VaR and expected shortfall in portfolios of dependent credit risks: Conceptual and practical insights

Rüdiger Frey*, Alexander J. McNeil

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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, Credit-Risk +) are presented as Bernoulli mixture models to facilitate their direct comparison. For homogeneous 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.

Original languageEnglish
Pages (from-to)1317-1334
Number of pages18
JournalJournal of Banking and Finance
Issue number7
Publication statusPublished - 2002


  • Bernoulli mixture models
  • Coherence
  • Expected shortfall
  • Portfolio credit risk models
  • Risk measures
  • Value-at-risk

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