How APR and APY actually work

APR is a rate you quote. APY is a rate you get.

An APR describes one period's rate multiplied up to a year. If a card charges 2 percent a month, its APR is 24 percent, and that multiplication ignores the fact that unpaid interest starts earning interest. APY accounts for that, which is why it is always the higher number whenever interest is added more than once a year.

To go from a quoted rate to the rate you actually get:

$$\text{APY} = \left(1 + \frac{\text{APR}}{n}\right)^{n} - 1$$

To go back the other way, when you know the yield and want the nominal rate behind it:

$$\text{APR} = n\left[(1 + \text{APY})^{\frac{1}{n}} - 1\right]$$

Both take $n$, the compounding frequency. An APY quoted with no frequency is complete on its own, because compounding is already inside it. An APR quoted with no frequency is not.

At 5 percent APR compounded monthly, the period rate is $0.05/12$. Twelve periods grow one dollar to 1.05116, so the APY is 5.116 percent. On $10,000 that is $511.62 of interest rather than $500. The extra $11.62 is interest that was itself earned by interest.

The APR against APY calculator on this page converts in both directions. The APR against APY page is which label a product is required to show you, and what fees sit inside a US loan APR. This page is the algebra.

The APR APY explorer is the same conversion as two bars: drag the APR and watch the yield pull away.

On a savings balance the gap is small and in your favour. On a revolving debt it is larger and against you, because card rates are high and the compounding is monthly or daily.

A card advertising 24.99 percent APR and compounding monthly has a monthly rate of $0.2499/12$. Twelve months of that is 1.280606, so the effective annual rate is 28.0606 percent. Carrying $5,000 for a year costs $1,403.03, which is 28.0606 percent of the balance rather than the 24.99 percent on the offer. On a card, compounding works against you and the gap is three full percentage points.

The gap widens as $n$ rises, but with a ceiling. At 5 percent, monthly compounding gives 5.116 percent and daily gives 5.127 percent. Compounding every instant, the mathematical limit, gives 5.127 percent as well to three decimal places. There is far less in the frequency than the marketing suggests. The rate itself moves the answer far more than how often it is added.

How credit cards charge interest is the daily-balance machinery that produces that 28.0606 percent. This page is why the advertised APR is not that number.

A bank advertises a 4.5 percent APY and compounds monthly. The APR behind it is $12\left[(1.045)^{1/12} - 1\right]$, which is 4.4098 percent. Both numbers describe the same account. This is why comparing one bank's APY against another's APR tells you nothing useful.

An account advertising 5.05 percent APY loses to one advertising 5 percent APR compounded monthly, because that 5 percent APR is worth 5.116 percent APY. The larger headline number is the worse account, and nothing on either page tells you so. Convert everything to APY before comparing anything, including debts.

On savings the two terms differ only by compounding. On US borrowing there is a second difference, and it runs the other way: a closed-end loan APR is required to fold in certain fees, which pushes it above the note rate, while compounding is still left out. APR against APY is that second difference. Do not fold it into this conversion. Fees that have already been packed into an APR cannot be unpacked by taking a root.

Savings account types is which account a quoted APY is attached to. A higher APY on a notice account is not a higher APY on money you can take tomorrow.

At 5 percent APR, monthly compounding gives a 5.116 percent APY and daily compounding gives 5.127 percent. Compounding every instant, the mathematical limit $e^{0.05} - 1$, is 5.127 percent to three decimal places as well. There is far less in the frequency than the marketing suggests.

A quarter of a point on the rate itself beats every possible improvement in frequency combined. Chasing daily compounding on a 5 percent account, rather than a 5.25 percent account that compounds monthly, is chasing hundredths while leaving tenths on the table.

On a card the picture flips because the rate is high. 24.99 percent compounded monthly is already 28.0606 percent effective. Daily compounding at that APR would sit a little higher still, and the extra is no longer hundredths of a point. The rate, not the calendar, is what opened the gap. How credit cards charge interest is the daily-balance machinery. This page is why the advertised APR is not that effective number.

The ceiling itself is how continuous compounding works: $e^{r} - 1$, on an 8 percent sheet rather than this 5 percent conversion. Monthly against continuous compounding is the finite schedule against that limit.

An account advertising 5.05 percent APY loses to one advertising 5 percent APR compounded monthly, because that 5 percent APR is worth 5.116 percent APY. The larger headline number is the worse account, and nothing on either page tells you so until you convert.

The same trap runs on debts. A card APR and a savings APY are not comparable until both are effective annual rates, and even then one is a cost and the other is a yield. Convert, then read the sign.

On US closed-end loans there is a second difference that this conversion does not unpack. A loan APR is required to fold in certain fees, which pushes it above the note rate, while compounding is still left out. APR against APY is that second difference. Fees packed into an APR cannot be unpacked by taking a root. Savings account types is which account a quoted APY is attached to.

It is not a Truth in Lending treatise, not a ranking of banks, and not a claim that more frequent compounding is always worth chasing. The frequency ceiling at ordinary deposit rates is a few hundredths of a point. A quarter of a point on the rate itself beats every possible improvement in frequency combined.

Reading APR as a nominal rate is a US convention. Consumer credit rules in the United Kingdom and the European Union define their APR as an effective annual rate, with the compounding already inside it, so a loan rate labelled APR outside the US may already be closer to what a US page would call an APY. Find out which convention a quote was written under before setting it against another.

The three sheets are a 5 percent APR compounded monthly (APY 5.116 percent, $511.62 on $10,000), a 24.99 percent card APR (28.0606 percent, $1,403.03 on $5,000), and a 4.5 percent APY taken back to a 4.4098 percent APR. This is educational material, not financial advice.