The Rule of 70 and 69
The 'Rule of 72' is famous. But why 72 of all numbers? In fact, the mathematically more accurate number is 69.3, and depending on the situation, 70 fits better.
The rules that measure doubling time
The time it takes for principal to double is called the 'doubling time.' The way to estimate this mentally is the Rule of 72 (or 70, or 69).
The method is the same. Divide the given number by the return (%). At annual 6%, that's 72÷6 = 12 years, 70÷6 ≈ 11.7 years, 69÷6 ≈ 11.5 years. It differs slightly depending on which number you use.
Why 69.3 is the 'real' one
If you calculate the exact time to double under compounding, the natural logarithm appears. The value needed to reach 2 is ln(2) ≈ 0.693, i.e., 69.3%. So the theoretically most accurate number is 69.3. In particular, 69 (or 69.3) is exact for 'continuous compounding,' where interest is applied very frequently.
So why is 72 more famous? Because 72 divides cleanly by 2, 3, 4, 6, 8, 9, 12, and so on, making mental math easy. And under ordinary annual compounding, where interest isn't applied often, 72 is actually closer to reality.
For continuous or daily compounding, 69–70 fits better; for low-rate annual compounding, 72 fits better. All three are 'rough estimates,' so precise calculation must be done with logarithms.
How to use it in practice
The difference among the three rules is very small in most cases, so when doing mental math, just pick whichever is easy to divide by. In the annual 7–10% return range, 72 is the most reliable.
Conversely, when estimating 'the time for prices to double,' as with inflation, 70 is often used. For example, if prices rise 3.5% each year, that's 70÷3.5 = 20 years until prices double, meaning the purchasing power of money is halved. These rules let you feel how compounding both grows assets and, through inflation, shaves down the value of your money.
常见问题
Q. Which should I memorize: 72, 70, or 69?
If you memorize just one, 72 is the most practical. It's easy to divide by and fits well in the common return ranges. If you frequently deal with doubling times for low rates such as inflation or population growth, use 70; if you need continuous-compounding calculations, use 69.
Q. Is the calculation from this rule accurate?
It's a rough estimate. The lower the return (roughly single digits), the better it matches reality; above 20%, the error grows. The exact doubling time must be calculated as log(2) ÷ log(1+return).
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