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

Mathematics of Operations Research, forthcoming

44 Pages Posted: 5 Dec 2019 Last revised: 18 Jan 2022

See all articles by Fangda Liu

Fangda Liu

University of Waterloo - Department of Statistics and Actuarial Science

Tiantian Mao

University of Science and Technology of China (USTC) - Department of Statistics and Finance

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Linxiao Wei

Wuhan University of Technology

Date Written: November 19, 2019

Abstract

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. Optimal allocations are obtained in the setting of elliptical models,
and several results are established for tail risk measures and risk sharing problems in the presence of model uncertainty. 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: Risk sharing, Pareto optimality, Value-at-Risk, Range-Value-at-Risk, Non-convex optimization

Suggested Citation

Liu, Fangda and Mao, Tiantian and Wang, Ruodu and Wei, Linxiao, Inf-convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures (November 19, 2019). Mathematics of Operations Research, forthcoming, Available at SSRN: https://ssrn.com/abstract=3490348 or http://dx.doi.org/10.2139/ssrn.3490348

Fangda Liu (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

200 University Ave.
Waterloo, Ontario N2L 3G1 N2L3G1
Canada

Tiantian Mao

University of Science and Technology of China (USTC) - Department of Statistics and Finance ( email )

96, Jinzhai Road
Hefei, Anhui 230026
China

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

Linxiao Wei

Wuhan University of Technology ( email )

Wuhan
China

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