A Technical Investigagion Into The Rule Of 72

The rule of 72 is a quick back-of-the-envelope investment calculation technique. Non-technical investors use the rule to estimate how long it would take to double an investment given a fixed rate of return. The rule of 72 has gained popularity among mainstream investors over the years primarily due to its simplicity.

For anyone just interested in the final result, it is an easy way to arrive at rough estimations. This way, they need not get involved in the mathematics behind the rule. However, there are certain costs involved in simplified mathematics that we cannot overlook.

In this article, I will dive into the mathematics behind this rule, and explore its limits in terms of the accuracy of results and assumptions involved.

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How Does the Rule of 72 Work?

Let us say that you are the typical non-technical investor. You have discovered an investment opportunity that gives you a fixed yearly compounded rate of return, say, 6%. You have an investment capital of 1000 monetary units at your disposal. You are now interested in (mentally) estimating how long it would take for this capital to double if you chose to invest.

You can take advantage of the rule of 72 by simply dividing 72 by the rate of return (in percentage). This would give you the time (in years in this case) it would take for your capital to double.

So, as an investor who is just interested in the final result, you get a quick estimate that it would take your initial capital about 6 years to double. Based on this back-of-the-envelope calculation, you may choose to invest or pass.

Now that we’ve seen how the rule of 72 works, it is time to get into the mathematics behind it. But before that, let us spend some time on the implicit assumptions the rule makes.

The Assumptions Behind the Rule of 72

The rule of 72 cannot just be applied to any investment. The first implicit requirement for the rule is that it can only be applied to investments that compound at a constant rate over time.

Will Kenton states another not so obvious implicit assumption from the rule of 72 as follows:

“The Rule of 72 applies to compounded interest rates and is reasonably accurate for interest rates that fall in the range of 6% and 10%.”

— Will Kenton (Investopedia).

Why is this the case? To understand the implicit assumptions behind the rule of 72, we will have to get into the mathematics behind the rule.

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The Mathematics Behind the Rule of 72

The formula for any investment return from a given principal amount (P) and a fixed rate of return (r) per compounding frequency over a total investment time (T) is defined as follows:

Now, let us say that we require the return be equal to 2P (double the principal amount). This would enable us to calculate the time taken (T) to double the initial investment as follows:

This gives us the precise formula to calculate the time taken to double any principal amount given a compounded rate of return.

The rule of 72 is a rough simplification of the above formula for people who are not interested in multiplicative dynamics or logarithms. It supposedly takes advantage of the following two facts:

1. ln(2) is approximately equal to 72%

2. ln(1+(r/100)) is approximately equal to r when the value of r ranges between 6% and 10%.

Applying these two approximations, non-technical investors transform the original formula as follows:

If you are the mathematical type, and your scepticism-meter is twitching, you are in good company. This does look like a recipe for poor results. So, let us proceed to investigate this approximation further.

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Technical Investigation into the Rule of 72

Let us now directly challenge the two major mathematical approximations of the rule of 72 that we just covered.

1. ln(2) = 0.69314718 (rounded to 8 digits).

2. ln(1+(6/100)) = 0.05826891; ln(1+(7/100)) = 0.06765865; ln(1+(8/100)) = 0.07696104; ln(1+(9/100)) = 0.08617770; ln(1+(10/100)) = 0.09531018.

From the first point, we see that ln(2) is much closer to 69% than 72%. The rule should have ideally been named ‘the rule of 69’. But why has it turned out to be the rule of 72?

It could be because of the fact that 72 has many more factors than 69. Hence, for the typical non-technical investor, it is probably easier to divide 72 by the various interest rates as compared to dividing 69.

From the second point, we see that there are potentially significant deviations in the interest rate consideration as well.

Combining these two factors, if we plot the error in investment-doubling-time approximation from the rule of 72 versus the accurate result from the actual formula (involving natural logarithms), we get the following result:

The error is the least (ranging between 0% and 1%) when we use the rule of 72 for interest rates ranging between 6% and 10%. So, it becomes quite clear why Investopedia suggests this as the acceptable range of applicability for the rule of 72.

From the error chart, we can safely say that the higher the interest rate on the right side of 10%, the higher the calculation error would be.

Having investigated the mathematics behind the rule of 72, there are a few more points that one needs consider to ensure its responsible usage.

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Responsible Application of the rule of 72

One of the primary issues with the application of the rule of 72 is the fact that most people using the rule are ill-aware of its implicit assumptions. The rule is often distributed via word-of-mouth or its social media equivalents. One does indeed come across as a clever investor when one is able to spit out quick estimates.

However, beyond the suggested (interest rate) percentage range, the application of this rule becomes very questionable. I’ve come across approaches where ‘experts’ suggest transforming the rule into ‘the rule of 73/74/75/…’ based on the interest rate being used.

These approaches revolve around the idea of minimizing the error by manipulating the numerator in the direction of the original formula’s result. In short, we arrive at a chart featuring different numerators that are applicable over different ranges of interest rates.

At this point, the calculation process becomes so cumbersome that the whole allure of the ‘rule of 72’ breaks down. I would suggest that you drop the rule altogether and just use the original formula (wherever applicable). In fact, if you could just get into terms with the idea of using natural logarithms, the original formula is not that complex at all.

Final Thoughts

Asa final thought, it is important to point out that the rule of 72 is meant to be applied to investments with a constant compounding rate. In the case of most real-world investments, a constant compounding rate is simply not available or possible. This naturally disqualifies the rule of 72 for most real-world investments.

It is at best just good enough for quick hypothetical estimates, but not much more. I have come across cases where people compute arithmetic averages of return rates over time and then use the rule of 72 to forecast doubling time.

I would also recommend against such practices. The average of a function is unfortunately not always equal to the function of the average. This leads to a bigger can of technical worms which is perhaps a topic for a future article. But for now, the knowledge that the rule of 72 applies only to constant rates of return should suffice.

In the end, when using any ‘rule’ that has a mathematical basis, the user should ideally understand the mathematics behind the so-called rule. As tedious as this may sound, it is the only way to ensure responsible application of such a rule, and thereby, ensure optimal results!

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Further reading that might interest you: Do Startups Really Need Venture Capital? and How To Quickly Calculate Percentages In The Head?