How long would it take for an investor to double his money at 10% interest per year compounded annually?

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 Rule of 72 is an easy way for an investor or advisor to approximate how long it will take an investment to double based on its fixed annual rate of return. Simply divide 72 by the fixed rate of return, and you’ll get a rough estimate of how long it will take for your portfolio to double in size.

The science isn’t exact, though, and you may want to use a different formula to account for rates of return that fall outside a certain range. 

Key Takeaways

• The Rule of 72 is a simple way to calculate how long it will take an investment to double based on the annualized rate of return.
• Investors can use the rule when planning for retirement, education expenses, or any other long-term financial goal.
• For more accuracy, investors can use a logarithmic formula to calculate the time for an investment to double.
• In some situations, investors might want to use the Rule of 70 instead.

What Is the Rule of 72?

The Rule of 72 is a rule of thumb that investors can use to estimate how long it will take an investment to double, assuming a fixed annual rate of return and no additional contributions.

If you want to dive even deeper, you can use the Rule of 115 to determine how long it will take to triple your investment. 

Both of these rules of thumb can help investors understand the power of compound interest. The higher the rate of return, the shorter the amount of time it will take to double or triple an investment. 

How To Use the Rule of 72 To Estimate Returns

Let’s say you have an investment balance of $100,000, and you want to know how long it will take to get it to $200,000 without adding any more funds. With an estimated annual return of 7%, you’d divide 72 by 7 to see that your investment will double every 10.29 years. 

Here’s an example of other rates of return and how the Rule of 72 affects your investment:

However, the calculation isn’t foolproof. If you have a little more time and want a more accurate result, you can use the following logarithmic formula:

T = ln(2) / ln(1+r)

In this equation, “T” is the time for the investment to double, “ln” is the natural log function, and “r” is the compounded interest rate. 

So, to use this formula for the $100,000 investment mentioned above, with a 6% rate of return, you can determine that your money will double in 11.9 years, which is close to the 12 years you'd get if you simply divided 72 by 6. 

Here's how the logarithmic formula looks in this case: 

T = ln(2) / ln(1+.06)

Note

If you don’t have a scientific calculator on hand, you can usually use the one on your smartphone for advanced functions. However, the basic calculation can give you a good ballpark figure if that’s all you need.

How To Use the Rule of 72 To Estimate Compound Interest

Like most equations, you can move variables around to solve for others that aren’t certain. If you’re looking back on an investment you’ve held for several years and want to know what the annual compound interest return has been; you can divide 72 by the number of years it took for your investment to double.

For example, if you started out with $100,000 and eight years later the balance is $200,000, divide 72 by 8 to get a 9% annual rate of return. 

Grain of Salt

The Rule of 72 is easy to calculate, but it’s not always the right approach. For starters, it requires a fixed rate of return, and while investors can use the average stock market return or other benchmarks, past performance doesn’t guarantee future results. So it’s important to do your research on expected rates of return and be conservative with your estimates.

Also, the simpler formula works best for return rates between 6% and 10%. The Rule of 72 isn’t as accurate with rates on either side of that range. 

For example, with a 9% rate of return, the simple calculation returns a time to double of eight years. If you use the logarithmic formula, the answer is 8.04 years—a negligible difference.   

In contrast, if you have a 2% rate of return, your Rule of 72 calculation returns a time to double of 36 years. But if you run the numbers using the logarithmic formula, you get 35 years—a difference of an entire year. 

As a result, if you’re looking to just get a quick idea of how long your investment will take to double, use the basic formula. But if you’re calculating the figure as part of your retirement or education savings plan, consider using the logarithmic equation to ensure that your assumptions are as accurate as possible.

Note

The Rule of 72 works best over long periods of time. If you’re nearing retirement, it may not be as helpful because short-term volatility can give your annual return rate less time to even out. 

Rule of 72 vs. 70

The Rule of 72 provides reasonably accurate estimates if your expected rate of return is between 6% and 10%. But if you’re looking at lower rates, you may consider using the Rule of 70 instead.

For example, take our previous example of a 2% return. With the simple Rule of 70 calculation, the time to double the investment is 35 years—exactly the same as the result from the logarithmic equation.

However, if you try to use it on a 10% return, the simple formula gives you seven years while the logarithmic function returns roughly 7.3 years, which has a wider discrepancy. 

As with any rule of thumb, the Rules of 72 and 70 aren’t perfect. But they can give you valuable information to help you with your long-term savings plan. Throughout this process, consider working with a financial advisor who can help you tailor an investment strategy to your situation.

Frequently Asked Questions (FAQs)

What is the Rule of 72 used for?

The Rule of 72 is a quick formula you can use to estimate the future growth of an investment. If you know the average rate of return, you can apply a simple formula to determine how long it will take to double your investment, assuming you don't put more money into it.

Who invented the Rule of 72?

The earliest known reference to the Rule of 72 comes from Luca Pacioli's 1494 book, "Summa de Arithmetica." This book went on to be used as an accounting textbook until the mid-1600s, granting Pacioli the title of the Father of Accounting.

When does money double every seven years?

To use the Rule of 72 to figure out when your money will double itself, all you need to know is the annual rate of expected return. If this is 10%, then you'll divide 72 by 10 (the expected rate of return) to get 7.2 years. Use this same formula to figure out the return on other investments by diving 72 with the expected annual rate of return.

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