The amortising loan formula worked by hand and in Excel, where each payment goes, and what a longer term or a small overpayment really does.
The payment is the amount that clears the loan exactly over its term: payment = loan × r ÷ (1 − (1 + r)−n), with r the monthly rate and n the number of months. A £10,000 loan at an illustrative 9.9 per cent over five years costs £211.98 a month, £2,718.72 of interest in all. In the first month £82.50 of the payment is interest and only £129.48 reduces the debt.
Worked in full in Personal Finance for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Lenders quote the monthly figure because it is the one people budget around. Knowing how it is built lets you check it, see what a longer term really costs, and work out what paying a little more each month is worth before you sign.
| Input | Value |
|---|---|
| Amount borrowed | £10,000 |
| Interest rate, a year, charged monthly at one twelfth | 9.9% |
| Monthly rate, r | 0.825% |
| Term | 5 years, 60 months |
| Fees | none |
Payment = L × r ÷ (1 − (1 + r)−n)
r = 9.9% ÷ 12 = 0.825% = 0.00825; n = 60
(1.00825)−60 = 0.6108, so the denominator is 1 − 0.6108 = 0.3892
£10,000 × 0.00825 ÷ 0.3892 = £211.98
Excel: =PMT(9.9%/12, 60, -10000) returns 211.98
The formula is the loan divided by the present value of £1 a month for 60 months at the monthly rate. It is the same arithmetic as a mortgage or a car loan; only the inputs change.
Each month the lender charges interest on the balance still owed, and whatever is left of the payment reduces that balance. Because the balance falls, the interest falls and more of the same payment goes to the debt.
Interest this month = balance × r; principal = payment − interest
Month 1: £10,000 × 0.00825 = £82.50 interest; £211.98 − £82.50 = £129.48 principal; balance £9,870.52
Excel: =IPMT(9.9%/12, 1, 60, -10000) and =PPMT(9.9%/12, 1, 60, -10000)
| Month | Interest | Principal | Balance after |
|---|---|---|---|
| 1 | 82.50 | 129.48 | 9,870.52 |
| 12 | 70.25 | 141.73 | 8,373.78 |
| 30 | 47.66 | 164.31 | 5,613.11 |
| 60 | 1.73 | 210.24 | 0.00 |
Over the first twelve payments the debt falls by only £1,626.22, while £917.52 goes to interest. Halfway through the term, at month 30, more than half the loan is still outstanding, £5,613.11. That is normal for an amortising loan and worth knowing if you plan to settle early.
60 payments of £211.98 come to £12,718.72. Take away the £10,000 borrowed and the interest is £2,718.72, or 21.4 per cent of everything you repay.
| Case | Monthly payment | Total interest |
|---|---|---|
| 9.9%, 3 years | 322.20 | 1,599.29 |
| 9.9%, 5 years | 211.98 | 2,718.72 |
| 9.9%, 7 years | 165.50 | 3,901.63 |
| 6.9%, 5 years | 197.54 | 1,852.43 |
| 14.9%, 5 years | 237.37 | 4,242.48 |
Stretching from five years to seven cuts the payment by £46.48 a month and adds £1,182.91 of interest. A lower monthly payment is not a cheaper loan; it is usually a more expensive one, spread thinner.
Overpaying goes straight to the balance, so every later month's interest is charged on less. Assuming the lender allows overpayments without a fee:
| Extra a month | Months to clear | Months saved | Total interest | Interest saved |
|---|---|---|---|---|
| 0 | 60 | 0 | 2,718.72 | 0.00 |
| 10 | 57 | 3 | 2,554.36 | 164.36 |
| 20 | 54 | 6 | 2,409.05 | 309.68 |
| 50 | 47 | 13 | 2,059.46 | 659.27 |
| 100 | 38 | 22 | 1,662.25 | 1,056.48 |
Fifty pounds more a month clears the loan thirteen months early and saves £659.27. The same logic drives the book's debt plan: money added to a payment is worth far more than it looks, because it stops interest that would otherwise compound for years. For revolving debt, where the payment falls as the balance falls, the effect is larger still; see what a minimum payment actually does.
Treating the quoted rate as a flat rate. 9.9 per cent of £10,000 for five years is £4,950, which would mean a payment of £249.17. On an amortising loan interest is charged only on the falling balance, so the true figure is £2,718.72. If a lender's figure looks like the flat calculation, ask why.
Ignoring how the rate is quoted. The calculation above charges 9.9 per cent ÷ 12 each month, which compounds to 10.36 per cent over a year. In the UK the representative APR on a loan is an annual equivalent that already includes compounding and fees, so the monthly rate is (1.099)1/12 − 1 = 0.7898 per cent and, with no fees, the payment would be £209.91. Use the lender's own schedule to confirm which convention applies.
Before taking a new loan to replace existing debts, compare the total interest, not the monthly payment. A consolidation loan at a lower payment over a longer term can cost more than the debts it replaces.
The last sheet of the free debt workbook for this book runs the same calculation on one debt of your own and shows what adding ten or twenty pounds a month would save. For ordering several debts, see debt snowball vs avalanche.
Use =PMT(annual rate/12, number of months, -amount). For £10,000 at 9.9 per cent over five years, =PMT(9.9%/12, 60, -10000) returns £211.98. IPMT and PPMT with the same arguments and a month number split that payment into interest and principal: £82.50 and £129.48 in month 1.
Interest is charged on the balance still owed, which is highest at the start. On £10,000 at 9.9 per cent the first month's interest is £82.50 of a £211.98 payment. By month 30 it is £47.66, and by the last month £1.73, so the principal share rises every month while the payment stays the same.
It saves interest whenever the lender charges no early repayment fee; whether it is the best use of spare money depends on any dearer debt and on savings. On £10,000 at 9.9 per cent over five years, £20 a month extra saves £309.68 and six months; £100 a month extra saves £1,056.48 and twenty-two months. The same pound saves more interest on a higher-rate card balance than on a cheaper loan.
Getting out of debt is the subject of chapter 7 of Personal Finance for Beginners, and its companion workbook prices one debt of your own. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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