The Rule of 72 in finance refers to a mathematical formula to illustrate compounding. Here we'll explore what it is, how it works, how to use it, and examples.
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What Is the Rule of 72?
The Rule of 72 is simply a mathematical formula to estimate the number of years for an investment to double based on its rate of return.
For a fixed income security like a bond, this would be its interest rate, called the coupon rate, for which the Rule of 72 illustrates the power of compound interest. For stocks, technically we'd call that compound returns, as stocks do not pay an interest rate.
It's important to note right off the bat that, as with most “rules” in finance such as the “4% Rule,” the Rule of 72 is not a rule but rather simply a rough estimate. Basically, the Rule of 72 allows you to calculate compound interest in your head without a calculator.
Next we'll look specifically at how the Rule of 72 works.
How the Rule of 72 Works
To explain how the Rule of 72 works, let's first briefly talk about interest and dividends.
Remember, a savings account or a bond will pay an interest rate, and stock will have a dividend yield. These are cash payments made to the investor for holding that investment.
The investor gets to choose what to do with those payments. She can withdraw them and buy something, or reinvest them on top of the initial principal.
That reinvesting is what creates what we call compounding. When reinvested, the investment's interest rate now applies to that greater principal amount, effectively earning interest on interest.
Rule of 72 Formula for Calculation
The formula for the Rule of 72 to calculate the number of years for an investment to double is as follows:
y = 72 / r
where y is the years to double and r is the expected rate of return or interest rate of the investment.
Math-savvy readers will recognize the Rule of 72 as being a rough approximation of the exponential function.
We can also use the Rule of 72 in reverse to see the interest rate required for an investment to double in a given number of years, such as:
r = 72 / y
where r is the rate of return and y is number of years.
Next we'll look at some examples of the Rule of 72.
Rule of 72 Examples
Now let's look at some examples of the Rule of 72 to help you fully understand it.
Suppose Investment A has an interest rate of 8%.
y = 72 / 8 = 9
So this investment will take 9 years for the principal to double in value.
Now suppose another investment, Investment B, has an interest rate of 6%.
y = 72 / 6 = 12
Investment B will take 12 years to double in value.
While this may be obvious, this illustrates that an investment with a higher interest rate will grow faster.
Using it in reverse, we can ask what rate of return is required for an investment to double in 4 years?
r = 72 / 4 = 18
We would require an investment with a rate of return of 18%.
Rule of 72 Flaws
Now let's cover some limitations of the Rule of 72.
First, note that the Rule of 72 is for annual compounding. It does not account for more or less frequent compounding. A stock might pay a monthly dividend that you can compound monthly, which means that investment would grow faster than what the Rule of 72 would suggest. In fact, using 69.3 as the numerator is a more accurate representation of continuous compounding, but of course that makes it impossible for most people to calculate in their head, and the “rule of 69.3” doesn't sound as nice.
Secondly, the simple calculation does not consider things like inflation, fees, and taxes, which can drastically influence the final realized return of the investment.
Thirdly, the Rule of 72's precision is worse at the extremes of very small or very large interest rates. It would estimate that an investment with a 1% interest rate would take 72 years to double, but that number would really be about 70.7 years. The calculation is most accurate in the range of about 5 to 8% for the rate.
Lastly, the Rule of 72 assumes a constant rate of return. In reality, virtually any investment – stocks, bonds, gold, etc. – will have a rate of return that fluctuates over time. For example, stocks might have a -10% return one year and a 30% return the next year.
How To Use the Rule of 72
So now that you've seen why the Rule of 72 is a very rough estimate in the context of investment returns, let's look at how one might actually use it in real life.
As discussed, the Rule of 72 is much more useful and reliable with a fixed rate. In finance, fees are much more constant than returns. The Rule of 72 can illustrate how damaging small differences in fees can be.
Suppose Investment A has a fee of 2% and Investment B has a fee of 3%. The Rule of 72 tells us Investment A's fee will cut the initial invested principal in half in 36 years (72 / 2 = 36), while Investment B's fee will cut the principal in half in 24 years (72 / 3 = 24).
Similarly, the Rule of 72 tells us that a credit card with an 18% interest rate will double the amount you owe in 4 years (72 / 18 = 4).
Inflation also tends to be much less volatile than stock market returns. Average annualized inflation has been about 3% historically. At a steady rate, the Rule of 72 tells us inflation will cut the purchasing power of our money in half in 24 years (72 / 3 = 24).
As you've seen, the Rule of 72 is a very rough approximation and is thus more of an illustration or a learning tool that realistically is of little practical use in the context of stock market investing other than showing how compound interest works.
However, the Rule of 72 is much more useful in the real world for rates with much less variability, such as fees, interest rates for credit cards or loans, and inflation.
What do you think of it? Let me know in the comments.
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