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Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio


  • M. A. Milevsky
  • S. D. Promislow
  • V. R. Young


We provide an overview of how the law of large numbers breaks down when pricing life-contingent claims under stochastic as opposed to deterministic mortality (probability, hazard) rates. In a stylized situation, we derive the limiting per-policy risk and show that it goes to a non-zero constant. This is in contrast to the classical situation when the underlying mortality decrements are known with certainty, per policy risk goes to zero. We decompose the standard deviation per policy into systematic and non-systematic components, akin to the analysis of individual stock (equity) risk in a Markowitz portfolio framework. Finally, we draw upon the financial analogy of the Sharpe Ratio to develop a premium pricing methodology under aggregate mortality risk. Copyright The Journal of Risk and Insurance, 2006.

Suggested Citation

  • M. A. Milevsky & S. D. Promislow & V. R. Young, 2006. "Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(4), pages 673-686.
  • Handle: RePEc:bla:jrinsu:v:73:y:2006:i:4:p:673-686

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    Cited by:

    1. Huang, Rachel J. & Tsai, Jeffrey T. & Tzeng, Larry Y., 2008. "Government-provided annuities under insolvency risk," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 377-385, December.
    2. Wang, Ting & Young, Virginia R., 2016. "Hedging pure endowments with mortality derivatives," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 238-255.
    3. 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.
    4. Anja De Waegenaere & Bertrand Melenberg & Ralph Stevens, 2010. "Longevity Risk," De Economist, Springer, vol. 158(2), pages 151-192, June.
    5. M. Martin Boyer & Joanna Mejza & Lars Stentoft, 2014. "Measuring Longevity Risk: An Application to the Royal Canadian Mounted Police Pension Plan," Risk Management and Insurance Review, American Risk and Insurance Association, vol. 17(1), pages 37-59, March.
    6. Matheus R Grasselli & Sebastiano Silla, 2009. "A policyholder's utility indifference valuation model for the guaranteed annuity option," Papers 0908.3196,
    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. Christophette Blanchet-Scalliet & Etienne Chevalier & Idris Kharroubi & Thomas Lim, 2015. "Max–Min Optimization Problem For Variable Annuities Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(08), pages 1-35, December.
    9. Helena Aro, 2013. "Systematic and non-systematic mortality risk in pension portfolios," Papers 1307.8020,, revised Jul 2013.
    10. Tsai, Jeffrey T. & Wang, Jennifer L. & Tzeng, Larry Y., 2010. "On the optimal product mix in life insurance companies using conditional value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 235-241, February.

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