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.
@article{bwmeta1.element.doi-10_2478_demo-2013-0003,
author = {Beatrice Acciaio and Gregor Svindland},
title = {Are law-invariant risk functions concave on distributions?},
journal = {Dependence Modeling},
volume = {1},
year = {2013},
pages = {54-64},
zbl = {1290.91072},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_demo-2013-0003}
}
Beatrice Acciaio; Gregor Svindland. Are law-invariant risk functions concave on distributions?. Dependence Modeling, Tome 1 (2013) pp. 54-64. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_demo-2013-0003/
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