# Are law-invariant risk functions concave on distributions?

Are law-invariant risk functions concave on distributions? AbstractWhile it is reasonable to assume that convex combinationson 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 correspondsto adding a risk factor. Hence, whereas asking forconvexity of risk functions defined on random variables makessense, convexity is not a good property to require on risk functionsdefined on distributions. In this paper we study the interplaybetween convexity of law-invariant risk functions on randomvariables and convexity/concavity of their counterpartson distributions. We show that, given a law-invariant convexrisk measure, on the level of distributions, if at all, concavityholds true. In particular, this is always the case under theadditional assumption of comonotonicity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Dependence Modeling de Gruyter

# Are law-invariant risk functions concave on distributions?

, Volume 1: 11 – Jan 1, 2013
11 pages

/lp/de-gruyter/are-law-invariant-risk-functions-concave-on-distributions-OxqseZIFL0
Publisher
de Gruyter
ISSN
2300-2298
eISSN
2300-2298
DOI
10.2478/demo-2013-0003
Publisher site
See Article on Publisher Site

### Abstract

AbstractWhile it is reasonable to assume that convex combinationson 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 correspondsto adding a risk factor. Hence, whereas asking forconvexity of risk functions defined on random variables makessense, convexity is not a good property to require on risk functionsdefined on distributions. In this paper we study the interplaybetween convexity of law-invariant risk functions on randomvariables and convexity/concavity of their counterpartson distributions. We show that, given a law-invariant convexrisk measure, on the level of distributions, if at all, concavityholds true. In particular, this is always the case under theadditional assumption of comonotonicity.

### Journal

Dependence Modelingde Gruyter

Published: Jan 1, 2013