# What Is the Rule of 72?

May 22, 2023
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The Rule of 72 is a simple, helpful equation that can be used to estimate the number of years it will take to double an investment. Alternatively, it can also be used to approximate the annual rate of return needed to double an investment.

Instead of relying on a financial calculator or making complex calculations, the Rule of 72 provides a quick, "back of the envelope" formula that gives an output that is reasonably close to the actual value. Not only can it be used by anyone, but the Rule of 72 can also be used for other applications that involved compound interest (such as those in economics).

## How Does the Rule of 72 Work?

The Rule of 72 works with two variables: one known and one unknown. Depending on which piece of information you have will determine the way in which you apply the rule.

### Years to Double

To calculate how many years it will take for an investment to double in value, the user needs to know the annual interest rate or rate of return. The equation can then be applied as follows:

Years to double = 72 / Annual rate of return

For example, if you'd like to put your money into an investment that has an average annual return of 10 percent, then you could estimate that it would take 72 / 10 = 7.2 years for the investment to double in value.

### Expected Rate of Return

The Rule of 72 can also be used inversely to calculate a desired rate of return. The equation can then be applied as follows:

Annual rate of return = 72 / Years to double

For example, suppose you wanted to invest your money in an asset that could potentially double within 8 years. Using the Rule of 72, you'd need to consider assets with an anticipated minimum annual return of 72 / 8 = 9 percent.

## How Accurate is the Rule of 72?

Despite the Rule of 72 being a simplification of a complex logarithmic equation, it can be surprisingly accurate.

In finance, the "future value" or FV is calculated as follows:

FV = PV*(1+r)^n

To get the same desired output as the Rule of 72 "n" (number of years), this equation would need to be rewritten into a more elaborate form:

n = [ln (FV/PV)] / [ln (1+r)]

Clearly, using the Rule of 72 would be handier and involve less, complex math.

To demonstrate the accuracy of the Rule of 72, let's do a side-by-side comparison. Suppose you have an investment that produces a 6 percent annual return. How long would it take to double your money?

• Rule of 72 = 12 years
• n = 11.8957 years
• Percent error = 0.88%

As a quick approximation, having less than 1% error is acceptable.

Math experts have said that the Rule of 72 works best when the input for the annual percentage is in the 5 to 12 percent range. They've also found that users may get more accurate results if they substitute 69.3 instead of 72.

## What Can the Rule of 72 Be Used For?

The Rule of 72 can be used for a variety of purposes. The following are a few common applications.

### Comparing Investment Opportunities

When you're trying to decide between two investment opportunities, it's easy to infer that the one with the greater anticipated annual return will be the better option. However, the Rule of 72 can help you to add some context to the situation.

Let's suppose you have \$10,000 and trying to decide between two options:

1. Parking it into a savings account with a 3 percent interest rate
2. Investing in a stock that has historically made a 9 percent average annualized return.

For the savings account, the Rule of 72 says it would take 24 years for your money to double to \$20,000.

Meanwhile, investing in the stock could potentially your money in 8 years.

Given that 24 years is three times as long as 8 years, that means your money could potentially double 3 times over with the equity investment: \$80,000 versus \$20,000.

### Screen Investments

The Rule of 72 can help you to dissect investment opportunities and determine if they are realistic.

For instance, let's say an investment professional tells you that he can guarantee to double your money in as short as five years. Using the Rule of 72, you could quickly determine that this would require an investment that reliably produces 14.4 percent per year.

Unfortunately, you'd be wise to doubt his claim since interest rates on bank products are nowhere close to this. Even assets that carry risk such as stocks classically return around 10 percent per year (though this is never guaranteed).

For these reasons, can reasonably assume that the investment is either highly speculative (and the investment professional is not fully disclosing how much risk you'd be undertaking) or it’s a scam.

### Inflation Rate Erosion

Not all uses of the Rule of 72 have to be limited to investments. Other applications that involve compound interest can also be explored such as inflation.

Inflation erodes the purchasing power of your money over time. As the price of everything increases, your dollars don't stretch as far.

We can put this context using the Rule of 72. If you assume the long-term average rate of inflation is 4 percent, then you can infer that it takes 18 years for the price of nearly everything to double in cost.

### Loan Interest Rate

Similar to the example with inflation, we can use this to illustrate other expenses that threaten your personal finances such as credit card interest.

Suppose you have a credit card with a 20 percent APR, and you've accumulated a balance of \$5,000. If you made no payments towards this amount, then we can estimate that your debt would double to \$10,000 after 3.6 years from the interest charges alone.

## Where Does the Rule of 72 Come From?

The Rule of 72 is not a new invention. It dates as far back as 1494 when it was referenced by the Italian mathematician Luca Pacioli in his mathematics textbook "Summa de Arithmetica". Pacioli is credited with being an early contributor to the field of what is now known as accounting.

## The Bottom Line

The Rule of 72 provides a quick and convenient way to calculate how long it takes for an investor to double their money. Alternatively, it can also be used to estimate the annual rate of return needed to double an investment. You can use this equation for a variety of applications, and it will serve as a reasonable approximation.