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Arbitrage Pricing Theory and Multifactor Models

of Risk and Return

- CHAPTER 10

Outline of the Chapter

- Arbitrage Pricing Theory
- Arbitrage
- Single factor APT
- The Security Market Lines
- Compare APT and CAPM
- Multifactor Models
- Multifactor APT

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 of a security. - 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

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.

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.

Arbitrage Pricing Theory (Continued)

- The solid line indicates a well diversified

portfolio, A with an expected return of 10 and

ßA1. - The dahsed line also indicates a well diversified

portfolio , B, with an expected return of 8 and

ßB1. - Could they coexisted?
- Arbitrage opportunity
- Well-diversified portfolios with equal betas must

have equal expected returns in market equilibrium.

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 ß. - 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.

Arbitrage Pricing Theory (Continued)

- Take M, market index portfolio as a

well-diversified portfolio. - The systematic factor, F, is the unexpected

return on that portfolio. - Since M is well-diversified, should be on the

line and its beta is 1. - Thus, the equation of the line is

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.

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.

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.

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.

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

Multifactor Models An Overview (Continued)

- 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.

Multifactor Models An Overview (Continued)

- Based on the same idea with CAPMs 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

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.

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

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.

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.

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

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