How credit cards charge interest

A card does not charge interest once a month on the number printed at the top of the statement. It charges every day, on whatever was owed that day.

The conversion is arithmetic. In the United States a card agreement takes the quoted purchase APR and divides it by 365 to get a daily periodic rate, and that is the rate the issuer actually applies. A 24 percent APR becomes 0.0658 percent a day. A few agreements divide by 360 instead, which makes the daily rate slightly higher, and the divisor is disclosed next to the rate in the agreement.

Applied day by day across a billing cycle, and as long as each day's interest is not itself added to the next day's balance, that rate produces the same charge as applying it once to the average of the daily balances:

$$\text{Interest} = \bar{B} \times \frac{\text{APR}}{365} \times d$$

where $\bar{B}$ is the average daily balance and $d$ is the number of days in the cycle. Issuers usually describe this on the statement as the average daily balance method including new purchases.

Two things follow straight away. A balance that existed for six days is charged for six days, whatever the statement shows later. And an issuer cannot reach back past the current cycle: double-cycle billing, where last month's balance was pulled into this month's calculation, was banned in the United States by the Credit CARD Act. The APR is the input. The average daily balance is the thing it multiplies, and that balance is rarely the number a cardholder has in mind.

The grace period is the reason a cardholder who pays in full pays no interest at all. It is the window running from the close of a billing cycle to the payment due date, and while it holds, the issuer charges no interest on the purchases from the cycle that just closed.

No federal rule in the United States requires a card to offer one. Nearly every card does on purchases, and Regulation Z, which implements the Truth in Lending Act, requires a card issuer to send the statement at least 21 days before the payment is due, and stops the issuer collecting a finance charge caused by the loss of a grace period when the statement went out late.

The condition attached is the part that gets missed. The grace period applies only when the previous statement balance was paid in full. Meet that condition every month and the chain never breaks: each cycle's purchases are cleared before interest can attach to them, so the rate is applied to nothing, month after month, however much was spent.

Two numbers on a statement matter here, and they are often different. The statement balance is what closed the cycle, and paying that in full is what preserves the grace period. The current balance includes purchases made since the cycle closed, which belong to the next statement and have their own grace period still ahead of them. Pay the statement balance on time and a purchase made the day after a cycle closes can sit unpaid for around 50 to 55 days at no cost: the rest of the cycle it fell into, plus that cycle's grace period.

Leave any part of the statement balance unpaid and the grace period stops applying. That switch is what makes a partial payment cost more than the arithmetic suggests.

Say a statement closes at $1,800 and $1,200 goes across before the due date. The obvious cost is interest on the $600 still owed. The larger cost is that new purchases now begin accruing interest from the day each one posts, with no interest-free window in front of them, and that continues until the balance is cleared in full again. A cardholder who keeps spending is then paying interest on almost everything bought, not on the leftover alone.

Agreements differ on how the grace period comes back. Many restore it as soon as one statement balance is paid in full; some ask for two consecutive cycles paid in full. That rule sits in the card agreement rather than in the law, so it varies from one issuer to the next.

The same mechanism produces the charge that surprises people most. Interest keeps accruing between the day a statement closes and the day the payment arrives, so paying a carried balance off in full still leaves a few days of interest, which lands on the next statement. It is usually called residual or trailing interest. The grace period is all or nothing, which is why the last dollar of a full payment does more than any of the dollars before it: it is the one that keeps the grace period alive for the cycle ahead.

The average daily balance is built by writing down what was owed at the end of each day, adding those figures up and dividing by the number of days in the cycle. Purchases raise the balance from the day they post. Payments and credits lower it from the day they are credited.

Take a 30 day cycle that opens at $1,200 and picks up a purchase on day 11:

The average is not $1,800, and it is not the midpoint of the two figures either. It is weighted by time: $\frac{(1200 \times 10) + (1800 \times 20)}{30} = \frac{48000}{30} = 1600$. At a 24 percent APR that cycle carries $31.56 of interest, worked through below.

Two consequences come out of the weighting. Timing changes the bill even when the closing balance does not: in the same 30 day cycle, a payment credited on day 5 is out of 26 daily balances, counted the same inclusive way as the purchase above, while the identical payment credited on day 25 is out of only 6. And a balance that is gone by the end of the cycle was still owed for the days it existed, so it is still charged for them.

This is also why a statement can show interest on a card that now reads zero. The rate met a balance that was there earlier in the month, and the statement figure was never the number the rate was applied to.

A United States card APR is a nominal annual rate. It is the daily periodic rate multiplied back out, and it says nothing about what happens to interest once it has been charged.

What happens is that unpaid interest is added to the balance, and from that point the daily rate is applied to it as well. That is compound interest pointed the other way. Interest is added to the balance at least once a cycle, and some agreements add each day's interest to the next day's balance. Over a year of carrying a balance with nothing repaid, the gap that opens between the quoted rate and the cost looks like this:

The effective annual rate is the comparable figure, and the APR against APY calculator above converts between the two.

The asymmetry with savings is a disclosure rule rather than a piece of maths. Deposit accounts in the United States advertise an annual percentage yield, which already contains the compounding. United States cards advertise an APR, which does not. Setting a card's 24 percent against a savings account's quoted yield compares a nominal rate with an effective one. A card APR also leaves out the annual fee and transaction fees, so it prices the balance rather than the card.

Which kind of number a headline APR is depends on where the card was issued rather than on arithmetic. A United Kingdom or European Union card APR is defined as an effective annual rate, with the compounding already inside it, so running the table above on one of those figures would count the same compounding twice.

Not every balance on a card is charged the same way, and the differences are wider than the headline APR.

• Cash advances. An ATM withdrawal, a cash-equivalent transaction such as a money order, and on many cards a wire or a wallet top-up. There is usually no grace period at all, so interest runs from the transaction date even for someone who pays in full every month. The APR is normally higher than the purchase APR, and a fee, commonly 3 to 5 percent of the amount with a small flat minimum, posts to the balance the same day and accrues interest alongside the cash.
• Balance transfers. Usually a fee set as a percentage of the amount moved, often with a promotional rate for a fixed period. While a transferred balance sits on the card, the grace period on purchases is normally gone until the whole balance, transfer included, is paid off.
• Deferred interest offers. Common on United States store cards. Interest accrues from the purchase date but is waived if the balance is cleared inside the promotional window. Miss that deadline and the whole accrued amount is billed at once.

Where a payment lands matters once several balances sit at different rates. Under Regulation Z the issuer may apply the minimum payment where it chooses, which is typically the cheapest balance, and anything above the minimum has to go to the highest APR balance first. The exception runs the other way: in the last two billing cycles before a deferred interest promotion expires, the excess has to go to the promotional balance instead, which is the balance about to be billed for everything it has quietly accrued. So a cash advance sitting behind a promotional purchase balance is paid off last unless the payment beats the minimum.