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Robust distortion risk measures

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  • Carole Bernard
  • Silvana M. Pesenti
  • Steven Vanduffel

Abstract

The robustness of risk measures to changes in underlying loss distributions (distributional uncertainty) is of crucial importance in making well‐informed decisions. In this paper, we quantify, for the class of distortion risk measures with an absolutely continuous distortion function, its robustness to distributional uncertainty by deriving its largest (smallest) value when the underlying loss distribution has a known mean and variance and, furthermore, lies within a ball—specified through the Wasserstein distance—around a reference distribution. We employ the technique of isotonic projections to provide for these distortion risk measures a complete characterization of sharp bounds on their value, and we obtain quasi‐explicit bounds in the case of Value‐at‐Risk and Range‐Value‐at‐Risk. We extend our results to account for uncertainty in the first two moments and provide applications to portfolio optimization and to model risk assessment.

Suggested Citation

  • Carole Bernard & Silvana M. Pesenti & Steven Vanduffel, 2024. "Robust distortion risk measures," Mathematical Finance, Wiley Blackwell, vol. 34(3), pages 774-818, July.
    • Handle: RePEc:bla:mathfi:v:34:y:2024:i:3:p:774-818
    DOI: 10.1111/mafi.12414
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    Cited by:

    1. Silvana M. Pesenti & Steven Vanduffel, 2023. "Optimal Transport Divergences induced by Scoring Functions," Papers 2311.12183, arXiv.org, revised Apr 2024.
    2. Ariel Neufeld & Matthew Ng Cheng En & Ying Zhang, 2024. "Robust SGLD algorithm for solving non-convex distributionally robust optimisation problems," Papers 2403.09532, arXiv.org.
    3. Mengshuo Zhao & Narayanaswamy Balakrishnan & Chuancun Yin, 2024. "Extremal cases of distortion risk measures with partial information," Papers 2404.13637, arXiv.org.
    4. Marlon Moresco & M'elina Mailhot & Silvana M. Pesenti, 2023. "Uncertainty Propagation and Dynamic Robust Risk Measures," Papers 2308.12856, arXiv.org, revised Feb 2024.
    5. Boonen, Tim J. & Jiang, Wenjun, 2024. "Robust insurance design with distortion risk measures," European Journal of Operational Research, Elsevier, vol. 316(2), pages 694-706.
    6. Baishuai Zuo & Chuancun Yin, 2024. "Worst-cases of distortion riskmetrics and weighted entropy with partial information," Papers 2405.19075, arXiv.org.
    7. Marcelo Righi, 2024. "Robust convex risk measures," Papers 2406.12999, arXiv.org.

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