How does compound interest work?
By Unlearnt
Compound interest is interest earned on both your original money and interest already added to it. The rate can stay constant while the amount earned rises, because the balance receiving that rate grows. The same mechanism can also increase a debt when unpaid interest is added to the balance.
The base changes, even when the rate does not
Suppose an account pays a fixed 5% each year and adds the interest to the account. After the first interest payment, next year's 5% applies to a larger balance. With simple interest calculated only on the original principal, that base would not grow.
For one deposit with a fixed annual rate and annual compounding, the balance after n years is principal × (1 + rate)ⁿ. Write 5% as 0.05. This formula assumes no contributions, withdrawals, fees or taxes, and that all interest stays in the account.
Worked example: £1,000 at a fixed 5%
Start with £1,000. Year one adds £50. Year two adds £52.50 because 5% now applies to £1,050. Year three adds £55.125 before rounding. The rate stays at 5%; earlier interest is now earning interest too.
These are hypothetical account terms, not an available product or a forecast. The illustration rounds displayed amounts to the nearest penny. Its final balance comes from £1,000 × 1.05³, approximately £1,157.63.
The same 5% earns more on a growing balance
Withdrawing interest changes what compounds
If you take out each £50 payment instead, the account remains at £1,000 under these assumptions. After three years you have the original £1,000 plus £150 withdrawn, or £1,150 in total if you kept that cash unspent. Compare total wealth, not only the balances left in the two accounts.
The £7.63 difference comes from earning interest on prior interest. If withdrawn money earns a return somewhere else, that needs to be included too. Extra deposits are another separate source of growth: an account can rise because you added money, because it earned a return, or both.
The Rule of 72 is a quick estimate, not a promise
The Rule of 72 is a mental shortcut for a fixed-rate example: divide 72 by the annual percentage rate to estimate how many years a balance takes to double, if the rate stays constant and interest is added once a year. At 5%, 72 ÷ 5 = 14.4 years. The exact calculation from £1,000 × 1.05ⁿ reaching £2,000 is a little over 14 years. The rule is a convenient estimate, and it drifts at high rates.
It is not a forecast, a guaranteed return or a reason to choose an account. A real balance can also be changed by deposits, withdrawals, fees, tax and inflation. Use the formula, or a calculator such as the Investor.gov tool linked below, when the number matters.
How often interest is added changes the result
Interest is often added more than once a year. For one starting amount at a fixed annual rate, split into equal periods, the usual formula is principal × (1 + rate/n)^(n × years), where n is the number of times interest is added each year. Write 5% as 0.05. This still assumes no contributions, withdrawals, fees or taxes, and that all interest stays in the account.
In the £1,000 example, annual compounding for three years gave about £1,157.63. With monthly compounding, the same assumed rate and term gives about £1,161.47, because interest starts earning interest sooner. The difference is modest over three years and would be larger over a longer term. These figures are hypothetical arithmetic, not an available product.
The same mechanism can increase a debt. If unpaid interest is added to a balance, later interest is charged on a larger amount. How often that happens, and whether payments are made, changes the result. This is a description of the arithmetic, not a recommendation or a description of a particular loan or card.
Misconception: an average return is a guaranteed growth rate
A fixed contractual interest rate in a simplified example differs from a risky investment's changing returns. For changing rates, multiply the successive growth factors: principal × (1 + first return) × (1 + second return), and so on. A loss reduces the base available for future gains.
Fees, taxes and inflation can further change the outcome or its purchasing power; a neat nominal balance is not the whole result. The Investor.gov calculator linked below lets you vary contributions, time, the assumed rate and how often interest is added. This page explains arithmetic rather than recommending an account or investment.
When you are ready, try today's question below. Explain a new case in your own words and the tutor will help you find the gaps.
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Optional: apply the idea to a new scenario.
A hypothetical investment starts at £2,000, gains 10% in year one and loses 10% in year two, with no deposits, withdrawals or costs. Is it back at £2,000? Explain using the amount each percentage applies to.
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