Transcript R i

Chapter Ten
Arbitrage Pricing Theory and
Multifactor Models of Risk and
Return
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Single Factor Model
• Returns on a security come from two
sources:
– Common macro-economic factor
– Firm specific events
• Possible common macro-economic factors
– Gross Domestic Product Growth
– Interest Rates
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Single Factor Model Equation
𝑅𝑖 = 𝐸 𝑅𝑖 + β𝑖 𝐹 + 𝑒𝑖
Ri = Excess return on security
βi= Factor sensitivity or factor loading or factor
beta
F = Surprise in macro-economic factor
(F could be positive or negative but has
expected value of zero)
ei = Firm specific events (zero expected value)
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Multifactor Models
• Use more than one factor in addition to
market return
– Examples include gross domestic product,
expected inflation, interest rates, etc.
– Estimate a beta or factor loading for each
factor using multiple regression.
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Multifactor Model Equation
Ri  E Ri   iGDPGDP  iIR IR  ei
Ri = Excess return for security i
βGDP = Factor sensitivity for GDP
βIR = Factor sensitivity for Interest Rate
ei = Firm specific events
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Interpretation
The expected return on a security is the sum of:
1.The risk-free rate
2.The sensitivity to GDP times the risk
premium for bearing GDP risk
3.The sensitivity to interest rate risk times the
risk premium for bearing interest rate risk
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Arbitrage Pricing Theory
• Arbitrage occurs if there is a zero
investment portfolio with a sure profit.
Since no investment is required,
investors can create large positions to
obtain large profits.
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Arbitrage Pricing Theory
• Regardless of wealth or risk aversion,
investors will want an infinite position in
the risk-free arbitrage portfolio.
• In efficient markets, profitable arbitrage
opportunities will quickly disappear.
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APT & Well-Diversified Portfolios
RP = E (RP) + bPF + eP
F = some factor
• For a well-diversified portfolio, eP
– approaches zero as the number of securities in
the portfolio increases
– and their associated weights decrease
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Figure 10.1 Returns as a Function of the
Systematic Factor
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Figure 10.2 Returns as a Function of the
Systematic Factor: An Arbitrage Opportunity
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Figure 10.3 An Arbitrage Opportunity
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No-Arbitrage Equation of APT
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the APT, the CAPM and the Index
Model
APT
• Assumes a welldiversified portfolio,
but residual risk is still
a factor.
• Does not assume
investors are meanvariance optimizers.
• Uses an observable,
market index
• Reveals arbitrage
opportunities
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CAPM
• Model is based on an
inherently unobservable
“market” portfolio.
• Rests on mean-variance
efficiency. The actions of
many small investors
restore CAPM
equilibrium.
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Multifactor APT
• Use of more than a single systematic factor
• Requires formation of factor portfolios
• What factors?
– Factors that are important to performance
of the general economy
– What about firm characteristics?
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Two-Factor Model
• The multifactor APT is similar to the onefactor case.
𝑅𝑖 = 𝐸 𝑅𝑖 + β𝑖1 𝐹1 + β𝑖2 𝐹2 + 𝑒𝑖
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Two-Factor Model
• Track with diversified factor portfolios:
– beta=1 for one of the factors and 0 for
all other factors.
• The factor portfolios track a particular
source of macroeconomic risk, but are
uncorrelated with other sources of risk.
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Fama-French Three-Factor Model
• SMB = Small Minus Big (firm size)
• HML = High Minus Low (book-to-market ratio)
• Are these firm characteristics correlated with
actual (but currently unknown) systematic risk
factors?
Rit   i  iM RMt  iSMBSMBt  iHMLHMLt  eit
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The Multifactor CAPM and the APT
• A multi-index CAPM will inherit its risk
factors from sources of risk that a broad
group of investors deem important enough
to hedge
• The APT is largely silent on where to look
for priced sources of risk
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