capm, arbitrage, and linear factor models · pdf file3.1: capm 3.2: arbitrage 3.3: apt 3.4:...

3.1: CAPM 3.2: Arbitrage 3.3: APT 3.4: Summary
CAPM, Arbitrage, and Linear Factor Models
George Pennacchi
University of Illinois
George Pennacchi University of Illinois
CAPM, Arbitrage, Linear Factor Models 1/ 41

Introduction
We now assume all investors actually choose mean-variancee cient portfolios.
By equating these investorsaggregate asset demands toaggregate asset supply, an equilibrium single risk factor pricingmodel (CAPM) can be derived.
Relaxing CAPM assumptions may allow for multiple riskfactors.
Arbitrage arguments can be used to derive a multifactorpricing model (APT)
Multifactor models are very popular in empirical asset pricing

Review of Mean-Variance Portfolio Choice
Recall that for n risky assets and a risk-free asset, the optimalportfolio weights for the n risky assets are
! = V1R Rf e
(1)
where Rp RfR Rf e
0 V1 R Rf e .The amount invested in the risk-free asset is then 1 e 0!.Rp is determined by where the particular investorsindierence curve is tangent to the e cient frontier.
All investors, no matter what their risk aversion, choose therisky assets in the same relative proportions.

Tangency Portfolio
Also recall that the e cient frontier is linear in p and Rp :
Rp = Rf +R Rf e
0 V1 R Rf e 12 p (2)
This frontier is tangent to the risky asset only frontier,where the following graph denotes this tangency portfolio as!m located at point m , Rm .

Graph of E cient Frontier

The Tangency Portfolio
Note that the tangency portfolio satises e 0!m = 1. Thus
e 0V1R Rf e
= 1 (3)
or = m
hR Rf e
0V1e
i1(4)
so that!m = mV1(R Rf e) (5)

Asset Covariances with the Tangency Portfolio
Now dene M as the n 1 vector of covariances of thetangency portfolio with each of the n risky assets. It equals
M = V!m = m(R Rf e) (6)
By pre-multiplying equation (6) by !m 0, we also obtain thevariance of the tangency portfolio:
2m = !m 0V!m = !m 0M = m!m 0(R Rf e) (7)
= m(Rm Rf )
where Rm !m 0 R is the expected return on the tangencyportfolio.

Expected Excess Returns
Rearranging (6) and substituting in for 1m =12m(Rm Rf )
from (7), we have
(R Rf e) =1mM =
M2m(Rm Rf ) = (Rm Rf ) (8)
where M2mis the n 1 vector whose i th element is
Cov (~Rm ;eRi )Var (~Rm)
.
Equation (8) links the excess expected return on the tangencyportfolio, (Rm Rf ), to the excess expected returns on theindividual risky assets, (R Rf e).

CAPM
The Capital Asset Pricing Model is completed by noting thatthe tangency portfolio, !m , chosen by all investors must bethe equilibrium market portfolio.
Hence, Rm and 2m are the mean and variance of the marketportfolio returns and M is its covariance with the individualassets.
Aggregate supply can be modeled in dierent ways(endowment economy, production economy), but inequilibrium it will equal aggregate demands for the riskyassets in proportions given by !m .
Also Rf < Rmv for assets to be held in positive amounts(!mi > 0).

CAPM: Realized Returns
Dene asset is and the markets realized returns as~Ri = Ri + ~ i and ~Rm = Rm + ~m where ~ i and ~m are theunexpected components. Substitute these into (8):
~Ri = Rf + i (~Rm ~m Rf ) + ~ i (9)= Rf + i (~Rm Rf ) + ~ i i ~m= Rf + i (~Rm Rf ) + e"i
where e"i ~ i i ~m . Note thatCov(~Rm ;e"i ) = Cov(~Rm ; ~ i ) iCov(~Rm ; ~m) (10)
= Cov(~Rm ; ~Ri )Cov(~Rm ; ~Ri )
Var(~Rm)Cov(~Rm ; ~Rm)
= Cov(~Rm ; ~Ri ) Cov(~Rm ; ~Ri ) = 0

Idiosyncratic Risk
Since Cov(~Rm ;e"i ) = 0, from equation (9) we see that thetotal variance of risky asset i , 2i , equals:
2i = 2i 2m +
2"i
(11)
Another implication of Cov(~Rm ;e"i ) = 0 is that equation (9)represents a regression equation.
The orthogonal, mean-zero residual, e"i , is referred to asidiosyncratic, unsystematic, or diversiable risk.
Since this portion of the assets risk can be eliminated by theindividual who invests optimally, there is no priceor riskpremiumattached to it.

What Risk is Priced?
To make clear what risk is priced, denote Mi = Cov(~Rm ; eRi ),which is the i th element of M . Also let im be thecorrelation between eRi and ~Rm .Then equation (8) can be rewritten as
Ri Rf =Mi2m
(Rm Rf ) =Mim
(Rm Rf )m
(12)
= mii(Rm Rf )
m= miiSe
where Se (RmRf )m
is the equilibrium excess return on themarket portfolio per unit of market risk.

What Risk is Priced? contd
Se (RmRf )m
is known as the market Sharpe ratio.
It represents the market priceof systematic ornondiversiable risk, and is also referred to as the slope of thecapital market line, where the capital market line is thee cient frontier that connects the points Rf and
Rm;m
.
Now dene !mi as the weight of asset i in the market portfolioand Vi as the i th row of V . Then
@m@!mi
=12m
@2m@!mi
=12m
@!mV!m
@!mi=
12m
2Vi!m =1m
nXj=1
!mj ij
(13)where ij is the i ; j th element of V .

What Risk is Priced? contd
Since ~Rm =nPj=1!mj~Rj , then Cov(~Ri ; ~Rm) =
nPj=1!mj ij . Hence,
(13) is@m@!mi
=1mCov(~Ri ; ~Rm) = imi (14)
Thus, imi is the marginal increase in market risk,m ,from a marginal increase of asset i in the market portfolio.Thus imi is the quantity of asset is systematic risk.
We saw in (12) that imi multiplied by the price ofsystematic risk, Se , determines an assets required riskpremium.

Zero-Beta CAPM (Black, 1972)
Does a CAPM hold when there is no riskless asset?Suppose the economy has I total investors with investor ihaving a proportion Wi of the economys total initial wealthand choosing an e cient frontier portfolio !i = a+ bRip ,where Rip reects investor is risk aversion.Then the risky asset weights of the market portfolio are
!m =IXi=1
Wi!i =IXi=1
Wia+ bRip
(15)
= aIXi=1
Wi + bIXi=1
Wi Rip = a+ bRm
where Rm PIi=1Wi Rip .

Zero-Beta CAPM contd
Equation (15) shows that the aggregate market portfolio, !m ,is a frontier portfolio and its expected return, Rm , is aweighted average of the expected returns of the individualinvestorsportfolios.
Consider the covariance between the market portfolio and anarbitrary risky portfolio with weights !0, random return ofeR0p , and mean return of R0p :
CoveRm ; eR0p= !m0V!0= a+ bRm0 V!0 (16)
=
&V1e V1 R
& 2 +V1 R V1e
& 2Rm
0V!0

=&e 0V1V!0 R 0V1V!0
& 2
+ Rm R 0V1V!0 Rme 0V1V!0
& 2
=& R0p + RmR0p Rm
& 2
Rearranging (16) gives
R0p =Rm & Rm
+ CoveRm ; eR0p & 2
Rm (17)
Recall from (2.32) that 2m =1 +
(Rm )2
&2George Pennacchi University of Illinois

Multiply and divide the second term of (17) by 2m , and addand subtract
2
from the top and factor out from thebottom of the rst term to obtain:
R 0p =
&
2
2Rm
+ CoveRm ; eR0p2m

capm, arbitrage, and linear factor models · pdf file3.1: capm 3.2: arbitrage 3.3: apt 3.4:...

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