Accumulated (Future) Value of an Annuity

The accumulated value (also called the future value or amount) of an annuity is what the stream of payments grows to, at the moment of the last payment, once each payment has earned compound interest.

Accumulated value — immediate annuity

For an immediate (ordinary) annuity of $n$ payments of $C$ at rate $i$ per period, the accumulated value is

$A=C\left[\frac{(1+i)^{n}-1}{i}\right].$

This is the sum of a geometric series: the first payment earns interest for $n-1$ periods, the last for $0$.

Accumulated value — annuity due

Every payment starts one period earlier, so it earns one extra period of interest:

$A_{\text{due}}=C\left[\frac{(1+i)^{n}-1}{i}\right](1+i).$

Where the formula comes from