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Robustness in the Optimization of Risk Measures

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  • Paul Embrechts
  • Alexander Schied
  • Ruodu Wang

Abstract

We study issues of robustness in the context of Quantitative Risk Management and Optimization. We develop a general methodology for determining whether a given risk measurement related optimization problem is robust, which we call "robustness against optimization". The new notion is studied for various classes of risk measures and expected utility and loss functions. Motivated by practical issues from financial regulation, special attention is given to the two most widely used risk measures in the industry, Value-at-Risk (VaR) and Expected Shortfall (ES). We establish that for a class of general optimization problems, VaR leads to non-robust optimizers whereas convex risk measures generally lead to robust ones. Our results offer extra insight on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the field of robust optimization, to which some interesting links are derived.

Suggested Citation

  • Paul Embrechts & Alexander Schied & Ruodu Wang, 2018. "Robustness in the Optimization of Risk Measures," Papers 1809.09268, arXiv.org, revised Feb 2021.
    • Handle: RePEc:arx:papers:1809.09268
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    References listed on IDEAS

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    1. Acharya, Viral V. & Cooley, Thomas & Richardson, Matthew & Walter, Ingo, 2010. "Manufacturing Tail Risk: A Perspective on the Financial Crisis of 2007–2009," Foundations and Trends(R) in Finance, now publishers, vol. 4(4), pages 247-325, April.
    2. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    3. Steven Kou & Xianhua Peng, 2016. "On the Measurement of Economic Tail Risk," Operations Research, INFORMS, vol. 64(5), pages 1056-1072, October.
    4. Paul Embrechts & Giovanni Puccetti & Ludger Rüschendorf & Ruodu Wang & Antonela Beleraj, 2014. "An Academic Response to Basel 3.5," Risks, MDPI, vol. 2(1), pages 1-24, 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. Paul Embrechts & Bin Wang & Ruodu Wang, 2015. "Aggregation-robustness and model uncertainty of regulatory risk measures," Finance and Stochastics, Springer, vol. 19(4), pages 763-790, October.
    7. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2017. "Domains of weak continuity of statistical functionals with a view toward robust statistics," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 1-19.
    8. 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.
    9. Paul Embrechts & Haiyan Liu & Tiantian Mao & Ruodu Wang, 2017. "Quantile-Based Risk Sharing with Heterogeneous Beliefs," Swiss Finance Institute Research Paper Series 17-65, Swiss Finance Institute, revised Jan 2018.
    10. Karthik Natarajan & Dessislava Pachamanova & Melvyn Sim, 2008. "Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization," Management Science, INFORMS, vol. 54(3), pages 573-585, March.
    11. Wolfram Wiesemann & Daniel Kuhn & Melvyn Sim, 2014. "Distributionally Robust Convex Optimization," Operations Research, INFORMS, vol. 62(6), pages 1358-1376, December.
    12. Shushang Zhu & Masao Fukushima, 2009. "Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management," Operations Research, INFORMS, vol. 57(5), pages 1155-1168, October.
    13. Steven Kou & Xianhua Peng & Chris C. Heyde, 2013. "External Risk Measures and Basel Accords," Mathematics of Operations Research, INFORMS, vol. 38(3), pages 393-417, August.
    14. Jun Sekine, 2004. "Dynamic Minimization of Worst Conditional Expectation of Shortfall," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 605-618, October.
    15. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    16. Susanne Emmer & Marie Kratz & Dirk Tasche, 2013. "What is the best risk measure in practice? A comparison of standard measures," Papers 1312.1645, arXiv.org, revised Apr 2015.
    17. Quaranta, Anna Grazia & Zaffaroni, Alberto, 2008. "Robust optimization of conditional value at risk and portfolio selection," Journal of Banking & Finance, Elsevier, vol. 32(10), pages 2046-2056, October.
    18. Joel Goh & Melvyn Sim, 2010. "Distributionally Robust Optimization and Its Tractable Approximations," Operations Research, INFORMS, vol. 58(4-part-1), pages 902-917, August.
    19. Georg Ch Pflug & Werner Römisch, 2007. "Modeling, Measuring and Managing Risk," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6478, August.
    20. Michel Baes & Pablo Koch-Medina & Cosimo Munari, 2017. "Existence, uniqueness and stability of optimal portfolios of eligible assets," Papers 1702.01936, arXiv.org, revised Dec 2017.
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    Cited by:

    1. Xia Han & Bin Wang & Ruodu Wang & Qinyu Wu, 2021. "Risk Concentration and the Mean-Expected Shortfall Criterion," Papers 2108.05066, arXiv.org, revised Apr 2022.
    2. Henryk Zähle, 2022. "A concept of copula robustness and its applications in quantitative risk management," Finance and Stochastics, Springer, vol. 26(4), pages 825-875, October.
    3. Tolulope Fadina & Yang Liu & Ruodu Wang, 2021. "A Framework for Measures of Risk under Uncertainty," Papers 2110.10792, arXiv.org, revised Sep 2023.
    4. Martin Herdegen & Nazem Khan, 2022. "$\rho$-arbitrage and $\rho$-consistent pricing for star-shaped risk measures," Papers 2202.07610, arXiv.org, revised May 2024.

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