The Rule of 72, and When It Lies to You

Divide 72 by a growth rate and you get the years it takes to double your money. The trick is accurate to a few weeks at 7% and off by months at 30%, and the error has a shape you can learn.

By Hutu Laurentiu · Published 1 November 2026 · 9 min read

Where 72 Comes From

Doubling means that (1 + r) raised to the power n equals 2. Take the natural log of both sides and you get n = ln 2 / ln(1 + r). The numerator is 0.693147. For small r, ln(1 + r) is very close to r itself, so n is roughly 0.693/r. Written in percentage terms, that is 69.3 divided by the rate.

So why 72 and not 69.3? Two reasons. The first is that ln(1 + r) is always slightly smaller than r, which makes the true n slightly larger than 69.3/r; nudging the constant up to 72 corrects for that in the range where most people do mental math. The second is that 72 divides evenly by 1, 2, 3, 4, 6, 8, 9, and 12, which matters when the entire point is doing the arithmetic in your head. A rule you will actually use beats a rule that is a few percent more accurate and requires long division.

The error is not random. It is a systematic curvature: 72/r runs long at low rates and short at high rates, crossing over near 8%. That shape is worth knowing, because it tells you which direction to correct when the rate in front of you is unusual.

Worked Examples at 4, 7, and 10 Percent

Start with $10,000 at 4%. The rule says 72 divided by 4, which is 18 years. The exact answer is ln 2 / ln 1.04, which is 17.67 years. Compounded at 4% for the full 18, $10,000 becomes $20,258, so the rule sends you about four months past the actual doubling point and the extra $258 of growth is the entire dollar error. For a household deciding whether a savings account keeps up with inflation, that is close enough to be harmless.

At 7% the rule gives 10.29 years against an exact 10.24, a gap of under three weeks. $10,000 at 7% for 10.29 years becomes $20,055. This is the band where the rule is at its best, which is convenient, because 7% is the long-run nominal return figure most retirement projections use and the rate that shows up in every 401(k) illustration.

At 10% the rule gives 7.2 years and the exact answer is 7.27. Now the rule is early rather than late: $10,000 at 10% for 7.2 years reaches $19,862, short of doubling by $138. The dollar error is still small, but the direction has flipped. Above about 8%, the rule starts promising growth that annual compounding does not quite deliver.

The Error Curve, Rate by Rate

The exact doubling time is ln 2 / ln(1 + r), and it can be computed for any rate. Set it beside 72/r and the shape of the error becomes obvious.

Every rate in the table is an effective annual rate. If a rate is quoted monthly, weekly, or continuously, convert it to an annual equivalent first, because the rule is calibrated to the rate that actually compounds over the year being measured.

Reading the Error Curve

Below 8% the rule runs long, above 8% it runs short, and the gap has a crossover at the point where the two errors balance. The shortfall in years widens with the rate up to about 20% and then flattens, settling around a quarter of a year. That plateau is why the rule is still usable at 30%, where it is wrong by under three months on a 2.64-year answer.

Relative error tells the more useful story. At 1% the rule misses by 3.4% of the correct answer. At 10% it misses by about 1%. At 20% it misses by 5%. At 30% it misses by 9%. The rule is at its worst at both ends, and for opposite reasons: at low rates the curvature it is trying to correct barely exists, and at high rates it exists in abundance.

For consumer finance that range is mostly academic. Savings accounts, mortgages, auto loans, and index-fund returns all sit between 2% and 12%, where the rule is accurate to within a couple of months. Where it starts to matter is in the small-print scenarios: a payday loan at an annualized triple-digit rate, a store card at 29.99%, or a speculative investment pitched on the promise of doubling in a few years.

A Version That Does Not Drift

If you want more accuracy without giving up mental arithmetic, the next term in the expansion provides it. Since ln(1 + r) is approximately r minus r squared over 2, the exact formula n = ln 2 / ln(1 + r) is approximated well by 69.3/R + 0.35, where R is the rate in percent.

Test it against the exact values. At 4%: 69.3/4 plus 0.35 is 17.68 against an exact 17.67. At 7%: 10.25 against 10.24. At 10%: 7.28 against 7.27. At 20%: 3.81 against 3.80. At 30%: 2.66 against 2.64. The correction term buys a second decimal place across the whole range for the cost of one addition.

That is the trade the rule of 72 makes. It gives up a small amount of accuracy for the ability to divide 72 by 6, 8, or 9 without pausing. For a rough check on a projection, that trade is excellent. For deciding whether a 25-year-old will have enough at 65, use the formula and a spreadsheet. The rule belongs in the conversation, not in the plan.

The Rule Running Backward: Inflation

The rule works on losses as well as gains. Divide 72 by the inflation rate and you get the years until a dollar buys half as much. At 3%, that is 24 years. At 2.5%, it is 28.8. At 2%, it is 36 years against an exact 35.0.

The useful insight is not any single number. It is that doubling the inflation rate halves the erosion time. Two percent inflation gives a dollar a 36-year half-life; four percent gives it 18. That is why a stretch of 4% inflation feels so different from a stretch of 2% even though the annual difference sounds small. The compounding does the work, not the annual rate.

Apply it to your own horizon. A 40-year-old planning to retire at 67 has 27 years. At 3% inflation, $100,000 of today's spending becomes about $222,000 of nominal spending by then, because 27 years is more than one 24-year doubling. A retirement plan built on today's dollars against a nominal account balance will look underfunded for exactly this reason, and the rule lets you see the size of the gap without a calculator.

The Rule Applied to Fees

Fees are a negative growth rate, and the rule handles them, though the obvious framing is misleading. A 1% annual fee divides 72 by 1 and gives 72 years for the fee alone to consume half your money. That statement is technically true and practically useless, because the portfolio is doing other things at the same time.

The better question is what the fee does to a portfolio that is also growing. At 7% gross, a 1% fee leaves 6% net. The rule says 7% doubles in 10.3 years and 6% doubles in 12. Over a 30-year career that is about 2.9 doublings at 7% and about 2.5 at 6%. You give up four-tenths of a doubling, and four-tenths of a doubling is roughly a quarter of the ending balance. $10,000 becomes about $76,000 gross and about $57,000 net.

That is why expense ratios matter more than the basis points sound. A fund at 0.03% and a fund at 1.03% differ by one percentage point, which sounds like nothing, and the rule of 72 turns it into the difference between doubling every 10.3 years and doubling every 11.9. Over a working life that compounds into a number large enough to change a retirement.

Cousins: Tripling and Continuous Compounding

The same construction handles any multiple. Tripling needs ln 3, which is 1.0986, so the constant is about 110 or 114 rather than 72. The rule of 114 is the one people quote, and it is remarkably good at 8%: 114 divided by 8 is 14.25 years against an exact 14.27. Quadrupling is just two doublings, so double the rule of 72's answer and you have it.

The other cousin is the rule of 70, which comes up when growth is continuous rather than annual. If a quantity grows at rate r with compounding applied constantly, the doubling time is exactly ln 2 / r, which in percentage terms is 69.3/R. Using 70 makes the arithmetic rounder and the answer very slightly longer than the truth. For inflation and for population growth, which are usually quoted as continuous rates, 70 is the better constant.

None of these variants is more correct in general. They are adjustments for the compounding convention and the divisibility of the constant. Matching the constant to the situation is what separates someone who understands the rule from someone who memorized a number.

A Sanity Check You Can Run in Your Head

The most valuable use of the rule is not projection. It is catching nonsense. Any claim about growth that reaches your inbox or your dinner table can be checked in about four seconds, and the check is usually decisive.

Suppose a pitch promises to quadruple your money in 18 years. Quadrupling is two doublings, so each doubling takes 9 years, so the implied return is 72 divided by 9, or 8% a year. That is a plausible equity return and the arithmetic is honest. Now suppose the same pitch promises to quadruple your money in 18 years and attributes it to a strategy returning 15% a year. Fifteen percent doubles in under five years, so 18 years of it is closer to twelve times your money, not four. One of the three numbers in the pitch is false, and the rule finds it without a spreadsheet.

The same check runs against your own plan. If a retirement projection says a balance triples between 45 and 67, that is 22 years, so the implied return is roughly 114 divided by 22, or a little over 5% a year. If the projection assumed 8%, the final balance should be about five and a half times the starting balance, and one of the assumptions has been mislabeled.

When the Rule Stops Being Useful

Five situations where the shortcut quietly fails.

The common thread is that the rule describes one lump sum growing at a constant rate with nothing added, nothing removed, and nothing taxed. Each case below breaks one of those assumptions.

• Contributions. The rule assumes one lump sum growing untouched. A portfolio fed by monthly deposits has no single doubling time, because every new dollar starts its own clock, and the shortcut cannot describe the mix.
• Negative returns. At -5% a year the exact halving time is 13.5 years. The rule has no clean sign convention for losses, and 72/5 = 14.4 is the wrong tool pointed at a real question.
• Taxes. A 7% return taxed annually at 22% is roughly a 5.5% return, and the rule should be applied to that number, not the headline.
• Variable returns. Seven percent averaged over 20 years with a 40% drawdown in the middle is not the same as 7% every year. The rule is indifferent to order. Your money is not.
• Debt. The rule describes compound growth on a lump sum. A mortgage or auto loan amortizes, so the balance falls as you pay. A 24% card does not double every three years unless you make no payments, which is not a payoff plan.