Backtesting expected shortfall for world stock index ETFs with extreme value theory and Gram–Charlier mixtures

This paper analyses risk quantification for three main stock market index exchange‐traded funds in world financial markets. We compare the relative performance of a set of parametric and semi‐nonparametric models in terms of both value‐at‐risk and expected shortfall backtesting techniques. To this end, we explore the result of the jointly elicitability of these two risk measures. We provide a new mixture of Gram–Charlier distributions that have been used in this framework for the first time and derive a close formula for directly computing expected shortfall. This model is compared to the Gaussian (benchmark model), Student's t, generalized Pareto (a case of the extreme value theory) and mixtures of Gram–Charlier distributions. The results show that peaks‐over‐threshold (extreme value theory) and flexible Gram–Charlier approximations are suitable to quantify market risk and mitigate concerns about possible financial instabilities generated by misuse of exchange‐traded funds trading.