Future Value of an Annuity

The future value (or amount) of an annuity is the total accumulated value of all its instalments, together with the compound interest each has earned, at the moment the very last instalment is paid.

The logic behind this formula: the first instalment earns compound interest for $(n-1)$ periods (since it is paid at the end of period 1 and the annuity runs to the end of period $n$), the second instalment earns interest for $(n-2)$ periods, and so on, down to the last instalment, which earns no interest at all (it is paid at the very moment the value is being found). Summing this geometric series of $n$ terms — $P(1+i)^{n-1} + P(1+i)^{n-2} + \cdots + P(1+i) + P$ — produces exactly the closed-form bracket above.

For an annuity due, since every payment falls one period earlier, the future value formula picks up one extra period of interest throughout:

$A_{\text{due}} = P\left[\dfrac{(1+i)^n - 1}{i}\right](1+i)$ …