Annuities

An annuity is a series of equal payments made at regular intervals — saving a fixed amount every month, or paying off a loan in equal instalments. Each payment earns compound interest from the moment it is made, but for a different length of time, so we cannot just multiply one payment by the number of payments.

Suppose you pay x at the end of each period and the interest rate per period is i. The first payment compounds for n - 1 periods, the next for n - 2, and so on. Adding the grown-up payments is a geometric series with common ratio 1 + i, and its sum collapses to a tidy formula for the future value F:

F = x\,\frac{(1 + i)^n - 1}{i}.

Present value: what it is worth today

Sometimes we want the lump sum today that is equivalent to the whole stream of future payments — the present value P. This is what a loan of P can be repaid by n equal instalments of x:

P = x\,\frac{1 - (1 + i)^{-n}}{i}.

For n equal payments of x at interest rate i per period:

future value (saving up): F = x\dfrac{(1 + i)^n - 1}{i};
present value (paying off): P = x\dfrac{1 - (1 + i)^{-n}}{i};
both are just a geometric series in disguise, ratio 1 + i;
the rate i is per period — match it to the payment interval (monthly payments need a monthly rate).

Saving, payment by payment

Below, a payment of 100 is made each period. The curve is the future value F after n payments; the straight line is the total cash actually paid in (100n). The gap between them is the interest the annuity has earned — and it widens fast as the rate climbs.