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Cailin Chen
Question 9: (10 Points)
1. A portfolio consists of the following securities:
Std.
Security Amount Invested Dev.
σi
Beta
βi
A
$250,000
15.50 1.45
B
$350,000
8.35
C
$200,000
21.22 1.85
D
$150,000
7.75
E
$50,000
13.75 1.25
0.75
0.45
What is the portfolio’s beta? Is the portfolio riskier than the market? Explain.
2. An investor’s portfolio two risky-asset portfolio is allocated as follows:
Asset Type
Amount Invested Standard Deviation Expected Return
Bonds
$45,000
15%
6.5%
Stocks
$55,000
22%
14.0%
The correlation coefficient between the two assets is zero (ρ b, s = 0). Based on this
information, calculate the portfolio’s variance of returns (σ2p), standard deviation (σp), and
expected return E (Rp).
3. The realized returns for the common stock of Company X are given as follows:
2009 2010
5.3%
2011
2012
2013
15.5% -35.0% 25.2% 12.7%
Based on this information calculate the stock’s average return, variance of returns, and
standard deviation.
Portfolio beta solution 1:
It is computed in a tabular form with formulas:
Std.
Dev.
σi
Amount
Security Invested
Beta
βi
Weights= invest/total
investment
Porfolio beta(weights*
beta)
A
$250,000
15.5
1.45
0.25
0.3625
B
$350,000
8.35
0.75
0.35
0.2625
C
$200,000
21.22
1.85
0.2
0.37
D
$150,000
7.75
0.45
0.15
0.0675
E
$50,000
13.75
1.25
0.05
0.0625
$1,000,000
Porfolio beta
1.125
The portfolio is riskier to market as the portfolio beta is more than 1 .Higher the beat higher the
risk of the portfolio
Solution 2:
Expected
Return
WEights
Expected
return
Difference Square
Variance
= square
*prob
Asset
Type
Amount Standard
Invested Deviation
Bonds
$45,000
15%
6.50%
$0.45
$0.03
-$0.04 00017015625 $0.00077
Stocks
$55,000
22%
14.00%
$0.55
$0.08
$0.03 00011390625 $0.00063
Portfolio
variance
$0.11
$0.00139
To compute the portfolio S.D = weight ^2 * variance ^2 + weight^2 * variance ^2 + 2* (0)
because correlation is zero
= .45^2 *.00077^2 + .55^2 *.00063^2
= .00024
Solution 3:
Year
Average return
return
Difference
Square
2009
5.30%
0.56%
0.0031%
2010
15.50%
10.76%
1.1578%
2011
-35.00%
-39.74%
15.7927%
2012
25.20%
20.46%
4.1861%
2013
12.70%
7.96%
0.6336%
0.0474 Variance
S.d = SQRT variance / n-1
= SQRT (.0544)
S.D = .233
Question 10: (10 Points)
1. Use the following scenario analysis for Stocks X and Y:
21.7733%
Scenario
Description
Probability of
Occurrence
Expected
Outcome (HPR)
Expected
Outcome (HPR)
Stock X
Stock Y
1
Bear Market
.25
-5.5%
-10.5%
2
Normal Market
.45
6.0%
8.0%
3
Bull Market
.30
11.5%
18.5%
Based on this information:
a. Calculate the expected returns for Stocks X and Y.
b. Calculate the standard deviations of returns on Stocks X and Y.
c. Which stock is riskier? Explain your answer. Hint: You may want to interpret and compare
the standard deviation for each stock.
d. Assume that 90% of your $100,000 portfolio is invested in Stock X, while the rest (or 10%) is
invested in Stock Y. What is the expected return of your portfolio?
2. Use the CAPM to calculate the required rate of return (expected return) to Stock X if: the
risk-free rate of return is 4%, the expected return of the market is 12% and the stock’s beta
coefficient is 1.25. What is the stock’s alpha if its actual return was 12.5%? Is the stock
overpriced or underpriced? Explain.
a.Expected returns for Stocks X and Y.
Stock X E(R ) = (0.25*-0.055)+(0.45*0.06)+(0.3*0.115) = -0.01375+0.027+0.0345 = 0.04775 or
4.775%
Stock Y E(R ) = (0.25*-0.105)+(0.45*0.08)+(0.3*0.185) = -0.02625+0.036+0.0555 = 0.06525 or
6.525%
b. Standard deviations of returns on Stocks X and Y.
Variance Stock X = 0.25(-0.055-0.04775)+0.45(0.06-0.04775)+ 0.3(0.115-0.04775)
=000263939+000006750+000135677=000406366
Standard deviation Stock X= 000406366)^0.5 =006374684 or 6.37%
Variance Stock Y = 0.25(-0.105-0.06525)+0.45(0.08-0.06525)+ 0.3(0.185-0.06525)
=000724627+000009810+000430202 = 001164638
Standard deviation Stock Y= 001164638)^0.5 =01079184 or 10.79%
c. Which stock is riskier?
Stock Y is more risker as it has Standard deviation of 10.79% compare to stock X 6.37%.
d. Assume that 90% of your $100,000 portfolio is invested in Stock X, while the rest (or 10%)
is invested in Stock Y.
Expected return of your portfolio =( 0.9 *0.04775) +(0.1*0.06525) = 0.0495 or 4.95%
2.Using CAPM
E(R) = Rf+Beta (Rm- Rf)
= 4% +1.25 ( 12%-4%)
= 14%
Stock’s alpha if its actual return was 12.5% = 12.5% -14% = -1.5%
Stock X is undervalued since forcasted return is higher than actual return.
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