A comparison of some univariate models for Value-at-Risk and expected shortfall

We compare in a backtesting study the performance of univariate models for Valueat-Risk (VaR) and expected shortfall based on stable laws and on extreme value theory (EVT). Analyzing these different approaches, we test whether the sum-stability assumption or the max-stability assumption, that respectively imply α-stable laws and Generalized Extreme Value (GEV) distributions, is more suitable for risk management based on VaR and expected shortfall. Our numerical results indicate that α-stable models tend to outperform pure EVT-based methods (especially those obtained by the so-called block maxima method) in the estimation of Value-at-Risk, while a peaks-over-threshold method turns out to be preferable for the estimation of expected shortfall. We also find empirical evidence that some simple semiparametric EVT-based methods perform well in the estimation of VaR.