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Vigilant measures of risk and the demand for contingent claims

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  • Ghossoub, Mario

Abstract

We examine a class of utility maximization problems with a non-necessarily law-invariant utility, and with a non-necessarily law-invariant risk measure constraint. Under a consistency requirement on the risk measure that we call Vigilance, we show the existence of optimal contingent claims, and we show that such optimal contingent claims exhibit a desired monotonicity property. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures and some classes of robust risk measures. As an illustration, we consider a problem of optimal insurance design where the premium principle satisfies the vigilance property, hence covering a large collection of commonly used premium principles, including premium principles that are not law-invariant. We show the existence of optimal indemnity schedules, and we show that optimal indemnity schedules are nondecreasing functions of the insurable loss.

Suggested Citation

  • Ghossoub, Mario, 2015. "Vigilant measures of risk and the demand for contingent claims," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 27-35.
  • Handle: RePEc:eee:insuma:v:61:y:2015:i:c:p:27-35
    DOI: 10.1016/j.insmatheco.2014.11.009
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    References listed on IDEAS

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    1. Dana, R. A., 2004. "Market behavior when preferences are generated by second-order stochastic dominance," Journal of Mathematical Economics, Elsevier, vol. 40(6), pages 619-639, September.
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    1. repec:gam:jrisks:v:4:y:2016:i:1:p:8:d:66161 is not listed on IDEAS
    2. Amarante, Massimiliano & Ghossoub, Mario & Phelps, Edmund, 2015. "Ambiguity on the insurer’s side: The demand for insurance," Journal of Mathematical Economics, Elsevier, vol. 58(C), pages 61-78.
    3. Mario Ghossoub, 2015. "Equimeasurable Rearrangements with Capacities," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 429-445, February.
    4. Massimiliano Amarante & Mario Ghossoub, 2016. "Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer," Risks, MDPI, Open Access Journal, vol. 4(1), pages 1-27, March.
    5. Mario Ghossoub, 2016. "Optimal Insurance with Heterogeneous Beliefs and Disagreement about Zero-Probability Events," Risks, MDPI, Open Access Journal, vol. 4(3), pages 1-28, August.

    More about this item

    Keywords

    Utility maximization; Optimal insurance design; Choquet integral; Distorted probabilities; Monotone Likelihood Ratio;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • D89 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Other
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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