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Pricing Longevity Bonds Using Affine-Jump Diffusion Models

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  • Jorge Bravo

    (University of Évora, Department of Economics and CEFAGEUE)

Abstract

Historically, actuaries have been calculating premiums and mathematical reserves using a deterministic approach, by considering a deterministic mortality intensity, which is a function of the age only, extracted from available (static) life tables and by setting a flat ("best estimate") interest rate to discount cash flows over time. Since neither the mortality intensity nor interest rates are actually deterministic, life insurance companies and pension funds are exposed to both financial and mortality (systematic and unsystematic) risks when pricing and reserving for any kind of long-term living benefits, particularly on annuities and pensions. In this paper, we assume that an appropriate description of the demographic risks requires the use of stochastic models. In particular, we assume that the random evolution of the stochastic force of mortality of an individual can be modelled by using doubly stochastic processes. The model is then embedded into the well known affine-jump framework, widely used in the term structure literature, in order to derive closed-form solutions for the survival probability. We show that stochastic mortality models provide an adequate framework for the development of longevity risk hedging tools, namely mortality-linked contracts such as longevity bonds or mortality derivatives.

Suggested Citation

  • Jorge Bravo, 2011. "Pricing Longevity Bonds Using Affine-Jump Diffusion Models," CEFAGE-UE Working Papers 2011_29, University of Evora, CEFAGE-UE (Portugal).
    • Handle: RePEc:cfe:wpcefa:2011_29
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    References listed on IDEAS

    as
    1. Renshaw, A. E. & Haberman, S., 2003. "On the forecasting of mortality reduction factors," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 379-401, July.
    2. Carlos Wong-Fupuy & Steven Haberman, 2004. "Projecting Mortality Trends," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(2), pages 56-83.
    3. Piet De Jong & Leonie Tickle, 2006. "Extending Lee-Carter Mortality Forecasting," Mathematical Population Studies, Taylor & Francis Journals, vol. 13(1), pages 1-18.
    4. Shripad Tuljapurkar, 1998. "Forecasting Mortality Change," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(4), pages 127-134.
    5. Ronald Lee, 2000. "The Lee-Carter Method for Forecasting Mortality, with Various Extensions and Applications," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(1), pages 80-91.
    6. Pitacco, Ermanno, 2004. "Survival models in a dynamic context: a survey," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 279-298, October.
    7. Andrew J. G. Cairns & David Blake & Kevin Dowd, 2006. "A Two‐Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(4), pages 687-718, December.
    8. Lawrence R. Carter & Alexia Prskawetz, 2001. "Examining structural shifts in mortality using the Lee-Carter method," MPIDR Working Papers WP-2001-007, Max Planck Institute for Demographic Research, Rostock, Germany.
    9. L. C. G. Rogers, 1997. "The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 157-176, April.
    10. Marek Rutkowski, 1997. "A note on the Flesaker-Hughston model of the term structure of interest rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(3), pages 151-163.
    11. Philippe Artzner & Freddy Delbaen, 1995. "Default Risk Insurance And Incomplete Markets1," Mathematical Finance, Wiley Blackwell, vol. 5(3), pages 187-195, July.
    12. Dahl, Mikkel, 2004. "Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 113-136, August.
    13. Cairns, Andrew J.G. & Blake, David & Dowd, Kevin, 2006. "Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk," ASTIN Bulletin, Cambridge University Press, vol. 36(1), pages 79-120, May.
    14. Shripad Tuljapurkar & Carl Boe, "undated". "Mortality Change and Forecasting: How Much and How Little Do We Know?," Pension Research Council Working Papers 98-2, Wharton School Pension Research Council, University of Pennsylvania.
    15. Shripad Tuljapurkar & Carl Boe, 1998. "Mortality Change and Forecasting," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(4), pages 13-47.
    16. Renshaw, A.E. & Haberman, S., 2006. "A cohort-based extension to the Lee-Carter model for mortality reduction factors," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 556-570, June.
    17. Ballotta, Laura & Haberman, Steven, 2006. "The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 195-214, February.
    18. Brouhns, Natacha & Denuit, Michel & Vermunt, Jeroen K., 2002. "A Poisson log-bilinear regression approach to the construction of projected lifetables," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 373-393, December.
    19. Milevsky, Moshe A. & David Promislow, S., 2001. "Mortality derivatives and the option to annuitise," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 299-318, December.
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    Cited by:

    1. Blake, David & Cairns, Andrew J.G., 2021. "Longevity risk and capital markets: The 2019-20 update," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 395-439.
    2. Bravo, Jorge M. & Nunes, João Pedro Vidal, 2021. "Pricing longevity derivatives via Fourier transforms," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 81-97.

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    More about this item

    Keywords

    Stochastic mortality intensity; Longevity risk; Affine models; Projected lifetables.;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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