Nonparametric versus Parametric Expected Shortfall

Abstract

The Basel III recommendation to replace value-at-risk with expected shortfall (ES) for capital adequacy risk management purposes is a welcome event that recognizes the intuitive and theoretical advantages of ES, and is likely to increase use of ES in portfolio construction and risk management generally. Correspondingly it seems timely to obtain a deeper understanding of the statistical properties ES estimators than currently exists, and in particular the comparative statistical properties of the simple non-parametric estimators versus the more complex parametric ES maximum-likelihood estimators. We analyze the statistical differences between these two basic types ES estimators in three ways for both normal and t-distributed returns. First we derive formulas for statistical influence functions for the two types of ES estimators for both normal and t-distributions and provide and use influence function displays to clearly reveal how asset returns impact the risk estimates. The influence function analysis immediately reveals the undesirable feature of the parametric ES estimators that positive returns contribute very substantially to risk, a behavior not present in the non-parametric ES estimator. We show how this undesirable feature of the parametric ES estimator can be substantially ameliorated through use of a semi-standard deviation (SSD) estimator as a scale estimator. Then we use the influence functions to derive asymptotic variance formulas of the ES estimators and compare their large sample variabilities and statistical efficiencies. The asymptotic variance formulas yield finite-sample standard error (S.E.) approximations, and our third analysis uses Monte Carlo to assess the accuracy of finite sample size standard error approximations, which turn out to be rather good for example using the Basel III recommended 2.5% tail probability and sample sizes of interest in risk management applications.