Analytical and easily calculated expressions for continuous commutation functions under Gompertz-Makeham mortality

TL;DR

This paper derives simple, easily computed formulas for life insurance calculations under Gompertz-Makeham mortality using common functions like exponential and gamma, avoiding complex numerical integration.

Contribution

It presents new expressions for commutation functions under Gompertz-Makeham mortality using standard functions available in common software.

Findings

Expressions use exponential, gamma, and gamma distribution functions.

Eliminates need for numerical integration in calculations.

Simplifies life insurance actuarial computations.

Abstract

It is known, but perhaps not well-known, that when the mortality is assumed to be of Gompertz-Makeham-type, the expected remaining life-length and the commutation functions used for calculating the expected values of various types of life insurances can be expressed with an incomplete gamma function with a negative shape parameter. This is not of much use if ones software cannot calculate these values. The aim of this note is to show that one can express the commutation functions using only the exponential function, the (ordinary) gamma function and the gamma distribution function, which are all implemented in common statistical and spreadsheet software. This eliminates the need to evaluate the commutation functions and expected remaining life-length with numerical integration.