How long will it take money to double itself if invested at 10% compounded annually?
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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 BasicsThe 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:
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. HistoryThe 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. Double Your Money: The Rule of 72The Rule of 72 is a quick and simple technique for estimating one of two things:
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). |
