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:

where the initial investment is ${\displaystyle A(0).}$

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.

Conversely:

reducing to

for constant ${\displaystyle \delta }$.

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