Unbiased estimation of risk

TL;DR

This paper introduces a new concept of unbiased risk estimation that corrects systematic bias in traditional methods, improving accuracy and backtesting performance for measures like VaR and ES.

Contribution

It proposes a novel, economically motivated notion of unbiasedness for risk estimators and provides bias correction techniques for common risk measures.

Findings

Unbiased estimators outperform biased ones in various scenarios.

Bias correction improves backtesting results.

Closed-form unbiased estimators are derived for normal distributions.

Abstract

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.