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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.

Suggested Citation

  • Carlo Acerbi, 2001. "Risk Aversion and Coherent Risk Measures: a Spectral Representation Theorem," Papers cond-mat/0107190, arXiv.org.
    • Handle: RePEc:arx:papers:cond-mat/0107190
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    References listed on IDEAS

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    1. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    2. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    3. Carlo Acerbi & Dirk Tasche, 2001. "Expected Shortfall: a natural coherent alternative to Value at Risk," Papers cond-mat/0105191, arXiv.org.
    4. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    5. Carlo Acerbi & Dirk Tasche, 2002. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 379-388, July.
    6. 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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    2. Omid Momen & Akbar Esfahanipour & Abbas Seifi, 2020. "A robust behavioral portfolio selection: model with investor attitudes and biases," Operational Research, Springer, vol. 20(1), pages 427-446, March.
    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. Choo, Weihao & de Jong, Piet, 2009. "Loss reserving using loss aversion functions," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 271-277, October.
    5. Pablo Lopez Sarabia, 2007. "Transmission And Impact Of The Financial Risk Of The European, Asian And American Stock Markets In The Return Of Mexican IPYC Index," EcoMod2007 23900054, EcoMod.
    6. Alexis Bonnet & Isabelle Nagot, 2005. "Methodology of measuring performance in alternative investment," Post-Print halshs-00196443, HAL.
    7. Mikhail Tselishchev, 2019. "On the Concavity of Expected Shortfall," Papers 1910.00640, arXiv.org.
    8. Alexis Bonnet & Isabelle Nagot, 2005. "Methodology of measuring performance in alternative investment," Cahiers de la Maison des Sciences Economiques b05078, Université Panthéon-Sorbonne (Paris 1).
    9. 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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