Are law-invariant risk functions concave on distributions?

Acciaio Beatrice () and Svindland Gregor ()
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Acciaio Beatrice: The London School of Economics and Political Science
Svindland Gregor: University of Munich

Dependence Modeling, 2013, vol. 1, issue 2013, 54-64

Abstract: While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on random variables makes sense, convexity is not a good property to require on risk functions defined on distributions. In this paper we study the interplay between convexity of law-invariant risk functions on random variables and convexity/concavity of their counterparts on distributions. We show that, given a law-invariant convex risk measure, on the level of distributions, if at all, concavity holds true. In particular, this is always the case under the additional assumption of comonotonicity.

Keywords: convexity; law-invariant risk measure; convex order; comonotonicity (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:1:y:2013:i::p:54-64:n:3

DOI: 10.2478/demo-2013-0003

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