Optimal risk allocation in a market with non-convex preferences

TL;DR

This paper investigates optimal risk allocation under non-convex preferences using distortion risk measures, providing conditions for existence, characterizing co-monotone solutions, and separating market preferences from total risk.

Contribution

It introduces an infimal representation for distortion risk measures in non-convex settings and characterizes co-monotone optimal allocations with a novel marginal risk property.

Findings

Existence conditions depend solely on preferences.

Co-monotone allocations have marginal risks of only zero or one.

Representation separates market preferences from total risk.

Abstract

The aims of this study are twofold. First, we consider an optimal risk allocation problem with non-convex preferences. By establishing an infimal representation for distortion risk measures, we give some necessary and sufficient conditions for the existence of optimal and asymptotic optimal allocations. We will show that, similar to a market with convex preferences, in a non-convex framework with distortion risk measures the boundedness of the optimal risk allocation problem depends only on the preferences. Second, we consider the same optimal allocation problem by adding a further assumption that allocations are co-monotone. We characterize the co-monotone optimal risk allocations within which we prove the "marginal risk allocations" take only the values zero or one. Remarkably, we can separate the role of the market preferences and the total risk in our representation.