# Accumulation function

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:

$A(t) = k \cdot a(t)$.

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:

$\delta_{t}=\frac{a'(t)}{a(t)}\,$

which is the rate of change with time of the natural logarithm of the accumulation function.

Conversely:

$a(t)=e^{\int_0^t \delta_u\, du}$

reducing to

$a(t)=e^{t \delta}$

for constant $\delta$.

The effective annual percentage rate at any time is:

$r(t) = e^{\delta_t} - 1$