Arbitrage Pricing Theory and
Multifactor Models of
Risk and Return
Chapter 11
McGraw-Hill/Irwin
Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
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
11-2
Single Factor Model Equation
Ri = E(ri) + Betai (F) + ei
Ri = Return for security i
Betai = 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
11-3
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.
They can provide better descriptions of
security returns.
11-4
Multifactor Model Equation
Ri = E(ri) + BetaGDP (GDP) + BetaIR (IR) + ei
Ri = Return for security i
BetaGDP= Factor sensitivity for GDP
BetaIR = Factor sensitivity for Interest Rate
ei = Firm specific events
11-5
Multifactor SML Models
E(r) = rf + BGDPRPGDP + BIRRPIR
BGDP = Factor sensitivity for GDP
RPGDP = Risk premium for GDP
BIR = Factor sensitivity for Interest Rate
RPIR = Risk premium for IR
11-6
Arbitrage Pricing Theory
It was developed by Stephen Rose in 1976.
Like CAPM, it predicts a SML but it is quite
different:
It relies on three propositions;
Security returns can be described by a factor
model.
There are sufficient number of securities to
diversify away.
Well-functioning security markets do not allow for
the persistance of arbitrage opportunities.
11-7
Fundamental principle of valuation is that investors
are compensated for assuming risk that cannot be
diversified away, they are not compensated for
assuming diversifiable risk.
In Arbitrage Pricing Theory, this principle is the
starting point of the theory.
The value of the stock is exposed to unexpected
changes in many factors. If you cannot hedge these
unexpected changes, then it is reasonable to assume
that you would like to be compensated for assuming
these risks.
11-8
An arbitrage portfolio is a portfolio that
requires zero wealth and eliminated all
unsystematic and systematic risk. The
basic principle is that such a portfolio
must also have zero return; otherwise, it
offers an arbitrage opportunity.
11-9
To create an arbitrage portfolio we must
eliminate all systematic and
unsystematic risk. Unsystematic risk
can be eliminated simply by having
many securities in the portfolio. To
achive this we must have a large
number of securities available.
11-10
The next requirement of a “zero wealth
portfolio; from your existing position you can
construct this portfolio without contributing or
withdrawing any cash. That is achievied by
taking both long and short positions that
offset each other in costs and revenues. This
requires you to be able to short sale.
In efficient markets, profitable arbitrage
opportunities will quickly disappear.
11-11
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. If investors observe a violation
of this Law, they will engage in arbitrage
activity-simultaneolusly buying the assets
where it is cheap and selling it where it is
expensive. In the process, they will bid up the
price where it is low and force it down where
it is high until the arbitrage opportunity is
eliminated.
11-12
The important property of the risk-free
arbitrage portfolio is that any investor,
regardless of risk-aversion or wealt, will
want to take an infinite position in it.
Because large positions will quickly
force prices up or down until the
opportunity vanishes.
11-13
According to CAPM, all investors hold meanvariance efficient portfolios. If a security is
mispriced, the investors will buy underpriced
and away from overpriced securities in their
portfolios. Pressure on equilibrium prices
results from many investors shifting their
porfolios, each by a relatively small enough.
According to the APT, a few investors who
identify an arbitrage opportunity will mobilize
large dollar amounts and quickly restore
equilibrium.
11-14
APT & Well-Diversified Portfolios
rP = E (rP) + bPF + eP
F = some factor
For a well-diversified portfolio:
eP approaches zero
Similar to CAPM
11-15
Portfolios and Individual Security
E(r)%
E(r)%
F
F
Portfolio
Individual Security
11-16
Suppose portfolio A has a beta of 1 and expected
return of 10%. Portfoio B has a beta of 1 and
expected return of of 8%. There is an arbitrage
opportunty:
If you sell short $1 milion of B and buy 1 million of A,
a zero net investment strategy, your riskless pay of
will be;
(0.10+1 F) x $1 million from long position in A
-(0.08+1F) x $1 million from short position in B.
0.02 x $1 million = $20,000 net proceeds
Your profit is riskless because the factor risk cancels out
across the long and short positions.
11-17
Disequilibrium Example
What about portfolios with different
betas?
Supose Rf = 4%.
Portfolio C has a beta of .5 and exp. Return of
6%.
Portfolio C plots below the line.
Use funds to construct an equivalent risk
higher return Portfolio D.
D is comprised of half of A & half of Risk-Free
Asset
11-18
Beta of D = .5 x 0 + .5x1 = .5
Exp return = .5x4 + .5x10 = 7%
Portfolio D has equal beta but greater
exp.return than that of C. There is an
arbitrage opportunity
Arbitrage profit of 1% (7%-6%)
11-19
Disequilibrium Example
E(r)%
10
7
6
A
D
C
Risk Free 4
.5
1.0
Beta for F
11-20
»APT with Market Index Portfolio
E(r)%
M
[E(rM) - rf]
Market Risk Premium
Risk Free
1.0
Beta (Market Index)
11-21
Suppose a market index is a well-diversified portfolio
with an exp.return 10%. Deviations of its return from
expectations serve as the systematic factor.
Rf = 4%.
SML implies that the expected return on a well
diversified portfolio E with beta of 2/3 should be
E(rp)= 0.04 + (0.10 – 0.04)2/3
= 8%
What if the exp.return actually is 9%. There will be an
arbitrage opportunity.
11-22
We can construct a new portfolio that has the
same Beta of E. 1/3 of Rf and 2/3 of market
portfolio.
B=1/3 x 0+ 2/3 x 1= 2/3
E(rp) = 1/3x0.04+2/3x0,10=0,08
Buy $1 of portfolio E and sell $1 portfolio
invested in 1/3 in rf and 2/3 in market index.
$1 x (0.09)
-$1 x (0,08)
$1 x 0.01
Deviation of expected return from the SML(9%-8%)
11-23
APT and CAPM Compared
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.
11-24
ri  E (ri )  b i1 F1  b i 2 F2  ei
A Multifactor APT
ri  E (ri )  b i1 F1  b i 2 F2  ei
A Multifactor CAPM (ICAPM)(Merton)

 
E (ri )  r f  b iM E (rM )  r f  b ie E (re )  r f

11-25