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Loss-Based Risk Measures

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  • Rama Cont

    () (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - CNRS : UMR7599 - Université Pierre et Marie Curie - Paris VI - Université Paris Diderot - Paris 7, IEOR Dept - Industrial Engineering and Operations Research Department - Columbia University)

  • Romain Deguest

    (EDHEC RIsk Institute - École des hautes études commerciales du Nord (EDHEC))

  • Xuedong He

    (IEOR Dept - Industrial Engineering and Operations Research Department - Columbia University)

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    Abstract

    Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We characterize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative robustness of risk estimators and compare this criterion with sensitivity analysis of estimators based on influence functions. Finally, we provide examples of statistically robust estimators for loss-based risk measures.

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    File URL: http://hal.archives-ouvertes.fr/docs/00/62/99/29/PDF/LossBasedRiskMeasures.pdf
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    Bibliographic Info

    Paper provided by HAL in its series Working Papers with number hal-00629929.

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    Date of creation: 2011
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    Handle: RePEc:hal:wpaper:hal-00629929

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00629929/en/
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    Web page: http://hal.archives-ouvertes.fr/

    Related research

    Keywords: risk measure; coherent risk measure; Fenchel-Legendre transform; Choquet capacity;

    This paper has been announced in the following NEP Reports:

    References

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    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    3. Carlo Acerbi, 2007. "Coherent measures of risk in everyday market practice," Quantitative Finance, Taylor and Francis Journals, vol. 7(4), pages 359-364.
    4. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Quantitative Finance, Taylor and Francis Journals, vol. 10(6), pages 593-606.
    5. Touzi, Nizar & Schachermayer, Walter & Jouini, Elyès, 2006. "Law Invariant Risk Measures Have the Fatou Property," Open Access publications from Université Paris-Dauphine urn:hdl:123456789/342, Université Paris-Dauphine.
    6. Robert Jarrow, 2002. "Put Option Premiums and Coherent Risk Measures," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 135-142.
    7. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    8. Song, Yongsheng & Yan, Jia-An, 2009. "Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 459-465, December.
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