On the link between monetary and star-shaped risk measures

TL;DR

This paper explores the relationship between monetary and star-shaped risk measures, showing that under mild conditions, monetary risk measures are closely related to star-shaped ones through translation, emphasizing the role of normalization.

Contribution

It clarifies the connection between monetary and star-shaped risk measures, highlighting the importance of normalization and showing that monetary measures are essentially translations of star-shaped measures.

Findings

Monetary risk measures are closely related to star-shaped measures via translation.

Normalization, specifically the acceptability of 0, is crucial in linking these classes.

Under mild conditions, the two classes are connected through simple transformations.

Abstract

Recently, Castagnoli et al. (2021) introduce the class of star-shaped risk measures as a generalization of convex and coherent ones, proving that there is a representation as the pointwise minimum of some family composed by convex risk measures. Concomitantly, Jia et al. (2020) prove a similar representation result for monetary risk measures, which are more general than star-shaped ones. Then, there is a question on how both classes are connected. In this letter, we provide an answer by casting light on the importance of the acceptability of 0, which is linked to the property of normalization. We then show that under mild conditions, a monetary risk measure is only a translation away from star-shapedness.