10-1

ARBITRAGE PRICING

THEORY AND

MULTIFACTOR MODELS OF

RISK AND RETURN

CHAPTER 10

10-2

Outline of the Chapter

• Arbitrage Pricing Theory

– Arbitrage

– Single factor APT

– The Security Market Lines

• Compare APT and CAPM

• Multifactor Models

– Multifactor APT

10-3

Arbitrage Pricing Theory

• Arbitrage Pricing Theory (APT) was developed by

Ross (1976).

• APT predicts a security market line as CAPM and

shows a linear relation with expected return and risk.

• According to APT:

– Security returns are described by a factor model

– There are sufficient securities to diversify away

idiosyncratic risk

– Well-functioning security markets do not allow for

the persistance of arbitrage opportunities

10-4

Arbitrage Pricing Theory

• An arbitrage opportunity arises when an investor can

earn riskless profits without making a net investment.

• The law of one price states that if two assets are

equivalent in all economically relevant respects, then

they should have the same market price.

• Otherwise there is a chance for arbitrage activity-

simultaneously buying the asset where it is cheap

and selling where it is expensive.

• During the arbitrage activity, investors will bid up the

price where it is low and force it down where it is

expensive. As a result they eliminate the arbitrage

opportunities.

• Security prices should satisfy a “no-arbitrage

condition”.

10-5

Arbitrage Pricing Theory (Continued)

• In a well-diversified portfolio nonsystematic risk across

firms cancels out. Thus only factor risk (systematic risk

of the portfolio) affects the risk premium on the security

in market equilibrium.

10-6

Arbitrage Pricing Theory (Continued)

•The solid line indicates a well

diversified portfolio, A with an

expected return of 10% and

βA=1.

•The dahsed line also indicates

a well diversified portfolio , B,

with an expected return of 8%

and βB=1.

• Could they coexisted?

•Arbitrage opportunity

•Well-diversified portfolios with

equal betas must have equal

expected returns in market

equilibrium.

10-7

Arbitrage Pricing Theory (Continued)• The risk-premiums of well-

diversified portfolios with

different betas should be

proportional to their betas.

•The risk premium

(difference between the

expected return on the

portfolio and the risk-free

rate) increases in direct

proportion to β.

•The expected return on all

well-diversified portfolios must

lie on the straight line from the

risk-free asset.

• The equation of the line will

also show the expected return

on all well-diversified

portfolios.

10-8

Arbitrage Pricing Theory (Continued)

• Take M, market index

portfolio as a well-

diversified portfolio.

•Since M is well-

diversified, should be on

the line and its beta is 1.

•Thus, the equation of

the line is:

PfMfP rrErrE ])([)(

10-9

Arbitrage Pricing Theory (Continued)

• The no-arbitrage condition leads us to the equation

that shows an expected return-beta relationship,

which is identical to that of the CAPM.

• There are only three assumptions employed this time

to obtain the same relationship as CAPM:

– A factor model describing security returns

– A sufficient number of securities to form well-

diversified portfolios

– Absence of arbitrage opportunities

• This approach under new assumptions is called

Arbitrage Pricing Theory.

10-10

Arbitrage Pricing Theory (Continued)

• In addition,

– APT does not require the benchmark portfolio

on SML to be the true market portfolio.

– Thus, the problems related to have an

unobservable market portfolio in CAPM are

not problems in APT.

– Also, the index portfolio can easily be

employed as a benchmark portfolio since it is

well-diversified in APT even though it is not

true market portfolio.

10-11

Individual Assets and the APT

• Remember:

– Imposing no-arbitrage condition on a single-

factor security market implies maintenance of

the expected return-beta relationship for all

well-diversified portfolios and for all but

possibly a small number of individual

securities.

10-12

Individual Assets and the APT (Continued)

• APT vs CAPM

– APT applies to well diversified portfolios and not

necessarily to individual stocks.

– APT gives a benchmark rate of return to be

employed in capital budgeting, security valuation,

or investment performance evaluation such as

CAPM.

– APT is more general in that it gets to an expected

return and beta relationship without the

assumption of the market portfolio.

– Although CAPM holds even for securities, with APT it

is possible for some individual stocks to be mispriced -

not lie on the SML.

10-13

Multifactor Models: An Overview

• Factor models are employed to describe and

quantify the different factors that affect the rate of

return on a security.

• In multifactor models stocks exhibit different

sensitivities to the different components of

systematic risk.

• Two-factor Model:

– Suppose there are two most important

macroeconomic sources of risk are:

• Uncertainties surrounding the state of the business

cycle (unanticipated growth in GDP)

• Unexpected changes in interest rates

10-14

Multifactor Models: An Overview (Continued)

iiIRiGDPii eIRGDPrEr )(

• Factor sensitivities (loadings, betas): measure the

sensitivity of the security returns to the systematic

factors.

• By using these mutifactor models different

responses of the securities to varying sources of

macro economy are captured.

•The question is where E(r) comes from?

•Security Market Line of CAPM: shows the

relationship between expected return and the

asset risk

•This time we have more than one risk factors.

10-15

Multifactor Models: An Overview (Continued)

• Based on the same idea with CAPM’s SML we can

say that in the two factor model the expected rate of

return on a security will be the sum of:

– The risk-free rate of return

– The sensitivity to GDP risk (GDP beta) *the risk

premium for bearing GDP risk

– The sensitivity to interest rate risk (interest rate beta)

*the risk premium for bearing interest rate risk

IRIRGDPGDPf RPRPrrE )(

10-16

A Multifactor APT

• Factor Portfolios: A well-diversified portfolio

constructed to have a beta of 1 on one of the factors

and a beta of zero on any other factor.

– Returns on factor portfolios are correlated to one source of

risk but totally uncorrelated with the other sources of risk.

2F2AF1F1AFfA RPRPrrE )(

10-17

Where Should We Look for Factors?

• The mutlifactor APT does not say anything

about the determination of relevant risk

factors and their risk premiums.

• Still we want to narrow the set:

– Limit ourselves to the systematic factors with

considerable ability to explain security returns

– Choose factors that seem likely to be

important risk factors

10-18

Where Should We Look for Factors? (Continued)

• Chen, Roll and Ross (1986)

– % change in industrial production, % change in

expected inflation, % change in unanticipated

inflation, excess return of long-term corporate

bonds over long-term government bonds, and

excess return of long-term government bonds over

T-bills.

ittiGBtiCGtiUItiEItiIPiit eGBCGUIEIIPr

10-19

Where Should We Look for Factors? (Continued)

• Fama and French three-factor model (1996)

– They use firm characteristices to capture the

effects of systematic risk.

– They expect that the firm-specific variables proxy

for yet-unknown more fundamental variables.

ittiHMLtiSMBMtiMiit eHMLSMBRr

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