Question

Problem 24-6 XYZ's stock price and dividend history are as follows: Dividend Paid at Year-End Year 2016 2017 2018 2019 Beginning-of-Year Price $ 90 100 An investor buys three shares of XYZ at the beginning of 2016, buys another two shares at the beginning of 2017, sells one share at the beginning of 2018, and sells all four remaining shares at the beginning of 2019 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.) Arithmetic average rate of return Geometric average rate of return 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, 2016, to January 1, 2019. If your calculator cannot calculate internal rate of return, you will have to use trial and error) (Round your answers to 4 decimal places. Negative amount should be indicated by a minus sign.) Dolar-weighted rate of return

Answer 1

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

Return from 2016-2017 = {(100 - 90) + 3} /90 *100 = 14.44%

Return from 2017-2018 = {(80 - 100) + 3} /100 *100 = -17%

Return from 2018-2019 = {(90 - 80) + 3} /80 *100 = 16.25%

Arithmetic mean = (14.44% -17% +16.25%)/3 = 13.69%/3 = 4.56%

Geometric mean = [(1+14.44%)*(1-17%) *(1+16.25%)] ^ (1/3) -1

= (1.1444 *0.83 * 1.1625) ^1/3 – 1 = 0.0336 = 3.36%

The arithmetic average time-weighted rates of return for the investor is 4.56%

The geometric average time-weighted rates of return for the investor is 3.36%

b. What is the dollar-weighted rate of return?

Prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2016, to January 1, 2019

Time	Year	Cash flow form stock buying/selling	Cash flow from Dividend	Total Cash Flow
1/1/2016	0	3* (-$90) = -$270	0	-$270.00
1/1/2017	1	2 *-$100 = - $200	3*3 =9	-$191.00
1/1/2018	2	1* $ 80 = $80	5*3 =15	$95.00
1/1/2019	3	4* $90 = $360	4*3 = 12	$372.00
IRR				-0.17%

Note: Negative sign for cash outflow

The dollar-weighted return is the internal rate of return (IRR) that can be calculated by discounting from the rate which makes the sum of the present value of each net cash flow to zero.

0 = -$270/ (1+IRR) ^0 - $191/ (1+IRR) ^1 + $95/ (1+IRR) ^2 + $ 372/ (1+IRR) ^3

From trial and error method, we can calculate the value of IRR, which is

IRR = 0.5442%

The dollar-weighted return is 0.5442%