Doubling time is the number of years money takes to grow to twice its starting value. At an 8% yearly return, it is 9.01 years. The rule of 72 says to divide 72 by 8, which gives 9 years. At 30%, the exact answer is 2.64 years. The rule gives 2.4.
This post checks the shortcut against the exact calculation at every whole-number rate from 1% to 30%. Every figure is arithmetic on an assumed rate that stays constant, with interest added once a year and no deposits, tax or fees. None of the rates is a forecast.
What is the rule of 72? The rule of 72 estimates the doubling time, the years money takes to grow to twice its starting value. Divide 72 by the yearly rate of return in percent. At 8% a year it gives 9 years, against an exact 9.01 years.
The rule's answer is within 1% of the exact doubling time at every whole-number yearly rate from 6% to 10%.
At a 1% yearly rate, the rule's answer is 3.36% longer than the exact time. At a 30% yearly rate, it is 9.16% shorter.
In our own calculation, a constant that moves by one third per 1% of rate above or below 8% stays within 0.10% of the exact time from 1% to 30%.
What is the rule of 72?
The rule of 72 estimates the doubling time from a yearly return. Divide 72 by the rate written as a whole number. At 6% the answer is 12 years. At 8% it is 9 years. At 12% it is 6 years.
The exact figure needs logarithms.
Money grows by a factor of 1 + r each year, where r is the rate as a decimal. The doubling time n solves (1 + r) to the power n = 2, so n = ln 2 ÷ ln(1 + r). Here ln is the natural logarithm key on a calculator.
With interest added once a year, the balance first reaches double at the end of year 10 at 8%. The 9.01 years is the point inside year 10 where growth, treated as smooth, reaches double.
What makes 72 convenient?
Among the numbers 1 to 12, 72 divides evenly by eight of them: 1, 2, 3, 4, 6, 8, 9 and 12. The number 70 divides evenly by five of them. The number 69 divides evenly by two. Mental arithmetic is easy at many common rates.
The number 72 also sits near the exact answer in the middle of the range. Multiply the exact doubling time by the rate and you get the constant the rule should use. We call it the exact constant. It is 69.66 at 1%, 72.05 at 8% and 79.26 at 30%.
The rule uses 72 at every rate. The exact constant rises by about one third for each extra 1% of rate.
How accurate is the rule of 72 at each rate?
We tested all 30 whole-number rates from 1% to 30%. At each, we divided 72 by the rate and compared the result with the exact doubling time. Chart 1 shows six of them.
Source: YX Insights
At 8% the two bars are almost level, at 9.01 and 9.00 years. At 20% the rule gives 3.60 years against an exact 3.80.
The table lists fifteen of the 30 rates. For each it shows the exact doubling time, the rule's answer and the exact constant. The exact constant rises as the rate rises.
Yearly rate | Exact doubling time (years) | Rule of 72 (years) | Gap | Exact constant |
|---|---|---|---|---|
1% | 69.66 | 72.00 | 3.36% too long | 69.66 |
2% | 35.00 | 36.00 | 2.85% too long | 70.01 |
3% | 23.45 | 24.00 | 2.35% too long | 70.35 |
4% | 17.67 | 18.00 | 1.85% too long | 70.69 |
5% | 14.21 | 14.40 | 1.36% too long | 71.03 |
6% | 11.90 | 12.00 | 0.88% too long | 71.37 |
7% | 10.24 | 10.29 | 0.40% too long | 71.71 |
8% | 9.01 | 9.00 | 0.07% too short | 72.05 |
9% | 8.04 | 8.00 | 0.54% too short | 72.39 |
10% | 7.27 | 7.20 | 1.00% too short | 72.73 |
12% | 6.12 | 6.00 | 1.90% too short | 73.40 |
15% | 4.96 | 4.80 | 3.22% too short | 74.39 |
20% | 3.80 | 3.60 | 5.31% too short | 76.04 |
25% | 3.11 | 2.88 | 7.28% too short | 77.66 |
30% | 2.64 | 2.40 | 9.16% too short | 79.26 |
Interest added once a year at a constant rate, no deposits, tax or fees. Gap is the rule of 72 time against the exact time, as a share of the exact time. Source: YX Insights
The gap is how far the rule's answer is from the exact time, as a share of the exact time. Chart 2 shows the gap at each of those fifteen rates. The rule runs too long from 1% to 7% and too short from 8% to 30%.
Source: YX Insights
The gap is smallest at 8%, where the rule is 0.07% too short. It is largest on the long side at 1%, where the rule's 72 years is 3.36% above the exact 69.66. It is largest on the short side at 30%, where the rule's 2.4 years is 9.16% below the exact 2.64.
The rule's answer is within 1% of the exact time at rates from 6% to 10%. It is within 3% at rates from 2% to 14%. It is within 5% at rates from 1% to 19%.
What do the errors mean in dollars?
Chart 3 grows $10,000 for the number of years the rule gives at each rate. A perfect rule would show $20,000 every time. Fractions of a year use the same yearly growth factor, applied smoothly.
Source: YX Insights
At 20%, 3.6 years gives $19,278, which is $722 short of $20,000. At 1%, 72 years gives $20,471, which is $471 over.
What is the rule of 72 used for?
The rule also runs in reverse. Divide 72 by a number of years to find the rate that doubles money in that time.
For 10 years it gives 7.2%, against an exact 7.18%. For 3 years it gives 24%, against 25.99%.
Suppose prices rise 3% a year, an assumed rate. The rule gives 24 years for prices to double, against an exact 23.45. After 24 years, $10,000 buys what $4,919 buys today. Our explainer on how inflation erodes savings uses measured data.
A fee lowers the rate you keep. Suppose a fee takes 1 percentage point a year from a 7% return, so the rate falls from 7% to 6%. Doubling then takes 11.90 years instead of 10.24. That is about 1.7 years longer.
Is there a better number than 72?
We tested three constants, 69, 70 and 72, against the exact time over four ranges of whole-number rates.
The mean gap is the average size of the gap, ignoring direction. The number 72 has the lowest mean gap of the three in all four ranges: 1% to 10%, 5% to 12%, 1% to 20% and 1% to 30%.
Rate by rate, 70 is the closest of the three from 1% to 4%. The number 72 is the closest from 5% to 30%. The number 69 is never the closest.
The best single constant is lower for low rates and higher for high rates. That drift is what a sliding constant removes.
Is there a more accurate version of the rule of 72?
Yes. Add one third to 72 for each 1% of rate above 8%. Subtract one third for each 1% below 8%. This is our own calculation, not a published rule. We tested it at all 30 rates, 1% to 30%.
At 20% the rate is 12 above 8, so the constant is 72 + 12 ÷ 3 = 76. That gives 3.80 years against an exact 3.80. Across all 30 rates, the largest gap is 0.10%, at 30%.
At rates below 5%, 70 is the closer of the plain constants. At rates above 14%, the plain rule runs more than 3% short, while the sliding constant stays within 0.10%.
What does the rule of 72 leave out?
The rule needs one constant rate. Real returns change from year to year. Our explainer on the compound rate against the average return covers the gap between the two.
The rule has no input for tax, fees or new deposits. The fee example takes fees off the rate before applying the rule. Our post on investing $100 a month for 10 years adds regular deposits to the same arithmetic.
Our explainer on compound interest works through a full 30-year example.
The rule of 72 estimates doubling time in one line. It stays within 1% of the exact figure from 6% to 10%, then drifts to 9.16% too short by 30%. Our adjusted version stays within 0.10% across 1% to 30%.
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Common questions about the rule of 72
How do you use the rule of 72?
Divide 72 by the yearly rate of return written as a whole number. The answer is the years money takes to double. At 9%, 72 divided by 9 is 8 years, against an exact 8.04 years. The rule assumes the same rate every year, with interest added once a year.
Is the rule of 72 accurate?
Yes, from 6% to 10% a year, where it lands within 1% of the exact doubling time. The exact time comes from the natural logarithm formula. Below 8% the rule runs too long, by 3.36% at 1%. Above 8% it runs too short, by 9.16% at 30%.
Is the rule of 70 better than the rule of 72?
No. The rule of 70 is closer only from 1% to 4%, while the rule of 72 is closer from 5% to 30%. The rule of 70 divides 70 by the rate. At 2% it gives 35 years, against an exact 35.00 years. At 8% it gives 8.75 years, against 9.01.
What rate doubles money in 10 years?
A rate of 7.18% a year, with interest added once a year. The rule of 72 gives 7.2%, which is 72 divided by 10. The exact rate is the tenth root of 2, minus 1. The rule's answer is off by 0.02 percentage points.
DISCLAIMER: This article is intended for educational purposes only. Any information or analysis in this note does not constitute an offer to sell or a solicitation of an offer to buy any securities. Nothing in this note is intended to be investment advice, nor should it be relied upon to make investment decisions. Any opinions, analyses, or probabilities expressed in this note are those of the author as of the note's date of publication and are subject to change without notice.
