CHAPTER 10
Arbitrage Pricing
Theory and
Multifactor
Models of Risk
and Return
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
10-2
Single-Factor Model
r i = αi + β i f + e i ,
ri = stock i’s return
f = a macro economic factor
βi = sensitivity of stock i’s return to the macro
economic factor
ei = return component due to stock specific events
10-3
Single-Factor Model in Alternate Form
(1)
ri = αi + βi f + ei ,
Taking expectation of (1), we have
(2)
E(ri)= E(αi + βi f + ei) = αi + βi E(f)
Subtract (2) from (1)
ri - E(ri) = βi f - βi E(f) + ei = βi [f - E(f)] + ei
ri = E(ri) + βi [f - E(f)] + ei
(10.1)
ri = E(ri) + βi F + ei
10-4
Single Factor Model Equation
ri  E (ri )  i F  ei
ri = Return for security i
βi = Factor sensitivity or factor loading or factor
beta
F = Surprise in macro-economic factor
(F could be positive, negative or zero)
ei = Firm specific events
10-5
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.
10-6
Multifactor Model Equation
ri = E(ri) +  iGDP GDP + i IR IR + ei
ri
= Return for security i
i GDP= Factor sensitivity for GDP
iIR = Factor sensitivity for Interest Rate
ei
= Firm specific events
10-7
Multifactor SML Models
E(r) = rf +  iGDPRPGDP +  iIRRPIR
= Factor sensitivity for GDP
GDP
i
RPGDP = Risk premium for GDP
i IR = Factor sensitivity for Interest Rate
RPIR = Risk premium for Interest Rate
10-8
Arbitrage Pricing Theory
Arbitrage - arises if an investor can construct a
zero investment portfolio with a sure profit
• Since no investment is required, an investor
can create large positions to secure large
levels of profit
• In efficient markets, profitable arbitrage
opportunities will quickly disappear
10-9
APT & Well-Diversified Portfolios
rP = E (rP) + PF + eP
F = some factor
• For a well-diversified portfolio:
eP approaches zero
Similar to CAPM,
10-10
Figure 10.1 Returns as a Function of the
Systematic Factor
10-11
Figure 10.2 Returns as a Function of the
Systematic Factor: An Arbitrage
Opportunity
10-12
Figure 10.3 An Arbitrage Opportunity
10-13
Figure 10.4 The Security Market Line
10-14
APT and CAPM Compared
• APT applies to well diversified portfolios and
not necessarily to individual stocks
• With APT it is possible for some individual
stocks to be mispriced - not lie on the SML
• APT is more general in that it gets to an
expected return and beta relationship without
the assumption of the market portfolio
• APT can be extended to multifactor models
10-15
Multifactor APT
• Use of more than a single factor
• Requires formation of factor portfolios
• What factors?
– Factors that are important to performance
of the general economy
– Fama-French Three Factor Model
10-16
Two-Factor Model
ri  E (ri )  i1F1  i 2 F2  ei
• The multifactor APR is similar to the onefactor case
– But need to think in terms of a factor portfolio
• Well-diversified
• Beta of 1 for one factor
• Beta of 0 for any other
10-17
Example of the Multifactor Approach
• Work of Chen, Roll, and Ross
– Chose a set of factors based on the ability
of the factors to paint a broad picture of the
macro-economy
10-18
Another Example:
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
rit  i  iM RMt  iSMB SMBt  iHML HMLt  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
10-19
The Multifactor CAPM and the APM
• 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
10-20