Rating Based Modeling of Credit Risk: Theory and Application of Migration Matrices (Academic Press Advanced Finance)

256 Pages·2008·1.92 MB·English

by  Stefan Trueck

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08 09 10 9 8 7 6 5 4 3 2 1
To my parents and Prasheela
(S.T.)
To Svetlozar Todorov Iotov
(S.T.R)
Preface
Credit risk has become one of the most intensely studied topics in quanti-
tativefinanceinthelastdecade.Alargenumberofbooksonthetopichave
beenpublishedinrecentyears,whileontheexcellenthomepagemaintained
by Greg Gupton there are more than 1200 downloadable working papers
related to credit risk. The increased interest in modeling and management
of credit risk in academia seems only to have started in the mid-1990s.
However, due to the various issues involved, including the ability to effec-
tively apply quantitative modeling tools and techniques and the dramatic
rise of credit derivatives, it has become one of the major fields of research
in finance literature.
As a consequence of an increasingly complex and competitive finan-
cialenvironment,adequateriskmanagementstrategiesrequirequantitative
modeling know-how and the ability to effectively apply this expertise and
its techniques. Also, with the revision of the Basel Capital Accord, various
credit risk models have been analyzed with respect to their feasibility, and
a significant focus has been put on good risk-management practices with
respecttocreditrisk.AnotherconsequenceofBaselIIisthatmostfinancial
institutions will have to develop internal models to adequately determine
the risk arising from their credit exposures. It can therefore be expected
that in particular the use and application of rating based models for credit
risk will be increasing further.
On the other hand, it has to be acknowledged that rating agencies
are at the center of the subprime mortgage crisis, as they failed to pro-
vide adequate ratings for many diverse products in the credit and credit
derivativemarketslikemortgagebonds,assetbackedsecurities,commercial
papers, collateralized debt obligations, and derivative products for compa-
nies and also for financial institutions. Despite some deficiencies of the
current credit rating structure—recommendations for their improvements
are thoroughly analyzed in Crouhy et al. (2008) but are beyond the scope
of this book—overall, rating based models have evolved as an industry
standard. Therefore, credit ratings will remain one of the most important
variables when it comes to measurement and management of credit risk.
The literature on modeling and managing credit risk and credit deriva-
tives has been widely extended in recent years; other books in the area
include the excellent treatments by Ammann (2002), Arvanitis and Gre-
gory (2001), Bielecki and Rutkowski (2002), Bluhm et al. (2003), Bluhm
and Overbeck (2007b), Cossin and Pirotte (2001), Duffie and Singleton
(2003), Fabozzi (2006a,b), Lando (2004), Saunders and Allen (2002), and
Sch¨onbucher (2003), just to mention a few. However, in our opinion, so
far there has been no book on credit risk management mainly focusing
xii Preface
on the use of transition matrices, which, while popular in academia, is
even more widely used in industry. We hope that this book provides a
helpful survey on the theory and application of transition matrices for
creditriskmanagement,includingmostofthecentralissueslikeestimation
techniques, stability and comparison of rating transitions, VaR simula-
tion, adjustment and forecasting migration matrices, corporate-yield curve
dynamics, dependent migrations, and the modeling and pricing of credit
derivatives. While the aim is mainly to provide a review of the existing
literature and techniques, a variety of very recent results and new work
have also been incorporated into the book. We tried to keep the presenta-
tion thorough but also accessible, such that most of the chapters do not
require a very technical background and should be useful for academics,
regulators, risk managers, practitioners, and even students who require
an introduction or a more extensive and advanced overview of the topic.
The large number of applications and numerical examples should also help
thereadertobetteridentifyandfollowtheimportantimplementationissues
of the described models.
Intheprocessofwritingthisbook,wereceivedalotofhelpfromvarious
people in both academia and industry. First of all, we highly appreciated
feedback and comments on the manuscript by many colleagues and
friends. We would also like to thank various master, research, and PhD
students who supplied corrections or contributed their work to several of
the chapters. In particular, we are grateful to Arne Benzin, Alexander
Breusch, Jens Deidersen, Stefan Harpaintner, Jan Henneke, Matthias
Laub, Nicole Lehnert, Andreas Lorenz, Christian Menn, Jingyuan Meng,
Emrah O¨zturkmen, Peter Niebling, Jochen Peppel, Christian Schmieder,
Robert Soukup, Martin Sttzel, Stoyan Stoyanov, and Wenju Tian for their
contributions. Finally, we would like to thank Roxana Boboc and Stacey
Walker at Elsevier for their remarkable help and patience throughout the
process of manuscript delivery.
Stefan Trueck and Svetlozar T. Rachev
Sydney and Karlsruhe, August 2008
1
Introduction: Credit Risk Modeling,
Ratings, and Migration Matrices
1.1 Motivation
The aim of this book is to provide a review on theory and application of
migration matrices in rating based credit risk models. In the last decade,
rating based models in credit risk management have become very popular.
These systems use the rating of a company as the decisive variable and
not—like the formerly used structural models the value of the firm—when
it comes to evaluate the default risk of a bond or loan. The popularity is
duetothestraightforwardnessoftheapproachbutalsotothenewCapital
Accord(BaselII)oftheBaselCommitteeonBankingSupervision(2001),a
regulatory body under the Bank of International Settlements (BIS). Basel
II allows banks to base their capital requirements on internal as well as
external rating systems. Thus, sophisticated credit risk models are being
developed or demanded by banks to assess the risk of their credit port-
folio better by recognizing the different underlying sources of risk. As a
consequence, default probabilities for certain rating categories but also the
probabilities for moving from one rating state to another are important
issuesinsuchmodelsforriskmanagementandpricing.Systematicchanges
inmigrationmatriceshavesubstantialeffectsoncreditValue-at-Risk(VaR)
of a portfolio but also on prices of credit derivatives like Collaterized Debt
Obligations(CDOs).Therefore,ratingtransitionmatricesareofparticular
interest for determining the economic capital or figures like expected loss
and VaR for credit portfolios, but can also be helpful as it comes to the
pricing of more complex products in the credit industry.
This book is in our opinion the first manuscript with a main focus in
particular on issues arising from the use of transition matrices in model-
ing of credit risk. It aims to provide an up-to-date reference to the central
problems of the field like rating based modeling, estimation techniques,
stability and comparison of rating transitions, VaR simulation, adjust-
ment and forecasting migration matrices, corporate-yield curve dynamics,
dependent defaults and migrations, and finally credit derivatives modeling
and pricing. Hereby, most of the techniques and issues discussed will be
illustrated by simplified numerical examples that we hope will be helpful
2 1. Introduction: Credit Risk Modeling, Ratings, and Migration Matrices
to the reader. The following sections provide a quick overview of most of
the issues, problems, and applications that will be outlined in more detail
in the individual chapters.
1.2 Structural and Reduced Form Models
This book is mainly concerned with the use of rating based models for
credit migrations. These models have seen a significant rise in popula-
rity only since the 1990s. In earlier approaches like the classical structural
models introduced by Merton (1974), usually a stochastic process is used
to describe the asset value V of the issuing firm
dV(t)=μV(t)dt+σV(t)dW(t)
where μ and σ are the drift rate and volatility of the assets, and W(t)
is a standard Wiener process. The firm value models then price the bond
as contingent claims on the asset. Literature describes the event of default
whentheassetvaluedropsbelowacertainbarrier.Thereareseveralmodel
extensions,e.g.,byLongstaffandSchwartz(1995)orZhou(1997),including
stochastic interest rates or jump diffusion processes. However, one fea-
ture of all models of this class is that they model credit risk based on
assuming a stochastic process for the value of the firm and the term struc-
ture of interest rates. Clearly the problem is to determine the value and
volatility of the firm’s assets and to model the stochastic process driving
the value of the firm adequately. Unfortunately using structural models,
especially short-term credit spreads, are generally underestimated due to
default probabilities close to zero estimated by the models. The fact that
both drift rate and volatility of the firm’s assets may also be dependent on
the future situation of the whole economy is not considered.
The second major class of models—the reduced form models—does not
condition default explicitly on the value of the firm. They are more gen-
eralthanstructuralmodelsandassumethatanexogenousrandomvariable
drivesdefaultandthattheprobabilityofdefault(PD)overanytimeinter-
val is non-zero. An important input to determine the default probability
and the price of a bond is the rating of the company. Thus, to determine
the risk of a credit portfolio of rated issuers one generally has to consider
historical average defaults and transition probabilities for current rating
classes. The reduced form approach was first introduced by Fons (1994)
and then extended by several authors, including Jarrow et al. (1997) and
Duffie and Singleton (1999). Quite often in reduced form approaches the
migrationfromoneratingstatetoanotherismodeledusingaMarkovchain
modelwithamigrationmatrixgoverningthechangesfromoneratingstate
to another. An exemplary transition matrix is given in Table 1.1.
1.3 Basel II, Scoring Techniques, and Internal Rating Systems 3
TABLE 1.1. Average One-Year Transition Matrix of Moody’s Corporate Bond
Ratings for the Period 1982–2001
Aaa Aa A Baa Ba B C D
Aaa 0.9276 0.0661 0.0050 0.0009 0.0003 0.0000 0.0000 0.0000
Aa 0.0064 0.9152 0.0700 0.0062 0.0008 0.0011 0.0002 0.0001
A 0.0007 0.0221 0.9137 0.0546 0.0058 0.0024 0.0003 0.0005
Baa 0.0005 0.0029 0.0550 0.8753 0.0506 0.0108 0.0021 0.0029
Ba 0.0002 0.0011 0.0052 0.0712 0.8229 0.0741 0.0111 0.0141
B 0.0000 0.0010 0.0035 0.0047 0.0588 0.8323 0.0385 0.0612
C 0.0012 0.0000 0.0029 0.0053 0.0157 0.1121 0.6238 0.2389
D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
Besides the fact that they allow for realistic short-term credit spreads,
reduced form models also give great flexibility in specifying the source of
default. We will now give a brief outlook on several issues that arise when
migration matrices are applied in rating based credit modeling.
1.3 Basel II, Scoring Techniques, and Internal
Rating Systems
As mentioned before, due to the new Basel Capital Accord (Basel II) most
oftheinternationaloperatingbanksmaydeterminetheirregulatorycapital
based on an internal rating system (Basel Committee on Banking Super-
vision, 2001). As a consequence, a high fraction of these banks will have
ratings and default probabilities for all loans and bonds in their credit
portfolio.Therefore,Chapters2and3ofthisbookwillbededicatedtothe
newBaselCapitalAccord,ratingagencies,andtheirmethodsandareview
on scoring techniques to derive a rating. Regarding Basel II, the focus
will be set on the internal ratings based (IRB) approach where the banks
are allowed to use the results of their own internal rating systems. Conse-
quently, it is of importance to provide a summary on the rating process of
a bank or the major rating agencies. As will be illustrated in Chapter 6,
internal and external rating systems may show quite a different behavior
in terms of stability of ratings, rating drifts, and time homogeneity.
While Weber et al. (1998) were the first to provide a comparative study
ontheratingandmigrationbehavioroffourmajorGermanbanks,recently
more focus has been set on analyzing rating and transition behavior also
in internal rating systems (Bank of Japan, 2005; Euopean Central Bank,
2004). Recent publications include, for example, Engelmann et al. (2003),
Aratenetal.(2004),BaselCommitteeonBankingSupervision(2005),and
4 1. Introduction: Credit Risk Modeling, Ratings, and Migration Matrices
Jacobson et al. (2006). Hereby, Engelmann et al. (2003) and the Basel
Committee on Banking Supervision (2005) are more concerned with the
validation, respectively, classification of internal rating systems. Araten
et al. (2004) discuss issues in evaluating banks’ internal ratings of borrow-
erscomparingtheex-postdiscriminationpowerofaninternalandexternal
rating system. Jacobson et al. (2006) investigate internal rating systems
and differences between the implied loss distributions of banks with equal
regulatoryriskprofiles.Weprovidedifferenttechnologiestocomparerating
systems and estimated migration matrices in Chapters 2 and 7.
Another problem for internal rating systems arises when a continuous-
time approach is chosen for modeling credit migrations. Since for bank
loans, balance sheet data or rating changes are reported only once a year,
there is no information on the exact time of rating changes available.
While discrete migration matrices can be transformed into a continuous-
time approach, Israel et al. (2000) show that for several cases of discrete
transition matrices there is no “true” or valid generator. In this case, only
an approximation of the continuous-time transition matrix can be chosen.
Possible approximation techniques can be found in Jarrow et al. (1997),
KreininandSidelnikova(2001),orIsraeletal.(2000)andwillbediscussed
in Chapter 5.
1.4 Rating Based Modeling and the Pricing
of Bonds
A quite important application of migration matrices is also their use for
determining the term structure of credit risk. In 1994, Fons (1994) devel-
opedareducedformmodeltoderivecreditspreadsusinghistoricaldefault
ratesandarecoveryrateestimate.Heillustratedthatthetermstructureof
credit risk, i.e., the behavior of credit spreads as maturity varies, depends
on the issuer’s credit quality, i.e., its rating. For bonds rated investment
grade, the term structures of credit risk have an upward sloping struc-
ture. The spread between the promised yield-to-maturity of a defaultable
bond and a default-free bond of the same maturity widens as the matu-
rity increases. On the other hand, speculative grade rated bonds behave in
the opposite way: the term structures of the credit risk have a downward-
slopingstructure.Fons(1994)wasabletoprovidealinkbetweentherating
of a company and observed credit spreads in the market.
However, obviously not only the “worst case” event of default has influ-
enceonthepriceofabond,butalsoachangeintheratingofacompanycan
affect prices of the issued bond. Therefore, with CreditMetrics JP Morgan
provides a framework for quantifying credit risk in portfolios using histor-
ical transition matrices (Gupton et al., 1997). Further, refining the Fons
model, Jarrow et al. (1997) introduced a discrete-time Markovian model
1.5 Stability of Transition Matrices 5
to estimate changes in the price of loans and bonds. Both approaches
incorporatepossibleratingupgrades,stableratings,andratingdowngrades
in the reduced form approach. Hereby, for determining the price of credit
risk, both historical default rates and transition matrices are used. The
model of Jarrow et al. (1997) is still considered one of the most important
approaches as it comes to the pricing of bonds or credit derivatives and
will be described in more detail in Chapter 8.
Both the CreditMetrics framework and Markov chain approach heavily
relyontheuseofadequatecreditmigrationmatricesaswillbeillustratedin
Chapters 4 and 5. Further, the application of migration matrices for deriv-
ing cumulative default probabilities and the pricing of credit derivatives
will be illustrated in Chapter 11.
1.5 Stability of Transition Matrices, Conditional
Migrations, and Dependence
Asmentionedbefore,historicaltransitionmatricescanbeusedasaninput
for estimating portfolio loss distributions and credit VaR figures. Unfor-
tunately, transition matrices cannot be considered to be constant over a
longer time period; see e.g., Allen and Saunders (2003) for an extensive
review on cyclical effects in modeling credit risk measurement. Further,
migrations of loans in internal bank portfolios may behave differently than
the transition matrices provided by major rating agencies like Moody’s or
Standard&Poor’swouldsuggest(Kru¨geretal.,2005;Weberetal.,1998).
Nickell et al. (2000) show that there is quite a big difference between tran-
sition matrices during an expansion of the economy and a recession. The
resultsareconfirmedbyBangiaetal.(2002)whosuggestthatforriskman-
agement purposes it might be interesting not only to simulate the term
structure of defaults but to design stress test scenarios by the observed
behavior of default and transition matrices through the cycle. Jafry and
Schuermann(2004)investigatethemobilityinmigrationbehaviorusing20
years of Standard & Poor’s transition matrices and find large deviations
through time. Kadam and Lenk (2008) report significant heterogeneity in
default intensity, migration volatility, and transition probabilities depend-
ing on country and industry effects. Finally, Trueck and Rachev (2005)
show that the effect of different migration behavior on exemplary credit
portfolios may lead to substantial changes in expected losses, credit VaR,
or confidence sets for probabilities of default (PDs). During a recession
period of the economy the VaR for one and the same credit portfolio can
be up to eight times higher than during an expansion of the economy.
As a consequence, following Bangia et al. (2002), it seems necessary
to extend transition matrix application to a conditional perspective using
additionalinformationontheeconomyorevenforecasttransitionmatrices