a shortcut on trial

The Wrong Constant That Keeps Being Right

Everyone who digs one layer down learns the smart-sounding correction: doubling time is really ln 2 ≈ 69.3 divided by the rate, and 72 was just picked because it divides nicely. But the correction is the misconception: 69.3 is exact only if interest compounds continuously, and in the annual world where real accounts live, doubling time runs ≈ 69.3/R + 0.35, a line that 72/R rides almost perfectly. Drag a rate slider and watch exact, 72/R, and 69.3/R recompute with signed errors: at 8%, 72 is off by 0.07% while the 'true' 69.3 misses by 3.8%. Flip the compounding toggle to continuous and the winner flips with it.

The exact annual-compounding curve is t = ln(2) / ln(1 + R/100). Expand its denominator near zero and the first two terms are 69.3147/R + 0.3466. That small added shelf is why a slightly larger numerator can win. The neat integer is not the derivation, but in the practical 4% to 12% band it shadows the corrected curve remarkably well.

The regime is the trick

Flip the bench to continuous compounding. Now the exact curve is t = ln(2)/(R/100), so 69.3/R is nearly exact at every rate and 72/R is about 3.87% high. The familiar shortcut is not a universal law. It is a well-placed approximation for periodic compounding.

At 18% annual compounding, a balance with no payments or new charges doubles in 4.1878 years; 72/18 says 4.0000, understating the wait by 4.49%. This is arithmetic, not a forecast or financial advice. At a constant 3% annual loss of purchasing power, the exact halving time is 22.7566 years, not 23.4. Decline follows ln(0.5)/ln(1 − r), so it must not borrow the growth error table.