Unbiased Estimation of Risk

26 Pages Posted: 27 Dec 2016 Last revised: 2 Sep 2017

See all articles by Marcin Pitera

Marcin Pitera

Jagiellonian University in Krakow - Department of Mathematics

Thorsten Schmidt

University of Freiburg

Date Written: December 25, 2016

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.

Keywords: value-at-risk, tail value-at-risk, expected shortfall, risk measure, estimation of risk measures, bias, risk estimation, elicitability, backtest, unbiased estimation of risk measures

JEL Classification: C52, C60, G18, G28

Suggested Citation

Pitera, Marcin and Schmidt, Thorsten, Unbiased Estimation of Risk (December 25, 2016). Available at SSRN: https://ssrn.com/abstract=2890034 or http://dx.doi.org/10.2139/ssrn.2890034

Marcin Pitera

Jagiellonian University in Krakow - Department of Mathematics ( email )

Collegium Novum
Krakow
Poland

Thorsten Schmidt (Contact Author)

University of Freiburg ( email )

Fahnenbergplatz
Freiburg, D-79085
Germany

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