capm and apt (rohit)

8/9/2019 Capm and APT (Rohit)

1/24

Return and Risk :CAPM and APT

Reference: RWJ Chp. 11


2/24

Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct azero investment portfolio with a sure profit.

Since no investment is required, an investorcan create large positions to secure largelevels of profit.

In efficient markets, profitable arbitrage

opportunities will quickly disappear.


3/24

Factor Models: Announcements,Surprises, and Expected Returns

The return on any security consists of two parts.

First the expected returns

Second is the unexpected or risky returns.

A way to write the return on a stock in thecoming month is:

returntheofpartunexpectedtheis

returntheofpartexpectedtheis

where

U

R

URR +=


4/24

Factor Models: Announcements,Surprises, and Expected Returns

Any announcement can be broken down into twoparts, the anticipated or expected part and thesurprise or innovation:

Announcement = Expected part + Surprise. The expected part of any announcement is part of

the information the market uses to form theexpectation, Rof the return on the stock.

The surprise is the news that influences theunanticipated return on the stock, U.


5/24

Risk: Systematic and Unsystematic

A systematic riskis any risk that affects a large number ofassets, each to a greater or lesser degree.

An unsystematic riskis a risk that specifically affects a singleasset or small group of assets.

Unsystematic risk can be diversified away.

Examples of systematic risk include uncertainty about generaleconomic conditions, such as GNP, interest rates or inflation.

On the other hand, announcements specific to a company,

such as a gold mining company striking gold, are examples ofunsystematic risk.


6/24

Risk: Systematic and Unsystematic

Systematic Risk; m

Nonsystematic Risk;

n

Total risk; U

We can break down the risk,U

, of holding a stock into twocomponents: systematic risk and unsystematic risk:

riskicunsystemattheis

risksystematictheis

where

becomes

m

mRR

URR

++=

+=


7/24

Systematic Risk and Betas

The beta coefficient, , tells us the response of thestocks return to a systematic risk.

In the CAPM, measured the responsiveness of asecuritys return to a specific risk factor, the return on themarket portfolio.

)(

)(2

,

M

Mii

R

RRCov

=

We shall now consider many types of systematic risk.


8/24

Systematic Risk and Betas For example, suppose we have identified three

systematic risks on which we want to focus:1. Inflation

2. GDPgrowth

3. The dollar-euro spot exchange rate, S($,)

Our model is:

riskicunsystemattheis

betarateexchangespottheis

betaGDPtheisbetainflationtheis

FFFRR

mRR

S

GDP

I

SSGDPGDPII ++++=

++=


9/24

Systematic Risk and Betas: Example

Suppose we have made the following estimates:

1. I= -2.30

2. GDP

= 1.50

3. S= 0.50.

Finally, the firm was able to attract a superstar CEO

and this unanticipated development contributes 1% tothe return.

FFFRR SSGDPGDPII ++++=

%1=

%150.050.130.2+++=

SGDPI FFFRR


10/24


We must decide what surprises took place in the systematicfactors.

If it was the case that the inflation rate was expected to be by3%, but in fact was 8% during the time period, then

FI = Surprise in the inflation rate

= actual expected

= 8% - 3%= 5%

%150.050.130.2 +++= SGDPI FFFRR

%150.050.1%530.2+++=

SGDP FFRR


11/24


If it was the case that the rate ofGDPgrowthwas expected to be 4%, but in fact was 1%,

then

FGDP

= Surprise in the rate ofGDPgrowth

= actual expected

= 1% - 4%

= -3%

%150.050.1%530.2 +++= SGDP FFRR

%150.0%)3(50.1%530.2 +++= SFRR


12/24


If it was the case that dollar-euro spot exchange rate,

S($,), was expected to increase by 10%, but in factremained stable during the time period, then

FS= Surprise in the exchange rate

= actual expected

= 0% - 10%

= -10%

%150.0%)3(50.1%530.2 +++= SFRR

%1%)10(50.0%)3(50.1%530.2 +++= RR


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Finally, if it was the case that the expected returnon the stock was 8%, then

%150.0%)3(50.1%530.2 +++= SFRR

%12

%1%)10(50.0%)3(50.1%530.2%8

=

+++=

R

R

%8=R


14/24

Portfolios and Factor Models

Now let us consider what happens to portfolios of stocks wheneach of the stocks follows a one-factor model.

We will create portfolios from a list ofNstocks and will capturethe systematic risk with a 1-factor model.

The ith stock in the list have returns:

iiii FRR ++=


15/24

Relationship Between the Return onthe Common Factor & Excess Return

Excess

return

The return on the factor F

i

iiii FRR +=

If we assumethat there is no

unsystematic

risk, then i=

0


16/24


Excess

return


If we assumethat there is no

unsystematic

risk, then i=

0

FRR iii =


17/24


Excess

return


Differentsecurities will

have different

betas

0.1=B

50.0=C

5.1=A


18/24

Portfolios and Diversification We know that the portfolio return is the

weighted average of the returns on theindividual assets in the portfolio:

NNiiP RXRXRXRXR +++++= 2211

)(

)()( 22221111

NNN

N

P

FRX

FRXFRXR

+++

++++++=

NNNNNN

P

XFXRX

XFXRXXFXRXR

+++

++++++=

222222111111

iiii FRR ++=


19/24

Portfolios and DiversificationThe return on anyportfolio is determined by three sets of parameters:

In a large portfolio, the third row of this equation

disappears as the unsystematic risk is diversified away.

NNPRXRXRXR +++= 2211

1. The weighed average of expected returns.

FXXXNN)( 2211 ++++

2. The weighted average of the betas times the factor.

NNXXX ++++ 2211

3. The weighted average of the unsystematic risks.


20/24

Portfolios and Diversification

So the return on a diversifiedportfolio isdetermined by two sets of parameters:

1. The weighed average of expected returns.

2. The weighted average of the betas timesthe factorF.

FXXX

RXRXRXR

NN

NNP

)( 2211

2211

++++

+++=

In a large portfolio, the only source of uncertainty is the

portfolios sensitivity to the factor.


21/24

Betas and Expected Returns

The return on a diversified portfolio is the sum of the expectedreturn plus the sensitivity of the portfolio to the factor.

FXXRXRXR NNNNP )( 1111 +++++=

FRRP

P

P

+=

NNP RXRXR ++= 11

thatRecall

NNP XX ++= 11

and

PR P


22/24

Relationship Between & ExpectedReturn

If shareholders are ignoring unsystematicrisk, only the systematic risk of a stock canbe related to its expectedreturn.

FRR PPP +=


23/24

Relationship Between & ExpectedReturn

Ex

pec

ted

return

FR

AB

C

D

SML

)( FPF RRRR +=


24/24

The Capital Asset Pricing Model andthe Arbitrage Pricing Theory

APT applies to well diversified portfolios and notnecessarily to individual stocks.

With APT it is possible for some individual

stocks to be mispriced - not lie on the SML. APT is more general in that it gets to an

expected return and beta relationship without theassumption of the market portfolio.

APT can be extended to multifactor models.

capm and apt (rohit)

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