# Rule of 72 explained

Consider this fantasy: A person strides into a Las Vegas casino and slaps a stack of money down at the roulette table, putting it all on black. The wheel starts to spin, the ball lands on black, and boom, that money is doubled.

If only making money with investments was so easy, but it often takes years before investment gains reach that point.

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.

Related: A beginner’s guide to investing in your 20s

## What is the Rule of 72?

The Rule of 72 helps investors understand how different types of investments might figure into their investment plans. The 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. And that return compounds regularly.

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.

## Where did the Rule of 72 come from?

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.

## What is compound interest?

To understand how the Rule of 72 works, it’s important to get a clear idea of how the interest rate in the equation functions. 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.

## What can be learned from the Rule of 72?

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.

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 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.

For example, if a 401(k) plan includes investments that offer a 6% return, the investment will double in 12 years. Is that fast enough according to the investor’s time horizon? If not, they may need to take a closer look at their retirement plan to figure out how to make up the difference.

## Digging a little deeper

The Rule of 72 is really just a convenient short-hand that can give investors an idea of the effects of compounding interest over time.

In addition to an initial investment, interest rate and compounding period, these forumlas allow investors to input a monthly contribution rate. The resulting calculation will be an accurate picture of how much the investor will save over that given period.

For example, an investor makes an initial investment of \$1,000 and a subsequent investment of \$100 per month for the next 30 years. At a 6% rate of return compounded monthly, the investor would have saved more than \$106,000.

Their total contribution would only have been \$37,000, meaning through returns and the power of compound interest, the investor would have made about \$69,000 in returns.

These tools can help investors understand how much to change their monthly contribution to reach higher savings goals.

## What else can the Rule of 72 be used for?

The Rule of 72 can be used in other scenarios that use the principle of compounding interest. For example, a borrower that has credit card debt can figure out at what point their debt will double.

If the borrower owes \$1,000 on their credit card with a 14% interest rate, they will double what they owe in a little over five years — and that’s if they stop using their card altogether in the interim.

The Rule of 72 can also be used to see the effects of things like inflation or fees that can take a bite out of an individual’s buying power.

For example, at an inflation rate of 2%, an individual’s money will lose half its buying power in 36 years (72 divided by two). Bump the inflation rate up to 3% and it would only take 24 years for the value of an individual’s money to be cut in half.

Similarly, the Rule of 72 can help you understand the effects of investment fees. If you invest in a mutual fund that charges 3% fees, after 12 years (72 divided by three), your investment principal will theoretically be cut in half.

Note, this is just the principal, the amount you initially invested, and does not account for gains in value or compounded returns the investment might have achieved.

Finally, the Rule of 72 doesn’t have to be used just for money. It can be used to help understand anything that grows exponentially, such as populations.

Hypothetically speaking, if a country’s population is growing at a constant 2%, the population would double in 72 divided by two, or 36 years. Of course, this calculation is only an estimate and doesn’t take into account other factors, such as birth rate, that might affect population growth.

## Variations on the Rule of 72

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. One variation is the Rule of 69.3.

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.

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.

Investors may find this calculation more difficult to do without a calculator than the Rule of 72.

As we’ve seen, the Rule of 72 works best at interest rates between 6% and 10%, which individuals are likely to achieve through investment accounts. Investors may consider applying the Rule of 72 to an existing account to help estimate the future value of the investment.

For those who don’t already have an investment account, they might consider opening one through their bank or a brokerage firm. Some firms may have minimum deposit requirements to open an account, while others may not.

After a brokerage account is funded, an investor can then start placing orders to buy and sell investments online or through a stockbroker who will execute trades on behalf of the investor.