Question

1. XYZ stock price and dividends history are as follows: Year Beginning-of-year price Dividend paid at year- end $4 $4 $4 $4 2005 $100 $110 $90 $95 2006 2007 2008 An investor buys three shares of XYZ at the beginning of 2005, buys another two shares at the beginning of 2006, sells one share at the beginning of 2007, and sells all four remaining shares at the beginning of 2008. Hint When the investor buys three shares at the beginning of 2005, she cannot receive dividends paid at the end of 2004. She can receive dividend payments on these three shares at the end of 2005. A. Computer HPR for each period (05-06, 06-07, and 07-08) (6pts) B. Compute the arithmetic return. (3pts) C. Compute the geometric return. (3pts) D. Computer the dollar-weighted rate of return. Draw the time line to show cash flows. Write down the formula and calculate the dollar-weighted rate of return. The time line includes four dates corresponding to the turns of the year for January 1, 2005 to January 1, 2008. (8pts)

Answer 1

2005-2006:

• Shares bought at the beginning of 2005 = 3, sold = 0
• Buy value = 3 * $100 = $300
• Dividend received at the end of 2005 on 3 shares = 3 * $4 = 12
• End of the year 2005 shares value = $110 * 3 = $330
• Holding period for 2005-06 = Capital appreciation + Dividend yield
  • HPR(2005-06) = [ ($330- $300)/$300] + $12/$100
  • HPR(2005-06) = 0.10 + 0.04 =14%

2006-07

• Shares bought in 2005 = 3
• Shares bought at the beginning of 2006 = 2, sold = 0
• Buy value = 2 * $110 = $220
• Total shares value as on 2006 beginning = $300 + $220 = $520
• Dividend received at the end of 2006 on 5 shares = 5 * $4 = $20
• End of the year 2006 shares value = $90 * 5 = $450
• Holding period for 2006-07: [($450 - $520)/$520 ] + $20/$520 = -0.135 + 0.038 = -0.096 = -9.60%

2007-08

• Shares bought in 2005 = 3, 2006 = 2 and sold in 2007 = 1. Net shares held at the beginning of 2007 = 5 - 1 = 4. Shares value = 4 * $90 = $360
• Dividend received at the end of 2007 on 5 shares held in the previous year = 5* $4 = $20
• 2007 End value of shares = 4 * $95 = $380
• HPR(2007-08)
  • HPR(2007-08) = [ ($380 - $360)/$360 ] + $20/$360 = 0.056 + 0.056 = 0.111 = 11.10%

Year	HPR
2005-06	14%
2006-07	-9.60%
2007-08	11.10%

B. Arithmetic return = 14% + (-9.60%) + 11.10% = 16%

C. Geometric return = [ { (1+0.14) * (1-0.096) * (1+0.111)}  ^ (1/3) ] - 1

Geometric return = [ {1.14 * 0.904 * 1.111 } ^ (1/3) ] - 1

Geometric return = [ {1.1449} ^ (1/3) ] - 1

Geometric return = [ 1.046154 ] - 1

Geometric return = 0.046154 = 4.615%

D. Dollar weighted return

At the beginning of each year, the below $ transactions happened:

	2005	2006	2007	2008
Invested	$300	$520	0	0
Sold	$0	$0	$90	$380

Dollar weighted return is:

$380/ [(1+r)^3] + $90 / [(1+r)^2] = $300 /[(1+r)^0] + $220 /[(1+r)^1]