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Dependent Defaults in Models of Portfolio Credit Risk
R¨diger Frey∗ u Department of Mathematics University of Leipzig Augustusplatz 10/11 D-04109 Leipzig frey@mathematik.uni-leipzig.de Alexander J. McNeil∗ Department of Mathematics Federal Institute of Technology ETH Zentrum CH-8092 Zurich mcneil@math.ethz.ch

16th June 2003

Abstract We analyse the mathematical structure of portfolio creditrisk models with particular regard to the modelling of dependence between default events in these models. We explore the role of copulas in latent variable models (the approach that underlies KMV and CreditMetrics) and use non-Gaussian copulas to present extensions to standard industry models. We explore the role of the mixing distribution in Bernoulli mixture models (the approach underlyingCreditRisk+ ) and derive large portfolio approximations for the loss distribution. We show that all currently used latent variable models can be mapped into equivalent mixture models, which facilitates their simulation, statistical fitting and the study of their large portfolio properties. Finally we develop and test several approaches to model calibration based on the Bernoulli mixture representation;we find that maximum likelihood estimation of parametric mixture models generally outperforms simple moment estimation methods. J.E.L. Subject Classification: G31, G11, C15 Keywords: Risk Management, Credit Risk, Dependence Modelling, Copulas

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Introduction

A major cause of concern in managing the credit risk in the lending portfolio of a typical financial institution is the occurrence ofdisproportionately many joint defaults of different counterparties over a fixed time horizon. Joint default events also have an an important impact on the performance of derivative securities, whose payoff is linked to the loss of a whole portfolio of underlying bonds or loans such as collaterized debt obligations (CBOs, CDOs, CLOs) or basket credit derivatives. In fact, the occurrence ofdisproportionately many joint defaults is what could be termed “extreme credit risk” in these contexts. Clearly, precise mathematical models for the loss in a portfolio of dependent credit risks are needed to adequately measure this risk. Such models are also a prerequisite for the active management of credit portfolios under risk-return considerations. Moreover, given improved availability of data on creditlosses, refined versions of current credit risk models might also be used for the determination of regulatory capital for credit risk, much as internal models are nowadays used for capital adequacy purposes in market risk management. The main goal of the present paper is to present a framework for analysing existing industry models, and various models proposed in the academic literature, withregard to the mechanisms they use to model dependence between defaults. These mechanisms are at
We wish to thank Dirk Tasche, Mark Nyfeler, Filip Lindskog and Philipp Sch¨nbucher for useful discuso sions.
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least as important in determining the overall credit loss of a portfolio under the model, as are assumptions regarding default probabilities of the individual obligors in the portfolio.In previous papers (Frey, McNeil, and Nyfeler 2001, Frey and McNeil 2002) we have shown that portfolio credit models can be subject to considerable model risk. Small changes to the structure of the model or to the model parameters describing dependence can have a large impact on the resulting credit loss distribution and in particular its tail. This is worrying because credit models are extremelydifficult to calibrate reliably, due to the relative scarcity of good data on credit losses. In our analysis of mechanisms for dependent credit events we divide existing models into two classes: latent variable models such as KMV or CreditMetrics which essentially descend from the firm-value model of Merton (Merton 1974); Bernoulli mixture models such as CreditRisk+ where default events have a...