# Accumulation function

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The **accumulation function** *a*(*t*) is a function defined in terms of time *t* expressing the ratio of the value at time *t* (future value) and the initial investment (present value). It is used in interest theory.

Thus *a*(0)=1 and the value at time *t* is given by:

- .

where the initial investment is *k*.

Examples:

In the case of a positive rate of return, as in the case of interest, the accumulation function is an increasing function.

## Variable rate of return

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.

Conversely:

reducing to

for constant .

The effective annual percentage rate at any time is: