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

We develop a theory for valuing non-diversifiable mortality risk in an incomplete market 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 identical to the upper good deal bound of Cochrane and Saa-Requejo (2000) and of Bjork and Slinko (2006) applied to our setting. A second result of our paper is that the value per contract solves a 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, and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market's price of mortality risk