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Are law-invariant risk functions concave on distributions?

Author

Listed:
  • Acciaio Beatrice

    (The London School of Economics and Political Science)

  • Svindland Gregor

    (University of Munich)

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.

Suggested Citation

  • Acciaio Beatrice & Svindland Gregor, 2013. "Are law-invariant risk functions concave on distributions?," Dependence Modeling, De Gruyter, vol. 1, pages 54-64, December.
    • Handle: RePEc:vrs:demode:v:1:y:2013:i::p:54-64:n:3
    DOI: 10.2478/demo-2013-0003
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    References listed on IDEAS

    as
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