Multi-asset return risk measures

TL;DR

This paper introduces multi-asset return risk measures (MARRMs), extending return risk measures by incorporating multiple assets for risk management, analyzing their properties, and comparing them through case studies in financial markets.

Contribution

It develops the concept of MARRMs, explores their mathematical properties, connects them to existing risk measures, and provides practical case studies for risk mitigation analysis.

Findings

Positively homogeneous MARRMs are quasi-convex iff they are convex.

Conditions to prevent inconsistent risk evaluations are identified.

Case studies compare RRMs, MARMs, and MARRMs in financial markets.

Abstract

We revisit the recently introduced concept of return risk measures (RRMs) and extend it by incorporating risk management via multiple so-called eligible assets. The resulting new class of risk measures, termed multi-asset return risk measures (MARRMs), introduces a novel economic model for multiplicative risk sharing. We analyze properties of these risk measures. In particular, we prove that a positively homogeneous MARRM is quasi-convex if and only if it is convex. Furthermore, we provide conditions to avoid inconsistent risk evaluations. Then, we point out the connection between MARRMs and the well-known concept of multi-asset risk measures (MARMs). This is used to obtain various dual representations of MARRMs. Moreover, we conduct a series of case studies, in which we use typical continuous-time financial markets and different notions of acceptability of losses to compare RRMs, MARMs, and MARRMs and draw conclusions about the cost of risk mitigation.