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Comparative and qualitative robustness for law-invariant risk measures

  • Volker Kr\"atschmer
  • Alexander Schied
  • Henryk Z\"ahle
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    When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel's classical notion of qualitative robustness is not suitable for risk measurement and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz space. This concept of robustness captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results that are of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for {\psi}-weak convergence.

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    Paper provided by in its series Papers with number 1204.2458.

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    Date of creation: Apr 2012
    Date of revision: Jan 2014
    Handle: RePEc:arx:papers:1204.2458
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    1. Alexander Cherny & Dilip Madan, 2009. "New Measures for Performance Evaluation," Review of Financial Studies, Society for Financial Studies, vol. 22(7), pages 2371-2406, July.
    2. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    3. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 593-606.
    4. Denis Belomestny & Volker Krätschmer, 2010. "Central limit theorems for law-invariant coherent risk measures," SFB 649 Discussion Papers SFB649DP2010-052, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    5. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Optimization of Convex Risk Functions," Risk and Insurance 0404001, EconWPA, revised 08 Oct 2005.
    6. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214.
    7. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    8. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    9. Carlo Acerbi & Dirk Tasche, 2001. "On the coherence of Expected Shortfall," Papers cond-mat/0104295,, revised May 2002.
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