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Aggregate Markov models in life insurance: Properties and valuation

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  • Ahmad, Jamaal
  • Bladt, Mogens
  • Furrer, Christian

Abstract

In multi-state life insurance, an adequate balance between analytic tractability, computational efficiency, and statistical flexibility is of great importance. This might explain the popularity of Markov chain modelling, where matrix analytic methods allow for a comprehensive treatment. Unfortunately, Markov chain modelling is unable to capture duration effects, so this paper presents aggregate Markov models as an alternative. Aggregate Markov models retain most of the analytical tractability of Markov chains, yet are non-Markovian and thus more flexible. Based on an explicit characterization of the fundamental martingales, matrix representations of the expected accumulated cash flows and corresponding prospective reserves are derived for duration-dependent payments with and without incidental policyholder behaviour. Throughout, special attention is given to a semi-Markovian case. Finally, the methods and results are illustrated in a numerical example.

Suggested Citation

  • Ahmad, Jamaal & Bladt, Mogens & Furrer, Christian, 2023. "Aggregate Markov models in life insurance: Properties and valuation," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 50-69.
    • Handle: RePEc:eee:insuma:v:113:y:2023:i:c:p:50-69
    DOI: 10.1016/j.insmatheco.2023.07.006
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    References listed on IDEAS

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    1. Kamille Sofie TÅgholt Gad & Jeppe Woetmann Nielsen, 2016. "Reserves and cash flows under stochastic retirement," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2016(10), pages 876-904, November.
    2. Kristian Buchardt & Thomas Møller & Kristian Bjerre Schmidt, 2015. "Cash flows and policyholder behaviour in the semi-Markov life insurance setup," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2015(8), pages 660-688, November.
    3. Kristian Buchardt & Christian Furrer & Mogens Steffensen, 2019. "Forward transition rates," Finance and Stochastics, Springer, vol. 23(4), pages 975-999, October.
    4. K. Buchardt & C. Furrer & M. Steffensen, 2018. "Forward transition rates," Papers 1811.00137, arXiv.org, revised Apr 2019.
    5. Christian Furrer, 2022. "Scaled insurance cash flows: representation and computation via change of measure techniques," Finance and Stochastics, Springer, vol. 26(2), pages 359-382, April.
    6. Jamaal Ahmad, 2022. "Multivariate higher order moments in multi-state life insurance," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2022(5), pages 399-420, May.
    7. Marcus Christiansen, 2012. "Multistate models in health insurance," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(2), pages 155-186, June.
    8. Milbrodt, Hartmut & Stracke, Andrea, 1997. "Markov models and Thiele's integral equations for the prospective reserve," Insurance: Mathematics and Economics, Elsevier, vol. 19(3), pages 187-235, May.
    9. Kristian Buchardt & Thomas Møller, 2015. "Life Insurance Cash Flows with Policyholder Behavior," Risks, MDPI, vol. 3(3), pages 1-28, July.
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    Cited by:

    1. Marcus C. Christiansen & Christian Furrer, 2024. "Canonical insurance models: stochastic equations and comparison theorems," Papers 2411.12522, arXiv.org.

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    More about this item

    Keywords

    Multi-state modelling; Duration dependence; Product integrals; Expected cash flows; Phase-type distributions;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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