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Ch25 Mini Case.xlsx Mini Case
10/28/2015
Expected return of a portfolio:
Standard deviation of a portfolio:
Asset A Asset B
Expected return, r hat 10% 16%
Standard deviation, s20% 40%
Correlation = 0.35
Proportion of Portfolio in
Security A
(Value of wA)
Proportion of
Portfolio in
Security B
(Value of 1-
wA)
rpsp
1.00 0.00 10.00% 20.00%
0.90 0.10 10.60% 19.76%
0.80 0.20 11.20% 20.24%
0.70 0.30 11.80% 21.39%
0.60 0.40 12.40% 23.12%
0.50 0.50 13.00% 25.30%
0.40 0.60 13.60% 27.83%
0.30 0.70 14.20% 30.62%
0.20 0.80 14.80% 33.61%
0.10 0.90 15.40% 36.75%
0.00 1.00 16.00% 40.00%
Chapter 25. Mini Case
a. Suppose Asset A has an expected return of 10 percent and a standard deviation of 20 percent. Asset
B has an expected return of 16 percent and a standard deviation of 40 percent. If the correlation
between A and B is 0.35, what are the expected return and standard deviation for a portfolio comprised
of 30 percent Asset A and 70 percent Asset B?
Using the equations above, we can find the expected return and standard deviation of a
portfolio with different percentages invested in each asset.
You have been hired at the investment firm of Bowers & Noon. One of its clients doesn’t understand the
value of diversification or why stocks with the biggest standard deviations don’t always have the highest
expected returns. Your assignment is to address the client’s concerns by showing the client how to
answer the following questions.
BAABAA
2
B
2
A
2
A
2
Ap )W1(W2)W1(W
sssss
��= �
�+ �
�
Michael C. Ehrhardt Page 1 9/22/2019
Ch25 Mini Case.xlsx Mini Case
Correlation = 1
Proportion of Portfolio in
Security A
(Value of wA)
Proportion of
Portfolio in
Security B
(Value of 1-
wA)
rpsp
1.00 0.00 10.00% 20.00%
0.90 0.10 10.60% 21.59%
0.80 0.20 11.20% 23.32%
0.70 0.30 11.80% 25.18%
0.60 0.40 12.40% 27.13%
0.50 0.50 13.00% 29.15%
0.40 0.60 13.60% 31.24%
0.30 0.70 14.20% 33.38%
0.20 0.80 14.80% 35.55%
0.10 0.90 15.40% 37.76%
0.00 1.00 16.00% 40.00%
b. Plot the attainable portfolios for a correlation of 0.35. Now plot the attainable portfolios for
correlations of +1.0 and -1.0.
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Expected return
Risk, sp
AB = +0.35: Attainable Set of Risk/Return
Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Expected return
Risk, sp
AB = +1.0: Attainable Set of Risk/Return
Combinations
Ch25 Mini Case.xlsx Mini Case
Correlation = -1
Proportion of Portfolio in
Security A
(Value of wA)
Proportion of
Portfolio in
Security B
(Value of 1-
wA)
rpsp
1.00 0.00 10.00% 20.00%
0.90 0.10 10.60% 14.63%
0.80 0.20 11.20% 9.80%
0.70 0.30 11.80% 6.78%
0.67 0.33 12.00% 6.67%
0.60 0.40 12.40% 8.00%
0.50 0.50 13.00% 12.25%
0.40 0.60 13.60% 17.44%
0.30 0.70 14.20% 22.93%
0.20 0.80 14.80% 28.57%
0.10 0.90 15.40% 34.26%
0.00 1.00 16.00% 40.00%
Asset A Risk-free Asset
Expected return, r hat 10% 5%
Standard deviation, s20% 0%
Correlation = 0
Proportion of Portfolio in
Security A
(Value of wA)
Proportion of
Portfolio in
Risk-free
Asset
(Value of 1-
wA)
rpsp
1.00 0.00 10.00% 20.00%
Using the equations above, we can find the expected return and standard deviation of a
portfolio with different percentages invested in each asset.
c. Suppose a risk-free asset has an expected return of 5 percent. By definition, its standard deviation is
zero, and its correlation with any other asset is also zero. Using only Asset A and the risk-free asset, plot
the attainable portfolios.
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Expected return
Risk, sp
AB = -1.0: Attainable Set of Risk/Return
Combinations
Michael C. Ehrhardt Page 3 9/22/2019
Ch25 Mini Case.xlsx Mini Case
0.90 0.10 9.50% 18.00%
0.80 0.20 9.00% 16.00%
0.70 0.30 8.50% 14.00%
0.60 0.40 8.00% 12.00%
0.50 0.50 7.50% 10.00%
0.40 0.60 7.00% 8.00%
0.30 0.70 6.50% 6.00%
0.20 0.80 6.00% 4.00%
0.10 0.90 5.50% 2.00%
0.00 1.00 5.00% 0.00%
FEASIBLE AND EFFICIENT PORTFOLIOS
The feasible set of portfolios represent all portfolios that can be constructed from a given set of stocks.
An efficient portfolio is one that offers: the most return for a given amount of risk or the least risk for a
given amount of return.
d. Construct a reasonable, but hypothetical, graph that shows risk, as measured by portfolio standard
deviation, on the X axis and expected rate of return on the Y axis. Now add an illustrative feasible (or
attainable) set of portfolios, and show what portion of the feasible set is efficient. What makes a
particular portfolio efficient? Don't worry about specific values when constructing the graph-merely
illustrate how things look with "reasonable" data.
Expected
Portfolio
Return, r p
Risk, sp
Efficient Set
Feasible Set
Feasible and Efficient Portfolios
0%
5%
10%
15%
0% 5% 10% 15% 20%
Expected return
Risk, sp
Attainable Set of Risk/Return Combinations
with Risk-Free Asset
Michael C. Ehrhardt Page 4 9/22/2019
Ch25 Mini Case.xlsx Mini Case
OPTIMAL PORTFOLIOS
An investor's optimal portfolio is defined by the tangency point between the efficient set and the investor's
indifference curve. The indifference curve reflect an investor's attitude toward risk as reflected in his or
her risk/return trade off function.
f. Now add the risk-free asset. What impact does this have on the efficient frontier?
EFFICIENT SET WITH A RISK-FREE ASSET
When a risk free asset is added to the feasible set, investors can create portfolios that combine this asset
with a portfolio of risky asset. The straight line connecting rrf with M, the tangency point between the line
and the old efficiency set, becomes the new efficient frontier.
e. Now add a set of indifference curves to the graph created for part b. What do these curves represent?
What is the optimal portfolio for this investor? Finally, add a second set of indifference curves which
leads to the selection of a different optimal portfolio. Why do the two investors choose different
portfolios?
.
s
Feasible and Efficient Portfolios
IB2IB1
IA2
IA1
Optimal Portfolio
Optimal Portfolio
Investor B
Expected
Return, r p
Z
Efficient Set with a Risk-Free Asset
Expected
Return, r p
Ch25 Mini Case.xlsx Mini Case
OPTIMAL PORTFOLIO WITH A RISK-FREE ASSET
The optimal portfolio for any investor is the point of tangency between the CML and the investors indifferen
curve.
Capital Market Line
The capital market line is all linear combinations of the risk free asset and portfolio M.
rhat= rrf + { [(rm-rrf)/sm]xσp }
Intercept Slope Risk Measure
The CML gives the risk and return relationship for efficient portfolios
The SML , also part of CAPM, gives the risk and return relationship for individual stocks.
SML =
ri + [ (RPM)xb ]
g. Write out the equation for the Capital Market Line (CML) and draw it on the graph. Interpret the CML.
Now add a set of indifference curves, and illustrate how an investor's optimal portfolio is some
combination of the risky portfolio and the risk-free asset. What is the composition of the risky portfolio?
h. What is the Capital Asset Pricing Model (CAPM)? What are the assumptions that underlie the model?
What is the Security Market Line? See PowerPoint Show.
.
.
rRF
sMRisk, sp
New Efficient Set
.
rRF
sMRisk, sp
I1
I2
CML
R = Optimal
Portfolio
.
R.
M
r
R
rM
sR
^
^
Expected
Return, r p
Michael C. Ehrhardt Page 6 9/22/2019
Ch25 Mini Case.xlsx Mini Case
Beta Calculation
Run a regression line of past returns on Stock I versus returns on the market. The regression line is the cha
Year
rMri
115% 18%
2-5% -10%
312% 16%
i. What is a characteristic line? How is this line used to estimate a stock's beta coefficient? Write out
and explain the formula that relates total risk, market risk, and diversifiable risk.
-5%
0%
5%
10%
15%
20%
25%
-10% -5% 0% 5% 10% 15% 20%
Stock Return
Beta Calculation
beta calculation
Linear (beta
calculation)
