What Exactly Is the ‘Rule of 72’?
The “Rule of 72” is a simple and practical formula that is commonly used to estimate the number of years necessary to double the amount of money invested at a particular annual rate of return.
Alternately, it can compute the annual rate of compounded return from an investment by using the number of years necessary to double the investment as input.
Although calculators and spreadsheet programs like Microsoft Excel have functions that can accurately calculate the precise amount of time required to double the amount of money invested, the Rule of 72 is useful for performing mental calculations in order to quickly gauge an approximate value.
As a result of this, the Rule of 72 is frequently taught to new investors because it is simple to comprehend and calculate. In resources geared toward grade-level financial literacy, the Securities and Exchange Commission (SEC) also mentions the Rule of 72.
KEY POINTS
- The simplified formula known as the Rule of 72 is able to determine, based on an investment’s rate of return, how long it will take for the value of the investment to increase by a factor of two.
- The Rule of 72 is used to calculate compound interest rates, and it provides results that are accurate to within a reasonable margin of error for interest rates that fall within the range of 6% to 10%.
- The so-called “Rule of 72” can be applied to anything that rises at an exponential rate, such as GDP or inflation; it can also be used to indicate the effect that annual fees will have over the long term on the growth of an investment.
- This estimation tool can also be used to estimate the rate of return required for an investment to double given the investment period.
- Utilizing the Rule of 69, Rule of 70, or Rule of 73 is typically the most effective strategy when dealing with various scenarios.
Mathematical Expression for the 72-Hour Rule
There are two ways to use the Rule of 72 to calculate a break-even point or a minimum acceptable rate of return.
1. Years to Double: 72/ ROI Expectation
Divide the anticipated rate of return by 72 to get the time in which an investment will double. The formula uses a single average rate calculated over the entire duration of the investment. Since each decimal represents a different fraction of a year, the results are just as accurate when expressed in fractional form.
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2. Rate of Return Expected: 72 / Years to Double
Divide 72 by the number of years it will take for your initial investment to double to get your annualized rate of return. The formula works with decimal values of the number of years as well as whole years.
The resulting expected rate of return also takes into account the impact of compounding interest throughout an investment’s holding period.
FACT:
Compound interest, and not simple interest, is what the Rule of 72 is for. Interest accrued using the simple interest method is calculated by multiplying the principal by the interest rate per day and then dividing by the number of days until the next payment is due.
When determining how much interest to add to a deposit each period, compound interest is applied not only to the initial deposit but also to the interest that has accrued from previous periods.
A Guide to Applying the Rule of 72.
Things like population, macroeconomic numbers, charges, and loans can all grow at a compounded rate and thus fall under the purview of the Rule of 72. It is estimated that the economy will double in size in 72 / 4% = 18 years if GDP growth averages 4% per year.
The Rule of 72 can be applied to illustrate the compounding impact of the fee that reduces investment returns over time. It would take about 24 years for the annual expense ratio of a mutual fund that charges 3% per year to eat away at the initial investment by half.
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To put it another way, if you borrow money and pay interest at a rate of 12% per year on your credit card (or any other loan that charges compound interest), your debt will double in just six years.
The rule can also be used to calculate how long it will take for the purchasing power of a currency to decrease by half as a result of inflation. An amount of money’s purchasing power will be cut in half after about 12 years if inflation averages 6% (72 divided by 6 equals 12). The average investment is expected to lose half its value after 18 years instead of 12 years if inflation is reduced from 6% to 4%.
In addition, as long as the rate of return is compounded annually, the Rule of 72 can be applied to investments with any duration. It would take (72 / 4) = 18 quarters, or 4.5 years, to double the principal if the interest rate was 4% per quarter but interest was only compounded once a year. When a country’s population grows by 1% per month, it will double in size in just over six years.
Who Came Up With the Rule of 72?
As mentioned by Luca Pacioli in his comprehensive mathematics text Summa de Arithmetica, the Rule of 72 has its origins in 1494. Some have hypothesized that the rule predates Pacioli’s novel because he provides no derivation or explanation for why it might work.
How to apply the Rule of 72 in calculation.
The 72-Hour Rule is explained below. Divide 72 by the expected annual return on the investment. If you double your initial investment every year, the result will be the approximate number of years it will take.
If an investment plan promises a return of 8% per year after taxes, it will take about nine years (72 / 8 = 9) for your money to double in value. Take into account that a CAGR of 8% is entered as 8, not 0.08, yielding a result of nine years (and not 900).
If you invest $1,000 and it doubles every nine years, you’ll have $2,000. If it doubles every 18 years, you’ll have $40,000. If it doubles every 27 years, you’ll have $8,000.
Just How Reliable Is the Rule of 72?
Given that it is a simplification of a more complex logarithmic equation, the Rule of 72 formula yields a time estimate that is close enough for most practical purposes. You’d have to do the whole calculation to get the correct doubling time, though.
For an investment yielding r percent compound interest per period, the formula for determining when it will double is as follows:
Here’s the formula you can use to calculate how long it will take for your investment to double if you earn 8% per year:
T = ln(2 / 1) + (8 / 100) = 9.006 years
This is a very close approximation to the value obtained by dividing (72 by 8) to get 9 years.
Difference Between Rules 72 and 73?
For best results, use an interest rate or rate of return between 6% and 10% when applying the rule of 72. Interest rates outside this range can be handled by either adding or subtracting 1 from 72 for every 3 points the interest rate deviates from the 8% threshold. For illustration, a compound interest rate of 11% per year is 3% higher than a rate of 8% per year.
Therefore, the rule of 73 can be applied for greater accuracy by adding 1 (for the 3 points higher than 8%) to 72. The rule of 74 applies to a ROI of 14% (by adding 2 for 6 percentage points higher), while the rule of 71 applies to a ROI of 5% (by subtracting 1 for 3 percentage points lower).
Take the hypothetical case of an extremely alluring investment yielding a rate of return of 22% as an illustration. An investment will have multiplied in value after 3.27 years, according to the simple Rule of 72. To make up for this, the numerator of the modified rule should now be set to 72 + 5 = 77 (22 – 8 = 14; 14 3 = 4.67 5).
In comparison to the result of 3.27 years obtained from the basic rule of 72, this yields a value of 3.5 years, indicating that you will have to wait an additional quarter to double your money. Adjusting the rule yields a more precise result because the period given by the logarithmic equation is 3.49.
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