Credit-Risk Modelling

Credit-Risk Modelling

David Jamieson Bolder

Credit-Risk ModellingTheoretical Foundations, Diagnostic Tools,Practical Examples, and Numerical Recipesin Python

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David Jamieson BolderThe World BankDistrict of ColumbiaWashington, DC, USA

ISBN 978-3-319-94687-0 ISBN 978-3-319-94688-7 (eBook)https://doi.org/10.1007/978-3-319-94688-7

Library of Congress Control Number: 2018948195

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https://doi.org/10.1007/978-3-319-94688-7

À Thomas, mon petit loup etla fièreté de ma vie.

Foreword

As there are already many good technical books and articles on credit-risk mod-elling, one may well question why there needs to be another one. David’s new bookis different and will be useful to those interested in a comprehensive developmentof the subject through a pedagogical approach. He starts from first principlesand foundations, deftly makes connections with market-risk models, which weredeveloped first, but shows why credit-risk modelling needs different tools andtechniques. Building on these foundational principles, he develops in extraordinarydetail the technical and mathematical concepts needed for credit-risk modelling.This fresh, step-by-step, build-up of methodologies is almost like someone helpingthe readers themselves develop the logic and concepts; the technique is unique andyields a solid understanding of fundamental issues relating to credit risk and themodelling of such risk.

David goes beyond analytical modelling to address the practical issues associatedwith implementation. He uses a sample portfolio to illustrate the developmentof default models and, most interestingly, provides examples of code using thePython programming language. He includes detailed treatment of issues relating toparameter estimation and techniques for addressing the thorny problem of limiteddata which, in some areas of credit risk, can be severe. This is especially true whencompared to market-risk models. In addition, as one who is currently working onmodel validation and model risk, David draws on his rich experience to elucidatethe limitations and consequences of modelling choices that the practitioner is oftenrequired to make. He also provides valuable insights into the use and interpretationof model results.

This book is deeply technical, draws on David’s skill and experience in educatingquantitative modellers, and will appeal to serious students with a strong mathemati-cal backgroundwho want a thorough and comprehensive understanding of the topic.

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While this book is not light reading, those who put in the effort will find it wellworth it; they will also be rewarded with surprising doses of humour, which lightenthe path, but also further aid the understanding.

Washington, DC, USA Lakshmi Shyam-Sunder2018

Preface

It’s a dangerous business, Frodo, going out your door. You step onto theroad, and if you don’t keep your feet, there’s no knowing where youmight be swept off to.

(J.R.R. Tolkien)

The genesis for this book was a series of weekly lectures provided to acollection of World Bank Group quantitative analysts. The intention of this effortwas to provide an introduction to the area of credit-risk modelling to a group ofpractitioners with significant experience in the market-risk area, but rather less inthe credit-risk setting. About half of the lectures were allocated to the models, withthe remaining focus on diagnostic tools and parameter estimation. There was astrong emphasis on the how, but there was also an important accent on the why.Practical considerations thus shared equal billing with the underlying rationale.There developed a challenging and stimulating environment characterized by activediscussion and reflection. A high level of participant interest was evident. Thisexperience suggested a certain demand for knowledge and understanding of themodels used in measuring and managing credit risk.

My Motivation

As sometimes happens, therefore, this innocent series of lectures evolved into some-thing more. In my spare time, I tinkered with the organization of the various topics,expanded my notes, worked out a number of relationships from first principles, andadded to my rapidly increasing Python code library. Eventually, I realized that Ihad gradually gathered more than enough material for a reasonably comprehensivebook on the subject. At this point in most technical endeavours, every prospectiveauthor asks: do we really need another book on this topic? Hamming (1985) makes

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an excellent point in this regard, borrowed from de Finetti (1974), which I will, inturn, use once again,

To add one more [book] would certainly be a presumptuous undertaking if I thought interms of doing something better, and a useless undertaking if I were to content myself withproducing something similar to the “standard” type.

Wishing to be neither presumptuous nor useless, I hope to bring a differentperspective to this topic.

There are numerous excellent academic references and a range of usefulintroductions to credit-risk modelling; indeed, many of them are referenced in thecoming chapters. What distinguishes this work, therefore, from de Finetti’s (1974)standard type? There are three main deviations from the typical treatment of theunderlyingmaterial. Motivated by the tone and structure of the original lectures—aswell as my own practical experience—I have tried to incorporate transparency andaccessibility, concreteness, and multiplicity of perspective. Each of these elementswarrants further description.

Transparency and Accessibility

In virtually every subject, there are dramatic differences between introductory andprofessional discussion. In technical areas, this is often extreme. Indeed, advancedbooks and articles are typically almost incomprehensible to the uninitiated. At thesame time, however, introductory works often contain frustratingly little detail.Credit-risk modelling, unfortunately, falls into this category. The quantitativepractitioner seeking to gain understanding of the extant set of credit-risk modelsis thus trapped between relatively introductory treatment and mathematically denseacademic treatises.

This book, in response to this challenge, attempts to insert itself in betweenthese two ends of the spectrum. Details are not taken for granted and virtuallyevery relationship is derived from either first principles or previously establishedquantities. The result is a text with approximately 1,000 equations. This level oftransparency may, for certain highly qualified practitioners, seem like tedium oroverkill. To attain the desired degree of accessibility, however, I firmly believe in thenecessity of methodical, careful, and complete derivations of all models, diagnostictools, and estimation methodologies. In this manner, it is my sincere hope that thiswork will help to bridge the inevitable distance between elementary introductionsand highly complex, original, academic work.

Concreteness

Cautious and detailed derivation of key relationships is an important first step,but this is not a theoretical pursuit. Quite the opposite, credit-risk modelling is

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an applied discipline that forms a critical element of many institutions’ risk-management framework. Merely discussing the material is insufficient. It is nec-essary to actually show the reader how the computations are performed. Demon-stration of key concepts is precisely what is meant by the notion of concreteness.

To permit the incorporation of this practical element, a compact, but reasonablyrich portfolio example is introduced in the first chapter. It is then repeatedly used—in the context of the models, diagnostics, and parameter estimations—throughoutthe remainder of the book. This provides a high level of continuity while bothinviting and permitting intra-chapter comparisons. The practical element in thiswork is thus a fundamental feature and not merely a supplementary add-on. Toaccomplish this, the following chapters contain almost 100 Python algorithms,130-odd figures, and more than 80 tables. Examination of how various modellingcomponents might be implemented, investigating potential code solutions, andreviewing the results are, in fact, the central part of this work.

All of the practical computer code in the subsequent chapters employs the Pythonprogramming language. While there are many possible alternatives, this choicewas not coincidental. Python offers three important advantages. First of all, itprovides, among other things, a comprehensive and flexible scientific computingplatform. Credit-risk models require a broad range of numerical techniques and, inmy experience, Python is well suited to perform them. Second, there is a growingpopulation of Python developers in finance. This popularity should imply a higherdegree of familiarity with the proposed code found in the coming chapters. Finally,it is freely available under the Open Source License. In addition to the benefitsassociated with an open development community, the resource constraints thisrelaxes should not be underestimated.

The Python libraries employed in all of the practical examples found in this bookare freely available, also under the GNU Open Source License, at the followingGitHub location:

https://github.com/djbolder/credit-risk-modelling

My sincere hope is that these libraries might help other practitioners start and extendtheir own modelling activities.

Multiplicity of Perspective

In recent years, after decades working as a quantitative practitioner developing andimplementing models, I moved into a model oversight and governance function. Inmany ways, my day-to-day life has not dramatically changed. Validating complexmathematical and statistical models requires independent replication and mathe-matical derivations. The lion’s share of my time, therefore, still involves workingclosely with models. What has changed, however, is my perspective. Working witha broad range of models and approaches across my institution, a fact that was

https://github.com/djbolder/credit-risk-modelling

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already intellectually clear to me became even more evident: there are typicallymany alternative ways to model a particular phenomenon.

This, perhaps obvious, insight is simultaneously the source and principalmitigantof model risk. If there is no one single way to model a particular object of interest,then one seeks the best model from the universe of alternatives. Best, however,is a notoriously slippery concept. Determination of a superior model is both anobjective and subjective task. In some areas, the objective aspect dominates. Thisis particularly the case when one can—as in the case of forecasting or pricing—compare model and actual outcomes. In other cases, when actual outcomes areinfrequent or only indirectly observed, model choice becomes more subjective.

What is the consequence of this observation? The answer is that model selectionis a difficult and complex decision, where the most defensible choice variesdepending on the application and the context. In all cases, however, multiple modelsare preferable to a single model. Model validation thus involves extensive use ofalternative approaches—often referred to as benchmarkmodels—and gratuitous useof sensitivity analysis. Not only do these techniques enhance model understanding,but they also help highlight the strengths and weaknesses of one’s possible choices.This work adopts this position. Rarely, if at all, is a single approach presented forany task. Instead, multiple methodologies are offered for every model, diagnostictool, or estimation technique. In some cases, the alternative is quite simple andinvolves some questionable assumptions. Such models, despite their shortcomings,are nonetheless useful as bounds or sanity checks on more complex alternatives.In other cases, highly technically involved approaches are suggested. They may beexcessively complex and difficult to use, but still offer variety of perspective.

The punchline is that, in real-life settings, we literally never know the rightmodel. Moreover, we do know that our model is likely to be flawed, often inimportant ways. To mitigate this depressing fact, the best approach is to examinemany models with differing assumptions, fundamental structures, or points of entry.By examining and using a multiplicity of approaches, we both decrease our relianceon any one model and increase the chances of making useful insights about ourunderlying problem. This is nevertheless not a book on model risk, but it wouldbe naive and irresponsible—in light of the advice provided by Derman (1996),Rebonato (2003), Morini (2011) and many others—to ignore this perspective.Recognition of this key fact, through my recent experience in model oversight, hasresulted in an explicit incorporation of many competing perspectives in this book.

Some Important Caveats

This is not an academic work; indeed, there is relatively little new or original—beyond perhaps organization and presentation—in the following chapters. Oneshould not, however, conclude that the following pages are devoid of academicconcepts. On the contrary, it is chock-full of them. This is a challenging, technicalarea of study. The point I am trying to make is that this work is practitioner

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literature. Written by a practitioner for other practitioners, the principal focus ison the practicalities of implementing, using, and communicating credit-risk models.Ultimately, the following chapters were designed to support quantitative financialanalysts—working in banking, portfolio management, and insurance—inmeasuringand managing default risk in their portfolios. This practical focus will also hopefullyappeal to advanced undergraduate and graduate students studying this area offinance.

The following chapters cover a significant amount of ground in substantial detail.This has two important consequences. First, the more material and the greater thedepth of discussion, the higher the probability of errors. Much time and effort wasinvested carefully reviewing the text to ensure clear, correct, and cogent treatmentof all topics, but it is inevitable that some small errors and inconsistencies remain.I ask for the reader’s patience and understanding and note that cautious use of anysource is an integral part of every quantitative analyst’s working life.

The second, and final, point is that, no matter how large or detailed a work, it isliterally impossible to cover all aspects of a given field. This book is no exception.Models, diagnostic tools, and parameter estimation cover a lot of ground, but noteverything. Two topics, in particular, receive rather meagre attention: severity ofdefault and migration risk. Both are addressed and all of the necessary tools areprovided to movemuch further into these areas, but these themes often receive muchgreater scrutiny. To a large extent, underplaying these areas was a conscious choice.Both are, in my view, generalizations of the basic framework and add complexitywithout enormous additional insight. This is not to say that these are unimportantareas, far from it, but rather that I have downplayed them somewhat for pedagogicalreasons.

Washington, DC, USA David Jamieson Bolder2018

References

Bolder, D. J. (2015). Fixed income portfolio analytics: A practical guide toimplementing, monitoring and understanding fixed-income portfolios. Heidelberg,Germany: Springer.de Finetti, B. (1974). Theory of probability. New York: Wiley.Derman, E. (1996). Model risk. Goldman Sachs Quantitative Strategies ResearchNotes.Hamming, R. W. (1985). Methods of mathematics applied to calculus, probability,and statistics. Upper Saddle River, NJ, USA: Prentice-Hall, Inc.Morini, M. (2011). Understanding and managing model risk: A practical guide forquants, traders, and validators. West Sussex, UK: Wiley.Rebonato, R. (2003). Theory and practice of model risk management. QuantitativeResearch Centre (QUARC) Technical Paper.

Acknowledgements

I would, first of all, like to sincerely thank Ivan Zelenko for his valuable support.This project was made possible by his decision to bringme into his team and throughhis ability to create a consistent atmosphere of intellectual curiosity. I would alsolike to thank my colleague, Marc Carré, for offering a steady stream of fascinatingmodelling challenges that stimulated my interest and the ultimate choice of subjectmatter for this work. I also owe a heartfelt debt of gratitude to Jean-Pierre Matt,a former colleague and mentor, for introducing me to the Python programminglanguage and gently advocating its many benefits.

I would also like to warmly acknowledge the World Bank Group participantswho attended my lecture series in early 2017. Despite the early time slots, thesededicated individuals provided the key motivation for the production of this workand helped, in many ways, to lend it a pragmatic tone. I would like to particularlysingle outMallik Subbarao andMichal Certik, who participated in numerousweeklydiscussions, replicated some of the examples, and proofread large sections of thetext. These contributions were invaluable and have certainly led to a higher-qualitywork.

Finally, and perhaps most importantly, I would like to thank my wife and sonfor their patience and understanding with the long hours required to produce thiswork. After Bolder (2015), I promised my wife that I would never write anotherbook. The dual lesson, it seems, is that she is a patient woman and that it wouldbe best to refrain from making promises that I cannot keep. I should also probably,only half facetiously, thank the many ice-hockey rinks and arenas where much ofthis book was written. Through my son, an avid ice-hockey player, we’ve travelledto a myriad of places over the last few years along the East Coast of the UnitedStates for both games and practice. My laptop—along with notes and drafts of theforthcoming chapters—has been my constant companion on these travels.

All of my thanks and acknowledgement are, quite naturally, entirely free ofimplication. All errors, inconsistencies, shortcomings, coding weaknesses, designflaws, or faults in logic are to be placed entirely on my shoulders.

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Contents

1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Alternative Perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Pricing or Risk-Management? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Minding our P’s and Q’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Instruments or Portfolios? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 The Time Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.5 Type of Credit-Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.6 Clarifying Our Perspective .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 A Useful Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Modelling Implications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Rare Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Seeing the Forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.1 Modelling Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.2 Diagnostic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.3 Estimation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.4 The Punchline.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 Our Sample Portfolio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.6 A Quick Pre-Screening.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6.1 A Closer Look at Our Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6.2 The Default-Loss Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6.3 Tail Probabilities and Risk Measures . . . . . . . . . . . . . . . . . . . . . . 301.6.4 Decomposing Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.6.5 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.7 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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Part I Modelling FrameworksReference .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 A Natural First Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1 Motivating a Default Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1.1 Two Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.1.2 Multiple Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.1.3 Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2 Adding Formality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2.1 An Important Aside. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2.2 A Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2.3 An Analytical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.3 Convergence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.4 Another Entry Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.5 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3 Mixture or Actuarial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.1 Binomial-Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.1.1 The Beta-Binomial Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . 923.1.2 The Logit- and Probit-Normal Mixture Models. . . . . . . . . . . 101

3.2 Poisson-Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.2.1 The Poisson-Gamma Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.2.2 Other Poisson-Mixture Approaches . . . . . . . . . . . . . . . . . . . . . . . 1253.2.3 Poisson-Mixture Comparison .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.3 CreditRisk+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.3.1 A One-Factor Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.3.2 A Multi-Factor CreditRisk+ Example . . . . . . . . . . . . . . . . . . . . . 141


4 Threshold Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.1 The Gaussian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.1.1 The Latent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.1.2 Introducing Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524.1.3 The Default Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.1.4 Conditionality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.1.5 Default Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.1.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.1.7 Gaussian Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.2 The Limit-Loss Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.2.1 The Limit-Loss Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.2.2 Analytic Gaussian Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.3 Tail Dependence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.3.1 The Tail-Dependence Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.3.2 Gaussian Copula Tail-Dependence . . . . . . . . . . . . . . . . . . . . . . . . 179

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4.3.3 t-Copula Tail-Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.4 The t-Distributed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.4.1 A Revised Latent-Variable Definition . . . . . . . . . . . . . . . . . . . . . 1824.4.2 Back to Default Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1864.4.3 The Calibration Question. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.4.4 Implementing the t-Threshold Model . . . . . . . . . . . . . . . . . . . . . 1904.4.5 Pausing for a Breather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.5 Normal-Variance Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934.5.1 Computing Default Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.5.2 Higher Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.5.3 Two Concrete Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.5.4 The Variance-GammaModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2014.5.5 The Generalized Hyperbolic Case . . . . . . . . . . . . . . . . . . . . . . . . . 2024.5.6 A Fly in the Ointment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.5.7 Concrete Normal-Variance Results . . . . . . . . . . . . . . . . . . . . . . . . 206

4.6 The Canonical Multi-Factor Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.6.1 The Gaussian Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.6.2 The Normal-Variance-Mixture Set-Up . . . . . . . . . . . . . . . . . . . . 214

4.7 A Practical Multi-Factor Example .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.7.1 Understanding the Nested State-Variable Definition.. . . . . 2194.7.2 Selecting Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.7.3 Multivariate Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224


5 The Genesis of Credit-Risk Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2295.1 Merton’s Idea .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5.1.1 Introducing Asset Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335.1.2 Distance to Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2365.1.3 Incorporating Equity Information.. . . . . . . . . . . . . . . . . . . . . . . . . 238

5.2 Exploring Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2405.3 Multiple Obligors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.3.1 Two Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.4 The Indirect Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

5.4.1 A Surprising Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495.4.2 Inferring Key Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2515.4.3 Simulating the Indirect Approach.. . . . . . . . . . . . . . . . . . . . . . . . . 252

5.5 The Direct Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.5.1 Expected Value of An,T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2575.5.2 Variance and Volatility of An,T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2595.5.3 Covariance and Correlation of An,T and Am,T . . . . . . . . . . . . 2615.5.4 Default Correlation Between Firms n and m . . . . . . . . . . . . . . 2635.5.5 Collecting the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.5.6 The Task of Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.5.7 A Direct-Approach Inventory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

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5.5.8 A Small Practical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2705.6 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Part II Diagnostic ToolsReferences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

6 A Regulatory Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876.1 The Basel Accords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

6.1.1 Basel IRB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2906.1.2 The Basic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2926.1.3 A Number of Important Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2956.1.4 The Full Story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

6.2 IRB in Action .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3036.2.1 Some Foreshadowing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

6.3 The Granularity Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096.3.1 A First Try . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.3.2 A Complicated Add-On . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126.3.3 The Granularity Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3176.3.4 The One-Factor Gaussian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3186.3.5 Getting a Bit More Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3256.3.6 The CreditRisk+ Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3286.3.7 A Final Experiment .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3406.3.8 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

7 Risk Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3517.1 The Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3527.2 A Surprising Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

7.2.1 The Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3577.2.2 A Direct Algorithm .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3617.2.3 Some Illustrative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3647.2.4 A Shrewd Suggestion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

7.3 The Normal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3687.4 Introducing the Saddlepoint Approximation .. . . . . . . . . . . . . . . . . . . . . . . 372

7.4.1 The Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3737.4.2 The Density Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.4.3 The Tail Probability Approximation .. . . . . . . . . . . . . . . . . . . . . . 3787.4.4 Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.4.5 A Bit of Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

7.5 Concrete Saddlepoint Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3847.5.1 The Saddlepoint Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3887.5.2 Tail Probabilities and Shortfall Integralls. . . . . . . . . . . . . . . . . . 3917.5.3 A Quick Aside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3927.5.4 Illustrative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

7.6 Obligor-Level Risk Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

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7.6.1 The VaR Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3957.6.2 Shortfall Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

7.7 The Conditionally Independent Saddlepoint Approximation . . . . . . 4067.7.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4127.7.2 A Multi-Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4157.7.3 Computational Burden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

7.8 An Interesting Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.9 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

8 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4298.1 Brains or Brawn? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4308.2 A Silly, But Informative Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4318.3 The Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

8.3.1 Monte Carlo in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4408.3.2 Dealing with Slowness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

8.4 Interval Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4458.4.1 A Rough, But Workable Solution .. . . . . . . . . . . . . . . . . . . . . . . . . 4458.4.2 An Example of Convergence Analysis . . . . . . . . . . . . . . . . . . . . 4478.4.3 Taking Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

8.5 Variance-Reduction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4508.5.1 Introducing Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 4518.5.2 Setting Up the Problem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4538.5.3 The Esscher Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4578.5.4 Finding θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4608.5.5 Implementing the Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4628.5.6 Shifting the Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4678.5.7 Yet Another Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4748.5.8 Tying Up Loose Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4768.5.9 Does It Work?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479


Part III Parameter Estimation

9 Default Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4919.1 Some Preliminary Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

9.1.1 A More Nuanced Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4939.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

9.2.1 A Useful Mathematical Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4999.2.2 Applying This Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5069.2.3 Cohort Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5089.2.4 Hazard-Rate Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5119.2.5 Getting More Practical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5129.2.6 Generating Markov-Chain Outcomes . . . . . . . . . . . . . . . . . . . . . 5139.2.7 Point Estimates and Transition Statistics . . . . . . . . . . . . . . . . . . 518

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9.2.8 Describing Uncertainty .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5239.2.9 Interval Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5389.2.10 Risk-Metric Implications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

9.3 Risk-Neutral Default Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5449.3.1 Basic Cash-Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5449.3.2 Introducing Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5479.3.3 Incorporating Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5539.3.4 Inferring Hazard Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5579.3.5 A Concrete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

9.4 Back to Our P’s and Q’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5679.5 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

10 Default and Asset Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57510.1 Revisiting Default Correlation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57610.2 Simulating a Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

10.2.1 A Familiar Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58110.2.2 The Actual Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

10.3 The Method of Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58910.3.1 The Threshold Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

10.4 Likelihood Approach.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59510.4.1 The Basic Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59610.4.2 A One-Parameter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59810.4.3 Another Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60210.4.4 A More Complicated Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

10.5 Transition Likelihood Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61510.5.1 The Elliptical Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61610.5.2 The Log-Likelihood Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62210.5.3 Inferring the State Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62610.5.4 A Final Example .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628


A The t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637A.1 The Chi-Squared Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638A.2 Toward the t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639A.3 Simulating Correlated t Variates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643A.4 A Quick Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

B The Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649B.1 Changing Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649B.2 Solving the Stochastic Differential Equation . . . . . . . . . . . . . . . . . . . . . . . 653B.3 Evaluating the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655B.4 The Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

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C Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659C.1 Some Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659C.2 Some Useful Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661C.3 Ergodicity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

D The Python Code Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667D.1 Explaining Some Choices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668D.2 The Library Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669D.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671D.4 Sample Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

List of Figures

Fig. 1.1 A (stylized) default-loss distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Fig. 1.2 Pricing credit risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Fig. 1.3 Market vs. credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Fig. 1.4 Brownian motion in action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Fig. 1.5 Brownian motion in credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Fig. 1.6 A stylized comparison.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Fig. 1.7 A sample portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Fig. 1.8 A bit more perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Fig. 1.9 Default-loss distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Fig. 1.10 Tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Fig. 1.11 Obligor VaR contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Fig. 1.12 VaR contribution perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Fig. 2.1 Four-instrument outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Fig. 2.2 Pascal’s triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Fig. 2.3 Many instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Fig. 2.4 A schematic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Fig. 2.5 The loss distribution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Fig. 2.6 Impact of homogeneity assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Fig. 2.7 Identifying λn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Fig. 2.8 Indistinguishable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Fig. 2.9 The law of rare events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Fig. 3.1 The role of default dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Fig. 3.2 Simulating beta random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Fig. 3.3 The logistic and probit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Fig. 3.4 Comparing default-probability densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Fig. 3.5 Comparing default probability mass functions.. . . . . . . . . . . . . . . . . . . . 108Fig. 3.6 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Fig. 3.7 Comparing tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

xxv

xxvi List of Figures

Fig. 3.8 CreditRisk+ conditional default probabilities . . . . . . . . . . . . . . . . . . . . . . 134Fig. 3.9 A default-correlation surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Fig. 3.10 CreditRisk+ tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Fig. 4.1 Conditional default probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Fig. 4.2 pn(G) dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Fig. 4.3 Comparative tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Fig. 4.4 Different limit-loss densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Fig. 4.5 The analytic fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Fig. 4.6 Tail-dependence coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Fig. 4.7 Calibrating outcomes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Fig. 4.8 Various tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Fig. 4.9 The modified Bessel function of the second kind .. . . . . . . . . . . . . . . . . 202Fig. 4.10 Visualizing normal-variance-mixture distributions . . . . . . . . . . . . . . . . 209Fig. 4.11 Multiple threshold tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Fig. 5.1 A model zoology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Fig. 5.2 Default-event schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Fig. 5.3 ODE behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Fig. 5.4 SDE behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244Fig. 5.5 Inferred quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252Fig. 5.6 Comparing tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Fig. 5.7 Inferring the capital structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Fig. 5.8 Equity and asset moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Fig. 5.9 Direct Merton-model default-loss distribution .. . . . . . . . . . . . . . . . . . . . 280

Fig. 6.1 Basel asset correlation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Fig. 6.2 Maturity correction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299Fig. 6.3 Risk-capital coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301Fig. 6.4 Obligor tenors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303Fig. 6.5 Comparative tail probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305Fig. 6.6 Ordered risk-capital contributions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308Fig. 6.7 Basel II’s pillars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310Fig. 6.8 Gaussian granularity adjustment functions. . . . . . . . . . . . . . . . . . . . . . . . . 324Fig. 6.9 Concentrated vs. diversified tail probabilities . . . . . . . . . . . . . . . . . . . . . . 326Fig. 6.10 Testing the granularity adjustment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Fig. 6.11 Choosing δα(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Fig. 6.12 Gα(L) accuracy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Fig. 6.13 Revisiting economic-capital contributions . . . . . . . . . . . . . . . . . . . . . . . . . 347

Fig. 7.1 Monte-Carlo obligor contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365Fig. 7.2 Regional contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366Fig. 7.3 VaR-matched comparison .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368Fig. 7.4 Accuracy of normal VaR approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Fig. 7.5 Saddlepoint positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380Fig. 7.6 Cumulant generating function and derivatives . . . . . . . . . . . . . . . . . . . . . 389Fig. 7.7 Independent-default density approximation.. . . . . . . . . . . . . . . . . . . . . . . 390

List of Figures xxvii

Fig. 7.8 Independent-default tail-probability and shortfall-integralapproximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Fig. 7.9 Independent-default VaR-contribution approximation . . . . . . . . . . . . 399Fig. 7.10 VaR contribution perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399Fig. 7.11 Independent-default shortfall-contribution approximation . . . . . . . . 405Fig. 7.12 Integrated saddlepoint density approximation . . . . . . . . . . . . . . . . . . . . . 408Fig. 7.13 Multiple-model visual risk-metric comparison . . . . . . . . . . . . . . . . . . . . 416Fig. 7.14 Saddlepoint risk-measure contributions .. . . . . . . . . . . . . . . . . . . . . . . . . . . 418Fig. 7.15 Proportional saddlepoint shortfall contributions . . . . . . . . . . . . . . . . . . . 419Fig. 7.16 Back to the beginning: The t-threshold model . . . . . . . . . . . . . . . . . . . . . 419

Fig. 8.1 A silly function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432Fig. 8.2 A simple, but effective, approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433Fig. 8.3 A clever technique.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434Fig. 8.4 A randomized grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436Fig. 8.5 The eponymous location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440Fig. 8.6 The role of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447Fig. 8.7 Understanding convergence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448Fig. 8.8 Variance-reduction techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451Fig. 8.9 Twisting probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463Fig. 8.10 Conditional VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468Fig. 8.11 J1 behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471Fig. 8.12 Twisting and shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473Fig. 8.13 The shifted mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474Fig. 8.14 Raw vs. IS VaR comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Fig. 8.15 A tighter comparison .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

Fig. 9.1 VaR sensitivity to default probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Fig. 9.2 Obligor differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497Fig. 9.3 Calculating forward Markov-Chain probabilities .. . . . . . . . . . . . . . . . . 503Fig. 9.4 Simulating a Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515Fig. 9.5 Transition overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518Fig. 9.6 Type of transitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519Fig. 9.7 Simulated transition counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522Fig. 9.8 The profile likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524Fig. 9.9 Likelihood curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525Fig. 9.10 Binomial likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534Fig. 9.11 Binomial bootstrap distribution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536Fig. 9.12 Comparing approaches .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Fig. 9.13 Default-probability confidence bounds .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 540Fig. 9.14 Bootstrap results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541Fig. 9.15 An assumed risk-free spot curve .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562Fig. 9.16 Key ingredients and outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566Fig. 9.17 Key CDS spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568Fig. 9.18 Approximate risk-premium evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

xxviii List of Figures

Fig. 10.1 Creating transition data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583Fig. 10.2 Choosing ρG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588Fig. 10.3 Aggregate default data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588Fig. 10.4 Single-parameter likelihood function.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Fig. 10.5 Single-parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602Fig. 10.6 The CreditRisk+ likelihood surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603Fig. 10.7 Fixed a CreditRisk+ likelihood functions .. . . . . . . . . . . . . . . . . . . . . . . . . 605Fig. 10.8 Regional default data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610Fig. 10.9 The regional likelihood surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611Fig. 10.10 Regional profile likelihoods.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612Fig. 10.11 Regional estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615Fig. 10.12 Multiple log-likelihoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630Fig. 10.13 Averaging log-likelihoods.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630Fig. 10.14 Lost in translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

Fig. A.1 Bivariate scatterplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

Fig. D.1 The library structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670Fig. D.2 A sample library .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

List of Tables

Table 1.1 Topic overview.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 1.2 Summary risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Table 1.3 VaR-contribution breakdown.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Table 2.1 A single instrument example .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Table 2.2 A two-instrument example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Table 2.3 Four instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Table 2.4 Numerical independent default results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Table 2.5 Analytic vs. numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Table 2.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Table 2.7 Analytical independent-default results . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Table 3.1 Analytic beta-binomial results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Table 3.2 Numerical beta-binomial mixture results . . . . . . . . . . . . . . . . . . . . . . . . . 100Table 3.3 Binomial calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Table 3.4 Analytic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Table 3.5 Numerical mixture results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Table 3.6 Analytic Poisson-Gamma results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Table 3.7 Numerical Poisson-Gamma mixture results . . . . . . . . . . . . . . . . . . . . . . 124Table 3.8 Poisson calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Table 3.9 Analytic Poisson-mixture results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Table 3.10 Numerical Poisson-mixture results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Table 3.11 Possible CreditRisk+ calibrations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Table 3.12 A one-factor CreditRisk+ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Table 3.13 Possible multi-factor CreditRisk+ parameterization . . . . . . . . . . . . . 145Table 3.14 Multi-factor CreditRisk+ default-correlation matrix .. . . . . . . . . . . . 146Table 3.15 A multi-factor CreditRisk+ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Table 4.1 Gaussian threshold results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Table 4.2 Gaussian threshold analytic-numerical comparison.. . . . . . . . . . . . . 174Table 4.3 Threshold calibration comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Table 4.4 t-threshold results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

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Table 4.5 One-factor summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Table 4.6 Normal-variance mixture calibration comparison.. . . . . . . . . . . . . . . 207Table 4.7 Simulated vs. theoretical moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Table 4.8 Multiple threshold results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210Table 4.9 Canonical multi-factor summary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Table 4.10 Sample parameter structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Table 4.11 Default correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Table 4.12 Multivariate threshold results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Table 5.1 Indirect Kn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Table 5.2 Our portfolio in the Merton world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Table 5.3 Indirect Merton results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Table 5.4 Challenges of the direct approach.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Table 5.5 Key Merton quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Table 5.6 A low-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Table 5.7 Asset correlations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275Table 5.8 Default correlations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Table 5.9 Direct Merton results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Table 5.10 A litany of default triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Table 6.1 Benchmark models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Table 6.2 Benchmarking results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306Table 6.3 Risk-capital contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Table 6.4 Two differing portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Table 6.5 The numerical reckoning .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328Table 6.6 CreditRisk+ parameter choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Table 6.7 Testing the CreditRisk+ granularity adjustment. . . . . . . . . . . . . . . . . . 346

Table 7.1 Key saddlepoint approximations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Table 7.2 Computational direction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392Table 7.3 Numerical saddlepoint results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394Table 7.4 VaR contribution results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398Table 7.5 Shortfall contribution results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405Table 7.6 Multiple-model saddlepoint comparison . . . . . . . . . . . . . . . . . . . . . . . . . 416Table 7.7 Multiple-model top-ten VaR contributions . . . . . . . . . . . . . . . . . . . . . . . 417Table 7.8 Comparing computational times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Table 8.1 Integration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Table 8.2 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449Table 8.3 A different perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465Table 8.4 A first try . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466Table 8.5 Variance-reduction report card . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Table 9.1 Ranking VaR default-probability sensitivities . . . . . . . . . . . . . . . . . . . . 496Table 9.2 Counting Markov-Chain outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500Table 9.3 Numerical parameter uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540Table 9.4 Categorizing obligors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

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Table 9.5 Parameter uncertainty and VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543Table 9.6 Bootstrapping results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566Table 9.7 Comparing P and Q estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Table 10.1 Simulated data at a glance .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589Table 10.2 Mixture method of moments results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592Table 10.3 Threshold method of moments results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595Table 10.4 Single-dataset results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600Table 10.5 Single-parameter simulation study results . . . . . . . . . . . . . . . . . . . . . . . . 601Table 10.6 CreditRisk+ MLE results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606Table 10.7 Regional single-dataset results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612Table 10.8 Regional simulation study results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614Table 10.9 T -threshold model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631Table 10.10 t-threshold model simulation study results . . . . . . . . . . . . . . . . . . . . . . . 631Table 10.11 t-threshold model full information estimators . . . . . . . . . . . . . . . . . . . 632

Table D.1 Function count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

List of Algorithms

2.1 Independent defaults by Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2 Numerically computing risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.3 Monte Carlo binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.4 Analytic binomial implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.5 Approximated expected shortfall in the analytic model . . . . . . . . . . . . . . . 652.6 Independent Poisson defaults by Monte Carlo. . . . . . . . . . . . . . . . . . . . . . . . . 762.7 Analytic Poisson implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.1 Analytic beta-binomial.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.2 Simulation beta-binomial .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.3 The logit- and probit-normal mixture density functions . . . . . . . . . . . . . . 1063.4 Analytic logit- and probit-normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.5 Logit- and probit-normal model calibration system .. . . . . . . . . . . . . . . . . . 1093.6 Logit- and probit-normal model Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . 1123.7 Analytic Poisson-Gamma mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.8 Analytic Poisson-Gamma moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.9 Poisson-Gamma model calibration system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.10 Poisson-Gamma numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.11 Poisson-mixture integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.12 Poisson-mixture analytic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.13 Poisson-mixture model moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.14 One-factor CreditRisk+ Monte-Carlo implementation .. . . . . . . . . . . . . . . 1393.15 Multi-factor CreditRisk+ Monte-Carlo implementation . . . . . . . . . . . . . . 146

4.1 Computing pn(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1574.2 The Gaussian joint default probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.3 A one-factor calibration objective function .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1624.4 Generating Gaussian latent variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634.5 The numerical Gaussian one-factor threshold implementation . . . . . . . 1634.6 An analytical approximation to the Gaussian threshold model . . . . . . . 1734.7 An analytical expected-shortfall computation . . . . . . . . . . . . . . . . . . . . . . . . . 173

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4.8 Computing the t-distribution tail coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.9 Calibrating the t-threshold model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1894.10 Computing the joint default probability from the t-threshold model . 1904.11 Generating t-threshold state variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.12 The t-threshold Monte-Carlo implementation .. . . . . . . . . . . . . . . . . . . . . . . . 1914.13 The normal-variance mixture density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2054.14 The normal-variance mixture distribution function . . . . . . . . . . . . . . . . . . . 2054.15 The normal-variance mixture inverse distribution function .. . . . . . . . . . 2064.16 Generating normal-variance mixture state variables . . . . . . . . . . . . . . . . . . 2084.17 Constructing the regional default-correlation matrix . . . . . . . . . . . . . . . . . . 223

5.1 Determining Kn and δn,T−t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495.2 Simulating the indirect Merton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.3 Finding An,t and σn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2755.4 Computing default correlations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.5 Simulating the direct Merton model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

6.1 Asset-correlation and maturity adjustment functions.. . . . . . . . . . . . . . . . . 3006.2 Basel IRB risk capital functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3026.3 A simple contribution computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3076.4 The granularity adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3186.5 The density functions for the Gaussian threshold granularity

adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3196.6 Conditional-expectation functions for the Gaussian threshold

granularity adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3226.7 Conditional variance functions for the Gaussian threshold

granularity adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3246.8 Key CreditRisk+ granularity adjustment functions . . . . . . . . . . . . . . . . . . . . 3396.9 Computing δα(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3426.10 Simulating the one-factor CreditRisk+ with stochastic

loss-given-default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3446.11 CreditRisk+ granularity adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.1 Monte-Carlo-based VaR and expected-shortfall decompositions .. . . . 3647.2 The Jk function for j = 0, 1, 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3847.3 Moment and cumulant generating functions .. . . . . . . . . . . . . . . . . . . . . . . . . . 3867.4 First and second derivatives of KL(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3877.5 Finding the saddlepoint: t̃� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.6 The saddlepoint density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3907.7 Computing the saddle-point tail probabilities and shortfall integrals . 3927.8 Identifying a specific saddlepoint VaR estimate . . . . . . . . . . . . . . . . . . . . . . . 3937.9 Computing the independent-default VaR contributions . . . . . . . . . . . . . . . 3977.10 Computing the independent-default shortfall contributions .. . . . . . . . . . 4047.11 A general saddlepoint function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4127.12 A general integral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4137.13 Integrated densities, tail probabilities and shortfall integrals . . . . . . . . . 413

List of Algorithms xxxv

7.14 Computing integrated VaR risk contributions . . . . . . . . . . . . . . . . . . . . . . . . . 4147.15 Determining the numerator for integrated VaR contributions .. . . . . . . . 414

8.1 Evaluating our silly function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4328.2 Two numerical integration options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4358.3 Monte-Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.4 Identifying θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4628.5 Computing twisted probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4638.6 Computing the Esscher-transform Radon-Nikodym derivative .. . . . . . 4638.7 Importance sampling in the independent default case . . . . . . . . . . . . . . . . . 4648.8 Finding the mean-shift parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.9 The Full Monte threshold-model importance-sampling algorithm . . . 480

9.1 Simulating a four-state Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5179.2 Counting transitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5209.3 Cohort transition-matrix estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5219.4 Generating our bootstrap distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5379.5 Hazard-function implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5639.6 The survival probability.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5639.7 Theoretical CDS prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5649.8 Pricing the protection payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

10.1 Computing Q and Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58510.2 Simulating correlated transition data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58610.3 Accounting for counterparties, defaults, and probabilities .. . . . . . . . . . . 58910.4 Simultaneous system of mixture model method of moments

equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59210.5 Computing threshold model moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59410.6 Simultaneous system of threshold model method of moments

equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59410.7 Single-parameter log-likelihood functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59910.8 Two-parameter regional log-likelihood functions . . . . . . . . . . . . . . . . . . . . . 61110.9 t-copula, transition, log-likelihood kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62510.10 Building a one-parameter, correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 62510.11 Inferring latent-state variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

A.1 Simulating correlated Gaussian and t-distributed random variates . . . 647

Credit-Risk Modelling

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