The one-sentence idea
Simple interest pays you on your original deposit. Compound interest pays you on your original deposit and on the interest already paid — so each period starts from a slightly larger base. Over one year the difference is trivial. Over thirty it is most of your balance.
Everything below follows from that. There is no second trick.
Formula 1: a single lump sum
For an amount you deposit once and leave alone:
FV = P x (1 + r/n)^(n x t) FV = future value P = the amount you start with r = annual interest rate (as a decimal: 6% = 0.06) n = compounding periods per year t = number of yearsWorked example — $10,000 at 6% for 10 years
Compounded once a year (n = 1): 10,000 x 1.0610 = 10,000 x 1.790847 = $17,908.48.
Compounded monthly (n = 12): 10,000 x (1 + 0.06/12)120 = 10,000 x 1.819397 = $18,193.97 — about $285 more, purely from compounding more often.
That is why the effective annual rate matters more than the headline one: 6% compounded monthly is really 6.1678% a year, because (1 + 0.06/12)12 − 1 = 0.061678.
When you compare a savings account, a bond or a loan, compare effective annual rates. Two products quoting the same nominal rate with different compounding frequencies are not the same product.
Formula 2: adding money every month
Most people do not invest once — they invest monthly. That is an ordinary annuity: a fixed payment at the end of each period. Its future value is
FV = C x [ ((1 + i)^m - 1) / i ] C = the contribution each period i = rate per period (annual rate / 12 for monthly) m = number of periods (years x 12 for monthly)If you also start with a lump sum, the two formulas simply add together: grow the lump sum with formula 1, grow the contributions with formula 2, add the results.
Worked example — $400 a month at 8%
i = 0.08 / 12 = 0.0066667 per month.
- 10 years (m = 120): 400 x [(1.0066667120 − 1) / 0.0066667] = 400 x 182.946 = $73,178. You contributed $48,000.
- 20 years (m = 240): 400 x 589.020 = $235,608. You contributed $96,000.
- 30 years (m = 360): 400 x 1,490.36 = $596,144. You contributed $144,000 — so roughly $452,000 of the balance is growth.
Look at what each decade adds: about $73k, then $162k, then $361k. Same contribution, same rate. The third decade earns more than the first two combined, because it starts from a much bigger base. This is the entire argument for starting early rather than starting big.
You can reproduce all three figures in the investment growth calculator — set the starting amount to $0, the monthly contribution to $400 and the return to 8%. It runs exactly this calculation.
The assumptions that change the answer
The formulas are exact. The inputs are where estimates go wrong.
- Contribution timing. The formula above assumes money arrives at the end of each period. If you invest at the start instead (an “annuity due”), multiply the result by (1 + i) — worth about 0.7% more at an 8% annual rate. Small, but know which one a calculator uses.
- Compounding frequency. Monthly compounding of an annual rate is a convention, not a law. Real funds do not pay a fixed monthly amount; the maths is a smoothed stand-in for lumpy reality.
- A constant rate is fiction. Markets deliver their average through wildly uneven years. Two portfolios with the same average return but different sequences end up in the same place if nothing is added or withdrawn — but not if you are contributing or drawing down. That is sequence-of-returns risk, and no compound-interest formula captures it.
- Fees compound too. A 1% annual fee on $400 a month for 30 years turns $596,144 into about $487,988 — roughly $108,000, or 18% of the final balance, for one percentage point. See what an expense ratio means.
- Inflation is not included. These are nominal figures. To see them in today's money, use a real return: roughly the nominal rate minus expected inflation.
- Taxes are not included and depend entirely on your country and account type.
The Rule of 72, and where it breaks
To estimate a doubling time in your head, divide 72 by the annual percentage rate. At 6%, that is 72 / 6 = 12 years. The exact answer is ln(2) / ln(1.06) = 11.9 years — close enough for conversation.
It degrades at the extremes: at 1% the rule says 72 years against a true 69.7, and at 25% it says 2.9 against a true 3.1. Use it as a sanity check, never as a projection.
What to actually do with this
- Start now, at any size. The time variable sits in the exponent; the contribution does not.
- Automate the contribution so it happens whether or not you feel like investing that month.
- Raise it with your income. Every worked example above holds the contribution flat for 30 years, which almost nobody does — it is a deliberately conservative floor.
- Compare effective annual rates, not headline rates, on both savings and debt.
- Watch fees, because they compound against you with exactly the same mathematics.
And remember which direction this runs: the same formula prices your credit-card balance. A 22% APR compounding monthly is an effective 24.4% a year working against you — which is why clearing expensive debt usually beats investing alongside it.
Sources and further reading
Primary sources are preferred: regulators, central banks, statistical agencies, tax authorities, index providers and original research. Links open on the publisher's own site; FinSage has no commercial relationship with any of them.
- U.S. Securities and Exchange Commission, Investor.gov — Compound Interest Calculator. The SEC's own tool, useful for cross-checking the figures above.
- U.S. Consumer Financial Protection Bureau — financial education glossary, for the standard definitions of compound interest, APR and APY.
- Dimson, Marsh & Staunton, Global Investment Returns Yearbook — long-run real returns across 20+ markets since 1900, the standard reference for what equity returns have actually been.