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http://ssrn.com/abstract=2053654
 
 

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


Walter Farkas


University of Zurich, Department of Banking and Finance; Swiss Finance Institute; ETH Zürich - Department of Mathematics

Pablo Koch-Medina


University of Zurich - Department of Banking and Finance

Cosimo Munari


ETH Zürich - Department of Mathematics

September 12, 2013

Finance Stochastics, 18(1), 145-173 (2014)
Swiss Finance Institute Research Paper No. 13-67

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.

Number of Pages in PDF File: 26

Keywords: risk measures, acceptance sets, general eligible assets, defaultable bonds, cash subadditivity, quasiconvexity, Value-at-Risk, Tail Value-at-Risk, shortfall risk

JEL Classification: C60, G11, G22

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Date posted: May 9, 2012 ; Last revised: March 11, 2014

Suggested Citation

Farkas, Walter and Koch-Medina, Pablo and Munari, Cosimo, Beyond Cash-Additive Risk Measures: When Changing the Numeraire Fails (September 12, 2013). Finance Stochastics, 18(1), 145-173 (2014); Swiss Finance Institute Research Paper No. 13-67. Available at SSRN: http://ssrn.com/abstract=2053654 or http://dx.doi.org/10.2139/ssrn.2053654

Contact Information

Walter Farkas (Contact Author)
University of Zurich, Department of Banking and Finance ( email )
Plattenstrasse 14
CH-8032 Zurich, Zurich 8032
Switzerland
+41-44-634 3953 (Phone)
+41-44-634 4345 (Fax)
Swiss Finance Institute ( email )
c/o University of Geneve
40, Bd du Pont-d'Arve
1211 Geneva, CH-6900
Switzerland

Swiss Federal Institute of Technology Zurich - Department of Mathematics ( email )
ETH Zentrum HG-F 42.1
Raemistr. 101
CH-8092 Zurich, 8092
Switzerland
Pablo Koch-Medina
University of Zurich - Department of Banking and Finance ( email )
Plattenstrasse 14
Zürich, 8032
Switzerland
Cosimo Munari
Swiss Federal Institute of Technology Zurich - Department of Mathematics ( email )
ETH Zentrum HG-F 42.1
Raemistr. 101
CH-8092 Zurich, 8092
Switzerland
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