A note on the statistical robustness of risk measures

The question of robustness in risk measurement emerged only fairly recently, but it has already attracted considerable attention. The problem has been studied using various approaches, and several methods aiming at robustifying the risk measures have been proposed. However, a general robustness theory is still missing. We focus on the parametric estimators of risk measures and use Hampel’s infinitesimal approach to derive the robustness properties. We derive the influence functions for the general parametric estimators of the value-at-risk and expected shortfall. For various distributions, the classical estimators, such as maximum likelihood, have unbounded influence functions and are not robust. Using the expression for the influence function, we propose a general strategy to construct robust estimators and explore their properties. The use of the methodology is demonstrated through several illustrative examples. Finally, we discuss an operational risk application and highlight the importance of the complementary information provided by nonrobust and robust estimates for regulatory capital calculation.