IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-01983433.html
   My bibliography  Save this paper

Prevention efforts, insurance demand and price incentives under coherent risk measures

Author

Listed:
  • Sarah Bensalem

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Nicolás Hernández Santibáñez

    (Department of Mathematics - University of Michigan - University of Michigan [Ann Arbor] - University of Michigan System)

  • Nabil Kazi-Tani

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

This paper studies an equilibrium model between an insurance buyer and an insurance seller, where both parties' risk preferences are given by convex risk measures. The interaction is modeled through a Stackelberg type game, where the insurance seller plays first by offering prices, in the form of safety loadings. Then the insurance buyer chooses his optimal proportional insurance share and his optimal prevention effort in order to minimize his risk measure. The loss distribution is given by a family of stochastically ordered probability measures, indexed by the prevention effort. We give special attention to the problems of self-insurance and self-protection. We prove that the formulated game admits a unique equilibrium, that we can explicitly solve by further specifying the agents criteria and the loss distribution. In self-insurance, we consider also an adverse selection setting, where the type of the insurance buyers is given by his loss probability, and study the screening and shutdown contracts. Finally, we provide case studies in which we explicitly apply our theoretical results.

Suggested Citation

  • Sarah Bensalem & Nicolás Hernández Santibáñez & Nabil Kazi-Tani, 2019. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Working Papers hal-01983433, HAL.
  • Handle: RePEc:hal:wpaper:hal-01983433
    Note: View the original document on HAL open archive server: https://hal.science/hal-01983433
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01983433/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Christian Gollier & James Hammitt & Nicolas Treich, 2013. "Risk and choice: A research saga," Journal of Risk and Uncertainty, Springer, vol. 47(2), pages 129-145, October.
    3. Courbage, Christophe & Rey, Béatrice & Treich, Nicolas, 2013. "Prevention and precaution," IDEI Working Papers 805, Institut d'Économie Industrielle (IDEI), Toulouse.
    4. Ian Jewitt, 1989. "Choosing Between Risky Prospects: The Characterization of Comparative Statics Results, and Location Independent Risk," Management Science, INFORMS, vol. 35(1), pages 60-70, January.
    5. Rogerson, William P, 1985. "The First-Order Approach to Principal-Agent Problems," Econometrica, Econometric Society, vol. 53(6), pages 1357-1367, November.
    6. Jewitt, Ian, 1988. "Justifying the First-Order Approach to Principal-Agent Problems," Econometrica, Econometric Society, vol. 56(5), pages 1177-1190, September.
    7. Doherty, Neil A & Eeckhoudt, Louis, 1995. "Optimal Insurance without Expected Utility: The Dual Theory and the Linearity of Insurance Contracts," Journal of Risk and Uncertainty, Springer, vol. 10(2), pages 157-179, March.
    8. Lee, Kangoh, 1998. "Risk Aversion and Self-Insurance-cum-Protection," Journal of Risk and Uncertainty, Springer, vol. 17(2), pages 139-150, November.
    9. Han Bleichrodt & Louis Eeckhoudt, 2006. "Willingness to pay for reductions in health risks when probabilities are distorted," Health Economics, John Wiley & Sons, Ltd., vol. 15(2), pages 211-214, February.
    10. Shogren, Jason F. & Crocker, Thomas D., 1991. "Risk, self-protection, and ex ante economic value," Journal of Environmental Economics and Management, Elsevier, vol. 20(1), pages 1-15, January.
    11. 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.
    12. Susan Athey, 2002. "Monotone Comparative Statics under Uncertainty," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 117(1), pages 187-223.
    13. Johanna Etner & Meglena Jeleva, 2013. "Risk Perception, Health Prevention and Diagnostic Test," Post-Print hal-01385815, HAL.
    14. Kai A. Konrad & Stergios Skaperdas, 1993. "Self-Insurance and Self-Protection: A Nonexpected Utility Analysis," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 18(2), pages 131-146, December.
    15. Jakša Cvitanić & Xuhu Wan & Huali Yang, 2013. "Dynamics of Contract Design with Screening," Management Science, INFORMS, vol. 59(5), pages 1229-1244, May.
    16. Ehrlich, Isaac & Becker, Gary S, 1972. "Market Insurance, Self-Insurance, and Self-Protection," Journal of Political Economy, University of Chicago Press, vol. 80(4), pages 623-648, July-Aug..
    17. Donald Meyer & Jack Meyer, 2011. "A Diamond-Stiglitz approach to the demand for self-protection," Journal of Risk and Uncertainty, Springer, vol. 42(1), pages 45-60, February.
    18. Yuliy Sannikov, 2008. "A Continuous-Time Version of the Principal-Agent Problem," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 75(3), pages 957-984.
    19. Pauline Barrieu & Nicole El Karoui, 2005. "Inf-convolution of risk measures and optimal risk transfer," Finance and Stochastics, Springer, vol. 9(2), pages 269-298, April.
    20. Christophe Courbage & Henri Loubergé & Richard Peter, 2017. "Optimal Prevention for Multiple Risks," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 84(3), pages 899-922, September.
    21. Georges Dionne (ed.), 2013. "Handbook of Insurance," Springer Books, Springer, edition 2, number 978-1-4614-0155-1, January.
    22. repec:hal:spmain:info:hdl:2441/2iocb3a66198q8ovlu1gfkqie7 is not listed on IDEAS
    23. J. A. Mirrlees, 1999. "The Theory of Moral Hazard and Unobservable Behaviour: Part I," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(1), pages 3-21.
    24. Johanna Etner & Meglena Jeleva, 2013. "Risk Perception, Prevention And Diagnostic Tests," Health Economics, John Wiley & Sons, Ltd., vol. 22(2), pages 144-156, February.
    25. 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.
    26. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    27. repec:dau:papers:123456789/361 is not listed on IDEAS
    28. Barrieu, Pauline & El Karoui, Nicole, 2005. "Inf-convolution of risk measures and optimal risk transfer," LSE Research Online Documents on Economics 2829, London School of Economics and Political Science, LSE Library.
    29. Christophe Courbage, 2001. "Self-Insurance, Self-Protection and Market Insurance within the Dual Theory of Choice," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 26(1), pages 43-56, June.
    30. repec:dau:papers:123456789/342 is not listed on IDEAS
    31. L. Eeckhoudt & C. Gollier & H. Schlesinger, 2005. "Economic and financial decisions under risk," Post-Print hal-00325882, HAL.
    32. Eric Langlais, 2005. "Willingness to Pay for Risk Reduction and Risk Aversion without the Expected Utility Assumption," Theory and Decision, Springer, vol. 59(1), pages 43-50, August.
    33. Nicolás Hernández Santibáñez & Dylan Possamaï & Chao Zhou, 2017. "Bank monitoring incentives under moral hazard and adverse selection," Working Papers hal-01435460, HAL.
    34. Mark J. Machina, 1995. "Non-Expected Utility and The Robustness of the Classical Insurance Paradigm," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 20(1), pages 9-50, June.
    35. Griselda Deelstra & Guillaume Plantin, 2014. "Risk Theory and Reinsurance," Post-Print hal-03256838, HAL.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bensalem, Sarah & Santibáñez, Nicolás Hernández & Kazi-Tani, Nabil, 2020. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 369-386.
    2. Christian Gollier & James Hammitt & Nicolas Treich, 2013. "Risk and choice: A research saga," Journal of Risk and Uncertainty, Springer, vol. 47(2), pages 129-145, October.
    3. Sarah Bensalem, 2020. "Self-insurance and Non-concave Distortion Risk Measures," Working Papers hal-02936349, HAL.
    4. Courbage, Christophe & Rey, Béatrice & Treich, Nicolas, 2013. "Prevention and precaution," TSE Working Papers 13-445, Toulouse School of Economics (TSE).
    5. Gabriela Zeller & Matthias Scherer, 2023. "Risk mitigation services in cyber insurance: optimal contract design and price structure," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 48(2), pages 502-547, April.
    6. Sarah Bensalem & Nicolás Hernández Santibáñez & Nabil Kazi-Tani, 2022. "A Continuous-Time Model of Self-Protection," Working Papers hal-02974961, HAL.
    7. Richard Peter, 2024. "The economics of self-protection," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 49(1), pages 6-35, March.
    8. Sarah Bensalem & Nicolás Hernández-Santibáñez & Nabil Kazi-Tani, 2023. "A continuous-time model of self-protection," Finance and Stochastics, Springer, vol. 27(2), pages 503-537, April.
    9. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2024. "Inf-convolution and optimal risk sharing with countable sets of risk measures," Annals of Operations Research, Springer, vol. 336(1), pages 829-860, May.
    10. Nabil Kazi-Tani, 2018. "Inf-Convolution of Choquet Integrals and Applications in Optimal Risk Transfer," Working Papers hal-01742629, HAL.
    11. Han Bleichrodt, 2022. "The prevention puzzle," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 47(2), pages 277-297, September.
    12. Li, Peng & Lim, Andrew E.B. & Shanthikumar, J. George, 2010. "Optimal risk transfer for agents with germs," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 1-12, August.
    13. Tsanakas, Andreas, 2009. "To split or not to split: Capital allocation with convex risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 268-277, April.
    14. Nicol'as Hern'andez Santib'a~nez & Dylan Possamai & Chao Zhou, 2017. "Bank monitoring incentives under moral hazard and adverse selection," Papers 1701.05864, arXiv.org, revised Jan 2019.
    15. Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.
    16. Bellini, Fabio & Rosazza Gianin, Emanuela, 2008. "On Haezendonck risk measures," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 986-994, June.
    17. Daniel Lacker, 2018. "Liquidity, Risk Measures, and Concentration of Measure," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 813-837, August.
    18. Mitja Stadje, 2018. "Representation Results for Law Invariant Recursive Dynamic Deviation Measures and Risk Sharing," Papers 1811.09615, arXiv.org, revised Dec 2018.
    19. Benoît Sévi & Fabrice Yafil, 2005. "A special case of self-protection: The choice of a lawyer," Economics Bulletin, AccessEcon, vol. 4(6), pages 1-8.
    20. Jana Bielagk & Arnaud Lionnet & Gonçalo dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Working Papers hal-01245812, HAL.

    More about this item

    Keywords

    Prevention; Self-insurance; Self-protection; Coherent risk measures; Stackelberg game;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-01983433. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.