Conditional Expectation as Quantile Derivative

By Dirk Tasche. 2001

63 citations recorded by the source corpus.

For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of the variables for the so-called expected shortfall (first or higher moments) of the combination. Finally, we study in some more detail the coherence properties of the expected shortfall in case it is defined as a first conditional moment. Key words: quantile; value-at-risk; quantile derivative; conditional expectation; expected shortfall; conditional value-at-risk; coherent risk measure.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/588ec94c896ff2019df2f39fbb26cded213885d9

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In Management Science and Operations Research

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Citations in the corpus

Information, Divergence and Risk for Binary Experiments (2009, background): “…2007),or development of results which although not explicitly called reductions are effectively so (Brown et al., 2002; Brown and Low, 1996; Brown and Zhao, 2003; Chaudhuri and Loh, 2002; Cossock and Zhang, 2006; Cuevas and Fraiman, 1997; Domingos, 1999; Steinwart et al., 2005; Tasche, 2001).”
Expected shortfall and beyond (2002, background): “Moreover, VaR and the risk measure ES to be dened below (Denition 3.2) satisfy the conditions of Proposition 2.5 u nder quite general assumptions on the joint distribution of (X;Y ) (cf. Tasche, 2000 ).”
Dynamic capital allocation with distortion risk measures (2004, unknown): “applying quantile derivatives (Tasche, 2000b) and using the quantile representation of the Choquet integral (Denneberg, 1994), Tsanakas and Barnett (2002) showed that ρ(Zu) is differentiable in u and, by direct calculation, obtained the following formula for the unique allocation in the fuzzy core:”
Risk Capital Allocation and Cooperative Pricing of Insurance Liabilities (2003, unknown): “ciples Explicit calculations of the Aumann-Shapley value have been given in the bibliography for the cases that the cost functional is derived from the risk measure Expected Shortfall (Tasche, 2000a) and a risk measure used by the Securities and Exchange Commission (Denault, 2001).”
A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules (2003, background): “Gourieroux et al. (2000), Tasche (2000) and others show how to take partial first derivatives of VaR.”
Expected Shortfall: a natural coherent alternative to Value at Risk (2001, background): “In this paper and henceforth we are going to take these axioms seriously as many other groups of researchers [4, 5, 6, 7, 9, 10], practitioners and regulators [11] have begun to do.”