CHAPTER 10CHAPTER 10 Arbitrage Pricing Arbitrage Pricing Theory and Theory and Multifactor Multifactor Models of Risk Models of Risk and Returnand Return
10-2
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
10-3
Single-Factor Model
ri = αi + βi f + ei ,
ri = stock i’s return
f = a macro economic factorβi = sensitivity of stock i’s return to the macro
economic factorei = return component due to stock specific events
10-4
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
10-5
Single Factor Model Equation
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
( )i i i ir E r F e
10-6
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.
10-7
Multifactor Model Equation
ri = E(ri) + GDP GDP + IR IR + ei
ri = Return for security i
GDP= Factor sensitivity for GDP
IR = Factor sensitivity for Interest Rate
ei = Firm specific events
i
i
i
i
10-8
Multifactor SML Models
E(r) = rf + GDPRPGDP + IRRPIR
GDP = Factor sensitivity for GDP
RPGDP = Risk premium for GDP
IR = Factor sensitivity for Interest Rate
RPIR = Risk premium for Interest Rate
i i
i
i
10-9
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
10-10
APT & Well-Diversified Portfolios
rP = E (rP) + PF + eP
F = some factor
• For a well-diversified portfolio:
eP approaches zero
Similar to CAPM,
10-11
Figure 10.1 Returns as a Function of the Systematic Factor
10-12
Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage
Opportunity
10-13
Figure 10.3 An Arbitrage Opportunity
10-14
Figure 10.4 The Security Market Line
10-15
• 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
APT and CAPM Compared
10-16
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
10-17
Two-Factor Model
• The multifactor APR is similar to the one-factor 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
1 1 2 2( )i i i i ir E r F F e
10-18
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
10-19
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
• 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
it i iM Mt iSMB t iHML t itr R SMB HML e
10-20
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