1. Arbitrage Pricing Theory and Multifactor Models of Risk and Return
CHAPTER 10
Arbitrage Pricing Theory and Multifactor Models of Risk and Return

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

3. Single-Factor Model ri = αi + βi f + ei , 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

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

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

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.

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

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

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. APT & Well-Diversified Portfolios
rP = E (rP) + bPF + eP
F = some factor
For a well-diversified portfolio:
eP approaches zero
Similar to CAPM,

11. Figure 10.1 Returns as a Function of the Systematic Factor

12. Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

13. Figure 10.3 An Arbitrage Opportunity

14. Figure 10.4 The Security Market Line

15. 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

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

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

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

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

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