Compound interest is interest paid on your interest as well as on your original money. That one sentence is the whole idea; the formula is just bookkeeping for it.
What is the compound interest formula?
The working
A = P (1 + r/n)^(n × t)
| Letter | Means | Example |
|---|---|---|
| A | The amount you end up with | what you are solving for |
| P | Principal — what you start with | £5,000 |
| r | Annual rate as a decimal | 5% → 0.05 |
| n | Times per year interest is added | monthly → 12 |
| t | Years | 20 |
A worked example
£1,000 at 5%, compounded once a year, for 10 years:
The working
A = 1000 × (1 + 0.05/1)^(1 × 10) A = 1000 × 1.05^10 A = 1000 × 1.62889… A = £1,628.89 Interest earned: £628.89
Simple interest on the same money would have paid £50 a year — £500 over the decade. The extra £128.89 is the interest your interest earned.
Why is 5% compounded monthly worth more than 5%?
Because each month’s interest immediately starts earning interest itself. The headline rate is the *nominal* rate; what you actually earn over a year is the *effective* rate.
The working
Effective rate = (1 + r/n)^n − 1 5% monthly: (1 + 0.05/12)^12 − 1 = 5.116% 5% daily: (1 + 0.05/365)^365 − 1 = 5.127% 5% yearly: 5.000%
The effective rate is the number to compare between savings accounts — in the UK it is usually published as the AER. Two accounts advertising 5% are not the same account if one compounds monthly and one annually.
What regular deposits do
For most savers the monthly payment matters more than the rate. £5,000 left alone at 5% for 20 years becomes about £13,500. Add £150 a month and it is over £75,000 — and only a fraction of that difference is interest on the deposits.
Timing matters too, more than people expect. Money paid in at the start of each period earns one extra period of interest every single time, which over decades compounds into a real gap.
Using it
- Convert the rate to a decimalDivide the percentage by 100. 4.5% becomes 0.045. This is where most by-hand attempts go wrong.
- Set n to how often interest is addedNot how often you pay in. Yearly is 1, quarterly 4, monthly 12, daily 365. Your account’s terms will say.
- Do the bracket first, then the power(1 + r/n) before the exponent. On a phone calculator the power key is usually x^y or ^.
- Keep the full precision until the endRound only the final answer. Rounding the balance to pennies at each step and feeding it forward drifts, and the drift grows with the term — exactly where the answer matters most.
Common questions
What is the difference between simple and compound interest?
Simple interest is always calculated on the original amount, so it adds the same figure every year and grows in a straight line. Compound interest is calculated on the balance including previous interest, so the amount added grows each year and the line bends upward.
What is AER?
Annual Equivalent Rate — the effective rate, standardised so UK savings accounts can be compared like for like regardless of how often they compound. If you are choosing between accounts, AER is the number to look at rather than the headline rate.
What is the rule of 72?
A shortcut for how long money takes to double: divide 72 by the interest rate. At 6%, roughly 12 years. It is an approximation that works well between about 4% and 12% and drifts outside that range.
Is the interest taxed?
It depends on the account and your allowances — interest inside an ISA is not taxed, and outside one there is a Personal Savings Allowance that depends on your tax band. Any projection you make from the formula is a gross figure, so treat it as a ceiling.
Does inflation change the answer?
It changes what the answer is worth, not what it is. A balance projected 30 years out is in today’s pounds with no adjustment, and will not buy what the same number buys now. Treat a long projection as a shape rather than a promise.
Check this before you rely on it. A free guide to everyday arithmetic — not tax, accounting, financial, legal or medical advice. The working is shown above so you can verify it against your own figures. See our terms of use.