Rule of 72
The Rule of 72 is a simple way to determine how long it will take for an investment or price level to double, given a fixed annual rate of return.
The rule states that the number of years required to double your money is approximately equal to 72 divided by the annual expected return, expressed as an integer (not a percentage).
- So, if you expect your investments to grow at a rate of 8 percent per year, it will take approximately 9 years (72/8) for them to double.
- If you earn 12 percent per year trading, you’d expect to double your returns every 6 years (72/12).
- If you expect inflation to increase 3 percent per year, you’d anticipate prices of goods and services to double every 24 years (72/3).
Of course, this is just a rule of thumb and not an exact science. It’s essentially a mental math shortcut.
Nevertheless, it can be a helpful tool for quickly estimating how long it will take your money to grow or price level to increase.
And, it can be especially useful when comparing different investment or trading opportunities.
For example, let’s say you’re trying to decide between investing in a stock that you expect will return 10 percent per year and a bond that you expect will return 5 percent per year.
Using the Rule of 72, you can estimate that it would take approximately 7.2 years for your money to double with the stock investment (72/10) and 14.4 years for the bond investment (72/5).
So, in this case, the stock investment would be the better option taking return as the only criteria because it would take less time for your money to grow.
Of course, there are other factors to consider (e.g., risk, goals) when making investment or trading decisions.
But, the Rule of 72 can be a helpful shortcut for quickly estimating how long it will take your money to grow.
There are other variations of the same rule we’ll cover below (e.g., Rule of 70).
Key Takeaways – Rule of 72
- Quick Doubling Estimate
- The Rule of 72 provides a simple way to estimate how long it will take for an investment, trade, price level, or asset price to double by taking 72 and dividing by the annual return rate.
- Derivation
- The Rule of 72 is derived from the compound interest formula by solving for the doubling time when an investment/trade/asset price/price level grows exponentially.
- Specifically, using the natural logarithm equation n = ln(2) / ln(1 + r).
- The term ln(2) represents the natural logarithm of 2 (since doubling means reaching 2 times the original value), which approximately equals 0.693. This is why 69-70 is mathematically more precise, but 72 is chosen for easier divisibility across a broader range of whole numbers.
- Comparing Investment Options
- Traders can use it to quickly compare different returns.
- For example, a 10% return doubles money in about 7.2 years, while a 5% return takes 14.4 years.
- The Rule of 70 would estimate 7 years and 14 years, respectively, which makes it a better fit for certain divisions.
- Inflation & Market Impact
- It also applies to inflation, showing how long it takes for prices to double (e.g., at 3% inflation, prices double in 24 years).
- Not Exact, Just a Shortcut
- While not perfectly precise, it offers an easy mental math shortcut for estimating compounding effects without more complex calculations.
- Easier Than Other Rules
- The Rule of 72 is favored over the Rule of 70 because it’s more divisible (by 1, 2, 3, 4, 6, 8, 9, and 12), making quick calculations simpler for traders.
How Does the Rule of 72 Work?
The Rule of 72 works by taking the number 72 and dividing it by the expected annual return on your investment.
The resulting number is approximately equal to the number of years it will take for your money to double.
For example, if you expect your investments to grow at a rate of 8 percent per year, it would take approximately 9 years (72/8) for them to double.
Similarly, if you expect your investments to return 6 percent per year, it would take approximately 12 years (72/6) for them to double.
How Is the Rule of 72 Derived?
The Rule of 72 is based on the compounding interest formula:
A = P(1+r/n)^nt
- P is the original investment (the principal).
- r is the annual interest rate.
- n is the number of compounding periods per year; for example, if you’re investing in a savings account that compounds interest monthly, n would be 12 (12 compounding periods per year).
- t is the number of years the investment is held.
- A is the final value of the investment after t years.
The Rule of 72 can be derived from this formula by solving for t when A = 2P (i.e., when the investment has doubled).
The more accurate number for the Rule of 72 would be 69 or 70 (hence why there exists a Rule of 69 and a Rule of 70), but 72 is commonly used because 72 is easily divisible by more numbers than 69 or 70.
Rule of 72 Formula
The formula for the Rule of 72 is:
T = 72 / r
Where T is the number of years it will take for an investment to double and r is the annual rate of return.
For example, if you expect your investments to return 10 percent per year, you would plug 10 into the formula like this:
- T = 72 / 10
Which would give you a result of 7.2 (it would take approximately 7.2 years for your money to double at a 10 percent annual return).
Similarly, if you expect your investments to grow at a rate of 8 percent per year, you would plug 8 into the formula like this:
- T = 72 / 8
Which would give you a result of 9 (it would take approximately 9 years for your money to double at an 8 percent annual return).
As you can see, the Rule of 72 is simply a shortcut for calculating the amount of time it will take for an investment to double.
And, it can be a helpful tool for quickly estimating how long it will take your money to grow.
How Accurate is the Rule of 72?
The Rule of 72 is only a quick rule of thumb and not as exact as what will be produced by accurately following the compound interest formula.
For example, if you wanted to know how long it would take for an 8 percent investment to double, the rule of 72 would predict 9 years.
To calculate the number of years it will take for an investment to double using the compound interest formula, we can rearrange the equation as follows:
n = ln(FV/PV) / ln(1+r)
substituting in the values from our example:
- n = ln(2) / ln(1 + 0.08)
- n = 9.006
The Rule of 72, in this case, is very accurate.
Let’s take a different number.
Let’s take the example of many central banks targeting an inflation rate of two percent.
Based on the Rule of 72, the price level of goods and services would double every 36 years.
Plugging into the compound interest formula:
- n = ln(2) / ln(1.02)
- n = 35.002
In this case we can see that the Rule of 72 is inaccurate by about one year, but is still less than three percent off.
This is why the Rule of 70 is considered more accurate, as it would have gotten it exactly. But the Rule of 72 generally enables easier mental calculations.
What Is the Rule of 70?
Similarly, the Rule of 70 is a simple way to determine how long it will take for an investment or price level to double given a certain annual rate of return.
The rule states that you can divide the number 70 by the annual rate of return to get the approximate number of years it will take for your money to double.
For example, if you are earning a 7 percent annual rate of return on your investment, it will take approximately 10 years for your investment to double (70/7 = 10).
It can also be used for retirement planning purposes.
For instance, if you expect inflation to average 2 percent annually, you might expect your cost of living to double every 35 years.
How Does the Rule of 70 Work?
The Rule of 70 is based on the power of compounding interest.
Compounding interest occurs when the interest earned on an investment is reinvested and begins to earn additional interest.
This process can continue for years, resulting in exponential growth.
The Rule of 70 is a quick way to estimate the number of years it takes for an investment to double without having to do any complicated math.
How Is the Rule of 70 Derived?
The Rule of 70 is derived from the compound interest formula.
This formula states that the value of an investment will grow at a rate equal to the annual interest rate raised to the power of the number of years the investment is held.
For example, if you have an investment that earns a 7 percent annual return and you plan to hold it for 10 years, the future value of your investment can be calculated using the compound interest formula as follows:
FV = PV (1+r)^n
where:
- FV = future value of investment
- PV = present value of investment
- r = annual interest rate
- n = number of years invested
Substituting in the values from our example:
- FV = $1,000 * (1 + 0.07) ^ 10
- FV = $1,967.15
To calculate the number of years it will take for an investment to double using the compound interest formula, we can rearrange the equation as follows:
n = ln(FV/PV) / ln(1+r)
substituting in the values from our example:
- n = ln($1,967.15/$1,000) / ln(1 + 0.07)
- n = 9.999… or approximately 10 years
As can be observed, the compound interest formula produces a very similar result to the Rule of 70.
The Rule of 70 is a quick and easy way to estimate how long it will take for an investment/price level/trade to double, which can be helpful when planning for future goals.
However, it’s important to keep in mind that the rule is not exact and will produce slightly different results than the compound interest formula.
Why Is the Rule of 70 Important?
The Rule of 70 can be used as a tool to help you reach your financial goals. By understanding how long it will take for your money to double, you can develop a savings plan that allows you to reach your goals in a specific timeframe.
For example, if you know you can get 3.5 percent in real returns per year, then you know your investment will double (in inflation-adjusted terms) in 20 years.
The Rule of 70 can also be used as a way to compare investment options.
The one with the shorter doubling time is the better investment, holding risk constant.
You can also use the Rule of 70 for reverse engineering what you need.
For example, if you want your investment to double every 10 years, you know you need 7 percent per year to achieve your goal. That could pertain to a standard passive investment, such as a stock or bond, or a personal project.
The Rule of 70 can be a helpful tool in personal finance, but it’s important to remember that it’s only an estimate. In reality, investments can take longer or shorter than predicted to double.
Investors should also keep in mind that the Rule of 70 doesn’t account for inflation on its own. Over time, inflation can reduce the purchasing power of your money, even if your investment is growing at a healthy rate.
For example, let’s say you have $10,000 invested at a 7 percent annual rate of return. In 10 years, your investment will be worth approximately $20,000.
However, if inflation is 3 percent per year, the purchasing power of your $20,000 will be equivalent to $14,802 today based on a real return of four percent per year.
Real returns are what are most important over time as the idea of an investment is to preserve or grow the purchasing power of money, not simply grow money itself.
Rule of 70 and Inflation
The rule of 70 can also be applied to other phenomena like inflation.
For example, if inflation is 3 percent, you can calculate how many years it would take of 3 percent inflation for the price level for goods and services (as an average) to double.
This would be about 23 years, taking 70 divided by 3 (23.33).
It is also commonly used for things in other fields, such as population doubling time.
Should I use the Rule of 72 or the Rule of 70?
The Rule of 70 is usually more accurate, but the Rule of 72 enables easy division by 3, 4, 6, 8, 9, and 12, for example, while 70 does not.
In general, the Rule of 72 will give you an answer that is close enough for many purposes.
Rule of 72 – FAQs
How Can I Use the Rule of 72?
The Rule of 72 can be a helpful tool for giving a rough estimate of how long it’ll take for your money to double given a certain return.
It can also be useful for comparing different investment opportunities.
For example, if you’re trying to decide between two investments, one with an 8 percent annual return and one with a 10 percent annual return, the rule of 72 can help you quickly estimate which one will grow your money faster.
Plugging 8 percent into the formula, we get:
T = 72 / 8 = 9 years
And plugging 10 percent into the formula, we get:
T = 72 / 10 = 7.2 years
As you can see, the investment with the higher annual return (10 percent) will grow your money faster (in 7.2 years vs. 9 years).
Keep in mind that the Rule of 72 is only an estimate.
Why Is the Rule of 72 Important?
The Rule of 72 is important because it can help you understand how compound interest works and how long it will take for your money to grow.
Compound interest is often called the “miracle of compounding” because it can cause your money to grow very quickly over time.
The rule of 72 can help you understand how quickly compound interest can work.
For example, if you invest $1,000 at an 8 percent annual return, your investment will be worth $2,000 in 9 years (72 / 8 = 9).
When Is the Rule of 72 Not Accurate?
The Rule of 72 makes some simplifying assumptions that might not always be realistic.
For example, it assumes that:
The investment will compound continuously
In reality, most investments don’t compound continuously, but rather on a yearly, monthly, or quarterly basis.
Nevertheless, the Rule of 72 can still be a helpful tool for estimating how long it will take your money to grow.
The interest rate stays constant
In reality, interest rates are often variable and can change over time.
Nevertheless, the Rule of 72 can still be a helpful tool for estimating how long it will take your money to grow.
It emphasizes divisibility by a range of common whole integers, not accuracy
The Rule of 72 prioritizes ease of divisibility by a broad range of common whole integers rather than mathematical accuracy.
The key integers it divides cleanly by are 1, 2, 3, 4, 6, 8, 9, and 12.
For example, if an investment earns a one percent rate of return, it will double in a bit under 70 years, not in 72 years as the Rule of 72 would suggest, giving an estimation error of a bit over two years.
What is the Rule of 70?
The Rule of 70 is a quick and easy way to estimate how long it will take for an investment to double.
How is the Rule of 70 calculated?
To calculate the number of years it will take for an investment to double using the Rule of 70, divide the number 70 by the annual rate of return.
What are some limitations of the Rule of 70?
The Rule of 70 is only an estimate and will produce slightly different results than the compound interest formula.
Like the Rule of 72, the Rule of 70 is used because of ease of divisibility by a broad range of common integers rather than mathematical accuracy.
The key integers for the Rule of 70 are 1, 2, 4, 5, 7, 10, 14, and 35.
It’s nonetheless slightly more accurate than the Rule of 72.
How can I use the Rule of 70 to reach my financial goals?
By understanding how long it will take for your money to double, you can develop a savings plan that allows you to reach your goals in a specific timeframe.
What is the difference between the Rule of 70 and the compound interest formula?
The Rule of 70 is a quick and easy way to estimate how long it will take for an investment to double, while the compound interest formula is more exact.
How can I account for inflation when using the Rule of 70?
Inflation can reduce the purchasing power of your money, even if your investment is growing in dollar terms.
To account for inflation, use a real rate of return rather than a nominal rate of return.
How is the Rule of 70 different from the Rule of 72?
The Rule of 72 is a similar calculation, but it uses the number 72 instead of 70, mostly because the mental math is easier (more numbers will evenly divide when using the Rule of 72 vs. the Rule of 70).
The Rule of 72 will produce slightly different results than the Rule of 70, but they will be close.
What is the history of the Rule of 70?
The Rule of 70 has been used by economists and mathematicians for centuries as a way to estimate exponential growth.
A similar calculation, known as the Rule of 72, has been used since the 18th century.
Summary – Rule of 72
The Rule of 72 is a quick and easy way to estimate how long it will take your money to double given a certain return.
You can perform the calculation by taking 72 and dividing it by the return percentage.
So, for example, if your investment returned 4 percent per year, it would take 18 years to double according to the Rule of 72.
It can also be used for things like inflation (how long will it take for the price level to double at X percent inflation) or for determining population growth.
In trading, investing, and financial market contexts, it can be helpful for comparing different investment opportunities and for understanding how quickly compound interest can work.