Inefficiency of Modified VaR and ES

Abstract

Modified Value-at-Risk (mVaR) and Modified Expected Shortfall (mES) are risk estimators that can be calculated without modelling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to reduce bias in risk measurement for non-normal return distributions. However, the use of skewness and kurtosis estimators that are needed to implement mVaR and mES can lead to highly inflated mVaR and mES estimator standard errors. To assess the magnitude of standard error inflation we derive formulas for the large sample standard errors of mVaR and mES using multivariate delta method and compare them against standard errors of parametric VaR and ES estimators, under both normal and t-distributions. Our asymptotic results show that mVaR and mES estimators can have standard errors considerably larger than those of parametric VaR and ES estimators, and small-sample Monte Carlo confirms that the asymptotic results are approximately correct in sample sizes commonly used in practice.