Compound interest

What is compound interest?

Suppose you put money in an account that pays interest. It is tempting to think your money grows by the same amount each year. But does it? What happens to the interest you earned last year — does it just sit there, or does it start working too? The answer decides whether we are looking at a straight line or a curve.

Say you earn interest on your savings. The next year, do you earn interest only on your original deposit, or also on the interest already added? If the interest itself earns interest, then each year you are earning on a slightly larger base than the year before. What shape does a thing take when its growth feeds its own growth?

Follow it arithmetically for a moment, because the arithmetic is the whole argument. Put 100 somewhere paying 7 percent a year. After a year you hold 107. The second year's 7 percent is charged not on 100 but on 107, so it brings 7.49 instead of 7.00 — a difference nobody would notice on a statement. The third year is charged on 114.49, and every year's base has all the previous years' surpluses folded inside it. Multiplying by 1.07 thirty times is not the same act as adding 7 thirty times: it arrives at about 761, against the 310 you would hold if each year's interest were paid out and spent. Those are the two lines in the figure below, and the gap between them is not the rate — the rate is identical.

Is there a way to feel that pace without a spreadsheet? Divide 72 by the rate as a percentage and you get, roughly, the years to double: 72 over 7 is a shade over ten, and 100 at 7 percent does reach 200 in about 10.2 years. Why 72? Doubling means multiplying by 2, so the time it takes is ln2 divided by ln(1+r), and for small rates ln(1+r) is nearly r itself — leaving about 0.693/r, or 69.3 divided by the percentage. We say 72 because it divides cleanly by 2, 3, 4, 6, 8 and 9, and because the small error leans the right way for once-a-year compounding. Notice what the rule quietly asserts: doubling time depends on the rate alone. How much you have does not enter it.

Now turn the mechanism around, because nothing in the arithmetic cares which way you are facing. A debt left unpaid compounds against you by precisely the same rule, usually at a rate well above anything savings pay, and often more frequently than yearly — which raises it further still. A card advertising 20 percent a year but charging monthly actually costs about 21.9 percent, because each month's charge joins the balance and is charged upon in turn.

And there is a third turn, the one that costs people most and gets discussed least. A fee is a subtraction from the rate, so a fee compounds too. Take that same 100 at 7 percent over thirty years and let one percentage point a year go in charges, leaving 6. You do not lose a thirtieth of anything. You finish near 574 rather than 761 — roughly a quarter of the final balance gone, to a number small enough to look like a rounding detail on any single statement.

This is why the early years feel disappointing and the later years feel magical. The snowball is tiny at first, so little sticks. Given enough turns, the same steady rate produces a curve that bends sharply upward — not because the rate changed, but because the base kept enlarging.

The misconception worth naming is that compounding is a property of clever products rather than of arithmetic. It is not: any quantity that grows by a percentage of itself compounds — a savings account, a debt, a population, a fee. What varies is only the rate and the number of turns, and of those two it is the turns that people systematically undervalue, probably because the first several look like nothing is happening.

Two honesties are owed. The first is that 7 percent is an illustration, not a forecast: no real investment delivers the same number every year on schedule, and if prices rise 3 percent while your balance rises 7, the part of the growth that buys anything new is nearer 4. The second matters more than it sounds. Because compounding multiplies rather than adds, an average return is not the return you get. A balance that falls 50 percent and then rises 50 percent has averaged zero and is nevertheless down a quarter, because the rise was applied to the smaller number. Order and volatility are part of the outcome, which the smooth curve below deliberately hides.

• Why starting ten years earlier often beats saving twice as much later.
• How the same mechanism works against you with debt.
• Why the order of good and bad years matters when you are drawing money out rather than paying it in.

Every term the collection defines is gathered in the glossary.