Annuities & Loan Amortization

Value a stream of equal payments with one geometric series — pensions, perpetuities, and loan payments.

The idea

An annuity is a contract that pays the same amount $C$ at the end of each of the next $n$ periods — a pension, a rent, the payments on a car loan. Its value today has a single closed form.

Theorem (Annuity formula).

If money earns interest at rate $r > 0$ per period, then a stream of $n$ payments of $C$, one at the end of each of the next $n$ periods, is worth today $\mathrm{PV} = C\,\frac{1 - (1+r)^{-n}}{r}.$

The formula values the whole stream at one moment, so it can be compared directly with a price, a loan amount, or a lump sum offered in its place.

Two consequences follow. A perpetuity pays $C$ at the end of every period forever; let $n$ grow without bound, and the powers $(1+r)^{-n}$ shrink to $0$, leaving $\mathrm{PV} = \frac{C}{r}$. And the same formula prices a loan: a lender who hands over $L$ today sets the repayment $C$ so that the present value of the $n$ payments equals $L$.

Ways to work on it

• Walkthrough. Derive PV = C 1-(1+r)^-nr from the geometric sum, take the perpetuity limit, and flip it to price a loan.
• Practice. Perpetuity values, short annuities, and loan payments with clean numbers.
• Hardest. Long-horizon annuities in exact form, and the balance remaining partway through a loan.

Not sure where to start? Take the ten-question placement test.