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VaR and Expected Shortfall in Portfolios of Dependent Credit Risks: Conceptual and Practical Insights

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

January 23, 2002

Abstract: In the first part of this paper we address the non-coherence of value-at-risk (VaR) as a risk measure in the context of portfolio credit risk, and highlight some problems which follow from this theoretical deficiency. In particular, a realistic demonstration of the non-subadditivity of VaR is given and the possibly nonsensical consequences of VaR-based portfolio optimisation are shown. The second part of the paper discusses VaR and expected shortfall estimation for large balanced credit portfolios. All standard industry models (CreditMetrics, KMV, CreditRisk+ ) are presented as Bernoulli mixture models to facilitate their direct comparison. For homogeneuous groups it is shown that measures of tail risk for the loss distribution may be approximated in large portfolios by analysing the tail of the mixture distribution in the Bernoulli representation. An example is given showing that, for portfolios of lower quality, choice of model has some impact on measures of extreme risk.

JEL Classification: G31, G11, C15.

Keywords: risk measures, value-at-risk, coherence, expected shortfall, portfolio credit risk models, Bernoulli mixture models.

Published in: Journal of Banking & Finance, Vol. 26, No. 7, (July 2002), pp. 1317-1334.

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