Chapter 08 1
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Johns Hopkins University *
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Course
180.367
Subject
Finance
Date
Jan 9, 2024
Type
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56
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1.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities.
Required:
a.
How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?
b.
If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?
a. Estimates of expected returns
60
a. Estimates of variances
60
a. Estimates of covariances
1,770
a. Total estimates
1,890
b. Estimates
182
Explanation:
a.
To optimize this portfolio, one would need:
n
= 60 estimates of means
n
= 60 estimates of variances
(
n
2
− n) ÷ 2 = 1,770 estimates of covariances
Therefore, in total: (
n
2
+ 3n) ÷ 2 = 1,890 estimates
b.
In a single index model:
r
i
−
r
f
=
i
+
i
(
r
M
−
r
f
) +
e
i
Equivalently, using excess returns:
R
i
=
i
+
i
R
M
+
e
i
The variance of the rate of return can be decomposed into the components:
(1) The variance due to the common market factor:
i
2
2
M
(2) The variance due to firm specific unanticipated events:
2
(
e
i
)
In this model:
Cov(
r
i
, r
j
) =
i
j
2
M
The number of parameter estimates is:
n
= 60 estimates of the mean
E
(
r
i
)
n
= 60 estimates of the sensitivity coefficient
i
n
= 60 estimates of the firm-specific variance
2
(
e
i
)
1 estimate of the market mean
E
(
R
M
)
1 estimate of the market variance
2
M
Therefore, in total, 182 estimates.
The single index model reduces the total number of required estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from:
to (3n + 2)
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
2.
Award:
10.00
points
Problems?
Adjust credit
for all students.
The following are estimates for two stocks.
Stock
Expected
Return
Beta
Firm-Specific Standard
Deviation
A
13%
0.8
30
B
18
1.2
40
The market index has a standard deviation of 22% and the risk-free rate is 8%.
Required:
a.
What are the standard deviations of stocks
A
and
B
?
b.
Suppose that we were to construct a portfolio with proportions:
Stock A
0.30
Stock B
0.45
T-bills
0.25
Compute the expected return, beta, nonsystematic standard deviation, and standard deviation of the portfolio.
Required A
Required B
Complete this question by entering your answers in the tabs below.
What are the standard deviations of stocks
A
and
B
?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Stock A
34.78
%
Stock B
47.93
%
Explanation:
a.
The standard deviation of each individual stock is given by:
Since
A
= 0.8,
B
= 1.2, (
e
A
) = 30%, (
e
B
) = 40%, and
M
= 22%, we get:
A
= (
(0.8)
2
×
(0.22)
2
+
(0.30)
2
)
1/2
= 34.78%
B
= (
(1.2)
2
×
(0.22)
2
+
(0.40)
2
)
1/2
= 47.93%
b.
The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities:
E
(
r
P
) =
w
A
×
E
(
r
A
) +
w
B
×
E
(
r
B
) +
w
f
×
r
f
E
(
r
P
) = (0.30 × 13%) + (0.45 × 18%) + (0.25 × 8%) = 14.00%
The beta of a portfolio is similarly a weighted average of the betas of the individual securities:
P
=
w
A
×
A
+
w
B
×
B
+
w
f
×
f
P
= (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78
The variance of this portfolio is:
where
P
2
M
2
is the systematic component and
2
(
e
P
)
is the nonsystematic component.
Since the residuals
(
e
i
)
are uncorrelated, the nonsystematic variance is:
= (
(0.30)
2
×
(0.30)
2
) + (
(0.45)
2
×
(0.40)
2
) + (
(0.25)
2
× 0) = 0.0405
where
2
(
e
A
)
and
2
(
e
B
)
are the firm-specific (nonsystematic) variances of Stocks A and B, and
2
(e
f
)
, the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus:
(
e
P
) = (0.0405)
1/2
= 20.12%
The total variance of the portfolio is then:
2
P
= (
(0.78)
2
×
(0.22)
2
) + 0.0405 = 0.0699
P
= 0.2645
The total standard deviation is 26.45%.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
2.
Award:
10.00
points
Problems?
Adjust credit
for all students.
The following are estimates for two stocks.
Stock
Expected
Return
Beta
Firm-Specific Standard
Deviation
A
13%
0.8
30
B
18
1.2
40
The market index has a standard deviation of 22% and the risk-free rate is 8%.
Required:
a.
What are the standard deviations of stocks
A
and
B
?
b.
Suppose that we were to construct a portfolio with proportions:
Stock A
0.30
Stock B
0.45
T-bills
0.25
Compute the expected return, beta, nonsystematic standard deviation, and standard deviation of the portfolio.
Required A
Required B
Complete this question by entering your answers in the tabs below.
Suppose that we were to construct a portfolio with proportions:
Stock A
0.30
Stock B
0.45
T-bills
0.25
Compute the expected return, beta, nonsystematic standard deviation, and standard deviation of the portfolio.
Note: Do not round intermediate calculations. Enter your answer for Beta as a number, not a percent. Round your answers to
2 decimal places.
Required A
Required B
Show less
Expected return
14.00
%
Standard deviation
26.45
%
Beta
0.78
Nonsystematic standard deviation
20.12
%
Explanation:
a.
The standard deviation of each individual stock is given by:
Since
A
= 0.8,
B
= 1.2, (
e
A
) = 30%, (
e
B
) = 40%, and
M
= 22%, we get:
A
= (
(0.8)
2
×
(0.22)
2
+
(0.30)
2
)
1/2
= 34.78%
B
= (
(1.2)
2
×
(0.22)
2
+
(0.40)
2
)
1/2
= 47.93%
b.
The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities:
E
(
r
P
) =
w
A
×
E
(
r
A
) +
w
B
×
E
(
r
B
) +
w
f
×
r
f
E
(
r
P
) = (0.30 × 13%) + (0.45 × 18%) + (0.25 × 8%) = 14.00%
The beta of a portfolio is similarly a weighted average of the betas of the individual securities:
P
=
w
A
×
A
+
w
B
×
B
+
w
f
×
f
P
= (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78
The variance of this portfolio is:
where
P
2
M
2
is the systematic component and
2
(
e
P
)
is the nonsystematic component.
Since the residuals
(
e
i
)
are uncorrelated, the nonsystematic variance is:
= (
(0.30)
2
×
(0.30)
2
) + (
(0.45)
2
×
(0.40)
2
) + (
(0.25)
2
× 0) = 0.0405
where
2
(
e
A
)
and
2
(
e
B
)
are the firm-specific (nonsystematic) variances of Stocks A and B, and
2
(e
f
)
, the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus:
(
e
P
) = (0.0405)
1/2
= 20.12%
The total variance of the portfolio is then:
2
P
= (
(0.78)
2
×
(0.22)
2
) + 0.0405 = 0.0699
P
= 0.2645
The total standard deviation is 26.45%.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the two (excess return) index model regression results for
A
and
B
:
R
A
= 1% + 1.2
R
M
R
-square = 0.576
Residual standard deviation = 10.3%
R
B
= −2% + 0.8
R
M
R
-square = 0.436
Residual standard deviation = 9.1%
Required:
a.
Which stock has more firm-specific risk?
b.
Which stock has greater market risk?
c.
For which stock does market movement explain a greater fraction of return variability?
d.
If
r
f
were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock
A
?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Which stock has more firm-specific risk?
Required A
Required B
Required C
Required D
Which stock has more firm-specific risk?
Stock A
Explanation:
a.
Firm-specific risk is measured by the residual standard deviation. Thus, Stock A has more firm-specific risk: 10.3% > 9.1%
b.
Market risk is measured by beta, the slope coefficient of the regression. Stock A has a larger beta coefficient: 1.2 > 0.8
c.
R
2
measures the fraction of total variance of return explained by the market return. A's
R
2
is larger than B’s: 0.576 > 0.436
d.
Rewriting the SCL equation in terms of total return (
r
) rather than excess return (
R
):
The intercept is now equal to:
Since
r
f
= 6%
, the intercept would be: 1% + 6%(1 − 1.2) = 1% − 1.2% = −0.2%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the two (excess return) index model regression results for
A
and
B
:
R
A
= 1% + 1.2
R
M
R
-square = 0.576
Residual standard deviation = 10.3%
R
B
= −2% + 0.8
R
M
R
-square = 0.436
Residual standard deviation = 9.1%
Required:
a.
Which stock has more firm-specific risk?
b.
Which stock has greater market risk?
c.
For which stock does market movement explain a greater fraction of return variability?
d.
If
r
f
were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock
A
?
Required A
Required C
Complete this question by entering your answers in the tabs below.
Which stock has greater market risk?
Required A
Required B
Required C
Required D
Which stock has greater market risk?
Stock A
Explanation:
a.
Firm-specific risk is measured by the residual standard deviation. Thus, Stock A has more firm-specific risk: 10.3% > 9.1%
b.
Market risk is measured by beta, the slope coefficient of the regression. Stock A has a larger beta coefficient: 1.2 > 0.8
c.
R
2
measures the fraction of total variance of return explained by the market return. A's
R
2
is larger than B’s: 0.576 > 0.436
d.
Rewriting the SCL equation in terms of total return (
r
) rather than excess return (
R
):
The intercept is now equal to:
Since
r
f
= 6%
, the intercept would be: 1% + 6%(1 − 1.2) = 1% − 1.2% = −0.2%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the two (excess return) index model regression results for
A
and
B
:
R
A
= 1% + 1.2
R
M
R
-square = 0.576
Residual standard deviation = 10.3%
R
B
= −2% + 0.8
R
M
R
-square = 0.436
Residual standard deviation = 9.1%
Required:
a.
Which stock has more firm-specific risk?
b.
Which stock has greater market risk?
c.
For which stock does market movement explain a greater fraction of return variability?
d.
If
r
f
were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock
A
?
Required B
Required D
Complete this question by entering your answers in the tabs below.
For which stock does market movement explain a greater fraction of return variability?
Required A
Required B
Required C
Required D
For which stock does market movement explain a greater fraction of return variability?
Stock A
Explanation:
a.
Firm-specific risk is measured by the residual standard deviation. Thus, Stock A has more firm-specific risk: 10.3% > 9.1%
b.
Market risk is measured by beta, the slope coefficient of the regression. Stock A has a larger beta coefficient: 1.2 > 0.8
c.
R
2
measures the fraction of total variance of return explained by the market return. A's
R
2
is larger than B’s: 0.576 > 0.436
d.
Rewriting the SCL equation in terms of total return (
r
) rather than excess return (
R
):
The intercept is now equal to:
Since
r
f
= 6%
, the intercept would be: 1% + 6%(1 − 1.2) = 1% − 1.2% = −0.2%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
Your preview ends here
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3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the two (excess return) index model regression results for
A
and
B
:
R
A
= 1% + 1.2
R
M
R
-square = 0.576
Residual standard deviation = 10.3%
R
B
= −2% + 0.8
R
M
R
-square = 0.436
Residual standard deviation = 9.1%
Required:
a.
Which stock has more firm-specific risk?
b.
Which stock has greater market risk?
c.
For which stock does market movement explain a greater fraction of return variability?
d.
If
r
f
were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock
A
?
Required C
Required D
Complete this question by entering your answers in the tabs below.
If
r
f
were constant at 6% and the regression had been run using total rather than excess returns, what would have been the
regression intercept for stock
A
?
Note: Negative value should be indicated by a minus sign. Round your answer to 1 decimal place.
Required A
Required B
Required C
Required D
Intercept
(0.2)
%
Explanation:
a.
Firm-specific risk is measured by the residual standard deviation. Thus, Stock A has more firm-specific risk: 10.3% > 9.1%
b.
Market risk is measured by beta, the slope coefficient of the regression. Stock A has a larger beta coefficient: 1.2 > 0.8
c.
R
2
measures the fraction of total variance of return explained by the market return. A's
R
2
is larger than B’s: 0.576 > 0.436
d.
Rewriting the SCL equation in terms of total return (
r
) rather than excess return (
R
):
The intercept is now equal to:
Since
r
f
= 6%
, the intercept would be: 1% + 6%(1 − 1.2) = 1% − 1.2% = −0.2%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
1.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the index model for stocks
A
and
B
is estimated from excess returns with the following results:
R
A
= 3% + 0.7
R
M
+
e
A
R
B
= −2% + 1.2
R
M
+
e
B
σ
M
= 20%;
R
-square
A
= 0.20;
R
-square
B
= 0.12
What is the standard deviation of each stock?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Standard Deviation
Stock A
31.30
%
Stock B
69.28
%
Explanation:
The standard deviation of each stock can be derived from the following equation for
R
2
:
Explained variance ÷ Total variance
Therefore:
A
= 31.30%
For stock B:
B
= 69.28%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
4.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the index model for stocks
A
and
B
is estimated from excess returns with the following results:
R
A
= 3% + 0.7
R
M
+
e
A
R
B
= −2% + 1.2
R
M
+
e
B
σ
M
= 20%;
R
-square
A
= 0.20;
R
-square
B
= 0.12
Break down the variance of each stock to the systematic and firm-specific components.
Note: Do not round intermediate calculations. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
Risk for A
Risk for B
Systematic
196
576
Firm-specific
784
4,224
Explanation:
The standard deviation of each stock can be derived from the following equation for
R
2
:
= Explained variance ÷ Total variance
Therefore:
For stock B:
The systematic risk for A is:
The firm-specific risk of A (the residual variance) is the difference between A’s total risk and its systematic risk:
980 − 196 = 784
The systematic risk for B is:
B’s firm-specific risk (residual variance) is:
4,800 − 576 = 4,224
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
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5.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the index model for stocks
A
and
B
is estimated from excess returns with the following results:
R
A
= 3% + 0.7
R
M
+
e
A
R
B
= −2% + 1.2
R
M
+
e
B
σ
M
= 20%;
R
-square
A
= 0.20;
R
-square
B
= 0.12
What are the covariance and the correlation coefficient between the two stocks?
Note: Do not round intermediate calculations. Calculate using numbers in decimal form, not percentages. Round your answers to 4 decimal places.
Covariance
0.0336
Correlation coefficient
0.1549
Explanation:
The standard deviation of each stock can be derived from the following equation for
R
2
:
R
2
i
=
β
2
i
σ
2
M
σ
2
i
= Explained variance ÷ Total variance
Therefore:
σ
2
A
=
β
2
A
σ
2
M
R
2
A
=
0.7
2
× 0.20
2
0.20
= 0.098
For stock B:
σ
2
B
=
β
2
B
σ
2
M
R
2
B
=
1.2
2
× 0.20
2
0.12
= 0. 48
The covariance between the returns of A and B is (residuals are uncorrelated):
COV
r
A
,
r
B
= β
A
, β
B
, σ
2
M
= 0.70 × 1.20 × 0.04 = 0.0336
The correlation coefficient between the returns of A and B is:
ρ
AB
=
COV
r
A
,
r
B
σ
A
σ
B
=
0.0336
0.3130 × 0.6928
= 0.1549
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
(
)
(
)
References
6.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the index model for stocks
A
and
B
is estimated from excess returns with the following results:
R
A
= 3% + 0.7
R
M
+
e
A
R
B
= −2% + 1.2
R
M
+
e
B
σ
M
= 20%;
R
-square
A
= 0.20;
R
-square
B
= 0.12
What is the covariance between each stock and the market index?
Note: Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%. Do not round your intermediate calculations.
Covariance
Stock A
280
Stock B
480
Explanation:
The standard deviation of each stock can be derived from the following equation for
R
2
:
Explained variance ÷ Total variance
Therefore:
σ
A
=
0.3130
For stock B:
B
=
0.6928
Note that the correlation is the square root of
R
2
:
COV(r
A
, r
M
)
=
ρσ
A
σ
M
= 0.20
1/2
× 31.30 × 20 = 280
COV(r
B
, r
M
)
=
ρσ
B
σ
M
= 0.12
1/2
× 69.28 × 20 = 480
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
7.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the index model for stocks
A
and
B
is estimated from excess returns with the following results:
R
A
= 3% + 0.7
R
M
+
e
A
R
B
= −2% + 1.2
R
M
+
e
B
σ
M
= 20%;
R
-square
A
= 0.20;
R
-square
B
= 0.12
Assume you create portfolio
P
with investment proportions of 0.60 in
A
and 0.40 in
B
.
Required:
a.
What is the standard deviation of the portfolio?
Note: Do not round your intermediate calculations. Round your answer to 2 decimal places. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard
deviation is provided as 20%.
b.
What is the beta of your portfolio?
Note: Do not round your intermediate calculations. Round your answer to 2 decimal places. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard
deviation is provided as 20%.
c.
What is the firm-specific variance of your portfolio?
Note: Do not round your intermediate calculations. Round your answer to 2 decimal places. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard
deviation is provided as 20%.
d.
What is the covariance between the portfolio and the market index?
Note: Do not round your intermediate calculations. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
a. Standard deviation
35.81
%
b. Portfolio beta
0.90
c. Firm-specific
958.08
d. Covariance
360
Explanation:
The standard deviation of each stock can be derived from the following equation for
R
2
:
Explained variance ÷ Total variance
Therefore:
A
=
0.3130 = 31.30%
For stock B:
B
= 0.6928 = 69.28%
The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated):
The correlation coefficient between the returns of A and B is:
Note that the correlation is the square root of
R
2
:
Cov(
r
A
,
r
M
) =
ρ
A
M
= 0.20
1/2
× 31.30 × 20 = 280
Cov(
r
B
,
r
M
) =
ρ
B
M
= 0.12
1/2
× 69.28 × 20 = 480
a.
For portfolio
P
we can compute:
P
= [(
(0.6)
2
× 980) + (
(0.4)
2
× 4800) + (2 × 0.4 × 0.6 × 336)]
1/2
= [1282.08]
1/2
= 35.81%
b.
P
= (0.6 × 0.7) + (0.4 × 1.2) = 0.90
c.
2
(
e
P
) =
2
P
−
2
P
2
M
= 1282.08 − (
(0.90)
2
× 400) = 958.08
d.
Cov(
r
P
,
r
M
) =
P
2
M
= 0.90 × 400 = 360
This same result can also be attained using the covariances of the individual stocks with the market:
Cov(
r
P
,
r
M
) = Cov(0.6
r
A
+ 0.4
r
B
,
r
M
) = 0.6 × Cov(
r
A
,
r
M
) + 0.4 × Cov(
r
B
,
r
M
)
Cov(
r
P
,
r
M
) = (0.6 × 280) + (0.4 × 480) = 360
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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8.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the index model for stocks
A
and
B
is estimated from excess returns with the following results:
R
A
= 3% + 0.7
R
M
+
e
A
R
B
= −2% + 1.2
R
M
+
e
B
σ
M
= 20%;
R
-square
A
= 0.20;
R
-square
B
= 0.12
Assume you create a portfolio
Q
, with investment proportions of 0.50 in a risky portfolio
P
, 0.30 in the market index, and 0.20 in T-bill. Portfolio
P
is composed of 60% Stock
A
and 40% Stock
B
.
Required:
a.
What is the standard deviation of portfolio
Q
?
Note: Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%. Do not round intermediate calculations. Round your answer to 2
decimal places.
b.
What is the beta of portfolio
Q
?
Note: Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%. Do not round intermediate calculations. Round your answer to 2
decimal places.
c.
What is the "firm-specific" risk of portfolio
Q
?
Note: Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%. Do not round intermediate calculations. Round your answer to 2
decimal places.
d.
What is the covariance between the portfolio
Q
and the market index?
Note: Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%. Do not round intermediate calculations. Round your answer to
nearest whole number.
a. Standard deviation
21.55
%
b. Portfolio beta
0.75
c. Firm-specific
239.52
d. Covariance
300
Explanation:
The standard deviation of each stock can be derived from the following equation for
R
2
:
= Explained variance ÷ Total variance
Therefore:
A
=
0.3130
For stock B:
B
= 0.6928
The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated):
The correlation coefficient between the returns of A and B is:
Note that the correlation is the square root of
R
2
:
Cov(
r
A
,
r
M
) =
ρσ
A
M
= 0.20
1/2
× 31.30 × 20 = 280
Cov(
r
B
,
r
M
) =
ρ
B
M
= 0.12
1/2
× 69.28 × 20 = 480
For portfolio
P
we can compute:
P
= [(
(0.6)
2
× 0.098) + (
(0.4)
2
× 0.48) + (2 × 0.4 × 0.6 × 0.0336)]
1/2
= [0.1282]
1/2
= 35.81%
P
= (0.6 × 0.7) + (0.4 × 1.2) = 0.90
2
(
e
P
) =
2
P
−
2
P
2
M
= 1282.08 − (
(0.90)
2
× 400) = 958.80
Cov(
r
P
,
r
M
) =
P
2
M
= 0.90 × 400 = 360
This same result can also be attained using the covariances of the individual stocks with the market:
Cov(
r
P
,
r
M
) = Cov(0.6
r
A
+ 0.4
r
B
,
r
M
) = 0.6 × Cov(
r
A
,
r
M
) + 0.4 × Cov(
r
B
,
r
M
)
Cov(
r
P
,
r
M
) = (0.6 × 280) + (0.4 × 480) = 360
a.
Note that the variance of T-bills is zero, and the covariance of T-bills with any asset is zero. Therefore, for portfolio
Q
:
σ
Q
= [
w
²
P
σ²
P
+
w
²
M
σ²
M
+ 2 ×
w
P
×
w
M
× Cov(
r
P
,
r
M
)]½
σ
Q
= [((0.5)
2
× 1282.08) + ((0.3)
2
× 400) + (2 × 0.5 × 0.3 × 360)]
1/2
= 21.55%
b.
Q
=
w
P
P
+
w
M
M
= (0.5 × 0.90) + (0.3 × 1) + (0.20 × 0) = 0.75
c.
2
(
e
Q
) =
2
Q
–
2
Q
2
M
= 464.52 − (
(0.75)
2
× 400) = 239.52
d.
Cov (
r
Q
,
r
M
) =
Q
2
M
= 0.75 × 400 = 300
Worksheet
Difficulty: 2 Intermediate
References
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
9.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A stock recently has been estimated to have a beta of 1.24:
Required:
a.
What is the “adjusted beta” of this stock?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
b.
Suppose that you estimate the following regression describing the evolution of beta over time:
t
= 0.3 + 0.7
t
−1
What would be your predicted beta for next year?
Note: Do not round intermediate calculations. Round your answer to 3 decimal places.
a. Adjusted beta
1.16
b. Predicted beta
1.168
Explanation:
a.
Beta books adjusts beta by taking the sample estimate of beta and averaging it with 1, using the weights of 2/3 and 1/3, as follows:
Adjusted beta = [(2/3) × 1.24] + [(1/3) × 1] = 1.160
b.
If you use your current estimate of beta to be
t
−1
= 1.24
, then
t
= 0.3 + (0.7 × 1.24) = 1.168
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
a.
Calculate expected excess returns, alpha values, and residual variances for these stocks.
b.
Compute the proportion in the active portfolio and the passive index.
c.
What is the Sharpe ratio for the optimal portfolio?
d.
By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy?
e.
What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of risk aversion of 2.8?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Calculate expected excess returns, alpha values, and residual variances for these stocks.
Note: Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round "Alpha values" to 1
decimal place. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation if standard
deviation is provided as 20%.
Required A
Required B
Required C
Required D
Required E
Stock A
Stock B
Stock C
Stock D
Excess returns
12
%
10
%
9
%
4
%
Alpha values
1.6
%
(4.4)
%
3.4
%
(4.0)
%
Residual variances
3,364
5,041
3,600
3,025
Explanation:
a.
Alpha (α)
Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)]
E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(e
A
) =
58
2
= 3,364
2
(e
B
) =
71
2
= 5,041
2
(e
C
) =
60
2
= 3,600
2
(e
D
) =
55
2
= 3,025
b.
To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:
A
0.000476
− 0.6142
B
− 0.000873
1.1265
C
0.000944
− 1.2181
D
− 0.001322
1.7058
Total
− 0.000775
1.0000
Be unconcerned with the negative weights of the positive α stocks-the entire active position will be negative, returning everything to good order.
With these weights, the forecast for the active portfolio is:
α = [−0.6142 × 1.6] + [1.1265 × (−4.4)] − [1.2181 × 3.4] + [1.7058 × (−4.0)]
α = −16.90%
β = [−0.6142 × 1.3] + [1.1265 × 1.8] − [1.2181 × 0.70] + [1.7058 × 1] = 2.08
The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.
2
(
e
) = [
(−0.6142)
2
× 3,364] + [
1.1265
2
× 5,041] + [
(−1.2181)
2
× 3,600] + [
1.7058
2
× 3,025]
2
(
e
) = 21,808.83
2
(
e
) = 147.68%
The levered position in B [with high
2
(
e
)] overcomes the diversification effect and results in a high residual standard deviation.
The optimal risky portfolio has a
proportion
w
*
in the active portfolio, computed as follows:
The negative position is justified for the reason stated earlier.
The adjustment for beta is:
∗
Since
w
* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:|
1 − (−0.0486) = 1.0486
c.
To calculate the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio for the active portfolio is computed as
follows:
A
2
= 0.0131
Hence, the square of the Sharpe measure (S) of the optimized risky portfolio is:
S = 0.3662
d.
Compare S = 0.3662 to the market’s Sharpe ratio:
S
M
= 8/23 = 0.3478
→ A difference of 0.0183
The only moderate improvement in performance results from only a small position taken in the active portfolio
A
because of its large residual variance.
Note: Small differences may result from rounding intermediate steps.
e.
To calculate the makeup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:
P
=
w
M
+ (
w
A
×
A
) = 1.0486 + [(−0.0486) × 2.08] = 0.9475
E
(
R
P
) =
P
+
P
E
(
R
M
) = [(−0.0486) × (−16.90%)] + (0.9475 × 8%) = 8.40%
P
= 22.94%
Since
A
= 2.8, the optimal position in this portfolio is:
In contrast, with a passive strategy:
The final positions are (M may include some of stocks A through D):
Bills
1 − 0.5701 =
42.99%
M
0.5701 × 1.0486 =
59.77%
A
0.5701 × (−0.0486) × (−0.6142) =
1.70%
B
0.5701 × (−0.0486) × 1.1265 =
− 3.12%
C
0.5701 × (−0.0486) × (−1.2181) =
3.37 %
D
0.5701 × (−0.0486) × 1.7058 =
− 4.72%
(subject to rounding error)
100.00%
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
a.
Calculate expected excess returns, alpha values, and residual variances for these stocks.
b.
Compute the proportion in the active portfolio and the passive index.
c.
What is the Sharpe ratio for the optimal portfolio?
d.
By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy?
e.
What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of risk aversion of 2.8?
Required A
Required C
Complete this question by entering your answers in the tabs below.
Compute the proportion in the active portfolio and the passive index.
Note: Negative values should be indicated by a minus sign. Do not round intermediate calculations. Enter your answer as
decimals rounded to 4 places. Calculate using numbers in decimal form, not percentages. For example use "20" for calculation
if standard deviation is provided as 20%.
Required A
Required B
Required C
Required D
Required E
Show less
Proportion in Active Portfolio
(0.0486)
Proportion in Passive Index
1.0486
Explanation:
a.
Alpha (α)
Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)]
E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(e
A
) =
58
2
= 3,364
2
(e
B
) =
71
2
= 5,041
2
(e
C
) =
60
2
= 3,600
2
(e
D
) =
55
2
= 3,025
b.
To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:
A
0.000476
− 0.6142
B
− 0.000873
1.1265
C
0.000944
− 1.2181
D
− 0.001322
1.7058
Total
− 0.000775
1.0000
Be unconcerned with the negative weights of the positive α stocks-the entire active position will be negative, returning everything to good order.
With these weights, the forecast for the active portfolio is:
α = [−0.6142 × 1.6] + [1.1265 × (−4.4)] − [1.2181 × 3.4] + [1.7058 × (−4.0)]
α = −16.90%
β = [−0.6142 × 1.3] + [1.1265 × 1.8] − [1.2181 × 0.70] + [1.7058 × 1] = 2.08
The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.
2
(
e
) = [
(−0.6142)
2
× 3,364] + [
1.1265
2
× 5,041] + [
(−1.2181)
2
× 3,600] + [
1.7058
2
× 3,025]
2
(
e
) = 21,808.83
2
(
e
) = 147.68%
The levered position in B [with high
2
(
e
)] overcomes the diversification effect and results in a high residual standard deviation.
The optimal risky portfolio has a
proportion
w
*
in the active portfolio, computed as follows:
The negative position is justified for the reason stated earlier.
The adjustment for beta is:
∗
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Since
w
* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:|
1 − (−0.0486) = 1.0486
c.
To calculate the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio for the active portfolio is computed as
follows:
A
2
= 0.0131
Hence, the square of the Sharpe measure (S) of the optimized risky portfolio is:
S = 0.3662
d.
Compare S = 0.3662 to the market’s Sharpe ratio:
S
M
= 8/23 = 0.3478
→ A difference of 0.0183
The only moderate improvement in performance results from only a small position taken in the active portfolio
A
because of its large residual variance.
Note: Small differences may result from rounding intermediate steps.
e.
To calculate the makeup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:
P
=
w
M
+ (
w
A
×
A
) = 1.0486 + [(−0.0486) × 2.08] = 0.9475
E
(
R
P
) =
P
+
P
E
(
R
M
) = [(−0.0486) × (−16.90%)] + (0.9475 × 8%) = 8.40%
P
= 22.94%
Since
A
= 2.8, the optimal position in this portfolio is:
In contrast, with a passive strategy:
The final positions are (M may include some of stocks A through D):
Bills
1 − 0.5701 =
42.99%
M
0.5701 × 1.0486 =
59.77%
A
0.5701 × (−0.0486) × (−0.6142) =
1.70%
B
0.5701 × (−0.0486) × 1.1265 =
− 3.12%
C
0.5701 × (−0.0486) × (−1.2181) =
3.37 %
D
0.5701 × (−0.0486) × 1.7058 =
− 4.72%
(subject to rounding error)
100.00%
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
a.
Calculate expected excess returns, alpha values, and residual variances for these stocks.
b.
Compute the proportion in the active portfolio and the passive index.
c.
What is the Sharpe ratio for the optimal portfolio?
d.
By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy?
e.
What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of risk aversion of 2.8?
Required B
Required D
Complete this question by entering your answers in the tabs below.
What is the Sharpe ratio for the optimal portfolio?
Note: Do not round intermediate calculations. Enter your answer as decimals rounded to 4 places. Calculate using numbers in
decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
Required A
Required B
Required C
Required D
Required E
Sharpe ratio
0.3662
Explanation:
a.
Alpha (α)
Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)]
E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(e
A
) =
58
2
= 3,364
2
(e
B
) =
71
2
= 5,041
2
(e
C
) =
60
2
= 3,600
2
(e
D
) =
55
2
= 3,025
b.
To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:
A
0.000476
− 0.6142
B
− 0.000873
1.1265
C
0.000944
− 1.2181
D
− 0.001322
1.7058
Total
− 0.000775
1.0000
Be unconcerned with the negative weights of the positive α stocks-the entire active position will be negative, returning everything to good order.
With these weights, the forecast for the active portfolio is:
α = [−0.6142 × 1.6] + [1.1265 × (−4.4)] − [1.2181 × 3.4] + [1.7058 × (−4.0)]
α = −16.90%
β = [−0.6142 × 1.3] + [1.1265 × 1.8] − [1.2181 × 0.70] + [1.7058 × 1] = 2.08
The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.
2
(
e
) = [
(−0.6142)
2
× 3,364] + [
1.1265
2
× 5,041] + [
(−1.2181)
2
× 3,600] + [
1.7058
2
× 3,025]
2
(
e
) = 21,808.83
2
(
e
) = 147.68%
The levered position in B [with high
2
(
e
)] overcomes the diversification effect and results in a high residual standard deviation.
The optimal risky portfolio has a
proportion
w
*
in the active portfolio, computed as follows:
The negative position is justified for the reason stated earlier.
The adjustment for beta is:
Since
w
* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:|
1 − (−0.0486) = 1.0486
∗
c.
To calculate the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio for the active portfolio is computed as
follows:
A
2
= 0.0131
Hence, the square of the Sharpe measure (S) of the optimized risky portfolio is:
S = 0.3662
d.
Compare S = 0.3662 to the market’s Sharpe ratio:
S
M
= 8/23 = 0.3478
→ A difference of 0.0183
The only moderate improvement in performance results from only a small position taken in the active portfolio
A
because of its large residual variance.
Note: Small differences may result from rounding intermediate steps.
e.
To calculate the makeup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:
P
=
w
M
+ (
w
A
×
A
) = 1.0486 + [(−0.0486) × 2.08] = 0.9475
E
(
R
P
) =
P
+
P
E
(
R
M
) = [(−0.0486) × (−16.90%)] + (0.9475 × 8%) = 8.40%
P
= 22.94%
Since
A
= 2.8, the optimal position in this portfolio is:
In contrast, with a passive strategy:
The final positions are (M may include some of stocks A through D):
Bills
1 − 0.5701 =
42.99%
M
0.5701 × 1.0486 =
59.77%
A
0.5701 × (−0.0486) × (−0.6142) =
1.70%
B
0.5701 × (−0.0486) × 1.1265 =
− 3.12%
C
0.5701 × (−0.0486) × (−1.2181) =
3.37 %
D
0.5701 × (−0.0486) × 1.7058 =
− 4.72%
(subject to rounding error)
100.00%
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
a.
Calculate expected excess returns, alpha values, and residual variances for these stocks.
b.
Compute the proportion in the active portfolio and the passive index.
c.
What is the Sharpe ratio for the optimal portfolio?
d.
By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy?
e.
What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of risk aversion of 2.8?
Required C
Required E
Complete this question by entering your answers in the tabs below.
By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy?
Note: Do not round intermediate calculations. Enter your answer as decimals rounded to 4 places. Calculate using numbers in
decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
Required A
Required B
Required C
Required D
Required E
Improvement in Sharpe ratio
0.0184
Explanation:
a.
Alpha (α)
Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)]
E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(e
A
) =
58
2
= 3,364
2
(e
B
) =
71
2
= 5,041
2
(e
C
) =
60
2
= 3,600
2
(e
D
) =
55
2
= 3,025
b.
To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:
A
0.000476
− 0.6142
B
− 0.000873
1.1265
C
0.000944
− 1.2181
D
− 0.001322
1.7058
Total
− 0.000775
1.0000
Be unconcerned with the negative weights of the positive α stocks-the entire active position will be negative, returning everything to good order.
With these weights, the forecast for the active portfolio is:
α = [−0.6142 × 1.6] + [1.1265 × (−4.4)] − [1.2181 × 3.4] + [1.7058 × (−4.0)]
α = −16.90%
β = [−0.6142 × 1.3] + [1.1265 × 1.8] − [1.2181 × 0.70] + [1.7058 × 1] = 2.08
The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.
2
(
e
) = [
(−0.6142)
2
× 3,364] + [
1.1265
2
× 5,041] + [
(−1.2181)
2
× 3,600] + [
1.7058
2
× 3,025]
2
(
e
) = 21,808.83
2
(
e
) = 147.68%
The levered position in B [with high
2
(
e
)] overcomes the diversification effect and results in a high residual standard deviation.
The optimal risky portfolio has a
proportion
w
*
in the active portfolio, computed as follows:
The negative position is justified for the reason stated earlier.
The adjustment for beta is:
Since
w
* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:|
1 − (−0.0486) = 1.0486
∗
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c.
To calculate the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio for the active portfolio is computed as
follows:
A
2
= 0.0131
Hence, the square of the Sharpe measure (S) of the optimized risky portfolio is:
S = 0.3662
d.
Compare S = 0.3662 to the market’s Sharpe ratio:
S
M
= 8/23 = 0.3478
→ A difference of 0.0183
The only moderate improvement in performance results from only a small position taken in the active portfolio
A
because of its large residual variance.
Note: Small differences may result from rounding intermediate steps.
e.
To calculate the makeup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:
P
=
w
M
+ (
w
A
×
A
) = 1.0486 + [(−0.0486) × 2.08] = 0.9475
E
(
R
P
) =
P
+
P
E
(
R
M
) = [(−0.0486) × (−16.90%)] + (0.9475 × 8%) = 8.40%
P
= 22.94%
Since
A
= 2.8, the optimal position in this portfolio is:
In contrast, with a passive strategy:
The final positions are (M may include some of stocks A through D):
Bills
1 − 0.5701 =
42.99%
M
0.5701 × 1.0486 =
59.77%
A
0.5701 × (−0.0486) × (−0.6142) =
1.70%
B
0.5701 × (−0.0486) × 1.1265 =
− 3.12%
C
0.5701 × (−0.0486) × (−1.2181) =
3.37 %
D
0.5701 × (−0.0486) × 1.7058 =
− 4.72%
(subject to rounding error)
100.00%
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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- Unlimited textbook solutions
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10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
a.
Calculate expected excess returns, alpha values, and residual variances for these stocks.
b.
Compute the proportion in the active portfolio and the passive index.
c.
What is the Sharpe ratio for the optimal portfolio?
d.
By how much did the position in the active portfolio improve the Sharpe ratio compared to a purely passive index strategy?
e.
What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of risk aversion of 2.8?
Required D
Required E
Complete this question by entering your answers in the tabs below.
What should be the exact makeup of the complete portfolio (including the risk-free asset) for an investor with a coefficient of
risk aversion of 2.8?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places. Calculate using numbers in decimal
form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
Required A
Required B
Required C
Required D
Required E
Show less
Final Positions
Bills
42.99
%
M
59.77
%
A
1.70
%
B
(3.12)
%
C
3.37
%
D
(4.72)
%
Total
100
%
Explanation:
a.
Alpha (α)
Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)]
E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(e
A
) =
58
2
= 3,364
2
(e
B
) =
71
2
= 5,041
2
(e
C
) =
60
2
= 3,600
2
(e
D
) =
55
2
= 3,025
b.
To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:
A
0.000476
− 0.6142
B
− 0.000873
1.1265
C
0.000944
− 1.2181
D
− 0.001322
1.7058
Total
− 0.000775
1.0000
Be unconcerned with the negative weights of the positive α stocks-the entire active position will be negative, returning everything to good order.
With these weights, the forecast for the active portfolio is:
α = [−0.6142 × 1.6] + [1.1265 × (−4.4)] − [1.2181 × 3.4] + [1.7058 × (−4.0)]
α = −16.90%
β = [−0.6142 × 1.3] + [1.1265 × 1.8] − [1.2181 × 0.70] + [1.7058 × 1] = 2.08
The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.
2
(
e
) = [
(−0.6142)
2
× 3,364] + [
1.1265
2
× 5,041] + [
(−1.2181)
2
× 3,600] + [
1.7058
2
× 3,025]
2
(
e
) = 21,808.83
2
(
e
) = 147.68%
The levered position in B [with high
2
(
e
)] overcomes the diversification effect and results in a high residual standard deviation.
The optimal risky portfolio has a
proportion
w
*
in the active portfolio, computed as follows:
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The negative position is justified for the reason stated earlier.
The adjustment for beta is:
Since
w
* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:|
1 − (−0.0486) = 1.0486
c.
To calculate the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio for the active portfolio is computed as
follows:
A
2
= 0.0131
Hence, the square of the Sharpe measure (S) of the optimized risky portfolio is:
S = 0.3662
d.
Compare S = 0.3662 to the market’s Sharpe ratio:
S
M
= 8/23 = 0.3478
→ A difference of 0.0183
The only moderate improvement in performance results from only a small position taken in the active portfolio
A
because of its large residual variance.
Note: Small differences may result from rounding intermediate steps.
e.
To calculate the makeup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:
P
=
w
M
+ (
w
A
×
A
) = 1.0486 + [(−0.0486) × 2.08] = 0.9475
E
(
R
P
) =
P
+
P
E
(
R
M
) = [(−0.0486) × (−16.90%)] + (0.9475 × 8%) = 8.40%
P
= 22.94%
Since
A
= 2.8, the optimal position in this portfolio is:
In contrast, with a passive strategy:
The final positions are (M may include some of stocks A through D):
Bills
1 − 0.5701 =
42.99%
M
0.5701 × 1.0486 =
59.77%
A
0.5701 × (−0.0486) × (−0.6142) =
1.70%
B
0.5701 × (−0.0486) × 1.1265 =
− 3.12%
C
0.5701 × (−0.0486) × (−1.2181) =
3.37 %
D
0.5701 × (−0.0486) × 1.7058 =
− 4.72%
(subject to rounding error)
100.00%
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
∗
References
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- Access to all documents
- Unlimited textbook solutions
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11.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
Calculate the following for a portfolio manager who is not allowed to short sell securities. If allowed to short sell securities, the manager's Sharpe ratio is 0.3662.
a.
What is the cost of the restriction in terms of Sharpe’s measure?
b.
What is the utility loss to the investor (
A
= 2.8) given his new complete portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Calculate the following for a portfolio manager who is not allowed to short sell securities. If allowed to short sell securities, the
manager's Sharpe ratio is 0.3662. What is the cost of the restriction in terms of Sharpe’s measure?
Note: Do not round intermediate calculations. Enter your answer as decimals rounded to 4 places. Calculate using numbers in
decimal form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
Required A
Required B
Show less
Cost of restriction
0.0127
Explanation:
Alpha (α) Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)] E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(
e
A
) =
58
2
= 3,364
2
(
e
B
) =
71
2
= 5,041
2
(
e
C
) =
60
2
= 3,600
2
(
e
D
) =
55
2
= 3,025
a.
If a manager is not allowed to sell short, he will not include stocks with negative alphas in his portfolio, so he will consider only A and C:
A
σ
2
(
e
)
a
a
/
σ
2
(
e
)
σ
2
(
e
)
Sa
/
σ
2
(
e
)
A
1.6
3,364
0.000476
0.3352
C
3.4
3,600
0.000944
0.6648
0.001420
1.0000
The forecast for the active portfolio is:
= (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80%
= (0.3352 × 1.3) + (0.6648 × 0.7) = 0.90
2
(e) = (
(0.3352)
2
× 3,364) +
(0.6648
2
× 3,600) = 1,969
σ(e) = 44.37%
The weight in the active portfolio is:
w
0
= (
÷
2
(
e
)
) ÷ (E(R
M
) ÷
2
M
) = (2.80 ÷ 1,969.03) ÷ (8 ÷
23
2
) = 0.0939 = 9.39%
Adjusting for beta:
w
*
= w
0
÷ 1 + (1 − β) w
0
= 0.094 ÷ 1 + [(1 − 0.90) × 0.094] = 0.0931 = 9.31%
The information ratio of the active portfolio is:
A
=
α
÷
σ
(
e
) = 2.80 ÷ 44.37 = 0.0630
Hence, the square of the Sharpe ratio is:
Therefore: S = 0.3535
The market's Sharpe ratio is:
S
M
= 0.347826
When short sales are allowed, the manager’s Sharpe ratio is higher (0.3662). The reduction in the Sharpe ratio is the cost of the short sale restriction. The cost of the short sale restriction is 0.0127.
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The characteristics of the optimal risky portfolio are:
β
P
=
w
M
+
w
A
×
β
A
= (1 − 0.0931) + (0.0931 × 0.9) = 0.99
E
(
R
P
) =
α
P
+
β
P
×
E
(
R
M
) = (0.0931 × 2.8%) + (0.99 × 8%) = 8.18%
σ
2
P
=
β
2
P
×
σ
2
M
+
σ
2
(
e
P
) = (0.99 ×
23)
2
+
(0.0931
2
× 1,968.9998) = 536.36
σ
P
= 23.14%
With
A
= 2.8, the optimal position in this portfolio is:
y
= 8.18 ÷ (0.01 × 2.8 × 536.36) = 0.5455 = 54.51%
The final positions in each asset are:
Bills
1 − 0.5455 =
45.42%
M
0.5455 × (1 − 0.0931) =
49.50%
A
0.5455 × 0.0931 × 0.3352 =
1.70%
C
0.5455 × 0.0931 × 0.6648 =
3.38%
100.00%
b.
The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases, and for the passive strategy are:
E(R
C
)
σ
2
C
Unconstrained
0.5685 × 8.42% = 4.79
0.5685
2
× 528.94 = 170.95
Constrained
0.5455 × 8.18% = 4.46
0.5455
2
× 536.36 = 159.77
Passive
0.5401 × 8.00% = 4.32
0.5401
2
× 529.00 = 154.31
The utility levels below are computed using the formula:
E
(
r
C
) − 0.005
A
2
C
Unconstrained
8% + 4.79% − (0.005 × 2.8 × 170.95) = 10.40%
Constrained
8% + 4.46% − (0.005 × 2.8 × 159.77) = 10.23%
Passive
8% + 4.32% − (0.005 × 2.8 × 154.31) = 10.16%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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- Access to all documents
- Unlimited textbook solutions
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11.
Award:
10.00
points
Problems?
Adjust credit
for all students.
A portfolio manager summarizes the input from the macro and micro forecasters in the following table:
Micro Forecasts
Asset
Expected
Return (%)
Beta
Residual
Standard
Deviation (%)
Stock A
20
1.3
58
Stock B
18
1.8
71
Stock C
17
0.7
60
Stock D
12
1.0
55
Macro Forecasts
Asset
Expected
Return (%)
Standard
Deviation
(%)
T-bills
8
0
Passive equity portfolio
16
23
Required:
Calculate the following for a portfolio manager who is not allowed to short sell securities. If allowed to short sell securities, the manager's Sharpe ratio is 0.3662.
a.
What is the cost of the restriction in terms of Sharpe’s measure?
b.
What is the utility loss to the investor (
A
= 2.8) given his new complete portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Calculate the following for a portfolio manager who is not allowed to short sell securities. If allowed to short sell securities, the
manager's Sharpe ratio is 0.3662. What is the utility loss to the investor (
A
= 2.8) given his new complete portfolio?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places. Calculate using numbers in decimal
form, not percentages. For example use "20" for calculation if standard deviation is provided as 20%.
Required A
Required B
Show less
Cases
Utility Levels
Unconstrained
10.40
%
Constrained
10.23
%
Passive
10.16
%
Explanation:
Alpha (α) Expected Excess Return
α
i
= r
i
− [r
f
+ β
i
× (r
M
− r
f
)] E(r
i
) − r
f
α
A
= 20% − [8% + 1.3 × (16% − 8%)] = 1.6%
20% − 8% = 12%
α
B
= 18% − [8% + 1.8 × (16% − 8%)] = −4.4%
18% − 8% = 10%
α
C
= 17% − [8% + 0.7 × (16% − 8%)] = 3.4%
17% − 8% = 9%
α
D
= 12% − [8% + 1.0 × (16% − 8%)] = −4.0%
12% − 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
2
(
e
A
) =
58
2
= 3,364
2
(
e
B
) =
71
2
= 5,041
2
(
e
C
) =
60
2
= 3,600
2
(
e
D
) =
55
2
= 3,025
a.
If a manager is not allowed to sell short, he will not include stocks with negative alphas in his portfolio, so he will consider only A and C:
A
σ
2
(
e
)
a
a
/
σ
2
(
e
)
σ
2
(
e
)
Sa
/
σ
2
(
e
)
A
1.6
3,364
0.000476
0.3352
C
3.4
3,600
0.000944
0.6648
0.001420
1.0000
The forecast for the active portfolio is:
= (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80%
= (0.3352 × 1.3) + (0.6648 × 0.7) = 0.90
2
(e) = (
(0.3352)
2
× 3,364) +
(0.6648
2
× 3,600) = 1,969
σ(e) = 44.37%
The weight in the active portfolio is:
w
0
= (
÷
2
(
e
)
) ÷ (E(R
M
) ÷
2
M
) = (2.80 ÷ 1,969.03) ÷ (8 ÷
23
2
) = 0.0939 = 9.39%
Adjusting for beta:
w
*
= w
0
÷ 1 + (1 − β) w
0
= 0.094 ÷ 1 + [(1 − 0.90) × 0.094] = 0.0931 = 9.31%
The information ratio of the active portfolio is:
A
=
α
÷
σ
(
e
) = 2.80 ÷ 44.37 = 0.0630
Hence, the square of the Sharpe ratio is:
Therefore: S = 0.3535
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The market's Sharpe ratio is:
S
M
= 0.347826
When short sales are allowed, the manager’s Sharpe ratio is higher (0.3662). The reduction in the Sharpe ratio is the cost of the short sale restriction. The cost of the short sale restriction is 0.0127.
The characteristics of the optimal risky portfolio are:
β
P
=
w
M
+
w
A
×
β
A
= (1 − 0.0931) + (0.0931 × 0.9) = 0.99
E
(
R
P
) =
α
P
+
β
P
×
E
(
R
M
) = (0.0931 × 2.8%) + (0.99 × 8%) = 8.18%
σ
2
P
=
β
2
P
×
σ
2
M
+
σ
2
(
e
P
) = (0.99 ×
23)
2
+
(0.0931
2
× 1,968.9998) = 536.36
σ
P
= 23.14%
With
A
= 2.8, the optimal position in this portfolio is:
y
= 8.18 ÷ (0.01 × 2.8 × 536.36) = 0.5455 = 54.51%
The final positions in each asset are:
Bills
1 − 0.5455 =
45.42%
M
0.5455 × (1 − 0.0931) =
49.50%
A
0.5455 × 0.0931 × 0.3352 =
1.70%
C
0.5455 × 0.0931 × 0.6648 =
3.38%
100.00%
b.
The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases, and for the passive strategy are:
E(R
C
)
σ
2
C
Unconstrained
0.5685 × 8.42% = 4.79
0.5685
2
× 528.94 = 170.95
Constrained
0.5455 × 8.18% = 4.46
0.5455
2
× 536.36 = 159.77
Passive
0.5401 × 8.00% = 4.32
0.5401
2
× 529.00 = 154.31
The utility levels below are computed using the formula:
E
(
r
C
) − 0.005
A
2
C
Unconstrained
8% + 4.79% − (0.005 × 2.8 × 170.95) = 10.40%
Constrained
8% + 4.46% − (0.005 × 2.8 × 159.77) = 10.23%
Passive
8% + 4.32% − (0.005 × 2.8 × 154.31) = 10.16%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Problems - Static and Algorithmic
References
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12.
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points
Problems?
Adjust credit
for all students.
Consider the single-index model. The alpha of a stock is 1.00%. The return on the market index is 12.00%. The risk-free rate of return is 6.00%. The stock earns a return that exceeds the risk-free rate by 6.00%, and there are no
firm-specific events affecting the stock performance.
What is the beta of the stock?
Note: Round your answer to 2 decimal places.
Beta
0.83
Explanation:
The expected return-beta relationship of the single index model is given by
6.00% = 1.00% + b(12.00% − 6.00%)
b = 0.83
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Additional Algorithmic Problems
References
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points
Problems?
Adjust credit
for all students.
The index model has been estimated from the excess returns for stock A with the following results:
R
A
= 12.00% + 1.80
R
M
+
e
A
σ
M
= 24.00%
σ(
e
A
) = 21.00%
What is the standard deviation of the return for stock A?
Note: Round your answer to 2 decimal places.
Standard deviation
48.00
%
Explanation:
Then standard deviation of security A can be calculated as σ
A
= [β
A
2
σ
2
M
+ σ
2
(
e
A
)]
½
.
Therefore, σ
A
= [(1.80)
2
(0.24)
2
+ (0.210)
2
]
½
= 0.4800 = 48.00%.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Additional Algorithmic Problems
References
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points
Problems?
Adjust credit
for all students.
The index model has been estimated for stocks A and B with the following results:
R
A
= 0.12 + 0.630
R
M
+
e
A
R
B
= 0.04 + 1.448
R
M
+
e
B
σ
M
= 0.290
σ(
e
A
) = 0.20
σ(
e
B
) = 0.10
What is the covariance between each stock and the market index?
Note: Round your answers to 4 decimal places.
Stock A covariance
0.0530
Stock B covariance
0.1218
Explanation:
The covariance between stock A and the market is β
A
× σ
M
2
, so Cov(
r
A
,
r
M
) = (0.630)(0.0841) = 0.0530.
The covariance between stock B and the market is β
B
× σ
M
2
, so Cov(
r
B
,
r
M
) = (1.448)(0.0841) = 0.1218
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Additional Algorithmic Problems
References
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Problems?
Adjust credit
for all students.
The index model has been estimated for stocks A and B with the following results:
R
A
= 0.12 +0.675
R
M
+
e
A
R
B
= 0.04 + 1.520
R
M
+
e
B
σ
M
= 0.335
σ(
e
A
) = 0.20
σ(
e
B
) = 0.10
What is the correlation coefficient between the two stocks?
Note: Round your answer to 4 decimal places.
Correlation coefficient
0.7350
Explanation:
The correlation coefficient between stocks A and B is (β
A
×
β
B
× σ
M
2
) ÷ (σ
A
× σ
B
), or ρ
A
,
B
= [(0.675)(1.520)(0.112)] ÷ [(0.3019)(0.5189)] = 0.7350.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 08: Index Models > Chapter 08 Additional Algorithmic Problems
References
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1.
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1.00 point
2.
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3.
Award:
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4.
Award:
10.00 points
As diversification increases, the total variance of a portfolio approaches:
0.
1.
the variance of the market portfolio.
infinity.
None of the options are correct.
As more and more securities are added to the portfolio, unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance of the market portfolio.
References
Multiple Choice
Difficulty: 1 Basic
As diversification increases, the standard deviation of a portfolio approaches:
0.
1.
infinity.
the standard deviation of the market portfolio.
None of the options are correct.
As more and more securities are added to the portfolio, unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance (or standard deviation) of the market portfolio.
References
Multiple Choice
Difficulty: 1 Basic
As diversification increases, the firm-specific risk of a portfolio approaches:
0.
1.
infinity.
(
n
−
1) ×
n.
n
.
As more and more securities are added to the portfolio, unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance (or standard deviation) of the market portfolio.
References
Multiple Choice
Difficulty: 1 Basic
As diversification increases, the unsystematic risk of a portfolio approaches:
1.
0.
infinity.
(
n
−
1) ×
n.
n
.
As more and more securities are added to the portfolio, unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance (or standard deviation) of the market portfolio.
References
Multiple Choice
Difficulty: 1 Basic
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5.
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6.
Award:
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7.
Award:
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8.
Award:
10.00 points
As diversification increases, the unique risk of a portfolio approaches:
1.
0.
infinity.
(
n
−
1) ×
n.
n
.
As more and more securities are added to the portfolio, unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance (or standard deviation) of the market portfolio.
References
Multiple Choice
Difficulty: 1 Basic
The index model was first suggested by:
graham.
markowitz.
miller.
sharpe.
malkiel.
William Sharpe, building on the work of Harry Markowitz, developed the index model.
References
Multiple Choice
Difficulty: 1 Basic
A single-index model uses _________ as a proxy for the systematic risk factor.
a market index, such as the S&P 500
the current account deficit
the growth rate in GNP
the unemployment rate
None of the options are correct.
The single-index model uses a market index, such as the S&P 500, as a proxy for the market and thus for systematic risk.
References
Multiple Choice
Difficulty: 1 Basic
A single-index model uses _________ as a proxy for the systematic risk factor.
S&P 500
Russell 2000
Vanguard Growth Fund
inflation rates
None of the options are correct.
The single-index model uses a market index, such as the S&P 500, as a proxy for the market and thus for systematic risk.
References
Multiple Choice
Difficulty: 1 Basic
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9.
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10.
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11.
Award:
10.00 points
12.
Award:
10.00 points
The index model has been estimated for stocks A and B with the following results:
R
A
= 0.03 + 0.7
R
M
+
e
A
R
B
= 0.01 + 0.9
R
M
+
e
B
σ
M
= 0.35
σ
(
e
A
) = 0.20
σ
(
e
B
) = 0.10.
The covariance between the returns on stocks A and B is:
0.0384.
0.0406.
0.1920.
0.0772.
0.4000.
Cov
(
R
A
,
R
B
) =
β
A
β
B
σ
2
M
= 0.7 × 0.9 × 0.35
2
= 0.0772.
References
Multiple Choice
Difficulty: 3 Challenge
According to the index model, covariances among security pairs are:
due to the influence of a single common factor represented by the market index return.
extremely difficult to calculate.
related to industry-specific events.
usually positive.
due to the influence of a single common factor represented by the market index return and usually positive.
Most securities move together most of the time and move with a market index, or market proxy.
References
Multiple Choice
Difficulty: 1 Basic
The intercept in the regression equations calculated by beta books is equal to:
α
in the CAPM.
+
r
f
× (1 + ).
+
r
f
× (1 − ).
1
−
α
.
α
−
1.
The intercept that beta books call alpha is really, using the parameters of the CAPM, an estimate of
a + r
f
(1 −
b
)
. The apparent justification for this procedure is that, on a monthly basis,
r
f
(1 −
b
)
is small and is apt to be
swamped by the volatility of actual stock returns.
References
Multiple Choice
Difficulty: 2 Intermediate
Analysts may use regression analysis to estimate the index model for a stock. When doing so, the slope of the regression line is an estimate of:
the
α
of the asset.
the
β
of the asset.
the
σ
of the asset.
the
δ
of the asset.
None of the options are correct.
The slope of the regression line,
β
, estimates the volatility of the stock versus the volatility of the market, and the
α
estimates the intercept.
References
Multiple Choice
Difficulty: 2 Intermediate
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13.
Award:
10.00 points
14.
Award:
10.00 points
15.
Award:
10.00 points
16.
Award:
10.00 points
Analysts may use regression analysis to estimate the index model for a stock. When doing so, the intercept of the regression line is an estimate of:
the
α
of the asset.
the
β
of the asset.
the
σ
of the asset.
the
δ
of the asset.
None of the options are correct.
The slope of the regression line,
β
, estimates the volatility of the stock versus the volatility of the market, and the
α
estimates the intercept.
References
Multiple Choice
Difficulty: 2 Intermediate
In a factor model, the return on a stock in a particular period will be related to:
firm-specific events.
macroeconomic events.
the error term.
both firm-specific events and macroeconomic events.
neither firm-specific events nor macroeconomic events.
The return on a stock is related to both firm-specific and macroeconomic events.
References
Multiple Choice
Difficulty: 2 Intermediate
Rosenberg and Guy found that _________ helped to predict a firm's beta.
the firm's financial characteristics
the firm's industry group
firm size
the firm's financial characteristics and the firm's industry group
All of the options are correct.
Rosenberg and Guy found that after controlling for the firm's financial characteristics, the firm's industry group was a significant predictor of the firm's beta.
References
Multiple Choice
Difficulty: 2 Intermediate
If the index model is valid, _________ would be helpful in determining the covariance between assets
GM
and
GE
.
GM
GE
M
All of the options
None of the options are correct.
If the index model is valid
GM
,
GE
, and
M
are determinants of the covariance between
GE
and
GM
.
References
Multiple Choice
Difficulty: 2 Intermediate
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17.
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18.
Award:
10.00 points
19.
Award:
10.00 points
20.
Award:
1.00 point
If the index model is valid, _________ would be helpful in determining the covariance between assets HPQ and KMP.
HPQ
KMP
M
All of the options
None of the options are correct.
If the index model is valid
HPQ
,
KMP
, and
M
are determinants of the covariance between
HPQ
and
KMP
.
References
Multiple Choice
Difficulty: 2 Intermediate
If the index model is valid, _________ would be helpful in determining the covariance between assets
K
and
L
.
β
k
β
L
σ
M
All of the options
None of the options are correct.
If the index model is valid,
β
k
,
β
L
, and
σ
M
are determinants of the covariance between K and L.
References
Multiple Choice
Difficulty: 2 Intermediate
Rosenberg and Guy found that _________ helped to predict firms' betas.
debt/asset ratios
market capitalization
variance of earnings
All of the options
None of the options are correct.
Rosenberg and Guy found that debt/asset ratios, market capitalization, and variance of earnings were determinants of firms' betas.
References
Multiple Choice
Difficulty: 2 Intermediate
If a firm's beta was calculated as 0.6 in a regression equation, a commonly-used adjustment technique would provide an adjusted beta of:
less than 0.6 but greater than zero.
between 0.6 and 1.0.
between 1.0 and 1.6.
greater than 1.6.
zero or less.
Betas, on average, equal one; thus, betas over time regress toward the mean, or 1. Therefore, if historic betas are less than 1, adjusted betas are between 1 and the calculated beta.
References
Multiple Choice
Difficulty: 2 Intermediate
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21.
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22.
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23.
Award:
1.00 point
24.
Award:
10.00 points
If a firm's beta was calculated as 0.8 in a regression equation, a commonly-used adjustment technique would provide an adjusted beta of:
less than 0.8 but greater than zero.
between 1.0 and 1.8.
between 0.8 and 1.0.
greater than 1.8.
zero or less.
Betas, on average, equal one; thus, betas over time regress toward the mean, or 1. If historic betas are less than 1, adjusted betas are between 1 and the calculated beta.
References
Multiple Choice
Difficulty: 2 Intermediate
If a firm's beta was calculated as 1.3 in a regression equation, a commonly-used adjustment technique would provide an adjusted beta of:
less than 1.0 but greater than zero.
between 0.3 and 0.9.
between 1.0 and 1.3.
greater than 1.3.
zero or less.
Betas, on average, equal one; thus, betas over time regress toward the mean, or 1. If historic betas are greater than 1, adjusted betas are between 1 and the calculated beta.
References
Multiple Choice
Difficulty: 2 Intermediate
The beta of Facebook stock has been estimated as 1.8 using regression analysis on a sample of historical returns. A commonly-used adjustment technique would provide an adjusted beta of:
1.20.
1.32.
1.13.
1.53.
None of the options are correct.
Adjusted beta = 2 ÷ 3 sample beta + ((1 ÷ 3) × 1) = ((2 ÷ 3) × 1.8) + (1 ÷ 3) = 1.53.
References
Multiple Choice
Difficulty: 2 Intermediate
The beta of Amazon stock has been estimated as 2.6 using regression analysis on a sample of historical returns. A commonly-used adjustment technique would provide an adjusted beta of:
2.20.
2.07.
2.13.
1.66.
None of the options are correct.
Adjusted beta = 2 ÷ 3 sample beta + ((1 ÷ 3) × 1) = ((2 ÷ 3) × 2.6) + (1 ÷ 3) = 2.07.
References
Multiple Choice
Difficulty: 2 Intermediate
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27.
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28.
Award:
10.00 points
The beta of Boeing stock has been estimated as 0.72 using regression analysis on a sample of historical returns. A commonly-used adjustment technique would provide an adjusted beta of:
1.20.
1.14.
0.81.
0.68.
None of the options are correct.
Adjusted beta = 2 ÷ 3 sample beta + ((1 ÷ 3) × 1) = ((2 ÷ 3) × 0.72) + (1 ÷ 3) = 0.81.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 150 stocks in order to construct a mean-variance efficient portfolio constrained by 150 investments. They will need to
calculate _________ expected returns and _________ variances of returns.
150; 150
150; 22500
22500; 150
22500; 22500
None of the options are correct.
The expected returns of each of the 150 securities must be calculated. In addition, the 150 variances around these returns must be calculated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 100 stocks in order to construct a mean-variance efficient portfolio constrained by 100 investments. They will need to
calculate _________ expected returns and _________ variances of returns.
100; 100
100; 4950
4950; 100
4950; 4950
None of the options are correct.
The expected returns of each of the 100 securities must be calculated. In addition, the 100 variances around these returns must be calculated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 150 stocks to construct a mean-variance efficient portfolio constrained by 150 investments. They will need to calculate
_________ covariances.
12
150
22,500
11,175
None of the options are correct.
Covariances Calculated
(
n
2
−
n
) ÷ 2 = (22,500 − 150) ÷ 2 = 11,175
References
Multiple Choice
Difficulty: 2 Intermediate
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30.
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31.
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32.
Award:
10.00 points
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 125 stocks to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate
_________ covariances.
125
7,750
15,625
11,750
None of the options are correct.
Covariances Calculated =
(n
2
−
n) ÷ 2
= (15,625 − 125) ÷ 2 = 7,750
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 100 stocks to construct a mean-variance efficient portfolio constrained by 100 investments. They will need to calculate
_________ covariances.
45
100
4,950
10,000
None of the options are correct.
Covariances Calculated
(
n
2
−
n
) ÷ 2 = (10,000 − 100) ÷ 2 = 4,950
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do follow a single-index structure. An investment fund analyzes 175 stocks to construct a mean-variance efficient portfolio constrained by 175 investments. They will need to calculate _________
estimates of expected returns and _________ estimates of sensitivity coefficients to the macroeconomic factor.
175; 15,225
175; 175
15,225; 175
15,225; 15,225
None of the options are correct.
For a single-index model,
n
= 175, expected returns and
n
= 175 sensitivity coefficients to the macroeconomic factor must be estimated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do follow a single-index structure. An investment fund analyzes 125 stocks to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate _________
estimates of expected returns and _________ estimates of sensitivity coefficients to the macroeconomic factor.
125; 15,225
15,625; 125
7,750; 125
125; 125
None of the options are correct.
For a single-index model,
n
= 125, expected returns and
n
= 125 sensitivity coefficients to the macroeconomic factor must be estimated.
References
Multiple Choice
Difficulty: 2 Intermediate
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34.
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35.
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36.
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Assume that stock market returns do follow a single-index structure. An investment fund analyzes 200 stocks to construct a mean-variance efficient portfolio constrained by 200 investments. They will need to calculate ________
estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor.
200; 19,900
200; 200
19,900; 200
19,900; 19.900
None of the above are correct.
For a single-index model,
n
= 200, expected returns and
n
= 200 sensitivity coefficients to the macroeconomic factor must be estimated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do follow a single-index structure. An investment fund analyzes 500 stocks to construct a mean-variance efficient portfolio constrained by 500 investments. They will need to calculate _________
estimates of firm-specific variances and _________ estimate/estimates for the variance of the macroeconomic factor.
500; 1
500; 500
124,750; 1
124,750; 500
250,000; 500
For the single-index model,
n
= 500 estimates of firm-specific variances must be calculated and 1 estimate for the variance of the common macroeconomic factor.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the single-index model. The alpha of a stock is 2%. The return on the market index is 16%. The risk-free rate of return is 5%. The stock earns a return that exceeds the risk-free rate by 11%, and there are no firm-specific
events affecting the stock performance. The
β
of the stock is:
0.67.
0.75.
1.00.
1.33.
1.50.
11% = 2% +
β
× 11%
→
β
= 1.0
References
Multiple Choice
Difficulty: 2 Intermediate
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.30 and
M
was 0.16, the
β
of the portfolio would be approximately:
0.64.
1.80.
1.88.
1.56.
None of the options are correct.
References
Multiple Choice
Difficulty: 3 Challenge
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37.
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38.
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39.
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40.
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Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.22 and
M
was 0.17, the
β
of the portfolio would be approximately:
1.34.
1.29.
1.25.
1.56.
None of the options are correct.
References
Multiple Choice
Difficulty: 3 Challenge
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.18 and
M
was 0.22, the
β
of the portfolio would be approximately:
0.82.
0.56.
0.07.
1.03.
None of the options are correct.
References
Multiple Choice
Difficulty: 3 Challenge
Suppose the following equation best describes the evolution of
β
over time:
t
= 0.36 + 0.85 ×
t
−
1
.
If a stock had a
β
of 0.6 last year, you would forecast the
β
to be _________ in the coming year.
0.45
0.60
0.87
0.75
None of the options are correct.
β
t
= 0.36 + 0.85 ×
β
t
−
1
= 0.36 + 0.85
× 0.6 = 0.87.
References
Multiple Choice
Difficulty: 1 Basic
Suppose the following equation best describes the evolution of
β
over time:
t
= 0.49 + 0.77 ×
t
−
1
.
If a stock had a
β
of 0.9 last year, you would forecast the
β
to be _________ in the coming year.
0.88
0.82
0.31
1.18
None of the options are correct.
β
t
= 0.49 + 0.77 ×
β
t
−
1
= 0.49 + 0.77 × 0.9 = 1.18
References
Multiple Choice
Difficulty: 1 Basic
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41.
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42.
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43.
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44.
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Suppose the following equation best describes the evolution of
β
over time:
β
t
= 0.15 + 0.72 ×
β
t
−
1
If a stock had a
β
of 1.05 last year, you would forecast the
β
to be _________ in the coming year.
0.91
0.18
0.63
0.81
None of the options are correct.
β
t
= 0.15 + 0.72 ×
β
t
−
1
= 0.15 + 0.72 × 1.05 = 0.91.
References
Multiple Choice
Difficulty: 1 Basic
An analyst estimates the index model for a stock using regression analysis involving total returns. The estimated intercept in the regression equation is 6% and the
β
is 0.5. The risk-free rate of return is 12%. The true
β
of the stock
is:
0%.
3%.
6%.
9%.
None of the options are correct.
6% =
a
+ 12% × (1
−
0.5)
→
a
= 0%
References
Multiple Choice
Difficulty: 3 Challenge
The index model for stock A has been estimated with the following result:
R
A
= 0.01 + 0.9
R
M
+ e
A
.
If
M
=
0.25
and
R
2
A
=
0.25
,
the standard deviation of return of stock A is:
0.2025.
0.2500.
0.4500.
0.8100.
None of the options are correct.
R
2
A
=
2
2
M
÷
2
A
0.25 = (0.9
2
× 0.25
2
) ÷
2
A
A
= 0.45
References
Multiple Choice
Difficulty: 3 Challenge
The index model for stock B has been estimated with the following result:
R
B
= 0.01 + 1.1
R
M
+
e
B
.
If
M
= 0.20
and
R
2
B
= 0.50
, the standard deviation of the return on stock B is:
0.1111.
0.2111.
0.3111.
0.4111.
None of the options are correct.
R
2
B
=
2
2
M
÷
2
B
0.5 = (1.1
2
× 0.2
2
) ÷
2
B
B
= 0.31
References
Multiple Choice
Difficulty: 3 Challenge
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45.
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46.
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Suppose you forecast that the market index will earn a return of 12% in the coming year. Treasury bills are yielding 4%. The unadjusted
β
of Mobil stock is 1.10. A reasonable forecast of the return on Mobil stock for the coming year is
_________ if you use a common method to derive adjusted betas.
15.0%
14.5%
13.0%
12.5%
None of the options are correct.
Adjusted
β
= ((2 ÷ 3) × 1.1) + (1 ÷ 3) × 1.00 = 1.067
r = r
f
+
β
× (r
M
−
r
f
)
= 0.04 + 1.067 × 0.08 = 0.125
References
Multiple Choice
Difficulty: 3 Challenge
The index model has been estimated for stocks A and B with the following results:
R
A
= 0.01 + 0.8
R
M
+
e
A
.
R
B
= 0.02 + 1.2R
M
+
e
B
.
M
= 0.30; (
e
A
) = 0.20; (
e
B
) = 0.10.
The covariance between the returns on stocks A and B is:
0.0384.
0.0864.
0.1920.
0.0050.
0.4000.
Cov
(
R
A
,
R
B
) =
β
A
β
B
σ
2
M
= 0.8 × 1.2 × 0.3
2
= 0.0864
References
Multiple Choice
Difficulty: 3 Challenge
The index model has been estimated for stocks A and B with the following results:
R
A
= 0.01 + 0.6
R
M
+
e
A
.
R
B
= 0.02 + 1.2
R
M
+
e
B
.
M
= 0.30; (
e
A
) = 0.20; (
e
B
) = 0.10.
The standard deviation for stock A is:
0.0656.
0.0676.
0.3499.
0.2600.
None of the options are correct.
A
= [0.6
2
× 0.3
2
+ 0.3
2
]
1/2
= 0.3499.
References
Multiple Choice
Difficulty: 3 Challenge
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49.
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50.
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The index model has been estimated for stock A with the following results:
R
A
= 0.01 + 1.2
R
M
+
e
A
.
M
= 0.15; (
e
A
) = 0.10.
The standard deviation of the return for stock A is:
0.0356.
0.1862.
0.1600.
0.6400.
None of the options are correct.
B
= [1.2
2
× 0.15
2
+ (× 0.1)
2
]
1/2
= 0.2059
References
Multiple Choice
Difficulty: 3 Challenge
Security returns:
are based on both macro events and firm-specific events.
are based on firm-specific events only.
are usually positively correlated with each other.
are based on firm-specific events only and are usually positively correlated with each other.
are based on both macro events and firm-specific events and are usually positively correlated with each other.
Stock returns are usually highly positively correlated with each other. Stock returns are affected by both macroeconomic events and firm-specific events.
References
Multiple Choice
Difficulty: 1 Basic
The single-index model:
greatly reduces the number of required calculations relative to those required by the Markowitz model.
enhances the understanding of systematic versus nonsystematic risk.
greatly increases the number of required calculations relative to those required by the Markowitz model.
greatly reduces the number of required calculations relative to those required by the Markowitz model and enhances the understanding of systematic versus nonsystematic risk are correct.
enhances the understanding of systematic versus nonsystematic risk and greatly increases the number of required calculations relative to those required by the Markowitz model are correct.
The single index model both greatly reduces the number of calculations and enhances the understanding of the relationship between systematic and unsystematic risk on security returns.
References
Multiple Choice
Difficulty: 1 Basic
The security characteristic line (SCL):
plots the excess return on a security as a function of the excess return on the market.
allows one to estimate the beta of the security.
allows one to estimate the alpha of the security.
All of the options.
None of the options are correct.
The security characteristic line, which plots the excess return of the security as a function of the excess return of the market, allows one to estimate both the alpha and the beta of the security.
References
Multiple Choice
Difficulty: 1 Basic
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53.
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54.
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55.
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The
expected
impact of unanticipated macroeconomic events on a security's return during the period is:
included in the security's expected return.
zero.
equal to the risk-free rate.
proportional to the firm's beta.
infinite.
The expected value of unanticipated macroeconomic events is zero, because by definition it must average to zero or it would be incorporated into the expected return.
References
Multiple Choice
Difficulty: 2 Intermediate
Covariances between security returns tend to be:
positive because of SEC regulations.
positive because of Exchange regulations.
positive because of economic forces that affect many firms.
negative because of SEC regulations.
negative because of economic forces that affect many firms.
Economic forces, such as business cycles, interest rates, and technological changes, tend to have similar impacts on many firms.
References
Multiple Choice
Difficulty: 2 Intermediate
In the single-index model represented by the equation
r
i
=
E
(
r
i
) +
i
F
+
e
i
, the term
e
i
represents:
the impact of unanticipated macroeconomic events on security i's return.
the impact of unanticipated firm-specific events on security i's return.
the impact of anticipated macroeconomic events on security i's return.
the impact of anticipated firm-specific events on security i's return.
the impact of changes in the market on security i's return.
The textbook discusses a model in which macroeconomic events are used as a single index for security returns. The
e
i
term represents the impact of unanticipated firm-specific events. The
e
i
term has an expected value of zero.
Only unanticipated events would affect the return.
References
Multiple Choice
Difficulty: 2 Intermediate
Suppose you are doing a portfolio analysis that includes all of the stocks on the NYSE. Using a single-index model rather than the Markowitz model:
increases the number of inputs needed from about 1,400 to more than 1.4 million.
increases the number of inputs needed from about 10,000 to more than 125,000.
reduces the number of inputs needed from more than 125,000 to about 10,000.
reduces the number of inputs needed from more than 5 million to about 10,000.
increases the number of inputs needed from about 150 to more than 1,500.
This example is discussed in the textbook. The main point for the students to remember is that the single-index model drastically reduces the number of inputs required.
References
Multiple Choice
Difficulty: 2 Intermediate
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One "cost" of the single-index model is that it:
is virtually impossible to apply.
prohibits specialization of efforts within the security analysis industry.
requires forecasts of the money supply.
is legally prohibited by the SEC.
allows for only two kinds of risk—macro risk and micro risk.
One "cost" of the single-index model is that it allows for only two kinds of risk—macro risk and micro risk.
References
Multiple Choice
Difficulty: 2 Intermediate
The security characteristic line (SCL) associated with the single-index model is a plot of:
the security's returns on the vertical axis and the market index's returns on the horizontal axis.
the market index's returns on the vertical axis and the security's returns on the horizontal axis.
the security's excess returns on the vertical axis and the market index's excess returns on the horizontal axis.
the market index's excess returns on the vertical axis and the security's excess returns on the horizontal axis.
the security's returns on the vertical axis and Beta on the horizontal axis.
The student needs to remember that it is the excess returns that are plotted and that the security's returns are plotted as a dependent variable.
References
Multiple Choice
Difficulty: 2 Intermediate
The idea that there is a limit to the reduction of portfolio risk due to diversification is:
contradicted by both the CAPM and the single-index model.
contradicted by the CAPM.
contradicted by the single-index model.
supported in theory, but not supported empirically.
supported both in theory and by empirical evidence.
The benefits of diversification are limited to the level of systematic risk.
References
Multiple Choice
Difficulty: 2 Intermediate
In their study about predicting beta coefficients, which of the following did Rosenberg and Guy find to be factors that influence beta?
I. Industry group
II. Variance of cash flow
III. Dividend yield
IV. Growth in earnings per share
I and II
I and III
I, II, and III
I, II, and IV
I, II, III, and IV
All of the factors mentioned, as well as variance of earnings, firm size, and debt-to-asset ratio, were found to help predict betas.
References
Multiple Choice
Difficulty: 2 Intermediate
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61.
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62.
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63.
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If a firm's beta was calculated as 1.6 in a regression equation, a commonly-used adjustment technique would provide an adjusted beta of:
less than 0.6 but greater than zero.
between 0.6 and 1.0.
between 1.0 and 1.6.
greater than 1.6.
zero or less.
Betas, on average, equal one; thus, betas over time regress toward the mean, or 1. Therefore, if historic betas are more than 1, adjusted betas are between 1 and the calculated beta.
References
Multiple Choice
Difficulty: 2 Intermediate
The beta of a stock has been estimated as 1.8 using regression analysis on a sample of historical returns. A commonly-used adjustment technique would provide an adjusted beta of
1.80.
1.53.
0.50.
1.00.
None of the options are correct.
Adjusted
β
= ((2 ÷ 3) × 1.8) + ((1 ÷ 3) × 1.00) = 1.53
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 40 stocks in order to construct a mean-variance efficient portfolio constrained by 40 investments. They will need to calculate
_________ expected returns and _________ variances of returns.
100; 100
40; 40
4950; 100
4950; 4950
None of the options are correct.
The expected returns of each of the 40 securities must be calculated. In addition, the 40 variances around these returns must be calculated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 40 stocks in order to construct a mean-variance efficient portfolio constrained by 40 investments. They will need to calculate
_________ covariances.
45
780
4,950
10,000
None of the options are correct.
Covariances Calculated =
(
n
2
−
n
) ÷ 2 = (1,600 − 40) ÷ 2 = 780
References
Multiple Choice
Difficulty: 2 Intermediate
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65.
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66.
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67.
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Assume that stock market returns do follow a single-index structure. An investment fund analyzes 60 stocks in order to construct a mean-variance efficient portfolio constrained by 60 investments. They will need to calculate
_________ estimates of expected returns and _________ estimates of sensitivity coefficients to the macroeconomic factor.
200; 19,900
200; 200
60; 60
19,900; 19.900
None of the options are correct.
For a single-index model,
n
= 60, expected returns and
n
= 60 sensitivity coefficients to the macroeconomic factor must be estimated.
References
Multiple Choice
Difficulty: 2 Intermediate
Consider the single-index model. The alpha of a stock is 0%. The return on the market index is 10%. The risk-free rate of return is 3%. The stock earns a return that exceeds the risk-free rate by 11%, and there are no firm-specific
events affecting the stock performance. The
β
of the stock is:
1.57.
0.75.
1.17.
1.33.
1.50.
11% = 0% +
β
× 7%
→
β
= 1.57
References
Multiple Choice
Difficulty: 2 Intermediate
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.25 and
σ
M
was 0.21, the
β
of the portfolio would be approximately
_________.
0.64
1.19
1.25
1.56
None of the options are correct.
β
2
P
=
σ
2
P
÷
σ
2
M
=
(0.25
2
÷ 0.21
2
) = 1.417
β
p
= 1.19
References
Multiple Choice
Difficulty: 3 Challenge
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.18 and
σ
M
was 0.22, the
β
of the portfolio would be approximately:
0.64.
1.19.
0.82.
1.56.
None of the options are correct.
β
2
P
=
σ
2
P
÷
σ
2
M
=
(0.18
2
÷ 0.22
2
) = 0.669
β
P
= 0.82
References
Multiple Choice
Difficulty: 3 Challenge
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69.
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70.
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71.
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Suppose the following equation best describes the evolution of
β
over time:
β
t
= 0.40 + 0.60 ×
β
t
−
1
If a stock had a
β
of 0.9 last year, you would forecast the
β
to be _________ in the coming year.
0.45
0.60
0.70
0.94
None of the options are correct.
β
t
= 0.40 + 0.60 ×
β
t
−
1
= 0.40 + 0.60 × 0.90 = 0.94
References
Multiple Choice
Difficulty: 1 Basic
Suppose the following equation best describes the evolution of
β
over time:
β
t
= 0.30 + 0.20 ×
β
t
−
1
If a stock had a
β
of 0.8 last year, you would forecast the
β
to be _________ in the coming year.
0.46
0.60
0.70
0.94
None of the options are correct.
β
t
= 0.30 + 0.20 ×
β
t
−
1
= 0.30 + 0.20 × 0.80 = 0.46.
References
Multiple Choice
Difficulty: 1 Basic
The index model for stock A has been estimated with the following result:
R
A
= 0.01 + 0.94
R
M
+
e
A
If
σ
M
= 0.30
and
R
2
A
= 0.28
, the standard deviation of return of stock A is:
0.2025.
0.2500.
0.4500.
0.5329.
None of the options are correct.
R
2
A
= (
β
2
×
σ
2
m
)
÷
σ
2
A
0.28 = (0.94
2
× 0.3
2
) ÷
σ
2
A
σ
A
= 0.5329
References
Multiple Choice
Difficulty: 3 Challenge
Suppose you forecast that the market index will earn a return of 12% in the coming year. Treasury bills are yielding 4%. The unadjusted
β
of Mobil stock is 1.50. A reasonable forecast of the return on Mobil stock for the coming year
is _________ if you use a common method to derive adjusted betas.
Note: Do not round your intermediate calculations.
15.0%
15.5%
16.0%
14.7%
None of the options are correct.
Adjusted
β
= (2 ÷ 3) × 1.5 + (1 ÷ 3) × 1.00 = 1.33
r = r
f
+
β
× (r
m
−
r
f
)
= 0.04 + 1.33 × 0.08 = 0.147
References
Multiple Choice
Difficulty: 3 Challenge
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73.
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74.
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75.
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The index model has been estimated for stocks A and B with the following results:
R
A
= 0.01 + 0.8
R
M
+
e
A
.
R
B
= 0.02 + 1.1R
M
+
e
B
.
M
= 0.30; (
e
A
) = 0.20; (
e
B
) = 0.10.
The covariance between the returns on stocks A and B is:
0.0384.
0.0406.
0.1920.
0.0050.
0.0792.
Cov
(
R
A
,
R
B
) =
β
A
β
B
σ
2
M
= 0.8 × 1.1 × 0.30
2
= 0.0792
References
Multiple Choice
Difficulty: 3 Challenge
If a firm's beta was calculated as 1.35 in a regression equation, a commonly-used adjustment technique would provide an adjusted beta of:
equal to 1.35.
between 0.0 and 1.0.
between 1.0 and 1.35.
greater than 1.35.
zero or less.
Betas, on average, equal one; thus, betas over time regress toward the mean, or 1. Therefore, if historic betas are more than 1, adjusted betas are between 1 and the calculated beta.
References
Multiple Choice
Difficulty: 2 Intermediate
The beta of a stock has been estimated as 1.4 using regression analysis on a sample of historical returns. A commonly-used adjustment technique would provide an adjusted beta of:
1.27.
1.32.
1.13.
1.00.
None of the options are correct.
Adjusted
β
= ((2 ÷ 3) × 1.4) + ((1 ÷ 3) × 1.00) = 1.27
References
Multiple Choice
Difficulty: 2 Intermediate
The beta of a stock has been estimated as 0.85 using regression analysis on a sample of historical returns. A commonly-used adjustment technique would provide an adjusted beta of:
1.01.
0.95.
1.13.
0.90.
None of the options are correct.
Adjusted
β
= ((2 ÷ 3) × 0.85) + ((1 ÷ 3) × 1.00) = 0.90
References
Multiple Choice
Difficulty: 2 Intermediate
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76.
Award:
10.00 points
77.
Award:
10.00 points
78.
Award:
10.00 points
79.
Award:
10.00 points
Assume that stock market returns follow a single-index structure. An investment fund analyzes 125 stocks to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate _________
expected returns and _________ variances of returns.
125; 125
125; 15,625
15,625; 125
15,625; 15,625
None of the options are correct.
The expected returns of each of the 125 securities must be calculated. In addition, the 125 variances around these returns must be calculated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 132 stocks to construct a mean-variance efficient portfolio constrained by 132 investments. They will need to calculate
_________ covariances.
100
132
4,950
8,646
None of the options are correct.
Covariances Calculated =
(n
2
−
n)
÷ 2 = (17,424 − 132) ÷ 2 = 8,646
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do follow a single-index structure. An investment fund analyzes 217 stocks to construct a mean-variance efficient portfolio constrained by 217 investments. They will need to calculate _________
estimates of expected returns and _________ estimates of sensitivity coefficients to the macroeconomic factor.
217; 47,089
217; 217
47,089; 217
47,089; 47,089
None of the options are correct.
For a single-index model,
n
= 217, expected returns and
n
= 217 sensitivity coefficients to the macroeconomic factor must be estimated.
References
Multiple Choice
Difficulty: 2 Intermediate
Assume that stock market returns do follow a single-index structure. An investment fund analyzes 750 stocks to construct a mean-variance efficient portfolio constrained by 750 investments. They will need to calculate _________
estimates of firm-specific variances and _________ estimate/estimate(s) for the variance of the macroeconomic factor.
750; 1
750; 750
124,750; 1
124,750; 750
562,500; 750
For the single-index model,
n
= 750 estimates of firm-specific variances must be calculated and 1 estimate for the variance of the common macroeconomic factor.
References
Multiple Choice
Difficulty: 2 Intermediate
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80.
Award:
10.00 points
81.
Award:
10.00 points
82.
Award:
10.00 points
83.
Award:
10.00 points
Consider the single-index model. The alpha of a stock is 0%. The return on the market index is 10%. The risk-free rate of return is 5%. The stock earns a return that exceeds the risk-free rate by 5%, and there are no firm-specific
events affecting the stock performance. The
β
of the stock is:
0.67.
0.75.
1.00.
1.33.
1.50.
5% = 0% +
β
×
5%
→
β
= 1.0.
References
Multiple Choice
Difficulty: 2 Intermediate
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.24 and
M
was 0.18, the
β
of the portfolio would be approximately:
0.64.
1.33.
1.25.
1.56.
None of the options are correct.
β
2
P
=
σ
2
P
÷
σ
2
M
= 0.24
2
÷ 0.18
2
= 1.78
β
= 1.33
References
Multiple Choice
Difficulty: 3 Challenge
Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the
σ
of your portfolio was 0.14 and
M
was 0.19, the
β
of the portfolio would be approximately:
0.74.
0.80.
1.25.
1.56.
None of the options are correct.
β
2
p
=
σ
2
p
÷
σ
2
M
= 0.14
2
÷ 0.19
2
= 0.54
β
= 0.74
References
Multiple Choice
Difficulty: 3 Challenge
Suppose the following equation best describes the evolution of
β
over time:
β
t
= 0.30 + 0.70 ×
β
t
−
1
If a stock had a
β
of 0.82 last year, you would forecast the
β
to be _________ in the coming year.
0.91
0.77
0.63
0.87
None of the options are correct.
β
t
= 0.30 + 0.70 ×
β
t
−
1
= 0.30 + 0.70 × 0.82 = 0.874
References
Multiple Choice
Difficulty: 1 Basic
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84.
Award:
10.00 points
85.
Award:
10.00 points
86.
Award:
10.00 points
87.
Award:
10.00 points
KMW Incorporated has an estimated beta of 1.45. Given a forecasted market return of 12% and a T-bill rate 3%, using the index model and the adjusted beta, what is the forecasted return?
Note: Do not round your intermediate calculations.
14.7%
15.2%
16.2%
17.8%
None of the options are correct.
Adjusted
β
= (2 ÷ 3) × 1.45 + (1 ÷ 3) × 1.00 = 1.30
r = r
f
+
β
× (r
M
−
r
f
)
= 0.03 + 1.30 × 0.09 = 0.147
References
Multiple Choice
Difficulty: 2 Intermediate
HAW Incorporated has an estimated beta of 0.87. Given a forecasted market return of 9% and a T-bill rate 3%, using the index model and the adjusted beta, what is the forecasted return?
Note: Do not round your intermediate calculations.
11.7%
10.2%
8.5%
7.8%
None of the options are correct.
Adjusted
β
= (2 ÷ 3) × 0.87 + (1 ÷ 3) × 1.00 = 0.91
r = r
f
+
β
× (r
M
−
r
f
)
= 0.03 + 0.91 × 0.06 = 0.085
References
Multiple Choice
Difficulty: 2 Intermediate
VM Incorporated has an estimated beta of 1.08. Given a forecasted market return of 10% and a T-bill rate 2%, using the index model and the adjusted beta, what is the forecasted return?
Note: Do not round your intermediate calculations.
12.75%
10.40%
9.54%
7.88%
None of the options are correct.
Adjusted
β
= (2 ÷ 3) × 1.08 + (1 ÷ 3) × 1.00 = 1.05
r = r
f
+
β
×
(r
m
−
r
f
)
= 0.02 + 1.05 × 0.08 = 0.1040
References
Multiple Choice
Difficulty: 2 Intermediate
Using the single index model, what is the alpha of a stock with beta of 1.2, a market return of 14%, risk free rate of 3% and the actual return of the stock is 18%?
−
1.37%
0.75%
1.80%
2.11%
−
3.12%
α
= r
−
[r
f
+
β
×
(r
m
−
r
f
)]
= 0.18
−
[ 0.03 + 1.2 × (0.14
−
0.03)] = 0.0180
References
Multiple Choice
Difficulty: 2 Intermediate
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88.
Award:
10.00 points
Using the single index model, what is the alpha of a stock with beta of 1.3, a market return of 14%, risk free rate of 4% and the actual return of the stock is 14%?
−
1.37%
0.75%
1.80%
2.11%
−
3.00%
α
= r
−
[r
f
+
β
×
(r
M
−
r
f
)]
= 0.14
−
[ 0.04 + 1.3 × (0.14
−
0.04)] =
−
0.03
References
Multiple Choice
Difficulty: 2 Intermediate
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Related Questions
QUESTIONS:
1) Assuming that the risk-free rate of return is currently 3,2%, the market risk premium is 6%
whereas the beta of HelloFresh SH. stock is 1.8, compute the required rate of return using
CAPM.
2) Compute the value of each investment based on your required rate of return and interpret
the results comparing with the market values.
3) Which investment would you select? Explain why using appropriate financial jargon
(language).
4) Assume HelloFresh SH's CFO Mr. Christian Gaertner expects an earnings upturn resulting
increase in growth (rate) of 1%. How does this affect your answers to Question 2 and 3?
5) AACSB Critical Thinking Questions:
A) Companies pay rating agencies such as Moody's and S&P to rate their bonds, and
the costs can be substantial. However, companies are not required to have their
bonds rated in the first place; doing so is strictly voluntary. Why do you think they do
it? (Textbook page: 198)
B) What are the difficulties in using the PE ratio to value stock?…
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Questions:
a. Compute the expected return for stock X and for stock Y
b. Compute the standard deviation for stock X and for stock Y.
c. Determine the best course to take for investing.
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b. As an equity portfolio manager, you may use certain risk-adjusted performance
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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Suppose you are given the following inputs for the Fama-Frech-3-Factor model. Required Return for Stock i: bi=0.8, kRF=8%, the market risk premium is 6%, ci=-0.6, the expected value for the size factor is 5%, di=-0.4, and the expected value for the book-to-market factor is 4%.
Task: Estimate the required rate of return of this asset using the Capital asset pricing model and compare it with the Fama-French-3-factor model.
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Using the stock price data for any two companies provided below carry out the following tasks:
1.Compute, for each asset:
i.Total Returns
ii.Expected returns
iii.standard deviation
iv.Correlation Coefficient
2.Construct the variance-covariance matrix
3.Construct equally weighted portfolio and calculate Expected Return, Standard Deviation and Sharpe ratio.
4.Reconstruct equally weighted portfolio and calculate Expected Return, Standard Deviation and Sharpe ratio.
5.Use Solver to determine optimal risky portfolio.
6.Create hypothetical portfolios (commencing from Weight A=0 and weight B=100)
7.Calculate Expected return and Standard Deviation for all the above combinations
8.Graph the efficient frontier
9.Graph the optimal portfolio
10.Assuming that the investors prefers lower level of risk than what a portfolio of risky assets offer, introduce a risk free asset in the portfolio with a return of 3%
11.Using hypothetical weights (A= Portfolio of Risky Assets, B= 1 Risk Free…
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Please help me answer this question in Excel Form
2. The risk-free rate is 4%, and the required return on the market is 12%.
b. A portfolio invests 40% in the asset in (a) and the rest in a market portfolio. What is the required return on this portfolio?
Thanks :)
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A portfolio management organization analyzes 75 stocks and constructs a mean-variance efficient portfolio that is constrained to these 75 stocks. How many estimates of expected returns, variances and covariances are needed to optimize this portfolio? (Using Markowitz Model)
75, 150, 2625
75, 75, 2700
75, 75, 2775
75, 75, 2925
75, 150, 2775.
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Using the spreadsheet answer question (c) please
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6) Assume that stock market returns do follow a single-index structure. An investment fund analyzes 1000 stocks in order to construct a mean-variance efficient portfolio constrained by 500 investments. They will need to calculate ________ estimates of firm-specific variances and ________ estimate/estimates for the variance of the macroeconomic factor.
A) 500; 1 B) 1000; 1 C) 124,750; 1 D) 124,750; 500 E) 250,000; 500
Choose the correct option th justification.
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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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A portfolio management organisation analyses 200 stocks and constructs a mean- variance efficient portfolio using only 150 out of those 200 securities. (i) how many estimates of expected returns, variances and covariances are needed to optimise this portfolio? (ii) If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?
In light of your answer to the above questions, discuss the advantages and disadvantages of the index model compared to the Markowitz procedure for obtaining an efficient diversified portfolio.
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5) Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 150 stocks in order to construct a mean-variance efficient portfolio constrained by 300 investments. They will need to calculate ____________ covariances.
A) 44,850 B) 150 C) 22,500 D) 11,175
Choose the correct answer with justification.
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Please show all work and formulas in excel please! The Table for the problem is attached.
Table below shows the historical returns for Companies A, B and C
If one investor has a portfolio consisting of 50% Company A and 50% Company B, what are the average portfolio return and standard deviation? What is Sharpe ratio if the risk-free rate is 3.8%?
If another investor has a portfolio consisting of 1/3 Company A, 1/3 Company B and 1/3 Company C, what are the average portfolio return and standard deviation? What is Sharpe ratio if the risk-free rate is 3.8%
What would happen to the portfolio risk if more and more randomly selected stocks were added?
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Jason Jackson is attempting to evaluate two possible portfolios consisting of the same five assets but held in different proportions. He is particularly interested in using beta to compare the risk of the portfolios and, in this regard, has gathered the following data:
LOADING...
.
a. Calculate the betas for portfolios A and B.
b. Compare the risk of each portfolio to the market as well as to each other. Which portfolio is more risky?
Question content area bottom
Part 1
Data table
(Click on the icon here
in order to copy its contents of the data table below into a spreadsheet.)
Portfolio Weights
Asset
Asset Beta
Portfolio A
Portfolio B
1
1.35
17%
29%
2
0.69
26%
8%
3
1.24
10%
22%
4
1.06
11%
20%
5
0.87
36%
21%
Total
100%
100%
a. The beta of portfolio A is
enter your response here.
(Round to three…
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d. Interpret your results in (c) above, assuming that the historical average return
of 8.5% from the stock is a good benchmark.
e. Critically evaluate the strengths and weaknesses of applying the implied rate of
return from the RIVM as a proxy of the expected return.
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Can you please answer and kindly show detailed human working.
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Compute the residual risk measure for portfolio A. Round off your final answer to three digits after the decimal point.
Compute the appraisal ratio for portfolio B. Round off your final answer to three digits after the decimal point.
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Can you answer please and kindly show detailed human working.
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Can you accurately answer these, please? show detailed human working out.
It's for financial managment
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I need to calculate the cost of equity with the following data:
The current appropriate risk-free rate is 6% and the return on the market is 13.5%. levered beta is 1.29. Using the CAPM, estimate DE’s cost of equity. Be sure to state any additional assumptions
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You are evaluating various investment opportunities currently available and you have calculated expected returns and standard deviation for five different well-diversified portfolios of risky assets.
Portfolio
Expected return (%)
Standard deviation (%)
Q
7.8
10.5
R
10
14
S
4.6
5
11.7
18.5
6.2
7.5
a. For each portfolio, determine the Sharpe ratio. Assume that the risk free rate is 3%.
b. Using your analysis in part (a), explain which of these five portfolios is most likely to be the market portfolio. Draw up the Capital Market Line Equation.
c. If you are willing to make an investment with a standard deviation of 7%, is it possible for you to earn 7%?
d. What is the minimum risk level that would be necessary for an investment to earn 7%. What is the composition of the portfolio along the CML that will generate the 7% return.
e. Suppose you are now willing t
f. Devise how would you apply portfolio performance evaluation and attribution analysis in practice. Illustrate with suitable…
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(Expected rate of return and risk) Syntex, Inc. is considering an investment in one of two common stocks. Given the information that follows, which investment is better, based on the risk (as
measured by the standard deviation) and return?
Common Stock A
Probability
0.20
0.60
0.20
Probability
0.15
0.35
0.35
0.15
(Click on the icon in order to copy its contents into a spreadsheet.)
ew an example Get more help.
T
3
a. Given the information in the table, the expected rate of retum for stock A is 15.6 %. (Round to two decimal places.)
The standard deviation of stock A is %. (Round to two decimal places.)
E
D
80
73
Return.
12%
16%
18%
U
с
$
4
R
F
288
F4
V
Common Stock B
%
5
T
FS
G
6
Return
-7%
7%
13%
21%
B
MacBook Air
2
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&
7
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8
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delete
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shift
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Related Questions
- QUESTIONS: 1) Assuming that the risk-free rate of return is currently 3,2%, the market risk premium is 6% whereas the beta of HelloFresh SH. stock is 1.8, compute the required rate of return using CAPM. 2) Compute the value of each investment based on your required rate of return and interpret the results comparing with the market values. 3) Which investment would you select? Explain why using appropriate financial jargon (language). 4) Assume HelloFresh SH's CFO Mr. Christian Gaertner expects an earnings upturn resulting increase in growth (rate) of 1%. How does this affect your answers to Question 2 and 3? 5) AACSB Critical Thinking Questions: A) Companies pay rating agencies such as Moody's and S&P to rate their bonds, and the costs can be substantial. However, companies are not required to have their bonds rated in the first place; doing so is strictly voluntary. Why do you think they do it? (Textbook page: 198) B) What are the difficulties in using the PE ratio to value stock?…arrow_forwardQuestions: a. Compute the expected return for stock X and for stock Y b. Compute the standard deviation for stock X and for stock Y. c. Determine the best course to take for investing.arrow_forwardKaren Kay, a portfolio manager at Collins Asset Management, is using thecapital asset pricing model (CAPM) for making recommendations to her clients.Her research department has developed the information shown in the followingexhibit.Forecast Returns, Standard Deviations, and BetasForecast Return Standard Deviation BetaLow β stock (X) 14.0% 36% 0.8High β stock (Y) 17.0% 25% 1.5Market index 14.0% 15% 1.0Risk-free rate 5.0%(a) Calculate expected return and alpha for each stock.(b) Identify and justify which stock would be more appropriate for an investorwho wants to add this stock to a well-diversified equity portfolio.(c) Now consider Karen employs the “Betting Against Beta” strategy and let, , and denote the portfolio weights of the investmentin each of the asset classes (e.g. risk-free asset, low beta stock, market index,high beta stock, respectively) such that .According to its investment mandate, Collins Asset Management shouldtarget a gross leverage of 2.3. How much does she have to…arrow_forward
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- Essentials Of InvestmentsFinanceISBN:9781260013924Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.Publisher:Mcgraw-hill Education,
- Foundations Of FinanceFinanceISBN:9780134897264Author:KEOWN, Arthur J., Martin, John D., PETTY, J. WilliamPublisher:Pearson,Fundamentals of Financial Management (MindTap Cou...FinanceISBN:9781337395250Author:Eugene F. Brigham, Joel F. HoustonPublisher:Cengage LearningCorporate Finance (The Mcgraw-hill/Irwin Series i...FinanceISBN:9780077861759Author:Stephen A. Ross Franco Modigliani Professor of Financial Economics Professor, Randolph W Westerfield Robert R. Dockson Deans Chair in Bus. Admin., Jeffrey Jaffe, Bradford D Jordan ProfessorPublisher:McGraw-Hill Education
Essentials Of Investments
Finance
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Mcgraw-hill Education,
Foundations Of Finance
Finance
ISBN:9780134897264
Author:KEOWN, Arthur J., Martin, John D., PETTY, J. William
Publisher:Pearson,
Fundamentals of Financial Management (MindTap Cou...
Finance
ISBN:9781337395250
Author:Eugene F. Brigham, Joel F. Houston
Publisher:Cengage Learning
Corporate Finance (The Mcgraw-hill/Irwin Series i...
Finance
ISBN:9780077861759
Author:Stephen A. Ross Franco Modigliani Professor of Financial Economics Professor, Randolph W Westerfield Robert R. Dockson Deans Chair in Bus. Admin., Jeffrey Jaffe, Bradford D Jordan Professor
Publisher:McGraw-Hill Education