Whether it’s a car loan, a personal loan or a 30-year mortgage, the maths behind your repayment is the same: interest is charged on what you still owe, and a fixed payment is set so the balance reaches zero exactly at the end of the term. This guide explains that formula, how APR differs from the headline rate, and why compound interest works for you in savings and against you in debt. Try any scenario in our loan calculator.
This guide is educational and not financial advice. Loan terms vary; read the lender’s key facts and, for large decisions such as a mortgage, consider speaking to a regulated adviser.
The repayment formula (amortisation)
For a loan with a fixed rate and equal monthly payments:
M = P × r × (1 + r)^n / ((1 + r)^n − 1)- M = monthly payment
- P = principal (amount borrowed)
- r = monthly interest rate (annual rate ÷ 12, as a decimal)
- n = number of monthly payments
Worked example: a personal loan
Borrow £10,000 at 6% a year for 3 years.
- r = 0.06 ÷ 12 = 0.005
- n = 36
- (1.005)^36 ≈ 1.19668
- M = 10,000 × 0.005 × 1.19668 ÷ (1.19668 − 1) = 59.834 ÷ 0.19668 ≈ £304.22
Total repaid: 36 × £304.22 = £10,951.92, so the interest cost is about £951.92.
How each payment splits
Interest each month = outstanding balance × r.
| Month | Payment | Interest | Capital repaid | Balance after |
|---|---|---|---|---|
| 1 | £304.22 | £50.00 | £254.22 | £9,745.78 |
| 2 | £304.22 | £48.73 | £255.49 | £9,490.29 |
| 3 | £304.22 | £47.45 | £256.77 | £9,233.52 |
The payment stays the same, but the interest part shrinks and the capital part grows every month. This schedule is called an amortisation table.
Worked example: a US mortgage
Borrow $300,000 at 6.5% for 30 years (360 payments):
- r = 0.065 ÷ 12 ≈ 0.0054167
- Monthly payment ≈ $1,896.20
- Total repaid ≈ $682,633, so total interest ≈ $382,633: more than the amount borrowed.
In the first month, interest is $300,000 × 0.0054167 ≈ $1,625, so only about $271 reduces the balance. That’s why early overpayments on long loans are so powerful. (Property taxes and insurance, often included in US monthly mortgage payments, are extra.)
Interest rate vs APR
The interest rate (sometimes called the nominal or borrowing rate) only covers the cost of the money. The APR (annual percentage rate) adds compulsory costs such as arrangement fees, expressed as one yearly rate, so you can compare offers fairly.
- In the UK, lenders must show the APR, and adverts quote a representative APR that at least 51% of successful applicants are expected to receive. Your personal rate may be higher.
- In the EU, the equivalent is the APRC (annual percentage rate of charge) under the Consumer Credit and Mortgage Credit Directives.
- In the US, the Truth in Lending Act requires lenders to disclose the APR. Note that a US APR is generally the periodic rate × 12, without monthly compounding, while UK and EU APRs are calculated on a compounding basis.
Example: two £10,000, 3-year loans. Loan A: 6% with no fees. Loan B: 5.5% with a £300 arrangement fee. Loan B’s headline rate is lower, but the fee pushes its APR above Loan A’s. Always compare APRs and the total amount repayable.
Simple vs compound interest
Simple interest is calculated only on the original amount:
Interest = P × rate × years$5,000 at 5% simple for 10 years: 5,000 × 0.05 × 10 = $2,500 interest, total $7,500.
Compound interest is calculated on the amount plus interest already added:
A = P × (1 + rate / k)^(k × years)where k is how many times a year interest is compounded. $5,000 at 5% compounded annually for 10 years: 5,000 × 1.05^10 ≈ $8,144.47, about $644 more than simple interest. Our compound interest calculator shows the growth year by year.
Effective annual rate
Compounding more often increases the true yearly cost or return. A nominal 6% compounded monthly gives an effective rate of (1 + 0.06/12)^12 − 1 ≈ 6.17%. For savings this is shown as AER in the UK or APY in the US; for credit cards, monthly compounding is why a balance grows faster than the headline rate suggests.
What changes the cost of a loan
- Rate: on the $300,000 example, every 0.5 percentage point makes a difference of roughly $95–100 a month.
- Term: longer terms lower the monthly payment but greatly increase total interest.
- Fees: arrangement, broker and early repayment fees.
- Fixed vs variable rate: variable payments move with a reference rate such as the Bank of England base rate, SONIA or SOFR.
- Credit score: better scores usually get lower rates.
Ways to pay less interest
- Shop around using APR and total repayable, not the monthly payment alone.
- Shorten the term if you can afford a higher payment.
- Overpay when allowed. Many UK fixed-rate mortgages permit up to 10% a year without penalty; check yours.
- Avoid paying fees with the loan itself, since you’ll then pay interest on the fees too.
- Clear high-rate debt first (credit cards, overdrafts) before low-rate debt.
For the percentage maths behind rate changes, see our guide to percentage formulas.
Checklist
- Know the principal, rate, term and all fees.
- Calculate the monthly payment with the amortisation formula or a calculator.
- Compare offers on APR and total amount repayable.
- Check the interest share of early payments and the effect of overpaying.
- Understand whether the rate is fixed or variable and what happens when a fixed period ends.