Main Structured Credit Portfolio Analysis, Baskets and CDOs (Chapman & Hall Crc Financial Mathematics Series)

Structured Credit Portfolio Analysis, Baskets and CDOs (Chapman & Hall Crc Financial Mathematics Series)

, ,
The financial industry is swamped by credit products whose economic performance is linked to the performance of some underlying portfolio of credit-risky instruments, like loans, bonds, swaps, or asset-backed securities. Financial institutions continuously use these products for tailor-made long and short positions in credit risks. Based on a steadily growing market, there is a high demand for concepts and techniques applicable to the evaluation of structured credit products. Written from the perspective of practitioners who apply mathematical concepts to structured credit products, Structured Credit Portfolio Analysis, Baskets & CDOs starts with a brief wrap-up on basic concepts of credit risk modeling and then quickly moves on to more advanced topics such as the modeling and evaluation of basket products, credit-linked notes referenced to credit portfolios, collateralized debt obligations, and index tranches. The text is written in a self-contained style so readers with a basic understanding of probability will have no difficulties following it. In addition, many examples and calculations have been included to keep the discussion close to business applications. Practitioners as well as academics will find ideas and tools in the book that they can use for their daily work.
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Year: 2006
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CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES

Structured Credit
Portfolio Analysis,
Baskets & CDOs

CHAPMAN & HALL/CRC
Financial Mathematics Series
Aims and scope:

The field of financial mathematics forms an ever-expanding slice of the financial sector. This
series aims to capture new developments and summarize what is known over the whole
spectrum of this field. It will include a broad range of textbooks, reference works and handbooks
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and concrete real-world examples is highly encouraged.

Series Editors
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Centre for Financial
Research
Judge Business School
University of Cambridge

Dilip B. Madan
Robert H. Smith School
of Business
University of Maryland

Rama Cont
CMAP
Ecole Polytechnique
Palaiseau, France

Published Titles
An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and
Christoph Wagner
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and
Ludger Overbeck

Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor and Francis Group
24-25 Blades Court
Deodar Road
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CHAPMAN & HALL/CRC FINANCIAL MATHEMATICS SERIES

Structured Credit
Portfolio Analysis,
Baskets & CDOs

Christian Bluhm
Ludger Overbeck

Boca Raton London New York

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Taylor & Francis Group, an informa business

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Bluhm, Christian.
Structured portfolio analysis, baskets and CDOs / Christian Bluhm, Ludger
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2006049133

Preface

The financial industry is swamped by structured credit products whose economic performance is linked to the performance of some underlying portfolio of
credit-risky instruments like loans, bonds, credit default swaps, asset-backed
securities, mortgage-backed assets, etc. The market of such collateralized
debt or synthetic obligations, respectively, is steadily growing and financial
institutions continuously use these products for tailor-made risk selling and
buying.
In this book, we discuss mathematical approaches for modeling structured
products credit-linked to an underlying portfolio of credit-risky instruments.
We keep our presentation mathematically precise but do not insist in always
reaching the deepest possible level of mathematical sophistication. Also, we do
not claim to present the full range of possible modeling approaches. Instead,
we focus on ideas and concepts we found useful for our own work and projects
in our daily business. Therefore, the book is written from the perspective of
practitioners who apply mathematical concepts to structured credit products.
As pre-knowledge, we assume some facts from probability theory, stochastic
processes, and credit risk modeling, but altogether we tried to keep the presentation as self-contained as possible. In the bibliography, the reader finds a
collection of papers and books for further reading, either for catching up with
facts required for an even deeper understanding of the model or for following
up on further investigations in alternative approaches and concepts found useful by other colleagues and researchers working in the field of portfolio credit
risk. A helpful pre-reading for the material contained in this book is the
textbook [25], which introduces the reader to the world of credit risk models.
However, there are many other suitable textbooks in the market providing
introductory guidance to portfolio credit risk and we are sure that for every
reader’s taste there is at least one ‘best matching’ textbook out there in the
book market.
It is our hope that after reading this book the reader will find the challenge
to model structured credit portfolios as fascinating as we find it. We worked
in this area for quite some years and still consider it as a great and intriguing
field where application of mathematical ideas, concepts, and models lead to
value-adding business concepts.
Zurich and Munich, August 2006
Christian Bluhm and Ludger Overbeck

v

vii
Acknowledgments
Christian Bluhm thanks his wife, Tabea, and his children, Sarah Maria and
Noa Rebeccah, for their continuous support and generous tolerance during
the writing of the manuscript. Their love and encouragement is his most
important source of energy.
Ludger Overbeck is most grateful to his wife, Bettina, and his children,
Leonard, Daniel, Clara, and Benjamin for their ongoing support.
We highly appreciate feedback and comments on the manuscript by various
colleagues and friends.
We owe special thanks to Christoff Goessl (HypoVereinsbank, London),
Christopher Thorpe (FitchRatings, London), Walter Mussil (Mercer Oliver
Wyman, Frankfurt), and Stefan Benvegnu (Credit Suisse, Zurich) with whom
we had the honor and pleasure to work for quite some years.

Disclaimer
This book reflects the personal view of the authors and does not provide
information about the opinion of Credit Suisse and HypoVereinsbank. The
contents of this book have been written solely for educational purposes. The
authors are not liable for any damage arising from any application of the
contents of this book.

About the Authors
Christian Bluhm is a Managing Director at Credit Suisse in Zurich. He heads the Credit
Portfolio Management unit within the Credit Risk Management Department of Credit Suisse. Before that, he headed the team Structured Finance Analytics in HypoVereinsbank’s
Group Credit Portfolio Management in Munich, where his team was responsible for the
quantitative evaluation of asset-backed securities with a focus on collateralized debt and
loan obligations, collateralized synthetic obligations, residential and commercial mortgagebacked securities, default baskets, and other structured credit instruments, from origination
as well as from an investment perspective. His first professional position in risk management was with Deutsche Bank in Frankfurt. In 1996, he earned a Ph.D. in mathematics
from the University of Erlangen-Nuremberg and, in 1997, he was a post-doctoral fellow at
the mathematics department of Cornell University, Ithaca, New York. He has coauthored a
book on credit risk modeling, together with Ludger Overbeck and Christoph Wagner, and
coauthored a book on statistical foundations of credit risk modeling, together with Andreas
Henking and Ludwig Fahrmeir. He regularly publishes papers and research articles on credit
risk modeling and management in journals and books, including RISK magazine and RISK
books. During his academic time, he published several research articles on harmonic and
fractal analysis of random measures and stochastic processes in mathematical journals. He
frequently speaks at conferences on risk management.
Ludger Overbeck holds a professorship in mathematics and its applications at the University of Giessen in Germany. His main interests are quantitative methods in finance,
risk management, and stochastic analysis. Currently, he also is a Director at HypoVereinsbank in Munich, heading the Portfolio Analytics and Pricing Department within the
Active Portfolio Management unit in HypoVereinsbank’s Corporates & Markets Division.
His main tasks are the pricing of structured credit products and the risk management of the
investment credit portfolio. Until June 2003, he was Head of Risk Research & Development
in Deutsche Bank’s credit risk function located in Frankfurt. His main responsibilities included development and implementation of the internal group-wide credit portfolio model,
the operational risk model, specific (market) risk modeling, the EC/RAROC-methodology,
integration of risk types, backtesting of ratings and correlations, quantitative support for
portfolio management and all other risk types, and risk assessment of credit derivatives
and portfolio transactions like collateralized debt and loan obligations. Prior to that, he
worked for the Banking Supervision Department in the main office of Deutsche Bundesbank
in Duesseldorf, Germany, mainly concerned with internal market risk models and inspections of the banks in line with the so-called ‘minimum requirements for trading businesses’.
He frequently publishes articles in various academic and applied journals, including RISK,
and is coauthor of a book on credit risk modeling, together with Christian Bluhm and
Christoph Wagner. He is a regular speaker at academic and financial industry conferences.
Ludger holds a Ph.D. in probability theory and habilitations in applied mathematics from
the University of Bonn and in economics from the University of Frankfurt.

ix

A Brief Guide to the Book

Before we send the reader on a journey through the different chapters
of this book, it makes sense to briefly summarize the contents of the
different sections so that readers can define their own individual path
through the collected material.
Chapter 1 is intended to be an introduction for readers who want
to have a brief overview on credit risk modeling before they move on
to the core topics of the book. For experienced readers, Chapter 1
can serve as a ‘warming-up’ and an introduction to the notation and
nomenclature used in this book. The following keywords are outlined:
• Single-name credit risk measures like ratings and scorings, default
probabilities, exposures, and loss given default
• Modeling of default risk via latent variable and threshold models
• Decomposition of credit risks into systematic and idiosyncratic
components by factor models
• Credit portfolio loss distributions and their summary statistics
like expected loss, unexpected loss, and quantile-based and expected shortfall-based economic capital
• Comments and remarks regarding the trade-off between accuracy
and practicability of credit risk methodologies
It is not necessary to read Chapter 1 in order to understand subsequent
chapters. But experienced readers will need not more than an hour to
flip through the pages, whereas non-experienced readers should take
the time until all of the keywords are properly understood. For this
purpose, it is recommended to additionally consult an introductory
textbook on credit risk measurement where the mentioned keywords
are not only indicated and outlined but fully described with all the
technical details. There are many textbooks one can use for crossreferences; see Chapter 5 for recommendations.
xi

xii
Chapter 2 treats the modeling of basket products. The difference
between basket products and collateralized debt obligations (CDOs)
covered in Chapter 3 is the enhanced cash flow richness in CDO products, whereas baskets are essentially credit derivatives referenced to
a portfolio of credit-risky names. The content of Chapter 2 can be
divided into two broad categories:
• Modeling techniques:
Essential tools for credit risk modeling are introduced and discussed in detail, including term structures of default probabilities,
joint default probabilities, dependent default times and hazard
rates, copula functions with special emphasis on the Gaussian,
Student-t, and Clayton copulas, and dependence measures like
correlations, rank correlations, and tail dependence.
• Examples and illustrations:
Chapter 2 always tries out the just-developed mathematical concepts in the context of fictitious but realistic examples like duo
baskets, default baskets (first-to-default, second-to-default), and
credit-linked notes. One section in the context of examples is dedicated to an illustration on scenario analysis showing how modeling results can be challenged w.r.t. their plausibility.
In general, we present techniques rather than ready-to-go solutions, because, based on our own experience, we found that general principles
and techniques are more useful due to a broader applicability. In many
cases, it is straightforward to adapt modeling principles and techniques
to the particular problem the modeler has to solve. For instance, we
do not spend much time with pricing concepts but provide many techniques in the context of the modeling of dependent default times, which
can be applied in a straightforward manner to evaluate cash flows of
credit-risky instruments. Such evaluations can then be used to derive a
model price of credit-risky instruments like default baskets or tranches
of collateralized debt obligations.
Chapter 3 is dedicated to collateralized debt obligations (CDOs).
In a first part of our exposition, we focus on a non-technical description
of reasons and motivations for CDOs as well as different types and applications of CDOs in the structured credit market. For the modeling of
CDOs we can apply techniques already elaborated in Chapter 2. This
is exercised by means of some CDO examples, which are sufficiently

xiii
realistic to demonstrate the application of models in a way ready to be
adopted by readers for their own work. In addition to the dependent
default times concept, Chapter 3 also includes a discussion on alternative modeling approaches like multi-step models and diffusion-based
first passage times. Techniques applicable for the reduction of modeling efforts like analytic and semi-analytic approximations as well as
a very efficient modeling technique based on the comonotonic copula
function conclude our discussion on modeling approaches.
At the end of Chapter 3, a comprehensive discussion on single-tranche
CDOs (STCDOs) and index tranches as important examples for STCDOs is included. In this section, we also discuss pricing and hedging
issues in such transactions.
The two last topics we briefly consider in Chapter 3 are portfolios of
CDOs and the application of securitization-based tranche spreads as
building blocks in a cost-to-securitize pricing component.
Chapter 4 is a collection of literature remarks. Readers will find
certain guidence regarding the access to a rich universe of research
articles and books. The collection we present does by no means claim
to be complete or exhaustive but will nevertheless provide suggestions
for further reading.
The Appendix at the end of this book contains certain results from
probability theory as well as certain credit risk modeling facts, which
are intended to make the book a little more self-contained. Included
are also certain side notes, not central enough for being placed in the
main part of the book, but nevertheless interesting. One example for
such side notes is a brief discussion on entropy-maximizing distributions
like the Gaussian distribution and their role as standard choices under
certain circumstances.
As mentioned in the preface of this book, the collection of material
presented in this book is based on modeling techniques and modeling
aspects we found useful for our daily work. We very much hope that
every reader finds at least some pieces of the presented material to be
of some value for her or his own work.

Contents

Preface

v

About the Authors

ix

A Brief Guide to the Book

xi

1 From Single Credit Risks to Credit Portfolios
1.1 Modeling Single-Name Credit Risk . . . . . . . .
1.1.1 Ratings and Default Probabilities . . . . .
1.1.2 Credit Exposure . . . . . . . . . . . . . .
1.1.3 Loss Given Default . . . . . . . . . . . . .
1.2 Modeling Portfolio Credit Risk . . . . . . . . . .
1.2.1 Systematic and Idiosyncratic Credit Risk
1.2.2 Loss Distribution of Credit Portfolios . .
1.2.3 Practicability Versus Accuracy . . . . . .

1
2
2
10
14
17
17
20
24

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2 Default Baskets
2.1 Introductory Example: Duo Baskets . . . . . . . . . . .
2.2 First- and Second-to-Default Modeling . . . . . . . . .
2.3 Derivation of PD Term Structures . . . . . . . . . . . .
2.3.1 A Time-Homogeneous Markov Chain Approach .
2.3.2 A Non-Homogeneous Markov Chain Approach .
2.3.3 Extrapolation Problems for PD Term Structures
2.4 Duo Basket Evaluation for Multi-Year Horizons . . . .
2.5 Dependent Default Times . . . . . . . . . . . . . . . . .
2.5.1 Default Times and PD Term Structures . . . . .
2.5.2 Survival Function and Hazard Rate . . . . . . .
2.5.3 Calculation of Default Time Densities and Hazard Rate Functions . . . . . . . . . . . . . . . . .
2.5.4 From Latent Variables to Default Times . . . . .
2.5.5 Dependence Modeling via Copula Functions . . .
2.5.6 Copulas in Practice . . . . . . . . . . . . . . . .

27
27
34
39
40
52
57
59
67
67
68
69
78
85
93
xv

xvi
2.5.7

2.6

2.7

Visualization of Copula Differences and Mathematical Description by Dependence Measures . .
2.5.8 Impact of Copula Differences to the Duo Basket
2.5.9 A Word of Caution . . . . . . . . . . . . . . . . .
Nth-to-Default Modeling . . . . . . . . . . . . . . . . .
2.6.1 Nth-to-Default Basket with the Gaussian Copula
2.6.2 Nth-to-Default Basket with the Student-t Copula
2.6.3 Nth-to-Default Basket with the Clayton Copula .
2.6.4 Nth-to-Default Simulation Study . . . . . . . . .
2.6.5 Evaluation of Cash Flows in Default Baskets . .
2.6.6 Scenario Analysis . . . . . . . . . . . . . . . . . .
Example of a Basket Credit-Linked Note (CLN) . . . .

3 Collateralized Debt and Synthetic Obligations
3.1 A General Perspective on CDO Modeling . . . . . . . .
3.1.1 A Primer on CDOs . . . . . . . . . . . . . . . . .
3.1.2 Risk Transfer . . . . . . . . . . . . . . . . . . . .
3.1.3 Spread and Rating Arbitrage . . . . . . . . . . .
3.1.4 Funding Benefits . . . . . . . . . . . . . . . . . .
3.1.5 Regulatory Capital Relief . . . . . . . . . . . . .
3.2 CDO Modeling Principles . . . . . . . . . . . . . . . . .
3.3 CDO Modeling Approaches . . . . . . . . . . . . . . . .
3.3.1 Introduction of a Sample CSO . . . . . . . . . .
3.3.2 A First-Order Look at CSO Performance . . . .
3.3.3 Monte Carlo Simulation of the CSO . . . . . . .
3.3.4 Implementing an Excess Cash Trap . . . . . . . .
3.3.5 Multi-Step and First Passage Time Models . . .
3.3.6 Analytic, Semi-Analytic, and Comonotonic CDO
Evaluation Approaches . . . . . . . . . . . . . . .
3.4 Single-Tranche CDOs (STCDOs) . . . . . . . . . . . . .
3.4.1 Basics of Single-Tranche CDOs . . . . . . . . . .
3.4.2 CDS Indices as Reference Pool for STCDOs . . .
3.4.3 ITraxx Europe Untranched . . . . . . . . . . . .
3.4.4 ITraxx Europe Index Tranches: Pricing, Delta
Hedging, and Implied Correlations . . . . . . . .
3.5 Tranche Risk Measures . . . . . . . . . . . . . . . . . .
3.5.1 Expected Shortfall Contributions . . . . . . . . .
3.5.2 Tranche Hit Contributions of Single Names . . .
3.5.3 Applications: Asset Selection, Cost-to-Securitize
3.5.4 Remarks on Portfolios of CDOs . . . . . . . . . .

99
113
118
120
121
127
127
129
136
140
147
165
166
167
172
178
184
186
190
194
194
199
202
210
213
220
250
250
253
259
271
287
288
292
294
299

xvii
4 Some Practical Remarks

303

5 Suggestions for Further Reading

307

6 Appendix
6.1 The Gamma Distribution . . . . . . . . . . . . . . . . .
6.2 The Chi-Square Distribution . . . . . . . . . . . . . . .
6.3 The Student-t Distribution . . . . . . . . . . . . . . . .
6.4 A Natural Clayton-Like Copula Example . . . . . . . .
6.5 Entropy-Based Rationale for Gaussian and Exponential
Distributions as Natural Standard Choices . . . . . . .
6.6 Tail Orientation in Typical Latent Variable Credit Risk
Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 The Vasicek Limit Distribution . . . . . . . . . . . . . .
6.8 One-Factor Versus Multi-Factor Models . . . . . . . . .
6.9 Description of the Sample Portfolio . . . . . . . . . . .
6.10 CDS Names in CDX.NA.IG and iTraxx Europe . . . .

311
311
312
312
314

References

339

Index

349

315
318
320
322
329
332

Chapter 1
From Single Credit Risks to Credit
Portfolios

We begin our exposition by a brief non-informal tour through some
credit risk modeling concepts recalling some basic facts we will need
later in the book as well as making sure that we are all at the same
page before Chapters 2 and 3 lead us to the ‘heart’ of our topic. A
good starting point is a saying by Albert Einstein who seemingly
made the observation
As far as the laws of mathematics refer to reality they are
not certain; and as far as they are certain they do not refer
to reality.1
We feel free to interpret his statement in the following way. Most of
the time we cannot work with deterministic models in order to describe
reality. Instead, our models need to reflect the uncertainty inherent in
the evolution of economic cycles, customer behavior, market developments, and other random components driving the economic fortune of
the banking business. Thanks to the efforts of probabilists over many
centuries, we have a full range of tools from probability theory at our
disposal today in order to model the uncertainties arising in the financial industry. By means of probability theory randomness can never be
eliminated completely in order to overcome the uncertainty addressed
in Einstein’s statement, but randomness can be ‘tamed’ and conclusions and forecasts with respect to a certain level of confidence can be
made on which business and investments decisions can be based. It
is the aim of the following sections to perform a quick but thorough
walk through concepts exploiting probability theory to finally arrive at
meaningful conclusions in structured finance.
1

Albert Einstein in his address Geometry and Experience to the Prussian Academy
of Sciences in Berlin on January 27, 1921.

1

2

Structured Credit Portfolio Analysis, Baskets & CDOs

As a reference for the sections in this chapter we refer to the book [25]
where most of the introductory remarks made in this chapter are elaborated in much greater detail. In order to keep the exposition fluent,
we do not interrupt the text too often to make bibliographic remarks.
Instead, suggestions for further reading are provided in Chapter 5. The
main purpose of Chapter 1 is to build up the basis of nomenclature,
risk terms, and basic notions required for understanding the subsequent
chapters of this book.

1.1

Modeling Single-Name Credit Risk

Based on Einstein’s observation regarding uncertainty in real world
problems, we have to rely on probabilistic concepts already at the level
of single borrowers. In the following brief exposition, we focus on ratings and default probabilities, loss quotes, and exposures.

1.1.1

Ratings and Default Probabilities

As a standard tool for the lending business, all banks have rating
systems in place today ranking the creditworthiness of their clients by
means of a ranking, e.g., AAA, AA, A, BBB, BB, B, CCC, or Aaa,
Aa, A, Baa, Ba, B, Caa, respectively, for best, 2nd-best, 3rd-best, ...,
and worst credit quality where the first row of letter combinations is
used by the rating agencies Standard & Poor’s2 (S&P) and Fitch3 and
the second row of letter combinations is used by Moody’s.4 Banks and
rating agencies assign default probabilities (short: PD for ‘probability
of default’) with respect to certain time horizons to ratings in order
to quantify the likelihood that rated obligors default on any of their
payment obligations within the considered time period. In doing so,
the ranking (with respect to creditworthiness) of obligors in terms of
ratings (letter combinations) is mapped onto a metric scale of numbers
(likelihoods) in the unit interval. Throughout the book we use one-year
default probabilities assigned to ratings (in S&P notation) according to
2

See www.standardandpoors.com
See www.fitchratings.com
4
See www.moodys.com
3

From Single Credit Risks to Credit Portfolios

3

TABLE 1.1:

One-year
PDs for S&P ratings; see
[108], Table 9
AAA
AA
A
BBB
BB
B
CCC

D
0.00%
0.01%
0.04%
0.29%
1.28%
6.24%
32.35%

Table 1.1. Table 1.1 can be read as follows in an intuitive way. Given a
portfolio of 10,000 B-rated obligors it can be expected that 624 of these
obligors default on at least one of their payment obligations within a
time period of one year.
Kind of ‘non-acceptable’ is the zero likelihood of default for AAArated borrowers. This would suggest that AAA-rated customers never
default and, therefore, can be considered as risk-free asset for every
investor. It is certainly very unlikely that a firm with a AAA-rating fails
on a payment obligation but there always remains a small likelihood
that even the most creditworthy client defaults for unexpected reasons.
Moreover, as addressed5 in research on PD calibration for low default
portfolios, the zero default frequency for AAA is just due to a lack
of observations of AAA-defaults. In order to assign a non-zero oneyear PD to AAA-rated customers, we make a linear regression of PDs
on a logarithmic scale in order to find a meaningful estimated PD for
AAA-rated borrowers.
This yields a one-year PD for AAA-rated obligors of 0.2 bps, which
we will assume to be the ‘best guess PD’ for AAA-rated clients from
now on.
Default probabilities are the ‘backbone’ of credit risk management.
In the new capital accord6 (Basel II), the so-called Internal Ratings
Based (IRB) approach (see [15]) allows banks to rely on their internal
5

See Pluto and Tasche [100] and Wilde and Jackson [114].
Note that the Bank for International Settlements (BIS) is located in the Swiss city
Basel such that the capital initiative, which led to the new regulatory framework is
often addressed under the label ‘Basel II’; see www.bis.org
6

4

Structured Credit Portfolio Analysis, Baskets & CDOs

estimates for default probabilities for calculating regulatory capital.
Therefore, the better the quality of a bank’s rating systems, the more
appropriate are the bank’s regulatory capital figures. But not only for
regulatory purposes but, even more important, for economic reasons
like competitive advantage in the lending market it is important that
banks pay a lot of attention to their internal rating systems. It is
beyond the scope of this book and beyond the purpose of this introductory chapter to go into great details here but the following remarks
will at least provide a link to the practical side of ratings.
Internal ratings can be obtained by various methodologies. In this
book, where the focus is on structured credit products like default baskets and collateralized debt obligations (CDOs; see Chapter 3), ratings
and PDs are typically modeled via so-called causal rating methods. In
such models, underlying drivers of the default event of a client or asset are explicitly modeled as well as the functional link between risk
drivers and the default event. We will come back to this principle
over and over again in subsequent chapters. For a first simple example, we refer to Figure 1.3. Causal rating models are a conceptually
preferred type of ratings because nothing is more desirable for a bank
than understanding the ‘explicit mechanism’ of default in order to define appropriate measures of default prevention. Unfortunately, it is
not always possible to work with causal rating models. For example,
private companies (Small and Medium Enterprises; SME) are difficult
to capture with causal models because underlying default drivers are
not given in up-to-date explicit form as it is in the case for listed corporate clients where public information, for instance, by means of stock
prices, is available.
In cases where causal models are not implementable, we need to
work with more indirect approaches like so-called balance sheet scorings
where balance sheet and income statement informations (as ‘explaining
variables’) are used to explain default events. In such rating systems,
the bank optimizes the discriminatory power (i.e., the ability of the
rating system to separate defaulting from non-defaulting clients w.r.t.
a certain time horizon) by means of a scheme as illustrated in Figure
1.1. Starting with a long list of financial ratios, the rating modelers
attempt to find the optimal selection of ratios and optimal weights for
the combination of ratios to finally arrive at a rating score best possible
identifying potential defaulters in the bank’s credit approval process.
An important working step in the development of scoring models

From Single Credit Risks to Credit Portfolios

FIGURE 1.1:

5

Rating/scoring system optimization (illustrative)

is the calibration of the scores to actual default probabilities. This
procedure is briefly called the PD calibration of the model. A major
challenge in PD calibration and rating/scoring model development is
to find a healthy balance between the so-called error of first kind (αerror) and the error of second kind (β-error), where the first mentioned
refers to a defaulting client approved by the scoring system and the
latter-mentioned refers to a non-defaulting client erroneously rejected
by the scoring system. The error of first kind typically contributes to
the bank’s P&L with the realized loss (lending amount multiplied by
the realized loss quote) of the engagement, whereas the second error
contributes to the bank’s P&L in the form of a missed opportunity,
which in monetary terms typically sums up to the lending amount
multiplied by some margin. It is obvious that the first error typically
can be expected to be more heavy in terms of P&L impact. However,
banks with too conservative PDs will have a competitive disadvantage
in the lending market. If too many clients are rejected based on ratings
or scorings, the earnings situation of a bank can severely suffer, in terms
of a negative P&L impact as well as in terms of lost market share.
There are other rating system categories, besides causal rating and
scoring systems, which are common in credit risk management. For
instance, in some cases ratings are based on a hybrid methodology

6

Structured Credit Portfolio Analysis, Baskets & CDOs

combining a scoring component and a causal model component into
a common framework. We will not comment further on ratings in
general but conclude our discussion with an example illustrating the
competitive advantage banks have from better rating systems. This
remark also concerns our structured credit applications since banks
with more sophisticated models for, e.g., the rating or evaluation of
tranches in collateralized debt obligations (CDOs), have a chance of
being more competitive in the structured credit market.

FIGURE 1.2:

Illustration of the competitive advantage arising from
discriminatory power in rating systems
For our example we consider a portfolio with 100 assets or clients.
We assume that the average default quote in the portfolio is 1%. If the
rating/scoring model of the bank has no discriminatory or prediction
power at all, then the rating outcome for a client is purely based on
chance. In Figure 1.2, we indicate such a worst case rating model by
a so-called receiver operating characteristic (ROC)7 on the diagonal of

7

See [43] for an introduction to receiver operating characteristic (ROC) curves. In
our chart, the x-axis shows the false alarm rate (FAR) and the y-axis shows the hit
rate (HR) of the scoring system. Let us briefly explain the meaning of FAR and HR.
Given that credit approval is based on a score S satisfying S > c, where c denotes

From Single Credit Risks to Credit Portfolios

7

the unit square. The AUROC is a common measure for discriminatory
power; see, e.g., [43]. The AUROC for the worst case rating system
(see Figure 1.2) equals 0.5, which corresponds to the area under the
diagonal in the unit square. This means that the credit score in the
credit approval process of the bank does a job comparable to a coinflipping machine with a fair coin: ‘head’ could mean approval of the
loan and ‘tail’ could mean rejection of the loan. If the bank has such a
rating/scoring system, then clients cannot be distinguished regarding
their default remoteness. Therefore, given a portfolio default quote of
1%, every client will get assigned a PD of 1%. This is extreme case I in
Figure 1.2. Extreme case II refers to a ‘crystal ball’ rating/scoring system where the model with certainty identifies the one defaulting client
out of 100 clients in the portfolio who is responsible for the portfolio
default quote of 1%. In such an unrealistically lucky case, it is natural
to assign a zero PD to all non-defaulting clients and a PD of 100% to
the one defaulting client.
In extreme case I as well as in extreme case II, the average of assigned PDs in the portfolio equals 1%. If the bank, starting from the
worst case rating system in extreme case I, manages to improve their
ratings toward the ‘crystal ball’ rating system from extreme case II,
then 99 PDs experience a significant reduction (from 1% down to 0%)
and one PD is significantly increased (from 1% to 100%). Our example is purely illustrative because typical rating/scoring systems have
an AUROC between, e.g., 70% and 90%, but rating revisions, which
improve the discriminatory power of the system exhibit the same behavior as our extreme case evolution in the example: The benefit of
improved predictive power in rating/scoring systems typically leads to
lower PDs for assets/clients with higher credit quality and to higher
PDs for assets/clients with worse credit quality. In this way, improved
ratings allow the bank to offer a more competitive or aggressive pricing
for high quality assets and, at the same time, lead to an improved rejection mechanism for low quality assets. It can be observed in practice
that better ratings have a potential to substantially contribute to an
a pre-specified critical cutoff, then a false alarm occurs if S < c for a non-defaulting
client. In contrast, the scoring system achieves a hit, if the score S shows S < c
for a defaulting client, because the comparison of score and cutoff values helped us
indentifying a dangerous credit applicant. Now, the ROC curve is a plot of the curve
(FAR(c), HR(c))c∈{cutoffs} . A ‘crystal ball’ scoring system yields an area under the
ROC curve (AUROC) of 100% because FAR=0 and HR=1 are best case situations.

8

Structured Credit Portfolio Analysis, Baskets & CDOs

improved P&L distribution.
After our brief excursion to the ‘practical side’ of ratings, we now
come back to the development of the framework used in this book
for modeling structured credit products. As already indicated, we are
working in a causal modeling world when it comes to default baskets
and CDOs; see Figure 3.8. But even at single-asset level, we can think
of default events in a causal model context. For this purpose, we introduce for obligor i in a portfolio of m obligors a Bernoulli variable for
indicating default within the time interval [0, t],
(t)

Li

(t)

(t)

= 1{CWI(t) < c(t) } ∼ B(1; P[CWIi < ci ])
i

(1.1)

i

following a so-called threshold model in the spirit of Merton [86],
Black & Scholes [23], Moodys KMV’s PortfolioManager,8 and
CreditMetrics9 from the Risk Metrics Group (RMG) for modeling
(t)
default of credit customers. Here, (CWIi )t≥0 denotes a creditworthiness index (CWI) of obligor i indexed by time t triggering a payment
default of obligor i in case the obligor’s CWI falls below a certain crit(t)
(t)
ical threshold ci within the time period [0, t]. The threshold (ci )t≥0
is called the (time dependent) default point of obligor i. Equation (1.1)
expresses the causality between the CWI of obligor i and its default
(t)
event. Note that - despite its notation, we do not consider (CWIi )t≥0
as a stochastic process but consider it as a time-indexed set of latent
random variables w.r.t. time horizons t where t ≥ 0 denotes continuous
time. It is important to keep this in mind in order to avoid misunderstandings. It needs some work10 (see Sections 3.3.5.1 and 3.3.5.2) to
get from a sequence of latent variables to a stochastic process reflecting
a certain time dynamics. For most of the applications discussed in this
book, the ‘fixed time horizon view’ is completely sufficient. However,
for the sake of completeness, we discuss in Section 3.3.5.2 a particular model based on the first passage time w.r.t. a critical barrier of a
8

See www.kmv.com
See www.riskmetrics.com
10
The step from fixed horizon CWIs to a stochastic process can be compared with
the construction of Brownian motion: just considering a sequence of normal random variables (Xt )t≥0 with Xt ∼ N (0, t) does not mean that (Xt )t≥0 already is a
Brownian motion. It needs much more structure regarding the time dynamics and
quite some work to choose (Xt )t≥0 in a way making it a Brownian motion.
9

From Single Credit Risks to Credit Portfolios

9

stochastic process instead of a sequence of (fixed horizon) latent variables w.r.t. a sequence of default points; see Figure 1.3 illustrating the
concept of stochastic processes as default triggers. The process introduced in Section 3.3.5.2 will be called an ability to pay process (APP)
because it reflects a stochastic time dynamics, whereas the sequence
of CWIs can be seen as a perfect credit score indicating default with
certainty according to Equation (1.1).

FIGURE 1.3:

Ability to pay process (APP) as default trigger

CWIs are difficult or may be even impossible to observe in practice.
In the classical Merton model, the ability to pay or distance to default
of a client is described as a function of assets and liabilities.. However,
the concept of an underlying latent CWI triggering default or survival
of borrowers is universally applicable: not only listed corporate clients
but any obligor has its own individual situation of wealth and financial
liabilities w.r.t. any given time horizon. If liabilities exceed the financial
power of a client, no matter if the client represents a firm or a private
individual,bankruptcy and payment failure will follow. In this way, the

10

Structured Credit Portfolio Analysis, Baskets & CDOs

chosen approach of modeling default events by means of latent variables
is a concept meaningful for corporate as well as for private clients and
can be applied to the total credit portfolio of banks.
For the ‘fixed time horizon’ CWI approach, the time-dependent de(t)
fault points (ci )t≥0 are determined by the obligor’s PD term structure
(see Section 2.3)
(t)

(t)

(t)

(pi )t≥0 = (P[CWIi < ci ])t≥0

(1.2)

in a way such that
(t)

ci

= F−1

(t)

(t)

CWIi

(pi )

(1.3)

where FZ denotes the distribution function of any random variable Z
and F−1
Z denotes the respective (generalized) quantile function
F−1
Z (z) = inf{q ≥ 0 | FZ (q) ≥ z}.
The derivation of PD term structures is elaborated in Section 2.3. Note
that the before-mentioned term structures and CWIs so far reflect only
the one-dimensional flow of the marginal distributions of single-name
credit risks, not the joint distributions of CWIs in the sense of a multivariate distribution. Later, we will catch-up in this point and spend
a lot of time with multi-variate CWI vectors reflecting dependencies
between single-name credit risks.

1.1.2

Credit Exposure

Despite default probabilities we need to know at least two additional
informations for modeling single-name client risk, namely the exposure
at default (EAD) outstanding with the considered obligor and the loss
given default (LGD) reflecting the overall default quote taking place if
obligors default. One could easily dedicate a separate chapter to each
of both risk drivers, but again we have to restrict ourselves to a few
remarks roughly indicating the underlying concepts.
Exposure measurement has several aspects. For drawn credit lines
without further commitment, exposure equals the outstanding11 notional exposure of the loan, whereas already for committed undrawn
11

Taking amortizations into account.

From Single Credit Risks to Credit Portfolios

11

lines exposure starts to involve a random component, in this case depending on the drawing behavior of the client. Another exposure notion
appears in the context of counterparty credit risk in derivative products
where exposure equals potential exposure (PE; typically applied in the
context of limit setting, etc.) or expected positive exposure (EPE; exposure often used for economic capital calculation, etc.); see, e.g., [79]
as well as [13] for an introductory paper12 on exposure measurement.
In this book we will not need PE or EPE techniques and can restrict
our exposure notion to EAD; see also the new capital accord [15].
The addendum ‘at default’ in EAD refers to the fact that for the
determination of realized loss we have to take into account all monetary
amounts contribution to the realized loss. Hereby, it is important that
we think of EAD as a conglomerate of principal and interest streams;
see Figure 1.4 where the interest stream consists of coupon payments
and the principal stream is made up by the repayment of capital at the
maturity of the bond. In asset-backed transactions as discussed later in
this book, we have to deal with EAD at two different levels, namely at
the level of the underlying assets (e.g., some reference credit portfolio)
as well as at the level of the considered structured security, e.g., some
CDO tranche. Actually, the same two-level thinking has to take place
for LGDs where the LGD at structured security level typically depends
on the LGDs of the underlying assets, loans, or bonds, respectively. We
will come back to this point later in the book.
Another aspect of EAD, mentioned for reasons of practical relevance
in the same way as our excursion on ratings, is the potential uncertainty
regarding the actual outstanding exposure at the default time of an asset. In CDOs, the amount of outstandings typically is well defined and
controlled by the offering circular or the term sheet, respectively, describing the structure of the considered transaction. Nevertheless, one
could easily think of situations where uncertainties in the outstanding
exposure can occur, e.g., in case of residential mortgage backed securities (RMBS) where loans can be prepayed13 or in cases where the
12

Note that the nomenclature in exposure measurement exhibits some variability
in the literature due to a certain lack of standardization of notions although the
concepts discussed in different papers are most often identical or at least close to
each other.
13
In RMBS transactions, prepayed loans are typically replaced by other mortgagebacked loans matching the eligibility criteria of the transaction; in such cases, the
offering circular defines clear rules how and in which cases replenishments can be

12

Structured Credit Portfolio Analysis, Baskets & CDOs

FIGURE 1.4:

Exposure as a conglomerate of principal and interest

streams
underlying assets are PIKable where ‘PIK’ stands for payment in kind
addressing more or less the option to exchange current interest payments against principal or capital (e.g., in a bond where the issuer
has the option to capitalize interest at any payment date by issuing
additional par amount of the underlying security instead of making
scheduled interest payments). The most common uncertainty in lending exposures at single loan level is due to unforeseen customer behavior
in times of financial distress; see Figure 1.5 where the well-known fact
that obligors tend to draw on their committed but so far undrawn lines
in times of trouble. The most sophisticated approach to deal with exposure uncertainties is by means of causal modeling in a comparable way
to causal rating models but focussing on exposure instead of default
events, hereby taking the underlying drivers for changes in the outstanding exposure into account and modeling the relationship between
these underlying exposure drivers and EAD, e.g., by means of Monte
Carlo simulation techniques. As an example, prepayment behavior of
obligors in RMBS portfolios is strongly coupled with interest rates such
that the interest rate term structure as underlying driver is a reasonable starting point for a causal modeling of prepayments in an RMBS
done by the collateral or asset manager in charge of the underlying asset pool.

From Single Credit Risks to Credit Portfolios

13

model. In a comparable way, prepayments in corporate lending can be
tackled, taking into account that in corporate lending prepayments are
a function of interest rates, borrower’s credit quality, and prepayment
penalties (depending on the domestic lending market); see, e.g., [68].

FIGURE 1.5:

Exposure uncertainty due to unforeseen customer be-

havior
To mention another example, the new capital accord deals with exposure uncertainties for off-balance sheet positions by means of so-called
credit conversion factors in the formula
EAD = OUTSTANDINGS + CCF × COMMITMENT
where CCF denotes the credit conversion14 factor determined w.r.t. the
considered credit product; see [15], §82-89 and §310-316. For instance,
in the so-called Foundation Approach a CCF of 75% has to be applied
to credit commitments regardless of the maturity of the underlying
facility, whereas a CCF of 0% applies to facilities, which, e.g., are
uncommitted or unconditionally cancellable. In practical applications
and in the sequel of this book, we will always consider EAD w.r.t.
some time horizon t, written as EAD(t) (where necessary) addressing
the outstanding exposure at time t.

14

In general, credit conversion can address the conversion of non-cash exposures
into cash equivalent amounts or the conversion of non-materialized exposures into
expected exposures at default (for example, quantifying the potential draw down of
committed unutilized credit lines), etc.

14

1.1.3

Structured Credit Portfolio Analysis, Baskets & CDOs

Loss Given Default

We now turn our attention to LGD. In general, LGD as a concept is
simply explained but far from being trivial regarding modeling and calibration. In the early times of quantitative credit management, many
banks defined ‘standard LGDs’ for broad loan categories, e.g., claiming
that corporate loans on average admit, e.g., a 40% recovery quote such
that the LGD of such products equals 60%. Today we see an increasing
trend toward more sophisticated LGD approaches, to some extent due
to the new capital accord [15] where banks approved for the alreadymentioned IRB approach in its advanced form can base their capital
figures not only on internally estimated PDs but also on their internal
LGD estimates. In the sequel, we briefly summarize some basic facts
about LGDs.
The most advanced LGD approach is, in the same way as mentioned
in the context of PDs and EADs - the causal modeling approach again.
We will later see that, e.g., the LGD of CDO tranches is a good example
for an LGD derived by a causal modeling approach; see Section 3.1.
Here, the loss of a CDO tranche is linked to the performance of the
underlying asset pool such that in a Monte Carlo simulation one can
model the economics of the underlying pool and the causality between
the pool and the credit-linked securities at the liability side of the CDO
structure. By transforming each asset pool scenario into a scenario at
the CDO tranche level one obtains a loss distribution for the CDO
tranche from which one can derive the LGD as the mean loss quote of
the full distribution of possible loss severities.
At single loan level, a good compromise between sophistication and
practicability is the following approach. Let us assume that the bank
built up a collateral database containing comprehensive historic experience regarding the achieved value in selling collateral for loss mitigation
in the context of defaulted credit-risky instruments. For example, such
a database will contain information about the average market value decline of residential mortgages, different types of commercial mortgages,
single stocks, bonds, structured products, cars for leasings, aircraft vessels for aircraft finance, ships for ship financing, etc. The database will
also contain information about recovery quotes on unsecured exposures
w.r.t. different seniorities, and so on. Based on such a database, value
quotes (VQ) w.r.t. different types of collateral can be derived where
the VQ incorporates the expected market value decline of the consid-

From Single Credit Risks to Credit Portfolios

15

ered collateral category. In addition, the collateral value volatility can
be derived, useful for the calibration of stochastic approaches to LGD
modeling; see Section 3.3. Given such a database exists, AIRB15 banks
need to have such databases for regulatory approval of their internally
calibrated LGDs, we can proceed as in the following example.

FIGURE 1.6:

Illustration of LGD determination

Let us assume that we want to determine the LGD in a situation
where some client has two loans secured by three types of collateral;
see Figure 1.6. Out of the bank’s collateral database we obtain value
quotes VQ1, VQ2, VQ3 (in percentage) for the collateral values. Given
the market value of the collaterals is given by MV1, MV2, and MV3,
the overall, on average achievable, estimated collateral value (ECV) for
loss mitigation is given by
ECV = VQ1 × MV1 + VQ2 × MV2 + VQ3 × MV3.
Note that in most of the cases, collateral selling to the market takes
place quite some time after the actual default time of the obligor such
15

Banks in the so-called advanced internal ratings based (AIRB) approach; see [15].

16

Structured Credit Portfolio Analysis, Baskets & CDOs

that an appropriate discounting of collateral values reflecting the time
value of money has to be incorporated into the value quotes. Denoting
the exposure of the two loans approved for the client by EAD1 and
EAD2, the total exposure allocated to the client equals
EAD = EAD1 + EAD2.
Then, the realized loss, here denoted by LOSS, is given by


LOSS = max (1 − VQunsecured) × (EAD − ECV), 0

where VQunsecured equals the percentage average recovery quote on
unsecured exposures, also to be calibrated based on the collateral/loss
database of the bank. Note that in Figure 1.6 we assume (for reasons
of simplicity) that we lose 100% of the unsecured exposure in case of
default, i.e., VQunsecured = 0. Because the value quotes already incorporate the potential market value decline of the considered collateral,16
LOSS expresses an expectation regarding the realized loss amount after
taking into account all available collateral securities for loss mitigation
as well as the typical recovery rate on unsecured exposures. Then,
the severity of loss in case the obligor defaults can be expressed as a
percentage loss quote, namely the LGD, by
LGD =

LOSS
.
EAD

(1.4)

Because LOSS represents the realized loss amount in units of money,
Equation (1.4) is ‘compatible’ with the well-known formula for the expected loss (EL) on single-names
EL[$] = PD × EAD × LGD.

(1.5)

Note that the EL can be written in the simple form of Equation (1.5)
only if the Bernoulli variable indicating default of the obligor and the
two quantities EAD and LGD, e.g., considered as realizations of corresponding random variables due to certain uncertainties inherent in
these quantities, are stochastically independent. If EAD and LGD are
used as fixed values, they can be considered as expectations of corresponding random variables. Also note that, in general, in the same
16

Equivalently to the chosen approach one could have defined LGDs for collateral
securities and an LGD for the overall exposure net of recoveries and then aggregated
both figures into an estimate for the ‘realized loss’ accordingly.

From Single Credit Risks to Credit Portfolios

17

way as in the case of EAD, we have an additional degree of complexity
in the time dimension. Because Definition 1.4 involves exposures, and
exposures (EADs) are time dependent, we will write LGD(t) in the sequel where necessary and meaningful, in line with our time-dependent
notion of exposures, EAD(t) .
Under all circumstances it is essential that LGD modeling clearly
reflects the recovery and workout process of the bank. Deviations from
the bank-internal workout practice will lead to distortions in the LGD
calibration and will cause unwanted long-term deviations of provisions
or realized losses, respectively, from forecasts based on the expected
loss of the bank’s portfolio. Advanced IRB banks have to make sure
that PDs and LGDs can be backtested and validated in order to achieve
regulatory approval of their internal estimates.

1.2

Modeling Portfolio Credit Risk

Turning from single-name to portfolio credit risk is a challenging step
in credit risk modeling. For the topics covered in this book, portfolio
credit risk is not only one of several issues but rather the fundamental
basis for modeling structured credit products. This is confirmed by the
notion correlation products sometimes used as a headline for default
baskets and CDOs, addressing the fact that these products trade correlations and interdependencies between single-name credit risks in a
tailor-made way leading to interesting (portfolio-referenced) risk selling
and buying opportunities.

1.2.1

Systematic and Idiosyncratic Credit Risk

The need for a sound modeling of correlations and ‘tail dependencies’
in the context of portfolio loss distributions is the main reason why for
structured credit products we prefer latent variable models indicated in
Equation (1.1) enabling an explicit modeling of correlations in contrast
to other well-known models where dependencies between single-name
credit risks are modeled implicitly, for example, by means of default rate
volatilities in systematic sectors; see CreditRisk+ [35] and Chapter 4
in [25]. According to Equation (1.1), every obligor i is represented by

18

Structured Credit Portfolio Analysis, Baskets & CDOs

a default indicator

(t)

Li = 1{CWI(t) < c(t) }
i

i

such that the obligor defaults if and only if its CWI falls below its
default point, always measured w.r.t. to some time interval [0, t].
Now, the starting point of dependence modeling is a decomposition of
the client’s risk into a systematic and an idiosyncratic risk component;
see Figure 1.7.

FIGURE 1.7:

Decomposition of two firm’s credit risk into systematic
and idiosyncratic parts (separation approach; illustrative)
In Figure 1.7, the credit risk of two German automotive manufacturers is decomposed into systematic and idiosyncratic components, here
with a 100% country weight in a factor representing Germany and a
100% industry weight in a factor representing the automobile industry. The basic assumption in such models is that dependence takes
place exclusively via the systematic components of counterparties. After normalizing the overall firm risk (say, the volatility of the CWI)
to 1, the so-called R2 (R-squared) of the firm captures the systematic
risk, whereas the quantity 1−R2 quantifies the residual or idiosyncratic
risk of the firm, which cannot be explained by means of systematic risk

From Single Credit Risks to Credit Portfolios

19

drivers like countries and industries; see [25], Chapter 1 for a detailed
discussion about systematic and idiosyncratic credit risk.
In more mathematical terms, Figure 1.7 suggests to decompose the
CWI of obligor i in Equation (1.1) into systematic and idiosyncratic
components,
(t)
(t)
(t)
CWIi = βi Φi + εi
(t ≥ 0)
(1.6)
(t)

(t)

where we assume that the residuals (ε1 )t≥0 , ..., (εm )t≥0 are independent and for every fixed t identically distributed as well as independent
(t)
of the systematic variables Φi = (Φi )t≥0 . Here, m denotes the number of obligors in the portfolio and the index Φi is called the composite
factor of obligor i because it typically can be represented by a weighted
sum of indices,
(t)
Φi

=

N
X

wi,n Ψ(t)
n

n=1

(i = 1, ..., m; t ≥ 0),

(1.7)

with positive weights wi,n . Regarding the indices
(t)

(t)

Ψ1 = (Ψ1 )t≥0 , ... , ΨN = (ΨN )t≥0
we find two major approaches in the market and ‘best practice’ industry
models, respectively.
• The separation approach: Here, Ψ1 , ..., ΨNC denote country
indices and ΨNC +1 , ..., ΨN are industry indices. The number of
industry factors is then given by NI = N − NC , whereas NC
denotes the number of country factors. An industry example
for such a model is the so-called Global Correlation Model by
Moodys KMV.17
• The combined approach: Here, every index Ψn refers to an
industry within a country. CreditMetrics by the RiskMetrics
Group (RMG) follows such an approach18 via MSCI indices.19
There are pros and cons for both approaches and it seems to us that
both factor models are well accepted in the market. There are other differences in the way Moodys KMV and RMG model correlations, e.g.,
17

See www.kmv.com
See www.riskmetrics.com
19
Morgan Stanley Capital International Inc.
18

20

Structured Credit Portfolio Analysis, Baskets & CDOs

KMV derives asset value correlations from a non-disclosed optiontheoretic model transforming equity processes and related information
into model-based asset value processes, whereas RMG works with equity
correlations as a proxy for asset value correlations. For more information on factor modeling we refer to Chapter 1 in [25].
In this context, it is worthwhile to mention that correlations are subject to certain controversial discussions. One can find research papers
in the market where people suddenly seem to discover that correlations
are negligible. However, we are convinced from our practical experience that correlations are not only inherent/omnipresent in the credit
market but play a fundamental role in the structuring and design of
default baskets and CDOs. If correlations would be negligible, these
kind of correlation products would not be as successful and growing
as they are in the credit market. Moreover, in Section 3.4.4 we discuss implied correlations in index tranches. Here, investment banks
actually quote correlations in the same way as they quote spreads and
deltas. This provides strong support for the importance of correlations and dependencies in the structured credit market. For further
study, we recommend to interested readers two very valuable discussion papers on the correlation topic. The first paper is from Moody’s
KMV [116] and the second paper is from Fitch Ratings [6]. In
these papers, the reader finds strong evidence that correlations are far
from being negligible. Moreover, correlations can be measured within
systematic sectors like industries (intra-industry) but also between different systematic sectors (e.g., inter-industry). The type of correlation
considered in the two papers is the asset correlation referring to the
correlation between underlying latent variables, in our case the CWIs
of clients. We come back to this issue in Chapter 2 where we introduce
correlations in a more formal way, distinguishing between asset or CWI
correlations and the default correlation, which refers to the correlation
of Bernoulli variables as in (1.1) indicating default. As we will see later,
default correlations typically live at a smaller order of magnitude than
CWI or asset correlations; see (2.5) and thereafter.

1.2.2

Loss Distribution of Credit Portfolios

In the early times of credit risk modeling, the ‘end product’ of credit
risk models typically was the portfolio loss distribution w.r.t. some fixed
time horizon. Today, and especially in structured finance, people are

From Single Credit Risks to Credit Portfolios

21

often more interested in looking at dependent default times. We will
come back to this issue in Section 2.5. Obviously, the fixed time horizon
point of view is a ‘side product’ of default time models just by restricting the default time τ of some obligor to a fixed time interval [0, T ] and
considering the Bernoulli variable 1{τ <T } . For reasons of completeness,
we illustrate in Figure 1.8 the loss distribution of a credit portfolio.

FIGURE 1.8:

The loss distribution of a credit portfolio

Based on the loss distribution of the credit portfolio, all relevant risk
quantities can be identified as ‘summary statistics’ of this distribution.
For instance, the expected loss (EL) of the portfolio20 (in percentage)

20

Recall Equation (1.5) for the EL of single-name credit risk in case of independence
of the default indicator, exposure, and severity of loss.

22

Structured Credit Portfolio Analysis, Baskets & CDOs

equals the sum of single-name ELs weighted by the total portfolio EAD,
EL(t) [%] = P
m

1

m
X

(t)
j=1 EADj i=1
m
X

= P
m

1

(t)
j=1 EADj i=1

(t)

(t)

ELi

(t)

(1.8)
(t)

(t)

E[Li × LGDi ] × EADi

where Li is defined in (1.1). Note that Equation (1.8) allows for
stochastic (non-deterministic) LGDs but assumes EAD to be a deterministic (non-random) fixed quantity.
The economic capital (EC) of a credit portfolio typically is defined as
the quantile of the loss distribution w.r.t. to some given target level of
confidence, e.g., 99.9%, minus the EL of the portfolio, which is supposed
to be fully priced-in by the front office of the bank,
hP
i
(t)
(t)
(t)
m
qα
L
×
EAD
×
LGD
i=1 i
i
i
− EL(t) [%] (1.9)
EC(t)
Pm
α [%] =
(t)
j=1 EADj
where qα [X] denotes the α-quantile of some random variable X. Note
that the quantile function qα [X] in Equation (1.9) is a much more
‘tricky’ object than the expectation E[·] arising in Equation (1.8). For
instance, expectations are linear functions in their arguments whereas
quantile functions are highly non-linear in general.

For capital allocation purposes better suitable than quantile-based
risk measures are so-called shortfall measures; see Chapter 5 in [25]
and the literature mentioned in the last chapter of this book for an
explanation21 why shortfall measures are superior to quantile-based
measures. Figure 1.8 illustrates a typical shortfall measure. Let us say
the senior management considers a certain loss threshold q as ‘critical’
for the economic future of the bank. Such a threshold typically will be
much lower than the α-quantile of the loss distribution. Then, it makes
sense to calculate the expectation of all losses exceeding q, weighted
via the loss likelihoods illustrated by the loss distribution. Such a
conditional expectation is called the expected shortfall (ESF) w.r.t. q of
the credit portfolio. It is the mean loss or expected loss conditional
21

Shortfall measures are the most prominent examples of so-called coherent risk
measures.

24

Structured Credit Portfolio Analysis, Baskets & CDOs

So far, we kept the presentation fairly generic without making explicit
assumptions regarding the distribution of involved random variables.
In the subsequent sections, we will be more explicit in tail risk modeling
mainly involving four different so-called copula functions, namely the
Gaussian, Student-t, Clayton, and comonotonic copulas. We will see
that copula functions as dependence modeling tools have a major impact
on portfolio-based risk measures like EC and ESF. We will also see
how copula functions influence the distribution of joint default times
in credit portfolios. The consequences for the modeling of structured
credit products will be discussed in great detail later on.

1.2.3

Practicability Versus Accuracy

We close our brief walk through credit risk modeling aspects with
Figure 1.9 illustrating the different trade-offs credit modelers are exposed to when balancing between practicability and accuracy of their
models, parameterizations and estimates.

FIGURE 1.9:

Finding a balance between practicability and accuracy
in credit risk evaluation
In vertical direction, Figure 1.9 shows at the top corner the wish for
steadily improved risk parameter estimates. For instance, in the previous section we discussed how positive an increase of the discriminatory

From Single Credit Risks to Credit Portfolios

25

power of a rating system can impact the bank’s P&L. The more accurate the rating systems and PD estimates of the bank the more reliable
are credit decisions. In opposite vertical direction we find drivers of
efficiency in the bank’s valuation and credit approval processes. For
instance, reducing the number of qualitative factors in a rating system
can significantly accelerate credit processes. Banks have to balance
between efficiency and accuracy/sophistication in credit risk estimates,
at single-name as well as at portfolio level.
In horizontal direction, Figure 1.9 shows some other diametral forces
in banking. At the left-hand side we find the clear demand for consequent risk/return steering. As soon as a bank has recognized the potential of sound credit risk models and steering measures these intiatives
should be applied to the day-to-day business of the bank. The classical ‘relationship banking paradigm’ points in the opposite direction.
In times of over-banked loan markets, banks will think about it twice
before they shock a long-term well-known client by an increased risk
premium as credit price component due to an increased exposure concentration with this client. The way out of the dilemma of well-justified
diametral demands is active credit portfolio management (ACPM). Creating tailor-made credit products for important clients and offering attractive conditions due to an active management of credit risks is a key
factor of success in today’s credit business.

Chapter 2
Default Baskets

A default basket is a portfolio of not too many obligors, e.g., not more
than 10, although there is no ‘officially agreed hard limit’ regarding the
number of obligors allowed in order to speak of a basket. At least it is
clear that a basket contains more than one obligor or asset, respectively,
such that in case of moderate or low correlation between the intruments
in the basket diversification effects will reduce the overall portfolio or
basket risk. In this chapter, we model default baskets and certain
related products. This chapter also serves as a preparation for the
slightly more complicated Chapter 3 on CDOs. As we will see later,
default baskets and CDOs are ‘close relatives’.

2.1

Introductory Example: Duo Baskets

We start our discussion by the most simple basket one can think of,
namely, a duo basket consisting of two credit-risky instruments only.
The purpose of the following introductory discussion is twofold. First,
we want to review the idea of diversification and its interplay with correlation. Second, we want to elaborate a first example illustrating the
necessary balance between risk and return, a fundamental way of thinking in portfolio management and especially in the field of structured
credit products.
Let us assume that we consider the following two loans:
• Loan A has a one-year default probability of pA = 100 bps, loan
B has a one-year default probability of pB = 50 bps.
• Both loans have a bullet-type exposure profile, where ‘bullet’
refers to a non-amortizing loan with 100% of the exposure outstanding from the first to the last day of the term of the loan.
27

28

Structured Credit Portfolio Analysis, Baskets & CDOs
• Both loans have an LGD of 100% such that in case of default
the full outstanding exposure amount will be lost for the lending
bank.

Let us restrict ourselves to the one-year horizon for the time being.
According to our threshold model (see Equation (1.1)) explained in
Chapter 1 we assume the existence of CWIs for obligors A and B and
corresponding Bernoulli variables
(1)

LA

= 1{CWI(1) < c(1) }
A

and

A

(1)

LB

= 1{CWI(1) < c(1) }
B

(2.1)

B

indicating default or survival within one year. In line with Equation
(1.3), the default points of the obligors are determined by
(1)

cA

= F−1

(1)

(1)

CWIA

(pA ) and

(1)

cB

= F−1

(1)

(1)

CWIB

(pB ).

(2.2)

In order to come up with explicit numbers, let us assume that the
CWIs of obligors A and B are standard normally distributed with a
CWI correlation of 10%. If we want to embed the CWI correlation
of two assets into a factor model as outlined in Section 1.2 then we
can introduce a standard normally distributed random variable Y ∼
N (0, 1) and two independent variables εA , εB ∼ N (0, 1), independent
of Y , and write (dropping the time index for a moment for reasons of
a simplified notation)
p
√
(2.3)
CWIA = ̺ Y + 1 − ̺ εA
p
√
CWIB = ̺ Y + 1 − ̺ εB
such that CWIA , CWIB ∈ N (0, 1), and

Corr[CWIA , CWIB ] = ̺A,B = ̺
where Corr[· , ·] denotes correlation. A CWI correlation of 10% determines ̺ = 0.1 for our working example. The default points of A and
B can then easily be calculated by application of the standard normal
quantile function N −1 [·],
cA = N −1 [pA ] = −2.33

and

cB = N −1 [pB ] = −2.58

confirming the natural expectation that the obligor with lower PD
should be more bankruptcy remote; see also Appendix 6.6.

Default Baskets

29

Now we are ready for studying the interplay of the two loans in our
duo basket regarding default behavior. Let’s say the bank wants to allocate a certain amount of money to the duo basket, namely w percent
of the available amount to obligor A and (1 − w) percent to obligor B.
The question comes up which breakdown of capital by means of exposure weights wA = w and wB = (1 − w) will lead to a maximum benefit
for the lending institute. This question is the classical starting point
for balancing between risk and return of an investment. Regarding
returns we have to make a reasonable working assumption1 . For the
moment let us assume that the gross margin the bank earns with each
loan is given by the formulas
MarginA = w × (pA + 0.2 × pA (1 − pA ) + 0.005)
MarginB = (1 − w) × (pB + 0.2 × pB (1 − pB ) + 0.005).

At the one-year horizon, pA and pB are the percentage ELs and pA (1 −
pA ) and pB (1 − pB ) are the percentage default variances of loans A and
B, assuming an LGD of 100%. Therefore, we assume that loans earn a
gross margin incorporating a 100% charge on the expected loss plus a
20% charge on the variance of losses2 plus a uniform margin of 50 bps.
For w = 1 we obtain the gross margin of loan A
MarginA = pA + 0.2 × pA (1 − pA ) + 0.005 = 0.017
and for w = 0 we obtain the gross margin for loan B
MarginB = pB + 0.2 × pB (1 − pB ) + 0.005 = 0.011.
Due to the low default probabilities of the assets, the gross margin
formula approximately yields the EL scaled by 1.2 plus an absolute
offset of 50 bps. We can now calculate the portfolio profit µ as a
weighted sum of margins,
µ = µ(w) = MarginA + MarginB
= w × (pA + 0.2 × pA (1 − pA ) + 0.005)

+ (1 − w) × (pB + 0.2 × pB (1 − pB ) + 0.005).
1

In the credit market, margins or spreads are not only determined by the magnitude
of credit risk inherent in the considered credit instrument but also by its liquidity
complemented, in case of structured investments, by some ‘complexity premium’.
2
In a portfolio context, one would rather price-in the economic capital cost based
on the EC-contribution of the loan. Comparable approaches are used in RAROC
(risk-adjusted return over capital) and transfer pricing concepts.

30

Structured Credit Portfolio Analysis, Baskets & CDOs

For measuring diversification effects in our duo basket we have to choose
a risk measure sensitive to portfolio effects. The three standard measures matching this condition are EC, ESF, and UL where UL stands
for unexpected loss as a synonym for the standard deviation of the portfolio’s loss distribution. For a two-name portfolio, EC and ESF are not
the most useful measures so that in this particular case we decide in
favor of the portfolio UL (here denotes by σ) given by the square root
of the portfolio variance σ 2 w.r.t. the chosen weight w,
σ 2 = σ 2 (w) = w2 pA (1 − pA ) + (1 − w)2 pB (1 − pB )
p
+ 2 r w(1 − w) pA (1 − pA )pB (1 − pB )

(2.4)

where r = rA,B denotes the default correlation of the two obligors,


(2.5)
r = rA,B = Corr 1{CWI(1) < c(1) } , 1{CWI(1) < c(1) }
A

A

B

B

N2 [N −1 [pA ], N −1 [pB ]; ̺A,B ] − pA pB
p
=
pA (1 − pA )pB (1 − pB )

based on the normal distribution assumption we made for the obligor’s
CWIs; see also Equation (2.3). In Equation (2.5), N2 [ · , · ; ̺] denotes
the standard bi-variate normal distribution function with a correlation
of ̺. For ̺ = 10%, pA = 0.01 and pB = 0.005 we obtain a default
correlation of 74 bps. The order of magnitude of the default correlation
compared to the CWI correlation is not unusual. Note that in typical
credit model approaches, default correlations are much lower than the
correlation between underlying latent variables.
Figure 2.1 illustrates the dependence of the default correlation r =
rA,B on the PDs pA and pB and on the CWI correlation ̺ = ̺A,B .
Table 2.1 shows the corresponding numbers.
For every w ∈ [0, 1], we obtain a duo basket with a mix of a w-portion
of risk/return due to loan A and a (1−w)-portion of risk/return arising
from loan B. The portfolio’s risk/return can graphically be illustrated
by plotting the curve of points

σ(w), µ(w) 0≤w≤1
in 2-dimensional risk/return space; see Figure 2.2.

The case w = 0 leads to a duo basket consisting of loan B only, the
case w = 1 is a basket with loan A only. Obviously, these cases are

Default Baskets

31

FIGURE 2.1:

Default correlation for different levels of CWI correlation and pairs of PDs
not interesting because one can hardly call a single asset a basket or
portfolio. For 0 < w < 1, both assets are contributing to the duo
basket’s performance. With w increasing from 0 to 1 we move from
the B-only portfolio to the A-only portfolio along the curve plotted in
Figure 2.2. An interesting point on this curve is the duo basket where
the portfolio UL attains its minimum. In line with classical portfolio
theory we call this portfolio the minimum variance portfolio.
Calculating the derivative of the portfolio variance (i.e., UL2 ) w.r.t.
the weight w based on Equation (2.4),
∂ 2
σ (w) = 2 w pA (1 − pA ) − 2 (1 − w) pB (1 − pB ) +
∂w

32

Structured Credit Portfolio Analysis, Baskets & CDOs

TABLE 2.1:

Default correlation as a function of CWI correlation and
default probabilities

p
+ 2 r (1 − 2w) pA (1 − pA )pB (1 − pB ),

we obtain (∂/∂w)σ 2 (w) = 0 for the minimum variance weight
wmin =
pB (1 − pB ) − r

p

(2.6)

pA (1 − pA )pB (1 − pB )
p
pA (1 − pA ) + pB (1 − pB ) − 2 r pA (1 − pA )pB (1 − pB )

yielding wmin = 0.33 if we insert pA , pB , and r into Equation (2.6).

In Figure 2.2, one can see that the minimum variance portfolio corresponding to w = 0.33 offers more return at less risk (if risk is identified
with volatility) than the single asset B. The benefit of w = 0.33 (basket) compared to w = 0 (single asset B) is due to the diversification
effect arising from investing in two instead of in one single asset only.
We also see that the diversification potential depends on the correlation of the assets in the basket. In Figure 2.2, we draw a return level
(horizontal line) at µ = 150 bps. The lower the correlation of the two
obligors the lower the risk (in terms of UL) of the 2-asset portfolio
and the higher the diversification benefit. If we consider the basket
exclusively from a portfolio-UL perspective, then the perfect correlation case (̺ = 1) is the worst situation one can obtain because the risk
corresponding to a return of 150 bps attains its maximum. Increasing

Default Baskets

33

FIGURE 2.2:

Different duo baskets consisting of w-mixed loans A
and B in risk/return space

the weight w from 0 to 1 yields the dashed (̺ = 1) or solid (̺ = 0.1)
line or curve, respectively, in Figure 2.2 starting in asset B (w = 0) and
ending in asset A (w = 1). In the perfect correlation case, we have no
diversification benefit at all in the basket. Due to the same argument,
the best possible diversification benefit can be achieved if we combine
independent3 assets in a basket.
Figure 2.2 is well-known in portfolio theory and everyone involved
in risk management has seen comparable pictures in the literature.
Nevertheless, the duo basket case study will help us in the sequel to
explore techniques for basket analysis in an easy ‘test environment’
before we turn our attention to more complicated portfolios.

3

Note that based on our model setup we always assume non-negative CWI correlations. We do not allow for negative correlations leading to offsets in terms of
hedging.

34

2.2

Structured Credit Portfolio Analysis, Baskets & CDOs

First- and Second-to-Default Modeling

Based on our duo basket example from Section 2.1, we can make some
first steps in typical basket products. In Chapter 3, we will see that
so-called equity investors in a CDO take the first loss of some reference
portfolio consisting of credit-risky instruments. In the same way, the
first-to-default buyer/taker in a basket structure bears the loss arising
from the first default in the basket. Analogously, the second-to-default
refers to the loss caused by the second default in the basket. Later, we
will investigate first- and second-to-defaults regarding their dependence
on the chosen time horizon, hereby relying on dependent default times.
For the moment, we restrict ourselves to the one-year horizon in order
to keep the exposition simple and reduced to certain aspects of these
products. As a consequence, we suppress the time index for CWIs.
For the duo basket, the question arises what first- and second-todefault means in terms of event probabilities and corresponding losses
in the context of our example. In the sequel, we systematically investigate this question. First of all, we define some notation by writing


(1)
p1st = P 1{CWIA < cA } + 1{CWIB < cB } > 0
(2.7)

for the one-year first-to-default probability and


(1)
p2nd = P 1{CWIA < cA } × 1{CWIB < cB } > 0

(2.8)
(1)

for the one-year second-to-default probability. Note that p1st is the
(1)
hitting probability of the basket, whereas p2nd is the wipe-out probability
of the basket. Equations (2.7) and (2.8) can also be written in terms
of sets (i.e., ‘events’ is the language of probability theory) as


(1)
p1st = P {CWIA < cA } ∪ {CWIB < cB }
(2.9)


(1)
p2nd = P {CWIA < cA } ∩ {CWIB < cB }
(2.10)
(1)

(1)

Let us now calculate p1st and p2nd for our duo basket example.

2.2.1 Proposition The one-year first-to-default probability equals
(1)

p1st =

Z∞

−∞


gpA ,̺ (y) + gpB ,̺ (y)[1 − gpA ,̺ (y)] dN (y)

(2.11)

Default Baskets

35

where the conditional one-year PD of A (analogously for B) is given by
h c − √̺ y i
A
gpA ,̺ (y) = N √
.
(2.12)
1−̺
Here, N [·] denotes the standard normal distribution function and
cA = N −1 [pA ]
represents the default point or default threshold of asset A.
Proof. First, let us recall the derivation of the conditional PD gpA ,̺ (y),
gpA ,̺ (y) = P[1{CWIA < cA } = 1 | Y = y]
p
√
= P[ ̺ Y + 1 − ̺ εA < cA | Y = y]
√
h
i
cA − ̺ Y
√
= P εA <
|Y =y
1−̺
h c − √̺ y i
A
= N √
1−̺
(see also Appendices 6.6 and 6.7). Then, Equation (2.11) can be explained as follows. The first summand is the likelihood that A defaults,
the second summand is the likelihood that B defaults and A survives.
(1)
In set-theoretic notation,4 the formula for p1st reflects the equation
{CWIA < cA } ∪ {CWIB < cB } =


= {CWIA < cA } ∪ {CWIB < cB }\{CWIA < cA }

where M \N addresses all elements in M which are not in N . 2
Equation (2.12) can be more generally described in the context of
Bernoulli mixture models; see [58], [51], Chapter 2 in [25] and Appendices 6.6 and 6.7. We will come back to this and related formulas again
in the following sections.
2.2.2 Proposition The one-year second-to-default probability equals
(1)

p2nd

Z∞
=
gpA ,̺ (y)gpB ,̺ (y) dN (y).
−∞

4

See also Figure 2.41 for a comparable situation with three events.

(2.13)

36

Structured Credit Portfolio Analysis, Baskets & CDOs

Proof. Conditional on Y = y the default probabilities multiply for
the joint default probability due to conditional independence in the
mixture model. As usual in mixture models, we have to integrate the
product of conditional PDs w.r.t. the mixing variable Y . Again, we
could have worked with a set-theoretic (event-based) approach as in
the proof of Proposition 2.2.1. 2
2.2.3 Remark The one-year second-to-default probability coincides
with the joint default probability (JDP) of the two assets defined by
(1)

JDPA,B = JDPA,B = P[1{CWIA < cA } = 1, 1{CWIB < cB } = 1] (2.14)

=

2π

p

1
1−

̺2

−1
N −1
Z (pA ) N Z (pB )

1

2

2

2

e− 2 (xA −2̺ xA xB +xB )/(1−̺ ) dxA dxB

−∞

−∞

based on the joint normal distribution of the obligor’s CWIs in line
with Equation (2.3).
In explicit numbers, we obtain in our example
(1)

• p1st = 0.0149 for the one-year first-to-default probability and
(1)

• p2nd = 0.0001 for the one-year second-to-default probability.
As a cross-reference, we can check that
(1)

(1)

p1st + p2nd = pA + pB .
(1)

(2.15)

(1)

Given p1st and p2nd we can now easily calculate the one-year EL and
other risk quantities, incorporating LGDs of the underlying assets and
the exposure weight w in our duo basket.
Before we close this section we want to briefly comment on the influ(1)
(1)
ence of PDs and the correlation impact on p1st and p2nd . Figures 2.3 and
(1)
(1)
2.4 illustrate the dependence of p1st and p2nd on the CWI correlation ̺.
Here, we make an important structural observation based on Figures
2.3 and 2.4.
2.2.4 Remark The first-to-default probability attains its maximum in
case of zero CWI correlation, whereas the second-to-default probability
attains its maximum in case of perfect correlation.

Default Baskets

FIGURE 2.3:

37

(1)

Influence of the CWI correlation on p1st

Later in this book we will re-discover this fact and interpret it by saying that an investor taking the first default or loss in a basket or CDO
is in a worst case scenario if assets default in a completely independent
way because that makes defaults in a basket most unpredictable. In
contrast, a second-to-default event will only take place if both assets
default. In addition, the higher the CWI correlation the higher the
probability of a joint default event; see Figure 2.5. In the extreme case
of perfect correlation, both default events are linearly related leading
(1)
to the highest possible value of p2nd . Again we have parallels to CDOs:
investors in senior tranches suffer from high correlations because they
make tail events (joint defaults) more likely.
For reasons of completeness we briefly discuss the extreme cases ̺ = 0
and ̺ = 1 in terms of formulas.
• ̺ = 0: According to Equation (2.12), conditioning on Y has no
effect in case of zero correlation. Equations (2.11) and (2.13) then
yield
(1)
p1st = pA + pB (1 − pA ) = 0.01495
(1)

p2nd = pA × pB = 0.00005.

38

Structured Credit Portfolio Analysis, Baskets & CDOs

FIGURE 2.4:

(1)

Influence of the CWI correlation on p2nd

In other words, the zero correlation case corresponds to a situation where the systematic risk component is switched-off completely.
• ̺ = 1: In this case, idiosyncratic deviations from the systematic
factor Y in Equation (2.3) are no longer possible. Then, we can
replace CWIA and CWIB in Equation (2.14) by Y such that
JDPA,B = P[1{Y < cA } = 1, 1{Y

< cB }

= 1]

= P[Y < min{cA , cB }]


= N min{N −1 [pA ], N −1 [pB ]}
= min[pA , pB ].
(1)

Therefore, we have p2nd = min[pA , pB ]. Since Equation (2.15)
(1)
must be fulfilled, we can conclude that p1st = max[pA , pB ].
We will come back to these and related aspects later in the text when
we consider baskets and CDOs over multi-year periods rather than with
respect to the one-year horizon. Typically, baskets and CDOs have
multi-year terms. A standard maturity for ‘real life’ baskets is 3 or 5

Default Baskets

FIGURE 2.5:

39

Joint default probability (JDP) as a function of CWI

correlation
years so that we are motivated to leave the one-year horizon perspective
behind and turn our attention to time horizons longer than a year.
In order to incorporate the time dimension, we have to introduce a
concept for modeling the term structure of default probabilities. This
is exercised in the following section, which then directly will lead us to
dependent default times in Section 2.5.

2.3

Derivation of PD Term Structures

In this section, we explain the calibration of PD term structures by
means of an example based on published data from Standard and Poor’s
(S&P) [108]. Note that PD term structures sometimes are also called
credit curves, just to mention another keyword for the same object in
order to make any search in the literature easier for interested readers.
Note that, as long as nothing else is said, time in this section is

40

Structured Credit Portfolio Analysis, Baskets & CDOs

measured in years.
Typically, every bank has its own way to calibrate credit curves. We
proceed in our exposition in three steps. First, we calibrate a so-called
generator or Q-matrix Q w.r.t. an average one-year migration matrix
from S&P, describing migrations of a continuous-time Markov chain
at ‘infinitesimal small’ time intervals. Next, we calculate the Markov
PD term structure generated by Q and compare the result with empirical PD term structures from S&P. We will find what other people before us already discovered: although time-homogeneous Markov
chains are very popular in credit risk modeling, their ability to fit empirical term structures is limited to some extent. Therefore, we leave
the time-homogeneous Markov approach behind and turn our attention to non-homogeneous Markov chains in continuous time. We will
see that this approach is more fruitful than our first attempt. The
empirical term structures published by S&P can be approximated by
a non-homogeneous Markov chain with surprising quality; see [29].

2.3.1

A Time-Homogeneous Markov Chain Approach

Time-homogeneous Markov chain approaches for the calibration of
PD term structures are well-known in the literature and have been
applied over and over again by financial practitioners; see Israel,
Rosenthal, and Wei [61], Jarrow, Lando, and Turnbull [62],
and Kreinin and Sidelnikova [70], just to mention a few examples
in the literature. In these papers, one finds several alternatives for
the calibration of continuous-time Markov chains to a given one-year
migration matrix. For our examples in the sequel, we work with the
so-called diagonal adjustment approach discussed in [70] and apply it
to S&P data; see also [26]. Alternative approaches can be worked out
analogously.
In Section 1.1 we introduced the notion for PD term structures in
(1.2) for some obligor i by the curve
(t)

(t)

(t)

(pi )t≥0 = (P[CWIi < ci ])t≥0 .
(t)

In other words, pi is the probability that obligor i defaults within the
time interval [0, t], and the curve of these cumulative default probabilities, naturally increasing with increasing time, constitutes the PD term
structure of the considered obligor. Going back to Table 1.1 in Section

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41
(t)

(t)

1.1, we find that one-year default probabilities pi and pj for differently rated obligors i and j are naturally different so that we expect to
see the same dependence of PD term structures on ratings of the corresponding obligors not only at the one-year horizon but over the whole
time axis. In fact, we will define term structures in a rating-monotonic
way such that the credit curve of obligor i is above the credit curve
of obligor j at all times if obligor i has a higher one-year PD (corresponding to a worse rating) than obligor j; see also Figure 2.13 and the
discussion in the corresponding section.
As mentioned at the beginning of this section, we now provide a
step-by-step calibration of credit curves based on the so-called diagonal
adjustment approach to time-continuous homogeneous Markov chains.
In practical applications, one can, in principal, follow a comparable approach but has to make sure that calibrated credit curves comply with
the multi-year default history of the bank’s portfolio. We will come
back to this point later in this section when we compare the calibrated
credit curves with empirically observed cumulative default frequencies.
Another difficulty we see in practice in this context are certain client
segments, e.g., private clients, for which migration history cannot as
easily build up as it is the case for corporate clients where balance
sheet data flows into the bank at least once a year. For such segments,
PD term structures are more difficult to obtain and have to be derived
from historical multi-year defaults rather than from estimates based
on credit migration data. A last argument worthwhile to be mentioned
is the discussion about how realistic the Markov assumption for PD
evolution over time possibly can be. Here, we clearly look at the problem through the ‘glasses of practitioners’. During recent years our jobs
forced us to look at various approaches in the context of PD term structure calibration. Without going into details, we found, although the
Markov assumption is somewhat questionable, that Markov approaches
can yield fairly well approximations to historically observed cumulative
default frequencies. In Section 2.3.2, we will see an example of a Markov
approach that fits empirical data with outstanding quality.
Rating agencies publish discrete credit curves based on cohorts of historically observed default frequencies annually, [108] is an example for
such an agency report. The problem with historically observed cohorts
is that the resulting multi-year PDs have a tendency to look kind of
‘saturated’ at longer horizons due to lack-of-data problems. Comparing corporate bond default reports from 5 years ago with reports as of

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Structured Credit Portfolio Analysis, Baskets & CDOs

TABLE 2.2:

Modified S&P average one-year migration matrix

today, one certainly recognizes major improvements on the data side.
For ratings containing many clients (like BBB for instance), curves
are smoother and do not imply zero forward PDs after a few years.
Nevertheless, data quality still needs improvements in order to rely
on historical data purely without ‘smoothing’ by some suitable model.
The following Markov chain approach is an elegant way to overcome
this problem and to generate continuous-time credit curves.
Let us now start with the adjusted average one-year migration matrix
M = (mij )i,j=1,...,8 shown in Table 2.2 from S&P (see [108], Table 9).
As said, we overcome the zero default observation problem for AAAratings by assuming an 0.2 bps default probability for AAA-rated customers. In addition, the matrix in Table 2.2 has been row-normalized
in order to force row sums to be equal to 1 such that M is a stochastic
matrix.
Next, we need the following theorem based on a condition, which
will be fulfilled in most of the cases, although we saw a few examples
of migration matrices (typically calibrated ‘point-in-time’ instead of
‘through-the-cycle’) for which the condition was hurt for certain rating
classes. We denote by I the identity matrix and by N the number of
credit states.
2.3.1 Theorem If a migration matrix M = (mij )i,j=1,...,N is strictly
diagonal dominant, i.e., mii > 12 for every i, then the log-expansion
Q̃n

n
X
(M − I)k
=
(−1)k+1
k
k=1

(n ∈ N)

converges to a matrix Q̃ = (q̃ij )i,j=1,...,N satisfying
1.

PN

j=1 q̃ij

= 0 for every i = 1, ..., N ;

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43

2. exp(Q̃) = M .
The convergence Q̃n → Q̃ is geometrically fast.
Proof. See Israel et al. [61]. 2
Theorem 2.3.1 leads us to Condition 1 of the following remark, which
serves, complemented by two other conditions, as definition for a special
type of square matrices known as generators or Q-matrices in Markov
chain theory; see Chapter 6 in [25].
2.3.2 Remark The generator of a time-continuous Markov chain is
given by a so-called Q-matrix Q = (qij )1≤i,j≤N satisfying the following
properties:
PN
1.
j=1 qij = 0 for every i = 1, ..., N ;
2. 0 ≤ −qii < ∞ for every i = 1, ..., N ;

3. qij ≥ 0 for all i, j = 1, ..., N with i 6= j.
For background on Markov chains we refer to the book by Noris [94].
Before we continue, let us briefly recall Theorem 2.3.1. If we calculate
the log-expansion of M only for k = 1, then we obtain
M −I
as a first-order approximation to the log-expansion of M . Because M is
a stochastic matrix, it is immediately clear that the row sums of M − I
are zero such that Condition 1 in Remark 2.3.2 naturally is fulfilled.
Conditions 2 and 3 are also obvious. Therefore, M − I is the most
simple generator we can obtain from a migration matrix M .
Now, how are generators related to migration matrices? The next
theorem, well known in Markov theory, gives an answer to this question.
2.3.3 Theorem The following conditions are equivalent (here we have
the special situation Q ∈ R8×8 ):
• Q satisfies Properties 1 to 3 in Remark 2.3.2.
• exp(tQ) is a stochastic matrix for every t ≥ 0.

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Structured Credit Portfolio Analysis, Baskets & CDOs

TABLE 2.3:

Calculation of 1st-order generator approximation M − I

TABLE 2.4:

Exponential exp(M − I) of 1st-order generator

approximation

Proof. See Noris [94], Theorem 2.1.2. 2
Above we mentioned that M − I is the most simple generator, which
can be derived from M . Let us see what we get when we calculate the
matrix exponential exp(M − I) corresponding to the situation t = 1
in Theorem 2.3.1. For this purpose, we first calculate M − I. Table
2.3 shows the result of this exercise. Next, we calculate the matrix
exponential of M − I. For this, recall that the matrix exponential is
given by the expansion
exp(A) =

∞
X
Ak
k=0

k!

for any square matrix A. Table 2.4 shows the result. We find that
exp(M − I) already shows some similarities to the one-year migration
matrix M we started with, although the approximation is very rough.
Based on this little exercise, one can imagine that with increasing k in
Theorem 2.3.1 the logarithmic expansion of M applied as argument in
the matrix-valued exponential function approaches the original matrix
M with better and better approximation quality. However, as we will
see in a moment, this does not necessarily mean that, in contrast to
M − I, the log-expansion yields a generator at all.

Default Baskets

TABLE 2.5:

Log-expansion of one-year migration matrix M

TABLE 2.6:

Approximative generator (Q-matrix) for M

45

Let us now see what we get from the log-expansion (n → ∞ in Theorem 2.3.1) rather than looking at M −I (case n = 1) only. We calculate
the log-expansion Q̃ = (q̃ij )i,j=1,...,8 of the one-year migration matrix
M = (mij )i,j=1,...,8 according to Theorem 2.3.1. This can be done with
a calculation program like Mathematica or Matlab, but can as easily also be implemented in Excel/VBA. Table 2.5 shows the resulting
matrix Q̃.
Theorem 2.3.1 guarantees that Q̃ fulfills Condition 1 of generators
listed in Remark 2.3.2. Condition 2 is also not hurt by Q̃, but Condition 3 is not fulfilled. So we see confirmed what we already indicated,
namely, that (in contrast to M − I) the log-expansion of M not necessarily yields a Q-matrix. However, there are only three migration rates
in Table 2.5 not in line with generator conditions:
• q̃AAA,B = −1 bps,
• q̃B,AAA = −1 bps,
• q̃CCC,AA = −2 bps.
Only these three entries disable Q̃ from being a generator matrix. Because these three values are very small numbers, we can follow the
diagonal adjustment approach announced already several times and set
the three negative migration rates equal to zero. In order for still being
in line with the zero row sum condition, we then need to decrease the diagonal elements of rows AAA, B and CCC by an amount compensating

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Structured Credit Portfolio Analysis, Baskets & CDOs

for the increased row sums by setting negative entries to a zero value.
The evident name diagonal adjustment for this procedure is mentioned
in [70]. As a result we obtain a generator matrix Q = (qij )i,j=1,...,8 as
shown in Table 2.6.
From Theorem 2.3.1 we know that we get back the original migration
matrix M from Q̃ by exp(Q̃). But what about getting M back from
Q? Because we manipulated Q̃ in order to arrive at a generator Q,
exp(Q) will not exactly equal M . What is the error distance between
M and exp(Q)? Because the manipulation we did is kind of negligible,
we already expect the result described in the following proposition.
2.3.4 Proposition M has an approximate Q-matrix representation
by Q. The mean-square error can be calculated as
v
u 8
uX
(mij − (exp(Q))ij )2 ≈ 0.00023.
kM − exp(Q)k2 = t
i,j=1

Proof. Just calculate the distance. 2

We conclude from Proposition 2.3.4 that we can work with Q instead
of Q̃, hereby accepting the minor approximation error.
Finding a Q-matrix representation Q for a time-discrete Markov
chain represented by a transition matrix M is called an embedding of the
time-discrete Markov chain represented by M into a time-continuous
Markov chain represented by its generator or Q-matrix Q in Markov
theory. Of course, our embedding only holds in an approximate manner
(see Proposition 2.3.4), but the error is negligibly small. Probabilists
know that the existence of such embeddings is far from being obvious,
and to some extent we have been very lucky that it worked so well with
the S&P-based migration matrix M . In [62], [61], and [70] one finds
other ways to manage such embeddings, but as already said we found
that the diagonal adjustment approach is well working in many situations. Readers interested in actually applying such embeddings in the
context of their credit risk engines should take the time to experiment
with different approaches in order to find their own ‘best way’ to deal
with the challenge of PD term structure calibrations.
For the time-homogeneous case we achieved our target. Our efforts
have been rewarded by a generator Q such that exp(Q) ≈ M . The

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credit curves can be read off from the collection of matrices (exp(tQ))t≥0
by looking at the default columns. More precisely, for any asset i in
the portfolio with rating R = R(i) we obtain
(t)

pi

= (etQ )row(R),8

(2.16)

where row(R) denotes the transition matrix row corresponding to the
given rating R. Figure 2.6 shows a chart of credit curves from t = 0 to
t = 50 years on a quarterly base, calibrated as just described.

FIGURE 2.6:

Calibrated credit curves based on a time-continuous
and time-homogeneous Markov chain approach

The curves in Figure 2.6 are kind of typical. Credit curves assigned to
subinvestment grade ratings have a tendency to slow down their growth,
because conditional on having survived for some time the chances for
further survival improve. For good ratings, we see the opposite effect.
For instance, assets with excellent credit quality have no upside potential because they are already outstanding but have a large downside
risk of potential deterioration of their credit quality over time.

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Structured Credit Portfolio Analysis, Baskets & CDOs

If we calculate quarterly forward PDs via
(t)

p̃i

(t)

=

(t−0.25)

pi − pi

(t−0.25)

1 − pi

(t ≥ 0.25)

(2.17)

we find a mean reversion effect in PD term structures. Moody’s KMV
writes in the documentation of their rating system RiskCalc for private
companies the following statement:
We noticed that obligors appear to exhibit mean reversion in
their credit quality. In other words, good credits today tend
to become somewhat worse credits over time and bad credits (conditional upon survival) tend to become better credits
over time. Default studies on rated bonds by Moodys Investors Services support this assertion, as do other studies.
We found further evidence for mean reversion in both our
proprietary public firm default databases and in the Credit
Research Database data on private firms. [85]
The homogeneous Markov approach, focussing on the aspect of mean
reversion, is in line with findings of Moody’s KMV. Mean reversion effects drive the forward or conditional PDs over time, respectively; see
Figure 2.7. Therefore, we can hope to have at least a somewhat realistic model at disposal. However, if we compare PD term structures on
a purely empirical base as published by S&P in [108] with the just calibrated term structure, we find as a disappointing surprise that curves
do not match very well; see Figure 2.8.
When we compare the homogeneous continuous-time Markov chain
(HCTMC-based) credit curves with observed multi-year default frequencies from S&P we immediately get the impression that HCTMC
model-based PDs systematically overestimate empirically observed default frequencies (exception: AAA). Here are some remarks:
• For good rating classes, AAA is the extreme case, the