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On the free boundary of an annuity purchase

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  • Tiziano De Angelis
  • Gabriele Stabile

Abstract

It is known that the decision to purchase an annuity may be associated to an optimal stopping problem. However, little is known about optimal strategies, if the mortality force is a generic function of time and if the `subjective' life expectancy of the investor differs from the `objective' one adopted by insurance companies to price annuities. In this paper we address this problem considering an individual who invests in a fund and has the option to convert the fund's value into an annuity at any time. We formulate the problem as a real option and perform a detailed probabilistic study of the optimal stopping boundary. Due to the generic time-dependence of the mortality force, our optimal stopping problem requires new solution methods to deal with non-monotonic optimal boundaries.

Suggested Citation

  • Tiziano De Angelis & Gabriele Stabile, 2017. "On the free boundary of an annuity purchase," Papers 1707.09494, arXiv.org, revised Jul 2018.
    • Handle: RePEc:arx:papers:1707.09494
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    References listed on IDEAS

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    1. E. Chevalier, 2006. "Optimal Early Retirement Near the Expiration of a Pension Plan," Finance and Stochastics, Springer, vol. 10(2), pages 204-221, April.
    2. Hainaut, Donatien & Deelstra, Griselda, 2014. "Optimal timing for annuitization, based on jump diffusion fund and stochastic mortality," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 124-146.
    3. Russell Gerrard & Bjarne Højgaard & Elena Vigna, 2012. "Choosing the optimal annuitization time post-retirement," Quantitative Finance, Taylor & Francis Journals, vol. 12(7), pages 1143-1159, September.
    4. Gabriele Stabile, 2006. "Optimal Timing Of The Annuity Purchase: Combined Stochastic Control And Optimal Stopping Problem," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 151-170.
    5. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    6. Menahem E. Yaari, 1965. "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 32(2), pages 137-150.
    7. E. Chevalier, 2006. "Optimal Early Retirement Near the Expiration of a Pension Plan," Finance and Stochastics, Springer, vol. 10(2), pages 204-221, April.
    8. Milevsky, Moshe A. & Young, Virginia R., 2007. "Annuitization and asset allocation," Journal of Economic Dynamics and Control, Elsevier, vol. 31(9), pages 3138-3177, September.
    9. Avner Friedman & Weixi Shen, 2002. "A variational inequality approach to financial valuation of retirement benefits based on salary," Finance and Stochastics, Springer, vol. 6(3), pages 273-302.
    10. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
    11. Tiziano De Angelis & Erik Ekstrom, 2016. "The dividend problem with a finite horizon," Papers 1609.01655, arXiv.org, revised Nov 2017.
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    Citations

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    Cited by:

    1. Zhenya Liu & Yuhao Mu, 2022. "Optimal Stopping Methods for Investment Decisions: A Literature Review," IJFS, MDPI, vol. 10(4), pages 1-23, October.
    2. Maria B. Chiarolla & Tiziano Angelis & Gabriele Stabile, 2022. "An analytical study of participating policies with minimum rate guarantee and surrender option," Finance and Stochastics, Springer, vol. 26(2), pages 173-216, April.
    3. Ferrari, Giorgio & Zhu, Shihao, 2023. "Optimal Retirement Choice under Age-dependent Force of Mortality," Center for Mathematical Economics Working Papers 683, Center for Mathematical Economics, Bielefeld University.
    4. Tiziano De Angelis & Alessandro Milazzo & Gabriele Stabile, 2024. "On variable annuities with surrender charges," Papers 2405.02115, arXiv.org.
    5. Giorgio Ferrari & Shihao Zhu, 2023. "Optimal Retirement Choice under Age-dependent Force of Mortality," Papers 2311.12169, arXiv.org.
    6. Jason S. Anquandah & Leonid V. Bogachev, 2019. "Optimal Stopping and Utility in a Simple Model of Unemployment Insurance," Papers 1902.06175, arXiv.org, revised Sep 2019.
    7. Giorgio Ferrari & Shihao Zhu, 2022. "On a Merton Problem with Irreversible Healthcare Investment," Papers 2212.05317, arXiv.org, revised Dec 2023.
    8. Ferrari, Giorgio & Zhu, Shihao, 2022. "Consumption Descision, Portfolio Choice and Healthcare Irreversible Investment," Center for Mathematical Economics Working Papers 671, Center for Mathematical Economics, Bielefeld University.
    9. Jason S. Anquandah & Leonid V. Bogachev, 2019. "Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance," Risks, MDPI, vol. 7(3), pages 1-41, September.

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    JEL classification:

    • J26 - Labor and Demographic Economics - - Demand and Supply of Labor - - - Retirement; Retirement Policies

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