Question

XYZ stock price and dividend history are as follows:

Year	Beginning-of-Year Price				Dividend Paid at Year-End
2015		$	114			$	5
2016			120				5
2017			100				5
2018			105				5

An investor buys four shares of XYZ at the beginning of 2015, buys another two shares at the beginning of 2016, sells one share at the beginning of 2017, and sells all five remaining shares at the beginning of 2018.

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

b-1. Prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2015, to January 1, 2018.

Answer 1

a. The arithmetic average return is simple average returns of stocks, calculated in following manner

Arithmetic average return = Sum of returns/ number of years

While geometric average return is calculated in following manner

Geometric average return = number of years√ {(1+r1)*(1+r2)*(1+r3)…} -1

Where r1, r2, r3…are the returns for year 1, 2, 3….

Yearly Return = {(capital gains + dividend)/price}*100

Return from 2015-2016 = {(120 - 114) + 5} /114 *100 = 9.65%

Return from 2016-2017 = {(100 - 120) + 5} /120 *100 = -12.50%

Return from 2017-2018 = {(105 - 100) + 5} /100 *100 = 10.00%

Arithmetic mean = (9.65% -12.50% +10.00%)/3 = 7.15%/3 = 2.38%

Geometric mean = [(1+9.65%)*(1-12.50%) *(1+10.00%)] ^ (1/3) -1

= (1.0965 *0.8750 * 1.10) ^1/3 – 1 = 0.01813 = 1.813%

The arithmetic average time-weighted rates of return for the investor is 2.28%

The geometric average time-weighted rates of return for the investor is 1.813%

(Note: the answer can slightly vary because of rounding off)

b-1. Chart of cash flows

Time	Year	Cash flow form stock buying/selling	Cash flow from Dividend	Total Cash Flow
1/1/2015	0	4* -$114 = -$456	0	-$456.00
1/1/2016	1	2 *-$120= -$240	4*5 =20	-$220.00
1/1/2017	2	1* $ 100 = $100	6*5 =30	$130.00
1/1/2018	3	5* $105 = $525	5*5 = 25	$550.00

Note: Negative sign for cash outflow