Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities

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 identical to the upper good deal bound of Cochrane and Saá-Requejo (2000) and of Björk 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 (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.

Indifference Pricing of Pure Endowments Via BSDEs under Partial Information

“…In the case where she/he sells the insurance contract the information at her/his disposal is given by the filtration G defined in (2.3), whereas in the case of pure investment by the filtration F given in (2). The set of admissible strategies is defined below.…”

“…In these papers the Black & Scholes pricing methodology is applied under the hypotheses of market completeness and independence between financial and insurance setting. Since then, many efforts have been done to relax the assumption of completeness and several approaches have been proposed, for instance in Møller [38], Ludkovski and Young [37], Bayraktar et al [2], Delong [22], Blanchet-Scalliet et al [8]. However the problem of incorporating some kind of dependence between the financial and the insurance market, which is empirically observed, has started to be addressed only recently.…”

Canonical Valuation of Mortality‐Linked Securities

“…The standard deviation is derived from an assumed process for the evolution of mortality. This approach has also been considered by Young (2008) and Bayraktar et al (2009). • Risk-neutral dynamics of death/survival rates This method is based on a stochastic mortality model, which is, at the very beginning, defined in the real-world measure and fitted to past data.…”

A Model-Point Approach to Indifference Pricing of Life Insurance Portfolios with Dependent Lives

“…In fact, the insurance market is inherently incomplete to the extent that no unique pricing probability measure is available. Thus, various pricing procedures have been proposed in the literature: using indifference pricing as in Ludkovski and Young (2008), the Sharpe ration as in Bayraktar et al (2009) or characterizing the price via the good deal bounds in Delong (2012). In Ludkovski and Young (2008), an indifference pricing procedure is developed using a fully stochastic framework that extends the original pricing approach introduced by Hodges and Neuberger (1989) to mortality-contingent claims.…”