Sharp convex bounds on the aggregate sums--An alternative proof

TL;DR

This paper provides a new proof for the characterization of comonotonic and mutually exclusive random vectors with given marginals, using distortion risk measures and expected utility theory, offering an alternative to geometric proofs.

Contribution

It introduces an alternative proof method for known bounds on aggregate sums, leveraging distortion risk measures and utility theory.

Findings

New proof of bounds on aggregate sums using risk measures

Characterization of comonotonic risks with maximal sums

Identification of mutually exclusive risks with minimal sums

Abstract

It is well known that a random vector with given marginal distributions is comonotonic if and only if it has the largest sum with respect to the convex order [ Kaas, Dhaene, Vyncke, Goovaerts, Denuit (2002), A simple geometric proof that comonotonic risks have the convex-largest sum, ASTIN Bulletin 32, 71-80. Cheung (2010), Characterizing a comonotonic random vector by the distribution of the sum of its components, Insurance: Mathematics and Economics 47(2), 130-136] and that a random vector with given marginal distributions is mutually exclusive if and only if it has the minimal convex sum [Cheung and Lo (2014), Characterizing mutual exclusivity as the strongest negative multivariate dependence structure, Insurance: Mathematics and Economics 55, 180-190]. In this note, we give a new proof of this two results using the theories of distortion risk measure and expected utility.