🤖 AI Summary
This paper investigates the Pareto-optimal risk-sharing problem among multiple agents under nonmonotonic, nonconvex distortion-based risk measures—specifically, law-invariant, comonotonically additive functionals. Focusing on three canonical variability measures—Gini deviation, mean–median deviation, and interquantile range—we develop an analytical framework grounded in extremal negative dependence structures and Pareto optimization. We establish that, for Gini and mean–median deviations, the optimal allocation remains comonotonic; in contrast, the interquantile range induces a novel *pairwise antimonotonic mixed structure*—a departure from classical comonotonicity assumptions and the first identification of such extreme negative dependence patterns in optimal risk sharing. We derive closed-form optimal allocations for all three measures, thereby extending the theoretical scope of distortion risk measures and offering a new paradigm for insurance reserving and portfolio risk management.
📝 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.
