CHAPTER

8 8-1

PORTFOLIO THEORY AND THE

CAPITAL ASSET PRICING MODEL

Brealey, Myers, and Allen

Principles of Corporate Finance
13th Edition
Slides by Matthew Will
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 Overview
8-2

• This is a very important chapter, as it deals with

portfolio theory and the capital asset pricing
model (CAPM).
• The concepts of efficient portfolios and the risk-
free asset are explained clearly.
• The chapter concentrates on the Markowitz
portfolio selection model and CAPM and builds
on the previous chapter.
• Other theories like arbitrage pricing theory
(APT) and the three-factor model are discussed.
• These form the basis for understanding the
trade-off between risk and return in theory and
in practice
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 Learning Objectives
8-3

• To understand the portfolio theory and the

related concepts of efficient portfolios, the
market portfolio, and the risk-free asset.
• To understand CAPM, the security market line
(SML), and its implications for risk-return trade-
offs.
• To understand other theories like the arbitrage
pricing theory and the three-factor model.

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 Topics Covered
8-4

• Harry Markowitz and the Birth of Portfolio

Theory
• The Relationship between Risk and Return
• Validity and Role of the the Capital Asset
Pricing Model
• Some Alternative Theories

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 Topics Covered
8-5

• Harry Markowitz and the Birth of Portfolio Theory

➢Starts by taking a close look at daily returns and
compares them with the normal distribution
➢The risk and return of a portfolio is reviewed
➢The idea of an efficient portfolio is introduced
➢Shows that when risk-free borrowing or lending is
allowed, the investor can easily identify the best
portfolio of risky assets
• The Relationship between Risk and Return
• Validity and Role of the the Capital Asset Pricing
Model
• Some Alternative Theories
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 Topics Covered
8-6

• The Relationship between Risk and Return

➢ Describes the security market line SML and shows it can be
used to estimate the required rate of return on a common
stock
➢ Describes the capital asset pricing model CAPM and its
usefulness
• Validity and Role of the the Capital Asset Pricing
Model
• Some Alternative Theories

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 Topics Covered
8-7

• Validity and Role of the the Capital Asset Pricing Model

➢ Reviews assumptions underlying the CAPM and provides
the empirical evidence supporting it
➢ An important point is that while the CAPM is not perfect, it
does seem to capture important elements of risk
➢ Worth emphasizing the role of the market portfolio in the
CAPM
➢ For any efficient portfolio, the expected risk premium on
each stock (r-rf) must be proportional to its beta measured
relative to that efficient portfolio
➢ If the market portfolio is efficient, then (r-rf) will be
proportional to beta measure relative to the market portfolil
(i.e, the CAPM will hold).
• Some Alternative Theories
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 Topics Covered
8-8

• Validity and Role of the the Capital Asset Pricing

Model
➢CAPM is equivalent to the statement that the market
portfolio is mean-variance efficient. This makes the model
difficult to test conclusively.
➢ The market portfolio should, in principle, included all
assets; the available market indexes fall short of that ideal.
• Some Alternative Theories

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 Topics Covered
8-9

• Validity and Role of the the Capital Asset Pricing Model

➢Assumptions behind the CAPM are:
o Investors maximize the expected utility of terminal wealth at
the end of one period.
o Investors choose between portfolios on the basis of the
mean and variance or return. This implies that investors
have quadratic utility function or (more realistically) that
returns are normally distributed.
o Investors can borrow or lend at an exogenously (i.e. derived
externally) determined rfr of interest
o There are no restriction on short selling.
o Investors have homogeneous probability beliefs.
o The market is perfect. All assets are marketable, no
transaction costs or taxes, and all investors are price takers.
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 Topics Covered
8-10

• Some Alternative Theories/Models of risk and return

➢ Explain Arbitrage pricing theory (APT) and three-factor
model
➢ Show comparative analysis of CAPM and APT
➢ Show comparison of the three-factor model and CAPM
➢ Point out the difficulties inherent in testing and using these
models in practice

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 Intro
8-11

• A thorough understanding of capital asset pricing model

(CAPM) and the security market line (SML) is essential for
understanding the trade-off between risk and return. These
have enormous applications in finance.
• This chapter deals with 1) portfolio theory, 2) capital asset
pricing model, 3) arbitrage pricing theory, and 4) three-factor
model.
• These form the basis for understanding the trade-off between
risk and return in theory and practice. Numerical examples
are shown for a good understanding of these topics.

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 Harry Markowitz and the Birth of Portfolio
Theory (1950s)
8-12

• Combining stocks into portfolios can reduce standard

deviation, below the level obtained from a simple
weighted average calculation. Correlation coefficients
make this possible
• The various weighted combinations of stocks that create
this standard deviation constitute a set of efficient
portfolios
• If the correlation coefficient < +1, then there is some
reduction in risk by forming portfolios.
• Maximum reduction in risk is possible if the correlation
coefficient is -1.
• By changing the proportion of funds invested in stocks,
one can change the risk-return characteristics of a
portfolio.
• Efficient portfolios provide the highest return for a given
level of risk or the least risk for given level of returns.
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 Figure 8.1 Daily Price Changes for IBM
8-13

Price changes vs. Normal distribution

1997–2017

A histogram of daily price changes resembles a normal distribution.

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 Figure 8.2 Investment A
8-14

Standard Deviation vs. Expected Return

The next three slides show the distribution of returns for three different stocks. While
each has a different standard deviation, they are all normally distributed. This is a
relatively realistic condition; however, reality is more lognormal than normal. Either
way, it does not change the validity of the following theory.
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 Figure 8.2 Investment B
8-15

Standard Deviation vs. Expected Return

This is the second of three histogram examples related to returns.

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 Figure 8.2 Investment C
8-16

Standard Deviation vs. Expected Return

This is the third of three histogram examples related to returns.

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 Figure 8.3 Southwest and Amazon
8-17

The curved line

illustrates how
expected return
and standard
deviation change
as you hold
different
combinations of
two stocks.
Diversification
reduces risk.

The slide shows the relationship between risk and return for a portfolio of two stocks. This takes
the math learned in the previous chapter to a new level of understanding. Recall in the prior
chapter the risk-reducing ideas.
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 Effect of Correlation Coefficient on
Portfolio Risk
8-18

Southwest Amazon Ave Correlation Portfolio Portfolio

Airlines Return Coefficient Variance SD
Proportion 60% 40%
Expected 15% 10% 13%
Return
Standard 26.6% 27.9% 1.0 749.7 27.4%
Deviation
0.26 486.1 22.0%
-1.0 37.2 6.1%

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 Table 8.1 Examples of Efficient Portfolios
Chosen from 10 Stocks
8-19

42.2
The table displays another big-picture example of how diversification can reduce risk.
At this point, the math from the prior chapter should be much clearer.
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 Figure 8.4 Efficient Portfolios
8-20

Each dot shows the expected return and standard deviation of

stocks. These are efficient portfolios, denoted A, B, and C.
Newmont

J&J

An efficient portfolio is the one that gives the highest return for a given level of risk. The set of
efficient portfolios is called the efficient frontier. These are all risky portfolios. There are three
efficient portfolios all generated from the same ten stocks.
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 Figure 8.5 Lending and Borrowing
8-21

Lending or borrowing at the risk-free rate (rf) allows us to

exist outside the efficient frontier.

By introducing a risk-free asset (rf) that allows investors to lend or borrow at the risk-free rate,
one can expand the risk-return opportunities available for investment. The risk-free asset plots
on the y-axis. One can invest along the line rf T and beyond by lending and borrowing at the
risk-free rate. But the tangential line along rf S constitutes the new efficient frontier that is
better than the earlier frontier without the risk-free asset. The new efficient frontier provides
higher returns for a given level of risk at all points except at the tangential point S.
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 Efficient Frontier Example
8-22

Example Correlation coefficient = .19

Stocks  % of Portfolio Avg Return
JNJ 13.2 60% 15%
Ford 31.0 40% 21%

Standard deviation = weighted avg = 20.3% wrong

Standard deviation = portfolio = 15.9 % CORRECT
Return = weighted avg = portfolio = 12.3%

This slide illustrates an example in the same format as the prior chapter. The goal is to draw
a link between the prior math and the Markowitz theory. The upcoming slides are actually
reproductions of the prior chapter and used for the same purpose.

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 Efficient Frontier Example Continued
8-23

Example Correlation coefficient = .4

Stocks  % of Portfolio Avg Return
ABC Corp 28 60% 15%
Big Corp 42 40% 21%
Standard deviation = weighted avg = 33.6% wrong
Standard deviation = portfolio = 28.1 % CORRECT
Return = weighted avg = portfolio = 17.4%
Additive standard deviation (common sense):
= .28 (60%) + .42 (40%) = 33.6% WRONG
Real standard deviation: = x 2 σ 2 + x 2 σ 2 + 2( x x ρ σ σ )
1 1 2 2 1 2 12 1 2

= .60 2.282 + .40 2.42 2 + 2(.6)(.4)(.4)(.28)(.42)

= .281 or 28.1% CORRECT

The slide is a reproduction from the prior chapter. The math is presented, not merely the results.
The goal is to strengthen the link between the math and the concept of portfolio theory.
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 Efficient Frontier Example 2
8-24

Previous Example Correlation coefficient = .3

Stocks  % of Portfolio Avg Return
Portfolio 28.1 50% 17.4%
New Corp 30 50% 19%

NEW standard deviation = weighted avg = 31.80 % wrong

NEW standard deviation = portfolio = 23.43 % CORRECT
NEW return = weighted avg = portfolio = 18.20%
NOTE: Higher return & lower risk
How did we do that? DIVERSIFICATION

As with the previous slides, this is a reproduction intended to help learn portfolio theory.

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 CAPM
8-25

• The CAPM is equivalent to the statement that the

market portfolio is mean-variance efficient.
• But it is difficult to test because the market
portfolio, in principle includes all assets; the
available market indexes clearly fall short of that
ideal

In the mid-1960s three economists- William Sharpe, John Lintner, and

Jack Treynor developed the CAPM theory separately.

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 CAPM
8-26

Assumption behind the CAPM:

• Investors maximize the expected utility of terminal wealth at
the end of one period.
• Investors choose between portfolios on the basis of the mean
and variance of return. This implies that investors have
quadratic utility functions or (more realistically) that returns are
normally distributed.
• Investors can borrow or lend at a exogenously determined risk-
free rate of interest.
• There are no restrictions on short selling.
• Investors have homogeneous probability beliefs.
• The market is perfect. Thus, all assets are marketable, there are
no transaction costs or taxes, and all investors are price takers.
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 Sharpe Ratio
8-27

Goal is to move
The ratio of the risk premium to up and left.
the standard deviation is called WHY?
the Sharpe ratio:

r - rf
Sharpe ratio =
s
The efficient ratio at the tangency point is better than all the others. Note
that it offers the highest ratio of risk premium to standard deviation.

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 The Relationship between Risk and Return
8-28

Security Market Line

Return
Market return = rm .
Market Portfolio

Risk-free return = rf
(Treasury bills)
Beta
By introducing a risk-free asset, a new efficient frontier is created. (The risk-free asset is
located on the y-axis). This line will give a higher return for every level of risk except at the
tangential point. A market portfolio is a tangential portfolio. The market portfolio is an
efficient portfolio and is also a risky portfolio.
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 Capital Asset Pricing Model
8-29

r – rf = +b (rm - rf )
CAPM
Every line has an equation. Recall from high school math the concept of y-intercept, slope,
and independent variables.
The equation to SML is: r = rf + (rm – rf).
Note that (r-rf) is the security risk premium and (rm-rf) is the market risk
premium.
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 Figure 8.6 Capital Asset Pricing Model
8-30

By changing the scale of the x-axis you can get the security market line. The new measure of
risk is beta. A market portfolio beta is assigned a value of 1. A risk-free asset has a beta of zero.
Risks of all other securities and portfolios are measured against this scale.

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 Table 8.2 Expected Returns
8-31

These estimates of the returns expected by investors in December

2017 were based on the capital asset pricing model. We assumed 2%
for the interest rate rf and 7% for the expected risk premium rm − rf.

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 Figure 8.7 Equilibrium
8-32

In equilibrium no stock can lie below the security market line. For example,
instead of buying stock A, investors would prefer to lend part of their money
and put the balance in the market portfolio. And instead of buying stock B,
they would prefer to borrow and invest in the market portfolio.

The difference between the return of the market and the risk free interest rate (often with Treasury bill as
proxy) = market risk premium (Rm-RFR) which since 1900 averaged 7.7% per year.
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 Basic principles of portfolio selection
8-33

1. Investors like high expected return and low standard deviation

• Common stock portfolios that offer the highest E(R) for a given sd →
efficient portfolios
2. If the investor can lend or borrow at the RFR, one efficient
portfolio is better than all the others: the portfolio that offers the
highest ratio of risk premium, i.e. r - RFR to sd
• A risk-averse investor will put part of his money in this efficient portfolio
and part in the risk-free asset
• A risk-tolerant investor may put all her money in this portfolio or she may
borrow and put in even more
3. The composition of this best efficient portfolio depends on the
investor’s assessments of expected returns, standard deviation
and correlations
• But suppose everybody has the same information and same
assessments. If there is no superior information, each investor should
hold the same portfolio as everybody else, i.e. everyone should hold the
market portfolio

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 Basic principles of portfolio selection
8-34

Risk of individual assets

4. Do no look at the risk of a stock in isolation but at its
contribution to portfolio risk
• This contribution depends on the stock’s sensitivity to changes in the
value of the portfolio
5. A stock’s sensitivity to changes in the value of the market
portfolio is known as beta.
• Beta, therefore, measures the marginal contribution of a stock to the
risk of the market portfolio

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 Figure 8.8 CAPM Test (1931–2017)
8-35

Beta vs. Average Risk Premium

Efficient markets will result in returns that plot near the security market line.
Dots show the actual average risk premiums from 10 portfolios with different betas vs. market
line
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 Figure 8.9 Relationship between Beta and
Average Return (mid-1960s) (1931-1965)
8-36

This presents the average risk premium versus beta. It is important to know the accuracy
of beta in predicting returns.

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 Figure 8.9 Relationship between Beta and
Average Return (1966–2017)
8-37

This is another example of the same; however, note the imperfections in beta and the
CAPM. While very good, it is not perfect.

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 Figure 8.10 Return vs. Book-to-Market
8-38

Value stocks high

BV/MV ratio
Growth stocks low
BV/MV ratio

Buy small market

capitalization, sell
large market
capitalization

Shows cumulative
difference between
returns

This is the updated version of the Fama-French Study showing the relationship between return and book
values.
Since 1926 average annual difference between small cap and large cap= 3.5% and between value stocks and
growth stocks = 4.8%
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
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 Figure 8.10 Return vs. Book-to-Market
8-39

Figure does not fit well with CAPM, which predicts that beta is the only reason that expected returns
differ. The plausibility of the CAPM theory will have to be weighed along with empirical “facts”.

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
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 Arbitrage Pricing Theory
8-40

Alternative to CAPM

Return = a + b1 (rfactor1 ) + b2 (rfactor 2 ) + b3 (rfactor 3 ) + .... + noise

Expected risk premium = r − rf

= b1 (rfactor1 − rf ) + b2 (rfactor 2 − rf ) + ...

This slide introduces the APT as a variation of CAPM. It is important to note that CAPM
is not abandoned but merely “improved.”

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 Arbitrage Pricing Theory
8-41

• APT does not ask which portfolios are efficient

➢ Instead, it starts by assuming that each stock’s return
depends partly on pervasive macroeconomic influences
or “factors” and partly on “noise”, i.e events that are
unique to that company
• The theory does not say what the factors are.
• For an individual stock there are two sources of
risk
➢ Risk that stems from pervasive macroeconomic factors
that cannot be eliminated by diversification
➢Risk arising from possible events that are specific to the
company
• The expected risk premium on a stock should
depend on the expected risk premium associated
with each factor and the stock’s sensitivity to each
of the factors (b1, b2, etc.)
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 Arbitrage Pricing Theory Continued
8-42

Estimated risk premiums for taking on risk factors

(1978–1990)
Estimated Risk Premium
Factor
(rfactor - rf )
Yield spread 5.10%
Interest rate –.61
Exchange rate –.59
Real GNP .49
Inflation –.83
Market 6.36
APT is derived assuming that stock returns are based partly on “factors” and partly on “noise.” For any
individual stock there are two sources of risk: the risk based on factors and the risk based on noise.
Diversification eliminates risk based on noise and only risk based on factors is relevant. This is an
important contribution.

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 Three-Factor Model
8-43

Steps to Identify Factors

1. Identify a reasonably short list of
macroeconomic factors that could affect
stock returns
2. Estimate the expected risk premium on
each of these factors (rfactor 1 − rf, etc.)
3. Measure the sensitivity of each stock to
the factors (b1, b2, etc.)

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 Three-Factor Model
8-44

Steps to Identify Factors 1- Applying Fama French 3 factor model

1. Identify a reasonably 1. Market factor (return on market index minus
short list of risk-free interest rate
macroeconomic factors 2. Size factor (return on small firm stocks less
that could affect stock return on large firm stocks
returns 3. Book to market factor (return on high book-
to market-ratio stocks less return on low
2. Estimate the expected book-to market- ratio stocks
risk premium on each of
these factors (rfactor 1 − rf,
etc.) 2- market risk premium = 7%, S-L cap = 3.5%.
H-L book to market ratios = 4.8%
3. Measure the sensitivity of
each stock to the factors
3- factor sensitivities of 10 industry groups for
(b1, b2, etc.) 60 months ending November 2014

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 Table 8.3 Estimates of Expected Equity Returns
Using Three-Factor Model and CAPM
8-45

<

<

<

<

This provides a comparative analysis of various industry returns using the three-factor
model and the CAPM.
Computer stocks? Largely because computer stocks are growth stocks with a low
exposure to the book-to market factor.

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 Table 8.3 Estimates of Expected Equity Returns
Using Three-Factor Model and CAPM
8-46

<

<

<

<

This Fama-French APT model is not widely used in practice to estimate cost of equity or WACC. It
requires 3 betas and 3 risk premiums vs. 1 for CAPM. All 3 APT betas are not as easy to predict and
interpret. Its widest use as a way of measuring performance of mutual funds, pension funds and
other professionally managed portfolios.

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 Arbitrage Pricing Theory
8-47

• CAPM is criticized because of

o The many unrealistic assumptions
o The difficulties in selecting a proxy for the market
portfolio as a benchmark
• An alternative pricing theory with fewer
assumptions was developed: Arbitrage Pricing
Theory (APT)

47
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Arbitrage Pricing Theory
8-48

• Three Major Assumptions:

o Capital markets are perfectly competitive
o Investors always prefer more wealth to less
wealth with certainty
o The stochastic process generating asset returns
can be expressed as a linear function of a set of
K factors or indexes
• In contrast to CAPM, APT does not assume
o Normally distributed security returns
o Quadratic utility function
o A mean-variance efficient market portfolio

48
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Arbitrage Pricing Theory
8-49

• The APT Model

E(Ri)=λ0+ λ1bi1+ λ2bi2+…+ λkbik
where:
λ0=the expected return on an asset with zero
systematic risk
λj=the risk premium related to the j th common
risk factor
bij=the pricing relationship between the risk
premium and the asset; that is, how
responsive asset i is to the j th common
factor
49
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 Empirical Tests of the APT
8-50

• The APT and Stock Market Anomalies

o Small-firm Effect
▪ Reinganum: Results inconsistent with the APT
▪ Chen: Supported the APT model over CAPM
o January Anomaly
▪ Gultekin and Gultekin: APT not better than
CAPM
▪ Burmeister and McElroy: Effect not captured by
model, but still rejected CAPM in favor of APT

50
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 Key Terms and Concepts
8-51

Portfolio theory
Efficient portfolio
Market risk premium
Risk-free asset
Capital asset pricing model (CAPM)
Security market line (SML)
Arbitrage pricing theory (APT)
Three-factor model

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 Challenge Areas
8-52

• Efficient frontier
• Security market line (SML)
• Capital asset pricing model (CAPM)
• Arbitrage pricing model (APT)

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 Additional Reference
8-53

Reilly, F. K. & Brown, K. C. (2012). Analysis of investments and management of

portfolios (10th international edition). Canada: South-Western, Cengage Learning.

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 CHAPTER
8 8-54

PORTFOLIO THEORY AND THE

CAPITAL ASSET PRICING MODEL

THE END
Brealey, Myers, and Allen
Principles of Corporate Finance
13th Edition
Slides by Matthew Will
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