1/5
Chapter 12
EQ # 1: Returns to an investment
You purchased 333 shares of Outel stock at a price of $30.00
per share. One year later, the shares are selling for $33.33 per share. In addition, a dividend of
$3.00 per share is paid at the end of the year. Calculate:
Income
= dividend * # of shares = $3x333=
$999
Capital gain
= [($33.33 – $30.00)/ $30.00] *
333 = $1108.89
The total dollar return
= $999 + $1108.89
=
$2107.89
Dividend yield
= dividend/original price =
3/30=10%
Capital gain yield =
($33.33  $30)/$30 =11.1%
Total percentage return
= dividend yield +
capital gain yield = 10% + 11.1% = 21.1%
EQ # 2: Variance and standard deviation of Returns
Given the following historical annual
returns: R1 in Year 1 = 5%; R2 = 15%; R3 = 8%; R4=3%; R5=  6%. What are the average
return, variance, and standard deviation?
*Average Return = [(.05) + (.15) + (.08) + (.03) + (.06)]/5 = .05 / 5 =  0.01
*Deviation form the Mean = Actual Return – Average Return
Year
Actual
Return
Average
Return
Deviation from
the Mean
Squared
Deviation
1
0.05
0.01
.06
36 (%)
2
2
 0.15
0.01
.14
196(%)
2
3
0.08
0.01
.09
81(%)
2
4
0.03
0.01
.04
16(%)
2
5
 0.06
0.01
.05
25(%)
2
Total
354(%)
2
Variance =354 x (%)
2
/ (5 – 1) =88.5 (%)
2
, standard deviation =
2
(%)
5
.
88
= 9.41%
EQ # 3: Historical Returns
Given the following historical data over the 1926 – 2000 period,
Asset
Average Return
Standard Deviation
Largecap stocks
13.0%
20.2%
Smallcap stocks
17.3%
33.4%
Longterm gov. bonds
5.7%
9.4%
Tbills
3.9%
3.2%
The historical risk premium on largecap stocks
= 13 – 3.9 = 9.1%
The historical risk premium of large stocks over LT gov. bonds
= 13 – 5.7 = 7.3%
The reward for bearing the risk of owning small rather than large stocks
= 17.3 – 13 = 4.3%
EQ # 4: Return Distribution
Using the data in EQ # 3, assume the return on each asset is normally distributed.
1.
With 68% confidence, what is the highest return you would expect to earn on Tbills?
2.
With 68% confidence, what is the lowest return you would expect to earn on small stock?
3.
With 99% confidence, what is the highest return you would expect to earn on Tbills?
4.
With 95% confidence, what is the lowest return you would expect to earn on large stock?
Using the normal distribution table (page A10 of the textbook),
5.
What is the probability of earning
at most
4.5% on Tbills?
6.
What is the probability of earning
at least
20% on small stocks?
Solution:
1.
68% confidence is within 1 standard deviation
Æ
3.9% +/– 3.2%
Î
the range is (0.7%,
7.1%), thus the highest is 7.1%;
2.
17.3% +/– 33.4%;
the lowest return on small stock is –16.1%;
3.
3.9% +/– (3.2% x 2.58); the highest is 3.9%+3.2 x 2.58 = 12.156%;
4.
13.0%  2 x 20.2% = 27.4%; or 13.0%  1.96 x 20.2% = 26.59%
5.
d=(4.5 – 3.9)/3.2 = .1875; N(d) = .575.
6.
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 Spring '05
 GPOTTER
 Normal Distribution, Standard Deviation, Variance, Capital Asset Pricing Model, Probability theory, Prof. Q. Ma

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