Corporate Finance, Module 6, “Risk, Return, and the Opportunity Cost of Capital”
Geometric Average vs Arithmetic Average
(The attached PDF file has better formatting.)
Illustration: Geometric vs Arithmetic Average
An investment returns either 0% or 20% each year, with a 50% probability of each. What is the expected annual return?
Question: Consider the expected return over two years. The investment may earn 0% the first year and 20% the next year, or 20% the first year and 0% the next year. Either way, the two year return is 20%, which is equivalent to 1.20½ –1 = 9.54% per annum. We use the geometric average, not the arithmetic average.
Answer: You are correct that 20% over two years is equivalent to 9.54% over one year. But the expected return over two years is not 20%. The investment has four possible returns, not two:
Returns
Scenario Year 1 Year 2 Two Year Probability
1 0% 0% 0% 25%
2 0% 20% 20% 25%
3 20% 0% 20% 25%
4 20% 20% 44% 25%
Total 84% 100%
The expected return is 84% / 4 = 21%. The expected annual return is 1.21½ – 1 = 10%. This is the arithmetic average of 0% and 20%.