Modelling Dependent Defaults

by Rüdiger Frey of the University of Zurich, and
Alexander J. McNeil of the Federal Institute of Technology

August 13, 2001

Abstract: We consider the modelling of dependent defaults in large credit portfolios using latent variable models (the approach that underlies KMV and CreditMetrics) and mixture models (the approach underlying CreditRisk+). We explore the role of copulas in the latent variable framework and show that for given default probabilities of individual obligors the distribution of the number of defaults in the portfolio is completely determined by the copula of the latent variables. We present results from a simulation study showing that, even for fixed asset correlations, assumptions concerning the latent variable copula can have a profound effect on the distribution of credit losses. In the mixture models defaults are conditionally independent given a set of common economic factors affecting all obligors and we explore the role of the mixing distribution of the factors in these models. In homogeneous, one-factor mixture models we find that the tail of the mixing distribution essentially determines the tail of the overall credit loss distribution. We discuss the relationship between latent variable models and mixture models and provide general conditions under which these models can be mapped into each other. Our contribution can be viewed as an analysis of the model risk associated with the modelling of dependence between individual default events.

JEL Classification: G31, G11, C15.

Keywords: Portfolio Credit Risk Models, Model Risk, Dependence Modelling, Copulas, Mixture Models.

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See also a closely related paper:

Modelling Dependent Defaults: Asset Correlations Are Not Enough!

by Rüdiger Frey of the University of Zurich,
Alexander J. McNeil of the Federal Institute of Technology, and
Mark A. Nyfeler of UBS Zurich

March 9, 2001

Introduction: In this article we focus on the latent variable approach to modelling credit portfolio losses. This methodology underlies all models that descend from Merton firm-value model (Merton 1974). In particular, it underlies the most important industry models, such as the model proposed by the KMV corporation and CreditMetrics.

In these models default of an obligor occurs if a latent variable, often interpreted as the value of the obligor's assets, falls below some threshold, often interpreted as the value of the obligor's liabilities. Dependence between default events is caused by dependence between the latent variables. The correlation matrix of the latent variables is often calibrated by developing actor models that relate changes in asset value to changes in a small number o economic actors. For further reading see papers by Koyluoglu and Hickman (1998)REF, Gordy (2000) and Crouhy, Galai, and Mark (2000).

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