# Are law-invariant risk functions concave on distributions?

Beatrice Acciaio; Gregor Svindland

Dependence Modeling (2013)

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

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topBeatrice 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 -

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