L6: CAPM & APT 1

Lecture 6: CAPM & APT

• The following topics are covered:– CAPM– CAPM extensions– Critiques– APT

L6: CAPM & APT 2

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.

L6: CAPM & APT 3

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

– (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:– (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 with respect to the percentage of the portfolio as follows.

)~

()~

()

~(

mip REREa

RE

assetsallofvaluemarket

assetindividualtheofvaluemarketwi

)~

()1()~

()~

( mip REaRaERE

2/12222 ])1(2)1([)~

( immip aaaaR

pR

L6: CAPM & APT 4

Derivation of CAPM

• Evaluating the two equations where a=0:

• The slope of the risk-return trade-off:

• Recall that the slope of the market line is:

;

• Equating the above two slopes:

]42222[])1(2)1([2

1)~

( 2222/12222imimmmiimmi

p aaaaaaaa

R

)~

()~

()

~(

0 miap REREa

RE

m

mimimmma

p

a

E

2

22/120 )22()(

2

1)~

(

mmim

mia

p

p RERE

aR

aRE

/)(

)~

()~

(

/)~

(

/)~

(20

m

fm RRE

)

~(

mmim

mi

m

fm RERERRE

/)(

)~

()~

()~

(2

2])

~([)

~(

m

imfmfi RRERRE

L6: CAPM & APT 5

Extensions of CAPM

1. No riskless assets

2. Forming a portfolio with a% in the market portfolio and (1-a)% in the minimum-variance zero-beta portfolio.

3. The mean and standard deviation of the portfolio are:–

–

4. The partial derivatives where a=1 are:– ;

– ;

5. Taking the ratio of these partials and evaluating where a=1:–

6. Further, this line must pass through the point and the intercept is . The equation of the line must be:

–

)()1()()( zmp REaRaERE 2/12222 ])1(2)1([)

~( mzzmzmp raaaaR

m

zm

p

p RERE

aR

aRE

)()(

/)(

/)(

)(),( mm RRE

)( zRE

pm

zmzp

RERERERE

]

)()([)()(

)()()(

zmp REREa

RE

]222[])1([2

1)(2222/12222zzmzm

p aaaaa

R

L6: CAPM & APT 6

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

L6: CAPM & APT 7

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.

L6: CAPM & APT 8

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.

L6: CAPM & APT 9

Arbitrage Pricing Theory• Assuming that the rate of return on any security is a linear function of k

factors:

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 factorFk=the mean zero kth factor common to the returns of all assetsεi=a random zero mean noise term for the ith asset

• We create arbitrage portfolios using the above assets. • • No wealth -- arbitrage portfolio

• Having no risk and earning no return on average

ikikiii FbFbRER ...)( 11

01

n

iiw

Deriving APT• Return of the arbitrage portfolio:

• To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks. I.e.,

L6: CAPM & APT 10

iii

ikiki

iii

iii

n

iiip

wFbwFbwREw

RwR

...)( 11

1

i

ikii factorsallforbwnn

w 0,,1

Deriving APT

L6: CAPM & APT 11

i

iip REwR )(

0)( i

ii REw

How does E(Ri) look like? -- a linear combination of the sensitivities

keachforbwi

iki 0As:

L6: CAPM & APT 12

APT

• There exists a set of k+1 coefficients, such that,– (6.57)

• If there is a riskless asset with a riskless rate of return Rf, then b0k =0 and Rf = – (6.58)

• In equilibrium, all assets must fall on the arbitrage pricing line.

0

ikkii bbRE ...)~

( 110

ikkifi bbRRE ...)( 11

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.

L6: CAPM & APT 13

L6: CAPM & APT 14

Example

• Page 182

• Empirical tests– Gehr (1975)– Reinganum (1981)– Conner and Korajczyk (1993)

FF 3-factor Model

• http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_factors.html

L6: CAPM & APT 15