A mathematics teacher stands in front of a googly-eyed crowd of students.

These students are about to learn one of the most important lessons in their lives.

The teacher holds up a blank piece of paper which he folds in half. He then poses the problem: I guess all of you would agree that there is practically no difference in the thickness of this paper after it’s folded in half. It’s still a thin piece of paper, as paper is. But let’s now imagine that I did this forty-five times. How thick do you think it would become?

The students, knowing that you can’t *actually* fold a piece of paper more than 7 times, looked bewildered and started to throw out all kinds of answers. Some answered the thickness of their mathematics textbook. Some answered the height of the teacher’s desk. A few answered the height of the classroom ceiling. But when the teacher asked, higher than this three-story school building?

, all were silent.

The correct answer threw the students aback.

You were all off by hundreds of thousands of miles

, the teacher said. Because the correct answer is that it would stack all the way from here to the moon.

## Compound Interest Is Always Surprising

The lesson from the teacher’s story is that compounding is an unnatural thing for the human mind to process. If you haven’t read the story before, I bet you were flabbergasted. I was too.

The problem is, we’re taught from an early age to understand mathematics linearly.

In introductory mathematics, students are taught to add, subtract, multiply, and divide numbers in their heads or using tricks and rules on a piece of paper.

But as soon as exponential functions get introduced, the mind shuts down and the calculator or computer takes over. If you asked the average student or working adult what’s seven to the eighth power, most would be clueless but everyone could pound it into the calculator.

And therefore, it always leaves us astounded if someone presents us with a compound interest problem with a long runway. We always get surprised by the effect of compound interest because we simply don’t ** think** about compounding enough.

This is my purpose with this note. I want to teach you a little trick that allows you to think about compounding problems every single day, all in your head. As soon as you’ve learned this trick, you will begin thinking about money and investing differently, and not only in the stock market. Another side effect is that you’ll likely become a more long-term investor.

I’ll unveil the trick in a minute, but let me first pound in why this stuff is so important to you.

## Compound Interest Is the Lifeblood of Finance

Compounding is at the top of the list of the most important concepts in finance. But it’s not talked about enough because it’s become a given that everyone understands what it means. But not everyone does.

It’s important because everything in the financial system works on a compound basis.

Think about it. Unless it’s your account statement, you rarely look at the absolute change of your investment holdings. You always look at the percentage change. It makes little sense to compare an absolute change in the quote of the Dow Jones Industrial Average today at $34,000 with when it stood at $1,000 in 1982 because a 2% percentage change today would equal a 68% percentage change in 1982.

More specifically, compounding matters a whole lot in any valuation because the returns a company makes compound based on the reinvestment rate and runway for investment opportunities.

Consider Company A that earns a cash return on invested capital (CROIC) of 15% and can reinvest all its free cash flows continuously for 20 years at those returns. Let’s say Company A generated $1,000 in cash flows last year, and for the sake of simplicity, let’s say Company A ceases operations after 20 years.

$1,000 cash flows compounded at 15% for 20 years will grow more than 16-fold to $16,367 in 20 years. (Adding just a single year would generate $18,822 in cash flows in year 21, increased by more than double the original $1,000). If we discount the 20 years of cash flows to present value using a rate of 10%, Company A would be worth $32,954, or 33 times trailing cash flows.

Now also consider Company B which also earns a 15% return but can only reinvest 50% of its cash flows in high return opportunities. Thus, it will only grow its cash flows at 7.5%.

$1,000 cash flows compounded at 7.5% for 20 years will comparatively only grow to $4,248. And Company B would only be valued at $15,849, or 15.8 times trailing cash flows.

Successful long-term investing is about compound interest and time. And it’s about not letting anything get in the way of these vital components.

## A Mental Trick for Calculating Compound Interest

There are three things to know about the mental trick I’m about to show you.

- It’s not precise,
- but it’s really effective,
- and it mostly works for annual rates of return under 20%.

The trick is called the Rule of 72.

Often when dealing in finance, we ask about the potential of doubling our investment and the time horizon for how soon that might happen. We will have to pose the same question here and this is where the Rule of 72 comes in handy.

All you have to do is divide 72 with the annual rate of return to see how long it takes to double.

*Years to Double = 72 / Annual Rate of Return*

For example, a 12% annual rate of return would double an investment in 6 years (72 / 12 = 6). A 4% annual rate of return would double an investment in 18 years.

Alternatively, you can divide 72 with the number of years it takes to double to get the compound rate of return.

*Compound Annual Rate of Return = 72 / Years to Double*

So from now on, you will always remember that a 10% annual rate of return takes 7.2 years to double and that for an investment to double in 10 years you would need a 7.2% annual rate of return (7.18%, to be precise).

Let’s now take our Company A from the previous section and calculate our cash flows in year 20 using the Rule of 72. We said that Company A’s cash flows would grow by 15% a year for 20 years.

We first follow the logic that the cash flows will double every 4.8 years by dividing 72 by 15. This means that the cash flows will double around four times over the next 20 years. All we have to do now is keep doubling until year 20. So in five years, we’ll have $2,000; $4,000 in 10 years; $8,000 in 15 years; and finally, $16,000 in 20 years—close to the correct amount of $16,367.

We said that Company B’s cash flows would grow by 7.5% a year for 20 years. Hence, Company B’s cash flows will double every 9.6 years—a bit over two times during the 20 year period. In 10 years, we’ll have $2,000 and $4,000 in year 20. The correct amount was $4,248.

This is the trick. Simple and effective.

To get the nature of compounding pounded into your head, use this method frequently. You will gain an immense competitive advantage in your life.

Because when you understand compounding, it can work for you with unimaginable wonders. And when you don’t, it can work against you with equal force in the opposite direction.

Or as the great Albert Einstein said:

He who understands it, earns it; he who doesn’t, pays it.

Cordially,

Oliver Sung

## 7 Responses

What an excellent post Oliver. I will use this trick without a question.

That’s great to hear, Thomas.

What did I miss regarding company A?

It has only one relevant cashflow (FCF) to derive its NPV in year 20 (it ceases operations thereafter, assume without liquidation value; reinvest all cash until year 20). I think I misunderstood your example.

I see why the phrasing may be confusing. Of course, reinvested cash flows accrue to book value over the 20 year period and are owed to shareholders upon liquidation.

Thx for clarifying the example/your thoughts

Interesting read Oliver

I’m glad you liked it.