How it works
Why compound interest is the only maths that matters for long-term saving
Simple interest grows linearly — a flat fee for lending the bank your money. Compound interest grows exponentially because it pays you interest on the interest. Over 10 or 20 years the gap between the two is enormous, which is why "time in the market" is the most reliable lever a private saver has.
Einstein is often quoted as calling compound interest "the eighth wonder of the world". Whether he did or not, the numbers speak for themselves.
The formula, explained
A = P × (1 + r/n)^(n×t)
- A — the final amount (principal + interest).
- P — the principal, i.e. the starting balance.
- r — the annual interest rate as a decimal (5% → 0.05).
- n — compounding periods per year (annually = 1, monthly = 12, daily = 365).
- t — time in years.
Worked examples
£10,000 at 5% for 10 years, compounded annually
A = 10,000 × (1 + 0.05)^10 = 10,000 × 1.62889 = £16,289
Interest earned = £6,289.
Same sum, compounded monthly
A = 10,000 × (1 + 0.05/12)^(12 × 10) = 10,000 × (1.004167)^120 ≈ £16,470
Monthly compounding adds roughly £181 compared to annual at the same headline rate — a small but free bonus.
Adding £200 a month
Now start with £10,000 and add £200 every month at 5% compounded monthly for 10 years.
Final balance ≈ £47,500 — of which £24,000 is contributions and £13,500 is interest. The habit beats the lump sum.
The rule of 72
A classic mental-maths shortcut: divide 72 by the annual interest rate to estimate how many years it takes for money to double.
| Rate | Approx doubling time | Exact years |
|---|---|---|
| 3% | 24 years | 23.4 |
| 5% | 14.4 years | 14.2 |
| 7% | 10.3 years | 10.2 |
| 10% | 7.2 years | 7.3 |
ISAs: compound interest without the tax drag
In the UK, the Individual Savings Account (ISA) wrapper removes income tax and capital gains tax on the interest and growth you earn inside it. Every adult has an annual allowance (£20,000 for 2025/26), split across Cash, Stocks & Shares, Lifetime and Innovative Finance ISAs.
Over long horizons, protecting gains from tax materially boosts the compounding outcome. A 5% return inside an ISA fully reinvests into the next period; the same 5% return outside, for a higher-rate taxpayer, might lose 40% of the interest to HMRC each year — slashing the effective compound rate.
Things the formula does not capture
- Inflation — A 5% return when inflation is 3% is only a 2% real return. Over 25 years, inflation halves the purchasing power of a pound.
- Fees — Investment platforms, fund charges and advisers skim compound growth. A 1% annual fee over 25 years at 6% gross cuts the final pot by roughly 20%.
- Volatility — Real-world investments don't return a clean percentage each year. Sequencing matters, especially near retirement.
- Tax changes — ISA rules, allowances and dividend taxation have all shifted over the last decade and will again.
Practical savings strategies
- Start early — a £100/month saver who starts at 25 beats a £200/month saver who starts at 45 by retirement age, assuming equal returns.
- Automate contributions — salary-sacrifice pensions and direct debits remove the willpower tax.
- Prioritise employer match — if your employer matches pension contributions, that's an instant 100% return before compounding even starts.
- Use tax wrappers first — ISA allowance and pension allowance are use-it-or-lose-it each year.