TVM payment (loan or annuity instalment)

What a level payment actually reconciles

A time-value-of-money problem has five quantities — present value, payment, rate, number of periods, and future value — and any four of them determine the fifth. This page solves for the payment, which is the one people usually need and the one most often quoted without the assumptions that produced it.

The rate is a PERIODIC rate, not an annual one. If payments are monthly, the rate is the rate per month. Supplying an annual rate against a monthly period count is the most common way to get an answer that looks plausible and is wrong by roughly a factor of twelve.

The target ending balance is a real input, not a formality. Set it to zero and the problem is a fully amortising loan: pay the balance down to nothing. Set it to a positive amount and the problem becomes a sinking fund: accumulate to a goal. The same instrument answers both, which is why it is worth understanding as one calculation rather than two.

Timing matters. A payment made at the beginning of each period earns or saves one extra period’s interest compared with the same payment made at the end. Leases and rents are usually beginning-of-period; loan instalments are usually end-of-period. The difference is small per payment and compounds over a term.

The sign convention is the one the underlying relation uses: money you receive and money you pay carry opposite signs. A payment returned with the opposite sign to the present value is the calculation telling you which direction the cash moves, not an error.

When this calculation is used

• Sizing an instalment when the rate, term and amount are known and the balance must reach zero.
• Sizing a contribution when the goal is a target balance rather than a zero balance.
• Comparing beginning-of-period and end-of-period timing on the same terms.
• Checking a quoted payment against the assumptions it was supposedly derived from.

Worked example

Take a borrowing arrangement with a known amount, a periodic rate matching the payment frequency, a fixed number of periods, and a target ending balance of zero — the ordinary fully amortising case.

The payment returned is the level amount that drives the balance to exactly the target over exactly that many periods. Change the target ending balance away from zero and watch the payment move: that is the same instrument answering a savings question instead of a debt one.

Then switch the timing to beginning-of-period without changing anything else. The payment falls, because each one now earns an extra period. The size of that fall is the value of the timing convention, and it is the detail most quoted figures omit.

What each input represents

Assumptions and limits

• The rate is constant across every period. A variable-rate arrangement is not this calculation.
• Payments are level — the same amount every period — and occur exactly once per period.
• The rate and the period count are expressed in the same unit. Nothing here can detect a mismatch between an annual rate and a monthly term.
• No fees, insurance, taxes or charges are included. This is the financial core of the arrangement, not its total cost.

What the guards protect against

• A periodic rate of −100% or lower is refused. It would mean a period in which the entire balance disappears, and the relation has no meaningful solution there.
• The period count must be greater than zero: there is no level payment across no periods.
• Present value and target ending balance are bounded to a realistic magnitude. The bound is about keeping the arithmetic meaningful, not a judgement about the size of an arrangement.

Provenance

Standard time-value-of-money relation

The level-payment form of the annuity relation, with an annuity-due factor applied when payments fall at the beginning of the period.

Educational reference, not financial advice. The signed pack states its own derivation provenance, and the page reports the verification state of the release it mounted rather than asserting one.