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 the value at time t where the initial investment is k is given by:

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

Accumulation functions can be expressed for complex functions (not merely linear) using integration, in the following set up

$A(t)=\int _{0}^{t}f(x)\,dx$

where "t" is the finishing point. Visually, the total amount of accumulation is the area between the function and the x-axis between the bounds given.

The accumulation function has the following two properties:

• $a(0)=1.$
• In the case that the accumulation is due to interest it is an increasing function.

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