How many years does it take to double your money if the continuously compounded interest rate is 10 %?

Double Your Money: The Rule of 72

The Rule of 72 is a quick and simple technique for estimating one of two things:

Inhaltsverzeichnis Show

Double Your Money: The Rule of 72
What is the Rule of 72?
Time (Years) to Double an Investment
Rule of 72 Formula
Example of the Rule of 72
Deriving the Rule of 72
Rules of 72, 69.3, and 69
More Resources
How long will it take money to double if it is invested at 10% compounded continuously?
How many years does it take to double your money if the continuously compounded interest rate is 6 %?
How long will it take money to double if compounded continuously?
How long will it take money to double if it is invested at 5% compounded continuously?

The time it takes for a single amount of money to double with a known interest rate.
The rate of interest you need to earn for an amount to double within a known time period.

The rule states that an investment or a cost will double when:

[Investment Rate per year as a percent] x [Number of Years] = 72.

When interest is compounded annually, a single amount will double in each of the following situations:

The Rule of 72 indicates than an investment earning 9% per year compounded annually will double in 8 years. The rule also means if you want your money to double in 4 years, you need to find an investment that earns 18% per year compounded annually.

You can confirm the rationality of the Rule of 72 as follows: Find factors on the FV of 1 Table that are close to 2.000. (The factor of 2.000 tells you that the present value of 1.000 had doubled to the future value of 2.000.) When you find a factor close to 2.000, look at the interest rate at the top of the column and look at the number of periods (n) in the far left column of the row containing the factor. Multiply that interest rate times the number of periods and you will get the product 72.

To use the Rule of 72 in order to determine the approximate length of time it will take for your money to double, simply divide 72 by the annual interest rate. For example, if the interest rate earned is 6%, it will take 12 years (72 divided by 6) for your money to double. If you want your money to double every 8 years, you will need to earn an interest rate of 9% (72 divided by 8).

Here's another way to demonstrate that the Rule of 72 works. Assume you make a single deposit of $1,000 to an account and wish for it to grow to a future value of $2,000 in nine years. What annual interest rate compounded annually will the account have to pay? The Rule of 72 indicates that the rate must be 8% (72 divided by 9 years). Let's verify the rate with the format we used with the FV Table:

To finish solving the equation, we search only the "n = 9" row of the FV of 1 Table for the FV factor that is closest to 2.000. The factor closest to 2.000 in the row where n = 9 is 1.999 and it is in the column where i = 8%. An investment at 8% per year compounded annually for 9 years will cause the investment to double (8 x 9 = 72).

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

What is the Rule of 72?

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

Time (Years) to Double an Investment

The Rule of 72 gives an estimation of the doubling time for an investment. It is a fairly accurate measurement, and more so when using lower interest rates rather than higher ones. It is used for situations involving compound interest. A simple interest rate does not work very well with the Rule of 72.

Below is a table showing the difference between the Rule of 72 calculation and the actual number of years required for an investment to double in value:

Rule of 72 Formula

The Rule of 72 formula is as follows:

Example of the Rule of 72

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 have 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 the said amount on the presumption that he will get a 12% annual rate of return on his investment, compounded yearly. He wants to know how long it will take for his investment in your company to double in value.

Using the Rule of 72:

It will take approximately six years for John’s investment to double in value.

Deriving the Rule of 72

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 our money ($1) to double at a certain interest rate.

Suppose we have a yearly interest rate of “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 get:

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

If we want to determine how long it takes to double our money, turning $1 into $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, not 72. Although 69.3 is more accurate, it is not easily divisible. Therefore, the Rule of 72 is used for the sake of simplicity. The number 72 also provides more factors (2, 3, 4, 6, 12, 24…).

Rules of 72, 69.3, and 69

Rules of 69.3 and of 69 are also methods of estimating an investment’s doubling time. The rule of 69.3 is considered more accurate than the Rule of 72, but can be much more troublesome to calculate. Therefore, investors typically prefer to use a 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, the rule of 69.3 yields more accurate results at lower interest rates. However, as the interest rate increases, the rule of 69.3 loses some of its predictive accuracy.

The Rule of 72 is a simple, helpful tool that investors can use to estimate how long a specific compound interest investment will take to double their money.

More Resources

Thank you for reading CFI’s guide on the Rule of 72. Below are additional free resources from CFI:

Investing: A Beginner’s Guide
Hurdle Rate
Return on Investment (ROI) Formula
Financial Modeling Courses Collection

How long will it take money to double if it is invested at 10% compounded continuously?

A 10% interest rate will double your investment in about 7 years (72 ∕ 10 = 7.2); an amount invested at a 12% interest rate will double in about 6 years (72 ∕ 12 = 6). Using the Rule of 72, you can easily determine how long it will take to double your money.

How many years does it take to double your money if the continuously compounded interest rate is 6 %?

How long will it take money to double if compounded continuously?

The rule says that to find the number of years required to double your money at a given interest rate, you just divide the interest rate into 72. For example, if you want to know how long it will take to double your money at eight percent interest, divide 8 into 72 and get 9 years.