How compound interest works: the formula A = P(1 + r/n)ⁿᵗ
Compound interest is the mechanism by which interest earned on an investment is automatically reinvested, so that in the next period it earns interest of its own. This "snowball" effect is why investing early and staying invested for many years has such a large impact on the final result, even with modest contributions.
The classic compound interest formula for a single lump sum is:
A = P × (1 + r/n)n×t
where A is the final accumulated value, P is the initial capital invested, r is the annual interest rate expressed as a decimal, n is the number of times interest compounds per year (12 for monthly, 1 for annual) and t is the number of years the investment runs. When regular contributions are added on top of the initial deposit, the future value of those contributions is added to this result using the future value of an annuity formula: each contribution compounds interest for whatever time remains until the end of the horizon, so earlier contributions earn more accumulated interest than later ones.
This calculator applies that exact logic month by month (if you choose monthly contributions) or year by year (if you choose annual contributions), adding the interest earned on the accumulated balance each period and then the new contribution, before charting how much of the final balance comes from your own pocket versus compound interest.
The Rule of 72: estimate how long it takes to double your money
The Rule of 72 is a popular mental shortcut among investors for quickly estimating, without a calculator, how many years it will take an investment to double at a constant annual compound return. The formula is simple:
Years to double ≈ 72 / annual return (%)
For example, at a 4% annual return, capital would double in roughly 72 / 4 = 18 years; at a 9% annual return, in roughly 72 / 9 = 8 years. This rule works especially well for interest rates between 6% and 10% a year, and loses some accuracy outside that range, but it's a handy way to compare investment scenarios at a glance without logarithms or a spreadsheet.
How interest and investment income are taxed in the UK
Most UK taxpayers have a Personal Savings Allowance that lets them earn a certain amount of interest tax-free each tax year: £1,000 for basic rate taxpayers, £500 for higher rate taxpayers, and £0 for additional rate taxpayers (who must pay tax on all savings interest). There is a separate Dividend Allowance of £500 a year for dividend income from shares held outside a tax wrapper. Beyond these allowances, interest and dividends are taxed at your marginal Income Tax rate.
The main way UK savers and investors shelter growth from tax entirely is the ISA (Individual Savings Account). You can put up to £20,000 a year (the current annual ISA allowance) into a Cash ISA, a Stocks & Shares ISA, or a mix of both, and any interest, dividends or capital gains earned inside an ISA are completely free of UK tax — no allowance limits to track, and nothing to report to HMRC. This calculator shows your gross final balance before tax; if your investment is outside an ISA and exceeds your allowances, your actual take-home growth will be lower once tax is applied.