Rule of 72

Exploration Essay

The Rule of 72 is a quick, useful formula that is popularly used to estimate the number of years required to double the invested money at the estimated annual rate of return. On the other hand, it can also calculate the annual rate of compounded return from an investment given how many years it will take to double the investment. While calculators and spreadsheet programs like Microsoft Excel have functions to accurately calculate the precise time required to double the invested money, the Rule of 72 comes is mostly used for calculations in your head to quickly get an approximate value. For this reason, the Rule of 72 is often taught to beginning investors as it is easy to understand and calculate. The Rule of 72 can be used in two different ways to determine an expected doubling period or required rate of return. To calculate the time period an investment will double, divide 72 by the expected rate of return. The formula relies on a single average rate over the life of the investment. The answer usually has decimals as all decimals represent an additional portion of a year. To calculate the expected rate of interest, divide 72 by the number of years required to double your investment. The number of years does not need to be a whole number, the formula can be in fractions or portions of a year. In addition, the expected rate of return also assumes compounding interest at that rate over the entire holding period of an investment. Here's how the Rule of 72 works. You take the number 72 and divide it by the investment's projected annual return. The result is the number of years, approximately, it'll take for your money to double. For example, if an investment project promises an 8% annual compounded rate of return, it will take approximately nine years (72 / 8 = 9) to double the invested money. A compound annual return of 8% is plugged into this equation as 8, and not 0.08, giving a result of nine years (and not 900). If it takes nine years to double a $1,000 investment, then the investment will grow to $2,000 in year 9, $4,000 in year 18, $8,000 in year 27, and so on. Another example is if the gross domestic product (GDP) grows at 4% annually, the economy will be expected to double in 72 / 4% = 18 years. The Rule of 72 can also be used to demonstrate the long-term effects of these costs. For example, a mutual fund that charges 3% in annual expense fees will reduce the investment principal to half in around 24 years. A borrower who pays 12% interest on their credit card (or any other form of loan that is charging compound interest) will double the amount they owe in six years. The rule can also be used to find the amount of time it takes for money's value to lose due to inflation. If inflation is 6%, then a given purchasing power of the money will be worth half in around 12 years (72 / 6 = 12). If inflation decreases from 6% to 4%, an investment will be expected to lose half its value in 18 years, instead of 12 years. Additionally, the Rule of 72 can be applied across all kinds of durations provided the rate of return is compounded annually. If the interest per quarter is 4% (but interest is only compounded annually), then it will take (72 / 4) = 18 quarters or 4.5 years to double the principal. If the population of a nation increases at the rate of 1% per month, it will double in 72 months, or six years. The Rule of 72 dates all the way back to 1494 when Luca Pacioli referenced the rule in his comprehensive mathematics book called Summa de Arithmetica. Pacioli makes no explanations of why the rule may work, so some say the rule was already made before the novel. The Rule of 72 formula provides an almost accurate but approximate answer due to the fact that it's a simplification of a more complex logarithmic equation. To get the exact doubling time, you'd need to do the entire calculation. The precise formula for calculating the exact doubling time for an investment earning a compounded interest rate of % per period is: To find out exactly how long it would take to double an investment that returns 8% annually, you would use the following equation: T = ln(2) / ln (1 + (8 / 100)) = 9.006 years As you can see, this result is very close to the approximate value obtained by (72 / 8) = 9 years. The rule of 72 mostly works with interest rates or rates of return that fall in the range of 6% and 10%. When dealing with rates outside this range, the rule can be adjusted by adding or subtracting 1 from 72 for every 3 points the interest rate diverges from the 8% threshold. For example, the rate of 11% annual compounding interest is 3 percentage points higher than 8% so adding 1 (for the 3 points higher than 8%) to 72 leads to using the rule of 73 for higher precision. For a 14% rate of return, it would be the rule of 74 (adding 2 for 6 percentage points higher), and for a 5% rate of return, it will mean reducing 1 (for 3 percentage points lower) to lead to the rule of 71. For example, let’s say you have an investment offering a 22% rate of return. The basic rule of 72 says the initial investment will double in 3.27 years. However, since (22 – 8) is 14, and (14 ÷ 3) is 4.67 ≈ 5, the adjusted rule should use 72 + 5 = 77 for the numerator. This gives a value of 3.5 years, indicating that you'll have to wait an additional quarter to double your money compared to the result of 3.27 years obtained from the basic rule of 72. The period given by the logarithmic equation is 3.49, so the result obtained from the adjusted rule is more accurate. For daily or continuous compounding, using 69.3 in the numerator gives a more accurate result. Some people adjust this to 69 or 70 for easier calculations