# 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