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

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.

Risk Sharing with Lambda Value at Risk

“…Finally, we compute some examples with the risk measures being EU and RDEU. Our results in this section generate the corresponding results in Liu et al (2022), where the inf-convolution of VaR and a monetary ε-tail risk measure is considered.…”

“…X, Y ∈ X . One can refer to Liu and Wang (2021) and Liu et al (2022) for more details on the definition, properties and applications of tail risk measures.…”

“…These results are extended to the mixture of left and right VaR, and tail-risk measures in Liu et al (2022) and VaR-type risk measures in Weber (2018). Moreover, Embrechts et al (2019) study the risk sharing problem with VaR and expected shortfall (ES) as risk measures under heterogeneous beliefs.…”

Risk sharing, measuring variability, and distortion riskmetrics

“…These formulas may be compared with the quantile inf-convolutions formulas obtained by Embrechts et al (2018) and Liu et al (2022)…”

“…It remains unclear to us whether our analysis can be generalized to distortion riskmetrics other than IQD, which are not convex (i.e., with non-concave distortion functions), and how large the class of such tractable risk functionals is. As far as we are aware, the unconstrained risk sharing problems for non-convex risk measures and variability measures have very limited explicit results (e.g., Embrechts et al (2018), Weber (2018) and Liu et al (2022)), and further investigation is needed for a better understanding of the challenges and their solutions.…”