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.

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.

Calculators that use this method

• Mortgage Payment CalculatorWork out the monthly payment on a repayment mortgage, and see how much of the term goes on interest.
• Mortgage Affordability CalculatorWork out the loan a given income and deposit supports, using the debt-to-income limits lenders actually apply.
• Home Improvement Loan Payment CalculatorEstimate the monthly payment on a fixed-rate home improvement loan from its amount, rate, and term.