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Comparative and qualitative robustness for law-invariant risk measures

Author

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  • Volker Kratschmer
  • Alexander Schied
  • Henryk Zahle

Abstract

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.

Suggested Citation

  • Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
  • Handle: RePEc:arx:papers:1204.2458
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    File URL: http://arxiv.org/pdf/1204.2458
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    References listed on IDEAS

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    1. Andrzej Ruszczynski & Alexander Shapiro, 2004. "Optimization of Convex Risk Functions," Risk and Insurance 0404001, EconWPA, revised 08 Oct 2005.
    2. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    3. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    4. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    5. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214.
    6. 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.
    7. 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.
    8. Denneberg, Dieter, 1990. "Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 20(02), pages 181-190, November.
    9. 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.
    10. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 26(01), pages 71-92, May.
    11. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
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    Citations

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    Cited by:

    1. Koch-Medina Pablo & Munari Cosimo, 2014. "Law-invariant risk measures: Extension properties and qualitative robustness," Statistics & Risk Modeling, De Gruyter, vol. 31(3-4), pages 1-22, December.
    2. repec:eee:ejores:v:262:y:2017:i:2:p:720-732 is not listed on IDEAS
    3. Zähle, Henryk, 2016. "A definition of qualitative robustness for general point estimators, and examples," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 12-31.
    4. Lauer Alexandra & Zähle Henryk, 2016. "Nonparametric estimation of risk measures of collective risks," Statistics & Risk Modeling, De Gruyter, vol. 32(2), pages 89-102, March.
    5. Matteo Burzoni & Ilaria Peri & Chiara Maria Ruffo, 2016. "On the properties of the Lambda value at risk: robustness, elicitability and consistency," Papers 1603.09491, arXiv.org, revised Feb 2017.
    6. Freddy Delbaen & Fabio Bellini & Valeria Bignozzi & Johanna F. Ziegel, 2014. "Risk measures with the CxLS property," Papers 1411.0426, arXiv.org.
    7. repec:eee:jmvana:v:158:y:2017:i:c:p:1-19 is not listed on IDEAS
    8. Krätschmer Volker & Schied Alexander & Zähle Henryk, 2015. "Quasi-Hadamard differentiability of general risk functionals and its application," Statistics & Risk Modeling, De Gruyter, vol. 32(1), pages 25-47, April.
    9. repec:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0328-4 is not listed on IDEAS
    10. Marcelo Brutti Righi, 2017. "A risk measure that optimally balances capital determination errors," Papers 1707.09829, arXiv.org.
    11. repec:spr:finsto:v:22:y:2018:i:1:d:10.1007_s00780-017-0347-1 is not listed on IDEAS
    12. Ruodu Wang & Johanna F. Ziegel, 2014. "Distortion Risk Measures and Elicitability," Papers 1405.3769, arXiv.org, revised May 2014.
    13. Niushan Gao & Denny H. Leung & Cosimo Munari & Foivos Xanthos, 2017. "Fatou Property, representations, and extensions of law-invariant risk measures on general Orlicz spaces," Papers 1701.05967, arXiv.org, revised Sep 2017.
    14. Daniel Lacker, 2015. "Law invariant risk measures and information divergences," Papers 1510.07030, arXiv.org, revised Jun 2016.
    15. repec:eee:insuma:v:74:y:2017:i:c:p:99-108 is not listed on IDEAS
    16. Tobias Fissler & Johanna F. Ziegel, 2015. "Higher order elicitability and Osband's principle," Papers 1503.08123, arXiv.org, revised Sep 2015.
    17. Xue Dong He & Xianhua Peng, 2017. "Surplus-Invariant, Law-Invariant, and Conic Acceptance Sets Must be the Sets Induced by Value-at-Risk," Papers 1707.05596, arXiv.org, revised Jan 2018.
    18. Freddy Delbaen & Fabio Bellini & Valeria Bignozzi & Johanna F. Ziegel, 2016. "Risk measures with the CxLS property," Finance and Stochastics, Springer, vol. 20(2), pages 433-453, April.

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