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Minimal representation of insurance prices

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  • Pichler, Alois
  • Shapiro, Alexander

Abstract

This paper prices insurance contracts by employing law invariant, coherent risk measures from mathematical finance. We demonstrate that the corresponding premium principle enjoys a minimal representation. Uniqueness–in a sense specified in the paper–of this premium principle is derived from this initial result. The representations are derived from a result by Kusuoka, which is usually given for nonatomic probability spaces. We extend this setting to premium principles for spaces with atoms, as this is of particular importance for insurance.

Suggested Citation

  • Pichler, Alois & Shapiro, Alexander, 2015. "Minimal representation of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 184-193.
    • Handle: RePEc:eee:insuma:v:62:y:2015:i:c:p:184-193
    DOI: 10.1016/j.insmatheco.2015.03.011
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    References listed on IDEAS

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    1. Nilay Noyan & Gábor Rudolf, 2013. "Optimization with Multivariate Conditional Value-at-Risk Constraints," Operations Research, INFORMS, vol. 61(4), pages 990-1013, August.
    2. Alexander Shapiro, 2013. "On Kusuoka Representation of Law Invariant Risk Measures," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 142-152, February.
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    9. Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    10. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2010. "Kusuoka representation of higher order dual risk measures," Annals of Operations Research, Springer, vol. 181(1), pages 325-335, December.
    11. Alois Pichler, 2013. "Premiums And Reserves, Adjusted By Distortions," Papers 1304.0490, arXiv.org.
    12. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
    13. Deprez, Olivier & Gerber, Hans U., 1985. "On convex principles of premium calculation," Insurance: Mathematics and Economics, Elsevier, vol. 4(3), pages 179-189, July.
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    15. repec:dau:papers:123456789/342 is not listed on IDEAS
    16. Rose‐Anne Dana, 2005. "A Representation Result For Concave Schur Concave Functions," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 613-634, October.
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    Cited by:

    1. Wei Wang & Huifu Xu, 2023. "Preference robust distortion risk measure and its application," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 389-434, April.
    2. Alois Pichler, 2024. "Higher order measures of risk and stochastic dominance," Papers 2402.15387, arXiv.org.
    3. Egging, Ruud & Pichler, Alois & Kalvø, Øyvind Iversen & Walle–Hansen, Thomas Meyer, 2017. "Risk aversion in imperfect natural gas markets," European Journal of Operational Research, Elsevier, vol. 259(1), pages 367-383.
    4. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    5. Alexander Shapiro, 2016. "Rectangular Sets of Probability Measures," Operations Research, INFORMS, vol. 64(2), pages 528-541, April.
    6. 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.

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