# Rule of 72

The length of time required for an investment to double in value at a fixed annual rate of return

The length of time required for an investment to double in value at a fixed annual rate of return

In finance, Rule of 72 is a formula that estimates the amount of time it takes for an investment to double in value, at a fixed annual rate of return. Rule of 72 is a shortcut or back-of-the-envelope calculation to determine the amount of time for an investment to double in value by taking the number 72 and dividing it by the interest rate.

Rule of 72 gives an estimation of the doubling time of an investment. It is a fairly accurate measurement of doubling time and is even more accurate when using lower interest rates than higher ones. It is used for situations involving compound interest; simple interest rate does not work very well with rule of 72. Below is a table showing the difference between rule of 72 calculation and the actual number of years for an investment to double in value:

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The rule of 72 formula for doubling an investment is as follows:

You are the owner of a coffee machine manufacturing company. Due to the large capital needed to establish a factory and warehouse for coffee machines, you turned to private investors to fund the expenditure. You meet with John, who is a high net-worth individual willing to contribute $1,000,000 to your company. However, John is only willing to contribute said amount on the presumption that he will get a 12% annual rate of return on his investment compounded yearly. He wants to first know how long it would take for his investment in your company to double in value.

Using the rule of 72:

Therefore, it will take approximately six years for John’s investment to double in value.

Let us derive the rule of 72 by starting with a beginning arbitrary value: $1. Our goal is to determine how long it will take for money ($1) to double at a certain interest rate.

Suppose we have a yearly interest rate (r). After one year, we will get:

**$1 x (1+r)**

At the end of two years, we will get:

**$1 x (1+r) x (1+r)**

Extending this year after year, we will get:

**$1 x (1+r)^n**, where n = number of years

If we want to determine how long it takes to double, that is, getting $1 to $2:

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

**Solving for 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 of both sides)

Step 4: n x ln(1+r) = .693

Step 5: n x r = 0.693 (Approximation that ln(1+r) = r)

Step 6: n = .693 / r

Step 7: n = 69.3 / r (Turning r into an integer rather than a decimal)

Notice that after deriving the formula, we end up with 69.3 and not 72. Although 69.3 is more accurate, as will be shown in the table below, it is not easily divisible. Therefore, the rule of 72 is used as it provides more factors (2, 3, 4, 6, 12, 24…).

Rule of 69.3 and Rule of 69 are also methods of estimating an investment’s doubling time. However, rule of 69.3 is considered more accurate than rule of 72. As a result, rule of 69.3 can be harder to calculate without the aid of a calculator. Therefore, investors may prefer to use rule of 69 or 72 rather than the rule of 69.3. Comparing the doubling time for rules of 69, 69.3 and 72 to actual years:

As you can see from the table above, rule of 69.3 yields more accurate results at lower interest rates. However, as the interest rate increases, rule of 69.3 loses predictive power.

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