Appendix 4. Life annuities and annuities certain
In appendix 1 we denoted the time to death of an individual now aged x by the random variable T. Approximate formulae for the expectation and standard deviation of T under the Gompertz “law” were given by (A1.7) and (A1.8) respectively.
The present value at force of interest of an annuity payable continuously until the time of
death is
a_{x }
= E(a_{T }) = E(1  e^{T})/ .
(A4.1)
This is the expectation of a nonlinear transform of the random variable T, and using standard
statistical formulae we are able to deduce that
a_{x }
a_{ }

1/2 v^{ }/ [k(1 + e^{k(xm) }
)]^{2 },
(A4.2)
where is the complete expectation of life e_{x}. In other words, we can approximate the life annuity by calculating an annuity for a term certain equal to the complete expectation of life and adjust for the bias by subtracting the second term in (A4.2). Applied directly, (A4.2) produces poor results at all but the older ages. The reason for this is that it takes account of the heavily discounted olderage mortality and ignores the nonGompertz mortality at the younger ages.
Formulae for the standard deviation of a life annuity can also be derived, but these also produce poor estimates at all except the older ages, for the same reason.