# What is the Rule of 72? Robinhood Learn
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Definition:

The rule of 72 is a simple formula to estimate how long it will take to double your investment or how long it will take for your money to lose half its value due to inflation.

## 🤔 Understanding the rule of 72

The rule of 72 is a simple formula that can help estimate the effect of exponential growth, such as on a savings account with compounded interest (interest added back to the principal at fixed intervals). It can also estimate the effect of exponential decay (like how your money can lose value due to inflation). This calculation is a simplified version of the original logarithmic formula –- The rule of 72 lets you get a rough estimate of how long it will take to double or halve your money without the need for a scientific calculator or log tables. It’s important to remember that the rule of 72 doesn’t take into account any fees or taxes that affect your returns if you’re calculating growth.

The formula is: ## Takeaway

The rule of 72 is like measuring a gemstone with your hand…

You’re making an estimate. You want to get an idea of what value it might have, but you should probably bring it to a gem laboratory (or do a more sophisticated calculation) before you assume what it could be worth (make your investment decisions). Certain limitations apply

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## History of the rule of 72

In 1494, the Italian mathematician Luca Pacioli first mentioned the importance of the number 72 in his book, “Summary of Arithmetic, Geometry, Proportions, and Proportionality” (“Summa de arithmetica geometria, proporzioni et proporzionalità.”) Pacioli said that you could use the number 72 to deduce the number of years it would take your money to double.

The rule of 72 was written nearly a century later. It is based on the standard compound interest formula: A = P (1 + r/n) nt. ‘A’ represents the interest you’ve earned plus your principal (your final investment total). ‘P’ is the principal or original investment. The ‘r’ is the interest rate in decimal form. The ‘n’ is the number of compounding periods. And ‘t’ stands for the time in years.

If we want to double our money, we can substitute A = 2 and P = 1. That leaves us with 2 = 1 ( 1 + r/n) nt.

If we assume our interest rate compounds annually, we can also replace n for 1. Now, we have 2 = 1 ( 1 + r/1)1*t. We can simplify this equation to 2 = (1 + r)t.

Now, let’s take the logarithm of both sides to simplify the equation further: ln 2 = ln (1 + r )nt.

Next, use the power rule to bring down the exponent. ln 2 = t * ln (1 + r).

The natural logarithm of 2 is about 0.693. And for small values of r, ln ( 1 + r ) ≈ r. In other words, we can say, 0.693 ≈ t * r.

We can multiply both sides by 100 so that we can use the interest rate as a whole number, instead of a decimal. So, we have 69.3 ≈ t * r (where r is a percentage).

Finally, to isolate t as the number of years it’ll take to double our investment, we can divide by 100r to get 69.3 / r ≈ t (where r is a percentage).

Since 69.3 is a difficult number to divide into, statisticians and investors agreed on using the next nearest integer with many divisible factors – 72. So 72 divided by the interest rate (expressed as a percentage) gives you the approximate time (number of years) it’ll take to double your investment.

## What does the rule of 72 tell you?

People like to see how their money grows — especially how their investment doubles. The calculation to figure out how much time it will take to double your money is related to the compound interest formula. Since most people can’t do that formula without a calculator, the rule of 72 is a useful shortcut to give a rough estimate of an investment’s doubling time.

An important distinction of this rule is that it doesn’t use the simple interest rate (your initial investment amount multiplied by the rate of interest multiplied by time). Instead, the rule of 72 uses compound interest (interest on your original investment plus the interest earned on your previous interest). In other words, the rule of 72 assumes that every time your investment pays interest, you reinvest that money. Your interest is also working to earn more interest.

Compound interest helps your investment grow faster. The rule of 72 tells you approximately how long it’ll take you to get there.

## How do you calculate the rule of 72?

While deriving the rule of 72 requires a bit more math, the rule of 72 only involves division. You can estimate the doubling time of nearly any investment by dividing 72 by the annual growth rate. You should use the interest rate’s whole number, not the percentage or decimal.

For example, let’s say you have a \$1 investment that has a 6% annual fixed interest rate. 72 divided by 6 is 12. So it would take 12 years for your \$1 to grow to \$2.

The rule of 72 can also tell you about money decay. For instance, if inflation is 8%, then 72 divided by 8 tells you that your money will be worth about half its current value in about 9 years (72 / 8). On the other hand, if inflation decreases to 6%, your money would then lose half its value in 12 years (72 / 6).

## What is the difference between the rule of 69 vs. the rule of 70 vs. the rule of 72?

The rule of 72 is best for annual interest rates.

On the other hand, the rule of 70 is better for semi-annual compounding. For example, let’s suppose you have an investment that has a 4% interest rate compounded semi-annually or twice a year.

According to the rule of 72, you’ll get 72 / 4 = 18 years. If you use the rule of 70, you’ll get 70 / 4 = 17.5 years.

Finally, if you do the original logarithm calculation, it’ll actually take you about 17.501 years to double your money. So, the rule of 70 is a better estimate.

The rule of 69 gives more accurate results for continuous compounding (extreme compounding where you reinvest the interest continuously as often as possible), such as monthly or daily. For instance, let’s compare the rules on an investment that has a 3% interest rate compounded daily.

According to the rule of 72, you’ll double your money in 24 years (72 / 3 = 24). According to the rule of 70, you’ll double your money in about 23.3 years (70 / 3 = 23.3). But, the rule of 69 says that you’ll double your money in 23 years (69 / 3 = 23).

Finally, the compound interest formula says that you’ll actually double your money in about 23.1 years. So, the rule of 69 is closest to the original logarithm calculation.

## When would you need to use the rule of 72?

For example, an investment with a 3% annual interest rate will take about 24 years to double your money. On the other hand, an investment with a 4% yearly rate of return will take around 18 years. A 1% difference in percentage points can mean a difference of 6 years.

Both investments likely carry different levels of risk. However, the rule of 72 can help you plan whether these investments fit with your retirement timeline and goals.

## Does the rule of 72 work?

The rule of 72 is a rough estimate of the compound interest formula to double your money. Here’s a break down to see how accurate the rule is.

Annual Interest RateDoubling Time (Compound Interest Formula)Rule of 72 Estimated Doubling Time
1%69.6672.00
2%35.0036.00
3%23.4524.00
4%17.6718.00
5%14.2114.40
6%11.9012.00
7%10.2410.29
8%9.019.00
9%8.048.00
10%7.277.20
11%6.646.55
12%6.126.00
13%5.675.54
14%5.295.14
15%4.964.80

If you compare the rule of 72 to the original formula, you’ll see that the rule of 72 is best for annual interest rates between 6% and 10%.

For lower interest rates, the rule of 72 tends to slightly overestimate how long it will take to double your money. For higher interest rates, the rule of 72 tends to slightly underestimate how long it will take to double your money.

The free stock offer is available to new users only, subject to the terms and conditions at rbnhd.co/freestock. Free stock chosen randomly from the program’s inventory. Securities trading is offered through Robinhood Financial LLC.

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