Accumulation function

Accumulation function
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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.

Examples:

simple interest: $a(t)=1+t \cdot i$
compound interest: $a (t) = (1 + i) t$

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 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:

$a(t)=e^{\int_0^t \delta_u\, du}$

reducing to

a (t) = e t δ

for constant $δ$ .

The effective annual percentage rate at any time is:

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

Variable rate of return

See also