Lecture 6: CAPM & APT
• The following topics are covered:
–
–
–
–
CAPM
CAPM extensions
Critiques
APT
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CAPM: Assumptions
• Investors are risk-averse individuals who maximize the
expected utility of their wealth
• Investors are price takers and they have homogeneous
expectations about asset returns that have a joint normal
distribution (thus market portfolio is efficient – page 148)
• There exists a risk-free asset such that investors may borrow or
lend unlimited amount at a risk-free rate.
• The quantities of assets are fixed. Also all assets are
marketable and perfectly divisible.
• Asset markets are frictionless. Information is costless and
simultaneously available to all investors.
• There are no market imperfections such as taxes, regulations,
or restriction on short selling.
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Derivation of CAPM
• If market portfolio exists, the prices of all assets must adjust until all are
held by investors. There is no excess demand.
• The equilibrium proportion of each asset in the market portfolio is
–
wi 
m arket value of the individual asset
m arket value of all assets
(6.1)
• A portfolio consists of a% invested in risky asset I and (1-a)% in the
market portfolio will have the following mean and standard deviation:
–
–
~
~
~
E ( R p )  aE( Ri )  (1  a) E ( Rm )
~
 ( R p )  [a 2 i2  (1  a) 2  m2  2a(1  a) im ]1 / 2
(6.2)
(6.3)
• A portfolio consists of a% invested in risky asset I and (1-a)% in the
market portfolio will have the following mean and standard deviation:
• Find expected value and standard deviation of R p with respect to the
percentage of the portfolio as follows.
~
E ( R p )
a
~
~
 E ( Ri )  E ( Rm )
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Derivation of CAPM
~
 ( R p )
a

1 2 2
[a  i  (1  a) 2  m2  2a(1  a) im ]1/ 2  [2a i2  2 m2  2a m2  2 im  4a im ]
2
• Evaluating
the two equations where a=0:
~
E ( Rp )
a
~
 ( E p )
a 0
~
~
 E ( Ri )  E ( Rm )
a
a 0

   m2
1 2 1 / 2
( m ) (2 m2  2 im )  im
2
m
• The slope of the risk-return trade-off:
~
E ( R p ) / a
~
 ( R p ) / a
a 0
~
~
E ( Ri )  E ( Rm )

( im   m2 ) /  m
• Recall~ that the slope of the market line is:
E ( Rm )  R f
;
m
• Equating the above two slopes:
~
E ( Rm )  R f
~
~
E ( Ri )  E ( Rm )

m
( im   m2 ) /  m

~
~
E ( Ri )  R f  [ E ( Rm )  R f ] im2
m
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Extensions of CAPM
1.
2.
No riskless assets
Forming a portfolio with a% in the market portfolio and (1-a)% in the
minimum-variance zero-beta portfolio.
The mean and standard deviation of the portfolio are:
3.
–
–
4.
E(Rp )  aE(Rm )  (1  a)E(Rz )
~
 ( R p )  [a 2 m2  (1  a) 2  z2  2a(1  a)rzm z m ]1 / 2
The partial derivatives where a=1 are:
–
–
5.
E ( R p )
a
 ( R p )
a
;
 E ( Rm )  E ( R z )
;
1
 [a 2 m2  (1  a) 2  z2 ]1 / 2 [2a m2  2 z2  2a z2 ]
2
Taking the ratio of these partials and evaluating where a=1:
–
E( R p ) / a
 ( R p ) / a

E( Rm )  E ( Rz )
m
Further, this line must pass through the point E(R
is E ( Rz ) . The equation of the line must be:
6.
m
–
E ( R p )  E ( Rz )  [
E ( Rm )  E ( Rz )
m
), ( Rm )
and the intercept
] p
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Extensions of CAPM
• The existence of nonmarketable assets
– E.g., human capital; page 162
• The model in continuous time
– Inter-temporal CAPM
• The existence of heterogeneous expectations and taxes
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Empirical tests of CAPM
• Test form -- equation 6.36
– the intercept should not be significantly different
from zero
– There should be one factor explaining return
– The relationship should be linear in beta
– Coefficient on beta is risk premium
• Test results – page 167
• Summary of the literature.
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Roll (1977)’s Critiques
• Roll’s (1977) critiques (page 174)
• The efficacy of CAPM tests is conditional on the
efficiency of the market portfolio.
• As long as the test involves an efficient index, we are
fine.
• The index turns out to be ex post efficient, if every
asset is falling on the security market line.
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Arbitrage Pricing Theory
• Assuming that the rate of return on any security is a linear function of k
factors:
Ri  E( Ri )  bi1F1  ... bik Fk   i
Where Ri and E(Ri) are the random and expected rates on the ith asset
Bik = the sensitivity of the ith asset’s return to the kth factor
Fk=the mean zero kth factor common to the returns of all assets
εi=a random zero mean noise term for the ith asset
• We create arbitrage portfolios using the above assets.
•
• No wealth
n
w  0
i 1
i
-- arbitrage portfolio
• Having no risk and earning no return on average
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Deriving APT
• Return of the arbitrage portfolio:
R  w R
n
p
i 1
i
i
  wi E ( Ri )  wi bi1F1 ...   wi bik Fk  wi i
i
i
i
i
• To obtain a riskless arbitrage portfolio, one
needs to eliminate both diversifiable and
nondiversifiable risks. I.e.,
1
wi  , n  ,  wi bik  0 for
n
i
all factors
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Deriving APT
Rp   wi E( Ri )
i
 w E( R )  0
i
i
i
As:
wb
i ik
 0 for each k
i
How does E(Ri) look like? -- a linear combination
of the sensitivities
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APT
• There exists a set of k+1 coefficients, such that,
–
~
E(Ri )  0  1bi1  ...  k bik
(6.57)
• If there is a riskless asset with a riskless rate of
return Rf, then b0k =0 and Rf = 0
– E(Ri )  R f  1bi1  ...  k bik
(6.58)
• In equilibrium, all assets must fall on the arbitrage
pricing line.
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APT vs. CAPM
• APT makes no assumption about empirical
distribution of asset returns
• No assumption of individual’s utility function
• More than 1 factor
• It is for any subset of securities
• No special role for the market portfolio in APT.
• Can be easily extended to a multiperiod framework.
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Example
• Page 182
• Empirical tests
– Gehr (1975)
– Reinganum (1981)
– Conner and Korajczyk (1993)
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FF 3-factor Model
• http://mba.tuck.dartmouth.edu/pages/faculty/ken.fren
ch/Data_Library/f-f_factors.html
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