ASSET VALUATION MODELS - CAPM & APT doc

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 ass

CHAPTER FIVE: ASSET VALUATION

MODELS - CAPM & APT

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)

• 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.

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

–

• 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:

–

–

• 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

)

~ ( )

~ ( )

~ (

m i

p

R E R E a

R E









assets all

of value market

asset individual

the of value market

w i 

)

~ ( ) 1 ( )

~ ( )

~ (R p aE R i a E R m

2 / 1 2

2 2

2

] ) 1 ( 2 )

1 ( [

)

~ (R p a i a m a aim

p

R

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:

] 4 2

2 2

2 [ ]

) 1 ( 2 )

1 ( [

2

1 )

~

im im

m m

i im

m i

p

a a

a a

a a

a a

R









































)

~ ( )

~ ( )

~ (

a p

R E R E a

R E











m

m im im

m m

a p

a

E













2 2 / 1 2

2

1 )

~











 



m m im

m i

a p

a R

a R E







 ( ) /

)

~ ( )

~ ( /

)

~ (

/ )

~ (

2 0

















m

f

R E





)

~ (

m m im

m i

m

f

R E







)

~ ( )

~ ( )

~ (

2









2

] )

~ ( [ )

~ (

m

im f m f

R E











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 ( ) ( )

(R p aE R m a E R z

E   

2 / 1 2

2 2

2

] )

1 ( 2 )

1 ( [

)

~ (R p a m a z a a r zmzm

m

z m

p

a R

a R E





) ( ) ( /

) (

/ )













E(R m),  (R m)

) (R z E

p m

z m

z p

R E R E R E R

) ( ) ( [ ) ( )





) ( ) ( ) (

z m

p

R E R E a

R E









] 2 2 2

[ ] ) 1 ( [

2

1 )

z z

m z

m p

a a

a a

a

R

























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 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 arbitrage portfolio

• Having no risk and earning no return on average

i k ik i

i

0

1







n

i i

w

Deriving APT

• Return of the arbitrage portfolio:

• To obtain a riskless arbitrage portfolio, one

needs to eliminate both diversifiable and

nondiversifiable risks I.e.,

























i

i i i

k ik i i

i i i

i i

n

i

i i p

w F

b w F

b w R

E w

R w R



)

1







i

ik i

i n w b for all factors

n

w 1 , , 0

Deriving APT





i

i i

0 )



i

i

w

How does E(Ri) look like? a linear combination

of the sensitivities

k each

for b

w

i

ik

 As:

• There exists a set of k+1 coefficients, such that,

–

• If there is a riskless asset with a riskless rate of

–

• In equilibrium, all assets must fall on the arbitrage

pricing line.

0



ik k i

R

E(~ )  0  1 1   

ik k i

f

i R b b R

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.