Mortality derivatives and the option to annuitise

Abstract

Most US-based insurance companies offer holders of their tax-sheltered savings plans (VAs), the long-term option to annuitise their policy at a pre-determined rate over a pre-specified period of time. Currently, there is approximately one trillion dollars invested in such policies, with guaranteed annuitisation rates, in addition to any guaranteed minimum death benefit. The insurance company has essentially granted the policyholder an option on two underlying stochastic variables; future interest rates and future mortality rates. Although the (put) option on interest rates is obvious, the (put) option on mortality rates is not. Motivated by this product, this paper attempts to value (options on) mortality-contingent claims, by stochastically modelling the future hazard-plus-interest rate. Heuristically, we treat the underlying life annuity as a defaultable coupon-bearing bond, where the default occurs at the exogenous time of death. From an actuarial perspective, rather than considering the force of mortality (hazard rate) at time t for a person now age x, as a number μx(t), we view it as a random variable forward rate μ˜x(t), whose expectation is the force of mortality in the classical sense ( μx(t)=E[μ˜x(t)]). Our main qualitative observation is that both mortality and interest rate risk can be hedged, and the option to annuitise can be priced by locating a replicating portfolio involving insurance, annuities and default-free bonds. We provide both a discrete and continuous-time pricing framework.