A math teacher stands in front of a googly-eyed crowd of students who are about to learn about compound interest.
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 threw all kinds of answers. Some said the thickness of their textbook. Some said the height of the teacher’s desk. A few said 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
Compounding is unnatural for the mind to process. From an early age, we’re taught to understand math linearly. In introductory math, 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 seven to the eighth power is, most would be clueless but everyone could pound it into the calculator.
It leaves us astounded if someone presents us with a compound interest problem with a long runway. We get surprised by the effect of compound interest because we simply don’t think about compounding enough.
Compounding is at the top of the list of the most important concepts in finance. Everything in the financial systems works on a compound basis.
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% change today would equal a 68% change in 1982.
Compounding matters in any valuation because the returns a company makes compound based on the reinvestment rate and runway for investment opportunities. Consider Company A which earns an ROIC 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 compounded at 15% for 20 years will grow more than 16x 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 33x trailing cash flows.
Now consider Company B which also earns a 15% return but can only reinvest 50% of its cash flows at those rates, meaning it will grow its cash flows at 7.5%/ year. $1,000 cash flows compounded at 7.5% for 20 years will only grow to $4,248. And Company B would only be valued at $15,849, or 15.8x 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 two components.
A mental trick for calculating compound interest
There are three things to know about this mental trick.
- It’s not precise,
- but it’s effective,
- and it mostly works for annual returns <20%.
It’s called the rule of 72.
In finance, you often ask about the potential of doubling your investment and the time horizon for how soon that might happen. This is where it 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 return
For instance, a 12% annual return would double an investment in 6 years (72 / 12 = 6). A 4% annual return would double it in 18 years.
Alternatively, you can divide 72 by the number of years it takes to double to get the compound rate of return.
Compound annual return = 72 / years to double
From now on, you’ll 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 Company A from the example earlier and calculate the 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. First we follow the logic that cash flows will double every 4.8 years (or roughly 5) by dividing 72 by 15. This means that the cash flows will double roughly four times over the next 20 years (20 / 5). All we gotta do now is keep doubling until year 20. So in 5 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.
For company B, its cash flows would grow by 7.5%/ year for 20 years. Hence, its 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.
When you understand compounding, it works wonders. When you don’t, it can work against you with equal force in the opposite direction. Or as Albert Einstein said: “He who understands it, earns it; he who doesn’t, pays it.”