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MGFB10H3 Principles of Finance
Tutorial 7
Solutions to Exercises
1. Suppose that the assumptions of MPT are satisfied. There are two risky assets and a
risk-free asset. You observe expected returns and standard deviations of portfolios of
two investors that have chosen their portfolios according to the MPT:
E
(
R
1
) = 10%,
E
(
R
2
) = 15%,
σ
1
= 20%,
σ
2
= 40%.
(a) A third investor comes into the market. This investor is highly risk averse, and
wants to have as low standard deviation as possible. What is the return that this
third investor should expect to earn?
(b) A fourth investor can stomach a portfolio with standard deviation of returns of
50%. What is the return that this fourth investor should expect to earn?
(c) The two risky assets enter equally into the market (tangency) portfolio (i.e., weight
of each one is 0
.
50). Expected returns of these two assets are
E
(
R
A
) = 0
.
08 and
E
(
R
B
) = 0
.
12. What is the standard deviation of the market portfolio?
(d) Suppose that the standard deviations of the two risky assets are
σ
A
= 15% and
σ
B
= 29
.
411%. What is the covariance
σ
AB
and correlation
ρ
AB
of returns of the
two assets?
(e) Suppose that you cannot invest in the risk-free asset but only in the two risky
assets. Describe the portfolio that will give the lowest standard deviation. What
is the expected return and standard deviation of this portfolio.
Solution:
(a) We know that these two investors must have portfolios on the capital market line,
which is described by
E
(
R
P
) =
R
f
+
E
(
R
M
)
-
R
f
σ
M
σ
P
Plugging in the numbers, we get
0
.
10
=
R
f
+
E
(
R
M
)
-
R
f
σ
M
·
0
.
20
0
.
15
=
R
f
+
E
(
R
M
)
-
R
f
σ
M
·
0
.
40
These are two equations with two unknowns (risk-free rate and Sharpe ratio (
SR
),
or intercept and slope). We can solve them to find
R
f
= 0
.
05 and
SR
= 0
.
25.
The minimum risk the third investor can achieve is
σ
= 0, which would earn
him/her the risk-free rate, so this investor’s return will be 5%.
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(b) All we have to do is plug the numbers into the CML formula:
E
(
R
P
) =
R
f
+
E
(
R
M
)
-
R
f
σ
M
σ
P
= 0
.
05 + 0
.
25
·
0
.
50 = 0
.
175 = 17
.
5%
(c) Expected return on the market portfolio is
E
(
R
M
) = 0
.
50
·
0
.
08 + 0
.
50
·
0
.
12 = 0
.
10
We can now find the standard deviation of the market portfolio:
E
(
R
M
)
-
R
f
σ
M
= 0
.
25, or
0
.
10
-
0
.
05
σ
M
= 0
.
25, or
σ
M
= 0
.
20
(d) We just have to use the variance formula:
σ
2
M
=
w
2
A
σ
2
A
+
w
2
B
σ
2
B
+ 2
·
w
A
·
w
B
·
σ
AB
Plugging in the numbers, we get
0
.
20
2
= 0
.
50
2
·
0
.
15
2
+ 0
.
50
2
·
0
.
29411
2
+ 2
·
0
.
50
·
0
.
50
·
σ
AB
Solving this is easy, and gives
σ
AB
= 0
.
0255. To find correlation, we have to recall
the correlation formula
ρ
AB
=
σ
AB
σ
A
σ
B
=
0
.
0255
0
.
15
·
0
.
29411
= 0
.
578
(e) We know that the weight in stock A is
w
MV P
A
=
σ
2
B
-
σ
A
σ
B
ρ
AB
σ
2
A
+
σ
2
B
-
2
σ
A
σ
B
ρ
AB
=
σ
2
B
-
σ
AB
σ
2
A
+
σ
2
B
-
2
σ
AB
=
0
.
29411
2
-
0
.
0255
0
.
15
2
+ 0
.
29411
2
-
2
·
0
.
0255
= 1
.
0517
So you would have to have a weight of 1
.
0517 in the first risky asset, and
-
0
.
0517
in the second risky asset. The expected return of this portfolio is
E
(
R
P
) = 1
.
0517
·
8%
-
0
.
0517
·
12% = 7
.
7932%
Standard deviation of this portfolio is
σ
P
=
√
1
.
0517
2
·
0
.
15
2
+ 0
.
0517
2
·
0
.
29411
2
-
2
·
1
.
05217
·
0
.
0517
·
0
.
0255 = 14
.
95%
2. All assumptions of MPT hold and you choose your portfolio following the MPT. Sup-
pose that you can borrow or lend money to the bank at the risk-free rate
R
f
= 2%.
The capital market line that starts at this risk-free rate and is tangent to the effi-
cient frontier has a slope of 0
.
40 and the tangency portfolio
M
has standard deviation
σ
M
= 10%.
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(a) What is the expected return
E
(
R
M
) of this tangency portfolio?
(b) Suppose you are quite risk averse and can tolerate standard deviation of just 5%.
How much of your wealth should you allocate to the tangency portfolio
M
and
how much should you deposit to the bank at rate
R
f
?
What is your expected
return?
(c) How much do you need to allocate to the tangency portfolio and the risk-free
asset to achieve an expected return of 5%? What is the standard deviation of the
resulting portfolio?
(d) If you cannot borrow and want to create a portfolio with the highest possible
Sharpe ratio, what is the highest expected return that you can achieve?
Solution:
(a) The slope of the capital market line is the Sharpe ratio:
SR
=
E
(
R
M
)
-
R
f
σ
M
=
E
(
R
M
)
-
0
.
02
0
.
10
= 0
.
40
,
which can be easily solved to get
E
(
R
M
) = 0
.
06 = 6%.
(b) You know that the standard deviation of a portfolio consisting of a risk-free and
a risky asset is
σ
P
=
w
M
σ
M
=
w
M
·
0
.
10 = 0
.
05
So you should allocate
w
M
= 0
.
5 of your money to the risky asset, and the
remaining half to the risk-free asset. Expected return of the resulting portfolio is
E
(
R
P
) = 0
.
5
·
6% + 0
.
5
·
2% = 4%
(c) Expected return of the portfolio is
E
(
R
P
) =
w
M
·
6% + (1
-
w
M
)
·
2% = 5%
Solving this, we get
w
M
= 0
.
75. So you need to allocate 75% of your money to the
tangency portfolio, and put the remaining 25% of it in the bank. The standard
deviation of the resulting portfolio is
σ
P
=
w
M
σ
M
= 0
.
75
·
0
.
10 = 0
.
075 = 7
.
5%
(d) If you cannot borrow, the highest expected return you can achieve is
E
(
R
M
) = 6%,
which is what you’d get if you put all your money in the tangency portfolio. The
standard deviation of this portfolio is obviously equal to
σ
M
= 10%.
3. All assumptions of MPT hold and you choose your portfolio following the MPT. Sup-
pose you have $100
,
000 in cash and you borrow another $15
,
000 from the bank at
the annual risk-free rate of 4%.
You invest everything in a portfolio
P
with annual
expected return of 15% and standard deviation of 25%.
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(a) What is the Sharpe ratio of your overall portfolio?
(b) What is your realized return if the portfolio
P
goes up by 25% over a year?
(c) What is your realized return if the portfolio
P
goes down by 25% over a year?
Solution:
(a) The easy way of finding the Sharpe ratio is to simply observe that Sharpe ratio of
your portfolio of risky and risk-free assets is the same as the Sharpe ratio of any
other portfolio on the capital market line such as portfolio
P
you are investing in:
SR
P
=
0
.
15
-
0
.
04
0
.
25
= 0
.
44
A slightly longer way is to calculate expected return and standard deviation of
your overall investment. The expected return is
E
(
R
P
) = 1
.
15
·
15%
-
0
.
15
·
4% = 16
.
65%
Standard deviation is
σ
P
= 1
.
15
·
25% = 28
.
75%
So Sharpe ratio is
SR
P
=
16
.
65
-
4
28
.
75
= 0
.
44
(b) The value of your portfolio in one year if the portfolio goes up by 25% is $115
,
000
·
1
.
25 = $143
,
750. Out of this amount, you have to pay the bank back $15
,
000
·
1
.
04 = $15
,
600. So the money available to you is $143
,
750
-
$15
,
600 = $128
,
150.
And hence your realized return was 28
.
15%.
(c) The value of your portfolio in one year if the portfolio goes down by 25% is
$115
,
000
·
0
.
75 = $86
,
250. Out of this amount, you have to pay the bank back
$15
,
000
·
1
.
04 = $15
,
600. So the money available to you is $86
,
250
-
$15
,
600 =
$70
,
650. And hence your realized return was
-
29
.
35%. So borrowing to invest
amplifies your return both in the up markets and down markets (this is known as
leverage).
4. All assumptions of MPT are satisfied. There are two risky assets: stocks
A
and
B
, and
a risk-free asset. An investor has $15,000 to invest, and optimally allocates $7,500 of
it to stock
A
, $4,500 to stock
B
, and deposits the rest in the risk-free asset. Expected
returns on the two stocks are
E
(
R
A
) = 5% ,
E
(
R
B
) = 10%. The overall portfolio of
the investor has an expected return of 6%.
(a) Find the weights of stocks
A
and
B
in the market (tangency) portfolio. What is
the expected return of the market portfolio?
(b) A second investor wants to have a portfolio with minimum possible risk. What is
the expected return on this portfolio?
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(c) Suppose the standard deviation of the market portfolio is
σ
M
= 20%.
A third
investor wants his overall portfolio to have a standard deviation of 15% and has
$10,000 to invest. Describe how much money this investor will allocate to stocks
A
and
B
and the risk-free asset. What is the expected return of his portfolio?
Solution:
(a) The weights that the investor allocates to stocks
A
and
B
are
w
A
= 0
.
5,
w
B
= 0
.
3.
Denote by
w
M
A
and
w
M
B
= 1
-
w
M
A
the weights of stocks
A
and
B
in the market
portfolio.
We know that investor’s weight in the underlying assets
A
and
B
depends on how much you invest in the market portfolio
M
,
w
M
, and on the
composition of the portfolio
M
itself,
w
M
A
and
w
M
B
:
w
A
=
w
M
w
M
A
(1)
w
B
=
w
M
w
M
B
=
w
M
(
1
-
w
M
A
)
(2)
From equations (
1
) and (
2
), we see that
w
M
A
=
w
A
w
M
=
w
A
w
A
+
w
B
,
which implies that
w
M
A
= 0
.
625 and therefore
w
M
B
= 1
-
0
.
625 = 0
.
375. The
expected return of the market portfolio is thus
E
(
R
M
) =
w
M
A
E
(
R
A
) +
w
M
B
E
(
R
B
) = 0
.
625
·
5% + 0
.
375
·
10% = 6
.
875%
.
(b) The lowest possible risk this investor can achieve is zero. The return he’ll receive
is equal to the risk-free rate. To calculate it, recall that the expected return of
the overall portfolio of the first investor is
E
(
R
P
)
=
w
A
E
(
R
A
) +
w
B
E
(
R
B
) + (1
-
w
A
-
w
B
)
R
f
, or
6
=
0
.
5
·
5 + 0
.
3
·
10 + 0
.
2
·
R
f
.
Solving this gives
R
f
= 2
.
5%, which is the return the second investor will earn.
(c) His allocation into the market portfolio can be most easily found from
σ
P
=
w
M
·
σ
M
15
=
w
M
·
20
,
which gives
w
M
= 0
.
75, and therefore
w
R
f
= 1
-
w
M
= 0
.
25,
w
A
=
w
M
w
M
A
=
0
.
75
·
0
.
625 = 0
.
46875, and
w
B
=
w
M
(
1
-
w
M
A
)
= 0
.
75
·
(1
-
0
.
625) = 0
.
28125. So
the third investor should allocate $4,687.50 into stock
A
, $2,812.50 into stock
B
and depost the remaining $2,500 into the risk-free asset. The third investor will
earn an expected return of
E
(
R
P
)
=
R
f
+
E
(
R
M
)
-
R
f
σ
M
σ
P
= 2
.
5 +
6
.
875
-
2
.
5
20
·
15 = 5
.
7813%
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21092923891
or equivalently
E
(
R
P
)
=
w
M
E
(
R
M
) +
w
RF
R
f
= 0
.
75
·
6
.
875 + 0
.
25
·
2
.
5 = 5
.
7813%
,
or equivalently
E
(
R
P
)
=
w
A
E
(
R
A
) +
w
B
E
(
R
B
) +
w
RF
R
f
= 0
.
46875
·
5 + 0
.
28125
·
10
+0
.
25
·
2
.
5 = 5
.
7813%.
5. Consider three asset classes: stocks, bonds and riskless cash and assume the following
Portfolio
Expected return
Standard deviation
Risk-free (
R
f
)
0.0%
0.0%
Bonds (
R
B
)
5.0%
8.0%
Stocks (
R
S
)
10.0%
20.0%
(a) Assume that an equally weighted portfolio of cash, stocks and bonds has a stan-
dard deviation of 7
.
0%. What must be the correlation between stocks and bonds?
(b) Assume a mean-variance investor’s optimal portfolio (this is the investor that
chooses the optimal portfolio according to the MPT) is invested 55
.
36% in bonds,
18
.
26% in stocks, has a standard deviation of 5
.
53% and a Sharpe ratio of 0
.
83.
Given this information, compute the composition, standard deviation and ex-
pected return on the tangency portfolio.
(c) Given the information from the previous point, compute the composition and
standard deviation of the mean-variance portfolio with 10% expected return.
(d) Consider the mean-variance frontier without the riskless asset and assume (for this
question only) that the correlation between stocks and bonds is equal to zero. Is
the portfolio that is 100% invested into bonds a mean-variance efficient portfolio?
Explain.
Solution:
(a) We can compute the correlation by using the portfolio variance formula:
σ
2
(
R
B
/
3 +
R
S
/
3)
=
1
/
3
2
σ
2
(
R
B
) + 1
/
3
2
σ
2
(
R
S
) + 2
/
9
σ
(
R
B
, R
S
)
.
Solving for covariance gives
σ
(
R
B
, R
S
) =
-
0
.
00115. Now using the definition of
correlation we have
ρ
(
R
B
, R
S
) =
σ
(
R
B
, R
S
)
σ
(
R
B
)
σ
(
R
S
)
=
-
0
.
071875
.
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(b) Using the optimal portfolio of the mean-variance investor we can find the weights
of the tangency portfolio
w
M
Bonds
=
0
.
5536
0
.
5536 + 0
.
1826
= 0
.
75197
w
M
Stocks
=
1
-
0
.
75197 = 0
.
24803
.
Using the weights and the information we have about the two risky assets we have
E
(
R
M
) = 0
.
75197
×
0
.
05 + 0
.
24803
×
0
.
1 = 0
.
0624
.
To find the volatility we can either use the volatility of the portfolio formula
σ
(
R
M
) =
p
0
.
75197
2
×
0
.
08
2
+ 0
.
24803
2
×
0
.
1
2
+ 2
×
0
.
75197
×
0
.
24803
×
(
-
0
.
00115)
= 0
.
0752
.
Alternatively, we can use the fact that all the efficient portfolios lie on the CML.
We know that the slope of this line is
max
SR
=
E
(
R
M
)
-
R
f
σ
(
R
M
)
= 0
.
83
,
therefore
σ
(
R
M
) =
0
.
0624015
0
.
83
= 0
.
0752
(c) The CML is described by the following equation
E
(
R
P
)
=
R
f
+ max
SR
×
σ
(
R
P
)
,
or, equivalently
E
(
R
P
)
=
w
M
E
(
R
M
) + (1
-
w
M
)
R
f
=
R
f
+
w
M
(
E
(
R
M
)
-
R
f
)
σ
P
=
w
M
σ
M
Plugging in the values we have
0
.
1
=
w
M
0
.
0624015
⇒
w
M
=
0
.
1
/
0
.
0624015 = 1
.
60253
.
The standard deviation of this portfolio can be found using the
w
M
and
σ
M
σ
P
= 1
.
60253
×
0
.
0752 = 0
.
1205 = 12
.
05%
,
or, alternatively, using the equation of the CML
σ
P
= 0
.
1
/
0
.
83 = 0
.
1205 = 12
.
05%
.
The composition of this portfolio is
w
Bonds
=
1
.
60253
×
0
.
75197 = 1
.
20505
,
w
Stocks
=
1
.
60253
×
0
.
24803 = 0
.
397475
.
(d) No, a portfolio that is fully invested in bonds is not a mean-variance efficient
portfolio. The all-bond portfolio lies on the inefficient part of the mean-variance
frontier.
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Related Documents
Related Questions
QUESTION 2
Exhibit 6.2
USE THE INFORMATION BELOW FOR THE FOLLOWING PROBLEM(S)
Asset (A)
Asset (B)
E(RA) = 25%
E(RB) = 15%
(σA) = 18%
(σB) = 11%
WA = 0.75
WB = 0.25
COVA,B = −0.0009
Refer to Exhibit 6.2. What is the expected return of a portfolio of two risky assets if the expected return E(Ri), standard deviation ( σ i ), covariance (COVi,j), and asset weight (Wi) are as shown above?
a.
18.64%
b.
22.5%
c.
11%
d.
13.65%
e.
20.0%
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QUESTION 4
Below are three different investment alternatives, along with information on their expected return and
return standard deviation. Suppose investors have the following utility function:
U = E(R) - 12 Ao²
The coefficient of risk aversion (A) for Alice is 1.5, and that for John is 2.5. Which investment will each
investor pick? (hint: Find the investment giving the investor the highest utility level).
Investment Expected Return E[r]
Standard Deviation o
0.10
0.02
0.25
0.05
0.30
0.15
0.35
0.40
For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac).
2
3
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Calculate the expected return on the asset on these financial accounting question
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a. The risk-free rate of return, r;
b. The required returns for investments A and B
Review Only
Click the icon to see the Worked Solution.
a. The risk-free rate of return is
%. (Round to one decimal place.)
b. The required return for investment A is
%. (Round to one decimal place.)
The required return for investment B is
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2. Assuming the following:
Average Return (Risky Portfolio)
3.86%
Standard Dev (Risky Portfolio)
10.56%
Average Risk Free Rate
2.18%
Return on Risk Free Asset Avg
4.15%
Using the formula: E(rc)=rf + y* (E(rp) - rf)
Solve for:
1. % of Risky Assets (y):
2. % of Risk Free Assets (1-y):
Note: You wish to generate a 7% return for your complete portfolio E(rc)
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<Urgent>
please answer M9,M10,M11
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Finance
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QUESTION 4
There are only 2 assets to invest in, both being risky. AAPL has an expected
return of 10% and volatility of 15%. MSFT has an expected return of 12% and
volatility of 18%. Which one of the following statements is correct?
On the expected return-volatility space, the set of all feasible risky
portfolios is a straight line.
On the expected return-volatility space, the set of all feasible risky
portfolios is a curve that does not go through any of the 2 risky assets.
100% invested in MSFT is an efficient portfolio.
100% invested in AAPL is an efficient portfolio.
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Question 1 Fill the parts in the above table that are shaded in yellow. You will notice that there are nineline items.
Question 2Using the data generated in the previous question (Question 1);a) Plot the Security Market Line (SML) b) Superimpose the CAPM’s required return on the SML c) Indicate which investments will plot on, above and below the SML? d) If an investment’s expected return (mean return) does not plot on the SML, what doesit show? Identify undervalued/overvalued investments from the graph
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Question 2:
If the simple CAPM is valid, which of the following situations are possible? Consider each
situation separately.
a)
b)
c)
d)
f)
Portfolio
A
B
Portfolio
A
B
Expected Return
20
25
Expected Return
30
40
Portfolio
Risk-free 7
Market
A
Portfolio
Risk-free
Market
A
Expected Return
16
14
Portfolio
Risk-free 10
Market
18
A
20
Expected Return
Expected Return
7
17
13
Portfolio
Risk-free 4
Market
A
Expected Return
17
13
Beta
1.4
1.2
Standard Deviation
35
25
Standard Deviation
0
30
15
Standard Deviation
0
24
22
Beta
0
1.5
2
Standard Deviation
0
31
16
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3.1 Problem 3: You have access to two investment opportunities. Mutual Fund A, which promises 20% expected return with a variance of 0.36, and Mutual Fund B, which promises 15% expected return with a variance 0f 0.12. The covariance between the two is 0.084.
Suppose that you seek to construct a portfolio with an expected return equal to 18%. What proportions of your wealth should you invest in A and B? What is the standard deviation of such portfolio?
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What is the expected return of a portfolio of two risky assets if the expected return
E(Ri), standard deviation (SDi), covariance (COVij), and asset weight (Wi) are as
shown below?
Asset (A)
E(R₂) = 25%
SDA = 18%
WA = 0.75
COVA, B = -0.0009
Select one:
A.
13.65%
B.
20 U ODN
20.0%
C.
18.64%
D.
22.5%
Asset (B)
E(R₂) = 15%
SDB = 11%
WB = 0.25
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What is the expected return of a portfolio of two risky assets if the expected return
E(Ri), standard deviation (SDi), covariance (COVij), and asset weight (Wi) are as
shown below?
Asset (A)
E(R₂) = 10%
SDA = 8%
WA = 0.25
COVAB = 0.006
Select one:
A.
13.75%
B.
7.72%
C.
12.5%
D.
8.79%
Asset (B)
E(RB) = 15%
SDB = 9.5%
WB = 0.75
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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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Jm. 157.
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2
Problem 2: Consider a market composed of only two risky assets, A and B, with the following properties: a
StockA Expected return 25%, SD 20%
StockB Expected return 15%, SD 25%
a) Suppose their correlation is 0.8. What is the expected return of the following portfolios: one that invests 0% in A, one that invests 10% in A, 20% in A and so on until 100%? Plot the portfolio frontier formed by these portfolios.b) Repeat part (1) under the assumption that the correlation is -0.8.c) Explain in detail what is the intuition behind the difference across the answers you found for part (1) and part (2).
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A4 a.
Suppose we have two risky assets, Stock I and Stock J, and a risk-free asset. Stock I has an expected return of 25% and a beta of 1.5. Stock J has an expected return of 20% and a beta of 0.8. The risk-free asset’s return is 5%.
a. Calculate the expected returns and betas on portfolios with x% invested in Stock I and the rest invested in the risk-free asset, where x% = 0%, 50%, 100%, and 150%.
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