Sensitivity Analysis of Distortion Risk Measures

This paper provides a uni¯ed statistical framework for the analysis of distortion riskmeasures (DRM) and of their sensitivities with respect to parameters representing riskaversion and/or pessimism. We derive the general formula for calculating the functionalasymptotic distribution of the nonparametric estimator of the functional distortion riskmeasures. Closed form expressions are provided for special examples such as VaR, Tail-VaR and Proportional Hazard distortion risk measure. Moreover, we analyze the linkbetween Value-at-Risk and Tail-VaR and characterize the underlying distributions underwhich the two risk measures are linearly related through their risk levels. We apply theresults to currency portfolios and observe that this linearity relationship between Value-at-Risk and Tail-VaR is a surprisingly common phenomenon for the portfolios considered.

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