4.

# Period and generational life expectancies under constant uniform mortality change

The actuary is often asked the expectation of life of a member of the community. A common approach to finding the answer is to consult the most recent national life table and quote the complete expectation of life corresponding to the enquirer’s age. Assuming that the individual is in average health, the answer obtained in this manner is usually conservative, for two reasons: first, the national life table is based on data some years prior to the enquiry and is therefore out of date, and second, the approach uses period data and makes no allowance for further improvements in mortality during the individual’s lifetime.

It is of course possible to make assumptions about mortality trends and construct a special generational life table for the enquirer and so deduce his/her life expectancy. Such a process is computationally straightforward, but somewhat of a nuisance for a single enquiry. Furthermore, different generational tables have to be produced for different ages of enquirers and for different assumptions about mortality improvement. The observations of section 2 allow the actuary to produce an accurate approximation to the generational life expectancy after only a few lines of arithmetic.

To demonstrate the approach, consider an Australian male aged 50 in 1997 and in average health. The most recent national life table is the Australian Life Table 1990-92, which is six years out of date at the time the life expectancy of the 50-year-old male is required.

Over the decade to 1991, Australian male mortality in the important age range 75-95 improved by about 1.5% per annum (table 3). The force of mortality at the modal age of the Australian Life Table 1990-92 (Males) is 0.09312 (table 4). If the rate of mortality improvement in the decade to 1991 has continued in subsequent years, then, under a Gompertz approximation and observation (b), one can treat the male aged 50 in 1997 as equivalent to one aged

50 + 6 ln(0.985)/0.09312 = 49.03

or about 49 in 1991. The life expectancy of a 49-year-old male according to the Australian Life Table 1990-92 is 28.37. This will be the life expectancy for a male aged 50 under the synthesised 1997 period life table.

We now need to determine the life expectancy of the 50-year old under the generational life table pertaining to males aged 50 in 1997. To do this, we note (from table 4) that _{49 }= 0.00358. According to formulae (2) and (3), therefore, the first two derivatives of the complete expectation of life are

d(e_{49})/dk = [1 - (0.00358 + 0.09312)28.37]/(0.09312)^{2 }= - 201 ;

d^{2}(e_{49})/dk^{2 }= - [(0.00358+30.09312)(-201) + 28.37]/(0.09312)^{2 }= 3,288.

Invoking observation (c) of section 2 and applying the Taylor series expansion, the generational life expectancy of the male aged 50 in 1997 will be