the need to build a separate factor for each market.


31


EXHIBIT 10

Emerging Market Spread Volatilities as of November 30, 2002

SPECIFIC RISK Specific returns are residual returns not explained by common factors. Common factors returns are typically larger than specific returns for higher quality investment-grade instruments; this is no longer the case in the lower portion of the investment grade segment and for high-yield instruments. One option is to use a CreditMetrics-like model based on transition probabilities reported by rating agencies. The model assumes that specific return variance of any bond can be written as: 2

σ spec =

∑ pi → j [ D ( sj – si ) – rm ] j

2

+ pi → d ( 1 – R – rm )

2

(6)


32


with pi → j D si R

= = = =

rm

=

the one-month probability of transitioning from rating i to j the bond’s spread duration the average spread level observed amongst bonds with rating i the recovery rate

∑ pi → j D ( si – sj )

+ pi → j ( 1 – R )

j

= the average expected return Transition probabilities are a crucial ingredient to this formula, and more generally, to any credit portfolio model based on ratings. Although agencies such as Standard and Poor’s do report European-specific rates, they are based on a small number of credit events, particularly for low quality ratings, yielding poorly constrained values. Global transition rates are statistically more robust because they are derived from a dataset that covers far more obligors and a longer time period. The model uses average spread levels observed within each rating category. Since these levels are market-dependent, so is specific risk. Another consequence is that this approach can only be implemented in highly liquid markets, where there are enough bonds to robustly estimate average spread levels—in practice, markets for which we can construct sector-by-rating credit factors. In markets where there is not enough data to construct a detailed model, a simple solution is to write the specific risk forecast as: σ spec = ( a + b ⋅ s )D

(7)

where s = the bond’s spread D = the bond’s duration The two constants a and b are fitted in each market using observed residual returns. Typical values for Swiss francs-denominated bonds are on the order of 5×10–4 bp/yr and 5×10–6 bp/yr, respectively, if spreads are expressed in basis points.

CURRENCY RISK Currency risk is potentially a large source of risk for global investors that can be handled with a multifactor model with one factor per currency. Yet, special attention has to be paid in forecasting the variances


33


and covariances of currency factors due to their high volatility and rapidly changing risk characteristics. The goal is to obtain a model that quickly adjusts to new risk regimes and responds to new data. Various forms of General Auto-Regressive Conditional Heteroskedastic (GARCH) models have been used to estimate return volatility. Such models express current volatility as a function of previous returns and forecasts. For instance, the GARCH(1,1) model takes the form: 2

2

2

2

2

2

σt = ω + β ( σt – 1 – ω ) + γ ( rt – 1 – ω )

(8)

where 2

= the conditional variance forecast at time t

2

= = = =

σt

ω β γ rt – 1

the the the the

unconditional variance forecast persistence sensitivity to new events observed return from t – 1 to t

The constants β and γ must be positive to insure a positive variance, even if large events occur. For the same reason, the condition β + γ < 1 must hold. The higher the sensitivity, the more responsive the model is. The weight given to past forecasts increase with the persistence constant. Using daily exchange rates as opposed to weekly or monthly exchange rates insures the convergence of GARCH parameters and minimizes standard errors.10 The aggregation formula for monthly GARCH forecasts is n

– (β + γ ) σ t, n = nω + 1 ----------------------------- ( σ t – ω ) 1 – (β + γ ) 2

2

2

2

(9)

where n is the number of business days in a month, typically 20 or 21. Exhibit 11 shows U.S. dollar versus euro returns from 1994 to 2000. Note how volatility forecasts (gray lines) quickly adjust to periods of small or large returns. The overall currency risk is large compared to interest rate risk. Consider a German government bond with a duration equal to five years. A U.S. investor holding this asset is facing an additional currency risk of at least 8% per year in addition to an interest risk of roughly 70 bp × 5 = 3.5%.11 The volatilities of several European currencies are plotted in Exhibit 12, and typically range from roughly 6.5% to 10% per year. 10 This is because daily exchange rate returns represent a much larger dataset than weekly and monthly returns. 11 70 basis points is the German shift volatility reported in Exhibit 4.


34


EXHIBIT 11

U.S. Dollar Against Euro Currency Returns and Volatility

EXHIBIT 12

Examples of European Currency Volatilities



35

GARCH volatilities can be combined with correlations computed independently, for instance from weekly returns, to produce the covariance matrix of currency factors. This approach has the advantage of combining accurate and highly responsive estimates of exchange rate volatilities with correlations computed over a longer time horizon and which are typically more robust.

PUTTING IT TOGETHER Common factors, returns, exposures, and a specific risk model: Everything is there except for one last critical ingredient: the covariance matrix. Building a sensible covariance matrix for more than a few factors is a complicated task that involves solving several problems.

Coping with Incomplete Return Series Factor return series often have different lengths, some series starting earlier than others. Return series can also have holes. As a result, what works well for two factors is here useless. That is, filling the factor covariance matrix row i and column j using the usual formula12 produces a nonpositive definite matrix. A statistical approach known as the EM algorithm is the conventional workaround. Details on the algorithm can be found in Dempster, Laird and Rubin,13 and for the purpose of this discussion, we only need to know that there exists a tool that can use incomplete series to produce an optimal estimate of the true covariance matrix.

Global Integration With a model that has on the order of 180 factors, we need to solve for over 16,000 covariances. Factor returns series include, in many cases, less than 30 to 40 periods. With such a small sample size compared to the number of factors, we have a severely under-determined problem and are virtually assured that the covariance forecasts will show a large degree of spurious linear dependence among the factors. One consequence is that it becomes possible to create portfolios with artificially low risk forecasts.14 The structure of these portfolios would be pecu12

Cov ( i, j ) =

( ri – ri ) ( rj – rj )

∑ -----------------------------------(N – 1)

13 A. P. Dempster, N.M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data using the EM algorithm,” Journal of the Royal Statistical Society, Series B Volume 39, 1977, pp. l–38. 14 For example, by use of an optimizer.


36


liar—for example, they might be overweight U.K. AA financials, apparently hedged by an underweight in euro industrial and telecom. Reducing the number of factors would compromise the accuracy of our risk analysis at the local level. However, we have seen for instance that the euro and sterling credit markets are to a large extent independent so that we do not need 34 × 16/2 = 272 covariances to describe the coupling between these two markets. Using our knowledge of the market in a more systematic fashion could go a long way in reducing the spurious correlations amongst factors. The structured approach presented in Stefek provides a solution to this problem.15 In this method, factor returns are decomposed into a global component and a purely local component, exactly as we already decomposed asset returns into systematic and nonsystematic returns. For instance, in the sterling market we can write f UK = X f

UK

⋅ g UK + ε f

UK

(10)

where fUK gUK XUK εUK

= = = =

the vector of factor returns for the sterling market the vector of global factor returns for the sterling market the exposure matrix of the local factors to the global factors the vector of residual factor returns not explained by global factors, or purely local returns

Local factors in each market include the shift, twist, butterfly, and spread factors. Currency and emerging market factors form two independent sets of local factors. The choice of global factors is based on econometric considerations. To a large degree, sterling credit factors behave independently of factors in other markets. As a result, we know a priori that we will gain very little by choosing more than one or two sterling global credit factors. Once the global factors are chosen, exposures are determined and also based on structural arguments. Equation (9) can then be easily extended to all the original factors in the model. Assuming now that purely local returns are uncorrelated across markets and uncorrelated with global returns, the covariance matrix can be written as T

F = XG X + Λ where 15

Dan Stefek, “The Barra Integrated Model,” Barra Research Insight, 2002.

(11)



37

G = the covariance matrix of global factors X = the exposure matrix of the local factor to the global factors Λ = the covariance matrix of local factors Global factors could typically include: ■ Shift, twist, and butterfly factors (including ipb) except in euro legacy ■ ■ ■ ■

markets Swap spread factors An average credit spread factor in the euro and sterling market An average emerging market spread Currency factors

Far less unknowns than before now separate us from the covariance matrix. For one, there are much fewer global factors than local factors (33 against 160 if we except currencies). The local covariance matrix Λ is also block diagonal, with only on the order of 10,000 nonzero entries. Unfortunately, we cannot stop there and use equation (11). The benefit of using global factors is that they help compute cross-market terms and constitute the skeleton of the matrix. The drawback is a loss of resolution at the local level. A solution to this problem is to replace local blocks by a local covariance matrix computed using the full set of original local factors. Off-diagonal blocks need to be adjusted in the process to insure that the final matrix is positive definite. A more detailed discussion of how the local covariance blocks are replaced can be found in Stefek.16 Local covariance blocks can be computed individually for each market, but also for emerging markets spread factors and currency factors. As a result, shorter half-life can be used for return series that are typically more volatile, such as currency and emerging market returns. At this point, we have a method for building a model that reconciles two conflicting goals, that is, provide a wide coverage of markets and securities while permitting an accurate and insightful analysis, particularly at the local level. In Exhibit 13, we compare correlations obtained using the standard and structured integration methods. As expected, running the EM algorithm on over 180 factors produces a large number of spurious correlations, notably between emerging market and euro spread factors. These artifacts disappear in the structured integration. Most of the cross-market coupling happens at the interest rate level, and yet, not for all markets. For instance, Czech and German Treasury yields vary in concert, but rather independently of Swiss yields. All other factors are clearly currency or market-dependent. 16

Stefek, “The Barra Integrated Model.”


38 EXHIBIT 13


Examples of Factor Correlations in the Standard and Structured

Integrations

THE MODEL IN ACTION European fixed income portfolios are now often managed against a broad index. The risk characteristics of an index like the Merrill Lynch EMU Corporate Large Cap are presented in Exhibit 14. This index tracks the performance of large investment grade corporate issues denominated in euro and is balanced to reflect the contributions of each market sector to the total outstanding corporate debt. Fluctuations in the euro exchange rate constitute the dominant source of risk, and would amount to about 8% per year for a U.S. based investor. This risk disappears for investors based in the euro zone. Local market risk originates for the most part from interest rate risk, with spread risk only



39

EXHIBIT 14 Merrill Lynch EMU Large Cap Risk Decomposition as of November 30, 2002 Risk Source

Exposure

Return Volatility (%/yr)

Interest rate Shift Twist Butterfly

4.1 0.35 –1.9

2.9 0.09 0.22

Total Spread Specific Currency Total

2.9 4.3 1

0.9 0.1 8.1 8.5

responsible for about 90 basis points per year. Specific risk is small because all assets in the index are rated BBB or above. Hedging currency and interest rate risk is relatively straightforward. This is not the case for spread risk so that understanding exposures to spread factors is a critical aspect of risk control. In Exhibit 15, we show a “risk map” of the Merrill Lynch EMU Corporate Large Cap index. Each row i column j entry represents the fraction of the total index return variance due to covariance between factor i and factor j. Credit factors are ordered by market sector and ratings. Lines indicate entries corresponding to negative covariances. This representation takes into account factor volatilities and correlations, the assets exposures to each factor as well as the index weight in each sector and rating category. Because sector-by-rating spreads are relative to swaps, all assets are exposed to the swap factor, yielding a large swap risk. This risk would be transferred to the other credit factors in a model where spreads are computed relative to Treasury. The negative covariances between corporate spread returns and both the shift and swap factors can be interpreted as follows. Inspection of the correlation matrix in Exhibit 13 shows that correlations between credit factors and the shift factor are negative with a magnitude that tends to be small for AAA and AA like factors but can reach large values for A and BBB like factors. Swap spreads move nearly independently of corporate spreads to government. As a result, swap spreads and corporate spreads to swap are negatively correlated.


40


EXHIBIT 15 Risk Map of the Euro Component of the Merrill Lynch Large Cap Index as of November 30, 2002

Most of the remaining credit risk originates from five clusters (C1 to C5 in Exhibit 15): ■ A large contribution from Financial securities, which represent over

65% of the portfolio (C1). ■ An unusually large contribution of Telecom assets, considering their

weight in the index (~7%), that is due to very high Telecom volatilities (C2). ■ A Pfandbrief “hotspot,” due to a high Pfandbrief portfolio weight of about 8% (C3). ■ Two clusters of high covariances created by high Financial/Pfandbrief, Financial/Telecom and Financial/Utility cross-sector correlations (C4, C5).


41


Such an analysis clearly identifies risk clusters and provides important clues on how to further diversify the portfolio.

COMPARISON WITH OTHER MARKETS In today’s asset management industry, organizations’ operations often extend beyond the European fixed income market. Controlling risk firm wide therefore calls for a detailed understanding of what the levels of risk in each market are and how markets interact with each other. The purpose of this last section is to provide a few elements of comparison between the euro and U.S. dollar fixed income markets. We compare in Exhibit 16 the volatilities of a few selected euro and U.S. dollar factors. The common denominator is that euro volatilities are less than their U.S. dollar counterparts. This is true for all factors if we ignore the volatility bursts sometimes observed over a few months for some factors (for instance the Industrial A factor in Exhibit 16). The average level of systematic risk observed amongst euro-denominated fixed income instruments is more generally low compared to other markets. Exhibit 16 shows one case where euro volatilities seem to be catching up with U.S. levels. A more systematic analysis of how euro volatilities have recently evolved since 2002 would show that this is an exception. On average, euro volatilities have remained low with respect to U.S. ones. Note that this is consistent first with the predictions of the swap factor model, euro spread levels and swap volatility being low compared to other markets. Examples of correlations between shift, twist, and butterfly factors in selected markets are given in Exhibit 16. Not surprisingly, changes in EXHIBIT 16

Selected Interest Rate Factor Correlations Germany Shift

Australia Canada Japan New Zealand South Africa United States

Twist

11/30/1999

11/30/2002

0.4 0.4 0.3 0.4 –0.04 0.6

0.65 0.65 0.1 0.5 0.25 0.8

11/30/1999 0.6 0.6 0.064 0.2 0.25 0.6

11/30/2002 0.5 0.5 0.02 0.15 0.08 0.5


42


European sovereign rates are generally comparable with those in other developed markets and especially the United States. In 2001 and 2002, correlations have been particularly high due to the global decrease in interest rates stemming from a general economic slowdown. Exceptions correspond to emerging markets such as South Africa, and markets where the economic cycle is at a different phase, such as Japan. Conversely, high-grade credit spreads show on average very little correlation across markets. High correlations do appear, but temporarily, and for market sectors that are globally experiencing distress such as Telecommunications companies.

SUMMARY In a complex European market, adequately measuring risk requires sophisticated methods and considerable care. A good risk model should provide a broad coverage without sacrificing accuracy, retain details but remain parsimonious, be responsive to ever-changing conditions, and so on. Certainly, there is no shortage of challenges. The typical euro investment grade corporate index is perhaps halfway between the conservative and speculative ends of the risk spectrum. We have seen that it has very specific credit risk characteristics, such as being heavily exposed to Financials and Telecommunications. European fixedincome instruments are on average less risky than their US dollar equivalent, which by no means implies that a sound risk management is less relevant. Building a reasonable risk model is fortunately not an elusive task as long as we know how to design or where to find the right tools.

APPENDIX—PRINCIPAL COMPONENT ANALYSIS Factor Shapes Changes in benchmark yields17 for different terms are highly correlated regardless of the market, which constitutes a strong incentive to step away from a key rate model in which the factors are rate changes at the term structure vertices. The principal component analysis consists in extracting a set of linear combinations of key rate changes that capture 17

For sovereign benchmarks, domestic government bond returns are used to compute term structures and key rate returns. For LIBOR/swap benchmarks, key rate returns can be computed directly from market yields.



43

most of the variations in a market’s benchmark. This is done mathematically by diagonalizing18 the key rate covariance matrix, each eigenvalue being a measure of how much of the benchmark variance is explained by the corresponding shape or eigenvector. The covariance matrices of principal components and key rates returns are such that T

C PC = Π C KR Π where the columns of matrix Π are the principal components, and the covariance matrix CPC is diagonal. In most markets, over 95% of changes in term structures can be captured with only three principal components usually called shift, twist, and butterfly, to reflect how term structures actually change. Interest rates tend to increase or increase simultaneously, which can be described as a shift of the term structure. The second most important effect is a twist that alters the slope of the term structure. The third factor is a butterfly that reflects a change in the term structure’s curvature.

Factor Returns Factor returns (hereafter called (STB) returns) are computed by regressing government bond’s returns or LIBOR/Swap key rate returns onto the shift, twist, and butterfly principal components. STB returns and other factor returns then go into the computation of the covariance matrix of all common factors. The shift, twist, and butterfly shapes are stable over time and only need to be reestimated periodically.

18

The diagonalization is always possible because the KR covariance matrix is symmetric.


3-Lloyd-Tracking Error Page 45 Wednesday, July 23, 2003 10:24 AM

Tracking Error William Lloyd Managing Director Barclays Capital Bharath Manium Associate Director Barclays Capital Mats Gustavsson Associate Barclays Capital

growing number of money managers in Europe have adopted a “beat the benchmark” approach to measure the performance of their fixed-income portfolios. This approach has been given a further boost by the rapid acceptance of the iBoxx indices by European investors. Increasingly, managers are compensated based on the performance of their funds relative to a benchmark, so straightforward risk measures are required to ensure they do not take on excessive risk. Many ways of measuring portfolio risk have been brought into the fixed-income market despite questions about their accuracy and suitability. We believe there is no one magical number that can capture the entire risk profile of the portfolio, and investors should not place too

A

We would like to specifically thank Ashok Varikooty and Naum Krochik for their contributions to the creation of this research. This article would not have been possible without their efforts.

45


46


much faith on measures that claim to do so. We would instead advocate the use of more in-depth and forward-looking analyses that lay the foundations of understanding portfolio risk. In this article, we examine several techniques used in the market for estimating relative risk in a portfolio, their assumptions, advantages and pitfalls. We also look at what we believe are the more accurate procedures for understanding and estimating this risk, and the tools that allow managers to do so quickly and easily.

TRACKING ERROR—THE FUNDAMENTALS Tracking error is the standard deviation of the difference in returns between a portfolio and a selected benchmark, which is usually a suitable bond index. Assuming a normal distribution of returns, a portfolio manager can expect to deviate by no more than the tracking error amount for 68% of the time during a selected period. Tracking error calculations have a relatively long history in the equity markets of measuring the relative risk of a portfolio against an index. The popularity of this methodology in equities has led many fixed-income managers to adopt the same approach, but we believe it is not as appropriate for the fixed-income markets. The following are some of the fundamental assumptions common to all tracking error calculations: 1. A static portfolio and index: We know that indices are not static and composition changes (especially for the euro market) can lead to significant changes in duration and other factors. In the following section, we illustrate how index characteristics in various markets have been far from static in the past, and we believe they will continue to vary. 2. Rely on historical data: As the standard disclaimer states, we know that past performance does not guarantee future results. While history can be a useful guide, the importance of historical data is often overemphasised. For example, the correlations on which these models depend tend to break down during periods of high volatility. Historical data largely ignore the current economic and political realities and can exclude shocks that have not occurred during the observed period, but are very likely to occur at some point (e.g., convergence of interest rates prior to the introduction of the euro). 3. A normal distribution of returns: This point is more amenable to mathematical testing, and so we have included it in the appendix. Most common tracking error models assume a multivariate normal distribu-


Tracking Error

47

tion of returns.1 The long and short of all the fancy footwork is that bond returns tend not to be normally distributed.

CHANGING MARKETS The belief that markets will change in structure and composition over time is almost axiomatic among market participants. These changes are often more pronounced in fixed income land than in equities, primarily due to the fact that the number of securities in fixed income is much larger, and the churn associated with new issues and bonds maturing is significant.

Duration Unlike bonds, equities do not mature. Equity portfolios and their benchmarks have the same expected duration (i.e., that of a perpetual security) at the end of a certain period as they did at the beginning. The duration of a fixed-income portfolio comes down over time through a process known as duration drift, which does not occur in equities. Duration drift is an important consideration when looking at a portfolio over a one-year horizon given that returns are predominantly duration-driven in fixed income markets. For example, on January 31, 2002, the average modified duration of the iBoxx Euro Index was 4.81. Assuming there was no change in interest rates, its duration (holding the constituents constant) would have been 4.41 in 12 months. In reality, the constituents of the index do change, and they change every month. In fact, the average modified duration of the index was 4.95 on January 31, 2003. So it is unrealistic to assume that the duration of a portfolio and its benchmark index will change at the same rate. Exhibits 1 and 2 show how the iBoxx indices have evolved during 2002. Notice the rebalancing spikes at the turn of each month, caused by bonds falling under a year to maturity, the addition of new, longerdated issues and the removal of cash.

HISTORICAL VIEW OF DURATION Exhibit 3 shows the average duration of bonds in the sterling market from December 1990 to January 2003, we find that it has increased 1

In this article, further references to normal distribution should be interpreted as multivariate normal distribution unless otherwise stated.


48


from 4.8 to 8.0. This increase represents more than a 60% extension of duration. Average life increased from 9.1 to 12.7 over the same period. The main driver of the increase in duration until 1998 was the falling interest rate environment. During this period the yield on the index fell from 11.4% to 6.6%. This fall in interest rates resulted in an increase in supply of longer-dated nongilts (see Exhibit 4). During this period the yield on the longest gilt fell from 10.1% to 4.4%. EXHIBIT 1

iBoxx Euro Index—Modified Duration January 31, 2002 to January 31, 2003

Source: Barclays Capital, iBoxx.

EXHIBIT 2

iBoxx GBP Index—Modified Duration January 31, 2002 to January 31, 2003

Source: Barclays Capital.


49

Tracking Error

EXHIBIT 3

Barclays Sterling Bond Index Duration


EXHIBIT 4

Market Value of Long-dated Sterling Issuance 1991–2002 (£ million)


Due to the youth of the euro credit market, an accurate comparison with the sterling market is difficult to make, but we can still observe an increased issuance of longer-dated bonds.

Sectors and Ratings As we have already discussed, the constituents of an index usually change every month, but the impact is not limited to duration. The sec-


50


tor weightings and credit ratings of an index can change dramatically as well. This is especially true in markets where corporate bond issuance is rapidly expanding, such as Europe. Given the number of issues in the broad indices, it is necessary to categorize them by rating, sector, and maturity. In 2002 credit quality deteriorated, reflected in the increased share of lower-rated bonds in the indices (see Exhibits 5 and 6). In particular, we note that the proportion of triple-B rated bonds in the iBoxx Euro Index has risen from 2.8% to 3.3% by duration contribution. This change in itself would have made a tracking error calculated at the beginning of the period a poor predictor of relative performance throughout the year. Even the more developed sterling market can be subject to substantial movements, and indeed we see the same pattern as in the euro market, albeit a less dramatic increase in triple-B rated credit. EXHIBIT 5

iBoxx Euro Index from January 31, 2002 to January 31, 2003 MV (%)

iBoxx Euro index Sovs SubSovs Collateralized Corp Total Corp triple-B

Duration Contribution (%)

1/31/02

1/31/03

1/31/02

1/31/03

72.2 6.6 10.8 10.4 100.0 3.7

69.1 7.9 10.9 12.1 100.0 4.9

75.1 6.3 9.2 9.3 100.0 2.8

74.5 7.0 8.6 9.9 100.0 3.3


EXHIBIT 6

iBoxx GBP Index from January 31, 2002 to January 31, 2003 MV (%)

Duration Contribution (%)

iBoxx GBP index

1/31/02

1/31/03

1/31/02

1/31/03

Gilts Sovs and subsovs Collateralized Corp Total Corp triple-B

48.0 19.5 4.5 28.0 100.0 7.2

46.5 18.3 5.4 29.9 100.0 7.4

46.9 20.4 6.0 26.7 100.0 6.1

44.7 19.7 6.9 28.7 100.0 6.1



Tracking Error

51

The data support the argument that sector and rating characteristics of markets and indices change significantly over time. This could lead to larger actual performance differences between a portfolio and an index than would be expected from a tracking error estimate at the beginning of that period.

DIFFERENT WAYS TO LOOK AT TRACKING ERROR An increasing number of institutional investors are measuring their fixed-income managers’ performance relative to major indices. At the same time, these institutions want to be sure that their managers are not taking excessive risks that could ultimately be hazardous to their clients and shareholders. Consequently, many managers are now required to regularly report a tracking error figure. However, there is still some uncertainty over the best way to calculate tracking error. There are three commonly used methods for calculating tracking error: (1) variance-covariance model, (2) historical simulation value at risk, (3) Monte Carlo simulations. We discuss each model in this article. Before we proceed, a distinction should be made between ex post and ex ante tracking error. Ex ante tracking error is the expected difference over a given future time period, whereas ex post tracking error is the actual difference measured between the portfolio and benchmark, or in other words, the divergence that actually occurred. The absolute value of the ex post tracking error should be less than or equal to the ex ante tracking error approximately 68% of the time.

THE VARIANCE-COVARIANCE MODEL The variance-covariance model approach extracts volatility information from historical returns and builds a model intended to predict divergence in performance. For this type of model, the underlying data are typically a time series of yields, spreads or returns. The model relies heavily on historical data and assumes both stable correlations and a normal distribution of returns.

Full Covariance Method The first and most straightforward way to calculate tracking error is by using the full covariance model. This method depends heavily on past data for every single instrument in the index and portfolio. Using matrix algebra, the covariance of daily total returns between every pair of


52


assets in the index can be calculated, as well as the variance of total returns for every single asset in the index. A variance-covariance matrix of the bond returns is then constructed based on these calculations. The variance-covariance matrix is then multiplied by the exposures’ vector, as shown in the equation below: 2

T

σ = Φ ⋅V⋅Φ

(1)

σ = tracking error Φ = exposures vector (ΦT= transposed exposures vector) V = variance-covariance matrix The exposure vector is the difference in the weight vector of the portfolio and the index. The weight vector is simply the percentage weight of each instrument in the index or the portfolio. Naturally, the sum of the weights in each vector equals one and the sum of the exposure vector is zero. The tracking error can then be calculated by multiplying the transposed exposures vector by the variance-covariance matrix, and then by the exposures vector, as shown in equation (1). As an example with Φ

T

= p1 p2 p3

v11 v12 v13 V = v21 v22 v23 v31 v32 v33 σ is calculated as v11 v12 v13 p 1 σ = Φ ⋅ V ⋅ Φ = p1 p2 p3 v21 v22 v23 p 2 v31 v32 v33 p 3 2

T

Building and Testing the Model In the following example, we illustrate how a full covariance model (FCM) can be constructed, and its performance measured during a oneyear verification period using corporate bonds. We selected a universe of corporate bonds with maturities (on May 31, 2001) between two and six years from the iBoxx Euro Corporate Non-financial Index. We chose a cut-off at two years to ensure the bonds


Tracking Error

53

would still be in the index at the end of the one-year test period. This produced 97 nonfinancial corporate bonds, which we aggregated into their own index. By using a random number generator to assign varying bond weights, we then constructed 1,500 different test portfolios from these 97 instruments, thereby creating a large number of different exposure vectors. We then calculated the ex ante tracking errors for each portfolio as at May 31, 2002. We verified the accuracy of the predicted tracking error by comparing ex post daily tracking errors with ex ante daily tracking errors for the months of June and July 2002. The ex post exceeded the ex ante tracking error for 39.8% of the time, which is significantly higher than the expected 32% for a normal distribution. The results of these comparisons indicate that the method is not accurate in this example (remember we expect the deviation to exceed the tracking error no more than 100% – 68% = 32% of the time), and this technique is often criticized in practice. Construction of the variance-covariance matrix can be extremely burdensome and requires large amounts of data. A covariance matrix for n instruments must have n × n elements. For an index with 1,000 instruments, the corresponding variance-covariance matrix would contain 1,000,000 cells. This would make it less suitable for an automatic portfolio optimizer as the computational difficulty of inverting such a large matrix is great and prone to numerical errors. Furthermore, if a security is issued during the time period for which the matrix is constructed, then it may be difficult to calculate accurate covariances for that security. One possible solution would be to use the total returns of a proxy bond in terms of sector, maturity and rating on the trading days for which there is no available data for the newer bond.

Multifactor Model Another form of the variance-covariance model incorporates a multifactor approach. Instead of looking at covariances between individual bonds the multifactor model aggregates information into common factors. These so-called principal factors can be obtained either by regression or by a market participant using qualitative modelling. In our model, we use the latter approach. The multifactor model (MFM) stipulates that a portfolio’s return is a function of a number of factors, and its exposures to those factors. The factors are designed to extrapolate information regarding past movements of the yield curve and selected spread factors relating to rating bands and sector classifications. Effectively, the factors are grouped


54


into those related to yield curve movements or credit spreads, and allow the portfolio manager to distinguish between the tracking error generated by each factor (though it is not possible to add these two figures together because the factors may be correlated). While this also uses a variance-covariance matrix much like the full covariance method, the actual matrix is much more condensed. As an example, the matrix used in a 20-factor model would have a size of (20 × 20) 400 cells, which is moderate compared with the one-million-cell matrix mentioned previously for the full variance-covariance model. The advantages of using a multifactor model are that it easily allows for mapping a new issue into past data for similar bonds by looking at its descriptive characteristics, and it can be inverted for use in a portfolio optimizer without too much effort. The multifactor model is also more tolerant to pricing errors in individual securities since prices are averaged within each factor bucket. An “m”-factor returns model takes the following form: R i = β i1 F 1 + β i2 F 2 + … + β im F m + ε i

(2)

where Ri βim Fm εi

= = = =

return on security i sensitivity of security i to factor m value of factor m residual error of security i

The returns model stipulates that return is function of m-factors. Splitting up return into several factors allows us to approach the model in the following way: m

2

σp =

m

∑∑

k = 1l = 1

n

2

α k α l σ kl +

∑ εi wip 2

2

i=1

where 2

σp

= variance of portfolio p n

αk 2

=

∑ βik wi = portfolio p’s sensitivity to factor k

i=1

σ kl = covariance between factors k and l εi = residual error of security i wip = weight of security i in portfolio p

(3)


55

Tracking Error

βik n m

= sensitivity of security i to factor k = number of securities in the portfolio = number of factors in the model

Using the MFM described in equation (3) to model, expected tracking error takes the following form: m

∑∑

σ TE =

n

m

k = 1l = 1

2 γ k γ l σ kl

+

∑ εi ( wip – wib ) 2

2

(4)

i=1

where σTE = tracking error of portfolio p with respect to benchmark b n

γk

=

∑

i=1 2 σ kl

εi wip wib βik n o m

= = = = = = = =

o

β ik w ip –

∑ βjk wjb = the net (portfolio – benchmark) sensitivity

j=0

to factor k

covariance between factors k and l residual error of security i weight of security i in portfolio p weight of security i in benchmark b sensitivity of security i to factor k number of securities in the portfolio number of securities in the benchmark number of factors in the model

An Illustrative Example: The Sterling Multifactor Model For the following example, we utilize the Barclays Capital Portfolio Analytics System XQA, which incorporates the aforementioned multifactor model. Again, this model incorporates factors that include points on the yield curve as well as factors related to credit spreads. We took the yield curve data in the sterling model from gilts and for the euro model from Bunds. The credit-spread factors consist of “buckets” by sector and rating, among other factors. Our sterling multifactor model consists of 32 factors reflecting changes in yield curve and credit spreads. We obtained historical monthly data on the above factors in order to create a variance-covariance matrix for the monthly changes in the various factors. We refer to this as the “Static VcV Matrix.” The exposures vector for the portfolio and index consist of the duration contributions of bonds that fall into the various buckets. For instance, if a portfolio contains a triple-A rated/Financial zero-coupon bond with a duration of 10, and the bond’s contribution to


56


the portfolio’s market value is 20%, the bond will be reflected in the exposure vector by placing 2.0 (20% × 10) into the 10-year interest rate bucket, and 2.0 into the triple-A rated/Financial bucket. However, for a bond that makes coupon payments, each individual cash flow must be placed into the appropriate interest rate bucket based on its own contribution to duration. The difference in the exposures vectors between the index and portfolio is used for calculating the tracking error. We created a random portfolio with a duration of 8.05 containing 50 corporate bonds for evaluating the sterling model, and selected the Barclays Non-Gilt All Maturities Index of duration 7.33 as a benchmark. We calculated the daily ex ante tracking error at the beginning of each month, and then compared it with the daily ex post tracking error for each day of that particular month, for 12 consecutive months until July 31, 2002. An analysis of the results indicates that the absolute value of the difference in price returns between the index and portfolio (ex post tracking error) exceeded the ex ante tracking error on 36.3% (91/251) of the trading days. Based on the aforementioned test, this method produced results relatively close to the expected 32% for this trial. We recalculated the exposure vectors at the beginning of each month to reflect any changes in the portfolio and index holdings. We ran identical tests for the same portfolios and index using a monthly updated variance-covariance matrix. We refer to this as the “Monthly Updated VcV Matrix.” This means that the matrix used for each month includes new data that did not exist at the beginning of the previous month. An analysis of the results indicates that the absolute value of the difference in price returns between the index and portfolio exceeded the tracking error on 36.7% (92/251) of the days. The effect of updating the covariance matrix on a monthly basis had little or no effect on the model’s accuracy in this particular case. However, looking at a longer time period, it is important to update the matrix regularly in order to get an adequate representation of changing markets. We ran the previous tests using simply weighted variance-covariance matrices. This means that each piece of data was equally weighted when constructing the matrix. For instance, we gave the one-year Gilt rate on January 31, 2001 the same consideration as the one on April 30, 2002. In order to compensate for recent changes in volatility of the various factors, it is possible to construct an exponentially weighted variance-covariance matrix. Under this technique, recent data is more heavily weighted than older data. Basically, weights are assigned to observations, based on their order of occurrence, using an exponential formula. Assuming the weighting factor to be used is 0.99, the calculation works as follows:


57

Tracking Error

Let N = the total number of observations. The nth observation is therefore assigned a weight of 0.99(N–n). The Nth (final) observation is assigned a weight of 1 (0.990). The sixth out of 10 observations would be assigned a weight of 0.994. Selecting a weighting factor can be somewhat arbitrary. A “forgetting factor” of 0.99 implies that an earlier value is 1% less important than its successor. The value chosen is usually between 0.96 and 0.99. When dealing with daily data, we would normally expect a higher weighting factor than when dealing with monthly data, which in turn would have a larger factor than quarterly data. This is because data that are closer in time by nature are more closely related. Once the weights are calculated, it is possible to construct a diagonal matrix, whereby the upper left-hand corner contains the weight that was assigned to the earliest observation, and the lower right-hand corner represents the weight assigned to the most recent observation. The simple covariance matrix is constructed using the following formula: T

Y ⋅ YV = -------------n–k where Y = value of each observation minus the average value for that particular column. Each column represents a different factor. (T = matrix transpose) n = number of observations k = number of factors (number of columns) An example of a covariance matrix containing three factors (in this case, yield curve movements for three different tenors): Tenor

1y

2y

3y

1y 2y 3y

0.049898774 0.040937125 0.039697223

0.040937125 0.046663569 0.046349494

0.039697223 0.046349494 0.052633651

Calculating the exponentially weighted covariance matrix is done by T

Y ⋅ W ⋅ YV = -----------------------------( n – k ) wi

∑ i


58


EXHIBIT 7

GBP Portfolio Tracking Error Performance (% Returns Exceed Daily TE)

Source: Barclays Capital. where

W = matrix containing weight factors on diagonal (as described above) wi = individual weight factors from matrix above We again calculated ex ante daily tracking error at the beginning of each month, and then compared it with ex post daily tracking error for each trading day of the same month. The results were better than those produced by the simply weighted covariance matrix—ex post tracking error exceeded ex ante for 34.7% of the days observed using exponential weighting (recall that the result for the unweighted covariance matrix was 36.7%). Therefore, exponentially weighting the matrix marginally improved the accuracy of the calculated tracking error. Exhibit 7 shows the percentage of times return exceeded the daily tracking error for different month ending using the Static VcV Matrix, Monthly Updated VcV Matrix, and Exponentially Weighted VcV Matrix.

Size of Maximum Deviations As previously mentioned, one of the assumptions made for calculating tracking error is that of normally distributed returns. However, return distributions in reality show “fat tails,” which means that there are more extreme (large positive or negative) events occurring than would be expected from a normal distribution. Furthermore, even though the occurrence of deviations larger than one standard deviation (i.e., the tracking error) is in line with predictions, the size of these deviations is


Tracking Error

59

EXHIBIT 8 Multifactor Model—Daily Returns versus ex ante Daily Tracking Error Corridor % (Sterling Model Monthly Updated VcV)


of great interest. If these over- or under-shoots are excessively large, then the portfolio’s performance is still subject to a large amount of risk. To illustrate this point, we plotted the daily price returns during a one-year period with the tracking error (see Exhibit 8). Notably, the tracking error that was updated monthly stays constant through onemonth periods. In the exhibit we see that tracking error is shown as a corridor bounded by the positive and negative tracking error values calculated monthly. The scatter plot shows that many data points lie far away (several tracking errors) from the predicted tracking errors. This highlights the fact that large deviations could occur even though your portfolio’s performance against the index is adhering to the prescribed tracking error limits. Therefore, in order to overcome this weakness of the tracking error model, practitioners turn to historical simulations, which we look at in the next section.

Advantages of the Variance-Covariance Method The advantages of using the variance-covariance method are: ■ The factor models allow for analysis of risk due to distinguishable fac-

tors (e.g., yield and spread curve movements). ■ New issues can be mapped into an existing framework using descrip-

tive characteristics. ■ The least computationally intensive of the three methods considered.


60


Disadvantages of the Variance-Covariance Method The disadvantages of using the variance-covariance method are: ■ ■ ■ ■

A normal distribution of returns has to be assumed. Future correlations are assumed to be equal to historical correlations. The size of deviations larger than the tracking error is not considered. The variance-covariance is not suitable for options.

HISTORICAL SIMULATION Historical simulation and value at risk (VaR) emerged in the 1980s as large derivatives houses sought an innovative method for measuring and controlling their risk positions. The need for change came about due to the increased complexity in their books during this decade of growth. These firms were looking for a measure that was accurate, and at the same time could be easily communicated within the organization. Therefore, by applying past market movements to their positions to simulate current risk, the value at risk methodology was born.

Definition of VaR The historical simulation approach uses the historical distribution of returns from the instruments in a portfolio to simulate the portfolio’s VaR. VaR is always defined for a certain probability α and time horizon h. Alternatively, we could refer to the 1 – α quantile (or confidence level) of the loss distribution. For instance, we could say that for a particular portfolio the one-day (time period) 5% (α) probability VaR is $100,000, or that the one-day 95% (1 – α) confidence level VaR is $100,000, which would mean that there is a 5% chance that the portfolio will lose $100,000 in one day. For calculating tracking error, historical simulations can be used by considering a position of being long the portfolio and short the index. Therefore, the difference in returns between the portfolio and the index is the variable for which the VaR is calculated. To get a tracking error measure using this methodology we have to calculate the 16% VaR as illustrated in Exhibit 9). It is worth noting that the VaR refers only to the potential loss whereas the tracking error refers to the absolute value deviation between portfolio and benchmark. Also note that in order to calculate this conversion into tracking error, the underlying assumption is that of a symmetric distribution of bond returns.


61

Tracking Error

EXHIBIT 9

Relationship Between Tracking Error and Symmetric Distribution VaR


Observation Period and Histogram The first step in trying to perform a historical simulation analysis is to select an observation period (e.g., 250 trading days). The observation period should be selected carefully; a longer period could be detrimental because it might include older and less appropriate information in the simulation whereas too short an observation period would limit the data. A good example of too long an observation period is the case of telecom bonds in Europe. In early 1999, most incumbent telecom operators in Europe were rated high double-A, and have steadily migrated downward to triple-B by 2002. The daily changes in the yield curve and credit spreads are reenacted during this 250-day period. The portfolio would then be revalued under each of these 250 scenarios, thereby producing a daily profit and loss for each scenario. This profit and loss figure can be easily converted to a daily percentage profit and loss. The same process is then used for the benchmark index. We can calculate the daily differences in percentage profit and losses between the portfolio and the index by subtracting one from the other. We can then create a histogram for the daily differences in profit and loss. The VaR can be determined by looking at the histogram (for instance, the value representing the second percentile would be the 2% VaR). In the example shown in Exhibit 10, 5% (the area to the left of the vertical line) of the expected variations would cause a daily loss of more than 0.04%. This is referred to as the “95% confidence level VaR.”


62


EXHIBIT 10

Daily Return for a Portfolio Holding versus a Benchmark (%)


When the x% probability VaR (portfolio versus index) is calculated for the desired confidence level, the portfolio’s daily return should not lag that of the index by that amount on more than x% of trading days. For instance, if the 5% daily VaR equals 0.04%, the portfolio’s daily return should lag the index’s daily return by 0.04% or more on approximately 5% of the future trading days.

Advantages of Historical Simulation The advantages of using the historical simulation method are: ■ Returns do not need to have any particular distribution (e.g., normally

■ ■ ■ ■ ■

distributed); it is a nonparametric approach so it is possible to use skewed distributions. However, to get a tracking error number that is symmetric, we need to assume a symmetric distribution. Higher confidence intervals can easily be calculated; the only limitation is the number of historical observations available. Easy to understand; the risk can be expressed as a loss amount or a loss percentage. Unlike the variance-covariance model, it is suitable for options and other derivatives. Can be used on a single universe with short/long positions as well as in the relative approach of being long a portfolio and short an index. Crisis periods can be used for incorporating known extreme events.


Tracking Error

63

Disadvantages of Historical Simulation The disadvantages of using the historical simulation method are: ■ Assumption that distribution of spreads and market correlations

remain the same in the future implies that volatility stays the same too (we address this issue in the following section). ■ Problem of data quality. Since tails are the primary focus of interest, and they are represented by very few samples, it might be difficult to get an accurate estimate of the loss figure. ■ Data that go back several years are of limited value. ■ Limitations in analyzing complex products such as baskets of credit default swaps as the availability of historical data is limited.

Further Developments of the VaR Methodology The standard VaR approach assumes that the volatility of the markets today is the same as it is during the period under consideration. Therefore, if the current market were more volatile than the previous one, this model would underestimate risk; conversely, if markets experienced a peak in volatility recently, the model would overestimate risk. There are several methods of weighting VaR calculations to rebalance the calculations.

Overweighting Current Values A simple solution to the problem of not reflecting current market conditions consists of trying to assign more weight to recent observations. This could be thought of as ordering the profit and loss series chronologically and then artificially increasing the frequency of the recent observations at the expense of the older values.

Volatility Update A more widely used approach first normalizes all historical returns with respect to the corresponding historical volatilities and then calculates return projections by multiplying the standardized returns by the current volatility. The advantage of using this method is that the original correlation structure is left intact while at the same time reflecting changing market conditions in the results. Disadvantages are that we have to assume linearity in the profit and loss values, which means it cannot be applied to portfolios containing options. To conclude, historical simulation with all its variants is another method that relies on past data to predict the future. It has problems coping with complex instruments, instruments with no history, and where the number of observations is limited. We look at a method that


64


uses numerous computer simulations to overcome this, the Monte Carlo simulation, in the next section.

MONTE CARLO SIMULATION Monte Carlo simulations are an alternative to parametric and historical approaches to risk measurements. They approximate the behavior of financial prices by using computer-generated simulations of price paths. The underlying idea is that bond prices are determined by factors that each have a specific distribution. As soon as these distributions (e.g., normal distributions) have been selected, a sequence of values for these factors can be generated. By using these values to calculate bond prices (and thus portfolio returns), the method creates a set of simulation outcomes that can be used for estimating value at risk. The approach is similar to the historical simulation method, except that it creates the hypothetical changes in prices by random draws from a stochastic process. It consists of simulating various outcomes of a state variable (or more than one in case of multifactor models), whose distribution has to be assumed, and pricing the portfolio with each of the results. A state variable is the factor underlying the price of the asset we want to estimate. It could be specified as a macroeconomic variable, the short-term interest rate or the stock price, depending on the economic problem. For fixed-income portfolios, this method theoretically represents an improvement because it takes into account all the factors affecting bond prices (and could also include pull-to-par) and it is a very powerful method for calculating VaR.

Calculation Framework The simulation involves the following steps: 1. For each state variable choose a stochastic process and corresponding parameters. 2. Generate a sequence of values e1, e2,…, en, from which prices are computed St+1, St+2,…, St+n In multifactor models, it is also important that the random variables generated have the desired2 correlation. 2 This is achieved through Cholesky factorization, which is a method to simulate multivariate normal returns, based on the assumption that the covariance matrix is symmetric and positive-definite. It is used to ensure the simulated series have a certain desired correlation.


65

Tracking Error

3. Calculate FT, the value of the asset (the portfolio) under this particular sequence of prices at the target horizon. 4. Repeat steps 2 and 3 as many times as necessary, say 10,000, to obtain a distribution of values, FT1,… FT10,000, from which the VaR can be generated. At the selected confidence level c, the VaR is the portfolio value exceeded in c times 10,000 replications. The number of iterations should reflect the usual trade-off between accuracy and computation cost. However, there are other techniques to increase the speed of convergence. There is always some error in the simulation estimate due to sample variability. As the number of replications increases, the estimate converges to the true value at a speed proportional to the square root of the number of replications. More replications bring about more precise estimates but take longer to estimate. In fast-moving markets, or with complex securities, speed may be more important than accuracy. If the underlying process is normal, the simulated distribution must converge to a normal distribution. In this situation, Monte Carlo analysis theoretically should yield exactly the same result as the multifactor variance-covariance method. The VaR estimated from the sample quantile must (not considering sampling variation) converge to the value of ασ, where α = quantile corresponding to the desired level of confidence (e.g., 1.96 for 95%) σ = standard deviation of the distribution

Examples of Simulation Models In order to illustrate the application of Monte Carlo simulation, we present two methods in detail below. The first considers price movements, and the second, which also handles pull-to-par, is a short-term interest rate model.

Model I: Small Price Movements Geometric Brownian motion is a commonly used model, which assumes that changes in asset prices are uncorrelated over time and that small movements in prices can be described by dS t = µ t S t dt + σ t S t dz where St

= the asset price


66


dSt = change in asset price dz = a random variable distributed normally with mean zero and variance dt; this variable drives the random shocks to the price and does not depend on past information µ = parameter representing the instantaneous drift at time t σ = parameter representing the instantaneous volatility at time t In practice, the process with an infinitesimally small increment dt is approximated by discrete moves of size ∆t. Integrating dS/S over a finite interval, we have approximately ∆S t = S t – 1 ( µ t ∆t + σ t ∆t ∆t ) where ε = a standard normal random variable (mean zero, unit variance). Using this linearization, it is possible to simulate the price path for S, starting from St and generating a sequence of ε to calculate St+1 , St+2 , …, St+n The Monte Carlo method, however, is prone to model risk. If the stochastic process chosen for the underlying variable is unrealistic, so will be the estimate of VaR. This is why the choice of the underlying model is particularly important. The geometric Brownian motion model described above adequately describes the behavior of some financial variables, but certainly not that of short-term fixed-income securities. In the Brownian motion, shocks on prices are never reversed. This does not represent the price process for default-free bonds, which must converge to their face value at expiration.

Model II: Dynamics of Interest Rates Another approach, which was used by Cox, Ingersoll, and Ross to model the term structure in a general equilibrium environment, consists of a model of the dynamics of interest rates.3 This process provides a simple description of the stochastic nature of interest rates that is consistent with the empirical observation that interest rates tend to be mean reverting. It is a one-factor model of interest rates, which is driven by movements in the short-term rates drt. In this model, movements in longerterm interest rates are perfectly correlated with movements in the shortterm rate through dz. dr t = κ ( θ – r t )dt + σ r t dz 3

John Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, March 1985, pp. 385–498.


Tracking Error

67

where κ < 1 defines the speed of mean reversion towards the long-run value q. Situations where the current interest rates are high, such as rt > θ, imply a negative drift κ(θ – rt) until the rates reverts to θ and vice versa for rt < θ. Also, r can never fall below 0 because while it decreases its variance σ r t also decreases, and as the limit of rt goes to zero, the variance goes to zero. The Monte Carlo experiment consists of first simulating movements in short-term interest rates, then using the simulated term structure to price the securities at the target rate. This interest rate process can be extended to a multicurrency environment, incorporating correlations across interest rates and exchange rates. For currencies, the drift can be based on short-term uncovered interest parity, which defines the expected return as the difference between the domestic and foreign interest rates. This creates a large system with interactions that provide realistic modelling of global fixedincome portfolios. For more precision, additional factors can be added. Longstaff and Schwartz extend the Cox-Ingersoll-Ross model to a twofactor model, using the short-term rate and its variance as variables.4

Advantages of Monte Carlo Simulation The advantages of the Monte Carlo simulation method are: ■ It overcomes the problems encountered when measuring the risk of a

portfolio comprised of instruments nonlinearly dependent on the underlying factors (e.g., baskets of credit default swaps). ■ It can be used for pricing some types of path-dependent instruments, such as barrier options or other exotic derivatives.

Disadvantages of Monte Carlo Simulation The disadvantages of the Monte Carlo simulation method are: ■ Because the method is based on the same kind of assumptions on the

behavior of financial prices made in the variance-covariance method, it is prone to model risk. If the stochastic process chosen for the underlying variables is unrealistic, the estimate of tracking error will be incorrect.

4

Francis Longstaff and Eduardo S. Schwartz, “Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model,” Journal of Finance, September 1992, pp. 1259–1282.


68


■ If corporate bonds were to be included in the analysis, a model for the

credit factors (sector, issuer, rating) would have to be developed to take into account credit risk and default probability. ■ It is computationally intensive, although there are methods to increase the speed of convergence to the solution.

SCENARIO ANALYSIS Fortunately for the investment community, there are alternatives to calculating tracking error that give an accurate idea of where a portfolio’s risks lie. These methods start with understanding the exposures of a portfolio relative to its benchmark, along several dimensions such as duration, term structure, rating, sector, and issuer. They then create interest rate and credit spread scenarios for different future time periods and perform a “what-if” analysis on the portfolio and the benchmark for these scenarios. These scenarios should encompass both expected and extreme conditions (best and worst case) in order to generate a return profile, both absolute and relative to the index, as well as to identify key thresholds. We have discussed the weaknesses of the tracking error calculations in previous sections, but there is an alternative way to estimate risk in a portfolio—scenario analysis. We explain the method in detail below. It is imperative to fully understand the following parts of the contribution to risk in terms of: (1) portfolio composition and (2) index composition. To better understand the future relative performance, we should carry out extensive stress tests on user-defined assumptions. A portfolio can be stress-tested by analyzing the performance under different scenarios comprising various interest rate and credit curve assumptions. A scenario analysis should be conducted using the following steps: 1. Start by identifying the composition of an index and portfolio in terms of sectors, duration, rating, etc. 2. Determine the time period over which a risk estimate is desired (e.g., one year, three months or instantaneous shocks). 3. Choose a select number of interest curve scenarios in terms of shifts and twists. 4. Take a credit view (widening or narrowing of credit spreads during the selected period). For this purpose views could be taken by sector, rating (e.g., A, triple-B) or even on issue level. 5. Run the scenarios and calculate horizon returns.


Tracking Error

69

EXHIBIT 11 XQA Scenario Analysis—Yield Curve Forecasting Choose any horizon yield


6. Interpret the results, change the bets, and by iteration come up with a portfolio that meets your requirements (or simply identify a range of performance figures for your portfolio against the benchmark under the chosen conditions). Scenario analysis is illustrated in Exhibits 11, 12, 13, and 14 using Barclays Capital Portfolio Analytics System XQA. The beauty of the scenario analysis method is that the fund manager has full control of the process in which the estimates are produced. The effect of changing the individual variables can be seen on relative performance. It requires more articulate views on possible scenarios than a tracking error number but the results are traceable (which leads to a more straightforward way of identifying the weak points and take appropriate countermeasures based on them).


70


EXHIBIT 12 XQA Scenario Analysis—Credit Spread Forecasting by Rating Change credit spread by rating or sector . . .


EXHIBIT 13 XQA Scenario Analysis—Credit Spread Forecasting by Issue and Sector . . . or by issuer to identify fallen angel risk



Tracking Error

71

EXHIBIT 14 XQA Scenario Analysis—Interpreting the Results Analyze results at the portfolio and the individual bond level


CONCLUSION After reviewing the various methodologies that can be used to calculate tracking error, it is apparent that such a measure is incomplete for assessing portfolio risk. The root causes of this interpretation lie in the underlying assumptions used across all the methodologies. Major banks have traditionally used historical simulations to assess firm-wide portfolio risk, and relied heavily on empirical data and historical distributions. Monte Carlo simulations, despite being computationally intensive, work well for instruments that have little or no history, and for more exotic derivative instruments. However, both methods suffer from the same assumption: whether you utilize a historical return distribution (historical simulation) or choose to model an arbitrary distribution (Monte Carlo), the dependence on such a distribution prevailing in the future can be dangerous, if not disastrous (WorldCom, Enron, need we go on?). The resolution lies


72


in analytics that make no such assumptions, and are flexible in modelling various scenarios or outcomes. We believe that scenario analysis offers such a solution by providing fund managers with a more comprehensive picture of the risk in their portfolios against a benchmark. Whereas tracking error provides just one number, scenario analysis generates several different return outcomes, including anything from recent trends to extreme price changes. It also allows fund managers to test their market forecasts, which is a necessity for any market participant.

APPENDIX: THE NORMALITY OF BOND RETURNS If we use the concept of tracking error—standard deviation of performance difference between your portfolio and a benchmark—and combine it with the assumption of normally distributed returns, we get the easy-tounderstand interpretation from elementary statistics: Tracking error predicts the maximum amount of deviation (positive or negative) from the benchmark we can expect with a certainty of approximately 68%. However, in making this statement, we have assumed that bond returns are normally distributed, when in reality they are often not. Bond returns, among other instruments, have “fat tails” (i.e., a higher proportion of large changes than the normal distribution predicts) and they are often skewed from the mean.

Statistical Test of Normality There are several tests that determine whether a population can be said to have a certain statistical distribution. In particular, the KolmogorovSmirnov test can be applied to test if a population is normally distributed. This test works as follows: 1. We plot the cumulative probability distribution of the observed population (here bond returns) with the cumulative density function of a normal distribution that has the same mean and standard deviation (see Exhibit A1). 2. We observe the vertical distances between the two curves, and determine the maximum values of these distances. This is called the D-statistic. 3. Depending on the size of the D-statistic, we can with varying degrees of certainty deduce if the observed population is in line with the reference distribution (the normal distribution). Using D-statistic, we are able to calculate a statistical value that tells us the level of significance (e.g., 1%) the empirical is equal to the normal distribution.


73

Tracking Error

EXHIBIT A1 Total Return Index for Nongilts versus Gilts 5–10 Year Maturity (%)


Application to Corporate Bond Returns In order to investigate whether we can consider corporate bond returns normally distributed, we focused on the sterling market due to its long series of quality data (relative to the euro market). We looked at the total return of nongilts for four different maturity classes: 1–5 years, 5– 10 years, 10–15 years, and more than 15 years. In our analysis, we used the complete time series for the Sterling Bond Index from December 1995 to December 2002 (equivalent to approximately 3,200 measurement points per variable). Days showing no change (weekends and holidays) have been stripped out, leaving a series of 1,800 points for each variable. We calculated (logarithmic) daily changes in the total return index as follows: ∆TRI D = ln ( δ NongiltD ) – ln ( δ NongiltD – 1 ) δ NongiltD – 1 = Total Return Index for Nongilts in day ( D – 1 ) As an example, we have included the daily difference in Total Return Index and the cumulative distribution for the 5- to 10-year maturity bucket in Exhibits A2 and A3. Results of the test show that we can reject the “null hypothesis” that says “there is no difference between the variables and a normal dis-


74


EXHIBIT A2

Daily Changes in Total Return Index for Nongilts versus Gilts 5–10 Year Maturity (log n today/yesterday)


EXHIBIT A3 Cumulative Probabilities for Nongilts versus Gilts 5–10 Year Maturity and the Normal Distribution (%)


tribution with a level of significance of 1% for this example.” In other words, we cannot with certainty say that any of the variables are normally distributed.


75

Tracking Error

Comment on the Result In order to better understand the results above, we calculated the skewness and kurtosis of the distribution (see Exhibit A4). Skewness shows how well a distribution is centred on its mean, so differing tail lengths give positive or negative skew numbers. For a perfectly centred distribution (such as the normal distribution), the skew is zero. All of the series in this example are moderately negatively skewed (i.e., they have a significant tail to the left of their mean). Furthermore, all are “leptokurtic,” meaning they have a shape that is steeper than the normal distribution (for the normal distribution the kurtosis is three). To illustrate this, Exhibit A5 shows the results for the 1- to 5-year maturity. From the shape of the sample distribution in the exhibit, we can see that the empirical distribution (i.e., the time series data) is steeper than a normal distribution of the same mean and standard distribution. EXHIBIT A4 Statistical Properties of Annual Yield Spread for Nongilts versus Gilts Maturity (Daily change in AYS)

Skewness

Kurtosis

1–5 5–10 10–15 15 +

–0.47 –0.32 –0.26 –0.07

4.83 4.80 4.93 5.80


EXHIBIT A5 Distribution Comparison for Maturity 1- to 5-year and a Normal Distribution



76


Implications for Tracking Error Measures As we have just demonstrated empirically, we cannot with any certainty say that sterling bond returns are normally distributed, and so the simple interpretation of the tracking error given at the beginning of this appendix is no longer valid. Therefore, this reinforces the view that tracking error is a questionable measure of bond portfolio risk.

4-Ilmanen_Fumagalli Page 77 Thursday, July 24, 2003 10:20 AM

Consistency of Carry Strategies in Europe Antti Ilmanen, Ph.D. Managing Director Citigroup Roberto Fumagalli Director Citigroup with the assistance of Rory Byrne, Heinz Gunasekera, Robert Minikin, Rafey Sayood, Miikka Tauren, and Etienne Varloot of Citigroup.

he advent of European Monetary Union (EMU) coincided with major changes in European investor behavior, including reduced homecountry bias and greater performance orientation. Yield-seeking and risk-taking strategies grew more popular—investors shifted money from governments to credits, extended duration across the curve, and raised their equity allocations. All these strategies were implemented to raise the expected return of the portfolio, but they also increased the risk of capital losses—which soon became painfully apparent. In this article, we focus on strategies in the European bond market that involve shifting money from government bonds to higher-yielding credits. Most carry strategies—overweighting high-yielding assets and underweighting low-yielding assets—seem to add value in the long run, but some strategies appear more risky than others. Specifically, we show that carry strategies are especially consistently profitable at short matu-

T

77


78


rities.1 Indeed, we argue that among various structural tilts that realmoney investors can make in their portfolios, replacing short-dated government debt with safe credits offers the best reward for risk. The superior performance consistency may appear surprising because credit spreads tend to be quite narrow at short maturities. Exhibit 1 suggests that this consistency follows from large break-even spread widening cushions at the front end of the curve, despite relatively narrow spreads. Only if yield spreads rise one-for-one with duration—a pattern some theoretical models imply but that we do not find in any market—would the break-even cushions be equal across maturities.2 The break-even cushions reflect the ex ante reward (spread) per duration while Sharpe ratios and information ratios measure the ex post reward per risk. Thus, it should not be surprising that wider break-even cushions at short maturities “translate” into superior Sharpe ratios and information ratios at short maturities. We find similar patterns in all markets we examine—we present results from European and U.S. swap-government spread markets and credit markets.3 However, we will discuss two caveats below: (1) the results are not as compelling for leveraged investors than for real-money investors because the former need to factor in funding spreads and (2) the consistency of outperformance is not as robust when investors go further down the credit curve than when they only shift from governments to highest-grade credits. The low risk and consistency of outperformance makes a defensive strategy attractive for “beginners” in the credit area—investors making their first reallocations of capital from governments to credits. Even experienced credit investors may find a systematic overweighting of credits at short durations an efficient core position, given the superior reward per spread duration at the front end. In contrast, investors should implement their active credit views (overweight a particular issue they are bullish on or overweight the whole credit sector when they expect general spread narrowing) typically with long-duration assets. 1

For an early study on the consistency of front-end credit outperformance in the United States, see Ray Iwanowski’s Application of Spread Immunization in the U.S. Corporate Bond Market (1995), Salomon Smith Barney research report. Our own empirical studies also show that front-end carry strategies can perform extremely well in cross-country trading (currency-hedged assets across countries). 2 In an equation, the break-even spread widening can be expressed as the ratio of annual yield spread divided by duration at horizon. (The duration at horizon times the relevant yield/spread change determines the size of capital gains or losses.) For horizons other than one year, the annual yield spread is multiplied by the fraction of year. 3 Moreover, our colleagues at Nikko Salomon Smith Barney in Tokyo find similar patterns in Japanese bond markets.


Consistency of Carry Strategies in Europe

EXHIBIT 1

79

Impact of Spread Curve Shape on the Break-Even Cushions (a) Stylized Shapes of Empirically Typical and Theoretical High-Grade Spread Curves

(b) Corresponding Break-Even Spread Widening Cushions Across Maturities


80


CONSISTENCY OF CARRY STRATEGIES IN EURO SWAP MARKET We start by examining the evidence in EMU markets, and show that swap-Bobl carry trades have been the most consistently profitable at short maturities. German Bundesobligationen (Bobls) are government bonds that until recently are issued regularly once a quarter (or more recently, twice a year) with roughly five-year maturity.4 When analyzing carry trades across the curve and over time, we should use homogeneous series with high data quality. These arguments guide us to use couponadjusted swap spreads5 (CAS) of Bobls. Although our empirical data involve swap-Bobl spreads, the idea can be applied to many similar trades: overweighting high-yielding euro governments (Italy, Greece), Jumbo Pfandbriefe and other swap-related credit products against lowyielding core governments (Germany, France). The investment implication for real-money managers is that they can improve their performance with very limited risk by employing front-end carry strategies.

Spread Curve in Mid-2001 and Historical Experience Exhibit 2a shows the curve of prevailing swap-Bobl spreads in mid-2001, together with their one-year break-even values. Forward spreads show how much CASs can widen before the swap-Bobl position suffers capital losses that just offset the annual yield advantage. For example, one arrow shows that today’s 2-year spread can double from 21 bp to 42 bp over the next year before the break-even point is reached. Recall the approximation that duration times yield change equals percentage price change, and note that at the end of the 12-month horizon today’s 2-year asset will have a 1-year maturity. In this case, a 21 bp yield advantage (is divided by 1-year duration and) translates to a 21 bp break-even spread widening cushion. Another arrow shows that if nothing happens to the spread curve, the spread actually narrows from 21 bp to 9 bp. This rolling-down-the-spread-curve feature adds 12 bp to the cushion for a constant-maturity (CM) spread, and the sum of the two components measures the CM break-even spread widening (third arrow). That is, the 4 We prefer to use bond-specific data rather than fitted curves because the former tend to have smaller measurement errors. Regular issuance pattern allows us to create quarterly constant-maturity series of swap spreads for Bobls. 5 Coupon-adjusted swap spread (CAS) is roughly speaking the spread between a government bond and matching swap rate (although we switch the sign here to make the spread positive). CAS is a better value measure than a simple yield spread because it adjusts for coupon and maturity discrepancies between the bond and the swap. More exactly, CAS is the difference between a bond’s market yield and its fitted yield if its cash flows are present-valued based on discount factors derived from the swap curve.



81

CM 1-year spread needs to widen by more than 33 bp over the next year before today’s 2-year swap-Bobl position underperforms.6 The gap between the two curves narrows with maturity, Exhibit 2b highlights this difference (top line), that is, the basis-point cushion against CM spread widening, plotted on horizon maturity. The other two lines show the “carry” and “roll” components across maturities. It may be counterintuitive that short-duration positions offer a larger spread widening cushion even though their spreads are narrower. The key intuition comes from the way duration translates yield/spread changes into capital gains/losses. For short-duration assets, a 10–20 bp spread widening year-on-year (YoY) will not easily cause capital losses that offset the initial spread advantage, whereas for long duration assets, a 10–20 bp widening easily causes annual losses. Say, a 50 bp yield spread for 10-year maturity assets (roughly 7-year duration) corresponds to only 7 bp (≈ 50/7) break-even spread widening cushion. EXHIBIT 2 Term Structure of Swap-Bobl Spreads and Break-Even Cushions (a) Current and One-Year Forward Swap-Bobl Spread Curve in Mid-2001

6

For funded investors, the yield spread should be adjusted for the funding spread (LIBOR-repo). As discussed below, we focus here on real-money investors and ignore funding spreads. Incidentally, we could express the rolling yield advantage of the swap spread position as a percentage return but instead focus here on the break-even market move needed to offset this advantage.


82


EXHIBIT 2 (Continued) (b) Total Spread Widening Cushion for Various Maturities, Split into Carry and Rolldown Components

Exhibit 3a shows the time series of 1–3-year CM swap-Bobl spreads between 1996 and 2001, and their forward paths over the subsequent year. Strikingly, the forward 1-year and 2-year spreads are well above historical ranges, while the 3-year spread is within it. Exhibit 3b shows that also on average the break-even spread widening cushion has been larger at the front end than at longer maturities. Meanwhile, front-end spreads have been more stable, as depicted by rising realized volatility across maturities. The combination of wide breakeven spreads and low volatilities strongly hints at superior reward to risk at short durations—a result we can soon confirm. Exhibit 4 addresses the outperformance consistency question by comparing break-even YoY spread widening with subsequent realized spread widening. Historically, the 1-year swap-Bobl spread never widened enough YoY to offset the break-even cushion, implying that a 2year swap-Bobl position never underperformed over annual evaluation periods.7 (We show below that over quarterly evaluation periods, it was 7

It may be confusing how 1-year and 2-year maturities are discussed interchangeably. The reason is that trades are put on for a 1-year horizon and end-of-horizon duration determines capital gains/losses. Thus, if a 2-year swap-Bobl reverse asset position is initiated today, the position will have 1-year maturity at the end of horizon, and the CM spread changes (actual or break-even) that are relevant are those of 1-year maturity.



83

unprofitable 25% of the time.) In contrast, panel B shows that the 3year spread widened more than the break-even cushion between August 1997–August 1998 and between February/May 1999–February/May 2000, giving rise to YoY underperformance. EXHIBIT 3 Wider Cushions and Less Volatility in the Front End (a) History and Forward Path of Constant-Maturity Swap-Bobl Spreads

(b) Average Spread Widening Cushion and Spread Volatility for Various Maturities, Mid-1996–Mid-2001


84


EXHIBIT 4

Comparing Break-Even YoY Spread Changes with Realized YoY Spread Changes (a) 1-Year Constant-Maturity Swap-Bobl Spread

(b) 3-Year Constant-Maturity Swap-Bobl Spread


85


EXHIBIT 5

Performance of Swap-Bobl Carry Strategies (with Quarterly Rebalancing) for Various Starting Maturities, Mid-1996–Mid-2001 Starting Maturity

2 yr

2.5 yr

3 yr

3.5 yr

4 yr

Annual Spread (bp) Annual Outperformance (bp) Volatility of Outperformance Information Ratioa Frequency of Qtrly Outperformanceb

18.1 24.1 18.5 1.30 0.70

20.8 27.5 26.6 1.03 0.60

20.9 20.4 36.7 0.56 0.65

21.2 24.1 42.9 0.56 0.60

24.0 41.2 53.6 0.77 0.70

a

Information ratio is a ratio of the average annualized outperformance (of swap position versus a Bobl benchmark) to its volatility. b “Frequency…” shows in how many quarters the swap position outperformed the Bobl benchmark.

Quantifying Carry Trades’ Past Outperformance To directly measure the performance of carry strategies, we create a constant-maturity time series of relative returns for the carry positions. For example, a 2-year swap-Bobl position is initiated; a year (quarter) later it is closed as a 1-year (1.75-year) maturity position, given annual (quarterly) rebalancing, and a new 2-year position is established. Realized outperformance of the swap over Bobl reflects both initial yield advantage and subsequent capital gains/losses due to spread changes. Exhibit 5 focuses on quarterly rebalanced trades (ignoring funding spreads). It shows that annualized ex ante spread and ex post outperformance increase slightly and irregularly with maturity, whereas volatility (tracking error) increases sharply with maturity. As a result, the information ratio—reward to volatility measure comparable to Sharpe ratio—is much higher at short maturities. The inverted term structure of information ratios is consistent with the term structure of break-even spread cushions in Exhibit 3b.8 Exhibit 6 shows the cumulative outperformance of carry strategies across maturities—for German Bobls and for U.S. Treasuries, as a comparison. Both graphs show that the (initially) 2-year maturity swap-government trade offers more consistent outperformance than longer maturity trades. 8

Incidentally, the tail of both term structures rises because of some Bobl-specific technical features. The swap-Bobl spread curve is especially steep near four years and quite flat near three years. Lack of rolldown gains (probably due to rolling into the deliverable basket of the 2-year futures) explains the relatively poor performance of the 3-year swap-Bobl trade, while the converse is true for four years. The spread curve rolldown is so pronounced near four years because Bobls gradually lose their newness (liquidity?) premium and repo specialness. The consequent cheapening tendency makes 4–5-year Bobls poor long-run investments (especially if one does not exploit their liquidity or repo advantages, as we do not) and makes it easy for the swap position to outperform.


86


EXHIBIT 6

Cumulative Outperformance of Swap-Government Carry Strategies Across (Starting) Maturities (a) German Data, Mid-1996–Mid-2001

(b) U.S. Data, Mid-1996–Mid-2001

Interpretations and Implications Ultimately the results in this and similar studies follow from the shape of the spread curve. Wider break-even spreads at shorter maturities in Exhibits 2b and 3b give the same shape to the term structure of information ratios in Exhibit 5.



87

What explains this pattern? As noted in Exhibit 1, as long as the spread curve does not increase one-for-one with duration, the breakeven widening cushion is higher at shorter durations.9 Spread curves for high-grade debt typically slope upward but rarely are so steep that the spread doubles with duration. Viewed from another angle, the flat spread curves reflect the positive intercept when plotting spread curves on duration. It follows that reward per duration in money market instruments is very high. Many theoretical credit spread models counterfactually imply a zero intercept—no spread at the very front end. One explanation is that for actual default exposure, nominal amount at risk is more relevant than spread duration exposure (in a default it hardly matters whether you own a 1-month asset or a 10-year asset). However, explaining the positive intercept by pure default risk (rather than spread duration risk) makes sense for speculative issuers but less so for the AAA/AA debt we discuss here. Market segmentation or illiquidity are more likely explanations for the existing front-end spreads. Positive TED spreads or LIBOR-GC spreads at very short maturities mean that our findings are much more relevant for unfunded real-money managers than for funded investors (banks, leveraged traders). After subtracting financing spreads from carry strategy profits, the riskadjusted return is much less compelling. This lower incentive for funded investors partly explains why this opportunity has not been arbitraged away. Thus, while financing spreads prevent leveraged arbitrage, their existence retains for real-money investors the opportunity to systematically outperform the benchmark with quite a high degree of consistency. So far, we have shown that systematic “passive” overweighting of front-end credit products has been consistently profitable (75% of quarterly evaluation periods and 100% of annual periods). We conclude by noting that an active carry-seeking strategy may provide even better results; say, putting on carry positions only when the break-even spread is above historical average.10 A simple data sorting exercise (not shown here) indicates that carry strength has been a useful guide to subsequent performance, at least at short maturities. Thus, using information about 9

This is just math: The break-even widening cushion is a ratio of the spread advantage over horizon duration. 10 Readers may find the following analogue useful. There is, on average, a reward for extending duration—thanks to a typically positive bond risk premium. However, it does not follow that the expected near-term reward for duration extension is always positive (for example, if the yield curve is inverted). In fact, duration extensions are profitable over long samples in just 51–55% of months. An active forecasting model based on curve shape and other predictors can predict monthly market-directional moves with a higher (about 60%) accuracy. Similarly, this study could be extended by developing an active forecasting model for carry trades.


88


the current carry environment, perhaps combined with other predictors (say, recent trend in market risk aversion indicators), might improve performance with accuracy that is sufficient to justify increased trading costs.

EXTENSION TO U.K. MARKET AND TO LEVERAGED TRADER’S PERSPECTIVE In this section, we extend the analysis to U.K. swap-gilt spreads. The triple whammy of global crises (and consequent increase in market risk aversion), large fiscal surpluses (both in the U.K. and in the U.S.), and strong institutional demand for long gilts (mainly due to the Minimum Funding Requirement) made the 1998–2000 period an extremely challenging environment for U.K. carry strategies. For example, the 10-year swap-gilt spread widened from 40 bp to 116 bp in the 24 months to May 2000, although the spread reverted to much narrower level one year hence (near 60 bp). Unlike the previous section, we present results for both real-money investors (excluding the impact of LIBOR-GC repo spread) and leveraged traders (including the impact of this funding spread). Real-money investors who just replace government bonds in their portfolio with higher-yielding alternatives enjoy a larger break-even cushion against spread widening than do leveraged traders. ■ For these real-money investors, swaps in our analysis are proxies for

credit bonds with a swap spread near zero. Alternatively, investors could place their money in short-term deposits and enter into a receivefixed, pay-floating interest rate swap. Since these investors do not fund their positions, we effectively use a zero funding spread in break-even and return calculations. ■ For leveraged traders, the natural carry trade is reverse asset swapping of gilts. The performance of these positions includes the handicap of paying LIBOR rate in the swap while receiving a lower repo rate for the short government position.

Spread Curve in Mid-2001 and Historical Experience When analyzing carry trades across the curve and over time, it is helpful to use homogeneous series with high data quality. In the U.K., gilt issuance has been so irregular and coupon differences are so large that we cannot use bond-specific swap spreads as useful raw material. Instead, we use the differences between our fitted sterling swap curve and government par curve. We analyze maturities up to ten years but ignore the ultra-long gilts that have been especially technical in recent years.



89

EXHIBIT 7 Term Sturcture of Swap-Gilt Spreads and Break-Even Cushions (a) Current and One-Year Forward Swap-Gilt Spread Curve in Mid-2001

(b) Total Spread Widening Cushion for Various Maturities, Split into Carry and Rolldown Components

Exhibit 7a shows the prevailing curve of swap-gilt spreads in mid2001, together with their one-year break-even values. The upward arrow shows that today’s 10-year spread can widen over the next year (as it also shortens into a 9-year spread) before the break-even point is reached. The downward arrow shows that if nothing happens to the


90


spread curve, the spread actually narrows over time (rolling down the spread curve). The break-even spread widening cushion for a constantmaturity spread includes both the carry and rolldown components. We compute break-even spread curves with and without including funding rate spreads. Real-money investors earn the full swap-gilt spread advantage over time, whereas leveraged traders “only” earn the excess of swap-gilt spread over the LIBOR-GC spread (currently, the 3-month LIBOR-GC spread is near 20 bp). The spread widening cushion is correspondingly slimmer for the leveraged trader. Exhibit 7b shows the total spread widening cushion curves for both real money and leveraged traders (top two lines). The latter line is also decomposed into the carry and rolldown components (bottom two lines). Both cushion curves are inverted but more steeply for the realmoney investor, consistent with our argument that particularly the realmoney investor can gain from front-end carry trades. Exhibit 8a shows that the swap-gilt curve has been upward-sloping during our sample, with wider spreads and steeper rolldown than in Continental Europe. However, spreads also have been more volatile than in the Continent, especially at longer maturities. In our 1996–2001 sample as a whole, short-dated spreads were broadly stable, while longer-dated spreads widened by about 30 bp. EXHIBIT 8

Typical Swap-Gilt Spread Curves Over Time (a) Average Swap-Gilt Spread Curve Shape, Mid-1996–Mid-2001



91

EXHIBIT 8 (Continued) (b) Average Spread Widening Cushion and Spread Volatility for Various Maturities, Mid-1996–Mid-2001

Exhibit 8b shows that also on average the break-even spread widening cushion has been larger at the front end than at longer maturities. At the same time, front-end spreads have been more stable, as depicted by rising realized volatility up to 3-year maturity. The combination of wide break-even spreads and low volatilities strongly hints at superior reward to risk at short durations. This message from ex ante indicators will soon be confirmed in our ex post analysis of realized returns.

Quantifying Carry Trades’ Past Outperformance To directly measure the performance of carry strategies, we create constant-maturity time series of relative returns for the carry positions. For example, a 2-year swap-gilt position is initiated; a quarter later it is closed as a 1.75-year maturity position, and a new 2-year position is established. Realized outperformance of the swap over gilt reflects both initial yield advantage and subsequent capital gains/losses due to spread changes. Exhibit 9 presents results separately for real-money investors and leveraged traders; the latter earn lower ex ante spread and ex post return due to the LIBOR-GC spread (28 bp on average). Annualized ex ante spread and ex post outperformance (for real money) increase with


92


maturity, but not as fast as volatility, resulting in much higher rewardto-volatility ratios11 at short maturities. The levels of average profits and reward-to-volatility ratios are lower for leveraged traders (as LIBOR-GC spread needs to be deducted) but both display an inverted term structure. The within-sample widening of long swap-gilt spreads exaggerates the poor performance of longer-maturity carry trades; yet, the superior reward-to-risk at short maturities would hold even over a more neutral sample period. Exhibit 10a shows the time series of 2-, 4-, and 8-year CM swap-gilt spreads between 1996 and 2001, and their forward paths over the subsequent year. Exhibit 10b highlights the smoothness of annual returns of shorter-dated carry strategies, compared to longer ones. EXHIBIT 9

Performance of Swap-Gilt Carry Strategies (with Quarterly Rebalancing) for Various Starting Maturities, Mid-1996–Mid-2001

Starting Maturity

2 yr

4 yr

6 yr

8 yr

10 yr

For Real-Money Investor (excl. LIBOR-GC) Annual Spread (bp) Annual Outperformance (bp) Volatility of Outperformance Information Ratioa Frequency of Qtrly Outperformanceb

37.6 49.7 52.9 57.0 61.9 53.1 54.2 44.6 40.5 38.2 43.0 124.9 166.1 207.1 229.1 1.23 0.43 0.27 0.20 0.17 0.70 0.55 0.50 0.45 0.50

For Leveraged Trader (incl. LIBOR-GC) Annual Spread Over LIBOR-GC (bp) Annual Average Profit (bp) Volatility of Profit Sharpe Ratioa Frequency of Qtrly Outperformanceb

10.2 22.3 25.5 29.6 34.5 25.2 26.3 16.7 12.7 10.4 42.0 123.9 165.1 206.1 228.2 0.60 0.21 0.10 0.06 0.05 0.60 0.50 0.45 0.40 0.40

a

Information/Sharpe ratio is a ratio of the average annualized outperformance/profit to its volatility. b “Frequency…” shows in how many quarters the swap position outperformed the government bond. 11

For real-money investors, information ratio measures the average outperformance versus a benchmark, divided by the volatility of outperformance (tracking error). For leveraged traders, the relevant yardstick is the absolute P/L of a self-funded position, instead of a benchmark portfolio’s return. Thus, the Sharpe ratio measures the average profit divided by its volatility. We express information ratios and Sharpe ratios in annualized form.



93

EXHIBIT 10 Consistent Profits at Short Maturities in the U.K. (a) History and Forward Path of Constant-Maturity Swap-Gilt Spreads

(b) Annual Outperformance of 2/4/8-Year Swaps Over Gilts Between 1996 and 2001


94


EXHIBIT 11 Cumulative Outperformance of Quarterly Swap-Gilt Carry Strategies Across Maturities, Mid-1996–Mid-2001 (a) Excluding LIBOR-GC Funding Spread

(b) Including the Funding Spread

As usual, cumulative profit lines are most informative. Exhibit 11 shows the cumulative outperformance of carry strategies for four maturities, without (panel A) and with the funding spread (panel B). Both



95

graphs show that the (initially) 2-year maturity swap-gilt trade offers more consistent outperformance than longer-maturity trades. The 8- and 10-year swap-gilt spread trades were much more risky, suffering serious losses between 1998 and 2000 (but recouped them in 2000–2001).

CONSISTENCY OF CORPORATE CARRY TRADES AT SHORT/LONG MATURITIES In the final section, we extend the analysis to actual corporate bonds. The above analysis used swaps as proxies for credit products. However, many investors are not allowed to use interest rate swaps but take their credit exposure using corporate bonds. In the U.S. market, we have the benefit of reasonably long data histories (here, January 1985 to February 2002). We use SSB BIG index subsector data to assess cumulative profits of carry trades across maturities (1–3-year and 7–10-year) and across the credit curve (AAA/AA rated Corporates versus Treasuries and BBB/A rated Corporates versus AAA/ AA rated Corporates). The lowest and most volatile line in Exhibit 12a shows that at long maturities, the Treasuries to high-grade corporates trade did not perform well. In contrast, the highest and smoothest line shows that at short maturities, the same trade performed extremely well. Going down the credit curve did not provide nearly as good results, presumably as actual default risk and/or spread volatility grew. Exhibit 12b indicates that the front-end credit carry trade that corresponds to the top line in Exhibit 12a gave much better reward-to-risk than any other static risk-bearing strategy (extending duration, buying equities versus Treasuries, long-end credit carry trade)—information ratio near one.

Euro Corporate Credit Spreads Since 1999 In the euro markets, we have much shorter data histories and smaller bond universes. Yet, broadly speaking, the available evidence looks very similar to that in the United States. Given the immature market, we face the following tradeoff: ■ If we use the whole credit section of SSSB EuroBIG index, the universe

is too heterogeneous.12 ■ If we use more homogeneous subindices, these often are too sparsely populated.

12 For example, the evolution of yield spreads over time may reflect changing sector composition of the index, while the spread differences across the curve could reflect different rating levels (say, if lower-rated issuers could only issue short-dated debt).


96


EXHIBIT 12 Front-End Carry Strategies in the U.S. (a) Cumulative Profits of Various Credit Bearing Strategies in the U.S.

(b) Ex Post Reward-to-Risk Ratios for Bearing Duration, Stock and Credit Risks, January 1985–February 2002

Notes: Information Ratios measure average excess returns over volatility for 7–10 year Treasury returns funded by 1-month general collateral repo, for equities over 7– 10 year Treasuries, for 7–10 year AAA/AA rated Corporates over duration-matched Treasuries and for 1–3 year AAA/AA rated Corporates over matched Treasuries.



97

We do the best we can with the available data. We classify bonds into relatively homogeneous maturity/rating/industry subsectors and then identify those that are reasonably well populated. First we exclude from the EuroBIG credit sector any bonds that were issued more than three years ago to reduce illiquidity and stale-price problems. We split bonds each month into a short-maturity universe of 1–5-year bonds and a long-maturity universe of 5–10-year bonds. Then we examine which rating/industry sectors have a sufficient number of issues to mitigate excessive sensitivity to single issuers. We conclude that with our two maturity subsectors, at most two rating/industry sectors are so well populated as to provide good diversification. Thus, our analysis focuses on AA rated financials (FIN AA) and A rated industrials/utilities (INDUT A).13 Finally, bonds are weighted so that the short-dated and long-dated universes have constant durations of 3 and 6, respectively.14 Exhibit 13a shows the spreads of these subsectors versus euro governments (SSSB’s relative value curve based on German, French, and Dutch bonds), telling the story of sharp spread widening in 2000 and a partial reversal in 2001. The last observation in each graph shows the break-even spread in six months starting from summer 2001. As noted before, the break-even spread widening cushion is larger for short-dated bonds than for long-dated bonds. Given the short duration, spreads would have to widen a lot to offset even a relatively slim carry advantage.15

Consistency of Outperformance As we did with swap-government spreads, we could examine the relation between the average break-even spread widening cushion and realized volatility of spread changes for corporate bonds. Their ratio, the ex ante reward-to-volatility ratio, is higher for the short-maturity bonds than for the long-maturity bonds, 0.7 versus 0.4 for FIN AAs and 0.7 versus 0.5 for INDUT As. 13

In European credit markets, rating and industry sector classifications are fortunately quite overlapping. The majority of financial issues in EGBI have a double-A rating, while the majority of industrials have a single-A rating. 14 Whichever universe we use, we only have a short data history since 1999, and in fact, we start the analysis only in mid-1999, when our subindices are deemed to be sufficiently diverse. In addition, the problems of unreliable index prices and of investors facing limited investment alternatives and nonnegligible trading costs may be especially pronounced among short-maturity bonds. 15 We take the perspective of a real-money investor and ignore funding spreads in our calculations. See above for a comparison of real-money and leveraged perspectives. Funding spreads make these carry strategies much less attractive for leveraged investors, which partly explains why these opportunities have not been “arbitraged” away.


98


EXHIBIT 13 Euro Corporate Carry Strategies (A) Euro Credit Subsector Spreads Over Bunds, June 1999 to August 2001 and Forward Path

(b) Outperformance of Credit Subsectors Over Duration-Matched German Bonds



99

When we move to realized (ex post) returns of carry trades in Exhibit 13b, we include the capital gains or losses from any spread widening trend that occurred within sample, as well as small gains from rolldown spread changes16 and small losses from the downgrading drift.17 Exhibit 13b shows that the outperformance of short-maturity credit bonds (both FIN AAs and INDUT As) over governments has been greater than that of long-maturity bonds—as evidenced by higher ending-level of cumulative excess returns. In addition, it has been much more consistent—as evidenced by smoother path of cumulative excess returns. Given higher excess returns and lower volatility, short-dated corporate bonds have much higher reward-to-volatility ratios18 than longer-dated bonds (see Exhibit 14). EXHIBIT 14

Euro Credit Outperformance Over Bunds, Mid-1999 to Mid-2001

16 As spread curves typically are upward-sloping for high-grade debt, if nothing happens to the spread curve during the horizon, simple passage of time will narrow a bond’s spread. 17 Rating downgrades are more common than upgrades in our sample. Thus, the actual spread of a given bond basket (say INDUT AA) tends to widen slightly more than the quoted spread for this subsector. The quoted spread change misses the effect that more bonds tend to be downgraded than upgraded within the month, and the associated slight net widening of the spread. In contrast, our index returns include the impact of any rating changes within the month. 18 Information ratio for a fund with a government benchmark, Sharpe ratio for a selffunded trade.


100


EXHIBIT 14 (Continued)

INTERPRETATIONS AND CONCLUDING REMARKS The spread-widening trend during our sample may overstate the superior return of front-end carry trades.19 In a spread-narrowing environment (and often in stable spread environment), we would expect longdated carry trades to provide higher returns. But there are good reasons to expect that the superior reward-to-risk for short-dated bonds holds in the long run. Recall that as long as the spread curve does not increase one-for-one with duration, the break-even widening cushion is higher at shorter durations. Spread curves for high-grade debt typically slope upward but rarely are so steep that the spread doubles with duration. In general we find that the ratio of ex ante spread over duration is higher for shorter-dated assets than longer-dated assets. This pattern might be justified by greater spread volatility at short durations. However, empirical spread volatilities are typically lower at shorter durations, at least for the highest credit ratings (not shown). This evidence is consistent with our tentative view that front-end carry trades offer the best risk-adjusted deal in highest-grade assets. 19

This sample may be especially negative for long-dated credit bonds as it includes the sharp spread widening trend in 2000. The long-term INDUT spreads widened overall by over 30 bp during the sample, contributing to almost 2% losses.



101

■ One explanation for the “excessively” wide spreads at short maturities

is that for actual default exposure, nominal amount at risk is more relevant than spread duration exposure. However, explaining the positive spread curve intercept by pure default risk (rather than spread duration risk) makes sense for speculative issuers but less so for the high-quality debt we discuss here. Historically, less than 0.3% of AAA/AA rated issues have fallen below investment-grade status, let alone defaulted, over a one-year horizon.20 Therefore, market segmentation or illiquidity are more likely explanations for the apparently excessive front-end spreads on AAA/AA rated debt. ■ The puzzle is smaller for lower-rated assets as their spread volatility can be so large that break-even cushions are more frequently broken— also recall U.S. evidence in Exhibit 12a. This has certainly been the case around the turn of the millennium. ■ So we argue that carry strategies tend to add value; that the consistency of carry strategies is better with short-dated bonds (the theme of this paper); and that the superior reward-to-risk for front-end carry trades is most pronounced for highest credit ratings. Among simple investment rules, we know few better than the idea of overweighting spread products at short maturities. At least, this rule has provided more consistent profits than other similar rules, such as “always overweight equities (or credit products or duration).” Of course, we have focused in this article on just one way to add value. Active credit investors should be able to do even better if they are successful in spread timing or in sector/security selection. We end with an important practical reservation: Many short-dated corporate bonds are not actively traded, and this lack of liquidity will inevitably limit available opportunities for front-end carry trades.21 Primary issuance is more concentrated on 5–10-year bonds, although growing investor demand for short-dated corporates may over time induce larger supply. Meanwhile, three alternative assets for carry trades are worth mentioning. Noncore euro governments and “quasigovernment” agencies issue large amounts of short-dated bonds and even their old 5-year bonds can be very liquid. Second, there is the huge 20 According to the S&P rating transition matrix (per notch data) estimated from 1970–1999, the probability of S&P rating falling below BBB– (the probability of defaulting) in one year is 0.03% (0.00%) for AAA, 0.21% (0.00%) for AA, 0.84% (0.05%) for A, and 4.86% (0.24%) for BBB. 21 For a given liquidity, bid-ask spreads in bp-terms may be somewhat wider at shorter maturities, but this effect appears small compared to shorter bonds’ performance advantage. The scarcity of available investment material at shorter maturities is a more acute problem for most investors.


102


euro floating-rate note market. Most importantly, investors can use interest rate swap market if short-dated bonds are too scarce or expensive. Indeed, receiving 2–4-year swaps as a part of this carry strategy may be a source of competitive advantage to those real-money investors who are not prohibited from using swaps.

5-Martellini Page 103 Wednesday, July 23, 2003 10:34 AM

The Euro Benchmark Yield Curve: Principal Component Analysis of Yield Curve Dynamics Lionel Martellini, Ph.D. Assistant Professor in Finance Marshall School of Business—University of Southern California and Research Associate EDHEC Risk and Asset Management Research Center Philippe Priaulet, Ph.D. Fixed Income Strategist Treasury and Capital Markets Department HSBC-CCF and Associate Professor Department of Mathematics—University of Evry Val d’Essonne Stéphane Priaulet Senior Index Portfolio Manager Structured Asset Management Department AXA Investment Managers

he term structure of interest rates is defined as the graph mapping interest rates corresponding to their respective maturity. The term structure of interest rates can take at any point in time various shapes

T

103


104


and the key question from a risk management perspective is to understand how the term structure of interest rates evolves over time. In this paper, we perform a factor analysis of the zero-coupon euro interbank yield curve, and also of zero-coupon Treasury yield curves from five individual countries, France, Germany, Italy, Spain, and the Netherlands, so as to isolate the key aspects of the dynamics of term structures of interest rates in the Euro zone.

THE PRINCIPAL COMPONENTS ANALYSIS MODEL Using a principal components analysis (PCA) has become a popular way to study movements of the term structure because it allows one to aggregate the risks in a non arbitrary way.1 The concepts behind this powerful statistical technique are fairly straightforward: ■ Different interest rates for different maturities are highly correlated

variables. A limited set of common economic, monetary, and financial factors affect money bond markets of different maturities. As a result, interest rates for various maturities tend to move in the same direction.2 ■ Highly correlated variables provide redundant information one with respect to another. As a consequence, it is tempting to try and identify a set of independent factors that would account for most of the information contained in the time-series of interest rate variations. This is exactly what a PCA does.

The Formal Model More formally, the PCA of a time-series consists in studying the correlation matrix of successive shocks. Its purpose is to explain the behavior of observed variables using a smaller set of unobserved implied variables. From a mathematical standpoint, it consists in transforming a set of K correlated variables into a set of orthogonal variables which reproduce the original information present in the correlation structure. The method allows one to express the interest rate variations as ∆R(t,θk), which are highly correlated across different maturities, in terms of new random variables Fti which are statistically uncorrelated. 1

We summarize in Appendix A the results of some of the most popular studies on that matter. 2 Even though they are highly correlated, interest rates of different maturities are not perfectly correlated, as can be inferred from the occurrence of nonparallel shifts of the yield curve.


The Euro Benchmark Yield Curve: Principal Component Analysis of Yield Curve Dynamics

105

We start with K variables, i.e., spot rates for K different maturities, and N observations for these variables. We consider the daily, weekly or monthly interest rate changes ∆R(t,θk) = R(t + 1,θk) – R(t,θk), and proceed to a PCA of the centered and reduced data, which amounts to using the correlation matrix of the interest rate changes.3 The idea is to define  ∆R ( t, θ k ) – ∆R ( ., θ k ) ∆R = ( ∆R tk ) 1 ≤ t ≤ N =  -----------------------------------------------------  1≤t≤N Nσ ∆R ( ., θ ) 1≤k≤K k

1≤k≤K

where ∆R ( ., θ k ) and σ ∆R ( ., θ ) are respectively the average value and the k standard-deviation of changes in the interest rate with maturity θk. We want to describe each variable as a linear function of a reduced number of factors as follows I

∆R tk =

∑ sik Fti + εtk

(1)

i=1

where sik = sensitivity of the kth variable to the ith factor defined as ∆( ∆R tk ) --------------------- = s ik ∆( F ti )

Fti εtk

that amounts to individually applying, for example, a 1% variation to each factor, and computing the absolute sensitivity of each zero-coupon yield curve with respect to that unit variation = value of the ith factor at date t. Note that I < K = residual part of ∆Rtk that is not explained by the factor model PCA enables us to decompose ∆Rtk as follows:4

3

PCA results are dependent on the methodology used to implement that PCA. Lardic, Priaulet, and Priaulet have shown that it should be implemented with interest rate changes that are stationary as original variables, and second that these original variables should be centered and variance-reduced [Sandrine Lardic, Philippe Priaulet, and Stéphane Priaulet, “PCA of the Yield Curve Dynamics: Questions of Methodologies,” Journal of Bond Trading and Management, 1 (4) April 2003, pp. 327– 349]. 4 For an explanation of this decomposition, see for example Joel R. Barber and Mark L. Copper, “Immunization Using Principal Component Analysis,” Journal of Portfolio Management, Fall 1996, pp. 99–105.


106


K

∆R tk =

I

∑ ∑

λ i U ik V ti + ε tk =

i=1 I

∆R tk =

K

λ i U ik V ti =

∑

λ i U ik V ti +

i=1

∑

λ i U ik V ti

i = I+1 I

∑ sik Fti + εtk

i=1

i=1

where (U)

= ( U ik ) 1 ≤ i, k ≤ K is the matrix of the K eigenvectors of ∆RT∆R

(UT) = ( U ki ) 1 ≤ k, i ≤ K is the transposed of U (V)

= ( V ti ) 1 ≤ t ≤ T is the matrix of the K eigenvectors of ∆R∆RT 1≤i≤K

Note that these K eigenvectors are orthonormal. λi is the eigenvalue (ordered by degree of magnitude) corresponding to the eigenvector Ui and we denote λ i U ik

s ik =

V ti = F ti sik is called the principal components sensitivity of the kth variable to the ith factor. The total dataset variance percentage explained by the first I factors is given by I

∑

λi i=1 -------------K

∑ λi

i=1

and the degree to which the ith factor explains the variance of the kth variable is calculated by 2 R ik

2

s ik = --------------K

∑

i=1

2 s ik

(2)



107

Use of the PCA Model for Interest Rate Risk Hedging We now explain that one can implement an improved hedging technique for a bond portfolio by focusing on principal components sensitivities (or dollar durations). Let us denote by Vt the value at date t of a portfolio made of, for example, Treasury bonds that we want to immunize against changes in the yield curve, using a set of hedging instruments. The value Vt of that portfolio at date t is given by K

Vt =

∑

K

C t + θ B ( t, t + θ k ) = k

k=1

Ct + θ

∑ --------------------------------------θ-

k = 1 [1

k

+ R ( t, θ k ) ]

k

where ( C t + θ ) k = 1, …, K = the cash flows of the portfolio V to be received at k dates t + θk B(t,t + θk) = the discount factors at date t for maturities t + θk R(t,θk) = the corresponding zero-coupon rates A first-order approximation to the change ∆Vt in the portfolio value between dates t and t + 1 is K

∆V t = V t + 1 – V t =

∂V k

- ∆R ( t, θ k ) ∑ ---------------------∂R ( t, θ k )

k=1

  –θk Ct + θ k  --------------------------------------------- ∆R ( t, θ ) = k  θ k + 1 k = 1  [ 1 + R ( t, θ k ) ]  K

∑

which transforms into I

∆V t =

 – Tσ  ∆R ( ., θ k ) s ik θ k C t + θ k  -------------------------------------------------------------- F ti =  θk + 1 k = 1  [ 1 + R ( t, θ k ) ] K

∑ ∑

i=1

I

∑ Sti Fti

i=1

where Sti are the principal components sensitivities of the portfolio V to the ith factor Fti. Let us consider at date t the hedging portfolio Ht containing assets j j with price H t and quantities φ t . The idea is to set the sensitivity of the global portfolio to zero (where the global portfolio is the portfolio to be hedged plus the hedging portfolio) with respect to the factors. A sufficient condition for this is: ∆V t + ∆H t = 0


108


We also impose a self-financing constraint Vt + Ht = 0 Note that the hedging portfolio should contain as many hedging instruments as there are risk factors, plus one because of the self-financing constraint. The number of factors, here denoted as I, is typically equal to 2 or 3.

THE DATA We now apply the PCA methodology both to the zero-coupon euro interbank yield curve, and to zero-coupon Treasury yield curves from five individual countries—France, Germany, Italy, Spain, and the Netherlands.

Deriving the Treasury Yield Curves We derive daily zero-coupon yield curves from five countries of the euro zone (France, Germany, Italy, Spain, and the Netherlands) during the period from January 2, 2001 to August 21, 2002, using zero-coupon rates with 26 different maturities ranging from one month to 30 years.5 The yield curves are extracted from daily Treasury bond market prices by using a standard cubic B-splines method. Our input baskets are composed of ■ Euribor rates with maturities between one month and one year. ■ Homogeneously liquid bonds with maturities exceeding one year.

We use Bloomberg Generic closing prices (BGN prices) for bonds.6 The inputs of the model are market gross prices of all instruments in the basket. The model we use falls into the category of discount function fitting models.7 First, we define the following N

5

= number of bonds used for the estimation of the zero-coupon yield curve

Maturities are 1 to 6 months, 9 months, 1 to 12, and 15, 18, 20, 22, 25, 27, and 30 years. 6 BGN (Bloomberg Generic Price) is Bloomberg’s market consensus price for government bonds. 7 For more details, see Lionel Martellini, Stéphane Priaulet, and Philippe Priaulet, Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies (Hoboken, NJ: John Wiley & Sons, Inc., 2003).



n

Pt n Pˆt Tn (n) Fθ

109

= = = =

market price at date t of the nth bond theoretical price at date t of the nth bond maturity of the nth bond (in years) coupon and/or principal payment of the nth bond at time t+θ≥t αn = number of cash flows for the nth bond B(t,t + θ) = discount function (price at date t of a zero-coupon bond paying $1 at date ) n

The price vectors are P t = ( P t ) n = 1, …, N and n Pˆ t = ( Pˆt ) n = 1, …, N

respectively. According to Steeley,8 the discount function B(t,t + θ) is a sum of cubic B-splines that can adapt any market curve configuration 4

B ( t, t + θ ) =

∑

k + 4 k + 4

4

3 βk βk ( θ )

=

k = –3

∑

βk

k = –3

   3 1 --------------- ( θ – λ j ) +  – λ λ  j j = k i=k i  i≠j 

∑ ∏

3

where β k ( θ ) is the kth cubic B-spline, β = (β–3,...,β4) is a vector of eight parameters and λi is the ith knot point. Taking into account common market segmentation, we place the knots at 1-, 2-, 5-, 10-, and 30-year maturities, roughly corresponding to very short, short, medium, long-term, and very long-term money. Then we define λ0 = 0, λ1 = 1, λ2 = 2, λ3 = 5, λ4 = 10, and λ5 = 30 to 3 3 obtain the central cubic B-splines β 0 ( θ ) , and β 1 ( θ ) . For technical purposes, we also define λ–3 = –3, λ–2 = –2, λ–1 = –1, λ6 = 31, λ7 = 32, and 3 3 3 3 3 3 λ8 = 33 to obtain B –3 ( θ ) , B –2 ( θ ) , B –1 ( θ ) , B 2 ( θ ) , B 3 ( θ ) , and B 4 ( θ ) . Within this framework and under the no-arbitrage hypothesis, the gross price of each instrument equals the present value of its cash flows, each being discounted at the appropriate zero-coupon rate n Pˆt =

Tn

∑

(n)

F θ B ( t, t + θ )

t + θ = Tn – αn + 1 t+θ>t 8

James M. Steeley, “Estimating the Gilt-Edged Term Structure: Basis Splines and Confidence Intervals.” Journal of Business Finance and Accounting (June 1991), pp. 512–529.


110


The model is expressed as P t = Pˆ t + ε , where the residuals ε capturing market frictions such as pricing errors or inaccuracies, liquidity, and 2 tax effects, satisfy ∀( n, n′ ) ∈ { 1, …, N } the condition  E ( εn ) = 0  2 2  Var ( ε n ) = σ ω n   Cov ( ε n, ε n′ ) = 0 for n ≠ n′ In accordance with Vasicek and Fong,9 we define n

2 n  dP t  2 D n ( t )P t 2 ω n =  --------------- = ---------------------------2  dr n ( t ) [ 1 + rn ( t ) ]

where rn(t) and Dn(t) are respectively the yield to maturity and the duration of the nth bond at date t. The rationale for this choice follows the intuition that the longer the maturity of a given bond, the more difficult is its price estimation. The coefficients vector β of the cubic B-spline discount function are the solutions of a generalized least squares program that minimizes the sum of the squared residuals given by N

 P n – Pˆn min t  t β n=1

∑

2

such that B(t,t) = 1.

Deriving the Euro Interbank Yield Curve We compute PCA with the zero-coupon euro interbank yield curve for the period from January 2, 2001 to August 21, 2002. We use zero-coupon rates with 17 different maturities from one month to ten years.10 The basket of inputs contains three kinds of instruments: money market rates, futures contracts, and swaps. ■ We consider Euribor rates with maturities ranging from one day to one

year. These rates, expressed on an Actual/360 basis, are first converted 9

Oldrich A. Vasicek and H. Gifford Fong, “Term Structure Modeling using Exponential Splines,” Journal of Finance 37(2) (May 1982), pp. 339–348. 10 Maturities are 1 to 6 months, 9 months, and 1 to 10 years.



111

into equivalent zero-coupon rates on an Actual/365 basis. For example, on January 1, 1999, the 1-month Euribor rate was equal to 2.5%. Using the Actual/365 basis, the equivalent zero-coupon rate (denoted by R(0,1/12)) is given by 31- × 2.5% R ( 0, 1 ⁄ 12 ) =  1 + -------- 360

365 ---------31

–1

■ We consider futures 3-month Euribor futures contracts and find zero-

coupon rates from raw data. The price of a 3-month LIBOR contract is given by 100 minus the underlying 3-month forward rate. For example, on March 15, 1999, the 3-month LIBOR rate was 3%, and the 3month LIBOR contract with maturity date June 1999 had a price equal to 96.5. Hence on March 15, 1999, the 3-month forward rate, starting on June 15, 1999 is 3.5%. The 6-month spot rate (denoted by R(0,6/ 12)) is obtained as follows 92- × 3.5% 92- × 3%  1 + --------R ( 0, 6 ⁄ 12 ) =  1 + --------  360 360

365 ---------184

–1

■ We consider three-or-six-month Euribor swap yields with maturities

ranging from one year to ten years and find recursively equivalent zerocoupon rates. Swap yields are par yields; so the zero-coupon rate with maturity two years R(0,2) is obtained as the solution to the following equation SR ( 2 ) 1 + SR ( 2 ) ---------------------------- + ------------------------------------ = 1 1 + R ( 0, 1 ) [ 1 + R ( 0, 2 ) ] 2 where SR(2) is the 2-year swap yield, and R(0,1) is equal to SR(1). Spot rates R(0,3), ..., R(0,10) are obtained recursively in a similar fashion. Least squared methods used to derive the current interbank curve are very similar to those used to derive the current nondefault Treasury curve. After converting market data into equivalent zero-coupon rates, the zero-coupon yield curve is derived using a two-stage process, first writing zero-coupon rates as a B-spline function, and then fitting them through an ordinary least squared method.


112


We have 17 equivalent zero-coupon rates denoted by R(0,θ), for θ = ¹₁₂,...,10. We write theoretical zero-coupon rates as a B-spline function using the following splines: [0,0.5], [0.5,1], [1,2], [2,3], [3,4], [4,5], [5,6], [6,8], and [8,10]. In that case we write specifically 8

ˆ ( 0, θ ) = R

∑

3

al Bl ( θ )

l = –3

l + 4 l + 4      3 1 al  --------------- ( θ – λ j ) + =   j = 1  i = 1 λ i – λ j  l = –3   i≠j   8

∑

∑ ∏

where λ–3 = –3, λ–2 = –2, λ–1 = –1, λ0 = 0, λ1 = ¹₂, λ2 = 1, λ3 = 2, λ4 = 3, λ5 = 4, λ6 = 5, λ7 = 8, λ8 = 8, λ9 = 10, λ10 = 11, λ11 = 12, λ12 = 13. Recall that the parameters λ–3, λ–2, λ–1, λ10, λ11, λ12 must only satisfy λ–3 < λ–2 < λ–1 < 0 < ... < 10 < λ10 < λ11 < λ12, and are defined as 3 mathematical conditions to write B-spline functions B l ( θ ) . Then we minimize the sum of the squared spreads between market rates and theoretical rates according to the following program θ = 10

ˆ ( 0, θ ) ] 2 [ R ( 0, θ ) – R min a l θ = ¹⁄₁₂

∑

This method allows for the fitting function to be a quasi-perfect match for all the R(0,n) points. Note that it is possible to be fully consistent with all market points by choosing as many splines as market points.11

PCA OF THE TREASURY AND EURO INTERBANK YIELD CURVES IN THE EURO ZONE We now apply the methodology just described to study the dynamics of the Treasury yield curve for selected individual countries from the euro zone, as well as the dynamics of the euro Interbank yield curve.

Percentage of Explanation by the Factors We first discuss the percentage of explanation power on the whole period and then analyze the breakdown on different subperiods. 11

For more details on that matter, see Martellini, Priaulet, and Priaulet, FixedIncome Securities: Valuation, Risk Management and Portfolio Strategies.



113

Percentage of Explanation by the Factors on the Entire Period We first consider the global fraction of the total variance of the zerocoupon yield curve changes that is accounted for by the five first factors (see Exhibit 1). For each factor, that percentage is given by λi/26 (λi/17 when we perform PCA of the euro interbank yield curve) where λi is the ith eigenvalue of the correlation matrix of the raw data (daily interest rate changes), where 26 (respectively 17) is the number of variables. ■ PCA of the Treasury yield curves. The first five factors account for

98.74% to 99.40% of interest rate changes for the examined period (see Exhibit 1) depending on the country we consider. The first three factors, typically interpreted as level, slope, and curvature factors, account for 91.05% to 97.07%. The results are very homogeneous from one country to another, since the explanation power from the first factor ranges from 62.22% to 66.87%, while it ranges from 21.87% to 22.61% for the second factor and from 6.76% to 9.60% for the third factor. The nontrivial weights on factors four and five signals the presence of nonnegligible residuals. For the period under consideration, these two factors account sometimes for more than 5% of the interest rate changes (in particular 7.69% for France). ■ PCA of the Interbank yield curve. Because they do not apply to the same variables, it should be expected that results obtained with the Interbank yield curve be different from results obtained with Treasury yield curves. This is confirmed by the numbers in Exhibit 1. The first factor only accounts for 47.54% of the interest rate changes, while the second factor explains 25.63%. The inferior percentage of explanation by the first factor can be related to the fact that eight out of the 17 variables we use relate to the short-term segment of the curve. The three first factors account for 84.08% of the yield curve deformations while the first five factors account for 93.34%, which means that residuals are not negligible.

Percentage of Explanation by the Factors on Selected Periods We provide in Exhibit 2 the percentage of explanation by the factors for the years 2001 and 2002. ■ PCA of the Treasury yield curves. The percentage of explanation for

each factor can vary substantially with the selected period (see Exhibit 2). The first factor is predominant in 2002 while its weight declines in 2001 (below 60% in 2001). Of course, such variations can be explained not only by changes in the dynamic behavior of term structures but also by sample fluctuations. In 2001, Treasury yield curves


114


EXHIBIT 1

Global Percentage of Explanation by the Five First Factors—

2001–2002 France

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

16.228 62.42% 62.42%

5.685 21.87% 84.29%

1.758 6.76% 91.05%

1.422 5.47% 96.52%

0.577 2.22% 98.74%

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

17.386 66.87% 66.87%

5.796 22.29% 89.16%

2.057 7.91% 97.07%

0.373 1.44% 98.51%

0.232 0.89% 99.40%

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Eigenvalue % Explained % Cumulative

17.191 66.12% 66.12%

5.842 22.47% 88.59%

2.154 8.28% 96.87%

0.439 1.69% 98.56%

0.187 0.72% 99.28%

The Netherlands

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Eigenvalue % Explained % Cumulative

16.909 65.03% 65.03%

5.797 22.30% 87.33%

1.933 7.44% 94.77%

0.734 2.82% 97.59%

0.388 1.49% 99.08%

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

16.176 62.22% 62.22%

5.879 22.61% 84.83%

2.496 9.60% 94.43%

0.732 2.82% 97.25%

0.402 1.55% 98.80%

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

8.082 47.54% 47.54%

4.357 25.63% 73.17%

1.854 10.91% 84.08%

0.986 5.80% 89.88%

0.589 3.46% 93.34%

Eigenvalue % Explained % Cumulative Germany Eigenvalue % Explained % Cumulative Italy

Spain Eigenvalue % Explained % Cumulative Euro Interbank Eigenvalue % Explained % Cumulative

were more affected by steepening and flattening moves than in 2002. That is why the second factor is less than 19.2% in 2002 while it is higher than 22% in 2001, whatever the country. The presence of nonnegligible residuals is particularly obvious in France in 2001, where the weight of residuals are higher than 10%. We note that these residuals



115

EXHIBIT 2 Global Percentage of Explanation by the First Three Factors Depending on the Selected Period Curve

France

Factor

2001

2002

Factor 1

59.06%

73.59%

Factor 2

22.37%

18.09%

Factor 3 Total

7.82%

5.34%

89.25%

97.02%

Factor 1

64.05%

74.35%

Germany Factor 2

23.47%

16.66%

Factor 3

Italy

Curve

The Netherlands

Factor

2002

Factor 1

62.26% 73.16%

Factor 2

23.37% 16.50%

Factor 3 Total

Spain

2001

8.97%

5.82%

94.60% 95.48%

Factor 1

59.93% 69.80%

Factor 2

23.38% 19.17%

9.29%

6.44%

Factor 3

11.06%

Total

96.81%

97.45%

Total

94.37% 95.46%

Factor 1

66.12%

73.83%

Factor 1

59.09% 42.70%

Factor 2

22.47%

16.75%

Factor 2

27.16% 25.49%

Factor 3

8.28%

6.53%

96.87%

97.11%

Total

Euro Interbank

Factor 3 Total

6.49%

6.43% 13.70% 92.68% 81.89%

are not homogeneous from one country to another and from one period of time to another. ■ PCA of the Interbank yield curve. The percentage of explanation by the first factor is relatively low, and even gets lower than 50% in 2002 to reach 42.70%. The residuals account for a fairly significant fraction (almost 20%) of the total variation in interest rates in 2002. This suggests that a strategy intended at immunizing the value of a bond portfolio with respect to small changes in the level, slope, and curvature of the term structure would have failed to properly hedge the portfolio.

Percentage of Explanation by the Factors for Each Maturity For France, we summarize in Exhibit 3 the extent to which the ith factor explains the variance of the kth variable (see equation (2)). We provide the same table for the other countries in Appendix B. The first factor is more significant (greater than 70%) for maturities between 2 years and 20 years and very significant (greater than 84%) for the maturities ranging from 4 years to 12 years. The short rates (with maturities ranging from one to six months) are significantly affected by the second factor (greater than 60%), while that factor is virtually negligible for medium-term maturities from two years to eight years. That the medium-term segment of the yield curve remains stable under changes in the second factor suggests that it can be interpreted as


116


EXHIBIT 3

Percentage of Explanation by the Factors for Each Maturity—France (01/02/2001–08/21/2002) Maturity

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6–26

1M 2M 3M 4M 5M 6M 9M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 11Y 12Y 15Y 18Y 20Y 22Y 25Y 27Y 30Y

6.15% 11.14% 17.12% 23.27% 28.94% 33.85% 45.14% 66.12% 72.44% 75.55% 84.62% 90.25% 93.52% 95.48% 95.18% 93.01% 90.59% 88.65% 87.10% 81.94% 73.33% 68.33% 67.07% 69.19% 42.44% 22.40%

70.11% 76.33% 78.00% 75.16% 69.11% 34.52% 39.62% 12.49% 0.11% 0.36% 0.22% 0.10% 0.30% 1.09% 2.62% 4.55% 6.31% 7.72% 8.78% 9.85% 8.55% 7.47% 7.03% 8.07% 7.41% 5.64%

11.31% 7.43% 3.94% 1.46% 0.21% 0.05% 3.29% 13.89% 19.76% 16.87% 12.32% 7.40% 3.79% 1.51% 0.34% 0.00% 0.18% 0.60% 1.13% 2.86% 4.53% 6.00% 8.62% 16.60% 18.47% 13.30%

0.00% 0.00% 0.01% 0.02% 0.03% 0.03% 0.00% 0.56% 2.56% 1.50% 0.23% 0.01% 0.01% 0.02% 0.19% 0.37% 0.29% 0.07% 0.03% 3.89% 13.42% 17.10% 14.27% 0.24% 29.25% 58.23%

12.04% 4.98% 0.92% 0.05% 1.57% 4.29% 11.50% 6.11% 0.85% 2.85% 1.19% 0.17% 0.04% 0.09% 0.30% 0.62% 0.85% 0.94% 0.89% 0.29% 0.03% 0.47% 1.43% 3.12% 1.71% 0.39%

0.39% 0.12% 0.01% 0.03% 0.14% 0.26% 0.44% 0.83% 4.29% 2.87% 1.43% 2.08% 2.34% 1.80% 1.38% 1.46% 1.78% 2.02% 2.07% 1.16% 0.32% 0.63% 1.58% 2.77% 0.73% 0.04%

Mean

62.42%

21.87%

6.76%

5.47%

2.22%

1.27%

a slope factor. The curvature effect, related to the third factor, traditionally opposes end segments of the curve to the medium-term segment. Factors 4 and 5 are not negligible. Factor 4 has only an effect on the very long-term segment (longer than 15 years) whereas factor 5 is a kind of curvature factor affecting more the short-term and the long-term ends of the curve. These results illustrate once more that:12 12

See Lardic, Priaulet, and Priaulet, “PCA of the Yield Curve Dynamics: Questions of Methodologies.”



117

■ The second factor explains most of the short-term segment. ■ The long-term segment is explained much more by the first factor, but

it is the medium segment with maturities between 1 year and 10 years, which is best explained by the first factor. ■ The third factor seems to have a major impact first on the short-term segment and then on the long-term segment; when we eliminate the short-term and long-term segments, its percentage of variance explained decreases dramatically. 2

Note finally that the mean of R ik with respect to maturity k gives the percentage of variance explained by the ith factor (see Exhibit 3), and that results are homogeneous from one country to another (see Appendix B).

Factors Correlation Using equation (1), we compute the correlation between factor 1 (as obtained from PCA) of the different Treasury yield curves and the Interbank yield curve. We do the same for factor 2 and factor 3. Results are first detailed for the whole period, and then on each year (see Appendix C). Generally speaking, the five Treasury yield curves prove to be very correlated with each other. This evidence is consistent with the fact that there is uniqueness of monetary policy (same short term rate for all countries); and that selected countries are fairly homogeneous in terms of credit quality and liquidity. These correlations have strengthened from 2001 to 2002, which supports the notion of increased financial integration in the euro zone. The core countries seem to be France, Germany, the Netherlands, and Italy. Indeed, France, Germany, and the Netherlands have the same credit quality, namely AAA; French, German, and, especially, Italian Treasury bonds are very liquid. Italy has now the highest weight in the euro area. To some extent, Spain seems to be slightly less correlated with the other countries (presumably because of the liquidity premium). The swap yield curve is correlated with the Treasury yield curves, but this correlation has decreased from 2001 to 2002. This may be explained by the increase in the investors’ risk aversion as continuing poor performance of equity markets in 2002 has triggered a search for liquidity and quality. The correlation with the Treasury yield curves is high for the first factor (see Exhibit 4), but weak for the second and third factors (see Exhibits 5 and 6).


118


EXHIBIT 4 Correlation Matrix between the PCA Factor 1 of the Different Treasury Curves and the Interbank Curve: 2001–2002 France Germany France Germany Italy The Netherlands Spain Euro Interbank

1

0.9929 1

Italy 0.9831 0.9891 1

The Netherlands 0.9940 0.9906 0.9825 1

Spain 0.9796 0.9865 0.9843 0.9830 1

Euro Interbank 0.8511 0.8542 0.8431 0.8445 0.8411 1


1

0.9669 1

Italy 0.9641 0.9892 1

The Netherlands 0.9932 0.9725 0.9698 1

Spain 0.5733 0.6031 0.5931 0.5687 1

Euro Interbank 0.2164 0.2047 0.1949 0.2245 0.0898 1


1

0.9027 1

Italy 0.9066 0.9428 1

The Netherlands 0.9372 0.8904 0.9156 1

Spain 0.6491 0.7063 0.6639 0.6484 1

Euro Interbank 0.1363 0.1743 0.1779 0.1411 0.0785 1



EXHIBIT 7

119

Sensitivity of Zero-Coupon Rate Changes with Respect to Factor 1

SENSITIVITIES OF THE ZERO-COUPON RATES Exhibits 7–9 display the sensitivities sik as functions of the interest rate maturities k for factors 1, 2, and 3. We can see that the sensitivities are fairly similar whatever the country under consideration. For the period as a whole, the first factor (see Exhibit 7) may actually be regarded as a level factor since it affects similarly all zero-coupon rates, except for the portion (1 month–1 year), which moves differently. Displaying the sensitivity of interest rates with respect to the second factor, Exhibit 8 shows a decreasing shape, first positive for short-term maturities then negative beyond. Hence, the second factor may be regarded as a rotation factor around a medium maturity between two and four years depending on the country we consider. The third factor (see Exhibit 9) has different effects on intermediate maturities as opposed to extreme maturities (short and long). Hence, it may be interpreted as a curvature factor.


120


EXHIBIT 8


EXHIBIT 9




121

CONCLUSION In this article we present an empirical analysis of the term structure dynamics in the euro zone, based on daily data over the period January 2, 2001 to August 21, 2002. We study both the zero-coupon euro interbank yield curve, and zero-coupon Treasury yield curves from five individual countries, France, Germany, Italy, Spain, and the Netherlands. Using principal components analysis, we find that three main factors typically explain more than 90% of the changes in the yield curve, whatever the country and the period under consideration. These factors can be interpreted as changes in the level, the slope, and the curvature of the term structure. We also find strong evidence of homogeneity in the dynamics of the yield curve for different countries in the euro zone, signaling an increasing financial integration.

APPENDIX A In the past few years there have been several many studies on the topic of PCA of interest rate curves by both academics and practitioners. Exhibit A summarizes the main results of these studies.

122

71/18/4

66.64/20.52/6.96

Note: M stands for month and Y for year. For example, “88.04/8.38/1.97” means that the first factor explains 883.04% of the yield curve deformations, the second 8.38%, and the third 1.97%. Sometimes, we also provide the total amount by adding up these terms.

L. Martellini and P. Priaulet, Fixed-Income Securities : Dynamic Methods for Interest France (1995–98)—Spot ZC Rate Risk Pricing and Hedging (New York: John Wiley & Sons, 2000).

42.8/25.5/17.1/6/4.9

50.6/17.3/13.5/8.8/5.8

56.5/17.4/9.86/8.12/4.3

92.8/4.8/1.27

75/16/3

63.5/6.3/7.5/8.1/5.3

3

5

3

3

Total: 97/98/98 80.93/11.85/4.36

Japan (1987–95) 1M–10Y

3 3

93.91/5.49/0.42

93.7/6.1

88.04/8.38/1.97

% of Explanation

UK (1987–95)

Germany (1987–95) 1 Year Forward

1Y–9Y

I. Lekkos, “A Critique of Factor Analysis of Interest Rates,” Journal of Derivatives, Fall 2000, pp. 72–83.

USA (1984–95)

3M–30Y

(1988–96)—Spot ZC 1M–10Y

Golub, B. W., and L. M. Tilman, “Measuring Yield Curve Risk Using Principal Com- RiskMetrics–09/30/96—Spot ZC ponents Analysis, Value at Risk, and Key Rate Durations,” Journal of Portfolio Management, Summer 1997, pp. 72–84.

A. Bühler and H. Zimmerman, “A Statistical Analysis of the Term Structure of Inter- Germany est Rates in Switzerland and Germany,” Journal of Fixed Income, 6(3) (December Switzerland 1996), pp. 55–67.

3M–10Y 1M–20Y

J. Kärki and C. Reyes, “Model Relationship,” Risk 7(12) (December 1994), pp. 32–35. Germ./Switz./USA (1990–94)—Spot ZC

J.R. Barber and M.L. Copper, “Immunization Using Principal Component Analysis,” USA (1985–91)—Spot ZC Journal of Portfolio Management, Fall 1996, pp. 99–105.

3

6M–7Y

R.L. D’Ecclesia and S.A. Zenios, “Risk Factor Analysis and Portfolio Immunization in the Italian Bond Market,” Journal of Fixed Income 4(2) (September 1994), pp. 51– 58. Italy (1988–92)—Spot ZC

2

1Y–25Y

Factors

C. Kanony and M. Mokrane, “Reconstitution de la courbe des taux, analyse des fac- France (1989–90)—Spot ZC teurs d’évolution et couverture factorielle,” Cahiers de la Caisse Autonome de Refinancement 1, June 1992.

Range 3

Country (Period)—Kind of Rates

Robert Litterman and José Scheinkman, “Common Factors Affecting Bond Returns,” USA (1984–88)—Spot Zero-Coupon (ZC) 6M–18Y Journal of Fixed Income, June 1991, pp. 54–61.

Authors

EXHIBIT A1 Results of Some Popular Studies on PCA of the Yield Curve Dynamics




123

APPENDIX B The exhibits in this appendix provide the percentage of explanation by the factors for each maturity for Germany, Italy, the Netherlands, Spain, and the euro Interbank for during the period from 01/02/01 to 08/21/02. EXHIBIT B1 Percentage of Explanation by the Factors for Each Maturity: Germany (02/01/2001–08/21/2002) Maturity

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6–26

1M 2M 3M 4M 5M 6M 9M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 11Y 12Y 15Y 18Y 20Y 22Y 25Y 27Y 30Y Mean

0.82% 3.41% 8.36% 15.78% 24.97% 34.57% 55.34% 67.96% 74.82% 80.28% 84.90% 88.80% 92.36% 95.03% 96.01% 95.12% 92.95% 90.52% 88.35% 84.19% 81.99% 80.63% 79.23% 77.57% 75.25% 69.42% 66.87%

81.05% 84.69% 85.42% 82.05% 74.36% 63.55% 31.24% 7.46% 1.72% 0.43% 0.11% 0.00% 0.11% 0.56% 1.38% 2.46% 3.63% 4.69% 5.59% 7.38% 7.98% 7.84% 7.42% 6.65% 6.08% 5.72% 22.29%

15.64% 10.73% 5.95% 2.15% 0.18% 0.36% 8.66% 20.60% 21.95% 18.20% 13.45% 8.86% 5.03% 2.20% 0.51% 0.00% 0.42% 1.35% 2.47% 5.73% 8.02% 8.96% 9.59% 10.45% 11.74% 12.51% 7.91%

1.42% 0.70% 0.19% 0.00% 0.20% 0.71% 2.54% 2.45% 0.57% 0.03% 0.03% 0.23% 0.59% 1.11% 1.70% 2.21% 2.50% 2.51% 2.29% 0.92% 0.01% 0.35% 1.57% 4.15% 5.00% 3.35% 1.44%

0.64% 0.28% 0.05% 0.01% 0.18% 0.50% 1.47% 1.13% 0.06% 0.25% 0.89% 1.31% 1.19% 0.73% 0.27% 0.02% 0.05% 0.27% 0.60% 1.55% 1.85% 1.49% 0.78% 0.00% 1.88% 5.71% 0.89%

0.44% 0.19% 0.03% 0.01% 0.11% 0.30% 0.76% 0.40% 0.88% 0.82% 0.61% 0.80% 0.71% 0.37% 0.14% 0.19% 0.45% 0.66% 0.70% 0.22% 0.15% 0.73% 1.41% 1.18% 0.04% 3.29% 0.60%


124


EXHIBIT B2 Percentage of Explanation by the Factors for Each Maturity: Italy (01/02/2001–08/21/2002) Maturity

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6–26


1.47% 4.39% 9.41% 16.54% 25.04% 33.70% 52.16% 65.57% 73.99% 79.93% 85.00% 88.72% 91.98% 94.52% 95.46% 94.54% 92.31% 89.78% 87.49% 82.74% 79.69% 77.61% 75.44% 73.26% 74.17% 74.23% 66.12%

76.94% 81.33% 82.98% 80.65% 74.02% 64.20% 33.53% 9.38% 2.61% 0.78% 0.23% 0.01% 0.10% 0.59% 1.49% 2.70% 3.99% 5.19% 6.23% 8.44% 9.41% 9.41% 8.97% 7.91% 6.88% 6.22% 22.47%

18.49% 12.84% 7.27% 2.79% 0.32% 0.26% 8.82% 21.03% 20.66% 16.91% 13.10% 9.18% 5.63% 2.79% 0.90% 0.06% 0.15% 0.85% 1.88% 5.72% 9.42% 11.31% 12.48% 12.73% 11.10% 8.69% 8.28%

1.60% 0.80% 0.22% 0.00% 0.20% 0.73% 2.59% 2.20% 0.27% 0.01% 0.00% 0.02% 0.17% 0.63% 1.42% 2.36% 3.15% 3.51% 3.46% 1.78% 0.12% 0.21% 1.63% 5.21% 6.93% 4.66% 1.69%

0.94% 0.43% 0.09% 0.01% 0.22% 0.66% 2.04% 1.73% 0.14% 0.26% 1.11% 1.69% 1.50% 0.88% 0.29% 0.01% 0.08% 0.34% 0.64% 1.20% 1.03% 0.64% 0.22% 0.03% 0.79% 1.74% 0.72%

0.55% 0.21% 0.02% 0.02% 0.19% 0.45% 0.86% 0.08% 2.35% 2.12% 0.56% 0.38% 0.61% 0.60% 0.44% 0.33% 0.32% 0.33% 0.30% 0.11% 0.33% 0.82% 1.26% 0.86% 0.14% 4.46% 0.72%



125

EXHIBIT B3 Percentage of Explanation by the Factors for Each Maturity: The Netherlands (01/02/2001–08/21/2002) Maturity

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6–26


5.21%

70.25%

13.69%

5.69%

4.96%

0.20%

9.91% 15.77% 22.04% 27.99% 33.21% 44.95% 64.46% 73.31% 76.32% 83.35% 88.48% 92.26% 94.93% 95.51% 94.16% 92.09% 90.00% 88.01% 81.65% 73.61% 68.88% 66.97% 72.67% 76.33% 58.79% 65.03%

76.31% 78.19% 75.75% 70.07% 62.72% 41.04% 15.02% 1.14% 0.00% 0.00% 0.01% 0.16% 0.81% 2.11% 3.79% 5.44% 6.89% 8.11% 10.27% 10.32% 9.65% 8.96% 8.64% 8.10% 5.92% 22.30%

9.26% 5.17% 2.12% 0.43% 0.01% 3.21% 14.12% 21.20% 18.87% 14.36% 9.41% 5.49% 2.69% 0.92% 0.10% 0.07% 0.65% 1.69% 6.72% 12.47% 15.23% 16.41% 13.96% 4.79% 0.31% 7.44%

2.50% 0.55% 0.00% 0.56% 1.74% 5.15% 3.33% 0.03% 0.25% 0.00% 0.13% 0.06% 0.02% 0.35% 0.87% 1.11% 0.97% 0.60% 0.15% 3.09% 5.84% 6.91% 2.36% 4.74% 26.41% 2.82%

1.96% 0.31% 0.06% 0.87% 2.18% 5.42% 2.38% 0.89% 2.61% 1.59% 0.56% 0.19% 0.09% 0.07% 0.08% 0.10% 0.13% 0.17% 0.31% 0.32% 0.18% 0.02% 0.56% 4.62% 8.17% 1.49%

0.06% 0.01% 0.02% 0.08% 0.15% 0.23% 0.69% 3.44% 1.95% 0.70% 1.42% 1.83% 1.46% 1.05% 1.00% 1.18% 1.36% 1.42% 0.89% 0.19% 0.22% 0.73% 1.82% 1.42% 0.40% 0.92%


126


EXHIBIT B4 Percentage of Explanation by the Factors for Each Maturity: Spain (02/01/2001–08/21/2002) Maturity

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6–26


0.42% 0.03% 0.20% 1.69% 5.91% 14.60% 46.52% 47.16% 56.87% 71.36% 81.80% 86.82% 91.26% 94.13% 94.31% 93.60% 92.95% 92.62% 92.57% 92.28% 88.42% 83.72% 78.91% 74.69% 72.13% 62.67% 62.22%

96.40% 98.36% 99.42% 97.99% 90.95% 74.23% 5.30% 9.39% 9.46% 1.95% 0.08% 0.05% 0.14% 0.12% 0.08% 0.03% 0.01% 0.00% 0.02% 0.26% 0.70% 0.95% 1.03% 0.67% 0.03% 0.27% 22.61%

2.03% 0.96% 0.15% 0.19% 2.35% 8.41% 36.93% 36.50% 30.50% 20.19% 11.62% 5.57% 2.08% 0.42% 0.00% 0.27% 0.89% 1.68% 2.56% 5.27% 7.65% 9.00% 10.42% 13.68% 19.09% 21.16% 9.60%

0.18% 0.06% 0.00% 0.12% 0.60% 1.69% 5.77% 4.30% 1.88% 0.12% 0.25% 1.35% 2.97% 4.53% 5.30% 5.27% 4.68% 3.76% 2.68% 0.17% 1.07% 3.67% 6.79% 9.44% 5.70% 0.90% 2.82%

0.75% 0.45% 0.17% 0.00% 0.15% 0.87% 4.66% 2.50% 0.00% 3.24% 5.05% 3.20% 0.90% 0.01% 0.25% 0.71% 1.00% 1.03% 0.86% 0.06% 0.48% 1.49% 2.26% 1.31% 0.53% 8.26% 1.55%

0.22% 0.14% 0.06% 0.02% 0.04% 0.20% 0.83% 0.15% 1.28% 3.14% 1.21% 3.02% 2.65% 0.79% 0.06% 0.11% 0.47% 0.91% 1.32% 1.96% 1.68% 1.17% 0.59% 0.21% 2.52% 6.73% 1.21%


127


EXHIBIT B5 Percentage of Explanation by the Factors for Each Maturity: Euro Interbank (02/01/2001–08/21/2002) Maturity

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6–26

1M 2M 3M 4M 5M 6M 9M 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y Mean

0.12% 0.66% 5.72% 17.11% 25.63% 29.49% 39.38% 59.24% 71.54% 64.31% 84.72% 83.15% 85.52% 83.42% 78.53% 75.14% 4.50% 47.54%

35.12% 55.74% 74.93% 77.60% 61.25% 46.04% 25.52% 2.88% 3.05% 4.19% 6.35% 6.59% 8.08% 8.61% 8.55% 10.07% 1.11% 25.63%

45.17% 42.62% 12.79% 0.14% 10.38% 21.39% 30.76% 9.71% 0.62% 0.42% 1.08% 1.09% 1.73% 2.13% 2.23% 2.87% 0.28% 10.91%

0.10% 0.02% 0.02% 0.12% 0.17% 0.15% 0.02% 0.61% 3.29% 3.16% 0.04% 0.00% 0.02% 0.58% 0.00% 0.37% 89.96% 5.80%

2.33% 0.05% 0.49% 0.95% 0.60% 0.22% 0.12% 4.24% 11.36% 15.97% 0.11% 0.03% 1.32% 3.07% 6.24% 7.69% 4.05% 3.46%

17.16% 0.90% 6.05% 4.08% 1.97% 2.70% 4.20% 23.31% 10.13% 11.95% 7.69% 9.13% 3.33% 2.20% 4.45% 3.86% 0.10% 6.66%

APPENDIX C The exhibits in this appendix show the factors correlations for the periods identified. EXHIBIT C1 Correlation Matrix between the PCA Factor 1 of the Different Treasury Curves and the Interbank Curve—2001 France Germany France Germany Italy The Netherlands Spain Euro Interbank

1

0.9901 1

Italy 0.9776 0.9865 1

The Netherlands 0.9919 0.9854 0.9754 1

Spain 0.9755 0.9836 0.9806 0.9792 1

Euro Interbank 0.8738 0.8786 0.8718 0.8625 0.8591 1


128


EXHIBIT C2 Correlation Matrix between the PCA Factor 2 of the Different Treasury Curves and the Interbank Curve—2001 France Germany France Germany Italy The Netherlands Spain Euro Interbank

1

0.9688 1

Italy 0.9676 0.9918 1

The Netherlands 0.9953 0.9738 0.9716 1

Spain 0.5668 0.5937 0.5852 0.5654 1

Euro Interbank 0.2707 0.2590 0.2572 0.2767 0.1219 1


1

0.8864 1

Italy 0.9022 0.9434 1

The Netherlands 0.9282 0.8700 0.9130 1

Spain 0.5854 0.6549 0.6092 0.5868 1

Euro Interbank 0.1837 0.2201 0.2209 0.1801 0.1268 1


1

0.9974 1

Italy 0.9918 0.9929 1

The Netherlands 0.9974 0.9981 0.9927 1

Spain 0.9867 0.9908 0.9897 0.9886 1

Euro Interbank 0.7975 0.7967 0.7735 0.8028 0.7998 1



129


1

0.9587 1

Italy 0.9404 0.9654 1

The Netherlands 0.9789 0.9616 0.9533 1

Spain 0.7354 0.7791 0.7502 0.6702 1

Euro Interbank 0.0297 0.0034 –0.0394 0.0469 –0.0763 1


1

0.9684 1

Italy 0.9255 0.9414 1

The Netherlands 0.9744 0.9680 0.9261 1

Spain 0.9130 0.9030 0.8786 0.8954 1

Euro Interbank 0.0761 0.1273 0.1359 0.0998 0.0054 1


6-HoGoodman-Dollar Rolling Page 131 Wednesday, July 23, 2003 10:33 AM

Dollar Rolling—Does It Pay? Jeffrey Ho Executive Director UBS Warburg Laurie Goodman, Ph.D. Managing Director UBS Warburg

n this article, we look at dollar rolling TBAs versus the performance of a mortgage market index. We show that on average, since 1992, dollar rolling a portfolio of TBAs outperformed a mortgage market index by 50– 60 basis points. The margin of outperformance is highest during prepay waves, while it’s most limited during periods of limited supply. Thus, in general, over the past 10¹₂ years, dollar rolling has been a very powerful addition to returns. However, this strategy is not a universal panacea, because there are definitely times when dollar rolling just does not pay.

I

DOLLAR ROLLING When we look at the profitability of dollar rolling, we usually look at: 1. Selling the mortgage for TBA settlement this month, buying the mortgage for TBA settlement next month, and holding the money in cash; versus 2. Holding the mortgage, receiving the coupon, and realizing the paydown.

131


132 EXHIBIT 1


Historical Financing Advantage—Current Coupon FNMA TBAs versus

LIBOR

That first strategy, in which the securities are dollar rolled from one month into the next, usually outperforms holding the TBA position. Exhibit 1 shows the historical financing advantage of dollar rolling TBA FNMA current coupons from 1998–2002. As can be seen, the implied repo rate on the cash position was one-month LIBOR minus 49 basis points. However, for an index-based investor, the question is not just whether to roll TBA mortgages or hold them outright. Clearly, one rolls when the roll is special. The question is: How much one should reconstruct their portfolio to take advantage of dollar rolls? For example, rather than owning seasoned paper, should investors who dollar roll place the entire coupon allocation into the TBA bucket and roll it? Or, taking that one step further, should index-based investors just overweight a few coupons which roll well? To answer this question, we examined the performance of three difference strategies, each centered on current coupons versus a broad mortgage index.

CURRENT COUPON MORTGAGE To begin, we first selected a current coupon mortgage at each point in time for each of 30-year FNMAs, 30-year GNMAs, and 15-year FNMAs.


Dollar Rolling—Does It Pay?

EXHIBIT 2

133

Perfect and Assumed Current Coupon

These three sectors were chosen because they are representative of, and capture, the overwhelming majority of the MBS market. Currently, 30year conventionals comprise 61% of fixed-rate Agency mortgages outstanding; 30-year GNMAs are 20%; while 15-year conventionals represent 18%. (By contrast, 15-year GNMAs and conventional balloons each only consist of about 1% of the market.) To avoid switching the current coupon mortgage so quickly that there was insufficient production in the new current coupon, we used a 60-day moving average of the perfect current coupon. Thus, in November 2001, the perfect current coupon hit a low of 5.65%, but rates did not stay that low sufficiently long so that the 6.0s, let alone the 5.5s, ever became the current coupon. Exhibit 2 compares the assumed current coupon for 30year FNMAs to the perfect current coupon. As can be seen from this analysis, the 6.5 coupon has been the assumed current coupon for over a year.

DOLLAR ROLL STRATEGIES We used three current coupon-based dollar roll strategies compared to an aggregate mortgage index: Strategy #1: This strategy involves our “liquid 9.” That is, being long 9 TBA coupons at any one time (3 GNMAs + 3 FNMAs + 3 Dwarfs). Of


134


that total notional amount of contracts held, we allocated (60% FNMAs, 25% GNMAs, 15% Dwarfs). Within each sector, 40% is held in the current coupon, with 30% in each of the coupons above and below. Strategy #2: This strategy uses only three TBA coupons. We used 60% FNMA current coupon, 25% GNMA current coupon, and 15% Dwarf current coupon. Strategy #3: This stategy involves rolling only the FNMA current coupon. We used 100% FNMA current coupon. This is the simplest of the three strategies. To implement each strategy, we assumed that an investor is long the TBA contracts, and has invested the cash at 1-month LIBOR flat. We also assumed that we maintained the TBA contract for settlement in the next month. Thus in April, we held a position in May TBA contracts. On May 1, we rolled into the June contract, and used that TBA for the entire month of May. Then, on June 1, we rolled into the July contract. Thus, our total return on the rolling strategy is the return on the cash plus the return on the TBA position.

RESULTS We compare the results of each of these three strategies to the actual return from holding a broad mortgage index in Exhibit 3. Note that our data for this analysis begins at year end 1991, and this exhibit shows annual returns over a 10.5-year period. Thus, as can be seen in the exhibit, from 1992–2002, Strategy #1 outperformed the index by 47 basis points. The second strategy outperformed by 52 basis points, while the simplest, Strategy #3 (rolling only the current coupon), outperformed on average by 60 basis points. While aggregate performance of the dollar roll strategies was very strong, there were five years during which that strategy underperformed the aggregate mortgage index. These consisted of 1994, 1995, 1996, 1999 and 2000. Thus, for example, in 1994, Strategy #3 (rolling the current coupon FNMA only) underperformed the index by 203 basis points. The periods of underperformance were long enough and severe enough to suggest that while dollar rolling is on average an excellent strategy, there still are times during which it does not pay.


135


EXHIBIT 3

a b

TRORs of Various Roll Strategies Strategy

Advantage

Year

Mtg Index

1

2

3

1

2

3

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002a 92–02b

6.99 6.86 –1.61 16.80 5.34 9.53 6.98 1.87 11.15 8.25 2.59 6.96

9.27 11.55 –3.37 16.61 3.90 10.92 9.00 0.60 10.30 8.41 3.02 7.43

9.23 11.70 –3.57 16.66 3.97 11.03 9.17 0.66 10.28 8.60 3.06 7.48

9.55 11.66 –3.64 16.56 4.07 11.35 9.17 0.70 10.37 8.62 3.19 7.56

2.28 4.69 –1.76 –0.19 –1.45 1.39 2.02 –1.27 –0.85 0.16 0.43 0.47

2.24 4.84 –1.96 –0.14 –1.37 1.51 2.19 –1.21 –0.87 0.35 0.48 0.52

2.57 4.79 –2.03 –0.24 –1.27 1.82 2.19 –1.17 –0.78 0.36 0.61 0.60

Thru 5/14/02. Annualized BEY.

It is important to realize that the over- or underperformance of our dollar roll strategies is not explained simply by specialness of the rolls. The dollar roll strategy will underperform during periods in which seasoned collateral does very well versus TBAs. It will also underperform when super premium collateral outperforms current coupon collateral.

TRACKING ERROR Exhibit 4 shows historical month-by-month tracking error on Strategy #3 versus the mortgage index. (Since the three series have 99.9% total rate-of-return correlation, we don’t need to show all three.) As can be seen from the exhibit, monthly tracking error in the later period appears to be substantially lower than in the earlier period. The middle section of Exhibit 5 shows that this is indeed the case. This exhibit calculates both the average monthly excess total rate of return, as well as the standard deviation of excess return, for the entire period 1992–2002, plus for two subperiods, 1992–1996 and 1997–2002. As can be seen, both average excess return and the volatility of excess returns in recent years is approximately one half that of earlier years.


136


EXHIBIT 4

Historical Monthly Excess Returns of Strategy #3

EXHIBIT 5

Monthly Excess TROR

Period

Strat1

Strat2

Strat3

Average 92-02 92-96 97-02

0.041 0.055 0.027

0.045 0.055 0.035

0.051 0.059 0.044

Std Dev 92-02 92-96 97-02

0.427 0.573 0.225

0.435 0.581 0.235

0.434 0.581 0.231

Average/Std Dev 92-02 92-96 97-02

0.095 0.097 0.119

0.103 0.095 0.149

0.117 0.101 0.189



137

The bottom panel of Exhibit 5 (which is simply average excess return/standard deviation of excess return) indicates that the risk/return trade-off has actually improved slightly. That is, for the earlier period, Strategy #3 produced risk-adjusted returns of 0.101, versus 0.189 in the later period. We believe that the standard deviation in the earlier period may be overstated due to data quality (which is lower than what we currently have). However, there is no question that the market is more efficient now, which drives both average excess returns and standard errors lower than what we used to see.

WHY DOLLAR ROLLS ARE SPECIAL? We originally decided to take a close look at the degree to which excess returns can be determined by the level of rates, the shape of the curve, and volatility. Our thought was that if we could explain when investors should be dollar rolling TBAs versus holding a broad mortgage index, we would actually be able to dispense some pretty interesting investment advice in this boringly slow market. But, unfortunately, our regression results had little explanatory power (a 4% R-squared). But then we went back to basics. Intuitively, it always seems that dollar rolls are the most robust during periods of heavy collateralized mortgage obligation (CMO) activity and heavy mortgage production. Mortgage bankers/originators usually sell their production a month or two forward, corresponding to the date when those new mortgages will close. By contrast, investors usually prefer to take delivery on investments as soon as possible. Thus, there is a timing mismatch—originators selling in the back months and CMO dealers trying to close deals in the front months. This configuration generally produces the most favorable dollar roll environment. It also explains why the dollar rolls are most special in the current coupon—because that is the most common coupon for both mortgage banker selling and CMO production. Exhibit 6 indicates that the MBA Index should have some explanatory power. So, we once again tried our hand at analytics. It turned out that the MBA Refi Index itself could not explain variations in excess returns. However, modeling excess return as a function of a (1,–1) binary variable as a function of whether the index was above 500 or not gave us an R-squared of 39%. This suggests that higher levels of the refi index are loosely correlated with periods in which the “dollar rolling” strategy was successful.


138 EXHIBIT 6


Excess Returns versus MBA Index

CALENDAR EFFECTS We have shown that on average, dollar rolls have added substantially to excess returns. We further showed that periods in which dollar rolls were special are those of heavy CMO production and heavy originator selling. We now show that there is a seasonal effect, with dollar rolls more special during the summer months. Exhibit 7 shows actual and forecast production over the 7-year period 1996–2002. As can be seen from this exhibit, summer production is generally heavier than the rest of the year. As a result, Exhibit 8 shows that dollar rolls are generally slightly special over the summer months. In fact, Months 4–7 are the most special using this analysis. Month 4 (April) reflects the May/June roll; Month 5 (May) the June/ July roll, and so on.

CONCLUSION Dollar rolls have been very important in consistently adding to the total rate-of-return on mortgage portfolios. Since none of the mortgage indices include this, it is a great way, on average, to outperform the index. However, during periods in which seasoned collateral generates great excess



139

returns, dollar rolling TBAs will underperform. In fact, in our analysis, dollar rolling a TBA portfolio did underperform for 5 of the 10.5 years in our sample. So clearly, that strategy cannot be blindly applied. The periods in which dollar rolls are very special can contribute so substantially to the long-term performance record that any money manager that ignores the benefits does so at their own peril. EXHIBIT 7

MBS Production Model

EXHIBIT 8

Calendar Effect—Roll Specialness


7-Mahadevan-Valuing Page 141 Wednesday, July 23, 2003 10:32 AM

Valuing Corporate Credit: Quantitative Approaches versus Fundamental Analysis Sivan Mahadevan Vice President Morgan Stanley Young-Sup Lee Vice President Morgan Stanley David Schwartz Associate Morgan Stanley Stephen Dulake Executive Director Morgan Stanley Viktor Hjort Associate Morgan Stanley

n this article, we compare fundamental approaches to valuing corporate credit with quantitative approaches, commenting on their relative merits and predictive powers. On the quantitative front, we first review structural models, such as KMV and CreditGrades™, which use infor-

I

141


142


mation from the equity markets and corporate balance sheets to determine default probabilities or fair market spreads. Second, we describe reduced form models, which use information from the fixed income markets to directly model default probabilities. Third, we review simple statistical techniques such as factor models, which aid in determining relative value. With respect to fundamental approaches, we examine rating agency and credit analyst methodologies in detail.

QUANTITATIVE APPROACHES TO VALUING CORPORATE CREDIT Quantitative approaches for analyzing credit have existed for decades but have surged in popularity over the last few years. This is due, in large part, to several trends in the credit markets: ■ As credit spreads have widened and default rates have increased, inves-

tors have looked to increase their arsenal of tools for analyzing corporate bonds. Quantitative models can be used to provide warning signals or to determine whether the spread on a corporate bond adequately compensates the investor for the risk. ■ The number of investors interested in credit products has grown worldwide. In part, this can be attributed to declining yields on competing investments and the expansion of the European corporate bond market following the introduction of the euro. Commercially available credit models have been developed to meet the growing investor demand. ■ The rapidly expanding credit derivatives market, which includes credit default swaps and collateralized debt obligations, has spurred a new generation of quantitative models. For derivative products, quantitative techniques are critical for valuation and hedging. ■ Risk management has become increasingly important for financial institutions. The need to compute “value at risk” and determine appropriate regulatory capital reserves has led to the development of sophisticated quantitative credit models. In this section, we introduce some popular quantitative techniques for analyzing individual credits. (We discuss quantitative methods for portfolio products later in this article.) The goal of these methods is to estimate default probabilities or fair market spreads. Although many different quantitative techniques are practiced in the market, we focus on two different approaches for modeling default: structural models and reduced form models.


Valuing Corporate Credit: Quantitative Approaches versus Fundamental Analysis

EXHIBIT 1

143

Stylized Balance Sheet

Assets

Claims on Assets

Assets of the firm

Liabilities (Debt) 1-year zero-coupon bond with face value of $100 million Equity Common shares

Source: Morgan Stanley.

Structural models use information from the equity market and corporate balance sheets to model a corporation’s assets and liabilities. Default occurs when the value of the corporation’s assets falls below its liabilities. Structural models are used to infer default probabilities and fair market spreads. KMV and CreditGrades are two commercial examples of this approach. Unlike structural models, reduced form models rely on information from the fixed income market, such as asset swap spreads or default swap spreads. In these models, default probabilities are modeled directly, similar to the way interest rates are modeled for the purpose of pricing fixed income derivatives. These models are particularly useful for pricing credit derivatives and basket products. For comparison to default-based models, we briefly present a simple factor model of corporate spreads. It focuses on the relative pricing of credit, using linear regression to determine which bonds are rich or cheap. The factors used in the model include credit rating, leverage (total debt/EBITDA), duration, and recent equity volatility.

Structural Models In the structural approach, we model the assets and liabilities of a corporation, focusing on the economic events that trigger default. Default occurs when the value of the firm’s assets falls below its liabilities. The inputs to the model are the firm’s liabilities, as projected from its balance sheet, as well as equity value and equity volatility. An option pricing model is used to infer the value and volatility of the firm’s assets. To see why an option pricing model is at the heart of the structural approach, consider a simple firm that has issued a single one-year zerocoupon bond with a face value of $100 million. A stylized balance sheet for this firm is shown in Exhibit 1. The key insight comes from examining the values of the equity and debt in one year, when the debt matures. If in one year the value of assets is $140 million, then the $100 million due to bondholders will be


144


paid, leaving the value of equity at $40 million. On the other hand, if in one year the value of assets is $60 million, equity holders can “walk away,” turning over the $60 million in assets to the bondholders. Because equity holders have limited liability, the value of equity is $0. The payoff diagram for equity and debt holders in one year as a function of assets is shown in Exhibit 2. From the “hockey stick” shape of the payoff diagram for equity holders, it is clear that equity can be thought of as a call option on the assets of the firm. In this example, the strike is the face value of the debt, $100 million. Similarly, the zero-coupon corporate bond is equivalent to being long a risk-free zero-coupon bond and short a put option on the assets of the firm. With the key insight that equity can be considered a call option on the assets of the firm, the rest of the structural approach falls into place. Exhibit 3 shows the steps involved in implementing a structural model. Equity value and volatility, along with information on the firm’s liabilities, are fed into an option pricing model in order to compute the implied value and volatility of the firm’s assets. Having computed the value and volatility of the firm’s assets, we can determine how close the firm is to default. This “distance to default” can be translated into a probability of default, or it can be used to determine the fair spread on a corporate bond. EXHIBIT 2

Value of Equity and Debt in One Year




EXHIBIT 3

145

Implementation of a Structural Model


Example: Merton’s Original Model To illustrate the calculations behind structural models, we consider the original structural model described by Robert Merton.1 We revisit our simple firm, which has a single one-year zero-coupon bond outstanding with a face value of $100 million. Furthermore, assume that the equity is valued at $30 million and has a volatility of 60%, and that the riskfree interest rate is 4%. These parameters are summarized in Exhibit 4. Step 1: Computing Asset Value and Volatility In Merton’s original approach, equity is valued as a call option on the firm’s assets using the Black-Scholes option pricing formula (N refers to the cumulative normal distribution function): E = AN ( d 1 ) – Fe

– rT

N ( d2 )

where 1

Robert C. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance 29 (1974), pp. 449–470.


146


EXHIBIT 4

Parameters for Structural Model Example

Inputs Value of Equity Volatility of Equity Face Value of Debt Maturity of Debt Risk-free Interest Rate

E = $30 million σE = 60%

F = $100 million T = 1 year r = 4%

Outputs Value of Assets Volatility of Assets

A=? σA = ?

Source: Morgan Stanley. 2

log ( A ⁄ F ) + ( r + σ A ⁄ 2 )T' d 1 = --------------------------------------------------------------σA T and d2 = d1 – σA T In the Black-Scholes framework, there is also a relationship between the volatility of equity and the volatility of assets:2 A σ E = σ A N ( d 1 ) ---E The Black-Scholes formula and the relationship between equity volatility and asset volatility provide two equations, which we must solve for the two unknown quantities: the value of assets (A) and the volatility of assets (σA). Solving the equations yields A = $125.9 million and σA = 14.7%.3 Step 2a: Computing Fair Market Spreads Having computed the implied asset value and volatility, we can now determine the implied spread on the zero-coupon bond over the risk-free rate. To do this, we note that the value of the debt is equal 2 This equation is derived from Ito’s lemma. For details, see John C. Hull, Options Futures and Other Derivatives: Third Edition (Upper Saddle River, NJ: Prentice Hall, 1997). 3 These two equations can be solved simultaneously in a spreadsheet by an iterative procedure (e.g., Goal Seek or Solver in Excel).



147

to the value of the assets minus the value of the equity. That is, the value of the debt equals $125.9 million – $30 million = $95.9 million. Since the face value of the debt is $100 million, we can easily determine that the yield on the zero-coupon bond is 4.22%, which corresponds to a spread of 22 basis points over the risk-free rate. At this point, it is worth noting that it is difficult to get “reasonable” short-term spreads from Merton’s original model. In part, the reason for this is that the asset value is assumed to follow a continuous lognormal process, and the probability of being significantly below a static default threshold after only a short amount of time is low. In this example, the spread of 22 basis points probably underestimates what would be the observed spread in the market. In practice, adjustments are made to Merton’s basic structural model in order to produce more realistic spreads. Step 2b: Computing Distance to Default and Probability of Default One popular metric in the structural approach is the “distance to default.” Shown graphically in Exhibit 5, the distance to default is the difference between a firm’s asset value and its liabilities, measured in units of the standard deviation of the asset value. In short, it is the number of standard deviations that a firm is from default. In the Black-Scholes-Merton framework, the distance to default is equal to d2, from above. Using the values of A and σA computed earlier, we calculate the distance to default to be 1.76. In other words, the projected asset value is 1.76 standard deviations above the default threshold. EXHIBIT 5

Distance to Default



148


The distance to default, d2, is important because it is used to compute the probability of default. In the Black-Scholes-Merton framework, the risk-neutral probability of default is N(–d2). In our example, the risk-neutral probability of default is N(–1.76), which equals 3.1%. Recovery Rates In Merton’s model, recovery rates are determined implicitly. In this example, if the value of assets in one year is $80 million, then the corporation defaults, and bondholders recover $80 million. We can also compute the expected recovery rate (under the risk neutral measure). Conditional on the default of the company, the expected value of assets to be recovered by debtholders is given by A N(–d1)/N(–d2). In this example, expected recovery value is $90.7 million. This is higher than we would likely observe, for the same reason that the model underestimates short-term spreads.

Extending Merton’s Original Model The original Merton model outlined above features a firm with a single zero-coupon bond and a single class of equity. Models used in practice will be more elaborate, incorporating short-term and long-term liabilities, convertible debt, preferred equity, and common equity. In addition, models used in practice are more sophisticated, in order to produce more realistic spreads, default probabilities and recovery rates. The following list of modeling choices is representative of some of the more popular extensions to Merton’s original model: ■ The default threshold need not be a constant level. It can be projected

to increase or decrease over time. ■ Default can occur at maturity, on coupon dates or continuously.

Exhibit 6 shows three possible paths for a firm’s asset value over the next year. In Merton’s original model, where default can only occur at maturity, the firm defaults only in asset value path C, where the recovery rate is 80%. If the default barrier is continuous, the firm defaults in asset value paths B and C, as soon as the asset value hits the default barrier. The recovery rate would be determined separately. ■ The default threshold can have a random component, reflecting imperfect information about current and future liabilities. Indeed, current liabilities may not be observable with sufficient accuracy, for example, because the balance sheet is out of date. Similarly, it is not easy to predict how management will refinance debt or adjust debt levels in the future in response to changing economic conditions.



EXHIBIT 6

149

Sample Asset Value Paths

Source: Morgan Stanley. ■ Asset value need not follow a lognormal distribution. For example, it

can have jumps, reflecting unanticipated surprises that cause asset value to decrease sharply. The option pricing model can be different from the Black-Scholes model, and equity can be modeled as a perpetual option. In addition, asset value and volatility can be inferred from the equity markets in a more robust way, using an iterative procedure that incorporates time series information. ■ Firm behavior can be incorporated into a structural model. One example is a “target leverage” model, in which the initial capital structure decision can be altered. The level of debt changes over time in response to changes in the firm’s value, so that the Debt/Assets ratio is meanreverting. In this model, the firm tends to issue more debt as asset values rise.4 ■ In a “strategic debt service” model, there is an additional focus on the incentives that lead to voluntary default and the bargaining game that occurs between debt and equity holders in the event of distress. These models acknowledge the costs associated with financial distress and the possibility of renegotiation before liquidation.5 4

Pierre Collin-Dufresne and Robert Goldstein, “Do Credit Spreads Reflect Stationary Leverage Ratios?” Journal of Finance 56(5), 2001, pp. 1928–1957. 5 For a simple example, see Suresh Sundaresan, Fixed Income Markets and Their Derivatives, Second Edition (Cincinatti, OH: South-Western, 1997).


150


Commercial Implementations of the Structural Approach Commercial implementations, such as KMV and CreditGrades, have refined the basic Merton model in different ways. Each strives to produce realistic output that can be used by market participants to evaluate potential investments. KMV has extended the basic structural model according to the Vasicek-Kealhofer (VK) model. The primary goal of the model is to compute real-world probabilities of default, which are referred to as Expected Default Frequencies, or EDF™s. The model assumes that the firm’s equity is a perpetual option, and default occurs when the default barrier is crossed for the first time. A critical feature of KMV’s implementation is the sophisticated mapping between the distance to default and the probability of default (EDF). The mapping is based on an extensive proprietary database of empirical default and bankruptcy evidence. As such, the model produces real-world, not risk-neutral, probabilities.6 CreditGrades, a more recent product, is an extension of Merton’s model that is primarily focused on computing indicative credit spreads. In the CreditGrades implementation, the default barrier has a random component, which is a significant driver of short-term spreads. Default occurs whenever the default threshold is crossed for the first time. Parameters for the model have been estimated in order to achieve consistency with historical default swap spreads.7

Advantages of Structural Models There are seven advantages of structural models: ■ Equity markets are generally more liquid and transparent than corpo-

rate bond markets, and some argue that they provide more reliable information. Using equity market information allows fixed income instruments to be priced independently, without requiring credit spread information from related fixed income instruments. ■ Structural models attempt to explain default from an economic perspective. They are oriented toward the fundamentals of the company, focusing on its balance sheet and asset value. ■ Credit analysts’ forecasts can be incorporated into the model to enhance the quality of its output. For example, balance sheet projections can be used to create a more realistic default threshold. The model can also be run under different scenarios for future liabilities.

6 7

Modeling Default Risk, KMV LLC, January 2002. CreditGrades Technical Document, RiskMetrics Group, Inc., May 2002.



151

■ Structural models are well-suited for handling different securities of the

same issuer, including bonds of various seniorities and convertible bonds. ■ A variety of structural models are commercially available. They can be used as a screening tool for large portfolios, especially when credit analyst resources are limited. ■ Structural models can be enhanced, for example, to incorporate firm behavior. Examples include target leverage models and strategic debt service models. ■ Default correlation can be modeled quite naturally in the structural framework. In a portfolio context, correlation in asset values drives default correlation.

Disadvantages of Structural Models The disadvantages of structural models are as follows: ■ If equity prices become irrationally inflated, they may be poor indica-

■

■

■

■

■

tors of actual asset value. The Internet and telecom bubbles of the past few years are perhaps the most striking examples. Generally, users of structural models must believe that they can reasonably imply asset values from equity market information. This can become a significant issue when current earnings are low or negative and equity valuations are high. Bond prices and credit default swap spreads, which arguably contain valuable information about the probability of default, are outputs of the model, not inputs. In Merton’s structural model, implied credit spreads on short-term debt and very high quality debt are very low when compared to empirical data. Refinements to the model have alleviated this problem, at the expense of simplicity. The determination of a unique arbitrage-free option price implicitly assumes that the value of the whole firm is tradable and available as a hedge instrument, which is a questionable assumption. In addition, it may not be clear how to best model a firm’s asset value. Structural models can be difficult to calibrate. In practice, asset values and volatilities are best calibrated using time series information. Assumptions for equity volatility can have a significant impact on the model. Structural models can be complex, depending on the capital structure of the issuer and the level of detail captured by the model. An issuer may have multiple classes of short-term and long-term debt, convertible bonds, preferred shares and common equity.


152


■ It can be difficult to get reliable, current data on a firm’s liabilities.

■

■

■

■

Issues regarding transparency and accounting treatment are, of course, not unique to structural models. In addition, once adequate information on the liabilities is obtained, the information must be consolidated to project a default barrier. Notwithstanding innovations such as target leverage models and strategic debt service models, it is difficult to model future corporate behavior. It can be difficult to model a firm that is close to its default threshold, since firms will often adjust their liabilities as they near default. Firms will vary in terms of their ability to adjust their leverage as they begin to encounter difficulties. (For this reason, KMV reports a maximum EDF of 20%.) Financial institutions should be modeled with caution, since it can be harder to assess their assets and liabilities. In addition, since financial institutions are highly regulated, default may not be the point where the value of assets falls below the firm’s liabilities. Structural models are generally inappropriate for sovereign issuers.

Reduced Form Models In the reduced form approach, default is modeled as a surprise event. Rather than modeling the value of a firm’s assets, here we directly model the probability of default. This approach is similar to the way interest rates are modeled for the purpose of pricing fixed income derivatives. Unlike the structural models described above, the inputs for reduced form models come from the fixed income markets in the form of default swap spreads or asset swap spreads. The quantity we are actually modeling in the reduced form approach is called the hazard rate, which we denote by h(t). The hazard rate is a forward probability of default, similar to a forward interest rate. The hazard rate has the following interpretation: Given that a firm survives until time t, h(t)∆t is the probability of default over the next small interval of time ∆t. For example, assume that the hazard rate is constant, with h = 3%. Conditional on a firm surviving until a given date in the future, its probability of default over the subsequent one day (0.0027 years) is approximately h∆t = 3% × 0.0027 = 0.008%. Letting τ represent the time to default, the hazard rate is defined mathematically as follows: Prob ( τ ≤ t + ∆t τ > t ) h ( t ) = ----------------------------------------------------∆t



EXHIBIT 7

153

Cumulative Probability of Default—3% Hazard Rate


Three features of hazard rates make them particularly useful for modeling default. Even though the hazard rate is an instantaneous forward probability of default, it tells us the probability of default over any time horizon. For example, assume a constant hazard rate. The probability of a bond defaulting in the next t years is 1 – e–ht. If h = 3%, the probability of the firm defaulting in the next two years is 1 – e–0.03(2) = 5.82%. A graph of the cumulative default probability when h = 3% is shown in Exhibit 7. Hazard rates can be inferred from the fixed income markets, in the form of default swap spreads or asset swap spreads. For example, assuming a constant hazard rate, the default swap premium is approximately equal to h × (1 – Expected Recovery Rate). If the default swap premium is 180 basis points and the expected recovery rate is 40%, we can set h = 1.80%/(1 – 0.40) = 3%. Hazard rates are convenient for running simulations to value derivative and credit portfolio products. In a portfolio context, a simulation would allow for defaults to be correlated. Assuming a constant hazard rate, we can simulate the time to default as follows: We can repeatedly generate values between 0 and 1 for the uniform random variable U and use the relation τ = –log(U)/h for the time to default. For example, with h = 3%, if in the first path of a simulation U = 0.757, the corresponding time until default is –log(0.757)/0.03 = 9.28 years. In the examples above, we have assumed that hazard rates are constant. The real exercise, however, is to model the hazard rates. Like interest rates, hazard rates are assumed to have a term structure, and


154


they are assumed to evolve randomly over time. Models for interest rates, such as a lognormal model or the Cox-Ingersoll-Ross model, can be used to model hazard rates. In addition, it is not uncommon for models of hazard rates to incorporate jumps that occur at random times. Hazard rate models are typically calibrated to a term structure of default swap spreads or asset swap spreads.

Advantages of Reduced Form Models The advantages of reduced form models are as follows: ■ Reduced form models are calibrated to the fixed income markets in the

■

■ ■ ■

■

form of default swap spreads or asset swap spreads. It is natural to expect that bond markets and credit default swap markets contain valuable information regarding the probability of default. Reduced form models are extremely tractable and well-suited for pricing derivatives and portfolio products. The models are calibrated to correctly price the instruments that a trader will use to hedge. In a portfolio context, it is easy to generate correlated hazard rates, which lead to correlated defaults. Hazard rates models are closely related to interest rate models, which have been widely researched and implemented. Reduced form models can incorporate credit rating migration. However, for pricing purposes, a risk-neutral ratings transition matrix must be generated. Reduced form models can be used in the absence of balance sheet information, e.g., for sovereign issuers.

Disadvantages of Reduced Form Models The disadvantages of reduced form models are as follows: ■ Reduced form models reveal limited information about the fixed

income securities that are used in their calibration. ■ Reduced form models can be sensitive to assumptions, such as the vol-

atility of the hazard rate and correlations between hazard rates. ■ Even if hazard rates are highly correlated, the occurrences of default

may not be highly correlated. For this reason, practitioners pay careful attention to which particular process hazard rates are assumed to follow. Models with jumps have been used to ameliorate this problem. ■ Whereas there is a large history on interest rate movements that can be used as a basis for choosing an interest rate model, hazard rates are not directly observable. (Only the events of default are observable.) Thus, it may be difficult to choose between competing hazard rate models.



155

Factor Models For comparison to the default-based pricing models described above, we include a brief discussion of a simple factor model of investment grade corporate spreads.8 Unlike the structural and reduced form models, the factor model does not attempt to model default in order to gain insight into fair market prices. Rather, it is a simple statistical approach to the relative pricing of credit and used to determine which bonds are rich or cheap. This factor model uses linear regression to attribute spreads to various characteristics of the bonds being analyzed. The idea is to quantify the importance of various drivers of corporate bond spreads. The residual from the regression is used to indicate rich and cheap securities. Some potential factors for investment grade credit are shown in Exhibit 8. Later in this article, in the section on Historical Analysis of Quantitative and Fundamental Approaches, we review the performance of this factor model, along with other quantitative and fundamental approaches. EXHIBIT 8

Sample Factor Model Inputs Factor

Type

Description

Total Debt/EBITDA Rating Watchlist Duration Stock Returns Stock Volatility Quintile of Debt Outstanding 10- to 15-year Maturity Gaming Cyclical Finance Technology Global AAA/AA Yankee

Numeric Numeric {–2,–1,0,1,2} Numeric Numeric Numeric {1,2,3,4,5} Numeric {0,1} {0,1} {0,1} {0,1} {0,1} {0,1} {0,1}

Measure of leverage Scaled to a numeric value On watchlist, negative or positive Modified duration 1-year total return Price volatility over last 90 days E.g., top 20% = 5th quintile Years to maturity >10 but 10 but < 15 E.g., casinos E.g., retail, autos E.g., banks, finance, brokerage E.g., software, hardware Global bond Rated Aaa/AA or Aa/AA or split Yankee bond



168 EXHIBIT 17


Factor Model Performance


The study included a universe of 2,000 investment grade corporate bonds. A linear regression was conducted each month where we calculated a residual (i.e., actual spread minus the model’s predicted spread) for each bond in the universe. A positive residual value indicates cheapness of the credit, while a negative value suggests richness. Rich-cheap residuals are not statistically significant unless their magnitudes are at least twice the standard error of the regression (standard deviation of all the residuals), which, in our experience, can be over 30 bp in a given month. The results of our study show that the factor model is quite successful at determining relative value among bonds. The factor model’s cheapest decile tightened significantly more than other bonds in nine of 16 months. Similarly, its richest decile significantly widened in nine of the 16 months. In Exhibit 17 we show the cumulative spread changes for richest and cheapest deciles (which are recomputed every month) and for the entire universe. The cheapest decile tightened an average of 160 bp versus the entire universe, while the richest decile widened 70 bp over that same period.

KMV EDFs Are Not as Useful for Relative Value Since many market participants are attempting to use KMV EDF data to predict relative spread changes, we studied how well this worked. It is important to note, however, that KMV is meant to be a predictor of default, not spreads. In studying how well KMV predicted spread changes, we determined richness and cheapness by comparing KMV EDFs to marketimplied probabilities of default. These implied default probabilities are derived from the market spread and an assumed recovery rate.



EXHIBIT 18

169

KMV Spread Model Performance

Source: Morgan Stanley, KMV.

Similarly to our factor model study, we observed the ensuing month’s spread change for the cheapest and richest deciles of this EDF-based relative value measure. The results for the EDF signals, shown in Exhibit 18, are not as compelling as the factor model. In the EDF study, the cheapest bonds rallied by 68 bp, while the richest widened by only 24 bp. The fact that KMV EDFs are poorer predictors of relative spread movements than our factor model does not surprise us. EDFs are designed to be predictors of default probability, not spread movement. To test this hypothesis, we conducted a default probability study using over 800 investment-grade and high-yield issuers covered by KMV for the years 2000 and 2001. We ranked all companies by their prior year-end EDFs, divided the universe into deciles based on absolute EDFs, and calculated the average EDF for each decile. If EDFs are a good predictor of the actual probability of default, companies in each decile should default over the next year by roughly that same average EDF. Exhibit 19 shows the results for our study for years 2000 and 2001. Our conclusions are as follows: ■ KMV default predictions were within 0% to 3% of actual default

experience within each decile. ■ During 2001, a more active year for corporate defaults than 2000,

KMV default predictions were remarkably close to actual default experience, particularly in the highest deciles (those with the highest default probabilities). We believe these results are robust, demonstrating that KMV EDFs are good predictors of default, at least over this period. Furthermore, our study did not show that KMV EDFs raised too many false negatives (high


170

EXHIBIT 19


EDF as Predictor of Default Year 2000

EDF as Predictor of Default Year 2001


EDFs that were disproportionate to default experience), a common market criticism. Default experience was consistent with default probability.

Spreads Were Less Reliable Predictors of Default For comparison, we investigated whether the market itself was a good predictor of default. If this were true, then tools such as KMV might not be as useful, since the information would be already priced into the market.



EXHIBIT 20

171

Spread-Implied Default Probability as Predictor of Default Year 2000

Spread-Implied Default Probability as Predictor of Default Year 2001


To answer this question, we conducted a study comparing one-year market-implied default rates with actual default experience, where market-implied rates are derived from market spreads and a recovery rate assumption. Our study included over 1,200 issuers over the 2000 and 2001 periods. As in the KMV study, we ranked each year’s starting implied default probabilities and divided the population into deciles. We compared each average to the actual default rate experienced over the following year. Exhibit 20 shows the results of our study. Our conclusions are as follows. First, market-implied default rates overestimated


172


default for most of the high-risk deciles by 5–8% and by 1–3% for the low-risk deciles. The overestimation is understandable, given that the market has priced in an additional risk premium and liquidity premium. However, during 2001, market-implied default rates for the highest risk decile actually underestimated default despite the risk premium.

Free Cash Flow Good at Relative Value Empirically testing the fundamental approach to credit analysis is not a straightforward task given the subjective nature of the output. Instead, we focus our empirical testing on a simple metric that captures some of what analysts attempt to understand: free cash flow generation. We first tested the hypothesis that free cash flow generation is a good predictor of relative spread in 2001.12 Results from that study are presented in Exhibits 21 and 22, based on a universe of approximately 200 nonfinancial U.S. corporate issuers. The study was backward looking in the sense that the universe was sorted into quintiles based on spread performance during calendar year 2000 (see Exhibit 21), and then free cash flow dynamics were observed for these quintiles from 1998 through 1999 (see Exhibit 22). We observed that companies within the poorest performing quintile experienced lower levels of free cash flow generation in 1999 relative to 1998. The best performers through 2000, on the other hand, generated more cash in 1999 relative to 1998. EXHIBIT 21 Calendar 2000 Median Spread Performance: Spread Widening versus Treasuries


See “The Bottom Line,” Morgan Stanley Fixed Income Research, February 27, 2001.



EXHIBIT 22

173

Median Free Cash Flow/Debt Changes: 1998 versus 1999


EXHIBIT 23

Sector 1998-1999 Free Cash Flow and 2000 Spread Performance


Exhibit 23 shows median spread performance versus free cash flow trends for the major sectors. Again, prior free cash flow trends are reasonably descriptive of subsequent performance.


174 EXHIBIT 24


Sector 1999–2000 Free Cash Flow Changes versus 2001 Spread

Performance


We conducted a similar study for European issuers more recently (spread changes in 2001 based on free cash flow dynamics from 1999 to 2000). In Exhibit 24 we show the free cash flow sector relationships based on a universe of the top 50 nonfinancial European corporate bond issuers, which account for about 70–80% of all European corporate debt outstanding. Again, we believe that free cash flow generation was a good predictor of spread change.

Ratings Are Consistent with Historical Experience Ratings agencies have been criticized for being both too slow and too quick in their ratings decisions. The agencies, for their part, consider it their job to produce ratings that, over time, match a default rate (expected loss), which in turn is based on historical experience. Hence, when judging the performance of the agencies, one needs to focus on the historical relationship between ratings and default rates. Exhibit 26 shows average cumulative default rates by rating using Moody’s historical data from 1970–2001. The data show a strong correlation between ratings and default rates. Over a five-year horizon, for instance, the cumulative default rate of Baa-rated companies is almost 14 times that of Aaa-rated companies. Similarly, the cumulative default rate of speculative grade companies is almost 23 times that of investment grade companies.



EXHIBIT 25

175

S&P Average Cumulative Default Rates (1987–2000)

Outlook

Rating

Stable Negative Positive Stable Negative Positive Stable Negative Positive Stable Negative Positive Stable Negative Positive Stable Negative Positive Stable Negative

AAA AAA AA AA AA A A A BBB BBB BBB BB BB BB B B B CCC CCC CCC

Year 1 (%)

Year 2 (%)

Year 3 (%)

0.00 0.00 0.00 0.00 0.10 0.00 0.03 0.07 0.10 0.15 0.19 0.12 0.34 2.64 2.42 2.76 9.65 2.08 7.84 29.18

0.00 0.00 0.00 0.03 0.22 0.00 0.05 0.21 0.33 0.20 0.52 1.30 1.72 6.86 7.55 8.45 18.05 2.08 15.16 37.95

0.00 0.00 0.00 0.07 0.35 0.00 0.07 0.29 0.33 0.39 1.04 2.35 3.59 10.44 12.63 12.80 23.72 6.25 20.42 44.53

Source: S&P.

Exhibit 25 illustrates the relationship between ratings outlooks and subsequent defaults. Speculative-grade issuers with negative outlooks are, on average, nearly five times more likely to default than those with positive outlooks. The multiple is highest for the one-year default rate, in which companies with negative outlooks are over nine times more likely to default.

QUANTITATIVE APPROACHES TO VALUING CREDIT PORTFOLIO PRODUCTS While we have focused our efforts on understanding how to value corporate credit in a single-name context, the portfolio perspective is important as well. At one level, a portfolio can simply be thought of as an aggregation of individual investments. In this respect, nearly every investor in the credit markets is managing a credit portfolio, some relative to a benchmark (which is also an aggregation of single names), others to a set of liabilities or investment guidelines.

0.07

176 4.73 1.54

AllCorps

3.08

9.55

0.19

Source: Moody’s.

0.06

Speculative Grade

21.99 34.69

Investment Grade

Caa-C

6.66 13.99

3.57

0.46

1.27

Baa

B

0.02 0.15

A

0.04

—

2

Ba

— 0.02

Aa

1

4.46

13.88

0.38

44.43

20.51

6.20

0.97

0.21

0.08

—

3

5.65

17.62

0.65

51.85

26.01

8.83

1.44

0.35

0.20

0.04

4

6.67

20.98

0.90

56.82

31.00

11.42

1.95

0.51

0.31

0.14

5

7.57

23.84

1.19

62.07

35.15

13.75

2.54

0.68

0.44

0.25

6

8.34

26.25

1.50

66.61

39.11

15.63

3.16

0.87

0.56

0.37

7

9.04

28.42

1.81

71.18

42.14

17.58

3.75

1.07

0.69

0.49

8

9.71

30.40

2.15

74.64

44.80

19.46

4.40

1.32

0.79

0.64

9

10.37

32.31

2.15

77.31

47.60

21.27

5.09

1.57

0.89

0.79

10

11.03

34.19

2.51

80.55

49.65

23.23

5.85

1.84

1.01

0.96

11

11.70

36.05

2.89

80.55

51.23

25.36

6.64

2.09

1.18

1.15

12

7.42

2.38

1.37

1.36

13

12.36

37.83

3.30

80.55

52.91

27.38

Moody’s Average Cumulative Default Rates by Letter Rating, 1970–2001

Aaa

EXHIBIT 26

12.98

39.44

3.72

80.55

54.70

29.14

8.23

2.62

1.64

1.48

14

13.58

40.84

4.15

80.55

55.95

30.75

9.10

2.97

1.76

1.60

15

14.22

42.37

4.60

80.55

56.73

32.62

9.94

3.35

1.90

1.74

16

14.84

43.67

5.08

80.55

57.20

34.24

10.76

3.78

2.13

1.88

17

15.42

44.78

5.58

80.55

57.20

35.68

11.48

4.30

2.31

2.03

18

15.96

45.71

6.55

80.55

57.20

36.88

12.05

4.88

2.62

2.03

19

16.43

46.58

6.96

80.55

57.20

37.97

12.47

5.44

2.87

2.03

20




177

In the total return world, investors are focused on relative portfolio return with respect to their bogeys, and their exposures to individual credits and credit sectors are generally calculated in this relative framework. Given the relatively large size of investment portfolios, the weights of issuers and sectors tend to be proportional to the market. Asset-liability managers focus efforts on forecasting liabilities and finding the portfolios that most efficiently match these liabilities given investment guidelines. Their choices of individual credits and sectors can also be thought of in a relative framework with respect to their guidelines and tolerances for risk. Absolute return portfolio products, such as synthetic baskets and CDOs, require a somewhat different thought process. First, price volatility at the single-name level is not as important as the projected default behavior of the portfolio. Second, the portfolios themselves are generally small enough that issuer and sector weights do not have to be proportional to the market. Structurers and managers have much more freedom in constructing these portfolio products. Third, for tranched portfolio products, such as CDOs, valuation is not as simple as adding up the values of the individual credits. For these products, default correlation directly affects the value of a given tranche.

The Importance of Default Correlation The single-name models of default presented earlier in this article provide a starting point for understanding the default distribution of a portfolio. However, the models need to be extended to account for the expected interrelationship between these credits over the term of the portfolio product. Default correlation is the glue that defines these interrelationships. Why is default correlation so important? Default correlation does not affect the expected number of defaults in a portfolio, but it greatly affects the probability of experiencing any given number of defaults. For example, if default correlation is high, the probability of extreme events (very few defaults or many defaults) will increase, even if the expected number of defaults does not change. The effect of positive default correlations can be quite significant. In Exhibits 27–29, we show the default distribution given by our model for a portfolio of 100 assets, assuming that each has a 10% probability of default and that all pairs of assets have either a 1%, 4%, or 8% default correlation. The greater the positive default correlation, the greater the “fat tail” that the distribution will have. At high correlation levels, the default distribution stands in sharp contrast to the binomial distribution, which features a more symmetric, bell-shaped curve.


178 EXHIBIT 27


Number of Defaults: Default Correlation = 1%


EXHIBIT 28



EXHIBIT 29





179

The traditional measure of default correlation is a correlation between binary (0 or 1) random variables. Specifically, for an issuer X, we define the random variable dX as follows:  1, if issuer X defaults over a given horizon  dX =    0, otherwise  The default correlation between two issuers A and B over a given time horizon is the correlation between dA and dB. The formula is given by p A and B – p A p B ρ ( d A, d B ) = ----------------------------------------------------------p A ( 1 – p A )p B ( 1 – p B ) Here, pA is the probability that A defaults, pB is the probability that B defaults, and pA and B is the joint probability that both A and B default over the horizon being considered. Default probabilities and default correlations are the key ingredients for determining the distribution of the number of defaults. However, it is important to note that knowing default probabilities and correlations does not quite give us everything we need to construct a default distribution. For example, even if we knew the default probabilities for corporations A, B, and C, as well as all default correlations, we still would not know the probability that all three issuers default over the given time horizon.13

Inferring Default Correlation Default correlation cannot be easily observed from history. Historical data on defaults for various sectors is relatively scarce. Moreover, the historical data might not be easily applicable to the two unique companies being considered, or it may not give a satisfactory forward-looking estimate of default correlation. For these reasons, we examine the practice of inferring default correlation from observable market data. We assume that we have already computed the individual probabilities of default for issuers A and B, perhaps by the structural approach or the reduced form approach, both of which were outlined earlier in this article. In order to calculate default correlation using the formula given above, we need to compute pA and B, the joint probability that A and B both default over the time horizon. 13

For this reason, “copula functions” are often used in modeling defaults. A copula function generates a complete distribution given a set of individual probabilities and a set of correlations. There are a variety of copula functions that have been used in practice.


180


One simple approach to computing the joint probability of default is based on the structural model. However, rather than focusing on asset values and asset volatilities, we can exploit the fact that we already know the individual default probabilities for A and B and use standardized normal random variables to simplify the calculations. First, we compute the implied default threshold for each asset, i.e., the default barriers that match the default probabilities. These barriers are given by zA = N–1(pA) and zB = N–1(pB), respectively, where N–1 represents the inverse normal cumulative distribution function. For example, if pA = 5%, the implied default threshold would be N–1(0.05) = –1.645. In other words, a standard normal random variable has a 5% chance of being below –1.645. Second, we use the correlation between asset values in order to compute the joint probability that both A and B default. This probability is given by the bivariate normal distribution function, M. pA and B = M(zA, zB; Asset CorrelationA,B) For example, if pA = 5%, pB = 10%, and the asset correlation is 30%, then pA and B = 1.22%. The default correlation is given by p A and B – p A p B ρ ( d A, d B ) = ----------------------------------------------------------p A ( 1 – p A )p B ( 1 – p B ) Using this formula, we compute that the default correlation equals 11.1%. As in this example, default correlations will typically be substantially lower than asset correlations. It is important to note that asset correlation, like asset value, may be a difficult parameter to observe, so equity correlation is often used as a proxy. This approximation often works well. However, equity correlations may be markedly different from asset correlations for firms with asset values near their default points, or for highly leveraged firms that are very sensitive to interest rates, such as financials and utilities.

Using Default Correlation Default correlation has two key applications for absolute return portfolio products. First, default correlation is critical for the valuation of tranched portfolio products such as CDOs. For example, the reduced form models presented earlier in this article can be easily extended to account for default correlation, and a simulation could be run to determine the value of a given tranche. Second, default correlation can be used to construct a portfolio. For example, a default correlation matrix



181

would be a key input to a portfolio optimization process, where the objective is to minimize the variance of the fraction of the portfolio that defaults, subject to various constraints. For absolute return products, the use of these quantitative techniques is becoming increasingly widespread.

CONCLUSION Clearly the topics we have discussed in this article are individually worthy of much more in-depth research. Our purpose in juxtaposing them in this article is to help investors gain insight into valuing corporate credit and select the most appropriate approach, or combinations of approaches, for a given situation. These approaches each have their benefits and drawbacks, and we recommend that investors think about a given company along the three dimensions noted earlier to help decide which approach is best: ■ Distance to default ■ Leverage, or the ability to service debt from operations ■ The management option to change the capital structure

Another issue which can dictate the usefulness of the various approaches is investor profile. In particular, it is important to distinguish those investors who are sensitive to mark-to-market fluctuations from those who are focused on absolute return to maturity. The latter may find the long-term signals provided by credit analysts, rating agencies, and quantitative models to be more important than the near-term risks priced into the market. Furthermore, as we highlighted in our section on credit portfolios, an investor’s benchmark is also an important consideration, as those portfolios that are not forced to be anchored to the market can apply methods to find value relative to the market. Finally, it is important to understand that credit investors, traders, and analysts do not have to select a single approach to value corporate credit as combinations of approaches may prove to be particularly insightful. For example, credit analysts could find structural models very useful in measuring the sensitivity of company valuations to changes in balance sheet items and cash flow projections. Similarly, investors and traders may combine analysts’ projections for a company with structural models to understand the potential impact corporate actions could have on valuation. In conclusion, rather than idealistically selecting a single approach, we encourage market participants to understand all approaches and select the best method or combinations of methods for a given investment situation.


8-Dym-Maturity Page 183 Wednesday, July 23, 2003 10:31 AM

Maturity, Capital Structure, and Credit Risk: Important Relationships for Portfolio Managers Steven I. Dym, Ph.D. Principal Brocha Asset Management

I

nvestors in corporate securities are faced with a wide array of choices, among them: ■ ■ ■ ■

Equity or debt of the firm Maturity of the specific debt issue Level of seniority within the capital structure of the firm Direct exposure to the firm’s credit risk, or indirectly by way of credit derivative

While each of these alternatives contains its own set of risk variables, there is a good deal of commonalty among them, resulting in important interrelationships and comovements. The purpose of this article is to provide an intuitive explanation of these interrelationships and, with straightforward examples, to illustrate their comovements (or lack thereof). We begin the analysis with the simplest of capital structures and observe that various outcomes of the firm’s earnings—as manifested in the firm’s asset and book values—impact its debt. Interest rates, of

183


184


course, do matter greatly. But variations in the firm’s return on assets produce an important nonlinearity in bond prices, not usually captured in standard bond pricing formula, which focus almost exclusively on interest rate changes. We next introduce more complexity into the capital structure, allowing for two types of debt issues. This illustrates the higher level of correlation between subordinated debt and the firm’s equity value than with respect to senior debt. We then bring the structure even closer to reality by splitting the debt into short and long maturities. This succinctly captures the dilemma of a corporate bond portfolio manager: While duration is a decent measure of relative price sensitivity to interest rate movements, it breaks down when bonds trade near their recovery values. We introduce shifts in interest rate regimes, which capture the interrelationships between macroeconomic shocks in the form of yield curve shifts and individual firm performance as manifested in return on assets. And we compare the performance of the corporate securities with Treasuries as they respond to the same interest rate movements. At this point we also show the relationship between credit default swaps and the firm’s value. We prove that a credit default swap together with a corporate bond does not produce a riskless security (a “synthetic” Treasury). Rather, it is equivalent to a senior debt issue, properly defined. We conclude by expanding the analysis in three directions. First, we increase the number of possible asset rates of return and interest rate outcomes. In this context, we admit nonparallel shifts in the yield curve. Second, we upgrade the firm’s capital structure to include different levels of seniority as well as maturity. These serve to demonstrate the potential robustness of our approach. Finally, we suggest a simplified model of equity market value, as opposed to book asset value, which conforms to the overall approach. This clearly shows how equity price movements correlate with prices of lower rated bonds.

FIRM VALUATION AND BOND PRICING We begin with a simplified capital structure. Although unadorned, it establishes the basic relationships we are looking for. The firm owner purchases assets at a cost of 180, issuing one class of debt with a face value of 100. At this point, we permit only a one-year debt instrument and assume a coupon of 6%. Its value will be determined momentarily. The difference between its value and the cost of the physical assets is, of course, financed with equity.


Maturity, Capital Structure, and Credit Risk: Important Relationships for Portfolio Managers

185

EXHIBIT 1

End-of-Year Asset Values: Simplified Capital Structure One Class of Debt: One-year Maturity, 6% Coupon Initial Asset Value: 180 Rate-of-return on assets Physical asset value Value of debt Book value of equity

–50% 90 90 0

0 180 100 80

+50% 270 100 170

At the end of the year, the net return on the assets—net of all costs, including interest on debt—is –50%, 0 or +50%. Exhibit 1 displays the possible outcomes, including the resultant ending values of the bond. At a rate-of-return (ROR) of 0 or 50%, the firm can pay off the maturing par value of principal. At a –50% ROR, the firm cannot meet its debt obligations. The assets are liquidated, so that bondholders recover 90 of their principal (the coupon payment has been met together with the firm’s other costs).1 There is nothing new or surprising in Exhibit 1. However, for investors used to thinking about bond price changes in terms of interest rate movements, a couple of points are worth highlighting. Note the sharp non-linearity along the set of bond price outcomes. This is clearly not the result of interest rate changes, since the bond is at maturity. Rather, it is entirely due to the changing value of the firm’s assets, which determines its ability to pay off maturing debt. As a result, we observe a correlation between bond prices and both asset value and firm equity (book) value, a relationship—while intuitively compelling—absent in typical corporate bond price formulations which assume a fixed recovery value.2 What about the value of the bond at issue? Assuming a 3% risk-free (i.e., U.S. Treasury) one-year rate, risk-neutral pricing produces, ( 90 + 100 + 100 ) ⁄ 3 + 6 P debt = ------------------------------------------------------------- = 99.68 1.03 for a yield of 6.34%. The firm owner’s equity investment is, therefore, 180 – 99.68 = 80.32.

1

We assume no liquidation costs which may further reduce the recovery value to debt-holders. 2 We examine the correlation with firm market value in a later section.


186


UPGRADING THE CAPITAL STRUCTURE: EFFECT OF SUBORDINATION ON BOND PRICING Say the firm owner now issues two classes of debt: senior with a 5% coupon and subordinated with a 7% coupon. Both have one-year maturities. To produce a face value of 100 for each, and maintain a reasonably similar debt-equity ratio to the example above, we assume assets worth 360 are purchased by the firm owner. With the same rate-ofreturn possibilities as before, end-of-year physical asset, debt and equity values are presented in Exhibit 2. We assume at this point that as long as the senior debt’s face value is covered by the assets, it loses nothing in the event of the firm’s bankruptcy, as all losses in excess of equity are borne by the junior class. Again, no surprises here. Under our assumptions, the senior debt behaves as a riskless security, and should be priced accordingly. Only the subordinated issue correlates with the real asset value and with the firm’s equity (book) value. The two bonds, therefore, exhibit dramatically different risk-return combinations. Their values are (continuing with a 3% one-year risk-free rate): ( 100 + 100 + 100 ) ⁄ 3 + 5 P senior = ---------------------------------------------------------------- = 101.94 1.03 ( 80 + 100 + 100 ) ⁄ 3 + 7 P subord = ------------------------------------------------------------- = 97.41 1.03 The bonds yield, respectively, 3.01% and 9.78%. Their blended yield is exactly equal to that of the single bond in the previous case. This is not surprising—an investor purchasing both securities becomes the single debtholder, entitled to the firm’s assets at bankruptcy. EXHIBIT 2

Upgraded Capital Structure: End-of-Year Asset Values Two Classes of Debt: One-year Maturity, 5% (Senior) and 7% (Subordinated) Coupons Initial Asset Value: 360 Rate-of-return on assets Physical asset value Value of senior debt Value of subordinated debt Book value of equity

–50% 180 100 80 0

0 360 100 100 160

+50% 540 100 100 340



187

BOND MATURITY, CREDIT RISK, AND HEDGE RATIOS In this section we take a major step closer to reality. We introduce two maturities for the firm’s debt, each longer than a year, and compare their performances, hence their prices, to Treasury bonds. Along with the set of return on asset outcomes, we allow for different interest rate regimes at the end of the year. Note the contrast to the capital structure in the previous section. There, the two bonds were of the same maturity, but differed in their positions within the firm’s capital structure. Here, they are of the same seniority, but of different maturities. Nevertheless, the possibility of default forces them to display relative price dynamics far different from what their respective durations would suggest, one of the main results of our analysis.3 We also include a simple credit derivative in order to examine how it would be employed and priced in this situation. The firm owner purchases 360 worth of real assets, as in the previous section, and finances it with a 5% coupon, 6-year note and a 6% coupon, 11-year bond. They are of equal seniority. Our analysis involves thinking about the firm’s survival after one year. That is, we assume that should the firm survive the year, the risk of future default is nil. Although simplistic, it is enough to produce the crucial relationships. (Furthermore, it is straightforwardly extended to multiperiod risk.) We need, therefore, to consider the 5- and 10-year risk-free rates expected to prevail in one year’s time. In essence we have a “two-variable” model: We examine outcomes for various combinations of physical asset rates of return and endof-year interest rates. Exhibit 3 provides the information concerning the assumptions for interest rate and rate of return outcomes, and the resultant end-of-year valuations for the instruments we are concerned with. We allow for two possible interest rate regimes for each rate-of-return outcome. But we recognize that there is typically more volatility in shorter than longer-term rates, and that is incorporated in Exhibit 3. For comparison, Exhibit 3 presents pricing results for Treasury instruments of 5- and 10-year maturities, with coupons equal to those of their corporate counterparts. In nonbankruptcy situations—at rates of return of zero and above— the price performances of the two bonds are entirely determined by their respective maturities (more correctly, their durations).4 However, in the event of bankruptcy, the two bonds, being of equivalent seniority, share equally in the liquidation of the assets, hence are valued equally, which

3

In the last section we allow for different maturities as well as seniority levels. The entries in Exhibit 4 for the two bonds in these situations utilize the standard bond price-yield formula.

4

Physical asset value Value of 5-year note Value of 10-year note Value of 5-year Treasury Value of 10-year Treasury Credit default swap Book value of equity

Rate-of-return on assets 5-year rate 10-year rate 180 90 90 104.45 107.72 10 0

–50% 4% 5% 180 90 90 93.77 92.98 10 0

–50% 6.5% 7% 360 104.45 107.72 104.45 107.72 0 160

0% 4% 5% 360 93.77 92.98 93.77 92.98 0 160

0% 6.5% 7% 540 104.45 107.72 104.45 107.72 0 340

+50% 4% 5%

Upgraded Capital Structure: End-of-Year Asset Values One Class of Debt: 6-year Maturity, 5% Coupon, 11-year Maturity, 6% Coupon Initial Asset Value: 360

EXHIBIT 3

540 93.77 92.98 93.77 92.98 0 340

+50% 6.5% 7%


188



189

makes their different maturities irrelevant. This has profound implications for portfolio management decisions, as will be discussed shortly. The prices of the four bonds, continuing with our assumption of a 3% one-year risk-free rate, are ( 90 + 104.45 + 93.77 ) ⁄ 3 + 5 P 6yrcorp = -------------------------------------------------------------------------- = 98.13 (5.37% yield) 1.03 ( 90 + 107.72 + 92.98 ) ⁄ 3 + 6 P 11yrcorp = -------------------------------------------------------------------------- = 99.90 (6.01% yield) 1.03 ( 104.45 + 93.77 ) ⁄ 2 + 5 P 6yrtreas = ------------------------------------------------------------- = 101.08 (4.79% yield) 1.03 ( 107.72 + 92.98 ) ⁄ 2 + 6 P 11yrtreas = ------------------------------------------------------------- = 103.25 (5.60% yield) 1.03 which shows that the initial equity investment must be 360 – 98.13 – 99.90 = 161.97. How sensitive are the two corporate bonds to shifts in the key parameters—the risk-free rate expected to prevail in one year and the firm’s physical asset value (hence the bond recovery value)? Consider first the bonds’ durations as of their issue date. Given their 5% and 6% coupons, 6- and 11-year maturities, and the 5.37% and 6.01% yields, respectively, their durations are 5.32 and 8.35. Portfolio managers often substitute one bond for another of the same issuer, for relative value considerations, but wish to maintain the same interest rate (i.e., duration) exposure. In our case, the manager would purchase 8.35/5.32 = 1.57 units of the 6-year bond for every one 11-year bond. How will this work out? Suppose, compared to the initial end-of-year assumptions, the yield curve shifts up in a parallel fashion by 1%. That is, in place of 4% and 6.5% for the 5-year rate in Exhibit 3, and 5% and 7% for the 10-year rate, we use 5% and 7.5%, and 6% and 8%, respectively. We keep the 90 recovery value intact, since we are examining a pure interest rate shock. Employing the above methodology, the 6-year note declines in price by 2.70, the 10-year by 4.57. (See Exhibit 4.) Thus, 1.57 units of the 6-year produces essentially the same exposure as the 11-year, close to what the duration ratios predicted.5 5

The well-known inexactness of the results of this substitution is due to the different convexities of the two bonds.


190


EXHIBIT 4

Summary of Price Responses to Interest Rate and Credit Shifts (Figures are Absolute Values)

1% upward shift in yield curve 5% decline in recovery value

5-year Corp.

10-year Corp.

5-year Treas.

10-year Treas.

2.70 1.61

4.57 1.62

4.05 0

6.85 0

Consider now a stable interest rate environment, but a drop in the low end asset value to 170 from 180. This produces a recovery value of 85 for each of the bonds rather than 90. Because the two bonds share equally in the assets in a default, by virtue of their identical positions in the capital structure, their prices drop by essentially the same amount. In this scenario, the 1.57 ratio of the 6-year bond against the 11-year bond is totally wrong. The portfolio manager would find the position much more exposed to credit risk, despite correcting for the duration difference. Exhibit 4 also shows that although Treasuries can be used as a hedge for corporates with respect to pure interest rate changes, they break down when credit shocks occur.6 This is the well-known credit spread basis risk. The basis risk shown here between two corporate bonds of the same issuer, however, is not as well known. Turn finally to the credit default swap. In exchange for periodic payments (in our case, only for the first year), the swap pays the difference between the recovery value of the real assets and the face value of the bond. Consider a combined position in either of the corporate bonds together with the credit swap. The swap “insures” against credit risk, since it makes up to the holder the difference between the bond’s face value and its value in default. That is, the combination is still worth par in event of default. Is this combination equivalent to a riskless position; i.e., a synthetic Treasury bond? Absolutely not! In the event of default by the firm, bankruptcy law requires corporate debt repayment to be accelerated. Hence, regardless of the level of interest rates at the time of default, the bond is worth its recovery value, or par, if combined with a credit default swap. The Treasury bonds, on the other hand, with their remaining years to maturity, react to the prevailing interest rate environment, regardless of the firm’s survival or demise, and so can be significantly 6

The outsized movements in Treasury prices compared to the corporates are due to our assumption of the 33% probability of default, since our simplified analysis uses only three possible ROR outcomes. Thus, any pure interest rate shift is factored into corporate bond prices by a factor of only 0.67, compared to a factor of 1 for the Treasuries. Allowing for a wider range of outcomes would reduce the discrepancy dramatically.



191

away from par, as seen in Exhibit 3. Rather, a corporate bond/default swap combination is more like a special type of “senior” bond, one which responds to interest rate movements in the “normal” way when the firm is out of bankruptcy, but pays par in the event of bankruptcy, regardless of interest rate levels (much like the senior bond in Exhibit 2).

EXPANDING THE RANGE OF POSSIBLE OUTCOMES This section brings together many of the ideas developed above. At the same time, it includes a wider range of rate-of-return outcomes for the firm’s assets, a greater variety of instruments, and a more macroeconomically consistent interest rate dynamic. In particular (see Exhibit 5): ■ We allow intermediate RORs between –50% and +50%. ■ We continue to examine two regimes of interest rates within each

ROR outcome. But now we recognize the macroeconomic correlation between profits and interest rates, and assume that bond yields are lower when the firm’s returns are at their lowest point. ■ We include both senior and subordinated debt, and separate the subordinated class into two maturities. ■ Finally, we introduce a market value of equity to accompany the book value examined up to now, and use a simple model to estimate market value in a way which conforms with our overall approach. The resulting prices and underlying assumptions are summarized in Exhibit 5. With three bonds issued, each 100 in face value, the firm purchases 480 worth of assets. A senior 10-year bond has a 5.5% coupon. Subordinated debt of 5- and 10-year maturities carry 6% and 7% coupons, respectively. Observe that during liquidation, even though the firm has assets to pay off the senior debt completely, noteholders are assumed to receive only 90% of their face value. This is in keeping with common reorganizations where negotiations typically shift some of the burden to the senior class. We also assume that at a –25% return on assets, despite its ability to meet obligations, the firm liquidates, since its market value falls below book.7 Treasury notes of similar maturities 7

Our simplified market value of equity model assumes that at each asset ROR outcome (at or above 0), equity investors predict future earnings as an average of current earnings, with a 40% weight, and a 20% weighting to each of the other three outcomes (except for the –50%). The life of the firm is assumed to be 10 years (as the 10-year bonds), and expected earnings plus terminal value are discounted by either of the two 10-year rates, as the case may be.

–50% 3.5% 4.75% 240 90 75 75 103.39 102.93 25 0 0

Rate of return on assets 5-year rate 10-year rate

Physical asset value Value of senior note Value of 5-year junior note Value of 10-year junior note Value of 5-year Treasury Value of 10-year Treasury Credit default swap Book value of equity Market value of equity

240 90 75 75 96.75 93.56 25 0 0

–50% 5% 6% 360 100 100 100 101.11 100.97 0 60 0

–25% 4% 5% 360 100 100 100 90.65 86.83 0 60 0

–25% 6.5% 7% 480 103.86 108.90 115.44 101.11 100.97 0 180 481.15

0% 4% 5% 480 89.46 97.92 100 90.65 86.83 0 180 428.63

0% 6.5% 7%

Robust Capital Structure Two Classes of Debt: Senior 11-Year Maturity, 5.5% Coupon; Subordinated 6-Year Maturity, 6% Coupon, Subordinated 11-Year Maturity, 7% Coupon Initial Asset Value: 480

EXHIBIT 5

600 103.86 108.90 115.44 101.11 100.97 0 360 666.47

+25% 4% 5%

600 89.46 97.92 100 90.65 86.83 0 360 597.20

+25% 6.5% 7%

720 103.86 108.90 115.44 101.11 100.97 0 420 851.79

+50% 4% 5%

720 89.46 97.92 100 90.65 86.83 0 420 765.77

+50% 6.5% 7%


192



193

(4.25% and 5.125% coupons) are included for comparison. Using the above methodology, and a 3% risk-free one-year rate, the firm’s bonds have an initial value of 101.50, 100.04, and 103.53 for the senior, the 5and the 10-year subordinated, respectively. Exhibit 6 displays the correlation coefficients among the assets. A number of results and implications are worth noting. ■ The senior debt exhibits significant correlation with Treasuries, but

more so with the subordinated issues. ■ The subordinated debt shows little correlation with Treasuries, but

strong correlation with the firm’s book and equity values. This is as it should be, since when bonds trade near their recovery values due to high expectations of default, the firm’s assets largely determine those recovery values. This relationship, not captured in standard pricing models, suggests that the firm’s equity shares should be included in a hedging strategy for its subordinated debt. ■ Although negatively correlated with the senior debt, the credit default swap, not surprisingly, has a much stronger negative correlation with the junior debt. Hence, it is best used a hedge (or substitute) for lowerrated bonds of the same issuer.

RESPONSES TO SHOCKS Our approach is useful in showing how corporate bonds of different seniorities and maturities respond to various macroeconomic and financial shocks. Exhibit 7 displays price changes for the three securities examined in the previous section. It follows the same procedure, but makes the assumptions more realistic by including two additional pairs of outcomes: physical asset rates-of-return of +10% and –10%, each with the two regimes of interest rates as above. We look at the effects on bond prices of: shifts in overall yields, in either direction; improvement and deterioration in profits, caused by either individual firm successes or failures or macroeconomic business cycle disturbances; changes in the firm’s capitalization, as represented by its debt/equity ratio. While the relative price responses of the bonds to the shifts in interest rates seem to reflect their durations, their absolute responses are substantially lower than a simple duration measure would predict.8 As explained earlier, this is because the recovery value and the likelihood of default play an explicit role in our pricing framework, as they should. 8

Along with the shift in yields used in pricing the bonds on the 1-year horizon date, we also allow the one-year rate to shift accordingly.

194

0.716 0.717 1 0.991 –0.020 0.050 –0.927 0.646 0.643

5-year Junior

98.72 93.04 105.01 97.61 99.83 98.26 100.11

Senior Bond

Price Responses to Economic Shocks

0.218 1 0.717 0.733 0.524 0.569 –0.464 0.169 0.188

Senior Debt

Initial price 1% increase in interest rates 1% decrease in interest rates 20% deterioration in rate-of-return 20% improvement in rate-of-return 25% higher leverage ratio 25% lower leverage ratio

EXHIBIT 7

1 0.218 0.716 0.738 –0.229 –0.179 –0.707 0.994 0.963

Physical Assets –0.229 0.524 –0.020 0.049 1 0.997 0.323 –0.202 –0.122

5-year Treasury

101.81 97.89 105.98 99.04 104.58 99.02 105.28

5-Year Subordinated

0.738 0.733 0.991 1 0.049 0.118 –0.879 0.678 0.697

10-year Junior

Correlation Coefficients of Assets in Exhibit 5

Physical assets Senior debt 5-year Junior 10-year Junior 5-year Treasury 10-year Treasury Default swap Book value Market value

EXHIBIT 6

–0.707 –0.464 –0.927 –0.879 0.323 0.252 1 –0.625 –0.574

Default Swap

105.77 99.67 112.50 103.00 108.54 102.39 109.24

10-Year Subordinated

–0.179 0.569 0.050 0.118 0.997 1 0.252 –0.158 –0.080

10-year Treasury 0.994 0.169 0.646 0.678 –0.202 –0.158 –0.625 1 0.974

Book Value 0.963 0.188 0.643 0.697 –0.122 –0.080 –0.574 0.974 1

Market Value




195

And since recovery value is insensitive to interest rate regimes (ignoring possible macroeconomic correlations), the price responses to shifts in rates are muted. Our ROR shocks (+25% becomes +30%, +50% becomes +60%, etc.) have the expected consequences for the bonds. To be consistent with the assumptions of the previous section, we assume the senior bond shares in one-sixth of the loss should the asset values fall below the bonds’ face values during liquidation. Although of different maturities, the two subordinated issues respond by the same dollar (although not percentage) change, a multiple of the response of the senior issue. Finally, from an initial debt/capitalization ratio of 62.5%, we examine the implications of 50% and 75% ratios. At 75%, the firm liquidates even after a –10% rate-of-return. By contrast, in the base case of 62.5%, a –10% R.O.R allows the firm to continue, and the bond prices reflect only the risk-free rate, as above. At a 50% ratio, the bonds pay off in whole even at –50% ROR, compared to their recovery values in the base case. This explains the asymmetry between the reactions of the two subordinated bonds, and between those and the senior, to the pair of leverage shifts.

CONCLUDING REMARK We have built and used a simplified pricing model. Yet, because it encompasses capital structure and rate-of-return variables, it brings out important relationships among corporate bond values on the one hand and firm asset and equity values on the other hand, relationships largely absent in typical pricing equations. The next step is to graduate from single to multiperiod analysis.


9-Dym-Unified Page 197 Wednesday, July 23, 2003 10:27 AM

A Unified Approach to Interest Rate Risk and Credit Risk of Cash and Derivative Instruments Steven I. Dym, Ph.D. Principal Brocha Asset Management

ith the proliferation of new cash and derivative instruments, especially in the credit realm, it has become easy to lose sight of what these instruments are supposed to accomplish. Further, their complexities and interactions tend to mask their price sensitivities, hence their risk. In this article I present an integrated, yet simple, approach to understanding the true nature of many of the instruments and strategies employed by portfolio managers. The approach provides important insights into the value determinants and price dynamics of many classes of cash and derivative instruments. Of course, any attempt to encompass a broad range of instruments will neglect many of the complexities that distinguish instruments from each other. Still, this approach is worthwhile, as it provides the portfolio manager with the means to keep sight of the forest even as she or he concentrates on the trees. It has the added benefit that, in the course of developing the intuitive representation of value, it clears up a number of relationships that have traditionally seemed counter-intuitive to market participants.

W

197


198


CASH BONDS, NO CREDIT RISK Consider first a “plain vanilla,” coupon-paying Treasury bond. It is well known that its price may be written as 100 – Coup ⁄ y P = P ( Coup; y; m ) ≡ Coup ⁄ y + ------------------------------------m (1 + y)

(1)

where Coup, y, and m are the coupon rate, yield to maturity, and term to maturity, respectively.1 The following almost simplistic procedure opens up a world of insight into the nature of a bond. Add and subtract 100 to produce a new representation of a bond’s value: 1 P ( Coup; y; m ) ≡ 100 + ( Coup ⁄ y – 100 ) 1 – -------------------m (1 + y)

(2)

The first term, 100, is the return of principal. It also is the bond’s market value on the issue date (assuming, as we will, that it is issued at par). The second term reflects the effect of changes in market conditions on the bond’s price. At issue, the coupon rate is set to equal the market’s required yield-to-maturity, or Coup/100 = y, so that the second term vanishes. As the bond seasons, the yield may diverge from the coupon rate, causing the bond to trade at a premium or discount. Notice that the distance from par depends on the term in brackets, in turn a positive function of the bond’s maturity. In short, we can rewrite the basic bond valuation as Pbond = 100 + B

(3)

with B defined conformably with equation (2) as 1  Coup - – 100 ⋅ 1 – -------------------- ------------m y (1 + y) A Treasury bond’s price equals the sum of two factors: its face, or par, value; plus or minus a factor, B, reflecting the divergence of the market’s risk-free yield from the bond’s coupon rate. This simple insight will provide us with a good deal of mileage. 1

This assumes, for simplicity, annual coupon payments and annual compounding.


A Unified Approach to Interest Rate Risk and Credit Risk of Cash and Derivative Instruments

199

With no credit risk, the bond’s price dynamics is entirely a function of changes in the risk-free rate, represented here by the yield to maturity, y. Changes in y will enhance or reduce the bond’s premium, enhance or reduce the bond’s discount, cause it to move to discount or premium from par, etc., as is well known from standard “bond math.” Since this obviously does not affect the principal term in equation (3), we may write2 ∆P bond = ( ∆B ⁄ ∆y ) ∆y

(4)

where (∆B/∆y) is the issue’s (modified) duration. For example, a 5-year, 6% coupon government bond earns a price of 104.3295 when the market requires a yield to maturity of 5% (i.e., B = 4.3295). A 50-basis-point increase in yield reduces the premium to 102.1351, hence ∆B = –2.1944.

PURE INTEREST RATE DERIVATIVES This formulation of a bond’s value adapts easily to derivatives. Nonoption interest rate derivatives, such as forwards, futures, and swaps, typically begin life with a net present value of zero. What does this mean in our context? First, the 100 in equation (3) is replaced by a 0, as no cash is “lent” at issue and there is no principal to “pay back.” Second, the parameters of the derivative are typically set so that there is no up-front payment; that is, B = 0.3 Over time, as interest rates move, the derivative will acquire value, positive or negative, depending on the side (short or long) of the transaction. Just as a cash bond, the coupon on the bond underlying the forward or future is fixed,4 but market rates move. With respect to swaps, the fixed payment is constant, whereas swap rates constantly change. Thus, we can very simply write Pderiv = B

(5)

Allowing y to represent the yield on the forward or future’s underlying security, or the fixed side of the swap, and assuming no basis risk between the derivative and the cash instrument, we can further write 2

This admittedly unusual formulation of the effect of yield changes on price can also be generated by a Taylor approximation of the bond’s price around 100. 3 “Off-market” derivatives are those which begin life with B ≠ 0. 4 This ignores the possible dynamics of changing cheapest-to-deliver issues for futures contracts.


200


∆P deriv = ( ∆B ⁄ ∆y ) ∆y

(6)

Here, (∆B/∆y) is the duration of the underlying bond (or that of the fixed payment on the swap). Notice that whereas the cash and derivative instruments obviously have different values [compare equation (3) to equation (5)], their price dynamics are equivalent [comparing equation (4) and equation (6)]. This is what makes the derivative so amenable to being employed to hedge, leverage, and synthesize the risk of the underlying cash bond.5

CASH BOND WITH CREDIT RISK Consider now a bond with credit risk, such as a corporate issue. It must yield the Treasury rate plus a spread to compensate for risk of default. When the bond is issued, its coupon typically includes a premium above the risk-free yield to bring it to par. Once it is seasoned, its price will be above, below or equal to par according to whether its coupon is above, below or equal to the sum of the current market risk-free yield and credit risk premium, with the distance from par dependent on the bond’s remaining maturity. We can summarize these parameters by writing its price, parallel to the Treasury bond above, as P = P ( Coup T + Coup R ; y T + y R ; m ) = ( Coup T + Coup R ) ⁄ ( y T + y R ) 100 – ( Coup T + Coup R ) ⁄ ( y T + y R ) + ----------------------------------------------------------------------------------------m ( 1 + yT + yR )

(7)

where CoupT and CoupR are the risk-free and risk premium coupons, respectively, on the issue date, yT and yR are the risk-free yields and credit spreads, respectively, as of the current market, and m the maturity. A little rewriting works wonders: P ( Coup T + Coup R ; y T + y R ) = P ( Coup T ; y T ) – [ P ( Coup T ; y T ) – P ( Coup T ; y T + y R ) ] + A ( Coup R ; y T + y R )

5

(8)

This discussion assumes no performance risk from the derivative’s counterparty (i.e., no counterparty risk).



201

with the m parameter omitted for convenience. This is a very enlightening result. Every credit-risky bond can now be seen to have three components: 1. A pure Treasury, or risk-free, component, which discounts riskless cash flows simply by the time value of money, P(CoupT; yT). 2. A pure credit risk correction to the riskless cash flows, since P(CoupT; yT) – P(CoupT; yT + yR) > 0. 3. Compensation for taking this credit risk, in the form of an annuity: Coup ⁄ y A ( Coup; y; m ) ≡ Coup ⁄ y – -------------------m (1 + y) To be consistent with the above formulation, rewrite this further as Pcorpbond = 100 + B – D + R

(9)

B reflects the relationship between the bond’s risk-free coupon and the risk-free yield, as it does for a riskless Treasury bond, P(CoupT; yT), in the same way as equation (3). D represents the market’s further discounting of the Treasury cash flows—coupon plus principal—since they are no longer risk-free, [P(CoupT; yT) – P(CoupT; yT + yR)]. And R equals the present value of the additional cash flow provided by the corporate as compensation for credit risk, or the annuity A(CoupR; yT + yR). Suppose CT = yT, so that B = 0. If, and only if, CR = yR will the bond be priced at par, because R will then exactly offset D; i.e., the borrower is providing enough additional expected cash flow to compensate for the additional discounting due to default risk. Movements away from par reflect the relationship between the risk-free cash flow and the risk-free yield—summarized in the B term—and the relationship between the credit risk premium and the additional cash flow promised by the risky borrower. An example is in order. Assume a corporate bond is issued simultaneously with the Treasury examined above, with a similar 5-year maturity.6 The Treasury yield is 6%. The bond carries a 7% coupon. Let’s break it apart into 6% for the risk-free rate plus 1% additional to compensate for the credit risk at the issue date. B is 0, as the Treasury component is par, since CT = yT. Assume the market requires a 1% risk premium. Then D = 4.1002, which is subtracted from the pure Treasury component of the bond. R, however, equals exactly 4.1002, bringing the issue to par. 6

We continue with our assumption of annual coupons and annual compounding.


202


Suppose market conditions change. The risk-free yield for 5-year Treasuries falls to 5%. But investors now require a 1.5% spread for this issuer’s default risk. At a 6.5% yield-to-maturity, the bond’s price is 102.0779. Why? 1. The pure Treasury component of the bond has risen to 104.3295, since B is 4.3295, as above. This reflects the drop in the risk-free rate. 2. Because of the increase in perceived default risk, the market discounts the pure Treasury cash flows of this corporate bond more deeply, as D is 6.4073. 3. The bond’s compensation for credit risk has now risen in absolute value. R is 4.1557 rather than 4.1002 because the general structure of yields—as reflected in the risk-free rate—has fallen. However, relative to the new discount of 6.4073, this is not enough, since the risk environment has now changed. Thus, the net effect of the credit risk in the new interest rate environment is –2.3071. In short, changes in interest rates—risk-free and credit spread—affect a corporate bond’s price by changing the three factors determining the bond’s value: ∆P corpbond = ∆B ⁄∆y T – ( ∆D ⁄∆y T + ∆D ⁄∆y R ) + (∆R ⁄∆y T + ∆R ⁄∆y R )

(10)

The components are summarized in Exhibit 1 for the bond in our example. It is important to note that the drop in the risk-free yield has exacerbated the effect of the increase in the risk premium on the bond’s price. Had the Treasury yield remained at 6% even as the required risk premium rose to 1.5%, the D factor would subtract only 6.0689, and the R compensation would be worth 4.0459, for a net cost of the credit risk of 2.023, compared to the actual 2.3071. Default risk is now discounting a cash flow of greater value. Compare Exhibit 2 to Exhibit 1. The drop in the risk-free rate means that the cost of credit risk is more severe. This has similarly significant implications for credit derivatives, as we will see later.

FLOATING RATE NOTES Before continuing with credit derivatives, let’s see how this approach applies to floating rate notes (FRNs). FRNs are often considered a “pure credit play.” The fact is, they’re not, and our approach shows why.



203

EXHIBIT 1

Components of Corporate Bond Value 5-year, 7% annual coupon (6% initial risk-free rate, 1% credit risk premium) 5% market risk-free rate, 1.5% market required credit risk premium Risk-free value less Credit risk discounting plus Compensation for credit risk equals Bond value

104.3295 – 6.4073 + 4.1557 = 102.0779

EXHIBIT 2

Components of Corporate Bond Value 5-year, 7% annual coupon (6% initial risk-free rate, 1% credit risk premium) 6% market risk-free rate, 1.5% market required credit risk premium Risk-free value less Credit risk discounting plus Compensation for credit risk equals Bond value

100 – 6.0689 + 4.0459 = 97.9770

FRNs are structured to pay a base interest rate—usually LIBOR of a given maturity—which adjusts on the note’s repricing dates, plus a spread reflecting credit risk. For presentation purposes, let’s consider LIBOR the “riskless” rate, as issuers with credit risk pay spreads above LIBOR. The spread is typically fixed for the life of the note. Whether an FRN is priced at par or not depends on whether the spread paid by the note is equal to what the market requires of it. Let’s begin simply and assume the market requires an issuer to pays a spread of zero, and indeed the FRN pays a zero spread. The FRN’s price will then be par on each repricing date.7 We can easily incorporate this into our format: yT is the base rate and yR the market’s required spread. By nature of the floater, CT = yT, and CR is the spread the floater pays. In the example under discussion, CR = yR = 0, so that by equation (1), the price is par [or B = 0 in equation (3)]. As market rates—reflected in the base rate, yT—change, CT changes along, keeping the note at par. Now recognize that most issuers are required to pay a spread above LIBOR. The formula for a credit risky cash bond, summarized in equa7

The price may move away from par between repricing dates, but that is not our concern here.


204


tion (9), is now relevant—with one caveat, that B = 0, because of its floating rate nature. Thus,8 PFRN = 100 – D + R

(11)

with D = 100 – P(CoupT; yT + yR) and A defined as previously in equation (8). Suppose the market requires a 1% spread from a particular borrower, which issues a 5-year FRN paying so. LIBOR equals 6% on the issue date. As long as the market’s required spread remains 1%, the note will be priced at par regardless of the level of LIBOR. That is, CT = yR so that P = 100 because D = 4.1002 = R. What if the issuer’s credit worthiness deteriorates, so that yR = 1.5%. with LIBOR unchanged at 6%? Then D = 6.0689, but R is only 4.0459; that is, the issuer’s compensation for credit risk is not enough to satisfy the market so that P = 97.9771. Now allow market rates, as represented by LIBOR, to change, with the credit picture steady. Assume LIBOR declines to 5%, while the issuer’s required spread remains 1.5%. Because the base rate falls along with the discount factor, the riskless portion of the FRN’s value is unchanged; that is, B = 0 as always. However, the default risk discounting and compensation factors do not change equally. D increases from 6.0689 to 6.2335 while R rises only to 4.1557 from 4.0459. The FRN’s price, therefore, drops from 97.9771 to 97.9222. A decline in interest rates, as reflected in LIBOR, has produced a drop in price—the FRN has negative duration! What explains this? In our framework, an FRN has three components, and the response of each to market changes needs to be considered: 1. The pure interest rate component remains at par, by nature of the FRN’s coupon structure ⇒ B = 0. 2. However, the decline in LIBOR reduces the size of the expected future coupons. Because the FRN was trading at a discount, this raises the FRN’s price responsiveness to interest rate changes ⇒ ∆D is relatively large. 3. The present value of the credit risk compensation stream rises as well, but because there is no principal accompaniment—it is an annuity—the effect is relatively small ⇒ ∆R is less than ∆D. 8

This uses the approach of “discount margin,” well known in the FRN market.



205

EXHIBIT 3

Components of FRN Value Panel (a) 5-year, annual-pay floater, LIBOR plus 1% credit risk premium 6% LIBOR, 1.5% market required credit risk premium Principal FRN value less Credit risk discounting plus Compensation for credit risk equals Bond value

100 – 6.0689 + 4.0459 = 97.9771

Panel (b) 5-year, annual-pay floater, LIBOR plus 1% credit risk premium 5% LIBOR, 1.5% market required credit risk premium Principal FRN value less Credit risk discounting plus Compensation for credit risk equals Bond value

100 – 6.2335 + 4.1557 = 97.9222

Exhibit 3 compares these two situations. Were the FRN trading at a premium, the relative magnitudes of the two effects would be reversed, so that a decline in LIBOR has the usual effect of increasing the FRN’s price. For example, if the required credit spread is 0.5%, but the note pays a 1% spread, its price is 102.0779 with LIBOR at 6% (D = 2.0778, R = 4.1557). Should LIBOR fall to 5%, the FRN’s price increases to 102.1352 (D = 2.1351, R = 4.2703).

SIMPLE CREDIT DERIVATIVE We begin with what may be more correctly termed a “full” credit derivative. Similar to the relationship between the interest rate derivative above and a Treasury bond, this credit derivative mimics a bond with credit risk but lacks the cash investment. An example would be a forward contract to purchase a corporate bond. The contract starts life with a present value of zero, as no cash is lent. Over time, as either the risk-free rate or the market’s required credit premium change, the derivative will acquire value, positive or negative, depending on the positions of the counterparties. Thus, equation (9) is simply transformed to


206


Pcreditder = B – D + R

(12)

P will respond to changes in yT and yC just as the associated cash corporate bond does, summarized in equation (10) above, even though its value is not the same, since it lacks the cash principal. But the very fact that it is affected by the risk-free rate, not only the credit risk premium, disqualifies it from being classified as a “pure” credit derivative. These are introduced later.

DEFAULT Let’s move on to a default event. While what happens to cash assets in the event of default is thought to be well understood, this is not the case for derivatives or for cash/derivatives combinations. We begin by assuming no recovery value in event of default and, therefore, no collateral.

CASH INSTRUMENTS Return to the price formula for a corporate security given by equation (8), which is reproduced here: P corp = P ( Coup T ; y T ) – [ P ( Coup T ; y T ) – P ( Coup T ; y T + y R ) ] + A ( Coup T ; y T + y R ) In our framework, we can define a default as an infinite widening of the credit spread, yR. It is clear that yR → ∞ ⇒ P → 0. (Alternatively, using summary equation (9), yR → ∞ ⇒ D → (100 + B) and R → 0.) In other words, for either a fixed coupon bond or FRN, with no recovery value, default produces a zero value for the security; that is, its existence, along with future expected cash flows, ends. This may seem obvious, but it carries a deeper implication. Suppose default occurs when yT is very low, relative to CT. Then P(CoupT; yT) is very high. By owning a fixed coupon bond in an environment of low riskless rates, the investor has achieved a theoretical capital gain. A default not only wipes out the future cash flows of the bond, it erases the capital gain due to the fall in market rates. If assets do exist from which to draw some recovery value during bankruptcy, the bondholder can only petition the court for com-


207


pensation for the face value, not the market value of the Treasury component of the bond.

ASSET SWAPS Attaching an interest rate swap to a cash instrument produces an asset swap. Suppose an investor owns a fixed coupon corporate bond. Entering into a swap as a fixed-rate payer and floating-rate receiver changes the nature of the asset from fixed to floating rate. The swap is an interest rate derivative, and as such has no principal, described earlier in our framework by equation (5). We simply combine this with equation (8) for the corporate bond to write9 P assetswap = P bond – P swap = { P ( Coup T ; y T ) – [ P ( Coup T ; y T ) – P ( Coup T ; y T + y R ) ] + A ( Coup T ; y T + y R ) } – { P ( Coup T ; y T ) – 100 } which simplifies to P assetswap = 100 – [ P ( Coup T ; y T ) – P ( Coup T ; y T + y R ) ] + A ( Coup T ; y T + y R ) }

(13)

Contrary to common belief, this is not a FRN. At first blush it certainly seems like an FRN: The riskless interest rate is absent, evidenced by the 100 term in place of P(CoupT; yT), and the credit factors are present. However, should the bond default, the net result, assuming no recovery, is not 0. Rather, from equation (12) it is clear that yR → ∞ ⇒ P → 100 – P(CoupT; yT). Although the bond is erased, the swap remains, and may well have a value different from zero, should interest rates differ from their levels at the swap’s inception. We can say this another way. Bonds with credit risk may move away from par for two possible reasons: changes in riskless interest rates and/ or credit spreads. In the event of default, the bondholder gives up not only the bond’s par value but its deviation from par due to riskless rate shifts. Not so with an asset swap. Default erases the cash bond; the swap remains, preserving any effects of the shift in riskless interest rates. 9 An alternative derivation is to recognize that a fixed-for-floating swap is economically equivalent to being short a par fixed coupon bond with a coupon equal to the swap’s fixed payment and long a FRN paying LIBOR flat, or 100 – P(CoupT; yT).


208


A floating-rate note/interest rate swap combination converts the interest rate exposure from floating to fixed. Here, too, we can show that the result is not truly a fixed-rate bond. Should the FRN default, the investor retains the interest rate swap position.10 What about recovery value after default? A slight change is necessary. Instead of yR → ∞ ⇒ D → P(CoupT; yT), we have D → P(CoupT; yT) – Recovery. For either the bond/swap or FRN/swap position, default results in the recovery value plus the swap position.

DEFAULT SWAPS Credit derivatives have grown from next to nothing in volume just a few years back to an integral part of the capital markets today. A simple forward contract for a credit risky bond is not a pure credit derivative. As explained earlier, it responds similarly to fluctuations in either the riskfree interest rate or the credit spread. A pure credit derivative, on the other hand, removes, to the extent it can, the pure interest rate factor, retaining only the credit risk factor. Such is the goal of a default swap. In exchange for regular payments upon acceptance of default risk, the investor—referred to popularly as the “protection seller”—agrees to reimburse the counterparty—the “protection buyer”—for any shortfall from par should the reference credit default. Since the protection seller has not made any cash investment, there is no return of principal at maturity. For the same reason, the regular payment paid to the protection seller does not include the riskless interest rate. Instead, the cash flow is simply the credit risk spread. Our framework is perfectly suited for analyzing a default swap. We can represent it as long a corporate bond, from the protection seller’s perspective, and short a Treasury bond of equal maturity. The protection buyer is in the reverse position. Simply subtract (2) from (8) to produce P defaultwap = P corpbond – P govtbond = – [ P ( Coup T ; y T ) – P ( Coup T ; y T + y R ) ] + A ( Coup T ; y T + y R ) = –D+R 10

Combining a FRN with a floating-for-fixed swap produces P FRN + P swap = { 100 – [ 100 – P ( Coup T ; y T + y R ) ] + A ( Coup T ; y T + y R ) } + { P ( Coup T ; y T ) – 100 } = P ( Coup T ; y T ) – [ 100 – P ( Coup T ; y T + y R ) + A ( Coup T ; y T + y R ) ]

Here, yR → ∞ ⇒ P → P(CoupT; yT) – 100, again not 0.

(14)



209

This is an enlightening result. First, it shows that the default swap is not merely the value of an annuity paying the credit spreads, as represented by A(CoupT; yT + yR), or R. The protection seller receives the annuity in return for bearing the credit risk of the reference security, as represented by –[P(CoupT; yT) – P(CoupT; yT + yR)], or the –D term. Therefore, in default yR → ∞ ⇒ D → P(CoupT; yT) and R → 0. In short, Pdefaultswap → –P(CoupT; yT). That is, the protection seller is theoretically required to return a bond to the protection buyer in the event of default. Recognizing recovery value in default, D → P(CoupT; yT) – Recovery. The protection seller returns the difference in value between recovery and par, so that the protection buyer receives a whole bond. Second, in event of default, the protection buyer is not actually compensated with the cash flow of the reference bond. The compensation is not even a Treasury bond, which would be P(CoupT; yT). Rather, the payment is par. Thus, in default, the swap is really a long position in a corporate bond and a short position in a riskless floater: P defaultwap = P corpbond – P FRN = – [ 100 – P ( Coup T ; y T + y R ) ] + A ( Coup T ; y T + y R )

(15)

so that yR → ∞ ⇒ PFRN → –100. In short, a default swap is a hybrid: When the reference credit is alive, the swap behaves as if it were long the corporate, short the corresponding fixed coupon government bond; when it defaults, it is equivalent to being long the corporate, short a riskless floater.

BOND/DEFAULT SWAP COMBINATION Consider now an investor with a long position in a corporate bond, just as described earlier and summarized in equation (8). Fearing credit deterioration, the investor purchases protection via a default swap. Combining the cash and derivative positions, that is, equation (8) with the reverse of equation (4): P = P(CoupT; yT)

(16)

which suggests a long position in a riskless, fixed coupon security. Here, again, this is only true when the corporate bond is alive. In default, P = 100, since the investor is “made whole.” This is clearly not a synthetic government bond, as one might think. Were the investor to actually own a Treasury bond, a drop in riskless interest rates, while corporates are


210

EXHIBIT 4


Summary of Factors Determining Value

Treasury bond Swap, forward, futureb Corporate bond Floating rate note “Full” credit derivative Corporate bond with swap FRN with swap Default Swapd Bond with default swap

Principal

Riskless interest rate

Credit risk discountinga

Special default provision

Yes No Yes Yes No Yes Yes No Yes

Yes Yes Yes No Yes No Yes No Yes

No No Yes Yes Yes Yes Yes Yes No

No No No No No Yesc Yesc No Yese

a

Includes the compensation for credit risk factor Underlying credit riskless security c Swap position remains after default d Seller of protection e Receives par after default, regardless of interest rate levels b

defaulting, would push the bond above par (and below par for a rise in rates). Instead, the cash corporate/default swap combination returns par in default and is, therefore, a hybrid fixed-coupon/floater position.11 Exhibit 4 summarizes the factors, in our formulation, that determine the value of the various instruments and combinations discussed above.12

11

There is a parallel in the cash market. Upon default of the issuer, the corporate bondholder petitions the bankruptcy court for the face value of the bond, not its market value given its coupon, maturity, and market interest rates. 12 Exhibit 4 focuses on the direct effects of the value factors. It ignores any interaction terms, for example, the possibility that changes in the risk-free interest rate can enhance or reduce the effect of the credit risk factor, as explained in the text.

10-Phoa-MertonModel Page 211 Wednesday, July 23, 2003 10:16 AM

Implications of Merton Models for Corporate Bond Investors Wesley Phoa Vice President, Quantitative Research Capital Strategy Research The Capital Group Companies

n the 2000–2002 period, and to a lesser extent in 2003, participants in U.S. capital markets have observed a strong link between equity markets and corporate bond spreads:

I

■ When stock prices fall, bond spreads tend to widen. ■ However, the relationship seems nonlinear: It appears strongest when

stock prices are low. A typical example is shown in Exhibit 1, which plots the relationship between the daily stock price of Nextel and the daily spread on its cash pay bonds. When the stock price was over $20, the relationship appeared weak or absent; but when the stock price fluctuated below that level, there was a very strong link. Have equity and bond markets always behaved like this, or is the recent period anomalous? Exhibit 2 plots generic spreads on single-A and single-B rated corporates against the (log of the) U.S. equity index level, using data from the past ten years. In the period since 1998, there is a strong relationship; but in the 1992–1998 period the relationship seems weaker, if it exists at all. To repeat the question: is the period since 1998 unusual? The author thanks Eknath Belbase and Ellen Carr for their useful comments.

211


212


EXHIBIT 1

Nextel: Bond Spread versus Stock Price

EXHIBIT 2

U.S. Equity Market and Corporate Bond Spreads, 1992–2002


Implications of Merton Models for Corporate Bond Investors

EXHIBIT 3

213

U.S. Equity Market and Corporate Bond Spreads, 1919–1943

Exhibit 3 plots U.S. corporate bond spreads against the U.S. equity index level during the 1919–1943 period. The resemblance to Exhibit 1 is striking—there is a clear relationship, which was stronger when equity prices were lower. (Note that spreads on low quality bonds were tighter in 1921 than they were when the equity index revisited comparable levels after the Crash, presumably because firms accumulated more debt in the intervening period.) Exhibit 4 plots spreads on U.S. railroad bonds against the U.S. railroad stock index during the 1857–1929 period. (U.K. gilts are used as a risk-free yield benchmark, consistent with market practice during that period; note that there is no currency component to the spread, since both countries were on the gold standard for almost the whole period.) Exactly the same relationship appears in this graph. Furthermore, deviations from this relationship mostly have reasonable explanations; for example, spreads were unusually wide in the late 1860s/early 1870s, but this was a period when railroads’ capital structures were being dishonestly manipulated on a massive scale. It therefore seems that this link between the equity market and the corporate bond market has always existed. But it was only in the 1970s that a theoretical framework was developed, within which formal models of the relationship could be constructed.


214 EXHIBIT 4


U.S. Railroad Stocks and Railroad Bond Spreads, 1857–1929

This framework has been the subject of intense academic research, and has been widely adopted by commercial banks as a method for forecasting default rates and pricing loans. Interest from total return investors has been more recent. This article describes the role that equitybased credit risk models have begun to play in the mark-to-market world of a typical corporate bond investor.

MODEL OVERVIEW, USES AND CAVEATS Robert Merton proposed in 1974 that the capital structure of a firm can be analyzed using contingent claims theory: Debtholders can be regarded as having sold a put option on the market value of the firm; and equityholders’ claim on the firm’s value, net of its debt obligations, resembles a call option.1 In this framework, the meaning of default is that the value of the firm falls to a sufficiently low level that “the put option is exercised” by liquidating the firm or restructuring its debt. This idea led to the so-called “structural models” of credit risk, which assume that (1) default occurs when the market value of the firm falls below a clearly defined threshold, determined by the size of the 1

Robert C. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance 29 (May 1974), pp. 449–470.



215

firm’s debt obligations, and (2) the market value of the firm can be modeled as a random process in a mathematically precise sense. Taken together, these assumptions make it possible to calculate an estimated default probability for the firm. The precise estimate will depend on the assumed default threshold, the nature and parameters of the random process used to model the firm’s value, and possibly other technical details. Different choices lead to different models.2 However, all the different structural models have a strong family resemblance. First, they have similar inputs; the key inputs tend to be: 1. The capital structure of the firm. 2. The market value of the firm, usually derived from its stock price. 3. The volatility of the firm’s market value, usually derived from stock price volatility. Second, they make qualitatively similar predictions. In particular, they all imply that: ■ The credit risk of a firm rises as its stock price falls. ■ However, this relationship is nonlinear, and is most apparent when the

stock price is fairly low. This is precisely the pattern observed in Exhibits 1 through 4. Since the late 1990s there has been a dramatic rise in the popularity of structural models. KMV Corporation pioneered the approach and built a formidable global client base among commercial banks, but other models have recently gained substantial followings among other capital market participants; these include CSFB’s CUSP Model, RiskMetrics’ CreditGrades and Bank of America’s COAS. The Capital Group has developed a proprietary model along similar lines. This article discusses whether structural models should be of interest to corporate bond investors. It assumes some general familiarity with the theoretical details. In gauging the reliability and usefulness of structural models of credit risk, it is important to understand that they can be used in a number of different ways. For example, 2

For an overview of structural credit risk models, as well as the alternative “reducedform” approach commonly used to price credit derivatives, see Kay Giesecke, Credit Risk Modeling and Valuation: An Introduction, Humboldt-Universität zu Berlin, August 19, 2002. For a survey of empirical results, see Young Ho Eom, Jean Helwege, and Jingzhi Huang, Structural Models of Corporate Bond Pricing: An Empirical Analysis, EFA 2002 Berlin, February 8, 2002.


216


1. To estimate default risk. 2. To predict rating transitions (especially downgrades). 3. To identify relative value opportunities within a specific firm’s capital structure. 4. To predict changes in corporate bond spreads. 5. To identify relative value opportunities within the corporate bond market. 6. To assess the sensitivity of corporate bond spreads to equity prices. It may turn out that a model performs well in some of these applications, but is useless for others. And it is important to understand that in each case, the key premises differ. Every proposed application of a model assumes that: 1. The assumptions underlying the model are reasonably accurate. However, all applications except the first, make further strong assumptions about the way in which different market participants process new information relevant to credit risk. Remembering that the key input to these models is the stock price, the corresponding assumptions are: 2. Rating agencies sometimes lag equity markets. 3. Equity and bond markets sometimes process information in inconsistent ways. 4. Bond markets sometimes lag equity markets. 5. Bond markets sometimes process information less efficiently than equity markets. 6. Equity and bond markets eventually process information in consistent ways. Assumptions 1 and 6, and perhaps 2, are quite plausible; assumptions 3, 4, and 5 are more questionable. Research within the Capital Group has been mainly concerned with the last application: assessing the sensitivity of corporate bond spreads to equity prices. It is therefore assumptions 1 and 6 that play the most important role. To see why assessing equity sensitivity is important, it is helpful to adopt the perspective of asset allocation. One reason to own bonds is that they provide diversification versus equity returns: Bond investments should hold up well in periods where equities have poor returns. Therefore, it is not rational to invest an excessive amount in corporate bonds, which have a high correlation with equities. However, structural models imply, and experience shows, that this correlation varies with equity prices. Therefore, a prudent approach to corporate bond investment in a



217

portfolio context should take equity prices into account. This is the place where structural models can play a crucial role in credit risk management. A final important use for structural models is the estimation of default correlations (or joint default probabilities), which are crucial in applications such as portfolio credit risk aggregation and CDO modeling. Default correlations are hard to measure, particularly for the investment grade universe. For example, default data is far too sparse; both default and rating transition are hard to use directly because of timing problems; and the use of bond spread data tends to understate correlations. The Merton approach suggests that default correlations may be inferred from directly observable equity market data. Note that default correlations need not be equal to equity (or firm value) correlations. Nor can one estimate the default correlation just by measuring the correlation of changes in the estimated default probability. However, the calculations do turn out to be computationally tractable.3

SOME EMPIRICAL RESULTS Since the technical details of structural models are covered elsewhere, it seems more helpful to organize the discussion around some empirical illustrations. To begin with, Exhibits 5 to 9 indicate how the model works, using the example of Nextel Communications. Exhibit 5 shows the historical stock price and “distance to default.” The latter is derived from Nextel’s enterprise value (determined by its stock price) and the amount of debt in its capital structure; a distinction is made between long- and short-term debt. Note that Nextel’s leverage increased during this period, so the fact that the stock price was the same on two different dates does not imply that the distance to default was the same. Exhibit 6 shows the estimated historical volatility of Nextel’s enterprise value; this estimate has fluctuated between 35% and 60%, as equity volatility has varied, so it would clearly not be valid to use a constant volatility input. Note that this volatility is not directly observable, and different models estimate it in different ways. Most models derive it from equity volatility. For example, KMV uses historical equity volatilities computed using a fixed window; the Capital Group’s model uses historical volatilities computed using a simple exponentially weighted scheme; and CUSP uses option implied volatilities if they are available, 3

For a closed-form formula, see Chunsheng Zhou, Default Correlation: An Analytical Result, Finance and Economics Discussion Series 1997–27, Board of Governors of the Federal Reserve System, May 1, 1997.


218


EXHIBIT 5

Nextel: Stock Price and Distance to Default

EXHIBIT 6

Nextel: Volatility of Enterprise Value



EXHIBIT 7

Nextel: Stock Price and Credit Risk Measure

EXHIBIT 8

Nextel: Credit Risk Measure and Bond Spread

219


220

EXHIBIT 9


Nextel: Bond Spread versus Credit Risk Measure

and a GARCH estimate if they are not. There are a few models (such as COAS) that do not look at equity volatility, but use the volatility implied by the market price of debt instead; this is an interesting alternative, though it is impracticable if the bonds are illiquid and/or there is a substantial amount of bank debt in the capital structure. Exhibit 7 shows the historical daily stock price and credit risk measure, labeled “bond risk” on the graph. This is the probability that Nextel’s enterprise value will cross the default threshold within the next 12 months; however, since default need not be an automatic event, this risk measure should be interpreted as a “probability of distress,” or the probability of a severe financial crisis, rather than a literal default probability. Exhibit 8 shows the historical daily credit risk measure (probability of distress) and the historical daily spread on a specific Nextel cash pay bond. As expected, the bonds tend to widen when the probability of distress rises, and vice versa. Though it does seem that in 2000-2001 the bond market was somewhat slow to respond to an increase in risk. Exhibit 9 is a scatter plot of the bond spread against the probability of distress. In theory, this should be an upward sloping line or curve, and the observations do show this pattern. That is, the model “works.” However, there are two other interesting phenomena:



221

1. The curve slopes upward more sharply at high levels of risk. This is observed for many high-yield issuers, and may reflect declining recovery value assumptions. 2. In the more recent period, the curve as a whole has shifted upward. This is not a widespread phenomenon, and may reflect an increased level of investor risk aversion towards the high yield wireless sector in 2002, independent of current equity prices. Some additional findings emerge when looking at further examples, this time drawn from the investment-grade universe. Exhibit 10 shows, for Ford, the historical probability of distress and the historical bond spread; as before, the bonds tend to widen when the probability of distress rises, and vice versa. Rating actions are also marked on the graph: The larger white circles indicate Moody downgrades and the smaller circles mark dates when Ford was put on negative watch; gray circles mark more recent rating actions by S&P. In some cases the bonds seemed to widen in response, but in some cases the bonds had clearly widened in anticipation, and often the performance of the bonds was not related at all to a rating action. This example shows that for bond investors, predicting rating transitions is not the most important application. EXHIBIT 10

Ford: Credit Risk Measure, Bond Spread, and Rating Actions


222 EXHIBIT 11


Ford: Bond Spread versus Credit Risk Measure

Exhibit 11 is a scatter plot of the bond spread against the probability of distress. In this case the observations cluster nicely around an upward sloping straight line. There is an excellent relationship between the actual spread on the bonds and the model’s estimate of credit risk. Thus, the model provides a good way to estimate the equity sensitivity of Ford bonds. Exhibit 12 shows a different way of visualizing how this sensitivity has changed over time. The thick solid line shows Ford’s historical stock price. The thin dotted line marks the “critical range” where the equity sensitivity of the bonds rises significantly, while the thin solid line marks the point of maximum sensitivity. (Note that if both the capital structure and volatility were constant over time, these lines would be horizontal.) Exhibit 13 shows, for Sprint, a scatter plot of the bond spread against the probability of distress. Again, the observations cluster nicely around an upward sloping straight line, indicating that the model is very consistent with the market behavior of the debt; note that there is more scatter at higher levels of risk. Does scatter represent trading opportunities? Exhibit 14 shows the daily probability of distress, the daily bond spread, and the daily credit default swap spread (CDS) (plus the 5-year swap spread). An analysis of this data indicates that none of these three markets consistently leads the other two. The bond spread and CDS



EXHIBIT 12

Ford: Stock Price and Critical Range

EXHIBIT 13

Sprint: Bond Spread versus Credit Risk Measure

223


224


EXHIBIT 14 Sprint: Credit Risk Measure, Bond Spread, and Credit Default Swap Spread

spread are tightly linked. One market occasionally lags the other, but only by a day. The probability of distress (which reflects the equity market) is much less tightly coupled. It sometimes leads the bond and CDS markets by a week or more—and is for that reason a trading signal— but it also sometimes lags.4 Structural models will only give useful trading signals to a bond investor when bond markets are less efficient than equity markets (perhaps due to the constraints affecting bond investors, and/or their bounded rationality). An example might be when a credit event affects 4

Note that if there are cash instruments trading at a significant discount to par, both an implied default probability and an implied recovery rate (for bonds) can be computed from the observed credit default swap basis. Furthermore, a structural model can be used to estimate an “expected recovery rate” based on the conditional mean exceedance (i.e., the mathematical expectation of the firm value conditional on its dropping below the default threshold). A would-be arbitrageur might attempt to profit from discrepancies between the default probabilities and recovery rates implied by the equity and CDS markets. Unfortunately, neither estimate of the recovery rate is very robust. A further problem is that priority of claims is often violated in the event of default and restructuring, which complicates the analysis of specific debt securities.



225

an industry as a whole, causing bond investors to rapidly reduce their industry allocation by selling across the board. This may trigger a uniform widening in bonds across different issuers, even though the actual rise in credit risk may vary from firm to firm. Finally, the above analysis assumes a constant capital structure going forward. Investors with longer term forecasting horizons might find it useful to extend the model by incorporating discretionary capital structure choices triggered by exogenous factors, such as macroeconomic conditions.5

FEEDBACK LOOPS AND CAPITAL STRUCTURE DYNAMICS As the popularity of structural credit risk models has grown—and particularly since Moody’s acquired KMV—some market participants have suggested that the widespread use of these models creates excess volatility in debt markets and even leads to liquidity crises. The underlying complaint is that models that refer to stock prices are in some sense circular. This concern is most commonly expressed at the level of the individual firm. The use of the KMV model by commercial banks can lead to an unfortunate feedback loop affecting highly levered companies. When the stock price falls, the model says that the credit risk of the company has risen. This causes banks to restrict access to credit, and also raises financing costs by pushing up bond spreads. This in turn puts financial pressure on the company, leading to a further decline in the stock price. Of course, this can only occur if the firm has an insufficiently liquid balance sheet. During 2002, several high profile firms in the telecommunications and energy sectors are said to have been affected. It is hard to model this effect at the firm level, but stylized numerical simulations at the industry level provide some interesting results. Consider the following ideal industry model: ■ The industry begins with a fixed allocation of debt and equity. ■ Banks provide all the debt, and in doing so maintain fixed capital

ratios. ■ Banks fund themselves at a constant rate (e.g., a fixed spread over a

risk-free rate). ■ The industry receives a growing revenue stream which is used to service

debt. 5 See, for example, Robert Korajczyk, and Amnon Levy, “Capital Structure Choice: Macroeconomic Conditions and Financial Constraints,” Journal of Financial Economics, 68 (April 2003), pp. 75–109.


226


■ Revenues in excess of debt service increase the industry’s equity base. ■ Individual issuer defaults occur at a rate predicted by a structural credit

risk model. ■ The stream of debt service payments causes bank capital to rise. ■ Individual issuer defaults trigger loan losses, causing bank capital to fall. ■ A fixed percentage of any increase in bank capital is allocated back to

the industry. Exhibits 15 and 16 compare the impact of two different loan pricing policies, static and risk based. More precisely, the two policies are: 1. New loans priced at the same rate as the original debt. 2. New loans continually repriced with reference to the structural credit risk model. Risk-based pricing is assumed to use a spread based on the model’s estimated default probability times a fixed expected loss rate. This is broadly consistent with the policy implied by the forthcoming Basel II regime. It is also assumed that the model is reliable, i.e., the actual default rate is equal to the model’s estimated default probability. EXHIBIT 15

Comparative Equity Return Dynamics: Static Loan Pricing



EXHIBIT 16

227

Comparative Equity Return Dynamics: Risk Based Loan Pricing

In each case the scatter plot compares the returns to industry equity investors and the return on that portion of bank tier-one capital allocated to the industry. The series of points shows how these returns evolve over time (the arrows indicate the direction of time). The model parameters are calibrated so that initial returns to both industry equity investors and bank tier-one capital are quite attractive. Exhibit 15 assumes static loan pricing. In this simulation, returns to both industry equity investors and bank tier-one capital remain high for a while. Then, as leverage rises, there is an increase in the industry default rate. Return on bank capital falls sharply because of credit losses. However, as banks continue to extend loans at the same interest rate, equity returns for industry investors remain high. The benefit of leverage for equity holders offsets the damage done by defaults. Finally an equilibrium is reached in which firms are more highly levered than initially, industry equity returns are higher, but banks suffer constant negative returns on tier-one capital due to the higher equilibrium rate of defaults. Exhibit 16 assumes risk based loan pricing. In this simulation, returns to both industry equity investors and bank tier one capital again remain high for a while. Then, as before, as leverage rises, there is an increase in the industry default rate. Banks adjust loan pricing, but return on bank capital still falls because there is always an existing


228


stock of debt whose interest rate is too low. Meanwhile, return on industry equity capital falls since debt service eats up a higher and higher proportion of revenue. This causes leverage to spiral even higher. Eventually debt service exceeds revenue. There is no equilibrium. Instead, returns on both industry equity and bank capital become increasingly negative until both are wiped out. These findings should not be taken too literally. The premises are not realistic. For example, it is implausible that banks would be willing to absorb credit losses forever, and it is implausible that the industry would never raise new equity in the capital markets (although this may become extremely difficult in periods of distress). So it is not yet possible to make quantitative real world predictions using this approach. The qualitative results remain a nagging worry rather than a concrete forecast.6 Structural models of credit risk can be powerful risk management tools, and perhaps even useful trading tools. Although they only became popular rather recently, they do seem to reflect timeless relationships between the corporate bond and equity markets. However, as the KMV model and its competitors exert an increasing influence on banks’ credit decisions, investor behavior and possibly even rating agency actions, it is legitimate to ask whether there is a tradeoff between capital market efficiency and market stability.

6 It is surprisingly difficult to devise more realistic models of capital structure dynamics. Even sophisticated models have trouble accounting for the mix of debt and equity observed in the real world; for example, see Mathias Dewatripont and Patrick Legros, Moral Hazard and Capital Structure Dynamics, CARESS Working Paper 0207, July 5, 2002.

11-Choudhry-CreditDerivs Page 229 Wednesday, July 23, 2003 10:15 AM

Some Issues in the Asset Swap Pricing of Credit Default Swaps Moorad Choudhry Senior Fellow Centre for Mathematical Trading and Finance CASS Business School, London

redit default swaps are well established in the debt capital markets. In addition to their use as hedging instruments against credit risk, default swaps enable market participants to establish a synthetic short position in a specific reference credit and implement trading strategies in the credit markets outside the cash markets. A common method of pricing default swaps is by recourse to the asset swap spread of the reference credit, as the default swap premium should (in theory) be equal to the asset swap spread of the reference asset. We first consider the use of this technique before looking at the issues that cause these two spread levels to differ.

C

ASSET SWAP PRICING Credit derivatives are commonly valued using the asset swap pricing technique. The asset swap market is a reasonably reliable indicator of the returns required for individual credit exposures, and provides a mark-to-market framework for reference assets as well as a hedging mechanism. A par asset swap typically combines the sale of an asset such as a fixed-rate corporate bond to a counterparty, at par and with

229


230


no interest accrued, with an interest-rate swap. The coupon on the bond is paid in return for LIBOR, plus a spread if necessary. This spread is the asset swap spread and is the price of the asset swap. In effect the asset swap allows market participants that pay LIBOR-based funding to receive the asset swap spread. This spread is a function of the credit risk of the underlying bond asset, which is why it may be viewed as equivalent to the price payable on a credit default swap written on that asset. The generic pricing is given by Y a = Y b – ir where Ya = the asset swap spread Yb = the asset spread over the benchmark ir = the interest-rate swap spread The asset spread over the benchmark is simply the bond (asset) redemption yield over that of the government benchmark. The interestrate swap spread reflects the cost involved in converting fixed-coupon benchmark bonds into a floating-rate coupon during the life of the asset (or default swap) and is based on the swap rate for that maturity. The theoretical basis for deriving a default swap price from the asset swap rate can be illustrated by looking at a basis-type trade involving a cash market reference asset (bond) and a credit default swap written on this bond. This is similar in approach to the risk-neutral or no-arbitrage concept used in derivatives pricing. The theoretical trade involves: ■ A long position in the cash market floating-rate note (FRN) priced at

par, and which pays a coupon of LIBOR + X basis points. ■ A long position (bought protection) in a default swap written on the

same FRN, of identical term-to-maturity and at a cost of Y basis points. The buyer of the bond is able to fund the position at LIBOR. In other words, the bondholder has the following net cash flow: ( 100 – 100 ) + [ ( LIBOR + X ) – ( LIBOR + Y ) ] or X – Y basis points. In the event of default, the bond is delivered to the protection seller in return for payment of par, enabling the bondholder to close-out the funding position. During the term of the trade the bondholder has


Some Issues in the Asset Swap Pricing of Credit Default Swaps

231

earned X – Y basis points while assuming no credit risk. For the trade to meet the no-arbitrage condition, we must have X = Y. If X ≠ Y, the investor would be able to establish the position and generate a risk-free profit. This is a logically tenable argument as well as a reasonable assumption. The default risk of the cash bondholder is identical in theory to that of the default seller. In the next section we illustrate an asset swap pricing example before looking at why, in practice, there exist differences in pricing between credit default swaps and cash market reference assets.

ASSET SWAP PRICING EXAMPLE XYZ plc is a Baa2-rated corporate. The 7-year asset swap for this entity is currently trading at 93 basis points. The underlying 7-year bond is hedged by an interest-rate swap with an Aa2-rated bank. The risk-free rate for floating-rate bonds is LIBID minus 12.5 basis points (assume the bid-offer spread is 6 basis points). This suggests that the credit spread for XYZ plc is 111.5 basis points. The credit spread is the return required by an investor for holding the credit of XYZ plc. The protection seller is conceptually long the asset, and so would short the asset to hedge its position. This is illustrated in Exhibit 1. The price charged for the default swap is the price of shorting the asset, which works out as 111.5 basis points each year. Therefore we can price a credit default written on XYZ plc as the present value of 111.5 basis points for seven years, discounted at the interest-rate swap rate of 5.875%. This computes to a credit default swap price of 6.25%. We list the terms below: Reference Term EXHIBIT 1

XYZ plc 7 years

Credit Default Swap and Asset Swap Hedge


232


Interest rate swap rate Asset swap

5.875% LIBOR plus 93 bps

Default swap pricing: Benchmark rate Margin Credit default swap Default swap price

LIBID minus 12.5 bps 6 bps 111.5 bps 6.252%

PRICING DIFFERENTIALS A number of factors observed in the market serve to make the price of credit risk that has been established synthetically using default swaps to differ from its price as traded in the cash market. In fact, identifying (or predicting) such differences gives rise to arbitrage opportunities that may be exploited by basis trading in the cash and derivative markets.1 These factors include the following: ■ Bond identity: The bondholder is aware of the exact issue that they are

■

■

■

■

■ 1

holding in the event of default, however default swap sellers may receive potentially any bond from a basket of deliverable instruments that rank pari passu with the cash asset, where physical settlement is required. This is the delivery option afforded the long swap holder. The borrowing rate for a cash bond in the repo market may differ from LIBOR if the bond is to any extent special. This does not impact the default swap price which is fixed at inception. Certain bonds rated AAA (such as U.S. agency securities) sometimes trade below LIBOR in the asset swap market. However, a bank writing protection on such a bond will expect a premium (positive spread over LIBOR) for selling protection on the bond. Depending on the precise reference credit, the default swap may be more liquid than the cash bond, resulting in a lower default swap price, or less liquid than the bond, resulting in a higher price. Default swaps may be required to pay out on credit events that are technical defaults and not the full default that impacts a cash bondholder. Protection sellers may demand a premium for this additional risk. The default swap buyer is exposed to counterparty risk during the term of the trade, unlike the cash bondholder.

This is known as trading the credit default basis and involves either buying the cash bond and buying a default swap written on this bond—or selling the cash bond and selling a credit default swap written on the bond.



233

For these and other reasons the default swap price often differs from the cash market price for the same asset. This renders continued reliance on the asset swap pricing technique a trifle problematic. Therefore banks are increasingly turning to credit pricing models, based on the same models used to price interest rate derivatives, when pricing credit derivatives.

ILLUSTRATION USING BLOOMBERG Observations from the market illustrate the difference in price between asset swaps on a bond and a credit default swap written on that bond, reflecting the factors stated in the previous section. We show this now using a euro-denominated corporate bond. The bond is the Air Products & Chemicals 6¹₂% bond due July 2007. This bond is rated A3/A as shown in Exhibit 2, the description page from Bloomberg. The asset swap price for that specific bond to its term to maturity as at January 18, 2002 was 41.6 basis points. This is shown in Exhibit 3. The relevant swap curve used as the pricing reference is indicated on the screen as curve 45, which is the Bloomberg reference number for the euro swap curve and is shown in Exhibit 4. EXHIBIT 2

Bloomberg DES Page for Air Products and Chemicals Bond

Source: Bloomberg Financial Markets.


234 EXHIBIT 3


Asset Swap Calculator Page ASW on Bloomberg, January 18, 2002


EXHIBIT 4

Euro Swap Curve on Bloomberg as at January 18, 2002




235

EXHIBIT 5

Default Swap Page CDSW for Air Products and Chemicals Bond, January 18, 2002


We now consider the credit default swap page on Bloomberg for the same bond, which is shown in Exhibit 5. For the similar maturity range the credit default swap price would be approximately 115 basis points. This differs significantly from the asset swap price. From the screen we can see that the benchmark curve is the same as that used in the calculation shown in Exhibit 3. However, the corporate curve used as the pricing reference is indicated as the euro-denominated U.S.-issuer A3 curve, and this is shown in Exhibit 6. This is page CURV on Bloomberg, and is the fair value corporate credit curve constructed from a basket of A3 credits. The user can view the list of bonds that are used to construct the curve on following pages of the same screen. For comparison we also show the Bank A3 rated corporate credit yield curve, in Exhibit 7. Prices observed in the market will invariably show this pattern of difference between the asset swap price and the credit default swap price. The page CDSW on Bloomberg uses the generic risky curve to calculate the default swap price, and adds the credit spread to the interest rate swap curve (shown in Exhibit 4). However, the ASW page is the specific asset swap rate for that particular bond, to the bond’s term to maturity. This is another reason why the prices of the two instruments will differ significantly.


236 EXHIBIT 6


Fair Market Curve, Euro A3 Sector


EXHIBIT 7

Fair Market Curve, Euro Banks A3 Sector




EXHIBIT 8

237

CDSW Page with Discounted Spreads Model Selected


On Bloomberg the user can select either the JPMorgan credit default swap pricing model or a generic discounted credit spreads model. These are indicated by “J” or “D” in the box marked “Model” on the CDSW page. Exhibit 8 shows this page with the generic model selected. Although there is no difference in the swap prices, as expected the default probabilities have changed under this setting. Our example illustrates the difference in swap prices that we discussed earlier, and can be observed for any number of corporate credits across sectors. This suggests that middle-office staff and risk managers who use the asset swap technique to independently value credit default swap books are at risk of obtaining values that differ from those in the market. This is an important issue for credit derivative market-making banks.


12-Hjort Page 239 Wednesday, July 23, 2003 9:59 AM

Exploring the Default Swap Basis Viktor Hjort Credit Strategist Morgan Stanley

he significant growth seen in the credit default swap market has been driven to a great extent by its role as a proxy for the bond and loan market. Although the default swap market adds value for many investors by providing access to credits not readily available in the cash markets, its main importance is as an instrument that allows investors to isolate credit risk from interest and funding risk. This assumes that the exposure achieved through a default swap position replicates that of a similar position in the cash market, but stripped of interest rate and funding risk. However, in the market we frequently observe divergence, often significant, between the premium paid on a default swap and that paid on a cash instrument of the same credit and maturity. This divergence arises for both fundamental and technical reasons. In this article, we aim to present an overview of the factors driving this basis and also analyze the relationship between the cash and derivatives markets at the market, sector, and individual credit level. The default swap market is often perceived as driven primarily by technical factors, particular to this market only. We find little evidence to support this view. Instead, the nature of the markets argues for a close correlation. It also argues for the default swap market effectively being a “high-beta” (i.e., positively correlated with, but more volatile) version of the underlying cash market. From this follows a number of investment implications. First, we think investors should aim to get exposure to credit in whichever market is cheaper. Trading the basis can allow investors to pick up signifi-

T

239


240


cant spread without changing the view on the credit. Second, and more importantly, we believe investors should use the high-beta character of the default swap market to position themselves for major rallies or selloffs. Being long the market that rallies the most can be as important as having the right call on the direction of the market itself.

A SIMPLIFIED WORLD: THE ARBITRAGE-FREE RELATIONSHIP In order to explore the reasons why cash and default swap markets can trade at different levels, we must first establish the relationship between the two markets. This can be illustrated by a theoretically risk-free trade (see Exhibit 1): ■ The investor buys a par-par asset swap, paying a coupon of LIBOR

plus a spread of S. The position is funded on balance sheet at a cost of LIBOR flat. ■ The investor buys credit protection with identical maturity on the notional amount, at a default swap premium of D. Consider the following two scenarios: ■ No default: In the absence of any default, the asset swap will redeem at

par at the maturity of the trade, and the redemption amount will be used to pay off the funding. The default swap will simply terminate at that time. ■ Default event: In the event of a default, the investor will deliver the defaulted asset to the protection seller, and receive par in return. The par amount will be used to pay off the funding, and the default swap will then terminate. EXHIBIT 1

The Theoretical Arbitrage Free Relationship

Source: Morgan Stanley Research.


Exploring the Default Swap Basis

241

Under both scenarios, the investor is fully hedged and has no exposure to the default of the asset—the trade is credit risk-neutral.1 The default swap basis, D-S, in this theoretical arbitrage-free framework must therefore be zero. If it is positive, i.e., the default swap premium is greater than the spread earned on the asset, the basis can be exploited by selling the asset and selling protection. If it is negative, the investor can buy the asset swap and buy protection. Both strategies would yield a positive margin for assuming no credit risk.2 The framework above does, however, require a number of assumptions, and these may not hold in the marketplace, hence causing a nonzero basis to exist. These factors may be of technical or fundamental nature. At any given point in time, the basis may be positive or negative, but most of the time it will be affected by more than one factor. Below, we discuss each of the factors in turn. Several factors, although listed as a driver of either positive or negative basis, can also work in the other direction if the opposite circumstance occurs. Coupon step-ups, for instance, will drive the basis in positive direction, while coupon stepdowns will drive it in a negative manner.

FUNDAMENTAL FACTORS Fundamental factors affecting the basis reflect the differences in the nature of the asset classes that can cause the basis to diverge from zero.

Fundamental Factors Causing a Positive Basis Exhibit 2 lists the fundamental factors that drive the basis in positive direction. EXHIBIT 2

Fundamental Drivers of a Positive Basis

The cheapest-to-deliver option Coupon step-ups in corporate bonds Default swap premiums floored at zero Bond price less than par Funding below LIBOR 1

In the event of default, the investor will have exposure to the unwind of the interest rate swap associated with the asset swap. If the asset is a par-floater, however, this will not be the case. 2 This arbitrage-free relationship will also hold in the event the asset is funded on repo at LIBOR. It does not take into account any mark-to-market effect due to convexity differences between asset swap and default swap position.


242


The Cheapest-to-Deliver Option A physically settled default swap contract entitles the protection buyer to deliver any of a potentially large number of assets into the contract upon the occurrence of a credit event. This option can potentially have material value, depending on the number of assets available and, to some extent, the type of credit event, as it allows the protection holder to switch out of one asset into a cheaper one. The possibility of being delivered a less-valuable asset should make protection sellers demand a higher premium than that of a comparable asset swap. If not, arbitrageurs would buy bonds and protection and get a free option. In theory, the option should have little value, as all pari passu ranked assets should trade on their recovery value following a default. In practice, however, this is not always the case. More structured assets may trade cheaper than more vanilla-type assets and less liquid assets cheaper than more liquid ones. In addition, loans should trade higher than bonds, as they have historically yielded better recovery values than bonds of the same seniority, due to covenants that often offer creditors better protection. Moody’s historical default price shows that senior secured loans trade at an average price of 71% compared to 37% for senior unsecured bonds.3 Importantly, the type of credit event matters. A “soft” credit event, which does not immediately cause default, may allow the assets to continue to trade with price differentials. The rapid development of the standard default swap contract has significantly reduced the scope for contracts to be triggered by soft credit events, but some issues, notably the restructuring credit event, still leave that possibility open. The most famous example of a soft credit event triggering a default swap contract was Conseco’s debt restructuring in 2000. Conseco renegotiated an extension of the maturity of some of its outstanding loans by three months, but simultaneously increased the coupon and enhanced the covenant protection. The net effect of the changes was roughly creditneutral and Moody’s, for instance, did not consider it a “diminished financial obligation.” The maturity extension, however, was clearly captured by the ISDA restructuring definition, and loss payments were triggered under default swap contracts. However, given that no “hard” default had occurred, there were major price differentials among Conseco debt assets; and many protection buyers, rather than delivering the restructured loans (which were actually trading up in the market due to the credit-improving features) delivered cheaper, long-dated bonds. The European and U.S. market trades under different standards regarding the Restructuring event. Whereas the European market uses the original 3

See Moody’s, Default and Recovery Rates of Corporate Bond Issuers 1970–2001, February 2002.



243

Restructuring definition, the U.S. market has moved to a modified version (ModR), which restricts the delivery of long-dated obligations, triggers by structural subordination, and bilateral loans.4 The ModR definition significantly reduces the scope for contracts being triggered by soft credit events and, all things equal, should reduce the basis. At the time of writing the European market looks likely to be moving towards a third definition, to be referred to as Modified-Modified Restructuring, which represents a move towards (albeit not identical to) the U.S. standard. In addition, U.S. contracts are increasingly often quoted without Restructuring as a credit event. In effect, the cheapest-to-deliver option adds an additional spread layer on top of the pure compensation for the credit risk. To the extent that the option has any value, it should increase the closer to the strike price, i.e., a potential credit event, the option gets. This has the potential not only to cause a positive basis per se, but also to cause the basis to widen and tighten with the overall direction of the market. For example, as credit quality deteriorates, investors will demand higher compensation for assuming credit risk and spreads will widen. However, the option value of the default swap will also increase, causing the basis to widen. Alcatel (BBB/Baa2) is a good example. In 2001, it was rated mid-A, but then saw a series of downgrades following a sharp deterioration in its operational cash flow as its largest customers, major European telecom operators, reduced capital expenditures. The basis consequently moved continuously wider throughout the period (see Exhibit 3). The reverse would be true as credit quality improves. KPN, for instance, suffered a sharp deterioration in its credit profile during autumn 2001, as merger talks with more highly rated Belgacom broke down and investors began questioning the liquidity situation of the € 5 company. Eventually, in November 2001, the company announced a A billion rights issue, fully underwritten by the Dutch government. Credit spreads rallied sharply following the announcement and the rating agencies changed their outlooks to stable from negative, removing the risk of downgrade to noninvestment grade. As a consequence, the basis tightened more than 300 bp (see Exhibit 4). This additional option premium creates a high-beta characteristic of the default swap in relation to the cash bond, an issue that will be explored later in this article.5 4

Restructuring Supplement to the 1999 ISDA Credit Derivatives Definitions, May 2001. 5 The cheapest-to-deliver option is only one of several factors contributing to the high-beta nature of the default swap. As is explained later in this article, a cash price less than par, bond coupon step-ups, difficulty of shorting the cash market, and relative liquidity all add to this nature.


244 EXHIBIT 3


Alcatel: Deteriorating Credit Profile Gives Option Value

Source: Morgan Stanley Research.

EXHIBIT 4

KPN: Recovery Causing Basis Tightening

Source: Morgan Stanley Research, MSCI.



245

Coupon Step-Ups in Corporate Bonds An increasing number of bonds carry clauses triggering the coupon to step up in the event of a ratings downgrade or similar protective covenants. The value of this embedded option is a function of the probability of the downgrade, but it can never be less than zero. Therefore, the existence of stepups will add a layer of protection for bondholders that is not reflected in a similar default swap position, and the basis should be positive as a result. A number of issues also have step-down clauses. This implies a higher risk in the bond position compared to the default swap position, and would be a force for a negative basis. Where both step-up and stepdown clauses are present, the basis would be a function of the probability of ratings action in either direction.

Default Swap Premiums Floored at Zero Default swap premiums must be positive, since they represent an insurance premium. Even for very highly rated reference entities, such as the German government or the World Bank, where the probability of default is very low, the premium for assuming the default risk must still be positive. However, as LIBOR merely reflects the interbank lending risk (roughly AA), it is obviously possible for higher rated entities to finance themselves at sub-LIBOR yields, and thus a AAA rated issue will normally asset swap to a negative spread, causing the basis to be positive. Look at Nestle (AAA/Aaa) for instance. The company announced a €A400 million issue in March 2002 yielding a spread of Bunds+3 bp! The default swap is one of the tightest quotes in the corporate universe at 15 bp (mid), and the basis has been very stable at between 35-40 bp (see Exhibit 5). An interesting effect of the positive basis at the high end of the ratings spectrum is the so-called basis smile (see Exhibit 6). This represents the default swap-cash basis, which is positive for lower-rated credits, for example due to the cheapest-to-deliver option, roughly zero in the “belly” of the curve, and then positive again at the very high end, due to the zero-floor for default swap premiums.

Bond Price Less than Par Bonds (fixed-rate) regularly trade at prices away from par and not infrequently significantly away from par. The equivalence of a cash bond and a default swap position, however, requires the bond to trade at par. If the price is below par, a bond investor is exposed to lower risk than the seller of protection, who is guaranteeing a par amount. An investor who owns a defaulting bond at a cash price of 50 with a recovery value of 40 loses 10% of the par amount. A seller of default protection, however, loses 60%. In the same way, if the bond position is


246

EXHIBIT 5


Nestle: Zero-Premium Floor Causing Positive Basis


EXHIBIT 6

The Basis Smile




EXHIBIT 7

247

Fundamental Drivers of a Negative Basis

Funding above LIBOR Counterparty risk Coupon step-downs in corporate bonds Bond price above par

above par, the risk is greater than the default swap position. This implies that where bonds are trading below par, the basis should be positive and vice versa (another factor contributing to the basis smile being positive at the lower and deteriorating end of the credit curve).

Fundamental Factors Causing a Negative Basis Exhibit 7 lists the fundamental factors that drive the basis in negative direction.

Funding above LIBOR The choice of strategy for an investor/hedger comparing the default swap and cash alternatives will be impacted by the funding cost. On balance, the greater the ratio of lower-rated/ higher-rated market participants, the more negative the basis. As demonstrated above, default swaps are unfunded transactions that effectively lock in a funding cost of LIBOR. Bond investors, however, would need to finance the investment amount or, alternatively, fund the position on repo, in which case the cost of funding would be the repo rate. The cost of funding therefore becomes crucial for the relative attractiveness of the alternative strategies. For example, an investor with a funding cost of LIBOR + 30 would be indifferent between buying an asset at LIBOR + 100 and selling default protection on the credit at 70 bp (ignoring counterparty risk, see below). The difference between the funding cost and LIBOR determines how far through the asset swap spread the default swap premium should trade for the investor to be indifferent between the two strategies. For higher-rated investors that are able to finance themselves at subLIBOR levels, the reverse is true. An AAA rated insurance company, for example, that finances itself at LIBOR – 10 would prefer to buy the asset, as long as the default swap premium was no more than 10 bp wider than the asset swap spread.

Counterparty Risk Unlike a bond trade, the two counterparties in a default swap transaction bear exposure to each other’s ability to fulfill their respective obli-


248


gations throughout the life of the trade. In particular, the protection buyer’s exposure can be significant, as it relies on the seller to redeem a defaulted asset at par. To compensate for the additional risk, the protection buyer will expect to pay less than the cash spread. The protection seller’s counterparty risk is the mark-to-market loss on the position should the protection buyer default following spread tightening. This risk, however, will generally be of a very different order of magnitude than that of the protection buyer. A long credit position at 100 bp with a price value of a basis point (PV01) of 5 that tightens 50 bp will have 2.5 points of profit at risk, quite unlike the protection buyer, who may lose 60 points or more, depending on recovery value, should the counterparty default following a default by the reference entity.6 The default of Enron highlights how important this exposure can be. It also highlights the importance of risk mitigation through collateral agreements, which protected many of Enron’s counterparties as the energy (and credit derivatives) trader collapsed. Counterparty risk also affects an asset swap position (but not a parfloating-rate note). The buyer of a par asset swap faces the mark-to-market risk of the interest rate swap in the package in the event of the asset defaulting. This may cause an investor to demand a higher spread on the asset swap relative to the default swap, and drive the basis in negative direction.

TECHNICAL FACTORS Technical factors reflect the different trading environments affecting the asset classes.

Technical Factors Causing a Positive Spread Exhibit 8 lists the technical factors that drive the basis in positive direction. EXHIBIT 8

Technical Drivers of a Positive Basis

New issues • Corporate bonds • Corporate loans • Convertible bonds The difficulty of short cash assets Relative liquidity • Market participants • Regulation 6

The degree of counterparty risk is also affected by the default correlation between the protection seller and the reference entity.



249

New Issues The impact of new issuance on default swap premiums illustrates the convergence of credit risk in different asset classes into the default swap market: in order to accommodate new credit risk, bank loan departments, bond investors, and convertible bond investors would all look to the default swap market to hedge their risk. In that sense, the default swap market is the alternative asset class, the “opportunity cost,” for all three markets and the default swap premium will widen with any new issuance, loans, bonds or convertibles. To the extent that the investor base in the three cash markets is segmented, i.e., does not trade across asset classes, spread widening in the default swap market should widen the basis relative to those markets where the new issuance did not take place. For example, the announcement of a new convertible bond issue would widen default swap premiums as investors, mainly hedge funds, hedge out the credit risk. Traditional investors in straight bonds, however, would not normally be investing in convertibles and would not sell existing positions to make room for new credit exposure. Spreads in that market would then remain stable, causing a widening in the basis. Corporate Bonds As new corporate issues are priced, new potential “cheapest-to-deliver” issues become available for protection buyers. In theory, this could have a positive impact on the basis, but given that investors will likely sell existing bonds to free up lines for new exposure, both markets tend to move in tandem, although the basis to the loan market may widen. Bank Loans New loans are often hedged in the default swap market. Banks have an incentive to hedge credit risk, not only to manage their exposure to a particular credit or enhance portfolio returns, but also to free up regulatory capital. In addition, banks use the default swap market to fill available credit lines when other assets are either unavailable or unattractively priced. Banks have therefore historically been the most active participants in the default swap market. The British Bankers’ Association estimated in 2002 that banks made up 52% of all buyers of protection and 47% of all sellers. Convertible Bonds Convertible bond arbitrage players make up roughly 50% of the new issue market, depending on the issue.7 They buy convertible bonds to source cheap equity volatility and tend to hedge out the credit exposure by buying credit protection. New convertible bond issues are normally priced with cheap equity call options. Arbitrage players seek to arbitrage the cheap volatility implied by the embedded call option relative to existing over-the-counter


250


equity options. Credit risk can be hedged out by either selling the bonds on a callable asset swap basis or purchasing credit protection. The decision is impacted by a number of factors, such as cost of funding, but the steadily improving liquidity in the default swap market has seen a migration toward this asset class as the “hedge of choice” for many arbitrage players. The situation can be illustrated by the Fiat exchangeable issued in December 2001. As part of its deleveraging strategy aiming to cut debt by $3.5 billion, Fiat issued a $2.2 billion 3.25% bond exchangeable into GM shares. The announcement caused immediate buying of protection among arbitrage funds and premiums gapped out by 80 basis points. Meanwhile, spreads in the Fiat 06s, tightly held by Italian retail, hardly moved at all, causing the basis to blow out. The concentration of convertible bond idduance in the short end of the market, with maturities or put options normally inside five years, can cause a positive basis in the short end of the credit curve of a frequent convertible issuer, such as Olivetti. A large number of convertible bonds outstanding will cause a technical shortage of protection sellers to match the demand by arbitrage players, which will flatten (or even invert) the default swap curve relative to the straight-bond curve.

The Difficulty of Shorting Cash Assets The relative difficulty of outright shorting the cash market, loans or bonds, and the relative ease of doing it through the default swap market, tend to push the basis wider for deteriorating credits. Investors who want to express a negative view on a credit by shortselling the bond will have to source the bond in an often very illiquid repo market, running the risk of being bought out. In addition, the funding is variable and affected by the specialness of the asset. This adds risk to short-selling. Buying credit protection, on the other hand, does not involve any asset, and also effectively locks in the cost of funding at execution, making it generally a more convenient and less risky way of going short. Large new issues will make short-selling easier by adding liquidity to the repo market and mitigate the basis-widening effect.

7

Even if hedge funds should get limited allocation in the new issue process, there is nothing preventing them from builiding up positions in the secondary market. In certain credits where the credit bid is strong and gamma (i.e., the rate of change of stock sensitivity to the underlying movements in the stock price) attractive this can bring the hedge fund share as high as 75%. Likewise, for issues with very little option value, the hedge fund presence would likely be much lower.



EXHIBIT 9

251

Technical Factors Driving a Negative Basis

The structured synthetic bid for credit Funding risk Market liquidity

Technical Factors Causing a Negative Basis Exhibit 9 lists the fundamental factors that drive the basis in negative direction.

The Structured Synthetic Bid for Credit The rise of large, structured synthetic portfolio trades has added a new dimension to the investment grade credit market. Investors taking on broad, diversified exposure to the credit market through investment grade synthetic CDOs create a demand for credit as they sell protection. This demand is usually hedged by brokers in the single-name default swap market; this pushes the default swap premium tighter for those names that are included in the basket, and may create a negative basis. Collateralized debt obligations (CDOs) pool and tranche credit risk and redistribute it to investors requiring different risk-return profiles. Investment grade synthetic CDOs will get done when the underlying collateral trades cheap relative to the cost of liabilities used to fund the structures. That generally happens when valuations in the investment grade market deviates significantly from the underlying default profile. When risk aversion runs high, as it did in September 2001, investors whose performance are judged on a mark-to-market basis may stay away from the market due to high perceived risk. However, this situation can create significant opportunities for investors with a hold-tomaturity setup, who can view the market as a whole as an asset, rather than individual risks. The bid for credit this creates can, at times, create very strong pressure on the default swap premiums for credits included in the baskets. The impact on individual credits, however, varies widely. In order to achieve a public rating, a given “diversity score” needs to be achieved. That requires all credits in the basket to be equally weighted, which means that the spread impact of sourcing the credit risk will vary significantly: ■ Less liquid names will be more impacted than more liquid ones; ■ Credits with high risk of downgrade to noninvestment grade and/or

outright default tend to get excluded and hence are less impacted.


252


Funding Risk Selling protection removes the risk of changes to the cost of funding. Removing this risk will make the default swap relatively more attractive than the cash alternative and will reduce the basis. A default swap effectively locks in LIBOR flat as the funding rate until maturity. Buying an asset and funding it on repo, however, usually requires regularly rolling the repo rate over—an extra risk the investor needs to be compensated for.

THE DEFAULT SWAP BASIS IN PRACTICE At any given point in time, the default swap basis will always be affected by more than one factor. Sometimes these factors neutralize each other. For example, valuable step-up coupons in a bond should, all things equal, push the basis wider, but if the credit frequently gets included in synthetic baskets, the net effect may be neutral. Often, however, several factors act to reinforce one and the same trend. Look, for instance, at the number of factors that conspire to drive the basis wider on a deteriorating credit: ■ The cheapest-to-deliver option increases in value as it approaches the

strike price, i.e., the credit event. ■ The cheapening up in cash terms as the credit trades down and starts

trading below par and so changes the risk profile relative the default swap contract. ■ Any coupon step-ups in the bonds will start to become valuable as the risk of ratings downgrades increases. ■ The difficulty of shorting the cash market will push investors who want to express their negative view on the credit to the default swap market. ■ Relative liquidity may become an issue as the number of protection buyers overwhelms the number of sellers when the credit starts to deteriorate rapidly.8 The important thing here is that all these factors will work in tandem to reverse the trend once the credit starts improving again. The delivery option will lose value, the cash price will recover, step-ups will be increasingly ignored (and investors may start focusing on any stepdowns instead), shorts will be covered and liquidity will improve. 8

As stated earlier, although absolute liquidity would decline with sharply deteriorating credit quality, relative liquidity between cash and derivatives markets is very difficult to make generalizations about.



EXHIBIT 10

253

High Correlation, High Beta—the CDS-Cash Basis


The fact that all these factors work in tandem and reinforce each other gives the default swap market an important high-beta character, vis-à-vis the underlying cash market. In a widening environment, one would expect the default swap market to underperform the cash market, and vice versa.

THE HIGH-BETA MARKET So how does this play out in reality? Exhibit 10 shows the performance of the European defaults swap and corporate bond markets from January 1 through June 2002.9 Three patterns clearly stand out: ■ The default swap market is more volatile than the cash market. Aver-

age spread volatility is 40% in the default swap market, while it is 38% in the cash market. 9 The comparison is of a basket of 60 European credits actively traded in both markets. All bonds have maturities in 4–6 years and are compared with the 5-year point on the default swap curve. All mid-levels. Each sector is weighted according to its weight in the MSCI index.


254


■ The correlation between the markets is very high—consistently above

90% throughout the period. ■ The default swap market is high beta. Simply as an effect of the factors

above, the basis is correlated with the overall direction of the market. That means that as the market widens, so does the basis, and vice versa. The default swap is a high-beta market in the sense that, in the aggregate, it trades as a leveraged play on the cash market.

13-Chen_Fabozzi_Okane Page 255 Wednesday, July 23, 2003 10:20 AM

The Valuation of Credit Default Swaps Ren-Raw Chen, Ph.D. Associate Professor Rutgers University Frank J. Fabozzi, Ph.D., CFA Frederick Frank Adjunct Professor of Finance School of Management Yale University Dominic O’Kane, Ph.D. Senior Vice President Lehman Brothers, Inc.

redit default swaps (CDSs), or simply default swaps, are the most popular of all the credit derivative contracts traded. Their purpose is to provide financial protection against losses incurred following a credit event for a reference obligation or reference issuer. As a result, they provide an efficient method for transferring of credit risk in a straightforward and increasingly standard manner. Their over-the-counter nature also makes them infinitely customizable, thereby overcoming many of the limitations of the traditional credit market instruments such as lack of availability of instruments with the required maturity or seniority.

C

The authors would like to thank Filippo Lanza for his valuable comments and suggestions in the preparation of this article.

255


256


Recent financial crises have spurred on the growth in the default swap market as banks and others exposed to concentrations of credit risk have sought to hedge and diversify their portfolios of securities and loans. Increasing standardization and familiarization with the legal framework has made capital market participants more willing to enter into default swap transactions as have developments in credit modeling and pricing that have made it possible to mark-to-market and hedge default swap positions. Bonds are the main source of liquidity in the credit markets, especially in the United States. Replication arguments that attempt to link CDSs to bonds are therefore generally used by the market as a first estimate for determining the price at which CDSs should trade. However the replication argument is not exact as it is based on a number of assumptions that often break down in practice. Market participants who wish to price CDSs and examine relative value opportunities need to understand replication and its assumptions. We discuss the replication approach in this article. However, replication only provides a starting point for quoting CDS spreads. It does not allow traders to actually mark to market their existing CDS positions. By definition, marking a CDS position to market must involve pricing it off the current market CDS spread curve—a set of CDS spreads quoted for different maturities. The main objective of this article will be to explain how to determine the CDS spread, what factors affect its pricing, and how to mark-to-market CDSs. We show that this requires a model and set out the standard model that is used by the market.

DEFAULT SWAPS In a standard CDS contract one party pays a regular fee to another to purchase credit protection to cover the loss of the face value of an asset following a credit event. The company (or sovereign) to which the triggering of the credit event is linked is known as the reference entity. This protection lasts until some specified maturity date, typically five years from the trade date. To pay for this protection, the protection buyer makes a regular stream of payments. These are quoted in terms of an annualized percentage known as the CDS spread. These payments are typically paid quarterly according to an Actual 360 basis convention and are collectively known as the premium leg. Payments occur until maturity of the contract or a credit event occurs, whichever happens first. If a credit event does occur before the maturity date of the contract, there is a payment by the protection seller, known as the protection leg. There are two ways to settle the payment of the protection leg: physical settlement and cash settlement. The form of settlement is specified at the initiation of the contract.


257

The Valuation of Credit Default Swaps

■ Physical Settlement: This is the most widely used settlement proce-

dure. It requires the protection buyer to deliver the notional amount of deliverable obligations to the protection seller in return for the notional amount paid in cash. In general there is a choice of deliverable obligations from which the protection buyer can choose. These deliverable obligations will satisfy a certain number of characteristics that typically include restrictions on the maturity of the deliverable obligations and the requirement that they be pari passu—most default swaps are linked to senior unsecured debt. Typically, they include both bonds and loans. If deliverable obligations trade with different prices following a credit event, which they are most likely to do if the credit event is a restructuring, the protection buyer can take advantage of this situation by buying and delivering the deliverable obligation. The protection buyer is therefore long a cheapest to deliver (CTD) option. ■ Cash Settlement: This is the alternative to physical settlement and is used less frequently. In cash settlement, a cash payment is made by the protection seller to the protection buyer equal to par minus the recovery rate. The recovery rate is the price of the cheapest to deliver and is calculated by referencing dealer quotes or observable market prices over some prespecified period after default has occurred. CDS spreads are typically quoted for a variety of maturities with most liquidity at the 5-year maturity followed by the 3-year and 7-year maturities. In Exhibit 1 we have presented some examples of CDS term structures for four different credits, three corporate and one sovereign. We have also shown the bid-offer. The bid is the spread at which the dealer is willing to buy protection, while the offer is the spread at which the dealer is willing to sell protection. Clearly the bid spread will be less than the offer spread. Note that this is opposite to the convention for bonds where the bid spread is the spread at which the dealer is willing to buy the bond EXHIBIT 1

Market CDS Spreads (bp): (Indicative Levels, April 2003)

Term

DCX

1Y 3Y 5Y 7Y 10Y

130/140 140/150 150/160 155/165 160/175

General Electric

Boeing Co.

Mexico

45/55 54/65 65/70 67/77 70/80

70/80 82/90 85/93 93/100 95/110

57/62 175/195 250/260 275/290 300/315

Source: Lehman Brothers Credit Trading.


258


and this will be higher than the offer spread. This is because the buyer of a bond is selling protection, while the buyer of a CDS is buying protection.

Illustration Suppose a protection buyer purchases 5-year protection on a company at a default swap spread of 200 bp. The face value of the protection is $10 million. The protection buyer therefore makes quarterly payments approximately1 equal to $10 million × 0.02 × 0.25 = $50,000. After a short period the reference entity suffers a credit event. Assuming that the cheapest deliverable asset of the reference entity has a recovery price of $35 per $100 of face value, the payments are as follows: ■ The protection seller compensates the protection seller for the loss on

the face value of the asset received by the protection buyer. This is equal to $10 million × (100% – 35%) = $6.5 million. ■ The protection buyer pays the accrued premium from the previous premium payment date to time of the credit event. For example, if the credit event occurs after a month then the protection buyer pays approximately $10 million × 0.02 × 1/12 = $16,666 of premium accrued. Note that this is the standard for corporate reference entity linked default swaps. For sovereign-linked default swaps, there may be no payment of premium accrued.

The Mechanics of Settlement The time-line around the physical settlement of a CDS following a credit event consists of three steps: 1. The protection buyer must give the protection seller a Credit Event Notice which informs the protection seller that a credit event has occurred. This is usually done by fax. The event must be evidenced by at least two sources of Publicly Available Information (e.g., a news article on Reuters, the Wall Street Journal, the Financial Times or some other recognized publication or electronic information service. The date that the Credit Event Notice is delivered to the protection seller is referred to as the Event Determination Date. 2. Within 30 calendar days of the Event Determination Date, the protection buyer must give Notice of Physical Settlement. If this deadline is not met, the protection is void. The date that the notice of Intended Physical Settlement is delivered is called the Conditions of Payment date. This must give information about what the deliverable obliga1

The exact payment amount is a function of the calendar and basis convention used.



259

tions will be. According to the new ISDA 2003, this notice may be updated at any time between the Conditions of Payment Date and the Physical Settlement date, with the last one being binding. 3. Within 30 business days of the Conditions of Payment date, the protection buyer must effect the physical delivery in return for par. This is the Physical Settlement Date. As a result, the maximum delay between notice of a credit event and the actual payment of the protection is approximately 72 calendar days.

CREDIT EVENTS The most important section of the documentation for a default swap is what the parties to the contract agree constitutes a credit event that will trigger a payment by the protection seller to the protection buyer. Definitions for credit events are provided by the International Swap and Derivatives Association (ISDA). First published in 1999, there have been periodic updates and revisions of these definitions.

ISDA Credit Event Definitions Of the eight possible credit events referred to in the 1999 ISDA Credit Derivative Definitions, the ones typically used within most contracts are listed in Exhibit 2. In terms of which are used, the market distinguishes between corporate and sovereign linked CDS. For corporate linked CDS the market standard is to use just three credit events—Bankruptcy, Failure to Pay, and Restructuring. For sovereign linked CDS, obligation acceleration/default and repudiation/moratorium are also included.

Restructuring Controversy Restructuring means a waiver, deferral, restructuring, rescheduling, standstill, moratorium, exchange of obligations, or other adjustment with respect to any obligation of the reference entity such that the holders of those obligations are materially worse off from either an economic, credit, or risk perspective. It has been the most controversial credit event that may be included in a default swap. In bankruptcy or failure to pay, pari passu assets trade at or close to the same recovery value. But restructuring is different. Following a restructuring, debt continues to trade. Short-dated bonds trade at higher prices than longer-dated bonds, bonds with large coupons trade at a higher price than bonds with low coupon. Loans, which are typically also deliverable, tend to trade at higher prices than bonds due to their additional covenants.


260

EXHIBIT 2


Credit Events Typically Used Within Most CDS Contracts

Credit Event

Description

Bankruptcy

Corporate becomes insolvent or is unable to pay its debts. The bankruptcy event is of course not relevant for sovereign issuers. Failure to pay Failure of the reference entity to make due payments, taking into account some grace period to prevent accidental triggering due to administrative error. Restructuring Changes in the debt obligations of the reference creditor but excluding those that are not associated with credit deterioration such as a renegotiation of more favorable terms. Obligation Obligations have become due and payable earlier than they Acceleration/ would have been due to default or similar condition Obligation Obligations have become capable of being defined due and payDefault able earlier than they would have been due to default or similar condition. This is the more encompassing definition and so is preferred by the protection buyer. Repudiation/ A reference entity or government authority rejects or challenges Moratorium the validity of the Obligations. Source: ISDA and Lehman Brothers Fixed Income Research.

This makes the delivery option which is embedded in a default swap potentially valuable. A protection buyer hedging a short-dated high coupon asset may find that following a restructuring credit event it is trading at say $80 while another longer-dated deliverable may be trading at $65. By selling the $80 asset, purchasing the $65 asset, and delivering it into the CDS, the protection buyer may make a $15 windfall gain out of the delivery option. However this gain is made at the expense of the protection seller who has to take ownership of the $65 asset in return for a payment of par. Such a situation arose in the summer of 2000 when the U.S. insurer Conseco restructured its debt. At that time, the range of deliverable obligations following a restructuring event was the same as those used for bankruptcy or failure to pay. This meant that bonds or loans with a maximum maturity of 30 years could be delivered. Protection sellers were displeased at being delivered long dated low priced bonds in the price range 65-80 by banks who held much higher priced short-term loans. In addition, it was believed that there was a conflict of interest—banks who exercised their default swaps had also been party to the restructuring of Conseco’s debt. The results of this experience led to the market discussing a restructuring supplement to the standard ISDA documentation. This was completed on May 11, 2001 and introduced a new restructuring definition called “Modified Restructuring” (“Mod-Re”). The essence of this was


261


EXHIBIT 3

Different Restructuring Standards as Proposed by ISDA 2003

Type

Description

Old Restructuring Modified Restructuring

This is the original standard for deliverables in default swaps in which the maximum maturity deliverable is 30 years. This is the current standard in the U.S. market. An exact description of the allowed deliverables is beyond the scope of this paper. Roughly speaking it limits the maturity of deliverable obligations to the maximum of the remaining maturity of the CDS contract and 30 months. ModifiedThis is the new standard for the European market. It limits the Modified maturity of deliverable obligations to the maximum of the Restructuring remaining maturity of the CDS contract and 60 months. It also allows the delivery of conditionally transferable obligations rather than only fully transferable obligations. This should widen the range of bonds/loans that can be delivered. No Restructuring This contract removes restructuring as a credit event.

Source: ISDA and Lehman Brothers Fixed Income Research.

to reduce the range of deliverable obligations following a restructuring event and so limit the value of the delivery option. This new Mod-Re standard has been adopted by the U.S. market. Due to regulatory issues Europe did not adopt Mod-Re, staying with the existing definitions known as “Old-Re.” It is expected that Europe will adopt a newer version to be called “Mod-Mod-Re,” published in the ISDA 2003 definitions in May 2003. A number of market participants including some commercial banks and insurance companies have also pushed the idea of a CDS contract without restructuring as a credit event. It remains to see whether this no restructuring (“No-Re”) contract will gain a following. A summary description of these different contracts is shown in Exhibit 3. Where the same credit trades with different restructuring conventions, these different contract standards should be reflected in the quoted market spreads. For example, modified-modified restructuring allows the protection buyer to have a broader range of deliverables than modified restructuring. This means that the value of the delivery option is greater for Mod-Mod-Re than for Mod-Re and so the protection should trade at a wider spread for the more valuable delivery option. More generally we can demonstrate2 that there should be a strict theoretical relationship between these spread levels of 2

See Dominic O’Kane, Claus Pedersen, and Stuart Turnbull, The Restructuring Clause in Credit Default Swap Contracts, Lehman Brothers Quantitative Credit Research Quarterly, April 2003.


262


SpreadOld-Re > SpreadMod-Mod-Re > SpreadMod-Re > SpreadNo-Re In this article, the aim is not to determine what the spread differences should be, but to price contracts of a given type given the corresponding curve of market spreads.

Credit Events and Implementation of Default Swap Pricing Models In the pricing model presented in this article, we refer to “default.” A default means one form of credit event that is agreed to by the parties. This means that the value of a contract will depend on which credit events are included in a particular trade (i.e., contract). While the model presented handles any of the credit events that may be selected by the parties to a trade, the data required are typically drawn from databases that collect defaults defined in a different way than those set forth by ISDA credit event definitions. For example, major studies regarding default rates and recovery rates, as well as default times, define default in terms of the legal definition of default. In contrast, consider restructuring. Suppose that full restructuring is included in a trade as a credit event. Then a reduction in a reference obligation’s interest rate that is material is a credit event. In fact, actions by lenders to modify the terms of a reference obligation without a bankruptcy proceeding are not uncommon. Yet, they are not included (or even known) to researchers who compile data on defaults. The key point is that in the implementation stage, the inputs must be modified based on the credit events included in a trade.

PRICING CREDIT DEFAULT SWAPS BY STATIC REPLICATION There is a fundamental relationship between the default swap market and the cash market in the sense that a default swap can be shown as being economically equivalent to a combination of cash bonds. This cash-CDS relationship means that determination of the appropriate default swap spread for a particular credit usually begins by observing the LIBOR spread at which bonds of that issuer trade. The usual comparison is to look at what is called the par asset swap spread of a bond of a similar maturity to the default swap contract. This is the spread over LIBOR paid by a package containing a fixed-rate bond and interest rate swap purchased at par. This spread can easily be calculated3 and is shown on the ASW screen for a specified corporate bond in Bloomberg. 3

See Dominic O’Kane, Credit Derivatives Explained, Lehman Brothers Fixed Income Research, March 2001.



263

In different markets the relative importance of cash to the CDS market varies. In the United States, the cash credit market is dominant and drives CDS pricing (albeit with some basis as described below). In Europe, the corporate cash market is not as dominant and neither market dominates the other. In Japan the CDS market has become significant. As one moves to non-Japan Asia it is possible to find markets where CDS are the most liquid credit format. The premium payments in a default swap contract are defined in terms of a default swap spread, S, which is paid periodically on the protected notional until maturity or a credit event. It is possible to show that the default swap spread can, to a first approximation, be proxied by a par floater bond spread (the spread to LIBOR at which the reference entity can issue a floating rate note of the same maturity at a price of par) or the asset swap spread of an asset of the same maturity provided it trades close to par. To see this, consider a strategy in which an investor buys a par floater issued by the reference entity with maturity T. The investor can hedge the credit risk of the par floater by purchasing protection to the same maturity date. Suppose this par floater (or asset swap on a par asset) pays a coupon of LIBOR plus F. Default of the par floater triggers the default swap, as both contracts are written on the same reference entity. With this portfolio the investor is effectively holding a defaultfree investment, ignoring counterparty risk. The purchase of the asset for par may be funded on balance sheet or on repo—in which case we make the assumption that the repo rate can be locked in to the bond’s maturity. The resulting funding cost of the asset is LIBOR plus B, assumed to be paid on the same dates as the default swap spread S. Consider what happens in the following scenarios: No Credit Event—The hedge is unwound at the bond maturity at no cost since the protection buyer receives the par redemption from the asset and uses it to repay the borrowed par amount. Credit Event—The protection buyer delivers the reference asset to the protection seller in return for par. If we assume that the credit event occurs immediately following a coupon payment date then the cost of closing out the funding is par which is repaid with this principal. The position is closed out with no net cost. Both scenarios are shown in Exhibit 4. As the hedged investor has no credit risk within this strategy they should not earn (or lose) any excess spread. This implies that S = F – B; that is, the default swap spread should be equal to the par floater spread minus the funding cost


264

EXHIBIT 4


Theoretical Default Risk-Free Hedge for an Investor Who Buys

Protection

Source: Lehman Brothers Fixed Income Research.

of the cash bond. For example, suppose the par floater pays LIBOR plus 25 bp and the protection buyer funds the asset on balance sheet at LIBOR plus 10 bp. For the protection buyer the breakeven default swap spread equals F – B = 25 – 10 = 15 bp. This analysis certainly shows that there should be a close relationship between cash and default swap spreads. However the argument is not exact as it relies on several assumptions that could result in small but observable differences. Some are listed below: 1. We have assumed the existence of a par floater with the same maturity date as the default swap to be priced. 2. We have assumed a common market-wide funding level of LIBOR + B. In practice, different market participants have different funding costs which therefore imply different spread levels. 3. We have assumed repo funding to term. Repo funding cannot usually be locked in to term but only for short periods of a couple of months only. One attraction of CDS is that unlike cash, they effectively lock in funding at LIBOR flat to maturity. 4. We have ignored accrued coupons. If the credit event occurs just before a coupon payment on the funding leg, the protection does not cover the loss of par plus coupon on the funding leg. We have also ignored the



265

EXHIBIT 5

Evolution of CDS Spread and Asset Swap Spread of the 5-Year CDS Spread for Citigroup and the Asset Swap Spread of a 5-Year Citigroup Bond

Source: LehmanLive.com.

5. 6. 7.

8.

9.

effect of the accrued CDS premium payment from the previous payment date. We have assumed that the par floater is the cheapest-to-deliver asset. We have ignored counterparty risk on the CDS. This is usually mitigated through the use of collateral. Due to the difficulty of shorting cash bonds, any widespread market demand to go short a particular credit will first impact CDS causing spreads to widen before cash. For asset swaps the initial price of the asset must be close to par. This is because the loss on an asset swap of a bond trading with a full price P is about P – R. The credit risk is then only comparable to a default swap when the asset trades close to par. We have ignored transaction costs.

Despite these assumptions, cash market spreads usually provide the starting point for where the default swap spreads should trade. An empirical example of this relationship is shown in Exhibit 5 where a time series of the spread of 5-year protection on Citigroup is plotted alongside the asset swap spread of a 5-year Citigroup bond. There is clearly a high correlation between the two spread levels. The difference between where default swap spreads and cash LIBOR spreads trade is known as the default swap basis, defined as


266


Default swap basis = S – F There are now a significant number of market participants who actively trade the default swap basis, viewing it as a new relative value opportunity.4

PRICING OF A SINGLE-NAME CREDIT DEFAULT SWAP To value credit derivatives it is necessary to be able to model credit risk. The two most commonly used approaches to model credit risk are structural models and reduced form models. The first structural model for credit-risky bonds was proposed by Fischer Black and Myron Scholes5 who explained how equity owners hold a call option on the firm. After that Robert Merton6 extended the framework and analyzed the behavior of risky debt using the model.7 The second type of credit models, known as reduced form models, are more recent.8 These models, most notably the Jarrow-Turnbull model9 and Duffie-Singleton model,10 do not look inside the firm. Instead, they model directly the likelihood of a default occurring. Not only is the current probability of default modeled, they also attempt to model a “forward curve” of default probabilities that can be used to price instruments of varying maturities. Characterizing default as an 4 For a discussion of the driving factors behind the basis, see Dominic O’Kane and Robert McAdie, Explaining the Basis: Cash versus Default Swaps, Lehman publication, March 2001. 5 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, May–June 1973, pp. 637–654. 6 Robert Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics, Spring 1973, pp. 141–183, and Robert Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, May 1974, pp. 449–470. 7 Robert Geske extended the Black-Scholes-Merton (BSM) model to include multiple debts (see, Robert Geske, “The Valuation of Corporate Liabilities as Compound Options,” Journal of Financial and Quantitative Analysis, November 1974, pp. 541– 552 and Robert Geske and Herbert Johnson, “The Valuation of Corporate Liabilities as Compound Options: A Correction,” Journal of Financial and Quantitative Analysis, June 1984, pp. 231–232. Recently many barrier models appear as an easy solution for analyzing the risky debt problem. 8 The name reduced form was first given by Darrell Duffie to differentiate from the structural form models of the BSM type. 9 Robert Jarrow and Stuart Turnbull, “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance, March 1995, pp. 53–86. 10 Darrell Duffie and Kenneth Singleton, “Modelling the term Structure of Defaultable Bonds,” Stanford University, working paper, 1997.



267

event which occurs with a modeled probability has the effect of making default a surprise—the default event is a random event which can suddenly occur at any time. All we know is its probability. Reduced form models are easy to calibrate to bond prices observed in the marketplace. This is known as working in an “arbitrage-free” framework. It is only by ensuring that a pricing model fits the market that a trader can be sure that he does not quote prices which expose him to any price arbitrages. The ability to quickly and easily calibrate to the entire CDS market is the major reason why reduced form models are strongly favored by real-world practitioners in the credit derivatives markets for pricing. Structural-based models are used more for default prediction and credit risk management.11 The basic framework that underlies the reduced form model is a binomial default process. There are two branches at each time point on the tree: default and survival. The branches that lead to default will terminate the contract and incur a recovery payment. The branches that lead to survival will continue the contract that will then face future defaults. This is a very general framework to describe how default occurs and contract terminates. Various models differ in how the default probabilities are defined and the recovery is modeled. Reduced form models use risk-neutral pricing to be able to calibrate to the market. In practice, we need to determine the risk-neutral probabilities in order to reprice the market and price other instruments not currently priced. In doing so, we do not need to know or even care about the real-world default probabilities. Since in reality, a default can occur any time, to accurately value a default swap, we need a consistent methodology that describes the following: (1) when defaults occur, (2) how recovery is paid, and (3) how discounting is handled.

Survival Probability Assume the risk-neutral probabilities exist. Then we can identify a series of risk-neutral default probabilities so that the weighted average of default and no-default payoffs can be discounted at the risk-free rate. 11 Increasingly, investors are seeking consistency between the markets that use different modeling approaches, as the interests in seeking arbitrage opportunities across various markets grows. Ren-Raw Chen has demonstrated that all the reduced form models described above can be regarded in a non-parametric framework. This nonparametric format makes the comparison of various models possible. Furthermore, as Chen contends, the non-parametric framework focuses the difference of various models on recovery. See Ren-Raw Chen, “Credit Risk Modeling: A General Framework,” working paper, Rutgers University, 2003.


268


The risk-free rate used in the pricing of CDS is LIBOR. This is because within a derivatives framework, the risk-free rate is actually the rate at which market dealers fund their hedges. For banks and other dealers in the credit derivative market, bonds are often used as hedges and these are usually funded at or close to LIBOR. LIBOR is an interest rate which reflects the funding costs of AA rated commercial banks. Assume Q(t) to be the survival probability from now till some future time t. Then Q(t) – Q(t + τ) is the default probability between t and t + τ (i.e., survive till t but default before t + τ). Assume defaults can only be observed at multiples of τ. Then the total probability of default over the life of the CDS is the sum of all the per period default probabilities: n

∑ Q [ ( j – 1 )τ ] – Q ( jτ )

= 1 – Q ( nτ ) = 1 – Q ( T )

j=1

where Q(0) = 1.0 and nτ = T, the maturity time of the CDS. It is no coincidence that the sum of the all the per-period default probabilities should equal one minus the total survival probability. The survival probabilities have a useful application. A $1 “risky” cash flow received at time t has a risk-neutral expected value of Q(t) and a present value of P(t)Q(t) where P is the risk-free discount factor. The value of the protection leg of a CDS is the present value of the payment of (1 – R) at default. To take into account the timing of the default payment (1 – R), we break the time to maturity into n intervals and consider the probability of defaulting in each. The probability of defaulting in a forward interval [(j – 1)τ,jτ] is given by Q [ ( j – 1 )τ ] – Q ( jτ )

(1)

We then discount the payment of (1 – R) back to today at the riskfree rate P(). We then consider the likelihood of default occurring in all of the intervals by summing over all intervals. We therefore have n

V = (1 – R)

∑ P ( jτ ) { Q [ ( j – 1 )τ ] – Q ( jτ ) }

(2)

j=1

where P(·) is the risk-free discount factor and R(·) is the recovery rate. In the above equation, it is implicitly assumed that the discount factor is independent of the survival probability. In reality, these two may be correlated—usually higher interest rates lead to more defaults because businesses suffer more from higher interest rates. To account


269


for this we would need to introduce a stochastic probability and interest rate model. However, the effect of this correlation is almost negligible on the valuation of CDS and is further reduced by calibration. Equation (2) has no easy solution. However, a continuous time version of the equation can be found in the appendix to this article. The value of the premium leg of the default swap is given by discounting each of the expected spread payments by risk-neutral discount factor weighted by the probability of surviving to each payment date. This is given by N

S

∑ ∆j P ( jτ )Q ( jτ )

j=1

where ∆j is the corresponding year fraction in the appropriate basis convention (typically Actual 360). By definition the value of the default swap spread is the value at which the premium and protection legs have the same present value. Hence we have N

V = S

∑ ∆j P ( jτ )Q ( jτ )

j=1

giving V S = -------------------------------------------n

(3)

∑ ∆j P ( jτ )Q ( jτ )

j=1

Exhibit 6 depicts the general default and recovery structure. The payoff upon default of a default swap can vary. In general, the owner of the default swap delivers the defaulted bond and in return receives principal. Many default swaps are cash settled and an estimated recovery is used. In either case, the amount of recovery is randomly dependent upon the value of the reference obligation at the time of default. Different models simply differ in how this recovery is modeled.12 In practice the portion of the premium payment which has accrued from the previous coupon payment date is paid by the protection buyer 12 In the appendix, we provide an example where the two variables are independent and the defaults follow a Poisson process. In the example presented in the appendix the simple solution exists under the continuous time assumption.


270

EXHIBIT 6


Payoff and Payment Structure of a Credit Default Swap

following the credit event. We have ignored it from our analysis since its effect on the calculated spread is small.13

Valuation of a Credit Default Swap The valuation of CDS can be broken down into two separate tasks. The first is the determination of the default swap spread which should be paid by a protection buyer at the initiation of a trade. This has already been discussed. The second is to determine the value of an existing CDS position, which we call the mark-to-market (MTM). At trade initiation, the MTM value of a CDS position is by definition zero—nothing has been paid and the expected present value of the protection leg is exactly equal to the expected present value of the premium leg. This is true since the CDS spread at which the trade was transacted, known as the contractual spread, was calculated with this purpose. Once a CDS position has been established, changes in the current market CDS spread will mean that the MTM begins to deviate from zero and must be determined by observing the current level of default swap spreads in the market. To see how this is done, consider the following example. An investor sells protection on a reference entity for five years at an agreed contractual spread of 250 bp. By selling protection the investor is assuming the credit risk of the reference entity as though he was buying one of the reference entity’s issued bonds. A year later the reference entity’s credit rating has improved and the corresponding default swap 13

See Dominic O’Kane and Stuart Turnbull, Valuation of Credit Default Swaps, Lehman Brothers Fixed Income Research, April 2003.


271


spreads have tightened so that 4-year default swaps trade at 100 bp. What is the MTM value of the position? To begin with, the MTM value of the contract to the investor is given by the difference between what the investor is expecting to receive minus what they are liable to pay. As a result we can write MTM = + Present value of premium payments of 250 bp – Present value of protection for the remaining 4 years We can also write that the current 4-year market spread of 100 bp is the current breakeven spread. By definition, the current value of a new 4 year contract is zero so we can write Present value of premium payments of 100 bp = Present value of protection for the remaining 4 years Substituting we write MTM = + Present value of premium payments of 250 bp – Present value of premium payments at 100 bp which can be rewritten as MTM = + Present value of premium payments of 150 bp To go any further we have to compute the discounted present value of these 150 bp payments. However these payments are only made until the maturity of the CDS or to the time of a credit event. To compute the MTM we therefore need to weight each premium payment by the probability that there is no credit event up until that payment date. We therefore write MTM = 150 bp × RPV01 where the RPV01 is the “risky” price value of a basis point (PV01). This is defined as the present value of a 1 bp payment made until the contractual maturity date of the position or to the date of a credit event, whichever sooner. Mathematically, we can write the RPV01 as n

RPV01 =

∑ ∆j P ( jτ )Q ( jτ )

j=1


272


where ∆j is the year fraction for the payment j in the appropriate basis (typically Actual 360). For quarterly paying CDS, ∆j is usually close to or equal to 0.25. Bringing this all together, we can write the MTM value of a long protection position as MTM = +[S(t,T) – S(0,T)] × RPV01[S(t,T),R] and that of a short protection position as MTM = –[S(t,T) – S(0,T)] × RPV01[S(t,T),R] where S(0,T) is the contractual spread of the contract, T is the contractual maturity date and S(t,T) is the current market breakeven spread to the contractual maturity date. It is essential to note that the RPV01 is a function of the market spread S(t,T) and the assumed recovery rate R since both are used to imply out the risk-neutral survival probabilities. To crystallize all of this theory, we present in Exhibit 7 the valuation of the trade introduced at the beginning of this section in which an investor sells $10 million of 5-year protection at 250 bp and then wishes to mark it to market one year later when the market has a flat term structure at 100 bp. For simplicity we have assumed a flat LIBOR term structure at 2.5%. We assume a recovery rate of 40%. In particular we show the quarterly coupon payment dates (we have ignored holidays and weekends for simplicity) and the corresponding values of P and Q, calibrated to reprice the term structure of default swap spreads. We see that the current market spread is 100 bp, and that the risky PV01 of the position is 3.717—the present value of four years of risky 1 bp payments is 3.717 bp. The resulting MTM value is $557,610. This makes sense. The investor sold protection at a wide spread, the market tightened by 150 bp so the investor has made money. Note that for investors who wish to apply this model to the valuation of real positions, it available under the CDSW function on Bloomberg.

CDS Risk and Sensitivities Market practitioners using CDS usually consider two risk measures. First is the Credit01 or Spread01. This is the change in the MTM value of a CDS position for a 1 bp parallel shift in the CDS curve. Then there is the Interest Rate 01 which is the change in the MTM value of a CDS position for a 1 bp change in the LIBOR rates. In practice the LIBOR sensitivity of a CDS is small, usually at least an order of magnitude less than that of the Credit01. This reflects the fact that a CDS is almost a pure credit play.


273


EXHIBIT 7

An Illustration of Calculation of the MTM Value Contract Details

Long or Short Protection Notional Contractual Spread (bp) Valuation Date Maturity Date Flat LIBOR Rate

Payment Dates 21-May-03 21-Aug-03 21-Nov-03 21-Feb-04 21-May-04 21-Aug-04 21-Nov-04 21-Feb-05 21-May-05 21-Aug-05 21-Nov-05 21-Feb-06 21-May-06 21-Aug-06 21-Nov-06 21-Feb-07 21-May-07

S 10,000,000 250 21-May-03 21-May-07 2.50%

Spread

1Y 3Y 5Y 7Y 10Y R

100 100 100 100 100 40.00%

Year Fraction

Q(t)

P(t)

0.256 0.256 0.256 0.250 0.256 0.256 0.256 0.247 0.256 0.256 0.256 0.247 0.256 0.256 0.256 0.247

0.99577 0.99155 0.98736 0.98327 0.97911 0.97496 0.97084 0.96686 0.96277 0.95869 0.95464 0.95073 0.94670 0.94270 0.93871 0.93486

0.99372 0.98748 0.98129 0.97526 0.96914 0.96305 0.95701 0.95120 0.94522 0.93929 0.93339 0.92773 0.92190 0.91612 0.91036 0.90484

Valuation Market Spread Risky PV01 MTM

Term

100.00 3.717 $557,610


274


It is actually possible to make some simple approximations that make clear the dependence of the MTM on these inputs. First, we can approximate the CDS spread in terms of the risk-neutral annualized default probability p, and assumed recovery rate R, using the equation S = p(1 – R). The interpretation is that the annualized spread received for assuming a credit risk should equal the annualized default probability times the loss on default, which in a CDS equals (100% – R). This approximation works very well in practice. If we assume a flat term structure of CDS spreads, approximate ∆ with ¹₄ then we can approximate the MTM of a long protection position as [ S ( t, T ) – S ( 0, T ) ] MTM = ---------------------------------------------4

N

∑ P ( jτ )

j=1

S ( t, T ) 1 – ----------------1–R

j⁄4

We can immediately draw a number of conclusions from this mathematical expression for the MTM value. First, the MTM value is not a linear function of the market spread S(t,T). In fact the MTM value of a short protection position is convex in the market spread, just as the price of a corporate bond is convex in the yield. Furthermore, it is also clear that the recovery rate sensitivity of the MTM value is large when the market spread is large. This means that where the market spread is below say 300 bp, one does not have to be so precise about the recovery rate assumption. However, if spreads become large (say 300 bp and above) the recovery rate sensitivity becomes increasingly significant and care must be taken in making a recovery rate assumption.

Calibrating the Recovery Rate Assumption To be precise, the recovery rate assumption, R, is the assumed price of the cheapest-to-deliver asset into the CDS contract within 72 calendar days of the notification of the credit event. This is not known today. Nor can it be extracted from any market prices.14 The usual starting point for calibrating recovery rates is to observe rating agency statistics. Both Moody’s and S&P maintain significant databases of U.S. corporate bond defaults. Care must be taken to adjust any average recovery rates for country and sector effects. Recovery rates 14

In theory, this would be possible given the existence of an active and liquid digital default swap market. A digital default swap is a contract which pays the face value in the event of default—it is like a standard default swap but instead assumes a fixed recovery rate of zero. The ratio of the normal CDS spread and the digital default swap spread would equal (1 – R). However the lack of liquidity of the digital market makes this calibration approach impractical.



275

also have a link to the economic cycle. In recent years, average recovery rates have fallen well below the long-term averages computed by rating agencies. One reason why this is so is that Moody’s, for example, defines the recovery rate of a bond as the price of that bond within some short period following the default. It is not the final value received by holders of the bond after going through the workout process. This means that the recovery rate is driven by the size of the bid for the bond in the distressed debt market. In periods of credit weakness, the distressed debt market is unable to absorb the oversupply of defaulted assets and the bid consequently falls. Another consideration when marking recovery rate assumptions is to take into account that following a restructuring event, which is not a full default, the deliverable obligations may trade at higher prices than in a full default. Since rating agencies do not consider restructuring as a full default, this effect is not accounted for in their statistics. Typical recovery rates being quoted in the market for good quality credits vary between 30% and 45%. When spreads are trading at very high levels of 1,000 bp and above, it is important to look to the bond market to see if bond prices are revealing any information about the expected recovery rate in the event of a default. For example, a recovery rate assumption of 40% would make no sense if one of the deliverable bonds into the CDS is trading at 30 cents on the dollar. In this case, the recovery rate assumption should clearly be moved below 30%.

The Practicalities of Unwinding a Credit Default Swap A CDS is an over-the-counter (OTC) derivative contract. This means that unlike some other derivatives contracts it is not exchange-traded. Instead it involves an agreement between two counterparties. As almost all CDS are traded within the framework of the ISDA Master Agreement, there is widespread standardization of the documentation of CDS and many counterparties are happy to trade these bilateral contracts in what is effectively a secondary market. To unwind a CDS before its maturity date, an investor may consider one of three courses of action: 1. Negotiate a cash unwind price with the original counterparty. The price should be the same as the MTM value calculated according to the model. In practice a bid-offer spread will have to be crossed. Part of this negotiation may involve some exchange of information as to the recovery rate assumptions used by both counterparties. 2. If the investor is shown a better unwind price by a counterparty different to the one with whom the initial trade was executed, they can ask


276


to have the contract reassigned to this other counterparty and then close it out for a cash unwind value. 3. They may choose to enter into an offsetting position. For example, an investor who has sold protection for five years may decide a year later to close out the contract by selling protection for four years. The value of this combined position should exactly equal the model market to market. Which one of these choices is made is usually determined by which is showing the best price. Note that option 3 is different from the other two since it does not result in the investor receiving a cash amount, realizing any profit and loss (P&L) and terminating the trade. Instead the P&L is unrealized and must be realized over the remaining life of the contracts. If the P&L is positive there is a risk that a credit event occurs and all remaining spread income is lost and the P&L is not realized. On the other hand, if no credit event occurs, the received income will be worth more than that received through a cash unwind.

RECENT DEVELOPMENTS IN THE CDS MARKET We conclude this article with a brief discussion of two recent developments in the CDS market: the rolling CDS contract and the up-front CDS contract.

Rolling CDS Contract One of the most recent developments in the CDS market has been the move to a rolling CDS contract. The idea is to ensure that all newly initiated contracts mature on one of four dates per year. The dates that have been agreed to are March 20, June 20, September 20, and December 20.15 The motivation for this development is to remove what is known as the “stub” problem. To see what this is, consider a dealer who buys 5-year protection from an investor on April 12, 2003, which therefore matures on the April 12, 2008. The dealer does not hedge this position for until April 25, 2003, when he sells 5-year protection to another dealer. This hedge matures on April 25, 2008. The dealer does not have a perfect hedge. He is left with a forward starting credit exposure between April 12 and April 25, 2008. Even though this exposure is well 15 These are loosely referred to as “IMM” dates (a convention used in the futures markets), though they are not. IMM dates are defined as falling on the third Wednesday of March, June, September, and December; so they are not guaranteed to always fall on the 20th of the month.



277

in the future, such exposures must be managed and monitored as they roll through time and in certain cases additional hedges may be needed. Also, bank regulators may not recognize this combined position as a full hedge and it may result in a regulatory capital charge. In the new rolling contract, any 5-year CDS executed on the April 12, 2003, will mature on the June 20, 2008. The same applies to all contracts executed up until June 19, 2003. From the June 20, 2003, all new contracts will then roll to the September 20, 2008. Contracts to other maturity dates can still be traded. However, they will have a lower liquidity.

Up-Front CDS The volatile credit markets of 2001 and 2002 have led dealers to quote default swaps on very distressed names in what is known as “up-front” format. In these contracts, the protection buyer pays for the protection to some specified maturity date in a single up-front payment. No further payments of premium are required. The protection or contingent leg of the default swap is exactly the same as in a standard default swap. To distinguish the two formats, the standard contract may be termed a “running” CDS as the premium is paid as a spread on the face value of the protection on payment dates running throughout the life of the contract. The reason why dealers prefer to quote distressed credits in upfront format is that the spread quoted becomes very sensitive to the timing of highly probable credit event. The outcomes can also be very skewed—if the protection seller quotes too low a spread they can lose significantly if there is a credit event. If the protection buyer quotes too high a spread then they can lose significantly if there is no credit event. Consider an example. Suppose dealers view that there is a 50% chance that a certain credit will default in the next month with a 20% recovery rate. They also estimate that there is a 50% chance that it will survive for another year. What is the correct value for one year upfront protection? Ignoring interest rates due to the short horizon, we can approximate the value of the protection leg by 50% times (100% – 20%) equals 40%. The up-front price of protection would therefore be $40 points on a $100 face value. What would the equivalent running spread be? With a 100% probability, this 40% would have to be paid out over a 1-month period (payments terminate following a credit event) and with a 50% probability it is paid over the full year. We approximate the breakeven spread as 40% divided by the Risky PV01. This we estimate at 100% times 1/12 plus 50% times 11/12 which is approximately equal to 0.54 and results in an annualized spread of 7,400 bp. Not only is this spread very high, it means that if the refer-


278


ence credit were to survive, the protection buyer who does a running CDS at this spread has to pay a total of $74 over the one year. This is much higher than the $40 up-front price. This may seem an extreme example but it is not. As such it highlights why the running format is not appropriate for distressed names. It is also possible to use up-front CDS as a way to implement new types of default swap basis trades. For example, consider a bond with a 5-year maturity that pays a 6% coupon and trades at a price of $85. How much should we pay for 5-year up-front protection? If the upfront protection trades for less than $15, then we have a free option as the package of bond plus up-front costs less than $100. If there is a credit event the package is worth par plus any coupons paid on the bond. If there is no credit event then we receive all the coupons on the bond plus principal. The cost of the up-front payment must therefore exceed $15. How much more than $15 depends on the size of the coupons and the expected timing of any credit event. It should in theory be equal to the value of the protection leg of a CDS, provided the model is calibrated to match the bond price of $85. Up-front CDS have enabled the default swap contract to adapt to the situation of highly distressed credits. Where both running and upfront contracts trade, sophisticated market players such as credit hedge funds have been presented with a new type of relative value basis trading opportunity.

APPENDIX Continuous Time Formalism In this appendix, we derive the continuous time counterparts of the discrete formulas in the text. The survival probability is labeled as Q ( t ) = E [ 1( u > t ) ] where 1 is the indicator function and u is the default time. Hence the instantaneous default probability is –dQ(t). The total default probability is the integration of the per period default probabilities: T

∫ –dQ ( t ) dt 0

= 1 – Q(T)


279


The “forward” default probability is a conditional default probability. Conditional on no default till time t, the default probability for the next instant is – dQ ( t ) ------------------Q(t) In a Poisson distribution, defaults occur unexpectedly with intensity λ. Hence, the survival probability can be written as t

  Q ( t ) = exp  – λ ( u ) du  

∫ 0

and the default probability is – dQ ( t ) = Q ( t )λ ( t )dt or the forward probability is – dQ ( t ) ------------------- = λ ( t )dt Q(t) This result states that the intensity parameter in the Poisson process is also an annualized forward default probability. In the case of constant λ, constant interest rate, and constant recovery rate, the CDS value can be simplified to give the following result: T

∫

V = ( 1 – R )λ e

–( r + λ ) u

du

0

–( r + λ ) T λ = ( 1 – R ) ------------ [ 1 – e ] r+λ

In a general case, the equation looks like T

V =

∫ [ 1 – R ( τ ) ]p ( τ ) [ –dQ ( τ ) ] 0

which is a continuous time counterpart of equation (2).


14-McElravey-no tables Page 281 Wednesday, July 23, 2003 10:19 AM

An Introduction to Residential ABS John N. McElravey, CFA Director Structured Debt Research Group Banc One Capital Markets, Inc.

uring the 1990s, the maturation of securitization combined with a dramatic growth in consumer credit and a secular decline in interest rates fueled the development of nonconforming mortgage products like home equity loans. These nonconforming mortgage products supply the collateral backing the residential ABS market. One of the factors drawing added attention to residential ABS (and to structured finance product overall) is a change to capital regulations for depository institutions that went into effect on January 1, 2002. The new risk-based capital rules give AAA and AA asset-backed securities a 20% risk weight (see Exhibit 1). This risk weight is now the same as that applied to agency debt, mortgage-backed securities, and collateralized mortgage obligations (CMOs). This regulatory change removes the capital advantage that securitized agency mortgages had over other structured finance product, including residential ABS. Thus, commercial banks and thrifts subject to these regulations now have another, complementary mortgage product available to them. This article describes the major features of the residential, or home equity loan, ABS market. Its intent is to provide the reader a foundation for understanding and analyzing residential ABS collateral and structures. Residential ABS offer the benefits of greater diversification for a mortgage portfolio and superior convexity attributes.

D

281


282

EXHIBIT 1


Risk-Based Capital Matrix for U.S. Depository Institutions

Product/Summary Treasuries, GNMAs Federally Related Institutions (e.g., EX-IM Bank) GSE Debt (FNMA, FHLMC, FHLB) GSE-backed MBS & CMOs (FNMA, FHLMC) AAA and AA Rated Securitizations (ABS, CMBS, Whole Loan/Private Label CMOs) A Rated Securitizations (ABS, CMBS, Whole Loan/Private Label CMOs) BBB Rated Securitizations (ABS, CMBS, Whole Loan/Private Label CMOs) BB Rated Securitizations B Rated or Unrated Securitizations

Risk Weight 0% 0% 20% 20% 20% 100% 100% 200% Dollar-for-dollar

Source: Banc One Capital Markets, Inc.

MARKET DEVELOPMENT The mortgage-related bond market is the largest segment of the U.S. fixed income markets.1 According to estimates from the Bond Market Association, more than $4 trillion of securities were outstanding at yearend 2001. This market segment consists primarily of agency mortgagebacked securities and CMOs, as well as private label MBS and CMOs. This market is “the mortgage market” in the minds of many investors. The origins of the residential ABS market lie in the development of the nonagency mortgage market beginning in the late 1970s and early 1980s. Many mortgages fell outside of the criteria developed by the agencies because of either their loan balance or underwriting criteria. The nonagency mortgage market developed as a means to securitize this product. The structures used in the residential ABS sector today, especially the senior/subordinate structures used by many issuers, echo the 1

Mortgage securities include: ■ Conforming mortgages/CMOs—conforming on balance and underwriting criteria with respect to agency guidelines. ■ Private label mortgages/CMOs—nonconforming due to balance or certain underwriting criteria. ■ Residential ABS—first lien mortgages, mainly to subprime borrowers, second lien mortgages to prime and subprime borrowers, home equity lines of credit (HELOCs), high loan-to-value mortgage loans.


An Introduction to Residential ABS

283

senior/subordingate structures developed for nonagency MBS transactions of the middle to late 1980s. Residential ABS are distinguished from the rest of the mortgage market by the purpose of the loan or the credit profile of the obligor base of the pool. Early residential ABS transactions were securitizations of second lien mortgages to prime borrowers with relatively low loan balances. Thus, the sector earned the name home equity loan ABS. During the middle 1990s, the residential ABS market started to evolve toward first lien loans as collateral. This trend toward first liens was driven by a falling interest rate environment, consumers’ use of the equity in their homes to consolidate debt, growing competition in the mortgage lending market, and a proliferation of subprime lending programs from major lenders. In addition, mortgage lenders preferred to take a first lien position when refinancing subprime borrowers. The “home equity” name stuck, but most transactions issued in the residential ABS market today are backed primarily by closed-end, first lien mortgages (90% to 100% of the original pool balance) to subprime borrowers. The “home equity loan” name still applies to this collateral because most lenders are making funding decisions based on the equity available in the home. Borrowers are most often refinancing existing mortgages to access the equity in their home, consolidate consumer debt, reduce their monthly payment, finance home improvements, or pay for education or medical expenses. Home equity loans may be used as a debt management tool for borrowers to improve their household balance sheet by reducing their monthly payments. The loans may be fixed rate, adjustable rate, or hybrid ARMs. These borrowers often have impaired credit histories or debt-to-income ratios that exceed agency guidelines, and may also be referred to as “B and C” borrowers. The subprime credit spectrum, however, extends from “A–” to “D” quality borrowers. A severe liquidity crunch during late 1998 squeezed the financial positions of a number of lenders in the residential ABS sector. Intense competition for refinancings during the falling interest rate environment led some issuers to weaken underwriting standards to maintain loan production and market share. In addition, access to the ABS market allowed originators an additional funding alternative. However, some of the firms that grew rapidly during this period were also weakly capitalized and overly reliant on the securitization market for funding. Access to the ABS market became constrained during the latter half of 1998 when liquidity evaporated as a consequence of the events surrounding the demise of Long-Term Capital Management. As a result, several of the weaker firms in the sector were forced into bankruptcy, or to merge with stronger firms. This shakeout left the sector with fewer, but stronger, subprime lenders, which inhabit the marketplace today.


284


CHARACTERISTICS OF SUBPRIME BORROWERS The agency mortgage securitization market developed well-established criteria based on loan balance and underwriting standards that transformed mortgages into something of a commodity. Credit performance is more or less uniform for prime borrowers. Indeed, the vast majority of mortgage borrowers fit into the prime category. The borrowers represented in the residential ABS market have credit profiles that are below this prime segment serviced by the agency mortgage sector. A number of established mortgage lenders have developed lines of business that target subprime obligors who have limited options in the traditional home loan market. In addition, new firms have entered the market over time. But what does it mean to say a borrower is subprime? Exhibit 2 shows Standard & Poor’s Rules-Based Credit Classifications for mortgage borrowers. This table provides a generalized view of the underwriting criteria used by many subprime mortgage lenders. In this matrix, borrowers with credit characteristics below the “A” category would be considered subprime. Subprime borrowers have had some mortgage EXHIBIT 2

Standard & Poor’s Rules-Based Credit Classifications Credit Grade

Characteristic Mortgage Credit

A 0 × 30

Consumer Credit

Revolving Installment Debt/Income Ratio Bankruptcy/ Notice of Default

2 × 30 1 × 30 36%

A–

B

C

D

2 × 30

3 × 30

4 × 30 1 × 60

5 × 30 2 × 60 1 × 90

2 × 30 1 × 60

3 × 30 2 × 60

4 × 30 3 × 60 1 × 90

4 × 30 3 × 60 2 × 90

45%

50%

55%

60%

None in None in None in None in None in past 7 yrs. past 5 yrs. past 3 yrs. past 2 yrs. past year

Note: Each cell indicates number of times a consumer is X days past due. Source: Standard & Poor’s Structured Finance Ratings Group.


285


EXHIBIT 3

FitchRatings FICO Score Tolerance Bands

Fitch Credit Grade

A+

A

A–

B

B–

Low High

720 900

680 720

620 680

590 620

575 590

Source: FitchRatings.

delinquencies, as well as some serious delinquencies on other consumer debt. Debt service-to-income ratios are higher than those of prime borrowers. Furthermore, better quality subprime borrowers would have no mortgage defaults or personal bankruptcies in the past few years. Many subprime mortgage lenders also make greater use of FICO scores than they have in the past. FICO scores have become of particular interest to investors because one index number is used to encapsulate the credit profile of a borrower. Making strict cutoff points on the FICO scale is more an art than a science, but some rules of thumb can be applied. In general, FICO scores above 680 correspond to prime borrowers. Borrowers with FICOs from 680 to 620 are considered “A–” borrowers. FICO scores below 620 place borrowers squarely in the subprime, or B&C, category. According to statistics published by Fair, Isaac and Company, about 20% of the population would be considered subprime borrowers based on their FICO score. Generalized FICO scores compared to credit grade used by Fitch in rating subprime mortgage ABS are listed in Exhibit 3. We offer a note of caution on the use of FICO scores. A number of factors beyond the FICO score are used by originators when making the lending decision. In addition, the credit profile of the borrower is only one of a number of factors used by the rating agencies when determining credit enhancement for a transaction. This is a statistic that should not be viewed in isolation, or as a substitute for understanding the underwriting criteria of an issuer.

PREPAYMENT SPEEDS The prepayment profile of subprime mortgage collateral differs in important ways from conforming, agency mortgages. These differences provide superior convexity compared to prime, conforming mortgages. As a result, residential ABS provides a good opportunity for an investor to diversify a mortgage portfolio. Prepayment rates for subprime mortgage pools have a faster seasoning ramp and reach a higher steady state


286


level than the prepayment rates on conforming, agency pools. These characteristics are true for both fixed rate and ARM collateral. For example, a baseline fixed rate home equity prepayment (HEP) curve ramps up to a steady state prepayment rate of 20% to 25% CPR over 10 to 12 months. This prepayment ramp compares to the baseline for conforming agency mortgages of 6% CPR over 30 months, corresponding to a 100% PSA (see Exhibit 4). The faster prepayment rates on subprime mortgage pools translate into shorter average lives on the ABS compared to agency mortgages and CMOs. Prepayment rates on ARM collateral show a similar pattern, usually peaking between 25% and 30% CPR after 12 months. Exhibit 5 shows prepayment curves for 2/28 and 3/27 hybrid ARMs. The prepayment spikes around 24 months for the 2/28 loans and around 36 months for the 3/27 loans coincide with the first reset dates for these mortgage products. When the fixed rate period ends, the interest rate on the loan resets to a higher level. The higher rate increases the monthly payment for the borrower, providing an incentive to refinance. Most borrowers will refinance within a few months of their reset date, thus creating the pattern of a sharp increase in prepayments followed by a gradual return to the steady state level as more borrowers seek out alternatives. The more muted effect of the prepayment spike on 3/27 hybrid ARMs is due to the longer seasoning period prior to the first reset date, which results in a greater “burnout” effect compared to the 2/28 product. EXHIBIT 4

Baseline Prepayment Curves—Fixed Rate Mortgages

Sources: Bloomberg, Bond Market Association.



EXHIBIT 5

287

Baseline Prepayment Curves—Adjustable Rate Mortgages


Lenders also may use prepayment penalties as a way to manage their prepayment risk. In many cases, lenders use prepayment penalties to make it costly for the borrower to refinance until the penalty period expires, which can be anywhere from one to five years after origination. For example, the dollar amount of the penalty to the borrower may be calculated as 80% of six months’ interest. Given the cash flow position of most subprime consumers, this amount is significant. Prepayment penalties can reduce observed prepayment rates by as much as 8% to 10% CPR while they are in effect. Loan pools with penalties will prepay more slowly than pools without penalties until the expiration of the penalties. At that point, pent up demand for refinancing takes over, and prepayments will be higher, on average, for the loan pool that had penalties attached to them. The major reason for faster average prepayment rates on subprime mortgage pools can be described as the “credit curing effect.” Borrowers who fall into the subprime credit categories have an opportunity to improve their credit quality over time. As their credit improves, they gain more refinancing options. Better credit means lower mortgage rates become available, and these borrowers can reduce their monthly mortgage payment significantly. Other major refinancing motivations include the consolidation of other consumer debt, term extension to reduce monthly payment amounts, and monetizing equity in the home to finance home improvements or to address temporary liquidity needs (such as for education or medical expenses).


288 EXHIBIT 6


Comparing Prepayment Rates

Sources: Banc One Capital Markets, Inc., Bloomberg.

Despite being faster on average, prepayment rates are generally more stable on subprime mortgages than they are on conforming mortgage pools. Prepayments on subprime mortgages are less sensitive to interest rate swings, and thus provide superior convexity compared to agency mortgages. Exhibit 6 compares prepayment rates over time for 1996 production loans for both FNMA pools and a group of subprime mortgage lenders. As expected, the subprime mortgages display the faster seasoning ramp. When interest rates fall, prepayments for both groups rise, but the conforming mortgages peak at a much higher level. When interest rates fall, prepayments for both groups slow, but they fall by less for the supreme group. Based on this data, conforming mortgage pools can have monthly prepayment rates that range from 10% to 60% CPR, while the subprime mortgages have prepayments that typically range between 15% and 40% CPR—a prepayment range that is only half as wide. As a result, the average lives of the residential ABS are more stable for a given change in interest rates. The main factor affecting prepayments on conforming mortgages is the prevailing mortgage rate for new loans compared to the rates on outstanding mortgages. However, the factors affecting prepayment rates on subprime mortgage pools are more varied, and interest rates are only part of the story. Subprime mortgage loans tend to have somewhat lower loan balances relative to conforming, agency pools. With a lower monthly payment, the incentive to refinance derived from falling mortgage rates is



289

more muted than it would be for prime borrowers. Furthermore, subprime borrowers tend to have fewer refinancing alternatives due to their credit history. This makes refinancing, even when interest rates are falling, that much more difficult than it is for prime borrowers.

RELATIVE VALUE CONSEQUENCES The prepayment profile of subprime mortgages and their superior convexity characteristics have consequences for their relative value compared to agency mortgage product (see Exhibit 7). As interest rates fell from June 2000 to June 2002, the relative attractiveness of residential ABS can be seen in a comparison of the yields on 5-year, sequential He’s and agency CMOs. Early in the period examined, HELs tracked agency CMOs, and offered a higher yield. When interest rates began to fall rapidly during 2001, and prepayment rates began to rise, the yield differential narrowed. At the point when prepayment rates were peaking, the yield on HELs moved inside the yield on the agency CMOs. The higher yield on the agency CMOs became necessary to compensate investors for the greater variability of conforming mortgage prepayment rates. The performance of the residential ABS product during this period suggests that the market recognizes the superior convexity of the subprime mortgage pools. Mortgage investors can improve the diversification of their portfolios by including the residential ABS product. EXHIBIT 7

Yields and Yield Difference—5-Year HELs and Agency CMOs


290 EXHIBIT 7


(Continued)


KEY ASPECTS OF CREDIT ANALYSIS For investors in the agency mortgage market, credit analysis is unnecessary because of the agency guarantee on the underlying collateral. Agency mortgages and CMOs derive their credit support from the agency guaranty. Credit analysis is necessary in residential ABS, just as it is in other ABS sectors, because residential ABS derive their credit support from internal sources (overcollateralization, subordination, cash, or excess spread) or external sources (monoline bond insurance). Stress scenarios on mortgage defaults and recoveries run by the credit rating agencies are based on the historical loss experience of the mortgage market over the past 70 to 75 years. Several episodes of real estate market stress have been incorporated into the default and loss severity outlook of the rating agencies. These episodes include the Great Depression of 1930s, the oil bust years of the middle and late 1980s in Texas, Louisiana, Oklahoma, and Alaska, and the recession of the early 1990s and its impact on real estate markets in New England and California. Loss coverage requirements calculated by the rating agencies for subprime mortgage pools start with a prime borrower, first lien loan pool as a benchmark. They increase the stress factors and credit enhancement



291

as necessary depending on the risk profile of each pool. The following list highlights the major risk factors reviewed by the rating agencies in the rating process. ■ Borrower credit quality: Credit quality is measured based on the

■

■

■

■

issuer’s underwriting criteria or FICO score distribution. Over the past few years, mortgage lenders have increasingly used FICO scores in their underwriting of new loans as a supplement to their traditional underwriting guidelines. LTV ratio: The loan-to-value ratio is a key indicator of default risk and loss severity. Loans with higher LTVs have greater default risk because the borrower has less equity in the property. This factor can affect the willingness of the borrower to pay. However, as a loan seasons equity grows because the loan amortizes and the underlying property tends to appreciate. As the equity builds, the borrower’s willingness to pay increases because there is more to lose in the event of a default. It should be noted, though, that the beneficial effect of seasoning is measured in years, not in months, because there is little in the way of principal that amortizes in the early years of a mortgage loan. Typical LTVs on subprime mortgages usually range between 80% and 100%. Dwelling type: Single family detached homes are the predominant type of property in most residential ABS pools. They present the lowest risk of default and offer the best recovery values. They are the largest part of the residential real estate market and the preferred property type for most homebuyers. As a result, their market value tends to be more stable than other dwelling types. High-rise condominiums, co-ops, and multifamily homes tend to be riskier properties because they have narrower appeal to buyers and have the potential for greater price volatility. Property maintenance becomes an important issue in multifamily dwellings. However, they tend to be only a small portion of a residential ABS transaction, if they are represented at all. Occupancy status: Subprime mortgage pools are composed primarily of owner-occupied homes, often exceeding 90%. Second homes, vacation homes, or investment properties make up the remainder. Owneroccupied homes tend to carry lower default risk because a homeowner is less likely to forfeit his or her primary residence than a second home or investment property. Rental income from the investment property, which may be needed to make mortgage payments, may not always be available. Lien status: Subprime mortgage pools are primarily composed of first lien mortgages. To the extent that second lien loans are included in the collateral pool, a combined LTV (CLTV) of the first and second mort-


292

■

■

■

■

■


gages would be used to determine the amount of equity each borrower has in their home. Geographic concentrations: Housing costs can vary greatly in different parts of the country. As a result, geographic concentrations in subprime mortgage pools can be relatively high in California and northeastern states compared to other types of ABS. Unusually high geographic concentrations will require extra credit enhancement to mitigate any additional risks. Loan purpose: New mortgage loans are used to purchase a home or to refinance an existing mortgage. Purchase loans may be viewed as less risky because a recent purchase price and an appraisal of the property customarily support them. Refinancings come in two types: a rate/term refinance or a cash-out refinance. A rate/term refinance replaces the current mortgage with a new mortgage that has a lower interest rate or a shorter maturity. The purpose of this loan is to reduce the monthly payment or to decrease the term of the loan. Cash-out refinancing replaces the current mortgage with a new mortgage loan in which the borrower is monetizing a portion of the equity built up in the property. In general terms, cash-out refinancing is more risky than purchase or rate/term refinancing because there is no sale by which to get an independent measure of the market value of the home. Mortgage seasoning: As noted above under the section on LTVs, seasoning of the collateral is beneficial, and more seasoning on the mortgage pool is preferred to less. The amount of seasoning in a subprime mortgage pool can be a significant mitigating factor for other risks present in the collateral pool. For example, significant amounts of seasoning will reduce current LTV ratios compared to their original values, which reduces the risk of default in the pool. Loan size: Loan size can be another important credit quality factor. Jumbo loans, which are greater than the $300,700 loan balance currently established by the agencies as a conforming loan, may be riskier because the underlying properties can suffer greater market value volatility due to a more limited universe of buyers. Loan documentation: Full documentation of borrower income, debt levels, and property valuation is required for prime, conforming mortgage pools sold to the agencies. Subprime mortgage pools are primarily composed of fully documented loans. Reduced or limited documentation programs may be offered by mortgage lenders to condense the amount of paperwork required of the borrower. For example, these programs may require more limited documentation of borrower income used to calculate debt-to-income ratios. The risk is that less qualified borrowers will be granted too much credit under limited documentation programs. However, lenders can adjust the required LTV,



293

meaning that borrowers using limited documentation programs must have more equity in their home. The rating agencies also make adjustments to their default frequencies (and credit enhancement) to compensate for lower documentation thresholds. ■ Loan type: Subprime mortgage pools may include a number of different loan types. Fixed rate loans, for example, may have terms of 15, 20, or 30 years. Furthermore, fixed rate mortgages may be offered with a balloon feature, where the mortgage has fixed payments of principal and interest based on a 30-year amortization schedule. After an amortization period (5, 7, 10, or 15 years), the unpaid principal balance becomes due in a lump sum payment. At this point, borrowers will repay the loan or need to refinance the remaining principal balance. The risk is that the borrower will not be able to refinance at an affordable rate, or that property values will not be adequate to support the desired loan balance. The rating agencies will require higher levels of credit enhancement for pools with higher levels of balloon loans. Adjustable rate mortgages (ARMs) come in a number of different varieties. Typically, ARMs have a low initial (or “teaser”) rate that adjusts periodically, for example every six months. Any rate adjustment is usually subject to periodic and lifetime caps. Periodic and lifetime caps may be 2% and 6%, respectively. Borrowers like ARMs because the low initial interest rate may allow them to qualify for a larger mortgage. However, the interest rate reset introduces the risk of higher future payments for the borrower. As a result, the rating agencies will usually require more credit enhancement to mitigate the interest rate risk inherent in adjustable rate loans. Another product offered by many lenders is a hybrid ARM. Hybrid ARMs offer a fixed rate period before the loan becomes fully adjustable. Fixed periods typically are for 2, 3, 5, 7, or 10 years. These loans also carry periodic and lifetime caps. The most common hybrid ARM terms found in residential ABS pools are 2/28 and 3/27 ARM collateral.

STRUCTURAL CONSIDERATIONS The collateral pools backing residential ABS transactions may include all fixed rate mortgages, all ARMs and hybrid ARMs, or distinct pools of fixed and adjustable rate collateral backing separate groups of fixed and floating rate securities. Like most new asset types in the ABS market, credit enhancement for residential ABS transactions started out with bond insurance. In


294


1997, the first senior/subordinate structures were introduced to the residential ABS market. Over time, more issuers moved to senior/subordinate structures as the market for AAA bonds became more liquid, more data on collateral performance for issuers became available, and a market developed for the subordinate bonds. Residential ABS transactions that utilize a senior/subordinate structure are typically tranched down to the BBB rating level. The issuer’s decision as to which type of credit enhancement to use will be based on the relative costs of executing each structure. From time to time, market dislocations will cause issuers to make greater use of bond insurance structures because of a lack of liquidity or demand in the subordinate bond sector. Since 2000, FNMA and FHLMC have become more active in the ABS market by wrapping pools of loans with conforming balances, which may be purchased by the agencies or sold to ABS investors. ABS investors also are sold the securities backed by the nonconforming loan pool segments of these transactions.

Bond Insurance Structures Bond insurance structures use internal credit enhancement in the form of overcollateralization and excess spread to support the bonds being issued at an investment grade rating level (BBB– equivalent or better). The bond insurer then guarantees timely payment of interest and ultimate payment of principal at maturity to achieve a credit rating of AAA. In residential ABS transactions utilizing bond insurance, principal is usually allocated sequentially to the AAA rated senior classes. This form of credit enhancement may be most economical when an issuer is new to the ABS market or has limited information on credit performance. Alternatively, the market from time to time may demonstrate weaker demand for subordinate ABS. This condition will manifest itself through relatively wide spread differentials because the relative value outlook of investors may have shifted, or some sort of market dislocation may have occurred. In those situations, a wrapped transaction may provide the most reliable execution.

Senior/Subordinate Structures The ratings on the residential ABS in senior/subordinate structures rely on a combination of the subordination of lower rated classes, overcollateralization, and excess spread. In addition, most transactions incorporate performance triggers based on the level of delinquencies or net losses that can redirect cash flow to support the senior bonds if credit performance of the collateral is weaker than expected.



295

Like most other ABS transactions, the first line of defense to protect investors from credit losses is excess spread. In the case of residential ABS, there is usually a substantial amount of excess spread available because the WAC on the mortgage loans is well in excess of the weighted-average coupon of the bonds issued. Excess spread may be used to accelerate the amortization of the AAA bonds in order to build up overcollateralization to a required target amount. The target amount is usually reached within the first several months of the transaction’s life. If overcollateralization is funded at the outset by the issuer, then excess spread can be used to maintain that target amount. The target overcollateralization amount is generally established as a percentage of the initial principal balance of the collateral pool.

Subordination and Shifting Interest Like other nonagency mortgage securities, residential ABS transactions utilize a “shifting interest” structure that increases the level of credit enhancement available to the senior bondholders. During the early stages of a transaction, all principal collections are paid to the senior bonds, and the subordinate bonds are locked out from receiving principal during this period. For example, consider a simple two-class transaction in which the AAA rated Class A is 88% of the bonds issued and the BBB rated Class B is 12% of the bonds issued. All principal collections would be paid to the Class A and the Class B would be locked out. Over time, the Class A bonds amortize, and their percentage interest in the collateral pool would decrease. At the same time, the percentage interest in the collateral pool of the Class B bonds would increase. In most cases, the subordinate bonds are locked out from receiving principal collections for the first 36 months of the transaction or until the credit enhancement level for the senior notes has doubled, whichever is later. A doubling of the initial credit enhancement level is equivalent to saying that the principal balance of the collateral pool has been reduced by 50% (a pool factor of 0.50). In our example above, when the percentage interest for Class B reaches 24% (alternatively, credit enhancement for Class A has doubled), and the transaction is at least 36 months old, then the Class B starts to receive principal payments. This point in the life of a residential ABS transaction is called the “Stepdown Date,” which refers to the reduction, or step down, of the dollar amount of subordination as credit enhancement from its thencurrent levels. For transactions with multiple classes carrying different ratings, the mezzanine and subordinate classes would receive their pro rata share of principal collections and begin to amortize.


296 EXHIBIT 8


Expected Defaults on Residential ABS Pools

Source: Standard & Poor’s.

Why a 36-Month Lockout? Based on observed default experience, a pool of subprime mortgage loans will experience about 60% of its total expected defaults by the 36th month of the transaction, with the majority of expected defaults occurring in years two, three, and four (see Exhibit 8). Less than 20% of the total amount of expected defaults occur in the first 12 months of a transaction. Therefore, the early lockout for the subordinate bonds increases the amount of credit enhancement during the period when the transaction most needs it. If the collateral pool performs as expected, then the subordinate bonds would receive principal payments as scheduled. If credit performance is below expectations, then sufficient credit enhancement should be available to withstand the additional stress.

Delinquency and Net Loss Triggers A reduction in the dollar amount of credit enhancement may occur on the step-down date as long as the collateral pool is performing as expected. Two tests, a delinquency test and a cumulative net loss test, have been designed to measure collateral performance. One or both of these tests may be used in a particular transaction. All of the relevant trigger events must be passed for the step down of credit enhancement to occur. Actual trigger levels will vary from one issuer to another, depending upon the actual credit performance of its mortgage loans. An issuer’s trigger levels may vary over time and across different transactions as well. Delinquency tests are typically based on the 3-month average of 60



297

plus-day delinquencies (including bankruptcy, foreclosure, and REO) being less than some threshold percentage of the outstanding credit enhancement. For example, the delinquency trigger event in Option One 2002-3 was set at 60 plus-day delinquencies exceeding 80% of the current credit enhancement percentage. For example, if current credit enhancement stands at 22%, then 60 plus-day delinquencies would have to be less than 17% after the step-down date for the subordinate bonds to receive principal payments. The cumulative net loss trigger is based on a percentage that steps up over time. As long as cumulative net losses are below the current threshold level, principal would be paid to the subordinate bonds after the step-down date. For example, Exhibit 9 shows the cumulative net loss threshold levels for Option One 2002-3. The dates for measuring this trigger begin with the step-down date, and the threshold for cumulative net losses increases in various increments over the next four years from 4% to 6%. The trigger tests are performed each month after the step-down date. If the triggers are passed in that month, then principal may be passed through to the subordinate classes in a transaction. If the triggers are breached, then principal cash flow is diverted from the subordinate classes and paid to the senior bonds. If a transaction is past its stepdown date, depending on whether or not the triggers are breached, the EXHIBIT 9

Cumulative Net Loss Trigger—Option One 2002-3

Note: Step-down date is May 2005. Source: Transaction Prospectus.


298


subordinate bonds may switch back and forth between receiving principal and being locked out. Because the breach of a trigger will lock out a subordinate bond and extend its average life, subordinate bonds in the secondary market will often be traded “to fail”—that is, to the average life implied by the transaction failing its triggers continuously after the step-down date.

Deep Mortgage Insurance An additional type of credit enhancement becoming more widely used since 2000 is mortgage insurance purchased by the issuer. This type of mortgage insurance is also referred to as “deep” mortgage insurance (or “deep MI”). The issuer pays an annual premium and the insurance covers losses on loans down to a prespecified LTV level. The mortgage insurance will usually cover only a portion of the mortgage pool for those loans that meet the insurer’s underwriting criteria. The mortgage insurance typically will cover the principal balance of the loan down to a 60% to 65% LTV. The presence of insurance has the effect of reducing realized losses on the mortgage pool. Overall, its effect on the credit performance acts as if the overall LTV of the pool has been reduced, and thus loss severities are lower. With more effective equity available, the rating agencies may reduce the amount of credit enhancement necessary to support the desired ratings on the bonds.

Available Funds Cap One of the key structural features found in floating rate residential ABS transactions is an available funds cap. Conceptually, the available funds cap says that investors will be paid interest on their bonds up to the amount that can be generated by the mortgage pool after transaction fees and expenses. Floating rate residential ABS are usually indexed to 1-month LIBOR and reset monthly. Adjustable rate mortgages included in these transactions may use several different indexes, such as 6-month LIBOR or the 1-year constant maturity Treasury. The loans may be offered to borrowers at a belowmarket teaser rate for the initial period. The loans reset less often than the bonds, so timing mismatch is present. Additionally, 2/28 or 3/27 hybrid ARMs, which have a 2-year or 3-year fixed rate period before adjusting, may be included in the pool. Adjustable rate mortgages also have periodic and lifetime caps that constrain the amount of adjustment of the mortgage rate each period. The available funds cap results from the mismatch between mortgage loan resets and the floating rate ABS liabilities. To calculate the initial level of the available funds cap, transaction expenses (servicing fee,


299


EXHIBIT 10

Calculating an Available Funds Cap and Life Cap Percentage

Initial Available Funds Cap WA Gross Coupon Less Servicing Fee Less Trustee Fee Less I/O Strip Less Mortgage Insurance Net Available Funds Cap WA Bond Coupon Initial Excess Spread Avail.

Life Cap 9.00% 0.50% 0.01% 0.38% 0.25% 7.86% 2.26% 5.60%

WA Life Cap Less Servicing Fee Less Trustee Fee Less I/O Strip Less Mortgage Insurance Net Available Funds Cap WA Bond Spread Maximum 1mL Rate to Cap Current 1m LIBOR Maximum LIBOR Increase

15.00% 0.50% 0.01% 0.38% 0.25% 13.86% 0.43% 13.43% 1.84% 11.59%


trustee fee, I/O strip, surety fee, etc.) are subtracted from the original weighted average coupon on the underlying mortgage loans. After the coupon on the bonds is accounted for, the available excess spread generated by the mortgage pool can be calculated (see Exhibit 10). Over time, the adjustable rate loans reach their reset dates, mainly at 24 and 36 months. Subprime mortgage pools have a variety of loan types and reset dates that are factored into the available funds cap. The available funds cap will increase from its initial level as the mortgage loans reset and the available WAC on the collateral increases. The life cap for the available funds cap also can be estimated (see Exhibit 11). Using the weighted average spread on the bonds, the maximum increase in LIBOR before hitting the life cap can be estimated on the fully-adjusted collateral pool. This adjustment process can be seen in the assumed available funds cap estimated for a home equity transaction, for example Option One 2002-3. The level of the cap is estimated assuming that LIBOR rates increase beyond the maximum rate obtainable on the mortgage loans, and is run at the pricing prepayment speed. The impact of the resets on 2/28 and 3/27 hybrid ARMs becomes clear in this representation (see Exhibit 11). The cap increases sharply between 24 and 48 months as the hybrid ARM loans reset. It is important to remember that the mix of loans in the mortgage pool can change over time due to prepayments or defaults. If the mix of loans is significantly different than assumed initially, then the interest cash flows from the collateral available to pay interest to investors also may be different. The available funds cap may be higher or lower than originally assumed due to the changing characteristics of the collateral pool.


300 EXHIBIT 11


Available Funds Cap Calculation – Option One 2002-3

Source: Transaction Prospectus.

Available Funds Cap Carryforward A carryforward mechanism may be established in a floating rate residential ABS transaction to mitigate the cap risk under the available funds cap. The difference between the coupon payable on the bonds and the available funds cap is carried forward to future periods. The carryforward amount is capitalized and accrues interest at the coupon on the bonds. These amounts will be repaid to investors with future excess spread when available. However, if excess cash is not available during the life of the transaction, then investors are still at risk to lose these payments at the clean-up call or at maturity. Some transactions do not provide for a carryforward, so investors should read the offering documents closely to verify the presence of any carryforward mechanism.

Step-Up Coupon Residential ABS transactions may also include a step-up in coupon on bonds with a longer average lives. In fixed rate transactions, the coupon may step up by 50 basis points. In floating rate transactions, the margin over the index may be increased by some multiple, for example 1.5× or 2× the original amount. Such an increase in coupon would come directly out of the excess spread that would normally flow back to the seller. The step-up coupon provides a powerful incentive for the seller/servicer to exercise its clean-up call option. This structural enhancement helps to mitigate the risk of extension in residential ABS transactions.



301

CONCLUSION Residential ABS offer significant benefits from greater diversification for a mortgage portfolio, including superior convexity attributes. These traits translate into good relative value for the investor. In addition, changes to risk-based capital regulations for depository institutions have removed the capital advantage held by agency mortgage product. Over time, the residential ABS sector has grown to be one of the largest and most liquid segments of the ABS market. Our expectation is that this sector will continue to evolve, making this one of the most dynamic parts of the market.


15-Bergantino-Nonagency Page 303 Wednesday, July 23, 2003 10:18 AM

Nonagency Prepayments and the Valuation of Nonagency Securities Steve Bergantino Vice President Mortgage Research Lehman Brothers

his article provides an overview of nonagency prepayments and an introduction to the valuation of nonagency securities using Lehman Brothers’ nonagency fixed rate prepayment model. The model covers 15- and 30-year fixed rate jumbos, jumbo alt-As, conforming balance alt-As, and jumbo relos, explicitly incorporating the effects on prepayments of loan size, borrower credit quality, prepayment penalties, and geographic distribution. In the sections that follow, we review the collateral characteristics and prepayment behavior of nonagency mortgages and present a series of valuation exercises highlighting the most important factors affecting prepayments and security valuation in each nonagency sector.

T

THE NONAGENCY MARKET The nonagency MBS market consists of three primary sectors: jumbos, alt-As, and jumbo relos.

303


304


The jumbo sector is by far the largest of the three nonagency sectors.1 Jumbos are mortgages that are excluded from agency pools primarily because of their nonconforming loan size. By law, Fannie Mae and Freddie Mac (the GSEs or agencies) are not allowed to underwrite or purchase mortgages with balances above a prespecified limit.2 Most nonconforming balance—or jumbo—mortgages are, therefore, purchased by private-label issuers and tranched into nonagency CMOs. Alt-A mortgages provide another source of collateral for the nonagency market. An Alt-A mortgage can have either a conforming or nonconforming balance (conforming alt-A versus jumbo alt-A) but is generally of lower credit quality than typical jumbo or agency collateral. The earliest alt-A securitizations were done in 1994. Prior to that time, mortgages with alt-A collateral characteristics were generally left unsecuritized, although a small portion was included in various jumbo deals. Since the development of a separate alt-A market in the mid-1990s, the volume of private label alt-A issuance has grown substantially. Jumbo relos are nonconforming balance purchase loans made to individuals who are being relocated by their employers, often temporarily. Jumbo relos have credit characteristics and loan size that are similar to standard jumbos but differ from them in one key respect: borrower horizon. The typical relo borrower has a significantly shorter expected stay in his/her home than the typical jumbo borrower and, therefore, displays noticeably different prepayment behavior. Jumbo relos are the smallest of the nonagency sectors. As for mortgage type, most nonagency MBS are backed by fully amortizing fixed-rate mortgages, although hybrid issuance has picked up significantly in recent years. In 2001, for example, 30-year fixed rate mortgages accounted for 71% of all securitized nonagency originations, with 15-year collateral accounting for an additional 11%. For alt-As and jumbo relos, the product mix is even more skewed toward 30-year fixed rate collateral. Roughly 90% of alt-A securities and virtually all jumbo relo securities are backed by 30-year fixed-rate mortgages. The remainder of this article will focus on the borrower characteristics and prepayment behavior of 30-year mortgages.

1

All information on nonagency origination volume, collateral composition and prepayment performance presented in this article has been calculated from loan-level data provided by LoanPerformance. 2 The agency loan size limit is adjusted each year in January to keep pace with measured housing price inflation. It was set to $275,000 for the 2001 calendar year and was increased to $300,700 in January 2002.


Nonagency Prepayments and the Valuation of Nonagency Securities

305

NONAGENCY COLLATERAL CHARACTERISTICS As can be seen in Exhibit 1, jumbos are large loan balance mortgages to residential borrowers with excellent credit. In particular, jumbo borrowers tend to be owner-occupants with low LTVs and high FICOs who provide full documentation of their income and assets. The high credit quality of jumbo borrowers causes the spread between jumbo and conforming mortgage rates to be fairly low, as evidenced by the 24-basispoint spread at origination (SATO) for jumbos in Exhibit 1.3 By comparison, the collateral characteristics of alt-As show obvious signs of credit impairment relative to jumbos. Compared to jumbos, altA pools have higher LTVs, lower FICOs, and greater concentrations of limited documentation loans and cash-out refinancings. These sources of credit impairment force alt-A borrowers to pay wide spreads over conforming mortgage rates, typically 80–100 basis points. Notice also that a much higher proportion of alt-A mortgages are subject to prepayment penalties than in the jumbo or jumbo relo sectors. The collateral characteristics of jumbo relos are very similar to those of standard jumbos. Like jumbos, relos are characterized by large loan sizes, high FICOs, and few investment properties. Unlike jumbos, however, relo pools tend to have high LTVs and a high proportion of limited documentation loans, reflecting the high proportion of purchase borrowers (100%) and the availability of streamlined application processes for corporate relocation programs. Additionally, the difference in jumbo and jumbo relo SATOs reflects the 50-basis-point rate reduction received by relo borrowers because of employer paid discount points.4 Loan size, credit quality, prepayment penalties, and borrower horizon are all important factors affecting nonagency prepayments. The existence of fixed origination costs such as underwriting fees, legal expenses, and the time cost of applying for and closing on a new mortgage make large loans prepay faster than smaller loans when faced with similar rate incentives. The worse the credit quality of a mortgage pool, the less responsive are its prepayments to observed refinancing opportunities. Prepayment penalties reduce refinancing activity by increasing the costs of refinancing, and borrowers who plan to move soon are less responsive to observed refinancing opportunities than are longer horizon borrowers. 3

SATO is defined as the difference between a borrower’s initial mortgage rate and the conforming mortgage rate prevailing at the time of origination. 4 Corporate relocation programs typically include an employer-paid purchase of two discount points, reducing the borrower’s mortgage rate by about 50 bp. Relo borrowers who refinance into a new mortgage, however, typically receive no further subsidies from their employer and so face standard jumbo mortgage rates.

306 1,410

Jumbo Relo

417

268 416 131

407

Loan Size ($ 000)

–24

94 87 114

24

SATOa (bp)

0

15 12 21

3

% Prepay Penalty

77

77 76 82

71

26

26 19 47

7

% % LTV LTV > 80

730

700 700 699

730

FICO

7

16 15 17

5

% FICO < 650

0

11 7 22

3

% NonOwner Occ.

62

71 71 71

24

% Lim Doc.

0

32 33 28

20

% Cash Out Refi.

93

70 72 64

78

23

42 49 21

45

% Sing. % Fam. Calif.

a SATO stands for spread at origination and is defined as the difference between a borrower’s initial mortgage rate and the conforming mortgage rate prevailing at the time of origination.

17,619 13,133 4,486

64,506

Orig. Balance ($ mn)

Collateral Characteristics of 2001 Nonagency Originations, 30-Year Fixed Rate Collateral

Alt-A Jumbo Alt-A Conforming Alt-A

Jumbo

EXHIBIT 1




307

The lack of uniformity in collateral characteristics across sectors, vintages, issuers, and even collateral groups within a given deal requires that a prepayment model explicitly incorporate these factors into its predictions if it is to be of any use in valuing nonagency securities. This is especially true of securities backed by alt-A collateral, for which loan size can range from jumbo to conforming, credit quality can vary from prime to subprime, and prepay penalty concentrations can be anywhere from 0–100%. In the sections that follow, we will examine the primary factors driving prepayments within each nonagency sector and demonstrate how these factors affect prepayment projections and security valuation using Lehman Brothers’ nonagency prepayment model.

JUMBO PREPAYMENTS AND SECURITY VALUATION The strong credit quality of both jumbo and agency collateral (with the exception of agency alt-As) means that differences in the refinancing behavior of jumbos and agencies are due primarily to differences in loan size. Loan size affects refinancing through its effect on the average cost to borrowers of originating a new loan. While some costs vary with loan size, others are fixed for all borrowers, regardless of the amount borrowed. These fixed costs, along with any caps placed on variable costs, make large loans more responsive to a given rate incentive than small loans. For a concrete example, consider a set of four borrowers with current loan balances of $50,000, $100,000, $200,000, and $400,000, all of whom could lower their mortgage rate by 1% if they refinance. Further, assume that they must all pay origination costs equal to 1% of the amount borrowed plus $1,000. If not for the $1,000 fixed cost, every borrower would face average origination costs of 1%. Because of the $1,000 charge, however, the average origination costs faced by the four borrowers are 3%, 2%, 1.5%, and 1.25%, respectively. Clearly, the 1% annual interest savings looks more attractive to the borrower with the $400,000 loan than to the borrower with the $50,000 loan since the former pays an up front cost of 1.25%, compared with the latter’s 3%. More generally, the fixed costs of origination create a positive relation between loan size and callability. Notice, however, that each doubling of loan size in this example reduces average cost by half as much as the previous increase, implying that the incremental effect of loan size on callability decreases as loan size increases. Exhibit 2 illustrates this point by plotting prepayments from the 2001 refinancing wave against loan size for various levels of rate incentive. As expected, the initially steep slope of the prepayment curves


308

EXHIBIT 2


Effect of Loan Size on Prepaymentsa

a

January 2001–March 2002 prepayments on agency and jumbo collateral with 12– 24 WALA.

begins to flatten as we move to higher and higher loan sizes. Among loans that saw a 100-basis-point rate incentive, for example, the difference in prepayments between $50,000 and $100,000 loans was 21% CPR, while the difference between $200,000 and $400,000 loans was just 8% CPR, despite the much larger loan size difference.

Jumbo Agency Prepayment Differences To see what these prepayment curves imply for the relation between jumbo and agency prepayments, consider what has happened to agency loan sizes and origination costs over the past decade. As Exhibit 3 illustrates, the rapid housing price appreciation of the 1990s and corresponding increases in agency loan size limits have had a significant effect on the loan size distribution of agency pools, increasing the average loan size of FNMA originations from $107,000 in 1992 to $130,000 in 2000. Over the same period, the average loan size of jumbo originations increased from $283,000 to $350,000. Even though the agency increase was smaller in absolute dollar terms, the prepayment curves in Exhibit 2 suggest that it should have had a larger effect on callability, assuming, that is, that the fixed costs of origination have risen less rapidly than loan sizes. Indeed, low overall inflation rates, the development of automated underwriting systems, streamlined refinancing programs, and increased competition among mortgage originators have kept the out-of-pocket expenses and time costs of refinancing well



EXHIBIT 3

309

Loan Size Distribution across FNMA Pools

EXHIBIT 4

Jumbo-Agency Prepayment Convergence (a) 1992 Originations in 1993

under control while, at the same time, rapid housing price appreciation has pushed average loan sizes steadily upward. The effect of these trends on the relative callability of jumbos versus agencies can be seen in Exhibits 4 and 5. Exhibit 4a presents prepayments on 1992 originated jumbos and agencies during the 1993 refinancing wave, while Exhibit 4b presents prepayments on 2000 originated jumbos and agencies during the 2001 refinancing wave. Both plots show prepayments on 8.5% WAC jumbos and 7.5% coupon FNMAs, which, after accounting for the jumboagency mortgage rate spread, experienced similar rate incentives. What


310


EXHIBIT 4 (Continued) (b) 2000 Originations in 2001

EXHIBIT 5

a

Jumbo and Agency Prepayments by Rate Incentivea

Prepayments are for collateral with 12–24 WALA.

is interesting to note about the two sets of prepayment curves is how modest the increase in the rate responsiveness of jumbo prepayments appears next to the dramatic increase in agency prepayments. Exhibit 5 provides another look at the same trend by plotting jumbo and agency



311

prepayments by rate incentive. At each level of rate incentive, prepayment differences are lower now than during the 1993 refinancing wave, due mainly to a significant increase in the rate responsiveness of agency prepayments.

Turnover Although available prepayment data do not separate out housing sales from other sources of prepayments, we can get a good indication of jumbo versus agency turnover rates by examining overall prepayments on current coupon and discount collateral during periods of high mortgage rates (i.e., when large portions of the mortgage universe are discounts and refinancing activity is low). Doing this leads one to conclude that jumbo turnover differs from what is observed on agency collateral in two ways. First, historical prepayment data suggest that base case jumbo turnover is higher than agency turnover. The primary reason for this difference is that high-income borrowers and borrowers who live in large metropolitan areas are both more mobile than other segments of the population and more likely to have jumbo mortgages. In Exhibit 6, for example, the fully seasoned prepayment rate on jumbo collateral with a 0 to –50-basis-point rate incentive was 12% CPR, compared with 9–10% CPR for agencies. EXHIBIT 6

Base Case Jumbo Turnover and Lock-in versus Agencies (July 1999–

December 2000)


312


Notice, however, that despite their higher base case prepayment rates, deep discount jumbos prepay at about the same rate as agency collateral, 7–8% CPR (see Exhibit 6). The drop off in prepayment rates between current coupon and discount collateral is called lock-in. It refers to the tendency of borrowers with below-market mortgage rates to become locked in to their current house because moving would entail replacing their existing mortgage with a new one at the current (higher) market rate. The reason for the greater lock-in of jumbos is, once again, loan size. That is, the opportunity cost of relinquishing a below market mortgage rate is greater for borrowers with large loan sizes. For deep discounts, the greater lock-in of jumbos offsets their higher mobility, causing them to prepay similar to agencies.

Economic Conditions and Jumbo Turnover Current economic conditions and, in particular, economic conditions in California are another important factor affecting jumbo turnover. Close to half the loans in most jumbo pools are from California, compared with 25% or less for typical agency pools. This makes jumbo turnover highly dependent on economic conditions in that state. When the California economy is booming, jumbo turnover rates are high, and when it is in recession, jumbo turnover rates fall. Exhibit 7 provides examples of both cases. In recent years, the California economy has grown rapidly, EXHIBIT 7

a

Effect of California Economy on Jumbo Turnovera

–50 bp < Rate Incentive < 0 bp.



313

producing high turnover rates in California and, consequently, across most jumbo pools.5 In contrast, during the 1994–1995 mortgage rate backup, California was suffering through a severe recession not felt by the rest of the country. As a result, turnover rates in California were lower than across the rest of the country, causing seasoned jumbo turnover rates to fall below turnover rates on agency aggregates.

Valuation of Jumbo Securities The first panel in Exhibit 8 (“2002 Loan Sizes”) displays collateral information, prepayment projections and constant OAS valuation statistics for new issue 5.25% coupon passthrough securities backed by representative jumbo and agency collateral. The model prices presented here and in later sections differ from market prices for several reasons. First, the model prices reflect model prepayment projections, whereas market prices reflect the prepayment expectations and risk aversion of market participants. Second, the model prices do not capture differences in liquidity across markets. Nor do they capture the effect on security demand and, hence, prices of prevailing regulatory restrictions or performance measurement conventions (e.g., greater demand and higher prices for securities included in a common benchmark index). Nevertheless, the valuation exercises presented in this and later sections provide useful illustrations of how loan size, credit quality, prepayment penalties, and borrower horizon affect the valuation of mortgage-backed securities. In generating the prepayment projections and pricing information for Exhibit 8, the loan sizes, SATOs, and geographic distributions of the underlying collateral were chosen to be consistent with recent jumbo and agency originations. The WACs, in turn, were set equal to base case jumbo and conforming mortgage rates so that base case projections reflect current coupon prepayments. Accordingly, negative rate shift scenarios illustrate the effect on prepayments of refinancing, while positive rate shift scenarios indicate the dampening effect of lock-in on housing turnover. Turning to the prepayment projections in Exhibit 8, notice the higher base case turnover, greater callability, and greater lock-in of jumbos. Projected base case jumbo prepayments, for example, are 2.2% CPR higher than for agencies. This prepayment difference increases to 18.2% CPR in response to a 100-basis-point drop in mortgage rates (greater callability) and shrinks to 1.2% CPR in response to a 100-basis-point increase in mortgage rates (greater lock-in). One would expect the steeper refinanc5 According to the 1999–2000 Cal and non-Cal seasoning curves in Exhibit 7, about 25% (0.6% CPR) of the observed difference between jumbo and agency turnover rates in Figure 8 is attributable to differences in California concentrations.

314

450 150 300

290 105 185

2002 Loan Sizes: Jumbo 6.05 Agency 5.80 Difference 0.25

1993 Loan Sizes: Jumbo 6.05 Agency 5.80 Difference 0.25 25 0 25

25 0 25

SATO

73.2 52.1 21.1

75.4 58.8 16.6

–150

59.2 38.7 20.5

62.2 44.0 18.2

–100

34.7 19.6 15.1

37.4 24.9 12.5

–50

7.8 5.6 2.2

7.8 5.6 2.2

0 bp

5.7 4.2 1.5

5.7 4.2 1.5

+50

4.7 3.5 1.2

4.7 3.5 1.2

+100

1-Year Prepayment Projections by Interest Rate Shift Scenario

3.9 3.1 0.8

3.9 3.1 0.8

+150

99-08 99-31+ (0-23+)

99-06 99-26 (0-20)

Price

0 0 0

0 0 0

OAS

4.66 4.74 –0.08

4.64 4.68 –0.04

OAD

–2.0 –1.5 –0.5

–2.0 –1.6 –0.4

OAC

97 76 21

98 79 19

Opt Cost

Constant LIBOR OAS Valuation

Note: Run date is 9/20/02. Model pricing is for a new issue 5.25% coupon passthrough. Base case conforming and jumbo mortgage rates are 5.80% and 6.05%, respectively.

Size

Collateral Characteristics

Valuation of New Issue Jumbos

WAC

EXHIBIT 8




315

ing and lock-in profiles of the jumbo collateral to imply a lower model price for the jumbo passthrough, since both are bad from the standpoint of the passthrough holder.6 This is exactly what happens, with the jumbo passthrough requiring a ²⁰⁄₃₂ price drop relative to the agency passthrough. The need for a price concession can also be seen by comparing the convexity and option cost of the two securities. The jumbo passthrough is more negatively convex and has higher option cost than the agency passthrough and is, therefore, worth less. The second panel in Exhibit 8 (“1993 Loan Sizes”) reprices the passthroughs in the first panel using 1993 loan sizes in an attempt to illustrate the effect of increased agency loan sizes on the relative callability of jumbo and agency collateral and on the relative valuation of jumbo and agency securities. Consistent with the prepayment data in Exhibit 2, the reduction in jumbo loan size from $450,000 to $290,000 has a lesser effect on prepayment projections and valuation than the much smaller reduction in agency loan size from $150,000 to $105,000. The result is an increase in the relative callability of jumbos, as captured by the larger prepayment, convexity, and option cost differences, and a ³·⁵⁄₃₂ widening of the jumbo-agency price drop.

ALT-A PREPAYMENTS AND SECURITY VALUATION The story of alt-A prepayments is one of initial credit impairment and gradual credit curing. The initial credit impairment of alt-A collateral, highlighted earlier in Exhibit 1, could be signaled by a low FICO score, a failure to provide full documentation of income and assets, a high LTV or DTI, or some combination of factors. Because of this credit impairment, prepayments on new issue alt-As are less responsive to a drop in mortgage rates than what is typically witnessed in either the agency or jumbo markets. Over time, however, alt-A borrowers can cure their credit by maintaining a clean pay history, gathering documentation on income and assets, lowering their LTVs through housing price appreciation, or making unscheduled principal payments, potentially allowing them to refinance into a jumbo or agency mortgage at a lower rate. 6

A steep refinancing profile is bad because it implies that prepayments are highest at exactly the times when one does not want them to occur, that is, when mortgage rates are low and reinvestment of the prepaid proceeds yields a lower coupon rate than on the prepaid mortgage. Similarly, a steep lock-in profile is bad because it implies that prepayment rates are lowest at exactly the times when one does want them to occur, when mortgage rates are high and reinvestment of the prepaid proceeds would yield a higher coupon rate than on the prepaid mortgage.


316

EXHIBIT 9


Alt-A Credit Impairment and Credit Curing

This process of gradual credit curing affects alt-A prepayments in two ways. First, because alt-A mortgages are originated at a significant premium over prime jumbo or agency collateral (typically 80-100 basis points), credit-related refinancings help maintain high alt-A prepayment rates even in the presence of unchanged or rising mortgage rates. As a result, base case alt-A prepayments are generally much higher than on similar vintage jumbos or agencies. In addition, the curing of credit issues by alt-A borrowers, in combination with the burnout of agency and jumbo pools, reduces differences in callability between seasoned alt-As and similar vintage jumbos or agencies. This life cycle of alt-A prepayments is illustrated in Exhibit 9, which compares prepayments over time between conforming balance alt-As originated in 1997 with a 9.0% WAC to prepayments on a lower WAC cohort of agency collateral originated in the same year. During the mortgage rate rally of 1998, the still unseasoned, lower credit quality alt-As produced a much weaker prepayment response to the apparent refinancing opportunity, generating a peak 3-month prepayment rate of 32% CPR, compared with 48% CPR for the agency cohort. This despite having an 85-basis-point higher WAC than the agency cohort. Notice, however, that just as the presence of credit-impaired borrowers muted the increase in alt-A prepayments during 1998, credit curing and a high WAC helped moderate the decline in alt-A prepayments during the mortgage rate backup of 1999–2000. Even with this outflow of good borrowers, by the time of the next mortgage rate rally in early 2001, the credit quality of the remaining collateral underlying the alt-A cohort had improved enough to cut the observed prepayment difference in half.


317


SATO as a Measure of Credit Quality So how does one capture credit quality in a prepayment model? One way would be to try to model the direct effect on prepayments of available collateral characteristics such as FICO, LTV, documentation type, loan purpose, and the like. The main problem with this approach is the availability of such data. When loan-level data are available for publicly traded securities, they usually exclude important measures of credit quality such as DTI, mortgage pay history, job tenure, and income. Additionally, a general lack of consistency in the coding of nonnumeric variables such as documentation and occupancy type makes their inclusion somewhat problematic.7 Finally, for agency alt-A pools, one typically has no loan-level collateral information at all. In an attempt to avoid some of these pitfalls and to produce a model that has easily met data requirements, we use the spread paid by a borrower over mortgage rates prevailing at the time of origination, also referred to as the borrower’s SATO, as an indicator of his/her credit quality. In addition to imposing no data requirements beyond the WAC and WALA necessary to amortize a mortgage, SATO has the advantage of incorporating all indicators of credit quality that actually affect a borrower’s mortgage rate, even indicators that are not directly observable by secondary market participants. The data in Exhibit 10 confirm that SATO does a reasonable job of capturing differences in credit quality across collateral cohorts, as well as the effect on prepayments of these differences in EXHIBIT 10

SATO as a Measure of Credit Qualitya

Jumbo

Jumbo Alt-A

WAC Cut

WAC

Size ($ 000)

SATO (bp)

1-Yr CPR

% 60+ Delb

8.0 8.5 9.0

7.96 8.40 8.86

349 328 273

–15 19 55

52 66 66

0.03 0.10 0.47

9.0 9.5 10.0 10.5

8.90 9.43 9.88 10.37

379 379 394 379

72 118 162 211

54 56 57 57

0.85 2.73 4.99 7.87

a

January 2001–December 2001 prepayments for 2000 origination jumbos and jumbo Alt-As. b % 60 plus-day delinquent as of 12/31/00. 7

Many big issuers of alt-A securities are loan aggregators who pass along multiple originator-level documentation coding systems to investors.


318


credit quality. In particular, notice that higher SATOs are associated both with higher 60 plus-day delinquency rates and a flattening out of the prepayment profile. As for the curing of alt-A collateral, this is done in the model by allowing both seasoning and cumulative housing price appreciation to lessen the effect of initial credit impairment on alt-A prepayments.

Loan Size and Prepayment Penalties In addition to credit quality, loan size and prepayment penalties are the most important variables affecting the responsiveness of alt-A prepayments to observed refinancing opportunities. The average loan size of alt-A pools varies dramatically across vintages, issuers, and even collateral groups within a given deal. As in the prime markets, large loan balance jumbo alt-As display much greater callability than conforming balance alt-As. Looking at the prepayment curves in Exhibit 11, for example, one sees that over the course of the 2001–2002 mortgage refinancing wave, jumbo alt-As with nominal rate incentives of 50 basis points or more have prepaid anywhere from 15–25% CPR higher than conforming balance alt-As with similar rate incentives. Exhibit 11 also illustrates the effect of prepayment penalties on altA prepayments. The most prevalent form of prepayment penalty in the prime and alt-A markets entails a penalty term of 1, 3, or 5 years and EXHIBIT 11 Effect of Loan Size and Prepayment Penalties on Alt-A Prepayments January 2001–March 2002 Prepayments for alt-As with 0–12 WALA



319

requires the borrower to pay six months’ interest on 80% of the prepaid balance of a refinanced or curtailed mortgage.8 That is, Prepayment penalty = 6 × Mortgage rate/1200 × 0.80 × Prepaid balance For example, a borrower who refinances a $100,000 mortgage with a 7.0% note rate would have to pay a $2,800 penalty, whereas a borrower who refinances a $400,000 mortgage with a 10% note rate would have to pay a $16,000 penalty. There are two interesting features of this form of penalty as relates to its effect on refinancing. First, because the penalty is proportional to the note rate on a mortgage, it tends to flatten out refinancing curves. Second, because the penalty is proportional to the prepaid balance, it tends to reduce the effect of loan size on refinancing. A quick look at Exhibit 11 confirms that the prepayment curves for prepay penalty alt-As are much lower, flatter, and closer together than for the non-penalty collateral.

Valuation of Alt-As We now examine the effect of credit quality, loan size, and prepayment penalties on the valuation of alt-A securities. Exhibit 12 displays prepayment projections and constant OAS valuation statistics for new issue passthrough securities backed by jumbo alt-A and conforming balance alt-A collateral with and without prepayment penalties. The loan sizes and geographic distributions of the underlying collateral were chosen to be consistent with recent alt-A originations, while the WACs were all set to be 85 basis points above base case conforming mortgage rates, consistent with the spread at origination on 2001 jumbo alt-A originations.

Credit Quality Looking first at the prepayment projections in the first panel of Exhibit 12, one sees that base case jumbo alt-A prepayments are significantly higher than for jumbos. The reason for this, of course, is that the higher alt-A WAC means that base case alt-A prepayments include not only turnover, but refinancing activity as well. For example, the base case jumbo alt-A prepayment rate of 23.5% CPR reflects the fact that over the course of the year, some borrowers will improve their credit characteristics enough to refinance into standard jumbo mortgages at significantly lower rates. Even 8

Prepayment penalties that include home sales are referred to as hard penalties, as opposed to soft penalties which exclude home sales. Some penalties may be hard for a short time after origination, typically 12 months, and then soft for the remainder of the penalty term. Most penalties in the prime and alt-A mortgage markets are soft or hard-into-soft penalties.

320 450 130 320

Prepayment Penalties:a Jumbo Alt-A 6.65 Conforming Alt-A 6.65 Difference 0 85 85 0

85 85 0

85 25 60

–100

26.3 23.2 3.1

57.7 47.6 10.1 20.7 14.8 11.8 8.7 18.2 13.5 11.2 8.5 2.5 1.3 0.6 0.2

52.0 40.5 23.5 8.9 38.8 28.3 15.8 8.6 13.2 12.2 7.7 0.3 6.7 6.5 0.2

6.7 6.5 0.2

6.7 4.7 2.0

5.6 5.4 0.2

5.6 5.4 0.2

5.6 3.9 1.7

–50 0 bp +50 +100 +150

57.7 52.0 40.5 23.5 8.9 75.4 62.2 37.4 7.8 5.7 –17.7 –10.2 3.1 15.7 3.2

–150


100-13 100-21+ (0-08+)

99-20+ 99-31+ (0-11)

99-20+ 99-06 0-14+

Price

0 0 0

0 0 0

0 0 0

5.10 –1.2 5.24 –1.1 –0.14 –0.1

4.47 –2.2 4.71 –1.8 –0.24 –0.4

4.47 –2.2 4.64 –2.0 –0.17 –0.2

86 81 5

100 93 7

100 98 2

OAS OAD OAC Opt Cost


Note: Run date is 9/20/02. Model pricing is for a new issue 5.25% coupon passthrough. Base case conforming and jumbo mortgage rates are 5.80% and 6.05%, respectively. a Assumes 5-year prepayment penalty term.

450 130 320

6.65 6.65 0

Loan Size: Jumbo Alt-A Conforming Alt-A Difference

450 450 0

6.65 6.05 0.6

WAC Size SATO


Valuation of New Issue Alt-As

Credit Quality: Jumbo Alt-A Jumbo Difference

EXHIBIT 12




321

if jumbo mortgage rates remain unchanged at 6.05%, these borrowers face a potential rate incentive of 60 basis points. Notice, however, that the model does not treat this 60-basis-point rate incentive the same way that it would for standard jumbo collateral. In particular, the base case jumbo alt-A prepayment rate of 23.5% CPR is 18.9% CPR lower than where jumbos would prepay in a –60-basis-point rate environment. Thus, the poorer credit quality of the jumbo alt-As, as captured by a higher SATO, makes them prepay lower than standard jumbos at any given level of rate incentive. Moreover, as further comparison of the prepayment projections in the first panel of Exhibit 12 reveals, the poorer credit quality of jumbo alt-As also implies a flatter refinancing profile relative to the jumbos, similar to what is seen in the actual prepayment data. In terms of pricing, the preferable prepayment characteristics of the jumbo alt-As, that is, higher prepayments in up rate scenarios and lower prepayments in down rate scenarios, generate a model payup of ¹⁴·⁵⁄₃₂ over the jumbo passthrough.

Loan Size The second panel of Exhibit 12 illustrates the impact of loan size on alt-A prepayment projections and security valuation. Similar to what was seen in the valuation exercise comparing jumbos to agencies, the greater callability of jumbo alt-As requires a price concession versus the conforming alt-A collateral. Specifically, the higher prepayments of the jumbo alt-As in the base case and down rate scenarios make the passthrough backed by this collateral more negatively convex than the passthrough backed by conforming balance alt-As. The jumbo alt-A passthrough would, therefore, need to trade at an ¹¹⁄₃₂ price concession in order to offer the same OAS as the conforming alt-A passthrough.

Prepayment Penalties The last panel of Exhibit 12 examines how prepayment penalties influence the value of alt-A securities. Recall that in addition to lowering overall prepayments, prepayment penalties tend to flatten refinancing curves and reduce the effect of loan size on prepayments. These features of the data are also present in model projections. Specifically, (1) the prepayment projection profile of prepay penalty alt-As is significantly flatter than for nonpenalty collateral, (2) base case and down rate scenario projections for prepay penalty jumbo alt-As are lower than for conforming balance alt-As with no penalty, and (3) in base case and down rate scenarios, jumbo alt-A versus conforming alt-A prepayment differences are lower for prepay penalty collateral than for non-penalty collateral. Accordingly, the model prices prepay-penalty jumbo alt-As at a premium to nonpenalty alt-As, both jumbo and conforming balance.


322


It also produces a jumbo alt-A versus conforming alt-A price concession that is tighter for prepay penalties than for nonpenalty collateral. It is important to keep in mind, however, that depending on the path of mortgage rates after origination, the payup for new issue prepay penalty collateral can easily become a price concession by the time the penalty term expires. The reason is that after accounting for differences in origination costs, a mortgage pool backed by prepay penalty collateral may be exposed to far less compelling refinancing opportunities than a similar pool of nonpenalty mortgages. More precisely, burnout on the prepay penalty pool will be no higher than on the non-penalty pool and will be strictly lower if the non-penalty pool has been exposed to refinancing opportunities any time prior to the penalty expiration date.

JUMBO RELO PREPAYMENTS AND SECURITY VALUATION The primary difference between new issue relo pools and standard jumbos is that relo pools contain a higher proportion of short-horizon borrowers. This seemingly minor difference in collateral composition causes a noticeable divergence in both turnover rates and refinancing behavior. In particular, the short horizon borrowers in relo pools increase base case turnover rates on these pools relative to jumbos. Also, because borrowers who plan to move soon are less likely to act on refinancing opportunities, new issue relo pools display less callability than otherwise similar jumbos. As relo pools season, however, these differences narrow. In particular, the exit of short-horizon borrowers from relo pools and burnout of jumbo pools can eliminate any differences in callability between sectors. Exhibit 13 illustrates these points with a comparison of prepayments on 1997 origination 7.5% WAC jumbos and relos. Despite having similar WAC and loan size, the relos displayed a more muted prepayment response to the mortgage rate rally of 1998 and maintained high turnover rates throughout the mortgage rate backup of 1999–2000. By the beginning of 2001, however, enough short-horizon borrowers had exited the relo cohort that its prepayment response to the ensuing mortgage rate rally was virtually identical to that of the seasoned jumbos.

Valuation of Jumbo Relos Exhibit 14 displays model prepayment projections and passthrough pricing for new issue relos relative to jumbos. In practice, new issue relo securities would have a 50 basis points lower coupon than same vintage jumbos because of the initial rate discount received by relocation borrowers. However, for ease of illustration, we consider relos and jumbos



EXHIBIT 13

323

Effect of Borrower Horizon on Jumbo Relo Prepayments

having the same WAC and passthrough coupon. As with earlier pricing exercises, the loan sizes and geographic distributions of the underlying collateral were chosen to be consistent with recent originations, and the WACs were set equal to base case jumbo mortgage rates. Thus, similar to the jumbo-agency pricing exercise earlier, base case projections reflect differences in current coupon prepayments, with negative rate shift scenarios illustrating differences in refinancing and positive rate shift scenarios illustrating differences in lock-in. Consistent with the historical prepayment experience of relos, the prepayment projections in Exhibit 14 indicate higher base case turnover, less lock-in, and less callability than for jumbos. These preferable prepayment characteristics, reflected in the better convexity and lower option cost of the relo passthrough, translate into a significant payup for the relo passthrough.

CONCLUSION Aside from loan age and rate incentive, the most important factors driving differences in prepayments across and within the various nonagency sectors are loan size, borrower credit quality, prepayment penalties, borrower horizon, and geographic distribution. The lack of uniformity in these collateral characteristics across sectors, vintages, issuers, and even collateral groups within a given deal requires that a prepayment model explicitly incorporate them into its predictions if it is to be of any use in valuing nonagency securities. This is especially true of securities backed by alt-A collateral, for which loan size can range from jumbo to conforming, credit quality can vary from prime to subprime, and prepay penalty concentrations can be anywhere from 0–100%.

324 6.05 6.05 0

WAC 450 450 0

Size 25 25 0

SATO


59.4 75.4 –16.0

–150

–50

47.5 28.2 62.2 37.4 –14.7 –9.2

–100 8.8 7.8 1.0

7.5 5.7 1.8

6.8 4.7 2.1

0 bp +50 +100


Valuation of New Issue Jumbo Relos

6.2 3.9 2.3

+150

101-17+ 99-06 2-11+

Price

0 0 0

OAS

3.38 4.64 –1.26

OAD

–1.7 –2.0 0.3

OAC


83 98 –15

Opt Cost

Note: Run date is 9/20/02. Model pricing is for a new issue 5.25% coupon passthrough. The base case jumbo mortgage rate is 6.05%.

Jumbo Relo Jumbo Difference

EXHIBIT 14


16-Bhattacharya Page 325 Wednesday, July 23, 2003 10:18 AM

The Role and Performance of Deep Mortgage Insurance in Subprime ABS Markets Anand K. Bhattacharya, Ph.D. Managing Director Countrywide Securities Corporation Jonathan Lieber First Vice President Countrywide Securities Corporation

he traditional role of private mortgage insurance in mortgage lending has been to protect lenders against losses resulting from real estate foreclosures. Historically, mortgage insurance (MI) has been an important component of the residential mortgage market. MI providers have written policies for potential homebuyers who have not had the ability or willingness to take out a mortgage with a loan-to-value ratio (LTV) of 80% or lower. In these cases, lenders have typically required a mortgage insurance policy to offset the amount of the loan over the 80% LTV level. This has been necessary in order to meet the requirements of the government-sponsored enterprises (GSEs), who have required this insurance for loans with LTVs greater than 80%. While the usage of mortgage insurance at the loan level to insure higher LTV loans against losses is fairly common, versions of this form of credit protection mechanism have also been used in structured transactions. Historically, mortgage insurance in the form of pool insurance has been used in the

T

325


326


conforming and jumbo markets. In this manifestation of credit protection, pool insurance covers losses in the entire mortgage pool up to a specified amount. It is only recently that a variant of mortgage insurance, known as deep mortgage insurance, has been used in subprime structured markets to cover foreclosure-related losses at the loan level. Prior to 2000, the usage of mortgage insurance was unheard of in the subprime mortgage market. Yet, by the middle of 2001, about half of all new home equity loan (HEL) issuance used deep MI as a component of credit enhancement, although this pace tailed off by year-end. MI continues to be a factor in this market; approximately 32% of new-issue deals in the fourth quarter of 2002 were structured with deep MI.1 The usage of deep MI in the structured markets coincides with a trend on the part of MI providers to diversify the revenue stream. Historically, providers of mortgage insurance had relied heavily upon the GSEs for a large portion of their revenues and to a lesser extent upon banks and thrifts. However, since 1999, there have been a few developments that may have led the MI providers to decrease their reliance on the GSEs. First, the GSEs changed their requirements for mortgage insurance on certain loan programs, with the implication that the GSEs were seeking to reduce their reliance on mortgage insurance and keep more of this risk on their own books. Second, the Office of Federal Housing Enterprise Oversight (OFHEO) has proposed some changes to its risk-based capital guidelines for the GSEs; the proposal would give favorable capital treatment to loans insured by AAA rated mortgage insurers. As a result, those MI providers that are not AAA rated would be at a distinct disadvantage with respect to GSE business. In this article, we will explain the mechanics of mortgage insurance, identify the major providers of MI, and consider the reasons for its increasing use in the subprime market. As part of this exercise, we also will compare HEL structures with and without deep MI and highlight the performance to date of MI as a form of credit enhancement in the subprime MBS market.

THE MECHANICS OF DEEP MI In terms of the manifestation of the credit protection, MI in subprime ABS deals insures a portion of the value of a number of specific loans in a pool of collateral. Each insured pool has an individual policy; there is no cross-collateralization of MI across an entire pool of loans. This is in 1

“Trends in Residential Mortgage Products: Fourth-Quarter 2002 LTV Ratios, FICO Scores, and Credit Support Levels,” Standard and Poor’s, January 31, 2003.


The Role and Performance of Deep Mortgage Insurance in Subprime ABS Markets

327

contrast to the use of pool insurance on jumbo mortgage deals used in the early 1990s. MI as used in this sector also differs from traditional MI policies in two important ways. First, the issuer usually pays for the insurance instead of the borrower with premiums typically paid for out of the excess spread of the entire securitization. Note that since the issuer is also typically the lender, this form of insurance also differs from lender-paid MI, where the borrower pays a higher mortgage rate to the lender, part of which is used by the lender to insure the losses on the mortgage. Second, the level of insurance is greater than the traditional use of MI (which is coverage down to 70% to 80% LTV); coverage of losses in lender-paid subprime issues usually goes down to a 50% or 60% LTV—hence, the term deep MI. It is instructive to understand the mechanics of deep MI with respect to loss coverage in subprime structured deals. When a loan that is covered by MI defaults, the insurer essentially has two options. The first option is to buy out the loan and acquire the property in the belief that the property could be sold at a favorable price. In some cases, the insurer may work with the lender to facilitate a sale. The second option is merely to pay the claim amount. The insurer must pay the lesser of the actual loss or the amount of the claim as calculated by the level of coverage and the unpaid balance of the loan. An example of deep MI coverage in different default scenarios is demonstrated below in Exhibit 1. The first important concept as it relates to deep MI is the coverage percentage. Coverage percentage quantifies the percentage of the original loan’s LTV that is protected from losses by the MI policy. Mathematically, this is defined as Original LTV – “Covered down to” LTV Covered percentage = ---------------------------------------------------------------------------------------------------------Original LTV To highlight this, in Exhibit 1, for scenarios 1 and 2 the coverage percentage is (80% – 50%)/80% = 37.5%. Therefore, the first 37.5% of the original loan balance is covered against losses by the MI policy. In scenario 3 the coverage percentage is (80% – 60%)/80% = 25%, and in scenario 4 the coverage percentage is (90% – 60%)/90% = 33%. As would be expected, the lower the “covered down to” LTV, the greater the coverage percentage (e.g., a loan with coverage down to 50% has a greater coverage percentage than the same loan with coverage down to 60%). Coverage percentage is a critical statistic because it indicates the level of benefit the policyholder will receive in case of a default. For example, scenarios 1 and 3 in the above example are identical in every way except the coverage percentage. Because of this difference in cover-


328

EXHIBIT 1


Calculating Mortgage Insurance Coverage Scenario 1 Scenario 2 Scenario 3 Scenario 4

Property: Original Property Value Original Loan Amount Current Loan Amount Original LTV MI Policy: Coverage Down to LTV Coverage Percentage Sale of Home: Gross Proceeds Accrued Interest (A/I) Other Expenses Net Sale Proceeds Loss Before MI MI Coverage = Coverage % × (Loan Balance + A/I + Expenses) Loss After MI

$150,000 $120,000 $100,000 80%

$150,000 $120,000 $100,000 80%

$150,000 $120,000 $100,000 80%

$150,000 $135,000 $100,000 90%

50% 37.50%

50% 37.50%

60% 25%

60% 33%

$75,000 $11,000 $5,000 $59,000 $41,000

$50,000 $11,000 $5,000 $34,000 $66,000

$75,000 $11,000 $5,000 $59,000 $41,000

$75,000 $11,000 $5,000 $59,000 $41,000

$43,500

$43,500

$29,000

$38,667

$0

$22,500

$12,000

$2,333

Source: Countrywide Securities Corporation.

age, the loss in scenario 3 (with a smaller coverage percentage) is $12,000, while in scenario 1 the coverage percentage is sufficiently high so as to avoid any loss. Thus, the greater the coverage percentage, the greater will be the benefit from MI. Note that the coverage percentage increases if the LTV is covered down to a greater degree (scenario 1 versus scenario 3) or if the original LTV of the loan is higher with the same covered down to percentage (scenario 3 versus scenario 4). It is also important to point out that, although it may be self-evident, the realized loss (if there is any) after MI is paid out will be affected by the severity of the decline in the value of the property (scenario 1 versus scenario 2). For two loans with the exact same level of coverage, the difference between whether or not a loss is realized after the payment of the MI policy often will be a factor of the market value of the property. If the loss on the sale of the property is severe enough, there may still be a loss despite the presence of deep MI. This will become important to remember as we consider the use of deep MI in ABS securitizations: The number of loans that default is not nearly as important as the magnitude of the loss on the defaulted loan.



329

MORTGAGE INSURANCE PROVIDERS The mortgage insurance industry is dominated by a small number of firms. The seven major MI providers are: Mortgage Guarantee Insurance Corp. (known as MGIC), PMI Mortgage Insurance, United Guaranty Corp., Radian Guaranty, Inc., GE Capital Mortgage, Republic Mortgage Insurance Corp., and Triad Guaranty Insurance Corp. According to data published by Inside Mortgage Finance,2 MGIC has been the market-share leader. In 2002, MGIC had $83.5 billion in primary MI written, for a 24.8% market share. PMI and United Guaranty were not far behind, with market shares of 18.0% and 15.2%, respectively. United Guaranty showed the largest increase in MI written from 2001 to 2002, with a growth rate of 40.2%. The two smallest firms in terms of market share—Republic Mortgage Insurance and Triad Guaranty Insurance (10.3% and 3.7% market share in 2002, respectively)— also had impressive growth in 2002, with growth rates of 37.7% and 22.2% respectively. In aggregate, primary MI written in 2002 by these seven companies was about $337 billion, about 19.1% greater than the total written in 2001. This is what would be expected in an economy that has been driven by explosive growth in the real estate market. These seven largest MI providers have varying degrees of business risk exposure to their mortgage insurance business. GE Capital Mortgage, United Guaranty, and Republic Mortgage Insurance are all subsidiaries of larger financial institutions (GE Capital, American International Group, and Old Republic, respectively) that are fairly diverse across product lines. In contrast, MGIC, PMI, Radian, and Triad have much more concentrated exposure to their MI business (although PMI does generate good revenues in title insurance). The credit quality of these firms is very strong according to the ratings agencies. Moody’s rates the claims-paying ability of all of these companies Aa3 or better; GE and United Guaranty (AIG) are both rated Aaa. These high ratings are critical for MI providers, especially within the context of the securitized MBS and ABS markets. MI cannot be considered a valid form of credit enhancement if the underlying insurance provider is not a very sound financial institution that can be expected to actually fulfill its claims-paying obligations. These companies have all maintained extremely good ratings stability over a prolonged period of time—only PMI has seen its ratings reduced (from Aaa to Aa2) in the past ten years. Given their historical track record and the need for MI providers to maintain sound credit ratings, it is not unreasonable to conclude that this history of ratings stability is likely to be maintained. 2

Source of data is Inside Mortgage Finance, February 7, 2003.


330


MORTGAGE INSURANCE VERSUS MONO-LINE INSURANCE Mortgage insurance differs in some important ways from mono-line insurance that has historically been used as a credit enhancement device in the HEL sector. Deep MI is a more limited form of insurance, both in terms of the scope of coverage on any particular loan and across an entire pool of loans. In addition, there are limits and exceptions to many MI policies that are not common in mono-line insurance. Since monoline insurance is the common form of insurance in the securitized HEL sector, it is important to contrast deep MI with mono-line insurance as a form of credit enhancement.

Depth of Coverage In the comparison of deep MI to mono-line insurance, the first obvious difference is the depth of coverage provided by each. Mono-line insurance basically protects an investor from any losses in the pool of assets; the coverage is generally a blanket assurance of “timely payment of interest and ultimate payment of principal.” In contrast, deep MI provides coverage only on specific loans in a pool of assets—it is not a blanket insurance policy on the entire pool. Most securitized deals need to have a large percentage of loans covered by deep MI to approach the depth of coverage provided by mono-line insurance in this area. Furthermore, deep MI only provides coverage to a certain level of losses on each insured loan; losses beyond the coverage level are not covered via MI. High loss severity, therefore, could still create losses to the investor, and thus the pool needs to be credit-enhanced by other means to create a AAA rating. However, the evaluation process in the deep MI insurance is customized to the actual parameters of the loan, while the blanket mono-line insurance coverage has typically been based upon pool level parameters, with adjustments for underwriting exceptions and outliers. In view of this observation, it appears that deep MI coverage is more likely to be tailored to the specific attributes of the loans.

Exclusions and Adjustments While mono-line insurance is basically an unconditional guarantee, MI policies are subject to exclusions and adjustments in specific cases. The provider of an MI policy may decline to make a payment on a policy in the case of fraud with respect to the defaulted mortgage. Fraud may occur in several ways, including misrepresentations made by the borrower or a flawed appraisal process. A claim may also be excluded in the event that there has been a breach of obligations on the part of the borrower or lender. Loans in default before the effective date of cover-



331

age are also excluded. Additionally, special hazards—losses due to earthquakes, war, and “acts of God”—are not covered by MI and must generally be supplemented by hazard insurance. However, in this respect deep MI coverage is not very different from mono-line insurance, where additional hazard protection also typically is required. There are other factors that may cause an MI claim to be excluded; these may vary in their specifics from policy to policy. In addition, MI claims may be adjusted, which would result in a lower payment than may have originally been indicated by the policy. Mortgage insurers can adjust a claim downward in case the servicer has not acted to minimize property losses. MI providers require that a servicer take specific actions within prescribed time periods so as to maximize resale value, and if these rules are not met, there may be claims adjustments. Data on claim adjustments and exclusions are not readily available due to the proprietary nature of this information. However, it is not unreasonable to argue that MI providers are likely to be more flexible in the claim evaluation process with originators with whom they transact substantial amounts of business. This may result in a higher amount of “split” claims (where the loss is shared equally by the guarantor and the originator) rather than in outright rejections, ultimately leading to lower severity for the originator in question.

Ratings Differential Mono-line insurance companies are rated almost exclusively AAA, while the majority of the MI providers are rated AA. Investors may take greater comfort with counter party credit exposure to a higher-rated institution, even though, as a practical matter, the rating difference is modest. Indeed, as we have noted, the MI companies have been very consistent in maintaining their ratings, and there is no reason to expect deterioration in this trend. In view of this consideration, deep MIinsured deals also provide investors the opportunity to diversify their credit exposure to alternative credit counter parties, particularly if the current portfolio exposure to mono-line insurers is high. This observation is particularly relevant as structures that use deep MI also employ other forms of credit enhancement to deal with this AA guarantee (as well as with the other shortcomings of MI) so as to gain AAA ratings on senior classes of MI-backed HEL deals.

Exposure to Different Markets With respect to diversification of counterparty credit exposure, the monoline insurers and mortgage insurers have exposures to somewhat different business risks as a result of historical involvement in different lines of busi-


332


ness. The mono-line companies have made their niche by providing insurance mostly to subprime markets, especially subprime mortgages. Until recently, mortgage insurers have had relatively little exposure to subprime borrowers and instead have been more exposed to the traditional mortgage borrower market. Although this is changing as MI providers attempt to enter the subprime mortgage arena, these two types of insurance providers are exposed to different default and risk parameters due to underlying differences in the fundamentals of the prime and subprime mortgage markets.

HEL STRUCTURES—WITH AND WITHOUT DEEP MI Since its introduction in the subprime ABS market in 2000, deep MI has presented another option for issuers in the structuring of new-issue subprime ABS. Prior to its introduction, subprime mortgage issues were generally structured either with an insurance wrap from a mono-line insurance company or with some combination of senior/subordination, over-collateralization, and cash reserve accounts. However, as noted above, deep MI must be supplemented with some other forms of enhancement to account for the fact that deep MI is a limited form of coverage against losses. Examples of different types of deal structures are presented below.

Mono-Line Wrap Structure The “simplest” structure for subprime mortgage issues is where the monoline insurance wrap provides the credit enhancement. This insurance policy, which is offered by a small number of AAA-rated financial firms, guarantees timely payment of interest and ultimate payment of principal. Investors look to the insurer for protection against any and all losses in the collateral. There is often no other form of credit enhancement except for the excess spread in the deal and occasionally a small amount of over-collateralization, which may start at zero and grow to a stated (albeit small) amount over time. Clearly, in this structure, the investor is dependent upon the creditworthiness of the insurer to protect against losses in the collateral. So far, the firms that offer this protection (e.g., MBIA, AMBAC) have long-term track records that are virtually spotless within this arena. Nevertheless, with respect to such structures, investor concerns may arise regarding reaching limits with respect to counter party credit exposure to a particular insurer or a group of insurers.

Senior/Subordinated Structure The traditional alternative to an insurance wrap in the structured products market has been a senior/subordinated structure. These kinds of



333

structures depend upon an estimation of expected losses and the correct sizing of subordination (plus any other forms of credit enhancement) to provide a senior rating of AAA. Senior/subordinated structures offer a very different type of risk exposure. There is no direct corporate insurance obligation on a subordinated deal; instead, the structure is selfinsured based upon the expected performance of the collateral and the structural integrity of the deal. The specific amount of credit enhancement needed to support a AAA senior rating is determined by the ratings agencies and varies from deal to deal depending on the agencies’ evaluation of the expected performance of the collateral. For example, according to Standard and Poor’s, the average level of credit enhancement required for the AAA rating on senior subprime tranches in the fourth quarter of 2002 was 19.62% for fixed rate collateral and 23.58% for adjustable rate collateral. This amount of credit enhancement may be in some combination of subordinated classes, over-collateralization, or cash reserve accounts. From the point of view of the issuer, while strategic considerations—such as maximization of securitization proceeds and cost—dictate the choice of mono-line insurance versus subordination, there are some general observations that can be made. Typically, mono-line insurance wraps are used by newer participants in the markets or until an asset class matures. On the other hand, as historical performance about collateral becomes available, which typically is associated with either the maturation of the issuer and/or the asset class, subordination of cash flows tends to become the structure of choice. Alternatively, senior/subordinated issuers might be forced to use mono-line insurance during periods when demand for rated subordinated risk is weak.

Deep MI Structures Deals structured with deep MI can be classified as a hybrid between insured deals and senior/subordinated structures. To highlight the manner in which MI is used in subprime ABS deals, a recent subprime issue from Countrywide Home Loans, Inc. is used as an example. The deal— CWL 2002-BC1, which was priced early in the first quarter of 2002—is collateralized with a combination of fixed and adjustable rate loans (approximately 26% fixed, 74% adjustable at the time of closing). Based on the average enhancement levels noted above, a senior/subordinated structure for this deal would have needed somewhere in the 22% range of credit enhancement to achieve a senior class rating of AAA (the actual level would be determined by specific collateral characteristics). However, in this deal, the trust acquired deep MI (through MGIC) for loans in the pool with original LTVs in excess of 50% for 98% of the


334


pool. As a result of the MI, only 5.5% of additional credit enhancement was necessary to achieve a senior rating of AAA. This 5.5% enhancement was achieved through the use of mezzanine and subordinated tranches, with the mezzanine tranches (one AA-rated and one A-rated) receiving support from the subordinated tranches (2.5%) as well as from the deep MI coverage. Thus, the inclusion of deep MI in this deal reduced the other forms of credit enhancement significantly. One twist on the deep MI structure involves the use of a corporate guaranty as supplemental credit enhancement. In this variant, the corporate guaranty covers losses that are not covered after applying all net liquidation proceeds and any proceeds from MI policies. The amount of the corporate guaranty generally decreases over time as it is used and is not replenished. In most of these structures, the corporate guaranty takes the place of other forms of enhancement beyond deep MI and subordination (no reserve accounts, over-collateralization, etc.) The corporate guaranty introduces incremental credit risk to all classes of the deal but most directly to the mezzanine and subordinated tranches. The rating agencies consider the credit rating of the guarantor in assigning ratings to the subordinated tranches; lower-rated tranches could be subject to a ratings change if the debt rating of the guarantor were to change.

PERFORMANCE OF MI TO DATE As noted earlier, deep MI as a means of credit enhancement is a recent innovation in the subprime ABS market. However, since the acceptance of this technology in the market, there is about two years of data available on subprime structures that have loans with deep MI in the ABS market. In order to assess the performance of deep MI structures, the loan characteristics of securitized loans with MI versus loans without MI in the subprime market are compared in Exhibit 2. Following this analysis, differences in historical performance of subprime loans with deep MI versus loans without MI are described in Exhibits 3 and 4. EXHIBIT 2

No MI With MI

Subprime Loan Characteristics With and Without MI, 2000–2002

WAC

WAM

Wtd LTV

Wtd FICO

Full Doc.

Low Doc.

No Doc.

9.52% 9.13%

314 mo. 336 mo.

77.85% 84.55%

605.26 615.2

72.77% 74.97%

26.07% 24.74%

0.33% 0.25%

Source: LoanPerformance (LP).



EXHIBIT 3

335

Credit Performance of Subprime Loans With and Without MI,

2000–2002 60+ 1-Month 12-Month Loss Delinquency Foreclosure REO Bankruptcy CDR CDR Severity No MI With MI

5.14% 4.37%

4.92% 4.13%

1.24% 1.02%

2.28% 1.15%

2.25% 1.93%

2.15% 1.83%

41.35% 31.50%

Source: LoanPerformance (LP).

EXHIBIT 4

Deep MI versus Non-MI, 2000–2001

Source: Moody’s Investors Service.

Exhibit 2 highlights the differences between securitized subprime loans with and without deep MI since the start of 2000.3 The objective of this analysis is to assess whether loans with MI have inherent differences that may lead to performance discrepancies. As noted in the data, the two cohorts do not exhibit significant differences in documentation styles. Loans with MI have a lower aggregate weighted average coupon (WAC) (39 basis points lower) and longer weighted average maturity (WAM) (22 months longer) than loans without MI. The timing of the deals and the usage of deep MI in securitized structures can explain both of these differences. MI became an accepted source of credit enhancement in subprime ABS in 2000, and most deals in the early part of 2000 did not use deep MI. Deep MI became a larger component of 3

The source of the data is Loan Performance.


336


issuance later in this time period as mortgage rates were trending lower4 (fixed rate HEL coupons declined by about 130 basis points from 2000 to 2001, and hybrid 2/28 subprime WACs declined by 75 to 100 basis points). Therefore, the lower WAC/longer WAM profile of deep MI loans is almost exclusively a function of the timing of the rise of MI in this time period. Finally, FICO scores appear to have been somewhat higher on loans with MI. The other distinction uncovered by this analysis is the difference in weighted average LTVs. Although this differential is not exceedingly large, the data do show that subprime loans that are backed by MI have had higher LTVs (84.55% versus 77.85%) than those loans without MI. This would be expected, as the key use of mortgage insurance is to insure those high-LTV loans against losses down to an acceptable level of LTV. However, since these subprime loans already tend to be “highLTV” in nature, the differential in LTV is not particularly large. A much greater difference in LTV may be found in the jumbo MBS market; data for jumbo MBS dating back to 1997 show that the average LTV for loans backed by MI is about 90% versus an average LTV of only 68% for those loans not backed by deep MI (data from Loan Performance). The results of this analysis suggest that there does not seem to be any overwhelming “selection bias” on the part of MI providers (or underwriters) to insure only a certain subset of subprime loans in a pool (e.g., only those loans with much higher FICO scores, etc.). The performance of securitized subprime loans backed by deep MI versus uninsured loans is highlighted in Exhibit 3. The data do show some noticeable differences. The most important benefit of deep MI as it relates to performance of subprime loans is in the area of loss severity. Because there does not seem to be any obvious bias in the credit quality of loans that carry deep MI in any particular pool of loans, the incidence of losses across MI versus non-MI loans should be about the same. However, the severity of losses on loans backed by deep MI should, on average, be less than the loss severity experienced by loans without deep MI. Deep MI should allow for a greater recovery of defaulted loans, even in cases where deep MI does not fully offset realized losses. There is still incremental recovery via deep MI down to the coverage percentage. Unless the number of exclusions and exceptions to MI is excessively high, loss severity should be lower for loans backed by MI. Exhibit 3 shows that this expectation has so far been realized, as the loss severity for loans backed by deep MI has been about 10% less than the loss severity for subprime loans without deep MI (31.50% versus 41.35%). This outcome is consistent with histor4

“2001 Review and 2002 Outlook: Home Equity ABS: Record Issuance Volume Dwarfs Prior Years,” Moody’s Investors Service, January 22, 2002.



337

ical data from the jumbo MBS market—loss severity on loans with deep MI in this sector has also been about 10% lower than loss severity for uninsured loans going back to 1997. Therefore, the decrease of loss severity is, perhaps, the greatest benefit of deep MI.5 The benefit of this reduction in loss severity may be seen in the cumulative loss performance of securitized subprime ABS issues. Exhibit 4 shows the cumulative loss performance of two aggregated groups. One group displays the aggregate cumulative loss performance from Countrywide subprime issues backed by deep MI from 2000–2001. This is compared to a subset of deals from other major issuers with no MI. Over this two-year time frame, the group with deep MI as credit enhancement shows lower cumulative losses than the group without deep MI (about 1% lesser losses). This is consistent with the intuitive goal of deep MI, which is to minimize loss severities and ultimately decrease losses in the pool. The lower loss severities noted above have the effect of translating into lower cumulative losses. Other performance differences (see Exhibit 3) are less intuitive. For example, the data show that the deep MI cohort has better historical performance in delinquencies, foreclosures, REO, and bankruptcies compared to uninsured subprime loans. Since there does not appear to be an obvious difference in the credit profile of MI versus non-MI loans in the subprime sector, these performance differences are not easily explainable and may be a function of either (1) the relatively short nature of the data available, or (2) somewhat higher FICO scores in the MI class. We note that the delinquency and foreclosure/REO experience in the jumbo MBS market since 1997 is actually superior for loans that do not have MI. This is probably the result of a combination of a longer history of price appreciation, lower LTVs, and noticeably higher average FICO scores for loans without MI in the jumbo market (731 versus 703, on average6—the reverse of the FICO experience in the subprime market noted earlier).

CONCLUSION As part of ongoing innovation in structured technology, recent structures used to monetize subprime loans have included a variant of mortgage insurance, labeled deep MI. This form of insurance coverage against 5

Note that the reduction in loss severity is attained at the deal level and therefore represents a positive attribute for deep MI-insured structures. However, this does not reflect the aggregate loss severity of the loans in the structure, a portion of which is borne by the MI provider. 6 The source of the data is Loan Performance.


338


defaults insures a portion of a number of specific loans in a pool of collateral. The coverage is loan-specific and typically deeper than traditional mortgage insurance. There is no cross-collateralization of MI coverage in the pool and premiums for the coverage are paid excess spread from the deal. In the absence of any strategic considerations, from the viewpoint of the issuer, the usage of deep MI is justified as long as the additional proceeds from lower subordination levels are greater than the cost of the insurance. While the usage of this technology is a fairly recent phenomenon in the structured markets, empirical data indicate that deep MI is successful in reducing the severity and the losses associated with the deal. While initial results are promising, the continued usage of this method of credit enhancement is heavily dependent upon the willingness of the MI providers to continue to underwrite such structures at costeffective prices.

17-Frank-GNMA Page 339 Wednesday, July 23, 2003 10:23 AM

Some Investment Characteristics of GNMA Project Loan Securities Arthur Q. Frank, CFA Director of MBS Research Nomura Securities International, Inc. James M. Manzi Analyst, MBS Research Nomura Securities International, Inc.

he National Housing Act of 1934 created the Federal Housing Administration (FHA) to encourage the construction, rehabilitation, and purchase of single-family and multifamily housing by backing certain loans with federal mortgage insurance. Today, the FHA insures mortgages for public, nonprofit, and private borrowers to build, improve, or buy multifamily housing projects. FHA insurance now covers a broad range of multifamily properties, including rental apartment complexes (many targeted toward low- and moderate-income families), condominiums and cooperatives, nursing homes, assisted living facilities for the elderly, and hospitals and other health care units. The GNMA multifamily mortgage market (also called the project loan market), representing the subset of FHA project loans that are secu-

T

The authors acknowledge the contributions of James Frohnhofer, who did the programming for this research effort, and of Nathaniel Jacob, our summer intern, who developed our GNMA project loan database.

339


340


ritized into GNMA pools, has also grown dramatically in recent years, to 4,145 pools outstanding having a total current principal balance of $22.885 billion (as of March 31, 2003). In this article we provide an overview of prepayment and default characteristics of these securities, as well as coverage of a key topical issue impacting value in this market: the prepayment and default behavior of health care loans issued under Section 232 compared to that of the apartment complex loans that have historically comprised the bulk of the GNMA project loan market.

GNMA PROJECT LOAN SECURITIES GNMA multifamily passthroughs are created when a mortgage originator makes an FHA project loan and then securitizes it as a GNMA pool. The originator pays a fee to GNMA (currently 13 basis points per year on the unpaid principal balance) to obtain GNMA’s guarantee, which backs with the full faith and credit of the U.S. government, the full and timely payment of principal and interest. The loan originator is not required to securitize through GNMA. Alternatively, the loan can be put into an FHAinsured passthrough participation certificate, which, in the case of a default, usually pays 99% of principal and interest at the FHA debenture rate, generally a bit lower than the passthrough rate. The timeliness of such payments is not guaranteed, and there is a loss of one month’s interest. Thus GNMA multifamily pools represent a subset of FHA project loans. As of December 31, 2002 (the most recent quarterly data release for FHA projects), 4,010 of the 13,519 FHA project loans were in GNMA pools, but on average the loans in GNMA pools were larger and more recently issued than those that those in FHA-insured passthrough participations. The average GNMA current loan balance as of year-end 2002 was $5.43 million, while the average current balance of the 9,664 FHA project loans not in GNMA pools was only $2.83 million. So only 29.7% of FHA outstanding project loans were in GNMA pools as of year-end 2002, but 44.3% of the total current balance of $48.916 billion outstanding FHA project loans was in GNMA pools. The proportion of new FHA project loans securitized into GNMA pools rose sharply after the GNMA guarantee fee was reduced from 45 to 13 basis points in March 1993, and in recent years the overwhelming majority of FHA project loans have been securitized into GNMA pools. GNMA multifamily pools each contain a single FHA project loan originated under the specific statutory authorization of a section of the National Housing Act of 1934, as subsequently amended several times by Congress. The FHA programs, designated by the section of the housing acts which defines them, differ as to type of project, loan purpose,


Some Investment Characteristics of GNMA Project Loan Securities

341

mortgage limits, etc. A brief description follows of the major programs that are currently active or historically important, with number of loans outstanding and total original balance of those loans as of December 31, 2002 summarized in Exhibit 1 based on information from the HUD insured multifamily database. The growing proportion of the GNMA project loan market (traditionally considered a funding mechanism for low and moderate income apartment complexes) that is comprised of loans for health care facilities, mostly for nursing homes and assisted living facilities, is a dramatic development for the GNMA project loan market. As of December 31, 2002, section 232 and 232/223(f) health care loans made up 12% of outstanding FHA project loans, but 21% of the outstanding FHA balance and 23% of the total GNMA project loan balance outstanding. This latter percentage is expected to grow steadily throughout 2003 and 2004. EXHIBIT 1

Major FHA Project Loan Insurance Programs

(1) Section 207: Rental Housing. This is the original FHA multifamily program created in 1934, with loans for the construction or rehabilitation of multifamily rental housing. Almost all of the FHA loans outstanding for this program are also 223(f) loans, which have been used to refinance or purchase existing projects; new multifamily projects in recent years have been originated under sections 221(d)(3) and 221(d)(4). These loans can have maturities up to 35 years. There are 1,838 section 207 and 207/223(f) loans outstanding, totaling $7.804 billion current balance; this comprises 13.6% of outstanding FHA project loans and 16.0% of FHA current balance. In the fiscal year ending September 30, 2002, FHA insured 149 section 207/223(f) projects with 22,106 units, totaling $758.4 million. (2) Section 213: Cooperative Housing. These loans are for construction, rehabilitation, acquisition, or conversion of cooperative housing projects. Investors can use this program to construct or rehabilitate multifamily projects that are then sold to nonprofit corporations who operate the projects as cooperatives. This is a relatively small program, with 218 loans totaling $0.23 billion current balance, representing 1.6% of FHA outstanding project loans and just 0.5% of FHA current balance. In the fiscal year ending September 30, 2002, FHA insured three section 213 projects with 174 units, totaling $16.9 million. (3) Section 220: Rental Housing for Urban Renewal and Concentrated Development Areas. These loans are for construction or rehabilitation of projects in locations designated by HUD as urban renewal or neighborhood development areas. This program has relatively few but large loans outstanding: 157 loans totaling $1.45 billion current balance, representing 1.2% of FHA outstanding loans but 3.0% of current balance. In the fiscal year ending September 30, 2002, FHA insured five section 220 projects with 499 units for a total of $76.9 million.


342


EXHIBIT 1 (Continued) (4) Section 221(d)(3) & 221(d)(4): Rental and Cooperative Housing for Low to Moderate Income and Displaced Families. This is by far the largest FHA program, comprising loans for the construction or substantial rehabilitation of rental and cooperative housing for low and moderate income families, as well as families displaced by urban renewal or disasters. The loans can have maturities up to 40 years. 221(d)(3) loans are offered to nonprofit sponsors, while 221(d)(4) loans are for profit-motivated sponsors. The latter predominate, as there are currently 4,890 221(d)(4) loans outstanding totaling $22.8 billion current balance, representing 36.2% of loans outstanding and 46.7% of total current balance. The 221(d)(3) loan program for nonprofits is much smaller, with 1,498 loans outstanding totaling $1.84 billion current balance, accounting for 11.1% of the FHA loans but only 3.8% of total current balance. The nonprofit loans are on average much smaller, as the average current balance of 221(d)(3) loans outstanding is just $1.23 million, compared to an average of $4.67 million for 221(d)(4) loans. In the fiscal year ending September 30, 2002, FHA insured 210 section 221(d)(3) or 221(d)4 projects with 40,000 units for a total of $2.5 billion. (5) Section 223(a)(7): Refinancing of FHA-Insured Mortgages. Under this program, the FHA refinances existing FHA-insured mortgages of any section of the Housing Act, resulting in the prepayment of the existing mortgage. The new loan is limited to the original balance of the existing mortgage loan, and the term is limited to the remaining term of the existing mortgage plus 12 years. There are no pure section 223(a)(7) loans; every loan in this program is also part of another program (e.g., a refinanced cooperative project is a 223(a)(7)/213 and a refinanced urban renewal project is a 223(a)(7)/220). In all, 1,036 loans totaling $2.95 billion current balance are outstanding 223(a)(7) loans of some kind, accounting for 7.7% of FHA loans and 6.0% of current balance. (6) Section 223(d): Two-year Operating Loss Loans. A small but significant FHA program, these loans augment the FHA-insured first mortgage financing the property and cover operating losses during the first two years of the project or any other two-year period within the first 10 years after completion of the project. These loans, used in recent years mainly to cover operating losses on nursing homes, help avoid defaults by insuring separate loans to cover operating losses. The maturity of the 223(d) loan is limited to the remaining term of the original mortgage. At present, this is a very small program, with 76 loans totaling just $50 million current balance, representing 0.6% of FHA outstanding loans and 0.1% of current balance. But the availability of these relatively small loans helps some borrowers, especially nursing home operators, avoid default as a consequence of temporary negative net cash flow from a project’s operation. In the fiscal year ending September 30, 2002, FHA insured four operating loss loans on nursing homes with 398 beds, totaling $2.7 million.



343

EXHIBIT 1 (Continued) (7) Section 223(f): Purchase or Refinancing of Existing Projects. An alternative refinancing mechanism to section 223(a)(7), as well as a means of financing purchases of existing projects. Almost all the 223(f) loans are either section 207 or section 232 loans. Unlike 223(a)(7) refinancings, 223(f) loans can be used to refinance conventional as well as FHA-insured mortgages, and the principal balance of the original loan is not restricted to the original balance of the refinanced loan. Instead, a 223(f) mortgage is limited to 85% of the FHA’s estimated project value, although this can be raised to 90% for projects in HUD-designated preservation areas. These loans can have maturities of up to 35 years. Like 223(a)(7) loans, every loan in this program is also part of another program. For example, most refinanced or purchased apartment complexes are 207/223(f) loans, while refinanced or purchased nursing homes are 232/223(f) loans. Much of the recent issuance of 223(f) loans is due to cashout refinancing of both FHA and conventional project loans. In all, 2,465 loans totaling $11.7 billion are outstanding 223 (f) loans of some kind, representing 18.2% of outstanding FHA loans and 23.9% of current balance. In the fiscal year ending September 30, 2002, FHA insured 207/223(f) mortgages for 149 apartment complexes with 22,106 units, totaling $758.4 million. (8) Section 232: Mortgage Insurance for Nursing Homes, Intermediate Care, Board & Care and Assisted Living Facilities. This program was enacted in 1959 to help meet the growing demand for nursing homes due to the rapid expansion of the U.S. elderly population. Borrowers under this program must be licensed in the state where the project is built, and to be financed by the FHA, a nursing home must have at least 20 patients who are classified as unable to live independently but not needing acute care. Assisted living facilities and board & care homes must contain a minimum of five units. Section 232 insures loans for construction or substantial rehabilitation, while Section 232/223(f) allows for the purchase or refinancing of existing projects not requiring substantial rehabilitation. Much of the 232/223(f) issuance is due to refinancing of conventional mortgages, as the private CMBS conduit market has backed away from financing and refinancing nursing homes. These loans are on average larger than other FHA project loans, with an average current loan balance of $5.98 million, compared to $3.28 million for all FHA non232 loans. This rapidly growing program has 1,695 loans outstanding totaling $10.13 billion current balance, representing 12.5% of FHA outstanding loans and 20.7% of current balance. In the fiscal year ending September 30, 2002, FHA insured 309 health care facilities with 35,403 beds, totaling $1.8 billion.


344


EXHIBIT 1 (Continued) (9) Section 236: Subsidized Rental Housing Projects. This program was instituted in 1968 but suspended in 1973 and never revived. While there are no new projects being insured under this section, refinancings under section 223(a)(7) and second mortgages to finance repairs under section 241(a) are authorized for existing projects from the 1968–73 period. Section 236 loans combined FHA mortgage insurance with federal subsidies to reduce property owners’ monthly payments, with the objective of reducing rents required of lowincome and elderly residents. Only projects renting exclusively to tenants with annual incomes below 80% of the median income of the area were eligible for section 236 loans. In spite of this program’s long period of inactive status for new projects, it remains of substantial size, with 2,456 loans outstanding totaling $2.8 billion current balance, representing 18.2% of FHA outstanding loans but only 5.7% of current balance. (10) Section 241(a): Supplemental Loans for Multifamily Projects. This program provides second mortgages for FHA-insured housing projects to finance property repairs, improvements or additions. These loans provide project owners with a method of extending the useful life of a project and to finance repairs and equipment replacement without having to refinance the existing mortgage. This offers an opportunity to increase the amount of FHA financing without refinancing the existing loan, and can thus be an attractive alternative for the borrower to an equity take-out refinancing when that would involve paying penalty points or a higher mortgage rate. The availability of these second mortgages also allows improvements to keep older projects competitive, extend their useful economic life, and finance the replacement of obsolete equipment, therefore holding down default risk from declining rents on deteriorating projects. These FHA second mortgages are considerably larger (average current balance of $2.68 million) than 223(d) operating loss loans (average current balance of $0.66 million), but on average smaller than FHA first mortgage loans (average current balance of $3.65 million). This program has 548 loans outstanding totaling $1.47 billion current balance, representing 4.1% of FHA outstanding loans and 3.0% of current balance. In the fiscal year ending September 30, 2002, FHA insured six 241(a) projects with 913 units/beds, totaling $11.8 million. Section 241(a) originations are likely to increase when interest rates are much higher than current rates and second mortgages are a more attractive alternative than refinancing.

Source: Various Federal Housing Administration sources. (Numbers calculated from the HUD multifamily database.)



345

PREPAYMENT BEHAVIOR OF GNMA MULTIFAMILY POOLS Except for a very small number of specially designated small loan pools, GNMA multifamily pools each contain a single FHA project loan. For an investor in the pool, this raises concerns about prepayment risk. But nearly all such loans provide some form of prepayment protection. Loans typically have lockout periods, a fixed period of time after loan origination when voluntary prepayments are forbidden; most often this runs for five or ten years. Loans with ten-year lockouts generally do not have post-lockout prepayment penalties, while loans with five year lockouts typically have a prepayment penalty schedule of 5 points during year six, declining a point a year to 1 point during year 10, after which voluntary prepayments are free of penalties. While these two prepayment protection variants are by far the most common, other lockout and penalty patterns exist in this marketplace. Under some rather stringent conditions specified in HUD Mortgage Letter 87-9, the FHA can override lockout and penalty provisions to avoid a credit loss to the U.S. government. Because such an override is only exercised to avoid an insurance claim to the FHA, these loans would have prepaid at par in any case if permitted to default. All such prepayments are classified as involuntary: for purposes of prepayment analysis, we define all prepayments during the lockout period and all those during the penalty period which do not result in the payment of penalties as “defaults,” whether or not they actually correspond to the ordinary meaning of a mortgage default. It should be noted that for these loans, “voluntary” prepayments are completely at the mortgagor’s discretion; even the sale of the property does not require prepayment of the mortgage, since all FHA project loans are assumable. Because there is only one loan in a GNMA multifamily pool, and generally fewer than 100 pools in REMICs backed by GNMA project loans, monthly prepayment speeds can be quite volatile. A typical $1 billion REMIC backed by single family agency mortgage pools has 5,000 to 10,000 loans represented in the collateral, while most REMICs backed by GNMA multifamily pools contain fewer than 100 loans. As noted in the introduction, the entire GNMA multifamily universe as of March 31, 2003 consisted of 4,145 pools totaling $22.885 billion current balance, for an average current pool balance of $5.52 million. The relatively small number of loans means that the monthly prepayments are much less consistent than in single family REMICs, and little information is garnered by a few months of actual prepayments from a single deal. We present below the prepayment history of the entire universe of GNMA multifamily pools for the past year (April 2002–March 2003), including all pools issued since the beginning of 1988. Only 40 GNMA


346


pools are outstanding that were issued in 1987 or earlier, totaling $121.9 million in current balance, too small to be helpful in making inferences about GNMA multifamily prepayment behavior. Exhibits 2 and 3 provide the prepayment history of the GNMA project loan market for the year encompassing April 2002–March 2003 in two different ways. Exhibit 2 gives the prepayment history of the EXHIBIT 2

Recent Prepayment History (April 2002–March 2003)

Production Year/ Net Coupon

Original $mm

Original # Pools

1988

2,113.7

338

1989

1,266.4

208

1990

1,151.2

1991

Outst. $mm

Current # Pools

Factor

Original Net WAC

Current Net WAC

1-yr CPR

158.2

42

0.0749

9.35

8.96

18.1

51.8

16

0.0409

9.02

8.78

39.3

164

91.0

19

0.0790

8.90

8.61

30.2

859.1

148

153.7

31

0.1789

8.78

8.36

10.1

1992

907.7

111

223.4

23

0.2462

8.48

9.01

24.9

1993

22.7

2,023.7

338

733.5

131

0.3624

7.42

7.12

6.000–6.499

168.5

25

113.2

21

0.6718

6.41

6.38

0.9

6.500–6.999

469.1

84

206.7

46

0.4405

6.82

6.79

29.1

7.000–7.499

619.5

118

302.0

44

0.4875

7.30

7.30

16.4

7.500–7.999

420.8

61

56.9

10

0.1352

7.87

7.96

37.8

1994

1,944.4

409

798.0

187

0.4104

7.60

7.42

19.5

6.500–6.999

512.8

85

232.4

42

0.4532

6.86

6.80

21.9

7.000–7.499

344.6

62

136.0

29

0.3945

7.30

7.30

18.7

7.500–7.999

284.1

69

135.5

34

0.4771

7.90

7.88

23.4

8.000–8.499

480.3

123

149.7

45

0.3116

8.34

8.34

25.2

1995

2,063.1

479

1,097.5

276

0.5320

7.81

7.71

22.5

6.500–6.999

219.4

37

166.5

31

0.7587

6.83

6.85

12.0

7.000–7.499

427.1

80

225.6

44

0.5283

7.36

7.33

17.0

7.500–7.999

551.6

133

236.9

64

0.4295

7.84

7.87

32.8

8.000–8.499

326.4

93

122.6

45

0.3757

8.31

8.29

42.5

8.500–8.999

301.8

92

159.0

58

0.5266

8.80

8.79

19.3

1996

2,324.0

406

1,273.3

260

0.5479

7.47

7.39

22.8

6.500–6.999

277.8

47

187.1

32

0.6735

6.87

6.84

21.3

7.000–7.499

477.0

84

277.3

55

0.5813

7.35

7.34

23.7

7.500–7.999

1,073.7

185

507.2

113

0.4723

7.80

7.80

26.0

8.000–8.499

202.3

42

97.9

24

0.4839

8.26

8.26

16.1

1997

2,007.5

391

1,470.8

303

0.7327

7.40

7.32

17.3

6.000–6.499

136.8

25

130.1

25

0.9507

6.22

6.22

0.0

6.500–6.999

250.6

38

200.5

35

0.8001

6.83

6.85

4.0

7.000–7.499

680.1

124

470.3

92

0.6915

7.36

7.35

22.1

7.500–7.999

641.4

141

478.2

105

0.7455

7.77

7.75

18.6

8.000–8.499

171.3

40

91.1

26

0.5318

8.24

8.25

36.3


347


EXHIBIT 2 (Continued) Production Year/ Net Coupon 1998

Original $mm

Original # Pools

Outst. $mm

Current # Pools

Factor

Original Net WAC

Current Net WAC

1-yr CPR

1,938.7

385

1,563.3

346

0.8064

6.91

6.88

8.7

5.000–5.499

133.3

28

118.1

27

0.8860

5.31

5.30

5.4

6.000–6.499

361.5

71

317.3

68

0.8775

6.37

6.36

6.9

6.500–6.999

609.6

157

473.8

147

0.7773

6.70

6.72

6.1

7.000–7.499

301.3

46

230.5

36

0.7652

7.36

7.35

9.5

7.500–7.999

333.5

47

264.3

39

0.7925

7.79

7.80

17.0 4.3

1999

2,626.9

473

2,371.0

433

0.9026

7.16

7.14

6.000–6.499

314.9

52

282.3

50

0.8964

6.35

6.34

7.0

6.500–6.999

539.6

142

503.2

134

0.9326

6.81

6.81

1.2

7.000–7.499

832.9

127

773.4

118

0.9285

7.30

7.30

3.2

7.500–7.999

613.7

107

527.2

95

0.8590

7.78

7.77

8.0

2000

2,717.3

437

2,474.5

402

0.9106

6.92

6.91

5.1

5.000–5.499

127.5

25

124.8

25

0.9783

5.43

5.43

0.0

6.000–6.499

412.0

50

392.2

48

0.9520

6.38

6.38

1.4

6.500–6.999

1,095.2

122

953.8

112

0.8709

6.75

6.74

9.1

7.000–7.499

328.1

54

304.7

50

0.9287

7.35

7.34

3.6

7.500–7.999

411.3

111

372.3

96

0.9051

7.77

7.77

4.3

8.000–8.499

192.9

49

183.7

47

0.9518

8.25

8.25

1.3

2001

3,643.8

559

3,448.4

544

0.9464

6.89

6.87

2.6

5.500–5.999

238.0

39

234.2

39

0.9841

5.75

5.75

0.1

6.000–6.499

554.2

83

547.6

83

0.9881

6.33

6.33

0.0

6.500–6.999

1,340.2

212

1,246.5

204

0.9301

6.79

6.79

4.1

7.000–7.499

782.7

121

728.4

117

0.9305

7.32

7.32

1.0

7.500–7.999

417.5

60

392.0

59

0.9390

7.75

7.75

5.3

8.000–8.499

151.7

25

143.0

23

0.9429

8.22

8.22

5.2 5.4

2002

5,462.4

876

5,297.6

868

0.9698

6.71

6.70

5.000–5.499

479.3

88

476.8

88

0.9949

5.36

5.36

0.0

5.500–5.999

726.4

144

697.2

142

0.9598

5.78

5.78

0.0

6.000–6.499

1,033.0

191

1,026.3

191

0.9936

6.32

6.32

0.0

6.500–6.999

1,388.4

242

1,332.9

241

0.9600

6.77

6.76

9.7

7.000–7.499

668.5

102

658.3

100

0.9847

7.33

7.33

1.2

7.500–7.999

932.2

83

872.5

80

0.9359

7.77

7.78

6.6

2003

1,559.2

224

1,557.2

224

0.9987

6.08

6.08

NA

5.000–5.499

619.0

100

618.1

100

0.9986

5.33

5.33

NA

5.500–5.999

275.9

61

275.5

61

0.9985

5.74

5.74

NA


348


entire universe of loans originated since 1988, including defaults counted as prepayments, as is customary in the single-family mortgage market. Exhibit 3 gives the prepayment history of the universe with all known defaults since the beginning of 1991 excluded. For a few paidEXHIBIT 3

Recent Prepayment History (Defaults Excluded, April 2002–March 2003) Production Year/ Net Coupon

Original $mm

1988

2,109.2

Original # Pools 337

Outst. $mm 158.2

Current # Pools 42

Factor 0.0750

Original Net WAC

Current Net WAC

1-yr CPR

9.35

8.96

18.1

1989

1,263.0

206

51.8

16

0.0410

9.02

8.78

1990

1,117.2

161

91.0

19

0.0814

8.92

8.61

39.3 30.2

1991

804.0

141

153.7

31

0.1912

8.77

8.36

10.1

1992

726.0

86

223.4

23

0.3077

8.54

9.01

24.9

1993

22.3

1,763.3

305

733.5

131

0.4160

7.35

7.12

6.000–6.499

163.7

24

113.2

21

0.6915

6.40

6.38

0.9

6.500–6.999

450.4

80

206.7

46

0.4588

6.82

6.79

27.6

7.000–7.499

581.8

109

302.0

44

0.5191

7.31

7.30

16.4

7.500–7.999

354.2

53

56.9

10

0.1606

7.87

7.96

37.8

1994

1,751.8

376

798.0

187

0.4556

7.56

7.42

18.0

6.500–6.999

501.8

83

232.4

42

0.4632

6.86

6.80

20.5

7.000–7.499

297.1

57

136.0

29

0.4576

7.30

7.30

18.7

7.500–7.999

265.7

66

135.5

34

0.5101

7.90

7.88

18.8

8.000–8.499

441.7

112

149.7

45

0.3389

8.35

8.34

24.2

1995

1,936.6

448

1,097.5

276

0.5667

7.80

7.71

20.5

6.500–6.999

204.0

36

166.5

31

0.8162

6.85

6.85

12.0

7.000–7.499

414.5

75

225.6

44

0.5443

7.35

7.33

17.0

7.500–7.999

526.9

126

236.9

64

0.4497

7.84

7.87

32.5

8.000–8.499

307.0

88

122.6

45

0.3994

8.31

8.29

40.1

8.500–8.999

268.6

84

159.0

58

0.5919

8.80

8.79

10.3

1996

2,005.8

379

1,273.3

260

0.6348

7.47

7.39

19.9

6.500–6.999

270.6

46

187.1

32

0.6913

6.87

6.84

19.1

7.000–7.499

426.1

79

277.3

55

0.6507

7.34

7.34

23.7

7.500–7.999

903.3

176

507.2

113

0.5615

7.81

7.80

25.4

8.000–8.499

162.3

37

97.9

24

0.6035

8.24

8.26

5.9

1997

1,881.1

369

1,470.8

303

0.7819

7.40

7.32

15.9

6.000–6.499

136.8

25

130.1

25

0.9507

6.22

6.22

0.0

6.500–6.999

214.1

36

200.5

35

0.9363

6.85

6.85

1.9

7.000–7.499

647.6

117

470.3

92

0.7262

7.35

7.35

22.1

7.500–7.999

603.9

134

478.2

105

0.7918

7.76

7.75

15.2

8.000–8.499

151.4

34

91.1

26

0.6017

8.25

8.25

36.3


349


EXHIBIT 3 (Continued) Production Year/ Net Coupon

Original $mm

Original # Pools

Outst. $mm

Current # Pools

Factor

Original Net WAC

Current Net WAC

1-yr CPR

1998

1,788.5

362

1,563.3

346

0.8741

6.89

6.88

4.0

5.000–5.499

126.1

27

118.1

27

0.9361

5.30

5.30

0.0

6.000–6.499

332.1

68

317.3

68

0.9553

6.36

6.36

0.0

6.500–6.999

584.7

152

473.8

147

0.8104

6.71

6.72

2.2

7.000–7.499

253.4

40

230.5

36

0.9097

7.35

7.35

5.0

7.500–7.999

304.7

42

264.3

39

0.8672

7.80

7.80

11.0 2.4

1999

2,516.0

450

2,371.0

433

0.9424

7.14

7.14

6.000–6.499

313.6

51

282.3

50

0.9000

6.35

6.34

6.7

6.500–6.999

528.2

137

503.2

134

0.9527

6.80

6.81

0.4

7.000–7.499

805.2

121

773.4

118

0.9605

7.30

7.30

1.6

7.500–7.999

562.7

100

527.2

95

0.9369

7.77

7.77

4.2

2000

2,541.8

406

2,474.5

402

0.9735

6.91

6.91

0.7

5.000–5.499

127.5

25

124.8

25

0.9783

5.43

5.43

0.0

6.000–6.499

403.5

48

392.2

48

0.9720

6.38

6.38

0.4

6.500–6.999

979.6

113

953.8

112

0.9737

6.74

6.74

0.9

7.000–7.499

312.5

51

304.7

50

0.9752

7.34

7.34

0.8

7.500–7.999

379.9

97

372.3

96

0.9799

7.77

7.77

0.2

8.000–8.499

190.0

48

183.7

47

0.9668

8.25

8.25

1.3

2001

3,505.6

544

3,448.4

544

0.9837

6.88

6.87

0.4

5.500–5.999

238.0

39

234.2

39

0.9841

5.75

5.75

0.1

6.000–6.499

554.2

83

547.6

83

0.9881

6.33

6.33

0.0

6.500–6.999

1,266.7

204

1,246.5

204

0.9840

6.79

6.79

0.3

7.000–7.499

736.8

117

728.4

117

0.9885

7.32

7.32

0.0

7.500–7.999

406.5

59

392.0

59

0.9642

7.75

7.75

2.8

8.000–8.499

143.8

23

143.0

23

0.9943

8.22

8.22

0.0 0.0

2002

5,334.8

868

5,297.6

868

0.9930

6.70

6.70

5.000–5.499

479.3

88

476.8

88

0.9949

5.36

5.36

0.0

5.500–5.999

700.5

142

697.2

142

0.9954

5.78

5.78

0.0

6.000–6.499

1,033.0

191

1,026.3

191

0.9936

6.32

6.32

0.0

6.500–6.999

1,341.8

241

1,332.9

241

0.9934

6.76

6.76

0.0

7.000–7.499

662.9

100

658.3

100

0.9931

7.33

7.33

0.0

7.500–7.999

882.8

80

872.5

80

0.9883

7.78

7.78

0.0

2003

1,559.2

224

1,557.2

224

0.9987

6.08

6.08

NA

5.000–5.499

619.0

100

618.1

100

0.9986

5.33

5.33

NA

5.500–5.999

275.9

61

275.5

61

0.9985

5.74

5.74

NA


350


off pools, mostly originated in the late 1980s and early 1990s, we are unable to determine whether the pay-offs were defaults or refinancings; we classify those as refinancings. These data show several ways in which the prepayment behavior of these loans is distinguished from the more familiar single-family mortgage market. First, given that in the last year we saw the largest refinancing wave in the history of the single-family mortgage market, with prepayment speeds from October 2002 through March 2003 exceeding anything seen in the 33-year history of the residential MBS market, GNMA multifamily prepayment speeds over the past year have been fairly tame. This 12-month period covers April 2002 through March 2003, a period of historically low interest rates. Assuming a two-month time lag from rate-lock to closing for a project loan refinancing, the corresponding period for rate locks was February 2002 through January 2003, a period in which the 10-year Treasury yield ranged from 3.57 to 5.43 and averaged just 4.50. No production years except 1989 and 1990, small vintages with only 16 and 19 loans outstanding, respectively, had a one-year speed exceeding 25% CPR. Further, while there was some tendency of the higher coupon pools in the 1993–1998 cohorts to prepay faster, that was not consistently true (e.g., 1995 production with coupons of 8.50–8.99 and 1996 production with coupons of 8.00–8.49 prepaid at 12 month CPRs of only 19.3 and 16.1, respectively, including defaults, and at only 10.3 and 5.9 CPR, excluding defaults); overall, prepayments were only modestly correlated with pool interest rate. The main motivation for refinancing in this market is the desire to remove equity from appreciated projects, with interest savings a secondary though not insignificant concern. Widespread increases in the value of multifamily real estate during the strong economy of the mid-to-late 1990’s led some low-rate borrowers to refinance, while higher-rate borrowers owning projects that have depreciated in value (often for location-specific reasons) had insufficient equity to qualify for refinancing. While the tax benefits of multifamily real estate partnerships are far less than before 1987, when the Tax Reform Act of 1986 became effective, tax syndication partnerships remain a common ownership type of forprofit projects. Under the 1986 tax reform, projects originated in 1987 or later must be depreciated over 27.5 years, using the straight-line depreciation method. The tax shelter provided by deducting mortgage interest payments, together with the tax advantages that remain even from the slower depreciation, typically generate passive tax losses for the first 10–15 years of a project, after which reduced interest deductions as the loan amortizes and higher revenue due to rent increases often generate net taxable income, even after the depreciation is deducted. So equity take-out refinancing via a Section 223(f) loan for a project that has



351

appreciated in market value can be economically advantageous to a partnership, even when the interest rate on the refinanced loan is somewhat higher than that of the original loan. The higher balance generates a larger tax deduction for mortgage interest, offsetting higher revenue as rents increase and avoiding the production of taxable income. The prepayment speeds for the year examined above demonstrate how GNMA multifamily pools prepay in a relatively low interest rate environment. To get an indication of how they might prepay in a much higher rate environment, we examine the one-year historical CPRs of the GNMA multifamily universe, broken down by production year and passthrough coupon, for the 12 months from July 1994 through June 1995. This period represents the interest rate peak of the past decade; the corresponding rate-lock period (assuming a 60-day lag) was May 1994 through April 1995, during which time the 10-year Treasury yield ranged from 6.91 to 8.03, and averaged 7.43. For that time period, 1983 and later production had enough pools outstanding to generate a somewhat meaningful prepayment history. We break down individual production years by coupon when there is at least $100 million current balance in a coupon-production year category; these results are seen in Exhibit 4. Notably in this high rate period, every production year except 1984 (a production year consisting of only 12 loans outstanding in June 1995) from 1983 through 1990 prepaid at over 10% CPR. Production later than 1990 was mostly locked out at this time. So the indication from this relatively high-rate year is that prepayments of seasoned GNMA multifamily pools do not slow dramatically when interest rates rise. The desire of investors to take equity out of properties that have appreciated in value, whether for project expansion or improvements or to realize a capital gain on the original investment, seems to prevent prepayment speeds from falling precipitously. This is in spite of the ability of project owners to extract equity for repairs, additions, or improvements with a section 241(a) second mortgage, which likely deters some refinancing in high rate environments. Analysis of historical conditional prepayment rates (CPRs) over other recent periods can help us draw generalizations about prospective prepayment behavior for these GNMA pools. In Exhibit 5, compiled from GNMA factor tapes, we show historical 1-year prepayments of the GNMA universe, broken down by production year, during the calendar years 1998 through 2002. For the corresponding 12-month rate-lock period (shifted two months earlier), the average 10-year Treasury yields were 5.44 for 1998, 5.40 for 1999, 6.13 for 2000, 5.11 for 2001, and 4.72 for 2002. It appears that production years with recently expired call protection (i.e., in the 11th or 12th year since origination) tend to have prepayment spikes, after which their speeds slow considerably. These pre-


352

EXHIBIT 4


July 1994—June 1995 Prepayment History

Production Year/ Original Original Current Current Net Coupon $mm # Pools $mm # Pools

Factor

1983

269.6

29

124.6

11

0.4622

8.56

8.28

1984

189.5

34

65.2

12

0.3441

10.30

9.78

5.9

1985

502.8

62

137.7

20

0.2739

9.94

10.15

24.2

1986

1,774.4

307

293.3

70

0.1653

9.30

9.44

14.3

899.1

158

131.1

30

0.1458

9.35

9.39

12.7

9.000–9.499 1987

Original Current Net WAC Net WAC

1-Year CPR 22.0

2,027.3

371

716.0

137

0.3532

8.93

8.57

10.6

8.000–8.499

536.9

83

252.7

42

0.4707

8.37

8.37

7.5

8.500–8.999

436.6

91

169.8

40

0.3889

8.88

8.86

6.5 17.2

1988

2,045.0

329

826.3

152

0.4041

9.36

9.17

8.000–8.499

211.2

30

115.4

22

0.5464

8.32

8.36

0.0

9.000–9.499

878.1

85

334.1

35

0.3805

9.34

9.34

17.2

9.500–9.999

559.6

122

187.9

48

0.3358

9.84

9.79

30.3

1,231.7

202

614.4

115

0.4988

9.03

8.92

15.3

230.5

54

130.4

31

0.5657

9.90

9.91

16.4

1989 9.500–9.999 1990

1,117.2

161

571.5

99

0.5115

8.92

8.77

10.5

8.500–8.999

221.8

33

143.4

23

0.6465

8.85

8.86

14.4

1991

811.2

142

630.7

119

0.7775

8.76

8.71

2.6

8.500–8.999

167.1

31

126.6

26

0.7576

8.91

8.93

0.0

1992

846.7

102

737.1

91

0.8706

8.52

8.54

4.0

7.500–7.999

169.8

24

147.3

23

0.8675

7.82

7.85

0.0

8.000–8.499

202.8

32

176.9

30

0.8723

8.35

8.34

11.5

1993

1,847.9

317

1,764.6

312

0.9549

7.40

7.35

0.0

6.000–6.499

168.5

25

165.8

25

0.9840

6.41

6.41

0.0

6.500–6.999

463.7

83

456.6

83

0.9847

6.82

6.82

0.0

7.000–7.499

578.9

111

570.6

111

0.9857

7.31

7.31

0.0

7.500–7.999

368.2

55

363.7

55

0.9878

7.88

7.88

0.0

1994

1,819.1

389

1,805.1

389

0.9923

7.57

7.57

0.0

6.500–6.999

507.7

84

502.8

84

0.9903

6.86

6.86

0.0

7.000–7.499

302.3

59

299.6

59

0.9911

7.30

7.30

0.0

7.500–7.999

284.1

69

282.4

69

0.9940

7.90

7.90

0.0

8.000–8.499

450.7

115

448.2

115

0.9945

8.35

8.35

0.0

8.500–8.999

137.2

45

136.7

45

0.9964

8.80

8.80

0.0


353


EXHIBIT 5 Production Year 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

CPR by Production Year

1998 CPR

1999 CPR

2000 CPR

2001 CPR

2002 CPR

39.0 31.2 23.2 25.9 24.1 11.6 11.0 1.7 2.4 5.1 0.8

8.9 33.1 54.2 11.2 15.4 5.5 7.9 14.1 1.3 4.1 2.1 0.0

24.5 8.2 15.5 9.0 11.3 17.3 9.4 5.5 5.2 2.1 0.6 0.3 0.3

11.8 7.3 26.1 28.6 31.8 15.5 12.6 18.7 13.7 11.2 2.7 5.6 1.5 0.0

36.6 16.0 42.3 31.5 9.4 26.5 19.1 20.5 20.1 17.1 11.5 7.9 3.5 4.5 2.6

payment spikes occurred for 1987–1988 production in 1998, for 1989 production in 1999, and for 1990–1991 production in 2001. However, 1989–1990 production exhibited no such rush to refinance in 2000, as the higher rates prevailing for most of that calendar year likely inhibited refinancing. The extensive refinancing of 1989 production during 1999 may have left few loans with a refinancing propensity remaining outstanding by 2000, while the higher rates of 2000 may have delayed some 1990 production refinancing until 2001, when that cohort saw a refinancing spike. The recent prepayment history of GNMA project loans provides support for our view that long run speeds above 25% CPR are quite unlikely, although faster speeds can be expected for a year or so after call protection expires if current interest rates are much lower than the loan coupon. Conversely, long-run speeds slower than 10% CPR are also unlikely, at least in an environment where Treasury yields are not far outside the range that has prevailed over the last decade. Prepayment behavior in the very high rate environment of the early 1980s seems to hold little relevance, as the tax benefits of real estate partnerships were dramatically different before the Tax Reform Act of 1986; expiration of real estate tax benefits was a driving force behind project


354


loan refinancing during that period. The post-1986 tax benefits from investments in multifamily housing are much smaller, but not insignificant, and declining tax benefits over time are still a force behind noneconomic refinancing, generally keeping speeds on seasoned pools from dropping into single-digit CPRs permanently. In summary, we think that the typical 15% CPR pricing assumption for voluntary prepayments on a diverse group of par-priced and slight premium GNMA multifamily pools backing a REMIC is reasonable given this prepayment history, and that long-term speeds are unlikely to exceed 25% CPR in a low rate environment or fall much below 10% CPR in a high rate environment. Of course, this conclusion covers the entire GNMA multifamily universe; subsets of the universe containing relatively few loans can prepay very differently for loan-specific reasons. Investors can limit their idiosyncratic prepayment risk by diversifying their GNMA project loan investments among a number of different REMICs.

DEFAULT BEHAVIOR OF GNMA MULTIFAMILY POOLS In the 1990s many of the REMICs backed by GNMA project loans were issued with pricing speeds that assumed no defaults would ever occur; i.e., that prior to lockout expiration, the collateral would experience no prepayments at all. Since the FHA was created in order to obtain financing for multifamily projects that would otherwise not be built at all (i.e., that the private sector would refuse to finance or would finance on such onerous terms that the project would not be economically viable), it would be very surprising if these securities experienced no defaults. In fact, we have identified 246 GNMA multifamily pools issued since the beginning of 1993 that have since defaulted, totaling $1.727 billion of original principal. However, gaps in the database make it impossible to know whether we have identified every default; of the 1,003 pools originated in 1993 or later that have paid off, approximately 40 have missing information so we cannot know whether it defaulted or was refinanced, although we think that most of these 40 represent refinancings. If we cannot clearly identify a prepayment as a default, it does not appear in our default statistics. Any loan that pays off in its lockout period is a default from the investor’s perspective, as is any loan that pays off during its prepayment penalty period but for which the borrower is not required to pay penalty points. As we noted before, the FHA can override lockout periods and penalty provisions under some conditions to avoid a credit loss to the U.S. government. We classify such prepayments as defaults, since the investor receives a cash flow



EXHIBIT 6

355

Number of Defaulted Pools by Years from Issuance to Default:

that is identical to what would be received if the borrower had actually defaulted. Because information is more likely to be missing on the oldest loans, we restricted our analysis to pools issued since January 1993, and hence cannot provide insight to historical default rates on loans more than 10 years old. In any case, underwriting standards for FHA project loans tightened in the early 1990s, so that pre-1993 history of default rates is probably of limited interest in projecting future defaults.1 Exhibits 6 and 7 shows defaults by number of loans and by original principal amount as a function of time from pool issuance to default. The principal amount pattern is skewed a bit by a single loan of approximately $75 million that defaulted in its second year since issuance. But the default pattern over time is fairly clear, and differs considerably from single-family “A quality” mortgage default patterns, in which defaults are rare in the first two years after origination. GNMA multifamily defaults occur with significant frequency in the very first year, rise in year two, then level off through year five, and decline fairly sharply after that. The early defaults probably reflect that many FHAinsured projects serve primarily social policy goals, and some projects funded under FHA regulations would be considered fairly high credit risk loans by private lenders. A small fraction of these projects fail to 1 Data in the monthly GNMA prepayment tapes allow easy identification of loans that have paid off, but considerable effort is required to determine whether each payoff represents a default or a refinancing, and for 40 paid-off pools the information to determine this is simply unavailable.


356

EXHIBIT 7


Dollar Balance of Defaulted Pools by Years from Issuance to Default:

become economically viable and default fairly soon after construction is completed. Those projects that generate sufficient income to service the mortgage during the first five years are likely to avoid default altogether.

CUMULATIVE DEFAULTS BY PRODUCTION YEAR AND THE GNMA PROJECT LOAN DEFAULT CURVE REMICs backed by GNMA project loans are priced using an industry convention known as the GNMA Project Loan Default Curve (GN PLD) to gauge the “involuntary” prepayment pricing speed. This list of default rates by year from origination is given in Exhibit 8. The typical pricing speed “15 CPJ” represents a voluntary or refinancing speed of 15% CPR plus the default pattern given by the GN PLD. This comes to an effective lifetime default rate of 15.45% over twenty years. As noted above, we have thoroughly analyzed the default behavior of GNMA project loans over the past ten years, and this default frequency data gives some insight into how closely the GN PLD curve fits the historical default experience. Exhibit 9 shows the number and percentage of loans that have defaulted by production year from 1993 forward, as well as the amount and percentage of original principal balance that defaulted. Only 1996 production has a significantly higher percentage of defaulted principal than predicted by the GN PLD curve, although 2000 and 2002 production are slightly above the GN PLD prediction. Both 1996 and 2002


357


EXHIBIT 8

GNMA Project Loan Default Curve

Year from Origination

Default Rate

1 2 3 4 5 6 7 8 9 10–14 15–20 21 on

EXHIBIT 9

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Total

1.30% 2.47% 2.51% 2.20% 2.13% 1.46% 1.26% 0.80% 0.57% 0.50% 0.25% 0.00%

Cumulative Default Rates by Production Year

Original Principal ($mm)

Original Number of Loans

Defaulted Original Face ($mm)

Percentage of Original Face Defaulted

Defaulted Number of Pools

Percentage of Original Number Defaulted

2,023.7 1,944.4 2,063.1 2,324.0 2,007.5 1,938.7 2,626.9 2,717.3 3,643.8 5,462.4 1,559.2 28,311.0

338 409 479 406 391 385 473 437 559 876 224 4,977

260.4 192.6 126.5 318.2 126.4 150.2 110.9 175.5 138.2 127.6 0 1,726.5

12.9 9.9 6.1 13.7 6.3 7.7 4.2 6.5 3.8 2.3 0.0 6.1

33 33 31 27 22 23 23 31 15 8 0 246

9.8 8.1 6.5 6.7 5.6 6.0 4.9 7.1 2.7 0.9 0.0 4.9

defaulted principal percentages are distorted by a few large loans; in both cases the percentage of loans that defaulted is much smaller than the percentage of principal that defaulted. For the most part, though, the default rates appear to be just slightly below the GN PLD curve, indicating that this default pattern used to price REMICs is perhaps slightly conservative, but overall fairly accurate.


358


EXHIBIT 10

Percentage of Original Balance Defaulted by Health Care and Nonhealth Care

A concern of many CMBS investors has been the relatively high defaults experienced by health care-related loans in CMBS conduit deals, especially in the late 1990s. In 1997, Congress changed Medicaid reimbursement policy for nursing homes from “cost plus” reimbursements, which guaranteed the nursing home owner a modest margin above actual patient care costs, to flat per diem payments regardless of provider costs. However, FHA-backed nursing homes have caps on the unit construction cost, so they are constructed more cheaply, with fewer amenities than the more luxurious nursing homes typically financed in the 1990s by CMBS conduits and other conventional financing. Consequently, the change in Medicaid reimbursement policy was less onerous for the health care projects in GNMA project loan pools. Exhibit 10 gives default rates by production year for Section 232 and 232/223(f) loans in GNMA pools versus the rest of the GNMA project loan universe, consisting almost entirely of apartment complexes of some kind. This does not show any clear pattern of health care facilities having consistently higher default rates than apartment complexes; it varies yearto-year, but overall default rates are comparable for health care and nonhealth care pools within the GNMA project loan market. In summary, GNMA multifamily pools experience a sufficiently high default rate to make it unreasonable to price REMICs backed by this collateral at a zero default assumption. The GNMA Project Loan Default (GN PLD) curve, which has recently been used to price most REMICs, is fairly close to the historical default pattern we found here. That curve, based upon historical default data since the mid-1980s, in our view slightly overestimates expected future default rates. The tight-



359

ening of FHA underwriting standards in the early 1990s is likely to keep future default rates somewhat below that of the past.2 We believe that the market practice of using the GN PLD curve to price REMICs backed by GNMA project loans is reasonable, but investors should also examine the impact of defaults running at about 75% of the GN PLD curve, which is a plausible scenario for the case of a relatively strong multifamily real estate environment. While we believe that Congress is more sensitive to the concerns of Medicaid providers now than it was in 1997, there remains some political risk embedded in projecting nursing home default rates. The above default history for Section 232 loans is reassuring, but hardly conclusive for the future. For the scenario where political decisions prevent Medicaid reimbursement rates from rising with the increase in operating costs (exacerbated at present by the national shortage of nurses), investors might examine the performance of REMIC tranches at default rates of about 120% of the GN PLD curve, at least for deals containing a high percentage of Section 232 loans. The 120% default scenario is also useful for examining the impact of a moderately severe real estate recession, like that of 1990– 1992. For all REMIC tranches other than Interest-Only securities, varying default rates from 75% to 120% of the GN PLD curve has little impact on the average life or yield.

RECENT BREAKDOWN OF GNMA PROJECT LOAN PREPAYMENTS Exhibit 11 shows the number of pools in the GNMA Multifamily universe paying off over the 15-month period January 2002 to March 2003, and the breakdown between those that were defaults and those that were refinancings with or without prepayment penalties. Those indicated as refinancing with penalties are non-defaulted loans whose penalty schedule indicates that penalties were owed; we were not able to confirm in all cases whether or not these penalties were actually paid, although the liens of the refinanced mortgages on the projects are not satisfied until penalty points not explicitly waived by the FHA are paid. An important point for investors is that the existence of penalty points does not seem to be a huge deterrent to refinancing, which often is based upon the desire of project owners to take equity out of a project which has appreciated in value. The expectation that once the lockout period ends refinancings will occur and penalty points paid to investors 2

For details of some of the underwriting changes, see the GNMA web site at: http://www.ginniemae.gov/multi/delinquency.asp?Section=MBSecurities


360

EXHIBIT 11


Breakdown of Recently Paid-off Pools

Mar 2003 # of Pools Current Face ($mm) Feb 2003 # of Pools Current Face ($mm) Jan 2003 # of Pools Current Face ($mm) Dec 2002 # of Pools Current Face ($mm) Nov 2002 # of Pools Current Face ($mm) Oct 2002 # of Pools Current Face ($mm) Sep 2002 # of Pools Current Face ($mm) Aug 2002 # of Pools Current Face ($mm) Jul 2002 # of Pools Current Face ($mm) Jun 2002 # of Pools Current Face ($mm) May 2002 # of Pools Current Face ($mm) Apr 2002 # of Pools Current Face ($mm) Mar 2002 # of Pools Current Face ($mm) Feb 2002 # of Pools Current Face ($mm) Jan 2002 # of Pools Current Face ($mm)

Total Outstanding

Total, Paid-Off Pools

Refi. (No Penalty Points)

Refi. (W/ Penalty Points)

Default

4,145 22,885.1

48 241.8

9 37.4

32 150.6

7 53.8

4,113 22,651.4

37 180.3

6 26.6

21 95.0

10 58.7

4,087 22,426.2

50 244.7

15 62.0

27 100.0

8 82.7

4,057 22,083.4

40 177.3

7 16.3

31 150.6

2 10.4

4,010 21,764.5

51 318.0

14 52.0

28 204.1

9 61.9

3,986 21,667.6

39 197.9

11 51.8

17 63.0

11 83.1

3,936 21,410.0

46 194.7

11 37.8

19 94.3

16 62.6

3,925 21,244.7

41 184.7

11 32.3

21 65.3

9 87.1

3,885 20,844.7

37 98.3

11 18.6

17 65.6

9 14.1

3,841 20,458.5

39 123.3

10 35.4

20 55.9

9 32.0

3,817 20,199.0

36 159.3

13 73.5

22 81.2

1 4.7

3,760 19,820.5

31 88.5

7 24.1

20 59.8

4 4.6

3,726 19,561.0

22 118.8

4 34.8

13 58.1

5 25.9

3,664 19,144.6

35 126.5

18 70.8

11 50.4

6 5.3

3,653 19,001.1

22 101.6

3 4.9

14 71.2

5 25.5



361

appears reasonable, although we would expect that the quantity of refinancing with penalties would decline in a period of slower real estate price appreciation than that of the past few years or in a period of much higher interest rates.

THE REFINANCING HISTORY OF HEALTH CARE LOANS COMPARED TO APARTMENT COMPLEXES While Section 232 and 232/223(f) health care related GNMA project loans have not shown a historical propensity to default much differently than apartment complex loans, they have consistently refinanced over the past five years at much lower rates than similarly seasoned nonhealth care loans. We examine below the 5-year prepayment history of the universe of GNMA projects loans that finance health care facilities, compared to the history of similarly seasoned nonhealth care loans. Each GNMA pool from each production year from 1993 through 1997 is classified as health care or nonhealth care using Nomura’s proprietary database of all GNMA pools by production year issued in 1993 or later. As noted earlier, we created this database from the GNMA multifamily Web site, from GNMA pool prospectuses and by matching of information from these sources with that in the HUD database. Exhibits 12 and 13 show the 5-year CPR history for the period January 1998 through December 2002 of health care versus nonhealth care GNMA project loans for each production year from 1993 through 1997. EXHIBIT 12 5-Year Prepayment History of Health Care-Related Loans versus Nonhealth Care


362


EXHIBIT 13 5-Year Prepayment History of Health Care-Related Loans versus Nonhealth Care (Defaults Excluded)

Pre-1993 pools are few and information needed to classify loans as health care or not is often missing, while from 1998 forward most loans were locked out, and in any case they had not been in existence for long enough to have a 5-year prepayment history. Exhibit 12 gives the 5-year CPRs including loans that defaulted, counting defaults as prepayments, while Exhibit 13 excluded from the calculation all pools that terminated in default. By either calculation, the universe of health care loans has consistently refinanced at only about half the rate of nonhealth care loans. Thus REMICs with a relatively high percentage of health care loans (40–50% in a few recent deals) clearly have less call risk to the investor than those with smaller percentages of such loans. For the investors purchasing long average life sequentials and last cash flow Z-bonds, as well as Interest Only (IO) tranches backed by GNMA project loans, there is clearly extra value in choosing REMIC tranches backed by collateral with a high percentage of health care loans. For buyers of short average life sequentials in a steep yield curve environment such as the present, a high percentage of health care loans slightly increases extension risk. However, short average life tranches in deals in which the sequentials receive accretions from the Z-bond as well as prepayments, have little extension risk in the first place. A typical 3.5 year average life sequential in a structure with a long Z-bond’s accretions directed to the sequential tranches only extends to about a 4.5 year even at the unrealistically slow speed of 5% CPJ. So the IO buyer, and to a lesser extent, the long tranche buyers benefit from a higher percentage of health care loans, while the short tranche buyer takes only slightly more extension risk.



363

Why have health care loans refinanced at so much slower rates than apartment complex loans? A major reason is likely the difficulty in operating FHA-financed nursing homes profitably in the environment of fixed per diem Medicaid reimbursement rates and rising compensation for skilled nurses due to their growing scarcity. Since a major factor in project loan refinancing is appreciation of property values leading to cash-out refinancing, even when prepayment penalty points are owed, the financial stress of the nursing home industry is preventing much price appreciation in this sector. Even if the financial stress is at present not severe enough to send default rates soaring, squeezed profit margins coupled with political uncertainty about the future of Medicaid reimbursement rates inhibit price appreciation for even successful nursing homes. Moreover, should nursing home operators succeed in obtaining higher per diem Medicaid reimbursements from Congress, the increasing competition from assisted living facilities for more affluent patients not dependent upon Medicaid will likely hold down nursing home prices. Hence, we think that slower price appreciation and the associated slower refinancing activity compared to apartment complexes will continue for the next several years.

ON THE INVESTMENT CHARACTERISTICS OF GNMA MULTIFAMILY POOLS AND REMICS The evidence from historical prepayment patterns shows that the interest-sensitivity of GNMA multifamily pools is quite modest, certainly far less than that of GNMA single-family mortgages. In a low-rate environment, prepayments increase dramatically in the year after call protections expire, but then settle down to more normal CPRs in the teens or low 20s, as the most interest-sensitive loans are refinanced and exit the pool. In a high rate environment, prepayments slow somewhat, but the declining value over time of tax advantages for project owners and the incentives to refinance to extract equity prevent long term speeds on seasoned pools with expired lockouts from falling much below 10% CPR. The explicit call protection on recently issued pools and the limited ratesensitivity of refinancings on seasoned pools provides the investor with much better convexity than that provided by single-family MBS. Defaults, which are not interest-sensitive, have risen slightly in the past year during 2002 and early 2003 (see Exhibit 11) after falling steadily in the late 1990s, but remain slightly below the assumptions of the GN PLD curve. Defaults create a little extra cash flow uncertainty in the early years after issuance but do not create negative convexity because


364


they are not significantly correlated with interest rates. Going forward, we believe that defaults are likely to be somewhat lower than 100% of the GN PLD curve, perhaps in the 75–85% range. The growing percentage of health care loans in recent GNMA project loan issuance tends to make this market a bit better call-protected, since health care loans have consistently refinanced at lower rates than nonhealth care project loans. While the growing health care percentage of issuance perhaps increases the near-term uncertainty about default rates, given the dependence of FHA-financed health care operators on political decisions about Medicaid reimbursement rates, to date GNMA health care loans have not defaulted at significantly different rates than apartment complex loans.

CONCLUSION The growth of the GNMA project loan market, with growing issuance and secondary market liquidity of GNMA REMICs backed by project loans, offers opportunities for structured finance investors to gain incremental yield compared to private label AAA rated CMBS tranches, at the cost of slightly worse convexity but better credit (full faith and credit U.S. government guarantee instead of a AAA rating which can later by downgraded). These REMIC tranches offer agency CMO investors the opportunity to gain significantly better convexity with, at present, only a small yield concession. While none of the popular bond indices used as benchmarks for fixed income portfolio managers yet include GNMA project loans, that may change in the future as the project loan market grows in the years ahead from its current size of just under $23 billion.

18-MahadevanSchwartz Page 365 Wednesday, July 23, 2003 10:22 AM

A Framework for Secondary Market CDO Valuation Sivan Mahadevan Vice President Morgan Stanley David Schwartz Associate Morgan Stanley

collateralized debt obligation is effectively an investment in a leveraged portfolio of credit instruments. The risk of this portfolio is disproportionately redistributed into the various tranches issued by the special purpose vehicle administering the investment portfolio. The principal risks investors face are price volatility, liquidity and default risk, and there are many metrics that measure these characteristics. Investors purchase CDO notes for a variety of reasons. The senior notes of a CDO offer substantial yield pickup over similarly rated instruments in the cash credit markets. These notes earn their high ratings because the CDO cash flow structure offers them substantial protection from defaults in the underlying collateral. The subordinated notes and residual components (equity) of CDOs offer investors unique risk and return characteristics not easily replicated in the cash or credit derivatives markets. CDOs are generally structured so that if default and recovery rates over their lifespan fall within a range that includes average historical rates, investors stand to benefit from the leverage of the structure and the resulting higher yields that the CDO tranches offer relative to investments in the corporate bond markets.

A

365


366


Many traditional CDO note and equity holders have been buy and hold investors. The bulk of arbitrage CDOs are structured as cash flow CDOs, meaning that the collateral managers focus their efforts on meeting the cash flow liabilities while conforming to the structural guidelines of the various tranches. In cash flow deals, collateral managers’ concerns about the market value or the price volatility of the underlying collateral are generally in the context of how such valuations relate to the credit quality of the collateral or the cost of transactions. This style is similar to that of asset/liability investment managers such as insurance companies. In arbitrage market value CDOs, the collateral managers focus on improving the market value of the underlying collateral, similar to how total-return oriented investment managers operate. Independent of the structure and collateral management style of CDOs, buyers and sellers of CDO notes and equity in the secondary market are interested in determining a fair (and agreed upon) value for the expected cash flows, price volatility, liquidity, and default risk that a given CDO tranche represents. The goal of this article is to discuss objective ways for valuing CDO tranches in the secondary market, and to apply subjective considerations including market technical information. In this article, we present several methodologies for secondary market CDO note and equity valuation with a focus on arbitrage cash flow CDO structures. The approaches differ in degrees of computational complexity and required market savvy, but ultimately they all help investors gauge the value of secondary market CDO investment opportunities.

CDO VALUATION FRAMEWORK Before we describe a valuation framework for CDOs, we make some observations on how related financial instruments are valued in the secondary markets. Investment grade corporate bonds are traditionally valued on a comparables basis. Liquid bonds from an issuer are compared to liquid bonds from other issuers with similar characteristics. The characteristics considered include credit quality, industrial sector and maturity, and from these comparisons, spread levels are determined. Bonds that have unique characteristics (such as optionality or special credit considerations) move away from their comparables, and spreads are adjusted accordingly, either independently, with respect to a new set of comparables, or with some sophisticated valuation techniques. For example, bonds with options can be valued in the Heath-Jarrow-Morton framework using a multipath simulation.


A Framework for Secondary Market CDO Valuation

367

Higher-rated noninvestment grade corporate bonds can be valued on a comparables basis as well, but reliance on these types of comparisons quickly disappears in favor of unique valuations derived from credit fundamentals and investor appetite. Valuation techniques for asset-backed securities vary given potential differences in security structure, but the comparables technique for securities with similar collateral and structure is common. Valuation techniques for collateralized mortgage obligations (CMOs) are also worth observing, given their conceptual similarity to CDOs. The least risky (i.e., most immune to prepayment risk) tranches of CMOs tend to be tested using various metrics and then compared to other similar CMO tranches. The residual components of CMOs, which carry the bulk of the collateral’s prepayment risk, are often valued using modeling techniques that generate and discount cash flows over multiple input paths and average these results. With this background in mind, we establish a framework for valuing CDO notes and equity by asking three fundamental questions: ■ What common market practices exist today for valuing CDOs? ■ What relevant input from the financial markets should influence CDO

valuations? ■ What financial valuation techniques can be applied to CDOs?

The answers to the above questions led us to organize our CDO valuation framework along three basic methodologies.

Three Valuation Methodologies Market practices for determining secondary market CDO note and equity prices exhibit a range of required market savvy and computational complexity. The computationally simpler approaches involve generalizing collateral characteristics and comparing CDO tranches to similar structures with a common credit rating and collateral type. Computationally more complex approaches exist as well, ranging from valuing tranches over a handful of input paths to full-blown simulation models run over a large distribution of paths. In this article, we describe three methodologies for valuing CDO notes and equity in the secondary market. ■ Rerating methodology, a computationally simple approach that involves

inferring a rating for a CDO tranche through a single set of assumptions and comparing it to other similarly rated tranches to derive a value.


368 EXHIBIT 1


Secondary Market CDO Valuation Methodologies

Source: Morgan Stanley. ■ Market value methodology, a technique that involves equating the

market value of the assets of a CDO (the collateral and interest rate hedges) with the market value of the liabilities (debt, equity, and management fees). Market value changes can be indicators of potential structural changes in a CDO as well. ■ Cash flow methodology, a technique that is based on financial engineering fundamentals, namely generating and discounting tranche cash flows along input paths. Variants of this approach include a single assumption set, several assumption paths, and a full simulation over a large distribution of events. Exhibit 1 summarizes the spectrum of inputs and valuation methodologies for determining CDO tranche values.

THE RERATING METHODOLOGY The rerating methodology is based on a simple, quantitative approach to identifying a credit rating for a CDO note. Once a credit rating is identified, the note is compared to similar structures in the marketplace to determine a spread value. Comparability is usually based on CDO characteristics such as collateral type and cohort year of issuance. The approach is particularly valuable when an investor has newly updated



369

information on a CDO, or would like to forecast changes to a CDO’s collateral. In such cases, the credit rating calculated by the rerating methodology can be different from those assigned by ratings agencies, giving the investor insight into the current status or evolution of the CDO. The approach is also useful in providing investors with a guide to the sensitivity of a tranche’s credit rating to changes in underlying assumptions. The rerating methodology is based on the well-known Moody’s Binomial Expansion Method for determining a CDO tranche rating.1 The Moody’s approach is a simple method that involves generalizing a CDO’s collateral pool, inferring a default rate for the portfolio and calculating the expected loss for a given CDO tranche, which leads to a credit rating. The rerating methodology is described in the flow chart in Exhibit 2. The expected loss of a CDO tranche is computed based on two inputs: the binomial distribution of defaults and the tranche loss function. From these two functions, we calculate the probability-weighted loss (the expected loss) for the tranche, which can then be mapped to a Moody’s rating using the Moody’s “Idealized” Cumulative Default Rates. EXHIBIT 2

The Rerating Methodology


The Binomial Expansion Method Applied to CBO/CLO Analysis, Moody’s Investors Services, December 13, 1996.


370


Generalizing the Collateral—The Diversity Score The rerating methodology gains its simplicity from Moody’s approach to generalizing the underlying CDO collateral. Moody’s simplifies the collateral portfolio by breaking it down into a set of uncorrelated assets. Correlation is defined as default correlation (not correlation of asset returns or changes in spreads). Given a portfolio of correlated assets, the diversity score represents the equivalent number of uncorrelated assets. If a portfolio has 30 uncorrelated assets, then it has a diversity score of 30. How does Moody’s simplify a portfolio to a collection of uncorrelated assets? The basic assumption is that assets in different industry categories are uncorrelated. If a portfolio has 30 credits (each from a different Moody’s industry category), then the portfolio has a diversity score of 30. Each of the 30 represented industry categories contributes 1.0 units to the total diversity. If more than one credit from the portfolio belongs to a given industry category, then that category’s contribution to the total increases, following a sliding scale depicted in Exhibit 3. This relationship implies that credits in the same industry group have a positive default correlation. In Exhibit 4, we show that the implied default correlation between credits in the same industry falls from 33.3% to 16.7% as the number of credits in the industry increases from 2 to 10. The implied correlation calculation is based on moment-matching, with the assumptions that each credit has the same weight in the portfolio and that the default correlations between any two assets in the same industry are equal. EXHIBIT 3

Moody’s Diversity Score Contribution for One Industry

Source: Moody’s.



EXHIBIT 4

371

Implied Default Correlation

Source: Morgan Stanley, Moody’s.

Modeling Defaults as a Binomial Distribution Moody’s uses the diversity score of a CDO to model the default behavior of the collateral. The approach assumes that the number of defaults in a portfolio of N uncorrelated assets (i.e., diversity score = N) follows a binomial distribution. The binomial distribution is derived as follows. First, an event of default is considered to be a binary operation: An asset either experiences default, or it does not. The probability of an asset defaulting is represented by p. The probability of an asset not defaulting is equal to 1 – p. The Moody’s method assumes that all assets in a portfolio have the same probability of default, derived from the WARF (numerical weighted-average ratings factor for the collateral). Defaults are assumed to occur independently. Given these simplifications, we can model the default behavior of a portfolio using a binomial tree. Consider a portfolio of three assets. There are several possible default behaviors for this portfolio: the portfolio can experience zero defaults, one default, two defaults, or three defaults. The binomial tree in Exhibit 5 depicts all of the possible default outcomes. There are eight possible paths in this binomial tree, and they are summarized in Exhibit 6. Given this representation of portfolio default behavior, we can compute the portfolio’s default probability distribution. The probability that a portfolio of N (uncorrelated) assets experiences j defaults is


372


EXHIBIT 5

Binomial Tree of Defaults for a Three-Asset Portfolio


EXHIBIT 6

Distribution of Defaults for a Three-Asset Portfolio

Number of Defaults

Frequency of Occurrence in Binomial Tree

Probability

Weighted Probability

0 1 2 3

1 3 3 1

(1 – p)3 p(1 – p)2 p2(1 – p) p3

(1 – p)3 3p(1 – p)2 3p2(1 – p) p3


N j (N – j) P j =   p ( 1 – p ) j The first parenthetical part of this expression is a combination or the number of groupings of j defaulted assets from a portfolio of N total assets (j ≤ N). This is mathematically equal to N!  N = --------------------- j j! ( N – j )! The notation N! (read “N factorial”) is equivalent to


373


EXHIBIT 7

Binomial Distribution of Defaults for a 20-Asset Portfolio


N × (N – 1) × (N – 2) × . . . × 1 The variable p is the generalized cumulative probability of default for any asset in the portfolio (derived from the portfolio WARF). The graph in Exhibit 7 shows the binomial distribution of defaults for a hypothetical 20-asset portfolio. The two curves represent different default probabilities, p = 0.5 and p = 0.15.

Computing a CDO Tranche’s Expected Loss We can use the CDO’s cash flow waterfall to determine the loss a CDO tranche will experience for a given number of defaults. In the example in Exhibit 8, the diversity score is 35 and p = 21.5%. The bell-shaped graph and the left-hand scale show the binomial distribution. The columns and right-hand scale show the losses that the hypothetical CDO tranche will experience under different default scenarios. We can now combine this tranche loss function with the binomial distribution to calculate the probability-weighted loss, otherwise known as the expected loss. N

Expected loss =

∑ Pj Lj

j=0

The variable Pj is the probability that the collateral portfolio experiences j defaults and the variable Lj represents tranche’s loss when the collateral experiences j defaults.


374


EXHIBIT 8

Binomial Distribution and Tranche Loss Function


Inferring a Moody’s Credit Rating We can use the CDO tranche’s expected loss to infer a Moody’s credit rating. To do this, we first assume a recovery rate for defaulted securities. Given this recovery rate, we can compute the implied default probability of the CDO tranche as follows: Implied default probability = Expected loss/(1 – Recovery rate) We can map the implied default probability to a Moody’s credit rating using the Moody’s “Idealized” Cumulative Default Rates (see Exhibit 9). Finally, given this rating, a value for the CDO tranche can be determined from a comparables analysis as discussed at the beginning of this section.

Tranche Rating Sensitivity Through implied default probability mappings, the graphs in Exhibits 10– 12 show the rating sensitivity of a hypothetical CDO tranche to changes in portfolio diversity score, WARF, and par loss (realized defaults). Such sensitivity analysis is an important use of the rerating methodology as it gives investors insights into the stability of current or forecasted ratings. For example, a change in the collateral WARF from 1850 to 1950 will move the hypothetical tranche from an investment grade Baa3 rating one notch lower to Ba1. Similarly, an immediate par loss increase from 6% to 7% will lower the rating one notch from Ba1 to Ba2.

375

Source: Moody’s.

0.000 0.003 0.008 0.019 0.037 0.070 0.150 0.280 0.470 1.050 2.020 3.470 5.510 8.380 11.670 16.610 32.500

0.001 0.010 0.026 0.059 0.117 0.222 0.360 0.560 0.830 1.710 3.130 5.180 7.870 11.580 15.550 21.030 39.000

0.002 0.021 0.047 0.101 0.189 0.345 0.540 0.830 1.200 2.380 4.200 6.800 9.790 13.850 18.130 24.040 43.880

0.003 0.031 0.068 0.142 0.261 0.467 0.730 1.100 1.580 3.050 5.280 8.410 11.860 16.120 20.710 27.050 48.750

5 Years 0.004 0.042 0.089 0.183 0.330 0.583 0.910 1.370 1.970 3.700 6.250 9.770 13.490 17.890 22.650 29.200 52.000

6 Years 0.005 0.054 0.111 0.227 0.406 0.710 1.110 1.670 2.410 4.330 7.060 10.700 14.620 19.130 24.010 31.000 55.250

7 Years 0.007 0.067 0.135 0.272 0.480 0.829 1.300 1.970 2.850 4.970 7.890 11.660 15.710 20.230 25.150 32.580 58.500

8 Years

0.000 0.001 0.001 0.003 0.006 0.011 0.039 0.090 0.170 0.420 0.870 1.560 2.810 4.680 7.160 11.620 26.000

4 Years

Aaa Aa1 Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3 Ba1 Ba2 Ba3 B1 B2 B3 Caa

3 Years

1 Year

Rating

2 Years

Moody’s Idealized Cumulative Default Rates by Rating and Number of Years

EXHIBIT 9

0.008 0.082 0.164 0.327 0.573 0.982 1.520 2.270 3.240 5.570 8.690 12.650 16.710 21.240 26.220 33.780 61.750

9 Years 0.010 0.100 0.200 0.400 0.700 1.200 1.800 2.600 3.600 6.100 9.400 13.500 17.660 22.200 27.200 34.900 65.000

10 Years



376 EXHIBIT 10


Tranche Ratings Sensitivity to Collateral Diversity Score


EXHIBIT 11

Tranche Ratings Sensitivity to Collateral WARF




EXHIBIT 12

377

Tranche Ratings Sensitivity to Immediate Collateral Par Losses


Valuing Equity as the Residual The rerating methodology, by definition, is useful only for determining value of CDO liabilities that carry a credit rating. How does one use this methodology to determine value for equity tranches that are not rated? The answer is that we can combine the valuations of rated CDO tranches in this approach with other methodologies (described in the following sections) to determine the residual value of the equity tranche.

THE MARKET VALUE METHODOLOGY One simple view of a CDO is to think of it as a portfolio with cash flows divided into many pieces. The sum of these parts must equal the whole portfolio. This equivalence property is the basis of the CDO valuation technique we call the market value methodology, the fundamental premise of which is: Market value of assets = Market value of liabilities The assets of a CDO are its collateral (adjusted for interest rate hedges that may be in the structure). The liabilities of a CDO are the notes issued by the special purpose vehicle, the equity and any fees collected by collat-


378


eral managers (over the life of a CDO) and by underwriters and administrators (at issuance). When CDOs are issued, this equivalence relationship between assets and liabilities exists (usually within a small band), but as CDOs age, we have found that the market value of assets does not always stay in line with the market value of liabilities. Absent are strong forces to drive equivalence, including: ■ The difficulty of going long one side of the asset liability equation and

short the other. Arbitrageurs are not easily able to drive the equivalence relationship. ■ The buy and hold behavior of traditional CDO note and equity investors. It should be noted that closed-end mutual funds experience similar behavior in the relationship between the NAV of the fund and the value of the fund’s shares. Nevertheless, market value equivalence is a fundamental relationship that can be used as a relevant metric in valuing CDO liabilities. Furthermore, we find that many CDO market participants rely on the market value equivalence, and in this section we describe a market value based approach for valuing CDOs.

The Role of Changing Market Value In theory, whenever we observe a change in the market value of the assets, we should observe an equivalent change in the value of CDO liabilities. For example, a 10 basis point widening in credit spreads of a CDO’s underlying collateral should have a commensurate impact on the market value of the CDO’s notes and equity. How can we measure this impact? The market value approach tells us that a decrease in value due to a 10 bp widening in the credit spread of the underlying collateral should be distributed to the various CDO tranches such that: ■ Each tranche is affected in a fair way. ■ The market value of assets continues to equal the market value of lia-

bilities. How do we determine a fair way of distributing the change in market value to the various tranches? One extreme is for the tranches to absorb the losses sequentially. The most subordinated tranche (equity) would absorb losses until it lost its full par value, at which point the next subordinated tranche would begin to absorb losses. Another extreme is to distribute the loss in a pro rata format to all tranches simultaneously (see Exhibit 13). Consider a simple hypothetical example, a CDO with three tranches as depicted in Exhibit 14. If the collateral loses 10% of its value, under

Market Value Impact: Sequential and Pro Rata Approaches


EXHIBIT 13


379


380 EXHIBIT 14


Sequential and Pro Rata Approaches: Examples 10% Collateral Loss

Tranche Senior Mezzanine Equity

30% Collateral Loss

Original Price

% of Par Value

Sequential Approach (Price)

Pro Rata Approach (Price)

Sequential Approach (Price)

Pro Rata Approach (Price)

100 100 100

75 15 10

100 100 0

90 90 90

93 0 0

70 70 70


the sequential approach, the equity tranche will absorb the entire loss (its price would fall to 0% of par, as equity comprises 10% of the par value of the structure). The prices of the other more senior tranches would remain at 100% of par. Under the pro rata approach, the loss would be distributed equally to each tranche (taking into consideration the tranche’s weight within the CDO). The price of each tranche would therefore fall to 90% of par. With a 30% loss of collateral market value, under the sequential approach, both subordinate tranches would lose all of their value (given that they make up 25% of the structure combined) and the remaining loss would be applied to the senior tranche, bringing its price as a percentage of par to 93.3. Under the pro rata approach, each tranche would lose 30% of its value, so all tranches would be priced at 70% of par. Is either extreme approach correct? Neither accurately reflects the risk of the various tranches, but the real values are likely bounded by these two approaches. For moderate declines in collateral value, the sequential approach, where the subordinate tranche absorbs the entire loss, is probably closer to what is practiced in the marketplace. In our view, to understand how the various tranches are affected, it is important to consider both the bounds and the nature of the market value changes. Because it can be difficult to assign market value changes to individual tranches, practitioners often focus instead on market value coverage measures. A more complex technique involves using a quantitative model to assess a tranche’s delta, which measures more precisely the change in a tranche’s value for a given change in the value of the underlying portfolio.

A Practical Approach: Categorizing Market Value Changes As we mentioned in the introduction, collateral managers of cash flow CDOs are primarily concerned about meeting the cash flow liabilities of the CDO, while keeping the CDO in compliance with credit ratings, par



381

coverage and interest coverage guidelines. Changes in market value of the underlying collateral are not necessarily of concern to collateral managers, unless these changes are large enough or specific enough in nature to have a structural impact on a cash flow CDO. Examples of changes in collateral that structurally affect a CDO include credit rating upgrades or downgrades, defaults, redemptions and reinvestment. From the perspective of cash flow CDOs, we can think of collateral market value changes as belonging to one of three categories: ■ Small market value changes that will not lead to structural changes in

cash flow CDOs. This includes small changes in the general level of interest rates and credit spreads. ■ Large market value changes that may not lead to structural changes in cash flow CDOs, but are large enough to change assumptions about the general level of interest rates and credit spreads. ■ Market value changes that are indicators of potential structural change in a CDO. We describe these three market value classifications along with the likely impact based on what we have observed in the marketplace. It is important to note, however, that the extent to which senior or mezzanine tranches are affected will depend on the amount of losses already incurred by the more subordinate tranches. For small changes in market value that are not indicators of future structural changes in the CDO, we have observed that the senior notes are likely unaffected, while the equity and mezzanine notes typically absorb the bulk of the impact (if any). A guide to the magnitude of this impact can be the new issue CDO market, where investors observe the value of comparable CDO tranches. For large changes in market value that are not indicators of future structural changes in the CDO, we have observed all tranches of a CDO being impacted. Senior notes generally take their cue from comparables or the general level of high credit quality instruments in the cash markets. Market savvy is required for the second priority or mezzanine notes, as we noted before. For equity, large changes in interest rates and credit spreads should influence fundamental assumptions like the yield (discount rate) required by investors (see the cash flow methodology).

Market Value as an Indicator of Structural Change One insightful use of the market value approach is to identify changes in market value that indicate potential structural change in a CDO. Examples of these indicators include:


382


■ The market prices in a ratings upgrade or downgrade for a particular

credit. ■ The market expects a credit to default or perceives an increased proba-

bility of default. ■ The collateral manager may be forced to sell a credit that falls below a

ratings threshold (e.g., CCC). In each of these examples, the investor may have an indication of the potentially changing characteristics of a CDO as a whole or a specific tranche. Investors can then use other methodologies (i.e., the rerating or cash flow methodologies) to value the tranche. For example, if the market begins to price in an upgrade or downgrade for a particular credit, an investor could forecast a different WARF for the collateral, and use the rerating methodology to value the tranche in question. If the market expects a credit to default (or if a credit falls to a ratings threshold that may force the collateral manager to sell it), then the investor can forecast a collateral sale and reinvestment and then revalue the tranche using the cash flow methodology.

THE CASH FLOW METHODOLOGY The cash flow approach draws its support from one of the most fundamental valuation techniques for fixed income securities: generating cash flows and computing their present value using yield or a series of discount factors. Rated notes of CDOs have scheduled cash flows, so valuing rated notes by generating and discounting cash flows is reasonable. CDO equity does not have contractually obligated cash flows, but for a given set of assumptions, there are definitive cash flows that can be valued as well.

Valuing Cash Flows of Rated CDO Notes What factors influence the cash flows of a rated note? For cash flow CDOs, default and recovery assumptions for the underlying collateral are clearly important. The underlying collateral for most CDOs is not static, so investment and reinvestment strategies and guidelines are important as well. Furthermore, any reasonable cash flow valuation based on default, recovery and reinvestment assumptions would require some measure of sensitivity to changes in these assumptions (that is, investors rarely have one view on these assumptions, and would rather see how altering assumptions can affect the value).



EXHIBIT 15

383

Cash Flow Methodology


In this section we describe two approaches to valuing a CDO tranche using cash flows. The first approach involves identifying a discount rate (yield) from comparable securities and testing its sensitivity to changes in input assumptions. The second approach involves projecting default-adjusted cash flows and discounting them at LIBOR over either a single path of assumptions or a distribution of assumptions. The cash flow approach is depicted generically in Exhibit 15.

The Cash Flow Approach Using a Comparable Yield From a corporate bond perspective, the discount rate we are most familiar with is yield. As we discussed earlier, corporate bond investors often value credits on a comparables basis, or making an educated guess at the right yield (based on comparable credits). In valuing CDO tranche cash flows, this comparables basis is used as well, along with tests of sensitivity to changes in input assumptions


384 EXHIBIT 16


Mezzanine Note PV of Cash Flows Sensitivity


A CDO note’s spread is most likely based on spread levels for comparable notes (new issue or secondary market) and possibly includes other market knowledge about the specific structure. However, the heart of the approach lies in testing the sensitivity of the note’s present value of cash flows (given a yield) to changes in underlying assumptions. Consider the example in Exhibit 16. If we assume that second priority notes on investment grade CDOs of recent vintage have a yield of 8%, then we can hold this yield constant and test the sensitivity of the present value of cash flows of the tranche to a set of default and recovery rate assumptions. For a recovery rate assumption of 40%, the PV is very stable for annual collateral default rates of 0% to 8%. At a recovery rate of 20%, the stability holds until approximately a 6% annual default rate. If these default and recovery rates are within ranges that cover collateral credit quality (with some degree of stressing), then we can say that the tranche’s PV is stable at a yield level of 8%.

How Do We Discount Equity Cash Flows? As we described earlier, the equity tranche of most CDOs do not have contractually obligated cash flows, and most don’t carry credit ratings either. However, for a given set of assumptions we can generate definitive cash flows for equity, so we can follow a similar cash flow discounting method as we do for rated notes. We can consider deriving a yield for equity from comparable securities. However, it is hard to find comparables for equity tranches, as


385


many secondary market equity tranches are unique. But referencing comparables is still an option, particularly for more recently issued deals (that could have similar collateral composition). The more common approach, however, continues to be driven by what the market requires for the yield of CDO equity. Many deals are structured to have internal rates of return in the 15–20% range if default and recovery assumptions follow historical averages for a given collateral type. As such, after computing equity cash flows under typical collateral default rate assumptions, discount rates of 15–20% are commonly used to compute present values for equity cash flows.

Projecting Default-Adjusted Cash Flows The first approach to using cash flows of a CDO note gives us a price only after we determine a value for the yield. How can we determine the price without assuming a yield? A simplified approach to valuing a security is to assume that an investor receives the expected cash flows of a security. We can calculate the expected cash flows by adjusting the scheduled cash flows based on the probability of default (see Exhibit 17). We then discount each adjusted cash flow by zero-coupon LIBOR to compute a present value. PV = EXHIBIT 17

Default adjusted cash flow

∑ -------------------------------------------------------------------i

Default-Adjusted Cash Flows


( 1 + Li )


386


In theory, the resulting present value is the price an investor would be willing to pay for a security with cash flows that have been adjusted for expected losses from defaults. We can now use this resulting present value to solve for the appropriate spread (or yield) on the original security (with the original scheduled cash flows). PV =

Scheduled cash flow

∑ ----------------------------------------------------------------i ( 1 + ( Tsy yld + Spread ) )

The above technique gives an implied spread, based on an assumed probability of default. The probability of default can be inferred from default swap spreads or asset swap spreads as well. When compared to historical default probabilities, the observed level in the default swap market probably more accurately reflects investors’ views on a credit. Is the simple one-path approach to valuing a CDO note described above valid? The approach of default-adjusting cash flows and discounting by zero-coupon LIBOR is used, but generally in a multipath context, which we describe in the next section.

Cash Flow Simulation Methodology We have discussed above computing a default-probability-implied yield for a tranche, given one path of input assumptions. We can extend this to a formal simulation where a large distribution of assumptions are generated programmatically, where for each path, cash flows are generated based on default and recovery rate assumptions and then discounted using risk-free rates. The average of these results can be considered a fair value for a CDO note.

Cash Flow Simulation Approach For each of thousands (or millions) of possible default scenarios, we repeat the following procedure: ■ For each period, determine which individual bonds in the collateral

pool default. Recovery and reinvestment assumptions need to be applied to defaulted bonds. ■ Use the cash flow waterfall to determine what payments are made to each tranche in each period. ■ Discount these cash flows at zero-coupon LIBOR. Having repeated this process over many different default possibilities, we can average the results for each tranche to determine a fair value.



EXHIBIT 18

387

Default Clustering Examples


Implementing the Simulation: Modeling the Timing of Defaults In theory this simulation is simple, but the implementation can be quite challenging. First, it can be computationally time consuming to simulate whether or not an individual bond defaults in each short interval of time. To overcome this computational burden, practitioners tend to directly model the amount of time it takes until a given bond defaults. The timing of defaults, not just the number of defaults, is critical for valuing CDOs. As demonstrated in Exhibit 18, there are many ways a portfolio can experience a given number of defaults. One extreme is that they are clustered early in the life of a CDO, while another extreme is that they are clustered late in the CDO life. The timing of these defaults affects the cash flow stream available to the various CDO tranches. Many early defaults can trigger the delevering of the structure by senior note holders, resulting in reduced cash flows (or no cash flows at all) to the subordinated note or equity holders.

Implementing the Simulation: Modeling Default Probability and Default Correlation To correctly value a CDO, we need to use market-implied (or “risk neutral”) default probabilities. A term structure of default probabilities can be implied from default swap spreads or asset swap spreads. Practitioners tend to model default probabilities using a stochastic process, which can be calibrated to default swap spreads or asset swap spreads.


388


The correlation of defaults must be accounted for as well. Correlation is critical, since individual bonds are affected by a number of common factors (see discussion in the Extensions to Valuation Methodologies section).

The Mathematics of a Simple Simulation To give the reader a sense of how the occurrence of default can be modeled, we mathematically describe a simplified model of default for a single asset. Suppose that, conditional on a bond surviving to a time t in the future (where time is measured in years), the probability of default over the next short time period ∆t is h∆t, where h is constant. Under this assumption, the time to default follows an exponential distribution; i.e., the probability of the bond defaulting within the next t years is 1 – e–ht. This is shown graphically in Exhibit 19 for h = 3%. For example, if h=3%, the probability of the bond defaulting within the next two years is 1 – e–0.03×2, or 5.82%. The parameter h is known as the hazard rate. Under this model, the probability of the bond surviving for the next t years is e–ht. Therefore, if we generated a uniform random variable U between 0 and 1, the relation t = –log(U)/h would give us the default time for the bond. For example, using h = 3% as above, if in our first path U = 0.757, the corresponding time until default would be –log(0.757)/0.03 = 9.28 years. EXHIBIT 19

Exponential Distribution: Cumulative Probability of Default




389

One aspect of this simplified simulation that is not realistic is that we have assumed that the hazard rate h is a constant. The hazard rate should be a stochastic function of time calibrated to current market levels. Moreover, to price a CDO tranche using this methodology, we would have one default process for each bond, and would have to combine them so as to account for the fact that defaults among bonds are correlated.

EXTENSIONS TO THE VALUATION METHODOLOGIES There are many natural extensions to the valuation methodologies that have been described in this report. In this section, we identify and summarize some of the extensions that are practiced in the marketplace.

Moody’s Double Binomial Method Moody’s applies the Double Binomial Method to CDOs that have collateral from two distinct pools (e.g., high-yield bonds and emerging markets debt).2 This technique allows each asset class to have a distinct default probability and diversity score. The two pools are assumed to be uncorrelated.

Distressed CDO Notes and Equity: Option Value Given the deep level of subordination that exists in many CDO structures, the marketplace has seen many subordinated tranches lose substantial value to a point where they trade like distressed securities. A distressed CDO does not imply that the entire CDO structure is in financial distress, it implies that the subordinated note of concern is in distress given its position in the cash flow waterfall. In distressed tranches, the collateral has experienced par losses (defaults) to a point where the likelihood of receiving future cash flows or full principal is small. It is difficult to value distressed CDO notes and equity using the valuation techniques we have described in this article. Any multipath approach will be particularly sensitive to the underlying assumptions. Yet these distressed securities do have some value to those investors who are willing to take the risk, similar to out of the money options. In fact, a distressed CDO tranche could be valued as a series of out of the money options, with each potential cash flow modeled as a separate option. The value of the security is the sum of the option value series. From an investor’s perspective, even if 2

The Double Binomial Method and its Application to a Special Case of CBO Structures, Moody’s Investor Services, March 1998.


390

EXHIBIT 20


Distressed Equity: Yield Sensitivity


one cash flow (coupon payment) is received, the investment could have a positive return if the price of the security is low enough. Consider a hypothetical equity tranche for a CBO. The graph in Exhibit 20 shows annual default rates versus yield for three different prices. At a price of 30, the equity has a yield of 7.4% at a future default rate of 0%, which is the best-case scenario for this residual tranche. However, the IRR quickly falls to a value of 0% if annual default rates climb to 2.25%. At a price of 10, the tranche yields nearly 25% for a 0% default rate. The tranche yields about 5% if annual default rates climb to 5%.

Nonbinomial Default Distributions and Default Correlation The rerating methodology described earlier in this article is based on two generalizations about the default behavior of assets. First, the collateral is generalized to be an N-asset portfolio (N = diversity score), where each asset has the same probability of default. Second, the default correlation among the N assets is assumed to be zero (i.e., their default behavior is independent). This general treatment of the collateral leads to a computationally simple approach to calculating the expected loss of a CDO tranche. However, a natural question to ask is: how different would our results be if we assumed that each credit had a unique probability of default and that the default correlation of assets were non-zero? In prac-



391

tice, computing an expected loss for a tranche in this manner is both data intensive and computationally complex, but it is practiced in the industry. There is commercially available software from vendors such as KMV LLC that provide default and correlation data and supports portfolio loss calculations based on this data. The KMV approach3 to modeling default risk is based on credit-specific expected default frequencies (EDFs™) and a model for default correlation. The EDFs are calculated for each modeled credit and are based on the notion that an issuer will default if the market value of its assets falls below its liabilities. The KMV model for default correlation is based the notion of a joint probability of default for two credits. KMV computes a default covariance matrix using a factor model approach as an alternative to directly using historical time series. The resulting loss distributions from the KMV approach are in practice much more skewed than the normal distribution or the binomial distributions we described in the ratings methodology section of this report (see Exhibit 21). The skewed behavior comes from positive default correlation among issuers and industries. What does a more skewed distribution imply? ■ More than half of the time, the loss is less than the average loss.

EXHIBIT 21

Comparing Distributions with Identical Mean and Variance


“Portfolio Management of Default Risk,” KMV LLC, 31 May, 2001.


392


■ The probability of large losses is higher (i.e., the skewed distribution

has a fat tail). In a normal distribution, the chance of a four standard deviation event occurring is very small. In a skewed distribution, it can be much higher.

Valuing the Equity Option As we discussed in the market value approach, a change in the market value of the underlying collateral pool is useful in valuing cash flow CDOs in several ways. Related to the market value of underlying assets, there is one feature of CDO equity that has been overlooked so far in this report: The value of the equity holders’ option have to call the entire structure. This call option allows equity holders to delever the structure by forcing a call on all of the CDO’s outstanding debt. Any remaining proceeds after the notes are retired (and after fees are paid) go to equity investors. The call option may be exercised after a specified noncall period on a discrete set of exercise dates. Equity holders may be motivated to call the structure for a variety of reasons including: ■ Capturing market value gains. ■ Limiting losses. ■ Reinvesting capital if more attractive opportunities arise.

Historically, not much attention has been paid to modeling the value of the equity option. There are three overriding reasons for this: ■ Relative to other factors, the impact of the call option on tranche valu-

ation is small. ■ The call option can be quite difficult to model. Whereas modeling col-

lateral defaults is the key to evaluating a CDO under the cash flow approach, in order to value the call option, collateral asset values would need to be modeled as well. ■ The call option is not necessarily exercised in an efficient manner. Investors typically think of the call option as a secondary consideration. As such, the call option is often analyzed heuristically, based on a scenario analysis.

SUMMARIZING THE VALUATION METHODOLOGIES In this article, we have presented three basic methodologies for secondary market CDO note and equity valuation, namely the rerating method-



393

ology, the market value methodology, and the cash flow methodology. We have presented variations and extensions for these methodologies as well, leaving CDO investors with a broad choice of valuation techniques. In our view, all three basic methodologies, and all of their variants have their valid uses, but they differ in the degrees of computational complexity and required market savvy. Which methodology should investors use? Investors and traders may never all agree on the best approach to value CDOs. In our view, a standardized approach to valuing CDOs would give investors a common language to speak and at the same time improve liquidity. As an analogy, in the mortgage-backed securities market, standardized prepayment assumptions and generalized collateral pools simplified the market tremendously, with sophisticated investors and traders relying on proprietary prepayment models to perform further valuations. One of the goals of this article is to encourage more dialogue among industry participants on the topic of standardization. In the absence of standardization, there are differences in the popularity of the various methodologies. Computational complexity has historically been a key factor, rendering some approaches more feasible than others. More recently, however, the cash flow approach has gained popularity despite its computational complexity. The cash flow approach allows investors to focus on the sensitivity of the tranche cash flows to changes in default rates, recovery rates and forward interest rates, while taking into consideration all of a given CDO’s structural features. Many prominent CDO dealers are offering cash flow analytics to clients over the Internet, and many third-party providers have developed platforms for analyzing CDO cash flows as well. Computationally simpler approaches, however, are still being used. Investors can assess the possibility of potential rating actions via the rerating approach or use the information embedded in underlying collateral prices through the market value approach. The choice of methodology will be a function of the tools available to the investor as well as the investor’s familiarity with CDO structures and the underlying collateral markets.


19-Lancaster Page 395 Wednesday, July 23, 2003 10:21 AM

Understanding Commercial Real Estate CDOs Brian P. Lancaster Managing Director Head of Structured Products Research Wachovia Securities

ince the first issue in 1999, commercial real estate collateralized debt obligations (CRE CDOs) have emerged as one of the most attractive asset classes in the CDO market. CRE CDOs are debt obligations typically collateralized by a combination of commercial mortgage-backed securities (CMBS) and senior unsecured real estate investment trust (REIT) debt.1 This article chronicles the rapid growth of the $13 billion CRE CDO market, the factors driving such growth, the market’s performance, issuer motivations in sponsoring CRE CDOs (with specific examples), and key factors for investors to consider in the purchase of CRE CDOs. We also analyze the relative value of CRE CDOs versus other fixed-income instruments, arguing that they benefit from the overly conservative nature of the rating agencies’ methodologies. Finally, we stress different types of CRE CDOs in light of the historic performance of the CRE markets and in so doing provide the investor with a methodology to discriminate among CRE CDOs.

S

A BRIEF HISTORY OF THE CRE CDO MARKET As in the CMBS market, the Russian debt crisis in the fall of 1998 was a milestone in the CRE CDO market. In addition to causing significant 1

Some CRE CDOs, such as CREST G-Dtar 2001-1, may have CRE whole loans as collateral although this is not as common.

395


396


losses for unhedged originators of unsecuritized CMBS collateral, it led to the demise of some of the most significant CMBS B-piece2 buyers, most notably CRIIMI MAE. These highly leveraged investors would typically finance their purchases of CMBS B-pieces and subordinated debt, at least in part, with short-term repo financing. The Russian debt crisis triggered margin calls, caused remarkable price volatility in the fixed-income markets, and led to a significant repricing of risk. More significantly, it led to the demise of heavily leveraged books with duration and/or basis mismatches (including CMBS B-piece buyers). The departure of key B-piece buyers compounded CMBS price volatility and prolonged the recovery period. However, it also created fertile ground for the development of the real estate CDO market. Real estate CDOs filled a key need primarily because they are a nonmark-to-market, long-term financing vehicle backed by B-pieces of CMBS transactions. The old issues of margin calls, price volatility, and the mismatch of shortterm financing were eliminated. Indeed, it is no coincidence that many current B-piece buyers, such as Arcap, Blackrock, GMAC, and Lennar, now use CDOs as a long-term, nonmark-to-market financing vehicle. Thus, although CDOs backed by corporate debt and emerging market debt (in that order) had existed for several years,3 CRE CDOs did not emerge as an asset class until October 1999, when Moody’s Investors Service, Inc., published its rating methodology for CDOs backed by commercial real estate.4 Since then, CRE CDO issuance has nearly doubled each year to $13 billion outstanding across 28 deals (see Exhibit 1).

WHAT ARE CRE CDOS? A CRE CDO is a special-purpose vehicle5 (SPV) that finances the purchase of CMBS, REIT debt, and other CRE assets by issuing, in 144A and “Reg. S” offerings, rated liabilities and equity. CRE CDOs are most frequently done as static pools6 but can also be actively managed. The benefits and implications of each are key considerations for an issuer. A static pool does not permit trading of the underlying collateral except in the case of an impaired security (generally defined as a security that has been downgraded or defaulted). Static pools qualify for off-balance-sheet treatment by being structured as qualified special purpose vehicles (QSPVs). 2

B-piece is the market moniker for subordinate debt that protects higher-rated asset classes from losses. 3 ABS CDOs had existed for less than a year when CRE CDOs emerged. 4 See H.Z. Remeza, J. Gluck, and E.J. Choi, The Inclusion of Commercial Real Estate Assets in CDOs, Moody’s Structured Finance Special Report, October 8, 1999. 5 Located in the Cayman Islands.


Understanding Commercial Real Estate CDOs

EXHIBIT 1

397

CRE CDO Issuance by Year

Source: Wachovia Securities, Inc.

Actively managed deals allow a manager to express views by trading the contributed collateral, which may allow the manager to achieve additional returns vis-à-vis a static deal. Managed deals are typically not structured as QSPVs, yet are still generally off-balance-sheet vehicles. Static pools are typically 100% “ramped up” or funded at closing (e.g., all the collateral securities have been purchased). Actively managed deals usually have a high percentage of assets purchased (greater than 75%) but are not necessarily fully “ramped up” in order to provide the manager flexibility in identifying relative value. The sponsor, or collateral manager, of a CRE CDO identifies the assets that will be sold into the trust and is paid an ongoing fee to monitor the performance of the collateral portfolio. Because of limitations placed on the collateral administrator by SPV guidelines, collateral bonds can only be sold when certain predefined events occur (referred to as trigger events) and any principal proceeds received by the trust cannot be used for reinvestment. Instead, proceeds must be applied to the redemption of the most senior class of notes outstanding. CRE CDOs are generally quarterly pay transactions. Typically, the bond tranches rated A3/A– and higher are issued as floating rate liabilities, whereas all other bonds are issued as fixed rate liabilities. Because the vast majority of the collateral is fixed rate, a fixed-floating swap must be entered into by the trust with a counterparty acceptable to the 6

A static pool CDO is collateralized by a pool of securities that remains predominantly unchanged over time. The asset manager of a static pool CDO may only sell a security if it has defaulted or is “credit impaired,” and may at no time purchase new collateral into the portfolio. Static pool CDOs are fully ramped as of the transaction’s closing date and are attractive because they allow investors to fully underwrite the portfolio before the closing date.


398 EXHIBIT 2


CRE CDO Diagram


rating agencies (see Exhibit 2). In addition, because the REIT bonds are semiannual pay, the trust also will generally enter into a timing swap to match the REIT payments with the payments on the liabilities of the CRE CDO. The CRE CDO bond tranches rated above BB generally receive “will” opinions7 for debt for tax purposes and are ERISA-eligible; the tranches rated BB and below (with the exception of the equity) generally receive “should” opinions8 and are restricted as to the amount that can be purchased with ERISA-eligible funds (24.9% of the tranche can be purchased with ERISA money). The senior notes in CRE CDOs are usually listed on an exchange, such as the Irish Stock Exchange Limited.

BORROWER MOTIVATIONS LEAD TO THE TWO BASIC TYPES OF CRE CDOS CRE CDO issuers generally fall into one of two groups: B-piece buyers and special servicers who need to obtain nonmark-to-market financing for subordinate CMBS (financing transactions) and money managers 7 A “will” opinion refers to the tax opinion issued by deal counsel withregard to the liabilities issued by a CDO. The opinion states that a security will be treated as debt for U.S. income tax purposes and is generally issued for a CDO’s investment-grade tranches. Investors should consult their accountants for further details. 8 Similar to a “will” opinion, a “should” opinion deals with the U.S. income tax treatment of a security issued by a CDO. The “should” opinion generally is issued with regard to the BB and B rated liabilities, tranches that, due to their subordinate position in the capital structure, may be viewed as equity interests in the CDO.


399


EXHIBIT 3

Collateral Composition of Selected CRE CDOs: Financing Transactions

Deal G-Force 2001-1 CREST G-Star 2001-1 CREST G-Star 2001-2 LNR CDO 2002-1 Anthracite RE CDO G-Force 2002-1 G-Star 2002-1

Collateral Contributor GMACCM GMACCM GMACCM LNR Anthracite GMACCM GMACCM

WARF

% below BB

AAA Class Size

% REIT

2,790 885 640 2,157 1,290 1,874 641

60% 3% 0% 65% 29% 41% 6%

16% 72% 76% 12% 41% 35% 75%

0% 50% 60% 0% 22% 0% 48%


seeking increased assets under management (asset management transactions). As noted previously, B-piece buyers’ and special servicers’ needs were the initial impetus for the creation of CRE CDOs and continue to be a powerful force in the market. However, money managers make up an important and rapidly growing second group. The CRE CDO issuer’s motivation generally drives the capital structure of the transaction, as the ratings of the CMBS collateral determine the subordination levels for the rated debt tranches and thus the return profile for the equity. The CMBS portion of financing transactions is primarily rated below investment grade and can include the nonrated, first-loss Bpiece of CMBS transactions. Although generally below investment grade, the collateral composition of financing transaction type CRE CDOs can vary considerably (see Exhibit 3). For example, one of GMAC’s CDOs, CREST G-Star 2001-1 shown in Exhibit 3, has a weighted average rating factor (WARF)9 of 885. On the other hand, LNR CDO 2002-1 contained more than 64% of single-B and below, a WARF of 2157 and no REIT debt. Reflecting their historic role in the formation of the market, financing transactions closed to date total almost two-thirds of the real estate CDO market, or about $7.7 billion, and include those shown in Exhibit 4. CREST G-Star 2001-1, a financing transaction for GMAC, is illustrative of the benefits of real estate CDOs for borrowers. Through this 9

Moody’s idealized cumulative default rate (ICDR) matrix serves to guage expected cumulative default rates for corporate credits depending on Moody’s rating and a given period of measurement (i.e., five years). The Moody’s applicable weighted average rating factor (WARF) is derived from the ICDR matrix by taking the expected cumulative default rate for a credit rated Baa2 is 3.60%, which equates to a Moody’s WARF of 360. For a complete display of Moody’s ICDR matrix and further information, see Exhibit B3 in Appendix B.

400

Fortress Commercial Mortgage Trust 1999-PC1 DRT 1999-1 Diversified REIT Trust 2000-1 Mach One CDO 2000-1 G-Force 2001-1 Ajax One, Ltd. CREST G-Star 2001-1 CREST G-Star 2001-2 G-Star 2002-1 CREST 2002-IG Anthracite RE CBO I G-Force 2002-1 Lennar CDO 2002-1 JER CDO 2002-1 Anthracite RE CBO II

Issuer

Wells Fargo Wells Fargo Bank One GMACCM ING Barings Capital Corp. GMAC I.A. GMAC I.A. GMAC I.A. Structured Credit Partners, LLC BlackRock GMACCM Lennar Partners, Inc. J.E. Roberts Cos. BlackRock

Fortress Investment Group LLC

Collateral Manager

Financing Financing Financing Financing Financing Financing Financing Financing Financing Financing Financing Financing Financing Financing

Financing

Type

Commercial Real Estate CDOs—Financing Transactions Issued as of Dec. 31, 2002


10/1/99 3/31/00 5/15/00 4/5/01 4/7/01 9/6/01 12/18/01 4/22/02 5/16/02 5/29/02 6/20/02 7/9/02 10/10/02 12/10/02

9/1/99

Closing Date

EXHIBIT 4

469 474 1,582 2,790 861 885 640 641 450 1,290 1,874 2,157 2,104 1,290 1,311

7,666.06

893

Original WARF

518.76 287.15 310.00 861.79 375.00 500.00 350.00 311.95 660.00 515.92 1,104.99 800.63 206.45 363.42

500.00

Size ($million)

52%

72% 68% 56% 16% 67% 72% 76% 75% 78% 41% 35% 12% NA 52%

65%

% AAA



401


EXHIBIT 5

Capital Structure for G-Star 2001-1—A Financing Transaction

Class

Amt. % of ($ Capital Moody’s S&P Fitch million) Structure Rating Rating Rating

Class A Notes Class B-1 Notes Class B-2 Notes Class C Notes Class D Notes Preferred Shares

360 60 15 20 15 30

72.0% 12.0% 3.0% 4.0% 3.0% 6.0%

Total

500

100.0%

Aaa A3 A3 Ba2 B2 NR

AAA A– A– BB B NR

AAA A– A– BB B NR

Rated Spread/ Coupon

Original WAL (years)

L + 47 bp 8.1 yrs. 7.08% 9.8 yrs. L + 120 bp 9.8 yrs. 9.00% 11.4 yrs. 9.00% 11.8 yrs. N/A N/A


transaction, GMAC was able to obtain an advance rate superior to that available in the repo market on its below-investment-grade CMBS portfolio. This attractive advance rate is achievable in part because of the diversity that the contributed REIT collateral provides to the aggregate portfolio. In addition, GMAC retained the vast majority of the capital structure below the BBB rated liability tranche, as do most issuers of financing transactions (an attractive feature to many investors). The CRE CDO market enabled GMAC to achieve off-balance sheet leverage, increasing its capital availability and diversifying its risk position while enhancing the yield on its below-investment-grade CMBS portfolio. The capital structure for G-Star 2001-1 is shown in Exhibit 5. Asset management transactions, the other primary type of real estate CDO, total about $4.3 billion closed to date, or just under onethird of CRE CDOs outstanding, and include those shown in Exhibit 6. A good example of an asset management transaction and the benefits to its issuer is CREST Clarendon Street 2002-1, on which MFS Investment Management (MFS) acted as collateral administrator. MFS, a predominantly total return mutual fund manager, was able to increase its assets under management while broadening its investor base, both in terms of geography (approximately one-third of the transaction was placed with European investors) and investor type (institutional versus retail). Because CRE CDOs, like all cash flow CDOs,10 are not marked to market, and because CREST Clarendon Street 2002-1 is a static pool that allows limited ongoing collateral administrator flexibility, MFS was 10

A cash flow CDO is a nonmark-to-market vehicle (either static or actively managed) that issues rated term liabilities and first-loss equity in order to finance portfolios predominantly consisting of corporate and/or structured credit products.

402


Duke Funding I CREST 2000-1 Sutter RE 2000-1 Pinstripe I CDO Putnam CDO 2001-1 Storrs CDO Ltd. Newcastle CDO TIAA RE CDO 2002-1 CREST Clarendon Street 2002-1 G-Star 2002-2 Charles River CDO

10/26/00 11/2/00 12/6/00 4/5/01 11/15/01 2/12/02 3/28/02 5/22/02 9/19/02 11/20/02 11/26/02

Ellington Capital Management, LLC Structured Credit Partners, LLC Wells Fargo Alliance Capital Management L.P. The Putnam Advisory Group David L. Babson & Co., Inc. Fortress Investment Group LLC Teachers MFS GMAC TCW

AUM AUM AUM AUM AUM AUM AUM AUM AUM AUM AUM

300.00 500.00 325.00 484.00 300.00 398.50 500.00 500.00 300.00 397.50 300.00 4,305.00

Size ($million)

Type

Issuer

Closing Date Collateral Manager

Commercial Real Estate CDOs: Asset Management Transactions Issued as of December 31, 2002

EXHIBIT 6

648 415 982 329 366 423 677 585 579 365 412 518

Original WARF

87% 79% 75% 66% 82% 81% 74% 75% 76% 87% 88% 78%

% AAA



403


EXHIBIT 7

Capital Structure for CREST Clarendon Street 2002-1—An Asset Management Transaction

Class Class A Notes Class B Notes Class C Notes Class D Notes Preferred Shares Total

Amt. % of Capital Original ($million) Structure Ratings Subordination WAL (years) 228 39 15 10 8 300

76.0% 13.0% 5.0% 3.3% 2.7% 100.0%

Aaa/AAA A3/A– NR/BBB NR/BBB NR/BB–a

24.0% 11.0% 6.0% 2.7% NA

7.3 9.9 10.0 10.7 NA

a Rated with respect to return of notional amount only. Source: Wachovia Securities, Inc.

able to focus on the roles of upfront underwriter and ongoing credit monitor rather than that of active trader. As with most asset management transactions, CREST Clarendon Street 2002-1 primarily consisted of investment-grade collateral (approximately 80%), resulting in the capital structure seen in Exhibit 7, of which MFS retained a percentage of the equity.

THE INVESTOR PROFILE: WHO BUYS REAL ESTATE CDOS The universe of CRE CDO investors includes traditional CMBS conduit buyers seeking relative value, floating-rate international buyers, and investors in other types of CDOs. Although there is no readily attainable data on investors for the entire real estate CDO market, we believe investors in Wachovia’s CRE CDOs, which account for more than 25% of CRE CDOs outstanding, are reasonably indicative of the investor breakdown. Almost two-thirds of CRE CDOs are purchased by financial institutions, such as banks, whereas another 25% are purchased by insurance companies (see Exhibit 8). Money managers (7%) and a variety of other investors make up the balance. Individual institutional investor patterns of CRE CDO investments may vary. Exhibit 9 shows a general breakdown of which financial institutions have bought what types of CRE CDO tranches. For example, although insurance companies and structured investment vehicles (SIVs) generally purchase the fixed and/or floating rate senior notes rated A or better, financial institutions (generally banks) have purchased these tranches but also lower-rated tranches down to BBs as well.


404 EXHIBIT 8


CRE CDO Investor Breakdown


EXHIBIT 9

Who Buys What in Commercial Real Estatae CDOs


Some CMBS investors have bought lower-rated CRE CDOs than they typically would in a CMBS deal. Their reasons range from the superior diversity offered by CRE CDOs over CMBS, the reputations of the CRE CDO sponsors, and the fact that CRE CDO sponsors, often keep a portion of the equity. Indeed, in deals backed by lower-rated collateral (i.e., higher WARFs), the sponsors usually retain all of the equity. A number of institutional investors prefer pure-play CRE CDOs—CRE CDOs backed exclusively by commercial real estate rather than a mix of asset types, such as asset-backed securities (ABS) and CMBS. Intuitively, this makes sense as many investors’ risk management groups are split by asset type (e.g., real estate groups and ABS groups), making the underwriting of homogenous collateral more efficient. Although it is difficult to say that this preference unequivocally translates into tighter pricing on pure-



EXHIBIT 10

405

Geographic Breakdown of Commercial Real Estate CDO Investors


play deals, CREST Clarendon Street, a pure-play CRE CDO with a WARF of 579, achieved pricing in late August 2002 of LIBOR plus 48 bp on the floating-rate 7.3-year AAA and swaps plus 135 bp on the single-A fixedrate tranche, which was regarded by market observers at the time as good. Geographically, the investor base in Wachovia’s CRE CDOs is nearly evenly split between U.S. domestic (48%) and international investors (52%) with a slight bias to the latter (Exhibit 10). Of the international investors, the British account for almost half (47%), followed by the Germans (34%), who are traditionally strong CRE investors. Belgian (4%), Austrian (4%) and Irish (3%) investors account for much of the European balance, however, investors from other European countries have also participated. Investors from around the world are included in the “Other” category, with Asian investors accounting for a growing share of the market.

INVESTOR CONSIDERATIONS While commercial real estate CDO performance has been outstanding with no CRE CDOs downgraded to date investors need to understand the rating agencies methodologies in rating CRE CDO tranches and how to stress CRE CDOs and thus judge for themselves the performance of these securities in the future.

Understanding WARF and D-Scores Two key concepts investors need to understand in evaluating CRE CDOs (and other CDOs) are a deal’s diversity, or D-Score,11 and the WARF of a 11

For a more detailed explanation of how the D-Score is derived, see Appendix A.


406


deal. Both are Moody’s concepts. The D-Score is Moody’s measure of the diversity of the CDO collateral. The WARF is a measure of the riskiness of the collateral. The WARF for each deal is derived numerically from Moody’s idealized corporate default rate (ICDR) matrix12 by taking the expected 10-year cumulative default rate for a credit and multiplying by 10,000. For instance, the 10-year expected cumulative default rate for a credit rated Baa2 is 3.60%, which equates to a Moody’s WARF of 360. Moody’s ICDR matrix serves to gauge expected cumulative default rates for corporate credits depending on Moody’s rating and a given period of measurement (e.g., five years). These two factors (WARF and D-Score) are important because they are the primary determinants of the subordination levels for CRE CDOs. The higher the WARF and the lower the DScore (diversity), the greater the subordination level and vice versa.

WARFs and D-Scores: Are CRE CDOs Unfairly Penalized? We believe Moody’s current incorporation of its WARF and D-Score methodologies for CRE CDOs (and the corresponding methodologies used by Fitch and S&P) is conservative, perhaps overly conservative. As noted above, the WARF and D-Score of a deal drives the deal’s subordination levels. One of Moody’s fundamental assumptions is that CMBS and REITs are, by their nature, diverse and therefore default probability correlations among each CMBS and REIT industry group should be relatively high. For example, as shown in Moody’s correlation matrix for CRE CDOs (Exhibit 11), Moody’s assumes significant levels of correlation for various CMBS and REITs. In contrast, Moody’s correlation matrix for corporate CDOs (not shown) assumes zero correlation among the various corporate sectors and 14% correlation between corporate assets within the same sector. Intuitively, one would think that the more diverse, the lower the correlation, not the higher. Moody’s reasoning is that because there are so many different properties backing CMBS and REIT cash flows that due to the law of large numbers, there is little diversification benefit from combining the tranches of different CMBS and different REIT debt in a CRE CDO. Moreover, they give little diversification “credit” for the different vintages of the CMBS in a CRE CDO even though research has historically demonstrated otherwise.13 Finally, the ICDR is based on an empirical study of corporate defaults, which have historically been far in excess of CMBS defaults. 12

For a complete display of Moody’s ICDR matrix, see Appendix B. According to Esaki, “The timing and total defaults of a cohort (or vintage) are highly dependent on its position in the real estate cycle.” See, Howard Esaki, “Commercial Mortgage Defaults 1972–2000,” Real Estate Finance (Winter 2002).

13

407 7.0% 7.0% 7.0% 7.0% 7.0% 7.0% 8.0%

REIT Multifamily

REIT Office

REIT Retail

REIT Industrial

REIT Healthcare

REIT Self-Storage

REIT Diversified

14.5%

8.0%

7.0%

7.0%

7.0%

7.0%

7.0%

7.0%

7.0%

13.0%

11.0%

13.0%

11.0%

17.0%

Below investment grade (BIG). Source: Moody’s Investors Service, Inc.

a

7.0%

11.0%

REIT Hotel

ABS-CMBS Large Loan BIGa

9.0%

ABS-CMBS Large Loan

9.0% 11.0%

ABS-CMBS CTL BIGa

ABS-CMBS CTL

12.0% 14.5%

ABS-CMBS Conduit BIGa

Conduit Conduit BIGa

Large Loan BIGa

9.0% 11.0%

Large Loan

9.0% 10.0%

6.0%

6.0%

6.0%

7.0%

6.0%

6.0%

6.0%

7.0%

6.0%

6.0%

6.0%

17.0% 17.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

7.0%

7.0%

Hotel

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

8.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0% 18.0%

10.0% 11.0% 21.5% 25.0%

9.0% 10.0% 18.0% 21.5%

21.5% 25.0% 10.0% 11.0%

18.0% 21.5%

11.0% 13.0% 11.0% 13.0%

9.0% 11.0%

CTL

CTL BIGa

ABS-CMBS

8.0%

6.0%

6.0%

6.0%

6.0%

6.0%

18.0%

6.0%

6.0%

6.0%

6.0%

6.0%

7.0%

7.0%

Multifamily

Moody’s Correlation Matrix—Commercial Real Estate Sectors

ABS-CMBS Conduit

EXHIBIT 11

7.0%

7.0%

Retail

6.0%

6.0%

6.0%

6.0%

6.0%

8.0%

6.0%

6.0%

6.0%

8.0%

6.0%

6.0%

6.0%

6.0% 18.0%

18.0%

6.0%

6.0%

6.0%

6.0%

6.0% 17.0%

6.0% 17.0%

7.0%

7.0%

Office

8.0%

6.0%

6.0%

18.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

7.0%

7.0%

8.0%

6.0%

18.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

7.0%

7.0%

8.0%

18.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

6.0%

7.0%

7.0%

13.0%

8.0%

8.0%

8.0%

8.0%

8.0%

8.0%

8.0%

6.0%

6.0%

7.0%

7.0%

8.0%

8.0%

SelfIndustrial Healthcare Storage Diversified

REITs



408


The D-Score and the WARF are important because they are the two major drivers of subordination in CDOs. Because rating agency methodologies assume CMBS and REITs are more correlated than corporates, a corporate CDO with low-quality collateral could have the same level of subordination for its AAA tranche as a CRE CDO with much better quality collateral.14 For example, a typical assets-under-management (AUM) CRE CDO will have a WARF of approximately 400–600 (between Baa2 and Baa3) and a D-Score of 8–10 (low diversity). This type of transaction would likely be able to attain a AAA tranche equal to approximately 80% of its entire capital structure, or 20% subordination to the AAA tranche. A corporate CDO with a WARF of about 2200 (B1), much lower quality collateral, could have a D-Score of 45–55 (highly diversified) and thus would also be able to achieve a AAA tranche equal to 80% of the deal, or 20% subordination. As is readily apparent, although the CRE CDO collateral has a much lower default probability (measured by its WARF) than the corporate collateral, the corporate CDO achieves a comparably sized AAA tranche because the rating agencies assume corporate collateral is so diverse. Looked at from another perspective, if a CRE CDO had low-rated collateral with a WARF of 2200, like the corporate high-yield CDO, Moody’s would require as much as 65% subordination to the AAA versus 20% for a high-yield corporate CDO with the same 2200 WARF because of the CRE CDO’s low diversity score (8 D-Score). Although a complete refutation of Moody’s methodology is beyond the scope of this article, the following analysis is indicative of why we believe this approach to rating CRE CDOs is overly conservative. In Exhibits 10 and 11, we show the break-even cumulative underlying collateral default rates necessary to “break” or cause discount margin impairment on the AAA rated and A- rated notes of a typical high-yield corporate CDO and a CRE CDO. We also show the break-even level of cumulative defaults necessary to “break” or cause yield impairment on the BBB rated notes of each CDO. As is apparent, the cumulative default rates necessary to break the high-yield corporate CDO are higher than those needed to break the CRE CDO (see column 4 in Exhibits 12 and 13). One might think at first glance that the high-yield corporate CDO is the more robust deal because the AAA tranche sizes are similar and the CRE CDO has less 14

Moreover, of the small number of industry groups (11) Moody’s assigns for REITS and CMBS, only about eight are common, reducing their diversity score even further. The other three groups, healthcare REIT, hotel REIT, and diversified REIT are used less commonly.


409


EXHIBIT 12

High-Yield Corporate CDO: Breakeven CDR Typical High-Yield Transactiona

Class Rating AAA A– BBB

Asset WAL

ICDR

Cumulative Default Rate

As a Multiple of ICDR

Tranche Size

3.3 3.9 4.1

16.2% 17.8% 18.3%

64.8% 53.1% 48.8%

4.0× 3.0× 2.7×

67.0% 10.7% 3.5%

a

Assumptions: (1) Collateral: 100% B2/B rated high-yield bonds; (2) Weighted average price: 91%; (3) Weighted average coupon: 10.75%; (4) Moody’s diversity score: 45; (5) Annual fees and expenses as a percentage of total collateral amount: 0.48%; (6) Defaults: Begin 0.5 year from closing date at the rates shown; (7) Immediate recoveries; (8) Constant annual prepayment rate: 10%; and (9) Reinvestment period: five years. Source: Wachovia Securities, Inc.

EXHIBIT 13

Commercial Real Estate CDO—Break-even CDR CRE CDO

Class Rating AAA A– BBB

Asset WAL

ICDR

Cumulative Default Rate

As a Multiple of ICDR

Tranche Size

7.7 7.8 7.9

4.5% 4.6% 4.6%

55.6% 29.6% 18.5%

12.3× 6.4× 4.0×

76.0% 13.0% 5.0%


equity than the high-yield corporate CDO. However, as shown in column 5 in each exhibit for each deal, the CRE CDO has a much higher multiple of ICDR15 than the high-yield corporate CDO at each tranche level. Moody’s ICDR is the expected-case estimate of how many cumulative defaults the respective portfolios should experience given the weighted average life (WAL) and WARF of the collateral. For example, the AAA bond in the high-yield corporate CDO breaks at a cumulative default rate of 64.8%, which is approximately four times Moody’s ICDR of 16.2%. The results show that the CRE CDO bonds are 1.5–3.1 times as strong as the high-yield corporate CDO bonds (column 4 in Exhibit 14).

15

For a complete display of Moody’s ICDR matrix, see Appendix B.


410


EXHIBIT 14 CRE CDO Quality versus Corporate High-Yield CDO Quality Based on Moody’s ICDR Class Rating

High-Yield Transaction ICDR Multiple (HYM)

CRE CDO ICDR Multiple (CRE CDO Multiple)

CRE Multiple/ HYM

4.0× 3.0× 2.7×

12.3× 6.4× 4.0×

3.1× 2.1× 1.5×

AAA A– BBB


EXHIBIT 15

Dual Subordination of a CRE CDO

Underlying CMBS

RE CDO

Rating Subordination

Rating Subordination

BBB+ BBB+ BBBBB+ BB

AAA A– BBB BB Equity

12% 10% 9% 7% 5% Initial Subordination

25% 10% 6% 3% NA Secondary Subordination


CRE CDOs versus High-Yield Corporate CDOs: Other Considerations CRE CDOs also offer a number of beneficial features that high-yield corporate CDOs lack. CRE CDOs, unlike high-yield corporate CDOs, offer investors two levels of subordination: structural and asset level. For example, the BBB tranche of a hypothetical real estate CDO (Exhibit 15) has 6% subordination at the structural level. Assuming this real estate CDO was backed 100% by BB+ CMBS, the CDO tranche would benefit further from the initial 7% subordination of the BB+ CMBS tranches at the asset level. As a result, before a real estate CDO tranche becomes impaired, subordination on the underlying CMBS must first be eaten through by losses on the underlying loans. Indeed, the CMBS tranche itself can withstand losses on the underlying loans before experiencing losses of its own. Thus, real estate CDO equity, unlike



411

high-yield corporate CBO equity, can withstand relatively stressed loan level losses without impairment due to the subordination of the underlying CMBS collateral. Moreover, unlike a high-yield CBO, the collateral in real estate CDOs is backed by tangible assets (the underlying properties) helping to support higher recovery rates when defaults occur. For example, various studies have shown average historic CRE loss severity rates of 31– 38%.16 In contrast, in the past three years corporate bond loss rates have generally been closer to the recovery rates of CMBS (60–70%).17 Corporate bonds also have greater event risk than CMBS.18

CRE CDOs versus Corporate CDOs: Downgrades to Date Although only a few years of performance history exist, the outperformance of CRE CDOs over corporate CDOs is telling. Although no CRE CDOs have been downgraded, significant numbers of high-yield corporate CDOs have been downgraded (see Exhibit 16). The relative performance of the collateral (Exhibits 17 and 18) has been important in the solid performance of CRE CDOs versus highyield corporate CDOs. However, we believe the rating agencies’ conservative approach to CRE CDOs, as well as the other factors noted previously, have played a key role in the outperformance of CRE CDOs and will continue to do so in the future.

Investor Considerations: CRE CDOs versus CMBS One of the biggest attractions of CRE CDOs to CMBS mezzanine investors is the extra spread offered by CRE CDO mezzanine tranches. Similarrated single A CRE CDOs typically trade 50 bp wider than similarly rated CMBS, whereas CRE CDO BBBs trade approximately 75 bp–100 bp wider. Although some of this certainly represents the lower liquidity in mezzanine CRE CDOs, a portion of it represents limited investor 16

See Howard Esaki, “Commercial Mortgage Defaults: An Update,” Real Estate Finance (Spring 1999). It is important to note that these severities are averages for all commercial real estate. Different property types typically exhibit significantly higher and lower severities. For example, hotels often exhibit average severities of around 50%, whereas apartment properties typically have 20% severities. However, CMBS have only had about 10% hotels historically. Following the events of 9/11, hotels have made up an even smaller percentage of CMBS deals. 17 Moody’s Investors Service, Inc. 18 Event Risk in the CMBS market up until the World Trade Center disaster in New York City had been virtually nonexistent other than several acts of fraud. However, in most cases in the CMBS market, with some notable exceptions, loans have been insured against terrorism.

412

0.50%

94.63% 80.95% 86.57%

0.67%

88.68%

3.98%

14.29%

Aa3

66.67%

7.55%

2.49%

A1

89.47%

1.99%

A2

91.67%

5.26%

16.67%

1.89%

0.50%

A3

85.71%

4.17%

0.50%

Baa1

93.38%

0.50%

Baa2

82.67%

2.21%

8.33%

1.89%

Baa3

60.00%

3.47%

0.74%

Ba1

83.33%

10.00%

5.94%

2.21%

2.08%

Ba2

92.31%

1.67%

1.49%

Ba3

71.19%

2.56%

1.67%

1.49%

1.47%

B1

Source: Moody’s Investors Service, Inc., “Credit Migration of CDO Notes, 1996–2001.”

Caa1

B3

B2

B1

Ba3

Ba2

Ba1

Baa3

Baa2

Baa1

A3

A2

A1

Aa3

Aa2

Aa1

Aaa

Aa2

B2

55.56%

1.28%

3.33%

20.00%

0.99%

Aa1

OneYear

Aaa

Arbitrage Cash Flow High Yield Corporate CBO Transition Matrix (1996–2001)

EXHIBIT 16

85.29%

11.11%

6.78%

1.67%

0.99%

B3

30.00%

2.94%

11.11%

10.17%

1.67%

0.99%

Caa1

20.00%

2.94%

1.69%

1.28%

1.67%

10.00%

14.29%

Caa2

10.00%

5.08%

1.67%

Caa3

40.00%

8.82%

22.22%

5.08%

1.28%

3.33%

Ca/C

1.28%

1.98%

2.08%

5.26%

8.33%

2.99%

4.76%

4.70%

W/R


413

88.24% 9.30% 1.55% 0.51% 0.22%

80.59% 13.17% 4.09% 1.59% 0.51%

100.00% 19.41% 4.70% 2.04% 0.79% 0.34%

AA

100.00% 11.76% 1.45% 1.06% 0.40% 0.11%

AAA

81.99% 9.19% 1.27% 1.02% 2.08%

89.16% 5.35% 1.01% 0.22%

A

0.13% 83.54% 7.95% 1.87%

0.09% 91.62% 4.65% 1.00%

BBB

CMBS Transition Matrices (1991–2001)

0.57% 86.65% 6.46% 4.17%

0.21% 92.73% 3.99% 2.17%

BB

Source: Fitch Ratings, “Structured Finance Rating Transition Study.”

One Year: AAA AA A BBB BB B CCC–C Two Year: AAA AA A BBB BB B CCC–C

EXHIBIT 17

0.34% 1.75% 84.86% 12.50%

0.07% 0.61% 92.58% 3.26%

B

4.76% 77.08%

0.23%

1.88% 92.39%

0.14%

CCC–C

0.17% 4.17%

2.17%

0.10%

D


414 92.08% 0.08% 0.10% 0.05%

95.80% 0.10% 0.08%

AAA

7.28% 86.35% 6.16% 0.28% 0.12%

3.84% 92.32% 3.03% 0.25%

AA

0.43% 12.83% 86.14% 10.70% 0.73% 0.27%

0.18% 7.23% 92.36% 5.38% 0.18%

A

0.21% 0.70% 6.84% 81.94% 10.68% 2.41%

0.18% 0.31% 4.20% 89.53% 6.88% 0.41%

BBB

Corporate Transition Matrices (1991–2001)

0.04% 0.43% 4.76% 79.00% 13.40%

0.17% 3.34% 83.57% 9.47%

BB

Source: Fitch Ratings, “Structured Finance Rating Transition Study.”

One Year: AAA AA A BBB BB B CCC–C Two Year: AAA AA A BBB BB B CCC–C

EXHIBIT 18

0.03% 0.94% 4.25% 79.62% 27.87%

0.03% 0.02% 0.76% 4.73% 84.77% 12.30%

B

0.13% 0.85% 2.79% 2.41% 63.93%

0.10% 0.44% 2.95% 3.50% 63.93%

CCC–C

0.18% 0.47% 2.43% 1.88% 8.20%

0.04% 0.29% 1.70% 1.85% 23.77%

D




415

understanding of mezzanine CRE CDOs—a perceived complexity premium, if you will. We would expect the extra spread offered by mezzanine CRE CDOs to narrow over time. Although pricing on AAA- rated CRE CDOs tracks that of AAArated CMBS,19 investors are attracted to CRE CDOs versus CMBS for other reasons. As in the case of CRE CDOs versus corporate high-yield CDOs, CRE CDOs offer CMBS investors an extra level of structural subordination.20 Investors also favorably regard the fact that the collateral behind CRE CDOs is more diverse than the collateral backing CMBS. A CRE CDO typically includes bonds of 15–25 CMBS conduit deals and a similar number of unsecured REIT bonds, which collectively have more than $100 billion of real estate supporting the bonds. This compares with the typical CMBS conduit deal that typically has approximately $1 billion of loans supporting the transaction and no REIT debt. As a result, CRE CDOs are more granular than CMBS deals, as the largest collateral positions rarely constitute more than 5% of a deal. In contrast, CMBS conduit deals frequently have single loans that constitute 7–15% of the deal. Because CRE CDOs are backed by a number of different deals, the collateral is not only diverse across a greater variety of real estate loans than the typical CMBS but also, and most important, across a variety of vintages—a key determinant of commercial mortgage defaults.21

Advantages and Disadvantages of REIT Debt Inclusion The REIT portion of CRE CDOs is nearly always investment grade and frequently constitutes 30–50% of a deal. Notably, CMBS conduit deals, 19

Later in this article we will see how CRE CDOs trade. The vast majority of CMBS collateral consists entirely of whole loans with no structural subordination. However, a few loans backing CMBS are occasionally split into an “A” and a subordinated “B” note with only the “A” note being placed in the CMBS deal and the subordinated “B” note being held by a third party outside the deal. However, the vast majority of CMBS collateral consists entirely of whole loans with no structural subordination. 21 Vintage is a key component of commercial real estate defaults as it determines when in the real estate cycle a commercial loan is originated. For example, a loan originated when rents are at a peak and vacancies low may exhibit more stress than one originated at the bottom of the cycle. Although rating agency and B-piece discipline have forced underwriters to originate loans more conservatively at all times, in the real estate cycle vintage is still likely to be an important factor affecting defaults. For an analysis of the variation in defaults experienced by commercial real estate loans originated in different vintages, see Esaki, “Commercial Mortgage Defaults 1972–2000.” 20


416


which contain no REIT debt, rarely have more than 15–20% of investment-grade loans. In most CRE CDOs, REIT senior unsecured debt is included in the collateral portfolio to enhance the diversity and therefore the stability of the transaction. Moody’s has only three principal CMBS categories it uses in its diversity score model, and thus the inclusion of REITs, for which there are eight categories for the purposes of diversity scores, reduces the aggregate portfolio default probability because REITs are appropriately assumed to be less correlated with CMBS. The primary purpose of including individual REITs is not to enhance the yield of the asset portfolio but rather to reduce the overall portfolio variance. However, the universe of investment-grade equity REITs that are appropriate for a CRE CDO (typically, healthcare and hospitality REITs are viewed as being too rating volatile for CRE CDOs) includes only 40–50 companies. Because inclusion of REITs outside of this universe creates an adverse selection problem, most CRE CDOs exhibit relatively substantive overlap across their REIT portfolios. Although we view this as an issue worthy of being addressed, we feel strongly that the benefits of including REITs in CRE CDOs greatly mitigate any overlap concerns. The positive credit fundamentals exhibited by the majority of investment-grade REITs and the debt covenant packages that mandate that such fundamentals be preserved are the primary reasons why there has never been a default by an investment-grade equity REIT. The weighted average debt to total assets for an investment-grade REIT is usually about 50%, whereas all REITs covenant to maintain debt to total asset ratios lower than 60% (and secured debt to total assets of less than 40%). This compares favorably with the average loan-to-value (LTV) of loans included in CMBS conduit transactions, which is typically 65–75% and we believe more than compensates investors for the unsecured nature of REIT debt. In addition, unlike CMBS, the portfolio of properties owned by a REIT can continually change, as management teams have the ability to reduce/increase exposure to out-of-favor/attractive markets and properties. REIT asset portfolios are also generally of higher quality (based on property values) than those of CMBS.

The Benefits of Overcollateralization and Interest Coverage Tests In addition to levels of structural subordination, CRE CDOs provide another level of protection that CMBS do not offer, overcollateralization (OC) and interest coverage (IC) tests. These performance tests further enhance interest and principal payments. When a CRE CDO breaches



417

either an OC or IC trigger, collateral interest proceeds that otherwise would have been used to pay interest on the subordinate notes are used, along with any collateral principal proceeds, to redeem senior notes in an amount sufficient to bring the breached test or tests back into compliance. Such interest diversion preemptively deleverages the transaction, thus reducing its risk profile while benefiting the senior noteholders. Although this would create negative convexity for the senior notes if they were fixed, 80% of the deal is floating rate and senior. Therefore, this is typically not an issue as the bonds being paid down after the breach of a trigger will likely be floaters. This mechanism can create situations in which the subordinate notes are forced to capitalize interest, or payment-in-kind (PIK), which is acceptable pursuant to the terms of such subordinate notes and does not constitute an event of default. The protection afforded to the senior notes in a CRE CDO by the OC and IC triggers is also available to the subordinate notes once the senior notes have been redeemed in full.

Investor Considerations: CRE CDOs versus ABS CDOs Beyond the fact that CMBS collateral has thus far outperformed ABS collateral, CRE CDOs exhibit greater average life stability and better convexity characteristics than ABS CDOs because of the significant prepayment protection in CMBS collateral.22 Some ABS collateral, such as home equity ABS, has greater prepayment sensitivity and negative convexity.

CRE CDO PROTECTION: A HISTORIC CONTEXT Rather than dwell on the protections afforded by CRE CDOs in the context of the solid performance of the CRE and CMBS markets for the past several years,23 we thought it more conservative to look at those protections in the context of the more severely stressed CRE markets of the 1980s and early 1990s. As noted previously, CRE CDOs, due to borrower motivations, are generally backed by, on average, investment-grade CMBS and REIT collateral (BBB– or higher) in the case of assets under management (AUM) 22

The vast majority of CMBS, particularly those issued after 1998, have strong prepayment protection in the form of lockout, Treasury defeasance, yield maintenance agreements, and other prepayment penalties. 23 For a current in-depth analysis of the CMBS and commercial real estate markets and their outlook, see Wachovia Securities, Inc.’s CMBS and Real Estate research reports, The Outlook for CMBS and Commercial Real Estate, October 2002 and The Outlook for Commercial Real Estate by Property Sector, October 2002.


418


CRE CDOs or noninvestment-grade collateral in the case of many financing transactions. In the case of CRE CDOs backed by investmentgrade CMBS, such as many AUM CRE CDOs, the subordination levels of the lowest investment-grade CMBS collateral in those CRE CDOs in nearly all cases would be sufficient to absorb the average 10-year historic CRE losses experienced over the past three decades. The average 10-year cumulative default rate for CRE loans according to an update of a prominent and widely referenced study was 18.4% beginning in 197224 (Exhibit 19). Using the severity calculations in that study, the average cohort lost 4.9% of its original balance.25 Because BBB– CMBS subordination levels range from 5% to 10% and BBB CMBS subordination levels range from 6% to 12%, “investment-grade CMBS would be EXHIBIT 19

Lifetime Default Rates by Origination Cohort (by loan count)

Source: Howard Esaki, “Commercial Mortgage Defaults 1972–2000,” Journal of Real Estate Finance, Winter 2002. 24

Esaki, “Commercial Mortgage Defaults: An Update.” To calculate losses, the study found that 59% of loans entering default were eventually liquidated, 39% were restructured, and 1% became current again. Foreclosed loans were found to have a 34% severity rate (including interest, expenses, and principal) and the assumption of 17% severity was made for restructured loans—the latter assumption sometimes being disputed with some market participants believing that restructured loans have close to a 0% severity and others arguing that it should be higher. Given the assumptions, the loss of 4.9% is calculated as follows (18.4% × 0.59% × 0.34%) + (18.4% × 0.39% × 0.17%) = 4.9% 25



EXHIBIT 20

419

Historic Losses versus CMBS Tranche Subordination

Source: Wachovia Securities, Inc

protected against the average loss of commercial real estate origination cohorts of the last 30 years”26 (Exhibit 20). In short, the investment-grade CMBS collateral backing CRE CDOs, in and of itself, without the additional structural subordination and excess spread of the real estate CDO, should be sufficient to absorb average historic losses as calculated previously. Moreover, even if we assume losses hit the extraordinarily high 8.5% level experienced by the 1986 cohort wiping out the BBB- subordination we would still be afforded the structural subordination in the deal to protect the higherrated real estate CDO tranches. This does not mean that some CRE CDO tranches would not experience losses in this case, but rather that the losses would be mitigated by the additional subordination in the structure. Moreover, more seasoned CMBS tranches have higher subordination levels due to deleveraging, increasing the ratings threshold that would be pierced by the level of defaults cited above.

A METHODOLOGY TO DISCRIMINATE AMONG CRE CDOS To provide the investor with a useful tool to help determine relative value and apply the historic losses and severities noted previously to specific deals and situations, we stressed two deals, CREST Clarendon Street 2002-1 and G-Force 2002-1, with the scenarios shown in Exhibit 21. “Curve” is the level and timing of annual defaults and severities as 26

Esaki, “Commercial Mortgage Defaults: An Update.”


420 EXHIBIT 21


Stress Methodology Assumptions CREST Clarendon Street 2002-1

G-Force 2002-1

Scenario 1

Base Case 0% GDR

Base Case (0 Defaults) 0% GDR

Scenario 2

CMBS: 0.5× Curve REIT: 0% GDR

CMBS: 0.5× Curve

Scenario 3

CMBS: 1.0× Curve REITs: 2.86% GDR (Moody’s ICDR)

CMBS: 1.0× Curve

Scenario 4

CMBS: 1.5× Curve REITs: 5.97% GDR (ICDR + 1 St. Dev.)

CMBS: 1.5× Curve

Scenario 5

CMBS: 2.0× Curve REITs: 9.08% GDR (ICDR + 2 St. Dev.)

CMBS: 2.0× Curve

Scenario 6


NA

Scenario 7


NA

Note: GDR: Gross default rate; ICDR: Idealized cumulative default rate. Source: Wachovia Securities, Inc.

defined in the study27 and the gross default rate (GDR) is based on Moody’s ICDR for REITs. For different scenarios, we assume various multiples of the curve to increase or decrease the stress. Indeed, this is a common methodology employed by many CRE CDO investors to determine the strength of a CRE CDO. The AAAs of CREST Clarendon Street 2002-1, which are backed mostly by investment-grade collateral with a WARF of 579, remain completely intact in Scenarios 2 and 3, where we assume stresses equal to or half of the historic averages found in the study (Exhibits 22 and 23). The same is true for the AAAs of G-Force 2002-1, which are backed by lower-quality collateral with a WARF of 1874 (Exhibit 23). However, the higher quality of the CREST Clarendon Street CRE CDO becomes apparent if we look at the mezzanine classes. Here, under one times the historic stress, the BBs of CREST Clarendon Street 2002-1 remain intact, whereas the BBBs of G-Force 2002-1 are hit. As we increase the stress levels further in Scenarios 4–7, the robustness of CREST Clarendon Street 2002-1 becomes more apparent. Indeed, its BBs remain intact under Scenario 4 when the AAAs of GForce 2002-1 begin to erode. 27

Source: Esaki, “Commercial Mortgage Defaults: An Update.”


421


EXHIBIT 22 Scenario

CREST Clarendon Street 2002-1 Default Stresses Tranche

WAL

DM/Yield

Principal Win

Principal Loss

1

AAA BB

7.31 10.42

0.48% 10.12%

6/05–6/12 9/12–12/13

— —

2

AAA BB

7.39 11.77

0.48% 10.12%

6/05–6/12 3/13–12/13

— —

3

AAA BB

7.46 11.21

0.48% 10.11%

9/03–6/12 9/13–12/13

— —

4

AAA BB

7.55 12.47

0.48% 10.05%

9/03–9/12 12/13–3/16

— —

5

AAA BB

7.63 NA

0.48% –5.39%

9/03–9/12 NA

— 100%

6

AAA BB

7.58 11.95

0.48% 10.07%

9/03–9/12 12/13–12/15

7

AAA BB

7.70 16.28

0.48% –1.01%

9/03–9/12 NA

— — — 100%

Source: Standard & Poor’s Conquest and Wachovia Securities, Inc.

EXHIBIT 23 Scenario

a

G-Force 2002-1 Default Stresses Tranche

WAL

DM/Yield

Principal Win

Principal Loss

1

AAAa BBB

8.04 10.70

0.46% 8.31%

7/10–6/11 11/12–3/14

— —

2

AAA BBB

8.43 13.12

0.46% 8.31%

7/10–4/12 8/15–11/16

— —

3

AAA BBB

9.62 NA

0.46% –8.37%

7/10–2/15 NA

— 100%

4

AAA BBB

10.91 NA

–1.17% –23.57%

8/10–3/25 NA

23% 100%

5

AAA BBB

10.65 NA

–7.25% –31.19%

12/10–3/25 NA

70% 100%

Represents “long” AAA of capital structure. G-Force 2002-1 also includes an AAA rated money market tranche that constitutes 16% of the capital structure. Source: Standard & Poor’s Conquest.


422


Although this type of analysis is useful for CRE CDO investors to distinguish between the quality of CRE CDOs, it is a conservative stress scenario. These stresses assume the same level of defaults and the timing of defaults occur simultaneously across all of the different CMBS deals backing the CRE CDOs. We know from history that defaults and losses of different CRE properties in different states, let alone vintages, do not occur at the same time. Furthermore, this assumption of simultaneous defaults and losses more negatively affects CRE CDOs with lower-rated collateral and higher subordination, such as G-Force 2002-1. If defaults and losses simultaneously rise to a level that breaches all of the subordination levels of all of the underlying CMBS collateral and consume all of the tranches making up the CRE CDO, then no level of secondary structural subordination in the CDO will protect the CRE CDO tranches. Thus, the lower collateral subordination characteristic of lower-rated CMBS collateral is more easily breached and the low-rated collateral tranches are entirely consumed by defaults and losses as we simultaneously increase defaults and losses. In reality, we know that defaults and losses of different CMBS may or may not occur at any time and indeed may never occur in some cases. In this more realistic scenario, lower-rated higher-subordination collateral will likely fare better than indicated in the aforementioned stress tests. It is also important to note that running one set of assumptions for all real estate loans ignores three key factors in commercial real estate: vintage, location and property type.

HOW CRE CDOs TRADE In 1999 and 2000, CRE CDOs priced behind high-yield bonds, leveraged-loan CDOs and ABS CDOs, however, as performance diverged and CRE CDOs outperformed, spreads on corporate-backed CDOs and ABS CDOs began to lag. Currently, CRE CDOs trade 10–15 bp tighter than leveraged loan and ABS CDOs. Floating rate CRE CDOS trade at LIBOR plus 47–50 bp versus LIBOR 60–65 bp for leveraged loans and ABS CDOs. The difference is more dramatic further down the yield curve. BBB CRE CDOs trade as much as 50– 75 bp tighter than leveraged-loan and ABS CDOs. The relative value of CRE CDOs is also notable in secondary trading. CRE CDO AAA bonds trade on top of new issue pricing (e.g., 48-discount margin) with a tight bid/ask spread that is somewhat wider than AAA conduit CMBS (e.g., 1–2 bp). On the other hand, high-yield and ABS CDO AAA product typically trade 8–10 bp wider than new issue product with an 8–10 bp bid/ask as the deal ages beyond one or two quarters.



EXHIBIT 24

423

AAA CRE CDO Spreads versus AAA CMBS Spreads (versus Swaps)


Secondary trading volume in CRE CDOs has been increasing, particularly in 2002, however, it remains a less liquid market than the CMBS conduit market. Nevertheless, according to our trading desk, floating rate CRE CDOs, particularly the AAA and A classes, have similar if not better liquidity than CMBS large-loan floating rate bonds. We expect liquidity in CRE CDOs to continue to improve as the sector matures further and transparency improves. For example, transparency both at primary issuance and thereafter has improved noticeably in many static pool transactions, allowing investors to reunderwrite each bond in the portfolio before making a purchase decision. Ongoing surveillance and modeling on third-party systems such as Trepp, Charter Research, and Intex Solutions, have helped facilitate tighter pricing relative to CMBS and substantial secondary trading support for CRE CDOs (Exhibit 24).

PERFORMANCE TO DATE OF THE CRE CDO SECTOR Since the inception of the CRE CDO market in 1999, the corporate bond market has faced a marked deterioration in credit that has resulted in increased default rates and decreased recovery rates. Since 1999, the Moody’s Corporate Default Index has averaged 3.3%, versus a 10-year trailing average default rate of 1.97% (see Exhibit 25). Although the Moody’s Corporate Default Index peaked at 5.34% in July 1991, in February 2002 it reached a recent high of 4.89%. The U.S. Speculative Grade Index has been even more volatile. The average since 1999 has been 7.52% compared with a 10-year trailing average of 4.73%. The recent high for this index was in February 2001 at 11.43%; although


424

EXHIBIT 25


U.S. Corporate Bond Defaults

Source: Moody’s Investors Service, Inc.

still shy of the 13.00% registered in July 1991, it is dramatically higher than recent averages. In addition, the volume of defaults has increased. The average per issuer default was $844 million in 2002 compared with $390 million in 2001 and $287 million in 2000. At the same time, corporate recovery rates have lagged. Over the past two years, recoveries for senior unsecured bonds have averaged 23% versus 40% in the three years before that. Although we are starting to see some improvement in the corporate bond sector, high default rates and low recovery rates have put many of the high-yield bond-backed and investment grade-backed CDOs under substantial pressure. In the first nine months of 2002, Moody’s downgraded 407 tranches of 146 CDOs in the United States alone. This activity exceeds all prior years combined. Based on Moody’s watch list for downgrades, negative rating actions will likely continue. As of September 30, 2002, 87 CDO deals and 167 CDO tranches were on watch for downgrade, primarily in the high-yield CBO sector. Given the volatile state of the corporate CDO market, the performance of CRE CDOs during this same period stands in stark contrast. As discussed earlier, there have been no CRE CDO downgrades to date. This is a direct result of the performance stability of CRE securities. Earlier in this report, we showed the rating migration stability that CMBS have exhibited. This explains why, based on a broad sample of 15 CRE CDOs issued between November 2000 and July 2002, the WARF migration


425


average was approximately 1% (990 WARF–1000 WARF). The diversity score exhibited similar robustness, which is partially due to the static pool nature of the CRE CDO transactions. In an environment in which equity in many high-yield and investment-grade CDOs has been locked out due to a breach of a cash flow diversion trigger, CRE CDO equity has averaged cash-on-cash returns of 16–20% based on our random sample.

CONCLUSION CRE CDOs have emerged as one of the most attractive asset classes in the CDO market. Although still a young market, the solid performance thus far has been encouraging. The strong performance of the underlying collateral (i.e., CMBS and REITs) has been important, however, the rating agencies’ particularly conservative approach to CRE CDOs has and should continue to benefit the sector. We hope this article will serve as a useful point of entry and analytical framework for investors to discover the opportunities offered by this new and rapidly growing fixed-income sector.

APPENDIX A: MOODY’S D-SCORE28 Moody’s assumes the default risk of ABS in different sectors is more highly correlated than that of two corporate credits in different sectors. The Moody’s Alternative Diversity Score model was developed to analyze bonds from sectors with correlated default risk. Each bond in the portfolio is assigned a default probability and expected loss rate based on its rating, WAL and applicable Moody’s recovery rate. The diversity score model is based on matching the mean and standard deviation of the portfolio’s expected loss profile. The formula used to calculate the alternative Diversity Score is as follows:  n   n  p i F i  q i F i  i = 1  i = 1  D = -------------------------- --------------------------------------n n pi qi pj qj ( Fi Fj ) ρ ij

∑

∑∑

i = 1j = 1

28

Moody’s Investors Service, Inc.

∑


426


where n equals the number of bonds in the portfolio; bond i has a face value of Fi, a default probability of Pi and a survival probability of Qi (Qi = 1 – Pi); and the correlation coefficient between bonds i and j is Pij. (The correlation coefficient is based on the Moody’s default correlation matrix provided earlier in this article.) Using this formula, the entire collateral pool can be expressed as D homogenous securities with uncorrelated default risk. This new pool has the following characteristics:  n  Average face value =  F i ⁄ D i = 1 

∑

 n  p i F i Average default probability =  i = 1 

∑

 n  ⁄ F i i = 1 

∑

APPENDIX B: MOODY’S WEIGHTED AVERAGE RATINGS FACTOR The weighted average ratings factor (WARF) is a Moody’s concept that is used to estimate the expected loss of a given pool of assets. The WARF is calculated based on the Moody’s Rating Factor Table (Exhibit B1) and the weighting of each security in a portfolio by its par amount. The lower the WARF, the fewer defaults the structure must withstand, resulting in greater leverage. For example, under Moody’s stress scenarios, a portfolio with a WARF of 2720 will be stressed to withstand 27.20% cumulative defaults over a 10-year period. EXHIBIT B1 Rating Aaa Aa1 Aa2 Aa3 A1 A2 A3 Baa1

Moody’s Rating Factors Rating Factor 1 10 20 40 70 120 180 260

Rating

Rating Factor

Ba1 Ba2 Ba3 B1 B2 B3 Caa1 Caa2


940 1350 1780 2220 2720 3490 4770 6500


427


Once the threshold in a structure has been breached, the asset manager must improve or maintain the WARF in all subsequent trades. Generally, this means the asset manager will either sell the asset that caused the breach and replace it with a higher-rated asset (but with potentially lower par) or sell a different asset and replace it with another higherrated asset to lower the WARF. Exhibit B2 shows shows typical WARF scores for select CDO types. EXHIBIT B2

WARF Scores of Select CDOs

CDO Type

WARF

Equivalent Rating

Plain Vanilla CDO ABS CDO Balance Sheet CDO

2500–2720 260–610 600–1600

B1/B2 Baa1–Baa3 Baa3–Ba3


EXHIBIT B3 Year

Assets—Moody’s Idealized Cumulative Default Rates (ICDR Matrix) 1

2

3

4

5

6

Aaa

0.00005%

0.00020%

0.00070%

0.00180%

0.00290%

0.00400%

Aa1

0.00057%

0.00300%

0.01000%

0.02100%

0.03100%

0.04200%

Aa2

0.00136%

0.00800%

0.02600%

0.04700%

0.06800%

0.08900%

Aa3

0.00302%

0.01900%

0.05900%

0.10100%

0.14200%

0.18300%

A1

0.00581%

0.03700%

0.11700%

0.18900%

0.26100%

0.33000%

A2

0.01087%

0.07000%

0.22200%

0.34500%

0.46700%

0.58300%

A3

0.03885%

0.15000%

0.36000%

0.54000%

0.73000%

0.91000%

Baa1

0.09000%

0.28000%

0.56000%

0.83000%

1.10000%

1.37000%

Baa2

0.17000%

0.47000%

0.83000%

1.20000%

1.58000%

1.97000%

Baa3

0.42000%

1.05000%

1.71000%

2.38000%

3.05000%

3.70000%

Ba1

0.87000%

2.02000%

3.13000%

4.20000%

5.28000%

6.25000%

Ba2

1.56000%

3.47000%

5.18000%

6.80000%

8.41000%

9.77000%

Ba3

2.81000%

5.51000%

7.87000%

9.79000%

11.86000%

13.49000%

B1

4.68000%

8.38000%

11.58000%

13.85000%

16.12000%

17.89000%

B2

7.16000%

11.67000%

15.55000%

18.13000%

20.71000%

22.65000%

B3

11.62000%

16.61000%

21.03000%

24.04000%

27.05000%

29.20000%

Caa1

17.38160%

23.23413%

28.63861%

32.47884%

36.31374%

38.96665%


428


EXHIBIT B3 Year

(Continued) 7

8

9

10

11

12

Aaa

0.00520%

0.00660%

0.00820%

0.01000%

0.01211%

0.01438%

Aa1

0.05400%

0.06700%

0.08200%

0.10000%

0.13067%

0.15933%

Aa2

0.11100%

0.13500%

0.16400%

0.20000%

0.24924%

0.30428%

Aa3

0.22700%

0.27200%

0.32700%

0.40000%

0.64366%

0.76341%

A1

0.40600%

0.48000%

0.57300%

0.70000%

1.03796%

1.22241%

A2

0.71000%

0.82900%

0.98200%

1.20000%

1.43226%

1.68140%

A3

1.11000%

1.30000%

1.52000%

1.80000%

2.33195%

2.67955%

Baa1

1.67000%

1.97000%

2.27000%

2.60000%

3.23138%

3.67740%

Baa2

2.41000%

2.85000%

3.24000%

3.60000%

4.13080%

4.67524%

Baa3

4.33000%

4.97000%

5.57000%

6.10000%

7.69682%

8.48789%

Ba1

7.06000%

7.89000%

8.69000%

9.40000%

11.26178%

12.29939%

Ba2

10.70000%

11.66000%

12.65000%

13.50000%

14.82673%

16.11089%

Ba3

14.62000%

15.71000%

16.71000%

17.66000%

19.61340%

21.07935%

B1

19.13000%

20.23000%

21.24000%

22.20000%

24.39863%

26.04633%

B2

24.01000%

25.15000%

26.22000%

27.20000%

29.18386%

31.01331%

B3

31.00000%

32.58000%

33.78000%

34.90000%

37.47788%

39.85510%

Caa1

41.38538%

43.65696%

45.67182%

47.70000%

49.97284%

52.06250%

Year

13

14

15

16

17

18

Aaa

0.01682%

0.01941%

0.02215%

0.02502%

0.02802%

0.03114%

Aa1

0.19101%

0.22571%

0.26340%

0.30402%

0.34755%

0.39389%

Aa2

0.36521%

0.43201%

0.50464%

0.58303%

0.66707%

0.75664%

Aa3

0.89248%

1.03046%

1.17691%

1.33136%

1.49334%

1.66236%

A1

1.41960%

1.62873%

1.84897%

2.07946%

2.31936%

2.56781%

A2

1.94672%

2.22700%

2.52103%

2.82756%

3.14538%

3.47326%

A3

3.04214%

3.41796%

3.80533%

4.20263%

4.60833%

5.02099%

Baa1

4.13723%

4.60857%

5.08925%

5.57728%

6.07084%

6.56825%

Baa2

5.23233%

5.79918%

6.37317%

6.95194%

7.53335%

8.11551%

Baa3

9.27384%

10.05209%

10.82050%

11.57735%

12.32125%

13.05114%

Ba1

13.31414%

14.30373%

15.26650%

16.20137%

17.10772%

17.98528%

Ba2

17.35444%

18.55537%

19.71250%

20.82540%

21.89418%

22.91943%

Ba3

22.47361%

23.79821%

25.05569%

26.24889%

27.38084%

28.45466%

B1

27.59125%

29.03949%

30.39727%

31.67075%

32.86586%

33.98824%

B2

32.70888%

34.28076%

35.73886%

37.09262%

38.35087%

39.52182%

B3

42.05837%

44.10091%

45.99560%

47.75471%

49.38972%

50.91127%

Caa1

53.94014%

55.63419%

57.16872%

58.56416%

59.83781%

61.00440%


429


EXHIBIT B3 Year

(Continued) 19

20

21

22

23

24

Aaa

0.03438%

0.03771%

0.04114%

0.04467%

0.04827%

0.05195%

Aa1

0.44298%

0.49474%

0.54907%

0.60588%

0.66505%

0.72649%

Aa2

0.85159%

0.95177%

1.05700%

1.16709%

1.28183%

1.40102%

Aa3

1.83794%

2.01958%

2.20682%

2.39916%

2.59616%

2.79735%

A1

2.82399%

3.08707%

3.35629%

3.63087%

3.91009%

4.19326%

A2

3.81003%

4.15456%

4.50576%

4.86257%

5.22402%

5.58917%

A3

5.43926%

5.86190%

6.28774%

6.71572%

7.14485%

7.57423%

Baa1

7.06800%

7.56872%

8.06919%

8.56831%

9.06510%

9.55869%

Baa2

8.69674%

9.27554%

9.85064%

10.42090%

10.98534%

11.54315%

Baa3

13.76618%

14.46580%

15.14956%

15.81722%

16.46864%

17.10380%

Ba1

18.83411%

19.65450%

20.44690%

21.21193%

21.95029%

22.66278%

Ba2

23.90204%

24.84320%

25.74424%

26.60664%

27.43195%

28.22175%

Ba3

29.47346%

30.44031%

31.35819%

32.22996%

33.05837%

33.84603%

B1

35.04322%

36.03575%

36.97045%

37.85159%

38.68311%

39.46862%

B2

40.61297%

41.63118%

42.58272%

43.47323%

44.30785%

45.09121%

B3

52.32914%

53.65224%

54.88868%

56.04583%

57.13036%

58.14828%

Caa1

62.07650%

63.06482%

63.97860%

64.82576%

65.61319%

66.34686%

Year

25

26

27

28

29

30

Aaa

0.05571%

0.05953%

0.06340%

0.06734%

0.07132%

0.07535%

Aa1

0.79008%

0.85570%

0.92324%

0.99259%

1.06362%

1.13622%

Aa2

1.52444%

1.65187%

1.78308%

1.91783%

2.05591%

2.19709%

Aa3

3.00231%

3.21061%

3.42185%

3.63564%

3.85161%

4.06941%

A1

4.47973%

4.76888%

5.06013%

5.35292%

5.64676%

5.94118%

A2

5.95716%

6.32715%

6.69840%

7.07021%

7.44192%

7.81294%

A3

8.00305%

8.43057%

8.85613%

9.27911%

9.69898%

10.11526%

Baa1

10.04834%

10.53337%

11.01320%

11.48734%

11.95536%

12.41689%

Baa2

12.09362%

12.63616%

13.17027%

13.69557%

14.21174%

14.71853%

Baa3

17.72276%

18.32566%

18.91270%

19.48411%

20.04017%

20.58119%

Ba1

23.35021%

24.01346%

24.65339%

25.27090%

25.86685%

26.44210%

Ba2

28.97766%

29.70125%

30.39409%

31.05770%

31.69353%

32.30300%

Ba3

34.59540%

35.30881%

35.98843%

36.63632%

37.25439%

37.84443%

B1

40.21146%

40.91468%

41.58109%

42.21327%

42.81358%

43.38419%

B2

45.82751%

46.52055%

47.17375%

47.79022%

48.37277%

48.92396%

B3

59.10505%

60.00560%

60.85438%

61.65543%

62.41242%

63.12865%

Caa1

67.03196%

67.67305%

68.27416%

68.83881%

69.37014%

69.87094%



20-Heberle-Aircraft Page 431 Wednesday, July 23, 2003 10:25 AM

Aircraft Valuation-Based Modeling of Pooled Aircraft ABS Mark A. Heberle Vice President Wachovia Securities, Inc.

he aircraft ABS market originated in 1992 with the issuance of the original Aircraft Lease Portfolio Securitization (ALPS 1992-1) by the GPA Group. Since that time a total of 25 deals have been brought to market worldwide for approximately $24 billion in total issuance of which, approximately $13 billion remains outstanding. To properly analyze pooled aircraft lease securitizations investors need to be able to make a forward-looking projection of individual aircraft values. This is necessary due to the use of portfolio valuation as drivers of the turbo feature in these deals. Typically, a pooled aircraft ABS transaction will have a threshold level of portfolio valuation stress which, when breached, will cause a diversion of funds in order to turbo the senior class of securities and bring the LTV of the deal back into equilibrium. In this article a valuation model that provides a robust means of analyzing pooled aircraft securitizations is explained. The methodology uses assumptions about an aircraft’s future value prospects to drive a forward looking portfolio valuation and related lease cash flows and should help investors in this asset-class to develop a more complete understanding of the correlation between aircraft values, lease revenue and deal structure.

T

431


432 EXHIBIT 1


Base Value versus Adjusted Base Value

Note: t: Time Source: Offering Memorandum and Wachovia Securities, Inc.

MODELING FUTURE AIRCRAFT VALUATIONS In 2001 and 2002 the aviation finance community became all too aware that the base values of aircraft can display volatility. This represents a change in the thought process because the concept of a base value is supposed to place values on aircraft under conditions of equilibrium in supply and demand. In 2002, this market condition has become difficult to imagine, and base values became unhinged from their long-term projections. To build a useful model for analyzing the susceptibility of a securitization structure to cash flow diversion, investors need to make a forwardlooking analysis of each aircraft in the deal. Our model uses these assumptions to determine whether a portfolio of aircraft will trigger a diversion of funds at a specific time in the future. At each point in time, we need to determine the dollar value of impairment, which equates to the difference between the base value depreciation at that point in time and the projected depreciation of the portfolio, given the most recent appraisals and our view on the recovery prospects for each aircraft in the deal. Exhibit 1 shows the base value depreciation curve as predicted in an example offering memorandum (OM) and the actual appraisal values recorded since the inception of the deal. From the most recent appraisal, we append an adjusted portfolio value curve that depreciates the last appraisal amount for each aircraft based on the depreciation assumptions in the OM. This adjusted portfolio value assumes evaluation of the latest appraisal value of the portfolio and then uses the original depreciation assumptions from the OM. There are two problems with relying


Aircraft Valuation-Based Modeling of Pooled Aircraft ABS

433

on the adjusted portfolio value to model the future expected value of the portfolio: ■ Simply depreciating the latest appraised value of aircraft implicitly

assumes aircraft values will not recover from their current depressed state. Certainly, most industry participants expect newer, narrowbody aircraft, such as the Airbus 320s and the Boeing 737 Next Gens, to recover much of the value reduction suffered in the current downturn. Our model must be able to build in recovery vectors for individual aircraft types. ■ Using the adjusted base value to model forward-looking portfolio valuations will overstate the forward valuations for aircraft for which the estimated useful life has been negatively affected by the events of 2001 and 2002. The parking of numerous older aircraft has caused many to speculate that some may not come back to service and similar aircraft still in use may have a reduced estimate of useful life. Our model allows for the reduction in the estimated useful life of individual aircraft. Any reduction in the estimated useful life will cause acceleration in the depreciation of that aircraft. This should give us a better indicator of the projected future appraisal value than that given by the adjusted portfolio value curve. The adjusted portfolio value, as modeled in a pooled aircraft securitization, is not intended to be a forward-looking valuation model. It is only used to calculate a depreciated portfolio valuation from the time of the last appraisal until the time it is delivered to the trustee and subsequently used to determine repayments of scheduled principal of the Class A notes. It is used for this purpose for a year until a new appraisal value is provided. Therefore, we have built a valuation model that corrects for the two main aforementioned problems and adds flexibility to adjust inflation and residual value assumptions. To accomplish this, we have modified the adjusted base value curve by building the following functionality: ■ Allow for the inclusion of recovery vectors for each aircraft in the

pool—this recovery is to the applicable initial depreciation curve for that aircraft. ■ Allow for adjustments to be made to the expected useful life of the aircraft—for aircraft that have suffered impairment to their expected useful lives due to the September 11, 2001 events and the subsequent early retirement of older aircraft types. ■ Allow for changing the assumption for inflation. ■ Allow for stressing of the aircraft residual value assumptions.


434


MODELING EXPECTED CASH FLOWS The key component of our cash flow model is that the stresses are driven off of the future projected value of the aircraft portfolio. The model projects cash flows that are intuitively consistent with our assumptions on aircraft valuations and our assumptions on the useful lives of aircraft. To do this, we break out the base case lease rental assumptions between the raw lease cash flows and the cash flows that are applicable to the assumed scrap values of the aircraft, as depicted in Exhibit 2. Once we separate these components, we stress them separately and then combine the results to get our projected cash flows that we will run through the liability waterfall of the deal. We use separate methodologies to stress these cash flow types: ■ Base lease revenue. To stress the base lease revenue, we compare the

relationship between the portfolio value in the OM with the reducEXHIBIT 2

Lease Rate Assumptions from Example Offering Memorandum

Source: Offering Memorandum and Wachovia Securities, Inc.



EXHIBIT 3

435

Model Projected Portfolio Valuation

Source: Offering Memorandum and Wachovia Securities, Inc.

tion and lease revenue reductions from the OM. We build the assumption for base lease rates by manually smoothing the cash flows as reported in the OM to take out noise created by projected aircraft residual values and quarterly payment cash flows from early in the deal. We then evaluate portfolio value stress at each point in time and stress our revenue assumption by an equal percentage. In this way, our assumptions about the early retirement of aircraft directly affect the revenue projections. In a stressed portfolio scenario, the revenue reduction is thus predicted to become more severe in the latter years of the deal (see Exhibit 3). ■ Aircraft residual values. To stress the residual values of aircraft, we model the residuals for each aircraft and then modify the projected cash flow to incorporate the changing assumptions of the useful life as set out in our projections. In this way, we can modify the scrap value amounts and the timing of those cash flows.

ADDITIONAL REVENUE HAIRCUTS Our asset valuation-based stress model incorporates lease revenue stress and residual value stress to project the forward cash flows and portfolio values. The base revenue assumptions in the aircraft-backed ABS deals generally include assumptions related to stress-related costs. These stresses include the following: ■ Lost rental revenue due to aircraft downtime following the termination

or expiration of a lease.


436

EXHIBIT 4


Asset Model Predicted Cash Flows

Source: Wachovia Securities, Inc. ■ Bad debts realized and/or provided for. ■ Aircraft repossession costs.

Within each pooled aircraft trust, the assumed case incorporates a 4.0–7.5% revenue haircut related to these types of stresses. Our valuation-based stress model incorporates this revenue stress as a part of the stress being applied to the base lease revenue stress. We have taken additional revenue haircuts that are applied to the cash flows generated by our asset model for each of our stress runs. Although the original model assumes net maintenance at zero, our additional revenue stresses can be assumed to cover the typical portfolio stresses as well as maintenance and additional sales, general and administrative costs. When we combine (1) our stressed cash flow from lease revenue, (2) the residual value cash flows, and (3) the haircut taken for additional stresses, we get our final predicted cash flows from our model shown in Exhibit 4.

CONCLUSION In each scenario, we evaluate the effect of the stresses by using the cash flows and the portfolio valuation curve that generated the cash flows as inputs to an aircraft deal’s liability model. In this way, we have analysis that is consistent with all of our views on aircraft values. This methodology captures the full nonparallel nature of aircraft stresses and uses a single set of assumptions to drive both aircraft values and cash flows. By running this type of analysis under a variety of different stresses a portfo-



437

lio manager can develop a better understanding of the relative strengths and weaknesses of a particular securitized structure. Further, by building this analysis across different deals assumptions related to individual aircraft prospects can be used to compare securities from separate deals under a similar stresses.


Index2 Page 439 Wednesday, July 23, 2003 3:40 PM

Index

Absolute liquidity, decline, 252 Adjustable-rate loans, 299 Adjustable-rate mortgages (ARMs), 286, 293, 298. See also Hybrid ARMs collateral, 293 Agency CMOs, 289 collateral characteristics. See Nonagency collateral characteristics debentures, 2 guidelines, 282 loan size limit, adjustment, 304 market. See Nonagency market mortgage securitization market, 284 origination volume. See Nonagency origination volume passthrough, 315 pools, 312 burnout, 316 prepayments. See Nonagency prepayments differences. See Jumbo agency prepayment differences securities, valuation. See Nonagency securities turnover rates, contrast. See Jumbo turnover Aircraft appraised value, depreciation, 433 Aircraft Lease Portfolio Securitization (ALPS), 431 Aircraft residual values, 435 Aircraft valuation, modeling, 432–433 Aircraft valuation-based modeling. See Pooled aircraft ABS Alt-A mortgages, 304 markets, 319 Alt-A originations, 319 Alt-A prepayments credit impairment, 318 security valuation, relationship, 315–322 value, 319 Alt-A securities, issuers, 317 AMBAC, 332

American International Group, 329 Arbitrage cash flow CDO structures, 366 considerations, 28 opportunities, 267 players, 249–250 Arbitrage-free option price, 151 Arbitrage-free relationship, 241. See also Default swaps Asset swap pricing, 229–231 example, 231–232 illustration, Bloomberg (usage), 233–237 issues. See Credit default swaps Asset-backed securities (ABSs), 1, 12. See also Residential ABS aircraft valuation-based modeling. See Pooled aircraft ABS CDOs, contrast. See Commercial real estate CDOs collateral, 281 Assets. See Uncorrelated assets classes, 13, 396 categorization. See Fixedincome returns risk/return trade-offs. See Fixed-income asset classes expected value, 148 portfolio, 390 swaps, 207–208 spread. See Par asset swap spread volatility, 146 computation, 145–146 Available funds cap, 298–300 carryforward, 300 B and C borrowers, 283, 285 Bankruptcy, 186, 259. See also Personal bankruptcies court, 210 event, 191 evidence, 150 protection (Chapter 11), 166 Barrier options, 67 Base case scenarios, 321 Base lease revenue, 434–435 Basis

convention, usage, 258 risk, 199 smile, 245, 247 Basis-type trade, 230 Baskets. See Synthetic baskets products, 143 Benchmark. See London Interbank Offered Rate swap account. See Czech benchmark account index, 47 yield curve. See Euro benchmark yield curve Binomial distribution. See Modeling defaults Binomial Expansion Method (Moody’s Investor Services), 369 Binomial method. See Moody’s Investor Services Black-Scholes option pricing formula, 145–146 Black-Scholes-Merton (BSM) model, 266 Bond returns application. See Corporate bond returns normality, 72–76 result, 75 symmetric distribution, 60 Bondholders, recovery, 158 Bonds. See Longer-dated bonds basket, 99 classification, 97 convexities, 189 coupon step-ups, 243 default swap, combination, 209–210 identity, 232 insurance structures, 294 markets. See European corporate bond market lag, 216 valuations, 167 maturity, 187–191 portfolio risk, 76 prices interest rate shift, factoring. See Corporate bonds

439


440 Bonds (Cont.) less than par, 245–247 price-yield formula, usage, 187 pricing, 184–185 subordination, effect, 186 ratings, 232 relative price responses, 193 risk, 220 premium, 87 spread, scatter plot, 222 Borrower. See Prime borrowers; Subprime borrowers credit quality, 291 mortgage rate, 305 motivations. See Commercial real estate CDOs B-piece, 396 buyers, 396 Break-even calculations, 88 Break-even cushions, 78, 82, 83 Break-even spread, 78, 81 average, widening, 97 cushions term structure, 85 widening, 91 Broad Investment Grade (BIG) Index (SSB), 2–3, 95 Burnout. See Agency; Jumbos effect, 286 Business risk, 156 Butterfly component, 22, 43 factors, 37, 41 shapes, 43 Buy-side credit analysts, 156 Calendar convention, usage, 258 Calendar effects, 138 Callable agencies, 12 performance, 6 Callable corporates, 13. See also Noncallable corporates Callable instruments, 11–13 Cap volatility, 8–11 significance, 12 Capital gains, 78, 91, 206 determination, 82 losses, 79, 81 determination, 82 markets, 165 reinvestment, 392 structural changes, 157–158 Capital structure, 183, 421. See also Firm change, motivation, 156 complexity, 184 dynamics, 225–228 subordinate position, 398 upgrading, 186 Carry components, 90 trade. See Long-end carry trade Carry strategies, 94

Index

consistency. See European swap market (Europe), consistency, 77 interpretations, 100–102 Carry trades consistency. See Short/long maturities forecasting model, 87 past outperformance, quantification, 85–86, 91–95 Carryforward. See Available funds cap mechanism, 300 Cash assets, shorting difficulty, 250–251 Cash bonds borrowing rate, 232 credit risk, inclusion, 200–202 credit risk, nonexistence, 198–199 purchase, 232 Cash flow. See Risk-free cash flows; Riskless cash flows approach, comparable yield (usage), 383–384 CDO, 401 discounting, 386 reasons. See Equity diversion trigger, 425 methodology, 368, 382–389 modeling. See Expected cash flows models, 434 projection. See Defaultadjusted cash flows valuation. See Rated CDO notes waterfall, 389. See also Collateralized debt obligation Cash instruments, 206–207 credit risk, unified approach, 197 interest rate risk, unified approach, 197 Cash market, 210 FRN, 230 reference assets, 231 shorting, 243 Cash settlement, 256, 257 Cheapest-to-deliver issues, 249. See also Futures contracts Cheapest-to-deliver option, 242–244, 252 Clean-up call, 300 option, 300 Collateral characteristics, 317, 367. See also Nonagency collateral characteristics uniformity, 307 composition, 304 generalizing, 370–371 pool, 291, 299, 392 repo, 96 type, 368 value, decline, 380

Collateralized debt obligation (CDO), 177. See also Cash flow; Investment grade synthetic CDOs; Static pool CDO age, 378 assets, 377 cash flow waterfall, 373 collateral, 370, 406 contrast. See Commercial real estate CDOs equity, 382 option value. See Distressed CDO notes/equity yield, 385 investment opportunities, 366 investment-grade tranches, 398 liabilities, 377–378 market, 381 notes, 365–367, 378, 384 cash flow valuation. See Rated CDO notes option value. See Distressed CDO notes/equity purchasers, characteristics. See Real estate CDOs structures. See Arbitrage understanding. See Commercial real estate CDOs valuation, 387 framework, 366–368. See also Secondary market CDO valuation Collateralized debt obligation (CDO) tranches, 365– 369, 387 expected loss computation, 373–374 pricing, 389 valuation, 383 Collateralized mortgage obligations (CMOs), 281–282. See also Nonagency CMOs activity, 137 production, 137–138 Combined LTV (CLTV), 291 Commercial mortgage defaults, 415 Commercial mortgage-backed securities (CMBSs), 408. See also Investmentgrade CMBS contrast. See Commercial real estate CDOs correlation, levels, 406 market, 411 portfolio, 401 prepayment protection, 417 recovery rates, 411 tranches, 410 transactions, 399 Commercial real estate (CRE) CDOs ABS CDOs, contrast, 417 borrower motivations, 398–403


441

Index

Commercial real estate (Cont.) CMBSs, contrast, 411–415 corporate CDOs, contrast, 411 definition, 396–398 discrimination methodology, 419–422 high-yield corporate CDOs, contrast, 410–411 market, history, 395–396 penalization, 406–410 protection, historic context, 417–419 quality, 422 sector, performance, 423–425 trading process, 415, 422–423 tranches, 422 understanding, 395 Commercial real estate (CRE) loss severity rates, 411 Common factors, 53 Conditional mean exceedance, 224 Conforming mortgages, 282 rates, 305 Constant maturity (CM) spread, 80, 90 changes, 82 swap-gilt spreads, 92 Convergence, speed, 68 Convertible bonds, 249–250 Convexity. See Bonds differences, 241 Copula functions, 179 Corporate bonds, 5, 29, 68, 249. See also Eurodenominated corporate bond; Euro-dominated corporate bond; Noninvestment grade corporate bonds; Short-dated corporate bonds coupon step-ups, 245 defaults. See U.S. corporate bond defaults investors, Merton models (implications), 211 prices, interest rate shift (factoring), 190 purchase, 205 returns, application, 73–74 sensitivity, 189 spreads changes, volatility, 97 connection. See Equity drivers, 155 Corporate carry trades consistency. See Short/long maturities outperformance, consistency, 97–100 Corporate CDOs, contrast. See Commercial real estate CDOs Corporate credit

fundamental approach, historical analysis, 166–175 products, 401 quantitative approach fundamental analysis, contrast, 141 historical analysis, 166–175 spreads. See Euro corporate credit spreads structural models, 143–152 valuation, 141, 175–177 fundamental approaches, 156–166 quantitative approaches, 142–155 Corporate spreads, factor model, 143 Corporate-specific factor, 10 Counterparty credit exposure, diversification, 331 Counterparty risk, 200, 247–248 degree, 248 Coupon-adjusted swap (CAS) spread, 80. See also German Bundesobligationen Coupons, 201 bond. See Fixed coupon bond; Par fixed coupon bond collateral, 135 dates, default occurrence, 148 mortgage, 133. See also Current coupon mortgage payments, 198 rates, 315 step-ups, 241, 252. See also Bonds; Corporate bonds Covariances blocks, 37 clusters, 40 construction, 19–20 forecasts, 35 matrix, 57, 64. See also KR covariance matrix method. See Full covariance method model. See Variance-covariance model Cox-Ingersoll-Ross model, 66– 67, 154 CPR, 286–288, 308, 316 history, 361 pricing assumption, 354 rate, 311–313, 321, 351–353 Credit analysts. See Buy-side credit analysts; Sell-side credit analysts forecasts, 150 curing effect, 287 curve, 247 derivative. See Pure credit derivative; Simple credit derivative inclusion, 187

market, 142 enhancement, 332–334 exposures, 229 impairment. See Alt-A prepayments market. See Euro credit markets portfolio products (valuation), quantitative approaches, 175–181 protection, purchase, 249, 250 quality, 319–321. See also Mortgages deterioration, 243, 252 measurement, SATO usage, 317–318 ratings, 50, 143 upgrades/downgrades, 381 relative pricing, 155 spread. See Euro corporate credit spreads factors, 26–29, 37 widening, 142 structured synthetic bid, 251 systematic overweighting, 78 usage. See Reference view, 68 Credit analysis key aspects, 290–293 principles, 156–157 Credit default basis, 232 Credit default swaps (CDSs), 190, 222. See also Up-front CDS asset swap pricing, issues, 229 attraction, 264 basis, 224 baskets, 67 developments, 276–278 markets, 224 premium, 265 pricing, 268 model. See JP Morgan static replication, usage, 262–266 risk, 272–274 sensitivities, 272–274 spread, 256, 270, 274 term structures, 257 unwinding, practicalities, 275–276 valuation, 255, 269–272 Credit events, 259–262. See also Soft credit event number, 32 occurrence, 263 Credit impaired security, 397 Credit risk, 68, 183, 187–191, 199 amount. See Cash bonds disaggregation, 156–157 factor, 210 present value, 204 structural models, 214–215 unified approach. See Cash instruments; Derivative instruments


442 Credit specific factor, 8 CreditGrades (RiskMetrics), 141, 150, 215 CreditMetrics, 31 Credit-specific EDF, 391 Credit-spread factors, 55 Creditworthiness, 165, 332 CREST Clarendon Street 20021, 395, 401–405, 420 CRIIMI MAE, 396 Cross-collateralization. See Mortgage insurance Cross-country trading, 78 Cross-market terms, computation, 37 Cubic B-splines method, 108 Cumulative defaults production year, 356–359 rates, 408 Cumulative net loss test, 296 trigger, 297 Currency exchange rates, 19 factors, 37 risk, 32–35 hedging, 39 Currency-hedged assets, 78 Current coupon mortgage, 132–133 Curvature effect, 116 Curvature factor, 119 Cushion curves, 90 CUSP Model (CSFB), 215, 217 Czech benchmark account, 24 Debt. See European debt fixed allocation, 225 inclusion. See Real estate investment trusts instrument, 184 levels, adjustment, 148 service payments, stream, 226 Debt/assets ratio, 149 Debt-holders, recovery value, 185 Debt-to-income (DTI), 315 ratios, 292 Deep mortgage insurance (MI), 298 mechanics, 326–328 role/performance. See Subprime ABS structures, 333–334 Deep mortgage insuranceinsured structures, 337 Default, 206 barriers, 180 cumulative probability, 373 distributions. See Nonbinomial default distributions event, 240 forward probability, 152 occurrence, 148, 206. See also Coupons; Maturities protection, seller, 245 rate. See Gross default rate

Index

risk, 200 estimation, 216 spread, prediction ability, 170–172 threshold, 148, 224. See also Implied default timing, modeling, 387 Default correlation, 151, 390–392. See also Protection seller importance, 177–179 inferring, 179–180 measurement, difficulty, 217 modeling, 387–388 usage, 180–181 Default probabilities, 68, 179 computation, 147–148 determination, 142 forward curve, 266 modeling, 387–388 recovery rates, discrepancies, 224 Default swap basis, 265. See also Credit default swaps arbitrage-free relationship, 240–241 exploration, 239 fundamental factors, 241–248 practice, 252–253 technical factors, 248–252 Default swaps, 208–209, 256– 259. See also Digital default swap; Sovereignlinked default swap buyer, exposure, 232 high-beta nature, 243 payout requirements, 232 premiums, 251 zero flooring, 245 pricing models, implementation, 262 Default-adjusted cash flows, projection, 385–386 Default-based pricing models, 155 Defaulted security, 397 Default-free bonds, 66 Deleveraging, 419 Deliquency, 296–298 rates, 318 test, 296 trigger event, 297 Delivery option, 252 valuation, 260 Depreciation, 350–351 Derivative contract. See Overthe-counter derivative contract Derivative instruments credit risk, unified approach, 197 interest rate risk, unified approach, 197 Digital default swap, 274 Discount margin, approach, 204 Distress. See Financial distress probability, 220–222

Distressed CDO notes/equity, option value, 389–390 Distressed credits, 278 Distribution. See Multivariate normal distribution; Returns Diversification, benefit, 406 Diversity score, 251, 370–371, 390, 406–408 robustness, 425 Documentation loans, 305 Dollar rolling, 131–132 results, 134–135 TBAs, 131, 137, 139 Dollar rolls, 137–138. See also Coupon-based dollar roll strategies, 133–135 performance, 134 Dollar rolls, advantages, 132 Domestic government bond returns, 42 Downgrades, 216, 411. See also Moody’s Investor Services D-Score (Moody’s Investor Services), 405–410, 425–426 importance, 408 understanding, 405–406 Duration, 47 contribution, 50 directionality, 6 measurement, 5 drift, 47 extensions, 87 historical view, 47–51 Duration-adjusted excess returns, 4–6 calculation, 2 Duration-adjusted returns, 10 Duration-based model, 24 Duration-equivalent portfolio. See On-the-run Treasuries Duration-matched Treasuries, 96 Dwarfs, 134 Dwelling type, 291 EBITDA, 143 Economic conditions change, 148 jumbo turnover, relationship, 312–315 Effective durations, 2, 5 EGBI, financial issues, 97 Embedded options, 18 Emerging market (EM) algorithm, 35 comparison, 41–42 spread, 37 factors, 29–31, 37 Emerging Markets Bond Index Global (EMBIG) (JP Morgan), 30 Enron, 71 Equity. See First-loss equity


443

Index

Equity (Cont.) cash flow, discounting reasons, 384–385 fixed allocation, 225 holders, payoff diagram, 144 markets, 150 corporate bond spreads, connection, 211 valuations, 167 model, market value, 191 option, 157–158. See also Over-the-counter equity options valuation, 293 value. See Distressed CDO notes/equity prices, 151 valuation, 377, 392 volatility, 143, 146 usage. See KMV Corporation Equity risk premium (ERP), 159 ERISA-eligible funds, 398 Euribor futures contracts, 111 rates, maturities, 108, 110 swap yields, 111 Euro benchmark yield curve, 103 data, 108–112 Euro Corporate Credit Index (MSCI), 161 Euro corporate credit spreads, 95–97 Euro credit markets, 49, 97 Euro interbank yield curve, 112, 113 derivation, 110–112 PCA (European zone), 112–119 Euro investment grade corporate index, 42 Euro spread factors, 37 Euro-denominated corporate bond, 25 European bond market, 27 European corporate bond market, 142 European currencies, volatilities, 33 European Monetary Union (EMU), 77 EMU Corporate Large Cap (Merrill Lynch), 38–39 EMU interest rate factors, construction, 23 markets, 80 European swap market carry strategies, consistency, 80–88 interpretations/implications, 86–88 historical experience, 80–85 spread curve (mid-2001), 80–85 European-denominated fixed income instruments, 41 Event. See Credit events; Default determination date, 258

distribution, 368 risk, 411 Ex ante tracking error, 51–53, 56, 58 Ex post tracking error, 51–53, 56 Excess returns. See Durationadjusted excess returns; Hedged excess returns capturing, 10–14 determination, 6 volatility, 135 Exchange rates, usage, 33 Exotic derivatives, 67 Expected cash flows, modeling, 434–435 Expected default frequency (EDF), 150. See also Credit-specific EDF; KMV Corporation Expected earnings, 191 Expected recovery rate, 153, 224 Exponential distribution, 388 Exponentially Weighted VcV Matrix, 58 Exposure vector, 52 difference, 56 External currencies, shortage, 29 External debt, 30 risk, 29 Extreme events, 62 Factors. See Common factors; Credit spread factors; Currency; Explanation power; Principal factors; Swaps correlation, 39, 117–119, 127–129 models, 59, 155. See also Corporate spreads historical study. See Statistical factor model study, 169 returns, 43 shapes, 42–43 volatilities, 39 Fair market spreads, computation, 146–147 Fair market value, 142 Fat tails, 58, 177, 392 Federal Housing Administration (FHA), 339 FHA-financed nursing homes, 358, 363 FHA-insured passthrough participations, 340 insurance, 339 override. See Lockout project loans, 340, 355 insurance programs, sections (listing), 341–344 regulations, 355 underwriting standards, 359

Federal National Mortgage Association (FNMA), 132–134 coupon, rolling, 134 pools, 288 Feedback loops, 225–228 FICO characteristic, 317 experience. See Subprime market reduction, 305 scores, 285, 315, 336 Financial distress, 149 Financial prices, behavior, 67 Financial securities, contribution, 40 Financial status, evaluation, 165 First lien mortgages, 282 First-loss equity, 401 Fixed coupon bond, 206 Fixed-income asset classes, 10–11 performance, 2 risk/return trade-offs/characteristics, 1 Fixed-income instruments. See Euro-denominated fixed income instruments Fixed-income returns, asset class categorization, 6–10 Fixed-income risk modeling. See Portfolio managers Fixed-rate payer, 207 Floating rate notes (FRNs), 202–205. See also Cash market appearance, 207 default, 208 floating-for-fixed swap, combination, 208 Floating rate residential ABS, 298 transaction, 300 Floating-for-fixed swap, combination. See Floating rate notes Floating-rate coupon, 230 Floating-rate international buyers, 403 Forward default probability, 279 Forward-looking analyses, 46 Forward-looking valuation model, 433 Free cash flow generation, 172 changes, 167 relationship. See Relative value sector relationships, 174 trends, 173 Front-end carry trades, 101 Front-end spreads, 82 Full covariance method (FCM), 51–53 Fundamental factors. See Default swap basis negative basis, 247–248 positive basis, 241–247


444 Funding risk, 252 Funding spread, 81, 97 Futures contracts, cheapest-todeliver issues, 199 GE Capital Mortgage, 329 General Auto-Regressive Conditional Heteroskedastic (GARCH) estimate, usage, 220 models, 33 volatilities, 35 Geographic concentrations, 292 Geometric Brownian motion, 65–66 German Bundesobligationen (Bobls) CAS spread, 80 liquidity, 85 reverse asset position. See Swap-Bobl reverse asset position spread curve. See Swap-Bobl spread curve swap spreads, 80 German shift volatility, 33 G-Force 2002-1, 421 Global integration, 35–38 Global portfolio, 107 GMAC, 399, 401 Gold standard, usage, 213 Government bond returns. See Domestic government bond returns Government National Mortgage Association (GNMA), 132–134 prepayment tapes, 355 Project Loan Default Curve (GN PLD), 356–359, 363–364 underwriting changes, 359 Government National Mortgage Association (GNMA) multifamily pools, 340 default behavior, 354–356 investment characteristics, 363–364 prepayment behavior, 345–354 Government National Mortgage Association (GNMA) project loan default curve, 356–359 prepayments, breakdown, 359–361 securities, 340–344 investment characteristics, 339 Government-sponsored enterprises (GSEs), 4, 304 Gross default rate (GDR), 420 Hard penalties, 319 Hazard rate, 152–154

Index

Health care loans, refinancing history, 361–363 Hedge ratios, 187–191 Hedged excess returns, 14 Hedging instruments, 107 High-beta, 239 character, 253 market, 253–254 Highest-grade assets, 100 High-grade corporate bonds, short-horizon spread returns, 18 High-grade debt, spread curve slope, 99 High-yield bonds, 27, 422 High-yield CBO, 411 sector, 424 High-yield corporate CDOs, 410 bonds, 409 contrast. See Commercial real estate CDOs High-yield instruments, 31 Historical equity volatilities, usage. See KMV Corporation Historical simulation, 60–64 advantages, 62 disadvantages, 63 Home equity lines of credit (HELOCs), 282 Home equity loans (HELs), 283, 289, 326 sector, 330 structures, MI inclusion/noninclusion, 332–334 Home equity prepayment (HEP), 286 Housing turnover, lock-in (dampening effect), 313 Hybrid ARMs, 293, 299 IBoxx Euro Corporate Nonfinancial Index, 52 IBoxx indices, 45, 47 Idealized Cumulative Default Rate (ICDR) (Moody’s Investor Services), 369, 374, 399, 420 matrix, 406, 409 Illiquidity problems, 97 Implied default probability mappings, 374 threshold, 180 Incomplete return series, usage, 35 Index composition, identification, 68 Index prices, 97 Index returns, 99 Index-based investors, 132 Inflation Protected Bonds (IPBs), 21 IPB-specific interest rate factors, 21–22 Information asymmetry, reduction, 163–165

Information ratio, 85, 92. See also Funds usage, 96 Instantaneous drift, 66 Instantaneous shocks, 68 Instantaneous volatility, 66 Interbank yield curve. See Euro interbank yield curve PCA, 113–115 Interest coverage (IC) tests, 416–417 trigger, 417 Interest curve scenarios, number (selection), 68 Interest rate. See Domestic interest rates; Foreign interest rates; Market; Maturities; Risk-free interest rate changes, 202 curves, PCA, 122 decline, 3 derivatives. See Pure interest rate dynamics, 66–67 environment, stability, 190 factors. See Inflation Protected Bonds; Pure interest rate; Sovereign-based interest rate factors construction. See European Monetary Union model, selection, 154 movements, 185 one-factor model, 66 process, 67 returns, computation, 42 shift, factoring. See Corporate bonds term structure, 103–104 variations, time series, 104 Interest rate risk, 20–31 hedging, 39 PCA model, usage, 107–108 unified approach. See Cash instruments; Derivative instruments Interest-only (I/O) securities, 359 Interest-only (I/O) strip, 299 Internal debt, risk, 29 International Swap and Derivatives Association (ISDA) 2003, 259 Credit Derivatives Definitions, 243 credit event definitions, 259, 262 documentation, 260 Mod-Mod-Re, 261 old definitions, (Old-Re), 261 restructuring definition, 242 Internet bubble, 151 Intex Solutions, 421 Investment grade corporate bond, 366


445

Index

Investment grade corporate index. See Euro investment grade corporate index Investment grade synthetic CDOs, 251 Investment grade-backed CDO, 424 Investment-grade CMBS, 417 collateral, 418–419 Investment-grade equity REITs, 416 Investment-grade REITs, 416 Investment-grade tranches. See Collateralized debt obligation Irish Stock Exchange Limited, 398 IRR, 390 Issuer defaults, 226 Iterative procedure, 146 usage, 149 Ito’s lemma, 146 JP Morgan. See Emerging Markets Bond Index Global; GBI Broad Index credit default swap pricing model, 237 Jumbo agency prepayment differences, 308–311 Jumbo mortgage rates, 305 Jumbo Pfandbriefe, 80 Jumbo prepayments increase, 310 security valuation, 307–315 Jumbo relos, 304 prepayments, security valuation, 322–323 valuation, 322–323 Jumbo securities, valuation, 313–315 Jumbo turnover rates, agency turnover rates (contrast), 313 relationship. See Economic conditions Jumbos, 304. See also Weighted average coupon collateral, relative callability, 315 lock-in, 312 pools, burnout, 316 refinancing behavior, 307 turnover rates, 312 KMV Corporation, 141, 150, 215 approach, 391 default predictions, 169 EDF, 166–169 report, 152 usefulness. See Relative value historical equity volatilities, usage, 217 LLC, 391 model, usage, 225, 228 KR covariance matrix, 43

Kurtosis. See Leptokurtosis calculation, 75 Legacy factors, 23 Legacy market. See Euro legacy markets idiosyncrasies, 23 Lehman Brothers. See Nonagency prepayments Leverage, 143 model. See Target leverage model Leveraged investors, 78 Leveraged-loan CDOs, 422 LIBID, 231 LIBOR. See London Interbank Offered Rate Lien mortgages. See First lien mortgages; Second lien mortgages status, 291–292 Liquidation costs, 185 Liquidity, 366, 411. See also Relative liquidity decline. See Absolute liquidity growth, 20–21 LoanPerformance data, 335, 337 loan-level data, 304 Loans. See Restructured loans aggregators, 317 documentation, 292–293 foreclosure, 418 identification, 355 origination, 415 purpose, 292 size, 292, 321 prepayment penalties, 318– 321 type, 293 Loan-to-value (LTV). See Combined LTV characteristic, 317 equilibrium, 431 level, 298 mortgage loans, 282 ratio, 291, 292 reduction, 315 Lock-in dampening effect. See Housing turnover profile, 315 Lockout, 296. See also 36-month lockout form, 417 patterns, 345 period, FHA override, 354 provisions, overriding, 345 Lognormal model, 154 London Interbank Offered Rate (LIBOR), 3, 422. See also Zero-coupon LIBOR benchmarks, 42 borrowing rate, difference, 232

contract (3-month), 111 cost, 240 effect, 205 funding cost, 247 relationship, 247 increase, 299 LIBOR-GC repo spread, 88 LIBOR-GC spreads, 87, 90, 91 deducting, 92 LIBOR-repo, 81 payment, 207, 230 London Interbank Offered Rate (LIBOR) swap benchmarks, 42 curve, 20–21, 25 key rate returns, 43 Long swap-gilt spreads, withinsample widening, 92 Long-end carry trade, 95 Longer-dated deliverable, 260 Longer-dated nongilts, 48 Long-Term Capital Management, 283 Long-term debt, 151 Long-term performance record, 139 Losses calculation, 418 coverage requirements, 290–291 severity, 336–337 Management evaluation, 165 fees, 368 option, framework (development), 158–160 Market correlations, 63 frictions, 110 idiosyncrasies, 19. See also Legacy market idiosyncrasies interest rates, 210 participants, 163, 197, 247, 418 segmentation, 109 Market value. See Equity changes categorization, 380–381 role, 378–380 methodology, 368, 377–382 usage. See Structural change indicator Market-adjusted ratings, usage, 27 Market-dependent factors, 37 Market-implied default probabilities, 167 rates, 172 Market-implied rates, 171 Market-lagging approach, 165 Market-wide common factors, 18 Mark-to-market (MTM), 270–273 basis, 251 effect, 241 fluctuations, 181


446 Mark-to-market (Cont.) value, 274–275 vehicle. See Nonmark-to-market vehicle Maturities, 183, 189 assets, 81 carry/roll components, 81 default, occurrence, 148 factors, 123 interest rates, 104 volatility, 82 MBA Refi Index, 137 MBIA, 332 Medium-term segment, 115, 116 Merton, Robert C., 145, 147, 214, 266 framework, 158 model. See Black-ScholesMerton model original structural model, 145–148 extension, 148–149 structural approach, commercial implementation, 150 Merton, Robert C., models empirical results, 217–225 implications. See Corporate bonds overview/uses/caveats, 214–217 Mezzanine CRE CDOs, 415 Mezzanine notes, 381 MFS Investment Management, 401, 403 Modeling defaults (binomial distribution), 371–373 Modified Restructuring (ModR/ ModRe), 243 Modified-Modified Restructuring (Mod-Mod-Re), 243. See also International Swap and Derivatives Association Moment-matching, 370 Money market tranche, 421 Mono-line insurance adjustments, 330–332 companies, ratings differential, 331 contrast. See Mortgage insurance coverage, depth, 330 exclusions, 330–331 selection, 333 Mono-line wrap structure, 332 Monte Carlo simulation, 51, 62, 64–68 advantages, 67 calculation framework, 64–65 disadvantages, 67–68 models, examples, 65–67 Monthly tracking error, 135 Moody’s Investor Services, 242, 336, 369, 411. See also Binomial Expansion

Index

Method; Idealized Cumulative Default Rate; Watch List; Weighted average rating factor Corporate Default Index, 423–424 credit rating, inferring, 374 databases, 274–275 Double Binominal Method, 389 downgrades, 221 historical data, 174 Rating Factor Table, 426 recovery rate, 425 Morgan Stanley, 156 Mortgage Guarantee Insurance Corporation (MGIC), 329, 333 Mortgage insurance (MI), 325– 326. See also Deep mortgage insurance adjustments, 330–331 coverage, depth, 330 cross-collateralization, 326 exclusions, 330–331 inclusion/noninclusion. See HEL structures markets, exposure, 331–332 MI-backed HEL deals, senior classes, 331 mono-line insurance, contrast, 330–332 performance, 334–337 providers, 329, 332, 337 ratings differential, 331 Mortgage-backed securities (MBSs), 1, 12, 282. See also Nonagency MBSs performance, 6 prepayment risk, exposure, 18 valuation, 313 Mortgage-related market, development, 282–283 Mortgages. See Prepaid mortgage default, 345 indices, 138 loans. See Loan-to-value markets, 282. See also Alt-A mortgages; Single-family mortgage market pool, credit quality, 305 portfolio, rate-of-return, 138 rates, 305. See also Borrower; Conforming mortgages; Jumbo mortgage rates refinancing, 305 seasoning, 292 MSCI. See Euro Corporate Credit Index index, 253 Multifactor model (MFM), 53–59 example. See Sterling multifactor model Multifamily pools

default behavior. See Government National Mortgage Association prepayment behavior. See Government National Mortgage Association Multivariate normal distribution, 47 National Housing Act of 1934, 339 Net loss test. See Cumulative net loss trigger, 296–298. See also Cumulative net loss No-arbitrage condition, 231 No-arbitrage hypothesis, 109 Nonagency CMOs, 304 Nonagency collateral characteristics, 305–307 Nonagency market, 303–304 Nonagency MBSs, 304 Nonagency origination volume, 304 Nonagency prepayments, 303 model (Lehman Brothers), 307 Nonagency securities, valuation, 303 Nonbinomial default distributions, 390–392 Noncallable agencies, 3 debentures, 1 Noncallable corporates, 13 Nondomestic sovereign bond, 27 Noninvestment grade corporate bonds, 367 Nonmark-to-market financing, 398 Nonmark-to-market vehicle, 396, 401 Nonnegligible trading costs, 97 Non-penalty pool, 322 No-restructuring (No-Re) contract, 261 Obligation acceleration/default, 259 Observation periods, 61–62 Occupancy status, 291 Off-balance-sheet vehicles, 397 Offering memorandum (OM), 432, 434–435 Office of Federal Housing Enterprise Oversight (OFHEO), 326 Off-market derivatives, 199 OFHEO. See Office of Federal Housing Enterprise Oversight Old Republic, 329 On-the-run Treasuries, durationequivalent portfolio, 4 Operating environment, industry analysis, 160–161 Opportunity cost, 249, 312


447

Index

Option value. See Distressed CDO notes/equity Option-adjusted spread (OAS), 321 models, 5 OAS-based models, 4 valuation statistics, 319 Optionality factor, 9 Origination, 319, 351. See also Alt-A originations; Federal National Mortgage Association costs, 305, 307 fixed costs, 308 volume. See Nonagency origination volume Originator-level documentation coding systems, 317 Overcollateralization (OC), 290, 334 benefits, 416–417 trigger, 417 Over-the-counter (OTC) derivative contract, 275 Over-the-counter (OTC) equity options, 249–250 Par asset swap spread, 262 Par fixed coupon bond, 207 Par loss, 374 Par-floater, 241 Pari passu ranked assets, 242, 253 Pari passu requirement, 257 Par-par asset swap, purchase, 240 Path-dependent instruments, 67 Pay-floating interest rate swap, 88 Penalties. See Hard penalties; Soft penalties provision, overriding, 345 Per period default probabilities, 268 Percentage profit/losses, differences (calculation), 61 Performance figures, range identification, 69 Performance risk, 200 Perpetual security, 47 Personal bankruptcies, 285 Pfandbrief. See Jumbo Pfandbriefe hotspot, 40 Physical assets. See End-of-year physical assets ROR, 193 Physical settlement, 256, 257 mechanics, 258–259 notice, 258 Plain vanilla coupon-paying Treasury bond, 198 PMI Mortgage Insurance, 329 Poisson distribution, 279 Poisson process, usage, 269 Pooled aircraft ABS, aircraft valuation-based modeling, 431 Pooled aircraft trust, 436 Portfolio. See Global portfolio

management, 17 performance, 59 products, valuation. See Tranched portfolio products quantitative approaches. See Credit risk, 67 WARF, 373 Portfolio managers fixed-income risk modeling, 17 usage, 38–41 relationships, 183 Premium leg, 256 payment, 269 zero flooring. See Default swaps Prepaid mortgage, 315 Prepayment assumption, 393 behavior. See Government National Mortgage Association multifamily pools breakdown. See Government National Mortgage Association project loan differences. See Jumbo agency prepayment differences history, 348, 351 penalties, 305, 319–322, 417. See also Loans usage, 287 performance, 304 protection, 345, 417. See also Commercial mortgagebacked securities risk, 367 exposure. See Asset-backed securities; Mortgagebacked securities security valuation. See Jumbo prepayments; Jumbo relos relationship. See Alt-A prepayments speeds, 285–289 Prepay-penalty collateral, 321 Pricing differentials, 232–233 errors, 54 issues. See Credit default swaps models, implementation. See Default swaps Prime borrowers, 282 Principal payment. See Unscheduled principal payments return, 198 Principal component analysis (PCA), 42–43. See also Interbank yield curve; Interest rate; Yield curve European zone. See Euro interbank yield curve; U.S. Treasury yield curve

model, 104–108 usage. See Interest rate risk Principal factors, 53 Private label mortgages, 282 Probability mappings. See Implied default Probability-weighted loss, 373 Production refinancing, 353 Profit & loss (P&L), 276 Profits/interest rates, macroeconomic correlation, 191 Project loan insurance programs. See Federal Housing Administration securities, investment characteristics. See Government National Mortgage Association project loan Protection. See Prepayment buyer, 258, 260 leg, 277 purchase, 263 seller, 209, 248. See also Default quotes, 277 reference entity, default correlation, 248 PSA, 286 Pull-to-par, 64, 65 Pure credit derivative, 206 Pure interest rate derivatives, 199–200 factor, 208 Qualified SPV (QSPV), 396–397 Radian Guaranty, Inc., 329 Rated CDO notes, cash flow valuation, 382–383 Rate-lock, 350–351 Rate-of-return (ROR), 185 money managers, 3–4 outcomes, 190 expansion, 191–193 possibilities, 186 shocks, 195 Ratings. See Sectors/ratings agencies, approaches, 163–166 change, 165–166 agencies, consistency, 174–175 assessment, 165 transitions, prediction, 216 Real estate CDOs, purchasers (characteristics), 403–405 Real estate cycle, 415 vintage, 415 Real estate investment trusts (REITs), 408. See also Investment-grade equity REITs; Investment-grade REITs asset portfolios, 416 bonds, 398 cash flows, 406 collateral, 401


448 Real estate investment trusts (Cont.) debt, 396, 399, 415 inclusion, advantages/disadvantage, 415–416 payments, 398 Real-money investors, 81, 90, 92 perspective, 97 Recovery rates, 148. See also Expected recovery rate assumption, calculation, 274– 275 discrepancies. See Default Recovery vectors, inclusion, 433 Reduced form models, 143, 152–154 advantages/disadvantages, 154 risk-neutral pricing, usage, 267 name, 266 Reference credit, usage, 232 entity, 256 default correlation. See Protection seller issuer, 255 obligation, 255 Refinancing. See Mortgages; Production refinancing (1993) wave, 309 history. See Apartment complexes; Health care loans loan size, effect, 319 motivations, 287 profile, 315 Relative value consequences, 289–290 determination, 161 free cash flow, relationship, 172–174 KMV EDFs, usefulness, 168–172 opportunities, identification, 216 Relo prepayments, security valuation. See Jumbo relos REMICs, 345 investment characteristics, 363–364 pricing, 357–359 tranches, 359, 362 Rental revenue, loss, 435 Repo. See Collateral advantages, 85 Republic Mortgage Insurance Corporation, 329 Repudiation/moratorium, 259 Rerating methodologies, 367–377 Reserve accounts, 334 Residential ABS, 282, 283, 285. See also Floating rate residential ABS credit support, 290 introduction, 281 product, 289 structural considerations, 293–300 transactions, 294

Index

Residual value. See Aircraft residual values cash flow, 436 Residuals, 114, 377. See also Rich-cheap residuals Restructured loans, 418 Restructuring, controversy, 259–262 Returns. See Multivariate normal returns data, 2 normal distribution, 46–47 variation, 9 Revenue. See Base lease revenue haircuts, 435–436 stream, 225 Reward-to-risk, 95 Reward-to-volatility ratios, 99 Rich-cheap residuals, 168 Risk analysis, 36 clusters, identification, 41 excess, 45 exposure, types, 333 factors, 108 management, 42, 142 group, asset type split, 404 model, 35 modeling/understanding framework, 18–20 nominal amount, 87 premiums, 167. See also Bonds profile, change, 252 underestimation, 63 understanding, 17 Risk-adjusted returns, 137 Risk-based loan pricing, 227–228 Risk-free cash flows, 201 RiskMetrics. See CreditGrades Risk-neutral default probabilities, 267 Risk-neutral pricing, 185 usage. See Reduced form Risk-neutral probabilities, 150 Risk/return measures, 1 Risk/return trade-offs. See Assets; Fixed-income asset classes Rolldown components, 90 gains, presence, 85 spread changes, 99 Rolling CDS contract, 276–277 Rolling yield advantage, 81 Rotation factor, 119 Russian debt crisis, 395–396 Salomon Smith Barney (SSB). See Broad Investment Grade data, Yield Book, 2 research report, 78 Sampling variation, 65 Scenario analysis, 68–71 conducting, 165

method, 69 Scholes, Myron, 266 model. See Black-ScholesMerton model Seasonal effect, 138 Seasoned collateral, 138 Seasoning. See Mortgages curves. See Non-Cal seasoning curves effect, 291 Second lien mortgages, 282 Second priority notes, 381 Secondary market CDO valuation framework, 365 methodologies, 367–368 Sector-by-rating credit factors, construction, 32 Sector-by-rating spreads, 39 Sectors groupings, 163 ratings, 49–51 contrast, 161–163 weightings, 49–50 Selection bias, 336 Self-financing constraint, 108 Self-funded position, absolute P/L, 92 Self-funded trade, Sharpe ratio, 99 Sell-side credit analysts, 156 Senior bondholders, 295 Senior/subordinate structures, 294–295 Senior/subordinated structure, 332–333 Servicing fee, 298 Settlement conventions, 19 Severity calculations, usage, 418 Sharpe ratios, 2–4, 78. See also Self-funded trade usage, 92 Shift component, 22, 43 factors, 37, 41 shapes, 43 Shifting interest, 295–298 structure, 295 Shocks, responses, 193–195 Short/long maturities, corporate carry trades (consistency), 95–100 Short-term interest rates, 67 movements, simulation, 67 Short-term uncovered interest parity, 67 Simulation. See Cash flow simulation models, 367 Single-family A quality mortgage default patterns, 355 Single-family MBS, 363 Single-family mortgage market, 348 Single-name credit default swap, pricing, 266–276 Skewed distribution, 391, 392


449

Index

Skewness, calculation, 75 Small price movements, 65–66 Soft credit event, 242 Soft penalties, 319 Sovereign benchmarks, 42 Sovereign governments, taxes (raising), 29 Sovereign issuer, 30, 154 Sovereign-based interest rate factors, 21 Sovereign-linked default swaps, 258 Special-purpose vehicle (SPV), 396. See also Qualified SPV guidelines, 397 Specific risk, 31–32 Speculative-grade issuers, 175 Spread. See Credit default swaps; Sector-by-rating spreads advantage, 87 change. See Constant maturity; Rolldown; Yield prediction, 168 curve. See Swap-Bobl spread curve rolldown, 85 slope. See High-grade debt determination, 161 distribution, 63 excess, 100–101, 290 factor, 23–24. See also Credit; Emerging market; Euro spread factors; Swap spread position. See Swap spread prediction ability. See Default risk, 24–31 modeling, 26 volatility, 95, 100–101 SSSB EuroBIG index, 95 relative value curve, 97 Stale-price problems, 97 Standard & Poor’s (S&P), 32, 326. See also Core Earnings databases, 274–275 rating actions, 221 rating transition matrix, 101 Rules-Based Credit Classifications, 284 S&P 500, 8–11, 166 Structured Finance Ratings Group, 284 targets. See CreditWatch listings Static loan pricing, 227 Static pool, 396 CDO, 396 Static portfolio/index, 46 Static replication, usage. See Credit default swaps Statistical factor model, 166 historical study, 167–168 Step-down clauses, 245 Stepdown date, 295 Step-up coupon, 300

Sterling Bond Index, 73 Sterling factors, 28–29 Sterling global credit factors, 36 Sterling market, 36, 47–50 Sterling multifactor model, example, 55–58 Sterling-denominated corporate bond, 25 Stochastic process, 64, 67 Strategy risk, 157 Stress tests. See User-defined assumptions Structural change indicator, market value (usage), 381–382 Structural credit risk model, 226 Structural models, 143, 224. See also Corporate credit; Credit risk; Merton advantages, 150–151 disadvantages, 151–152 Structural subordination, 415 Structural-based models, usage, 267 Structured credit products, 401 Structured synthetic bid. See Credit Stub problem, 276–277 Subordinate tranches, 380 Subordinated A note, 415 Subordinated B note, 415 Subordinated tranches, 334 Subordination, 290 interest, 295–298 Subordination, effect. See Bonds Subprime ABS deals, 333 markets, deep mortgage insurance (role/performance), 325 Subprime borrowers, 282 characteristics, 284–285 refinancing, 283 Subprime loans, 336 Subprime market, FICO experience, 337 Subprime mortgage pools, 288, 292–293 Surety fee, 299 Survival probability, 267–270, 278 Swap spread, 14. See also Bondspecific swap spreads; German Bundesobligationen; Par asset swap spread constant-maturity series, creation, 80 factors, 25–26, 37 position, 81 volatilities, 25 Swap-Bobl reverse asset position, 82 Swap-Bobl spread curve, 85 Swap-gilt curve, 90 Swap-gilt spreads, 89. See also Constant maturity

Swap-related credit products, 80 Swaps. See Asset swaps basis. See Credit default swaps; Default swap basis benchmarks, 42 combination. See Bonds; Floating rate notes curves, 80. See also London Interbank Offered Rate transparency, 20–21 factor model, 41 fixed payment, 200 gilt spreads. See U.K. swapgilt spreads market, carry strategies (consistency). See European swap market pricing. See Asset swap pricing yields. See Euribor Swiss francs-denominated bonds, 32 Synthetic baskets, 177, 252 Synthetic CDOs. See Investment grade synthetic CDOs Systematic passive overweighting, 87 Systematic risk, 41 Target leverage model, 149, 152 Tax effects, 110 Tax Reform Act of 1986, 350, 353 Tax shelter, 350 TBAs. See Dollar rolling contract, 134 portfolio, 131 position, 134 settlement, 131 Technical factors. See Default swap basis negative basis, 251–252 new issues, 249–250 positive spread, 248–251 TED spreads, 87 Telecom assets, contribution, 40 bubble, 151 Term structures. See Break-even spread; Interest rate computation, 42 risk, 22 shift, 43 Terminal value, usage, 191 Time-decay constant, 20 Total Return Index, 73 Tracking error, 45, 92, 135–137. See also Ex ante tracking error; Ex post tracking error; Monthly tracking error calculations, 46–47, 51, 60 estimate, 51, 67 fundamentals, 46–47 increase, 85 measures, implications, 76 model, 59


450 Tracking error (Cont.) prediction capability, 72 procedures, 51 Trading costs. See Nonnegligible trading costs increase, justification, 88 Tranche. See Money market tranche expected loss, computation. See Collateralized debt obligation tranches loss function, 373 rating sensitivity, 374–377 valuation. See Collateralized debt obligation tranches Tranched portfolio products, valuation, 180 Transaction expenses, 298–299 Transition probabilities, 32 Trepp, 421 Triad Guaranty Insurance Corporation, 329 Triggers. See Net loss triggers events, 397 Trustee fee, 299 Turnover, 311–312 lock-in, dampening effect. See Housing turnover rates, 313. See also Jumbos rates, contrast. See Jumbo turnover relationship. See Economic conditions Twist component, 22, 43 factors, 37, 41 shapes, 43 U.K. swap-gilt spreads, 88–95 historical experience, 88–91 spread curve (mid-2001), 88–91 Uncorrelated assets, 370 Underwriting changes. See Government National Mortgage Association criteria, 282 fees, 305 standards. See Federal Housing Administration United Guaranty Corporation, 329 Unscheduled principal payments, 315 Up-front CDS, 277–278 Up-front costs, 278 U.S. corporate bond defaults, 274–275 U.S. swap-government spread markets, 78 U.S. Treasury bond market prices, 108 defeasance, 417 notes, 191 prices, outsized movements, 190

Index

securities. See Benchmark yield curve derivation, 108–110 PCA (European zone), 112–119 yields, 5 User-defined assumptions, stress tests, 68 Valuation. See Credit default swaps; Equity; Firm valuation; Secondary market CDO valuation approaches. See Corporate credit framework. See Collateralized debt obligation methodologies extensions, 389–392 summarization, 392–393 quantitative approaches. See Credit Value at Risk (VaR). See Confidence level computation, 142 definition, 60–61 estimation, 64–66 historical simulation, 51 methodology, developments, 63 Variables, orthogonalization, 13 Variance-covariance matrix, 54, 56 Variance-covariance method advantages, 59 disadvantages, 60 Variance-covariance model, 51– 60, 62 building/testing, 52–53 Variance-reduced variables, 105 Vasicek-Kealhofer (VK) model, 150 Vintage. See Real estate cycle component, 415 Volatility, 22, 28, 63–64, 92. See also Euro currencies; General Auto-Regressive Conditional Heteroskedastic computation. See Assets peak, 63 Wachovia Securities, Inc., 403, 417 Watch List (Moody’s Investor Services), 165 Weighted average cost of capital (WACC), 159–160 Weighted average coupon (WAC), 313, 323, 335–336. See also Alt-A WAC jumbos, 309, 322 rate, 316 Weighted average life (WAL), 409 Weighted average maturity (WAM), 335–336 Weighted average rating factor (WARF) (Moody’s Inves-

tor Services), 371, 406– 410, 426–429. See also Portfolio amount, 399, 405, 420 changes, 374 forecast, 382 importance, 408 increase, 404 migration, 424–425 reduction, 426–427 understanding, 405–406 usage, 409 What-if analysis, 68 World Trade Center disaster, 411 WorldCom, 71 Wrap structure. See Mono-line wrap structure Yield advantage. See Rolling yield advantage maintenance agreements, 417 shift, 193 spread evolution, 95 variation, 78 usage. See Cash flow Yield curve change, 61, 107 derivation. See European interbank yield curve; U.S. Treasury dynamics, principal component analysis, 103, 122 environment, 362 movement, 53–54 nonparallel shifts, occurrence, 104, 184 short-term segment, 113 Z-bonds. See Cash flow Z-bonds accretions, 362 Zero funding spread, 88 Zero-coupon bond. See Riskfree zero-coupon bond face value, 157–158 implied spread, 146–147 one year, 145 Zero-coupon corporate bond, 144 Zero-coupon euro interbank yield curve, 104 computation, 110–111 Zero-coupon LIBOR, 385–386 Zero-coupon rates, 107, 109, 111–112 sensitivities, 119–120 usage, 119 Zero-coupon Treasury yield curves, 104 Zero-coupon yield curves, 108, 111 spread, 105

Professional Perspectives on Fixed Income Portfolio Management, Vol. 4 (Frank J. Fabozzi Series)

The Handbook of European Fixed Income Securities (Frank J. Fabozzi Series)

Introduction to Securitization (Frank J. Fabozzi Series)

Financial Management and Analysis (Frank J. Fabozzi Series)

Financial Management and Analysis (Frank J. Fabozzi Series)

SchweserNotes. 2010 CFA exam. Level 3, Book 3: Fixed Income portfolio management, fixed income derivatives, and equity portfolio management

Schweser Notes, 2011 CFA Exam, Level 3- Book 3 – Fixed Income Portfolio Management, Fixed Income Derivatives, and Equity Portfolio Management

Probability and Statistics for Finance (Frank J. Fabozzi Series)

The Handbook of Municipal Bonds (Frank J. Fabozzi Series)

Active Private Equity Real Estate Strategy (Frank J. Fabozzi Series)

Credit Derivatives: Instruments, Applications, and Pricing (Frank J. Fabozzi Series)

Subprime Mortgage Credit Derivatives (Frank J. Fabozzi Series)

The Handbook of Commodity Investing (Frank J. Fabozzi Series)

The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management (Frank J. Fabozzi Series)

The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management (Frank J. Fabozzi Series)

Fixed-Income Management for the 21st Century

Active Fixed Income and Credit Management

Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies (The Wiley Finance Series)

Fixed Income Mathematics

Fixed income securities

Fixed income attribution

Advanced Fixed Income Analysis

Fixed income analysis workbook

Fixed Income Modelling

Advanced fixed income analysis

Advanced fixed income analysis

Fixed-income securities

Fixed Income Attribution

Fixed Income Mathematics

Fixed income mathematics

Advanced Fixed Income Analysis

Professional Perspectives on Fixed Income Portfolio Management, Vol. 4 (Frank J. Fabozzi Series)

The Handbook of European Fixed Income Securities (Frank J. Fabozzi Series)

Introduction to Securitization (Frank J. Fabozzi Series)

Financial Management and Analysis (Frank J. Fabozzi Series)

Financial Management and Analysis (Frank J. Fabozzi Series)

SchweserNotes. 2010 CFA exam. Level 3, Book 3: Fixed Income portfolio management, fixed income derivatives, and equity portfolio management

Schweser Notes, 2011 CFA Exam, Level 3- Book 3 – Fixed Income Portfolio Management, Fixed Income Derivatives, and Equity Portfolio Management

Probability and Statistics for Finance (Frank J. Fabozzi Series)

The Handbook of Municipal Bonds (Frank J. Fabozzi Series)

Active Private Equity Real Estate Strategy (Frank J. Fabozzi Series)

Credit Derivatives: Instruments, Applications, and Pricing (Frank J. Fabozzi Series)

Subprime Mortgage Credit Derivatives (Frank J. Fabozzi Series)

The Handbook of Commodity Investing (Frank J. Fabozzi Series)

The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management (Frank J. Fabozzi Series)

The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management (Frank J. Fabozzi Series)

Fixed-Income Management for the 21st Century

Active Fixed Income and Credit Management

Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies (The Wiley Finance Series)

Fixed Income Mathematics

Fixed income securities

Fixed income attribution

Advanced Fixed Income Analysis

Fixed income analysis workbook

Fixed Income Modelling

Advanced fixed income analysis

Advanced fixed income analysis

Fixed-income securities

Fixed Income Attribution

Fixed Income Mathematics

Fixed income mathematics

Advanced Fixed Income Analysis

Recommend Documents