Beyond Cash-Additive Risk Measures: When Changing the Numeraire Fails

Walter Farkas, Pablo Koch-Medina and Cosimo Munari
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Walter Farkas: University of Zurich - Department of Banking and Finance; Swiss Finance Institute; ETH Zurich
Pablo Koch-Medina: University of Zurich - Department of Banking and Finance; Swiss Finance Institute
Cosimo Munari: University of Zurich - Department of Banking and Finance; Swiss Finance Institute

No 13-67, Swiss Finance Institute Research Paper Series from Swiss Finance Institute

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.

Keywords: risk measures; acceptance sets; general eligible assets; defaultable bonds; cash subadditivity; quasiconvexity; Value-at-Risk; Tail Value-at-Risk; shortfall risk (search for similar items in EconPapers)
JEL-codes: C60 G11 G22 (search for similar items in EconPapers)
Pages: 26 pages
Date: 2013-12
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:chf:rpseri:rp1367

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