The effect on the present value of a life annuity of changing the ageing parameter in the Gompertz “law”
If equation (A2.6) is multimplied by vt and integrated for values of t from 0 to , one finds that
d(ax)/dk = -[k (IA)x - Ax + x ax] / k2 ,
which may be rearranged as
d(ax)/dk = [k (Ia)x + 1 - (k++x) ax] / k2 .
Since the present value of an annuity payable annually in arrear (in advance) is effectively the present value of the continuous annuity minus (plus) 0.5, the right-hand side of (A3.2) is also the derivative of ax and ax.
If instead of integrating from 0 to as we did to derive (A3.1), we integrate from 0 to n, we obtain the obtain the derivative of the temporary annuity ax:n:
d(ax:n)/dk = -[k (IA)x:n - Ax:n + x ax:n ] / k2 .