How long a debt lasts at a chosen payment

This inverts the usual loan question. Instead of fixing a term and deriving the payment, it fixes the payment and derives the term: a logarithmic rearrangement of the same annuity relation, solved for the number of periods. The answer arrives as a count of periods, usually fractional — the whole part is full payments, and the fraction says the last one is smaller than the rest.

Every balance carrying a rate has an interest-only line: the payment that exactly covers the interest one period accrues and touches nothing else. At that line the debt is a treadmill — serviced forever, reduced never. Below it the balance grows despite the paying. The relation only produces a finite horizon strictly above the line, which is why the calculator refuses at or below it instead of returning something meaningless.

Near that line the horizon is violently sensitive. A payment barely above interest-only takes a startlingly long time to finish, and a small increase from there removes years; far above the line the arithmetic calms down and looks almost like simple division of balance by payment. That asymmetry is the quiet danger of minimum payments on revolving balances: a card minimum is typically set just above the interest-only line, which parks the horizon at its slowest, longest extreme while remaining technically finite.

The relation also runs in reverse, and the workspace offers that direction: name the horizon instead — the date the debt is supposed to end — and it solves for the payment that gets there. Between the forward and reverse directions this is the honest conversation a balance and a budget can have: what the affordable payment buys in time, or what a chosen finish date demands in money.

At a rate of nought the whole drama disappears: with no interest accruing there is no line to stay above, and the horizon is simply the balance divided by the payment. That branch is handled separately because the logarithmic form divides by quantities that vanish exactly there.

Lenders reason from the calendar to the wallet: choose a term, derive the payment that clears the debt across it. Real budgets often run the other way — a household knows what it can pay each month, and the open question is what that payment buys in time. Inverting the relation dignifies that direction: the payment becomes a fact, the horizon becomes the answer.

The inversion matters most where no quote exists at all. A card balance, a store account, money owed to family — debts without contractual terms still obey the arithmetic of balance, rate and payment, and this calculation is how such a debt is given the ending its paperwork never specified.

It also runs in reverse. Name the finish date instead — the anniversary by which the debt should be gone — and the same relation solves for the payment that reaches it. Between the two directions this is the honest conversation a balance and a budget can have: what the affordable payment buys in time, or what a chosen ending costs in money.

Each period, the balance accrues its interest. A payment first meets that accrual, and only the excess shrinks the debt. If the payment exactly equals the accrual, the balance is untouched — the same debt greets every payday, forever. If the payment falls short, the unpaid interest joins the principal and the debt grows while being paid. Neither case has a payoff horizon, and no rearrangement of the mathematics can produce one.

This is why the calculator refuses at or below the interest-only line rather than returning something. Any number it invented there would be a lie with units. The refusal is the answer: this payment cannot end this debt, and the honest response is to raise the payment, lower the rate, or restructure — not to reread the result.

Just above the line, the arithmetic is violently sensitive. A payment barely clearing the accrual takes a startlingly long time to finish, and small increases from there remove years at a time; far above the line, the horizon calms down toward what plain division of balance by payment would suggest. That asymmetry is the quiet danger of minimum payments — they live in the steep zone by design.

The horizon comes back as a count of periods, and the count is rarely whole. The whole part is the number of full payments; the fraction says the final payment is partial — the debt runs out of balance before the last period runs out of payment. On a monthly schedule the count is months, on a fortnightly one fortnights, so the same debt at the same annual pace reads differently depending on the rhythm of paying.

A horizon is also a forecast with assumptions attached: the payment holds every period, the rate does not move, and nothing new is borrowed. The last assumption is the one revolving balances break daily — a horizon computed on a card that keeps being used is obsolete before the next statement. The number is honest about the debt as it stands today; keeping it true is behavioural, not mathematical.

The immediate use is diagnosis: the payment currently going out, converted into the years it implies. For a card balance drifting along at the minimum, that conversion is routinely the moment the debt becomes real — the balance looked manageable, the horizon does not.

The deeper use is negotiation with yourself. Because the horizon is so sensitive near the interest-only line, modest payment increases buy outsized time reductions exactly where the situation is worst. The lesson after this one prices that trade precisely — what adding a fixed extra to every payment buys back, in months and in money — and the reverse direction here already hints at it: pick the ending you want, and see the payment it demands.

The engine below arrives pre-filled with the pack’s anchor example: a six-figure balance at an unremarkable rate, paid monthly at an amount sitting comfortably above the interest-only line. Put in your own balance and your own honest payment first. Then walk the payment slowly down toward the interest the balance accrues each month and watch the horizon stretch — and finally watch the calculator refuse, which is the lesson’s whole argument performed live.

Read the result as the number of periods the debt survives if the payment holds, the rate stays put and nothing new is borrowed. A refusal is not an error: it is the engine reporting that this payment cannot end this debt. Every figure is computed live by the verified engine — this page stores none.