The accumulation function a(t) is a function defined in terms of time t expressing the time value of money. It is used in interest theory.

The accumulation function assumes the initial investment to be 1. To obtain value of money where the initial investment is k, simply have

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

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

$A(t)=\backslash int\_0^x\; f(t)\backslash ,dt$

where "x" is the finishing point and the function is in terms of "t" or time. 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:

• $a(0)=1.$
• It is an increasing function, in most cases, because interest are rarely negative.

Common accumulation functions

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The accumulation function for the two common types of interest:

Simple interest

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

Compound interest

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

Mathematical finance

Funcţie de acumulare