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

26 Pages Posted: 9 May 2012 Last revised: 11 Mar 2014

Walter Farkas

University of Zurich, Swiss Finance Institute (SFI) at Department of Banking and Finance; ETH Zürich - Department of Mathematics

Pablo Koch-Medina

University of Zurich - Department of Banking and Finance

Cosimo Munari

University of Zurich - Department of Banking and Finance

Date Written: September 12, 2013

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

JEL Classification: C60, G11, G22

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: https://ssrn.com/abstract=2053654 or http://dx.doi.org/10.2139/ssrn.2053654

Walter Farkas (Contact Author)

University of Zurich, Swiss Finance Institute (SFI) at Department of Banking and Finance ( email )

Plattenstrasse 14
CH-8032 Zurich, Zurich 8032
Switzerland
+41-44-634 3953 (Phone)
+41-44-634 4345 (Fax)

HOME PAGE: http://https://people.math.ethz.ch/~farkas/

ETH Zürich - Department of Mathematics ( email )

ETH Zentrum HG-F 42.1
Raemistr. 101
CH-8092 Zurich, 8092
Switzerland

HOME PAGE: http://https://people.math.ethz.ch/~farkas/

Pablo Koch-Medina

University of Zurich - Department of Banking and Finance ( email )

Schönberggasse 1
Zürich, 8001
Switzerland

Cosimo Munari

University of Zurich - Department of Banking and Finance ( email )

Schönberggasse 1
Zürich, 8001
Switzerland

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