Inefficiency and Bias of Modified Value-at-Risk and Expected Shortfall

26 Pages Posted: 4 Aug 2017

See all articles by R. Douglas Martin

R. Douglas Martin

University of Washington

Rohit Arora

University of Texas at Austin

Date Written: August 2, 2017


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.

Keywords: modified value-at-risk (mVaR), modified expected shortfall (mES), standard error, efficiency, delta method, Basel III

Suggested Citation

Martin, R. Douglas and Arora, Rohit, Inefficiency and Bias of Modified Value-at-Risk and Expected Shortfall (August 2, 2017). Journal of Risk, Vol. 19, No. 6, 2017, Available at SSRN:

R. Douglas Martin (Contact Author)

University of Washington ( email )

Applied Mathematics & Statistics
Seattle, WA 98195
United States

Rohit Arora

University of Texas at Austin ( email )

2317 Speedway
Austin, TX 78712
United States

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