In actuarial mathematics, the accumulation function a(t) is a function of time t expressing the ratio of the value at time t (future value) and the initial investment (present value).
Thus a(0) = 1 and the value at time t is given by: where the initial investment is
For various interest-accumulation protocols, the accumulation function is as follows (with i denoting the interest rate and d denoting the discount rate): In the case of a positive rate of return, as in the case of interest, the accumulation function is an increasing function.
The logarithmic or continuously compounded return, sometimes called force of interest, is a function of time defined as follows: which is the rate of change with time of the natural logarithm of the accumulation function.
The effective annual percentage rate at any time is: