Compound Interest Explained: Formula, Examples, Rule of 72
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Compound interest means earning returns on your returns. The formula is A = P(1 + r/n)^(nt): $1,000 at 5% compounded yearly for 10 years grows to $1,000 × 1.05^10 = $1,628.89 — $628.89 of it pure compounding, not deposits. Run your own numbers in our compound interest calculator.
Time matters more than rate. Starting 10 years earlier beats a slightly higher return started late, because each year's growth becomes next year's base. That is why the first dollars you invest are the most powerful dollars of your life.
The fastest estimator is the Rule of 72: divide 72 by your annual rate to get the years needed to double. At 6%, money doubles in 72 ÷ 6 = 12 years.
The Formula, Piece by Piece
A = P(1 + r/n)^(nt). P is your starting principal, r is the annual rate as a decimal (5% → 0.05), n is how many times per year interest compounds, and t is years. The exponent nt counts every compounding event.
Compounding frequency matters but less than people think. Monthly compounding beats yearly compounding by a small margin at the same nominal rate — always compare APY (which includes compounding) rather than APR when choosing savings accounts.
Why Starting Early Beats Higher Returns
Compare $5,000 invested at 7% for 30 years ($38,061) versus the same $5,000 at 9% for 20 years ($28,022). The lower rate wins because it compounds a full decade longer. Every year of delay must be compensated by meaningfully higher returns or bigger deposits.
Contributions amplify the effect. Adding $200 monthly to that 7% account turns $38,061 into roughly $227,000 over 30 years — deposits provide fuel, compounding provides the engine.
The Fee Mistake That Costs Thousands
A 1% annual fee on a $100,000 portfolio growing at 7% for 20 years costs about $60,000 in lost growth versus a 0.1% fee — the fee compounds against you exactly like returns compound for you. Expense ratios deserve the same scrutiny as returns.
Verify any projection by checking the three inputs: real rate after inflation, compounding frequency, and whether contributions are included. Change one input at a time to see which drives the result.
APY vs APR: The Comparison That Matters
APR quotes a rate without compounding; APY bakes compounding in. A savings account at 4% compounded monthly pays an APY of about 4.07% — small on paper, meaningful across decades. For savings always compare APY; for loans compare APR plus fees.
Banks advertise whichever number looks better for them. Convert with APY = (1 + r/n)^n − 1 before deciding anything, or let our investment return calculator do the comparison with your real figures.
- Savings: compare APY (includes compounding)
- Loans: compare APR plus origination and annual fees
- Formula: APY = (1 + r/n)^n − 1
Real Returns: Subtracting Inflation
Nominal 7% growth at 3% inflation is roughly 4% real growth — the spending power you actually gain. A quick estimate is nominal minus inflation; the precise figure divides: (1.07 ÷ 1.03) − 1 = 3.88%.
This is why "high-yield" 4.5% savings at 3% inflation quietly earns 1.5% real. Plan retirements and house deposits in real terms, and measure every ROI calculation against inflation, not zero.
Worked Examples
1Basic: $1,000 at 5% for 10 years (yearly)
- Write the inputs: P = 1000, r = 0.05, n = 1, t = 10.
- Apply the formula: A = 1000 × (1 + 0.05/1)^(1×10) = 1000 × 1.05^10.
- Compute: 1.05^10 ≈ 1.62889, so A ≈ $1,628.89.
Balance $1,628.89 — interest earned $628.89.
2Monthly compounding: $5,000 at 4% for 5 years
- Inputs: P = 5000, r = 0.04, n = 12, t = 5 (60 periods).
- A = 5000 × (1 + 0.04/12)^60 ≈ 5000 × 1.22099.
Balance ≈ $6,105 — monthly compounding adds ~$8 over yearly.
3Rule of 72: doubling at 6%
- Divide: 72 ÷ 6 = 12.
- Interpretation: money doubles roughly every 12 years at 6%.
$10,000 at 6% ≈ $20,000 in 12 years.
What $10,000 becomes at different rates over time (yearly compounding).
| Rate | 10 years | 30 years |
|---|---|---|
| 3% | $13,439 | $24,273 |
| 5% | $16,289 | $43,219 |
| 7% | $19,672 | $76,123 |
| 9% | $23,674 | $132,676 |
Mistakes to Avoid
Comparing APR instead of APY
APR ignores compounding while APY includes it. Two accounts quoting 4% can pay different amounts — always compare APY for savings.
Ignoring fees
A 1% fee compounds against you for decades. A fund earning 7% with a 1% fee behaves like 6% — over 20 years that gap costs tens of thousands.
Forgetting inflation
Nominal 7% at 3% inflation is roughly 4% real growth. Plan spending power in real terms, not statement balances.
Frequently Asked Questions
What is the compound interest formula?
A = P(1 + r/n)^(nt): principal × (1 + annual rate ÷ compounds per year)^(compounds × years). $1,000 at 5% yearly for 10 years = 1000 × 1.05^10 = $1,628.89.
What is the Rule of 72?
Divide 72 by your annual rate to estimate doubling time. At 6%, money doubles in 72 ÷ 6 = 12 years. It is an approximation, most accurate for rates between 4% and 12%.
Is compound interest better monthly or yearly?
Monthly compounding pays slightly more at the same nominal rate ($5,000 at 4% for 5 years: ≈$6,105 monthly vs ≈$6,083 yearly). Compare APY to decide — it already reflects frequency.
How much does a 1% fee cost long term?
Enormously. On $100,000 at 7% over 20 years, a 1% annual fee versus 0.1% costs roughly $60,000 in lost compounding. Fees compound against you.
Should I compare APY or APR for savings?
APY — it includes compounding while APR does not. Convert with APY = (1 + r/n)^n − 1. A 4% nominal rate compounded monthly pays about 4.07% APY.
What is a real return after inflation?
Roughly nominal minus inflation: 7% growth at 3% inflation ≈ 4% real. Precisely: (1.07 ÷ 1.03) − 1 = 3.88%. Plan big goals in real terms.
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Written by PercentLab Editorial Team · Last updated: