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Pricing General Insurance Using Optimal Control Theory

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  • Emms, Paul
  • Haberman, Steven

Abstract

Insurance premiums are calculated using optimal control theory by maximising the terminal wealth of an insurer under a demand law. If the insurer sets a low premium to generate exposure then profits are reduced, whereas a high premium leads to reduced demand. A continuous stochastic model is developed, which generalises the deterministic discrete model of Taylor (1986). An attractive simplification of this model is that existing policyholders should pay the premium rate currently set by the insurer. It is shown that this assumption leads to a bang-bang optimal premium strategy, which cannot be optimal for the insurer in realistic applications. The model is then modified by introducing an accrued premium rate representing the accumulated premium rates received from existing and new customers. Policyholders pay the premium rate in force at the start of their contract and pay this rate for the duration of the policy. It is shown that, for two demand functions, an optimal premium strategy is well-defined and smooth for certain parameter choices. It is shown for a linear demand function that these strategies yield the optimal dynamic premium if the market average premium is lognormally distributed.

Suggested Citation

  • Emms, Paul & Haberman, Steven, 2005. "Pricing General Insurance Using Optimal Control Theory," ASTIN Bulletin, Cambridge University Press, vol. 35(2), pages 427-453, November.
    • Handle: RePEc:cup:astinb:v:35:y:2005:i:02:p:427-453_01
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    Citations

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    Cited by:

    1. Pantelous, Athanasios A. & Passalidou, Eudokia, 2015. "Optimal premium pricing strategies for competitive general insurance markets," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 858-874.
    2. Yves L. Grize, 2015. "Applications of Statistics in the Field of General Insurance: An Overview," International Statistical Review, International Statistical Institute, vol. 83(1), pages 135-159, April.
    3. Bingzheng Chen & Zongxia Liang & Shunzhi Pang, 2024. "Dynamic Investment-Driven Insurance Pricing: Equilibrium Analysis and Welfare Implication," Papers 2410.18432, arXiv.org.
    4. Asmussen, Søren & Christensen, Bent Jesper & Thøgersen, Julie, 2019. "Nash equilibrium premium strategies for push–pull competition in a frictional non-life insurance market," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 92-100.
    5. Hong Mao & Zhongkai Wen, 2020. "Optimal Decision on Dynamic Insurance Price and Investment Portfolio of an Insurer with Multi-dimensional Time-Varying Correlation," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 18(1), pages 29-51, March.
    6. Emms, Paul, 2007. "Pricing general insurance with constraints," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 335-355, March.
    7. Mourdoukoutas, Fotios & Boonen, Tim J. & Koo, Bonsoo & Pantelous, Athanasios A., 2021. "Pricing in a competitive stochastic insurance market," Insurance: Mathematics and Economics, Elsevier, vol. 97(C), pages 44-56.
    8. Boonen, Tim J. & Pantelous, Athanasios A. & Wu, Renchao, 2018. "Non-cooperative dynamic games for general insurance markets," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 123-135.
    9. Emms, P. & Haberman, S. & Savoulli, I., 2007. "Optimal strategies for pricing general insurance," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 15-34, January.
    10. Rajeev Rajaram & Nathan Ritchey, 2023. "Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy," Mathematics, MDPI, vol. 11(15), pages 1-16, July.
    11. Hu, Duni & Chen, Shou & Wang, Hailong, 2018. "Robust reinsurance contracts with uncertainty about jump risk," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1175-1188.
    12. Mao, Hong & Carson, James M. & Ostaszewski, Krzysztof M. & Wen, Zhongkai, 2013. "Optimal decision on dynamic insurance price and investment portfolio of an insurer," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 359-369.
    13. Søren Asmussen & Bent Jesper Christensen & Julie Thøgersen, 2019. "Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation," Risks, MDPI, vol. 7(2), pages 1-23, May.

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