A Word About Average Returns

In the world of investments, returns are often stated as average returns.

"The S&P 500 has averaged 10.3% per year since 1930."

That certainly seems like a meaningful way to express returns, but is it really? 

Quick math lesson

There are two types of averages, or means,  in math -- the arithmetic mean and the geometric mean.

Here are the formulas:

Arithmetic Mean = [R1 + R2…+ Rn] / n

Geometric Mean = [(1 + R1) (1 + R2)…(1 + Rn)]1/n− 1

Now, let's look at an example of why this matters.

Suppose you start out with $100. In the first year, it grows 100% (the balance goes to $200) and in the second year it falls 50% (it goes back down to $100). This is shown in the above table.

How well did your investment do?

Let's calculate your average return using the arithmetic mean formula.

Arithmetic mean = [100 + -50] / 2 = 25 (You averaged a 25% return over 2 years!)

Now, let's calculate your average return using the geometric mean formula.

Geometric mean = [(1 + 1) (1 + -0.5)]2 - 1 = 0 (You averaged a 0% return over 2 years.)

What does it all mean? 😜

Can we agree you experienced a 0% return? You have exactly what you started with. If you advertised this investment to others by stating that it has an average return of 25% per year, don't you think that may be misleading?!

Geometric mean returns are "real world" returns. They are the correct way to calculate investment returns. Importantly, the compound annual growth rates (CAGR) of an investment is the geometric mean return.