The Rule of 72: A Quick Way to Estimate Doubling Time

The Rule of 72 is a mental shortcut. Divide 72 by an annual interest rate, in percent, and the answer is roughly how many years it takes for money to double with compound interest. At 8% a year, 72 divided by 8 is 9 years. It is an estimate, not an exact figure, and it works best for ordinary rates.

How to use it

The method has one step: years to double ≈ 72 ÷ annual rate. Corporate Finance Institute gives an example in which an investor expects 12% a year, so 72 ÷ 12 = 6 years to double. At 6%, the answer is 12 years. At 9%, it is 8 years. The exact answer for 9% is about 8.04 years, according to Wikipedia, so the shortcut is very close.

The rule assumes the interest is compounded, meaning each year's interest earns interest of its own. Corporate Finance Institute notes it does not work well with simple interest, where the same amount is added every year.

Why 72?

The exact doubling time comes from solving (1 + r)^t = 2, which gives t = ln(2) ÷ ln(1 + r). For small rates, ln(1 + r) is close to r, so t is about 0.693 ÷ r. That is why the figure 69.3 is the mathematically natural number, and it is exact when interest is compounded continuously.

So why use 72? Wikipedia explains that 72 fits annual compounding at typical rates near 8% better, and that it has many small divisors: 1, 2, 3, 4, 6, 8, 9 and 12. This makes the mental arithmetic easy for many common rates. Corporate Finance Institute makes the same point, saying 72 is preferred over 69.3 because it is easier to divide.

How accurate is it?

Wikipedia compares the rule with the exact answer. Some of its figures, in years:

• At 2%: exact 35.0, Rule of 72 gives 36.0
• At 8%: exact 9.006, Rule of 72 gives 9.0
• At 10%: exact 7.273, Rule of 72 gives 7.2
• At 20%: exact 3.802, Rule of 72 gives 3.6
• At 50%: exact 1.710, Rule of 72 gives 1.44

The pattern is clear. Around 8% the rule is almost perfect. At very low rates, using 70 or 69.3 is closer, and at high rates the shortcut underestimates the time. Wikipedia suggests 69 or 70 for daily or continuous compounding.

Other things you can estimate

The same idea works whenever something grows at a steady percentage. Wikipedia gives two examples that go the other way. With inflation of 3.5% a year, 70 ÷ 3.5 gives 20 years for the buying power of money to halve. A fee of 3% a year would halve an account's value in about 72 ÷ 3 = 24 years, in the sense of the growth lost to it. You can also change the 2 in the formula to 3 to estimate tripling time.

An old rule

Wikipedia says the rule appears in Luca Pacioli's book Summa de arithmetica, printed in Venice in 1494. Pacioli states the rule without explaining where it comes from, which suggests it was known before him. His example is 6%: 72 divided by 6 is 12 years.

Limits to remember

• It assumes a fixed rate every year. Real investment returns vary, and no rate is guaranteed.
• It ignores taxes, fees and inflation unless you include them in the rate.
• It is a way to build a feel for numbers, not a forecast.

This is general information about arithmetic, not financial advice. For any real decision, use an exact calculator or the terms given by the provider.

Sources

• Rule of 72, Wikipedia
• Rule of 72, Corporate Finance Institute