APR vs APY: The Compounding Fine Print

0:00Two banks, the same 5% — and two different jars ▶

A bank on this street, and a bank across the road. Both windows show the same interest rate: 5%. You put $10,000 in each, and you leave it there for a year. The bank on the left pays you at the end of the year, and only then. The bank on the right pays you a little every single day. Same rate, same money, same year — so watch the jars fill side by side. At the end of the year the left jar holds exactly what the sign promised. The right jar holds more, but so little more that you need a magnifier to see it. That is twelve dollars and sixty-seven cents. That gap has a name: the rate they multiply by is called the APR, the Annual Percentage Rate. The rate your jar actually reached is the APY — the Annual Percentage Yield. On savings, US law makes the bank print the APY. On what you borrow, the sign shows the APR. Same fine print — it pays you in one direction and costs you in the other. Same 5% in both windows. So why did the jars end up different at all? All you need is one idea: interest that arrives early starts earning interest itself.

1:29Cut the year into pieces ▶

Cut the year into pieces: the left bank makes just one payment, right at the very end. Nothing arrives early, so nothing gets a head start. Now cut the same year into four. The first payment lands in March. And from March onward, that payment earns as well. Before I show you the number, guess how much of the whole gap those four payments already buy you,

1:57Where it comes from: APY = (1 + r ÷ m)^m − 1 ▶

and hold that guess. Here is where it comes from. Cut the year into twelve, and every month the bank pays a twelfth of the rate. Next month it pays again — but now on the bigger pile. Give both of them short names: the rate is r, and how often they pay is m. Then every payment hands you the same slice, and that slice is r divided by m. But the second slice is not applied to your $10,000. It is applied to what is already there. Adding would hand you the very same slice every single time. Stacking hands you a slice of something bigger, every single time — and that is the whole difference. They stack: (1 + r ÷ m), multiplied by itself m times. That is the whole formula: APY = (1 + r ÷ m)^m − 1. Four payments a year turn 5% on the sign into 5.09%. Four cuts already buy you three quarters of the whole gap. Twelve payments give 5.12%. Every day gives 5.13%. And what if the bank paid you every second, or every single instant? Then m runs off to infinity — and the stack does not run off with it. Slice it finer and finer, and the number does not run away: it walks up to one spot and stops. Mathematicians have a name for that spot: they call it e. And the ceiling here is that same e, raised to your own rate. So the ceiling is e^r − 1, and at 5% it lands just short of 5.13%. More often is better, but it runs into a wall, and the wall is close.

3:57Rule vs exact: 5.00, 5.09, 5.12, 5.13 — and the ceiling e^r − 1 ▶

So put the four of them side by side, and hold them against that ceiling. Once a year: exactly 5%. Four times, twelve times, every day: the numbers on the scale climb. And each step up buys less than the step before it. Which makes those twelve dollars look like nothing — until you leave them alone.

4:21Thirty years: the gap you no longer need a magnifier for ▶

Leave both jars alone for thirty years and touch nothing. The right one pulls away, and now you can see it without a magnifier. The gap is almost $1,600. The same rate in both windows, the same money in,

4:42The whole story ▶

and one single line of fine print. So, here is the whole story. The APR is the rate they multiply by. The APY is the rate your jar actually reaches. They only agree when the bank pays you exactly once a year. And the gap is never the bank being generous. It is your own interest, arriving sooner.

5:11What's next: Real vs Nominal Return ▶

Every other time, the fine print is doing the work — and that's the math behind the money. Next time: the number that quietly takes it all back — even at 5.13%. And for you: does your bank quote APR or APY? Go and look it up today. I'm the MoneyMathician, and if you subscribe, the next one will find you. Until then: do the math!