# The Rule of 72: What makes it tick?

The Rule of 72 is a guideline to estimate how long an investment that earns compound interest takes to double. This guideline helps you plan your investment mix in several ways. Much has been written on this topic.

A simple way to compare different investments is to look at their annual rates of return. The higher the rate, the higher the investment balance over time, for the most part.

A practical way to compare the effectiveness of investments is to compare the times they take to double the starting capital. It is the balance in the investment account that matters in the end, so there is a lot of interest in this metric. It is widely calculated using the formula, The Rule of 72.

This rule simply states that

(years to double an investment) = 72 / (its average annual percentage return)

For example, if an investment returns 4% each year, doubling it will take you 18 years (72 divided by 6). If that annual rate had been 8%, then it will only take you 9 years to do the same doubling. Manage a 12% return? You’ll race to the goal in just 6 years.

Pretty cool. Makes sense that larger annual returns let you double your investment faster.

When a rule is this simple, you may wonder: Is it some universal truth? What’s so special about the number 72 that makes this possible?

A Google search for “Rule of 72” produces several perspectives on the topic. Interestingly, you may also see references to Rule of 70 and even Rule of 69 or Rule of 69.3! What gives?

Let’s get to the bottom of this, so we know what we are doing with this rule. We need to juggle a few numbers in this quest.

The Rule of 72 can be restated as

(annual percentage return of the investment)*(years to double the investment)=72

We are trying to find out if this product is, indeed, the fixed number 72.

### Math behind the Rule of 72

Using some simple math concepts, it is possible to arrive at the following equation for the time to double a compound interest investment:

(time to double the investment) = (log 2)/log (1+r/100)

where r is (annual percentage return of the investment).

[If you are curious, full derivation of this equation is available here.]

This means that we can calculate time to double for different values of r. Multiplying the rate with its time-to-double should always give us 72, if the Rule of 72 is correct.

### Actual and Rule of 72 compared

The following table captures what we get from the real calculation using the formula.

For the percentage annual rate of return in column A, column B indicates the number of years it would take to double your investment. Column C is simply the product of these two. This is expected to be a constant 72, if Rule of 72 holds.

You can see that the sweet spot is an investment returning an average annual interest rate of 8%. This actually comes out with the product being about 72. The others come out differently.

### What does this mean for the Rule of 72?

Column C tells how close we are to 72, the expected value. Or, it tells you what ‘rule’ you need to apply instead of the Rule of 72.

For example,

• If the investment returns 1% per year, you must apply the Rule of 69.66.
• If it returns 2% per year, the Rule of 70.01 applies.
• A return of 12% warrants application of the Rule of 73.40.
• Found a way to extract a return of 20% per year? You should apply the Rule of 76.04.
• A 40% return will require the Rule of 82.40 to be applied.

I think you can see the pattern. As the interest rate goes up, the “Rule of” number also goes up.

What we really want to know is whether it is OK to apply the Rule of 72 even though column C indicates several other numbers. This means we want to understand what happens if we choose something other than 72.

Let’s pick some numbers and check.

### If not 72, what about: 69, 70, 80, or 100?

Consider an expanded table that includes the findings in the previous table but also calculates how much the actual deviates from the estimate calculated by the number chosen. Take, for example, 70. The number indicated in this column for an annual rate of return of 2% is 0.01. This suggests that the actual and estimate are very close and 70 is the right number to use for estimating the time to double for an investment with a 2% return. However, for other rates of return, it is not so perfect. Likewise, 72 is great for estimating when the rate of return is 8% and 100 is perfect for estimating when the rate of return is 100%.

No rule-of number is good for the full range of rates of return under consideration. So how do you choose?

A graphical plot of the table above is revealing and gives us the means.

### Plot the lines for 69, 70, 72, 80, 100

The graph below plots a series of lines representing how the actual time to double differs from estimates for the same using the numbers above for the ‘rule’. When a line is above 0.00, it means that the estimate is optimistic: the actual time is more than the estimate. Not that good. When it is below 0.00, the estimate is conservative: the actual is less than the estimate. Not great because it is not accurate, but safe. In this graph, the ‘Actual’ line of 0.00% represents perfection. None of the rules of thumb is aligned with the ideal.

So, the next best approach is to focus on the typical rates of return that most investors work with and figure out which rule works best in that range. I believe annual returns of 12% or less is this region of significance, as shown.

In this region, you can see that Rule of 72 is the line that offers a little of both below and above the Actual, keeping the amounts of deviation small. In others, we have either a too optimistic estimate (69 and 70) or an excessively conservative estimate (80, 90, and 100) with high deviations.

The most promising number is 72.

### How good is 72, really?

For the average investor who typically earns less than 12% annual rate of return: Rule of 72 rules!

On the high end, if you manage a 12% return on your investment, Rule of 72 says that you need 6 years (72/12) for the doubling. The real duration is 6.12 years, from the table. This means that the Rule of 72 underestimated the time to double by 0.12 years, or a month and a half. Actual time is about 2% longer than the estimate.

Towards the low end, consider an investment with 2% return. Rule of 72 says that you need to wait for 36 years to double your initial investment at that rate of return. But the table indicates that the doubling will actually happen in 35 years, an entire year ahead! Actual time here is about 3% shorter than the estimate.

If you go outside the region of our focus, think of a genius investor who somehow manages a 72% return on investment each year. The Rule of 72 says that investment will double in one year (72 divided by 72). But that’s not really true, is it?

Then again, when you get to be savvy enough to realize returns a lot higher than 12%, you’ll be savvy enough to use the more accurate chart to figure out your time-to-double duration.

We are catering to 0-12%, so all’s well. For the average investor who typically earns less than 12% annual rate of return: Rule of 72 rules!

### Some closing thoughts

• We started out framing the context of this rule to apply to annual returns. However, you can simply replace year with any other time span. Units like month, week, or even a single day, and the results will still work. Only, the rate of return and time to double will be in the same unit: months, weeks, or days. For example, an investment that fetches a 2% return per month will double the capital in 35 months and the Rule of 72 will indicate this doubling happening in 36 months.
• We focused on the financial investment scenario here. However, Rule of 72 applies to any situation where there is a consistent growth of anything. For example, if a country’s population is seen to grow a constant percentage each year, we can use the concept to project how many years it would take the population to double.
• A frequent concern in planning any long-term investment is the effect of inflation. You can account for inflation very simply. If you assume a constant annual rate of inflation, you can just subtract it from the investment rate. For example, if you hope to get a 10% return annually from your stock investment and the expected inflation rate is 3%, you can simply assume that your investment nets 7% per year and the calculations will give you today’s dollars.