Chapter 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 due to the straightforwardness of the approach but also to the new Capital Accord (Basel II) of the Basel Committee on Banking Supervision (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 portfolio 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 issues in such models for risk management and pricing. Systematic changes in migration matrices have substantial effects on credit Value-at-Risk (VaR) of a portfolio but also on prices of credit derivatives like Collaterized Debt Obligations (CDOs). Therefore, rating transition matrices are of particular 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 modeling 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, adjustment 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 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 popularity 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

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 when the asset value drops below a certain barrier. There are several model extensions, e.g., by Longstaff and Schwartz (1995) or Zhou (1997), including stochastic interest rates or jump diffusion processes. However, one feature 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 structure 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 general than structural models and assume that an exogenous random variable drives default and that the probability of default (PD) over any time interval 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 migration from one rating state to another is modeled using a Markov chain model with a migration matrix governing the changes from one rating state to another. An exemplary transition matrix is given in Table 1.1.

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 of the international operating banks may determine their regulatory capital based on an internal rating system (Basel Committee on Banking Supervision, 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, Chapters 2 and 3 of this book will be dedicated to the new Basel Capital Accord, rating agencies, and their methods and a review 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. Consequently, 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 on the rating and migration behavior of four major German banks, recently more focus has been set on analysing rating and transition behavior also in internal rating systems (Bank of Japan, 2005; European Central Bank, 2004). Recent publications include, for example, Engelmann et al. (2003), Araten et al. (2004), Basel Committee on Banking Supervision (2005), and 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 borrowers comparing the ex-post discrimination power of an internal and external rating system. Jacobson et al. (2006) investigate internal rating systems and differences between the implied loss distributions of banks with equal regulatory risk profiles. We provide different technologies to compare rating 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), Kreinin and Sidelnikova (2001), or Israel et al. (2000) and will be discussed 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) developed a reduced form model to derive credit spreads using historical default rates and a recovery rate estimate. He illustrated that the term structure of 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 structure. The spread between the promised yield-to-maturity of a defaultable bond and a default-free bond of the same maturity widens as the maturity increases. On the other hand, speculative grade rated bonds behave in the opposite way: the term structures of the credit risk have a downward-sloping structure. Fons (1994) was able to provide a link between the rating of a company and observed credit spreads in the market.

However, obviously not only the “worst case” event of default has influence on the price of a bond, but also a change in the rating of a company can affect prices of the issued bond. Therefore, with CreditMetrics JP Morgan provides a framework for quantifying credit risk in portfolios using historical transition matrices (Gupton et al., 1997). Further, refining the Fons model, Jarrow et al. (1997) introduced a discrete-time Markovian model to estimate changes in the price of loans and bonds. Both approaches incorporate possible rating upgrades, stable ratings, and rating downgrades 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 rely on the use of adequate credit migration matrices it will be illustrated in Chapters 4 and 5. Further, the application of migration matrices for deriving 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

As mentioned before, historical transition matrices can be used as an input for estimating portfolio loss distributions and credit VaR figures. Unfortunately, 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’s would suggest (Krüger et al., 2005; Weber et al., 1998). Nickell et al. (2000) show that there is quite a big difference between transition matrices during an expansion of the economy and a recession. The results are confirmed by Bangia et al. (2002) who suggest that for risk management 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) investigate the mobility in migration behavior using 20 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 depending 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 additional information on the economy or even forecast transition matrices using revealed dependencies on macroeconomic indices and interest rates. Based on the cyclical behavior of migration, the literature provides some approaches to adjust, re-estimate, or change migration matrices according to some model for macroeconomic variables or observed empirical prices. Different approaches suggest conditioning the matrix based on macroeconomic variables or forecasts that will affect future credit migrations. The first model developed to explicitly link business cycles to rating transitions was in the 1997 CreditPortfolioView (CPV) by Wilson (1997a, b). Kim (1999) develops a univariate model whereby ratings respond to business cycle shifts. The model is extended to a multifactor credit migration model by Wei (2003) while Cowell et al. (2007) extend the model by replacing the normal with an α-stable distribution for modeling the risk factors. Nickell et al. (2000) propose an ordered probit model which permits migration matrices to be conditioned on the industry, the country domicile, and the business cycle. Finally, Bangia et al. (2002) provide a Markov switching model, separating the economy into two regimes. For each state of the economy—expansion and contraction—a transition matrix is estimated such that conditional future migrations can be simulated based on the state of the economy.

To approach these issues, the a major concern is to be able to judge whether one has an adequate model or forecast for a conditional or unconditional transition matrix. It raises the question: What can be considered to be a “good” model in terms of evaluating migration behavior or risk for a credit portfolio? Finally, the question of dependent defaults and credit migration has to be investigated. Knowing the factors that lead to changes in migration behavior and quantifying their influence may help a bank improve its estimates about expected losses and Value-at-Risk. These issues will be more thoroughly investigated in Chapters 8, 9 and 10.

1.6. Credit Derivative Pricing

As mentiond before, credit migration matrices also play a substantial role in the modeling and pricing of credit derivatives, in particular collaterized debt obligations (CDOs). The market for credit derivatives can be considered as one of the fastest growing in the financial industry. The importance of transition matrices for modeling credit derivatives has been pointed out in several studies. Jarrow et al. (1997) use historical transition matrices and observed market spreads to determine cumulative default probabilities and credit curves for the pricing of credit derivatives. Bluhm (2003) shows how historical one-year migration matrices can be used to determine cumulative default probabilities. This so-called calibration of the credit curve can then be used for the rating of cash-flow CDO tranches.

In recent publications, the effect of credit migrations on issues like credit derivative pricing and rating is examined by several authors, by Bielecki et al. (2003), Hrvatin et al. (2006), Hurd and Kuznetsov (2005), and Picone (2005) among others. Hrvatin et al. (2006) investigate CDO near-term rating stability of different CDO tranches depending on different factors. Next to the granularity of the portfolio, in particular, credit migrations in the underlying reference portfolio are considered to have impact on the stability of CDO tranche ratings. Pointing out the influence of changes in credit migrations, Picone (2005) develops a time-inhomogeneous intensity model for valuing cash-flow CDOs. His approach explicitly incorporates the credit rating of the firms in the collateral portfolio by applying a set of transition matrices, calibrated to historical default probabilities. Finally, Hurd and Kuznetsov (2005) show that credit basket derivatives can be modeled in a parsimonious and computationally efficient manner within the affine Markov chain framework for multifirm credit migration while Bielecki et al. (2003) concentrate on dependent migrations and defaults in a Markovian market model and the effects on the valuation of basket credit derivatives. Both approaches heavily rely on the choice of an adequate transition matrix as a starting point.

Overall, the importance of credit transition matrices in modeling credit derivatives cannot be denied. Therefore, Chapter 11 is mainly dedicated to the application of migration matrices in the process of calibration, valuation, and pricing of these products.

1.7. Chapter Outline

Chapters 2, 3 and 4 provide a rather broad view and introduction to rating based models in credit risk and the new Basel Capital Accord. Chapter 2 aims to give a brief overview on rating agencies, rating systems, and an exemplary rating process. Then different scoring techniques like discriminant analysis, logistic regression, and probit models are described. Further, a section is dedicated to the evaluation of rating systems by using cumulative accuracy profiles and accuracy ratios. Chapter 3 then illustrates the new capital accord of the Basel Committee on Banking Supervision. Since 1988, when the old accord was published, risk management practices, supervisory approaches, and financial markets have undergone significant transformations. Therefore, the new proposal contains innovations that are designed to introduce greater risk sensitivity into the determination of the required economic capital of financial institutions. This is achieved by taking into account the actual riskiness of an obligor by using ratings provided by external rating agencies or internally estimated probabilities of default. In Chapter 4 we review a number of models for credit risk that rely heavily on company ratings as input variables. The models are focused on risk management and give different approaches to the determination of the expected losses, unexpected losses, and Value-at-Risk. We will focus on rating based models including the reduced-form model suggested by Fons (1994) and extensions of the approach with respect to default intensities. Then we will have a look at the industry models CreditMetrics and CreditRiskPlus. In particular the former also uses historical transition matrices to determine risk figures for credit portfolios.

Chapters 5, 6 and 7 are dedicated to various issues of rating transitions and the Markov chain approach in credit risk modeling. Chapter 5 introduces the basic ideas of modeling migrations with transition matrices. We further compare discrete and continuous-time modeling of rating migrations and illustrate the advantages of the continuous-time approach. Further, the problems of embeddability and identification of generator matrices are examined and some approximation methods for generator matrices are described. Finally, a section is dedicated to simulations of rating transitions using discrete time, continuous-time, and nonparametric techniques. In Chapter 6 we focus on time-series behavior and stability of migration matrices. Two of the major issues to investigate are time homogeneity and Markov behavior of rating migrations. Generally, both assumptions should be treated with care due to the influence of the business cycle on credit migration behavior. We provide a number of empirical studies examining the issues and further yielding results on rating drifts, changes in Value-at-Risk figures for credit portfolios, and on the stability of probability of default estimated through time. Chapter 7 is dedicated to the study of measures for comparison of rating transition matrices. A review classical matrix norms is given before indices based on eigenvalues and eigenvectors, including a recently proposed mobility metric, are described. The rest of the chapter then proposes some criteria that should be helpful to compare migration matrices from a risk perspective and suggests new risk-adjusted indices for measuring those differences. A simple simulation study on the adequacy of the different measures concludes the chapter.

Chapters 8 and 9 deal with determining risk-neutral and conditional migration matrices. While the former are used for the pricing of credit derivatives based on observed market probabilities of defaults, the latter focus on transforming average historical transition matrices by taking into account information on macroeconomic variables and the business cycle. In Chapter 8 we start with a review of the seminal paper by Jarrow et al. (1997) and then examine a variety of adjustment techniques for migration matrices. Hereby, methods based on a discrete and continuous-time framework as well as a recently suggested adjustment technique based on economic theory are illustrated. For each of the techniques we give numerical examples illustrating how it can be conducted. Chapter 9 deals with conditioning and forecasting transition matrices based on business cycle indicators. Hereby, we start with the approach suggested in the industry model CreditPortfolioView and then review techniques that are based on factor model representations and other techniques. An empirical study comparing several of the techniques concludes the chapter.

Chapters 10 and 11 deal with more recent issues on modeling dependent migrations and the use of transition matrices for credit derivative pricing. In Chapter 10 we start with an illustration on how dependency between individual loans may substantially affect the risk for a financial institution. Then different models for the dependence structure with a focus on copulas are suggested. We provide a brief review on the underlying ideas for modeling dependent defaults and then show how a framework for modeling dependent credit migrations can be developed. In an empirical study on dependent migrations we show that both the degree of dependence entering the model as well as the choice of the copula significantly affects determined risk figures for credit portfolios. Chapter 11 finally provides an overview on the use of transition matrices for the pricing of credit derivatives. The chapter illustrates how derived credit curves can be used for the pricing of single-named credit derivatives like, e.g., credit default swaps and further shows the use of migration matrices for the pricing of more complex products like collaterized debt obligations. Finally we also have a look at the pricing of step-up bonds that have been popular in particular in the Telecom sector.