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LECTURE 8 :
FACTOR MODELS
(Asset Pricing and Portfolio Theory)
Contents
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The CAPM
Single index model
Arbitrage portfolioS
Which factors explain asset prices ?
Empirical results
Introduction
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CAPM : Equilibrium model
– One factor, where the factor is the excess
return on the market.
– Based on mean-variance analysis
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Stephen Ross (1976) developed
alternative model  Arbitrage Pricing
Theory (APT)
Single Index Model
Single Index Model
Alternative approach to portfolio theory.
Market return is the single index.
Return on a stock can be written as :
Ri = ai + biRm
ai = ai + ei
Hence Ri = ai + biRm + ei
Equation (1)
Assume :
Cov(ei, Rm) = 0
E(eiej) = 0 for all i and j (i ≠ j)
Single Index Model
(Cont.)
Obtain OLS estimates of ai, bi and sei (using OLS)
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Mean return :
ERi = ai + biERm
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Variance of security return :
s2i = b2is2m + s2ei
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Covariance of returns between securities :
sij = bibjs2m
Portfolio Theory and the
Market Model
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Suppose we have a 5 Stock Portfolio
Estimates required
– Traditional MV-approach
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5 Expected returns
5 Variances of returns
10 Covariances
– Using the Single Index Model
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5 OLS regressions
– 5 alphas and 5 betas
– 5 Variances of error term
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1 Expected return of the market portfolio
1 Variance of market return
Factor Models
Single Factor Model
ER
Slope = b
a
Factor
Factor Model : Example
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Ri = ai + biF1 + ei
Example :
Factor-1 is predicted rate of growth in industrial
production
i
Stock 1
Stock 2
Stock 3
mean Ri
15%
21%
12%
bi
0.9
3.0
1.8
The APT : Some Thoughts
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The Arbitrage Pricing Theory
– New and different approach to determine asset
prices.
– Based on the law of one price : two items that
are the same cannot sell at different prices.
– Requires fewer assumptions than CAPM
– Assumption : each investor, when given the
opportunity to increase the return of his portfolio
without increasing risk, will do so.
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Mechanism for doing so : arbitrage portfolio
An Arbitrage Portfolio
Arbitrage Portfolio
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Arbitrage portfolio requires no ‘own funds’
– Assume there are 3 stocks : 1, 2 and 3
– Xi denotes the change in the investors holding
(proportion) of security i, then X1 + X2 + X3 = 0
– No sensitivity to any factor, so that b1X1 + b2X2 +
b3X3 = 0
– Example : 0.9 X1 + 3.0 X2 + 1.8 X3 = 0
– (assumes zero non factor risk)
Arbitrage Portfolio
(Cont.)
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Let X1 be 0.1.
Then
0.1 + X2 + X3 = 0
 0.09 + 3.0 X2 + 1.8 X3 = 0
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– 2 equations, 2 unknowns.
– Solving this system gives
X2 = 0.075
 X3 = -0.175
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Arbitrage Portfolio
(Cont.)
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Expected return
X1 ER1 + X2 ER2 + X3 ER3 > 0
Here 15 X1 + 21 X2 + 12 X3 > 0 (= 0.975%)
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Arbitrage portfolio is attractive to investors
who
– Wants higher expected returns
– Is not concerned with risk due to factors other than
F1
Portfolio Stats / Portfolio
Weights (Example)
Weights
Old Portf.
Arbitr. Portf.
New Portf.
X1
1/3
0.1
0.433
X2
1/3
0.075
0.408
X3
1/3
-0.175
0.158
ERp
16%
0.975%
16.975%
bp
1.9
0.00
1.9
sp
11%
small
approx 11%
Properties
Pricing Effects
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Stock 1 and 2
– Buying stock 1 and 2 will push prices up
– Hence expected returns falls
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Stock 3
– Selling stock 3 will push price down
– Hence expected return will increase
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Buying/selling stops if all arbitrage possibilities are
eliminated.
Linear relationship between expected return and
sensitivities
ERi = l0 + l1bi
where bi is the security’s sensitivity to the factor.
Interpreting the APT
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ERi = l0 + l1bi
l0 = r f
l1 = pure factor portfolio, p* that has unit
sensitivity to the factor
For bi = 1
ERp* = rf + l1
or l1 = ERp* - rf
(= factor risk premium)
Two Factor Model :
Example
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Ri = ai + bi1F1 + bi2F2 + ei
i
ERi
bi1
bi2
Stock 1
15%
0.9
2.0
Stock 2
Stock 3
Stock 4
21%
12%
8%
3.0
1.8
2.0
1.5
0.7
3.2
Multi Factor Models
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Ri = ai + bi1 F1 + bi2 F2 + … + bik Fk + ei
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ERi = l0 + l1 bi1 + l2 bi2 + … + lkbik
Identifying the Factors
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Unanswered questions :
– How many factors ?
– Identity of factors (i.e. values for lamba)
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Possible factors (literature suggests : 3 – 5)
Chen, Roll and Ross (1986)
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Growth rate in industrial production
Rate of inflation (both expected and unexpected)
Spread between long-term and short-term interest
rates
Spread between low-grade and high-grade bonds
Testing the APT
Testing the Theory
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Proof of any economic theory is how well it describes
reality.
Testing the APT is not straight forward
– theory specifies a structure for asset pricing
– theory does not say anything about the economic or firm
characteristics that should affect returns.
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Multifactor return-generating process
Ri = ai + S bijFj + ei
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APT model can be written as
ERi = rf + S bijlj
Testing the Theory
(Cont.)
bij : are unique to each security and represent
an attribute of the security
Fj : any I affects more than 1 security (if not all).
lj : the extra return required because of a
security’s sensitivity to the jth attribute of the
security
Testing the Theory
(Cont.)
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Obtaining the bij’s
– First method is to specify a set of
attributes (firm characteristics) : bij are
directly specified
– Second method is to estimate the bij’s
and then the lj using the equation shown
earlier.
Principal Component
Analysis (PCA)
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Technique to reduce the number of variables being
studied without losing too much information in the
covariance matrix.
Objective : to reduce the dimension from N assets to k
factors
Principal components (PC) serve as factors
– First PC : (normalised) linear combination of asset returns
with maximum variance
– Second PC : (normalised) linear combination of asset returns
with maximum variance of all combinations orthogonal to the
first component
Pro and Cons of Principal
Component Analysis
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Advantage :
– Allows for time-varying factor risk premium
– Easy to compute
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Disadvantage :
– interpretation of the principal components,
statistical approach
Summary
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APT alternative approach to explain
asset pricing
– Factor model requiring fewer assumptions
than CAPM
– Based on concept of arbitrage portfolio
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Interpretation : lamba’s are difficult to
interpret, no economics about the
factors and factor weightings.
References
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Cuthbertson, K. and Nitzsche, D. (2004)
‘Quantitative Financial Economics’,
Chapters 7
Cuthbertson, K. and Nitzsche, D. (2001)
‘Investments : Spot and Derivatives
Markets’, Chapter 10.5 (The Arbitrage
Pricing Theory)
END OF LECTURE