Question

5. Consider the rate of return of stocks ABC and XYZ.

Year                rABC               rXYZ

1                      20%                 30%

2                      12                    12

3                      14                    18

4                      3                      0

5                      1                      −10

a. Calculate the arithmetic average return on these stocks over the sample period.

b. Which stock has greater dispersion around the mean return?

c. Calculate the geometric average returns of each stock. What do you conclude?

d. If you were equally likely to earn a return of 20%, 12%, 14%, 3%, or 1% in each year (these are the five annual returns for stock ABC), what would be your expected rate of return?

e. What if the five possible outcomes were those of stock XYZ?

f. Given your answers to parts (d) and (e), which measure of average return, arithmetic or geometric, appears more useful for predicting future performance?

Answer 1

a. Arithmetic Average of ABC = (20%+12%+14%+3%+1%)/5 = 10%
Arithmetic Average of XYZ = (30%+12%+18%+0%-10%)/5 = 10%

b. Standard Deviation of ABC = ((20%-10%)^2+(12%-10%)^2+(14%-10%)^2+(3%-10%)^2+(1%-10%)^2)/(5-1))^0.5 = 7.91%
Standard Deviation of XYZ = ((30%-10%)^2+(12%-10%)^2+(18%-10%)^2+(0%-10%)^2+(-10%-10%)^2)/(5-1))^0.5 = 15.56%
Dispersion of XYZ is higher

c. Geometric mean of ABC = ((1+20%)*(1+12%)*(1+14%)*(1+3%)*(1+1%))^(1/5)-1 = 9.77%
Geometric mean of XYZ = ((1+30%)*(1+12%)*(1+18%)*(1+0%)*(1+10%))^(1/5)-1 = 13.58%
Geometric mean of XYZ is higher.

d. Expected Return = Arithmetic Average of ABC = (20%+12%+14%+3%+1%)/5 = 10% ( if equually likely)

e. Expected Return = Arithmetic Average of XYZ = (30%+12%+18%+0%-10%)/5 = 10%

f. Best method is standard deviation to predict future returns.

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