On a class of law invariant convex risk measures

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2011
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Angelsberg, Gilles
Delbaen, Freddy
Kaelin, Ivo
Näf, Joachim
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We consider the class of law invariant convex risk measures with robust representation ρh,p(X)=supf10[AV@Rs(X)f(s)−fp(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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ISO 690ANGELSBERG, Gilles, Freddy DELBAEN, Ivo KAELIN, Michael KUPPER, Joachim NÄF, 2011. On a class of law invariant convex risk measures. In: Finance and Stochastics. Springer. 2011, 15, pp. 343-363. ISSN 0949-2984. eISSN 1432-1122. Available under: doi: 10.1007/s00780-010-0145-5
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@article{Angelsberg2011class-55458,
  year={2011},
  doi={10.1007/s00780-010-0145-5},
  title={On a class of law invariant convex risk measures},
  volume={15},
  issn={0949-2984},
  journal={Finance and Stochastics},
  pages={343--363},
  author={Angelsberg, Gilles and Delbaen, Freddy and Kaelin, Ivo and Kupper, Michael and Näf, Joachim}
}
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    <dcterms:abstract xml:lang="eng">We consider the class of law invariant convex risk measures with robust representation ρ&lt;sub&gt;h,p&lt;/sub&gt;(X)=sup&lt;sub&gt;f&lt;/sub&gt;∫&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;0&lt;/sub&gt;[AV@R&lt;sub&gt;s&lt;/sub&gt;(X)f(s)−f&lt;sup&gt;p&lt;/sup&gt;(s)h(s)]ds, where 1≤p&lt;∞ 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 &lt;sub&gt;h,p&lt;/sub&gt;(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.</dcterms:abstract>
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