Analytic expressions for annuities based on Makeham–Beard mortality laws

Abstract

This note derives analytic expressions for annuities based on a class of parametric mortality “laws” (the so-called Makeham–Beard family) that includes a logistic form that models a decelerating increase in mortality rates at the higher ages. Such models have been shown to provide a better fit to pensioner and annuitant mortality data than those that include an exponential increase. The expressions derived for evaluating single life and joint life annuities for the Makeham–Beard family of mortality laws use the Gauss hypergeometric function and Appell function of the first kind, respectively.

Closed-form expressions to Gompertz-Makeham life expectancies: a historical note

“…Under the Gompertz-Makeham law, Milevsky (2006) proved that: , we can observe that expression (3) is equal to expression (2), reinforcing that those results were already well known and used by the actuarial science community. Nowadays, the work of Bowie (2021), which derives an analytical expression for annuities based on Makeham-Beard mortality law (RICHARDS, 2012), is a good example of the perpetuation of this practice in the field of actuarial science.…”