Months to clear a balance on a fixed minimum payment

How long a balance survives on the minimum

Each month, interest is charged on what is still owed, and only the part of the payment above that interest touches the balance. The payoff time follows from how fast that excess compounds the balance away — which is why the answer comes from logarithms rather than from dividing the balance by the payment.

Near the minimum, the arithmetic is brutally nonlinear. When the payment barely exceeds the month’s interest, almost all of it is interest, the balance shrinks at a crawl, and the months stretch toward years. The same balance with a slightly larger payment clears in a fraction of the time — the payoff curve is steepest exactly where the statement’s suggested number tends to sit.

When the payment does not exceed the month’s interest, there is no payoff time to report: the balance holds level or grows forever. The calculator refuses that case rather than printing an enormous figure, and the refusal is the finding — it is the “minimum-payment trap” this page is named for, stated as a fact about the inputs rather than a warning.

One honest caution about the model itself: it holds the payment fixed. Real statement minimums are usually a share of the balance, so they shrink as the balance shrinks — which stretches the payoff even further than this page shows. Someone who keeps paying the first month’s minimum as a fixed amount does better than the statement’s own schedule; someone who pays the declining minimum does worse.

When this calculation is used

• Reading a card statement’s minimum against the balance and rate it came with, and seeing the months — usually years — it implies.
• Choosing a fixed monthly amount to commit to a balance and seeing the payoff time it buys before committing.
• Working backwards through the declared reverse workflow: naming a payoff horizon and letting the engine find the fixed payment that meets it.
• Putting a number on whether a balance is being paid down at all, or merely serviced — the refusal case answers that question by itself.

Worked example

A mid-sized card balance at an ordinary card rate, with the payment fixed at roughly what the current statement suggests as the minimum.

The months returned are how long the balance survives — read them in years to feel the weight. In the early months nearly all of the payment is interest, which is why the figure is so much larger than the balance-divided-by-payment guess. The gap between those two numbers is the interest the arrangement quietly collects.

Nudge the payment upward a little and re-run: near the trap, the months collapse far faster than the payment grows, because every extra unit of payment goes entirely to principal. Push the payment down toward the month’s interest instead and the answer disappears into a refusal — the boundary of the trap, found exactly.

What each input represents

Assumptions and limits

• Nothing new is charged to the balance while it is being paid down; a single purchase restarts the clock.
• The payment is a fixed amount every month. Real statement minimums typically shrink with the balance, which stretches the payoff beyond what this page shows.
• The monthly rate is the annual rate divided by twelve — a nominal convention — and it stays fixed for the whole payoff; promotional expiries and penalty rates are not modelled.
• Fees, annual charges and insurance add-ons are not included; they would lengthen the answer or deepen the trap.
• The payoff time is returned as an exact, generally fractional count of months; the final payment is smaller than the fixed amount.

What the guards protect against

• A payment that does not exceed the month’s interest is refused, and the refusal is the finding: the balance would never reach zero, which is the trap in the page’s name stated as arithmetic.
• A zero or negative balance is refused — there is nothing to pay off, and a payoff time for nothing owed has no meaning.
• A zero or negative payment is refused: with nothing arriving each month, no payoff exists even in principle.
• The rate is bounded to a realistic card range and is refused outside it, because the result would not describe any credit arrangement that exists; zero is allowed and is handled as its own interest-free case.

Provenance

Fixed-payment payoff-time relation for a compounding revolving balance

The negative logarithm of one minus the balance times the monthly rate over the payment, divided by the logarithm of one plus the monthly rate — with the monthly rate taken as the quoted annual rate divided by twelve, and the zero-rate case handled as the balance divided evenly by the payment.

Educational reference, not financial advice, and not a reproduction of any issuer’s minimum-payment schedule. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.