Unbiased estimation and backtesting of risk in the context of heavy tails

While the estimation of risk is an important question in the daily business of banks and insurances, many existing plug-in estimation procedures suffer from an unnecessary bias. This often leads to the underestimation of risk and negatively impacts backtesting results, especially in small sample cases. In this article we show that the link between estimation bias and backtesting can be traced back to the dual relationship between risk measures and the corresponding performance measures, and discuss this in reference to value-at-risk and expected shortfall frameworks. Motivated by this finding, we propose a new algorithm for bias correction and show how to apply it for generalized Pareto distributions. In particular, we consider value-at-risk and expected shortfall plug-in estimators, and show that the application of our algorithm leads to gain in efficiency when heavy tails exist in the data.