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.

In the case that the accumulation is due to interest it is an increasing function,

Variable rate of return

The logarithmic or continuously compounded return 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:

$a(n)=e^{\int _{0}^{n}\delta _{t}\,dt}\ ,$

The effective annual percentage rate at any time is:

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

Common accumulation functions

The accumulation function for the two common types of interest:

Simple interest

$a(t)=1+t\cdot i.$

Compound interest

$a(t)=(1+i)^{t}.$

See also

• time value of money