On risk management problems related to a coherence property

Abstract

Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.

Keywords:

• Coherent measures
• Expected shortfall
• Value-at-risk
• Estimation
• Model risk

Notes

†Jorion (1996) appears to have provided one of the early discussions on model risk, specifically in the case of VaR; Derman (1996) discussed model risk on a broader basis. While a more recent review is contained in Dowd (2002), it is fair to say that in general the issue of model risk has not received an in depth treatment in the literature.

†Note that this measure can be different from ES or VaR.

‡As is pointed out in Acerbi (2004), working with the upper quantile for the empirical VaR is equivalent to using the order statistic.

§We thank one of the referees for this example.