Time × rate × consistency. The simplest formula in finance, the one that does almost all the heavy lifting.
| Start | Years invested | Total contributed | Final balance |
|---|
Compound growth is the most powerful and least intuitive force in personal finance. The intuition trap: people add up the contributions and add the rate, expecting roughly that. The reality: at meaningful horizons, accumulated interest dwarfs contributions. On a $500/month investment over 40 years at 7%, you contribute $240,000 and end with $1.3 million. The vast majority of the final balance is interest on interest.
P — initial principalC — periodic contribution (per compounding period)r — annual interest rate (decimal)n — compounding periods per yeart — time in yearsFor continuous compounding, the formula collapses to FV = P · ert, which is the theoretical maximum compounding can achieve at a given rate.
| Type | Formula | Use case |
|---|---|---|
| Annual compounding | FV = P(1 + r)t | Bonds, simple savings comparisons |
| Periodic (n times/yr) | FV = P(1 + r/n)nt | Most retail savings (monthly), CDs (daily/monthly) |
| Continuous | FV = P · ert | Theoretical / academic; some derivatives pricing |
For typical retail products at typical rates, the difference between monthly, daily, and continuous compounding is well under 0.2% of the final balance over a multi-decade horizon. Compounding frequency is a marketing knob, not a real driver of returns.
The Rule of 72 is a mental-math shortcut: years to double ≈ 72 ÷ annual rate (in percent). It's accurate to within ~1% for rates between 4% and 12%.
| Rate | Years to double (Rule of 72) | Exact (ln 2 / ln 1+r) |
|---|---|---|
| 4% | 18.0 | 17.7 |
| 6% | 12.0 | 11.9 |
| 7% | 10.3 | 10.2 |
| 8% | 9.0 | 9.0 |
| 10% | 7.2 | 7.3 |
| 12% | 6.0 | 6.1 |
At 7% real return (typical long-run S&P), purchasing power doubles every 10 years. $100,000 today becomes $200,000 of purchasing power in 10 years, $400,000 in 20, $800,000 in 30. That's the math behind why the FIRE movement targets 25× annual expenses — at 4% withdrawal, the portfolio replaces itself indefinitely if real returns hold.
A 7% nominal return at 3% inflation is only 4% real. Plans built on 7% nominal but spending in today's dollars systematically overestimate purchasing power. Either project nominal and discount when budgeting, or use real return throughout.
An expense ratio is a haircut on returns, compounded. 1% in fees over 30 years on a $10,000 starting balance reduces the final balance from $76,000 (at 7%) to about $58,000 (at 6%) — a $18,000 cost. Index funds at 0.03-0.10% are functionally free; actively managed funds at 0.75-1.5% give back 25-40% of compounded growth as fees.
Taxable account dividend and capital-gains tax create the same compounding drag as fees. Tax-advantaged accounts (Roth IRA, 401(k), HSA) shield growth — which is why asset location across account types matters as much as asset allocation.
Sources: Robert Shiller historical S&P 500 dataset (1871-present), Aswath Damodaran annual return updates, Vanguard / Fidelity / JP Morgan / Schwab capital-market expectations, Truth in Savings Act (12 CFR 1030) APR vs APY definitions.
The S&P 500 has averaged about 10% nominal / 7% real since 1928 (per Robert Shiller's dataset and Aswath Damodaran's annual updates). Vanguard, Fidelity, Schwab, and JP Morgan capital-market expectations for the next 10 years cluster between 6.5% and 7.5% nominal — somewhat below the historical average because starting valuations are elevated. Use 7% for conservative long-term real-dollar planning, 6% for nearer-term horizons, 9-10% only if stress-testing the optimistic case.
Less than people think. The difference between annual, monthly, and daily compounding at 7% over 30 years is small — roughly 0.1-0.2% of the final balance. Continuous compounding adds only about 0.05% on top of daily. Where it matters: short-term comparison via APY (4.0% APR compounded monthly = APY 4.07%). Where it matters far more: rate and time. A 1% return improvement is worth more than any compounding-frequency upgrade.
A mental-math approximation for doubling time. Years to double ≈ 72 ÷ rate. At 7%: 10.3 years. At 10%: 7.2 years. At 4%: 18 years. The exact formula is ln(2)÷ln(1+r) ≈ 0.693÷r, but 72 is close enough for rates between 4% and 12% — and 72 has many divisors, making mental math easy. Useful for quick reality checks.
Over decades, enormously. $500/month over 40 years grows to about $1.18M at 6%, $1.53M at 7%, and $2.00M at 8% — each percentage point added is worth roughly 25-30% more in final balance. This is why low-cost index funds (0.03-0.10% expense ratios) compound to so much more than higher-cost actively managed funds (0.50-1.5%) over a career.
At 7% annual return on $500/month, a one-year delay costs roughly $50,000-$100,000 of final balance over a 30-40 year horizon. Each decade of delay roughly doubles the monthly contribution required to land at the same number. The "cost of waiting" table on this page shows the same calculation for delays of 5, 10, and 15 years.
APR (Annual Percentage Rate) is the simple annual rate before compounding. APY (Annual Percentage Yield) includes the effect of compounding within the year. For an investment account, APY is always slightly higher than APR. Truth in Savings Act (12 CFR 1030) requires US banks to disclose APY on deposit accounts. Compare APY for investments, APR for borrowing.
Inflation eats nominal returns. At 7% nominal and 2.5% inflation, real return is roughly 4.5%. The calculator's "after-inflation" line shows ending balance in today's dollars at 3% assumed inflation. Two ways to handle: (1) project nominal and discount at expected inflation when budgeting; (2) use real return throughout. Method 2 is cleaner when comparing scenarios.
Per Vanguard's 2012 (and updated 2023) study, lump-sum investing beats DCA roughly 68% of the time for typical horizons — markets go up more often than they go down, so getting invested earlier captures more compounding. DCA wins only in years that start with material drawdowns. The honest framing: if the money is already on hand, lump sum is the higher-expected-return approach. DCA's value is behavioral.
There's no verified evidence Einstein ever said "compound interest is the eighth wonder of the world" or similar. The quote appears in personal finance writing starting in the 1980s without source attribution and was almost certainly invented post-hoc. The fact remains true even without a famous physicist's endorsement: compound growth is one of the most powerful forces in personal finance.
Simple interest is computed only on the principal: I = P × r × t. The interest doesn't earn interest. Compound interest applies the rate to (principal + accumulated interest), so the base grows each period. On $10,000 at 7% over 30 years: simple interest produces $31,000 total; compound interest produces $76,123. The gap widens with longer horizons.