Investment Returns and Geometric Mean

What if an investment returned 5%, 20%, and 50% over three years?
Year 1:  $100 * .05 = $105
Year 2:  $105 * .20 = $126
Year 3:  $126 * .50 = $189

What was the annual return?  Can you just average the percentages?
Let’s try it.  The average of 5%, 20%, and 50% is 25%
Did you average 25% a year?

Year 1:  $100 * .25 = $125
Year 2:  $125 * .25 = $156.25
Year 3:  $156.25 * .25 = $195.31

No, 25% for 3 years gets you a different amount!

You calculate the average annual return via the geometric mean (not the arithmetic mean)

\sqrt[3]{1.05 * 1.20 * 1.50} -1 = .236

Your annual return was actually 23.6% 

Year 1:  $100 * .236 = $123.6
Year 2:  $123.6 * .236 = $152.77
Year 3:  $152.77 * .236 = $188.82

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