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

Author

Listed:
  • Juan-Juan Cai
  • John H. J. Einmahl
  • Laurens Haan
  • Chen Zhou

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.

Suggested Citation

  • Juan-Juan Cai & John H. J. Einmahl & Laurens Haan & Chen Zhou, 2015. "Estimation of the marginal expected shortfall: the mean when a related variable is extreme," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(2), pages 417-442, March.
    • Handle: RePEc:bla:jorssb:v:77:y:2015:i:2:p:417-442
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    File URL: http://hdl.handle.net/10.1111/rssb.2015.77.issue-2
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    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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