Solving a loan for time instead of payment
An amortised loan has four quantities — balance, rate, payment and number of payments — and knowing any three fixes the fourth. Lenders quote you the payment because they set the term. When you are paying down a credit card, a line of credit or any loan faster than its schedule, you are the one setting the payment, and the term is what you want to know.
The mechanics are the same either way. Each period the balance is charged interest at the periodic rate, the payment is applied, and whatever is left of the payment reduces the principal. What makes the time answer feel counter-intuitive is that the split shifts every period: early payments are mostly interest, later ones are mostly principal, so the balance falls slowly at first and then collapses. That is why raising the payment by a quarter cuts the term by far more than a quarter.
This calculator runs the schedule period by period rather than only applying the closed-form formula, for two reasons. The formula returns a fractional number of payments, and real loans do not have fractional payments — the last one is smaller. And simulating lets the calculator handle a payment that fails to cover the interest by detecting it directly rather than returning a meaningless result.
The formula, and the threshold hiding inside it
Write the balance after n payments and set it to zero, and the number of payments comes out as n = −ln(1 − iB/M) ÷ ln(1 + i), where i is the periodic rate, B is the balance and M is the payment. Every term matters, but the one to look at is iB/M: the share of your payment that goes to interest in the very first period.
If iB/M is close to 1, the logarithm's argument is close to zero, the logarithm dives towards negative infinity, and n explodes. If iB/M reaches 1 exactly — the payment equals the interest charge — the expression is undefined, and correctly so: paying exactly the interest leaves the balance untouched forever. Above that point the balance grows and no finite n exists. This is the single most important number on the page, and the calculator reports it as the interest in the first period.
A worked instance: $5,000 at 24% APR accrues 5,000 × 0.02 = $100 in its first month. A $90 payment leaves the balance $10 higher than it started; a $100 payment holds it exactly level forever; a $110 payment retires it in 122 months and costs $8,319.96 in interest, well over the balance itself. The distance between "never" and "eight years" is twenty dollars a month, which is why the shape of this function matters more than any single result it produces.
Note also that the frequency is not neutral. Paying $125 every two weeks is 26 × 125 = $3,250 a year against $250 monthly, which is $3,000 — a fortnightly schedule sneaks in an extra month of payments each year, and most of the saving people attribute to "biweekly" schedules is really that extra payment. The biweekly mortgage calculator separates the two effects.
Worked example: $10,000 at 6% paying $200 a month
You owe $10,000 on a personal loan at 6% APR and can pay $200 a month.
- Periodic rate. 6% ÷ 12 = 0.005 per month.
- First month's interest. 10,000 × 0.005 = $50.00, so $150 of your first payment reduces the balance and the payment is comfortably above the $50 threshold.
- Interest share of the payment. iB/M = 50 ÷ 200 = 0.25.
- Apply the formula. n = −ln(1 − 0.25) ÷ ln(1.005) = 0.2876821 ÷ 0.0049875 = 57.68.
- Round to whole payments. 58 payments: fifty-seven of $200 and a final one of $136.14.
- Total interest. 57 × 200 + 136.14 − 10,000 = 11,400 + 136.14 − 10,000 = $1,536.14.
Now add $100 a month. At $300 the same balance clears in 37 payments with $966.80 of interest — 21 payments and $569.34 sooner and cheaper for a 50% larger payment. The saving is disproportionate because interest is charged on time as well as on money: cutting the term by a third removes a third of the periods in which any interest can accrue at all.
Reading the answer
Compare total interest against the balance. Paying $1,536.14 to borrow $10,000 for five years is an ordinary outcome for a 6% loan. Paying more in interest than you borrowed, which happens whenever the payment sits close to the interest-only threshold, is a signal to change something structural rather than to keep paying.
Then check the sensitivity table. It repeats the calculation at 75%, 100%, 125%, 150% and 200% of your payment, and the pattern is always the same shape: the term falls much faster than the payment rises. That convexity is the argument for putting any windfall against the balance rather than spreading it, and it is why the last extra dollar is worth more than the first when your payment is near the threshold, and less when it is far above it.
If the answer is unacceptable, there are only three levers. Raise the payment, which this calculator prices directly. Lower the rate — a balance transfer or a consolidation loan, weighed against their fees. Or reduce the balance with a lump sum. If you are juggling several debts rather than one, the ordering question comes first, and the snowball and avalanche calculators answer that.
Months to clear $10,000 at a fixed monthly payment
| Payment | 0% | 6% | 12% | 18.99% | 24% |
|---|---|---|---|---|---|
| $150 | 67 | 82 | 111 | never | never |
| $200 | 50 | 58 | 70 | 100 | never |
| $250 | 40 | 45 | 52 | 64 | 82 |
| $300 | 34 | 37 | 41 | 48 | 56 |
| $400 | 25 | 27 | 29 | 33 | 36 |
| $500 | 20 | 22 | 23 | 25 | 26 |
Read the $150 row: at 18.99% the first month's interest is $158.25 and at 24% it is $200, so a $150 payment never clears the balance at either rate, while at 6% it finishes in 82 months.
What this calculation assumes
- The payment never changes. Credit card minimums are recalculated each month as a percentage of the balance, so they fall as you pay down and the real payoff takes far longer than a fixed minimum implies. Use the minimum payment calculator for that case.
- Nothing is added to the balance. One new purchase on the card resets much of the progress, and the schedule here assumes none.
- Interest is charged once per period on the full balance. Card issuers accrue daily on the average daily balance, which lands within a few dollars a year of this for a balance being paid down steadily.
- The rate is fixed. A variable rate, a promotional rate that expires, or a penalty rate triggered by a late payment all change the answer, usually sharply.
- No fees. Annual fees, late fees and transfer fees are not modelled. A 3% transfer fee on $10,000 is $300, which has to be paid back before any saving begins.
- Payments are applied immediately and in full to the balance shown. Where a card carries several balances at different rates, federal rules direct amounts above the minimum to the highest-rate balance first, so a single-balance model understates how quickly a mixed card clears.
Where this sits among the other loan questions
Use this calculator when the payment is the thing you control. When the term is fixed and you want the payment instead, that is the standard amortisation formula in the personal loan payment calculator or the mortgage payment calculator. When you want to know what is still owed part-way through a scheduled loan, the remaining loan balance calculator solves that directly.
One caution on comparing offers. The rate you enter here is the nominal annual rate used to accrue interest, not necessarily the APR disclosed under Regulation Z, which folds in certain fees and is designed for comparing the cost of credit across products. For a card the two are the same because there are no financeable fees; for an instalment loan with an origination fee, the APR is higher than the note rate, and the payoff arithmetic runs on the note rate. Use the disclosed APR to choose the loan, and the note rate to model the schedule.