# 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:

${\displaystyle A(t)=A(0)\cdot a(t)}$.

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):

• simple interest: ${\displaystyle a(t)=1+t\cdot i}$
• compound interest: ${\displaystyle a(t)=(1+i)^{t}}$
• simple discount: ${\displaystyle a(t)=1+{\frac {td}{1-d}}}$
• compound discount: ${\displaystyle a(t)=(1-d)^{-t}}$

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:

${\displaystyle \delta _{t}={\frac {a'(t)}{a(t)}}\,}$

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

Conversely:

${\displaystyle a(t)=e^{\int _{0}^{t}\delta _{u}\,du}}$

reducing to

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

for constant ${\displaystyle \delta }$ .

The effective annual percentage rate at any time is:

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