Beyond cash-additive risk measures: when changing the num\'{e}raire fails

TL;DR

This paper explores risk measures beyond the traditional cash-additive framework, especially when the eligible asset is defaultable, providing new insights into their properties and limitations in such contexts.

Contribution

It extends the theory of risk measures to include general eligible assets, characterizes cash subadditivity, and analyzes the impact of non-linear pricing rules.

Findings

Cash subadditivity is rare when the eligible asset is defaultable.

Finiteness and continuity results are established for various risk measures.

Non-linear pricing rules restrict cash subadditivity to continuous cases.

Abstract

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numeraire. However, discounting does not work in all financially relevant situations, typically when the eligible asset is a defaultable bond. In this paper we fill this gap allowing for general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on Value-at-Risk and Tail Value-at-Risk on $L^p$ spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.