# As a further example, consider the female pensioner of section 3 aged 60 in 1998, we observed that her effective PA(90) age was 57.76, or 58 to the nearest integer. According to equation (3), the e_{58 }derivative with respect to k is

[1 - (0.00532 + 0.10123) 24.25) / (0.10123)^{2 }= - 155 .

The force of mortality _{58}, complete expectation of life e_{58 }and ageing parameter k required for this calculation can be found in table 2. A linear term Gompertz approximation to the generational expectation of life of the female pensioner is therefore

e_{57.76 }+ (-155) ln(.9875) = 26.40 .

The present value at 5% per annum interest of a life annuity to a female pensioner aged 60 is 12.40, and the life expectancy of such a period female is 22.52. Using the deferred annuity certain approach, therefore, the present value at 5% per annum interest of a generational life

annuity for this female is

12.40 + _{22.52} a

26.40-22.52

= 13.58,

which is close to the exact value of 13.54 calculated exactly using a specially computed generational life table. The annuity derivative method of section 3 produced an answer which was only slightly more accurate: 13.53.

6.

# Concluding remarks

The Gompertz “law” of mortality, so popular in years gone by, remains a very useful model for obtaining quick accurate approximations for many life table and related monetary functions, even though it no longer provides an exact representation of modern mortality on an age-by-age basis. In particular, it allows the rapid calculation of life expectancies and life annuity values in an environment of mortality change, and so avoids the need to prepare special generational life tables for different cohorts of lives and for different assumptions about future mortality change. Assurance functions can be obtained using standard premium conversion formulae.

The derivatives essential to the calculation of generational life annuities are easy to compute. Where a standard period life table is to be used frequently to obtain generational annuities, a series of columns of annuity derivatives at various rates of interest might be tabulated to make the calculations even easier.

All the special generational tables used in this paper for comparative purposes have been computed assuming that the mortality improvement factor applies to the force of mortality rather than the q mortality rate. Comparisons of generational life tables produced with adjustments to the q mortality rate with tables produced with the same adjustments to the force of mortality indicate that the distinction between the two is negligible and may be ignored for all practical purposes.

The Gompertz “law” is characterised by two parameters, and for this purpose the mode of the curve of deaths and the force of mortality at the mode are both convenient and