# Are law-invariant risk functions concave on distributions?

Dependence Modeling (2013)

• Volume: 1, page 54-64
• ISSN: 2300-2298

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## Abstract

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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.

## How to cite

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Beatrice Acciaio, and Gregor Svindland. "Are law-invariant risk functions concave on distributions?." Dependence Modeling 1 (2013): 54-64. <http://eudml.org/doc/266749>.

@article{BeatriceAcciaio2013,
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.},
author = {Beatrice Acciaio, Gregor Svindland},
journal = {Dependence Modeling},
keywords = {convexity; law-invariant risk measure; convex order; comonotonicity},
language = {eng},
pages = {54-64},
title = {Are law-invariant risk functions concave on distributions?},
url = {http://eudml.org/doc/266749},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Beatrice Acciaio
AU - Gregor Svindland
TI - Are law-invariant risk functions concave on distributions?
JO - Dependence Modeling
PY - 2013
VL - 1
SP - 54
EP - 64
AB - 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.
LA - eng
KW - convexity; law-invariant risk measure; convex order; comonotonicity
UR - http://eudml.org/doc/266749
ER -

## References

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