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

TL;DR

This paper introduces a new method for valuing non-diversifiable mortality risk in incomplete markets by using the instantaneous Sharpe ratio, applying it to life annuities, and connecting it to existing bounds and PDE solutions.

Contribution

It develops a novel valuation framework for mortality risk using the instantaneous Sharpe ratio, linking it to good deal bounds and PDE representations.

Findings

Annuity value equals the upper good deal bound.

The contract value satisfies a linear PDE as contracts increase.

The limiting value can be expressed as an expectation under an equivalent martingale measure.

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.