Variability in Modified Estimators of 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 modifided 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 degree of inflation we derive formulas for the large sample standard errors of mVaR and mES using multivariate delta method. Finally, we assess the goodness of our analytical result with small sample Monte-Carlo at Normal and t-ditributions. Our results show that projected standard errors in small sample underestimate the true standard error. Also, the effect is worse for mES than mVaR.