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Convex risk functionals: Representation and applications

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  • Liu, Fangda
  • Cai, Jun
  • Lemieux, Christiane
  • Wang, Ruodu

Abstract

We introduce the family of law-invariant convex risk functionals, which includes a wide majority of practically used convex risk measures and deviation measures. We obtain a unified representation theorem for this family of functionals. Two related optimization problems are studied. In the first application, we determine worst-case values of a law-invariant convex risk functional when the mean and a higher moment such as the variance of a risk are known. Second, we consider its application in optimal reinsurance design for an insurer. With the help of the representation theorem, we can show the existence and the form of optimal solutions.

Suggested Citation

  • Liu, Fangda & Cai, Jun & Lemieux, Christiane & Wang, Ruodu, 2020. "Convex risk functionals: Representation and applications," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 66-79.
    • Handle: RePEc:eee:insuma:v:90:y:2020:i:c:p:66-79
    DOI: 10.1016/j.insmatheco.2019.10.007
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    References listed on IDEAS

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    Cited by:

    1. Yuanying Guan & Zhanyi Jiao & Ruodu Wang, 2022. "A reverse ES (CVaR) optimization formula," Papers 2203.02599, arXiv.org, revised May 2023.
    2. Felix-Benedikt Liebrich & Cosimo Munari, 2022. "Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity," Mathematics and Financial Economics, Springer, volume 16, number 2, June.
    3. Silvana Pesenti & Qiuqi Wang & Ruodu Wang, 2020. "Optimizing distortion riskmetrics with distributional uncertainty," Papers 2011.04889, arXiv.org, revised Feb 2022.
    4. Xia Han & Ruodu Wang & Xun Yu Zhou, 2022. "Choquet regularization for reinforcement learning," Papers 2208.08497, arXiv.org.
    5. Nendel, Max & Riedel, Frank & Schmeck, Maren Diane, 2021. "A decomposition of general premium principles into risk and deviation," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 193-209.
    6. Chi, Yichun & Liu, Fangda, 2021. "Enhancing an insurer's expected value by reinsurance and external financing," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 466-484.
    7. Fabio Bellini & Tolulope Fadina & Ruodu Wang & Yunran Wei, 2020. "Parametric measures of variability induced by risk measures," Papers 2012.05219, arXiv.org, revised Apr 2022.
    8. Wakker, Peter P. & Yang, Jingni, 2021. "Concave/convex weighting and utility functions for risk: A new light on classical theorems," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 429-435.
    9. Bellini, Fabio & Fadina, Tolulope & Wang, Ruodu & Wei, Yunran, 2022. "Parametric measures of variability induced by risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 270-284.
    10. Xia Han & Ruodu Wang & Qinyu Wu, 2023. "Monotonic mean-deviation risk measures," Papers 2312.01034, arXiv.org.
    11. Ruodu Wang & Qinyu Wu, 2022. "Quasi-convexity in mixtures for generalized rank-dependent functions," Papers 2209.03425, arXiv.org.
    12. Tim J. Boonen & Xia Han, 2023. "Optimal insurance with mean-deviation measures," Papers 2312.01813, arXiv.org.

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