Inverse Rule of 72 (yield to double in N periods)

The inverse Rule of 72: finding the rate from doubling time

The Rule of 72 is the mental shortcut people use to guess how long money takes to double at a fixed compound rate. You divide: years ≈ 72 / rate. The inverse version flips the question. Your capital took N periods to double, so what rate was it earning? Just rearrange the same formula into rate ≈ 72 / N and you have it. Why 72 and not some other number? Because it divides cleanly by a lot of things (1, 2, 3, 4, 6, 8, 9, 12...), which keeps the math doable in your head, and it happens to follow real compounding fairly closely across the everyday 4%-12% range.

Two close relatives exist. The Rule of 70 uses 70, which also divides easily and runs a touch more accurate at low rates. The Rule of 69.3 is the mathematically honest version, since doubling under continuous compounding takes exactly ln(2) ≈ 0.6931, which becomes 69.31 once you scale it to percent. Reach for 69.3 or 70 when compounding happens daily or continuously. Stick with 72 for annual compounding and quick back-of-the-envelope estimates.

How the calculation works

Each approximation divides a constant by N, nothing more. For the exact per-period rate under discrete compounding, you solve (1 + r)^N = 2, which works out to r = (2^(1/N) − 1) × 100. The rule-of-X shortcuts drift away from this exact value as the rate climbs. They're at their best around 8% and start overstating the rate once returns get high.

Worked example

Say capital doubles in N = 8 periods. Rule of 72 gives 72 / 8 = 9.00%, Rule of 70 gives 70 / 8 = 8.75%, and Rule of 69.3 gives 69.3 / 8 = 8.66%. The exact answer is (2^(1/8) − 1) × 100 ≈ 9.05%, so 72 comes out closest. With N = 10 the exact rate is 7.18%, and Rule of 72 (7.20%) again edges out 70 (7.00%) and 69.3 (6.93%) under discrete annual compounding.

Frequently asked questions

When should I use 69.3 or 70 instead of 72? Switch to them when interest compounds continuously or very often (daily). In that case 69.3 is the precise constant, and 70 is a close stand-in that is easier to divide.

How accurate are these rules? In the 4%-12% band they land within a few tenths of a percent. Push past that into very high rates and the accuracy falls off, so lean on the exact 2^(1/N) − 1 formula instead.