Compound Interest Calculator
Compound interest grows interest on top of interest — not just on your principal. The frequency of compounding and how rates are disclosed differ between countries. US banks show APY, UK banks show AER — but they mean the same thing.
A = P × (1 + r/n)^(n×t) where P = principal, r = annual rate, n = compounding periods/year, t = years. US: APY = (1 + APR/n)^n − 1. UK: AER is the identical concept. Daily compounding yields ~0.1% more than annual at typical savings rates.
How Compound Interest Works
The compound interest formula is:
Where: P = principal, r = annual rate, n = compounds per year, t = years.
The more frequently interest compounds, the more you earn — even at the same stated rate.
APY vs APR vs AER: How Countries Disclose Rates
| Term | Country | What It Means | Includes Compounding? |
|---|---|---|---|
| APR | 🇺🇸 US, 🌍 Global | Annual Percentage Rate — stated rate before compounding | No |
| APY | 🇺🇸 US | Annual Percentage Yield — actual return after compounding | Yes |
| AER | 🇬🇧 UK | Annual Equivalent Rate — same as APY, UK/EU standard | Yes |
| EAR | 🌍 International | Effective Annual Rate — academic/international term for APY | Yes |
| Nominal Rate | 🌍 Global | The stated rate — same as APR | No |
Compounding Frequency Comparison (5% stated rate, $10,000, 10 years)
| Frequency | Effective APY | Final Balance | Total Interest |
|---|---|---|---|
| Annually (n=1) | 5.000% | $16,288.95 | $6,288.95 |
| Quarterly (n=4) | 5.095% | $16,436.19 | $6,436.19 |
| Monthly (n=12) | 5.116% | $16,470.09 | $6,470.09 |
| Daily (n=365) | 5.127% | $16,486.65 | $6,486.65 |
The Rule of 72: Quick Mental Math
Divide 72 by your interest rate to estimate how many years it takes to double your money:
- At 4%: 72 ÷ 4 = 18 years to double
- At 6%: 72 ÷ 6 = 12 years
- At 9%: 72 ÷ 9 = 8 years
- At 12%: 72 ÷ 12 = 6 years
This works because of the mathematical properties of compound growth and is accurate to within a few percent for rates between 2–20%.
Frequently Asked Questions
Does the UK use APY or APR?
The UK primarily uses AER (Annual Equivalent Rate) for savings accounts, which is equivalent to APY in the US. For borrowing, the UK uses APR. The EU's Consumer Credit Directive also uses APR for loans. When comparing savings rates between a US bank (showing APY) and a UK bank (showing AER), you can compare them directly — they're the same metric.
What is continuous compounding?
Continuous compounding is the theoretical limit where interest compounds infinitely often. The formula is A = P × e^(rt). In practice, no real financial product compounds continuously, but some academic analyses use it. Daily compounding (used by US banks) comes very close to continuous compounding in practice.
Why does compound interest matter for long-term investing?
Albert Einstein reportedly called compound interest "the eighth wonder of the world." A $10,000 investment at 7% annual return becomes $76,123 after 30 years — and $149,745 after 40 years. The difference between 30 and 40 years is nearly $74,000, all from the same initial investment. Starting 10 years earlier almost doubles the outcome.
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