Rule of 72 (Doubling Time)
Rule of 72 (Doubling Time)
A mental-arithmetic estimate of how many years it takes money to double at a given annual rate: 72 divided by the rate in percent. Accurate to within a few months for rates between roughly 4% and 12%.
When to use: Use for quick comparisons without a calculator. The secondary output gives the exact doubling time from the compound-interest formula for comparison.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| DoublingYears | Years to Double | Estimated years for the value to double | years |
| k | Interest Rate | Nominal annual interest rate as a decimal | % |
Real-Life Examples
Example 1: Eight Percent
How long does money take to double at 8% a year?
Given
Step-by-Step
The rule says 9 years; the exact answer is 9.006 years. At 8% the rule is almost perfect.
Example 2: Six Percent
Doubling time at 6%.
Given
Step-by-Step
The rule says 12 years against an exact 11.90. Every 1% of extra return shaves years off the doubling time, which is the whole point of the rule.
Frequently Asked Questions
The exact doubling time is ln 2 / ln(1 + k), which is close to 69.3 / (100k) for small rates. 72 is used instead because it is divisible by many small numbers and its slight overshoot happens to offset the approximation error near 8%.
At very low rates it overestimates (use 69 or 70) and at high rates it underestimates. Beyond about 20% the error grows quickly; the exact formula is the secondary output here.