Arithmetic mean is just a fancy way to say ‘average.’ So if you’ve ever calculated your car’s MPG, or a batting average, or your GPA, then you’ve used an arithmetic mean before. And for most things they work perfectly fine. But they’re completely useless for averaging stock returns.
That’s because arithmetic mean is not meant to be used on percent changes. Here, I’ll prove it. Let’s say you invest $100 into the stock market. The first year you lose 50%, the next year you gain 20%, and the last year you gain 33.33%. According to the arithmetic mean, your rate of return was 1.11%.
(-50% + 20% + 33.33%) / 3 years = 1.11% per year
So you didn’t make much money, but at least you didn’t lose money right? WRONG!
You lost 20%.
I’ll show you. Let’s say you started with $100 to keep the math simple:
Year 0: $100
Year 1: $50
Year 2: $60
Year 3: $80
So you started with $100 and ended with $80. In order to find out your actual rate of return, you’d want to use something called a geometric mean.
Use this formula:
($80 / $100) ^ (1/3) – 1
So it’s the quantity $80 over $100, to the 1/3rd power, minus 1.
Alternatively, you can use Excel’s GEOMEAN function, but it get’s thrown off by negative rates of change. To get around this, you’ll want to add a 1 to each percentage. So -50% becomes 50%; 20% becomes 120%; 33.33% becomes 133.33%. Then you’ll subtract 1 from your answer. I’ve created an Excel worksheet that may help: CAGR Calculator[link]. CAGR stands for Calculated Annual Growth Rate and it’s calculated using the geometric mean.
Written in 2015