The formulas
- Exact = ln 2 / ln(1 + r)
- Rule of 72 = 72 / rate in percent
- Rule of 69.3 = 69.3 / rate (exact for continuous compounding)
How it works
Doubling time solves (1 + r)^n = 2, so n = ln 2 / ln(1 + r). The rule of 72 is a mental shortcut. It is most accurate near 8 percent, where 72 / 8 = 9 years against an exact 9.006 years.
Below about 4 percent the rule of 70 is closer. For continuous compounding, 69.3 is exact, since ln 2 = 0.693. Above 15 percent all three rules drift, and the error table shows by how much.
Illustrative figures only. Check the result against your firm's methodology before relying on it.
Common questions
Why 72 and not 69.3?
72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, and annual compounding pushes the right number above 69.3.
Does it work for inflation?
Yes. At 3% inflation prices double in about 24 years, and a dollar loses half its buying power in that time.
What about tripling?
Use ln 3 / ln(1 + r), or the rule of 114 as a shortcut.
Formulas on this page are the standard definitions.
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