Chapter 2: Q7I. (page 142)

XYZ stock price and dividend history are as follows:
Year Beginning-of-Year Price Dividend Paid at Year-End
2010 $100 $4
2011 $110 $4
2012 $ 90 $4
2013 $ 95 $4
An investor buys three shares of XYZ at the beginning of 2010, buys another two shares at the beginning of 2011, sells one share at the beginning of 2012, and sells all four remaining shares at the beginning of 2013.
a. What are the arithmetic and geometric average time-weighted rates of return for the investor?
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, 2010, to January 1, 2013. If your calculator cannot calculate internal rate of return, you will have to use a spreadsheet or trial and error.).

Expert verified

a. Arithmetic mean = 3.15%, Geometric mean = 2.33%

b. Dollar-weighted return = Internal rate of return = –0.1661%

Step by step solution

Capital gain for 2010-2011 = Beginning of year price in 2011 - Beginning of year price in 2010= 110-100

Capital gain for 2011-2012 = Beginning of year price in 2012 - Beginning of year price in 2011= 90-110

Capital gain for 2012-2013 = Beginning of year price in 2013 - Beginning of year price in 2012= 95-90

Return = [(Capital gains + Dividend)/ Price]

Arithmetic mean

= Sum of return/ number of years

= (14.00%+ (-14.55%) + 10.00%)/3 = 3.15%

Geometric mean

= [(1+r1)*(1+r2)......(1+rn)}^1/n -1

[(1+14.00) (1 - 14.55) (1+ 10.00)]^1/3-1 = 1.2235.75 - 1 = 2.33%

Time	Cash-flow	Explanation
0	-300	Purchase of three shares at the rate of $100 per share
1	-208	Purchase of two shares at the rate of $110 per share plus dividend income on three shares held
2	110	Dividends on four shares plus the sale of four shares at the rate of $95 per share
3	396	Dividends on five shares plus the sale of one share at the rate of $90 per share