Unexpectedly Intriguing!
29 August 2024

If you spend any time working with the math of finance, you'll eventually encounter the Rule of 72. What is the Rule of 72? It is, quite simply, one of the most useful tools you can use to quickly approximate how long it will take to double your money in an investment that compounds annually. All you need is to take the percentage rate of return for your investment and divide it into the number 72. The result you get will be a good estimate of how many years it will take for the value of your investment to double.

Best of all, it's math you can do in your head. Which compared to the math formula you would need to use to get an exact answer otherwise, is lightning fast. Even if you were to use a calculator to do the exact math.

Sal Khan has a short video explainer describing the Rule of 72 that picks up after he uses the exact formula. What makes this video stand out from others is he goes the extra mile to answer the question of how good the Rule of 72 is at approximating the exact answer across a range of common rates of return that matter to investors.

Because the video picks up the discussion from examples of compounding interest, it might help to take a step back and start from the basic future value formula, which looks something like this for an investment that compounds annually:

Future Value = Present Value * (1 + Rate of Return)Time in Years

Since we want to find out how long it takes the value of the investment to double, we'll set "Future Value" to be equal to 2 times the "Present Value"

2*Present Value = Present Value * (1 + Rate of Return)Time in Years

At this point, since we now have "Present Value" on both sides of the equation, we can simplify the equation by dividing both sides by it. Doing that gives us the following result:

2 = (1 + Rate of Return)Time in Years

From here, if we want to solve this equation for "Time in Years", we'll need to deploy the magic of logarithms. Which for us, starts with taking the natural logarithm (ln) of both sizes of the equation:

ln(2) = ln[(1 + Rate of Return)Time in Years]

In this next step, we'll apply the power law of logarithms to transform the exponent math into simple multiplication on the right hand side of the equation:

ln(2) = Time in Years * ln(1 + Rate of Return)

We're almost there. We'll next solve for "Time in Years" by dividing both sides of the equation by ln(1 + Rate of Return). Here's the result:

ln(2)/ln(1 + Rate of Return) = Time in Years

In the final step, we'll swap the left and right hand side of the equation to make it easier to read by putting it into a more standard format:

Time in Years = ln(2)/ln(1 + Rate of Return)

To use this formula, the Rate of Return has to be written in a decimal format. For example, a rate of return of 6% would be written as 0.06 in the formula. And from here, you would be doing the same math Sal Khan was doing in the video, which we'll leave as an exercise for you if you really want to do it.

But why does the Rule of 72 work to closely approximate the exact result you get from the logarithm math we just showed? Steve Fiorino takes the explanation to the next level:

It may not be apparent at first glance how this exact equation is able to bring us to the rule of 72. For it to become clearer, input ln(2) into a calculator. It's an irrational number, but when you put it into the calculator by itself it will give you a number that it equals: 0.69314718056.

Or, phrased in another way, 69.3%.

That's how you get the rule of 69.3, but unless you're a math whiz who somehow memorized multiples of 69.3 it's still pretty difficult to do the equation. Thus, 70 and 72, which have more numbers that divide cleanly into them while still giving close approximations, became popular.

The Rule of 72 became the most popular version of this application because it has more whole number divisors that produce whole number results.

Perhaps more remarkably, the Rule of 72 for approximating the time it takes an investment to double in value predates the invention of logarithms! Here's WesBanco's description of its earliest use:

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.

Here's a link to Pacioli's 1494 book. Meanwhile, logarithms as we know them today weren't invented until John Napier developed them over a century later, publishing them in his 1614 book Mirifici logarithmorum canonis descriptio.

For what it's worth, the origin of the Rule of 72 can be attributed to earlier, unknown mathematicians who built it on a foundation of much older math that dates back centuries earlier, who recognized they could apply it to solve practical, real-world problems. However, it wasn't until after logarithms were invented that mathematicians could demonstrate why it works as well as it does.

Image credit: A number (72) of white letters on a black surface photo by iridial on Unsplash.

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