Are law-invariant risk functions concave on distributions?

Beatrice Acciaio; Gregor Svindland

Dependence Modeling (2013)

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

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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