# Rule of 72

## What Is the Rule of 72?

The Rule of 72 is a simple, effective formula that is widely employed to calculate the amount of time needed to double the investment at a certain year-to-year rate. In addition, it can calculate the annual compounded return on an investment about the time it takes to double the amount invested.

While spreadsheet and calculator programs such as Microsoft Excel have functions to precisely calculate the amount of time needed to double capital invested, the Rule of 72 comes in handy to help you mentally calculate and quickly determine the approximate value. This is why that Rule of 72 is often taught to investors just beginning their journey, as it is easy to grasp and calculate. The Security and Exchange Commission also cites it also mentions it as the Rule of 72 in grade-level financial literacy materials. 1

### KEY TAKEAWAYS

• The Rule of 72 is a simplified formula that calculates the amount of time it will take for a person’s investment to increase its value depending on the rates of returns.
• The Rule of 72 applies to compounded interest rates and is reliable for interest rates within the range of 10% and 6 percent.
• The Rule of 72 can be applied to any item that increases exponentially, like inflation or GDP. It could also be used to determine the long-term impact of the annual fee on investment growth.
• This tool is also able to calculate the amount of return required to make an investment double in the course of an investment.
• In different circumstances, It is usually better to apply one of the following rules: Rule of 69, Rule of 70, and the Rule of 73.

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### Rule of 72 Formula

This formula is known as the Rule of 72 formula is as the following:

### Example of the Rule of 72

You own an enterprise that manufactures coffee machines. Due to the huge investment required to build an office and warehouse for coffee machines, your company has looked to private investors to finance the expense. You meet John, a high-net-worth person willing to give \$1,000,000 to your organization.

But, John is only willing to make the required contribution on the assumption that he’ll earn an annual rate of 12% of the return on his investment multiplied every year. John would like to know how it will take the money he invests in your business to increase in value by a factor of.

Using the Rule of 72:

It could take six years for John’s money investment to double in value.

### Deriving the Rule of 72

Let’s find our Rule of 72 by starting with an undetermined amount of \$1. Our objective is to find the time it takes for our cash (\$1) to increase by a specific interest rate.

If we take a hypothetical annual percentage, that is “r.” In one year we receive:

\$1 x (1+r)

After two years, we’ll get:

\$1 x (1+r) x (1+r)

The cycle continues year following year, we receive:

\$1 1 (1+r)^n, where n is the number of years

If we are trying to figure out the amount of time needed to double our money changing the equivalent of \$1 to \$2

\$1 x (1+r)^n = \$2

Solving for many years (n):

Step 1: \$1 x (1+r)^n = \$2

Step 2: (1+r)^n = \$2

Step 3. ln((1+R)^n) = ln(2) (Taking the natural log on both sides)

Step 4: nx ln(1+r) equals .693

Step 5: nx (r) is 0.693 (Approximation of ln(1+r) is r)

Step 6: n = .693 / r

Step 7: n = 69.3 / r

Note that when we deduce from the formula, we come at 69.3 instead of 72. While 69.3 is more precise, however, it isn’t easy to divide. This is why this Rule of 72 is used for simplicity. It also offers additional factors (2 3, 4 6, 12, 24). …).

### Rules of 72, 69.3, and 69

The rules of 69.3 and 69 are also ways of estimating the time to double an investment. This Rule 69.3 is thought to be more precise than the Rule of 72. However it could be more difficult to determine. So, investors generally prefer to follow the Rule of 69 and 72 over using the rules of 69.3.

## How Accurate Is the Rule of 72?

It is believed that the Rule of 72 formula provides an reasonably precise, yet approximate, timeline. This reflects the fact that it’s a simplified version of a more complicated logarithmic equation. To find the exact time of doubling it is necessary to perform the whole calculation.

The exact formula to calculate the exact time to double to earn an interest rate compounded of r% for a period is:

To determine exactly the time it will take to double the value of an investment that earns 8percent per year then you can use an equation like this:

T = ln(2) * ln (1 + (8 100)) = 9.006 years

This figure is extremely close to the value that is obtained by (72 8) = 9 years.

## What Is the Difference Between the Rule of 72 and the Rule of 73?

The 72 rule generally is applicable to the rates of returns or interest rate that are within the range of 10% and 6 percent. For rates that are not within this limit, the Rule can be modified by subtracting or adding one from 72 for each 3 points at which the interest rate differs from the threshold of 8. For instance 11% compounding annual interest is three percentage points higher than 8%..