Estimation of the marginal expected shortfall: the mean when a related variable is extreme
Juan-Juan Cai, John H. J. Einmahl, Laurens de Haan, Chen Zhou
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2015 · Volume 77 · Issue 2
Abstract
Denote the loss return on the equity of a financial institution asXand that of the entire market asY. For a given very small value ofp>0, the marginal expected shortfall (MES) is defined as, whereQY(1−p) is the (1−p)th quantile of the distribution ofY. 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 whenp↓0, as the sample sizen→∞. Since we are in particular interested in the casep=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.
Keywords
Asymptotic normality, Conditional tail expectation, Extreme values
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