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