How long a debt lasts at a chosen payment

The line between a payment that finishes and one that never does

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.

When this calculation is used

• Working out how long a card or store balance lasts at the amount actually being paid, rather than the amount a quote once assumed.
• Testing whether a proposed payment amortises the debt at all before committing to it.
• Converting a sustainable monthly amount into a payoff horizon, in periods and therefore in years.
• Using the reverse direction to size the payment a chosen finish date requires.
• Seeing how sharply the horizon stretches as a payment drifts down toward the interest-only line.

Worked example

The pack’s anchor vector is a six-figure balance at an unremarkable annual rate, paid monthly at an amount sitting comfortably above the interest-only line; its companion vectors run a smaller balance at a higher rate, and an interest-free balance where the answer is plain division.

The output is the number of periods the debt survives — on a monthly schedule, months. The anchor lands on a horizon of several years with a fractional tail, meaning a final payment smaller than the rest. Nudge the payment down toward the interest-only amount and watch the horizon stretch disproportionately; nudge it up and watch years fall away faster than proportion suggests.

Every figure is produced by the certified engine when the calculator loads; none is stored in this page. The pack also declares the refusal worth seeing once: the same balance offered a payment exactly at the interest-only line is declined, because no count of such payments ever ends the debt.

What each input represents

Assumptions and limits

• The payment is the same amount every period, paid without fail — a horizon computed from a payment that sometimes slips is optimistic by construction.
• The rate is fixed for the whole horizon; variable-rate debt re-answers this question every time the rate moves.
• The periodic rate is the annual rate divided by the payments per year — the nominal convention, not an effective rate compounded down.
• Nothing new is borrowed along the way: a revolving balance that keeps revolving invalidates the horizon the moment it grows.
• The answer is a count of periods, and its fractional part means the final payment is partial rather than the horizon being approximate.

What the guards protect against

• A payment at or below the interest-only amount for the period is refused: such a payment services the debt without ever reducing it, so no finite horizon exists to report. The refusal is the answer — it says the payment cannot end the loan.
• The balance and the payment must both be greater than nought — a horizon needs a real debt and a real payment to measure between.
• The rate is bounded to a realistic range and the payments per year must be at least one, so the periodic arithmetic describes an actual repayment arrangement.

Provenance

Payoff-time inversion of the level-payment annuity relation

The number of periods solved logarithmically from the balance, the periodic rate and the chosen payment — the algebraic inverse of the annuity payment formula — with the zero-rate case reduced to plain division and payments at or below interest-only refused.

Educational reference, not financial advice. The signed pack carries its own citation — a standard annuity-inversion derivation — which displays from the verified leaf once the calculator loads; the page reports the verification state of the release it mounted rather than asserting one.