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Insurance premium-based shortfall risk measure induced by cumulative prospect theory

Author

Listed:
  • Sainan Zhang

    (The Chinese University of Hong Kong)

  • Huifu Xu

    (The Chinese University of Hong Kong)

Abstract

The risk measure of utility-based shortfall risk (SR) proposed by Föllmer and Schied (Finance Stoch 6:429–447, 2002) has been well studied in risk management and finance. In this paper, we revisit the concept from an insurance premium perspective. Under some moderate conditions, we show that the indifference equation-based insurance premium calculation can be equivalently formulated as an optimization problem similar to the definition of SR. Subsequently, we call the premium functional as an insurance premium-based shortfall risk measure (IPSR) defined over non-negative random variables. We then use the latter formulation to investigate the properties of the IPSR with a focus on the case that the preference functional is a distorted expected value function based on the cumulative prospect theory (CPT). Specifically, we exploit Weber’s approach (Weber in Math Finance Int J Math Stat Financ Econ 16:419–441, 2006) for characterization of the shortfall risk measure to derive a relationship between properties of IPSR induced by the CPT (IPSR-CPT) and the underlying value function in terms of convexity/concavity and positive homogeneity. We also investigate the IPSR-CPT as a functional of cumulative distribution function of random loss/liability and derive local and global Lipschitz continuity of the function under Wasserstein metric, a property which is related to statistical robustness of the IPSR-CPT. The results cover the premium risk measures based on the von Neumann-Morgenstern’s expected utility and Yaari’s dual theory of choice as special cases. Finally, we propose a computational scheme for calculating the IPSR-CPT.

Suggested Citation

  • Sainan Zhang & Huifu Xu, 2022. "Insurance premium-based shortfall risk measure induced by cumulative prospect theory," Computational Management Science, Springer, vol. 19(4), pages 703-738, October.
    • Handle: RePEc:spr:comgts:v:19:y:2022:i:4:d:10.1007_s10287-022-00432-0
    DOI: 10.1007/s10287-022-00432-0
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    References listed on IDEAS

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    1. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    2. Tiantian Mao & Jun Cai, 2018. "Risk measures based on behavioural economics theory," Finance and Stochastics, Springer, vol. 22(2), pages 367-393, April.
    3. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    4. Heilpern, S., 2003. "A rank-dependent generalization of zero utility principle," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 67-73, August.
    5. Denneberg, Dieter, 1990. "Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation1," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 181-190, November.
    6. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    7. Quiggin, John, 1982. "A theory of anticipated utility," Journal of Economic Behavior & Organization, Elsevier, vol. 3(4), pages 323-343, December.
    8. Itzhak Gilboa & David Schmeidler, 1992. "Additive Representation of Non-Additive Measures and the Choquet Integral," Discussion Papers 985, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    9. 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.
    10. Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    11. 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.
    12. Martina Nardon & Paolo Pianca, 2019. "Insurance premium calculation under continuous cumulative prospect theory," Working Papers 2019:03, Department of Economics, University of Venice "Ca' Foscari".
    13. Cai, Jun & Wang, Ying & Mao, Tiantian, 2017. "Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 105-116.
    14. Kaluszka, Marek & Krzeszowiec, Michał, 2012. "Pricing insurance contracts under Cumulative Prospect Theory," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 159-166.
    15. Stefan Weber, 2006. "Distribution‐Invariant Risk Measures, Information, And Dynamic Consistency," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 419-441, April.
    16. Wakker, Peter & Tversky, Amos, 1993. "An Axiomatization of Cumulative Prospect Theory," Journal of Risk and Uncertainty, Springer, vol. 7(2), pages 147-175, October.
    17. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    18. Marco Frittelli & Giacomo Scandolo, 2006. "Risk Measures And Capital Requirements For Processes," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 589-612, October.
    19. Daniel Cavagnaro & Mark Pitt & Richard Gonzalez & Jay Myung, 2013. "Discriminating among probability weighting functions using adaptive design optimization," Journal of Risk and Uncertainty, Springer, vol. 47(3), pages 255-289, December.
    20. Castaño-Martínez, Antonia & López-Blazquez, Fernando & Pigueiras, Gema & Sordo, Miguel Á., 2020. "A Method For Constructing And Interpreting Some Weighted Premium Principles," ASTIN Bulletin, Cambridge University Press, vol. 50(3), pages 1037-1064, September.
    21. Tsanakas, A. & Desli, E., 2003. "Risk Measures and Theories of Choice," British Actuarial Journal, Cambridge University Press, vol. 9(4), pages 959-991, October.
    22. Wei Wang & Huifu Xu & Tiejun Ma, 2021. "Quantitative statistical robustness for tail-dependent law invariant risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 21(10), pages 1669-1685, October.
    23. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    24. Fabio Bellini & Valeria Bignozzi, 2015. "On elicitable risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 725-733, May.
    25. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
    26. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    27. Drazen Prelec, 1998. "The Probability Weighting Function," Econometrica, Econometric Society, vol. 66(3), pages 497-528, May.
    28. 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.
    29. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    30. Daniela Escobar & Georg Pflug, 2018. "The distortion principle for insurance pricing: properties, identification and robustness," Papers 1809.06592, arXiv.org.
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