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 non-parametric. 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 well-known 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 .
The estimation of risk measures recently gained a lot of attention, partly because of the backtesting issues of expected shortfall related to elicitability. In this work we shed a new and fundamental light on optimal estimation procedures in terms of bias. We show that once the parameters of a model need to be estimated, one has to take additional care when estimating risks. The typical plug-in approach, for example, introduces a bias which leads to a systematic underestimation of risk. In this regard , we introduce a novel notion of unbiasedness to the estimation of risk which is motivated from economic principles. In general, the proposed concept does not coincide with the well-known statistical notion of unbiasedness. We show that an appropriate bias correction is available for many well-known estimators. In particular, we consider value-at-risk and expected shortfall (tail value-at-risk ). In the special case of normal distributions, closed-formed solutions for unbiased estimators can be obtained. We present a number of motivating examples which show the outperformance of unbiased estimators in many circumstances. The unbiasedness has a direct impact on backtesting and therefore adds a further viewpoint to established statistical properties .
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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 non-parametric. 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 well-known 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
measures recently gained a lot of attention, partly because of the backtesting issues of expected shortfall related to elicitability. In this work we shed a new and fundamental light on optimal estimation procedures in terms of bias. We show that once the parameters of a model need to be estimated, one has to take additional care when estimating risks. The typical plug-in approach, for example, introduces a bias which
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