Set-valued average value at risk and its computation

TL;DR

This paper introduces new set-valued average value at risk measures for multivariate risks, including a market-independent and a market-aware version, with properties proven and algorithms for computation demonstrated.

Contribution

It generalizes the average value at risk to set-valued forms for multivariate risks and provides algorithms for their computation in finite spaces.

Findings

Both risk measures are proven to have essential properties.

Algorithms for computing the measures are developed.

Examples illustrate the theoretical concepts.

Abstract

New versions of the set-valued average value at risk for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first "regulator" version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute the values of both version over finite probability spaces. Several examples illustrate various features of the theoretical constructions.