The Rule of 72: What It Is and How to Use It in Investing (2024)

Notice that although it gives an estimate, the Rule of 72 is less precise as rates of return increase.

The Rule of 72 and Natural Logs

The Rule of 72 can estimate compounding periods using natural logarithms. In mathematics, the logarithm is the opposite concept of a power; for example, the opposite of 10³ is log base 10 of 1,000.

$\begin{aligned} &\text{Rule of 72} = ln(e) = 1\\ &\textbf{where:}\\ &e = 2.718281828\\ \end{aligned}$​Ruleof72=ln(e)=1where:e=2.718281828​

The natural logarithm is the amount of time needed to reach a certain level of growth withcontinuous compounding.

The time value of money (TVM) formula is the following:

$\begin{aligned} &\text{Future Value} = PV \times (1+r)^n\\ &\textbf{where:}\\ &PV = \text{Present Value}\\ &r = \text{Interest Rate}\\ &n = \text{Number of Time Periods}\\ \end{aligned}$​FutureValue=PV×(1+r)nwhere:PV=PresentValuer=InterestRaten=NumberofTimePeriods​

To see how long it will take an investment to double, state the future value as 2 and the present value as 1.

$2 = 1 \times (1 + r)^n$2=1×(1+r)n

Simplify, and you have the following:

$2 = (1 + r)^n$2=(1+r)n

To remove the exponent on the right-hand side of the equation, take the natural log of each side:

$ln(2) = n \times ln(1 + r)$ln(2)=n×ln(1+r)

This equation can be simplified again because the natural log of (1 + interest rate) equals the interest rate as the rate getscontinuously closerto zero. In other words, you are left with:

$ln(2) = r \times n$ln(2)=r×n

The natural log of 2 is equal to 0.693 and, after dividing both sides by the interest rate, you have:

$0.693/r = n$0.693/r=n

By multiplying the numerator and denominator on the left-hand side by 100, you can express each as a percentage. This gives:

$69.3/r\% = n$69.3/r%=n

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.

The number 72, however, 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.

How toCalculate the Rule of 72 Using Matlab

The calculation of the Rule of 72 in Matlab requires running a simple command of "years = 72/return," where the variable "return" is the rate of return on investment and "years" is the result for the Rule of 72. The Rule of 72 is also used to determine how long it takes for money to halve in value for a given rate ofinflation. For example, if the rate of inflation is 4%, a command "years = 72/inflation" where the variable inflation is defined as "inflation = 4" gives 18 years. Matlab, short for matrix laboratory, is a programming platform from MathWorks used for analyzing data and more.

The Bottom Line

The Rule of 72 is a quick and easy method for determining how long it will take to double an investment, assuming you know the annual rate of return. While it is not precise, it does provide a ballpark figure and is easy to calculate. Investments, such as stocks, do not have a fixed rate of return, but the Rule of 72 still can give you an idea of the kind of return you'd need to double your money in certain amount of time. For example, to double your money in six years, you would need a rate of return of 12%.