The second derivative can be calculated from the first by means of the formula
d^{2}(e_{x})/dk^{2 }= [(_{x}+3k) d(e_{x})/dk + e_{x}]/k^{2 }.
(4)
Observations (a) and (b) are wellknown and are easily proved. Observation (c) is also easily proved. Observation (d) appears to be novel, and proofs of formulae (3) and (4) are
given in appendix 2. We note also that
d(a_{x})/dk d(a_{x})/dk d(a_{x})/dk = [k(Ia)_{x }+ 1  (k++_{x}) a_{x}] / k^{2 }.
(5)
This follows as a corollary from the proof of (3), and the derivation is set out in appendix 3.
3.
Life annuities under constant uniform mortality change
Let us imagine that we need to find the present value at 5% interest of an annuity to be issued to a female pensioner aged 60 in 1998. An investigation of the mortality of the female pensioners of the insurance company involved has revealed that the PA(90) table rates were about 10% too heavy in 1990. Since then, mortality has improved by about 1.5% per annum at all ages. The actuary expects that in respect of pensions issued in 1998, the ongoing improvement will be about 1.25% per annum. The observations of section 2 allow us to obtain an accurate approximation to the annuity value very quickly and easily using the Taylor series expansion.
Extracts from the PA(90) table for female pensioners are given in table 2. We note that
the mode of the curve of deaths is near age 85.5, so that k _{85.5 }
= 0.10123. The observations
of the previous section allow us to proceed as follows, using mortality adjustment factors 0.90 in 1990, 0.985 between 1990 and 1998, and 0.9875 from 1998 onwards:
1. Effective PA(90) age in 1998 = 60 + [ln(0.90) + 8 ln(0.985)]/0.10123 = 57.76 = 58 (to nearest integer)
2. Derivative of annuity = [k (Ia)_{58 }
+
1  (k++_{58}) a_{58}] / k^{2 }
= [k(S_{59}/D_{58 }+ 1/12) + 1  (k++_{58}) a_{58}] / k^{2 }= [0.10123 (0.04879)(1,124.959/8.126414 + 1/12) + 1

(0.10123+.04879+0.00532) (13.461)] / (0.10123)^{2 }
= 39.707 3. Generational annuity value a_{57.76 }

39.707 ln(0.9875)
= 13.03 + 0.50 = 13.53
Construction of a special generational life table for females aged 60 in 1998 leads to an exact answer of 13.54. The approximate method is clearly very accurate. As we would expect, the generational annuity value is rather larger than the PA(90) cross sectional value for age 60 of 12.403.