A note on the induction of comonotonic additive risk measures from acceptance sets

TL;DR

This paper characterizes when acceptance sets induce comonotonic additive risk measures, linking stability under convex combinations of comonotonic variables to the additive property of the induced measures.

Contribution

It provides necessary and sufficient conditions for acceptance sets to induce comonotonic additive risk measures, extending to specific dependence structures.

Findings

Acceptance sets induce comonotonic additive risk measures if and only if they are stable under convex combinations of comonotonic variables.

The results generalize to risk measures with specified dependence structures such as perfect correlation or independence.

The paper offers a theoretical framework connecting acceptance set stability to the additive properties of risk measures.

Abstract

We present simple general conditions on the acceptance sets under which their induced monetary risk and deviation measures are comonotonic additive. We show that acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables. A generalization of this result applies to risk measures that are additive for random variables with a priori specified dependence structures, e.g., perfectly correlated, uncorrelated, or independent random variables.