Albert Einstein probably never called compound interest the eighth wonder of the world, but the idea behind the quote is real: money that earns interest on its interest grows exponentially, not in a straight line. The same force works against you with debt. Understanding the formula helps you compare savings accounts, judge investment projections and see why starting early matters so much. Our compound interest calculator runs the numbers for any rate and period.
This guide is general information, not financial advice. Investment returns are not guaranteed and can be negative.
Simple vs compound interest
Simple interest is paid only on the original amount (the principal):
Interest = P × r × tCompound interest is paid on the principal plus all interest added so far:
A = P × (1 + r/n)^(n × t)- P = principal (starting amount)
- r = annual nominal rate as a decimal (5% = 0.05)
- n = compounding periods per year (1 annual, 12 monthly, 365 daily)
- t = years
- A = final amount
Example: £10,000 at 5% for 10 years
| Method | Calculation | Result |
|---|---|---|
| Simple interest | 10,000 + 10,000 × 0.05 × 10 | £15,000.00 |
| Compound, annually | 10,000 × 1.05^10 | £16,288.95 |
| Compound, monthly | 10,000 × (1 + 0.05/12)^120 | £16,470.09 |
Compounding earns an extra £1,289 compared with simple interest over ten years, and the gap widens every year. Over 30 years at 5% annual compounding, £10,000 becomes about £43,219, compared with £25,000 under simple interest.
How compounding frequency matters
The more often interest is added, the more you earn, but the gains shrink quickly:
| Compounding | Effective annual rate for 5% nominal |
|---|---|
| Annually | 5.000% |
| Quarterly | 5.095% |
| Monthly | 5.116% |
| Daily | 5.127% |
The effective rate is (1 + r/n)^n − 1. That effective rate is exactly what UK banks must show as the AER (Annual Equivalent Rate) and US banks as the APY (Annual Percentage Yield). Always compare savings accounts by AER or APY, not by the headline nominal rate.
For borrowing, the equivalent figure is APR, which also includes certain fees. Our loan interest guide covers how loan repayments are calculated.
The rule of 72
To estimate how long it takes money to double, divide 72 by the annual percentage rate:
| Rate | Rule of 72 | Exact doubling time |
|---|---|---|
| 2% | 36 years | 35.0 years |
| 4% | 18 years | 17.7 years |
| 5% | 14.4 years | 14.2 years |
| 8% | 9 years | 9.0 years |
| 12% | 6 years | 6.1 years |
It works in reverse too: to double money in 10 years you need about 7.2% a year. And it applies to inflation: at 3% inflation, prices double in about 24 years.
Regular savings: the real power
Most people save monthly rather than investing one lump sum. The future value of regular deposits (made at the end of each period) is:
FV = PMT × ((1 + i)^N − 1) / iwhere PMT is the regular payment, i the rate per period and N the number of periods.
Example: saving £200 a month at 5% a year, compounded monthly (i = 0.05/12), for 20 years (N = 240):
- Total paid in: 200 × 240 = £48,000
- Final value: about £82,207
- Interest earned: about £34,207
Why starting early wins
| Saver | Saves | Total paid in | Value at 65 (5% a year) |
|---|---|---|---|
| Starts at 25 | £200/month for 40 years | £96,000 | about £305,000 |
| Starts at 35 | £200/month for 30 years | £72,000 | about £166,000 |
Ten extra years of saving add only £24,000 of contributions but nearly double the final pot.
Real return: inflation matters
A 5% return when prices rise 3% a year does not make you 5% richer. The real return is:
real = (1 + nominal) / (1 + inflation) − 1At 5% nominal and 3% inflation: 1.05 ÷ 1.03 − 1 = 1.94%, not the 2% you might expect from simple subtraction (though subtraction is a fine quick estimate).
Tax reduces returns further. In the UK, interest inside an ISA (up to £20,000 a year) is tax-free, and outside an ISA the Personal Savings Allowance covers £1,000 of interest for basic-rate taxpayers (£500 for higher-rate). In the US, accounts such as 401(k)s and IRAs offer tax deferral or tax-free growth.
Compound interest on debt
Credit cards typically charge interest monthly on unpaid balances. At a 24% APR, a £2,000 balance with no repayments would grow to about £2,536 in a year with monthly compounding (2,000 × 1.02^12). Paying only the minimum can stretch repayment over many years. Paying off high-interest debt is often the best “investment” available.
Common mistakes
- Comparing accounts by nominal rate instead of AER/APY.
- Assuming investment returns are smooth: real markets go up and down, and averages hide volatility.
- Forgetting fees: a 1% annual fee on a 6% return cuts the result by a sixth each year, and the effect compounds.
- Ignoring inflation and tax when planning.
- Treating the rule of 72 as exact at very high rates.
Checklist
- Use A = P(1 + r/n)^(nt) for lump sums and the future value formula for regular saving.
- Compare savings by AER (UK) or APY (US).
- Estimate doubling time with the rule of 72.
- Adjust for inflation, tax and fees to see your real return.
- Start early and pay off expensive debt first.
For the percentage basics behind all of this, see how to calculate percentages.