Estimation of the Marginal Expected Shortfall: The Mean when a Related Variable is Extreme

J. Cai, J.H.J. Einmahl, L.F.M. de Haan, C. Zhou

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Abstract: Denote the loss return on the equity of a financial institution as X and that of the entire market as Y . For a given very small value of p > 0, the marginal expected shortfall (MES) is defined as E(X | Y > QY (1−p)), where QY (1−p) is the (1−p)-th quantile of the distribution of Y . The MES is an important factor when measuring the systemic risk of financial institutions. For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p ↓ 0, as the sample size n → ∞. Since we are in particular interested in the case p = O(1=n), we use extreme value techniques for deriving the estimator and its asymptotic behavior. The finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks.
Original languageEnglish
Place of PublicationTilburg
Number of pages28
Publication statusPublished - 2012

Publication series

NameCentER Discussion Paper


  • Asymptotic normality
  • extreme values
  • tail dependence


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