A composition between risk and deviation measures

TL;DR

This paper introduces a new framework for combining risk and deviation measures, demonstrating that the resulting composition can be a coherent risk measure under certain axioms, with practical examples illustrating its application.

Contribution

It proposes a novel composition framework for risk and deviation measures, emphasizing the role of the Limitedness axiom in ensuring coherence and extending to convex and co-monotone measures.

Findings

The composition is a coherent risk measure under the Limitedness axiom.

Examples include both known and new risk measures within this framework.

The approach highlights the importance of the Limitedness axiom in risk measure construction.

Abstract

The intuition of risk is based on two main concepts: loss and variability. In this paper, we present a composition of risk and deviation measures, which contemplate these two concepts. Based on the proposed Limitedness axiom, we prove that this resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. We also provide examples of known and new risk measures constructed under this framework in order to highlight the importance of our approach, especially the role of the Limitedness axiom.