Capital requirements with defaultable securities

• Published in: Insurance, mathematics & economics Vol. 55; pp. 58 - 67
• Main Authors: Farkas, Walter, Koch-Medina, Pablo, Munari, Cosimo
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2014; Elsevier Sequoia S.A
• Subjects: Acceptance sets; Assets; Bonds; Capital adequacy; Capital investments; Capital requirements; Default; Defaultable securities; Eligible assets; Financial analysis; Financial crisis; Measurement; Risk assessment; Risk measures; Studies; Tail Value-at-Risk; Value at risk; Tail Value-at-Risk; 91B32; Defaultable securities; G11; G22; Acceptance sets; Capital adequacy; Value-at-Risk; C60; 91B30; Risk measures; Eligible assets
• ISSN: 0167-6687; 1873-5959
• DOI: 10.1016/j.insmatheco.2013.11.009

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 numéraire. 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. •We study capital requirements with general acceptance sets and general eligible assets.•We show that general capital requirements need not be finitely valued or continuous.•We establish characterizations of finiteness and continuity.•We provide applications to capital requirements based on Value-at-Risk and Tail Value-at-Risk.•We show the nonexistence of “optimal” eligible assets.