Question

XYZ's stock price and dividend history are as follows:

Year	Beginning-of-Year Price				Dividend Paid at Year-End
2017	$	190			$	5
2018		200				5
2019		180				5
2020		190				5

An investor buys three shares of XYZ at the beginning of 2017, buys another two shares at the beginning of 2018, sells one share at the beginning of 2019, and sells all four remaining shares at the beginning of 2020.

a. What are the arithmetic and geometric average time-weighted rates of return for the investor? (Round your year-by-year rates of return and final answer to 2 decimal places. Do not round other calculations.)

b. What is the dollar-weighted rate of return? (Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2017, to January 1, 2020. If your calculator cannot calculate IRR, you will have to use trial and error or a spreadsheet program.) (Round your answers to 4 decimal places. Negative amount should be indicated by a minus sign.)

Answer 1

a)

return = year end price - year beginning price + dividend / year beginning price

2017 - 2018 = 200 - 190 + 5 / 190

= 15 / 190 = 7.89%(rounded of to 2 decimal places)

2018 - 2019 = 180 - 200 + 5 / 200

  = -15 / 200 = -7.5%

2019 - 20 = 190 - 180 + 5 / 180

= 15 / 180 = 8.33%

Arithmetic average time weighted return:

average return = 7.89% - 7.5% + 8.33% / 3

= 2.91% (rounded off to 2 decimal places)

Geometric average time weighted return:

Geometric return =

V(1 + R1)X(1 + R2) X(1 + R3).......(1 + Rn) - 1

Where R1 = Return of 1st year

R2 = Return of second year

Rn = Return of nth year

here n = 3

by using above formula

Geometric Return = [(1 + 7.89%) x (1 + (-7.5%) x (1+8.33%)]^1/3 - 1

= (1.0789 * 0.925 * 1.0833)^1/3 - 1

= 1.026338 - 1

= 0.026338 or 2.63%(rounded off to 2 decimal places)

b)

Chart of cash flows:

explanation:

at time 0(beginning of year 2017) investor purchases 3 shares @ 190 per share, so cash flow = -570 (3 x 190)

at time 1(beginning of year 2018) investor purchases 2 shares @ 200 and he gets dividend for 3 existing shares = -385

[(2 x 200) - (3 x 5)]

at time 2(beginning of year 2019) investor sells one share @ 180 and he gets dividend for 5 existing shares = 205

  [(1 x 180) + (5 x 5)]

at time 3 (beginning of year 2020) investor sells 4 shares @ 190 per share and he gets dividend for 4 existing shares = 780

[(4 x 190) + (4 x 5)]

Year	Net cash flows
0	-570 (net cash outflow)
1	-385 (net cash outflow)
2	205 (net cash inflow)
3	780 (net cash inflow)

Dollar - Weighted rate of return is the IRR :

0 = -570 - 385 / (1 + IRR) + 205 / (1 + IRR)² + 780 / (1 + IRR)³

Using spread sheet

year	cash flow
0	-570
1	-385
2	205
3	780
IRR	1.30299%