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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

    as
    1. Chi, Yichun & Tan, Ken Seng, 2013. "Optimal reinsurance with general premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 180-189.
    2. Cai, Jun & Wei, Wei, 2012. "Optimal reinsurance with positively dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 57-63.
    3. Furman, Edward & Wang, Ruodu & Zitikis, Ričardas, 2017. "Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks," Journal of Banking & Finance, Elsevier, vol. 83(C), pages 70-84.
    4. Cai, Jun & Tan, Ken Seng, 2007. "Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 93-112, May.
    5. repec:dau:papers:123456789/361 is not listed on IDEAS
    6. Stefan Weber, 2006. "Distribution‐Invariant Risk Measures, Information, And Dynamic Consistency," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 419-441, April.
    7. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    8. R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 51-74, January.
    9. Deprez, Olivier & Gerber, Hans U., 1985. "On convex principles of premium calculation," Insurance: Mathematics and Economics, Elsevier, vol. 4(3), pages 179-189, July.
    10. Cai, Jun & Tan, Ken Seng & Weng, Chengguo & Zhang, Yi, 2008. "Optimal reinsurance under VaR and CTE risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 185-196, August.
    11. 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.
    12. E. Jouini & W. Schachermayer & N. Touzi, 2008. "Optimal Risk Sharing For Law Invariant Monetary Utility Functions," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 269-292, April.
    13. 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.
    14. Bogdan Grechuk & Anton Molyboha & Michael Zabarankin, 2009. "Maximum Entropy Principle with General Deviation Measures," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 445-467, May.
    15. Chi, Yichun & Liu, Fangda, 2017. "Optimal insurance design in the presence of exclusion clauses," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 185-195.
    16. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    17. 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.
    18. Cai, Jun & Liu, Haiyan & Wang, Ruodu, 2017. "Pareto-optimal reinsurance arrangements under general model settings," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 24-37.
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    1. Tim J. Boonen & Yuyu Chen & Xia Han & Qiuqi Wang, 2024. "Optimal insurance design with Lambda-Value-at-Risk," Papers 2408.09799, arXiv.org.
    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, February.
    3. Xia Han & Ruodu Wang & Xun Yu Zhou, 2022. "Choquet regularization for reinforcement learning," Papers 2208.08497, arXiv.org.
    4. Baishuai Zuo & Chuancun Yin, 2024. "Worst-cases of distortion riskmetrics and weighted entropy with partial information," Papers 2405.19075, arXiv.org.
    5. 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.
    6. Cai, Jun & Liu, Fangda & Yin, Mingren, 2024. "Worst-case risk measures of stop-loss and limited loss random variables under distribution uncertainty with applications to robust reinsurance," European Journal of Operational Research, Elsevier, vol. 318(1), pages 310-326.
    7. Mengshuo Zhao & Narayanaswamy Balakrishnan & Chuancun Yin, 2024. "Extremal cases of distortion risk measures with partial information," Papers 2404.13637, arXiv.org, revised Oct 2024.
    8. Mengshuo Zhao & Chuancun Yin, 2024. "Best- and worst-case Scenarios for GlueVaR distortion risk measure with Incomplete information," Papers 2409.19902, arXiv.org.
    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. Yuanying Guan & Zhanyi Jiao & Ruodu Wang, 2022. "A reverse ES (CVaR) optimization formula," Papers 2203.02599, arXiv.org, revised May 2023.
    11. Silvana Pesenti & Qiuqi Wang & Ruodu Wang, 2020. "Optimizing distortion riskmetrics with distributional uncertainty," Papers 2011.04889, arXiv.org, revised Feb 2022.
    12. Boonen, Tim J. & Han, Xia, 2024. "Optimal insurance with mean-deviation measures," Insurance: Mathematics and Economics, Elsevier, vol. 118(C), pages 1-24.
    13. 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.
    14. 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.
    15. 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.
    16. Xia Han & Ruodu Wang & Qinyu Wu, 2023. "Monotonic mean-deviation risk measures," Papers 2312.01034, arXiv.org, revised Aug 2024.
    17. Ruodu Wang & Qinyu Wu, 2022. "Probabilistic risk aversion for generalized rank-dependent functions," Papers 2209.03425, arXiv.org, revised Sep 2024.
    18. Tim J. Boonen & Xia Han, 2023. "Optimal insurance with mean-deviation measures," Papers 2312.01813, arXiv.org.

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