IDEAS home Printed from https://ideas.repec.org/p/cfe/wpcefa/2011_29.html
   My bibliography  Save this paper

Pricing Longevity Bonds Using Affine-Jump Diffusion Models

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.cefage.uevora.pt/en/content/download/2905/38764/version/1/file/2011_29.pdf
    Download Restriction: no
    ---><---

    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Haberman, Steven & Renshaw, Arthur, 2011. "A comparative study of parametric mortality projection models," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 35-55, January.
    2. Jorge Bravo, 2011. "Modelling Mortality Using Multiple Stochastic Latent Factors," CEFAGE-UE Working Papers 2011_26, University of Evora, CEFAGE-UE (Portugal).
    3. Booth, Heather, 2006. "Demographic forecasting: 1980 to 2005 in review," International Journal of Forecasting, Elsevier, vol. 22(3), pages 547-581.
    4. Matheus R Grasselli & Sebastiano Silla, 2009. "A policyholder's utility indifference valuation model for the guaranteed annuity option," Papers 0908.3196, arXiv.org.
    5. LUCIANO, Elisa & VIGNA, Elena, 2008. "Mortality risk via affine stochastic intensities: calibration and empirical relevance," MPRA Paper 59627, University Library of Munich, Germany.
    6. Anja De Waegenaere & Bertrand Melenberg & Ralph Stevens, 2010. "Longevity Risk," De Economist, Springer, vol. 158(2), pages 151-192, June.
    7. Pitacco, Ermanno, 2004. "Survival models in a dynamic context: a survey," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 279-298, October.
    8. Jorge Bravo & Carlos Pereira da Silva, 2012. "Prospective Lifetables: Life Insurance Pricing and Hedging in a Stochastic Mortality Environment," CEFAGE-UE Working Papers 2012_01, University of Evora, CEFAGE-UE (Portugal).
    9. O'Hare, Colin & Li, Youwei, 2014. "Identifying structural breaks in stochastic mortality models," MPRA Paper 62994, University Library of Munich, Germany.
    10. Blake, David & El Karoui, Nicole & Loisel, Stéphane & MacMinn, Richard, 2018. "Longevity risk and capital markets: The 2015–16 update," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 157-173.
    11. Stéphane Loisel, 2010. "Understanding, Modeling and Managing Longevity Risk: Key Issues and Main Challenges," Post-Print hal-00517902, HAL.
    12. 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.
    13. Bravo, Jorge M. & Ayuso, Mercedes & Holzmann, Robert & Palmer, Edward, 2021. "Addressing the life expectancy gap in pension policy," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 200-221.
    14. Tickle Leonie & Booth Heather, 2014. "The Longevity Prospects of Australian Seniors: An Evaluation of Forecast Method and Outcome," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 8(2), pages 1-34, July.
    15. Colin O’hare & Youwei Li, 2017. "Modelling mortality: are we heading in the right direction?," Applied Economics, Taylor & Francis Journals, vol. 49(2), pages 170-187, January.
    16. Shen, Yang & Siu, Tak Kuen, 2013. "Longevity bond pricing under stochastic interest rate and mortality with regime-switching," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 114-123.
    17. Levantesi, Susanna & Menzietti, Massimiliano, 2012. "Managing longevity and disability risks in life annuities with long term care," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 391-401.
    18. Njenga Carolyn N & Sherris Michael, 2011. "Longevity Risk and the Econometric Analysis of Mortality Trends and Volatility," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 5(2), pages 1-54, July.
    19. Elisa Luciano & Elena Vigna, 2005. "Non mean reverting affine processes for stochastic mortality," ICER Working Papers - Applied Mathematics Series 4-2005, ICER - International Centre for Economic Research.
    20. Plat, Richard, 2009. "On stochastic mortality modeling," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 393-404, December.

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cfe:wpcefa:2011_29. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Angela Pacheco (email available below). General contact details of provider: https://edirc.repec.org/data/cfevopt.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.