Linear interpolation between the Gompertz e0 and e40 values was used to obtain the complete expectation of life below age 40. The necessary formulae ((A1.1), (A1.3) and (A1.7) in appendix 1) are easy to apply.
For each of the seven life tables, the Gompertz approximations to the median, quartiles and life expectancies are shown in table 1 alongside the exact values obtained directly from the life table. Based on the Gompertz distribution over (-, ), we would expect 57% of births to survive to an age equal to the complete expectation of life and 37% to survive to the modal age (appendix 1). These figures are also shown in table 1.
The comparisons are generally very good, and suggest that the Gompertz model should provide reliable approximations over broad sections of the life table encompassing the prime mortality ages. At the same time, it should be noted that the model is unlikely to be particularly helpful over limited ager ranges and should not, for example, be used to price short term life assurances.
Period and generational lifetables under constant uniform changes in mortality
Whilst no modern life table follows the Gompertz “law” closely, we saw in section 1 that the broad shape of most if not all life tables is still essentially of this form. The model allows useful accurate approximations of many functions based on the life table (Pollard, 1991). The relationship between period and generation life tables under constant uniform mortality change is another area in which such approximations can be made.
The methods outlined in this paper are dependent on four important observations concerning a life table of the Gompertz form.
The force of mortality at the mode of the curve of deaths is equal to k.
The effect on the life table of multiplying the force of mortality at every point throughout
the lifespan by the same factor rN (where r, the annual mortality improvement factor, is typically in the neighbourhood of 1.0) is to produce a new Gompertz life table in which a life aged x has mortality equivalent to a life in the original table aged x + N ln(r)/k.
(c) If a population has a cross sectional (period) life table of the Gompertz form with ageing parameter k, and all its members are subject to a constant on-going annual improvement factor r at every age, then a member of the population aged x with force of mortality x under the period table has a Gompertz generational life table with ageing parameter k+ln(r).
(d) If the force of mortality at exact age x is fixed at x and the ageing parameter k is allowed to change, then the change in the complete expectation of life at age x corresponding to the change in k is given by the derivative formula
d(ex)/dk = [1 - (x + k) ex]/k2 .