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Beyond cash-additive risk measures: when changing the numéraire fails

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  • Walter Farkas
  • Pablo Koch-Medina
  • Cosimo Munari

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 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. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2014. "Beyond cash-additive risk measures: when changing the numéraire fails," Finance and Stochastics, Springer, vol. 18(1), pages 145-173, January.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:1:p:145-173
    DOI: 10.1007/s00780-013-0220-9
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    References listed on IDEAS

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    1. Marco Frittelli & Giacomo Scandolo, 2006. "Risk Measures And Capital Requirements For Processes," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 589-612, October.
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    6. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2013. "Capital Requirements with Defaultable Securities," Swiss Finance Institute Research Paper Series 13-66, Swiss Finance Institute.
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    9. Konstantinides, Dimitrios G. & Kountzakis, Christos E., 2011. "Risk measures in ordered normed linear spaces with non-empty cone-interior," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 111-122, January.
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    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Risk measures; Acceptance sets; General eligible assets; Defaultable bonds; Cash subadditivity; Quasiconvexity; Value-at-risk; Tail value-at-risk; Shortfall risk; 91B30; 46B42; 46B40; 46A55; 06F30; C60; G11; G22;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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