Andrew J.G. Cairns
Actuarial Mathematics and Statistics
School of Mathematical and Computer Sciences
Edinburgh, EH14 4AS, United Kingdom
Prepared for the Encyclopaedia of Actuarial Science
In this article we will review the role of mathematics as a tool in the running
of pension funds. We will consider separately defined benefit (DB) and defined
contribution (DC) pension funds.
2 Defined benefit pension funds
2.1 Deterministic methods
We will concentrate here on final salary schemes, and in order to illustrate the
various uses of mathematics we will consider some simple cases. In this section we
will describe the traditional use of mathematics first, before moving on to more
recent developments using stochastic modelling. We will use a model pension
scheme with a simple structure:
• Salaries are increased each year in line with a cost-of-living index CLI(t).
• At the start of each year (t, t + 1) one new member joins the scheme at age
25, with a salary of C 10, 000× CLI(t)/CLI(0).
• All members stay with the scheme until age 65 and mortality before age 65
is assumed to be zero.
• At age 65 the member retires and receives a pension equal to 1/60 of final
salary for each year as a member of the scheme, payable annually in advance
for life. This pension is secured by the purchase of an annuity from a life
insurer, so that the pension fund has no further responsibility for making
payments to the member. Final salary is defined as the salary rate at age
65 including the cost-of-living salary increase at age 65.1
• Salaries are increased at the start of each year and include a combination of
age-related and cost-of-living increases.
Let S(t, x) represent the salary at time t for a member aged x at that time. The
scheme structure described above indicates that
S(t + 1,x + 1) = S(t, x)× w(x + 1)
× CLI(t + 1)
where the function w(x) is called the wage profile and determines age-re