The Rule of 72 is wrong. Here's why that's fine, and the exact rule when it isn't.

If you've taken any finance class, or read any investing book aimed at beginners, you've met the Rule of 72: at an annual return of r percent, your money doubles in approximately 72 / r years. At 6%, that's 12 years. At 8%, 9 years. At 12%, 6 years.

What you may not have been told is that the Rule of 72 is exactly correct at one specific rate, slightly wrong at every other rate, and increasingly wrong as the rate gets farther from that one specific point. The 72 is chosen for mental-arithmetic convenience (it has lots of divisors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), not for mathematical purity.

This piece walks through the actual derivation, identifies the one rate where 72 is exactly right, quantifies the error at other rates, and gives you the corrected rule for when you need a sharper answer.

The exact doubling-time formula

Starting from the compound interest formula, FV = P × (1 + r)^t, set FV = 2P:

2P = P × (1 + r)^t

Divide both sides by P, take the natural log of both sides, solve for t:

2 = (1 + r)^t

ln 2 = t × ln(1 + r)

t = ln 2 / ln(1 + r)

That's the exact doubling time for any compounding rate r. At r = 0.06 (6%), ln(2) / ln(1.06) = 0.6931 / 0.0583 = 11.896 years. The Rule of 72 says 12 years. The error is one month — close enough for mental math.

Where the 72 actually comes from

Here's the trick. For small r, ln(1 + r) ≈ r − r²/2 + r³/3 − ... — the Taylor series. The dominant term is just r. So:

t = ln 2 / ln(1 + r) ≈ ln 2 / r ≈ 0.6931 / r

Multiply both sides by 100 to express in percent: t ≈ 69.31 / (r × 100).

That means the mathematically pure mental rule, for continuously compounded interest, is the Rule of 69.3. It's the right answer in the limit of small r, where ln(1 + r) ≈ r.

But annual compounding isn't continuous, and for typical rates (4–12%), the ln(1+r) term diverges from r enough that 69.3 understates the doubling time. The 72 is a slightly inflated constant that happens to give a better fit across the practical range.

The exact value of "exactly right"

The Rule of 72 is exactly right when t × r = 72 satisfies the true formula. Solving numerically:

72 / r = ln 2 / ln(1 + r/100)

Which is true at r ≈ 7.85%. At that one specific rate, 72 / 7.85 = 9.17 years, and the exact formula also gives 9.17 years. Above and below 7.85%, the approximation drifts:

• r = 1%: Rule of 72 says 72 years. Exact: 69.66 years. Error: +2.34 years (3.4% overestimate).
• r = 3%: Rule of 72 says 24 years. Exact: 23.45 years. Error: +0.55 years.
• r = 5%: Rule of 72 says 14.4 years. Exact: 14.21 years. Error: +0.19 years.
• r = 7.85%: Rule of 72 says 9.17 years. Exact: 9.17 years. Error: ~0.
• r = 10%: Rule of 72 says 7.2 years. Exact: 7.27 years. Error: −0.07 years.
• r = 15%: Rule of 72 says 4.8 years. Exact: 4.96 years. Error: −0.16 years.
• r = 20%: Rule of 72 says 3.6 years. Exact: 3.80 years. Error: −0.20 years.
• r = 30%: Rule of 72 says 2.4 years. Exact: 2.64 years. Error: −0.24 years (a 9% underestimate).

For rates between 4% and 12% — which is most of the realistic personal-finance range — the Rule of 72 is accurate to within a few months. For very low rates (1–2%, where this matters anyway because anything growing that slowly may as well be flat), the rule overestimates the doubling time. For very high rates (above 20%), it underestimates.

The corrections that matter

If you need sharper accuracy and you're doing mental math, two refinements help:

The Rule of 70. Slightly more accurate than 72 for typical rates, slightly less convenient because 70 has fewer mental-friendly divisors. Used in some economics textbooks (especially for inflation, where the rate is usually low).

The Eckart rule. Add (rate − 8) / 3 to 72. So at 14%, use 72 + 2 = 74 / 14 = 5.29 years instead of 72/14 = 5.14. At 2%, use 72 − 2 = 70 / 2 = 35 years instead of 72/2 = 36. This is the simplest correction that closes the gap at the extremes.

For nontrivial calculations, just use the exact formula. ln 2 / ln(1+r) is two button presses on any calculator, and the result is exact.

Why the Rule of 72 survives anyway

You'd think a rule that's wrong at every rate except 7.85% would have been replaced by something better by now. It hasn't been, and for good reason: at the rates people actually care about, the error is in the noise.

If you're estimating "my retirement portfolio at 7% real return will double in roughly 10 years" (Rule of 72: 10.3 years, exact: 10.24 years), the seven-day error vs the exact formula is unmeasurable against:

• The uncertainty in your future return (could be 4%, could be 9%, you don't know).
• The uncertainty in the sequence of returns (averaging 7% with high variance is very different from averaging 7% with low variance).
• The uncertainty in your savings rate, contribution timing, taxes, fees, and behavior.

The Rule of 72 is approximately correct in a context where everything else is wildly uncertain. It would be silly to use a more precise approximation when the inputs themselves are this noisy.

The honest mental model

The Rule of 72 is a useful back-of-envelope tool because doubling time is a back-of-envelope thing. If you need to know whether $50,000 invested at 7% becomes roughly $200,000 by age 60 (two doublings in 20 years), the rule answers that question instantly. If you need to know whether your $50,000 becomes exactly $193,484 by some specific date, you're already at the calculator.

The right framing isn't "the Rule of 72 is wrong." It's "the Rule of 72 is a deliberately rounded version of ln 2 / ln(1+r), accurate enough for the rates that matter, off by enough at extreme rates that you should know when to switch to the exact formula." Like every useful approximation, its value comes from knowing where it stops working.

And if you want to see the exact answer for any rate, the compound interest calculator on this site does the full computation with one input. It's not the Rule of 72; it's the formula the Rule of 72 is approximating.