| Year | Balance | Contributions | Interest |
|---|
Each period’s gains join the balance and earn their own return, which is why the curve bends upward instead of climbing in a straight line.
Starting balance, contributions and rate are the three you choose. Most of the growth happens late, so starting early matters more than almost anything else.
A 7% nominal return at 3% inflation is closer to a 4% real return: the number that reflects what you can actually buy later.
They come out of the balance every period, so a 1% annual charge costs far more than the fees paid. Set the slider to what you actually pay.
Fees come out of the balance every period, so they compound against you exactly as returns compound for you. On the default plan a 1% annual charge takes roughly 11,000 in fees, and leaves you about 19,000 worse off, because the money taken would itself have grown. The gap between those two figures widens the longer you hold, and it is the reason a seemingly small percentage matters so much. Set the charges slider to what you actually pay, not to zero.
cost > fees paid, always
Early on nearly all the growth is money you paid in. There is a year, shown in the results, where twelve months of growth first exceeds twelve months of contributions, and after it the balance is pulling harder than you are. On the default plan that is around year seven. Raising your contributions each year pushes it later, which looks like a step backwards and is not: you are simply raising the bar the growth has to clear.
growth/yr > contributions/yr
Most calculators assume you pay in the same cash amount for thirty years. Almost nobody does, and if you are reading the results in real terms that assumption quietly shrinks your contributions every single year. Setting the yearly increase to about inflation keeps them level in real terms; setting it to your expected pay rises is closer to what actually happens. It moves the final number more than most people expect.
contribution × (1 + g)y
Interest earned on interest, not just on the original amount. Each period’s gains join the balance and start earning their own return the next period, which is what makes the curve bend upward instead of climbing in a straight line.
A = P(1 + r/n)nt
Adding a fixed amount on a schedule, on top of the starting balance. Even modest recurring contributions usually outweigh the starting amount over a long enough horizon, simply because more of the money has had more time to compound. End-of-period is the ordinary annuity; beginning-of-period deposits earn that period’s interest too.
future value of a series
A mental-math shortcut: divide 72 by the annual rate to estimate how many years it takes to double your money. No calculator needed, and it stays close to the exact answer across the rates most people actually deal with (the table below shows where it stops).
Nominal is the rate your statement shows. Real is what that growth is actually worth after inflation quietly erodes purchasing power. A 7% nominal return at 3% inflation is closer to a 4% real return, which is the number that matters for what you can actually buy later.
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.
Fees come out of the balance every period, so the money taken stops growing too. On this page’s default plan a 1% annual charge takes roughly 11,000 in fees and leaves you about 19,000 worse off. The cost is always more than the fees paid, and the gap widens the longer you hold. Set the charges slider to what you actually pay, not to zero.
Tripling time is ln 3 / ln(1 + r), or about 114 divided by the annual rate in percent, the way 72 gives doubling time and 144 quadrupling. At 7% that gives 16.3 years against an exact 16.2.
The stated annual rate is split into twelve monthly steps (rate ÷ 12) and each month’s interest is added to the balance, so it earns interest too. At 7% compounded monthly the effective annual yield is about 7.23%, a little above 7% compounded once a year.
Switch on real mode and set the inflation rate, and the page deflates every balance by it. As a rough rule, real return ≈ nominal − inflation, so 7% with 3% inflation is about 4%. Exactly, it is (1.07 ÷ 1.03) − 1 = 3.88%.
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.
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.