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Qualitative robustness of set-valued value-at-risk

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  • Giovanni Paolo Crespi

    (Universitá degli Studi dell’Insubria)

  • Elisa Mastrogiacomo

    (Universitá degli Studi dell’Insubria)

Abstract

Risk measures are defined as functionals of the portfolio loss distribution, thus implicitly assuming the knowledge of such a distribution. However, in practical applications, the need for estimation arises and with it the need to study the effects of mis-specification errors, as well as estimation errors on the final conclusion. In this paper we focus on the qualitative robustness of a sequence of estimators for set-valued risk measures. These properties are studied in detail for two well-known examples of set-valued risk measures: the value-at-risk and the maximum average value-at-risk. Our results illustrate, in particular, that estimation of set-valued value-at-risk can be given in terms of random sets. Moreover, we observe that historical set-valued value-at-risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as the maximum likelihood average value at-risk.

Suggested Citation

  • Giovanni Paolo Crespi & Elisa Mastrogiacomo, 2020. "Qualitative robustness of set-valued value-at-risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 25-54, February.
    • Handle: RePEc:spr:mathme:v:91:y:2020:i:1:d:10.1007_s00186-020-00707-9
    DOI: 10.1007/s00186-020-00707-9
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    1. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    2. Dobrislav Dobrev∗ & Travis D. Nesmith & Dong Hwan Oh, 2017. "Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors," JRFM, MDPI, vol. 10(1), pages 1-14, February.
    3. 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.
    4. Dominik Wied & Rafael Weißbach, 2012. "Consistency of the kernel density estimator: a survey," Statistical Papers, Springer, vol. 53(1), pages 1-21, February.
    5. 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.
    6. Andreas H. Hamel & Birgit Rudloff & Mihaela Yankova, 2012. "Set-valued average value at risk and its computation," Papers 1202.5702, arXiv.org, revised Jan 2013.
    7. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    8. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    9. Carlo Acerbi, 2007. "Coherent measures of risk in everyday market practice," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 359-364.
    10. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    11. Tasche, Dirk, 2002. "Expected shortfall and beyond," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1519-1533, July.
    12. Ilya S. Molchanov, 1998. "A Limit Theorem for Solutions of Inequalities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 235-242, March.
    13. Gabriella Salinetti & Roger J.-B. Wets, 1986. "On the Convergence in Distribution of Measurable Multifunctions (Random Sets) Normal Integrands, Stochastic Processes and Stochastic Infima," Mathematics of Operations Research, INFORMS, vol. 11(3), pages 385-419, August.
    14. Norberg, Tommy, 1987. "Semicontinuous processes in multi-dimensional extreme value theory," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 27-55.
    15. 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.
    16. repec:dau:papers:123456789/353 is not listed on IDEAS
    17. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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    Cited by:

    1. Elisa Mastrogiacomo & Matteo Rocca, 2021. "Set optimization of set-valued risk measures," Annals of Operations Research, Springer, vol. 296(1), pages 291-314, January.

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