Title: Arbitrage Pricing Theory and Multifactor Models of Risk and Return
1Arbitrage Pricing Theory and Multifactor Models
of Risk and Return
2Outline of the Chapter
- Arbitrage Pricing Theory
- Arbitrage
- Single factor APT
- The Security Market Lines
- Compare APT and CAPM
- Multifactor Models
- Multifactor APT
3Arbitrage 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
4Arbitrage 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.
5Arbitrage 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.
6Arbitrage 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.
7Arbitrage 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.
8Arbitrage 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
9Arbitrage 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.
10Arbitrage 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.
11Individual 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.
12Individual 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.
13Multifactor 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
14Multifactor 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.
15Multifactor 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
16A 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.
17Where 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
18Where 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.
19Where 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