To calculate compound interest, multiply the starting amount by 1 + r once for each year it is invested, where r is the annual rate written as a decimal.
The idea behind it is simple: each year, interest is added to the balance, and the next year's interest is worked out on that larger balance. Over a few years the difference from simple interest is small; over twenty or thirty years it becomes large, on savings and on debts alike. This guide gives the formula, a year-by-year example, the difference between simple and compound interest, how monthly compounding and the annual equivalent rate work on savings accounts, what happens on a loan left unpaid, and a quick way to estimate it in your head.
The compound interest formula
With a starting amount P, an annual rate r and a number of years n, the balance at the end is:
A = P × (1 + r)ⁿ, and the interest earned is A − P.
Take £2,000 at 5% a year for 5 years. The multiplier is 1.05, applied five times: 1.05⁵ is about 1.27628, so the balance becomes £2,552.56 and the interest is £552.56. The same method works for any period, as long as the rate matches it: a monthly rate with a number of months, an annual rate with a number of years.
This is also the formula pupils meet at school. The AQA GCSE Mathematics specification asks students to set up, solve and interpret growth and decay problems, including compound interest, and lists compound interest among the formulae students are expected to know: it is not given in the exam.
On a calculator, type the multiplier and use the power key: 2000 × 1.05 ^ 5. Without a calculator, multiply year by year, as in the table in the next section.
Simple vs compound interest, year by year
Simple interest is paid on the starting amount only, so it adds the same sum every year. Compound interest is paid on the whole balance, interest included, so each year adds a little more than the year before.
| Year | Simple interest balance | Compound interest balance | Compound interest added that year |
|---|---|---|---|
| Start | £2,000.00 | £2,000.00 | - |
| 1 | £2,100.00 | £2,100.00 | £100.00 |
| 2 | £2,200.00 | £2,205.00 | £105.00 |
| 3 | £2,300.00 | £2,315.25 | £110.25 |
| 4 | £2,400.00 | £2,431.01 | £115.76 |
| 5 | £2,500.00 | £2,552.56 | £121.55 |
Both columns use £2,000 at 5% a year. After five years, simple interest has added £500 (2,000 × 0.05 × 5) and compound interest £552.56. The gap of £52.56 is interest earned on interest. It looks modest here, but it grows every year: the extra amount is itself earning.
The year-by-year method is also the safest way to check your answer. If the result from the formula does not match the last line of a table like this one, the rate was probably entered as 5 instead of 0.05, or the number of periods is wrong.
Interest on savings and the annual equivalent rate
Many savings accounts add interest monthly rather than once a year. If an account pays 4% a year in twelve monthly instalments, each month adds a twelfth of 4% to a balance that already includes the previous months' interest. Over a year, that comes to (1 + 0.04 ÷ 12)¹² − 1, about 4.07%, slightly more than 4%. The more often interest is added, the higher the effective rate over a year:
| Interest added | Calculation for 4% a year | Effective rate over a year |
|---|---|---|
| Once a year | 1.04 − 1 | 4.00% |
| Every six months | 1.02² − 1 | 4.04% |
| Every quarter | 1.01⁴ − 1 | 4.06% |
| Every month | (1 + 0.04 ÷ 12)¹² − 1 | 4.07% |
| Every day | (1 + 0.04 ÷ 365)³⁶⁵ − 1 | 4.08% |
That figure is the annual equivalent rate, or AER: the rate an account would pay over a full year once compounding is taken into account. Because every account quotes it on the same basis, comparing AERs is the fair way to compare savings accounts that pay interest at different intervals.
Savings rates also move with the economy. The Bank of England kept Bank Rate at 3.75% at its decision published on 17 September 2026, with the next decision due on 5 November 2026. It explains that when it raises Bank Rate, banks usually increase the interest they offer on savings, and the reverse when it lowers it.
Tax can matter too. Most people can earn some savings interest tax free: the Personal Savings Allowance is £1,000 a year for basic rate taxpayers, £500 for higher rate taxpayers and nothing for additional rate taxpayers, according to GOV.UK. Interest on savings in an ISA is not usually taxed.
How compound interest works on loans and cards
Compounding works the same way on money you owe. If interest on a debt is charged monthly and nothing is repaid, each month's interest is added to the balance, and the next month's interest is charged on the larger amount.
Suppose £1,000 is owed at 2% a month and nothing is paid for a year. Twelve months of 2% do not add 24%: the balance becomes 1,000 × 1.02¹², about £1,268.24, an increase of 26.8%. The longer a balance is left, the faster it grows, for exactly the same reason savings grow.
Compound decay follows the same formula with a multiplier below 1. A car worth £12,000 that loses 15% of its value each year is worth 12,000 × 0.85³, or £7,369.50, after three years. Questions of this kind, growth and decay with a percentage multiplier, are standard at GCSE.
The habit to keep is the multiplier: a 5% increase is × 1.05, a 15% decrease is × 0.85. Our guide to percentages in your head shows how to find these multipliers quickly.
How to calculate compound interest in your head
For a quick estimate, the rule of 72 tells you roughly how long money takes to double: divide 72 by the annual rate in percent. At 4%, money doubles in about 72 ÷ 4 = 18 years; at 6%, in about 12 years. The exact figures are 17.7 and 11.9 years, so the rule is close enough to compare two options.
Over a short period, compound interest is only a little above simple interest. For two or three years, work out the simple interest first, then add a small amount: at 5% on £2,000, simple interest gives £200 over two years and compounding adds £5 more, the 5% earned on the first year's £100.
These estimates help with real decisions, such as whether a pay rise keeps up with prices, a question covered in our article on pay rises and inflation. Kalc trains times tables, fractions, percentages and algebra in about ten minutes a day, which is what makes estimates like these quick.
Compound interest at GCSE
At GCSE, simple interest belongs to the basic foundation content, inside percentage problems, while compound interest belongs to the additional foundation content, as part of growth and decay. In practice, a typical question gives an amount, a rate and a number of years and asks for the final value, or gives the final value and asks how many years it takes to pass a threshold.
For the second type, multiply year by year until you pass the target: at 5%, £2,000 passes £2,400 after four years (£2,431.01 at the end of year 4), not three. Write each year on its own line, so that a slip is easy to find.
Without a calculator, the year-by-year method is the one to use, because 5% or 10% of a round amount is quick to find. Our guide to the non-calculator GCSE paper covers the methods that make this kind of question manageable by hand.
Frequently asked questions
What is the formula for compound interest?
A = P × (1 + r)ⁿ, where P is the starting amount, r the rate per period as a decimal and n the number of periods. The interest earned is A − P.
How much is £1,000 worth after 5 years at 4% compound interest?
£1,000 × 1.04⁵, which is about £1,216.65. The interest earned is £216.65, against £200 with simple interest.
What is the difference between simple and compound interest?
Simple interest is paid on the starting amount only. Compound interest is paid on the balance including earlier interest, so it adds a little more each period.
What does AER mean on a savings account?
AER, the annual equivalent rate, shows what an account would pay over a year with compounding included, so accounts that pay monthly and yearly can be compared fairly.
How long does compound interest take to double money?
Divide 72 by the annual rate in percent for an estimate: about 18 years at 4% and about 12 years at 6%.
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