CHAPTER 9
Exercise 8. True or false? Explain or qualify as necessary.
a) Investors demand higher expected rates of return on stocks with more variable rates of
return.
False – investors demand higher expected rates of return on stocks with more non-diversifiable
risk (market risk).
b) The CAPM predicts that a security with a beta of 0 will offer zero expected return.
False – a security with a beta of zero will offer the risk-free rate of return (see SML).
c) An investor who puts $10,000 in Treasury bills and $20,000 n the market portfolio will have a
beta of 2.0.
False – the beta will be: (1/3 × 0) + (2/3 × 1) = 0.67
d) Investors demand higher expected rates of return from stocks with returns that are highly
exposed to macroeconomics risk.
True. To prove this you could also refer to the APT.
e) Investors demand higher expected rates of return from stocks with returns that are very
sensitive to fluctuations in the stock market.
True, actually this is the inverse statement of a).
Exercise 10. Mark Harrywitz proposes to invest in two shares, X and Y. He expects a return f 12%
from X and 8% from Y. The standard deviation of returns is 8% for X and, 5% for Y. The
correlation coefficient between the returns is 0.2.
a) Compute the expected return and standard deviation of the following portfolios:
Portfolio
1
2
3
Percentage in X
50
25
75
Percentage in Y
50
75
25
Solution:
Portfolio
1
2
Percentage in X
50
25
Percentage in Y
50
75
r
σ
10.0%
9.0
5.1%
4.6
1
3
75
X: 12%
X: 8%
Return
StDev
25
Y: 8%
Y: 5%
11.0
6.4
Where to calculate the standard deviation we used the squared root of the variance. Remember
that the variance of the portfolio is:
2
2
2
2
2
σ p = [ xI σI + x J σ J + 2(xI x JρIJσIσ J )]
b) Sketch the set of portfolios composed of X and Y.
The set of portfolios is represented by the curved line –below-. The five points are the three
portfolios from Part (a) plus the following two portfolios: one consists of 100% invested in X and
the other consists of 100% invested in Y.
0.15
Expected Return
2
0.1
3
1
g
g
g
g
g
0.05
0
0
0.02
0.04
0.06
0.08
0.1
Standard Deviation
c) Suppose that Mr. Harrywitz can also borrow or lend at an interest rate of 5%. Show
on your sketch how this alters his opportunities. Given that he can borrow or lend,
what proportions of the common stock portfolio should be invested in X and Y?
See the figure above. The best opportunities lie along the straight line. From the diagram, the
optimal portfolio of risky assets is portfolio 1, and so Mr. Harrywitz should invest 50 percent in X
and 50 percent in Y.
Exercise 11. Ebenezer Scrooge has invested 60% of his money in share A and the remainder in
share B. He assesses their prospects as follows:
Expected return (%)
Standard Deviation (%)
A
15
20
B
20
22
2
Correlation between returns
0.5
a) What are the expected return and standard deviation of returns on this portfolio?
Expected return = (0.6 × 15) + (0.4 × 20) = 17%
Variance = (0.62 × 202) + (0.42 × 222) + 2(0.6)(0.4)(0.5)(20)(22) = 0,0327
Standard deviation = 327.04(1/2) = 0,1808 = 18.08%
b) How would your answer change if the correlation coefficient were 0 or –0.5?
Correlation coefficient = 0 ⇒ Standard deviation = 14.88%
Correlation coefficient = –0.5 ⇒ Standard deviation = 10.76%
c) Is Mr. Scrooge’s portfolio better or worse than one invested entirely in share A, or is
it not possible to say?
The following table allow us to compare the two choices:
Expected return
St. Dev.
Portfolio (60/40)
17%
18,08%
100% A
15%
20%
His portfolio is clearly better because it has a higher expected return and a lower
standard deviation.
Exercise 15. The Treasury bill rate is 4%, and the expected return on the market portfolio is
12%. Using the Capital Asset Pricing Model:
a) Draw a graph similar to Figure 9.6 showing how the expected return varies with beta.
Expected Return
20
15
12%
10
5
0
0
0,5
1
1,5
2
Beta
3
b) What is the risk premium on the market?
Market risk premium = rm – rf = 0.12 – 0.04 = 0.08 = 8.0%
c) What is the required return on an investment with a beta of 1.5?
Use the security market line:
r = rf + β(rm – rf)
r = 0.04 + [1.5 × (0.12 – 0.04)] = 0.16 = 16.0%
Questions 20 and 21 – see BMA for the question text
20.
Stock P: r = 5% + (1.0 × 6.4%) + [(–2.0) × (–0.6%)] + [(–0.2) × 5.1%] = 11.58%
Stock P2: r = 5% + (1.2 × 6.4%) + [0 × (–0.6%)] + (0.3 × 5.1%) = 14.21%
Stock P3: r = 5% + (0.3 × 6.4%) + [0.5 × (–0.6%)] + (1.0 × 5.1%) = 11.72%
21.
a.
Factor risk exposures:
b1(Market) = [(1/3)×1.0] + [(1/3)×1.2] + [(1/3)×0.3] = 0.83
b2(Interest rate) = [(1/3)×(–2.0)] +[(1/3)×0] + [(1/3)×0.5] = –0.50
b3(Yield spread) = [(1/3)×(–0.2)] + [(1/3)×0.3] + [(1/3)×1.0] = 0.37
b.
rP = 5% + (0.83×6.4%) + [(–0.50)×(–0.6%)] + [0.37×5.1%] = 12.50%
Question 22 – see BMA for the question text CORRECTED
rC = 5.0% + (0.36 × 7.6%) + (-0.23 × 3.7%) + (0.38 × 5.2%) = 8.861%
rF = 5.0% + (2.00 × 7.6%) + (-0.03 × 3.7%) + (1.10 × 5.2%) = 25.809%
rP = 5.0% + (0.58 × 7.6%) + (–0.47 × 3.7%) + (–0.15 × 5.2%) = 6.89%
rM= 5.0% + (0.89 × 7.6%) + (0.07 × 3.7%) + (–1.17 × 5.2%) = 5.42%
4