Credit & Debt
How To Calculate Credit Card Payoff Time
Paying a fixed amount every month retires a revolving balance on a predictable schedule — and that schedule can be written as a closed-form equation rather than simulated line by line. The catch is a boundary condition the formula cannot cross: pay only the interest and the balance never falls.
Quick Answer
n = -ln(1 - (rm x B) / P) / ln(1 + rm)
- n
- Number of monthly payments to reach zero
- B
- Current balance owed
- rm
- Monthly periodic rate — APR divided by 12
- P
- Fixed monthly payment
- ln
- Natural logarithm
The number of monthly payments needed to retire a revolving balance equals minus the natural log of (1 minus the monthly rate times the balance divided by the payment) divided by the natural log of (1 plus the monthly rate). It requires the payment to exceed the first month's interest charges; if it does not, the balance never reaches zero and the logarithms have no real solution.
What Is Credit Card Payoff Time?
Every month a revolving balance grows by interest and shrinks by your payment. Written as a recurrence, the new balance equals the old balance times (1 + rm) minus the payment, where rm is the monthly periodic rate. Applying that recurrence thirty or forty times gives the answer, which is exactly what an amortization spreadsheet does.
Because the recurrence is geometric, it can be solved in closed form instead of iterated. Setting the balance after n months equal to zero and rearranging produces n = -ln(1 - rm·B / P) / ln(1 + rm). The result is rarely a whole number, so it is rounded up: paying for the whole final month is required even though only part of it is used.
The formula carries a condition that matters more than the algebra. The quantity inside the logarithm, 1 - rm·B / P, must be positive, which means P must exceed rm·B — the first month's interest on the current balance. On a $6,500 balance at 22.99% APR the monthly rate is 1.91583%, making the first month's interest $124.53. Any payment at or below that amount never reduces the principal, the logarithm's argument reaches zero or goes negative, and the equation returns no answer. That is not a failure of the formula; it is the arithmetic reality of paying only interest.
This is why the gap between two payment sizes is so violent. Retiring the same $6,500 balance costs $2,581.82 of interest at $250 a month over 37 months, and $15,202.63 at $130 a month over 167 months. Both figures come from identical principal, identical rate and identical terms — the only variable altered is how much cash leaves each month. Roughly the last $120 of each payment is pure interest, so every dollar above that threshold attacks principal.
One modelling caveat: most issuers accrue interest daily on the average daily balance rather than monthly on the statement balance. The formula above assumes monthly compounding, and on a static balance with no new purchases the two are very close — typically within a few dollars over the whole term. Where they diverge is when purchases continue: adding new charges changes B every month, which breaks the fixed-payment assumption entirely and turns a thirty-seven-month projection into something much longer.
Used as intended, though — a static balance retired by a fixed payment — the equation is exact. Reverse it to answer the question people actually have: what payment clears this in twenty-four months? Solving for P instead of n gives P = B·rm / (1 - (1 + rm)^-n), which on this balance returns $340.39.
Formula
n = -ln(1 - (rm x B) / P) / ln(1 + rm)
Requires P > rm x B. Round the result up, because a partial final month is still a full calendar month of payments.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| B | Balance owed | currency | The current outstanding balance. Assumes no new purchases during the payoff period. |
| r(m) | Monthly periodic rate | decimal | APR divided by 12, expressed as a decimal: 22.99% becomes 0.2299 / 12 = 0.0191583. |
| P | Monthly payment | currency | A fixed amount paid every month. Must exceed the first month's interest or the balance never clears. |
| n | Number of payments | months | Months until the balance reaches zero, rounded up to the next whole month. |
P = (B x rm) / (1 - (1 + rm)^-n)
The same relationship solved for payment instead of time. Pick the payoff deadline and read off what that costs each month.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Required monthly payment | currency | The fixed amount that retires the balance in exactly n months. |
| n | Target number of months | months | Any positive whole number. Larger targets give smaller payments but more total interest. |
Total Interest = (P x n) - B
Everything paid beyond the original balance is interest. Compare two payment sizes here — the difference is startling.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| I | Total interest | currency | Cumulative interest across the whole payoff period, not per month. |
How To Calculate Credit Card Payoff Time
- 1
Convert the APR to a monthly periodic rate
Divide by 12 and by 100. An APR of 22.99% becomes 0.2299 / 12 = 0.0191583. Many issuers accrue daily, but for a fixed balance without new purchases the monthly approximation lands within a few cents.
- 2
Check that the payment exceeds the monthly interest
Compute rm x B. Here 0.0191583 x $6,500 = $124.53. If your payment is at or below this, stop — the balance will not fall and the formula has no real result. This is exactly what the requirement that the logarithm's argument stay positive means.
- 3
Build the logarithm's argument
Divide the monthly interest by the payment and subtract from one: 1 - (0.0191583 x $6,500 / $250) = 1 - 0.4981167 = 0.5018833. The closer this sits to zero, the more slowly principal is falling.
- 4
Divide the two natural logarithms
Take ln of that argument and ln of (1 + rm): ln(0.5018833) = -0.6893876 and ln(1.0191583) = 0.0189771. Negate the first and divide: 0.6893876 / 0.0189771 = 36.3273 months.
- 5
Round up and price the total
Thirty-seven payments of $250 come to $9,250.00, of which $2,750.00 above the $6,500 balance is interest. Using the unrounded 36.3273 gives the theoretically exact $9,081.82 — the $168.18 difference is the overshoot in that final partial month.
Examples
Example 1: $6,500 at 22.99% APR paid at $250 a month
- Balance
- $6,500
- APR
- 22.99%
- Monthly payment
- $250
| Step | Calculation | Result |
|---|---|---|
| Monthly periodic rate | 0.2299 ÷ 12 | 0.0191583 |
| First month's interest (must be less than the payment) | $6,500 x 0.0191583 | $124.53 |
| Logarithm argument | 1 - (0.0191583 x $6,500 ÷ $250) | 0.5018833 |
| Natural logs | ln(0.5018833) = -0.6893876 ; ln(1.0191583) = 0.0189771 | 0.0189771 |
| Divide | 0.6893876 ÷ 0.0189771 | 36.3273 months |
| Total interest over 37 payments | (37 x $250) - $6,500 | $2,750.00 |
Result: 36.3273 months — 37 payments of $250, interest of $2,750.00
Example 2: The same balance paid at $130 a month
- Balance
- $6,500
- APR
- 22.99%
- Monthly payment
- $130
| Step | Calculation | Result |
|---|---|---|
| Interest still $124.53, leaving only $5.46 to principal | $130 - $124.53 | $5.46 |
| Logarithm argument — very close to zero | 1 - (0.0191583 x $6,500 ÷ $130) | 0.0420833 |
| Number of months | 0.0420833 natural log / 0.0189771 natural log = 3.1681035 ÷ 0.0189771 | 166.9433 months |
| Round up to whole payments | 167 payments | 167 months |
| Total interest | (167 x $130) - $6,500 | $15,210.00 |
Result: 166.9433 months — 167 payments of $130, interest of $15,210.00
Calculator
Months to pay off
36.3273
- First month's interest
- $124.53
- Total of all payments
- $9,081.82
- Total interest
- $2,581.82
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Credit Card Payoff Time calculator page.
Common Mistakes
Applying the formula to a payment at or below the monthly interest
Paying $124.53 or less on this balance retires nothing. The logarithm's argument becomes zero or negative, and there is no real solution because the balance never falls. The formula failing there is correct behavior, not a bug.
Plugging the card's minimum payment into P
Minimums are usually expressed as a percentage of the balance plus interest, so they shrink as the balance shrinks. This formula assumes a fixed payment and will badly overstate progress if you let the payment float downward.
Continuing to make purchases while projecting
The model assumes a static balance. A $200 purchase mid-payoff adds its own interest for every remaining month, and consistently using the card can push actual payoff far past the projected date.
Rounding months down instead of up
Thirty-six point three months is thirty-seven payments, not thirty-six. Rounding down leaves a small balance that continues accruing interest past the projected end date.
Ignoring daily accrual on the average daily balance
Most issuers charge interest daily rather than monthly. On a static balance the difference is small, but when payment timing varies within the cycle, the effective interest is higher than the monthly model predicts.
FAQ
Why does my minimum payment never seem to pay the card off?
Because minimums usually fall to roughly one or two percent of the balance plus interest, which keeps the payment near the monthly interest charge. Once the payment barely exceeds interest, nearly nothing reaches principal and the payoff horizon stretches to decades.
What payment clears a balance in a specific number of months?
Solve the same equation for payment instead of time: P = B x rm / (1 - (1 + rm)^-n). On a $6,500 balance at 22.99%, clearing it in twenty-four months takes $340.39 a month.
Does it matter which day of the cycle I pay?
Paying earlier in the cycle reduces the average daily balance on cards that accrue interest that way, lowering the month's interest slightly. It does not change the formula's structure, but it does shave real dollars off longer payoffs.
Is a balance transfer worth modelling separately?
Yes, and it often pays. A promotional 0% period stops interest entirely, so every payment attacks principal and the balance falls in B / P months. Always add the transfer fee to the balance and compare against paying down the original card.
What if my APR changes halfway through?
Treat the change as a new problem: recompute with the remaining balance as the new starting point and the new monthly rate. Projections made before the change describe a scenario that no longer exists.
References
- [1]Consumer Financial Protection Bureau, Truth in Lending Act — Regulation Z, Subpart B — https://www.consumerfinance.gov/rules-policy/regulations/1026/
- [2]Consumer Financial Protection Bureau, Ask CFPB, Choosing a credit card: how interest is calculated — https://www.consumerfinance.gov/ask-cfpb/how-do-credit-card-companies-calculate-interest-en-1073/
- [3]Board of Governors of the Federal Reserve System, Consumer Credit — G.19 Statistical Release — https://www.federalreserve.gov/releases/g19/