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# Dependence Modeling

Ed. by Puccetti, Giovanni

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# Are law-invariant risk functions concave on distributions?

Beatrice Acciaio
/ Gregor Svindland
Published Online: 2013-12-17 | DOI: https://doi.org/10.2478/demo-2013-0003

## 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.

MSC: 46N10; 60E15; 91B30

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Accepted: 2013-12-07

Published Online: 2013-12-17

Citation Information: Dependence Modeling, Volume 1, Pages 54–64, ISSN (Online) 2300-2298,

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©2013 Versita Sp. z o.o.. This content is open access.