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The Rule of 72 is a shortcut equation to help you figure out just how long it will take to double an investment at a given rate of return. Best of all, the math is easy to do without the help of a calculator.

In short, the Rule of 72 can help investors determine whether an investment may have a place in their overall investment strategy, and how to proceed.

What Is the Rule of 72?

As noted, the Rule of 72 helps investors understand how different types of investments might figure into their investment plans. The basic formula for the rule is:

Number of years to double an investment = 72 / Interest rate

In the case of investing, the interest rate is the rate of return on an investment. For example, an investor has $10,000 to invest in an investment that offers a 6% rate of return. That investment would double in 72 / 6 = 12 years. Twelve years after making an initial investment, the investor would have $20,000.

Notice that when making this calculation, investors divide by six, not 6% or 0.06. Dividing by 0.06 would indicate 1,200 years to double the investment, an outlandishly long time.

This shorthand allows investors to quickly compare investments and understand whether their rate of return will help them meet their financial goals within a desired time horizon.

Who Came Up with the Rule of 72?

The Rule of 72 is not new, in fact, it dates back to the late 1400s, when it was referenced in a mathematics book by Luca Pacioli. The Rule itself, though, could date even further back. Albert Einstein is often credited with its invention, however.


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The Formula and Calculation of the Rule of 72

The Rule of 72 is a shortened version of a logarithmic equation that involves complex functions you would need a scientific calculator to calculate. That formula looks like this:

T = ln(2) / ln(1 + r / 100)

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

This calculation is too complicated for the average investor to perform on the fly, and it turns out 72 divided by r is a close approximation that works especially well for lower rates of return. The higher the rate of return — as the rate nears 100% — the less accurate the Rule of 72 gets.

Example of the Rule of 72 Calculation

For a relatively simple equation, the Rule of 72 can help investors figure out a lot of helpful information. For one, it can help them compare different types of investments that offer different rates of returns.

For example, an investor has $25,000 to invest and plans to retire in 20 years. In order to meet a certain retirement goal, that investor needs to at least double their money to $50,000 in that time period.

The same investor is presented with two investment options: One offers a 3% return and one offers a 4% return. The investor can quickly see that at 3% the investment will double in 72 / 3 = 24 years, four years past their retirement date. The investment with a 4% return will double their money in 72 / 4 = 18 years, giving them two years of leeway before they retire.

The investor can see that when choosing between the two options, choosing the 4% rate of return will help them reach their financial goals, while the 3% return will leave them short.

Applying the Rule of 72 in Investment Planning

There are numerous instances in which the Rule of 72 can be applied to investment planning. But it’s also important to understand a bit about how simple and compound interest differ, and come into play when using the Rule to make projections.

Remember, there are two types of interest rates: simple interest and compound interest.

Simple interest is calculated using only the principal or starting amount. For example, an individual opens an account with $1,000 and a 1% simple interest rate. At the end of the year, they will have $1,010 in their bank account. But they’ll only earn 1% each year on their principal, aka that initial $1,000.

So even over a longer time period, the individual isn’t earning very much—after 10 years, for example, they will have accumulated a total of $1,100.

Simple interest may be even easier to conceptualize as a savings account from which an individual withdraws the interest each year.

In the example above the individual would withdraw $10 at the end of the year and start again with $1,000 the next year. Every year after that, they would start over with the same principal and earn the same amount in interest.

Compound interest, on the other hand, can help investments grow exponentially. That’s because it incorporates the interest earned on an investment in addition to the initial investment. In other words, an investor earns a return on their returns.

To get an idea of the power of compound interest it might help to explore a compound interest calculator, which allows users to input principal, interest rate, and compounding period.

For example, an individual invests that same $1,000 at a 6% interest rate for 30 years with interest compounding annually. At the end of the investment period, they will have made more than $5,700 without making any additional investments.

That fact is important to consider when conceptualizing the Rule of 72, because compound interest plays a big role in helping an investment double in value within a given time frame. It can help achieve high reward with relatively little effort.

Practical Uses in Financial Projections

Higher returns are often correlated with higher risk. So this rule can help investors gauge whether their risk tolerance — or their return on investment — is high enough to get them to their goal. Depending on what their time horizon is, investors can easily see whether they need to bump up their risk tolerance and choose investments that offer higher returns.

By the same token, this rule can help investors understand if their time horizon is long enough at a certain rate of return. For example, the investor in the above example is already invested in the instrument that offers 3%.

The Rule of 72 can illustrate that they may need to rethink their timeline for when they will retire, pushing it past 20 years. Alternatively, they could sell their current investments and buy a new investment that offers a higher rate of return.

It’s also important to understand that the Rule of 72 does not take into account additional savings that may be made to the principal investment. So if it becomes clear that the goal won’t be met at the current savings rate, an investor will be able to consider how much extra money to set aside to help reach the goal.

Estimating Investment Doubling Times

Using the Rule of 72 to estimate investment doubling times can be a little tricky, and perhaps inaccurate, unless an investor has a clear idea of what the expected rate of return for an investment will be. For instance, it may be very difficult to get an idea of an expected return for a particularly volatile stock. As such, investors may want to proceed with caution when using it to calculate investment doubling times.

Application in Stock Market Investments

As mentioned, stock market investments can be difficult to predict. But some are more predictable than others. For example, investors can probably use the historic rate of return for the S&P 500 to try and get a sense of an expected return for the market at large – which can help when applying the Rule of 72 to index funds or other broad investments.

For example, if a 401(k) plan includes investments that offer a 6% return, the investment will double in 12 years. Again, that’s an estimate, but it gives investors a ballpark figure to work with.

Use During Periods of Inflation

Money loses value during bouts of inflation, which means that the Rule of 72 can be used to determine how long it’ll take a dollar’s value to fall by half – the opposite of doubling in value.

Accuracy and Limitations of the Rule of 72

The Rule of 72 has its place in the investing lexicon, but there are some things about its accuracy and overall limitations to take into consideration.

Is the Rule of 72 Accurate?

Perhaps the most important thing to keep in mind about the Rule of 72’s accuracy is that it’s a derivation of a larger, more complex operation, and therefore, is something of an estimate. It’s not perfectly accurate, but will get you more of a “ballpark” figure that can help you make investing decisions.

Situations Where the Rule is Most Accurate

The Rule of 72 is only an approximation and depending on what you’re trying to understand there are a few variations of the rule that can make the approximation more accurate.

The rule of 72 is most accurate at 8%, and beyond that at a range between 6% and 10%. You can, however, adjust the rule to make it more accurate outside the 6% to 10% window.

The general rule to make the calculation more accurate is to adjust the rule by one for every three points the interest rate differs from 8% in either direction. So, for an interest rate of 11%, individuals should adjust from 72 to 73. In the other direction, if the interest rate is 5%, individuals should adjust 72 to 71.

Comparisons and Variations on the Rule

There are a few alternatives or variations of the Rule of 72, too, such as the Rule of 73, Rule of 69.3, and Rule of 69.

Rule of 72 vs. Rule of 73

The basic difference between the Rule of 72 and the Rule of 73 is that it’s used to estimate the time it takes for an investment’s value to double if the rate of return is above 10%. The Rule of 73 is only a slight tweak to the rule of 72, using different figures in the calculation.

Rules of 72, 69.3, and 69

Similarly to the Rule of 73, some people prefer to use the Rule of 69.3, especially when interest compounds daily, to get a more accurate result. That number is derived from the complete equation ln(2) / ln(1 + r / 100). When plugged into a calculator by itself, ln(2) results in a number that’s approximately 0.693147.

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FAQ

What are flaws of the Rule of 72>

There are a few key drawbacks to using the Rule of 72, including the fact that it’s mostly accurate only for a certain subset of investments, it’s only an estimation, and that unforeseen factors can cause the rate of return for an investment to change, rendering it useless.

Does the Rule of 72 apply to debt?

Yes, the Rule of 72 can apply to debt, and it can be used to calculate an estimate of how long it would take a debt balance to double if it’s not paid down or off.

Who created the Rule of 72?

Albert Einstein often gets credit, but Italian mathematician Luca Pacioli most likely invented, or introduced the Rule of 72 to the popular world in the late 1400s.

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