Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality

The work of this paper is motivated by the study in Gerber et al. (2012) and some following papers, in which equity-linked death benefits embedded in various variable annuity products are valuated for any time-until-death random variables whose density function can be approximated by a linear combination of densities of exponential random variables. Their analysis is made for the case where the time-until-death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The main purpose of our study is to show that the discounted density approach can also be used to obtain similar explicit results on life-contingent options under the assumption of piecewise constant forces of mortality. Moreover, we study a term insurance product with the payoff at the time of death being equity-linked and inflation-indexed, and investigate two types of annuity-immediate products whose annual payments are equity-indexed with a minimum guaranteed amount. We also illustrate approximations and numerical calculations for some results obtained in this paper, and analyze parameter sensitivities.