Accumulation function

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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

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.

Variable rate of return[edit]

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.


reducing to

for constant .

The effective annual percentage rate at any time is:

See also[edit]