Beyond cash-additive risk measures: when changing the numéraire fails
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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 numéraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing 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.
KeywordsRisk measures Acceptance sets General eligible assets Defaultable bonds Cash subadditivity Quasiconvexity Value-at-risk Tail value-at-risk Shortfall risk
Mathematics Subject Classification (2010)91B30 46B42 46B40 46A55 06F30
JEL ClassificationC60 G11 G22
Partial support through the SNF project 51NF40-144611 “Capital adequacy, valuation, and portfolio selection for insurance companies” is gratefully acknowledged. Financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FinRisk), project “Mathematical Methods in Financial Risk Management”, is gratefully acknowledged by W. Farkas and C. Munari. Part of this research was undertaken while P. Koch-Medina was employed by Swiss Re.
We gratefully acknowledge the careful review and constructive feedback provided by two anonymous referees. In particular, part of the proof of Proposition 4.6 could be significantly shortened following an idea of one of them.
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