Journal of Risk

Inefficiency and bias of modified value-at-risk and expected shortfall

R. Douglas Martin and Rohit Arora

  • Derive expressions for the asymptotic variance and bias of mVaR and mES estimators.
  • Compare the bias and RMSE efficiency of the mVaR and mES estimators relative to parametric VaR and ES.
  • Assess the finite sample behavior of mVaR and mES estimators.

Modified value-at-risk (mVaR) and modified expected shortfall (mES) are risk estimators that can be calculated without modeling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to obtain more accurate risk measurement for non-normal return distributions. Use of skewness and kurtosis corrections can result in reduced bias, but these also lead to inflated mVaR and mES estimator standard errors. We compare modified estimators with their respective parametric counterparts in three ways. First, we assess the magnitude of standard error inflation by deriving formulas for the large-sample standard errors of mVaR and mES using the multivariate delta method. Monte Carlo simulation is then used to determine sample sizes and tail probabilities for which our asymptotic variance formula can be reliably used to compute finite-sample standard errors. Second, to evaluate the large-sample bias, we derive formulas for the asymptotic bias of modified estimators for t-distributions. Third, we analyze the finite-sample performance of the modified estimators for normal and t -distributions using their root-mean-squared-error efficiency relative to the parametric VaR and ES maximum likelihood estimators using Monte Carlo simulation. Our results show that the modified estimators are inefficient for both normal and t-distributions: the more so for t-distributions.

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