Inf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures

Fangda Liu (), Tiantian Mao (), Ruodu Wang () and Linxiao Wei ()
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Fangda Liu: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Tiantian Mao: International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230026, China
Ruodu Wang: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Linxiao Wei: College of Science, Wuhan University of Technology, Wuhan 430070, China

Mathematics of Operations Research, 2022, vol. 47, issue 3, 2494-2519

Abstract: Inspired by the recent developments in risk sharing problems for the value at risk (VaR), the expected shortfall (ES), and 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. Optimal allocations are obtained in the settings of elliptical models and model uncertainty. In particular, several results are established for tail risk measures in the presence of model uncertainty, which may be of independent interest outside the framework of risk sharing. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk sharing.

Keywords: Primary: 91G70; secondary: 91A12; risk sharing; Pareto optimality; value at risk; range value at risk; nonconvex optimization (search for similar items in EconPapers)
Date: 2022
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