A sum of money can sometimes double in value almost unnoticed, and then double again in a far shorter time. The reason lies in compound interest and in a mental shortcut that makes its pace easy to estimate: the Rule of 72.
Compound Interest: Earning Interest on Interest
Compound interest is the interest earned on a principal that is then added back to that principal, so it goes on to earn interest of its own. The U.S. Securities and Exchange Commission's investor education site, Investor.gov, sums it up as "interest on interest." A deposit of 10,000 at 10% grows to 11,000 after the first year; in the second year the interest is calculated on 11,000 rather than 10,000, and the balance reaches 12,100. That extra 100 is interest earned on interest, and over time it snowballs.
The striking feature of this mechanism is that growth is exponential, not linear. With simple interest the same amount is added every year, but with compound interest the amount added grows a little more each period. As the years pass the curve steepens, which is why starting to invest early creates an advantage that is hard to make up later with larger sums.
Simple Interest Versus Compound Interest
With simple interest, interest is calculated only on the original principal. At 10% simple interest, 10,000 earns 10,000 in interest over 10 years and reaches 20,000. Left to compound at the same rate, the same money grows to roughly 25,937. The gap looks small in the early years but widens as the term lengthens: over 30 years simple interest quadruples the money, while compound interest takes it past 17 times the original.
This is exactly the kind of compound growth the Rule of 72 is meant to estimate; it does not work for simple interest.
What Is the Rule of 72?
The Rule of 72 is a practical method for estimating how many years it takes money to double at a fixed annual rate of return. The rule is simple: divide 72 by the expected annual percentage return, and the result is the approximate number of years needed for the money to double.
The formula is written as:
Doubling time (years) ≈ 72 / annual rate of return (%)
The rule also works in reverse: if you want your money to double within a certain number of years, divide 72 by that number of years to find the annual return you would need.
How to Calculate With the Rule of 72
A few examples make the rule concrete:
- 8% a year: 72 / 8 = 9. The money doubles in about 9 years.
- 6% a year: 72 / 6 = 12 years.
- 4% a year: 72 / 4 = 18 years.
- A deposit paying 3.5% a year: 72 / 3.5 ≈ 20.6 years.
In an example from the Nebraska Department of Banking and Finance, assuming a long-run average return of 10% gives 72 / 10 = 7.2 years. In reverse: to double your money in 6 years you need 72 / 6 = 12, or an average annual return of 12%.
The power of the rule is that it lets you grasp exponential growth in your head, without a calculator or a table of logarithms.
Why 72? The Math Behind the Formula
The exact doubling time is given by an expression derived from the compound interest equation:
t = ln(2) / ln(1 + r)
Here ln is the natural logarithm and r is the annual return as a decimal. The number ln(2) is about 0.6931, so at low rates a "Rule of 69.3" is actually more accurate. But 69.3 is awkward to divide by. The number 72, by contrast, divides evenly by 2, 3, 4, 6, 8, 9 and 12, which makes it far more useful in practice, and that is why it has been preferred for centuries.
The rule's origins reach back at least to the 15th century: the Italian mathematician Luca Pacioli mentions the Rule of 72 in his 1494 work Summa de arithmetica, though he does not derive it and is thought not to have invented it.
The Limits of the Rule and Its Close Relatives
The Rule of 72 is an estimating tool, not an exact calculation. It gives its most accurate results at rates between about 6% and 10%, and especially near 8%. The further the rate strays from that range, the larger the error: at very high returns the rule overstates the true time, and at very low rates it can understate it. One correction is to add or subtract 1 from 72 for every 3 percentage points the rate differs from 8%.
There are other versions of the same idea. For daily or continuous compounding the "Rule of 69.3" is more accurate; at very low rates the "Rule of 70" offers a practical middle ground. All three are different roundings of the same logarithmic formula.
The rule also applies only to compound, reinvested returns; if the interest earned is withdrawn, or the interest is simple, the result is misleading. Taxes, fees and variable rates of return can also lengthen the real time.
The Rule of 72 in Inflation and Growth
The Rule of 72 applies not only to investment returns but to anything that grows or shrinks exponentially. At 6% annual inflation, the purchasing power of money halves in 72 / 6 = 12 years. An economy or population growing 2% a year doubles in about 36 years. In the same way, 3% annual growth means national income doubles in 24 years.
Seen this way, the Rule of 72 is a mental compass for sensing what the percentages in economic news mean over the long run: even a small difference in a rate produces a large result when it is spread over decades.