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On a class of law invariant convex risk measures

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Abstract

We consider the class of law invariant convex risk measures with robust representation \(\rho_{h,p}(X)=\sup_{f}\int_{0}^{1} [AV@R_{s}(X)f(s)-f^{p}(s)h(s)]\,ds\), where 1≤p<∞ and h is a positive and strictly decreasing function. The supremum is taken over the set of all Radon–Nikodým derivatives corresponding to the set of all probability measures on (0,1] which are absolutely continuous with respect to Lebesgue measure. We provide necessary and sufficient conditions for the position X such that ρ h,p(X) is real-valued and the supremum is attained. Using variational methods, an explicit formula for the maximizer is given. We exhibit two examples of such risk measures and compare them to the average value at risk.

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Authors and Affiliations

  1. Commissariat aux Assurances, 2449, Luxembourg, Luxembourg

    Gilles Angelsberg

  2. Department of Mathematics, ETH Zurich, 8092, Zürich, Switzerland

    Freddy Delbaen & Ivo Kaelin

  3. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Kupper

  4. Institute for Theoretical Physics, University of Zurich, 8057, Zürich, Switzerland

    Joachim Näf

Authors
  1. Gilles Angelsberg
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  2. Freddy Delbaen
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  3. Ivo Kaelin
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  4. Michael Kupper
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  5. Joachim Näf
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Correspondence to Michael Kupper.

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Angelsberg, G., Delbaen, F., Kaelin, I. et al. On a class of law invariant convex risk measures. Finance Stoch 15, 343–363 (2011). https://doi.org/10.1007/s00780-010-0145-5

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  • DOI: https://doi.org/10.1007/s00780-010-0145-5

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