I really like it when people try to understand math. So I enjoyed when someone showed me a summary prospectus for the Vanguard Global Equity Fund, investor shares, and asked how one of the numbers was calculated.

Specifically, they wanted to know how the return before taxes was calculated for a 5 year return period. Now those individual returns in the past 5 years were:

• 2016 – 6.57%
• 2017 – 27.78%
• 2018 – -9.15%
• 2019 – 28.73%
• 2020 – 22.38%

My investment math student had done what they could think of, and come up with numbers ranging from 15.2% to 94%. The number from the prospectus is 14.28%. They wanted to know how one could possibly arrive at 14.28% from the above numbers.

Now, I know where the ~15.2% idea had come from. That’s the arithmetic mean, that “average” that you probably learned in elementary school. So (a1+a2+a3+a4+a5)/5. But the arithmetic mean doesn’t account for the effect of compounding, so this isn’t the measure we want.

I must admit, I’m not sure how they got 94% as a possibility. I’ve seen some claim you can just add the measures, which would be 76.3%; but nope, that’s not right either.

The correct tool to use is a geometric mean. This is the measure that properly takes into account the effect of compounding.  For 5 years, the formula would be the fifth root of the product of the returns, eg

(a1*a2*a3*a4*a5) ^ 1/5

But the extra tricky bit is that this formula doesn’t like zero or negative numbers, like the negative return observed in this prospectus in 2018. So instead of framing all of these returns around 0, we re-frame them to be around 100, eg what percentage of your starting year balance will you have at the end of the year.

So in this case our series of numbers would be (106.57 * 127.78 * 90.85 * 128.73 * 122.38)^ 1/5 = 114.28, or a positive 14.28 return.

Thanks for having fun with math with me 🙂