Rule of 72 Calculator

Doubling times from 2% to 12%

Annual return	Rule of 72	Exact	Error

The shortcut runs long at low rates and short at high rates, and crosses over near 8%.

Where the number 72 comes from

Doubling is a question about exponents. You want the number of years t that satisfies (1 + r)^t = 2, and taking logs of both sides gives the exact answer:

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

Under continuous compounding that collapses to 69.3 / r, because the natural log of 2 is 0.693. Annual compounding needs a slightly larger constant, and 72 is the number people settled on because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12. You can do it in your head on a train. The cost of that convenience is a small error that grows as you move away from the middle of the range.

The default numbers, worked through

At an 8% annual return on $10,000:

• Rule of 72: 72 / 8 = 9.00 years.
• Exact: ln(2) / ln(1.08) = 9.0065 years, which is nine years and about two days.
• The shortcut is off by 0.006 years. At 8% it is as good as it gets.
• Tripling: 114 / 8 = 14.25 years against an exact 14.27.
• Quadrupling: 144 / 8 = 18.00 years against an exact 18.01, which is simply two doublings.

So $10,000 becomes $20,000 in year nine, $40,000 in year eighteen and $80,000 in year twenty-seven. The third double adds $40,000 and took exactly as long as the first double, which added $10,000. Nothing about the rate changed. That is what compounding looks like from the inside, and it is why the last decade of a long horizon does more work than the first two combined.

Using it on the way down

The same arithmetic applies to anything that compounds against you. At 3% inflation, 72 divided by 3 says prices double in 24 years, so a dollar buys half as much. Run the same number against a fund fee and the effect is slower and just as certain, which the expense ratios guide puts in dollars.

This is why the rate you feed the calculator matters. A 7% nominal return with 3% inflation is a real return near 3.9%, which doubles buying power in about 18 years rather than the 10 the nominal figure suggests. Work the conversion with the inflation-adjusted return calculator and read inflation and CPI explained for where the rate itself comes from.

Quick uses worth remembering

Running the rule backwards turns a claim into a rate. A fund that says it doubled in six years is claiming about 12% a year, which you can then compare against the index it tracks. A house that doubled in 20 years grew at about 3.6% a year, which is closer to inflation than most people expect.

For anything more precise, the stock return and CAGR calculator gives the exact compound rate from a start value, an end value and a holding period. To see what regular contributions add to a doubling balance, use the dollar-cost averaging calculator, and for the arithmetic of getting back to even after a loss, the drawdown recovery calculator shows how long recovery takes at the same rate. What return you can reasonably assume in the first place depends on what you own, which starts with asset allocation.