Rule of 72

$$t_{double} \approx \frac{72}{r\%}$$

Variables

Symbol	Name	Unit	Description
$t_double$	Time to double	years	Approximate years for an investment to double in value.
$r%$	Annual interest rate	percent	Annual return rate expressed as a percentage (not decimal).

The Rule of 72

The Rule of 72 is a mental-math shortcut to estimate how long it takes for an investment to double at a given annual compound interest rate:

$$t_{double} \approx \frac{72}{r\%}$$

For example, at 6% annual return: 72 ÷ 6 = 12 years to double.

Why 72?

The mathematically precise formula uses the natural logarithm:

$$t = \frac{\ln(2)}{\ln(1+r)} \approx \frac{0.6931}{r}$$

Multiplying by 100 (to use percentage rates): t ≈ 69.3 / r%.

The number 69.3 is mathematically exact, but 72 is preferred because: 1. It divides evenly by many common interest rates (1, 2, 3, 4, 6, 8, 9, 12) 2. It slightly overestimates, providing a conservative buffer 3. It's easier to compute mentally

Accuracy at Common Rates

Rate	Rule of 72	Exact	Error
2%	36.0 years	35.0 years	+2.9%
4%	18.0 years	17.7 years	+1.7%
6%	12.0 years	11.9 years	+0.8%
8%	9.0 years	9.0 years	−0.1%
10%	7.2 years	7.3 years	−1.4%
15%	4.8 years	4.96 years	−3.2%

The rule is most accurate between 6–10%, the range of typical equity returns.

Reverse Application

The rule works in reverse too — to find the rate needed to double in a target number of years:

$$r\% \approx \frac{72}{t_{double}}$$

To double in 9 years: rate needed ≈ 72 ÷ 9 = 8% per year.

Rule of 72 for Inflation and Debt

The rule also estimates how long inflation takes to halve purchasing power, or how quickly high-interest debt doubles if unpaid: - 3% inflation → 72 ÷ 3 = 24 years to halve purchasing power - 18% credit card APR → 72 ÷ 18 = 4 years for unpaid balance to double

Derivation & History

The Rule of 72 is attributed to Luca Pacioli, who mentioned dividing 72 by the interest rate in Summa de arithmetica (1494) without formal proof. Albert Einstein is often (apocryphally) credited with the rule; the actual attribution to Pacioli is more historically reliable.

The mathematical basis is the Taylor series expansion of ln(1+r) ≈ r for small r, giving t = ln(2)/r ≈ 0.693/r. Converting to percentage rates: t ≈ 69.3/r%. Empirically, 72 works better over the practical range of interest rates (6–10%) than the more accurate 69.3, because the linearisation error and the rounding difference partially cancel.

Worked Examples

Savings account at 4%

t_double ≈ 72 ÷ 4 = 18 years
Exact: ln(2)/ln(1.04) = 0.6931/0.03922 = 17.67 years
Rule of 72 is off by only 0.33 years

Result: Approximately 18 years to double

Stock market (historical avg 7%)

t_double ≈ 72 ÷ 7 ≈ 10.3 years

Result: Approximately 10 years to double at 7%

Edge Cases & Limitations

Very high rates (>20%): Accuracy deteriorates significantly. At 25%, Rule of 72 gives 2.88 years vs actual 3.11 years.

Very low rates (<2%): The rule works but the large doubling time means the approximation error is also large in absolute years.

Continuous compounding: The exact formula uses ln(2)/r without needing an approximation; Rule of 72 is only useful for discrete annual compounding.

Real-World Applications

Financial advisors use the Rule of 72 to explain the long-term power of investing without calculators in client meetings. Teachers use it to make compound interest intuitive in personal finance courses. Investors quickly compare asset classes: at 4% bonds double in 18 years; at 10% equities in 7.2 years. Policy makers use it to communicate inflation's effect on purchasing power. The rule appears in the CFA curriculum as a standard mental-math tool.

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