Estimation of the marginal expected shortfall: the mean when a related variable is extreme

Juan-Juan Cai, John Einmahl, Laurens Haan and Chen Zhou ()

Journal of the Royal Statistical Society Series B, 2015, vol. 77, issue 2, 417-442

Abstract: type="main" xml:id="rssb12069-abs-0001">

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 &7C Y > Q Y ( 1 − p ) } , where Q Y (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 non-parametric 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 behaviour. The finite sample performance of the estimator and the relevance of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large US investment banks.

Date: 2015
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Working Paper: Estimation of the Marginal Expected Shortfall: The Mean when a Related Variable is Extreme (2012)
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