Future Value (Amount) of an Ordinary Annuity

The future value (or amount) of an annuity is the total accumulated value of all the payments, together with the compound interest they earn, computed at the time of the last payment. It answers: if I deposit the same sum every period, how much will I have built up by the end?

Consider $n$ equal payments of $P$, each made at the end of its period (an ordinary annuity), earning interest at rate $i$ per period. The last payment earns no interest (it is made at the very moment we value the fund), the second-last earns interest for one period, and so on back to the first payment, which earns interest for $(n-1)$ periods. Adding these up gives a geometric series whose sum is:

where $A$ is the future value (amount), $P$ is the periodic payment, $i$ is the interest rate per payment period (as a decimal), and $n$ is the number of payments. The bracketed factor $\\dfrac{(1+i)^n-1}{i}$ is called the future-value annuity factor; it is simply the amount to which a payment of ₹1 per period accumulates.

The formula is nothing more than compound interest applied payment-by-payment and then totalled — which is exactly the independent cross-check used to verify every future-value numerical in this chapter (Worked Example 1 does both, side by side). Because the very first deposit sits in the fund longest, it contributes the most interest; the final deposit contributes none at all. …