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Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach

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  • Krzysztof Burnecki

    (Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland)

  • Marek A. Teuerle

    (Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland)

  • Aleksandra Wilkowska

    (Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland)

Abstract

In this paper, we consider a two-dimensional risk process in which the companies split each claim and premium in a fixed proportion. It serves as a classical framework of a quota-share reinsurance contract for a given business line. Such a contract reduces the insurer’s exposure to the liabilities created through its underwriting activities. For the analyzed model, we derive a joint infinite-time ruin probability formula for exponentially distributed claims. To this end, we apply a change of measure technique. We illustrate the admissible range of parameters of the risk process. We also justify our result using Monte Carlo simulations and compare it with Theorem 2 in Avram, Palmowski and Pistorius [Insurance: Mathematics and Economics 42 (2008) 227], which was obtained by explicitly inverting a Laplace transform of the ruin probability. Our formula leads to a correction of that result. Finally, we note that the obtained formula leads to efficient approximation of the ruin probability for other claim amount distributions using De Vylder’s idea.

Suggested Citation

  • Krzysztof Burnecki & Marek A. Teuerle & Aleksandra Wilkowska, 2021. "Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach," Risks, MDPI, vol. 9(5), pages 1-10, May.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:5:p:86-:d:548228
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    References listed on IDEAS

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    1. Krzysztof Burnecki & Marek Teuerle, 2010. "Ruin Probability in Finite Time," HSC Research Reports HSC/10/04, Hugo Steinhaus Center, Wroclaw University of Technology.
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    Cited by:

    1. Khreshna Syuhada & Arief Hakim & Suci Sari, 2021. "The Combined Stop-Loss and Quota-Share Reinsurance: Conditional Tail Expectation-Based Optimization from the Joint Perspective of Insurer and Reinsurer," Risks, MDPI, vol. 9(7), pages 1-21, July.
    2. Krzysztof Burnecki & Zbigniew Palmowski & Marek Teuerle & Aleksandra Wilkowska, 2023. "Ruin probability for the quota share model with~phase-type distributed claims," Papers 2303.07705, arXiv.org.
    3. Krzysztof Burnecki & Marek A. Teuerle & Aleksandra Wilkowska, 2022. "Diffusion Approximations of the Ruin Probability for the Insurer–Reinsurer Model Driven by a Renewal Process," Risks, MDPI, vol. 10(6), pages 1-16, June.

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