Partial comonotonicity and distortion riskmetrics

TL;DR

This paper introduces partial comonotonicity, a new dependence concept linking it to additive subclasses of distortion riskmetrics, and characterizes Expected Shortfall through this framework.

Contribution

It develops the notion of partial comonotonicity, connecting dependence structures with additive subclasses of distortion riskmetrics, and characterizes Expected Shortfall via single-point concentration.

Findings

Partial comonotonicity generalizes comonotonicity and single-point concentration.

Additivity of distortion riskmetrics is characterized by partial comonotonicity.

Expected Shortfall is characterized using single-point concentration.

Abstract

We establish a connection between dependence structures and subclasses of distortion riskmetrics under which the latter are additive. A new notion of positive dependence, called partial comonotonicity, is developed, which nests the existing concepts of comonotonicity and single-point concentration. For two random variables, being comonotonic with a third one does not imply that they are comonotonic; instead, this defines an instance of partial comonotonicity. Any specific instance of partial comonotonicity uniquely characterizes a class of distortion riskmetrics through additivity under this dependence structure. An implication of this result is the characterization of the Expected Shortfall using single-point concentration.