chapter 10 arbitrage pricing theory and multifactor models of risk and return

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