Comonotonic measures of multivariate risks

TL;DR

This paper extends the concept of comonotonic risk measures to multivariate risks, introduces a new property called strong coherence, and reformulates risk measure computation as an optimal transportation problem with an algorithm.

Contribution

It introduces a multivariate extension of Kusuoka's risk measures, proposes strong coherence as a new axiom, and develops an optimal transportation approach with implementation.

Findings

Extended comonotonic risk measures to multivariate risks

Proposed strong coherence as an alternative to traditional axioms

Reformulated risk computation as an optimal transportation problem

Abstract

We propose a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invariance, subadditivity and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we reformulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.