Nonparametric versus Parametric Expected Shortfall

We use influence functions as a basic tool to study unconditional non-parametric and parametric expected shortfall (ES) estimators with regard to returns data influence, standard errors and coherence. Non-parametric ES estimators have a monotonically decreasing influence function of returns. ES maximum-likelihood estimator (MLE) influence functions are non-monotonic and approximately symmetric, resulting in large positive returns contributing to risk. However, ES MLE's have the lowest possible asymptotic variance among consistent ES estimators. Influence functions are used to derive large sample standard error formulas for both types of ES estimators for normal and t-distributions and evaluate non-parametric ES estimator inefficiency. Monte Carlo results determine finite-sample sizes for which the standard errors of which both types of ES estimators are sufficiently accurate to be used in practice. Non-monotonicity of ES MLE's led us to study modification of normal distribution MLE's in which standard deviation is replaced by semi-standard deviation (SSD). Influence function theory is used to establish a condition under which an SSD based ES risk estimator has a monotonic influence functions and the underlying risk measures is coherent. It is also shown that the SSD based estimator's asymptotic standard error is only slightly larger than that of the standard deviation-based estimator