Capital Asset Pricing Model and the Arbitrage Pricing

Theorem

Lecture XXV

Deriving the Capital Asset Pricing Model

E

tz

fr

t S

Using the results from the Expected Value-Standard Deviation frontier, we can derive the security market line (SML)

Security Market Line (SML) iE R

1 i

Which is consistent with the standard CAPM relationship

Starting with a two-asset portfolio, we construct a portfolio using investment and asset .

j mE r r E r r

1

2 22 2 2

1

2 1 1

p i m

p i im m

E R aE R a E R

R a a a a

Next, we examine the risk/return relationship based on changes in the share of asset .

1

22 2 2 2

2 2 2

12 1 1

2

2 2 2 2 4

p

i m

p

i im m

i m m im im

E RE R E R

a

Ra a a a

a

a a a

Consider what happens as the share held in asset becomes small

2

0

2

1

p im m

ma

im m mm

m m

R

a

The risk/return relationship as the share in asset becomes small is then computed as

0

1

1

1

p

i m

mp

a

i m

m

E RE R E Ra

Ra

E R E R

This relationship then yields

1

1

1

m f

m

m fi m

m m

i m m f

m f m f

i f m f

E R r

E R rE R E R

E R E R E R r

E R r E R r

E R r E R r

Empirical Tests of CAPM

The typical estimation procedure for empirically testing the CAPM is a two step model.

First, the annual returns are estimated as a function of the returns on the market portfolio:

jt j j mtR a b R

Using the estimated results from the first estimation, the SML is estimated across equations

0 1ˆ ˆj j jR b u

The “testable” implications of the CAPM are:

The intercept term 0 should be equal to zero.

Beta should be the only factor that explains the rate of return on a risky asset.

The relationship in beta should be linear.

The coefficient 1 should be equal to Rmt-Rft.

Uses of CAPM

Risk Adjusted Discount Rate

0

0

ej

P PR

P

0

0

0

0

0

1

1

1

e

j f m f

ef m f

ef m f

e

f m f

E P PE R R E R R

P

E PR E R R

P

E PR E R R

P

E PP

R E R R

Certainty Equivalent Approach

0

0

0

0

0

1

1

1

e

f m f

e m f

f

e m f

f

E PR E R R

P

E P E R R PR

P

E P E R R PP

R

Arbitrage Pricing Theorem

Again, the concept is to show under what conditions a riskless, wealthless trade provides no expected rate of return. The basic construct for this argument is the arbitrage portfolio w. The arbitrage portfolio is defined as that set of purchases and sales that leaves wealth unchanged:

1 0w

The vector form of the factor model is

where R is the vector of returns on assets

E is the vector of expected returns on the assets

F is a matrix of factor loadings relating changes in the common factors with the fluctuations in asset returns

is a vector of idiosyncratic risks .

t t tR E bF

t

{a diagonal matrix}

0

0

{a diagonal matrix}

t t

t s

t t

t t

E

E

E F

E F F

t t tw R w E w bF w

1 1 2i it i i i i t i i kt i iti i i i i

w R w E w b F w b F w

E w RR w E w R E w R

E w E w bF w E w F b w w w EE w

E w EE w w EF b w w E w w bFE w w bFF b w

w bF w w E w w F b w

w w w EE w

V w R w b b w w w

0w b

1

2

0

0

0

i ii

i ii

i iki

w b

w b

w b

1 0 0ii

w w No cost portfolio

0 0i ii

w E w E No profit

0w b

No Risk

The algebraic consequence of this statement is that

or the expected return on an asset is a linear function of the factor loadings. This result is identical to the CAPM results if there is a single factor.

0 1 1i i k ikE b b