Quantiles under ambiguity and risk sharing

TL;DR

This paper introduces Choquet quantiles, a generalization of probabilistic quantiles under ambiguity, providing axiomatic characterization, risk sharing solutions, and new risk measures with practical algorithms and financial data applications.

Contribution

It develops the theory of Choquet quantiles, characterizes them axiomatically, and applies them to risk sharing and new risk measures under ambiguity.

Findings

Choquet quantiles share properties with probabilistic quantiles.

Inf-convolution of Choquet quantiles remains a Choquet quantile.

Introduction of Choquet Expected Shortfall with properties similar to Expected Shortfall.

Abstract

Choquet capacities and integrals are central concepts in decision making under ambiguity or model uncertainty, pioneered by Schmeidler. Motivated by risk optimization problems for quantiles under ambiguity, we study the subclass of Choquet integrals, called Choquet quantiles, which generalizes the usual (probabilistic) quantiles, also known as Value-at-Risk in finance, from probabilities to capacities. Choquet quantiles share many features with probabilistic quantiles, in terms of axiomatic representation, optimization formulas, and risk sharing. We characterize Choquet quantiles via only one axiom, called ordinality. We prove that the inf-convolution of Choquet quantiles is again a Choquet quantile, leading to explicit optimal allocations in risk sharing problems for quantile agents under ambiguity. A new class of risk measures, Choquet Expected Shortfall, is introduced, which enjoys most properties of the coherent risk measure Expected Shortfall. Our theory is complemented by optimization algorithms, numerical examples, and a stylized illustration with financial data.