A = P × (1 + r ÷ n) ^ (n × t)
where P is the starting amount, r the annual nominal rate as a decimal, n the number of compounding periods per year, and t the number of years. If you add money regularly, a second term is needed for the stream of deposits:
future value of deposits = PMT × ((1 + i) ^ N − 1) ÷ i
with i the rate per period and N the number of periods. The two results are added.
10,000 invested for 20 years at 7% nominal, compounded annually:
| Step | Value |
|---|---|
| Growth factor | 1.07 ^ 20 = 3.8697 |
| Final balance | 10,000 × 3.8697 = 38,697 |
| Of which interest | 28,697 |
Now add 500 a month at the same rate compounded monthly. The rate per period is 0.0058333 and there are 240 periods:
| Component | Value |
|---|---|
| Initial 10,000 grown monthly | 40,375 |
| Annuity factor — (1.0058333 ^ 240 − 1) ÷ 0.0058333 | 520.71 |
| 500 × 520.71 | 260,355 |
| Total | 300,730 |
| Total paid in | 130,000 |
| Growth | 170,730 — about 57% of the balance |
Less than the marketing suggests. Same 10,000, same 7%, same 20 years:
| Compounded | Effective annual rate | Final balance |
|---|---|---|
| Annually | 7.000% | 38,697 |
| Semi-annually | 7.123% | 39,593 |
| Quarterly | 7.186% | 40,064 |
| Monthly | 7.229% | 40,375 |
| Daily | 7.250% | 40,532 |
| Continuously | 7.251% | 40,552 |
The whole span from annual to continuous is 4.8%, and almost all of it is captured by the time you reach monthly. Daily compounding is a rounding difference. Rate and time do the work; frequency is a detail.
That effective-rate column is the difference between APR and APY. APR is the nominal rate before compounding; APY is what you actually earn once compounding is applied. A 7% APR compounded monthly is a 7.229% APY, which is why a savings account advertising one number and a loan advertising the other are not directly comparable. Always compare like with like.
Dividing 72 by the percentage rate estimates the doubling time. It is a good shortcut in a narrow band:
| Rate | Rule of 72 | Exact | Error |
|---|---|---|---|
| 2% | 36.0 years | 35.0 | +2.9% |
| 7% | 10.3 years | 10.24 | +0.4% |
| 8% | 9.0 years | 9.01 | −0.1% |
| 20% | 3.6 years | 3.80 | −5.3% |
The exact figure is ln(2) ÷ ln(1 + r). The rule is calibrated near 8% and drifts either side; for continuous compounding the correct numerator is 69.3, and 72 was chosen historically because it divides neatly by 2, 3, 4, 6, 8, 9 and 12.
Two deductions turn an impressive projection into a realistic one, and most calculators show neither.
Fees compound too. A 1% annual charge does not cost 1%; it removes a percentage point from the growth rate for every year of the run. On the original example, 7% becomes 6%:
| Net rate | Balance after 20 years |
|---|---|
| 7.0% | 38,697 |
| 6.0% | 32,071 |
One percentage point of fee removed 6,626 — about 17% of the final balance, from a charge that reads as trivial on a statement. Over 30 years the same gap costs around 24%.
Inflation removes more. The 300,730 figure above is in future currency. With inflation at 3%, prices rise by a factor of 1.03 ^ 20 = 1.806, so that balance buys what 166,490 buys today. The real rate is not 7% minus 3% but 1.07 ÷ 1.03 − 1 = 3.88%. More than 40% of the apparent gain is the currency shrinking rather than the portfolio growing.
The practical fix is to run the projection in real terms — use an inflation-adjusted rate and read the answer as today money — or to run both and quote the pair. A projection stated only in nominal terms flatters itself.
You will see compound interest described as the eighth wonder of the world, or the most powerful force in the universe, attributed to Albert Einstein. There is no evidence he said either. Neither line appears in his writings, letters or recorded remarks, and the attributions surface in print decades after his death — most often in advertising for financial products. It is a marketing flourish that acquired a famous name, and it is worth noticing that the claim most often used to sell compounding is itself fabricated. The mathematics needs no celebrity endorsement.
The formula assumes a constant rate, and no investment delivers one. A portfolio averaging 7% arrives at that average through good and bad years, and the order of those years changes the outcome whenever money is being added or withdrawn — the same average return can produce materially different balances. Compound interest models a bank deposit accurately and a stock market only as a smoothed long-run average.
Other omissions to hold in mind: tax on interest, dividends or gains is not modelled, and where it applies annually it reduces the compounding base; contributions held flat in nominal terms shrink in real value each year, so a plan should escalate them with inflation; and the projection assumes you never interrupt the deposits, which real life rarely permits.
This is general information, not financial advice. Any rate entered here is an assumption, not a forecast, and decisions about your own money warrant a qualified adviser who can see your full position.
Whatever you assume, run a lower one beside it. The gap between a 5% and an 8% assumption over 30 years is larger than most people expect, and planning to the optimistic figure is how projections fail. Historical averages are not promises.
Usually APR against APY, or a different compounding basis, or an introductory rate that reverts. Ask for the effective annual yield and the date it changes.
Rarely on its own — see the frequency table, where the whole range is under 5% of the outcome. A meaningfully higher rate or a lower fee matters far more.
No. The projection runs in your browser and nothing you enter is transmitted or saved.
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