Credit card payoff calculator: months to clear and interest paid

A credit card payoff calculator answers two questions at once: how long a balance takes to clear at the payment you are making, and how much of what you send goes to interest rather than to the debt.

Card balances behave differently from installment loans. There is no scheduled end date, the required minimum shrinks as the balance falls, and interest is recalculated every cycle on whatever is still owed. Those three features are why the same balance can be gone in three years or still be there in twenty, with the only difference being the payment.

This page sets out the arithmetic the calculator uses, works one payoff schedule out by hand, explains where the model differs from the daily-balance method issuers actually use, and shows the payment level at which a balance stops shrinking altogether. Every rate here is an assumption -- your APR is on your statement.

#The model this calculator uses

There is no single formula for a card balance, because it is an iterative process. Each month:

Formula: interest = balance x r, then new balance = balance + interest - payment where r is the monthly rate, taken as the APR divided by 12 and expressed as a decimal.

The calculator repeats those two lines until the balance reaches zero, counting months and adding up interest. When the payment is fixed, that loop has a closed-form answer:

Formula: n = -ln(1 - r * B / P) / ln(1 + r) where n is the number of months, B is the starting balance, P is the fixed monthly payment and r is the monthly rate. If r * B is greater than or equal to P, the logarithm is undefined -- the payment never clears the balance.

The minimum-payment comparison uses a different rule, because a minimum is not fixed. It is modeled as 1% of the current balance plus that month's interest, with a floor of $25. Minimum-payment terms of roughly that shape are common, but the percentage, the dollar floor and whether interest and fees sit on top of it are set by each cardholder agreement and differ from issuer to issuer, so read yours.

#A worked example, done by hand

Take a $6,000 balance at an assumed 22.5% APR, paying $200 a month.

1. Monthly rate: 0.225 / 12 = 0.01875, or 1.875%.
2. Month 1 interest: 6,000 x 0.01875 = $112.50. Principal: 200 - 112.50 = $87.50. Balance: $5,912.50.
3. Month 2 interest: 5,912.50 x 0.01875 = $110.86. Principal: $89.14. Balance: $5,823.36.
4. Month 3 interest: 5,823.36 x 0.01875 = $109.19. Principal: $90.81. Balance: $5,732.55.
5. Closed form: -ln(1 - 0.01875 x 6,000 / 200) / ln(1.01875) = -ln(0.4375) / ln(1.01875) = 0.826679 / 0.018576 = 44.50, so the balance clears on the 45th payment, the last one being partial.

Worked example: At $200 a month the $6,000 balance is gone in 45 months and costs $2,900.78 in interest, for $8,900.78 repaid. Adding $100 a month clears it in 26 months with $1,590.93 of interest -- 19 months sooner and $1,309.85 less.

#Why the minimum payment behaves so differently

A fixed payment works like an installment loan: the amount stays put while the interest charge falls, so principal repayment accelerates every month. A minimum payment does the opposite. It is calculated from the balance, so as the balance falls the payment falls with it.

While the calculated minimum stays above the floor, the principal reduction is exactly 1% of the balance each month whatever the rate happens to be, because the interest part of the payment cancels the interest charge exactly. A balance shrinking 1% a month halves in about 69 months, which is most of the story of why minimum payments stretch a debt for decades.

The floor changes the endgame. The model never lets the minimum fall below $25, and once 1% plus interest drops under that figure -- below about $870 at the balance and rate used here, which takes 194 months -- the payment stops shrinking. From that point principal repayment accelerates, because a flat $25 covers a growing share of a falling balance, and the debt actually clears rather than decaying toward zero forever.

The first minimum on that balance is $172.50 -- $60 of principal and $112.50 of interest. Twelve months in, the payment has fallen to about $154.45 and the balance is still $5,318.31, after $1,278.17 of interest. The $25 floor only starts to bind after roughly 16 years, once the balance is below about $870.

Notice the shape of the fixed-payment rows. Going from $150 to $200 removes 30 months and $2,293.55 of interest. Going from $400 to $500 removes 4 months and $250.06. The early dollars matter most, because they are the ones that shorten the long tail where interest dominates.

#The payment level at which a balance stops shrinking

Multiply the balance by the monthly rate and you have the interest charge for the month. Any payment at or below that figure leaves the balance the same or larger.

On $6,000 at an assumed 22.5%, that break-even payment is $112.50. At exactly $112.50 the balance never moves. At $110 it grows quietly for as long as it takes for the minimum-payment rule to catch up. The calculator detects this case and says so rather than reporting a payoff date.

The same threshold explains why the difference between paying $115 and paying $200 is not 74% -- it is the difference between $2.50 and $87.50 of principal, a factor of 35 in the first month.

#Monthly interest here, daily interest on your statement

A card statement is generally not built by multiplying the balance by one-twelfth of the APR. The common convention is a daily periodic rate, applied to the average daily balance and multiplied by the number of days in the billing cycle. The specifics -- the divisor used, how the average is taken, which balances are included -- are set by your cardholder agreement and vary between issuers, so that document is the authority on how your interest is figured.

Formula: cycle interest = average daily balance x (APR / 365) x days in cycle

At an assumed 22.5% APR the daily periodic rate is 0.0616438%. The table shows what each method charges on a $6,000 balance held for a full cycle.

Two consequences. Cycle length makes each individual month differ by a few dollars, but over a year the totals converge: twelve months at 1.875% compounds to an effective 24.97% a year, and a run of 30-day cycles at the daily rate compounds to about 24.98%. The model here is accurate in aggregate and approximate in any single month.

The bigger difference is the average daily balance. Because the balance is averaged over the cycle, a payment made on day 5 reduces more of the cycle's interest than the same payment made on day 25. This calculator applies one payment at the end of each month, so it slightly overstates interest for anyone who pays early in the cycle.

#The grace period

Most cards charge no interest on new purchases if the previous statement balance was paid in full by the due date. That is the grace period, and it is why a card used as a payment tool and cleared monthly can cost nothing in interest.

Once a balance revolves, the grace period is typically lost. New purchases then start accruing interest from the transaction date rather than from the statement date, and it usually takes paying the balance in full -- sometimes for more than one cycle -- to restore it. That is the mechanism behind the common experience of a balance that seems to grow despite steady payments.

Cash advances normally have no grace period at all, and often a separate, higher rate. This calculator models a single balance at a single rate, so a mix of promotional, purchase and advance balances needs separate runs.

#Where each input comes from

Balance. Use the current statement balance, not the credit limit and not the amount charged this month.

APR. Read it off the statement's interest charge summary. Cards frequently carry different rates for purchases, transfers and advances, and a promotional rate that expires.

Monthly payment. The amount you actually send, not the minimum shown. If your payment varies, use a conservative figure -- the model treats it as fixed.

Extra payment. Anything above your usual amount. Because it applies to a balance that is already earning no compounding benefit for you, every extra dollar has the same effect as a dollar of principal reduction.

The result is far more sensitive to payment than to APR. Doubling the payment in the table above changes the outcome more than several points of rate would.

#How to read the result

The payoff length is the headline, but the interest figure is the price. On the $200 path, $2,900.78 of interest on a $6,000 purchase means the goods effectively cost 48% more.

The comparison against minimum-only payments is there to isolate one variable: it is the same balance and the same rate, with only the payment rule changed. If the fixed-payment row is dramatically shorter, the difference comes from holding the payment steady while the balance falls.

For a fixed-rate installment debt, the loan calculator does the same arithmetic with a contractual end date. To see the same compounding working in the other direction, the compound interest calculator uses an identical monthly loop.

#What this model leaves out

• New spending. The model assumes nothing is added to the card. Continued use is the most common reason real payoffs take longer than projected.
• Average daily balance and payment timing. Interest is applied once a month here, on the balance at the start of the month.
• Multiple balances and rates. Promotional, transfer and cash-advance balances are treated as one.
• Fees. Annual, late, over-limit, foreign transaction and balance transfer fees are not modeled, and a late fee can also trigger a penalty rate.
• Variable rates. Card APRs are usually tied to an index and can change during the payoff period.
• Rewards and statement credits. These offset cost but are not part of the interest math.
• Payment allocation rules. Issuers apply required minimums by their own rules; amounts above the minimum generally go to the highest-rate balance first.

#Common mistakes

Paying the minimum and assuming progress. Under the 1%-plus-interest rule the balance falls by exactly 1% a month, which is roughly 69 months to halve.

Treating the APR as the annual cost of carrying a balance. Because interest is added each cycle and then itself earns interest, 22.5% compounded monthly is an effective 24.97% a year.

Paying on the due date rather than early. With average daily balance billing, an earlier payment lowers the average and therefore the interest.

Charging while paying down. New purchases in a revolving account usually accrue from the transaction date, since the grace period is not in force.

Ignoring a promotional expiry. A 0% period ends on a fixed date, after which the remaining balance carries the standard rate. Run the calculator twice: once for the promotional window, once for the balance you expect to be left with. The full calculator index covers the surrounding budget questions, and the real hourly wage calculator puts an interest bill in terms of hours worked.