Loan Amortisation

The annuity payment

A repayment loan is an annuity: a fixed periodic payment that exactly retires the principal plus its accruing interest over the term. The payment is the principal times the periodic rate, divided by one minus the discount factor over the full number of periods.

Every part of the formula is doing something. The numerator is the interest on the whole principal for one period. The denominator scales that up to account for the principal being repaid gradually, so that later periods carry interest on a smaller balance.

The zero-rate branch is not hypothetical

At r = 0 the formula divides by zero, and code that does not handle it returns NaN. This is not an edge case worth ignoring: interest-free finance is a real product, offered routinely on cars, furniture and home improvements.

The correct behaviour at zero is obvious once stated — the payment is simply the principal divided by the number of periods — and every calculator here branches to it explicitly rather than relying on floating-point luck near zero.

The split between interest and principal is not constant

The payment is level, but what it is buying changes every month. Interest is charged on the outstanding balance, so early payments are mostly interest and late ones are mostly principal, and the transition between the two is gradual and slow.

On a long loan at an ordinary rate, the first payment can be three-quarters or more interest, and the point at which half of each payment goes to principal arrives well past the halfway mark of the term — often around two-thirds of the way through a twenty-five or thirty-year mortgage. Nothing is wrong when this happens; it is what a level payment against a declining balance produces.

The practical consequence is that the balance curve is CONVEX rather than straight. Someone five years into a thirty-year loan has repaid far less than a sixth of it, which is why early-years equity builds slowly and why selling early after a high-fee remortgage can leave less than the arithmetic of payments made would suggest.

Three different rates, and only one of them is in the formula

The rate in the annuity formula is a PERIODIC rate — the rate charged per payment period. Converting an annual rate to it by dividing by twelve is a convention, not a conversion: twelve periods at a twelfth of six per cent compound to about 6.17 per cent over a year, not six.

That gap is why three different numbers circulate for the same loan. The NOMINAL annual rate is the periodic rate multiplied back up and ignores compounding. The EFFECTIVE annual rate includes it. The APR, in most jurisdictions, is a regulated figure that also folds in fees and charges spread across the term, which is why an APR can exceed a headline rate on a loan whose interest has not changed at all.

Comparing two offers therefore means comparing the same kind of number. Two loans with identical nominal rates and different compounding frequencies are different loans, and two with identical APRs can have different monthly payments if their fee structures differ. A payment calculator answers the third question — what leaves the account each month — and says nothing about the first two.

Overpayment: why early money is worth so much more

An overpayment reduces the balance, and every subsequent interest charge is computed on that smaller balance. So the saving from an extra payment is not the payment — it is all the interest that payment would otherwise have attracted for the whole REMAINING term, compounded.

This makes the timing of an overpayment worth more than its size. The same sum paid in year one and in year twenty of a thirty-year loan produce vastly different savings, and on a long loan an early lump sum can remove several years from the term. It is also why the conventional advice to overpay before saving elsewhere holds whenever the loan rate exceeds the after-tax return available.

The same mechanism explains a marketing claim that is true for the wrong reason. A fortnightly payment schedule at half the monthly amount is usually presented as a benefit of paying more often; twenty-six half-payments a year is THIRTEEN monthly payments rather than twelve. Almost all of the advertised saving is the extra month, not the frequency.

Where the model stops describing the loan

The annuity formula assumes a fixed rate for the whole term, and most long loans are not that. A rate that changes mid-term is two annuities joined together: at the change, the outstanding balance is re-amortised over the remaining periods at the new rate, and the payment moves. A calculator showing a level payment for thirty years at today's rate is showing one scenario, not a schedule.

It also assumes the payment is exact. In practice the payment is rounded to the smallest currency unit, so the schedule accumulates a tiny error and the FINAL payment differs — a few units either way. That is normal, and a schedule whose last line is identical to every other line has been computed without rounding.

And it describes only the loan. Property taxes, buildings insurance, mortgage insurance, service charges and escrow payments are commonly bundled into what a lender collects, and they are not interest and not principal. A figure from this formula is the debt service, which is usually the largest part of a housing payment and rarely the whole of it.

Other structures have other shapes. Interest-only payments never touch the principal, so the balance at the end is the balance at the start. A balloon loan amortises on a long schedule and terminates early, leaving a large residual due. An offset arrangement charges interest on the balance net of a linked savings account, which changes the effective rate rather than the formula. Each is a different question from the one answered here.

What amortisation does not tell you

The payment is arithmetic on the figures entered. It is not an affordability assessment, an offer, or advice. Lending decisions rest on income, expenditure, credit history and a stress-tested rate materially above the headline one, and a comfortable-looking payment at today's rate says nothing about whether a lender will agree to it.

The stress test is the part most worth understanding, because it is where the lender's arithmetic diverges from the borrower's. Affordability is assessed at a rate above the one being offered — a reversion rate plus a margin, in many regimes — so the payment that decides whether a loan is granted is deliberately not the payment that will be made. A borrower who works only with today's rate is checking a different condition from the one the application will face.