A Simple and Robust Approach for Expected Shortfall Estimation

Abstract

In risk management, estimating Expected Shortfall (ES), though important and indispensable, is difficult when a sample size is small. This paper makes efforts to create a recipe for such challenge. A tail-based normal approximation with explicit formulas is derived by matching a specific quantile and mean excess square of the sample observations. To enhance the estimation accuracy, we then propose an adjusted tail-based normal approximation based on the sample's tail weight. The adjusted tailed-based normal approximation is robust and efficient in the sense that it can be applied to various heavy-tailed distributions, such as student's t, lognormal, Gamma, Weibull, etc., and the errors are very small. In addition, compared to two common ES estimators --- mean of excessive losses and extreme value theory estimator, the proposed approach achieves more accurate estimates with significantly smaller errors, especially at high confidence levels. Another appealing feature of the approach is that it works very well with small sample size. Effects of linear transformations on the ES estimator are also investigated to guarantee the practicality and further validate this new approach.