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.

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

Optimal Decision Making Under Distorted Expectation with Partial Distribution Information Decision makers who are not risk neutral may evaluate expected values by distorting objective probabilities to reflect their risk attitudes, a phenomenon known as distorted expectations. This concept is widely applied in behavioral economics, insurance, finance, and other business domains. In “Distributionally Robust Optimization Under Distorted Expectations,” Cai, Li, and Mao study how decision makers using distorted expectations can optimize their decisions when only partial information about objective probabilities is available. They show that decision makers who are ambiguity averse can optimize their decisions as if they are risk averse with their risk attitudes characterized by a convex distortion function. This finding demonstrates why even non–risk-averse decision makers, such as those studied in the celebrated cumulative prospect theory, may consider it optimal to take risk-averse decisions when facing uncertainty about objective probabilities. Leveraging this finding, the authors show that a large class of distributionally robust optimization problems involving the use of distorted expectations can be tractably solved as convex programs.

Generalized quantiles of a random variable were defined as the minimizers of a general asymmetric loss function, which include quantiles, expectiles and M-quantiles as their special cases. Expectiles have been suggested as potentially better alternatives to both Value-at-Risk and expected shortfall risk measures. In this paper, we first establish the first-order expansions of generalized quantiles for extreme risks as the confidence level α↑ 1, and then investigate the first-order and/or second-order expansions of expectiles of an extreme risk as α↑ 1 according to the underlying distribution belonging to the max-domain of attraction of the Fréchet, Weibull, and Gumbel distributions, respectively. Examples are also presented to examine whether and how much the first-order expansions have been improved by the second-order expansions.