Rule of 72

The Rule of 72 is a quick mental shortcut for estimating how long it takes an amount to double at a steady compound growth rate: divide 72 by the annual percentage rate. At 6% a year, money doubles in about 72 ÷ 6 = 12 years. It is used by savers, investors, teachers and economists to get a feel for compound interest without a calculator.

The rule is an approximation of the exact doubling-time formula, which uses logarithms. It works for anything that grows at a constant percentage rate, including investments, debts, prices under inflation and national output. It appears in Luca Pacioli's Summa de arithmetica, printed in Venice in 1494, where it is stated without proof, so it is probably older. Close variants include the Rule of 70 and the Rule of 69.3.

Picture a jar of sourdough starter that grows a little every day, and each day's growth also grows. At first it barely seems to change; later it suddenly fills the jar. Money earning interest on its interest behaves the same way. The Rule of 72 is a kitchen-table way to answer the obvious question: how long until the jar is twice as full?

The answer depends only on the growth rate. A faster rate means a shorter wait, and the rule turns that into one easy division. It also works in reverse: if a price is rising quickly, the same trick shows how soon it will cost twice as much, which is the same as saying how soon your money will buy half as much.

The rule is:

Here $t$ is the number of periods (usually years) needed to double, and $r$ is the growth rate per period written as a whole-number percentage (6 for 6%). The exact answer for annual compounding is:

where $\ln$ is the natural logarithm and $\ln 2 \approx 0.693$. At 6%, the exact result is $0.693 \div \ln(1.06) = 0.693 \div 0.0583 \approx 11.9$ years, very close to the rule's 12. The same division also works backwards: to double money in 8 years, a saver needs about $72 \div 8 = 9\%$ a year.

The precise constant for continuous compounding is 69.3 (100 times $\ln 2$), which is why economists working with continuously compounded growth rates often use the Rule of 70 or 69.3. With annual compounding, the exact doubling time is slightly longer than $69.3/r$, and 72 compensates for that over the range of everyday rates. It is also convenient: 72 divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, so most mental divisions come out cleanly. The rule is most accurate around 8%, and the error grows at very low or very high rates; above about 20% it understates the doubling time noticeably.

The same idea extends to other multiples. Dividing about 114 by the rate gives the time to triple, and about 144 the time to quadruple (two doublings).

The rule shows up wherever compound growth matters:

• Savings and investing. Comparing a 2% savings account or money market account (doubling in about 36 years) with a portfolio expected to grow 8% (about 9 years) makes the long-run gap vivid. The relevant rate is the effective yield, so the annual percentage yield (APY) is the better input than a quoted nominal rate.
• Fees. A 1% annual charge, such as a fund's expense ratio, takes a growing bite over decades because it compounds too.
• Debt. A balance on a credit card charging 24% that is left unpaid would roughly double in three years, ignoring payments and fees; see finance charge for how card interest is actually calculated.
• Inflation. At 3% inflation, prices double in about 24 years, so a fixed sum loses half its purchasing power in that time. This is the core of inflation risk for savers and retirees.
• Economic growth. An economy whose GDP grows 7% a year doubles in size in about a decade; one growing 2% takes about 35 years.

Growth rates measured over many years, such as a compound annual growth rate (CAGR), can be fed straight into the rule.

The Rule of 72 assumes a constant rate and full reinvestment. Real returns vary from year to year, and an investment that averages 8% with large swings will usually end with less than one that earns a steady 8%. Taxes, fees and withdrawals slow the doubling, and changes in interest rates mean a savings rate available today may not last. The rule also says nothing about risk: a higher expected rate shortens the doubling time on paper but usually comes with a greater chance of loss, the trade-off at the heart of investing. Finally, for planning that must be precise, such as loan schedules or retirement projections, the exact formula or a time value of money calculation should be used instead of the shortcut, and returns should be considered in real terms after inflation.

• Investor.gov (U.S. SEC): Compound Interest Calculator