Unbiased estimation of risk

The estimation of risk measured in terms of a risk measure is typically done in two steps: in the first step, the distribution is estimated by statistical methods, either parametric or nonparametric. In the second step, the estimated distribution is considered as true distribution and the targeted risk-measure is computed. In the parametric case this is achieved by using the formula for the risk-measure in the model and inserting the estimated parameters. It is wellknown that this procedure is not efficient because the highly nonlinear mapping from model parameters to the risk-measure introduces an additional biases. Statistical experiments show that this bias leads to a systematic underestimation of risk. In this regard we introduce the concept of unbiasedness to the estimation of risk. We show that an appropriate bias correction is available for many well known estimators. In particular, we consider value-at-risk and tail value-at-risk (expected shortfall). In the special case of normal distributions, closed-formed solutions for unbiased estimators are given. For the general case we propose a bootstrapping algorithm and illustrate the outcomes by several data experiments.