thesis scm nivi

7/23/2019 Thesis SCM Nivi

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EMPIRICAL STUDIES ON CREDIT MARKETS


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ISBN 90 5170 713 4

Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul

This book is number 320 of the Tinbergen Institute Research Series, es-

tablished through cooperation between Thela Thesis and the Tinbergen

Institute. A list of books which already appeared in the series can be found

in the back.

This PhD project was partly financed by the Risk Management & Modelling

department of Rabobank International.


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Empirical Studies on Credit Markets

(Empirisch onderzoek naar markten voor kredieten)

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Erasmus Universiteit Rotterdam

op gezag van de Rector Magnificus

Prof.dr.ir. J.H. van Bemmel

en volgens besluit van het College voor Promoties

De openbare verdediging zal plaatsvinden opvrijdag 3 oktober 2003 om 13.30 uur door

Patrick Houweling

geboren te Leiden


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Promotiecommissie

Promotor: Prof.dr. A.C.F. Vorst

Overige leden: Prof.dr. D. Lando

Prof.dr. A.A.J. Pelsser

Prof.dr. M.J.C.M. Verbeek


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Acknowledgements

Writing my PhD thesis has been the single largest project I ever conducted, so these

thank yous may be the most important ones I will ever say.

First, and foremost, I would like to thank my supervisor Ton Vorst. Your guidance in

finding my way in the world of financial research has been invaluable. You have taught

me a lot about doing research in general and about credit markets in particular. Your

comments on early versions of our papers were always relevant and helped me to improve

them again and again. I am equally thankful to Albert Mentink, with whom I have worked

on several papers, two of which now appear in this thesis. Finding out that you were also

doing a PhD on credit markets, and seeing your enthusiasm to start a joint project, has

definitely been the turning point in my time as a PhD student. I greatly benefited from

your knowledge, refreshing ideas and project management skills. I am also grateful to my

other co-authors Jaap Hoek and Frank Kleibergen. Working on our paper, which is now

a chapter in this thesis, has been a very instructive and rewarding experience.

Over the years, I shared my room at the university with various PhD students: David

zuur! Dekker, Jedid-Jah ik zou haast zeggen inteGENdeel Jonker, Erjen deeeze gast

van Nierop and Richard had ik dit eerder geweten Kleijn. I sure had a great time with

you guys. Although we kept the ball rolling, we also had time to relax with a cup of tea

with our other Friends: Dennis het was weer reuze gezellig Fok, Klaas mwuuuh Staaland Bjorn wheee! Vroomen. I also enjoyed the time I spent with the rest of the lunch

gang: Wilco van den Heuvel, Joost Loef, Ivo Nobel, Rutger van Oest, Richard Paap,

Kevin Pak and Pim van Vliet. Our conversations were sometimes serious, sometimes

hilarious, but always entertaining. I much enjoyed occasional chats with Reimer Beneder,

Marisa de Brito, Jeab Cumperayot, Anna Gutkovska, Dennis Huisman, Jos van Iwaarden,

Daina Konter, Roy Kouwenberg, Bert Menkveld, Robin Nicolai, Bart Oldenkamp, Emoke

Oldenkamp, Ioulia Ossokina, Alexander Otgaar, Lennie Pattikawa, Rom Phisalaphong,

Raol Pietersz, Lidewey van der Sluis, Jan-Frederik Slijkerman, Marielle Sonnenberg, YuliaVeld, Ingrid Verheul and Martijn van der Voort. Further, I would like to thank the staff at


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vi

the Tinbergen Institute and the Econometric Institute for excellent support. In particular,

I am indebted to Carine Horbach, Tineke Kurtz, Carien de Ruijter and especially Elli Hoek

van Dijke.

During most of my time as a PhD student, I held a part-time position at the Risk

Management & Modelling department of Rabobank International. Although I changed

rooms and roommates even more often than the department changed its name, I really

felt at home. I am especially grateful to Theo Kocken, who hired me at RI. You have

given me the unique opportunity to combine my academic work at the university with

more practical projects at the bank. I am also thankful to Kees van den Berg. You

were my guide in the bank and opened doors that would otherwise have remained closed

for me. My roommates made my four years at RI not only a very instructive, but alsoa very enjoyable period. I especially owe a big thank you to Freddy spa-n-sma van

Dijk, Walter copula Foppen, Mace kan ik hier even een kopietje van maken Mesters,

Joeri van der Tonnekreek Potters, Erik okay dan! van Raaij, Marion ik ben even

naar een meeting Segeren and Sacha homo! van Weeren. I further thank my other

colleagues of the Modelling & Research team: Natalia Borokyvh, Martijn Derix, Elles

Jongenelen, Estelle Jonkergouw, Adrian Kuckler, Frans Ligtenberg, Roger Lord, Erwin

Sandee, Harmen-Jan Sijtsma and Krishna Varu. I derived much pleasure from working

on the Specific Risk project with risk managers Gerben Hagedoorn and Micha Schipper.I have learnt a lot from our many discussions in meetings, talks, phone calls and e-mails.

From the London branch, I thank Andrew Gates. You have answered more questions on

credit markets and credit derivatives than I can ever thank you for. I liked my contacts

with Jan van den Bovenkamp, Sander van Geloven, Jan-Willem de Koning and Rene van

der Pol from the IT department. I appreciate our co-operation in developing, maintaining

and extending PHsim and Rates. Support from the data base team has been a big time-

saver. I particularly give thanks to Marit de Brouwer, Rebecca Groenhuis and Marcel

Molenaar. Finally, I thank Marjolijn Benneker-Faber and Annelie Lander for superb

assistance and equally superb chats.

Besides my research activities, I also conducted educational tasks at Erasmus Univer-

sity and Rabobank International. I especially enjoyed (co-)supervising Masters students

Victor Bellido, Georges Beukering, Maaike Duijts, Rob Groot-Zwaaftink, Wing-Hei Chan,

Nathalie van der Mheen, Chios Slijkhuis and Michel van der Spek. For the werkcollege

bedrijfseconometrie I co-supervised Arjan er staat toch geen punt achter?! van Dijk,

Martijn nu zijn alle bugs eruit Krijger and Joost vlookup Kromhout. I hope you

learnt as much from me, as I learnt from you.


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vii

I am grateful to Kees van den Berg, Albert Mentink and Martijn van der Voort for

reading one or more chapters of a preliminary version of this thesis. Your comments really

helped me improve the readability of the text. I also thank Dennis Fok, Richard Kleijn

and Erjen van Nierop for getting me started with LATEX.

Last, but never least, I thank my parents for believing in me, stimulating me to achieve

the best and for always being there for me. I love you.

Patrick Houweling

Rotterdam, May 2003


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Contents

Acknowledgements v

List of Figures xi

List of Tables xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Sources of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Credit Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Credit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Estimating Spread Curves 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Multi-Curve Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Model Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.A B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.B Variances and Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Measuring Corporate Bond Liquidity 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


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x Contents

3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Pricing Credit Default Swaps 75

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Pricing Step-Up Bonds 111

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2 Step-Up Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Summary 137

Nederlandse samenvatting (Summary in Dutch) 141

Author Index 147

Bibliography 151


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List of Figures

Chapter 2

2.1 Distribution of the bonds maturity dates by rating category. . . . . . . . . 29

2.2 Single-curve estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Single-curve and multi-curve estimates. . . . . . . . . . . . . . . . . . . . . 35

2.4 Spread curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Chapter 3

3.1 Liquidity premiums for different age thresholds. . . . . . . . . . . . . . . . 59

Chapter 4

4.1 Sensitivity of spreads and default swap premiums to the recovery rate. . . 85

4.2 Scatter plots of pricing errors versus default swap premiums per rating. . . 97

Chapter 5

5.1 Credit ratings history. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2 Deutsche Telecom step-up premiums. . . . . . . . . . . . . . . . . . . . . . 127

5.3 France Telecom step-up premiums. . . . . . . . . . . . . . . . . . . . . . . 129

5.4 KPN step-up premiums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.5 Recovery rate sensitivity analysis. . . . . . . . . . . . . . . . . . . . . . . . 132


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List of Tables

Chapter 22.1 Distribution of bonds in the data set by rating. . . . . . . . . . . . . . . . 29

2.2 Distribution of included bonds in the data set by rating and industry. . . 30

2.3 Model specifications for single-curve and multi-curve models. . . . . . . . . 31

2.4 Summary statistics of single-curve and multi-curve estimates. . . . . . . . . 33

2.5 Curve Similarity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3

3.1 Overview of liquidity measures from the empirical bond liquidity literature. 56

3.2 Overview of liquidity measures, their expected signs and the portfolio order. 62

3.3 Results for the entire sample . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 Results for the characteristics portfolios. . . . . . . . . . . . . . . . . . . . 67

3.5 Portfolio statistics P = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6 Results for model 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Portfolio statistics P = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.8 Results for model 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9 Results of the comparison tests. . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 4

4.1 Characteristics of the default swap data set. . . . . . . . . . . . . . . . . . 91

4.2 Performance of the direct comparison methods. . . . . . . . . . . . . . . . 96

4.3 Paired Z-tests of the direct comparison methods. . . . . . . . . . . . . . . . 98

4.4 Goodness of fit of the reduced form credit risk models. . . . . . . . . . . . 100

4.5 Parameter estimates for the reduced form credit risk models. . . . . . . . . 101

4.6 Performance of the reduced form credit risk models. . . . . . . . . . . . . . 104

4.7 Paired Z-tests of the reduced form credit risk models. . . . . . . . . . . . . 105

4.8 Analysis of absolute pricing errors from reduced form credit risk models. . 108


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xiv List of Tables

Chapter 5

5.1 Step-up bond types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Number of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3 Characteristics of the step-up bonds. . . . . . . . . . . . . . . . . . . . . . 121

5.4 Pricing errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.5 Paired Z-tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.6 Confidence interval coverage percentages. . . . . . . . . . . . . . . . . . . . 125

5.7 Volatility analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.8 Event analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135


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Chapter 1

Introduction

1.1 Motivation

The analysis of credit markets can be traced back to at least Fisher (1959) and the val-

uation of credit-risky securities made a significant step forward due to Black and Scholes

(1973) and Merton (1974). Nevertheless, this area of research has received relatively

little attention, until several developments in the last decade awoke academics and prac-

titioners, and brought about a wave of research on credit markets. The most important

developments1 were the following:

Companies increasingly raised capital directly from the capital markets by issuingbonds rather than borrowing money from their bank, especially in the United States

(US) and Europe. Since loans are privately held and bonds are publicly traded, this

changing behavior led to more publicly available data on credit markets.

The European Monetary Union (EMU) and the liberalization of the European cap-ital markets effectively integrated the markets of the participating countries into a

single European corporate bond market. Liquidity, transparency and competition

were greatly improved.

New derivatives were developed to take on and lay off credit risk in a flexible way.The market for these credit derivatives has grown tremendously over the last decade,

both in size and product range. This necessitated the development of new models

to price and hedge these new instruments.

The prospering economic conditions and the reduction of governments budgetdeficits in the US and EMU in the second half of the 1990s, drove yields and issuance

1This list draws on Schonbucher (1999).


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2 Introduction Chapter 1

of government bonds to historically low levels. Investors thus needed other securities

to enhance their portfolio yields. Bonds issued by corporations and emerging mar-

kets were found as alternatives. Later on, credit derivatives were used to reshape

portfolios risk profiles.

Several well-published derivatives losses (e.g. Barings, MetallGesellschaft and Or-ange County), financial turmoil (in Argentina, Asia and Russia), the near-collapse

of hedge funds (most notably Long Term Capital Management, LTCM), and the

actual defaults of several large companies (e.g. Enron, KPN Qwest and Worldcom)

all contributed to a growing awareness among investors and regulators of credit and

liquidity risk.

These events and trends led to a large growth of credit markets, and at the same

time to a growing need to understand them. Numerous models were developed, some

extending the classic models of the 1970s, some drawing on the default-free interest rate

literature. Especially the valuation of credit derivatives required the development of more

sophisticated credit risk models. Empirical studies on credit markets were for a long time

hampered by a lack of market data, and often restricted to US corporate bonds. Only

since the last few years, research started to appear that analyzed not only US but also

European and emerging markets, and not only bonds but also credit derivatives.

This thesis adds four studies to the empirical literature on credit markets. The studies

deal with credit risk, liquidity risk and credit derivatives. Before their contribution is

discussed in Section 1.5, first Section 1.2 lists the sources of risk to which an investor

in credit markets is exposed, Section 1.3 gives an introduction to credit instruments and

Section 1.4 describes the main approaches to credit risk modelling.

1.2 Sources of Risk

An investor who holds credit-risky securities is exposed to a number of risks, most im-

portantly market risk, credit risk and liquidity risk. By subtracting the yields of two

instruments with equal amounts of market risk, for example a defaultable bond and a

default-free, liquid (but otherwise similar) bond, we obtain the (yield) spread. The spread

compensates investors for being exposed to credit, liquidity and other risks. The remain-

der of this section discusses the main sources of risk. Most attention is paid to credit and

liquidity risk, since they are the focus of this thesis.


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Section 1.2 Sources of Risk 3

1.2.1 Market Risk

Market risk is the risk of losses resulting from adverse movements in the level or volatility

of market prices. In credit markets, changes in interest rates are the most important

market factor, although specific instruments may also be sensitive to fluctuations in eq-

uity prices or exchange rates. To measure the market risk of portfolios, most financial

institutions use the concept of Value-at-Risk (VaR): the potential loss that is associated

with a price movement of a given probability over a specified time horizon. For instance,

a portfolio with a 1-day VaR ofE

10 million at a 95% confidence level is expected to

suffer a loss in excess ofE

10 million in one out of 20 days. Under the internal models

approach of the Basel Committee on Banking Supervision (BCBS, 1996) of the Bank for

International Settlements (BIS), the amount of regulatory capital banks have to put aside

to cover market risks is based on their VaR level.

1.2.2 Credit Risk

Credit risk concerns the losses caused by the possibility that an entity will fail to fully

and timely meet its contractual obligations. With traditional debt instruments, such as

bonds and loans, the borrower is obliged to pay the coupons and the notional amount

in time. With derivatives, such as swaps and options, the amounts due depend on theprevailing market conditions, as specified by the contract. Credit risk can be separated

into two components: default risk is the uncertainty about whether or not the entity will

fail to meet its obligations; recovery risk is the uncertainty about the amount that will be

recovered in case of default. Together they determine the compensation investors receive

for bearing credit risk: the credit spread.2 The spread on a borrowers securities can be

seen as the markets assessment of its credit risk.

Another important indicator of a borrowers credit worthiness is its credit rating:

a subjective assessment of its credit or default risk, measured on an alphanumeric scale.

The scales of the two major independent rating agencies, Moodys and Standard & Poors

(S&P), are {Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C} and {AAA, AA, A, BBB, BB, B, CCC,CC, C}, respectively. Aaa/AAA indicates the highest credit quality, C the lowest. Therating agencies refine these major ratings into minor ratings by adding notches to the

letters: Moodys uses postfixes 1, 2 and 3 and S&P adds a + sign, no postfix, or a

sign. The first four ratings are collectively called investment grade and the remainingratings speculative grade. Many financial institutions assign internal credit ratings to their

counterparties as well. Sometimes rating migration risk is seen as a separate component

2Changes in credit spreads are sometimes considered part of market risk instead of credit risk.


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of credit risk. Also, some credit derivatives explicitly depend on the credit rating of the

underlying issuer.

While market risk can typically be measured and hedged on a day-to-day basis, credit

risk accumulates over longer-term time horizons, e.g. one year. Moreover, market prices

of many credit-risky securities are not daily updated, and defaults (and to a lesser extent

rating migrations) occur only infrequently. The data scarcity and the longer time horizon

make estimating and backtesting of credit risk models much more difficult than of market

risk models. Regulation of credit risk has long followed the Basel Capital Accord (BCBS,

1988), requiring banks to hold regulatory capital equal to at least 8% of a risk-weighted

basket of assets to cope with potential losses from these assets. In 1999 and 2001, the

BCBS issued consultative documents on the New Basel Capital Accord, proposing morerisk-sensitive weights based on external or internal credit ratings (BCBS, 1999, 2001).

The new accord is planned to replace the current accord in 2006.

1.2.3 Liquidity Risk

Liquidity risk, also called marketability risk, involves the possibility of not being able to

timely buy or sell an instrument in the desired quantity with little impact on prices.

Liquidity may differ between instruments (e.g. swaps are generally more liquid than

bonds), between instrument types (e.g. plain vanilla instruments are more liquid than

exotic instruments), between issuers (e.g. government bonds are typically more liquid than

corporate bonds) and between markets (e.g. the euro capital market is more liquid than

emerging markets). Liquidity risk tends to aggregate other sources of risk. For example,

markets tend to lose liquidity in times of crises and/or high volatility, e.g. the 1987

stock market crash or the financial turmoil in Asia and Russia in 1998. Also, lower rated

bonds are often less liquid than otherwise similar bonds with higher ratings, since many

large investors, like pension funds, are not allowed to hold speculative grade securities.

At the moment, there are no regulatory guidelines that cover liquidity risk, but the BISCommittee on the Global Financial System has started publishing on the topic; see CGFS

(1999, 2001).

1.2.4 Other Risks

This section discusses several other sources of risk, though without being exhaustive.

Although institutions and regulators put the most effort in managing and measuring

market, credit and liquidity risk, many of the major losses of the 1990s mentioned above

(including the collapses of Barings and Enron) were due to operational risk: the risk


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Section 1.3 Credit Instruments 5

of unexpected losses arising from deficiencies in a firms management information, proce-

dures, and control systems. Specifically, for banks a mistake or fraud in the trading or risk

management department can have more harmful effects than a market crash. Operational

risk will be covered in the New Basel Capital Accord (BCBS, 2001).

Another source of risk that is not limited to financial institutions is legal risk, which is

the risk that a transaction proves unenforceable in law or because it has been inadequately

documented. For example, in the early days of credit derivatives, legal risk was the largest

concern to participants of the biannual Credit Derivatives Survey by the British Bankers

Association (BBA, 1998), because documentation was not yet standardized and different

counterparties used different definitions and legal structures. Documentation disputes led

to several lawsuits, for instance after Russias default on its sovereign debt in 1998 andafter the restructuring of Consecos debt in 2000.

Systemic risk refers to the possibility of disruptions in the functioning of financial

markets that are severe enough to reduce economic activity. Such a systemic event is

typically hypothesized to occur after the initial bankruptcy of one large institution, fol-

lowed by a domino-style contagion that causes the bankruptcy of many more and, as a

worst-case outcome, the collapse of the financial system as a whole.

As mentioned above, the studies comprising this thesis focus on credit and liquidity

risk. While the studies take into account market risk, the other sources of risk are ignored;not because they are less important, but because they affect all instruments alike and are

much harder to quantify.

1.3 Credit Instruments

The empirical studies in this thesis use market data on two types of credit instruments:

bonds and credit derivatives. The remainder of this section discusses bonds only briefly

and credit derivatives in more detail, because the latter are more recently introduced andless well-known.

1.3.1 Bonds

A bond is an obligation on its issuer to its holder with the purpose of raising capital

by borrowing. Typically, the holder pays the amount borrowed (notional amount) to

the issuer at issuance and the issuer promises to repay this amount in the future, along

with interest payments (coupons). The frequency at which interest payments are made

varies across markets, but is usually annually or semi-annually. Various coupon types


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exist: a fixed-income bond pays a fixed percentage of its notional; the coupon percentage

of a floating-rate bond is reset periodically to a specified short-term interest rate plus

a specified spread; for a (rating-triggered) step-up coupon bond, the coupon percentage

depends on the credit rating of its issuer; a zero-coupon bond does not make interest

payments. More bond flavors can be created by varying the redemption method: if the

principal is repaid as a whole on the maturity date, the bond is called a bullet bond; with

a sinking bond the issuer repays the face value in several terms on a set of pre-specified

dates. Uncertainty regarding the bonds maturity is introduced when option-like features

are added: a callable bond gives the issuer the right to redeem the principal prematurely;

a puttable bond gives the holder the right to sell the bond back to the issuer early; an

extendiblebond enables the issuer to extend the life of the bond beyond the initially agreedredemption date; finally, with a perpetual bond the issuer never repays the principal.

Most chapters in this thesis analyze fixed-income or zero-coupon bullet bonds without

optionalities (usually called plain vanilla bonds), with the exception of Chapter 5, which

studies step-up bonds.

1.3.2 Credit Derivatives

A credit derivative allows the transfer of credit risk3 without transferring the ownership

of debt issued by the underlying borrower(s). The pay-out of a credit derivative can

depend on: (i) the occurrence of a credit event, e.g. bankruptcy, failure to make an

interest payment, debt restructuring or debt acceleration (jointly called default), or a

rating migration; (ii) the payments, price, and/or yield spread of one or more bonds of

the underlying borrower(s). Many credit derivatives are insurance-like contracts between

two parties, where one party buys protection from the other party to a deterioration of the

borrowers credit worthiness. The protection buyer will either make periodic payments or

pay an up-front premium to the seller, and the protection seller will, upon the occurrence

of the specified credit event, make a payment to the buyer. Hence, the buyer has reducedhis credit exposure to the underlying entity in return for a periodic fee. Several examples

of credit derivatives are given below.

Credit derivatives were first introduced on the annual meeting of the International

Swaps and Derivatives Association (ISDA) of 1992 and some trading began in 1992 as

well. Academic papers started to appear around 1995; see e.g. Howard (1995), Smithson

(1995) and Das (1996). Several books were published on the subject a few years later; see

e.g. Das (1998), Tavakoli (1998) and Francis, Frost and Whittaker (1999). Since 1992,

3Some credit derivatives, like total return swaps, also transfer market risk.


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the global credit derivatives market has experienced impressive growth. Whereas market

size (measured in total outstanding notional) amounted to no more than a few billion US

dollars in 1995, participants to the latest Credit Derivatives Survey(BBA, 2002) estimated

that the market has grown to US$ 2.0 trillion at the end of 2002; participants to the annual

Credit Derivatives Survey by Risk Magazine (Patel, 2003) estimated a market size of US$

2.4 trillion.4 To put these figures into perspective, as of June 2002 the total outstanding

notional of interest rate swaps amounted to US$ 68 trillion and of interest rate options to

US$ 13 trillion (BIS, 2002, Table 19). Although the market for credit derivatives is still

relatively small, it is catching up fast with annual growth rates of at least 50%.

The publication of the Credit Derivatives Definitions (ISDA, 1999) was a big move

to standardizing the terminology in credit derivatives transactions. The ISDA Defini-tions were amended in 2001 with the Restructuring Supplement (ISDA, 2001) following

disagreements in the market on which obligations can be delivered in physically settled

contracts in case of a debt restructuring; see also Tolk (2001). The Definitions established

a uniform set of definitions of important terms, such as the range of credit events that

could trigger payments or deliveries. In addition to the enhanced enforceability and inter-

pretation of the contracts, the Definitions increased flexibility and reduced the complexity

of administration and documentation. More than 90% of all credit derivative transactions

are being documented by the ISDA confirms (BBA, 2002, page 22).

Types

According to the BBA (2002), credit default swaps are the most popular type of credit

derivative (accounting for 45% of the market), followed by collateralized debt obligations

(22%), credit-linked notes (8%), total return swaps (7%), basket products (6%) and credit

spread options (5%). These contracts are briefly discussed below; see Tavakoli (1998) or

OKane (2001) for details and other credit derivatives.

Default swap: A (credit) default swap (CDS) is an insurance-like contract thatprotects the holder of the underlying asset(s) from the losses caused by the occur-

rence of a specified credit event to the reference entity. The protection buyer makes

periodic payments to the protection seller, typically a specified percentage of the

notional amount. If the credit event occurs, the protection seller reimburses the loss

incurred by the protection buyer, so that the value of the buyers asset(s) is restored

to the notional amount. Note that a default swap only pays out if the reference en-

4

Both surveys are based on interviews and estimations, and should therefore be treated as indicationsrather than hard numbers.


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tity defaults; reductions in value unaccompanied by default do not compensate the

buyer in any way.

Credit-linked notes: A credit-linked note is basically a combination of a bond anda default swap. At initiation of the contract, the protection buyer sells a bond to the

protection seller and thus receives the notional of the bond. The protection buyer

makes (fixed or floating) coupon payments on the bond during the contract period.

At the maturity date, the bond is to be redeemed at par, unless a credit event has

occurred to the reference entity, in which case the buyer only pays the recovered

amount. Although much like a default swap, there is an important difference. With

credit-linked notes the protection seller makes his payment in advance and receives

it back fully (if no event occurs), or partially (if the event does occur). With defaultswaps, the protection seller only makes a payment after the event has occurred.

Total return swap: In contrast to a default swap, which only transfers credit risk,a total return swap (TRS; also called a total rate of return swap, TRORS) transfers

both credit risk and interest rate risk. The buyer makes periodic payments to the

seller, usually specified as a spread over interbank interest rates (LIBOR). The seller

pays to the buyer the total return of the asset, comprising of interest payments and

change-in-values payments. The latter are defined as any appreciation or depreci-

ation in the market value of the reference obligation. Hence, a net depreciation invalue results in a payment to the seller. When entering into a total return swap on

an asset, the buyer has effectively removed all economic exposure to the underlying

asset. The seller on the other hand has gained exposure to the underlying without

the initial outlay required to purchase the reference obligation.

Credit spread option: A credit spread option is similar to a standard stockoption, except that the underlying is a credit spread rather than a stock price.

With a credit spread option, one party pays an up-front premium to the other in

return for a payment at the maturity date that is linked to the difference betweenthe actual spread and the specified strike. Credit spread options thus allow users to

bet on or hedge against future spread movements.

Collateralized debt obligations: A collateralized debt obligation (CDO) is astructure of fixed-income securities, called the tranches, whose cash flows are backed

by the payments of an underlying pool of debt instruments, the collateral, through

a set of rules, the waterfall structure. When the collateral is a pool of bonds, the

structure is called a collateralized bond obligation (CBO); with a pool of loans,

it is a collateralized loan obligations (CLO); with a pool of default swaps, it is asynthetic CDO. The tranches have different priorities: income from the collateral


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is first paid to the senior tranches, than to the mezzanine tranches and finally to

the equity tranches. A CDO allows the redistribution of the credit risk of a pool of

assets to create securities with a variety of risk profiles.

Basket products: An nth-to-default swap is similar to a regular default swap, butnow the credit event that triggers the payment to the protection buyer is the nth

default in a specified basket of borrowers. For instance, in a first-to-default swap the

first borrower to default, triggers the contract. Likewise, a basket total return swap

is just like a regular TRS, but instead of single underlying bond, the cash flows and

price changes of a portfolio of bonds are passed through.

Applications

Like any derivative, a credit derivative can be used to take on or lay off risk. Because

a derivative does not transfer the ownership of the underlying assets and often does not

require an initial investment, risk can be transferred more efficiently than in the cash

market: buyers can reduce credit exposure without physically removing assets from their

balance sheets, and sellers get the opportunity to run credit risk without actually buying

the reference asset. Specifically, credit derivatives have the following applications (ranked

in order of importance according to the BBA, 2002):

Managing credit lines: Banks are limited in the amount of business they cando with a particular borrower. Yet even if a credit line is full, a bank may want

to lend additional funds to a borrower to prevent a deterioration in its relationship

with the client. Credit derivatives offer a solution to this dilemma: the bank can

give the client a new loan and, without having to notify the client, simultaneously

buy default protection on the client in the credit derivatives market. Now the bank

has both fostered its relationship with the client and kept its credit risk within the

specified limits. Regulatory arbitrage: In the 1988 Capital Accord, corporations have a risk

weight of 100% in calculating the amount of regulatory capital, but banks from

OECD countries only 20%. Consequently, by using credit derivatives banks can

transfer credit risk on corporate loans to an OECD bank to reduce regulatory cap-

ital. These risk transfers are profitable to banks, because the freed capital can be

put to other uses again. Banks regulators are dissatisfied with such transactions,

because regulatory capital is not in line with actual risk taking behavior. The New

Capital Accord addresses this issue by proposing rating-based risk weights.


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Product structuring: Credit derivatives can be used to split the credit risk ofone or more assets and redistribute it into more risky or less risky forms that suit

the risk appetites of different investors. This also allows the creation of tailor-made

investment products.

Portfolio management: Credit derivatives allow investors to change their port-folio characteristics by reducing or taking on exposure to particular companies,

sectors, regions and/or maturities, which would be much harder or even impossible

just using bonds and loans. For example, the lack of a market for repurchase agree-

ments (repos) for most corporates makes shorting corporate bonds infeasible. So,

credit derivatives are the only viable way to go short corporate credit risk.5

1.4 Credit Models

In the literature, there are two classes of credit risk models: portfolio models and pric-

ing models. Portfolio models are primarily used for risk measurement and management

purposes, like calculating VaR-like risk measures and marginal risk contributions. In this

thesis, only instances of the class of pricing models are used, so we restrict ourselves to

mentioning some examples of the class of portfolio models: CreditMetrics (J.P. Morgan:

Gupton, Finger and Bhatia, 1997), CreditPortfolioView (McKinsey: Wilson, 1997a,b),

CreditRisk+ (Credit Suisse Financial Products, 1997) and Portfolio Manager (Moodys

KMV: Kealhofer, 2001); see Crouhy, Galai and Mark (2000) for a comparison of this class

of models or the books by Caouette, Altman and Narayanan (1998) and Saunders (1999)

for in-depth treatments.

Pricing models are mainly used for investment-related purposes, including pricing of

bonds and credit derivatives, calculating hedge ratios, and seeking favorable investment

opportunities. There are two types of pricing models: structural form models and reduced

form models. The distinction between these two types of models is blurring though, since

hybrid models have also started to appear (e.g. Madan and Unal, 2000). In fact, Duffie

and Lando (2001) showed that under asymmetric information, reduced form models can

be seen as the reduced form of a particular type of structural model.

The remainder of this section outlines structural and reduced form models; see also

Nandi (1998), Jeanblanc and Rutkowski (1999) and OKane and Schlogl (2001).

5Even if a bond can be shorted on repo, investors can only do so for short periods of time (one day

to one year), exposing them to changes in the repo rate, next to changes in credit spreads. On the otherhand, credit derivatives allow investors to go short credit risk at a known cost for long time spans.


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Section 1.4 Credit Models 11

1.4.1 Structural Models

In the class of structural models, also called firm value models or Merton models, a firm

defaults when the value of the firms assets drops below a certain threshold. If this

happens, bond holders get the residual value of the firm and share holders receive nothing.

If the firm survives, bond holders are paid off, and share holders receive the remaining

value of the assets. In this framework, both bonds and stocks are contingent claims on

the value of the firms underlying assets, so that option pricing theory can be used to

calculate theoretical debt and equity values.

The advantage of structural models is that they describe how default actually occurs

and that recovery is determined endogenously. However, structural models have diffi-

culties incorporating complex debt structures. Moreover, their parameters are hard to

estimate, because the assets market value and volatility are difficult to observe. Finally,

structural models may be better suited for corporate than for sovereign issuers, because

for countries the asset value concept is not applicable, and even though a country is able

to pay, it may not be willing to do so.

Structural models were first developed by Black and Scholes (1973) and Merton (1974).

In their model, a firm has a simple debt structure consisting of one zero-coupon bond. De-

fault can only occur at the bonds maturity date. The classic model has been extended by

several authors. Black and Cox (1976) introduced the possibility of intermediate default

into the model, as well as indenture clauses, safety covenants and subordination arrange-

ments. Geske (1977) and Geske and Johnson (1984) allowed a more general capital struc-

ture and considered coupon-bearing bonds instead of zero-coupon bonds. Shimko, Tejima

and van Deventer (1993) and Longstaff and Schwartz (1995a) relaxed the assumption of

deterministic interest rates and used a Vasicek (1977) model to describe the evolution of

the default-free term structure. Leland (1994), Leland and Toft (1996) and Mella-Barral

and Perraudin (1997) combined the model with strategic behavior models from corpo-

rate finance. Schonbucher (1996) and Zhou (1997) generalized the continuous asset valueprocess to a jump-diffusion process, so that defaults can also come as a surprise.

Moodys KMV commercially applies the structural approach (Crosbie, 2002). Fitting

a Merton-type model to balance sheet and equity price data, KMV calculates an Expected

Default Frequency measure, which is a firms probability of default for the next year.

1.4.2 Reduced Form Models

In reduced form models, also called intensity-based models, the direct reference to the

firms asset value process is abandoned. Instead, default is modelled as an exogenous


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event. In particular, default is linked to a counting process that literally counts the

number of defaults. Typically, we are only interested in the first default, so the default

time is defined as the time of the first jump of the counting process. A popular example

is the Poisson process, whose stochastic behavior is driven by a hazard process, also called

an intensity process. The hazard rate is the arrival rate of the default event and can

be interpreted as a conditional, instantaneous default probability. Theoretical prices for

bonds and derivatives are computed using equivalent martingale measures; see Bielecki

and Rutkowski (2001) for a detailed account of the mathematics of reduced form models.

The advantage of reduced form models is that their parameters are easy to estimate.

Also, they can be calibrated to the market prices of liquid instruments and subsequently

used for the pricing of credit derivatives; this is very similar to the calibration and pricingof default-free interest rate derivatives. Their drawback is that it is difficult to realistically

model the recovery process (see below).

The first reduced form model was developed by Litterman and Iben (1991), who

considered a simple setup without recovery in case of default. Jarrow and Turnbull (1995)

formalized their model using risk-neutral valuation and assumed a fixed recovery rate at

maturity and a Poisson process with a fixed hazard rate. Lando (1998) further generalized

the framework by making the hazard rate stochastic; this is called a Cox process or

doubly-stochastic Poisson process. Typically, a factor model is employed to drive both thedefault-free term structure and the hazard rate, so that default-free rates and the default

time are correlated. With Cox processes, hazard rates can also be made dependent on

equity prices, hence bringing in new information; see Jarrow and Turnbull (2000), Jarrow

(2001) and Pan (2001). In a somewhat different approach, due to Duffie and Singleton

(1999), there is no need to separately model the hazard and recovery components of credit

risk, but it suffices to model the spread process. Other implementations of this approach

include Longstaff and Schwartz (1995b) and Das and Sundaram (2000).

To use the information present in credit ratings and to value securities that explicitly

depend on ratings, Lando (1994) and Jarrow, Lando and Turnbull (1997, JLT) developed

a rating-based reduced form model. They used a Markov chain with ratings as states. The

model was generalized by Das and Tufano (1996) to incorporate stochastic recovery rates,

and by Lando (1998) and Arvanitis, Gregory and Laurent (1999) to make transition

intensities stochastic and possibly dependent on state variables. Schonbucher (1999),

Bielecki and Rutkowski (2000) and Acharya, Das and Sundaram (2002) embedded the

Markov chain in a Heath, Jarrow and Morton (1992) framework.

A difficult issue in reduced form models is the recovery assumption. Whereas in

structural models the amount recovered by bond holders in case of default is determined


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Section 1.5 Overview 13

endogenously, reduced form models have to specify the recovery process explicitly. Three

recovery assumptions are found in the literature:

Recovery of Treasury: This approach, first used by Jarrow and Turnbull (1995),assumes that at the default time, the defaulting bond is replaced by a default-free,

but otherwise similar, bond. The main advantage of this approach is that it leads to

a closed-form solution of defaultable bonds, so that risk-neutral default probabilities

can be easily backed out from default-free and defaultable term structures.

Recovery of market value: Under this assumption, introduced by Duffie andSingleton (1999), a bond loses a constant fraction of its market value. Duffie and

Singleton (1999) showed that this assumption allows credit-risky claims to be valued

as if they were default-free, but now discounted by risk-adjusted interest rates.

Recovery of face value: This assumption, applied by Jarrow and Turnbull (2000)and Schonbucher (2000), conforms best to real-world defaults, where investors re-

cover a fraction of the bonds face value (and sometimes accrued interest as well). It

also corresponds to the way Moodys and S&P publish their recovery rate estimates.

For the management of credit risk on a portfolio basis, and for the pricing of basket

credit derivatives and CDOs, models are required that describe the joint credit worthi-

ness of multiple issuers. This has been accomplished in the literature in several ways.

Duffie (1998) and Duffie and Singleton (1998) introduced correlations between the hazard

processes of issuers by making them dependent on common factors. Gupton et al. (1997)

and Hull and White (2001) extended the Merton (1974) approach to multiple issuers by

correlating the underlying asset processes. Finally, Li (1999, 2000) and Schonbucher and

Schubert (2001) used copula functions to model the dependency structure between the

marginal default densities. Since copula functions allow the univariate behavior to be

separated from the dependency structure, any correlation structure can be imposed, for

example equity correlations or spread correlations.

1.5 Overview

This thesis contributes four studies to the empirical literature on credit markets. Chap-

ters 2 and 3 are concerned with the measurement of yield spreads and corporate bond

liquidity, respectively. Chapters 4 and 5 contain two empirical studies on the pricing of

credit derivatives. The remainder of this section outlines the four chapters.

Chapter 2 presents a robust framework for the estimation of yield spreads. Spreads

are an important input for the pricing of defaultable bonds and credit derivatives and for


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risk management purposes. Inaccuracies or errors in the estimated spreads will result in

incorrect prices or risk measures. Traditionally, spread curves are calculated by subtract-

ing independently estimated default-free and defaultable term structures of interest rates.

It is illustrated that this results in twisting spread curves that alternately have positively

and negatively sloped segments. In Chapter 2, a new framework is presented for the joint

estimation of the default-free term structure and corporate spread curves. The model

is based on the decomposition of a defaultable term structure into a default-free part

and a spread part. The default-free curve is estimated from government bonds, so that

the model for the corporate term structure can focus on the spread curve only and can

thus be parsimonious. The performance of the new model is compared to the traditional

method by estimating them on a data set of German mark-denominated government andcorporate bonds.

Chapter 3 is concerned with the estimation of liquidity spreads of corporate bonds.

For an investor, it is important to know whether a bond is liquid or illiquid, because if

he needs to sell an illiquid bond before its maturity, he faces higher transaction costs,

due to a larger bid-ask spread and/or order processing costs, than for a comparable,

liquid bond. Direct liquidity measures, such as trading volume or trading frequency, are

not available for corporate bonds, since most transactions occur on the over-the-counter

market. Therefore, the literature has proposed numerous indirect liquidity measures thatare based on bond characteristics and/or market prices. In Chapter 3, an empirical

comparison is conducted of eight indirect measures of corporate bond liquidity, one of

which is new to the literature. Great care is taken to ensure that bond yields are corrected

for market and credit risk to properly identify the spread associated with liquidity risk.

For each liquidity measure, the significance of the liquidity effect is determined on a

data set of euro-denominated bonds. Moreover, a series of pairwise comparison tests is

conducted to establish the effectiveness of the liquidity measures relative to each other.

Chapter 4 contains an empirical study on the pricing of default swaps. Since default

swaps are the most popular credit derivative, it is important to know how market par-

ticipants price them. Moreover, default swap data provide an interesting challenge for

the credit risk models that have been developed for the pricing of credit derivatives. In

Chapter 4, a reduced form model is implemented with a deterministic hazard process and

a constant recovery of face value assumption. The model is estimated on market prices of

bonds and subsequently used to calculate theoretical default swap premiums. For com-

parison, a simple spread-based approach is also implemented, which directly compares

bonds yield spreads to default swap premiums. The chapter pays attention to the im-

plementation of the approaches, by considering several alternatives for the choice of the


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Section 1.5 Overview 15

default-free term structure and by testing the robustness with respect to the assumed

recovery rate.

Chapter 5 provides an empirical analysis of the pricing of rating-triggered step-up

bonds. Step-up bonds are basically fixed-income bonds with a built-in credit derivative,

whose payoff depends on the issuers credit rating. These bonds formed an important

source of financing for European telecom companies in recent years when they had diffi-

culty issuing plain vanilla bonds due to financial distress. Further, step-up bonds allow

empirical testing of rating-based credit risk models. In Chapter 5, three methods are

compared to value step-up bonds: (i) the Jarrow, Lando and Turnbull (1997) framework,

(ii) a similar framework using historical probabilities and (iii) as plain vanilla bonds. It

is tested which method provides the best approximation to market prices, and whetherstep-up bonds offer protection to investors in the form of superior returns or lower price

volatility.

Chapter 6 concludes the thesis.


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Chapter 2

Estimating Spread Curves1

2.1 Introduction

Many credit risk models require an accurate description of the term structures of inter-

est rates of different credit risk classes as input data. Measuring a term structure for a

particular credit rating class amounts to estimating its credit spread curve relative to the

government curve, which proxies the default-free curve. Traditionally, spread curves are

calculated by subtracting independently estimated government and corporate term struc-

tures. In this chapter, we present a new framework that jointly estimates the government

curve and credit spread curves. Unlike the twisting curves one gets from the traditional

method, the estimated spread curves are now smooth functions of time to maturity, and

are less sensitive to model settings. Moreover, we develop a novel test statistic that allows

us to determine the optimal settings of the new model.

An important application in which accurately estimated term structures of interest

rates are essential inputs is the pricing of defaultable bonds and credit derivatives. The

leading frameworks are the Jarrow, Lando and Turnbull (1997) Markov chain model,which extended the work of Litterman and Iben (1991) and Jarrow and Turnbull (1995)

to multiple credit ratings, and the Duffie and Singleton (1999) framework, which can be

cast into a defaultable Heath, Jarrow and Morton (1992, HJM) model. Similar to the

default-free interest rate models developed in the early 1990s most notably the extended

Vasicek (1977) models, such as Hull and White (1990), the lognormal short rate models,

like Black, Derman and Toy (1990), and the models in the HJM framework these credit

risk models provide an exact fit to todays default-free and defaultable term structures

1

This chapter is a slightly revised version of the article by Houweling, Hoek and Kleibergen (2001),which has been published in the Journal of Empirical Finance.


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18 Estimating Spread Curves Chapter 2

of interest rates. Any error in the input of such models will be amplified in the prices of

interest rate and credit derivatives that are subsequently priced with them.

Interest rates and spread curves are also required for risk management purposes, for

example in applying the historic simulation method to calculate the Value at Risk for

a corporate bond portfolio; see e.g. Saunders (1999, Chapter 11). Future scenarios are

generated by adding historical day-to-day movements in interest rates and spread curves

to todays curves. Since in each scenario the bond portfolio is revalued to obtain the

empirical distribution of the future portfolio value, inaccurate curves may lead to an

unnecessarily large Value at Risk and a too large amount of regulatory capital. Other

applications of default-free and defaultable interest rates include the pricing of new bond

issues and assessing counterparty risk in derivative products; see e.g. Hull and White(1995), Duffee (1996) and Caouette et al. (1998).

An obstacle in the above mentioned applications is that the term structures are not

directly observable in the market and have to be estimated from market prices of traded

instruments. Until now the literature has primarily focused on the estimation of the

default-free term structure from a data set of government bonds. The standard approach

originates from McCulloch (1971, 1975), who modelled the discount curve as a linear com-

bination of polynomial basis functions. Other approaches include the usage of Bernstein

polynomials (Schaefer, 1981), exponential splines (Vasicek and Fong, 1982), B-splines(Shea, 1985; Steeley, 1991), exponential forms (Nelson and Siegel, 1987) or a bootstrap-

ping procedure as employed on electronic information systems Bloomberg and Reuters;

Anderson, Breedon, Deacon, Derry and Murphy (1996, Chapter 2) provided an extensive

overview of these and other term structure estimation methods. After choosing one of

these methods, we could independently estimate a separate model for each credit class.

We illustrate that these calculations are likely to result in twisting spread curves that al-

ternately have positively and negatively sloped segments. Moreover, the level and shape

of the spread curve are shown to be sensitive to model misspecification.

Instead, we jointly estimate the default-free and defaultable interest rate curves. Our

joint estimation is based on the decomposition of a defaultable term structure into a

default-free curve and a credit spread curve. The default-free curve is estimated from

government bonds, so that our model for a corporate term structure focuses on the credit

spread only. Both the government curve and the corporate spread curve are modelled

as B-spline functions and all parameters are jointly estimated from a combined data

set of bonds. We apply the model to a data set of liquid, German mark-denominated

bonds, whose credit ratings range from Standard and Poors ratings AAA to B. We

obtain smooth and reliably estimated spread curves that are relatively robust to model


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Section 2.2 Multi-Curve Model 19

misspecification. Moreover, we demonstrate that these results can be attributed to both

the joint and the parsimonious modelling. Independently estimating the government

curve and a parsimoniously specified corporate curve model does not yield the same

results, nor does jointly estimating the government curve and a richly specified corporate

spread curve.

The remainder of this chapter is structured as follows. Section 2.2 presents the new

framework for the joint estimation of the government term structure and corporate credit

spread curves. The specification of the model is described in Section 2.3, whereas Sec-

tion 2.4 goes over several methods to choose between competing models, including a novel

statistic that is developed to compare spread curves obtained from alternative model

specifications. Section 2.5 describes our data set. Section 2.6 applies the new model andcompares jointly estimated term structures with independently estimated term structures.

Finally, Section 2.7 summarizes the chapter.

2.2 Multi-Curve Model

Ideally, we would like to use a different spread curve for each firm, reflecting the uniqueness

of a firms characteristics that determine its credit risk. Due to data constraints, however,

which are discussed in Section 2.5, we have to resort to grouping firms that have similarcredit worthiness and face similar operating environments. A disadvantage of grouping

bonds is that a particular type of heterogeneity2 may occur; see Helwege and Turner

(1999). Suppose we have created C categories of bonds, where category 1 corresponds to

government bonds and the other categories are formed by using, e.g., credit rating and

industry as criteria. The purpose is to estimate a spread curve for each category. Instead

of independently estimating term structures, we propose a joint estimation approach.

Since a corporate term structure consists of a default-free curve and a credit spread

curve, it seems natural to only model the spread and take the default-free part from thegovernment curve. Several representations of the term structure exist, e.g. as discount

factors or spot interest rates, but it is common practice to model the discount curve. We

use the following framework for jointly estimating the discount curves

D1(t) = d(t)

Dc(t) = d(t) + sc(t), c = 2, 3, . . . , C ,(2.1)

2Within a data set of bonds of the same rating, the longest maturity bonds usually have been issued by

the relatively most credit worthy firms. Therefore, credit spreads may decrease for the longest maturitiesin such a data set.


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where Dc() is the discount curve of category c, d() is the model for the government dis-count curve and sc() is the model for the discount spread curve of category c with respectto the government curve. We impose C constraints Dc(0) = 1, because a payment due

today does not need to be discounted. All parameters in the models for the government

curve and the discount spread curves are jointly estimated from a combined data set of

government and corporate bonds. We refer to this model as the multi-curve model as

opposed to a single-curve model that independently estimates a single term structure.

To model d() and sc(), we use spline functions, as introduced to the term structureestimation literature by McCulloch (1971). Some commonly used types of splines are

exponential splines by Vasicek and Fong (1982), and B-splines, as discussed by Shea

(1985) and Steeley (1991). Splines are tailored to approximate a scatter of data points bya continuous and preferably smooth function. Their main advantage is their flexibility:

there is no need to a priori impose a particular curvature, because the shape of the curve

is determined by the data. Bliss (1997) compared several non-parametric term structure

estimation models and found that spline models perform at least as good as competing

models, and outperform the other considered models if the data contains longer maturity

bonds (over 5 years). Jankowitsch and Pichler (2002) estimated our framework on a

data set of nine EMU governments using both cubic splines and Svensson (1994) curves.

They confirmed our results, and moreover, found that the multi-curve splines model andthe multi-curve Svensson model performed similarly. Jankowitsch and Pichler (2003)

extended our framework to multiple currencies.

Splines are basically piecewise polynomials. The approximation interval3 [a, b] is di-

vided into n subintervals [0, 1], [1, 2], . . . , [n1, n], where the knots i are chosen such

that a = 0 < 1 < .. . < n = b. The data points in each subinterval are modelled as a kth

degree polynomial. The n polynomials are constrained by the condition that the spline

has to be k 1 times continuously differentiable. This condition imposes k constraints onthe coefficients of two adjacent polynomials at the knots 1, 2, . . . , n1. In sum, we have

n(k + 1) coefficients minus (n 1)k constraints, leaving n + k degrees of freedom. A moreparsimonious way of representing splines is by means of basis functions; see e.g. Powell

(1981, page 228). Any kth degree spline function S() with knots = (0, . . . , n) can beexpressed as a linear combination of n + k basis functions f() = {f1(), f2(), . . . , f n+k()}

S(t) =n+ks=1

sfs(t) = f(t).

3

With term structure estimation the approximation interval runs from 0 to the longest bond maturityin the sample.


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Section 2.2 Multi-Curve Model 21

Once the basis is chosen and the degree k and the knots are set, the basis functions

are fully specified. The spline weights , however, are unknown and have to be estimated

from the data. Powell (1981) recommended the use of a basis of B-spline functions,

because of their efficiency and numerical stability. Steeley (1991) applied B-splines to

term structure estimation. Appendix 2.A contains a concise description of constructing a

basis of B-splines; see Powell (1981) for more details.

We use B-splines to model the government discount curve and corporate discount

spread curves in Equation (2.1). We set d(t) = g1(t)1 and sc(t) = gc(t)

c, where gi()contains ni + ki B-spline basis functions that span a spline of k

thi degree with knots i.

Section 2.3 discusses the specification of the degrees and knots. Using B-spline basis

functions, we rewrite the multi-curve model in Equation (2.1) as

D1(t) = g1(t)1 (2.2a)

Dc(t) = g1(t)1 + gc(t)

c, c = 2, 3, . . . , C . (2.2b)

To estimate the unknown spline weights 1, 2, . . . , C, we construct a data set, consisting

of Bc bonds of category c, and use the discounted cash flow (DCF) principle to link the

bond prices to a discount curve. According to the DCF principle, the price that an

investor is willing to pay for the bth bond of category c equals the sum of the present

values of the cash flows

PDCFcb =

Ncbi=1

CFcbiDc(tcbi), (2.3)

where PDCFcb is the DCF bond price, Ncb is the number of remaining cash flows and

CFcbi is the ith cash flow that is paid at time tcbi. By using the DCF equation as the

theoretical bond price model, we have to confine our data set to fixed-income bonds with

known redemptions and exclude any bonds with optional elements such as callable andputtable bonds and bonds with floating or index-linked coupons. The DCF method is

valid if we assume a perfectly competitive capital market, i.e. if all relevant information is

widely and freely available and no barriers, frictions and taxes exist. Brealey and Meyers

(1991, page 20) stated that even though these conditions are not fully satisfied, there is

considerable evidence that security prices behave almost as if they were.


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For category 1, i.e. for government bonds, we substitute Equation (2.2a) into Equa-

tion (2.3), yielding

PDCF1b =

N1bi=1

CF1bi

n1+k1s=1

1sg1s(t1bi)

=

n1+k1s=1

1s

N1bi=1

CF1big1s(t1bi)

n1+k1s=1

x1bs1s = x1b1,

(2.4a)

where xcbs =Ncb

i=1 CFcbig1s(tcbi). For categories 2, 3, . . . , C substitution of Equation (2.2b)

into Equation (2.3) results in

PDCFcb =n1+k1s=1

1s

Ncbi=1

CFcbig1s(tcbi)

+

nc+kcs=1

cs

Ncbi=1

CFcbigcs(tcbi)

n1+k1s=1

xcbs1s +nc+kcs=1

ycbscs = xcb1 + y

cbc,

(2.4b)

where ycbs =Ncb

i=1 CFcbigcs(tcbi). Note that these equations for the theoretical bond price

are linear in the unknown parameters, because all terms in x and y are either known

from the characteristics of the bond or the specification of the spline models. Also,the constraints Dc(0) = 1 on the discount functions are linear restrictions on the spline

weights; see Equation (2.2).

In order to estimate the spline weights, we substitute the theoretical prices PDCFcb by

observed market prices Pcb and add an error term cb to the equations. The error term is

necessary, because due to market imperfections the DCF method is not able to perfectly

explain bond prices.4 Using matrix notation, we obtain the following linear regression

model

P1

P2

P3...

PC

=

X1 0 0 . . . 0X2 Y2 0 . . . 0

X3 0 Y3 . . . 0...

......

. . ....

XC 0 0 . . . YC

1

2

3...

C

+

1

2

3...

C

, c i.i.d.(0,

2c ), (2.5)

where Xc is a (Bc(n1+k1)) matrix with rows {xc1, xc2, . . . , xcBc} and Yc is a (Bc(nc +kc)) matrix with rows {yc1, yc2, . . . , ycBc}. We allow the disturbances to have different

4

It is possible to obtain an arbitrary high goodness of fit by increasing the number of parameters.However, the resulting term structures are likely to have twisting shapes and wide confidence intervals.


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Section 2.3 Model Settings 23

variances for each category, because prices of lower rated bonds are generally more noisy

due to lower liquidity and a higher uncertainty about their perceived credit worthiness.

Also, the residuals of independently estimated single-curve models can be shown to have

significantly different variances using a heteroscedasticity test. Estimates 1, 2, . . . , C of

the spline weights are readily obtained by applying Restricted Feasible Generalized Least

Squares estimation to (2.5); see e.g. Greene (2000, page 473). Once we have estimated

the spline weights, we can evaluate the category-c discount curve Dc() for any desiredmaturity. It is important to emphasize, however, that discount factors for maturities

beyond the maximum maturity bond become unreliable.

2.3 Model Settings

Before we are able to estimate the regression model (2.5), we have to specify the exact

form of the basis functions. The functional form of the basis functions follows by choosing

the degree of the splines and the number and location of the knots. These choices reflect

the familiar trade-off between flexibility and smoothness. The degree of the splines should

not be chosen too high, to preclude the problems of higher order polynomials.5 If the order

is too low however, the estimated curve will not fit the data very well and thus will not

reflect the interest rates that are prevalent in the market. Similarly, if the number ofknots is chosen too low, the model will not be able to fit term structures with difficult

shapes. On the other hand, if it is too high, the estimated curve is sensitive to outliers.

For the spline model for the government discount curve, we can use results from the

term structure estimation literature. Almost all studies that employ spline functions to

model the discount curve use third degree splines. Only McCulloch (1971) used quadratic

splines in his pioneering study, resulting in knuckles in the forward curve, which made

him switch to cubic splines in his follow-up paper (McCulloch, 1975). Beim (1992) con-

ducted a simulation study and concluded that cubic splines are preferable. Poirier (1976,page 49) demonstrated that fitting a cubic spline minimizes the integral of the square

of the second derivative, which is an approximation of a functions smoothness; see also

Adams and van Deventer (1994). Consequently, cubic splines are a convenient compromise

between high goodness of fit and smooth curves.

With regard to the specification of the knots, there is less agreement. McCulloch (1971)

proposed to set the number of knots equal to the integer nearest to the square root of the

5Using higher order polynomials often results in spurious curvature between the data points; see e.g.

Shea (1984, page 255). This is especially true if such polynomials are fitted to data that are not uniformlydistributed over the approximation interval, as is the case with term structure estimation; see Figure 2.1.


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number of bonds in the sample. The knots are then located such that an approximately

equal number of bonds is placed in each segment. Litzenberger and Rolfo (1984) stated

that the McCulloch scheme is likely to result in a poor fit for longer maturities due

to the larger number of shorter maturity bonds. As an alternative, they suggested to

exogenously place the knots at 1, 5 and 10 years, roughly corresponding to an economic

segmentation into short, medium and long maturities. Langetieg and Smoot (1989) tested

the McCulloch knot placement scheme against the economic scheme and found that the

latter typically performed better. Steeley (1991) experimented with the specification of

the knots and recommended placing knots at 5 and 10 years as a starting point for future

research. The simulation study by Beim (1992) revealed that cubic splines with two

knots minimized the standard error of fit between the estimated and the simulated truediscount curves.

We also use spline functions to model discount spread curves, but there is no prior

evidence available on the specification of the degree and the knots. Given the disagree-

ment in the literature on the specification of the default-free discount curve, our task of

specifying the splines of the spread curve is not an easy one. Our choices are guided by

the observation that a spread curve generally has a less complicated shape than a term

structure. Therefore, we reduce the flexibility of the spline model for the spread curve by

reducing the degrees of freedom. Compared to the spline for the discount curve of categoryc in a single-curve model, we specify the discount spread curve sc() in the multi-curvemodel as a lower degree spline with a smaller number of knots. That is, we use a quadratic

spline function and the knots are chosen to be a subset of the knots of the single-curve

spline model. This still leaves us with several competing degree-knot combinations. To

choose the optimal combination we use a newly developed test statistic that allows us

to compare spot spread curves that are obtained from competing multi-curve models. We

describe this curve similarity test in the next section and apply it in Section 2.6.

2.4 Model Comparison

A problem in comparing different single- and multi-curve models amongst and against

one another is that there is no general estimable model that encompasses all other mod-

els. Therefore, we cannot use standard econometric testing procedures. Moreover, most

econometric tests only focus on goodness of fit, i.e. the ability of a model to fit the data.

In term structure estimation, however, practitioners are additionally interested in other

features of the models, such as smoothness and statistical reliability. For these reasons,

we compare single- and multi-curve models in three ways:


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Section 2.4 Model Comparison 25

Usage of statistics that reflect the goodness of fit, smoothness and reliability, suchthat models can be compared by confronting these statistics, though without being

able to determine the statistical significance of possible differences.

Usage of a newly developed test statistic that allows two curves from two differ-ent multi-curve models to be compared to one another. We focus here on spot

spread curves, because of their importance as inputs for pricing and risk manage-

ment models, but the statistic may also be employed to compare other curves that

can be calculated from the multi-curve models.

Visual inspection of the estimated term structures, most notably the spot curvesand the spot spread curves. Desirable features are smoothness and monotonicity.

The remainder of this section discusses the first two ways in more detail.

2.4.1 Statistics

Since interest rates are the main determinants of bond prices, any term structure model

should be able to explain market prices fairly accurately. Therefore, goodness of fit is a

useful criterion to compare models. We measure the fit as the Root Mean Squared Error

of the residuals

RMSEc =

1Bc

Bcb=1

e2cb

where RMSEc denotes the Root of Mean Squared Error for category c, Bc the number

of category c bonds and ecb the residual of the bth category c-bond, which is calculated as

the market price of the bond minus its theoretical DCF price (2.4a) or (2.4b).

Although a low value of the RMSE statistic is desirable, we run the risk of ending

up with a twisting curve. Therefore, we also measure the smoothness of estimated term

structures. Following Poirier (1976, page 49) and Powell (1981), the smoothness of afunction over an interval [t1, t2] is computed as the integral of the square of its second

derivative

s(, t1, t2) =

t2t1

(t)2dt.

We evaluate this statistic for both spot curves and spot spread curves.

Finally, we want to judge to what degree deviations between theoretical and market

prices is transformed into uncertainty about estimated interest rates and spreads. Thereliability of a point (maturity) on an estimated curve is indicated by its standard error.


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The reliability can be evaluated in a number of maturities to compare different segments

of the curve. Appendix 2.B derives the standard errors for a number of curves: discount

curve, discount spread curve, spot curve and spot spread curve.

2.4.2 Curve Similarity Test

The Curve Similarity Test described below helps in choosing between two curves that are

estimated with two different multi-curve models. The test especially guides in striking

a balance between goodness of fit and smoothness; see Section 2.3. Given our focus on

credit risk models, we describe the construction of the test for spot spread curves, but

the test is suitable for any other curve for which standard errors and covariances can be

computed.

Suppose we estimate a richly specified multi-curve model and compose a vector s1r,c(t)

of spot spread rates of category c evaluated in a q-vector of maturities t = (t1, . . . , tq).

We would like to know to what extent we can reduce this model to a more parsimonious

model, i.e. a smoother curve, without loosing too much on the goodness of fit criterion.

Consider therefore the vector s0r,c(t) that contains spot spread rates evaluated in the

same vector of maturities that result from a more parsimonious multi-curve model. This

alternative model contains less parameters due to a lower degree and/or less spline knots.

The Curve Similarity Test (CST) aims to test whether s0r,c(t) lies in the realm of s1r,c(t).

To compute the CST statistic we weigh differences between the spot spread vectors

with the covariance matrix 1c(t) of s1r,c(t)

CST = (s1r,c(t) s0r,c(t))(1c(t))1(s1r,c(t) s0r,c(t)).

The covariance matrix, which is constructed in Appendix 2.B, measures the uncertainty

in the spread estimates and by using it as weight matrix we put more emphasis on the

reliable maturities of the spread curve, and vice versa. We compare the CST value to crit-ical values from a 2 distribution with q degrees of freedom to determine whether s0r,c(t) is

approximately equal to s01,c(t) at the selected maturities. The testing procedure can only

be applied to spot spread vectors from multi-curve models. Spread curves from single-

curve models are obtained by subtracting independently estimated term structures, so

that we are unable to construct the covariance matrix of a spread vector. As s1r,c(t) curve

we choose the multi-curve model with the same degree-knot settings as the single-curve

model, because its spread curve resembles the spread curves obtained from single-curve

models the most; see Section 2.6. As s01,c(t) curves we consider several more parsimonious

multi-curve models, i.e. with a lower degree and/or less knots. These different parsimo-


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Section 2.5 Data 27

nious models are all compared to the most richly specified model. The results that stem

from such a model comparison should be interpreted with care as the testing procedure

is conceptually different from standard econometric testing procedures. For example, the

test statistic may prefer a model with low order splines that has appropriately selected

knots to a high order model with badly located knots. Therefore, the test may reject

a model that has a larger number of parameters than a competing model that is not

rejected; this outcome is not possible with traditional econometric tests that compare

nested models.

To make the test operational we have to specify the maturity vector t. Since the

covariance matrix of the spot spread vector s1r,c(t) is derived from the covariance matrix

of the estimators (11 ,

1c ) of the richly specified model, we cannot construct the covari-

ance matrix for an arbitrarily chosen maturity vector t. For example, if the number of

maturities q exceeds the number of parameters in the underlying regression model, the

covariance matrix of s1r,c(t) becomes singular. Another issue is the location of the maturi-

ties. Because of the smoothness of the curve, spreads for two adjacent maturities cannot

be very different from each other. Therefore, the grid points should not be chosen too

close to each other to preclude a near-singular covariance matrix. A final consideration

is the location of the maturities relative to the spline knots. Since each spline interval

corresponds to an extra parameter, we cannot place too much grid points of the test inone spline interval. Again, doing so would lead to a near-singular matrix. In practice, the

above mentioned conditions on the maturity vector t imply that we can only conduct a

joint comparison of the spot spreads at a limited number of maturities, which lie reason-

ably far apart. To determine the robustness of the results from the testing procedure we

can vary the maturity vector t while satisfying the conditions.

2.5 Data

To appraise the performance of the proposed multi-curve model and compare it to inde-

pendently estimated single-curve models, we use a data set of German mark-denominated

bonds. Their charact

thesis scm nivi

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