Capital requirements with defaultable securities

TL;DR

This paper develops a comprehensive framework for capital requirements involving defaultable securities, extending traditional models to include assets with non-bounded payoffs, and analyzes their properties and practical implications.

Contribution

It introduces a generalized risk measure framework accommodating defaultable assets, highlighting the limitations of cash-additive measures and exploring properties like finiteness and continuity.

Findings

Risk measures with defaultable assets are not cash-additive.

Finiteness and continuity depend on asset and acceptance set properties.

No optimal eligible asset exists under the proposed framework.

Abstract

We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numeraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.