## Valuation of mortality risk via the instantaneous Sharpe ratio: applications to life annuities.(English)Zbl 1170.91406

Summary: 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 J. Cochrane and J. Saá-Requejo [Beyond arbitrage: good deal asset price bounds in incomplete markets. J. Polit. Econ. 108, 79–119 (2000)] and of T. Björk and I. Slinko [Rev. Finance 10, No. 2, 221–260 (2006; Zbl 1125.91049)] 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.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 91B24 Microeconomic theory (price theory and economic markets) 93C20 Control/observation systems governed by partial differential equations 93C10 Nonlinear systems in control theory

Zbl 1125.91049
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### References:

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