| Year | Balance | Contributions | Interest |
|---|
Divide 72 by the rate and you get the doubling time. At 7% it gives 10.29 years against a true 10.24 — near enough perfect, which is why the rule has survived. But it is an approximation chosen to be good somewhere, and it is only that good near the middle. The exact answer is ln 2 / ln(1 + r).
| Rate | Exact | Rule of 72 | Error |
|---|---|---|---|
| 1% | 69.7 y | 72.0 y | +3.4% |
| 2% | 35.0 y | 36.0 y | +2.8% |
| 5% | 14.2 y | 14.4 y | +1.4% |
| 8% | 9.01 y | 9.00 y | −0.1% |
| 12% | 6.12 y | 6.00 y | −1.9% |
| 20% | 3.80 y | 3.60 y | −5.3% |
| 25% | 3.11 y | 2.88 y | −7.3% |
The error changes sign at about 8%, so it is not a simple bias you can correct for. Below that the rule is long — at 2% it says 36 years when the answer is 35. Above it the rule is short, and by 20% it is out by five months on a four-year answer.
The mathematically natural constant is 100 × ln 2 = 69.3, which is exact in the limit of continuous compounding. Yet 69.3 is worse at almost every rate anyone actually invests at: it is out by a flat 0.34 years everywhere, while 72 is out by 0.01 at 8%, 0.12 at 12% and 0.19 at 5%. Only below about 3% does 69.3 win.
And then there is the reason nobody writes down: 72 has twelve divisors and 69.3 has none worth having. You can do 72 ÷ 6 or 72 ÷ 8 or 72 ÷ 9 in your head. A mental shortcut that needs a calculator is not a mental shortcut, and the rule survives because it is divisible, not because it is the most accurate constant available.
The rate here is constant, and no real return is. That matters far more than it looks. Two portfolios averaging the same return can end in very different places depending on the order the good and bad years arrive — and for anyone drawing an income, a bad first decade is close to unrecoverable. This page cannot show you that, and the smooth curve it draws is the single most misleading thing on it.
There is no tax. Not on contributions, growth, dividends or withdrawal. Any real number depends on the wrapper the money sits in, and that is jurisdiction-specific.
Fees are one flat annual percentage, charged pro rata each period on the balance at the start of it. Real charges are layered — platform, fund, transaction, spread — and some are stepped or capped. A 1% fee here costs 1.06 points of effective yield and about a quarter of a 30-year pot.
Worth knowing, because it is not obvious: the second percentage point of fee costs less than the first. On the figures above, the first 1% takes about £21k off a 30-year pot and the second only a further £16k — there is simply less left to take it from. Fee damage is severe, but it saturates.
No contribution gaps, career breaks, lump-sum windfalls or changes of plan. Verified against closed-form annuity formulas in compound_model.py, which agrees with this page’s period-by-period loop to fifteen significant figures.
Compound interest is interest earning interest. Each period's gains join your balance and start earning their own return — which is why the growth curve bends upward instead of climbing in a straight line. The three levers are your starting amount, your regular contributions, and the rate, but the quiet fourth lever is time: most of the growth happens late, so starting early matters more than almost anything else.
Start with $10,000, add $200/mo at the end of each period, and assume a 7% annual return compounded monthly. After 20 years you'd have about $144,600.
You only contributed $58,000 of that — the other ~$86,600 is interest. Push the time horizon to 30 years and the interest portion grows dramatically faster than the contributions, because the earliest dollars have had the longest to compound.
A mental shortcut: divide 72 by your rate to estimate years to double your money. At 7%, that's about 10.3 years — very close to the exact 10.2, which is why the rule has stuck around.
A little. Daily vs. annual compounding at the same rate makes a modest difference over long horizons — real and worth knowing, but far smaller than changing the rate, contribution, or time. The effective annual yield (APY) converts any compounding schedule into a single apples-to-apples rate.
Nominal is the headline rate; real is what's left after inflation erodes purchasing power. A 7% nominal return at 3% inflation is closer to a 4% real return — the number that reflects what you can actually buy later. Set the inflation assumption yourself when real mode is on.
End-of-period (ordinary annuity) is the classic textbook default — deposits land after that period's interest. Beginning-of-period (annuity due) deposits first, so each contribution earns interest in the same period. Same cash in, slightly higher ending balance for beginning.
No — real investments vary year to year, sometimes sharply. This model assumes a steady average, which is useful for intuition but will never match any actual sequence of returns.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.