Journal of Risk

Nonparametric versus parametric expected shortfall

R. Douglas Martin and Shengyu Zhang

  • Expected shortfall (ES) non-parametric estimators and parametric maximum-likelihood estimators (MLEs) have very different statistical behaviors.
  • Influence functions are very useful for studying the ES estimator differences.
  • ES MLE’s have accurate standard errors performance but do not satisfy all the risk coherence axioms.
  • A particular semi-standard deviation modification of normal distribution ES is a coherent risk measure.

We use influence functions as a basic tool to study unconditional nonparametric and parametric expected shortfall (ES) estimators with regard to returns data influence, standard errors and coherence. Nonparametric ES estimators have a monotonically decreasing influence function of returns. ES maximum likelihood estimator (MLE) influence functions are nonmonotonic and approximately symmetric, resulting in large positive returns contributing to risk. However, ES MLEs 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 estimator for normal and t -distributions as well as to evaluate nonparametric ES estimator inefficiency. Monte Carlo results determine finite sample sizes for which the standard errors of both types of ES estimators are sufficiently accurate to be used in practice. The nonmonotonicity of ES MLEs leads us to study a modification of normal distribution MLEs 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 monotonic influence functions and the underlying risk measures are 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.

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