On Gamma-Gompertz-Makeham Assurances and Life Annuities

We focus on the gamma-Gompertz-Makeham model, and derive useful structural properties for this mortality model. We provide the basic properties like moments, remaining life expectancy, single life annuity, among many others, in closed form, and so it eliminates the need of evaluating them through numerical integration directly. The estimation of the gamma-Gompertz-Makeham model parameters is performed by using the maximum likelihood method under the traditional discrete Poisson distribution, as well as under the recently introduced discrete Bell distribution, which is an interesting alternative to the usual Poisson distribution, mainly in the presence of overdispersion. We illustrate the performance of the gamma-Gompertz-Makeham model in a human mortality database, and compute, based on the Poisson and Bell distributions, the remaining life expectancy, single life annuity and single assurance at ages 30, 55, and 80 in France, Italy, Japan, and Sweden, from 1947 to 2020 males and females separately.