The Rule of 72: Where the Shortcut Works and Where It Quietly Breaks

Divide 72 by the annual return and you get the number of years it takes money to double. At 8%, that is 9 years. At 6%, 12 years. It is quick enough to do in your head in a meeting, and it is close enough that people have been using it for roughly five hundred years. The earliest written mention usually cited is in Luca Pacioli's 1494 arithmetic textbook, which states the rule without explaining where it comes from.

It is also wrong, slightly, almost every time you use it. Whether the error matters depends on the rate you plug in, and that is what this post is about.

Where the 72 comes from

With annual compounding, money doubles when (1 + r)^t = 2. Solve for t and you get:

t = ln(2) / ln(1 + r)

For small r, ln(1 + r) is almost exactly r, so t is about ln(2) / r, and ln(2) is 0.693. That gives you the "Rule of 69.3", which is exact for continuous compounding. Nobody uses 69.3 because it divides badly. 72 is a little too big, but it splits evenly by 2, 3, 4, 6, 8, 9 and 12, which covers most of the rates people actually think about. The rule survived because it is easy, not because it is the most accurate constant.

How wrong is it? A table

Here are the exact doubling times (computed from the formula above) next to what the Rule of 72 says:

• 2% a year: exact 35.0 years, rule says 36.0 (too long by 3%)
• 4%: exact 17.7, rule says 18.0 (too long by 2%)
• 6%: exact 11.9, rule says 12.0 (too long by 1%)
• 8%: exact 9.01, rule says 9.0 (essentially perfect)
• 10%: exact 7.27, rule says 7.2 (too short by 1%)
• 12%: exact 6.12, rule says 6.0 (too short by 2%)
• 20%: exact 3.80, rule says 3.6 (too short by 5%)
• 40%: exact 2.06, rule says 1.8 (too short by 13%)

So the rule is sharpest around 8% and drifts either side. Between about 4% and 12% you will never be off by more than a couple of percent, which is fine for any back-of-envelope question. Outside that band, treat the answer as a rough guess. At the rates credit cards charge, the rule underestimates doubling time noticeably, and at savings-account rates it overestimates it.

A better constant if you want one

If you want a mental-math version that holds up at any rate, nudge the numerator as the rate rises. A common refinement is 69.3 plus one third of the rate in percent. At 8% that is 69.3 + 2.67 = 71.97, so you divide 71.97 by 8. At 20% it is 69.3 + 6.67 = 75.97, and 75.97 / 20 = 3.80, which matches the exact figure. It takes one extra step and fixes the high-rate problem.

The same idea works for tripling. The numerator for tripling is about 114 instead of 72, so at 7% money triples in roughly 114 / 7 = 16.3 years. The exact answer is 16.24.

Three places the rule is genuinely useful

• Inflation. At 3% a year, prices double in about 24 years (exact: 23.4). Run it backward and your cash loses half its purchasing power in the same time. That one fact does more to explain why savings need to earn something than any lecture on the topic.
• Fees. Compare 7% growth with 6% growth on 10,000 over 30 years. The first ends near 76,100 and the second near 57,400. A single percentage point of annual cost took about a quarter of the final balance. Doubling times show it too: 10.2 years at 7%, 11.9 at 6%.
• Sanity-checking a pitch. If someone promises 24% a year, the rule says doubling in 3 years. Over 30 years the rule counts ten doublings, and the exact figure (1.24 to the 30th power) is a gain of about 635 times. When the arithmetic produces something absurd, the promise probably is.

Where it breaks down in practice

The formula is only half the story, because the assumptions behind it rarely hold in real life.

• It assumes one lump sum. If you add money every month, the balance grows faster than the rule predicts, and the doubling idea stops being meaningful because you are not doubling the same money.
• It assumes a constant rate. Stock returns do not arrive as a smooth 8%. The average over decades can be near 8% while individual years swing from large gains to large losses, and the order of those years matters for anyone withdrawing money.
• It ignores tax. If you pay tax on interest or dividends each year, your effective rate is lower than the headline rate, sometimes by a lot.
• It does not work for loans in the way people hope. A loan with monthly payments is not doubling, because you are shrinking the principal as you go. Applying 72 to a mortgage rate tells you nothing useful.
• People mix up units. The rule uses the rate as a whole number of percent, so 8% means 8, not 0.08. Dividing 72 by 0.08 gives 900 years.

When to stop estimating

Use the rule to decide whether a question deserves a closer look. If an investment doubles in 9 years at your assumed rate and you need the money in 5, you have your answer without opening anything. If the decision is closer, say whether to keep 20,000 in an account paying 4.1% or move it to one paying 4.5%, put the real numbers into a compound interest calculator and set the compounding frequency to match the account. The difference in doubling time between those two rates is only a few months, but over a decade the dollar gap is worth seeing precisely.

For longer plans with regular contributions, the investment calculator handles monthly deposits, which the rule cannot. Keep the 72 for what it does well: giving you an instant feel for how violently small differences in rate compound over time.