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

Abstract

Results well known in the actuarial community about closed-form expressions to Gompertz and Gompertz-Makeham life expectancies for a person aged x are still being independently rediscovered to this day. This note seeks to acknowledge previous results about closed-form expressions to Gompertz-Makeham life expectancies, especially in the actuarial science field, hoping to stimulate interdisciplinarity and provide the background for further developments, especially since the derivation of closed-form expressions for life expectancy (and annuities) based on particular mortality laws are matters of interest for multiple fields such as actuarial science, biology, demography, statistics among others.