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Portfolio Management
3-228-07
Albert Lee Chun
Multifactor Equity Pricing
Models
Lecture 7
6 Nov 2008
0
Today’s Lecture
Single Factor Model
Multifactor Models
Fama-French
APT
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Alpha
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Alpha
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Suppose a security with a particular  is offering
expected return of 17% , yet according to the
CAPM, it should be 14.8%.
It’s under-priced: offering too high of a rate of
return for its level of risk
Its alpha is 17-14.8 = 2.2%
According to CAPM alpha should be equal to 0.
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Frequency Distribution of Alphas
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The CAPM and Reality
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Is the condition of zero alphas for all stocks as
implied by the CAPM met?
• Not perfect but one of the best available
Is the CAPM testable?
• Proxies must be used for the market portfolio
CAPM is still considered the best available
description of security pricing and is widely
accepted.
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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
ri  E (ri )  iF  ei
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Single Factor Model
ri  E (ri )  iF  ei
ßi = index of a security’s particular return to the factor
F= some macro factor; in this case F is unanticipated
movement; F is commonly related to security
returns
Assumption: a broad market index like
the S&P/TSX Composite is the common factor
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Regression Equation: Single Index Model
(ri  rf)  i  i(rM  rf)  ei
ai = alpha
bi(rM-ri) = the component of return due to market
movements (systematic risk)
ei = the component of return due to unexpected firmspecific events (non-systematic risk)
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Risk Premium Format
Let:
Ri = (ri - rf )
Rm = (rm - rf )
Risk premium
format
Ri = i + ßiRm + ei
The above equation regression is the single-index
model using excess returns.
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Measuring Components of Risk
2
i
2
i
2
M
  
 2(ei)
i2 = total variance
i2 m2 = systematic variance
2(ei) = unsystematic variance
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Index Models and Diversification
RP   P   P  eP
N
P  1N  P
i 1
N
 P  1 N  P
i 1
eP  1

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p
N
N
e
i 1
P
2
  P2 M
  2 (e P )
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The Variance of a Portfolio
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Security Characteristic Line for X
Excess Returns (i)
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SCL
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Excess returns
.on. market index
Ri =  i + ßiRm + ei
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Multi Factor Models
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More than 1 factor?
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CAPM is a one factor model: The only determinant
of expected returns is the systematic risk of the
market. This is the only factor.
What if there are multiple factors that determine
returns?
Multifactor Models: Allow for multiple sources of
risk, that is multiple risk factors.
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Multifactor Models
•
Use other factors in addition to market returns:
–Examples include industrial production,
expected inflation etc.
–Estimate a beta or factor loading for each
factor using multiple regression
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Example: 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

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Multifactor SML
E(r) = rf + BGDPRPGDP + BIRRPIR
BGDP = Factor sensitivity for GDP
RPGDP = Risk premium for GDP
BIR = Factor sensitivity for Interest Rates
RPIR = Risk premium for GDP
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Multifactor Models
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CAPM say that a single factor, Beta, determines the
relative excess return between a portfolio and the
market as a whole.
Suppose however there are other factors that are
important for determining portfolio returns.
The inclusion of additional factors would allow the
model to improve the model`s fit of the data.
The best known approach is the three factor model
developed by Gene Fama and Ken French.
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Fama French 3-Factor Model
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The Fama-French 3 Factor Model

Fama and French observed that two classes of stocks
tended to outperform the market as a whole:
(i) small caps
(ii) high book-to-market ratio
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Small Value Stocks Outperform
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Fama-French 3-Factor Model

They added these two factors to a standard CAPM:
Ri , t  rf , t = i  bi1( Rm, t  rf , t )  bi 2 SMBt  bi 3 HML t  i , t
SMB = “small [market capitalization] minus big”
"Size" This is the return of small stocks minus that of large stocks.
When small stocks do well relative to large stocks this will be
positive, and when they do worse than large stocks, this will be
negative.
HML = “high [book/price] minus low”
"Value" This is the return of value stocks minus growth stocks,
which can likewise be positive or negative.
The Fama-French Three Factor model explains over 90% of stock returns.
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Arbitrage Pricing Theory (APT)
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APT

Ross (1976): intuitive model, only a few
assumptions, considers many sources of risk
Assumptions:
1. There are sufficient number of securities to diversify
away idiosyncratic risk
2. The return on securities is a function of K different
risk factors.
3. No arbitrage opportunities
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APT

1.
2.
3.
APT does not require the following CAPM
assumptions:
Investors are mean-variance optimizers in the sense
of Markowitz.
Returns are normally distributed.
The market portfolio contains all the risky securities
and it is efficient in the mean-variance sense.
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APT & Well-Diversified
Portfolios
rP  E(rP )  PF  eP


F is some macroeconomic factor
For a well-diversified portfolio eP approaches zero
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Returns as a Function of the Systematic Factor
Well-diversified portfolio
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Single Stocks
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Returns as a Function of the Systematic Factor: An
Arbitrage Opportunity
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Example: An Arbitrage Opportunity
E(r)%
10
7
6
A
D
C
Risk Free = 4
.5
1.0
Beta for F
Risk premiums must be proportional to Betas!
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Disequilibrium Example
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Short Portfolio C, with Beta = .5
One can construct a portfolio with equivalent risk
and higher return : Portfolio D
• D = .5x A + .5 x Risk-Free Asset
• D has Beta = .5
Arbitrage opportunity: riskless profit of 1%
Risk premiums must be proportional to Betas!
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APT Security Market Line
E(rP )  rf   P ( Rm  rf )
This is CAPM!
Risk premiums must be proportional to Betas!
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APT and CAPM Compared
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APT applies to well diversified portfolios and not
necessarily to individual stocks
With APT it is possible for some individual stocks to
be mispriced – that is to 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
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A Multifactor APT
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A factor portfolio is a portfolio constructed so
that it would have a beta equal to one on a given
factor and zero on any other factor
These factor portfolios are the building blocks for
a multifactor security market line for an economy
with multiple sources of risk
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Where Should we Look for Factors?
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The multifactor APT gives no guidance on where to
look for factors
Chen, Roll and Ross
– Returns a function of several macroeconomic and
bond market variables instead of market returns
Fama and French
• Returns a function of size and book-to-market
value as well as market returns
9-36
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Generalized Factor Model

In theory, the APT supposes a stochastic process that generates returns and that may be
represented by a model of K factors, such that
Ri = E ( Ri )  Bi1F 1  Bi 2 F 2  ...  BikFk   i  i  1...n
where:
Ri = One period realized return on security i, i= 1,2,3…,n
E(Ri) = expected return of security i
Bij
= Sensitivity of the reutrn of the ith stock to the jth risk factor
Fj
= j-th risk factor
i

=captures the unique risk associated with security i
Similar to CAPM, the APT assumes that the idiosyncratic effects can be diversified
away in a large portfolio.
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Multifactor APT
APT Model
E ( Ri )  rf  Bi1( E ( R1)  rf )  Bi 2( E ( R2)  rf ).  ...  Bik ( E ( Rk )  rf )
The expected return on a secutity depends on
the product of the risk premiums and the factor betas (or factor
loadings)
E(Ri) – rf is the risk premium on the ith factor portfolio.
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Sample APT Problem
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Suppose that the equity market in a large economy
can be described by 3 sources of risk: A, B and C.
Factor
A
B
C
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Risk Premium
.06
.04
.02
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Example APT Problem
Suppose that the return on Maggie’s Mushroom Factory
is given by the following equation, with an expected
return of 17%.
r(t) = .17 + 1.0 x A + .75 x B + .05 x C + error(t)
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Sample APT problem

The risk free rate is given by 6%
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1. Find the expected rate of return of the mushroom
factory under the APT model.

2. Is the stock-under or over-valued? Why?
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Sample APT Problem
Factor
Risk Premium
A
.06
B
.04
C
.02
Risk-Free Rate = 6%
Return(t) = .17 + 1.0*A + 0.75*B + .05*C + e(t)
The factor loadings are in green.
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Sample APT Problem
Factor
Risk Premium
A
.06
B
.04
C
.02
Risk-Free Rate = 6%
Return = .17 + 1.0*A + 0.75*B + .05*C + e
So plug in risk-premia into the APT formula
E[Ri] = .06 + 1.0*0.06+0.75*0.04+0.5*0.02 = .16
16% < 17% => Undervalued!
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Quick Review of Underpricing
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
Undervalued = Underpriced = Return Too High
Overvalued = Overpriced = Return Too Low
P(t) = P(t+1)/ 1+ r
r = P(t+1)/P(t) – 1
where r is the return for a risky payoff P(t+1).
This is easy to remember if you think about the inverse
relationship between price (value) today and return.
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Examples 9.3 and 9.4
Factor portfolio 1: E(R1) = 10%
Factor Portfolio 2: E(R2) = 12%
Rf = 4%
Portfolio A with B1 = .5 and B2 = .75
Construct aPortfolio Q using weights of
B1 = .5 on factor portfolio 1 and a weight of
B2 = .75 on factor portfolio 2 and a weight of
1- B1 – B2 = -.25 on the risk free rate.
E(Rq) = B1E(R1) + B2 E(R2) + (1-B1-B2) Rf
= rf + B1(E(R1) –rf )+ B2(E(R2) – rf) =13%
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Example 9.4
Suppose that: E(RA) = 12% < 13%
Portfolio Q
Ponderation B1 = .5: facteur portefeuille 1
Ponderation B2 = .75: facteur portefeuille 2
Ponderation 1- B1 – B2 = -.25 : rf
E(Rq ) = 12%
$1 x E(Rq) - $1x E(RA)=1%
There is a riskless arbitrage
opportunity of 1%!
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Next Week
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
We will continue our lecture with Chapter 12
Market Efficiency (Chapter 10; Section 11.1)
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