Question

A manager buys three shares of stock today, and then sells one of those shares each year for the next 3 years. His actions and the price history of the stock are summarized below. The stock pays no dividends.

Time	Price			Action
0	$	150		Buy 3 shares
1		180		Sell 1 share
2		180		Sell 1 share
3		180		Sell 1 share

a. Calculate the time-weighted geometric average return on this portfolio. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

b. Calculate the time-weighted arithmetic average return on this portfolio. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

c. Calculate the dollar-weighted average return on this portfolio. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Answer 1

a.time-weighted geometric average return (g_n)

g_n = (1+r_c)^1/n-1 where r_c = cumulative return over the period; n = number of periods; 1/n = 1/3= 0.33

rate of return for year 1 = (180 x 3) - (150 x 3) / (150 x 3) = 20%

rate of return for year 2 = (180 x 2) - (150 x 2) / (150 x 2) = 20%, one share is sold at the end of year 1

rate of return for year 3 = (180 x 1) - (150 x 1) / (150 x 1) = 20%, two shared sold at the end of year 2.

r_c = 20% + 20% + 20% = 60% or 0.60

g_n = (1 + 0.60)^1/3 - 1 = 1.17 - 1 = 0.17 or 17%.

b.time-weighted arithmetic average return (a_n)

a_n = (r₁ + r₂ + r₃) / n where r_n is the rate of return for the respective year.

since the rate of return for each year is already calculated above, arithmetic mean is as below:

a_n = (20%+20%+20%) / 3 = 20% or 0.20.

c.dollar-weighted arithmetic average return (d_n)

Year	Return (r)	Weight (w)	r x w
1	90	1	90
2	60	2	120
3	30	3	90
	Total	6	300
	Weighted average		50
	Return %		33%