Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Life Annuities

Erhan Bayraktar, Moshe Milevsky, David Promislow and Virginia Young

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Abstract: We develop a theory for valuing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We apply our method to value life annuities. One result of our paper is that the value of the life annuity is {\it identical} to the upper good deal bound of Cochrane and Sa\'{a}-Requejo (2000) and of Bj\"{o}rk and Slinko (2006) applied to our setting. A second result of our paper is that the value per contract solves a {\it linear} partial differential equation as the number of contracts approaches infinity. One can represent the limiting value as an expectation with respect to an equivalent martingale measure (as in Blanchet-Scalliet, El Karoui, and Martellini (2005)), and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market's price of mortality risk.

Date: 2008-02
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Journal Article: Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities (2009)
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