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Optimal dividend strategies for a catastrophe insurer

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  • Hansjoerg Albrecher
  • Pablo Azcue
  • Nora Muler

Abstract

In this paper we study the problem of optimally paying out dividends from an insurance portfolio, when the criterion is to maximize the expected discounted dividends over the lifetime of the company and the portfolio contains claims due to natural catastrophes, modelled by a shot-noise Cox claim number process. The optimal value function of the resulting two-dimensional stochastic control problem is shown to be the smallest viscosity supersolution of a corresponding Hamilton-Jacobi-Bellman equation, and we prove that it can be uniformly approximated through a discretization of the space of the free surplus of the portfolio and the current claim intensity level. We implement the resulting numerical scheme to identify optimal dividend strategies for such a natural catastrophe insurer, and it is shown that the nature of the barrier and band strategies known from the classical models with constant Poisson claim intensity carry over in a certain way to this more general situation, leading to action and non-action regions for the dividend payments as a function of the current surplus and intensity level. We also discuss some interpretations in terms of upward potential for shareholders when including a catastrophe sector in the portfolio.

Suggested Citation

  • Hansjoerg Albrecher & Pablo Azcue & Nora Muler, 2023. "Optimal dividend strategies for a catastrophe insurer," Papers 2311.05781, arXiv.org.
    • Handle: RePEc:arx:papers:2311.05781
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    References listed on IDEAS

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    1. Liu, Wenyue & Cadenillas, Abel, 2023. "Optimal insurance contracts for a shot-noise Cox claim process and persistent insured's actions," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 69-93.
    2. Simon Pojer & Stefan Thonhauser, 2023. "The Markovian Shot-noise Risk Model: A Numerical Method for Gerber-Shiu Functions," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    3. Peter Grandits, 2019. "A two-dimensional dividend problem for collaborating companies and an optimal stopping problem," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(1), pages 80-96, January.
    4. Benjamin Avanzi, 2009. "Strategies for Dividend Distribution: A Review," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 217-251.
    5. Hansjörg Albrecher & Pablo Azcue & Nora Muler, 2023. "Optimal dividends under a drawdown constraint and a curious square-root rule," Finance and Stochastics, Springer, vol. 27(2), pages 341-400, April.
    6. Dassios, Angelos & Zhao, Hongbiao, 2012. "Ruin by dynamic contagion claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 93-106.
    7. Hansjörg Albrecher & Brandon Garcia Flores, 2023. "Optimal dividend bands revisited: a gradient-based method and evolutionary algorithms," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2023(8), pages 788-810, September.
    8. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    9. Onno Boxma & Michel Mandjes, 2021. "Shot-noise queueing models," Queueing Systems: Theory and Applications, Springer, vol. 99(1), pages 121-159, October.
    10. Hansjörg Albrecher & Søren Asmussen c, 2006. "Ruin probabilities and aggregrate claims distributions for shot noise Cox processes," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2006(2), pages 86-110.
    11. Jia-Wen Gu & Mogens Steffensen & Harry Zheng, 2018. "Optimal Dividend Strategies of Two Collaborating Businesses in the Diffusion Approximation Model," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 377-398, May.
    12. Dassios, Angelos & Jang, Jiwook, 2003. "Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity," LSE Research Online Documents on Economics 2849, London School of Economics and Political Science, LSE Library.
    13. Jang, Jiwook & Oh, Rosy, 2021. "A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes," Annals of Actuarial Science, Cambridge University Press, vol. 15(3), pages 623-644, November.
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