Reference

The Rule of 72, and where it breaks.

A pocket trick to estimate how long it takes money to double. Useful, occasionally wrong, and worth understanding deeply before you stake any decision on it.

The rule

To approximate the years it takes for a balance to double at a given annual rate of return, divide 72 by the rate (expressed as a whole number).

years to double ≈ 72 / rate

At 6 %: 72/6 = 12 years. At 8 %: 72/8 = 9 years. At 12 %: 72/12 = 6 years. The rule fits in your head and is good enough for back-of-envelope arithmetic.

Where 72 comes from

The exact doubling time is t = ln(2) / ln(1 + r). Since ln(2) ≈ 0.6931 and ln(1 + r) ≈ r for small r, we get t ≈ 0.6931 / r. Multiply by 100 to express r as a whole-number percentage and the constant becomes 69.31.

69.31 is annoying to divide by. 72 has many divisors (2, 3, 4, 6, 8, 9, 12) and is close enough — the constant 72 also nudges the approximation upward in a way that compensates for the curvature of the logarithm at moderate rates. It is not a mathematically derived constant; it is a memory aid that happens to be most accurate near 8 %.

Accuracy by rate

How the Rule of 72 compares to the true doubling time at various rates:

Annual rate	Rule of 72	Exact doubling time	Error
1%	72.0 yrs	69.66 yrs	+2.3 yrs (overstates)
3%	24.0 yrs	23.45 yrs	+0.55 yrs
5%	14.4 yrs	14.21 yrs	+0.19 yrs
7%	10.29 yrs	10.24 yrs	+0.05 yrs
8%	9.0 yrs	9.01 yrs	−0.01 yrs (best fit)
10%	7.2 yrs	7.27 yrs	−0.07 yrs
15%	4.8 yrs	4.96 yrs	−0.16 yrs
20%	3.6 yrs	3.80 yrs	−0.20 yrs (understates)
30%	2.4 yrs	2.64 yrs	−0.24 yrs

The rule is most accurate around 8 %. It overstates the true doubling time at lower rates (the answer is too pessimistic) and understates it at higher rates (the answer is too optimistic). For rates above roughly 20 % the rule should be replaced.

Better approximations

Rule of 70. Use 70 instead of 72 when rates are below 5 %. Better fit at low rates, slightly worse around 8 %.
Rule of 69.3. Theoretically optimal for continuous compounding. Awkward to divide.
The exact formula. Carry a calculator and use ln(2) / ln(1 + r) directly. The main calculator reports the exact doubling time at your chosen compounding frequency.

The hidden caveat

The Rule of 72 assumes a single, constant rate of return. Real-world investments do not deliver constant returns. A portfolio that returns 10 % one year and −5 % the next does not double in 7.2 years — the geometric mean return matters, not the arithmetic average. When applying the rule to volatile assets, use the compound annual growth rate (CAGR), not the simple average return.

The same logic in reverse: the Rule of 114 (tripling) and 144 (quadrupling)

By the same derivation, dividing 114 by the rate gives the years to triple (ln(3) × 100 ≈ 109.86, rounded up to 114 for divisibility), and dividing 144 gives the years to quadruple (ln(4) × 100 ≈ 138.63, rounded for memorability). They are useful for the same kinds of back-of-envelope estimates the original rule supports.