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Optimal Insurance Arrangements

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  • Borch, Karl

Abstract

In a recent paper on the theory of demand for insurance Arrow [1] has proved that the optimal policy for an insurance buyer is one which gives complete coverage, beyond a fixed deductible. The result is proved under very general assumptions, but its content can be illustrated by the following simple example.Assume that a person is exposed to a risk which can cause him a loss x, represented by a stochastic variable with the distribution F(x). Assume further that he by paying the premium P(y) can obtain an insurance contract which will guarantee him a compensation y(x), if his loss amounts to x. The problem of our person is to find the optimal insurance contract, i.e. the optimal function y(x), when the price is given by the functional P(y).In order to give an operational formulation to the problem we have outlined, we shall assume that the person's attitude to risk can be represented by a Bernoulli utility function u(x), and we shall write S for his “initial wealth”. His problem will then be to maximizewhen the functional P(y) is given, and y(x) є Y. The set Y can be interpreted as the set of insurance policies available in the market. It is, natural to assume that o ≤ y(x) ≤ x, but beyond this there is no need for assuming additional restrictions on the set Y.

Suggested Citation

  • Borch, Karl, 1975. "Optimal Insurance Arrangements," ASTIN Bulletin, Cambridge University Press, vol. 8(3), pages 284-290, September.
    • Handle: RePEc:cup:astinb:v:8:y:1975:i:03:p:284-290_01
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    Cited by:

    1. Hong Mao & Krzysztof Ostaszewski, 2021. "Optimal Claim Settlement Strategies under Constraint of Cap on Claim Loss," Mathematics, MDPI, vol. 9(24), pages 1-12, December.
    2. Elisa Luciano, 1999. "A Note on Loadings and Deductibles: Can a Vicious Circle Arise?," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 1999(2), pages 157-169.
    3. Dutang, Christophe & Albrecher, Hansjoerg & Loisel, Stéphane, 2013. "Competition among non-life insurers under solvency constraints: A game-theoretic approach," European Journal of Operational Research, Elsevier, vol. 231(3), pages 702-711.
    4. Corina Constantinescu & Alexandra Dias & Bo Li & David Šiška & Simon Wang, 2022. "Effect of Stop-Loss Reinsurance on Primary Insurer Solvency," Risks, MDPI, vol. 10(10), pages 1-15, October.
    5. José Daniel López-Barrientos & Ekaterina Viktorovna Gromova & Ekaterina Sergeevna Miroshnichenko, 2020. "Resource Exploitation in a Stochastic Horizon under Two Parametric Interpretations," Mathematics, MDPI, vol. 8(7), pages 1-29, July.
    6. Kaluszka, Marek, 2004. "An extension of Arrow's result on optimality of a stop loss contract," Insurance: Mathematics and Economics, Elsevier, vol. 35(3), pages 527-536, December.
    7. Bakker, Frank M. & van Vliet, Rene C. J. A. & van de Ven, Wynand P. M. M., 2000. "Deductibles in health insurance: can the actuarially fair premium reduction exceed the deductible?," Health Policy, Elsevier, vol. 53(2), pages 123-141, September.
    8. Li, Wenyuan & Tan, Ken Seng & Wei, Pengyu, 2021. "Demand for non-life insurance under habit formation," Insurance: Mathematics and Economics, Elsevier, vol. 101(PA), pages 38-54.
    9. Cheung, K.C. & Chong, W.F. & Yam, S.C.P., 2015. "The optimal insurance under disappointment theories," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 77-90.
    10. Burren, Daniel, 2013. "Insurance demand and welfare-maximizing risk capital—Some hints for the regulator in the case of exponential preferences and exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 551-568.
    11. Sung, K.C.J. & Yam, S.C.P. & Yung, S.P. & Zhou, J.H., 2011. "Behavioral optimal insurance," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 418-428.
    12. repec:hal:wpaper:hal-00746245 is not listed on IDEAS
    13. Wang, Shaun, 1996. "Ordering of risks under PH-transforms," Insurance: Mathematics and Economics, Elsevier, vol. 18(2), pages 109-114, July.

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