Question

Caswell Enterprises had the following end-of-year stock prices over the last five years and paid no dividends. Time Caswell 1 $12 2 9 3 7 4 6 5 12 Calculate the annual rate of return for each year from the above information. What is the arithmetic average rate of return earned by investing in Caswell's stock over this period? What is the geometric average rate of return earned by investing in Caswell's stock over this period? Considering the beginning and ending stock prices for the five-year period are the same, which type of average rate of return best describes the annual rate of return earned over the period (arithmetic or geometric)?

Answer 1

i)

Average return for each year = (Stock price of current year – Stock price of previous year)/ Stock price of previous year

Average return in year 2, r₂ = ($ 9 - $ 12)/$ 12 = - $ 3/$ 12 = - 0.25

Average return in year 3, r₃= ($ 7 - $ 9)/$ 9 = - $ 2/$ 9 = - 0.222222222

Average return in year 4, r₄ = ($ 6 - $ 7)/$ 7 = - $ 1/$ 7 = - 0.142857143

Average return in year 5, r₅ = ($ 12 - $ 6)/$ 6 = $ 6/$ 6 = 1

ii)

Arithmetic average return = (r₂ + r₃ + r₄ + r₅)/n = [- 0.25 + (- 0.222222222) + (- 0.142857143) + 1]/4

                                                = 0.38492063492/4 = 0.096230159 or 9.62 %

iii)

Geometric rate of return = ⁴√ [(1+ r₂) x (1+ r₃) x (1+ r₄) x (1+ r₅)]

                                             = ⁴ √ [(1 - 0.25) x (1+ - 0.222222222) x (1+ - 0.142857143) x (1+ 1)] – 1

                                             = [⁴ √ (0.75 x 0.777777778 x 0.857142857 x 2)] – 1

                                              = (⁴ √ 1) – 1

                                              = 1 – 1 = 0

iv)

Geometric rate of return better explains the annual rate of return earned over the period as it considers periodic compounding for a series of correlated returns.