# Are a credit card's interest charges a declining infinite series if you always pay the full balance shown on the statement, on the due date?

The question is just what the title says.

Why it might be so: Assume statements are provided monthly. The amount owing on the statement date may include a certain amount of interest. Paying the total amount shown on the statement, on the due date, would seem to constitute "payment in full" but the interest charge was probably not projected to the due date, which is some number of days after the statement date. Rather the interest charge is likely to be calculated as of the statement date. That means paying the amount shown on the statement on the due date leaves a debt outstanding and that debt incurs interest. If the same practice is followed each month, an interest charge will appear on each statement until the time comes that for the most recent period the accrued interest is a fraction of a cent that rounds to zero.

Are a credit card's interest charges a declining (semi) infinite series if you always pay the full amount shown on the statement, on the due date?

Aside: If this is true, then the advice commonly given that people should try to pay "in full" on or before the due date, is actually incomplete. It should say that if you incurred some interest charge, perhaps because you decided to take a cash advance which incurs interest (unlike purchases), you should estimate the amount of daily interest and try to pay it immediately rather than to wait for the monthly statement. Furthermore if you do have a statement in hand and it shows an interest charge, ignore the due date shown on the statement and try to pay sooner. Payment "in full" might only seem to be in full.

TL;DR: No.

If you always pay the full amount of the bill rather than carrying a balance into next month, many credit cards don't charge interest at all. That's how I use mine, especially given that credit card interest rates are moderately obscene.

Even if there is "some interest initially", if you pay the balance in full that includes paying the interest. So that doesn't carry over.

No declining series. Maybe one month's interest in the cash advance or whatever.

• @H2ONaCl For clarity, the question should be edited to spell out that assumption explicitly. Commented Jun 25 at 6:40
• @H2ONaCl: If so, we need to know what that interest is and why paying the bill in full doesn't eliminate it after the first payment. Commented Jun 25 at 8:39
• But that gets paid off in full in the next bill. Interest, yes, but no complicated decay curve. You're still overcomplicating it. Commented Jun 25 at 15:38
• Essentially I am asking if there will be a third interest charge (assuming interest is large enough to fail to round to zero). - then ask that. That's not what your question states. Commented Jun 25 at 16:30
• It's not a religion. There's nothing here to believe in. It either exists or doesn't. I just can't comprehend what's confusing to you. It's a question of fact, not belief. You're speaking about some philosophical debate where I need to convince you. It's not. Read the credit card agreement, it's all there, it's very simple. Commented Jun 27 at 1:28

This seems more of a math question than personal finance. You're making certain assumptions about how interest is being charged and for what. In any case, since money amounts are not real numbers but rather integer values of dollars and cents (or whatever currency you're using), the series cannot be infinite. It will round down to zero eventually.

If this is true, then the advice commonly given that people should try to pay "in full" on or before the due date, is actually incomplete

This advice also comes together with the advice to never ever have cash advances from your credit card.

At the end of the day, if you want to avoid interest charges you need to understand how your credit card issuer charges interest. In your question you're making several assumptions that may or may not be true, and these policies changes between different countries, even between different issuers within the same country.

You seem to insist that we address the premise of your question. The premise of your question is that the interest is charged as a declining infinite series, so yes - the interest described in your question is a declining infinite series.

In real life however your premise doesn't exist.

Interest is either charged per transaction, in which case it's paid off once it's paid off, or for the whole balance, in which case it's not a series.

In the case of a cash advance with the balance paid in full on due date, the interest is charged on the transaction until the full balance is paid, and then stops. Which means, that after two consecutive full statement balance payments there will be no more interest to pay. Which is why the advice you're contesting stands.

If the balance is not paid in full, then the interest is charged on the whole balance, until the statement balance is brought to 0. Which means that the interest is not an infinite declining series, or any series at all.

• As I said - the advice is not to incur interest to begin with. Usually this is achieved by paying the balance in full before due date. Cash advance is not a usual use of credit cards, and there's a separate advice to never ever do that. You've received the answer to the questions as stated, this is not a discussion forum. Commented Jun 25 at 5:31
• @H2ONaCl: If you have a better answer, feel free to answer your own question. But even for cash advance, payment in full will eliminate interest after the first payment, so there really isn't any complicated math here. Commented Jun 25 at 8:43
• If the premise of the question is that interest is charged as an infinite declining series, then that's what if is. You're being ridiculous. Commented Jun 25 at 15:28
• @H2ONaCl Your premise is unrealistic. If you want to ask a math question - go to a math forum. Credit cards don't work the way your question assumes. Commented Jun 25 at 16:17
• @H2ONaCl I addressed the personal finance aspect of your theoretical unrealistic scenario in my edit to the answer. Bottom line - what you're describing doesn't exist. Commented Jun 25 at 16:29

Your assumptions are wrong. If you pay the "statement balance" before the due date, then no further interest is changed on the purchases and cash advances made before the statement end date.

So paying the "full balance" on time will definitely stop the interest accrual and it will not be a "declining infinite series".

A practical example:

My billing cycle starts on Jan 1 and ends on Jan 31. I take out a cash advance on Jan 5 for \$100. I also make an Amazon purchase for \$50 on Jan 10. I do nothing else.

I get a statement on Feb 5 that reflects my activity from Jan 1 to Jan 31 with a payment due date of Feb 25. The cash advance starts accruing interest on the date it is made, and has accrued 2% in interest (\$2). The Amazon purchase does not accrue interest if I pay the statement balance, so my statement balance is \$100 + \$2 + \$50 = \$152. If I pay \$152 by the due date no more interest is charged. You seem to believe that the bank will continue to accrue interest on the cash advance, and thus the real balance will be more than \$152. That is not the case. At least not on any card I've ever had. It's only if you pay less that the full amount that the interest keeps accruing.

So the advice you hear is completely sound - pay off your statement balance by the due date and you will not get charged any additional interest (just the interest that accrues on the cash advance between the transaction date and the statement end date).

If I do not pay the full statement balance, then things get trickier and I am subjected to compounded interest. The bank keep track of the different types of charges (cash advances, balance transfers, "normal" purchases, promotional purchases, etc. and accounts for them each appropriately based on the terms of the card.But if you pay the statement balance in full you don't have to worry about any of that.

• That's how I used to think it works. However your answer is inconsistent with DJClayworth and DilipSarwate and quid, three contributors who say it takes 2 payments on or before the due date (presumably in the amount shown on the monthly statement) to eliminate continuing interest charges. Their answers are among others at this link. money.stackexchange.com/questions/104130/… Commented Jun 26 at 2:15
• It's not inconsistent. If you pay the full balance on the day statement closes then you don't need the second statement. The interest is charged on the transaction until it's been paid. Commented Jun 27 at 1:32
• @littleadv please define the term "the day the statement closes". If by that you mean the statement date (in other words, the date the statement was issued) then I agree. However, if you read the question carefully you will see that I am inquiring about the situation where people pay on the due date (and it's in the title). Commented Jun 27 at 14:39
• I think the mistake you're making is assuming interest is compounded daily. It isn't. No interest accrues on the billed balance in the window between billing and due dates. If you pay in full, you really have zeroed the account as of the billing date, even though that payment didn't reach them until the due date. Remember that this system was designed to work with mail, not EFT; a grace period for arrival of the bill and return of the payment is designed into it. Commented Jul 1 at 7:30

Technically, no. That's not how credit card interest works. But there is an element of truth to what you say.

First off, every card I've ever had, if you pay the balance in full by the due date, there is zero interest. I'm not sure if this is a law or just they have to do it to keep up with the competition. Peronally, I pay my credit cards in full just about every month so I almost never pay interest. The only time I leave a balance is for a rare big purchase where I can't afford to pay it off, or more often if I forget to pay a bill or make a mistake on the amount to pay.

If you don't pay in full, the amount of interest charged is based on when they closed the statement. Basically, the statement date. I suppose your idea that interest could then carry over is basically correct. The next month you will be charged interest based on the "average daily balance", which will include the unpaid interest from the previous month.

So if you never paid anything, ignoring the whole issue of late fees, for this purposes let's assume there are no late fees, yes, you would have an infinite series. Say the interest is 2% per month. You put \$100 on the credit card. You pay \$0 so they charge 2\$ of \$100 = \$2 interest. So your balance is now \$102. The next month the interest would be 2% of \$102 = \$2.04, so your new balance would be \$104.04. The amount of interest would increase every month, exponentially. After a year your balance would be \$126.82 (depending on how they handle rounding). After 3 years the bill would be ... churn churn ... I calculate \$248.59. The interest would be more than the original bill.

If you pay something every month but don't pay the full amount, and if the amount you pay is more than the interest, then each month the bill will go down. Credit card companies normally set a minimum payment that is more than the interest so you will pay it off eventually. But you'll be kind of racing against the interest. They keep adding interest and you keep paying it off.

This answer applies to US credit cards.

You missed a whole other source of interest in your scenario. Once you failed to pay in full, all new purchases start being charged interest immediately. That means when the 2nd bill arrives you owe interest on the outstanding balance, and all your new purchases. Even more important if you wait till the due date of the second bill even more interest will be charged during what you thought was a grace period.

This all but all but guarantees that even after you pay in full you will still have one more bill showing interest being charged. This trips people up all the time. You will save money by quickly sending in a payment that will not only pay everything but also extra money to cover the hidden interest.

• One interest cycle. The next payment in full zeros that out. No complicated calculations needed. Commented Jun 25 at 14:25
• Thank you, a contributor that I would note has 142k reputation, for actually reading my question and not ignoring the premise of the question. Commented Jun 25 at 15:12
• That's not exactly true for US credit cards, it's a bit trickier than that. Which is why the OP's "premise" cannot exist. Commented Jun 25 at 15:33
• @littleadv the premise of the question is that interest is being charged for whatever reason. That can exist. For example, interest on a cash advance. Apart from that, the original post contains an inquiry which is distinct from the premise. Commented Jun 27 at 1:03
• @H2ONaCl I don't think you understand how facts work. Facts exist whether you want them or not. You can premise your question any way you want, but this forum deals with reality, and reality doesn't adhere to your premise. Commented Jun 27 at 1:30