  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 of risk measures 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 by 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.
