# Appendix 3

# 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 v^{t }and integrated for values of t from 0 to , one finds that

d(a_{x})/dk = -[k (IA)_{x }- A_{x }+ _{x }a_{x}] / k^{2 },

(A3.1)

which may be rearranged as

d(a_{x})/dk = [k (Ia)_{x }+ 1 - (k++_{x}) a_{x}] / k^{2 }.

(A3.2)

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 a_{x }and a_{x}.

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 a_{x:n}:

d(a_{x:n})/dk = -[k (IA)_{x:n }- A_{x:n }+ _{x }a_{x:n }] / k^{2 }.

(A3.3)