How many years does it take to double my money if I get a 9% return on investment use the Rule of 72 )?

Calculating compound interest is complicated. Luckily, there’s a simple shortcut that helps you estimate how a fixed interest rate will affect your savings: the Rule of 72.

The Basics

The Rule of 72 is a tool used to estimate how long it will take an investment to double at a given interest rate, assuming a fixed annual rate of interest. All you need to use the tool is an interest rate, which means you can make estimates for your current account rate or use this rule to know what rate you should look for if you want to double your money by a specific deadline.

To figure out how long it will take to double your money, take the fixed annual interest rate and divide that number into 72. Let’s say your interest rate is 8%. 72 ∕ 8 = 9, so it will take about 9 years to double your money. A 10% interest rate will double your investment in about 7 years (72 ∕ 10 = 7.2); an amount invested at a 12% interest rate will double in about 6 years (72 ∕ 12 = 6).

Using the Rule of 72, you can easily determine how long it will take to double your money.

To figure out what interest rate to look for, use the same basic formula, but run it backward: divide 72 by the number of years. So if you want to double your money in about 6 years, look for an interest rate of 12%.

The basic algebraic formula looks like this, where Y is the number of years and r is the interest rate:

Y = 72 ∕ r and r = 72 ∕ Y

This rule works for interest rates from about 4% up to about 20%; after that, the error becomes significant and more straightforward math is required.

Illustration: Chelsea Miller

Why 72?

Here, we merely scrape the surface of that “more straightforward math.” To really dive deep into why the rule works, check out this article.

The Rule of 72 is itself an estimation. It uses a concept called natural logarithms to estimate compounding periods. In mathematics, the natural logarithm is the amount of time needed to reach a particular level of growth using continuous compounding.

For math enthusiasts out there: it is easiest to see how this works through continuously compounded interest. (The Rule of 72 addresses annually compounded interest, but we’ll get there in a minute.)

When dealing with continuously compounding interest, you can work out the exact time it takes an investment to double by using the time value of money formula (TVM) and simplifying the equation until eventually, you are left with something like this:

ln(2)= rY

The natural log (ln) of 2 is about 0.693. Solve for interest rate (r) or number of years (Y), and then multiply by 100 to express as a percentage or year, respectively.

Click here to read how this tool works, and for disclaimers.

Click here to read how this tool works, and for disclaimers.

Wait...

If our new formula is based on the number 69.3 (0.693 × 100), that begs the question: Why isn’t it called the Rule of 69.3?

First, that just doesn’t sound quite as good as “The Rule of 72.” Second, there are two points to remember:

1. The “Rule of 69.3” is not an estimation. It is the actual amount of time that it will take money to double, and works for any range of interest rates.

2. The Rule of 69.3 works for continuously compounded interest. The Rule of 72 works for a fixed annual rate of interest.

The math equation for fixed annual interest is slightly more complex, and simplifying it leaves us with approximately 72.7.

Normally, we would round up to 73. However, 72 is much easier to work with, as it is readily divisible by 2, 3, 4, 6, 8, 9, and 12. As we are already estimating, convenience wins out, and we are left with the Rule of 72.

History

The Rule of 72 was first introduced in the late fifteenth century by the Franciscan friar and Italian mathematician Luca Pacioli. A contemporary of Leonardo da Vinci, Pacioli is considered by many to be the father of accounting. The Rule of 72 was introduced in his book Summa de arithmetica, geometria, proportioni et proportionalita, published in 1494 for use as a textbook for schools in what is now northern Italy.

The length of time required for an investment to double in value at a fixed annual rate of return

What is the Rule of 72?

In finance, the Rule of 72 is a formula that estimates the amount of time it takes for an investment to double in value, earning a fixed annual rate of return. The rule is a shortcut, or back-of-the-envelope, calculation to determine the amount of time for an investment to double in value. The simple calculation is dividing 72 by the annual interest rate.

Time (Years) to Double an Investment

The Rule of 72 gives an estimation of the doubling time for an investment. It is a fairly accurate measurement, and more so when using lower interest rates rather than higher ones. It is used for situations involving compound interest. A simple interest rate does not work very well with the Rule of 72.

Below is a table showing the difference between the Rule of 72 calculation and the actual number of years required for an investment to double in value:

Rule of 72 Formula

The Rule of 72 formula is as follows:

Example of the Rule of 72

You are the owner of a coffee machine manufacturing company. Due to the large capital needed to establish a factory and warehouse for coffee machines, you have turned to private investors to fund the expenditure. You meet with John, who is a high net-worth individual willing to contribute $1,000,000 to your company.

However, John is only willing to contribute the said amount on the presumption that he will get a 12% annual rate of return on his investment, compounded yearly. He wants to know how long it will take for his investment in your company to double in value.

Using the Rule of 72:

It will take approximately six years for John’s investment to double in value.

Deriving the Rule of 72

Let us derive the Rule of 72 by starting with a beginning arbitrary value: $1. Our goal is to determine how long it will take for our money ($1) to double at a certain interest rate.

Suppose we have a yearly interest rate of “r”. After one year, we will get:

$1 x (1+r)

At the end of two years, we will get:

$1 x (1+r) x (1+r)

Extending this year after year, we get:

$1 x (1+r)^n, where n = number of years

If we want to determine how long it takes to double our money, turning $1 into $2:

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

Solving for years (n):

Step 1: $1 x (1+r)^n = $2

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

Step 3: ln((1+R)^n) = ln(2)              (Taking the natural log of both sides)

Step 4: n x ln(1+r) = .693

Step 5: n x r = 0.693                       (Approximation that ln(1+r) = r)

Step 6: n = .693 / r

Step 7: n = 69.3 / r                         (Turning r into an integer rather than a decimal)

Notice that after deriving the formula, we end up with 69.3, not 72. Although 69.3 is more accurate, it is not easily divisible. Therefore, the Rule of 72 is used for the sake of simplicity. The number 72 also provides more factors (2, 3, 4, 6, 12, 24…).

Rules of 72, 69.3, and 69

Rules of 69.3 and of 69 are also methods of estimating an investment’s doubling time. The rule of 69.3 is considered more accurate than the Rule of 72, but can be much more troublesome to calculate. Therefore, investors typically prefer to use a rule of 69 or 72 rather than the rule of 69.3.

Comparing the doubling time for rules of 69, 69.3, and 72 to actual years:

As you can see from the table above, the rule of 69.3 yields more accurate results at lower interest rates. However, as the interest rate increases, the rule of 69.3 loses some of its predictive accuracy.

The Rule of 72 is a simple, helpful tool that investors can use to estimate how long a specific compound interest investment will take to double their money.

More Resources

Thank you for reading CFI’s guide on the Rule of 72. Below are additional free resources from CFI:

• Investing: A Beginner’s Guide
• Hurdle Rate
• Return on Investment (ROI) Formula
• Financial Modeling Courses Collection

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