What is the Rule of 72?

What is the Rule of 72?

The Rule of 72 is a quick, useful formula that is popularly used to estimate the number of years required to double the invested money at a given annual rate of return. Alternatively, it can compute the annual rate of compounded return from an investment given how many years it will take to double the investment.

While calculators and spreadsheet programs like Microsoft Excel have functions to accurately calculate the precise time required to double the invested money, the Rule of 72 comes in handy for mental calculations to quickly gauge an approximate value.
For this reason, the Rule of 72 is often taught to beginning investors as it is easy to comprehend and calculate. The Security and Exchange Commission also cites the Rule of 72 in grade-level financial literacy resources.

How the Rule of 72 works

The actual mathematical formula is complex and derives the number of years until doubling based on the time value of money.

You’d start with the future value calculation for periodic compounding rates of return, a calculation that helps anyone interested in calculating exponential growth or decay:

FV = PV*(1+r)t

FV is future value, PV is present value, r is the rate and the t is the time period. To isolate t when it’s located in an exponent, you can take the natural logarithms of both sides. Natural logarithms are a mathematical way to solve for an exponent. A natural logarithm of a number is the number’s own logarithm to the power of e, an irrational mathematical constant that is approximately 2.718. With the example of a doubling of $10, deriving the Rule of 72 would look like this:

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20 = 10*(1+r)t

20/10 = 10*(1+r)t/10

2 = (1+r)t

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

ln(2) = r*t

The natural log of 2 is 0.693147, so when you solve for t using those natural logarithms, you get t = 0.693147/r.

The actual results aren’t round numbers and are closer to 69.3, but 72 easily divides for many of the common rates of return that people get on their investments, so 72 has gained popularity as a value to estimate doubling time.

For more precise data on how your investments are likely to grow, use a compound interest calculator that’s based on the full formula.

How the Rule of 72 Came About

Interest has existed since ancient times in mathematical and economic studies. In fact, it appears to date as far back as the Mesopotamian, Roman and Greek civilizations. The Quran even makes mention of it. Its roots stem from agriculture and the first incarnations of land and money loans.

The first individual to mention the rule of 72, though, is Luca Pacioli, a renowned mathematician from Italy. His impressive book, “Summa de arithmetica, geometria, proportioni et proportionalita” (“Summary of Arithmetic, Geometry, Proportions and Proportionality”), was published in 1494 and holds the first known reference of the rule, making him the closest we know to an inventor. Some credit Albert Einstein as the architect of the rule. There is no documentation to support this claim, though.

Rule of 72 during inflation

Investors can use the rule of 72 to see how many years it will take to cut in half their purchasing power due to inflation. For example, if inflation is around 8 percent (as during the middle of 2022), you can divide 72 by the rate of inflation to get 9 years until the purchasing power of your money is reduced by 50 percent.

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72/8 = 9 years to lose half your purchasing power.

The Rule of 72 allows investors to realize the severity of inflation concretely. Inflation might not remain elevated for such a long period of time, but it has done so in the past over a multi-year period, really hurting the purchasing power of accumulated assets.

How Accurate Is the Rule of 72?

The Rule of 72 formula provides a reasonably accurate, but approximate, timeline—reflecting the fact that it’s a simplification of a more complex logarithmic equation. To get the exact doubling time, you’d need to do the entire calculation.

The precise formula for calculating the exact doubling time for an investment earning a compounded interest rate of r% per period is:

To find out exactly how long it would take to double an investment that returns 8% annually, you would use the following equation:

T = ln(2) / ln (1 + (8 / 100)) = 9.006 years

As you can see, this result is very close to the approximate value obtained by (72 / 8) = 9 years.

How to Adjust the Rule of 72 for Higher Accuracy

The Rule of 72 is more accurate if it is adjusted to more closely resemble the compound interest formula—which effectively transforms the Rule of 72 into the Rule of 69.3.

Many investors prefer to use the Rule of 69.3 rather than the Rule of 72. For maximum accuracy—particularly for continuous compounding interest rate instruments—use the Rule of 69.3.

The number 72 has many convenient factors including two, three, four, six, and nine. This convenience makes it easier to use the Rule of 72 for a close approximation of compounding periods.