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%.