An expectation of life of 31.41 corresponds to the present value of an annuity payable annually in arrear at 0% interest of 30.91. According to the PA(90) tables, such an annuity value corresponds to a male life aged 44.17 years. The annuity value at 5% for such a life is 14.80. Construction of a special generational life table for the 50-year-old Australian male yields an exact value of 14.73. Comparisons at other ages using this approach are set out in columns (3) and (5) of table 6.
No entries are shown in column (3) of table 6 for ages below 30, because the generational expectations of life at the younger ages all exceed those in the PA(90) table, and reference to this standard table at these younger ages is not feasible.
The difficulty can be overcome by adopting an alternative approach which is applicable at all ages. The method relies on appendix equation (A4.2), which indicates that the value of a continuous annuity can be approximated by calculating the present value of a continuous annuity for a term certain equal to the complete expectation of life and then subtracting an adjustment term. The direct application of this approach produces relatively poor results, particularly at the “non-Gompertz” ages less than 50. But the formula does suggest a simpler approach, which is surprisingly accurate at all ages. If one were to apply the formula to calculate the continuous annuity value in both the basic period life table and in the generational life table, the difference according to (A4.2) would be equal to a deferred continuous annuity certain with term certain equal to the the difference between the life expectancies for the two tables and deferral period equal to the period life expectancy, plus the difference between the two adjustment terms. The latter difference is small and may be omitted.
To apply the method, therefore, one simply takes the annuity value from the original cross-sectional table (continuous, annually in arrear or annually in advance, as appropriate) and adds the value of a continuous deferred annuity with term certain equal to the difference of the life expectancies and deferral period equal to the the period table complete expectation of life.
As an example, consider the 50-year-old Australian male described earlier in this section. At age 50, the present value of an annuity payable annually in arrear to a 1990-92 period Australian male is 13.83. The expectation of life for this period male is 27.48. We have already used the Gompertz approach to estimate the expectation of life of an Australian male aged 50 in 1997, namely 31.41 years (using only the linear term). According to the deferred annuity approach, therefore, an approximation to the value at 5% of a generational life annuity payable annually in arrear to the Australian male aged 50 in 1997 is
13.83 + 27.48 a31.41-27.48 = 14.77 .
This is close to the exact value obtained by computing a special generational life table for this life (14.73).
Comparisons for other ages are shown in columns (4) and (5) of table 6. The method produces reliable answers over the entire age range.