Chapter 10
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180.367
Subject
Finance
Date
Jan 9, 2024
Type
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38
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1.
Award: 10.00
points
Problems? Adjust credit for all students.
Consider the one-factor APT. The standard deviation of returns on a well-diversified portfolio is 29.00%. The standard deviation on the factor portfolio is 26.00%.
What is the beta of the well-diversified portfolio?
Note: Note: Round your answer to 5 decimal places.
Beta
1.11538
Explanation:
The portfolio variance is the beta squared times the variance of returns on the factor portfolio.
Inserting the known values and rearranging, we have (0.29)
2
= β
2
(0.26)
2
. Therefore, β
2
= 1.2441 and β = 1.11538.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Additional Algorithmic Problems
References
2.
Award: 10.00
points
Problems? Adjust credit for all students.
Consider the multifactor APT with two factors. Stock A has an expected return of 15.20%, a beta of 1.2 on factor 1, and a beta of 0.6 on factor 2. The risk premium on the factor 1 portfolio is 4.00%. The risk-free rate of return is
6.20%.
What is the risk premium on factor 2 if no arbitrage opportunities exist?
Note: Round your answer to 2 decimal places.
Risk-premium
7.00
%
Explanation:
The return on the portfolio must equal the risk-free rate plus the sum of the products of each of the factor betas times the risk premium of that factor, or 15.20% = 6.20% + 1.20(4.00%) + 0.6(
F
2
). So F
2
= 7.00%.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Additional Algorithmic Problems
References
3.
Award: 10.00
points
Problems? Adjust credit for all students.
Assume that there are three stocks, A, B, and C, and that you can either invest in these stocks or short sell them. There are also three possible states of nature for economic growth in the upcoming year: strong, moderate, or
weak. The returns for the upcoming year on stocks A, B, and C for each of these states of nature are given below:
State of Nature
Stock
Strong Growth
Moderate Growth
Weak Growth
A
37%
17.50%
−8%
B
31%
12.00%
−4%
C
33%
15.00%
−6%
If you invested in an equally weighted portfolio of stocks A and C, what is your portfolio return if economic growth is moderate?
Note: Round your answer to 2 decimal places.
Portfolio return
16.25
%
Explanation:
The portfolio return would be an equally weighted average of the returns of the stocks assuming that the specified state of nature is realized, or E
(
r
p
) = 0.5%(17.50%) + 0.5%(15%) = 16.25%.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Additional Algorithmic Problems
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4.
Award: 10.00
points
Problems? Adjust credit for all students.
In the APT model, what is the nonsystematic standard deviation of an equally-weighted, well diversified portfolio of 260 securities that has an average value (across securities) of nonsystematic standard deviation, σ(
e
i
), equal to
22%?
Note: Round your answer to 2 decimal places.
Nonsystematic standard deviation
1.36
%
Explanation:
The nonsystematic standard deviation of an equally-weighted portfolio can be calculated as follows (the last term is the average nonsystematic variance):
The nonsystematic standard deviation of the portfolio is smaller than the average nonsystematic standard deviation of the securities and will approach zero as the number of securities increases.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Additional Algorithmic Problems
References
5.
Award: 10.00
points
Problems? Adjust credit for all students.
Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial production, IP, and the inflation rate, IR. IP is expected to be 3%, and IR 5%. A stock with a beta of 1 on IP and 0.5 on IR currently
is expected to provide a rate of return of 12%. If industrial production actually grows by 5%, while the inflation rate turns out to be 8%, what is your revised estimate of the expected rate of return on the stock?
Note: Do not round intermediate calculations. Round your answer to 1 decimal place.
Revised expected rate of return
15.5
%
Explanation:
The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor (industrial production and inflation) times the respective sensitivity
coefficient:
Revised estimate = 12% + [1 × (5% − 3%) + 0.5 × (8% − 5%)] = 15.5%
Note that the IP change is (5% − 3%), and the IR change is: (8% − 5%).
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
6.
Award: 10.00
points
Problems? Adjust credit for all students.
Suppose that there are two independent economic factors, F
1
and F
2
. The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 45%. Portfolios A
and B
are both well-
diversified with the following properties:
Portfolio
Beta on F
1
Beta on F
2
Expected Return
A
1.5
2.0
31%
B
2.2
−0.2
27%
Required:
What is the expected return-beta relationship in this economy? Calculate the risk-free rate, r
f
, and the factor risk premiums, RP
1
and RP
2
to complete the equation below.
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
E
(
r
P
)
= r
f
+ (
P
1
× RP
1
)
+ (
P
2
× RP
2
)
r
f
6.00 %
RP
1
10.00
%
RP
2
5.00
%
Explanation:
E
(
r
P
)
= r
f
+ P
1
[
E
(
r
1
)
− r
f
]
+ P
2
[
E
(
r
2
)
− r
f
]
We need to find the risk premium (
RP
) for each of the two factors:
RP
1
= [
E
(
r
1
)
− r
f
]
and RP
2
= [
E
(
r
2
)
− r
f
]
To do so, solve the following system of two equations with two unknowns:
0.31 = 0.06 + (1.5 × RP
1
) + (2.0 × RP
2
)
0.27 = 0.06 + (2.2 × RP
1
) + (−0.2 × RP
2
)
The solution to this set of equations is
RP
1
= 10.00% and RP
2
= 5.00%
Thus, the expected return-beta relationship is
E(r
P
) = 6% + (
P
1
× 10.00%) + (
P
2
× 5.00%)
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
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7.
Award: 10.00
points
Problems? Adjust credit for all students.
Assume that portfolios A
and B
are both well diversified and that E
(
r
A
) = 12%
, and E
(
r
B
) = 9%
. If the economy has only one factor, and
β
A
= 1.2
, whereas β
B
= 0.8
, what must be the risk-free rate?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Risk-free rate
3.00
%
Explanation:
Substituting the portfolio returns and betas in the expected return-beta relationship, we obtain two equations with two unknowns, the risk-free rate (
r
f
) and the factor risk premium (
RP
):
12% = r
f
+ (1.2 × RP
)
9% = r
f
+ (0.8 × RP
)
Solving these equations, we obtain:
RP
= 7.5% and
r
f
= 3.00%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
8.
Award: 10.00
points
Problems? Adjust credit for all students.
Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%.
Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of +2%, and the other half have an alpha of −2%. Suppose the analyst invests $1 million in an equally weighted portfolio of the positive
alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks.
Required:
a. What are the expected profit (in dollars) and standard deviation of the analyst’s profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What are the expected profit (in dollars) and standard deviation of the analyst’s profit?
Note: Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.
Required A
Required B
$
$
Expected profit (in dollars)
40,000
Standard deviation
134,164
Explanation:
a. Shorting an equally weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the 10 positive-alpha stocks eliminates the market exposure and creates a zero-
investment portfolio. Denoting the systematic market factor as R
M
, the expected dollar return is (noting that the expectation of nonsystematic risk, e
, is zero):
$1,000,000 × [0.02 + (1.0 × R
M
)] − $1,000,000 × [(−0.02) + (1.0 × R
M
)]
= $1,000,000 × 0.04 = $40,000
The sensitivity of the payoff of this portfolio to the market factor is zero since the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving R
M
sum to zero.) The systematic
component of total risk is also zero. The variance of the analyst’s profit is not zero, since this portfolio is not well diversified.
For n
= 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will have a $100,000 position (either long or short) in each stock. Net market exposure is zero, but firm-specific risk has not been fully diversified. The
variance of dollar returns from the positions in the 20 stocks is
20 × [(100,000 × 0.30)
2
] = 18,000,000,000
The standard deviation of dollar returns is $134,164.
b. If
n
= 50 stocks (25 stocks long and 25 stocks short), the investor will have a $40,000 position in each stock, and the variance of dollar returns is
50 × [(40,000 × 0.30)
2
] = 7,200,000,000
The standard deviation of dollar returns is $84,853, while profit remains the same at $40,000.
Similarly, if n
= 100 stocks (50 stocks long and 50 stocks short), the investor will have a $20,000 position in each stock, and the variance of dollar returns is
100 × [(20,000 × 0.30)
2
] = 3,600,000,000
The standard deviation of dollar returns is $60,000.
Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of = 2.23607 (from $134,164 to $60,000).
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
8.
Award: 10.00
points
Problems? Adjust credit for all students.
Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%.
Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of +2%, and the other half have an alpha of −2%. Suppose the analyst invests $1 million in an equally weighted portfolio of the positive
alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks.
Required:
a. What are the expected profit (in dollars) and standard deviation of the analyst’s profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?
Required A
Required B
Complete this question by entering your answers in the tabs below.
How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?
Note: Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.
Required A
Required B
$
$
50 stocks
100 stocks
Standard deviation
84,853
60,000
Explanation:
a. Shorting an equally weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the 10 positive-alpha stocks eliminates the market exposure and creates a zero-
investment portfolio. Denoting the systematic market factor as R
M
, the expected dollar return is (noting that the expectation of nonsystematic risk, e
, is zero):
$1,000,000 × [0.02 + (1.0 × R
M
)] − $1,000,000 × [(−0.02) + (1.0 × R
M
)]
= $1,000,000 × 0.04 = $40,000
The sensitivity of the payoff of this portfolio to the market factor is zero since the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving R
M
sum to zero.) The systematic
component of total risk is also zero. The variance of the analyst’s profit is not zero, since this portfolio is not well diversified.
For n
= 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will have a $100,000 position (either long or short) in each stock. Net market exposure is zero, but firm-specific risk has not been fully diversified. The
variance of dollar returns from the positions in the 20 stocks is
20 × [(100,000 × 0.30)
2
] = 18,000,000,000
The standard deviation of dollar returns is $134,164.
b. If
n
= 50 stocks (25 stocks long and 25 stocks short), the investor will have a $40,000 position in each stock, and the variance of dollar returns is
50 × [(40,000 × 0.30)
2
] = 7,200,000,000
The standard deviation of dollar returns is $84,853, while profit remains the same at $40,000.
Similarly, if n
= 100 stocks (50 stocks long and 50 stocks short), the investor will have a $20,000 position in each stock, and the variance of dollar returns is
100 × [(20,000 × 0.30)
2
] = 3,600,000,000
The standard deviation of dollar returns is $60,000.
Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of = 2.23607 (from $134,164 to $60,000).
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
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9.
Award: 10.00
points
Problems? Adjust credit for all students.
Assume that security returns are generated by the single-index model,
R
i
= α
i
+ β
i
R
M
+ e
i
where R
i
is the excess return for security i
and R
M
is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A
, B
, and C
, characterized by the following data:
Security
β
i
E(R
i
)
σ(e
i
)
A
0.8
10%
25%
B
1.0
12
10
C
1.2
14
20
Required:
a. If σ
M
= 20%
, calculate the variance of returns of securities A
, B
, and C
.
b. Now assume that there are an infinite number of assets with return characteristics identical to those of A
, B
, and C
, respectively. What will be the mean and variance of excess returns for securities A
, B
, and C
?
Required A
Required B
Complete this question by entering your answers in the tabs below.
If M
= 20%
, calculate the variance of returns of securities A
, B
, and C
.
Note: Do not round intermediate calculations. Round your answers to the nearest whole number.
Required A
Required B
Variance
Security A
881
Security B
500
Security C
976
Explanation:
a. 2
= 2
2
M
+ 2
(
e
)
2
A
= (0.8
2
× 20
2
) + 25
2
= 881
2
B
= (1.0
2
× 20
2
) + 10
2
= 500
2
C
= (1.2
2
× 20
2
) + 20
2
= 976
b. If there are an infinite number of assets with identical characteristics, then a well-diversified portfolio of each type will have only systematic risk since the nonsystematic risk will approach zero with large n.
Each variance is
simply 2
× market variance:
Well-diversified 2
A
= 256
Well-diversified 2
B
= 400
Well-diversified 2
C
= 576
The mean will equal that of the individual (identical) stocks.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
9.
Award: 10.00
points
Problems? Adjust credit for all students.
Assume that security returns are generated by the single-index model,
R
i
= α
i
+ β
i
R
M
+ e
i
where R
i
is the excess return for security i
and R
M
is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A
, B
, and C
, characterized by the following data:
Security
β
i
E(R
i
)
σ(e
i
)
A
0.8
10%
25%
B
1.0
12
10
C
1.2
14
20
Required:
a. If σ
M
= 20%
, calculate the variance of returns of securities A
, B
, and C
.
b. Now assume that there are an infinite number of assets with return characteristics identical to those of A
, B
, and C
, respectively. What will be the mean and variance of excess returns for securities A
, B
, and C
?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Now assume that there are an infinite number of assets with return characteristics identical to those of A
, B
, and C
,
respectively. What will be the mean and variance of excess returns for securities A
, B
, and C
?
Note: Enter the variance answers as a percent squared and mean as a percentage. Do not round intermediate calculations.
Round your answers to the nearest whole number.
Required A
Required B
Show less
Mean
Variance
Security A
10
%
256
Security B
12
%
400
Security C
14
%
576
Explanation:
a. 2
= 2
2
M
+ 2
(
e
)
2
A
= (0.8
2
× 20
2
) + 25
2
= 881
2
B
= (1.0
2
× 20
2
) + 10
2
= 500
2
C
= (1.2
2
× 20
2
) + 20
2
= 976
b. If there are an infinite number of assets with identical characteristics, then a well-diversified portfolio of each type will have only systematic risk since the nonsystematic risk will approach zero with large n.
Each variance is
simply 2
× market variance:
Well-diversified 2
A
= 256
Well-diversified 2
B
= 400
Well-diversified 2
C
= 576
The mean will equal that of the individual (identical) stocks.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
10.
Award: 10.00
points
Problems? Adjust credit for all students.
Consider the following multifactor (APT) model of security returns for a particular stock.
Factor
Factor Beta
Factor Risk Premium
Inflation
1.2
6%
Industrial production
0.5
8
Oil prices
0.3
3
Required:
a. If T-bills currently offer a 6% yield, find the expected rate of return on this stock if the market views the stock as fairly priced.
b. Suppose that the market expects the values for the three macro factors given in column 1 below, but that the actual values turn out as given in column 2. Calculate the revised expectations for the rate of return on the stock
once the “surprises” become known.
Factor
Expected
Value
Actual
Value
Inflation
5%
4%
Industrial production
3
6
Oil prices
2
0
Note: For all requirements, do not round intermediate calculations. Round your answers to 1 decimal place.
a. Expected rate of return
18.1
%
b. Expected rate of return
17.8
%
Explanation:
a. E
(
r
) = 6% + (1.2 × 6%) + (0.5 × 8%) + (0.3 × 3%) = 18.1%
b. Surprises in the macroeconomic factors will result in surprises in the return of the stock:
[1.2 × (4% − 5%)] + [0.5 × (6% − 3%)] + [0.3 × (0% − 2%)] = −0.3%
E
(
r
) = 18.1% − 0.3% = 17.8%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
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11.
Award: 10.00
points
Problems? Adjust credit for all students.
Suppose that the market can be described by the following three sources of systematic risk with associated risk premiums.
Factor
Risk Premium
Industrial production (
I
)
6%
Interest rates (
R
)
2
Consumer confidence (
C
)
4
Required:
The return on a particular stock is generated according to the following equation:
r
= 15% + 1.0
I
+ 0.5
R
+ 0.75
C
+ e
a-1. Find the equilibrium rate of return on this stock using the APT. The T-bill rate is 6%.
Note: Do not round intermediate calculations. Round your answer to 1 decimal place.
a-2. Is the stock over- or underpriced?
a-1. Equilibrium rate of return
16.0
%
a-2. Is the stock over- or underpriced?
Overpriced
Explanation:
a-1.
The APT required
(i.e., equilibrium) rate of return on the stock based on r
f
and the factor betas is
Required E
(
r
) = 6% + (1 × 6%) + (0.5 × 2%) + (0.75 × 4%) = 16.0%
a-2.
According to the equation for the return on the stock, the expected return on the stock is 15% (because the expected
surprises on all factors are zero by definition). Because the (actually) expected return based on risk is
less than the required return, we conclude that the stock is overpriced.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
12.
Award: 10.00
points
Problems? Adjust credit for all students.
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The only model that Orb's senior management has promoted in the past is the capital asset pricing model (CAPM). Now Orb’s
management has asked one of its analysts, Kevin McCracken, CFA, to investigate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk premium for real GDP is 8%,
while the factor risk premium for inflation is 2%. He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively. Using his APT results, he computes the equilibrium expected
return of the fund. For comparison purposes, he then uses fundamental analysis to compute the actually expected return of Orb’s High Growth Fund. McCracken finds that the two estimates of the Orb High Growth Fund’s
expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the expected return is 8.5% above
the risk-free rate. McCracken then applies the APT model to the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are 0.75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to construct a portfolio that has a unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT estimates for the High Growth
Fund and the Large Cap Fund. He then computes the sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken will use his APT results for these three funds to create a
portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees who live off the steady income of their
investments. McCracken does not agree with Stiles, but says that the fund would be a good choice if upcoming supply-side macroeconomic policies of the government are successful.
Required:
According to the APT, if the risk-free rate is 4%, what should be McCracken's estimate of the expected return of Orb's High Growth Fund?
Note: Do not round intermediate calculations. Round your answer to 1 decimal place.
Expected return
17.0
%
Explanation:
The formula is E
(
r
) = 0.04 + (1.25 × 0.08) + (1.5 × 0.02) = 0.17 = 17.0%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
13.
Award: 10.00
points
Problems? Adjust credit for all students.
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The only model that Orb’s senior management has promoted in the past is the capital asset pricing model (CAPM). Now Orb’s
management has asked one of its analysts, Kevin McCracken, CFA, to investigate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk premium for real GDP is 8%,
while the factor risk premium for inflation is 2%. He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively. Using his APT results, he computes the equilibrium expected
return of the fund. For comparison purposes, he then uses fundamental analysis to compute the actually expected return of Orb’s High Growth Fund. McCracken finds that the two estimates of the Orb High Growth Fund’s
expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the expected return is 8.5% above
the risk-free rate. McCracken then applies the APT model to the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are 0.75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to construct a portfolio that has a unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT estimates for the High Growth
Fund and the Large Cap Fund. He then computes the sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken will use his APT results for these three funds to create a
portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees who live off the steady income of their
investments. McCracken does not agree with Stiles, but says that the fund would be a good choice if upcoming supply-side macroeconomic policies of the government are successful.
Required:
With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the information Kwon provides, is an arbitrage opportunity available?
Is an arbitrage opportunity available?
No
Explanation:
If r
f
= 4%
and based on the sensitivities to real GDP (0.75) and inflation (1.25), McCracken would calculate the expected return for the Orb Large Cap Fund to be:
E(r)
= 0.04 + (0.75 × 0.08) + (1.25 × 0.02) = 0.04 + 0.085 = 8.5%
above the risk free rate
Therefore, Kwon's fundamental analysis estimate is congruent with McCracken's APT estimate. If we assume that both Kwon and McCracken's estimates on the return of Orb’s Large Cap Fund are accurate, then no arbitrage
profit is possible.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
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14.
Award: 10.00
points
Problems? Adjust credit for all students.
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The only model that Orb’s senior management has promoted in the past is the capital asset pricing model (CAPM). Now Orb’s
management has asked one of its analysts, Kevin McCracken, CFA, to investigate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk premium for real GDP is 8%,
while the factor risk premium for inflation is 2%. He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively. Using his APT results, he computes the equilibrium expected
return of the fund. For comparison purposes, he then uses fundamental analysis to compute the actually expected return of Orb’s High Growth Fund. McCracken finds that the two estimates of the Orb High Growth Fund’s
expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the expected return is 8.5% above
the risk-free rate. McCracken then applies the APT model to the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are 0.75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to construct a portfolio that has a unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT estimates for the High Growth
Fund and the Large Cap Fund. He then computes the sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken will use his APT results for these three funds to create a
portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees who live off the steady income of their
investments. McCracken does not agree with Stiles, but says that the fund would be a good choice if upcoming supply-side macroeconomic policies of the government are successful.
Required:
If the GDP Fund is constructed from the other three funds, which of the following would be its weight in the Utility Fund?
If the GDP Fund is constructed from the other three funds, which of the following would be its weight in the Utility Fund?
-2.2
Explanation:
In order to eliminate inflation, the following three equations must be solved simultaneously, where the GDP sensitivity will equal 1 in the first equation, inflation sensitivity will equal 0 in the second equation and the sum of the
weights must equal 1 in the third equation.
1. 1.25
w
x
+ 0.75
w
y
+ 1.0
w
z
= 1
2. 1.5
w
z
+ 1.25
w
y
+ 2.0
w
z
= 0
3. w
x
+ w
y
+ w
z
= 1
Here, "x" represents Orb's "High Growth Fund", "y" represents "Large Cap Fund" and "z" represents "Utility Fund". Using algebraic manipulation will yield:
w
x
= w
y
= 1.6 and w
z
= −2.2 Weight in Utility Fund = −2.2.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
15.
Award: 10.00
points
Problems? Adjust credit for all students.
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The only model that Orb’s senior management has promoted in the past is the capital asset pricing model (CAPM). Now Orb’s
management has asked one of its analysts, Kevin McCracken, CFA, to investigate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk premium for real GDP is 8%,
while the factor risk premium for inflation is 2%. He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively. Using his APT results, he computes the equilibrium expected
return of the fund. For comparison purposes, he then uses fundamental analysis to compute the actually expected return of Orb’s High Growth Fund. McCracken finds that the two estimates of the Orb High Growth Fund’s
expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the expected return is 8.5% above
the risk-free rate. McCracken then applies the APT model to the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are 0.75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to construct a portfolio that has a unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT estimates for the High Growth
Fund and the Large Cap Fund. He then computes the sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken will use his APT results for these three funds to create a
portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees who live off the steady income of their
investments. McCracken does not agree with Stiles, but says that the fund would be a good choice if upcoming supply-side macroeconomic policies of the government are successful.
Required:
With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund would be appropriate:
With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund would be appropriate:
McCracken is correct and Stiles is wrong.
Explanation:
Since retirees living off a steady income would be hurt by inflation, this portfolio would not be appropriate for them. Retirees would want a portfolio with a return positively correlated with inflation to preserve value, and less
correlated with the variable growth of GDP. Thus, Stiles is wrong. McCracken is correct in that supply side macroeconomic policies are generally designed to increase output at a minimum of inflationary pressure. Increased
output would mean higher GDP, which in turn would increase returns of a fund positively correlated with GDP.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 10: Arbitrage Pricing Theory and Multifactor Models of Risk and Return > Chapter 10 Problems - Algorithmic & Static
References
1.
Award: 10.00 points
2.
Award: 10.00 points
3.
Award: 10.00 points
4.
Award: 10.00 points
_________ a relationship between expected return and risk.
APT only stipulates
CAPM only stipulates
Both CAPM and APT stipulate
Neither CAPM nor APT stipulate
No pricing model has been found.
Both models attempt to explain asset pricing based on risk or return relationships.
References
Multiple Choice
Difficulty: 1 Basic
Consider the multifactor APT with two factors. Stock A has an expected return of 17.6%, a beta of 1.75 on factor 1, and a beta of 0.86 on factor 2. The risk premium on the factor 1 portfolio is 3.2%. The risk-free rate of return is 5%.
What is the risk-premium on factor 2 if no arbitrage opportunities exist?
8.14%
3.61%
4.25%
7.75%
None of the options are correct.
E
(
r
A
)
= β
1
×
RP
1
+
β
2
×
RP
2
+ r
f
17.6% = 1.75 × 3.20% + 0.86
×
RP
2
+ 5.00% →
RP
2
= 8.14%
References
Multiple Choice
Difficulty: 3 Challenge
In a multifactor APT model, the coefficients on the macro factors are often called:
systematic risk.
factor sensitivities and insensitivities.
idiosyncratic or diversifiable risk.
factor alphas or betas.
factor sensitivities or factor betas.
The coefficients are called factor betas, factor sensitivities, or factor loadings.
References
Multiple Choice
Difficulty: 1 Basic
In a multifactor APT model, the coefficients on the macro factors are often called:
systematic risk.
firm-specific risk.
idiosyncratic risk.
factor betas.
The coefficients are called factor betas, factor sensitivities, or factor loadings.
References
Multiple Choice
Difficulty: 1 Basic
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5.
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6.
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7.
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8.
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In a multifactor APT model, the coefficients on the macro factors are often called:
systematic risk.
firm-specific risk.
idiosyncratic risk.
factor loadings.
None of the options are correct.
The coefficients are called factor betas, factor sensitivities, or factor loadings.
References
Multiple Choice
Difficulty: 1 Basic
Which pricing model provides no guidance concerning the determination of the risk premium on factor portfolios?
The CAPM
The multifactor APT
Both the CAPM and the multifactor APT
Neither the CAPM nor the multifactor APT
None of the options are correct.
The multifactor APT provides no guidance as to the determination of the risk premium on the various factors. The CAPM assumes that the excess market return over the risk-free rate is the market premium in the single factor
CAPM.
References
Multiple Choice
Difficulty: 2 Intermediate
An arbitrage opportunity exists if an investor can construct a _________ investment portfolio that will yield a sure profit.
positive
negative
zero
All of the options are correct.
None of the options are correct.
If the investor can construct a portfolio without the use of the investor's own funds and the portfolio yields a positive profit, arbitrage opportunities exist.
References
Multiple Choice
Difficulty: 1 Basic
The APT was developed in 1976 by:
Lintner.
Modigliani and Miller.
Ross.
Sharpe.
Markowitz.
Ross developed this model in 1976.
References
Multiple Choice
Difficulty: 1 Basic
9.
Award: 10.00 points
10.
Award: 10.00 points
11.
Award: 10.00 points
12.
Award: 10.00 points
A _________ portfolio is a well-diversified portfolio constructed to have a beta of 1 on one of the factors and a beta of 0 on any other factor.
factor
market
index
factor and market
factor, market, and index
A factor model portfolio has a beta of 1 one factor, with zero betas on other factors.
References
Multiple Choice
Difficulty: 1 Basic
The exploitation of security mispricing in such a way that risk-free economic profits may be earned is called:
arbitrage.
capital-asset pricing.
factoring.
fundamental analysis.
None of the options are correct.
Arbitrage is earning of positive profits with a zero (risk-free) investment.
References
Multiple Choice
Difficulty: 1 Basic
In developing the APT, Ross assumed that uncertainty in asset returns was a result of:
a common macroeconomic factor.
firm-specific factors.
pricing error.
a common macroeconomic factor and firm-specific factors.
None of the options are correct.
Total risk (uncertainty) is assumed to be composed of both macroeconomic and firm-specific factors.
References
Multiple Choice
Difficulty: 2 Intermediate
The _________ provides an unequivocal statement on the expected return-beta relationship for all assets, whereas the _________ implies that this relationship holds for all but perhaps a small number of securities.
APT; CAPM
APT; OPM
CAPM; APT
CAPM; OPM
None of the options are correct.
The CAPM is an asset-pricing model based on the risk or return relationship of all assets. The APT implies that this relationship holds for all well-diversified portfolios, and for all but perhaps a few individual securities.
References
Multiple Choice
Difficulty: 2 Intermediate
13.
Award: 10.00 points
14.
Award: 10.00 points
15.
Award: 10.00 points
16.
Award: 10.00 points
Consider a single factor APT. Portfolio A has a beta of 1.0 and an expected return of 16%. Portfolio B has a beta of 0.8 and an expected return of 12%. The risk-free rate of return is 6%. If you wanted to take advantage of an arbitrage
opportunity, you should take a short position in portfolio _________ and a long position in portfolio _________.
A; A
A; B
B; A
B; B
A; the riskless asset
A: 16% = 1.0
F
+ 6% →
F
= 10%;
B: 12% = 0.8
F
+ 6% →
F
= 7.5%;
Thus, short B and take a long position in A.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the single factor APT. Portfolio A has a beta of 0.2 and an expected return of 13%. Portfolio B has a beta of 0.4 and an expected return of 15%. The risk-free rate of return is 10%. If you wanted to take advantage of an
arbitrage opportunity, you should take a short position in portfolio _________ and a long position in portfolio _________.
A; A
A; B
B; A
B; B
None of the options are correct.
A: 13% = 10% + 0.2
F →
F
= 15%;
B: 15% = 10% + 0.4
F
→
F
= 12.5%; therefore, short B and take a long position in A.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the one-factor APT. The variance of returns on the factor portfolio is 5. The beta of a well-diversified portfolio on the factor is 1.2. The variance of returns on the well-diversified portfolio is approximately:
3.6.
7.2.
8.3.
19.1.
None of the options are correct.
σ
2
P
= 1.2
2
× 5 = 7.2
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the one-factor APT. The standard deviation of returns on a well-diversified portfolio is 14%. The standard deviation on the factor portfolio is 10%. The beta of the well-diversified portfolio is approximately:
0.80.
1.40.
1.65.
1.82.
None of the options are correct.
σ
2
p
= 14%
2
= β
2
× 10%
2 β
= 1.4
References
Multiple Choice
Difficulty: 2 Intermediate
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17.
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18.
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19.
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20.
Award: 10.00 points
Consider the single-factor APT. Stocks A and B have expected returns of 12% and 19%, respectively. The risk-free rate of return is 3%. Stock B has a beta of 1.2. If arbitrage opportunities are ruled out, stock A has a beta of:
0.675.
1.000.
1.300.
1.675.
0.750.
E
(
r
B
)
= β
B
× RP + r
f
19% = 1.2 × RP
+ 3% →
RP
=13.3%
E
(
r
A
)
= β
A
× RP + r
f
12% = β
A
× 13.3% + 3% →
β
A
= 0.675
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the multifactor APT with two factors. Stock A has an expected return of 14%, a beta of 1.2 on factor 1, and a beta of 0.8 on factor 2. The risk premium on the factor-1 portfolio is 3%. The risk-free rate of return is 4%. What is
the risk-premium on factor 2 if no arbitrage opportunities exist?
2%
4%
6%
8%
E(r
A
) = β
1
,A
× RP
1
+ β
2
,A × RP
2
+ r
f
14% = 1.2 × 3% + 0.8 × RP
2
+ 4% RP
2 =
8.0%
References
Multiple Choice
Difficulty: 3 Challenge
Consider the multifactor model APT with two factors. Portfolio A has a beta of 1.20 on factor 1 and a beta of 1.50 on factor 2. The risk premiums on the factor-1 and factor-2 portfolios are 1% and 7%, respectively. The risk-free rate of
return is 4%. The expected return on portfolio A is _________ if no arbitrage opportunities exist.
13.5%
15.0%
15.7%
23.0%
E
(
r
A
)
= β
1
,A
× RP
1
+ β
2,A × RP
2 + r
f
= 1.2 × 1% + 1.5 × 7% + 4%
=
15.7%
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the multifactor APT with two factors. The risk premiums on the factor 1 and factor 2 portfolios are 5% and 6%, respectively. Stock A has a beta of 1.2 on factor-1, and a beta of 0.7 on factor-2. The expected return on stock A
is 17%. If no arbitrage opportunities exist, the risk-free rate of return is:
6.0%.
6.5%.
6.8%.
7.4%.
None of the options are correct.
E
(
r
A
)
= β
1,A
×
RP
1
+
β
2,A ×
RP
2 +
r
f
17% = 1.2 × 5% + 0.7 × 6% + r
f
r
f
= 6.8%
References
Multiple Choice
Difficulty: 2 Intermediate
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21.
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22.
Award: 10.00 points
23.
Award: 10.00 points
Consider a one-factor economy. Portfolio A has a beta of 1.0 on the factor, and portfolio B has a beta of 2.0 on the factor. The expected returns on portfolios A and B are 11% and 17%, respectively. Assume that the risk-free rate is 6%,
and that arbitrage opportunities exist. Suppose you invested $100,000 in the risk-free asset, $100,000 in portfolio B, and sold short $200,000 of portfolio A. Your expected profit from this strategy would be:
−
$1,000.
$0.
$1,000.
$2,000.
None of the options are correct.
$100,000 × 0.06 = $6,000 (risk-free position);
$100,000 × 0.17 = $17,000 (portfolio B);
−
$200,000 × 0.11 = −
$22,000 (short position, portfolio A);
1,000 profit.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the one-factor APT. Assume that two portfolios, A and B, are well diversified. The betas of portfolios A and B are 1.0 and 1.5, respectively. The expected returns on portfolios A and B are 19% and 24%, respectively.
Assuming no arbitrage opportunities exist, the risk-free rate of return must be:
4.0%.
9.0%.
14.0%.
16.5%.
None of the options are correct.
E
(
r
B
)
= β
B
×
RP
+
r
f
24% = 1.5 × RP
+ r
f
RP
= ((24% − r
f
) ÷ 1.5)
E
(
r
A
)
= β
A
×
RP
+
r
f
19% = 1 × ((24% − r
f
) ÷ 1.5) + r
f →
r
f = 9%
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the multifactor APT. The risk premiums on the factor 1 and factor 2 portfolios are 5% and 3%, respectively. The risk-free rate of return is 10%. Stock A has an expected return of 19% and a beta on factor 1 of 0.8. Stock A has
a beta on factor 2 of:
1.33.
1.50.
1.67.
2.00.
None of the options are correct.
E
(
r
A
)
= β
1
×
RP
1
+
β
2
×
RP
2
+
r
f
19% = 0.8 × 5% + β
2
× 3% + 10% β
2
= 1.67
References
Multiple Choice
Difficulty: 2 Intermediate
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25.
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26.
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Consider the single factor APT. Portfolios A and B have expected returns of 14% and 18%, respectively. The risk-free rate of return is 7%. Portfolio A has a beta of 0.7. If arbitrage opportunities are ruled out, portfolio B must have a
beta of:
0.45.
1.00.
1.10.
1.22.
None of the options are correct.
E
(
r
A
)
= β
A
× RP + r
f
14% = 0.7 × RP + 7% RP
= 10%
E
(
r
B
)
= β
B
× RP + r
f
18% = β
B
× 10% + 7% β
B
= 1.10
References
Multiple Choice
Difficulty: 2 Intermediate
There are three stocks: A, B, and C. You can either invest in these stocks or short sell them. There are three possible states of nature for economic growth in the upcoming year (each equally likely to occur); economic growth may
be strong, moderate, or weak. The returns for the upcoming year on stocks A, B, and C for each of these states of nature are given below:
Stock
State of Nature Strong
Growth
Moderate Growth
Weak Growth
A
39%
17%
−
5%
B
30%
15%
0%
C
6%
14%
22%
If you invested in an equally-weighted portfolio of stocks A and B, your portfolio return would be _________ if economic growth were moderate.
3.0%
14.5%
15.5%
16.0%
None of the options are correct.
E
(
r
p
)
= w
A
×
r
A
+
w
B
×
r
B
=
0.5 × 17% + 0.5 × 15% = 16%
References
Multiple Choice
Difficulty: 1 Basic
There are three stocks: A, B, and C. You can either invest in these stocks or short sell them. There are three possible states of nature for economic growth in the upcoming year (each equally likely to occur); economic growth may
be strong, moderate, or weak. The returns for the upcoming year on stocks A, B, and C for each of these states of nature are given below:
Stock
State of Nature Strong
Growth
Moderate Growth
Weak Growth
A
39%
17%
−
5%
B
30%
15%
0%
C
6%
14%
22%
If you invested in an equally-weighted portfolio of stocks A and B, your portfolio return would be _________ if economic growth were moderate.
17.0%
22.5%
30.0%
30.5%
None of the options are correct.
E(r
p
) = w
A
×
r
A
+
w
c
×
r
c
=
0.5 × 39% + 0.5 × 6% = 22.5%
References
Multiple Choice
Difficulty: 1 Basic
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27.
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28.
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29.
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There are three stocks: A, B, and C. You can either invest in these stocks or short sell them. There are three possible states of nature for economic growth in the upcoming year (each equally likely to occur); economic growth may
be strong, moderate, or weak. The returns for the upcoming year on stocks A, B, and C for each of these states of nature are given below:
Stock
State of Nature Strong
Growth
Moderate Growth
Weak Growth
A
39%
17%
−
5%
B
30%
15%
0%
C
6%
14%
22%
If you invested in an equally-weighted portfolio of stocks B and C, your portfolio return would be _________ if economic growth was weak.
−
2.5%
0.5%
3.0%
11.0%
None of the options are correct.
E(r
p
) = w
B
×
r
B
+
w
c
×
r
c
=
0.5 × 0% + 0.5 × 22% = 11%
References
Multiple Choice
Difficulty: 1 Basic
Consider the multifactor APT. There are two independent economic factors, F
1
and F
2
. The risk-free rate of return is 6%. The following information is available about two well-diversified portfolios:
Portfolio
on Factor
1
on Factor2
Expected Return
A
1.0
2.0
19%
B
2.0
0.0
12%
Assuming no arbitrage opportunities exist, the risk premium on the factor F
1
portfolio should be:
3%.
4%.
5%.
6%.
None of the options are correct.
E(r
B
) = β
1,B
×
F
1
+
β
2,B ×
F
2
+
r
f
12% = 2 × F
1
+ 0 × F
2
+ 6% F
1 = 3%
E(r
A
) = β
1,A
×
F
1
+
β
2,A ×
F
2
+
r
f
19% = 1 × 3%
+ 2 × F
2
+ 6% F
2 = 5%
References
Multiple Choice
Difficulty: 3 Challenge
Consider the multifactor APT. There are two independent economic factors, F
1
and F
2
. The risk-free rate of return is 6%. The following information is available about two well-diversified portfolios:
Portfolio
on Factor
1
on Factor
2
Expected Return
A
1.0
2.0
19%
B
2.0
0.0
12%
Assuming no arbitrage opportunities exist, the risk premium on the factor F
2
portfolio should be:
3%.
4%.
5%.
6%.
None of the options are correct.
E(r
B
) = β
1,B
×
F
1
+
β
2,B ×
F
2
+ r
f
12% = 2 × F
1
+ 0 × F
2
+ 6% F
1 = 3%
E(r
A
) = β
1,A
×
F
1
+
β
2,A ×
F
2
+
r
f
19% = 1 × 3%
+ 2 × F
2
+ 6% F
2 = 5%
References
Multiple Choice
Difficulty: 3 Challenge
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32.
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33.
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A zero-investment portfolio with a positive expected return arises when:
an investor has downside risk only.
the law of prices is not violated.
the opportunity set is not tangent to the capital-allocation line.
a risk-free arbitrage opportunity exists.
None of the options are correct.
When an investor can create a zero-investment portfolio (by using none of the investor's own funds) with a possibility of a positive profit, a risk-free arbitrage opportunity exists.
References
Multiple Choice
Difficulty: 1 Basic
An investor will take as large a position as possible when an equilibrium-price relationship is violated. This is an example of:
a dominance argument.
the mean-variance efficiency frontier.
a risk-free arbitrage.
the capital asset pricing model.
None of the options are correct.
When the equilibrium price is violated, the investor will buy the lower priced asset and simultaneously place an order to sell the higher priced asset. Such transactions result in risk-free arbitrage. The larger the positions, the greater
the risk-free arbitrage profits.
References
Multiple Choice
Difficulty: 2 Intermediate
The APT differs from the CAPM because the APT:
places more emphasis on market risk.
minimizes the importance of diversification.
recognizes multiple unsystematic risk factors.
recognizes multiple systematic risk factors.
None of the options are correct.
The CAPM assumes that market returns represent systematic risk. The APT recognizes that other macroeconomic factors may be systematic risk factors.
References
Multiple Choice
Difficulty: 2 Intermediate
The feature of the APT that offers the greatest potential advantage over the CAPM is the:
use of several factors instead of a single market index to explain the risk-return relationship.
identification of anticipated changes in production, inflation, and term structure as key factors in explaining the risk-return relationship.
superior measurement of the risk-free rate of return over historical time periods.
variability of coefficients of sensitivity to the APT factors for a given asset over time.
None of the options are correct.
The advantage of the APT is the use of multiple factors, rather than a single market index, to explain the risk-return relationship. However, APT does not identify the specific factors.
References
Multiple Choice
Difficulty: 1 Basic
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35.
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36.
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37.
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In terms of the risk or return relationship in the APT,
only factor risk commands a risk premium in market equilibrium.
only systematic risk is related to expected returns.
only nonsystematic risk is related to expected returns.
Both A & B
Both A & C
Nonfactor risk may be diversified away; thus, only factor risk commands a risk premium in market equilibrium. Nonsystematic risk across firms cancels out in well-diversified portfolios; thus, only systematic risk is related to expected
returns.
References
Multiple Choice
Difficulty: 1 Basic
Which of the following factors might affect stock returns?
The business cycle
Interest rate fluctuations
Inflation rates
All of the options.
None of the options are correct.
All of the options are likely to affect stock returns.
References
Multiple Choice
Difficulty: 1 Basic
Advantage(s) of the APT is (are):
that the model provides specific guidance concerning the determination of the risk premiums on the factor portfolios.
that the model does not require a specific benchmark market portfolio.
that risk need not be considered.
that the model provides specific guidance concerning the determination of the risk premiums on the factor portfolios, and that the model does not require a specific benchmark market portfolio.
that the model does not require a specific benchmark market portfolio, and that risk need not be considered.
The APT provides no guidance concerning the determination of the risk premiums on the factor portfolios. Risk must be considered in both the CAPM and APT. A major advantage of APT over the CAPM is that a specific benchmark
market portfolio is not required.
References
Multiple Choice
Difficulty: 1 Basic
An important difference between CAPM and APT is:
CAPM depends on risk-return dominance; APT depends on a no-arbitrage condition.
CAPM assumes many small changes are required to bring the market back to equilibrium; APT assumes a few large changes are required to bring the market back to equilibrium.
implications for prices derived from CAPM arguments are stronger than prices derived from APT arguments.
Both A & B
All of the options are true.
Under the risk-return dominance argument of CAPM, when an equilibrium price is violated many investors will make small portfolio changes, depending on their risk tolerance, until equilibrium is restored. Under the no-arbitrage
argument of APT, each investor will take as large a position as possible so only a few investors must act to restore equilibrium. Implications derived from APT are much stronger than those derived from CAPM.
References
Multiple Choice
Difficulty: 3 Challenge
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39.
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40.
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41.
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A professional who searches for mispriced securities in specific areas such as merger-target stocks, rather than one who seeks strict (risk-free) arbitrage opportunities is engaged in:
pure arbitrage.
risk arbitrage.
option arbitrage.
equilibrium arbitrage.
None of the options are correct.
Risk arbitrage involves searching for mispricing based on speculative information that may or may not materialize.
References
Multiple Choice
Difficulty: 2 Intermediate
In the context of the Arbitrage Pricing Theory, as a well-diversified portfolio becomes larger, its nonsystematic risk approaches:
one.
infinity.
zero.
negative one.
systematic risk.
As the number of securities, n, increases, the nonsystematic risk of a well-diversified portfolio approaches zero.
References
Multiple Choice
Difficulty: 1 Basic
A well-diversified portfolio is defined as:
one that is diversified over a large enough number of securities that the nonsystematic variance is essentially zero.
one that contains securities from at least three different industry sectors.
a portfolio whose factor beta equals 1.0.
a portfolio that is equally weighted.
None of the options are correct.
A well-diversified portfolio is one that contains a large number of securities, each having a small (but not necessarily equal) weight, so that nonsystematic variance is negligible.
References
Multiple Choice
Difficulty: 2 Intermediate
The APT requires a benchmark portfolio:
that is equal to the true market portfolio.
that contains all securities in proportion to their market values.
that need not be well-diversified.
that is well-diversified and lies on the SML.
that is unobservable.
Any well-diversified portfolio lying on the SML can serve as the benchmark portfolio for the APT. The true (and unobservable) market portfolio is only a requirement for the CAPM.
References
Multiple Choice
Difficulty: 2 Intermediate
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42.
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43.
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44.
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45.
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Imposing the no-arbitrage condition on a single-factor security market implies which of the following statements?
I. The expected return-beta relationship is maintained for all but a small number of well-diversified portfolios.
II. The expected return-beta relationship is maintained for all well-diversified portfolios.
III. The expected return-beta relationship is maintained for all but a small number of individual securities.
IV. The expected return-beta relationship is maintained for all individual securities.
I and III
I and IV
II and III
II and IV
Only I is correct.
The expected return-beta relationship must hold for all well-diversified portfolios and for all but a few individual securities; otherwise arbitrage opportunities will be available.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider a well-diversified portfolio, A, in a two-factor economy. The risk-free rate is 6%, the risk premium on the first factor portfolio is 4%, and the risk premium on the second factor portfolio is 3%. If portfolio A has a beta of 1.2 on
the first factor and .8 on the second factor, what is its expected return?
7.0%
8.0%
9.2%
13.0%
13.2%
E(r
A
) = β
1
×
RP
1
+
β
2
×
RP
2
+
r
f =
1.2 × 4% + 0.8 × 3% + 6% = 13.2%
References
Multiple Choice
Difficulty: 2 Intermediate
The term "arbitrage" refers to:
buying low and selling high.
short selling high and buying low.
earning risk-free economic profits.
negotiating for favorable brokerage fees.
hedging your portfolio through the use of options.
Arbitrage is exploiting security mispricing by the simultaneous purchase and sale to gain economic profits without taking any risk. A capital market in equilibrium rules out arbitrage opportunities.
References
Multiple Choice
Difficulty: 1 Basic
To take advantage of an arbitrage opportunity, an investor would
I. construct a zero-investment portfolio that will yield a sure profit.
II. construct a zero-beta-investment portfolio that will yield a sure profit.
III. make simultaneous trades in two markets without any net investment.
IV. short sell the asset in the low-priced market and buy it in the high-priced market.
I and IV
I and III
II and III
I, III, and IV
II, III, and IV
Only I and III are correct. II is incorrect because the beta of the portfolio does not need to be zero. IV is incorrect because the opposite is true.
References
Multiple Choice
Difficulty: 3 Challenge
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48.
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The factor F in the APT model represents:
firm-specific risk.
the sensitivity of the firm to that factor.
a factor that affects all security returns.
the deviation from its expected value of a factor that affects all security returns.
a random amount of return attributable to firm events.
F measures the unanticipated portion of a factor that is common to all security returns.
References
Multiple Choice
Difficulty: 2 Intermediate
In the APT model, what is the nonsystematic standard deviation of an equally-weighted portfolio that has an average value of σ
(
e
i
) equal to 25% and 50 securities?
12.5%
625%
0.5%
3.54%
14.59%
References
Multiple Choice
Difficulty: 2 Intermediate
In the APT model, what is the nonsystematic standard deviation of an equally-weighted portfolio that has an average value of σ
(
e
i
) equal to 20% and 20 securities?
12.5%
625%
4.47%
3.54%
14.59%
References
Multiple Choice
Difficulty: 2 Intermediate
In the APT model, what is the nonsystematic standard deviation of an equally-weighted portfolio that has an average value of σ
(
e
i
) equal to 20% and 40 securities?
12.5%
625%
0.5%
3.54%
3.16%
References
Multiple Choice
Difficulty: 2 Intermediate
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51.
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52.
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53.
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In the APT model, what is the nonsystematic standard deviation of an equally-weighted portfolio that has an average value of σ
(
e
i
) equal to 18% and 250 securities?
Note: Do not round your intermediate calculations.
1.14%
625%
0.5%
3.54%
3.16%
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following is true about the security market line (SML) derived from the APT?
The SML has a downward slope.
The SML for the APT shows expected return in relation to portfolio standard deviation.
The SML for the APT has an intercept equal to the expected return on the market portfolio.
The benchmark portfolio for the SML may be any well-diversified portfolio.
The SML is not relevant for the APT.
The benchmark portfolio does not need to be the (unobservable) market portfolio under the APT, but can be any well-diversified portfolio. The intercept still equals the risk-free rate.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following is false about the security market line (SML) derived from the APT?
The SML has an upward slope.
The SML for the APT shows expected return in relation to factor intensity.
The SML for the APT has an intercept that does not equal the expected return on the market portfolio.
The benchmark portfolio for the SML must be the CAPM market portfolio.
All of the options are correct.
The benchmark portfolio does not need to be the (unobservable) market portfolio under the APT, but can be any well-diversified portfolio. The intercept still equals the risk-free rate.
References
Multiple Choice
Difficulty: 2 Intermediate
If arbitrage opportunities are to be ruled out, each well-diversified portfolio's expected excess return must be:
inversely proportional to the risk-free rate.
inversely proportional to its standard deviation.
proportional to its weight in the market portfolio.
proportional to its standard deviation.
proportional to its beta coefficient.
For each well-diversified portfolio (P and Q, for example), it must be true that:
(E(r
P
) −
r
f
) ÷
β
P
=
(E(r
Q
) −
r
f
) ÷
β
Q
References
Multiple Choice
Difficulty: 2 Intermediate
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55.
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56.
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57.
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Suppose you are working with two factor portfolios, portfolio 1 and portfolio 2. The portfolios have expected returns of 15% and 6%, respectively. Based on this information, what would be the expected return on well-diversified
portfolio A, if A has a beta of 0.80 on the first factor and 0.50 on the second factor? The risk-free rate is 3%.
15.2%
14.1%
13.3%
10.7%
8.4%
E(r
P
) = β
1
× F
1
+ β
2
× F
2
+ r
f
= 0.8 × (15% − 3%) + 0.5 × (6% − 3%) + 3% = 14.1%
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following is(are) true regarding the APT?
I. The security market line does not apply to the APT.
II. More than one factor can be important in determining returns.
III. Almost all individual securities satisfy the APT relationship.
IV. It doesn't rely on the market portfolio that contains all assets.
II, III, and IV
II and IV
II and III
I, II, and IV
I, II, III, and IV
All except the first item are true. There is a security market line associated with the APT.
References
Multiple Choice
Difficulty: 2 Intermediate
In a factor model, the return on a stock in a particular period will be related to:
factor risk, only.
nonfactor risk, only.
standard deviation of returns, only.
factor risk and nonfactor risk.
None of the options are true.
Factor models explain firm returns based on both factor risk and nonfactor risk.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following factors did Chen, Roll, and Ross not include in their multifactor model?
Change in industrial production
Change in expected inflation but not unanticipated inflation
Change in unanticipated inflation but not expected inflation
Excess return of long-term government bonds over T-bills
All of the factors are included in the Chen, Roll, and Ross multifactor model.
Chen, Roll, and Ross included the four listed factors as well as the excess return of long-term corporate bonds over long-term government bonds in their model.
References
Multiple Choice
Difficulty: 2 Intermediate
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59.
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60.
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61.
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Which of the following factors did Chen, Roll, and Ross include in their multifactor model?
Change in industrial waste
Change in expected inflation
Change in unanticipated inflation
Change in expected inflation and unanticipated inflation
All of the factors were included in their model.
Chen, Roll, and Ross included the change in expected inflation and the change in unanticipated inflation as well as the excess return of long-term corporate bonds over long-term government bonds in their model.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following factors were used by Fama and French in their multifactor model?
Return on the market index
Excess return of small stocks over large stocks
Excess return of high book-to-market stocks over low book-to-market stocks
All of the factors were included in their model.
None of the factors were included in their model.
Fama and French included all three of the factors listed.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the single-factor APT. Stocks A and B have expected returns of 12% and 14%, respectively. The risk-free rate of return is 5%. Stock B has a beta of 1.2. If arbitrage opportunities are ruled out, stock A has a beta of:
0.67.
0.93.
1.30.
1.69.
None of the options are correct.
E(r
B
) = β
B
×
F +
r
f
14% = 1.2 × F + 5% →
F
= 7.5%
E(r
A
) = β
A
×
F +
r
f
12% = β
A
× 7.5% + 5% →
β
A
= 0.93
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the one-factor APT. The standard deviation of returns on a well-diversified portfolio is 19%. The standard deviation on the factor portfolio is 12%. The beta of the well-diversified portfolio is approximately:
1.58.
1.13.
1.25.
0.76.
None of the options are correct.
19%
2
= 12%
2
β
2
→
β
= 1.58
References
Multiple Choice
Difficulty: 2 Intermediate
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62.
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63.
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64.
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65.
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Black argues that past risk premiums on firm-characteristic variables, such as those described by Fama and French, are problematic because:
they may result from data snooping.
they are sources of systematic risk.
they can be explained by security characteristic lines.
they are more appropriate for a single-factor model.
they are macroeconomic factors.
Black argues that past risk premiums on firm-characteristic variables, such as those described by Fama and French, are problematic because they may result from data snooping.
References
Multiple Choice
Difficulty: 2 Intermediate
Multifactor models seek to improve the performance of the single-index model by:
modeling the systematic component of firm returns in greater detail.
incorporating firm-specific components into the pricing model.
allowing for multiple economic factors to have differential effects.
All of the options are correct.
None of the options are correct.
Multifactor models seek to improve the performance of the single-index model by modeling the systematic component of firm returns in greater detail, incorporating firm-specific components into the pricing model, and allowing for
multiple economic factors to have differential effects.
References
Multiple Choice
Difficulty: 1 Basic
Multifactor models, such as the one constructed by Chen, Roll, and Ross, can better describe assets' returns by:
expanding beyond one factor to represent sources of systematic risk.
using variables that are easier to forecast ex ante.
calculating beta coefficients by an alternative method.
using only stocks with relatively stable returns.
ignoring firm-specific risk.
The study used five different factors to explain security returns, allowing for several sources of risk to affect the returns.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the multifactor model APT with three factors. Portfolio A has a beta of 0.8 on factor 1, a beta of 1.1 on factor 2, and a beta of 1.25 on factor 3. The risk premiums on the factor 1, factor 2, and factor 3 are 3%, 5%, and 2%,
respectively. The risk-free rate of return is 3%. The expected return on portfolio A is _________ if no arbitrage opportunities exist.
13.5%
13.4%
16.5%
23.0%
None of the options are correct.
3% + 0.8 × 3% + 1.1 × 5% + 1.25 × 2% = 13.4%
References
Multiple Choice
Difficulty: 2 Intermediate
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66.
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67.
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68.
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69.
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Consider the multifactor APT. The risk premiums on the factor 1 and factor 2 portfolios are 6% and 4%, respectively. The risk-free rate of return is 4%. Stock A has an expected return of 16% and a beta on factor-1 of 1.3. Stock A has a
beta on factor-2 of:
1.33.
1.05.
1.67.
2.00.
None of the options are correct.
E(r
X
)
=
β
1
×
F
1
+
β
2
×
F
2
+
r
f
16% = 1.3 × 6% + β
2
× 4% + 4% β
2
= 1.05
References
Multiple Choice
Difficulty: 2 Intermediate
Consider a well-diversified portfolio, A, in a two-factor economy. The risk-free rate is 5%, the risk premium on the first-factor portfolio is 4%, and the risk premium on the second-factor portfolio is 6%. If portfolio A has a beta of 0.6 on
the first factor and 1.8 on the second factor, what is its expected return?
7.0%
8.0%
18.2%
13.0%
13.2%
E(r
A
)
= β
1
× F
1
+ β
2
× F
2
+ r
f
= 0.6 × 4% + 1.8 × 6% + 5% = 18.2%
References
Multiple Choice
Difficulty: 2 Intermediate
Consider a single factor APT. Portfolio A has a beta of 2.0 and an expected return of 22%. Portfolio B has a beta of 1.5 and an expected return of 17%. The risk-free rate of return is 4%. If you wanted to take advantage of an arbitrage
opportunity, you should take a short position in portfolio _________ and a long position in portfolio _________.
A; A
A; B
B; A
B; B
A; the riskless asset
E(r
A
) = 22% = 2.0 × F
+ 4% F
= 9.00%
E(r
B
) = 17% = 1.5 × F
+ 4% F
= 8.67%
Thus, short B and take a long position in A.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the single factor APT. Portfolio A has a beta of 0.5 and an expected return of 12%. Portfolio B has a beta of 0.4 and an expected return of 13%. The risk-free rate of return is 5%. If you wanted to take advantage of an
arbitrage opportunity, you should take a short position in portfolio _________ and a long position in portfolio _________.
A; A
A; B
B; A
B; B
None of the options are correct.
E(r
A
)
= 12% = 0.5 × F
+ 5% F
= 14.00%
E(r
B
)
= 13% = 0.4 × F
+ 5% F
= 20.00%
Therefore, short A and take a long position in B.
References
Multiple Choice
Difficulty: 2 Intermediate
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72.
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Consider the one-factor APT. The variance of returns on the factor portfolio is 9. The beta of a well-diversified portfolio on the factor is 1.25. The variance of returns on the well-diversified portfolio is approximately:
3.6.
6.0.
7.3.
14.1.
None of the options are correct.
σ
2
P
= 1.25
2
× 9 = 14.06
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the one-factor APT. The variance of returns on the factor portfolio is 11. The beta of a well-diversified portfolio on the factor is 1.45. The variance of returns on the well-diversified portfolio is approximately:
23.1.
6.0.
7.3.
14.1.
None of the options are correct.
σ
2
P
= 1.45
2
× 11 = 23.13
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the one-factor APT. The standard deviation of returns on a well-diversified portfolio is 22%. The standard deviation on the factor portfolio is 14%. The beta of the well-diversified portfolio is approximately:
0.80.
1.13.
1.25.
1.57.
None of the options are correct.
σ
2
P
= 22%
2
= β
2
× 14%
2
β
= 1.57
References
Multiple Choice
Difficulty: 2 Intermediate
The market return is 11% and the risk-free rate is 4%. Mammoth Incorporated has a market beta of 1.2, a SMB beta of −
0.78, and a HML beta of −
1.2. If the risk premium on HML and SMB are both 3%, using the Fama-French Three
Factor Model, what is the expected Return on Mammoth Incorporated stock?
4.66%
6.46%
12.3%
15.3%
None of the options are correct.
E
(
r
)
= β
M
×
F
M
+
β
SMB
×
F
SMB
+
β
HML
×
F
HML
+
r
f
= 1.2 × (11% −
4%) −
0.78 × 3% −
1.2 × 3% + 4% = 6.46%
References
Multiple Choice
Difficulty: 3 Challenge
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The market return is 12% and the risk-free rate is 4%. Smallish Incorporated has a market beta of 0.9, a SMB beta of 0.65, and a HML beta of 0.52. If the risk premium on HML and SMB are both 2%, using the Fama-French Three
Factor Model, what is the expected Return on Smallish Incorporated stock?
4.86%
7.46%
12.3%
13.54%
None of the options are correct.
E
(
r
)
= β
M
×
F
M
+
β
SMB
×
F
SMB
+
β
HML
×
F
HML
+
r
f
= 0.9 × (12% −
4%) + 0.65 × 2% + 0.52 × 2% + 4% = 13.54%
References
Multiple Choice
Difficulty: 3 Challenge
The market return is 10% and the risk-free rate is 3%. Rascals Incorporated has a market beta of 1.0, a SMB beta of −
0.60, and a HML beta of −
0.85. If the risk premium on HML and SMB are both 2%, using the Fama-French Three
Factor Model, what is the expected Return on Rascal Incorporated stock?
5.85%
7.10%
13.2%
15.3%
None of the options are correct.
E
(
r
)
= β
M
×
F
M
+
β
SMB
×
F
SMB
+
β
HML
×
F
HML
+
r
f
= 1.0 × (10% −
3%) −
0.6 × 2% −
0.85 × 2% + 3% = 7.10%
References
Multiple Choice
Difficulty: 3 Challenge
The market return is 11% and the risk-free rate is 4%. Mammoth Incorporated has a market beta of 1.2, a SMB beta of −
0.78, and a HML beta of −
1.2. The risk premium on HML and SMB are both 3%, using the Fama-French Three
Factor Model. If the single factor model generates a regression coefficient of 1.2, what is the different in returns between the Three-Factor model and the single factor model expected returns on Mammoth Incorporated stock?
5.66%
5.94%
11.3%
16.3%
None of the options are correct.
E
(
r
FF
)
= β
M
×
F
M
+
β
SMB
×
F
SMB
+
β
HML
×
F
HML
+
r
f
= 1.2 × (11% −
4%) −
0.78 × 3% −
1.2 × 3% + 4% = 6.46%
E
(
r
CAPM
)
= β
M
×
F
M
+
r
f
= 1.2 × (11% −
4%) + 4% = 12.4%
Difference = 12.4% −
6.46% = 5.94%
References
Multiple Choice
Difficulty: 3 Challenge
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The market return is 12% and the risk-free rate is 4%. Smallish Incorporated has a market beta of 0.9, a SMB beta of 0.65, and a HML beta of 0.52. The risk premium on HML and SMB are both 2%, using the Fama-French Three
Factor Model. If the single factor model generates a regression coefficient of 0.8, what is the different in returns between the Three-Factor model and the single factor model expected returns on Smallish Incorporated stock?
6.86%
5.46%
4.30%
3.14%
None of the options are correct.
E
(
r
FF
)
= β
M
×
F
M
+
β
SMB
×
F
SMB
+
β
HML
×
F
HML
+
r
f
= 0.9 × (12% −
4%) + 0.65 × 2% + 0.52 × 2% + 4% = 13.54%
E(r
CAPM
) = β
M
×
F
M
+
r
f
= 0.8 × (12% −
4%) + 4% = 10.4%
Difference = 13.54% −
10.40% = 3.14%
References
Multiple Choice
Difficulty: 3 Challenge
The market return is 10% and the risk-free rate is 3%. Rascals Incorporated has a market beta of 1.0, a SMB beta of −
0.60, and a HML beta of −
0.85. The risk premium on HML and SMB are both 2%, using the Fama-French Three
Factor Model. If the single factor model generates a regression coefficient of 1.3, what is the different in returns between the Three-Factor model and the single factor model expected returns on Rascal Incorporated stock?
2.8%
5.0%
5.8%
6.3%
None of the options are correct.
E
(
r
FF
)
= β
M
×
F
M
+
β
SMB
×
F
SMB
+
β
HML
×
F
HML
+
r
f
= 1.0 × (10% −
3%) −
0.6 × 2% −
0.85 × 2% + 3% = 7.10%
E(r
CAPM
) = β
M
× F
M
+ r
f
= 1.3 × (10% −
3%) + 3% = 12.1%
Difference = 12.1% −
7.10% = 5.00%
References
Multiple Choice
Difficulty: 3 Challenge
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Related Questions
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Security
A
B
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Component
Systematic Factor
Expected Return
5.65%
9.06%
€ A
EB
Variance
10.0365
0.0387
0.039
Beta
0.5
1.6
1. The beta of an equally weighted portfolio is: Number
2. The the variance of an equally weighted portfolio is (answer exactly): Number
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Scenario
Probability
M Return J Return
Strong
.30
18%
20%
Normal
.30
15%
12%
Weak
.40
9%
5%
1. What is the range for M?
2. What is the average exp. return for M ?
3. What is the standard deviation* M? 3.85 (given)
4. What is the CV for M?
5. What is the range for J?
6. What is the average exp. return for J?
7. What is the standard deviation J? 6.22 (given)
8. What is the CV for J?
9. Which is the better choice?
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Use the following information:
E[rXOM] = 15.6%, standard deviationyOM = 15.9%
%3D
E[IMSI=29.7%, standard deviationMS = 35.2%
Correlation of returns: PXOM.MS = 0.139, r=10%
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a. 85.00%
b. 15.00%
c. 34.00%
d. 17.00%
e. 9.00%
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Pls answer all questions with explanations. Rounded to four decimal places. Thx
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1 18% 30%
2 8% 10%
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1. Determine the expected return and the variance of the portfolio formed by the two
assets S₁, S₂ with weights ₁
=
0.6, x2
= 0.4. The assets returns are described by the
following scheme:
scenario
W1
2لا
W3
probability
0.1
0.4
0.5
T1
-20%
0%
20%
12
-10%
20%
40%
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(corrected problem)
NEW Problem 3: You have access to two investment opportunities. Mutual Fund A, which promises 20% expected return with a variance of 0.36, and Mutual Fund B, which promises 15% expected return with a variance 0f 0.12. The CORRELATION COEFFICIENT between the two is 0.084.
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Consider the following two assets:
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1 18% 30%
2 8% 10%
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Calculate the proportions of assets 1 and 2 that generate a portfolio with a standard deviation of zero.
What is the expected return of that portfolio?
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Please solve only part a of this question in 2 hours and get a thumbs up
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Suppose that there are four risky assets whose expected returns E(r) and variance-
covariance matrix (S) are shown in the spreadsheet below. We also consider the portfolio
weights of two portfolios x and y of risky assets (see Cells B8:E9):
1
8 Portfolio x
9 Portfolio y
A FOUR-ASSET PORTFOLIO PROBLEM
Variance-covariance, S
20 Portfolio variance,
21 Portfollo standard deviation o
0.10
0.01
0.03
0.05
11 Portfolio x and y statistics: Mean, variance, covariance, correlation
12 Mean, Ejr,
13 Variance,
14 Covariance()
15 Correlation P
16
17 Calculating returns of combinations of Portfolio x and Portfolio y
18 Proportion of x
19 Mean portfolio return, r
0.01
0.30
0.06
-0.04
0.20
0.20
10.50%
0.1216
0.0714
0.4540
?
?
?
0.3
0.03
0.06
0.40
0.02
0.30
0.10
?
0.05
0.02
0.50
0.40
0.10
0.10
0.60
Mean, Er
Variance, 0.2014
Question il
Question ili
Mean returns E(r)
?
7%
9%
11%
20%
Question i
i. Write the Excel formula used to estimate the mean and variance of portfolio y in cells
E12 and E13,…
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