Risk sharing, measuring variability, and distortion riskmetrics

TL;DR

This paper characterizes Pareto-optimal risk sharing allocations among agents using distortion riskmetrics, including measures like Gini deviation and inter-quantile difference, revealing complex dependence structures.

Contribution

It provides explicit solutions for Pareto-optimal allocations under general and comonotonic risk sharing problems with novel insights into dependence structures.

Findings

Explicit Pareto-optimal allocations for Gini and median deviations

Optimal allocations with inter-quantile difference involve counter-monotonic structures

Reveals complex dependence patterns in risk sharing scenarios

Abstract

We address the problem of sharing risk among agents with preferences modelled by a general class of comonotonic additive and law-based functionals that need not be either monotone or convex. Such functionals are called distortion riskmetrics, which include many statistical measures of risk and variability used in portfolio optimization and insurance. The set of Pareto-optimal allocations is characterized under various settings of general or comonotonic risk sharing problems. We solve explicitly Pareto-optimal allocations among agents using the Gini deviation, the mean-median deviation, or the inter-quantile difference as the relevant variability measures. The latter is of particular interest, as optimal allocations are not comonotonic in the presence of inter-quantile difference agents; instead, the optimal allocation features a mixture of pairwise counter-monotonic structures, showing some patterns of extremal negative dependence.