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Risk Aversion and Coherent Risk Measures: a Spectral Representation Theorem

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  • Carlo Acerbi

Abstract

We study a space of coherent risk measures M_phi obtained as certain expansions of coherent elementary basis measures. In this space, the concept of ``Risk Aversion Function'' phi naturally arises as the spectral representation of each risk measure in a space of functions of confidence level probabilities. We give necessary and sufficient conditions on phi for M_phi to be a coherent measure. We find in this way a simple interpretation of the concept of coherence and a way to map any rational investor's subjective risk aversion onto a coherent measure and vice--versa. We also provide for these measures their discrete versions M_phi^N acting on finite sets of N independent realizations of a r.v. which are not only shown to be coherent measures for any fixed N, but also consistent estimators of M_phi for large N. Finally, we find in our results some interesting and not yet fully investigated relationships with certain results known in insurance mathematical literature.

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File URL: http://arxiv.org/pdf/cond-mat/0107190
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Paper provided by arXiv.org in its series Papers with number cond-mat/0107190.

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Date of creation: Jul 2001
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Handle: RePEc:arx:papers:cond-mat/0107190

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  1. Carlo Acerbi & Dirk Tasche, 2001. "Expected Shortfall: a natural coherent alternative to Value at Risk," Papers cond-mat/0105191, arXiv.org.
  2. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
  3. 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.
  4. Carlo Acerbi & Claudio Nordio & Carlo Sirtori, 2001. "Expected Shortfall as a Tool for Financial Risk Management," Papers cond-mat/0102304, arXiv.org.
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Cited by:
  1. Alexis Bonnet & Isabelle Nagot, 2005. "Methodology of measuring performance in alternative investment," Cahiers de la Maison des Sciences Economiques, Université Panthéon-Sorbonne (Paris 1) b05078, Université Panthéon-Sorbonne (Paris 1).
  2. Choo, Weihao & de Jong, Piet, 2009. "Loss reserving using loss aversion functions," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 271-277, October.
  3. Cadogan, Godfrey, 2009. "On behavioral Arrow Pratt risk process with applications to risk pricing, stochastic cash flows, and risk control," MPRA Paper 20174, University Library of Munich, Germany.
  4. Cadogan, Godfrey, 2010. "Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures," MPRA Paper 22380, University Library of Munich, Germany.

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