Question

What's the difference between the arithmetic and geometric average return (conceptually, not mathematically), and when is it best to use each?

Answer 1

Suppose you own a Stock A and in the first year it earns 5%, second year 7%, third year 3%. Arithmetic return is used to calculate average of these returns. So Arithmetic return will give you the result (5+7+3)%/3 = 5%. Consider this as the mean of the normal distribution curve where there are n observations and the most likely value will be the Arithmetic mean of it. So if you invest in stock A your return for a year will likely to hover around 5%. Whereas the geometric average return tells you at what rate of return your investment has succeeded ovr the given time period.So, to calculate Geometric average return you calculate like this: Initial investment in Stock A = $100  

1st year return is =5%, 2nd year return is =7%, 3year return is = 3%    So this means in the first year your initial investment has grown to $105 which will serve as the base amount for 2nd year and the investment value at the end of 2nd year will be calculated as $105x(1.07) and similarly at the end of 3rd year will be $105x(1.07)x(1.03). Now when you take the average of these returns it will tell you what is the actual rate of return over that initial investment of $100. The investment value at the end of 3 year is 115.72 and your geometric return is simply    (115.72/100)^power(1/3) =1.0498. so your geometric return is nearly 4.98% which is less than Arithmetic mean and it will be always be lesser than that because arithmetic mean is calculated for return on initial investment of $100 for all investments but in geometric mean foe subsequent periods the base investment on which return is calculated is increased or decreased by the return on the last period. So in the second year 7% was the return on $ 105 and for the third year it was $112.35.