How long will it take money to double if it is invested at 9 compounded continuously

Calculating compound interest is complicated. Luckily, there’s a simple shortcut that helps you estimate how a fixed interest rate will affect your savings: the Rule of 72.

The Basics

The Rule of 72 is a tool used to estimate how long it will take an investment to double at a given interest rate, assuming a fixed annual rate of interest. All you need to use the tool is an interest rate, which means you can make estimates for your current account rate or use this rule to know what rate you should look for if you want to double your money by a specific deadline.

To figure out how long it will take to double your money, take the fixed annual interest rate and divide that number into 72. Let’s say your interest rate is 8%. 72 ∕ 8 = 9, so it will take about 9 years to double your money. 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.

To figure out what interest rate to look for, use the same basic formula, but run it backward: divide 72 by the number of years. So if you want to double your money in about 6 years, look for an interest rate of 12%.

The basic algebraic formula looks like this, where Y is the number of years and r is the interest rate:

Y = 72 ∕ r and r = 72 ∕ Y

This rule works for interest rates from about 4% up to about 20%; after that, the error becomes significant and more straightforward math is required.

Illustration: Chelsea Miller

Why 72?

Here, we merely scrape the surface of that “more straightforward math.” To really dive deep into why the rule works, check out this article.

The Rule of 72 is itself an estimation. It uses a concept called natural logarithms to estimate compounding periods. In mathematics, the natural logarithm is the amount of time needed to reach a particular level of growth using continuous compounding.

For math enthusiasts out there: it is easiest to see how this works through continuously compounded interest. (The Rule of 72 addresses annually compounded interest, but we’ll get there in a minute.)

When dealing with continuously compounding interest, you can work out the exact time it takes an investment to double by using the time value of money formula (TVM) and simplifying the equation until eventually, you are left with something like this:

ln(2)= rY

The natural log (ln) of 2 is about 0.693. Solve for interest rate (r) or number of years (Y), and then multiply by 100 to express as a percentage or year, respectively.

Click here to read how this tool works, and for disclaimers.

Click here to read how this tool works, and for disclaimers.

Wait...

If our new formula is based on the number 69.3 (0.693 × 100), that begs the question: Why isn’t it called the Rule of 69.3?

First, that just doesn’t sound quite as good as “The Rule of 72.” Second, there are two points to remember:

1. The “Rule of 69.3” is not an estimation. It is the actual amount of time that it will take money to double, and works for any range of interest rates.

2. The Rule of 69.3 works for continuously compounded interest. The Rule of 72 works for a fixed annual rate of interest.

The math equation for fixed annual interest is slightly more complex, and simplifying it leaves us with approximately 72.7.

Normally, we would round up to 73. However, 72 is much easier to work with, as it is readily divisible by 2, 3, 4, 6, 8, 9, and 12. As we are already estimating, convenience wins out, and we are left with the Rule of 72.

History

The Rule of 72 was first introduced in the late fifteenth century by the Franciscan friar and Italian mathematician Luca Pacioli. A contemporary of Leonardo da Vinci, Pacioli is considered by many to be the father of accounting. The Rule of 72 was introduced in his book Summa de arithmetica, geometria, proportioni et proportionalita, published in 1494 for use as a textbook for schools in what is now northern Italy.

How long will it take money to double if it is invested at​ (A) 9​% compounded​ continuously? (B) 11​% compounded​ continuously?

​(A) At 9​% compounded​ continuously, the investment doubles in ______ years.

​(Round to one decimal place as​ needed.)

​(B) At 11​% compounded​ continuously, the investment doubles in _____ years.

​(Round to one decimal place as​ needed.)

The doubling time formula with continuous compounding is the natural log of 2 divided by the rate of return.

The formula for doubling time with continuous compounding is used to calculate the length of time it takes doubles one's money in an account or investment that has continuous compounding.

It is important to note that this formula will return a time to double based on the term of the rate. For example, if the monthly rate is used, the answer to the formula will return the number of months it takes to double. If the annual rate is used, the answer will then reflect the number of years to double.

Example of the Doubling with Continuous Compounding Formula

An example of the doubling time with continuous compounding formula is an individual would like to calculate how long it would take to double his investment that earns 6% per year, continuously compounded. The individual could either calculate the number of years or calculate the number of months to double his investment by using the annual rate or the monthly rate. Using the doubling time for continuous compounding formula, the time to double at a rate of 6% per year would show

This equation would return a result of 11.55 years.

How is the Doubling Time with Continuous Compounding Formula derived?

The doubling time with continuous compounding formula can be found by first looking at the continuous compounding formula individually.

FV is the future value and PV is the present value. To double one's money would be to have the future value equal to twice the amount of the present value. Considering this, we can substitute 2 for FV and 1 for PV in the formula above. This would show

This formula can be adjusted as

Since we are solving for t, this formula can be rewritten as

The denominator of the formula above becomes simply r, which is the formula shown at the top of the page.

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• Formulas related to Doubling Time - Continuous Compounding
• Doubling Time
• Continuous Compounding
• Doubling Time - Simple Interest

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