The rule of 72: doubling time in your head

Divide 72 by the interest rate and you get the number of years it takes for money to double. At 6 % that is twelve years; at 9 %, eight. It is the one piece of compound interest arithmetic worth memorising, because it turns a percentage into a span of time — and time is what people can actually judge.

What the rule says

doubling time ≈ 72 / interest rate in percent

6 % → 72 / 6 = 12 years

8 % → 72 / 8 = 9 years

The rate goes in as a percentage, not a decimal

It assumes interest is credited once a year and nothing is added or taken out

It works in both directions and that is where most of its value lies. If a plan needs your money to double in ten years, the rule says you need roughly 7.2 % a year. Saying "I need 7.2 %" out loud is a much better test of a plan than saying "I want it to double".

How accurate it actually is

The rule is an approximation, and it is worth knowing which way it errs. The middle column is the exact doubling time; the last is the first full year in which a balance has genuinely doubled when interest is credited annually.

Between about 6 % and 10 % the rule is accurate to within a tenth of a year. Below that it overshoots — at 2 % it says 36 years where the answer is 35. Above it, it undershoots. For the rates that appear in real savings plans, the error is smaller than the uncertainty in the rate itself.

The last column carries a separate lesson. At 8 % the exact doubling time is 9.01 years, but with annual crediting the balance only crosses 2,000 € at the end of year ten — interest arrives in one lump, not continuously. The rule tells you when the maths doubles; the crediting schedule decides when you can spend it.

Where the 72 comes from

The exact doubling time is ln 2 / ln(1 + r). For small r, ln(1 + r) is close to r, which gives ln 2 / r — and ln 2 is 0.6931, so the numerator is really 69.31 rather than 72.

• 69.31 is the mathematically correct numerator, and it is most accurate at very low rates.
• 70 is easier to divide and is common in economics, particularly for growth and inflation rates.
• 72 is the least accurate at low rates and the most divisible: by 1, 2, 3, 4, 6, 8, 9 and 12. That is why it won.
• 72 also happens to compensate for the approximation error in the mid single digits, which is exactly where consumer interest rates live.

So the choice of 72 is not a mathematical result. It is a convenience that turns out to be well-calibrated for the range people actually use.

Using it backwards, and in multiples

Three variations cover most quick estimates.

1. Rate from time: rate ≈ 72 / years. Doubling in fifteen years needs about 4.8 %.
2. Quadrupling: two doublings. At 6 %, four times your money takes about twenty-four years.
3. Tripling: use 114 instead of 72 — the same derivation with ln 3. At 6 %, roughly nineteen years.

Chaining doublings is where the rule becomes genuinely useful for judging a horizon. Forty years at 6 % is a little over three doublings, so a deposit becomes roughly ten times itself. That is a claim you can sanity-check without a calculator, and it holds: 10,000 € at 6 % for forty years is 102,857 €.

Where it breaks down

• It only describes a lump sum. A monthly savings plan has no single doubling time, because every contribution starts its own clock.
• It assumes a constant rate. Applying it to a volatile portfolio using the average return will overstate the outcome, because averages and compounded returns are not the same thing.
• It ignores fees, taxes and inflation. Subtract those from the rate before dividing, or the answer is about a different rate than the one you will receive.
• It gets noticeably wrong above about 20 %. At 20 % the rule says 3.6 years against an exact 3.8; at 50 % it says 1.44 against 1.71, a 16 % underestimate.

The most useful application: inflation

Run the rule on an inflation rate and it tells you how long until prices double — or, equivalently, until your cash halves in purchasing power. At 3 % inflation, 72 / 3 gives twenty-four years. The exact figure is 23.4, so the estimate is slightly conservative and close enough.

This is the version worth carrying around. It converts an abstract percentage into a statement about your own money: at 3 %, a sum left in a current account for your child's twenty-fifth birthday will buy roughly half of what it buys today.

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