CAPM, Arbitrage, and Linear Factor Models

41 Pages·2016·0.31 MB·English

3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
CAPM, Arbitrage, and Linear Factor Models
George Pennacchi
University ofIllinois
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
Introduction
We now assume all investors actually choose mean-variance
e¢ cient portfolios.
By equating these investors(cid:146)aggregate asset demands to
aggregate asset supply, an equilibrium single risk factor pricing
model (CAPM) can be derived.
Relaxing CAPM assumptions may allow for multiple risk
factors.
Arbitrage arguments can be used to derive a multifactor
pricing model (APT)
Multifactor models are very popular in empirical asset pricing
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
Review of Mean-Variance Portfolio Choice
Recall that for n risky assets and a risk-free asset, the optimal
portfolio weights for the n risky assets are
! = (cid:21)V 1 R(cid:22) R e (1)
(cid:3) (cid:0) f
(cid:0)
(cid:0) (cid:1)
R R
p f
where (cid:21) (cid:0) .
(cid:17) R(cid:22) Rfe 0V(cid:0)1 R(cid:22) Rfe
(cid:0) (cid:0)
The amount invested in the risk-free asset is then 1 e ! .
(cid:0) (cid:1) (cid:0) (cid:1) 0 (cid:3)
(cid:0)
R is determined by where the particular investor(cid:146)s
p
indi⁄erence curve is tangent to the e¢ cient frontier.
All investors, no matter what their risk aversion, choose the
risky assets in the same relative proportions.
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
Tangency Portfolio
Also recall that the e¢ cient frontier is linear in (cid:27) and R :
p p
1
Rp = Rf + R(cid:22) Rfe 0V(cid:0)1 R(cid:22) Rfe 2 (cid:27)p (2)
(cid:0) (cid:0)
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
This frontier is tangent to the (cid:147)risky asset only(cid:148)frontier,
where the following graph denotes this tangency portfolio as
!m located at point (cid:27) , R .
m m
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
Graph of E¢ cient Frontier
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
The Tangency Portfolio
Note that the tangency portfolio satis(cid:133)es e !m = 1. Thus
0
e (cid:21)V 1 R(cid:22) R e = 1 (3)
0 (cid:0) f
(cid:0)
(cid:0) (cid:1)
or
1
(cid:21) = m R Rfe 0V(cid:0)1e (cid:0) (4)
(cid:17) (cid:0)
h i
so that (cid:0) (cid:1)
!m = mV 1(R(cid:22) R e) (5)
(cid:0) f
(cid:0)
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
Asset Covariances with the Tangency Portfolio
Now de(cid:133)ne (cid:27) as the n 1 vector of covariances of the
M
(cid:2)
tangency portfolio with each of the n risky assets. It equals
(cid:27) = V!m = m(R(cid:22) R e) (6)
M f
(cid:0)
By pre-multiplying equation (6) by !m , we also obtain the
0
variance of the tangency portfolio:
(cid:27)2 = !m V!m = !m (cid:27) = m!m (R(cid:22) R e) (7)
m 0 0 M 0 f
(cid:0)
= m(R(cid:22) R )
m f
(cid:0)
where R(cid:22) !m R(cid:22) is the expected return on the tangency
m 0
(cid:17)
portfolio.
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
Expected Excess Returns
Rearranging (6) and substituting in for 1 = 1 (R R )
m (cid:27)2m m (cid:0) f
from (7), we have
1 (cid:27)
(R(cid:22) R e) = (cid:27) = M(R(cid:22) R ) = (cid:12)(R(cid:22) R ) (8)
(cid:0) f m M (cid:27)2 m (cid:0) f m (cid:0) f
m
where (cid:12) (cid:27)M is the n 1 vector whose ith element is
(cid:17) (cid:27)2m (cid:2)
Cov(R~m;Ri).
Var(R~m)
Equatioen (8) links the excess expected return on the tangency
portfolio, (R(cid:22) R ), to the excess expected returns on the
m f
individual risky(cid:0)assets, (R(cid:22) R e).
f
(cid:0)
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
CAPM
The Capital Asset Pricing Model is completed by noting that
the tangency portfolio, !m, chosen by all investors must be
the equilibrium market portfolio.
Hence, R and (cid:27)2 are the mean and variance of the market
m m
portfolio returns and (cid:27) is its covariance with the individual
M
assets.
Aggregate supply can be modeled in di⁄erent ways
(endowment economy, production economy), but in
equilibrium it will equal aggregate demands for the risky
assets in proportions given by !m.
Also R < R for assets to be held in positive amounts
f mv
(!m > 0).
i
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3.1:CAPM 3.2:Arbitrage 3.3:APT 3.4:Summary
CAPM: Realized Returns
De(cid:133)ne asset i(cid:146)s and the market(cid:146)s realized returns as
R~ = R(cid:22) +(cid:23)~ and R~ = R(cid:22) +(cid:23)~ where (cid:23)~ and (cid:23)~ are the
i i i m m m i m
unexpected components. Substitute these into (8):
R~ = R +(cid:12) (R~ (cid:23)~ R )+(cid:23)~ (9)
i f i m m f i
(cid:0) (cid:0)
= R +(cid:12) (R~ R )+(cid:23)~ (cid:12) (cid:23)~
f i m f i i m
(cid:0) (cid:0)
= R +(cid:12) (R~ R )+"
f i m f i
(cid:0)
where " (cid:23)~ (cid:12) (cid:23)~ . Note that
i i i m
(cid:17) (cid:0) e
Cov(R~ ;" ) = Cov(R~ ;(cid:23)~ ) (cid:12) Cov(R~ ;(cid:23)~ ) (10)
em i m i (cid:0) i m m
Cov(R~ ;R~ )
= Cov(R~ ;R~ ) m i Cov(R~ ;R~ )
e m i (cid:0) Var(R~ ) m m
m
= Cov(R~ ;R~ ) Cov(R~ ;R~ ) = 0
m i m i
(cid:0)
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