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The strictest common relaxation of a family of risk measures

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  • Roorda, Berend
  • Schumacher, J.M.

Abstract

Operations which form new risk measures from a collection of given (often simpler) risk measures have been used extensively in the literature. Examples include convex combination, convolution, and the worst-case operator. Here we study the risk measure that is constructed from a family of given risk measures by the best-case operator; that is, the newly constructed risk measure is defined as the one that is as restrictive as possible under the condition that it accepts all positions that are accepted under any of the risk measures from the family. In fact we define this operation for conditional risk measures, to allow a multiperiod setting. We show that the well-known VaR risk measure can be constructed from a family of conditional expectations by a combination that involves both worst-case and best-case operations. We provide an explicit description of the acceptance set of the conditional risk measure that is obtained as the strictest common relaxation of two given conditional risk measures.

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Bibliographic Info

Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 48 (2011)
Issue (Month): 1 (January)
Pages: 29-34

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Handle: RePEc:eee:insuma:v:48:y:2011:i:1:p:29-34

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Web page: http://www.elsevier.com/locate/inca/505554

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Keywords: Nonconvex risk measures Value at risk Best-case operator;

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  1. Pauline Barrieu & Nicole El Karoui, 2005. "Inf-convolution of risk measures and optimal risk transfer," Finance and Stochastics, Springer, Springer, vol. 9(2), pages 269-298, 04.
  2. Berend Roorda & J. M. Schumacher & Jacob Engwerda, 2005. "Coherent Acceptability Measures In Multiperiod Models," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 15(4), pages 589-612.
  3. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2005. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, Springer, vol. 9(3), pages 369-387, 07.
  4. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  5. Frank Riedel, 2003. "Dynamic Coherent Risk Measures," Working Papers, Stanford University, Department of Economics 03004, Stanford University, Department of Economics.
  6. Susanne Klöppel & Martin Schweizer, 2007. "Dynamic Indifference Valuation Via Convex Risk Measures," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 17(4), pages 599-627.
  7. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(3), pages 203-228.
  8. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, Springer, vol. 9(4), pages 539-561, October.
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