Inf-convolution and Optimal Allocations for Tail Risk Measures
25 Pages Posted: 5 Dec 2019
Date Written: November 19, 2019
Inspired by the recent developments in risk sharing problems for the Value-at-Risk (VaR), the Expected Shortfall (ES), or the Range-Value-at-Risk (RVaR), we study the optimization of risk sharing for general tail risk measures. Explicit formulas of the inf-convolution and Pareto optimal allocations are obtained in the case of a mixed collection of left and right VaRs, and in that of a VaR and another tail risk measure. The inf-convolution of tail risk measures is shown to be a tail risk measure with an aggregated tail parameter, a phenomenon very similar to the cases of VaR , ES and RVaR. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Moreover, we find, via several results, that the roles of left and right VaRs are generally asymmetric in the optimization problems. Our analysis generalizes in several directions the recent work on quantile-based risk sharing.
Keywords: Risk sharing, Pareto optimality, Value-at-Risk, Range-Value-at-Risk, Non-convex optimization
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