IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v42y2008i2p691-703.html
   My bibliography  Save this article

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

Author

Listed:
  • Young, Virginia R.

Abstract

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.

Suggested Citation

  • Young, Virginia R., 2008. "Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 691-703, April.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:2:p:691-703
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-6687(07)00085-6
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Dahl, Mikkel & Moller, Thomas, 2006. "Valuation and hedging of life insurance liabilities with systematic mortality risk," Insurance: Mathematics and Economics, Elsevier, vol. 39(2), pages 193-217, October.
    2. Blanchet-Scalliet, Christophette & El Karoui, Nicole & Martellini, Lionel, 2005. "Dynamic asset pricing theory with uncertain time-horizon," Journal of Economic Dynamics and Control, Elsevier, vol. 29(10), pages 1737-1764, October.
    3. Tomas Björk & Irina Slinko, 2006. "Towards a General Theory of Good-Deal Bounds," Review of Finance, European Finance Association, vol. 10(2), pages 221-260.
    4. Schrager, David F., 2006. "Affine stochastic mortality," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 81-97, February.
    5. Virginia R. Young, 2007. "Pricing Life Insurance under Stochastic Mortality via the Instantaneous Sharpe Ratio: Theorems and Proofs," Papers 0705.1297, arXiv.org.
    6. 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.
    7. Erhan Bayraktar & Virginia Young, 2008. "Pricing options in incomplete equity markets via the instantaneous Sharpe ratio," Annals of Finance, Springer, vol. 4(4), pages 399-429, October.
    8. Carter, Lawrence R. & Lee, Ronald D., 1992. "Modeling and forecasting US sex differentials in mortality," International Journal of Forecasting, Elsevier, vol. 8(3), pages 393-411, November.
    9. Bayraktar, Erhan & Young, Virginia R., 2007. "Hedging life insurance with pure endowments," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 435-444, May.
    10. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
    11. 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.
    12. 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. Wang, Ting & Young, Virginia R., 2016. "Hedging pure endowments with mortality derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 238-255.
    2. Chen, Bingzheng & Zhang, Lihong & Zhao, Lin, 2010. "On the robustness of longevity risk pricing," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 358-373, December.
    3. Bahl, Raj Kumari & Sabanis, Sotirios, 2021. "Model-independent price bounds for Catastrophic Mortality Bonds," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 276-291.
    4. Huang, Rachel J. & Miao, Jerry C.Y. & Tzeng, Larry Y., 2013. "Does mortality improvement increase equity risk premiums? A risk perception perspective," Journal of Empirical Finance, Elsevier, vol. 22(C), pages 67-77.
    5. Huang, Yu-Lieh & Tsai, Jeffrey Tzuhao & Yang, Sharon S. & Cheng, Hung-Wen, 2014. "Price bounds of mortality-linked security in incomplete insurance market," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 30-39.
    6. Hua Chen & Michael Sherris & Tao Sun & Wenge Zhu, 2013. "Living With Ambiguity: Pricing Mortality-Linked Securities With Smooth Ambiguity Preferences," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 80(3), pages 705-732, September.
    7. Zimbidis, Alexandros A., 2014. "Insurance pricing using H∞-control," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 685-697.
    8. Johnny Siu‐Hang Li & Andrew Cheuk‐Yin Ng, 2011. "Canonical Valuation of Mortality‐Linked Securities," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 78(4), pages 853-884, December.
    9. Marcus C. Christiansen, 2013. "Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates," Risks, MDPI, vol. 1(3), pages 1-20, October.
    10. Kung, Ko-Lun & Liu, I-Chien & Wang, Chou-Wen, 2021. "Modeling and pricing longevity derivatives using Skellam distribution," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 341-354.
    11. Ballestra, Luca Vincenzo & Ottaviani, Massimiliano & Pacelli, Graziella, 2012. "An operator splitting harmonic differential quadrature approach to solve Young’s model for life insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 442-448.
    12. Li, Jing & Szimayer, Alexander, 2011. "The uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 471-486.

    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. Virginia R. Young, 2007. "Pricing Life Insurance under Stochastic Mortality via the Instantaneous Sharpe Ratio: Theorems and Proofs," Papers 0705.1297, arXiv.org.
    2. Ludkovski, Michael & Young, Virginia R., 2008. "Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 14-30, February.
    3. Wang, Ting & Young, Virginia R., 2016. "Hedging pure endowments with mortality derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 238-255.
    4. Chen, Bingzheng & Zhang, Lihong & Zhao, Lin, 2010. "On the robustness of longevity risk pricing," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 358-373, December.
    5. Ting Wang & Virginia R. Young, 2010. "Hedging Pure Endowments with Mortality Derivatives," Papers 1011.0248, arXiv.org.
    6. Jevtić, Petar & Regis, Luca, 2019. "A continuous-time stochastic model for the mortality surface of multiple populations," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 181-195.
    7. Bayraktar, Erhan & Milevsky, Moshe A. & David Promislow, S. & Young, Virginia R., 2009. "Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities," Journal of Economic Dynamics and Control, Elsevier, vol. 33(3), pages 676-691, March.
    8. Marcus C. Christiansen, 2013. "Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates," Risks, MDPI, vol. 1(3), pages 1-20, October.
    9. Homa Magdalena, 2020. "Mathematical Reserves vs Longevity Risk in Life Insurances," Econometrics. Advances in Applied Data Analysis, Sciendo, vol. 24(1), pages 23-38, March.
    10. 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.
    11. 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.
    12. Deelstra, Griselda & Grasselli, Martino & Van Weverberg, Christopher, 2016. "The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 205-219.
    13. Bauer, Daniel & Börger, Matthias & Ruß, Jochen, 2010. "On the pricing of longevity-linked securities," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 139-149, February.
    14. David Blake & Andrew Cairns & Guy Coughlan & Kevin Dowd & Richard MacMinn, 2013. "The New Life Market," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 80(3), pages 501-558, September.
    15. Rihab Bedoui & Islem Kedidi, 2018. "Modeling Longevity Risk using Consistent Dynamics Affine Mortality Models," Working Papers hal-01678050, HAL.
    16. Jevtić, Petar & Luciano, Elisa & Vigna, Elena, 2013. "Mortality surface by means of continuous time cohort models," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 122-133.
    17. Fadoua Zeddouk & Pierre Devolder, 2020. "Longevity Modelling and Pricing under a Dependent Multi-Cohort Framework," Risks, MDPI, vol. 8(4), pages 1-23, November.
    18. Li, Jing & Szimayer, Alexander, 2011. "The uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 471-486.
    19. Lin, Tzuling & Tzeng, Larry Y., 2010. "An additive stochastic model of mortality rates: An application to longevity risk in reserve evaluation," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 423-435, April.
    20. 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.

    More about this item

    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:eee:insuma:v:42:y:2008:i:2:p:691-703. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    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.