Optimal Risky Portfolio, CAPM,
and APT
 Diversification
 Portfolio of Two Risky Assets
 Asset Allocation with Risky and Risk-free Assets
 Markowitz Portfolio Selection Model
 CAPM
 APT (arbitrage pricing theory)
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Top-down process
 Capital allocation between the risky portfolio and
risk-free assets
 Asset allocation across broad asset classes
 Security selection of individual assets within each
asset class
Chapter 1: Overview
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Diversification Effect
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Systematic risk v. Nonsystematic
Risk
 Systematic risk, (nondiversifiable risk or market
risk), is the risk that remains after extensive
diversifications
 Nonsystematic risk (diversifiable risk, unique risk,
firm-specific risk) – risks can be eliminated
through diversifications
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Two-Security Portfolio: Return
rp = w1r1 + w2r2
w1 = Proportion of funds in Security 1
w2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2
n
w 1
i 1
i
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Two-Security Portfolio: Risk
p2 = w1212 + w2222 + 2w1w2 Cov(r1r2)
12 = Variance of Security 1
22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
security 1 and security 2
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Covariance
Cov(r1r2) = 1,212
1,2 = Correlation coefficient of returns
1 = Standard deviation of
returns for Security 1
2 = Standard deviation of
returns for Security 2
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Correlation Coefficients: Possible Values
Range of values for 1,2
+ 1.0 >  > -1.0
If = 1.0, the securities would be
perfectly positively correlated
If = - 1.0, the securities would be
perfectly negatively correlated
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Three-Asset Portfolio
E (rp )  w1E (r1 )  w2 E (r2 )  w3 E (r3 )
 p2  w1212  w22 22  w32 32
 2w1w2 1, 2  2w1w3 1,3  2w2 w3 2,3
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Expected Return and Portfolio Weights
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Expected Return and Standard Deviation
Look at ρ=-1, 0 or 1.
Minimum Variance
Portfolio
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The Effect of Correlation
 The relationship depends on correlation
coefficient.
 -1.0 <  < +1.0
 The smaller the correlation, the greater the risk
reduction potential.
 If  = +1.0, no risk reduction is possible.
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The Minimum Variance Portfolio
 The minimum variance
portfolio is the portfolio
composed of the risky
assets that has the
smallest standard
deviation, the portfolio
with least risk.
 See footnote 4 on page
204.
7-13
 When correlation is less
than +1, the portfolio
standard deviation may
be smaller than that of
either of the individual
component assets.
 When correlation is -1,
the standard deviation of
the minimum variance
portfolio is zero.
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Optimal Portfolio
 Given a level of risk aversion, on can determine
the portfolio that provides the highest level of
utility.
 See formula on page 205.
 Note: no risk free asset is involved.
Chapter 1: Overview
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Capital Asset Line
 A graph showing all feasible risk-return
combinations of a risky and risk-free asset.
 See page 206 for possible CAL
 Optimal CAL – what is the objective function in
the optimization?
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Sharpe ratio
 Reward-to-volatility (Sharpe ratio)
 Page 206
Chapter 1: Overview
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Optimal CAL and the Optimal Risky Portfolio
Equation 7.13, page 207
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Exercises
 A pension fund manager is considering 3 mutual funds.
The first is a stock fund, the second is a long-term
government and corporate bond fund, and the third is a Tbill money market fund that yields a rate of 8%. The
probability distribution of the risky funds is as following:
Exp(ret)
Std Dev
Stock fund
20%
30%
bond Fund
12%
15%
The correlation between the fund returns is 0.10
Answer Problem 4 through 6, page 225.
Also see Example 7.2 (optimal risky portfolio) on page 208
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Determination of the Optimal Overall Portfolio
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Markowitz Portfolio Selection
 Generalize the portfolio construction problem to
the case of many risky securities and a risk-free
asset
 Steps



Get minimum variance frontier
Efficient frontier – the part above global MVP
An optimal allocation between risky and risk-free asset
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Minimum-Variance Frontier
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Capital Allocation Lines and Efficient Frontier
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Capital Asset Pricing Model
(CAPM)
 Harry Markowitz laid down the foundation of
modern portfolio theory in 1952. The CAPM was
developed by William Sharpe, John Lintner, Jan
Mossin in mid 1960s.
 It is the equilibrium model that underlies all modern
financial theory.
 Derived using principles of diversification with
simplified assumptions.
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Assumptions





Individual investors are price takers.
Single-period investment horizon.
Investments are limited to traded financial assets.
No taxes and transaction costs.
Information is costless and available to all
investors.
 Investors are rational mean-variance optimizers.
 There are homogeneous expectations.
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Resulting Equilibrium Conditions
 All investors will hold the same portfolio for risky assets –
market portfolio
 Market portfolio contains all securities and the proportion
of each security is its market value as a percentage of total
market value
 Risk premium on the market depends on the average risk
aversion of all market participants
 Risk premium on an individual security is a function of its
covariance with the market
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Figure 9.1 The Efficient Frontier and the Capital
Market Line
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CAPM
E(R)=Rf+β*(Rm-Rf)
i 
Cov(ri , rM )

2
M
  R  E (R)
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Security Market line and a Positive-Alpha Stock
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CAPM Applications: Index Model
 To move from expected to realized returns—use
the index model in excess return form:
Ri=αi+βiRM+ei
 The index model beta coefficient turns out to be
the same beta as that of the CAPM expected
return-beta relationship
 What would be the testable implication?
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Estimates of Individual Mutual Fund Alphas
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CAPM Applications: Market
Model
 Market model
Ri-rf=αi+βi(RM-rf)+ei
 Test implication: αi=0
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Is CAMP Testable?
 Is the CAPM testable
 Proxies
must be used for the market
portfolio
 CAPM is still considered the best available
description of security pricing and is widely
accepted
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Other CAPM Models: Multiperiod
Model
 Page 303
 Considering CAPM in the multi-period setting
 Other than comovement with the market portfolio,
uncertainty in investment opportunity and changes
in prices of consumption goods may affect stock
returns
 Equation (9.14)
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Other CAPM Models: Consumption Based
Model
 No longer consider the comovements in returns of
individual securities with returns of market
portfolios
 Key intuition: investors balance between today’s
consumption and the saving and investments that
will support future consumption
 Page 305; Equation (9.15)
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Liquidity and CAPM
 Liquidity – the ease and speed with which an
asset can be sold at fair market value.
 Illiquidity Premium


The discount in security price that results from
illiquidity is large
Compensation for liquidity risk – inanticipated
change in liquidity
 Research supports a premium for illiquidity.
Amihud and Mendelson and Acharya and
Pedersen
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Illiquidity and Average Returns
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APT
 Arbitrage Pricing Theory

This is a multi-factor approach in pricing stock
returns. See chapter 10
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Fama-French Three-Factor Model
The factors chosen are variables that on past
evidence seem to predict average returns well
and may capture the risk premiums (page 335)
rit=αi+βiMRMt+βiSMBSMBt+βiHMLHMLt+eit
Where:
SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess
of the return on a portfolio of large stocks
HML = High Minus Low, i.e., the return of a portfolio of stocks with a high
book to-market ratio in excess of the return on a portfolio of stocks with a low
book-to-market ratio
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/
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