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Facing a Risk: To Insure or Not to Insure—An Analysis with the Constant Relative Risk Aversion Utility Function

Author

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  • M. Mercè Claramunt

    (Department of Mathematics for Economics, Finance and Actuarial Science, Faculty of Economics and Business, University of Barcelona, Avinguda Diagonal 690, 08034 Barcelona, Spain
    These authors contributed equally to this work.)

  • Maite Mármol

    (Department of Mathematics for Economics, Finance and Actuarial Science, Faculty of Economics and Business, University of Barcelona, Avinguda Diagonal 690, 08034 Barcelona, Spain
    These authors contributed equally to this work.)

  • Xavier Varea

    (Department of Mathematics for Economics, Finance and Actuarial Science, Faculty of Economics and Business, University of Barcelona, Avinguda Diagonal 690, 08034 Barcelona, Spain
    These authors contributed equally to this work.)

Abstract

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth.

Suggested Citation

  • M. Mercè Claramunt & Maite Mármol & Xavier Varea, 2023. "Facing a Risk: To Insure or Not to Insure—An Analysis with the Constant Relative Risk Aversion Utility Function," Mathematics, MDPI, vol. 11(5), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:5:p:1070-:d:1075240
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    References listed on IDEAS

    as
    1. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    2. Pitthan, Francisco & De Witte, Kristof, 2021. "Puzzles of insurance demand and its biases: A survey on the role of behavioural biases and financial literacy on insurance demand," Journal of Behavioral and Experimental Finance, Elsevier, vol. 30(C).
    3. Landsman, Zinoviy & Sherris, Michael, 2001. "Risk measures and insurance premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 29(1), pages 103-115, August.
    4. Christian Gollier, 2003. "To Insure or Not to Insure?: An Insurance Puzzle," The Geneva Risk and Insurance Review, Palgrave Macmillan;International Association for the Study of Insurance Economics (The Geneva Association), vol. 28(1), pages 5-24, June.
    5. Lourdes B. Afonso & Rui M. R. Cardoso & Alfredo D. Egídio dos Reis & Gracinda R. Guerreiro, 2020. "Ruin Probabilities And Capital Requirement for Open Automobile Portfolios With a Bonus‐Malus System Based on Claim Counts," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 87(2), pages 501-522, June.
    6. Dickson,David C. M., 2016. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9781107154605, October.
    7. Moore, Kristen S. & Young, Virginia R., 2006. "Optimal insurance in a continuous-time model," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 47-68, August.
    8. Christian Gollier & Harris Schlesinger, 1996. "Arrow's theorem on the optimality of deductibles: A stochastic dominance approach (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(2), pages 359-363.
    9. repec:bla:scandj:v:97:y:1995:i:1:p:123-35 is not listed on IDEAS
    10. Geweke, John, 2001. "A note on some limitations of CRRA utility," Economics Letters, Elsevier, vol. 71(3), pages 341-345, June.
    11. Boonen, Tim J., 2017. "Risk Sharing With Expected And Dual Utilities," ASTIN Bulletin, Cambridge University Press, vol. 47(2), pages 391-415, May.
    12. Levy, Moshe, 2019. "Stocks for the log-run and constant relative risk aversion preferences," European Journal of Operational Research, Elsevier, vol. 277(3), pages 1163-1168.
    13. Edward Furman & Ričardas Zitikis, 2009. "Weighted Pricing Functionals With Applications to Insurance," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(4), pages 483-496.
    14. 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.
    15. Wang, Shaun S. & Young, Virginia R., 1998. "Ordering risks: Expected utility theory versus Yaari's dual theory of risk," Insurance: Mathematics and Economics, Elsevier, vol. 22(2), pages 145-161, June.
    16. Chiu, Mei Choi & Wong, Hoi Ying, 2013. "Optimal investment for an insurer with cointegrated assets: CRRA utility," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 52-64.
    17. Wang, Shaun & Dhaene, Jan, 1998. "Comonotonicity, correlation order and premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 235-242, July.
    18. Michael Rothschild & Joseph Stiglitz, 1976. "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 90(4), pages 629-649.
    19. Ñíguez, Trino-Manuel & Paya, Ivan & Peel, David & Perote, Javier, 2012. "On the stability of the constant relative risk aversion (CRRA) utility under high degrees of uncertainty," Economics Letters, Elsevier, vol. 115(2), pages 244-248.
    20. Raviv, Artur, 1979. "The Design of an Optimal Insurance Policy," American Economic Review, American Economic Association, vol. 69(1), pages 84-96, March.
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