Some Further Experiments in the use of the Incomplete Gamma Function For the Calculation of Actuarial Functions

Abstract

The experiments described in this paper arose from some calculations made in connexion with the recent discussion on the mortality of life-office annuitants (J.I.A. lxxviii, 27) and are a natural development of the principles described in the paper submitted to the Centenary Assembly of the Institute (Proceedings, 11, 89). They fall into two groups. Those in the first, arising from an attempt to find an alternative method on which to base projections, have an affinity with those set out by Starke in his recent paper (J.I.A. lxxviii, 171), in that a formula for expressing mortality is derived essentially dependent on year of birth. There are, however, fundamental differences between the approaches, in that Starke is concerned mainly with an ad hoc expression for a derived function of qx without regard to the problem of computing functions, whereas the present experiments are concerned with (i) further developing actuarial technique based on the curve of deaths rather than on rates of mortality, and (ii) the calculation of actuarial functions without recourse to the usual methods of commutation columns. The second group of experiments is an extension of the gamma-function technique to the calculation of joint-life functions, including contingent assurances.