More Related Content Similar to Optimal Portfolio Choice and the Capital Asset Pricing Model (20) Optimal Portfolio Choice and the Capital Asset Pricing Model2. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Chapter Outline (1 of 2)
11.1 The Expected Return of a Portfolio
11.2 The Volatility of a Two-Stock Portfolio
11.3 The Volatility of a Large Portfolio
11.4 Risk Versus Return: Choosing an Efficient Portfolio
11.5 Risk-Free Saving and Borrowing
11.6 The Efficient Portfolio and Required Returns
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Chapter Outline (2 of 2)
11.7 The Capital Asset Pricing Model
11.8 Determining the Risk Premium
Appendix
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Learning Objectives (1 of 7)
• Given a portfolio of stocks, including the holdings in each
stock and the expected return in each stock, compute the
following:
– Portfolio weight of each stock (Eq. 11.1)
– Expected return on the portfolio (Eq. 11.3)
– Covariance of each pair of stocks in the portfolio
(Eq. 11.5)
– Correlation coefficient of each pair of stocks in the
portfolio (Eq. 11.6)
– Variance of the portfolio (Eq. 11.8)
– Standard deviation of the portfolio
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Learning Objectives (2 of 7)
• Compute the variance of an equally weighted portfolio,
using Eq. 11.12.
• Describe the contribution of each security to the portfolio.
• Use the definition of an efficient portfolio from Chapter 10 to
describe the efficient frontier.
• Explain how an individual investor will choose from the set
of efficient portfolios.
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Learning Objectives (3 of 7)
• Describe what is meant by a short sale, and illustrate how
short selling extends the set of possible portfolios.
• Explain the effect of combining a risk-free asset with a
portfolio of risky assets, and compute the expected return
and volatility for that combination.
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Learning Objectives (4 of 7)
• Illustrate why the risk-return combinations of the risk-free
investment and a risky portfolio lie on a straight line.
• Define the Sharpe ratio, and explain how it helps identify the
portfolio with the highest possible expected return for any
level of volatility, and how this information can be used to
identify the tangency (efficient) portfolio.
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Learning Objectives (5 of 7)
• Calculate the beta of investment with a portfolio.
• Use the beta of a security, expected return on a portfolio,
and the risk-free rate to decide whether buying shares of
that security will improve the performance of the portfolio.
• Explain why the expected return must equal the required
return.
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Learning Objectives (6 of 7)
• Use the risk-free rate, the expected return on the efficient
(tangency) portfolio, and the beta of a security with the
efficient portfolio to calculate the risk premium for an
investment.
• List the three main assumptions underlying the Capital
Asset Pricing Model.
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Learning Objectives (7 of 7)
• Explain why the CAPM implies that the market portfolio of
all risky securities is the efficient portfolio.
• Compare and contrast the capital market line with the
security market line.
• Define beta for an individual stock and for a portfolio.
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11.1 The Expected Return of a
Portfolio (1 of 3)
• Portfolio Weights
– The fraction of the total investment in the portfolio held
in each individual investment in the portfolio
The portfolio weights must add up to 1.00 or 100%.
i
X
Value of investment i
=
Total value of portfolio
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11.1 The Expected Return of a
Portfolio (2 of 3)
• Then the return on the portfolio, p
R , is the weighted average
of the returns on the investments in the portfolio, where the
weights correspond to portfolio weights.
P 1 1 2 2 n n i i
i
X X X X
R = R + R + ××× + R = R
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Textbook Example 11.1 (1 of 2)
Calculating Portfolio Returns
Problem
Suppose you buy 200 shares of Dolby Laboratories at $30
per share and 100 shares of Coca-Cola stock at $40 per
share. If Dolby’s share price goes up to $36 and Coca-Cola’ s
falls to $38, what is the new value of the portfolio, and what
return did it earn? Show that Eq. 11.2 holds. After the price
change, what are the portfolio weights?
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Textbook Example 11.1 (2 of 2)
Solution
The new of value of the portfolio is 200 × $36 + 100 × $38 = $11,000,
for a gain of $ 1000 or a 10% return on your $ 10,000 investment. Dolby’s
return was 36
-1 = 20%,
30
and Coca-Cola’s was
38
-1=-5%.
40
Given the initial portfolio weights of 60% Dolby and 40% Coca-Cola,
we can also compute the portfolio’s return from Eq. 11.2
P D D C C
X X
R = R + R =0.6×(20%)+0.4×(-5%)=10%
After the price change, the new portfolio weights are
D C
X X
200×$36 100×$38
= =65.45%, = =34,55%
$11,000 $11,000
Without trading, the weights increase for those stocks whose returns
exceed the portfolio’s return.
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Alternative Example 11.1 (1 of 4)
Problem
– Suppose you buy 500 shares of Ford at $11 per share
and 100 shares of Citigroup stock at $28 per share. If
Ford’s share price goes up to $13 and Citigroup’s rises
to $40, what is the new value of the portfolio, and what
return did it earn? Show that Eq. 11.2 holds.
– After the price change, what are the new portfolio
weights?
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Alternative Example 11.1 (2 of 4)
Solution
– The initial value of the portfolio is 500 × $11 + 100 × $28 = $8,300.
The new value of the portfolio is 500 × $13 + 100 × $40 = $10,500,
for a gain of $2,200 or a 26.5% return on your $8,300 investment.
Ford’s return was
$13
-1=18.18%,
$11
and Citigroup’s was
$40
-1=42.86%.
$28
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Alternative Example 11.1 (3 of 4)
Solution
– Given the initial portfolio weights of
$5,500
=66.3%
$8,300
for Ford and
$2,800
=33.7%
$8,300 for Citigroup, we can
also compute the portfolio’s return from Eq. 11.2:
R = X R + X R
P Ford Ford Citgroup Citigroup
= .663 × (18.2%) + .337 ×(42.9%)=26.5%
– Thus, Eq. 11.2 holds.
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Alternative Example 11.1 (4 of 4)
Solution
– After the price change, the new portfolio weights are
$6,500
=61.9%
$10,500
for Ford and
$4,000
=38.1%
$10,500
for Citigroup.
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11.1 The Expected Return of a
Portfolio (3 of 3)
• The expected return of a portfolio is the weighted average
of the expected returns of the investments within it.
P i i i i i i
i i i
X X X
E R = E R = E R = E R
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Textbook Example 11.2 (1 of 2)
Portfolio Expected Return
Problem
Suppose you invest $10,000 in Ford stock, and $30,000 in
Tyco International stock, you expect a return of 10% for Tyco.
What is your portfolio’s expected return?
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Textbook Example 11.2 (2 of 2)
Solution
You invested $40,000 in total, so your portfolio weights are
10,000
=0.25
40,000
in Ford and
30,000
=0.75 in Tyco.
40,000
Therefore, your portfolio’s expected return is
P F F T T
X X
E[R ]= E[R ]+ E[R ]=0.25×10%+0.75×16%=14.5%
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Alternative Example 11.2 (1 of 2)
Problem
– Assume your portfolio consists of $25,000 of Intel stock
and $35,000 of ATP Oil and Gas.
– Your expected return is 18% for Intel and 25% for ATP
Oil and Gas.
– What is the expected return for your portfolio?
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Alternative Example 11.2 (2 of 2)
Solution
– Total Portfolio = $25,000 + 35,000 = $60,000
‒ Portfolio Weights
Intel:
$25,000
=0.4167
$60,000
ATP:
$35,000
=0.5833
$60,000
– Expected Return
E R = 0.4167 0.18 + 0.5833 0.25
E R = 0.075006 + 0.145825 = 0.220885 = 22.1%
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11.2 The Volatility of a Two-Stock
Portfolio (1 of 4)
• Combining Risks
Table 11.1 Returns for Three Stocks, and Portfolios of Pairs
of Stocks
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11.2 The Volatility of a Two-Stock
Portfolio (2 of 4)
• Combining Risks
– While the three stocks in the previous table have the
same volatility and average return, the pattern of their
returns differs.
For example, when the airline stocks performed well,
the oil stock tended to do poorly, and when the
airlines did poorly, the oil stock tended to do well.
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11.2 The Volatility of a Two-Stock
Portfolio (3 of 4)
• Combining Risks
– Consider the portfolio which consists of equal
investments in West Air and Tex Oil.
The average return of the portfolio is equal to the average
return of the two stocks.
However, the volatility of 5.1% is much less than the
volatility of the two individual stocks.
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11.2 The Volatility of a Two-Stock
Portfolio (4 of 4)
• Combining Risks
– By combining stocks into a portfolio, we reduce risk
through diversification.
– The amount of risk that is eliminated in a portfolio
depends on the degree to which the stocks face
common risks and their prices move together.
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Determining Covariance and
Correlation (1 of 3)
• To find the risk of a portfolio, one must know the degree to
which the stocks’ returns move together.
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Determining Covariance and
Correlation (2 of 3)
• Covariance
– The expected product of the deviations of two returns from their
means.
‒ Covariance between Returns i j
R andR
i j i i j j
Cov(R ,R ) = E[(R - E[R ]) (R - E[R ])]
– Estimate of the Covariance from Historical Data
i j i,t i j,t j
t
1
Cov(R ,R ) = (R - R ) (R - R )
T - 1
If the covariance is positive, the two returns tend to move
together.
If the covariance is negative, the two returns tend to move in
opposite directions.
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Determining Covariance and
Correlation (3 of 3)
• Correlation
– A measure of the common risk shared by stocks that
does not depend on their volatility
i j
i j
i j
Cov(R ,R )
Corr(R ,R ) =
SD(R ) SD(R )
The correlation between two stocks will always be
between –1 and +1.
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Figure 11.1 Correlation
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Textbook Example 11.3 (1 of 2)
The Covariance and Correlation of a Stock with Itself
Problem
What are the covariance and the correlation of a stock’s
return with itself?
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Textbook Example 11.3 (2 of 2)
Solution
Let R be the stock’s return. Form the definition of the covariance,
2
S, S S S S S S S S
S
Cov(R R )=E[(R -E[R ])(R -E[(R -E[R ])]=E[R -E[R ]) ]
=Var(R )
where the last Eq. follows from the definition of the variance. That
is, the covariance of a stock with itself is simply its variance. Then,
S, S S
S, S 2
S S S
Cov(R R ) Var(R )
Corr(R R )= = =1
SD(R )SD(R ) SD(R )
Where the last Eq. follows from the definition of the standard
deviation. That is, a stock’s return is perfectly correlated with itself,
as it always moves together with itself in perfect synchrony.
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Table 11.2 Computing the Covariance and
Correlation Between Pairs of Stocks
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Table 11.3 Historical Annual Volatilities
and Correlations for Selected Stocks
Blank
Microsoft HP Alaska
Air
Southwest
Airlines
Ford
Motor
Kellogg General
Mills
Volatility (Standard
Deviation) Correlation
with32%
32% 36% 36% 31% 47% 19% 17%
Microsoft 1.00 0.40 0.18 0.22 0.27 0.04 0.10
HP 0.40 1.00 0.27 0.34 0.27 0.10 0.06
Alaska Air 0.18 0.27 1.00 0.40 0.15 0.15 0.20
Southwest Airlines 0.22 0.34 0.40 1.00 0.30 0.15 0.21
Ford Motor 0.27 0.27 0.15 0.30 1.00 0.17 0.08
Kellogg 0.04 0.10 0.15 0.15 0.17 1.00 0.55
General Mills 0.10 0.06 0.20 0.21 0.08 0.55 1.00
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Textbook Example 11.4 (1 of 2)
Computing the Covariance and Correlation
Problem
Using the data in Table 11.1, What are the correlation
between North Air and West Air? Between West Air and Tex
Oil?
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Textbook Example 11.4 (2 of 2)
Solution
Given the returns in Table 11.1, we deduct the mean return (10%) from
each and compute the product of these deviations between
the pairs of stocks. We then sum them and divide by T- 1 = 5
to compute the covariance, as in Table 11.2.
From the table, we see that North Air and West Air have a positive
covariance, indicating a tendency to move together, whereas West Air
and Tex Oil have a negative covariance, indicating a tendency to move
oppositely. We can assess the strength of these tendencies from
correlation, obtained by dividing the covariance by the standard deviation
of each stock (13.4%). The correlation for North Air and West Air is
62.4%; the correlation for West Air
and Tex Oil is - 71.3%.
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Textbook Example 11.5 (1 of 2)
Computing the Covariance from the Correlation
Problem
Using the data from Table 11.3, what is the covariance
between Microsoft and HP?
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Textbook Example 11.5 (2 of 2)
Solution
We can rewrite Eq. 11.6 to solve for the covariance
M, HP M, HP M HP
Cov( R R ) =Corr( R R ) SD( R ) SD( R )
=(0.40)(0.32)(0.36)=0.0461
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Alternative Example 11.5 (1 of 2)
Problem
– Using the data from Table 11.3, what is the covariance
between General Mills and Ford?
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Alternative Example 11.5 (2 of 2)
Solution
General Mills Ford General Mills Ford General Mills Ford
Cov(R ,R )=Corr(R ,R )SD(R )SD(R )
=(0.08)(0.17)(0.47)=.00639
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Computing a Portfolio’s Variance and
Volatility
• For a two security portfolio,
P P P
1 1 2 2 1 1 2 2
1 1 1 1 1 2 1 2 2 1 2 1 2 2 2 2
X X X X
X X X X X X X X
Var(R ) = Cov(R ,R )
= Cov( R + R , R + R )
= Cov(R ,R ) + Cov(R ,R ) + Cov(R ,R ) + Cov(R ,R )
– The Variance of a Two-Stock Portfolio
2 2
P 1 1 2 2 1 2 1 2
X X X X
Var(R ) = Var(R ) + Var(R ) + 2 Cov(R ,R )
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Textbook Example 11.6 (1 of 2)
Computing the Volatility of a Two-Stock Portfolio
Problem
Using the data from Table 11.3, what is the volatility of a
portfolio with equal amounts invested in Microsoft and
Hewett-Packard stock? What is the volatility of a portfolio with
equal amounts invested in Microsoft and Alaska Air stock?
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Textbook Example 11.6 (2 of 2)
Solution
• With portfolio weights of 50% each in Microsoft and Hewlett-
Packard stock, from Eq. 11.9, the portfolio’s variance is
2 2 2 2
P M M HP HP M HP M HP M HP
2 2 2 2
X X X X
Var(R ) = SD(R ) + SD(R ) +2 Corr(R ,R )SD(R )SD(R )
= (0.50) (0.32) +(0.50) (0.36) +2(0.50)(0.50)(0.40)(0.32)(0.36)
= 0.0810
The volatility is therefore P
SD(R )= Var(R)= 0.0810=28.5%
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Alternative Example 11.6 (1 of 2)
Problem
– Continuing with Alternative Example 11.2:
Assume the annual standard deviation of returns is
43% for Intel and 68% for ATPOil and Gas.
– If the correlation between Intel and ATPis 0.49, what
is the standard deviation of your portfolio?
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Alternative Example 11.6 (2 of 2)
Solution
2 2
P 1 1 2 2 1 2 1 2
X X X X
SD(R )= Var(R )+ Var(R )+2 Cov(R ,R )
2 2 2 2
P
SD(R )= (.4167) (.43) +(.5833) (.68) +2(.4167)(.5833)(.49)(.43)(.68)
P
SD(R )= (.1736)(.1849)+(.3402)(.4624)+2(.4167)(.5833)(.49)(.43)(.68)
P
SD(R )= .0321+.1573+.0696= 0.259=.5089=50.89%
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11.3 The Volatility of a Large Portfolio
• The variance of a portfolio is equal to the weighted average
covariance of each stock with the portfolio:
P P P i i P i i P
i i
X X
Var(R ) = Cov (R ,R ) = Cov R ,R = Cov(R ,R )
– which reduces to
P i i P i i j j j
i i
i j i j
i j
X X X
X X
Var(R ) = Cov (R ,R ) = Cov (R ,Σ R )
= Cov (R ,R )
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Diversification with an Equally
Weighted Portfolio
• Equally Weighted Portfolio
– A portfolio in which the same amount is invested in each
stock
• Variance of an Equally Weighted Portfolio of n Stocks
P
1
Var(R ) = (Average Variance of the Individual Stocks)
n
1
+ 1 - (Average Covariance between the Stocks)
n
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Figure 11.2 Volatility of an Equally Weighted
Portfolio Versus the Number of Stocks
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Textbook Example 11.7 (1 of 2)
Diversification using Different Types of Stocks
Problem
Stock within a single industry tend to have higher correlation
than stocks in different industries. Likewise, stocks in different
countries have lower correlation on average than stocks
within the United States. What is the volatility of a very large
portfolio of stacks within an industry in which the stocks have
a volatility of 40% and a correlation of 60%? What is the
volatility of a very large portfolio of international stacks with a
volatility of 40% and a correlation of 10%?
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Textbook Example 11.7 (2 of 2)
Solution
• From Eq.11.12, the volatility of the industry portfolio as
n is given by
Average Covariance= 0.60×0.40×0.40=31.0%
• This volatility is higher than when using stocks from different
industries as in Figure 11.2. Combining stocks from the same
industry that are more highly correlated therefore provides less
diversification. We can achieve superior diversification using
international stocks. In this case,
Average Covariance= 0.10×0.40×0.40=12.6%
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Textbook Example 11.8 (1 of 2)
Volatility when Risks Are Independent
Problem
What is the volatility of an equally weighted average of n
independent risks?
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Textbook Example 11.8 (2 of 2)
Solution
• If risks are independent, they are uncorrelated and their
covariance is zero. Using Eq. 11.12, the volatility of an equally
weighted portfolio of the risks is
P P
1 SD(individual Risk)
SD(R )= Var(R )= Var(individual Risk)=
n n
• This result coincides with Eq. 10.8, which we used earlier to
evaluate independent risks. Note that as ,
n the volatility
goes to 0—that is, a very large portfolio will have no risk. In
this case, we can eliminate all risks because there is no
common risk.
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Diversification with General Portfolios
• For a portfolio with arbitrary weights, the standard deviation
is calculated as follows:
– Volatility of a Portfolio with Arbitrary Weights
Security i’s contribution to the
volatility of the portfolio
P i i i p
i
Amount Total Fraction of i’
of i held risk of i
X
SD(R ) = × SD(R ) × Corr(R ,R )
- - -
s
risk that is
common to p
• Unless all of the stocks in a portfolio have a perfect positive
correlation of +1 with one another, the risk of the portfolio
will be lower than the weighted average volatility of the
individual stocks:
P i i i p i i
i i
X X
SD(R ) = SD(R ) Corr(R ,R ) < SD(R )
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11.4 Risk Versus Return: Choosing an
Efficient Portfolio (1 of 3)
• Efficient Portfolios with Two Stocks
– Identifying Inefficient Portfolios
In an inefficient portfolio, it is possible to find
another portfolio that is better in terms of both
expected return and volatility.
– Identifying Efficient Portfolios
Recall from Chapter 10, in an efficient portfolio
there is no way to reduce the volatility of the portfolio
without lowering its expected return.
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11.4 Risk Versus Return: Choosing
an Efficient Portfolio (2 of 3)
• Efficient Portfolios with Two Stocks
– Consider a portfolio of Intel and Coca-Cola.
Table 11.4 Expected Returns and Volatility for Different
Portfolios of Two Stocks
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Figure 11.3 Volatility Versus Expected Return
for Portfolios of Intel and Coca-Cola Stock
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11.4 Risk Versus Return: Choosing an
Efficient Portfolio (3 of 3)
• Efficient Portfolios with Two Stocks
– Consider investing 100% in Coca-Cola stock. As shown
in on the previous slide, other portfolios—such as the
portfolio with 20% in Intel stock and 80% in Coca-Cola
stock—make the investor better off in two ways: It has a
higher expected return, and it has lower volatility. As a
result, investing solely in Coca-Cola stock is inefficient.
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Textbook Example 11.9 (1 of 2)
Improving Returns with an Efficient portfolio
Problem
Sally Ferson has invested 100% of her money in coca–cola
stock and is seeking investment advice. She would like to
earn the highest expected return possible without in
increasing her volatility. Which portfolio would you
recommend?
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Textbook Example 11.9 (2 of 2)
Solution
In Figure 11.3, we can see that Sally can invest up to 40% in
Intel stock without increasing her volatility. Because stock has
a higher expected return than coca-cola stock, she will earn
higher expected returns by putting more money in Intel stock.
Therefore, you should recommend that sally put 40% of her
money in Intel stock, leaving 60% in coca-cola stock. This
portfolio has the same volatility of 25%, but an expected
return of 14% rather than the 6% she has now.
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Alternative Example 11.9 (1 of 2)
Problem
– Using Figure 11.3, what combination of Intel and Coca-
Cola provides the worst risk/return trade-off? What
combination provides the lowest amount of risk? The
greatest amount of risk?
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Alternative Example 11.9 (2 of 2)
Solution
– The worst risk/return trade-off is 100% in Coca-Cola
because you can earn a higher return with the same
amount of risk or the same return with a lower amount of
risk by moving to a more efficient portfolio.
– 20% in Intel and 80% in Coca-Cola provides the lowest
risk, while 100% in Intel offers the highest amount of
risk.
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The Effect of Correlation
• Correlation has no effect on the expected return of a
portfolio.
• However, the volatility of the portfolio will differ depending
on the correlation.
• The lower the correlation, the lower the volatility we can
obtain.
• As the correlation decreases, the volatility of the portfolio
falls.
• The curve showing the portfolios will bend to the left to a
greater degree as shown on the next slide.
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Figure 11.4 Effect on Volatility and Expected
Return of Changing the Correlation Between
Intel and Coca-Cola Stock
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Short Sales
• Long Position
– A positive investment in a security
• Short Position
– A negative investment in a security
– In a short sale, you sell a stock that you do not own and
then buy that stock back in the future.
– Short selling is an advantageous strategy if you expect a
stock price to decline in the future.
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Textbook Example 11.10 (1 of 3)
Expected Return and Volatility with a Short Sale
Problem
Suppose you have $20,000 in cash to invest. You decide to
short sell $10,000 worth of Coca-Cola stock and invest the
proceeds from your short sale, plus your $20,000, in Intel.
The expected return of Coca-Cola and Intel is 6% and 26%.
What is the expected return and volatility of your portfolio?
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Textbook Example 11.10 (2 of 3)
Solution
• We can think of our short sale as a negative investment of $10,000 in
Coca-Cola stock. In addition, we invested + $30,000
in Intel stock, for a total net investment of$30,000 - $10,000 = $20,000
cash. The corresponding portfolio weights are
1
Value of investment in Intel 30,000
x = = =150%
Total value of portfolio 20,000
C
Value of investment in Coca-Cola -10,000
x = = =-50%
Total valueof portfolio 20,000
• Note that the portfolio weights still add up to 100%. Using these
portfolio weights, we can calculate the expected return and volatility
of the portfolio using Eq. 11.3 and Eq. 11.8 as before:
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Textbook Example 11.10 (3 of 3)
p 1 1 C C
E[R ]=x E[R ]+x E[R ]=1.50×26%+(-0.50)×6%=36%
2 2
P P 1 1 C C 1 C 1 C
SD(R )= Var(R )= x Var(R )+x Var(R )+2x x Cov(R ,R )
2 2 2 2
= 1.5 ×0.50 +(-0.5) ×0.25 +2(1.5)(-0.5)(0)=76.0%
• Note that in this case, short selling increases the expected
return of your portfolio, but also its volatility, above those of
the individual stocks.
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Figure 11.5 Portfolios of Intel and
Coca-Cola Allowing for Short Sales
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Efficient Portfolios with Many Stocks
• Consider adding Bore Industries to the two-stock portfolio:
Blank Blank Blank Blank
Correlation with
Blank
Stock Expected
Return
Volatility Intel Coca-Cola Bore
Ind.
Intel 26% 50% 1.0 0.0 0.0
Coca-Cola 6% 25% 0.0 1.0 0.0
Bore
Industries
2% 25% 0.0 0.0 1.0
– Although Bore has a lower return and the same volatility as Coca-
Cola, it still may be beneficial to add Bore to the portfolio for the
diversification benefits.
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Figure 11.6 Expected Return and Volatility for
Selected Portfolios of Intel, Coca-Cola, and Bore
Industries Stocks
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Figure 11.7 The Volatility and Expected Return
for All Portfolios of Intel, Coca-Cola, and Bore
Stock
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Risk Versus Return: Many Stocks
• The efficient portfolios, those offering the highest possible
expected return for a given level of volatility, are those on
the northwest edge of the shaded region, which is called the
efficient frontier for these three stocks.
– In this case, none of the stocks, on its own, is on the
efficient frontier, so it would not be efficient to put all our
money in a single stock.
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Figure 11.8 Efficient Frontier with
Three Stocks Versus Ten Stocks
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11.5 Risk-Free Saving and Borrowing
• Risk can also be reduced by investing a portion of a
portfolio in a risk-free investment, like T-Bills.
– However, doing so will likely reduce the expected return.
• On the other hand, an aggressive investor who is seeking
high expected returns might decide to borrow money to
invest even more in the stock market.
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Investing in Risk-Free Securities (1 of 2)
• Consider an arbitrary risky portfolio and the effect on risk
and return of putting a fraction of the money in the portfolio,
while leaving the remaining fraction in risk-free Treasury
bills.
– The expected return would be
f P
f P f
xP
E [R ] = (1 - x)r + xE[R ]
= r +x(E[R ] - r )
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Investing in Risk-Free Securities (2 of 2)
• The standard deviation of the portfolio would be calculated
as follows:
– Note: The standard deviation is only a fraction of the
volatility of the risky portfolio, based on the amount
invested in the risky portfolio.
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Figure 11.9 The Risk–Return Combinations
from Combining a Risk-Free Investment and a
Risky Portfolio
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Borrowing and Buying Stocks on Margin
• Buying Stocks on Margin
– Borrowing money to invest in a stock
– A portfolio that consists of a short position in the risk-
free investment is known as a levered portfolio.
Margin investing is a risky investment strategy.
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Textbook Example 11.11 (1 of 2)
Margin Investing
Problem
Suppose you have $10,000 in cash, and you decide to borrow
another $10,000 at a 5% interest rate in order to invest
$20,000 in portfolio Q which has a 10% expected return and
volatility of your investment? What is your realized return if Q
goes up 30% over the year ? What if Q falls by 10%?
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Textbook Example 11.11 (2 of 2)
Solution
• You have doubled your investment in Q using margin, so x= 200%.
From Eq. 11.16, we see that you have increased both your
expected returns and your risk relative to the portfolio Q:
XQ f Q f
X
E(R )=r + (E[R ]-r )=5%+2×(10%-5%)=15%
XQ Q
X
SD(R ) = SD(R ) = 2×(20%) = 40%
• If Q goes up 30%, your investment will be worth $26,000, but you will
owe $10,000 × 1.05 = $10,500 on your loan, for a net payoff of $ 15,500
or a 55% return on your $10,000 initial investment. If Q drops by 10%,
you are left with with $18,000 - $10,500 = $7500, and your return is −25%.
Thus the use of margin doubled the range of your returns
55% - -25% = 80%
(
versus
30% - -10% = 40%), corresponding to the doubling of the volatility
of the portfolio.
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Identifying the Tangent Portfolio (1 of 4)
• To earn the highest possible expected return for any level of
volatility we must find the portfolio that generates the
steepest possible line when combined with the risk-free
investment.
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Identifying the Tangent Portfolio (2 of 4)
• Sharpe Ratio
– Measures the ratio of reward-to-volatility provided by a
portfolio
P f
P
E[R ] - r
Portfolio Excess Return
Sharpe Ratio = =
Portfolio Volatility SD(R )
The portfolio with the highest Sharpe ratio is the
portfolio where the line with the risk-free investment
is tangent to the efficient frontier of risky investments.
The portfolio that generates this tangent line is known
as the tangent portfolio.
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Figure 11.10 The Tangent or Efficient
Portfolio
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Identifying the Tangent Portfolio (3 of 4)
• Combinations of the risk-free asset and the tangent portfolio
provide the best risk and return trade-off available to an
investor.
– This means that the tangent portfolio is efficient and that
all efficient portfolios are combinations of the risk-free
investment and the tangent portfolio.
– Every investor should invest in the tangent portfolio
independent of his or her taste for risk.
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Identifying the Tangent Portfolio (4 of 4)
• An investor’s preferences will determine only how much to
invest in the tangent portfolio versus the risk-free
investment.
– Conservative investors will invest a small amount in the
tangent portfolio.
– Aggressive investors will invest more in the tangent
portfolio.
– Both types of investors will choose to hold the same
portfolio of risky assets, the tangent portfolio, which is
the efficient portfolio.
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Textbook Example 11.12 (1 of 2)
Optimal Portfolio Choice
Problem
Your uncle asks for investment advice. Currently, he has
$100,000 invested in portfolio p in Figure 11.10, which has an
expected return of 10.5% and a volatility of 8%. Suppose the
risk-free rate is 5%, and the tangent portfolio has an expected
return of 18.5% and a volatility of 13%. To maximize his
expected return without increasing his volatility, which
portfolio would you recommend? If your uncle prefers to keep
his expected return the same but minimize his risk, which
portfolio would you recommend?
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Textbook Example 11.12 (2 of 2)
Solution
• In either case the best portfolios are combination of the risk-free investment and
the tangent portfolio. T, using Eq. 11.16, the expected return and volatility are
f T f
X
X X
E[R ]=r + (E[R ]-r )=5%+ (18.5%-5%)
T
XT T
X X
SD(R )= SD(R )= (13%)
• So, to maintain the volatility at 8%, X
8%
= =61.5%.
13%
In this case, your uncle
should invest $61,500 in the tangent portfolio, and the remaining $38,500 in the
risk-free investment. His expected return will then be
5% + 61.5% 13.5% = 13.3%,
13.3%, the highest possible given his level of risk.
Alternatively, to keep the expected return equal to the current value of 10.5%, x
must satisfy
5% +x 13.5% = 10.5%, sox= 40.7%. Now your uncle should
invest $40,700 in the tangent portfolio and $59,300 in the risk-free investment,
lowering his volatility level to
40.7% 13% = 5.29%, the lowest possible given
his expected return.
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11.6 The Efficient Portfolio and
Required Returns (1 of 5)
• Portfolio Improvement: Beta and the Required Return
– Assume there is a portfolio of risky securities, P
To determine whether P has the highest possible
Sharpe ratio, consider whether its Sharpe ratio could
be raised by adding more of some investment i to the
portfolio.
The contribution of investment i to the volatility of the
portfolio depends on the risk that i has in common
with the portfolio, which is measured by i’s volatility
multiplied by its correlation with P.
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11.6 The Efficient Portfolio and
Required Returns (2 of 5)
• Portfolio Improvement: Beta and the Required Return
– If you were to purchase more of investment i by
borrowing, you would earn the expected return of i
minus the risk-free return.
– Thus adding i to the portfolio P will improve our Sharpe
ratio if
P f
i f i i P
P
Additional return Incremental volatility
from investment i from investment i Return per unit of volatilty
available from portfolio P
E[R ] - r
E [R ] - r >SD(R ) ×Corr(R ,R ) ×
SD(R )
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11.6 The Efficient Portfolio and
Required Returns (3 of 5)
• Portfolio Improvement: Beta and the Required Return
– Beta of Portfolio i with Portfolio P
P i i P i P
i
P P
SD(R ) × Corr(R ,R ) Corr(R ,R )
β = =
SD(R ) Var(R )
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11.6 The Efficient Portfolio and
Required Returns (4 of 5)
• Portfolio Improvement: Beta and the Required Return
– Increasing the amount invested in i will increase the
Sharpe ratio of portfolio P if its expected return i
E[R ]
exceeds the required return i
r, which is given by
P
i f i P f
r º r + β × (E[R ] - r )
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11.6 The Efficient Portfolio and
Required Returns (5 of 5)
• Portfolio Improvement: Beta and the Required Return
– Required Return of i
The expected return that is necessary to compensate
for the risk investment i will contribute to the portfolio.
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Textbook Example 11.13 (1 of 2)
The Required Return of a New Investment
Problem
You are currently invested in the omega Fund, a broad-based
fund with an expected return of 15% and a volatility of 20%,
as well as in rick-free Treasuries paying 3%. Your broker
suggests that you add a real estate fund to your portfolio. The
real estate fund has an expected return of 9%, a volatility of
35%, and a correlation of 0.10 with the omega fund. Will
adding the real estate fund improve your portfolio?
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Textbook Example 11.13 (2 of 2)
Solution
• Let re
R be the return of the real estate fund and 0
R
be the return of the Omega fund. Form Eq. 11.19, the beta of
the real estate fund with the Omega Fund is
O re re O
re
O
SD(R )Corr(R ,R ) 35%×0.10
β = = =0.175
SD(R ) 20%
• We can then use Eq. 11.20 to determine the required return that
makes the real estate fund an attractive addition to our portfolio:
O
re f re O f
r =r +β (E[R ]-r )=3%+0.175×(15%-3%)=5.1%
• Because its expected return of 9% exceeds the required
return of 5.1%, investing some amount in the real estate fund
will improve our portfolio’s Sharpe ratio.
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Alternative Example 11.13 (1 of 2)
Problem
– Assume you own a portfolio of 25 different “large cap”
stocks. You expect your portfolio will have a return of
12% and a standard deviation of 15%. A colleague
suggests you add gold to your portfolio. Gold has an
expected return of 8%, a standard deviation of 25%, and
a correlation with your portfolio of −0.05. If the risk-free
rate is 2%, will adding gold improve your portfolio’s
Sharpe ratio?
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Alternative Example 11.13 (2 of 2)
Solution
– The beta of gold with your portfolio is
Gold Gold Your Portfolio
Gold
Your Portfolio
SD(R )Corr(R ,R ) 25%×-0.05
β = = =-.08333
SD(R ) 15%
– The required return that makes gold an attractive
addition to your portfolio is
Gold f Gold Your Portfolio f
r =r +β (E[R ]-r )=2%-.08333×(8%-2%)=2.50%
– Because the expected return of 8% exceeds the
required return of 2.5%, adding gold to your portfolio will
increase your Sharpe ratio.
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Expected Returns and the Efficient
Portfolio
• Expected Return of a Security
eff
i i f i eff f
E[R ] = r º r + β × (E[R ] - r )
– A portfolio is efficient if and only if the expected return of
every available security equals its required return
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Textbook Example 11.14 (1 of 3)
Identifying the Efficient Portfolio
Problem
Consider the Omega Fund and real estate fund of Example
11.13. Suppose you have $100 million invested in the Omega
Fund. In addition to this position, how much should you invest
in the real estate fund to form an efficient portfolio of these
two funds?
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Textbook Example 11.14 (2 of 3)
Solution
• Suppose that for each $1 invested in the Omega Fund, we
borrow re
x dollars (or sell re
x worth of treasury bills) to invest
in the real estate fund. Then our portfolio has a return of P
R
0 re re f
=R +x R -r
( ), where o
R is the return of the Omega
Fund and re
R is the return of the real estate fund. Table 11.5
shows the change to the expected return and volatility of
our portfolio as we increase the investment re
x in the real
estate fund, using the formulas
2
p O re re O re re re re O
X X X
Var(R )=Var[R + (R -rf)]=Var(R )+ Var(R )+2 Cov(R ,R )
p O re re f
X
E[R ]=E[R ]+ (E[R ]-r )
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Textbook Example 11.14 (3 of 3)
• Adding the real estate fund initially improves the Sharpe ratio of the
portfolio, as defined by Eq. 11.17. As we add more of the real estate fund,
however, its correlation with our portfolio rises, computed as
re p re O re re f
re p
re p re p
X
Cov(R ,R ) Cov(R ,R + (R -r ))
Corr(R ,R )= =
SD(R )SD(R ) SD(R )SD(R )
re re re O
re p
X Var(R )+Cov(R ,R )
=
SD(R )SD(R )
• The beta of the estate fund—computed from Eq. 11.19—also rise,
increasing the required return. The required return equals the 9%
expected return of the estate fund at about re
x = 11%, which is the same
level of investment that maximizes the Sharpe ratio. Thus, the efficient
portfolio of these two funds includes $0.11 in the real estate fund per $1
invested in the Omega Fund.
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Table 11.5 Sharpe Ratio and Required Return for
Different Investments in the Real Estate Fund
X sub r e
E left bracket R sub P right bracket
S D left parenthesis R sub P right parenthesis
Sharpe
Ratio
Corr
left parenthesis R sub r e, R sub P right parenthesis
Beta super P sub r e
Required
return
r sub r e
0% 15.00% 20.00% 0.6000 10.0% 0.18 5.10%
4% 15.24% 20.19% 0.6063 16.8% 0.29 6.57%
8% 15.48% 20.47% 0.6097 23.4% 0.40 8.00%
10% 15.60% 20.65% 0.6103 26.6% 0.45 8.69%
11% 15.66% 20.74% 0.6104 28.2% 0.48 9.03%
12% 15.72% 20.84% 0.6103 29.7% 0.50 9.35%
16% 15.96% 21.30% 0.6084 35.7% 0.59 10.60%
re
X p
E[R ] ( )
p
SD R re p
(R ,R )
P
re
β
re
r
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11.7 The Capital Asset Pricing Model
• The Capital Asset Pricing Model (CAPM) allows us to
identify the efficient portfolio of risky assets without having
any knowledge of the expected return of each security.
• Instead, the CAPM uses the optimal choices investors
make to identify the efficient portfolio as the market
portfolio, the portfolio of all stocks and securities in the
market.
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The CAPMAssumptions (1 of 3)
• Three Main Assumptions
– Assumption 1
Investors can buy and sell all securities at
competitive market prices (without incurring taxes or
transaction costs) and can borrow and lend at the
risk-free interest rate.
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The CAPMAssumptions (2 of 3)
• Three Main Assumptions
– Assumption 2
Investors hold only efficient portfolios of traded
securities—portfolios that yield the maximum
expected return for a given level of volatility.
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The CAPMAssumptions (3 of 3)
• Three Main Assumptions
– Assumption 3
Investors have homogeneous expectations
regarding the volatilities, correlations, and expected
returns of securities.
Homogeneous Expectations
– All investors have the same estimates concerning
future investments and returns.
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Supply, Demand, and the Efficiency of
the Market Portfolio
• Given homogeneous expectations, all investors will demand
the same efficient portfolio of risky securities.
• The combined portfolio of risky securities of all investors
must equal the efficient portfolio.
• Thus, if all investors demand the efficient portfolio, and the
supply of securities is the market portfolio, the demand for
market portfolio must equal the supply of the market
portfolio.
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Textbook Example 11.15 (1 of 2)
Portfolio Weights and the Market Portfolio
Problem
Suppose that after much research, you have identified the
efficient portfolio. As part of your holdings, you have decided
to invest $10,000 in Microsoft, and $5000 in Pfizer stock.
Suppose your friend, who is a wealthier but more
conservative investor, has $2000 invested in Pfizer. If your
friend’s portfolio is also efficient, how much has she invested
in Microsoft? If all investors are holding efficient portfolios,
what can you conclude about Microsoft’s market
capitalization, compared to Pfizer’s?
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Textbook Example 11.15 (2 of 2)
Solution
Because all efficient portfolios are combination of the risk-free
investment and the tangent portfolio, they share the same
proportions of risky stocks. Thus, since you have invested
twice as much in Microsoft as in Pfizer, the same must be true
for your friend; therefore, she has invested $4000 in Microsoft
stock. If all investors hold efficient portfolios, the same must
be true of each of their portfolios. Because, collectively, all
investors own all shares of Microsoft and Pfizer, Microsoft’s
market capitalization must therefore be twice that of Pfizer’s.
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Optimal Investing: The Capital Market
Line (1 of 2)
• When the CAPM assumptions hold, an optimal portfolio is a
combination of the risk-free investment and the market
portfolio.
– When the tangent line goes through the market portfolio,
it is called the capital market line (CML ).
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Optimal Investing: The Capital Market
Line (2 of 2)
• The expected return and volatility of a capital market line
portfolio are
xCML f Mkt f Mkt f
E[R ] = (1 - x)r + xE[R ] = r + x(E [R ] - r )
xCML Mkt
SD(R ) = xSD(R )
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Figure 11.11 The Capital Market Line
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11.8 Determining the Risk Premium
• Market Risk and Beta
– Given an efficient market portfolio, the expected return
of an investment is
i i f i Mkt f
Risk premium for security i
E[R ] = r = r + β (E[R ] - r )
– The beta is defined as follows:
Volatility of i that is common with the market
i i Mkt i Mkt
i
Mkt Mkt
SD(R ) × Corr(R ,R ) Cov(R ,R )
β = =
SD(R ) Var(R )
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Textbook Example 11.16 (1 of 3)
Computing the Expected Return for a stock
Problem
Suppose the risk-free return is 4% and the market portfolio
has an expected return of 10% and a volatility of 16%. 3M
stock has a 22% volatility and a correlation with the market of
0.50. What is 3M’s beta with the market? What capital market
line portfolio has equivalent market risk, and what is the
expected return?
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Textbook Example 11.16 (2 of 3)
Solution
• We can compute beta using Eq. 11.23:
MMM MMM Mkt
MMM
Mkt
SD(R )Corr(R ,R ) 22%×0.50
β = = =0.69
SD(R ) 16%
• That is, for each 1% move of the market portfolio, 3M stock
tends to move 0.69%. We could obtain the same sensitivity
to market risk by investing 69% in the market portfolio, and
31% in the risk-free security. Because it has the same
market risk, 3M’s stock should have the same expected
return as this portfolio, which is (using Eq. 11.15 with
x= 0.69),
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Textbook Example 11.16 (3 of 3)
MMM f Mkt f
X
E[R ] = r + (E[R ]-r )=4%+0.69(10%-4%)
= 8.1%
• Because MMM
x =β , this calculation is precisely the CAPM
Eq. 11.22. Thus, investors will require an expected return
of 8.1% to compensate for the risk associated with 3M
stock.
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Alternative Example 11.16 (1 of 2)
• Problem
– Assume the risk-free return is 5% and the market
portfolio has an expected return of 12% and a standard
deviation of 44%.
– ATP Oil and Gas has a standard deviation of 68% and a
correlation with the market of 0.91.
– What is ATPM’s beta with the market?
– Under the CAPM assumptions, what is its expected
return?
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Alternative Example 11.16 (2 of 2)
• Solution
i i Mkt
i
Mkt
SD(R )×Corr(R ,R ) (.68)(.91)
β = = =1.41
SD(R ) .44
Mkt
i f i Mkt f
E[R ]=r +β (E[R ]-r )=5%+1.41(12% - 5%)=14.87%
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Textbook Example 11.17 (1 of 2)
A Negative-Beta stock
Problem
Suppose the stock of Bankruptcy Auction Services, Inc.(BAS),
has a negative beta of −0.30. How does its expected return
compare to the risk-free rate, according to the CAMP? Does
this result make sense?
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Textbook Example 11.17 (2 of 2)
Solution
• Because the expected return of the market is higher than the risk-
free rate. For example, if the risk-free rate is 4% and the
expected return on the market is 10%
BAS
E[R ]=4%-0.30(10%-4%)=2.2%
• This result seems odd: Why would investors be willing to accept
a 2.2% expected return on this stock when they can invest in a
safe investment and earn 4%? A savvy investor will not hold BAS
alone; instead, she will hold it in combination with other securities
fall, BAS portfolio. Because BAS will tend to rise when the market
and most other securities as part of a well-diversified provides
“recession insurance’’ for the portfolio, and investors pay for the
insurance by accepting an expected return below the risk-free
rate.
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Alternative Example 11.17 (1 of 2)
Problem
– Assume you own a security with a beta of 1.5 and you
have found a security that has a beta of -1.5. How
much market risk could be eliminated by adding this
security to create a two-security portfolio? Conceptually,
what would be the expected return of this portfolio?
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Alternative Example 11.17 (2 of 2)
Solution
– A portfolio with 50% in a security with a beta of 1.5 and
50% in another security with a beta of -1.5% would
– yield a portfolio beta of 0, indicating an elimination of
the market risk.
– The expected return on a portfolio with a beta of 0 will
be equal to the risk-free rate.
i f Portfolio Mkt f f Mkt f f
E[R ]=r +β (E[R ]-r )=r +0×(E[R ]-r )=r
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The Security Market Line (1 of 2)
• There is a linear relationship between a stock’s beta and its
expected return (See figure on next slide).
• The security market line (SML) is graphed as the line
through the risk-free investment and the market.
– According to the CAMP
, if the expected return and beta
for individual securities are plotted, they should all fall
along the SML.
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Figure 11.12 The Capital Market Line
and the Security Market Line
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Figure 11.12 The Capital Market Line
and the Security Market Line, Panel (a)
(a) The CML depicts portfolios combining the risk-free investment and the efficient
portfolio, and shows the highest expected return that we can attain for each level
of volatility. According to the CAMP
, the market portfolio is on the CML and all other
stocks and portfolios contain diversifiable risk and lie to the right of the CML, as
illustrated for Exxon Mobil (XOM ).
126. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Figure 11.12 The Capital Market Line
and the Security Market Line, Panel (b)
(b) The SMLshows the expected return for each security as a function of
its beta with the market. According to the CAMP, the market portfolio is
efficient, so all stocks and portfolios should lie on the SML
127. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
The Security Market Line (2 of 2)
• Beta of a Portfolio
– The beta of a portfolio is the weighted average beta of
the securities in the portfolio.
i i Mkt
i
P Mkt i Mkt
P i i i
i i
Mkt Mkt Mkt
X
X X
Cov R ,R
Cov(R ,R ) Cov(R ,R )
β = = = = β
Var(R ) Var(R ) Var(R )
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Textbook Example 11.18 (1 of 3)
Textbook Example 11.18
Problem
Suppose Kraft Foods’ stock has a beta of 0.50, whereas
Boeing’s beta is 1.25. If the risk-free rate is 4%, and the
expected return of the market portfolio is 10%, what is the
expected return of an equally weighted portfolio of Kraft
Foods and Boeing stocks, according to the CAPM?
129. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Textbook Example 11.18 (2 of 3)
Solution
• We can compute the expected return of the portfolio in two
ways. First, we can use the SML to compute the expected
return of Kraft Foods (KFT) and Boeing (BA) separately:
KFT f KFT Mkt f
E[R ]=r +β (E[r ]-r )=4%+0.50(10%-4%)=7.0%
BA f BA Mkt f
E[R ]=r +β (E[R ]-r )=4%+1.25(10%-4%)=11.5%
• Then, the expected return of the equally weighted portfolio
P is
P KFT BA
1 1 1 1
E[R ]= E[R ]+ E[R ]= (7.0%)+ (11.5%)=9.25%
2 2 2 2
130. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Textbook Example 11.18 (3 of 3)
• Alternatively, we can compute the beta of the portfolio using
Eq. 11.24:
P KFT BA
1 1 1 1
β = β + β = (0.50)+ (1.25)=0.875
2 2 2 2
• We can then find the portfolio’s expected return from the
SML:
P f P Mkt f
E[R ]=r +β (E[R ]-r )=4%+0.875(10%-4%)=9.25%
131. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Alternative Example 11.18 (1 of 2)
Problem
– Suppose the stock of the 3M Company (MMM) has a
beta of 0.69 and the beta of Hewlett-Packard Co. (HPQ)
stock is 1.77.
– Assume the risk-free interest rate is 5% and the
expected return of the market portfolio is 12%.
– What is the expected return of a portfolio of 40% of
3M stock and 60% Hewlett-Packard stock, according
to the CAPM?
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Alternative Example 11.18 (2 of 2)
Solution
P i i
i
X
β = β =(.40)(0.69)+(.60)(1.77)=1.338
Mkt
i f i Portfolio f
E[R ]=r +β (E[R ]-r )
i
E[R ]=5%+1.338(12%-5%)=14.37%
133. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Summary of the Capital Asset Pricing
Model
• The market portfolio is the efficient portfolio.
• The risk premium for any security is proportional to its beta
with the market.
134. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Chapter Quiz (1 of 2)
1. How do you calculate a portfolio return?
2. Conceptually, how are covariance and correlation
different?
3. How does adding stocks to a portfolio affect its volatility?
4. What is the efficient frontier?
5. What is the Sharpe ratio, and what does it measure?
6. How does beta affect an investment’s cost of capital?
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Chapter Quiz (2 of 2)
7. Is the market portfolio efficient? Justify your answer.
8. How are the capital market line and security market line
similar? How are they different?
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Appendix (1 of 3)
• The CAPM with Differing Interest Rates.
– If borrowing and lending rates differ, then investors with
different preferences will choose different portfolios of
risky securities.
– With different saving and borrowing rates, the CAPM
conclusion that the market portfolio is the unique
efficient portfolio of risky investments is no longer valid.
138. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Appendix (2 of 3)
Figure 11A.1 The CAMP with Different Saving and
Borrowing Rates
139. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
Appendix (3 of 3)
• The Security Market Line with Differing Interest Rates
– Determination of the security market line depends only
on the market portfolio being tangent for some interest
rate; the SMLstill holds in the following form:
* *
i i Mkt
E[R ]=r +β ×(E[R ]-r )
– The SML holds with some rate *
S B
r between r and r in
f
place of r , as illustrated in Figure 11A.1
140. Copyright © 2020, 2017, 2014 Pearson Education, Inc. All Rights Reserved
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Editor's Notes If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed:
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Slides in this presentation contain hyperlinks. JAWS users should be able to get a list of links by using INSERT+F7 1. Year 2013
North Air (Stock Returns): 21 percent
West Air (Stock Returns): 9 percent
Tex Oil (Stock Returns): minus 2 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): 15 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 3.5 percent
2. Year 2014
North Air (Stock Returns): 30 percent
West Air (Stock Returns): 21 percent
Tex Oil (Stock Returns): minus 5 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): 25.5 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 8 percent
3. Year 2015
North Air (Stock Returns): 7 percent
West Air (Stock Returns): 7 percent
Tex Oil (Stock Returns): 9 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): 7 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 8 percent
4. Year 2016
North Air (Stock Returns): minus 5 percent
West Air (Stock Returns): minus 2 percent
Tex Oil (Stock Returns): 21 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): minus 3.5 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 9.5 percent
5. Year 2017
North Air (Stock Returns): minus 2 percent
West Air (Stock Returns): minus 5 percent
Tex Oil (Stock Returns): 30 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): minus 3.5 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 12.5 percent
6. Year 2018
North Air (Stock Returns): 9 percent
West Air (Stock Returns): 30 percent
Tex Oil (Stock Returns): 7 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): 19.5 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 18.5 percent
7. Average return
North Air (Stock Returns): 10 percent
West Air (Stock Returns): 10 percent
Tex Oil (Stock Returns): 10 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): 10 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 10 percent
8. Volatility
North Air (Stock Returns): 13.4 percent
West Air (Stock Returns): 13.4 percent
Tex Oil (Stock Returns): 13.4 percent
1 by 2RN plus 1 by 2Rw (Portfolio Returns): 12.1 percent
1 by 2Rw plus 1 by 2RT (Portfolio Returns): 5.1 percent A horizontal bar depicts the degrees of correlation at different points. The horizontal bar displays the degrees from left to right as follows. At the left extreme, perfectly negatively correlated and the variables always move oppositely. Between negative 1 and 0. Tend to move oppositely. 0, Uncorrelated and there is no tendency between the two variables. Between 0 and positive 1, variables tend to move together. Positive 1 at the right extreme, perfectly positively correlated, and variables always move together. Between 0 and positive 1, Variables tend to move together. Positive 1 at the right extreme, perfectly positively correlated, and variables always move together. 1. Year: 20 13
R sub N minus R bar sub N) (Deviation from Mean): 11 percent
(R sub w minus R bar sub w) (Deviation from Mean): negative 1 percent
(R sub T minus R bar sub T) (Deviation from Mean): negative 12 percent
(R sub N minus R bar sub N) (R sub W minus R bar sub w) (North Air and West Air): negative 0.0011
(R sub w minus R bar sub w) (R sub T minus R bar sub T) (West Air and Tex Oil): 0.0012
2. Year: 20 14
R sub N minus R bar sub N) (Deviation from Mean): 20 percent
(R sub w minus R bar sub w) (Deviation from Mean): 11 percent
(R sub T minus R bar sub T) (Deviation from Mean): negative 15 percent
(R sub N minus R bar sub N) (R sub W minus R bar sub w) (North Air and West Air): negative 0.0220
(R sub w minus R bar sub w) (R sub T minus R bar sub T) (West Air and Tex Oil): negative 0.0165
3. Year: 20 15
R sub N minus R bar sub N) (Deviation from Mean): negative 3 percent
(R sub w minus R bar sub w) (Deviation from Mean): negative 3 percent
(R sub T minus R bar sub T) (Deviation from Mean): negative 1 percent
(R sub N minus R bar sub N) (R sub W minus R bar sub w) (North Air and West Air): 0.0009
(R sub w minus R bar sub w) (R sub T minus R bar sub T) (West Air and Tex Oil): 0.0003
4. Year: 20 16
R sub N minus R bar sub N) (Deviation from Mean): negative 15 percent
(R sub w minus R bar sub w) (Deviation from Mean): negative 12 percent
(R sub T minus R bar sub T) (Deviation from Mean): 11 percent
(R sub N minus R bar sub N) (R sub W minus R bar sub w) (North Air and West Air): 0.0180
(R sub w minus R bar sub w) (R sub T minus R bar sub T) (West Air and Tex Oil): negative 0.0132
5. Year: 20 17
R sub N minus R bar sub N) (Deviation from Mean): negative 12 percent
(R sub w minus R bar sub w) (Deviation from Mean): negative 15 percent
(R sub T minus R bar sub T) (Deviation from Mean): 20 percent
(R sub N minus R bar sub N) (R sub W minus R bar sub w) (North Air and West Air): 0.0180
(R sub w minus R bar sub w) (R sub T minus R bar sub T) (West Air and Tex Oil): negative 0.0300
6. Year: 20 18
R sub N minus R bar sub N) (Deviation from Mean): negative 1 percent
(R sub w minus R bar sub w) (Deviation from Mean): 20 percent
(R sub T minus R bar sub T) (Deviation from Mean): negative 3 percent
(R sub N minus R bar sub N) (R sub W minus R bar sub w) (North Air and West Air): negative 0.0020
(R sub w minus R bar sub w) (R sub T minus R bar sub T) (West Air and Tex Oil): negative 0.0060
Sum = sum of start expression left parenthesis R sub start expression i, t end expression minus R bar sub i right parenthesis, left parethesis R sub start expression i, t end expression minus R bar sub j right parenthesis end expression from t = 0.0558 for North Air and West Air; and negative 0.0642 for West Air and Tex Oil.
Covariance of R sub i, R sub j = start fraction 1 over T minus 1 end fraction sum = 0.0112 for North Air and West Air; and negative 0.0128 for West Air and Tex Oil.
Correlation of R sub i, R sub j = start fraction Covariance of R sub i, R sub j over S D of R sub i times S D of R sub j end fraction = 0.624 for North Air and West Air; and negative 0.713 for West Air and Tex Oil. The y-axis shows the portfolio velocity from 0 to 50 percent in increments of 10. The y-axis shows the number of stocks from 1 to 1000. The curve starts from 40 percent at 1 stock and slowly falls down to 20 percent somewhere between 100 and 100 stocks and remains the same thereafter. The part of graph from 0 to 20 percent is marked as correlated (market) risk and the part from 40 to 20 percent is marked as elimination of diversifiable risk. The columns have the following headings from left to right. Portfolio weights, x sub I, Portfolio weights, x sub C, Expected return in %, E left bracket R sub P right bracket, Volatility in %, S D left bracket R sub P right bracket. The row entries are as follows. Row 1. 1, 0, 26, 50. Row 2. 0.8, 0.2, 22, 40.3. Row 3. 0.6, 0.4, 18, 31.6. Row 4. 0.4, 0.6, 14, 25. Row 5. 0.2, 0.8, 10, 22.4. Row 6. 0, 1, 6, 25.
The x-axis shows the volatility (standard deviation) from 0 to 60 percent in increments of 10. The y-axis shows expected return from 0 to 30 in increments of 5. The graph shows that the curve starts at (0, 1) for Coca-Cola, then goes to (0.2, 0.8) showing inefficient portfolios, then to (0.4, 0.6), then (0.6, 0.4), (0.8, 0.2) all showing efficient and last at (1, 0) for Intel. The x-axis shows the volatility (standard deviation) from 0 to 60 percent in increments of 10. The y-axis shows expected return from 0 to 30 in increments of 5. The graph shows that the curve starts at (0, 1) for Coca-Cola and ends at (1, 0) for Intel. The figure shows that when correlation is plus 1, there is a straight line between Coca-Cola and Intel and when the correlation is minus 1, it shows two lines connected on the y-axis, while all those in between are curved. The x axis plots the volatility, standard deviation, from 0 to 80 percent in increments of 10. The y axis plots expected return from 0 to 40 percent in increments of 5. The graph shows that the curve starts at (negative 0.5, 1.5) and ends at (1.5, negative 0.5). Short Intel, Long Coca Cola is marked at (negative 0.5, 1.5), Coca Cola is marked at next, from where the curve bends and marks Long Intel, Long Coca Cola, after which Long Intel, Short Coca Cola is marked at (1.2, negative 0.2). The details of the graph are as follows:
The x-axis shows the volatility (standard deviation) from 0 to 60 percent in increments of 10. The y-axis shows expected return from 0 to 30 in increments of 5. The graph shows three points: Bore at (25, 2); Coca-Cola at (25, 0.5); and Intel at (50, 26). A curve between Bore and Intel is marked B plus I. Another curve between Bore and Coca-Cola is marked B plus C. A between Coca-Cola and Intel is marked I plus C, and passes through a point (50 percent I, 50 percent C). A curve between Bore and the point (50 percent I, 50 percent C) is marked B plus (50 percent I, 50 percent C). All curves are concave downward in nature. The details of the graph are as follows:
The x-axis shows the volatility (standard deviation) from 0 to 60 percent in increments of 10. The y-axis shows expected return from 0 to 30 in increments of 5. The graph shows three points: Bore at (25, 2); Coca-Cola at (25, 0.5); and Intel at (50, 26). A concave downward curve from the point (22, 0) to the point (57, 30) is marked Efficient Frontier with 3 Stocks. The area under this curve and above the x-axis is shaded and marked as Portfolios of Intel plus Coca-Cola plus Bore (including short sales).
The x-axis shows the volatility (standard deviation) from 0 to 40 percent in increments of 10. The y-axis shows expected return from 0 to 15 in increments of 5. The graph shows the points Tiffany at (35, 13.5); GE at (28, 11.5); Apple at (33, 9); Amazon at (38, 8); Nike at (22.5, 6.5); IBM at (18, 5.5); Molson-Coors at (27, 4.8); McDonald’s at (14.5, 4.8); Walmart at (16.5, 3); Newmont Mining at (32.5, 3). A concave downward curve from the point (0, 14) to the point (36, 15); turning at the point (10, 3.5) is marked as Efficient Frontier with all 10 Stocks. Another concave downward curve from the point (0, 24) to the point (49.5, 15) that turns at the point (14.4, 5) is marked as Efficient Frontier with Amazon, GE, and McDonald’s.
S D of R sub start expression x P end expression = square root of start expression left parenthesis 1 minus x right parenthesis squared times V a r of r sub f plus x squared V a r of R sub p plus 2 left parenthesis 1 minus x right parenthesis times x times C o v left parenthesis r sub f, R sub p right parenthesis = square root of start expression x squared v a r of R sub p end expression. = x S D of R sub p. left parenthesis 1 minus x right parenthesis squared times V a r of r sub f and x times C o v left parenthesis r sub f comma R sub p right parenthesis are labeled, 0. The x-axis shows the volatility (standard deviation) from 0 to 20 percent in increments of 2. The y-axis shows expected return from 0 to 25 in increments of 5. A concave downward curve starting from the point (0, 14.2) to the point (19.8, 25) turns at the point P (8, 10). A line from the point (0, 5) is marked Investing in P and the Risk-Free Investment. The point (0, 5) is marked Risk-free investment, the point (4, 7.5) is marked x equals 50 percent, and the point P (8, 10) is marked x equals 100 percent. Another line from the point P (8, 10) to the point (16, 16) is marked Buying P on Margin. The point (16, 16) is marked x equals 200 percent, the point (11.7, 12.5) is marked x equals 150 percent.
The x-axis shows the volatility (standard deviation) from 0 to 20 percent in increments of 2. The y-axis shows expected return from 0 to 25 in increments of 5. A concave downward curve starting from the point (0, 14.2) to the point (19.8, 25) turns at the point P (8, 10). A line starting from the point (0, 5) passes through the point P. Another line marked Efficient Frontier Including Risk-Free Investment starts from the point (0, 5) passes through the point (11, 17.5) on the curve. The point (0, 5) is marked as Risk-Free Investment, the point (11, 17.5) on the curve is marked Tangent or Efficient Portfolio. The curve is marked as Efficient Frontier of Risky Investments. The x-axis shows the volatility (standard deviation) from 0 to 40 percent in increments of 10. The y-axis shows expected return from 0 to 15 in increments of 5. The graph shows the points Tiffany at (35, 13.5); GE at (28, 11.5); Apple at (33, 9); Amazon at (38, 8); Nike at (22.5, 6.5); IBM at (18, 5.5); Molson-Coors at (27, 4.8); McDonald’s at (14.5, 4.8); Walmart at (16.5, 3); Newmont Mining at (32.5, 3). A concave downward curve from the point (0, 14) to the point (36, 15) turns at the point (10, 3.5) is marked as Efficient Frontier of All Risky Securities. 1. Capital market line
The x-axis shows the volatility (standard deviation) from 0 to 40 percent in increments of 10. The y-axis shows expected return from 0 to 15 in increments of 5. The graph shows the points Tiffany at (35, 13.5); GE at (28, 11.5); Apple at (33, 9); Amazon at (38, 8); Nike at (22.5, 6.5); IBM at (18, 5.5); Molson-Coors at (27, 4.8); McDonald’s at (14.5, 4.8); Walmart at (16.5, 3); Newmont Mining at (32.5, 3). A concave downward curve from the point (0, 14) to the point (36, 15) turns at the point (10, 3.5). A line starting at the point (0, 1) marked as T-bills, passes through the point (15, 7.5) on the curve is marked as capital market line. The point (15, 7.5) on the curve is marked as market portfolio equals efficient portfolio. The distance from the point (0, 4.8) to (7.5, 4.8) is marked as Market Risk of MCD Corr of (R sub MCD R sub Mkt) times SD of (R sub MCD). The distance between the point (7.5, 4.8) and (14.5, 4.8) is marked as Additional Diversifiable Risk of MCD. The distance from (0, 0) to (14.5, 0) is marked as Total Volatility of MCD SD (R sub MCD).
2. Security market line
The x-axis shows beta values ranging from negative 0.50 to 2.0 in increments of 0.5. The y-axis shows expected return from 0 to 15 percent in increments of 5. A line starting at the point (0, 1) marked as T-bills, passes through the point (1, 8.3) on the curve is marked as security market line. The point (1, 8.3) is marked as market portfolio. The distance between (0, 0) and (0.48, 0) is marked as beta of MCD. The y-axis shows expected return from 0 to 15 in increments of 5. The graph shows the points Tiffany at (35, 13.5); GE at (28, 11.5); Apple at (33, 9); Amazon at (38, 8); Nike at (22.5, 6.5); IBM at (18, 5.5); Molson-Coors at (27, 4.8); McDonald’s at (14.5, 4.8); Walmart at (16.5, 3); Newmont Mining at (32.5, 3). A concave downward curve from the point (0, 14) to the point (36, 15) turns at the point (10, 3.5). A line starting at the point (0, 1) marked as T-bills, passes through the point (15, 7.5) on the curve is marked as capital market line. The point (15, 7.5) on the curve is marked as market portfolio equals efficient portfolio. The distance from the point (0, 4.8) to (7.5, 4.8) is marked as Market Risk of MCD Corr of (R sub MCD R sub Mkt) times SD of (R sub MCD). The distance between the point (7.5, 4.8) and (14.5, 4.8) is marked as Additional Diversifiable Risk of MCD. The distance from (0, 0) to (14.5, 0) is marked as Total Volatility of MCD SD (R sub MCD).
The y-axis shows expected return from 0 to 15 percent in increments of 5. A line starting at the point (0, 1) marked as T-bills, passes through the point (1, 8.3) on the curve is marked as security market line. The point (1, 8.3) is marked as market portfolio. The distance between (0, 0) and (0.48, 0) is marked as beta of MCD.
If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed:
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Slides in this presentation contain hyperlinks. JAWS users should be able to get a list of links by using INSERT+F7 The x-axis shows the volatility (standard deviation) from 0 to 40 percent in increments of 10. The y-axis shows expected return from 0 to 14 percent in increments of 2. A concave downward curve from the point (40, 1.4) to the point (40, 14) turns at the point (12.5, 8) is marked as Efficient Frontier of Risky Investments. A line starting at the point (0, 3) passing through the point T sub s at (14, 8.8) on the curve is marked as T sub s and saving. A line from the point (0, 6) passing through the point T sub B (19, 10) on the curve is marked as T sub B and borrowing. A line from the point (0, 4.8) passed through the point (15.2, 9.2). The point (15.2, 9.2) is marked market portfolio.