A new view on risk measures associated with acceptance sets

TL;DR

This paper explores properties of risk measures linked to acceptance sets in finance, focusing on their geometric features, finiteness, and relaxed assumptions, providing new insights into their mathematical structure.

Contribution

It introduces new analyses of the sublevel sets and level lines of risk measures, extending understanding beyond traditional closedness assumptions and non-convex cases.

Findings

Analysis of sublevel sets and level lines of risk measures

Conditions for finiteness of risk measures

Relaxation of closedness assumptions in the study of risk measures

Abstract

In this paper, we study properties of certain risk measures associated with acceptance sets. These sets describe regulatory preconditions that have to be fulfilled by financial institutions to pass a given acceptance test. If the financial position of an institution is not acceptable, the decision maker has to raise new capital and invest it into a basket of so called eligible assets to change the current position such that the resulting one corresponds with an element of the acceptance set. Risk measures have been widely studied. The risk measure that is considered here determines the minimal costs of making a financial position acceptable. In the literature, monetary risk measures are often defined as translation invariant functions and, thus, there is an equivalent formulation as Gerstewitz-Functional. The Gerstewitz-Functional is an useful tool for separation and scalarization in multiobjective optimization in the non-convex case. In our paper, we study properties of the sublevel sets, strict sublevel sets and level lines of a risk measure defined on a linear space. Furthermore, we discuss the finiteness of the risk measure and relax the closedness assumptions.