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Robustness in the Optimization of Risk Measures

Author

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  • Paul Embrechts

    (Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland; ETH Risk Center, ETH Zurich, 8092 Zurich, Switzerland)

  • Alexander Schied

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L3G1, Canada)

  • Ruodu Wang

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L3G1, Canada)

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 nonrobust 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, 2022. "Robustness in the Optimization of Risk Measures," Operations Research, INFORMS, vol. 70(1), pages 95-110, January.
    • Handle: RePEc:inm:oropre:v:70:y:2022:i:1:p:95-110
    DOI: 10.1287/opre.2021.2147
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