Accumulation function

In actuarial mathematics, the accumulation function a(t) is a function of time t expressing the ratio of the value at time t (future value) and the initial investment (present value).^[1][2] 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.