# Why I love the Rule of 72

WARNING: This post contains math. (But you knew this with a title like that.)

How long will it take an investment to double?

It’s a good question. And while doubling itself isn’t the most important aspect of an investment, it’s a good way to give you an intuitive sense of how well an investment is likely to grow.

For example, when I say that a fund grows at an average annual rate of 4%, you may go “Hmmm” and leave it at that. But if I say that at that rate it will take approximately 18 years for your investment to double, that means something more to you, doesn’t it?

We feel years. We don’t feel percentages. (Despite what the mortgage brokers say.)

### What is the Rule of 72?

The Rule of 72 states:

For a given annual rate of return, the amount of time in years that the value will take to double is roughly equal to 72 divided by the number of that rate.

Or, in an equation form:

$\dpi{150} \textup{Years to double} = \tfrac{72}{100 \times \textup{Rate}}$

So, a 4% rate? 72 ÷ 4 = 18 years to double.

A 12% rate? 72 ÷ 12 = 6 years to double.

It’s pretty easy to remember and to do the math in your head.

It’s not an exact calculation, though. If you need convincing of this, what happens if the rate is 72%? Well, 72 ÷ 72 = 1, but something can’t double if it grows by…72%.

So why do we use this rule? And how accurate is it?

### Let’s do some math

What we in effect want to solve is this:

$\dpi{150} 2 = (1 + r)^n$

where r is the annual rate and n is the number of years.

(If you need more convincing of that term on the right, remember that after a year of a 4%, or 0.04, rate, you have 1 + 0.04 = 1.04 times what you started with. Every year you multiply that by itself, for a total of n times.)

Given r, what is n?

Well, let’s take that 4% example. There we have:

$\dpi{150} 2 = (1 + 4\%)^n$
$\dpi{150} 2 = (1 + 0.04)^n$
$\dpi{150} 2 = (1.04)^n$

We can take the natural log of both sides to get:

$\dpi{150} ln(2) = ln (1.04^n)$
$\dpi{150} ln(2) = n \times ln(1.04)$
$\dpi{150} n = \tfrac{ln(2)}{ln(1.04)}$

Now, if you’ve forgotten your log tables (that’s a joke), this gives us:

$\dpi{150} n = \tfrac{0.693}{0.0392}$

$\dpi{150} n = 17.679$

Now, remember that the Rule of 72 would tell us that the investment would double in 72 ÷ 4 = 18 years. And 17.679 is pretty darn close to 18, isn’t it?

### Rule of 69.3

So, “why 72?” I hear you cry.

There are good derivations online about how the rule of 72 comes about. This post is mathy enough without me going into Taylor Series, and besides, I think I may have lost most of you by dragging out the natural logarithm. (But it’s really cool!)

Suffice to say, we can take a non-linear expression and make it linear without much reduction in accuracy. (The first term of the Taylor Series…sorry couldn’t help myself.)

So that approximation allows you to do this:

$\dpi{150} ln (1 + r) \rightarrow r$

And so:

$\dpi{150} n \approx \tfrac{ln (2)}{r}$

$\dpi{150} n \approx \tfrac{0.693}{r}$

But since r is a percentage, you can multiply the fraction by 100:

$\dpi{150} n \approx \tfrac{69.3}{\textup{Number of the rate without the \%}}$

So there you have it. The rule of 72 is more accurately called the Rule of 69.3.

### So why 72?

Because once you start approximating, you can’t, you won’t, and you don’t stop.

Just kidding. But only sort of. 72 and 69.3 aren’t that far apart. And 72 has the advantage of being a number with many divisors in the range that we care about. The numbers 2, 3, 4, 6, 8, 9, 12, they all divide cleanly into 72.

So while dividing 69.3 by 8 is going to spin your mental wheels for a while, anyone who remembers their times tables will remember that 9 × 8 = 72, so 72 ÷ 8 = 9. It’s just easier, and when you’re dealing with a rule of thumb, that’s pretty important.

### Here’s what I especially like

As mentioned above, there are some places where the Rule of 72 is very inaccurate. (See the example above at 72%.)

But, what’s simply amazing, is that the rule is most accurate precisely at the interest rates that we’re likely to care about. And by this I mean reasonable rates of return for an investment product.

Check this. Here is a chart showing how far off the Rule of 72 is from the correct answer.

It’s lowest around 8%. And it’s almost never far off, except when dealing with super low interest rates.

But you don’t need math to tell you that it’s going to take several forevers for your savings account that pays interest at 0.25% to double.

Now, we don’t typically invest in a lump sum; we invest a little bit over a long time. And returns are variable. Some fund may have a long-term average of around 8%, but that doesn’t mean you’re going to get 8% a year.

But never mind all that. This is still a good rule of thumb to figure out whether an investment is sufficient, given your timeline. Sometimes, the math can actually be easy enough to do in your head.