Question

You are given the returns for the following three stocks:

Return
Year	Stock A	Stock B	Stock C
1	8%	7%	-22%
2	8%	15%	35%
3	8%	3%	15%
4	8%	12%	3%
5	8%	3%	9%

Calculate the arithmetic return, geometric return, and standard deviation for each stock. Do you notice anything about the relationship between an asset’s arithmetic return, standard deviation, and geometric return? Do you think this relationship will always hold?

Answer 1

STOCK- A

arithmetic return = (0.08+0.08+0.08+0.08+0.08) / 5 = 0.08 = 8%

variance = 1/4 [ (0.08-0.08)^2 + (0.08-0.08)^2 + (0.08-0.08)^2 + (0.08-0.08)^2 + (0.08-0.08)^2 ] = 0.0

standard deviation = (0.0)1/2 = 0.0

geometric return = ( (1+0.08) (1+0.08) (1+0.08) (1+0.08) (1+0.08) )^ 1/5 - 1 = 0.80

STOCK- B

arithmetic return = (0.07+0.15+0.03+0.12+0.03) / 5 = 0.08 = 8%

variance = 1/4 [ (0.07-0.08)^2 + (0.15-0.08)^2 + (0.03-0.08)^2 + (0.12-0.08)^2 + (0.03-0.08)^2 ] = 0.0029

standard deviation = (0.0029)1/2 = 0.053

geometric return = ( (1+0.07) (1+0.15) (1+0.03) (1+0.12) (1+0.03) )^ 1/5 - 1 = 0.08

STOCK- C

arithmetic return = (-0.22 + 0.35 + 0.15 + 0.03 + 0.09) / 5 = 0.08 = 8%

variance = 1/4 [ (0.22-0.08)^2 + (0.35-0.08)^2 + (0.15-0.08)^2 + (0.03-0.08)^2 + (0.09-0.08)^2 ] = 0.0426

standard deviation = (0.0426)1/2 = 0.206

geometric return = ( (1-0.22) (1+0.35) (1+0.15) (1+0.03) (1+0.09) )^ 1/5 - 1 = 0.08

in case of standard deviation , it gets larger when there is no difference between the arithmetic return and geometric return .