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

25 Pages Posted: 31 Jan 2009 Last revised: 25 Jan 2014

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Moshe A. Milevsky

York University - Schulich School of Business

S. D. Promislow

York University - Department of Mathematics & Statistics

V.R. Young

University of Michigan at Ann Arbor - Department of Mathematics

Date Written: March 3, 2008

Abstract

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.

Keywords: Stochastic mortality, pricing, annuities, Sharpe ratio, non-linear partial differential equations, market price of risk, equivalent martingale measures

JEL Classification: G13, G22, C60

Suggested Citation

Bayraktar, Erhan and Milevsky, Moshe A. and Promislow, S. D. and Young, V.R., Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Life Annuities (March 3, 2008). Journal of Economic Dynamics and Control, Vol. 33, No. 3, 2009. Available at SSRN: https://ssrn.com/abstract=1335476 or http://dx.doi.org/10.2139/ssrn.1335476

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Moshe Milevsky

York University - Schulich School of Business ( email )

4700 Keele Street
Toronto, Ontario M3J 1P3
Canada

S. D. Promislow

York University - Department of Mathematics & Statistics ( email )

4700 Keele Street
Toronto, Ontario M3J 1P3
Canada
(416)736-2100, ext. 33767 (Phone)

HOME PAGE: http://starcraft.ccs.yorku.ca/atlas/servlet/atlas/action/AtlasAction/template/person.vm?eventsubmit_

Virginia R. Young

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States
734-764-7227 (Phone)

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