Topic F Exercises
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May 1, 2024
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Exercises
Topic F CAPM, Single-Index Model and Multi-Factor Models
Exercises #1 (Slide 9)
CML
(a special case of CAL, see Topic E Slides 21,22,31)
SML (slide 7 of Topic F)
Relevant Risk
Measures total risk - Standard dev.
Measures systematic risk - beta
Horizontal Axis
Standard dev. - Complete portfolio
Beta – individual securities Vertical Axis
Expected return – complete portfolio
Expected return – individual securities Intercept
Rf (risk free rate)
Rf (risk free rate)
Slope
Sharpe ratio – market portfolio Expected market of risk premium Exercise #2 (Slide 11)
You invest $800 in security A with a Beta of 1.4 and $500 in security B with a beta of
0.7. What is the weighted average Beta of the portfolio?
A weight: $800 / ($800 + $500) = $800 / $1,300 = 0.6153 = 61.50%
B weight:
$500 / $1,300 = 0.3846 = 38.46%
Weighted average Beta of the Portfolio:
A weight * A beta + B weight * B beta 0.6153 * 1.4 + 0.3846 * 0.7 0.862 + 0.269 = 1.31
1
Exercise #3 (Slide 17)
Within the context of the CAPM, assume:
Expected Return on the market=12%
Risk-free rate=2%
Expected Return on the ABC stock=16%
Beta on the ABC stock=1.5
(a) What is the ABC stock’s fair expected rate of return according to CAPM?
2% + 1.5 (12% - 2%)
2% + 1.5 * 10% 2% + 15% = 17%
(b) Is the stock underpriced, overpriced or fairly priced? How much is the alpha?
The stock is overpriced because the ABC stock expected return is 16% which is more than 17%.
Alpha: 16% - 17% = -1%
Exercise #4 (Slide 28)
Suppose that the index model for stocks A and B is estimated from R
A
=
3%
+
.7
R
M
+
e
A
R
B
=−
2%
+
1.2
R
M
+
e
B
σ
M
=
20%;
RSquared
A
=
.20
; RSquared
B
=
.12
excess returns with the following results:
2
(a) What is the standard deviation of each stock?
A: .7^2 * .2^2 / .2 = 31.3%
B: 1.2^2 * 0.2^2 / .12 = 69.28%
(b) Break down the variance of each stock to the systematic and firm-specific components.
A Systematic: .7^2 * .2^2 = 0.0196
Unsystemtaic= .0980 - .0196 = 0.0784
B Systemtaic: 1.2^2 * .2^2 = 0.0576
Unsytematic: .4800 – 0.0576 = 0.4224
(c) What are the covariance and correlation between the two stocks?
0.0336 / 31.30% * 69.28% = 0.155 (d) What is the covariance between each stock and the market index? 0.70 * 0.0400 = 0.028
1.20 * 0.0400 = 0.048
Exercise #5 (Slide 34)
(a) Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 3,000 stocks in order to construct a mean-variance efficient portfolio constrained by 3,000 investments. They will need to calculate (N) 3,000____ expected returns, (N)_3,000__________
variances of returns and __4,498,500______ covariances.
(b) (b)Assume that stock market returns do follow a single-index structure. An investment fund analyzes 3,000 stocks in order to construct a mean-variance efficient portfolio constrained by 3,000 investments. They will need to calculate 3
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(N)_3,000_______ estimates of expected returns and (N)___3,000_____
estimates
of sensitivity coefficients to the market factor (betas). 4
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Related Questions
Exercises:
a. The standard deviation of returns is 0.30 for Stock A and 0.20 for Stock B. The covariance between
the returns of A and B is 0.006. The correlation of returns between A and B is:
b. Explain the differences between systemic risk and unsystematic risk, give additional examples
c. Compare and contrast the Capital Market Line and Security Market Line
d.
The covariance of the market's returns with the stock's returns is 0.008. The standard deviation of
the market's returns is 0.08, and the standard deviation of the stock's returns is 0. 11. What is the
correlation coefficient of the returns of the stock and the returns of the market?
e. According to the CAPM, what is the required rate of return for a stock with a beta of 0.7, when the
risk-free rate is 7% and the expected market rate of return is 14%
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Question 1 Fill the parts in the above table that are shaded in yellow. You will notice that there are nineline items.
Question 2Using the data generated in the previous question (Question 1);a) Plot the Security Market Line (SML) b) Superimpose the CAPM’s required return on the SML c) Indicate which investments will plot on, above and below the SML? d) If an investment’s expected return (mean return) does not plot on the SML, what doesit show? Identify undervalued/overvalued investments from the graph
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use attachments to answer question
This question relates to Diagrams 6 - 9 from the 9.2 diagrams, each of which shows a set of portfolios plotted on a set of risk/return axes.
Which diagram shows (in red) the set of efficient portfolios in the presence of a risk-free asset?
Select one:
a.
Diagram 6
b.
Diagram 7
c.
Diagram 8
d.
Diagram 9
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Question No. 1:
Explain the following Financial Terminology and then determined the relationship between its.
portfolio efficient
Beta Coefficient
frontier efficient
Diversification
Diversifiable Risk
Systematic Risk
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risk premium (RP)
expected return on a portfolio,r^P
realized rate of return, r¨
diversification
correlation coefficient, ρρ
firm-specific (diversifiable) risk
market (nondiversifiable) risk
relevant risk
beta coefficient, ββ
capital asset pricing model (CAPM)
security market line (SML)
market risk premium (RPM)
equilibrium
Define all terms
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ma.4
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The Capital Market Line (CML) expresses the risk-return trade-off for a portfolio as follows:
E(Rport )=RFR+Oport [(E(Rm)-RFR)/om ]
Required:
Extend this expression to allow for the evaluation of any individual risky Asset i. Explain the steps in details.
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Considering the attached set of securities and portfolio returns:
Find the combination of the weights that minimizes CV of the portfolio.
How does the CV of the optimal portfolio compare with the CVs of its constituents?
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Consider the following performance data for a portfolio manager:
Benchmark
Portfolio
Index
Portfolio
Weight
Weight
Return
Return
Stocks
0.65
0.7
0.11
0.12
Bonds
0.3
0.25
0.07
0.08
Cash
0.05
0.05
0.03
0.025
a.Calculate the percentage return that can be attributed to the asset allocation decision.
b.Calculate the percentage return that can be attributed to the security selection decision.
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measures.
Describe and discuss the following measures of performance evaluation!
Treynor Index, William Sharpe, Michael Jensen
Using the following table evaluate which is better than other using three different measure of
performance evaluation.
Asset
X
E(R)%
12
beta
Stdv
1.25
16
Y
11
1.0
12
Risk-free
3
0
0
Market index
12
1
12
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(a) The risk premium indicates the amount of compensation investors require for taking risks.
(b) The risk premium should be the same across different stocks.
(c) The greater common risks an investment has, the higher risk premium investors would require.
Group of answer choices:
1. (a)
2. (a) and (b)
3. (a) and (c)
4. (a), (b), and (c)
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Consider following information on a risky portfolio, risk-free asset and the market index.
What is the Sharpe ratio of the market index?
Risky portfolio
Risk-free asset
Market index
Average return
8.2%
2%
6%
Std. Dev.
26%
20%
Residual std. dev.
10%
Alpha
1.4%
Beta
1.2
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The table below contains the covariance matrix of stock returns and the market. Assume that the assumptions of CAPM hold.
1. Find the market risk.
2. Find the systematic risk of BlueChip.
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