= 28.37 + (-201) ln(0.985) + /2 (3,288) [ln(.985)]2 = 31.78
Full computation of a special generational life table for a male aged 50 in 1997 under the given mortality improvement assumptions yields an exact generational life expectancy of 32.00. The approximate method has therefore produced an adjustment (from the 1990-92 period value of 27.48) of 4.30 compared with an exact difference of 4.52.
Comparisons for other 1977 ages under the same scenario are shown in table 5. The effects of including only the linear term and including both linear and quadratic terms are shown in this table, and it is clear that the quadratic Gompertz-based method produces accurate approximations at all ages. Reasonable approximations are obtained using the linear term alone from about age 50 onwards, and might be expected to be obtained at even younger ages in situations where the annual rate of mortality change is smaller. For most purposes, therefore, only the first derivative is required. Calculations under different mortality improvement assumptions indicate that the discrepencies between the Gompertz approximations and the exact calculations are minor compared with the the effects of small changes in the very uncertain mortality improvement assumption.
The force of mortality in any real life table is high immediately after birth and falls away rapidly; it is usually considered indeterminate at age 0. By contrast, under the Gompertz “law”, the force of mortality continues to decline with reducing age. Under modern low mortality regimes, it is the mortality at the adult ages and improvements in this mortality which have the major impact on life expectancy. For the application of the above methods at age zero, therefore, the high, indeterminate 0 value is not used in evaluating the derivatives, but 1.
The adjustment for mortality improvement between 1990-92 and 1997 for a male aged
in 1997 leads to consideration of a life aged -1 in the 1990-92 table. In our calculations, the
expectation of life of a male aged -1 has been treated as being one year greater than the 1990- 92 expectation of life at birth (ie zero mortality before birth).
Generational life annuities in the absence of increasing annuity functions
In section 3, we demonstrated the use of the annuity derivative formula (5) in the calculation of the present value of a generational annuity. The formula relies on the availability of increasing annuity monetary functions in the basic cross-sectional life table. The necessary functions are not always provided. Where suitable monetary functions are not available, it is possible to obtain the generational annuity value by first determining the generational expectation of life.
Let us imagine, for example that we require the present value at 5% per annum interest of a life annuity on a 50-year-old male selected at random from the Australian population in 1997, assuming an annual mortality improvement factor of 0.985. The generational expectation of life of this life based on the Taylor series expansion using only linear terms is 31.41 (table 5). By reference to a standard table, it is easy to deduce an approximation for the life annuity.