Understanding the rule of 72: a popular rule that has little practical value

The Doubling Question: If your money grows at a certain rate, how long will it take to double?

You can surprisingly answer this question without doing much math. All you need to know is the Rule of 72, which states:

For instance, if an investment returns 6% annually, it will take 12 years for your money to double. Or if an investment returns 8% annually, it will take 9 years for your money to double.

You can also work in reverse. If you want your money to double in 7.2 years, then you would want an investment with an annual 10% return.

The Rule of 72 is easy to remember and doesn’t require a calculator. And it is a popular subject.

The Popularity of the Rule of 72

People have been talking about the Rule of 72 for centuries. The earliest record I can find is Italian mathematician Fra Luca Pacioli who wrote the rule down in 1494.

And the rule is widespread even today. Virtually all of the major personal finance blogs mention it:

• J.D. Roth of Get Rich admits he loves stuff like the rule of 72 because he is a stats geek
• My Money Blog encourages you to curb holiday spending since the rule of 72 shows how expensive credit card debt can be
• JLP of AllFinancialMatters shows the accuracy of the rule of 72 and introduces the rule of 114 and the rule of 144
• Ramit Sethi of IWillTeachYouToBeRich explains the rule of 72 is a cool trick to see how fast your money will grow
• Someone decided the rule was so great that a website should be named in its honor.

Does the Popularity Mean the Rule is Useful for a Sophisticated Investor?

The Rule of 72 is so simple you can explain it to a fifth grader. I mean, it’s amazing that you can use it to explain the power of compound interest and encourage people to invest, without using technical math. Financial advisers can use the rule loosely in presentations because it sounds cool and because it is very accurate for “reasonable investment returns” of 4 – 18%.

But before you start using the rule to make decisions, you might want to ask two important questions:

(1) When the “rule” not work?

(2) Why does this “rule” work?

In fact, those two questions are good to ask about any financial formula or rule. It is this critical thinking that helps you become a more sophisticated investor.

I asked myself those two questions when I first heard the Rule of 72. To my dismay, there are some important assumptions going into the rule that limit its application. And even when you can use the rule, it would probably be easier to use a calculator.

I’m going to explain some important caveats to the Rule of 72, and for those interested, I provide the mathematical derivation for the rule.

When the Rule of 72 Doesn’t Apply

The Rule of 72 assumes many important things, which unfortunately are not all that realistic. Here is a list of common scenarios where the rule does not apply:

1. You invest money on a regular basis, like monthly or biweekly in a 401(k) plan. [the Rule of 72 assumes a one-time investment]

2. You choose investments with highly variable returns like stocks, mutual funds, or bonds. [the Rule of 72 assumes a constant annual return]

3. Your return is below 4% or above 18%. For instance, if you use 72%, you get the amazingly stupid result that your money doubles in one year. This is of course only possible for a return of 100%. [the Rule of 72 is an approximation and it is less accurate for returns above 18%]

What a Sophisticated Investor Could Do Instead

The caveats make the Rule of 72 impractical for the sophisticated investor.

I can’t think of a single place where the Rule of 72 has been useful in my life besides as a way to tell people about compounding without using much math. It is a great way to explain a concept, but it is not a good way to make decisions.

If you really need a doubling estimate on the fly, you could use a calculator or spreadsheet and use the formula t = ln(2) / ln(1+r). If you are interested, see the derivation section for why this is the appropriate formula to use.

Most likely you will want to track a return on investment where you contribute regularly, say a monthly investment plan. Many brokerages will calculate a return on investment for you; for instance, Fidelity does this on its 401(k) accounts.

But not all brokerages will calculate returns in the way you want them. You might want an annualized return and they might give you a return on investment since you purchased the investment, which could be weeks, months, or years.

Stay tuned as I’ll be writing about a do-it-yourself investment tracking method in the near future.

Why the Rule of 72 Works (Some Math Involved, Optional)

Here is the math formulation of the “doubling question:”

Suppose you invest P dollars and are guaranteed a fixed annual return of r. How long will it take for money to double, or to grow to 2P?

Using the compound interest formula introduced in my APR and APY article, we need to solve for the time t such that:

Taking natural logs on both sides, and solving for t, we get:

The term ln(1+r) is approximately equal to r, because this is the first order Taylor expansion.

Finally, we multiply the denominator and numerator by 100 to get rid of the decimals, and we then have:

There you go! We could equally have a rule of 69.3, or the rule of the natural log of 2.

But 72 was chosen because it has several divisors, and when you crunch the numbers, 72 / r is a pretty good approximation of ln(2) / ln(1+r).