Chapter 06
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180.367
Subject
Finance
Date
Jan 9, 2024
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1.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Which of the following choices best completes the following statement? An investor with a higher degree of risk aversion, compared to one with a lower degree, will demand investment portfolios:
Select the best completes from the above statement?
with lower trading costs.
Explanation:
"with lower trading costs" - While a higher or lower trading costs are not necessarily an indication of an investor's tolerance for risk, risk-averse investors may prefer the portfolios associated with lower trading costs (less actively
managed, and therefore less risk). Investors with a higher degree of risk aversion will not want; "with higher risk premiums" - higher risk premiums (as they are associated with higher risks), "that are riskier (with higher standard
deviations)" - riskier portfolios (higher standard deviation), or "with lower Sharpe ratios" - lower Sharpe Ratios (any investor will always prefer investment portfolios with higher Sharpe ratios)
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
2.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Which of the following statements are true?
The higher the borrowing rate, the lower the Sharpe ratios of levered portfolios.
Explanation:
A higher borrowing rate is a consequence of the risk of the borrowers' default. In perfect markets with no additional cost of default, this increment would equal the value of the borrower's option to default, and the Sharpe
measure, with appropriate treatment of the default option, would be the same. However, there are costs to default so that this part of the increment lowers the Sharpe ratio. Also, notice that answer "With a fixed risk-free rate,
doubling the expected return and standard deviation of the risky portfolio will double the Sharpe Ratio." is not correct because doubling the expected return with a fixed risk-free rate will more than double the risk premium and
the Sharpe ratio.
Worksheet
Difficulty: 1 Basic
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of 0.5. The alternative risk-free investment in T-bills pays 2% per year.
Required:
a.
If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
b.
Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?
c.
Now suppose that you require a risk premium of 12%. What price are you willing to pay?
Required A
Required B
Complete this question by entering your answers in the tabs below.
If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
Note: Do not round your intermediate calculations. Round your answer to the nearest whole dollar amount.
Required A
Required B
Required C
$
Price
122,727
Explanation:
a.
The expected cash flow is: (0.5 × $70,000) + (0.5 × $200,000) = $135,000.
With a risk premium of 8% over the risk-free rate of 2%, the required rate of return is 10%. Therefore, the present value of the portfolio is:
$135,000 ÷ 1.10 = $122,727
b.
If the portfolio is purchased for $122,727 and provides an expected cash inflow of $135,000, then the expected rate of return [
E
(
r
)] is as follows:
$122,727 × [1 +
E
(
r
)] = $135,000
Therefore,
E
(
r
) = 10%. The portfolio price is set to equate the expected rate of return with the required rate of return.
c.
If the risk premium over T-bills is now 12%, then the required return is:
2% + 12% = 14%
The present value of the portfolio is now: $135,000 ÷ 1.14 = $118,421
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of 0.5. The alternative risk-free investment in T-bills pays 2% per year.
Required:
a.
If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
b.
Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?
c.
Now suppose that you require a risk premium of 12%. What price are you willing to pay?
Required A
Required C
Complete this question by entering your answers in the tabs below.
Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the
portfolio?
Note: Round your intermediate calculations and final answer to the nearest whole number.
Required A
Required B
Required C
Rate of return
10 %
Explanation:
a.
The expected cash flow is: (0.5 × $70,000) + (0.5 × $200,000) = $135,000.
With a risk premium of 8% over the risk-free rate of 2%, the required rate of return is 10%. Therefore, the present value of the portfolio is:
$135,000 ÷ 1.10 = $122,727
b.
If the portfolio is purchased for $122,727 and provides an expected cash inflow of $135,000, then the expected rate of return [
E
(
r
)] is as follows:
$122,727 × [1 +
E
(
r
)] = $135,000
Therefore,
E
(
r
) = 10%. The portfolio price is set to equate the expected rate of return with the required rate of return.
c.
If the risk premium over T-bills is now 12%, then the required return is:
2% + 12% = 14%
The present value of the portfolio is now: $135,000 ÷ 1.14 = $118,421
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
3.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of 0.5. The alternative risk-free investment in T-bills pays 2% per year.
Required:
a.
If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
b.
Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?
c.
Now suppose that you require a risk premium of 12%. What price are you willing to pay?
Required B
Required C
Complete this question by entering your answers in the tabs below.
Now suppose that you require a risk premium of 12%. What price are you willing to pay?
Note: Round your answer to the nearest whole dollar amount.
Required A
Required B
Required C
$
Price
118,421
Explanation:
a.
The expected cash flow is: (0.5 × $70,000) + (0.5 × $200,000) = $135,000.
With a risk premium of 8% over the risk-free rate of 2%, the required rate of return is 10%. Therefore, the present value of the portfolio is:
$135,000 ÷ 1.10 = $122,727
b.
If the portfolio is purchased for $122,727 and provides an expected cash inflow of $135,000, then the expected rate of return [
E
(
r
)] is as follows:
$122,727 × [1 +
E
(
r
)] = $135,000
Therefore,
E
(
r
) = 10%. The portfolio price is set to equate the expected rate of return with the required rate of return.
c.
If the risk premium over T-bills is now 12%, then the required return is:
2% + 12% = 14%
The present value of the portfolio is now: $135,000 ÷ 1.14 = $118,421
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
4.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider a portfolio that offers an expected rate of return of 7% and a standard deviation of 18%. T-bills offer a risk-free 2% rate of return.
What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Maximum level of risk aversion must be
less than
3.09
Explanation:
When we specify utility by
U
=
E
(
r
) − 0.5
A
2
, the utility level for T-bills is: 0.02
The utility level for the risky portfolio is:
U
= 0.07 − 0.5 ×
A
× (0.18)
2
= 0.07 − 0.0162 ×
A
In order for the risky portfolio to be preferred to T-bills, the following must hold:
0.07 − 0.0162 ×
A
> 0.02
A
< 0.05 ÷ 0.0162 = 3.09
A
must be less than 3.09 for the risky portfolio to be preferred to T-bills.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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5.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 95 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has
been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current T-bill rate is 2%.
Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as shown below.
Note: Round your "Expected Return" answers to 2 decimal places and "Variance" answers to 4 decimal places.
W
Bills
W
Index
Expected Return
Variance
0.0
1.0
10.00 %
0.0400 Example
0.2
0.8
8.40 %
0.0256
0.4
0.6
6.80 %
0.0144
0.6
0.4
5.20 %
0.0064
0.8
0.2
3.60 %
0.0016
1.0
0.0
2.00 %
0.0000
Explanation:
The portfolio expected return and variance are computed as follows:
(1)
W
Bills
(2)
r
Bills
(3)
W
Index
(4)
r
Index
r
Portfolio
(1)×(2)+(3)×(4)
σ
Portfolio
(3)×20%
σ
2
Portfolio
0.0
2%
1.0
10.0%
10.00%
20.00%
0.0400
0.2
2%
0.8
10.0%
8.40%
16.00%
0.0256
0.4
2%
0.6
10.0%
6.80%
12.00%
0.0144
0.6
2%
0.4
10.0%
5.20%
8.00%
0.0064
0.8
2%
0.2
10.0%
3.60%
4.00%
0.0016
1.0
2%
0.0
10.0%
2.00%
0.00%
0.0000
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
6.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 95 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has
been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current T-bill rate is 2%.
Calculate the utility levels of each portfolio for an investor with
A
= 2. Assume the utility function is
U
=
E
(
r
) − 0.5 ×
A
2
.
Note: Do not round intermediate calculations. Round your answers to 4 decimal places.
W
Bills
W
Index
U
(
A
= 2)
0.0
1.0
0.0600
0.2
0.8
0.0584
0.4
0.6
0.0536
0.6
0.4
0.0456
0.8
0.2
0.0344
1.0
0.0
0.0200
Explanation:
Computing utility from
U
=
E
(
r
) − 0.5 ×
A
2
=
E
(
r
) −
2
, we arrive at the values in the column labeled
U
(
A
= 2) in the following table:
W
Bills
W
Index
r
Portfolio
σ
Portfolio
σ
2
Portfolio
U
(
A
= 2)
0.0
1.0
10.00%
20.00%
0.0400
0.0600
0.2
0.8
8.40%
16.00%
0.0256
0.0584
0.4
0.6
6.80%
12.00%
0.0144
0.0536
0.6
0.4
5.20%
8.00%
0.0064
0.0456
0.8
0.2
3.60%
4.00%
0.0016
0.0344
1.0
0.0
2.00%
0.00%
0.0000
0.0200
The column labeled
U
(
A
= 2) implies that investors with
A
= 2 prefer a portfolio that is invested 100% in the market index to any of the other portfolios in the table.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
7.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 95 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has
been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current T-bill rate is 2%.
Calculate the utility levels of each portfolio for an investor with
A
= 3. Assume the utility function is
U
=
E
(
r
) − 0.5 ×
A
2
.
Note: Do not round intermediate calculations. Round your answers to 4 decimal places.
W
Bills
W
Index
U
(
A
= 3)
0.0
1.0
0.0400
0.2
0.8
0.0456
0.4
0.6
0.0464
0.6
0.4
0.0424
0.8
0.2
0.0336
1.0
0.0
0.0200
Explanation:
Computing utility from
U
=
E
(
r
) − 0.5 ×
A
2
=
E
(
r
) − 1.5
2
, we arrive at the values in the column labeled
U
(
A
= 3) in the following table:
W
Bills
W
Index
r
Portfolio
σ
Portfolio
σ
2
Portfolio
U(A = 3)
0.0
1.0
13.0% = 0.1300
20% = 0.20
0.0400
0.0400
0.2
0.8
11.4% = 0.1140
16% = 0.16
0.0256
0.0456
0.4
0.6
9.8% = 0.0980
12% = 0.12
0.0144
0.0464
0.6
0.4
8.2% = 0.0820
8% = 0.08
0.0064
0.0424
0.8
0.2
6.6% = 0.0660
4% = 0.04
0.0016
0.0336
1.0
0.0
5.0% = 0.0550
0% = 0.00
0.0000
0.0200
The more risk averse investors (A = 3) prefer the portfolio that is invested 60% in the market index to any of the other portfolios in the table.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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8.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Your client chooses to invest 70% of a portfolio in your fund and 30% in an essentially risk-free money market fund. What are the expected value and standard deviation of the rate of return on his portfolio?
Note: Round "Standard deviation" to 1 decimal place.
Expected return
9
%
Standard deviation
19.6
%
Explanation:
Expected return = (0.7 × 12%) + (0.3 × 2%) = 9%
Standard deviation = 0.7 × 28% = 19.6%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
9.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Suppose that your risky portfolio includes the following investments in the given proportions:
Stock A
25%
Stock B
32%
Stock C
43%
What are the investment proportions of each asset in your client’s overall portfolio, including the position in T-bills?
Note: Round your answers to 1 decimal place.
Investment
Proportions
T-Bills
30.0
%
Stock A
17.5
%
Stock B
22.4
%
Stock C
30.1
%
Explanation:
Investment proportions:
30.0% in T-bills
0.7 × 25% =
17.5% in Stock A
0.7 × 32% =
22.4% in Stock B
0.7 × 43% =
30.1% in Stock C
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
10.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund.
What is the reward-to-volatility (Sharpe) ratio (
S
) of your risky portfolio? Your client’s?
Note: Do not round intermediate calculations. Round your answers to 4 decimal places.
Your reward-to-volatility (Sharpe) ratio
0.3571
Client's reward-to-volatility (Sharpe) ratio
0.3571
Explanation:
Your reward-to-volatility (Sharpe) ratio:
S
= (0.12 − 0.02) ÷ 0.28 = 0.3571
Client's reward-to-volatility (Sharpe) ratio:
Expected return = (0.7 × 12%) + (0.3 × 2%) = 9%
Standard deviation = 0.7 × 28% = 19.6%
S
= (0.09 − 0.02) ÷ 0.196 = 0.3571
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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11.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Stock A
25%
Stock B
32%
Stock C
43%
Suppose that your client decides to invest in your portfolio a proportion
y
of the total investment budget so that the overall portfolio will have an expected rate of return of 10%.
Required:
a.
What is the proportion
y
?
b.
What are your client’s investment proportions in your three stocks and the T-bill fund?
c.
What is the standard deviation of the rate of return on your client’s portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What is the proportion
y
?
Required A
Required B
Required C
Proportion
y
80
%
Explanation:
a.
E
(
r
c
) =
r
f
+
y
× [
E
(
r
p
) −
r
f
] = 0.02 +
y
× (0.12 − 0.02)
If the expected return for the portfolio is 10%, then:
10% = 2% + 10% ×
y
y
= (0.10 − 0.02) ÷ 0.10 = 0.8
Therefore, in order to have a portfolio with expected rate of return equal to 10%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills.
b.
Client’s investment proportions:
20.0% in T-bills
0.8 × 25% =
20.0% in Stock A
0.8 × 32% =
25.6% in Stock B
0.8 × 43% =
34.4% in Stock C
c.
c
= 0.8 ×
p
= 0.8 × 28% = 22.4%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
11.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Stock A
25%
Stock B
32%
Stock C
43%
Suppose that your client decides to invest in your portfolio a proportion
y
of the total investment budget so that the overall portfolio will have an expected rate of return of 10%.
Required:
a.
What is the proportion
y
?
b.
What are your client’s investment proportions in your three stocks and the T-bill fund?
c.
What is the standard deviation of the rate of return on your client’s portfolio?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What are your client’s investment proportions in your three stocks and the T-bill fund?
Note: Round your answers to 1 decimal place.
Required A
Required B
Required C
Investment
Proportions
T-Bills
20.0
%
Stock A
20.0
%
Stock B
25.6
%
Stock C
34.4
%
Explanation:
a.
E
(
r
c
) =
r
f
+
y
× [
E
(
r
p
) −
r
f
] = 0.02 +
y
× (0.12 − 0.02)
If the expected return for the portfolio is 10%, then:
10% = 2% + 10% ×
y
y
= (0.10 − 0.02) ÷ 0.10 = 0.8
Therefore, in order to have a portfolio with expected rate of return equal to 10%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills.
b.
Client’s investment proportions:
20.0% in T-bills
0.8 × 25% =
20.0% in Stock A
0.8 × 32% =
25.6% in Stock B
0.8 × 43% =
34.4% in Stock C
c.
c
= 0.8 ×
p
= 0.8 × 28% = 22.4%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
11.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Stock A
25%
Stock B
32%
Stock C
43%
Suppose that your client decides to invest in your portfolio a proportion
y
of the total investment budget so that the overall portfolio will have an expected rate of return of 10%.
Required:
a.
What is the proportion
y
?
b.
What are your client’s investment proportions in your three stocks and the T-bill fund?
c.
What is the standard deviation of the rate of return on your client’s portfolio?
Required B
Required C
Complete this question by entering your answers in the tabs below.
What is the standard deviation of the rate of return on your client’s portfolio?
Note: Round your answer to 1 decimal place.
Required A
Required B
Required C
Standard deviation
22.4
%
Explanation:
a.
E
(
r
c
) =
r
f
+
y
× [
E
(
r
p
) −
r
f
] = 0.02 +
y
× (0.12 − 0.02)
If the expected return for the portfolio is 10%, then:
10% = 2% + 10% ×
y
y
= (0.10 − 0.02) ÷ 0.10 = 0.8
Therefore, in order to have a portfolio with expected rate of return equal to 10%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills.
b.
Client’s investment proportions:
20.0% in T-bills
0.8 × 25% =
20.0% in Stock A
0.8 × 32% =
25.6% in Stock B
0.8 × 43% =
34.4% in Stock C
c.
c
= 0.8 ×
p
= 0.8 × 28% = 22.4%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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12.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Suppose that your client prefers to invest in your fund a proportion
y
that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 12%.
Required:
a.
What is the investment proportion,
y?
b.
What is the expected rate of return on the complete portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What is the investment proportion,
y
?
Note: Round your answer to 2 decimal places.
Required A
Required B
Investment proportion
y
42.86
%
Explanation:
a.
c
=
y
× 28%
If your client prefers a standard deviation of at most 12%, then:
y
= 0.12 ÷ 0.28 = 0.4286 = 42.86% invested in the risky portfolio.
b.
E
(
r
c
) = 0.02 + 0.1 ×
y
= 0.02 + (0.1 × 0.4286) = 0.0629 = 6.29%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
12.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Suppose that your client prefers to invest in your fund a proportion
y
that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 12%.
Required:
a.
What is the investment proportion,
y?
b.
What is the expected rate of return on the complete portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What is the expected rate of return on the complete portfolio?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
Required A
Required B
Rate of return
6.29
%
Explanation:
a.
c
=
y
× 28%
If your client prefers a standard deviation of at most 12%, then:
y
= 0.12 ÷ 0.28 = 0.4286 = 42.86% invested in the risky portfolio.
b.
E
(
r
c
) = 0.02 + 0.1 ×
y
= 0.02 + (0.1 × 0.4286) = 0.0629 = 6.29%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
13.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Your client’s degree of risk aversion is
A
= 3.5.
Required:
a.
What proportion,
y
, of the total investment should be invested in your fund?
b.
What are the expected value and standard deviation of the rate of return on your client’s optimized portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What proportion,
y
, of the total investment should be invested in your fund?
Note: Round your answer to 2 decimal places.
Required A
Required B
Investment proportion
y
36.44
%
Explanation:
a.
y
* = (
E
(
r
P
) −
r
f
) ÷ (
A
2
P
) = (0.12 − 0.02) ÷ (3.5 × 0.28
2
) = 0.10 ÷ 0.2744 = 0.3644
Therefore, the client’s optimal proportions are: 36.44% invested in the risky portfolio and 63.56% invested in T-bills.
b.
E
(
r
C
) = 0.02 + 0.10 ×
y
* = 0.02 + (0.3644 × 0.1) = 0.0564 or 5.64%
C
= 0.3644 × 28% = 10.20%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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13.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You manage a risky portfolio with an expected rate of return of 12% and a standard deviation of 28%. The T-bill rate is 2%.
Your client’s degree of risk aversion is
A
= 3.5.
Required:
a.
What proportion,
y
, of the total investment should be invested in your fund?
b.
What are the expected value and standard deviation of the rate of return on your client’s optimized portfolio?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What are the expected value and standard deviation of the rate of return on your client’s optimized portfolio?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
Expected return
5.64
%
Standard deviation
10.20
%
Explanation:
a.
y
* = (
E
(
r
P
) −
r
f
) ÷ (
A
2
P
) = (0.12 − 0.02) ÷ (3.5 × 0.28
2
) = 0.10 ÷ 0.2744 = 0.3644
Therefore, the client’s optimal proportions are: 36.44% invested in the risky portfolio and 63.56% invested in T-bills.
b.
E
(
r
C
) = 0.02 + 0.10 ×
y
* = 0.02 + (0.3644 × 0.1) = 0.0564 or 5.64%
C
= 0.3644 × 28% = 10.20%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
14.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Refer the table below on the average excess return of the U.S. equity market and the standard deviation of that excess return. Suppose that the U.S. market is your risky portfolio.
Average Annual Returns
U.S. Equity Market
Period
U.S. equity
1-Month T-
Bills
Excess
return
Standard
Deviation
Sharpe Ratio
1927–2021
12.17
3.30
8.87
20.25
0.44
1927–1950
10.26
0.93
9.33
26.57
0.35
1951–1974
10.21
3.59
6.62
20.32
0.33
1975–1998
17.97
6.98
10.99
14.40
0.76
1999–2021
10.16
1.66
8.50
18.85
0.45
Required:
a.
If your risk-aversion coefficient is
A
= 4 and you believe that the entire 1927–2021 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to
equity? Assume your utility function is
U
=
E
(
r
) − 0.5 ×
A
2
.
b.
What if you believe that the 1975–1998 period is representative?
Required A
Required B
Complete this question by entering your answers in the tabs below.
If your risk-aversion coefficient is
A
= 4 and you believe that the entire 1927–2021 period is representative of future expected
performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity? Assume your utility
function is
U
=
E
(
r
) − 0.5 ×
A
2
.
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
T-bills
45.92
%
Equity
54.08
%
Explanation:
a.
If the period 1927–2021 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity:
A
= 4,
E
(
r
M
) −
r
f
= 8.87%
,
M
= 20.25%
(we use the
standard deviation of the risk premium from the table). Then
y
*
is given by:
y
* = (
E
(
r
M
) −
r
f
) ÷
A
2
M
= 0.0887 ÷ (4 × 0.2025
2
) = 0.5408
That is, 54.08% of the portfolio should be allocated to equity and 45.92% should be allocated to T-bills.
b.
If the period 1975–1998 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity:
A
= 4,
E
(
r
M
) −
r
f
= 11.00%
,
M
= 14.40%
and
y
* is
given by:
y
* = (
E
(
r
M
) −
r
f
) ÷
A
2
M
= 0.1100 ÷ (4 × 0.1440
2
) = 1.3262
Therefore, 132.62% of the complete portfolio should be allocated to equity by borrowing 32.62% in T-bills.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
14.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Refer the table below on the average excess return of the U.S. equity market and the standard deviation of that excess return. Suppose that the U.S. market is your risky portfolio.
Average Annual Returns
U.S. Equity Market
Period
U.S. equity
1-Month T-
Bills
Excess
return
Standard
Deviation
Sharpe Ratio
1927–2021
12.17
3.30
8.87
20.25
0.44
1927–1950
10.26
0.93
9.33
26.57
0.35
1951–1974
10.21
3.59
6.62
20.32
0.33
1975–1998
17.97
6.98
10.99
14.40
0.76
1999–2021
10.16
1.66
8.50
18.85
0.45
Required:
a.
If your risk-aversion coefficient is
A
= 4 and you believe that the entire 1927–2021 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to
equity? Assume your utility function is
U
=
E
(
r
) − 0.5 ×
A
2
.
b.
What if you believe that the 1975–1998 period is representative?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What if you believe that the 1975–1998 period is representative?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
Required A
Required B
T-bills
32.62
%
Equity
132.62
%
Explanation:
a.
If the period 1927–2021 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity:
A
= 4,
E
(
r
M
) −
r
f
= 8.87%
,
M
= 20.25%
(we use the
standard deviation of the risk premium from the table). Then
y
*
is given by:
y
* = (
E
(
r
M
) −
r
f
) ÷
A
2
M
= 0.0887 ÷ (4 × 0.2025
2
) = 0.5408
That is, 54.08% of the portfolio should be allocated to equity and 45.92% should be allocated to T-bills.
b.
If the period 1975–1998 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity:
A
= 4,
E
(
r
M
) −
r
f
= 11.00%
,
M
= 14.40%
and
y
* is
given by:
y
* = (
E
(
r
M
) −
r
f
) ÷
A
2
M
= 0.1100 ÷ (4 × 0.1440
2
) = 1.3262
Therefore, 132.62% of the complete portfolio should be allocated to equity by borrowing 32.62% in T-bills.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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15.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the following information about a risky portfolio that you manage and a risk-free asset:
E(r
P
)
= 8%,
P
= 15%,
r
f
= 2%.
Required:
a.
Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky
portfolio, P, and what proportion in the risk-free asset?
b.
What will be the standard deviation of the rate of return on her portfolio?
c.
Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse?
Required A
Required B
Complete this question by entering your answers in the tabs below.
Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return
on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky portfolio, P, and what
proportion in the risk-free asset?
Required A
Required B
Required C
Risky portfolio
100
%
Risk-free asset
0
%
Explanation:
a.
E
(
r
C
) = 8% = 2% +
y
× (0.08% − 0.02%)
⇒
y
= (0.08 − 0.02) ÷ (0.08 − 0.02) = 100%
Risk-free asset
1 − 1 = 0
b.
C
=
y
×
P
= 1 × 15% = 15%
c.
The first client is more risk averse, preferring investments that have less risk as evidenced by the lower standard deviation.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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15.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the following information about a risky portfolio that you manage and a risk-free asset:
E(r
P
)
= 8%,
P
= 15%,
r
f
= 2%.
Required:
a.
Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky
portfolio, P, and what proportion in the risk-free asset?
b.
What will be the standard deviation of the rate of return on her portfolio?
c.
Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse?
Required A
Required C
Complete this question by entering your answers in the tabs below.
What will be the standard deviation of the rate of return on her portfolio?
Required A
Required B
Required C
Standard deviation
15
%
Explanation:
a.
E
(
r
C
) = 8% = 2% +
y
× (0.08% − 0.02%)
⇒
y
= (0.08 − 0.02) ÷ (0.08 − 0.02) = 100%
Risk-free asset
1 − 1 = 0
b.
C
=
y
×
P
= 1 × 15% = 15%
c.
The first client is more risk averse, preferring investments that have less risk as evidenced by the lower standard deviation.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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15.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Consider the following information about a risky portfolio that you manage and a risk-free asset:
E(r
P
)
= 8%,
P
= 15%,
r
f
= 2%.
Required:
a.
Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky
portfolio, P, and what proportion in the risk-free asset?
b.
What will be the standard deviation of the rate of return on her portfolio?
c.
Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse?
Required B
Required C
Complete this question by entering your answers in the tabs below.
Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more
than 12%. Which client is more risk averse?
Required A
Required B
Required C
Which client is more risk averse?
First client
Explanation:
a.
E
(
r
C
) = 8% = 2% +
y
× (0.08% − 0.02%)
⇒
y
= (0.08 − 0.02) ÷ (0.08 − 0.02) = 100%
Risk-free asset
1 − 1 = 0
b.
C
=
y
×
P
= 1 × 15% = 15%
c.
The first client is more risk averse, preferring investments that have less risk as evidenced by the lower standard deviation.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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16.
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10.00
points
Problems?
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for all students.
Investment Management Incorporated (IMI) uses the capital market line to make asset allocation recommendations. IMI derives the following forecasts:
Expected return on the market portfolio: 12%
Standard deviation on the market portfolio: 20%
Risk-free rate: 5%
Samuel Johnson seeks IMI’s advice for a portfolio asset allocation. Johnson wants the standard deviation of the portfolio to equal half that of the market portfolio.
Using the capital market line, what expected return can IMI provide subject to Johnson’s risk constraint?
Note: Round your answer to 1 decimal place.
Expected return of the portfolio
8.5
%
Explanation:
Johnson requests the portfolio standard deviation to equal one half the market portfolio standard deviation. The market portfolio
M
= 20%
, which implies
P
= 10%
. The intercept of the CML equals
r
f
= 0.05
and the slope of
the CML equals the Sharpe ratio for the market portfolio (35%). Therefore, using the CML:
E
(
r
P
) =
r
f
+ [(
E
(
r
M
) −
r
f
) ÷
M
]
σ
P
= 0.05 + (0.35 × 0.10) = 0.085 = 8.5%
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
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17.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that
r
f
= 5%
.
What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which
y
= 1?
Note: Do not round intermediate calculations. Round your answers to 2 decimal places.
y = 1 for
0.64
≤ A ≤
1.28
Explanation:
For
y
to be less than 1.0 (that the investor is a lender), risk aversion (
A
) must be large enough such that:
y
= ((
E
(
r
M
) −
r
f
) ÷
A
2
M
) < 1
⇒
A
> ((0.10 − 0.02) ÷ 0.25
2
) = 1.28
For
y
to be greater than 1 (the investor is a borrower), A must be small enough:
y
= ((
E
(
r
M
) −
r
f
) ÷
A
2
M
) > 1
⇒
A
< ((0.10 − 0.06) ÷ 0.25
2
) = 0.64
For values of risk aversion within this range, the client will neither borrow nor lend but will hold a portfolio composed only of the optimal risky portfolio:
y
= 1 for 0.64 ≤
A
≤ 1.28
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
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18.
Award:
10.00
points
Problems?
Adjust credit
for all students.
Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%. Also assume that the risk-free rate is
r
f
= 5%
. Your fund
manages a risky portfolio, with the following details:
E
(r
p
) = 8%,
σ
p
= 15%
.
What is the largest percentage fee that a client who currently is lending (
y
< 1) will be willing to pay to invest in your fund? What about a client who is borrowing (
y
> 1)?
Note: Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 1 decimal place.
y < 1
1.2
%
y > 1
(0.4)
%
Explanation:
The maximum feasible fee, denoted
f
, depends on the reward-to-variability ratio. For
y
< 1, the lending rate, 2%, is viewed as the relevant risk-free rate, and we solve for
f
as follows:
((0.08 − 0.02 −
f
) ÷ 0.15) = ((0.10 − 0.02) ÷ 0.25)
f
= 0.06 − ((0.15 × 0.08) ÷ 0.25) = 0.012 = 1.2%
For
y
> 1, the borrowing rate, 6%, is the relevant risk-free rate. Then we notice that, even without a fee, the active fund is inferior to the passive fund because:
((0.08 − 0.06 −
f
) ÷ 0.15) = ((0.10 − 0.06) ÷ 0.25) →
f
= −0.004 or −0.4%
More risk tolerant investors (who are more inclined to borrow) will not be clients of the fund. We find that
f
is negative: that is, you would need to
pay
investors to choose your active fund. These investors desire higher risk−higher
return complete portfolios and thus are in the borrowing range of the relevant CAL. In this range, the reward-to-variability ratio of the index (the passive fund) is better than that of the managed fund.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
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19.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 18% with a standard deviation of 25%. You manage an active portfolio
with expected return 13% and standard deviation 28%. The risk-free rate is 3%.
Your client's degree of risk aversion is
A
= 3.5.
Required:
a.
If he chose to invest in the passive portfolio, what proportion,
y
, would he select?
b.
What is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation
decision (i.e., his choice of
y
)?
Required A
Required B
Complete this question by entering your answers in the tabs below.
If he chose to invest in the passive portfolio, what proportion,
y
, would he select?
Note: Round your answer to 2 decimal places.
Required A
Required B
Proportion of
y
68.57
%
Explanation:
a.
The formula for the optimal proportion to invest in the passive portfolio is:
y
* = (
E
(
r
M
) −
r
f
) ÷ (
A
2
M
)
Substitute the following:
E
(
r
M
) = 18%;
r
f
= 8%;
M
= 25%
;
A
= 3.5:
y
* = (0.18 − 0.03) ÷ (3.5 × 0.25
2
) = 0.21875 = 68.57%
in the passive portfolio
b.
The fee would reduce the reward-to-volatility ratio, i.e., the slope of the CAL. The client will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let
f
denote the fee:
Slope of CAL with fee = ((0.13 − 0.03 −
f
) ÷ 0.28) = ((0.10 −
f
) ÷ 0.28)
Slope of CML (which requires no fee) = ((0.18 − 0.03) ÷ 0.25) = 0.6
Setting these slopes equal we have:
((0.10 −
f
) ÷ 0.28) = 0.6
⇒
f
= 0.1680 = 6.8% per year
The fee that you can charge a client is the same regardless of the asset allocation mix of the client’s portfolio. You can charge a fee that will equate the reward-to-volatility
ratio
of your portfolio to that of your competition.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
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19.
Award:
10.00
points
Problems?
Adjust credit
for all students.
You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 18% with a standard deviation of 25%. You manage an active portfolio
with expected return 13% and standard deviation 28%. The risk-free rate is 3%.
Your client's degree of risk aversion is
A
= 3.5.
Required:
a.
If he chose to invest in the passive portfolio, what proportion,
y
, would he select?
b.
What is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation
decision (i.e., his choice of
y
)?
Required A
Required B
Complete this question by entering your answers in the tabs below.
What is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the
client indifferent between your fund and the passive strategy affected by his capital allocation decision (i.e., his choice of
y
)?
Note: Do not round intermediate calculations. Round your answer to 1 decimal place.
Required A
Required B
Fee
6.8
% per year
Explanation:
a.
The formula for the optimal proportion to invest in the passive portfolio is:
y
* = (
E
(
r
M
) −
r
f
) ÷ (
A
2
M
)
Substitute the following:
E
(
r
M
) = 18%;
r
f
= 8%;
M
= 25%
;
A
= 3.5:
y
* = (0.18 − 0.03) ÷ (3.5 × 0.25
2
) = 0.21875 = 68.57%
in the passive portfolio
b.
The fee would reduce the reward-to-volatility ratio, i.e., the slope of the CAL. The client will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let
f
denote the fee:
Slope of CAL with fee = ((0.13 − 0.03 −
f
) ÷ 0.28) = ((0.10 −
f
) ÷ 0.28)
Slope of CML (which requires no fee) = ((0.18 − 0.03) ÷ 0.25) = 0.6
Setting these slopes equal we have:
((0.10 −
f
) ÷ 0.28) = 0.6
⇒
f
= 0.1680 = 6.8% per year
The fee that you can charge a client is the same regardless of the asset allocation mix of the client’s portfolio. You can charge a fee that will equate the reward-to-volatility
ratio
of your portfolio to that of your competition.
Worksheet
Difficulty: 3 Challenge
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 06: Capital Allocation to Risky Assets > Chapter 06 Problems - Algorithmic & Static
References
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1.
Award:
1.00 point
2.
Award:
10.00 points
3.
Award:
10.00 points
4.
Award:
1.00 point
Which of the following statements regarding risk-averse investors is
true
?
They only care about the rate of return.
They accept investments that are fair games.
They only accept risky investments that offer risk premiums over the risk-free rate.
They are willing to accept lower returns and high risk.
They only care about the rate of return, and they accept investments that are fair games.
Risk-averse investors only accept risky investments that offer risk premiums over the risk-free rate.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following statements is(are)
true
?
I. Risk-averse investors reject investments that are fair games.
II. Risk-neutral investors judge risky investments only by the expected returns.
III. Risk-averse investors judge investments only by their riskiness.
IV. Risk-loving investors will not engage in fair games.
I only
II only
I and II only
II and III only
II, III, and IV only
Risk-averse investors consider a risky investment only if the investment offers a risk premium. Risk-neutral investors look only at expected returns when making an investment decision.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following statements is(are)
false
?
I. Risk-averse investors reject investments that are fair games.
II. Risk-neutral investors judge risky investments only by the expected returns.
III. Risk-averse investors judge investments only by their riskiness.
IV. Risk-loving investors will not engage in fair games.
I only
II only
I and II only
II and III only
III and IV only
Risk-averse investors consider a risky investment only if the investment offers a risk premium. Risk-neutral investors look only at expected returns when making an investment decision.
References
Multiple Choice
Difficulty: 2 Intermediate
In the mean-standard deviation graph, an indifference curve has a _________ slope.
negative
zero
positive
vertical
Cannot be determined.
The risk-return trade-off is one in which greater risk is taken if greater returns can be expected, resulting in a positive slope.
References
Multiple Choice
Difficulty: 1 Basic
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5.
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1.00 point
6.
Award:
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7.
Award:
10.00 points
8.
Award:
10.00 points
In the mean-standard deviation graph, which one of the following statements is
true
regarding the indifference curve of a risk-averse investor?
It is the locus of portfolios that have the same expected rates of return and different standard deviations.
It is the locus of portfolios that have the same standard deviations and different rates of return.
It is the locus of portfolios that offer the same utility according to returns and standard deviations.
It connects portfolios that offer increasing utilities according to returns and standard deviations.
None of the options are correct.
Indifference curves plot trade-off alternatives that provide equal utility to the individual (in this case, the tradeoffs are the risk-return characteristics of the portfolios).
References
Multiple Choice
Difficulty: 2 Intermediate
In a return-standard deviation space, which of the following statements is(are)
true
for risk-averse investors? (The vertical and horizontal lines are referred to as the expected return-axis and the standard deviation-axis, respectively.)
I. An investor's own indifference curves might intersect.
II. Indifference curves have negative slopes.
III. In a set of indifference curves, the highest offers the greatest utility.
IV. Indifference curves of two investors might intersect.
I and II only
II and III only
I and IV only
III and IV only
None of the options are correct.
An investor's indifference curves are parallel (thus they cannot intersect) and have positive slopes. The highest indifference curve (the one in the most northwestern position) offers the greatest utility. Indifference curves of investors
with similar risk-return trade-offs might intersect.
References
Multiple Choice
Difficulty: 2 Intermediate
Elias is a risk-averse investor. David is a less risk-averse investor than Elias. Therefore,
for the same risk, David requires a higher rate of return than Elias.
for the same return, Elias tolerates higher risk than David.
for the same risk, Elias requires a lower rate of return than David.
for the same return, David tolerates higher risk than Elias.
cannot be determined.
The more risk averse the investor, the less risk that is tolerated for a given rate of return.
References
Multiple Choice
Difficulty: 2 Intermediate
When an investment advisor attempts to determine an investor's risk tolerance, which factor would they be
least
likely to assess?
The investor's prior investing experience
The investor's degree of financial security
The investor's tendency to make risky or conservative choices
The level of return the investor prefers
The investor's feelings about loss
Investment advisors would be
least
likely to assess the level of return the investor prefers. The investor's investing experience, financial security, feelings about loss, and disposition toward risky or conservative choices will impact
risk tolerance.
References
Multiple Choice
Difficulty: 2 Intermediate
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9.
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10.
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11.
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12.
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10.00 points
Assume an investor with the following utility function:
U = E(r)
− 0.6(
2
)
.
To maximize her expected utility, she would choose the asset with an expected rate of return of _________ and a standard deviation of _________, respectively.
12%; 20%
10%; 15%
10%; 10%
8%; 10%
U(A)
= 0.12 − 0.60 × 0.20
2
= 0.096
Choice C dominates B and D, but only provides a utility of:
U(C)
= 0.10 − 0.60 × 0.10
2
= 0.094 < 0.096 =
U
(
A
)
References
Multiple Choice
Difficulty: 2 Intermediate
Assume an investor with the following utility function:
U = E(r) − 6
2
.
To maximize her expected utility, which one of the following investment alternatives would she choose?
A portfolio that pays 10% with a 60% probability or 5% with 40% probability.
A portfolio that pays 10% with 40% probability or 5% with a 60% probability.
A portfolio that pays 12% with 60% probability or 5% with 40% probability.
A portfolio that pays 12% with 40% probability or 5% with 60% probability.
U
(
C
) = 9.20%; highest utility of possibilities.
References
Multiple Choice
Difficulty: 3 Challenge
A portfolio has an expected rate of return of 0.15 and a standard deviation of 0.15. The risk-free rate is 6%. An investor has the following utility function:
U = E(r) − 0.5A
2
.
Which value of
A
makes this investor indifferent
between the risky portfolio and the risk-free asset?
5
6
7
8
r
f
=
U
(to be indifferent)
0.06 = 0.15 −
A
÷ 2 × 0.15
2
A
= 8
References
Multiple Choice
Difficulty: 3 Challenge
According to the mean-variance criterion, which one of the following investments dominates all others?
E
(
r
) = 0.15; Variance = 0.20
E(r)
= 0.10; Variance = 0.20
E(r)
= 0.10; Variance = 0.25
E(r)
= 0.15; Variance = 0.25
None of these options dominates the other alternatives.
E
(
r
) = 0.15; Variance = 0.20 gives the highest return with the least risk; return per unit of risk is 0.75, which dominates the reward-risk ratio for the other choices.
References
Multiple Choice
Difficulty: 3 Challenge
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14.
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15.
Award:
1.00 point
Consider a risky portfolio, A, with an expected rate of return of 0.15 and a standard deviation of 0.15, that lies on a given indifference curve. Which one of the following portfolios might lie on the same indifference curve for a risk
averse investor?
E(r)
= 0.15; Standard deviation = 0.20
E(r)
= 0.15; Standard deviation = 0.10
E(r)
= 0.10; Standard deviation = 0.10
E(r)
= 0.20; Standard deviation = 0.15
E(r)
= 0.10; Standard deviation = 0.20
Portfolio A has a reward to risk ratio of 1.0; portfolio C is the only choice with the same risk-return trade-off.
References
Multiple Choice
Difficulty: 3 Challenge
Use the below information to answer the following question.
Investment
Expected Return
E(r)
Standard Deviation
1
0.12
0.3
2
0.15
0.5
3
0.21
0.16
4
0.24
0.21
U = E
(
r
)
−
(1/2)
A
σ
2
where
A
= 4.0.
Based on the utility function above, which investment would you select?
1
2
3
4
Cannot be determined from the information given.
U(3)
= (0.21 − 4 ÷ 2)(0.16)
2
= 0.1588 (highest utility of choices).
References
Multiple Choice
Difficulty: 3 Challenge
Use the below information to answer the following question.
Investment
Expected Return
E(r)
Standard Deviation
1
0.12
0.3
2
0.15
0.5
3
0.21
0.16
4
0.24
0.21
Which investment would you select if you were risk neutral?
1
2
3
4
Cannot be determined from the information given.
If you are risk neutral, your only concern is with return, not risk.
References
Multiple Choice
Difficulty: 3 Challenge
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18.
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19.
Award:
10.00 points
Use the below information to answer the following question.
Investment
Expected Return
E(r)
Standard Deviation
1
0.12
0.3
2
0.15
0.5
3
0.21
0.16
4
0.24
0.21
U = E(r)
−
(1 ÷ 2)
A
σ
2
,
where
A
= 4.0.
The variable (
A
) in the utility function represents the:
investor's return requirement.
investor's aversion to risk.
certainty-equivalent rate of the portfolio.
minimum required utility of the portfolio.
A
is an arbitrary scale factor used to measure investor risk tolerance. The higher the value of
A
, the more risk averse the investor.
References
Multiple Choice
Difficulty: 2 Intermediate
The exact indifference curves of different investors:
can be known with perfect certainty.
can be calculated precisely with the use of advanced calculus.
are known with perfect certainty and allow the advisor to create more suitable portfolios for the client.
although not known with perfect certainty, do allow the advisor to create more suitable portfolios for the client.
cannot cross indifferences curves of other investors.
Indifference curves cannot be calculated precisely, but the theory does allow for the creation of more suitable portfolios for investors of differing levels of risk tolerance.
References
Multiple Choice
Difficulty: 1 Basic
The riskiness of individual assets:
should be considered for the asset in isolation.
should be considered in the context of the effect on overall portfolio viability.
should be combined with the riskiness of other individual assets in equal proportions.
should be considered in the context of the effect on overall portfolio volatility and should be combined with the riskiness of other individual assets in the proportions these assets constitute the entire portfolio.
is small compared to the riskiness of a portfolio of those individual assets.
The relevant risk is portfolio risk; thus, the riskiness of an individual security should be considered in the context of the entire portfolio.
References
Multiple Choice
Difficulty: 2 Intermediate
A fair game:
will be undertaken by a risk-averse investor.
is a risky investment with a positive-risk premium.
is a riskless investment.
will not be undertaken by a risk-averse investor and is a risky investment with a zero-risk premium.
will not be undertaken by a risk-averse investor and is a riskless investment.
A fair game is a risky investment with a payoff exactly equal to its expected value. Since it offers no risk premium, it will not be acceptable to a risk-averse investor.
References
Multiple Choice
Difficulty: 2 Intermediate
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22.
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23.
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10.00 points
The presence of risk means that:
investors will lose money.
more than one outcome is possible.
the standard deviation of the payoff is larger than its expected value.
final wealth will be greater than initial wealth.
terminal wealth will be less than initial wealth.
The presence of risk means that more than one outcome is possible.
References
Multiple Choice
Difficulty: 1 Basic
The utility score an investor assigns to a particular portfolio, other things equal,
will decrease as the rate of return increases.
will decrease as the standard deviation decreases.
will decrease as the variance decreases.
will increase as the variance increases.
will increase as the rate of return increases.
Utility is enhanced by higher expected returns and diminished by higher risk.
References
Multiple Choice
Difficulty: 1 Basic
The certainty equivalent rate of a portfolio is:
the rate that a risk-free investment would need to offer with certainty to be considered equally attractive as the risky portfolio.
the rate that the investor must earn for certain to give up the use of his money.
the minimum rate guaranteed by institutions such as banks.
the rate that equates "
A
" in the utility function with the average risk aversion coefficient for all risk-averse investors.
represented by the scaling factor "
−
0.005" in the utility function.
The certainty equivalent rate of a portfolio is the rate that a risk-free investment would need to offer with certainty to be considered equally attractive as the risky portfolio.
References
Multiple Choice
Difficulty: 2 Intermediate
According to the mean-variance criterion, which of the statements below is correct?
Investment
E(r)
Standard
Deviation
A
10%
5%
B
21%
11%
C
18%
23%
D
24%
16%
Investment B dominates investment A.
Investment B dominates investment C.
Investment D dominates all of the other investments.
Investment D dominates only investment B.
Investment C dominates investment A.
Investment B dominates investment C because investment B has a higher return and a lower standard deviation (risk) than investment C.
References
Multiple Choice
Difficulty: 2 Intermediate
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24.
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25.
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26.
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27.
Award:
10.00 points
Steve is more risk-averse than Edie. On a graph that shows Steve and Edie's indifference curves, which of the following is true? Assume that the graph shows expected return on the vertical axis and standard deviation on the
horizontal axis.
I. Steve and Edie's indifference curves might intersect.
II. Steve's indifference curves will have flatter slopes than Edie's.
III. Steve's indifference curves will have steeper slopes than Edie's.
IV. Steve and Edie's indifference curves will not intersect.
V. Steve's indifference curves will be downward sloping, and Edie's will be upward sloping.
I and V
I and III
III and IV
I and II
II and IV
This question tests whether the student understands the graphical properties of indifference curves and how they relate to the degree of risk tolerance.
References
Multiple Choice
Difficulty: 2 Intermediate
The capital allocation line can be described as the:
investment opportunity set formed with a risky asset and a risk-free asset.
investment opportunity set formed with two risky assets.
line on which lie all portfolios that offer the same utility to a particular investor.
line on which lie all portfolios with the same expected rate of return and different standard deviations.
The CAL has an intercept equal to the risk-free rate. It is a straight line through the point representing the riskfree asset and the risky portfolio, in expected-return ÷ standard deviation space.
References
Multiple Choice
Difficulty: 2 Intermediate
Which of the following statements regarding the capital allocation line (CAL) is
false
?
The CAL shows risk-return combinations.
The slope of the CAL equals the increase in the expected return of the complete portfolio per unit of additional standard deviation.
The slope of the CAL is also called the reward-to-volatility ratio.
The CAL is also called the efficient frontier of risky assets in the absence of a risk-free asset.
The CAL consists of combinations of a risky asset and a risk-free asset whose slope is the reward-to-volatility ratio; thus, all statements except The CAL is also called the efficient frontier of risky assets in the absence of a risk-free
asset. are true.
References
Multiple Choice
Difficulty: 2 Intermediate
Given the capital allocation line, an investor's optimal portfolio is the portfolio that:
maximizes her expected profit.
maximizes her risk.
minimizes both her risk and return.
maximizes her expected utility.
None of the options are correct.
By maximizing expected utility, the investor is obtaining the best risk-return relationships possible and acceptable for her.
References
Multiple Choice
Difficulty: 2 Intermediate
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31.
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10.00 points
An investor invests 40% of her wealth in a risky asset with an expected rate of return of 0.15 and a variance of 0.04 and 60% in a T-bill that pays 6%. Her portfolio's expected return and standard deviation are _________ and
_________ , respectively.
0.114; 0.12
0.096; 0.08
0.295; 0.06
0.087; 0.12
None of the options are correct.
E(r
P
) = w
Risky
× E(r
Risky
) + w
f
× r
f
= 0.4 × 0.15 + 0.6 × 0.06 = 0.096
= 0.4 × 0.04
0.5
= 0.08
References
Multiple Choice
Difficulty: 2 Intermediate
An investor invests 40% of his wealth in a risky asset with an expected rate of return of 0.13 and a variance of 0.03 and 60% in a T-bill that pays 6%. His portfolio's expected return and standard deviation are _________ and
_________, respectively.
0.114; 0.128
0.087; 0.063
0.295; 0.125
0.088; 0.069
E(r
P
) = w
Risky
× E(r
Risky
) + w
f
× r
f
= 0.4 × 0.13 + 0.6 × 0.06 = 0.088
σ
= 0.4 × 0.03
5
= 0.069
References
Multiple Choice
Difficulty: 2 Intermediate
An investor invests 25% of her wealth in a risky asset with an expected rate of return of 0.17 and a variance of 0.08 and 75% in a T-bill that pays 4.5%. Her portfolio's expected return and standard deviation are _________ and
_________, respectively.
0.114; 0.126
0.087; 0.068
0.076; 0.071
0.087; 0.124
None of the options are correct.
E(r
P
) = w
Risky
× E(r
Risky
) + w
f
× r
f
= 0.25 × 0.17 + 0.75 × 0.045 = 0.076
σ
= 0.25 × 0.08
5
= 0.071
References
Multiple Choice
Difficulty: 2 Intermediate
An investor invests 70% of his wealth in a risky asset with an expected rate of return of 0.15 and a variance of 0.04 and 30% in a T-bill that pays 5%. His portfolio's expected return and standard deviation are _________ and
_________, respectively.
0.120; 0.14
0.087; 0.03
0.295; 0.03
0.087; 0.14
E(r
P
) = w
Risky
× E(r
Risky
) + w
f
× r
f
= 0.7 × 0.15 + 0.3 × 0.05 = 0.120
= 0.7 × 0.04
0.5
= 0.14
References
Multiple Choice
Difficulty: 2 Intermediate
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33.
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34.
Award:
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You invest $100 in a risky asset with an expected rate of return of 0.12 and a standard deviation of 0.15 and a T-bill with a rate of return of 0.05.
What percentages of your money must be invested in the risky asset and the risk-free asset, respectively, to form a portfolio with an expected return of 0.09?
85% and 15%
75% and 25%
67% and 33%
57% and 43%
Cannot be determined.
E(r
P
) = w
risky
× E(r
risky
) + (1
−
w
risky
)× r
f
0.09 = w
risky
× 0.12 + (1− w
rishy
) × 0.05
= w
risky
× 0.07 + 0.05
w
risky
= 0.5717 or 57%
(1−w
risky
) = 0.4283 or 43%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $100 in a risky asset with an expected rate of return of 0.12 and a standard deviation of 0.15 and a T-bill with a rate of return of 0.05.
What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard deviation of 0.06?
30% and 70%
50% and 50%
60% and 40%
40% and 60%
Cannot be determined.
= w
risky
×
risky
0.06 = w
risky
× 0.15
w
risky
= 0.4
(1 − w
risky
) = 0.60
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $100 in a risky asset with an expected rate of return of 0.12 and a standard deviation of 0.15 and a T-bill with a rate of return of 0.05.
A portfolio that has an expected outcome of $115 is formed by:
investing $100 in the risky asset.
investing $80 in the risky asset and $20 in the risk-free asset.
borrowing $43 at the risk-free rate and investing the total amount ($143) in the risky asset.
investing $43 in the risky asset and $57 in the riskless asset.
Such a portfolio cannot be formed.
For $100: to grow to $115, r
p
= 15%
E(r
p
) = w
risky
× E(r
risky
) + (1
−
w
risky
) × r
f
0.15 =
w
risky
× 0.12 + (1 −
w
risky
) × 0.05
=
w
risky
× 0.7 + 0.05
w
risky
= 1.4286
(1 −
w
risky
) = − 0.4286
References
Multiple Choice
Difficulty: 3 Challenge
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36.
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37.
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You invest $100 in a risky asset with an expected rate of return of 0.12 and a standard deviation of 0.15 and a T-bill with a rate of return of 0.05.
The slope of the capital allocation line formed with the risky asset and the risk-free asset is equal to:
0.4667.
0.8000.
2.1400.
0.41667.
Cannot be determined.
Slope = SR = (r
risky
−
r
f
)
÷
σ
risky
= (0.12 − 0.05) ÷ 0.15 = 0.4667
References
Multiple Choice
Difficulty: 2 Intermediate
Consider a T-bill with a rate of return of 5% and the following risky securities:
Security A:
E(r)
= 0.15; Variance = 0.04
Security B:
E(r)
= 0.10; Variance = 0.0225
Security C:
E(r)
= 0.12; Variance = 0.01
Security D:
E(r)
= 0.13; Variance = 0.0625
From which set of portfolios, formed with the T-bill and any one of the four risky securities, would a risk-averse investor always choose his portfolio?
The set of portfolios formed with the T-bill and security A.
The set of portfolios formed with the T-bill and security B.
The set of portfolios formed with the T-bill and security C.
The set of portfolios formed with the T-bill and security D.
Cannot be determined.
Security C has the highest reward-to-volatility ratio.
References
Multiple Choice
Difficulty: 3 Challenge
You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio,
P
, constructed with two risky securities,
X
and
Y
. The weights of
X
and
Y
in
P
are 0.60 and 0.40, respectively.
X
has an expected rate of return of
0.14 and variance of 0.01, and
Y
has an expected rate of return of 0.10 and a variance of 0.0081.
If you want to form a portfolio with an expected rate of return of 0.11, what percentages of your money must you invest in the T-bill and
P
, respectively?
0.25; 0.75
0.19; 0.81
0.65; 0.35
0.50; 0.50
Cannot be determined.
E(r
risky
) = w
x
× E(r
x
) + w
y
× E(r
y
)
= 0.60 × 0.14 + 0.40 × 0.10 = 0.124
E(r
p
)
=
w
risky
× E(r
risky
) + (1
−
w
risky
) × r
f
0.11
=
w
risky
× 0.124 + (1 −
w
risky
) × 0.05
=
w
risky
× 0.074 + 0.05
w
risky
= 0.81
(1 −
w
risky
) = 0.19
References
Multiple Choice
Difficulty: 2 Intermediate
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40.
Award:
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You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio,
P
, constructed with two risky securities,
X
and
Y
. The weights of
X
and
Y
in
P
are 0.60 and 0.40, respectively.
X
has an expected rate of return of
0.14 and variance of 0.01, and
Y
has an expected rate of return of 0.10 and a variance of 0.0081.
If you want to form a portfolio with an expected rate of return of 0.10, what percentages of your money must you invest in the T-bill,
X
, and
Y
, respectively, if you keep
X
and
Y
in the same proportions to each other as in portfolio
P
?
0.25; 0.45; 0.30
0.19; 0.49; 0.32
0.32; 0.41; 0.27
0.50; 0.30; 0.20
Cannot be determined.
E
(
r
risky
) = w
x
× E(r
x
) + w
y
× E(r
y
)
= 0.60 × 0.14 + 0.40 × 0.10 = 0.124
E(r
p
) = w
risky
× E(r
risky
) + (1 − w
risky
) × r
f
0.10
=
w
risky
× 0.124 + (1 − w
risky
) × 0.05
= w
risky
× 0.074 + 0.05
w
risky
= 0.6757
(1 − w
risky
) = 0.32
W
x
= 0.6757 × 0.60 = 0.41
W
y
= 0.6757 × 0.40 = 0.27
References
Multiple Choice
Difficulty: 3 Challenge
You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio,
P
, constructed with two risky securities,
X
and
Y
. The weights of
X
and
Y
in
P
are 0.60 and 0.40, respectively.
X
has an expected rate of return of
0.14 and variance of 0.01, and
Y
has an expected rate of return of 0.10 and a variance of 0.0081.
What would be the dollar values of your positions in
X
and
Y
, respectively, if you decide to hold 40% of your money in the risky portfolio and 60% in T-bills?
$240; $360
$360; $240
$100; $240
$240; $160
Cannot be determined.
w
risky
× Initial Investment = 0.40 × $1,000 = $400
→
X
Investment = 0.60 × $400 = $240
Y
Investment = 0.40 × $400 = $160
References
Multiple Choice
Difficulty: 2 Intermediate
You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio,
P
, constructed with two risky securities,
X
and
Y
. The weights of
X
and
Y
in
P
are 0.60 and 0.40, respectively.
X
has an expected rate of return of
0.14 and variance of 0.01, and
Y
has an expected rate of return of 0.10 and a variance of 0.0081.
What would be the dollar value of your positions in
X
,
Y
, and the T-bills, respectively, if you decide to hold a portfolio that has an expected outcome of $1,120?
$568; $378; $54
$568; $54; $378
$378; $54; $568
$108; $514; $378
Cannot be determined.
For $1,000 to grow to $1,120
→
r
p
= 12%
E
(
r
risky
) = w
x
× E(r
x
) + w
y
× E(r
y
)
= 0.60 × 0.14 + 0.40 × 0.10 = 0.124
E(r
p
) = w
risky
× E(r
risky
) + (1 − w
risky
) × r
f
0.12
=
w
risky
× 0.124 + (1 − w
risky
) × 0.05
= w
risky
× 0.07 + 0.05
w
risky
= 0.946
w
x
= 0.60 × 0.946 = 0.568
$568 in X
W
y
= 0.40 × 0.946 = 0.378
$378 in Y
(1 − w
risky
) = 0.054
$54 in Risk-free asset
References
Multiple Choice
Difficulty: 3 Challenge
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A reward-to-volatility ratio is useful in:
measuring the standard deviation of returns.
understanding how returns increase relative to risk increases.
analyzing returns on variable-rate bonds.
assessing the effects of inflation.
None of the options are correct.
A reward-to-volatility ratio is useful in understanding how returns increase relative to risk increases.
References
Multiple Choice
Difficulty: 2 Intermediate
The change from a straight to a kinked capital allocation line is a result of:
reward-to-volatility ratio increasing.
borrowing rate exceeding lending rate.
an investor's risk tolerance decreasing.
increase in the portfolio proportion of the risk-free asset.
an investor's risk tolerance increasing.
The linear capital allocation line assumes that the investor may borrow and lend at the same rate (the risk-free rate), which obviously is not true. Relaxing this assumption and incorporating the higher borrowing rates into the model
results in the kinked capital allocation line.
References
Multiple Choice
Difficulty: 3 Challenge
The first major step in asset allocation is:
assessing risk tolerance.
analyzing financial statements.
estimating security betas.
identifying market anomalies.
All steps must be made simultaneously.
Assessing risk tolerance should be the first consideration in asset allocation. The other options refer to security selection.
References
Multiple Choice
Difficulty: 2 Intermediate
Based on their relative degrees of risk tolerance,
investors will hold varying amounts of the risky asset in their portfolios.
all investors will have the same portfolio asset allocations.
investors will hold varying amounts of the risk-free asset in their portfolios.
investors will hold varying amounts of the risky asset and varying amounts of the risk-free asset in their portfolios.
By determining levels of risk tolerance, investors can select the optimum portfolio for their own needs; these asset allocations will vary between amounts of risk-free and risky assets based on risk tolerance.
References
Multiple Choice
Difficulty: 1 Basic
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46.
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47.
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48.
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Asset allocation may involve:
the decision as to the allocation between a risk-loving asset and a risky asset.
the decision as to the allocation among different risky assets and considerable security analysis.
considerable security analysis.
the decision as to the allocation between a risk-free asset and a risky asset and the decision as to the allocation among different risky assets.
the decision as to the allocation between a risk-free asset and a risky asset and considerable security analysis.
The decision as to the allocation between a risk-free asset and a risky asset and the decision as to the allocation among different risky assets are possible steps in asset allocation. Considerable security analysis is related to
security selection.
References
Multiple Choice
Difficulty: 1 Basic
In the mean-standard deviation graph, the line that connects the risk-free rate and the optimal risky portfolio,
P
, is called:
the security market line.
the capital allocation line.
the indifference curve.
the investor's utility line.
the security allocation line.
The capital allocation line (CAL) illustrates the possible combinations of a risk-free asset and a risky asset available to the investor.
References
Multiple Choice
Difficulty: 2 Intermediate
Treasury bills are commonly viewed as risk-free assets because:
their short-term nature makes their values insensitive to interest rate fluctuations, only.
the inflation uncertainty over their time to maturity is negligible, only.
their term to maturity is identical to most investors' desired holding periods, only.
their short-term nature makes their values insensitive to interest rate fluctuations, and the inflation uncertainty over their time to maturity is negligible.
the inflation uncertainty over their time to maturity is negligible, and their term to maturity is identical to most investors' desired holding periods.
Treasury bills do not exactly match most investors' desired holding periods, but because they mature in only a few weeks or months they are relatively free of interest rate sensitivity and inflation uncertainty.
References
Multiple Choice
Difficulty: 1 Basic
Your client, Bo Regard, holds a complete portfolio that consists of a portfolio of risky assets (
P
) and T-Bills. The information below refers to these assets.
E(R
p
)
12.00%
Standard Deviation of P
7.20%
T-Bill rate
3.60%
Proportion of Complete Portfolio in P
80%
Proportion of Complete Portfolio in T-Bills
20%
Composition of P:
Stock A
40.00%
Stock B
25.00%
Stock C
35.00%
Total
100.00%
What is the expected return on Bo's complete portfolio?
10.32%
5.28%
9.62%
8.44%
7.58%
E(r
p
) = w
complete
× E(r
complete
) + w
f
× r
f
= 0.8 × 0.12 + 0.2 × 0.036 = 0.1032 = 10.32%
References
Multiple Choice
Difficulty: 1 Basic
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49.
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50.
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51.
Award:
10.00 points
Your client, Bo Regard, holds a complete portfolio that consists of a portfolio of risky assets (
P
) and T-Bills. The information below refers to these assets.
E(R
p
)
12.00%
Standard Deviation of P
7.20%
T-Bill rate
3.60%
Proportion of Complete Portfolio in P
80%
Proportion of Complete Portfolio in T-Bills
20%
Composition of P:
Stock A
40.00%
Stock B
25.00%
Stock C
35.00%
Total
100.00%
What is the standard deviation of Bo's complete portfolio?
7.20%
5.40%
6.92%
4.98%
5.76%
σ
= 0.8 × 0.072 = 0.0576 = 5.76%
References
Multiple Choice
Difficulty: 1 Basic
Your client, Bo Regard, holds a complete portfolio that consists of a portfolio of risky assets
(P)
and T-Bills. The information below refers to these assets.
E(R
p
)
12.00%
Standard Deviation of P
7.20%
T-Bill rate
3.60%
Proportion of Complete Portfolio in P
80%
Proportion of Complete Portfolio in T-Bills
20%
Composition of P:
Stock A
40.00%
Stock B
25.00%
Stock C
35.00%
Total
100.00%
What is the equation of Bo's capital allocation line?
E(r
C
)
= 7.2 + 3.6 × Standard Deviation of
P
E(r
C
)
= 3.6 + 1.167 × Standard Deviation of P
E(r
C
)
= 3.6 + 12.0 × Standard Deviation of P
E(r
C
)
= 0.2 + 1.167 × Standard Deviation of P
E(r
C
)
= 3.6 + 0.857 × Standard Deviation of P
The intercept is the risk-free rate (3.60%) and the
slope
= SR = (r
complete
−
r
f
) ÷
σ
complete
= (0.12
−
0.036) ÷ 0.072 = 1.167
References
Multiple Choice
Difficulty: 2 Intermediate
Your client, Bo Regard, holds a complete portfolio that consists of a portfolio of risky assets (
P
) and T-Bills. The information below refers to these assets.
E(R
p
)
12.00%
Standard Deviation of P
7.20%
T-Bill rate
3.60%
Proportion of Complete Portfolio in P
80%
Proportion of Complete Portfolio in T-Bills
20%
Composition of P:
Stock A
40.00%
Stock B
25.00%
Stock C
35.00%
Total
100.00%
What are the proportions of stocks A, B, and C, respectively, in Bo's complete portfolio?
40%, 25%, 35%
8%, 5%, 7%
32%, 20%, 28%
16%, 10%, 14%
20%, 12.5%, 17.5%
Proportion in
A
= 0.8 × 40% = 32%; proportion in
B
= 0.8 × 25% = 20%; proportion in
C
= 0.8 × 35% = 28%.
References
Multiple Choice
Difficulty: 2 Intermediate
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52.
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53.
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54.
Award:
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55.
Award:
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To build an indifference curve, we can first find the utility of a portfolio with 100% in the risk-free asset, then:
find the utility of a portfolio with 0% in the risk-free asset.
change the expected return of the portfolio and equate the utility to the standard deviation.
find another utility level with 0% risk.
change the standard deviation of the portfolio and find the expected return the investor would require to maintain the same utility level.
change the risk-free rate and find the utility level that results in the same standard deviation.
This question references the procedure described in the text. The authors describe how to trace out indifference curves using a spreadsheet.
References
Multiple Choice
Difficulty: 3 Challenge
The capital market line
I. is a special case of the capital allocation line.
II. represents the opportunity set of a passive investment strategy.
III. has the one-month T-Bill rate as its intercept.
IV. uses a broad index of common stocks as its risky portfolio.
I, III, and IV
II, III, and IV
III and IV
I, II, and III
I, II, III, and IV
The capital market line is the capital allocation line based on the one-month T-Bill rate and a broad index of common stocks. It applies to an investor pursuing a passive management strategy.
References
Multiple Choice
Difficulty: 2 Intermediate
An investor invests 35% of his wealth in a risky asset with an expected rate of return of 0.18 and a variance of 0.10 and 65% in a T-bill that pays 4%. His portfolio's expected return and standard deviation are _________ and
_________, respectively.
0.089; 0.111
0.087; 0.063
0.096; 0.126
0.087; 0.144
None of the options are correct.
E(r
P
) = w
Risky
× E(r
Risky
) + w
f
× r
f
= 0.35 × 0.18 + 0.65 × 0.04 = 0.089
= 0.35 × 0.10
0.5
= 0.111
References
Multiple Choice
Difficulty: 2 Intermediate
An investor invests 30% of his wealth in a risky asset with an expected rate of return of 0.11 and a variance of 0.12 and 70% in a T-bill that pays 3%. His portfolio's expected return and standard deviation are _________ and
_________, respectively.
0.086; 0.242
0.054; 0.104
0.295; 0.123
0.087; 0.182
None of the options are correct.
E(r
P
) = w
Risky
× E(r
Risky
) + w
f
× r
f
= 0.3 × 0.11 + 0.7 × 0.03 = 0.054
σ
= 0.3 × 0.12
0.5
= 0.104
References
Multiple Choice
Difficulty: 2 Intermediate
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57.
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58.
Award:
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You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.20 and a T-bill with a rate of return of 0.03.
What percentages of your money must be invested in the risky asset and the risk-free asset, respectively, to form a portfolio with an expected return of 0.08?
85% and 15%
75% and 25%
62.5% and 37.5%
57% and 43%
Cannot be determined.
E(r
P
) = w
risky
× E(r
risky
) + (1
−
w
risky
) × r
f
0.08 =
w
risky
× 0.11 + ( 1
−
w
risky
) × 0.03
=
w
risky
× 0.08 + 0.03
→
w
risky
= 0.625 = 62.5%
→
(1
−
w
risky
) = 0.375 = 37.5%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.20 and a T-bill with a rate of return of 0.03.
What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard deviation of 0.08?
30% and 70%
50% and 50%
60% and 40%
40% and 60%
Cannot be determined.
σ
= w
risky
× 0.20 = 0.08
→
w
risky
= 0.40 = 40%
→
(1
−
w
risky
) = 0.60 = 60%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.20 and a T-bill with a rate of return of 0.03.
The slope of the capital allocation line formed with the risky asset and the risk-free asset is equal to:
0.47.
0.80.
2.14.
0.40.
Cannot be determined.
slope = SR = (r
risky
−
r
f
)
÷
σ
risky
= (0.11 − 0.03) ÷ 0.20 = 0.40
References
Multiple Choice
Difficulty: 2 Intermediate
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60.
Award:
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61.
Award:
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You invest $1,000 in a risky asset with an expected rate of return of 0.17 and a standard deviation of 0.40 and a T-bill with a rate of return of 0.04.
What percentages of your money must be invested in the risky asset and the risk-free asset, respectively, to form a portfolio with an expected return of 0.11?
53.8% and 46.2%
75% and 25%
62.5% and 37.5%
46.2% and 53.8%
Cannot be determined.
E(r
P
) = w
risky
× E(r
risky
) + (1
−
w
risky
) × r
f
0.11 =
w
risky
× 0.17 +
(1
−
w
risky
) ×
0.04
=
w
risky
× 0.13 + 0.04
w
risky
= 0.538 or 53.8%
(1
−
w
risky
)
= 0.462 or 46.2%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $1,000 in a risky asset with an expected rate of return of 0.17 and a standard deviation of 0.40 and a T-bill with a rate of return of 0.04.
What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard deviation of 0.20?
30% and 70%
50% and 50%
60% and 40%
40% and 60%
Cannot be determined.
= w
risky
× 0.40 = 0.02
w
risky
= 0.50
(1 − w
risky
) = 0.50 or 50%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $1,000 in a risky asset with an expected rate of return of 0.17 and a standard deviation of 0.40 and a T-bill with a rate of return of 0.04.
The slope of the capital allocation line formed with the risky asset and the risk-free asset is equal to:
0.325.
0.675.
0.912.
0.407.
Cannot be determined.
Slope = SR = (r
risky
−
r
f
)
÷
σ
risky
= (0.17 − 0.04) ÷ 0.40 = 0.325
References
Multiple Choice
Difficulty: 2 Intermediate
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62.
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63.
Award:
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64.
Award:
10.00 points
You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.21 and a T-bill with a rate of return of 0.045.
What percentages of your money must be invested in the risky asset and the risk-free asset, respectively, to form a portfolio with an expected return of 0.13?
130.77% and
−
30.77%
−
30.77% and 130.77%
67.67% and 33.33%
57.75% and 42.25%
Cannot be determined.
E(r
P
) = w
risky
× E(r
risky
) + (1
−
w
risky
) × r
f
0.13 =
w
risky
×
0.11 +
(1
−
w
risky
) ×
0.045
=
w
risky
× 0.065 + 0.045
w
risky =
1.3077 or 130.77%
(1
−
w
risky
) =
−0.3077 or −30.77%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.21 and a T-bill with a rate of return of 0.045.
What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard deviation of 0.08?
30.1% and 69.9%
50.5% and 49.50%
60.0% and 40.0%
61.9% and 38.1%
Cannot be determined.
σ
= w
risky
× 0.21 = 0.08
w
risky
= 0.381 or 38.1%
(1−
w
risky
) = 0.619 or 61.9%
References
Multiple Choice
Difficulty: 2 Intermediate
You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.21 and a T-bill with a rate of return of 0.045.
A portfolio that has an expected outcome of $114 is formed by:
investing $100 in the risky asset.
investing $80 in the risky asset and $20 in the risk-free asset.
borrowing $46 at the risk-free rate and investing the total amount ($146) in the risky asset.
investing $43 in the risky asset and $57 in the risk-free asset.
Such a portfolio cannot be formed.
For $100 to grow to $114
r
p
= 0.14%
E(r
P
) = w
risky
× E(r
risky
) + (1
−
w
risky
) × r
f
0.14 =
w
risky
×
0.11 +
(1
−
w
risky
) ×
0.045
=
w
risky
×
0.065 + 0.045
w
risky
= 1.46
$146 in risky asset
(1
−
w
risky
)
=
− 0.46
borrow $46 at risk-free rate
References
Multiple Choice
Difficulty: 3 Challenge
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66.
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67.
Award:
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68.
Award:
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You invest $100 in a risky asset with an expected rate of return of 0.11 and a standard deviation of 0.21 and a T-bill with a rate of return of 0.045.
The slope of the capital allocation line formed with the risky asset and the risk-free asset is equal to:
0.4667.
0.8000.
0.3095.
0.41667.
Cannot be determined.
Slope = SR = (r
risky
−
r
f
)
÷
σ
risky
= (0.11
−
0.045) ÷ 0.21 = 0.3095
References
Multiple Choice
Difficulty: 2 Intermediate
The standard deviation of a two-asset portfolio with a correlation coefficient of 0.35 will be _________ the weighted average standard deviation of the portfolio.
below
above
equal to
incomparable to
None of the options are correct.
According to portfolio theory, the existence of a correlation coefficient creates a standard deviation below the simple weighted average.
References
Multiple Choice
Difficulty: 2 Intermediate
The expected return of a two asset portfolio with a correlation coefficient of 0.35 will be _________ the weighted average expected return of the portfolio.
below
above
equal to
incomparable to
None of the options are correct.
According to portfolio theory, the existence of a correlation coefficient has no impact on the expected return of a portfolio.
References
Multiple Choice
Difficulty: 2 Intermediate
For capital investments where the forecasted return is below the investor’s required return and above the capital market line, the investment is likely _________.
overvalued
undervalued
properly valued
ambiguous
None of the options are correct.
The CML is the expected return. If an investment if s forecasted to be above the CML, it will likely be overvalued.
References
Multiple Choice
Difficulty: 2 Intermediate
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69.
Award:
10.00 points
70.
Award:
10.00 points
For capital investments where the forecasted return is above the investor’s required return and below the capital market line, the investment is likely _________.
overvalued
undervalued
properly valued
ambiguous
None of the options are correct.
The CML is the expected return. If an investment if s forecasted to be below the CML, it will likely be undervalued.
References
Multiple Choice
Difficulty: 2 Intermediate
The reduction in standard deviation from a well-diversified portfolio of 100 stocks will _________ than that of a 200-stock portfolio.
not be statistically significantly different
be statistically significantly different
equal to
be materially different
None of the options are correct.
Once a portfolio exceeds 30 securities it is statistically significantly diversified and the additional securities to not provide a statistically significant reduction in risk.
References
Multiple Choice
Difficulty: 2 Intermediate
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Related Questions
Give typing answer with explanation and conclusion
The additional compensation that investors require to take on higher risk investments is the a. standard deviation. b. coefficient variation. c. Sharpe ratio. d. risk premium. e. risk aversion.
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Management has constructed the below table of estimates reflecting the possible returns and probabilities for pessimistic, most likely and optimistic results.
Possible outcomes probability return(n$)
Pessimistic 0.4 14.00
Most likely 0.2 34.00
Optimistic 0.4 6.00
a) Determine the expected value of return for the above company
b) What is the risk involved if the company chooses to invest in the above opportunity?
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An investor is consider four different opportunities, A, B, C, or D. The payoff for each opportunity will depend on the economic conditions, represented in the payoff table below.
Economic Condition
Investment
Poor
Average
Good
Excellent
(S1)
(S2)
(S3)
(S4)
A
50
75
20
30
B
80
15
40
50
C
-100
300
-50
10
D
25
25
25
25
What decision would be made under minimax regret?
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When describing the attitude of investors
towrds risk, which statement is correct?
A.Investors may behave as though they are
risk seekers for small investments
B.For a risk-averse investor, the standard
deviation of the return distribution is a
relevant measure of risk
C.Investors behave as though they are risk
averse for investments of significant size
D.All of the above
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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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Which of the following statements correctly describe characteristics of a risk averse investor?
Group of answer choices
A. A risk-averse investor may be willing to give up some expected return in order to be exposed to a higher level of risk.
B. Given a choice, a risk-averse investor will always choose the investment with the lower level of risk when deciding between two investments offering different levels of expected return.
C. More than one of the other statements is correct.
D. A risk-averse investor will demand compensation in the form of higher expected returns in order to take on investments with higher risk.
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As investors become more pessimistic (risk averse): Select one: a. they invest in a portfolio with a high beta. b. prices of securities fall in order to raise their expected rate of return. c. they require larger betas for taking risk. d. they require smaller premiums for taking risk.
arrow_forward
Investors require a _____ return as compensation for taking ____ risk.
A) higher, margin
B) higher, more
C) higher, convexity
D) lower, margin
E) lower, more
F) lower, convexity
arrow_forward
What does risk tolerance measure in the context of investment strategy?
This is a multiple choice question. Once you have selected an option, select the submit button below
The ability to take on higher risks for potentially higher returns
The willingness to accept losses in the short term
The capacity to afford potential losses
The preference for low-risk, low-return investments
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Correct answer needed. Explain also.
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No risk, no reward. Most people intuitively understand that they
have to bear some risk to achieve an acceptable return on their
investment portfolios.
But how much risk is right for you? If your investments turn
sour, you may put at jeopardy your ability to retire, to pay for
your kid's college education, or to weather an unexpected
need for cash. These worst-case scenarios focus our attention
on how to manage our exposure to uncertainty.
Assessing and quantifying-risk aversion is, to put it mildly,
difficult. It requires confronting at least these two big questions.
First, how much investment risk can you afford to take?
If you have steady high-paying job, for example, you have
greater ability to withstand investment losses. Conversely, if
you are close to retirement, you have less ability to adjust your
lifestyle in response to bad investment outcomes.
Second, you need to think about your personality and
decide how much risk you can tolerate. At what point will you
be unable to…
arrow_forward
An investor’s first step of investing in the financial markets is to establish an investment objective aligned with his or her long-term financial goals and needs. The critical part of the investment process is to earn the maximum return possible while minimizing risk. Portfolio diversification is the cornerstone of reducing risk in a portfolio. How would you use the Excel spreadsheet to quantify and reduce the risk in your risky asset investment portfolio?
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Wheelan's chapter 7, "Financial Markets" of the book, Naked Economics, states that "...basic economics can give us the sniff test. It provides us with a basic set of rules to which any decent investment advice must conform." These "set of rules" include all of the below EXCEPT THIS ONE. Which of the below is NOT one of these rules for wise investment?
Take risk, earn reward,.
Engage in high risk short-term trading.
Diversify your investments.
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Answer whether each of the following statements is correct and explain your argument. \
(a) According to CAPM, the expected return of a risky asset is larger than the risk free rate.
(b) According to CAPM, the expected return of a risky asset increases with its variance.
(c) According to the separation property, the optimal risky portfolio for an investor dependson the investor’s personal preference.
(d) A less risk-averse investor has a steeper indifference curve for the utility function.
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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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An estimation by marginal investor, a higher expected return is earned on A. more risky securities B. less risky securities C. less premium D. high premium (Don't use chatgpt otherwise give 10 downvotes)
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Please see image for question to answer.
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account
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In the standard model of investment management, investors care only for:
a. The return and the risk of their portfolio.
b. The return, the risk and the degree of ambiguity of their portfolio.
c. The return of their portfolio when the market is bullish.
d. The relative level of profit they will make in comparison to other investors.
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Required account answer
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