3. Arbitrage Pricing Model - univie.ac.at · Asset Management Youchang Wu 3. Number of...

3. Arbitrage Pricing Model

Dr Youchang WuDr. Youchang WuWS 2007

Asset Management Youchang Wu 1

Single factor modelSingle factor modelA ti A t t l t d ONLYAssumptions: Asset returns are correlated ONLY because they all respond to one common risk factor, which is usually the market return.factor, which is usually the market return.

jiifCOVrr jijtmtjjjt ≠=++= 0),(, εεεβα

Macro events affect returns of all assetsMicro events affect only individual assetsMicro events affect only individual assetsAfter controlling for the common risk factor, there is no correlations between returns of different assets.It follows that )(),( 2

mjiji rrrCOV σββ=


jj

Portfolio variancePortfolio varianceTotal variance = Systematic risk + idiosyncratic riskTotal variance = Systematic risk + idiosyncratic risk

+= rr jmjj )()()( 2222 εσσβσ

+=

∑∑

rrnn

pmpp

jmjj

)()()(

)()()(

2222

2222 εσσβσ

β

+∞→→

+=

∑

∑∑==

nasrw

wrw

n

ii

imii

i

)()(

)()()(

22

1

2222

1

σβ

εσσβ

Factor structure greatly simplies the calculation of portfolio variance

+∞→→ ∑=

nasrw mii

i )()(1

σβ

Factor structure greatly simplies the calculation of portfolio variance. This is the main motivation for Sharpe (1963) to introduce the factor model.


Number of parametersNumber of parameters

To estimate the variance of a portfolio with N assets, originally we would need to estimate– N variances– N*(N-1)/2 covariances

With the factor structure, we only need to estimate– N idiosyncratic variances– N betas– 1 factor variance


Multifactor modelsMultifactor models

Security returns are driven by multiple common risk factors

If th i k f tjtLtLjtjtjjjt FFFr εβββα +++++= ...2211

If the common risk factors are uncorrelated with each other, then

...)()()()()( 222

121

22

11

2n

iii

n

iiip FwFwr σβσβσ ∑∑

==

++=

)()()( 2

1

222

1

11

i

n

iiL

n

iLii

ii

wFw εσσβ ∑∑==

==

++


11 ii

Multifactor model: exampleMultifactor model: exampleE l 2 f tExample: 2 factors

β β Residualβ1 β2 Residual variance

Security I 0.6 0.2 0.05Security II 0.9 0.1 0.02

Assume that σF1²=0.12 and σF2²=0.10 and

y

COV(F1, F2)=0. Compute the variance of the following portfolio wI=wII=0.5


Constructing zero beta portfolioConstructing zero-beta portfolioE lExample:

Security β βSecurity β1 β2A -0.4 1.75B 1.6 -0.75C 0.667 -0.25C 0.667 0.25

How can you construct a zero-beta portfolio?


Rewriting the factor modelsRewriting the factor modelsF i li it d fi f th t dFor simplicity, define fi as the unexpected component of Fi,

)(FEFfBy definition E(f )=0

)( iii FEFf −≡By definition, E(fi)=0Rewrite the factor models as

Asset return = expected return + response to

jtLtLjtjtjjjt fffrEr εβββ +++++= ...)( 2211

Asset return = expected return + response to unexpected factor realization + idiosyncratic return


APTAPT

R (1976) d th t t tRoss (1976) argued that to prevent arbitrage, we must have

∑=

+=L

llljfj rrE

1

)( λβ

where λl is the risk premium of factor l, i.e., the expected excess return of an asset

l 1

the expected excess return of an asset with βl=1 and βi=0 for all i≠l.=>Expected returns increase linearly with the loadings on common risk factors


A simple derivation of APTA simple derivation of APT

For simplicity, consider just the single factor model without idiosyncratic risk

The return of a portfolio with two assets isfrEr jjj β+= )(

The return of a portfolio with two assets is

If we set

fwwrEwrwErwwrr jijijip ])1([)]()1()([)1( ββ −++−+=−+=

)/( βββIf we set then this portfolio is risk free, therefore its

)/( jijw βββ −−=

expected return must be rf.Otherwise there will be arbitrage opportunity


A simple derivation of APTA simple derivation of APTIt follows thatIt follows that

βββ

βββ

=++− fjj

ij rrErE )()1()(

ββ

ββββ

−=−⇒

−−

fijfji

jiji

EErrErrE

)()(])([])([

The derivation is similar for the case of multiple factors without

λββ

≡−

=−

⇒j

fj

i

fi rrErrE )()(

pidiosyncratic riskWhen they are idiosyncratic risk, APT hold only approximately.Key argument: If APT does not hold, then arbitrageurs can create a y g , gwell-diversified zero beta portfolios with an expected return different from the risk free rate.This would make arbitrage possible!


Arbitrage in expectationsArbitrage in expectations


Arbitrage in expectations (2)Arbitrage in expectations (2)


APT ExampleAPT ExampleSecurity β β Expected returnSecurity β1 β2 Expected return

A 1.0 1.6 16.2B 3.0 2.8 21.6rf 0 0 0 0 10 0rf 0.0 0.0 10.0

Suppose APT holdspp

What is the risk premium for factor 1?

What is the risk premium for factor 2?p


What are the risk factorsWhat are the risk factorsAPT d t if h t th f t t d fAPT does not specify what the factors stand forThe factor structure is ambiguous. If securities’ returns can be written using the k-factor vector f and the n x k beta matrix B, they

l b itt i f t t f* dcan also be written using a factor vector f* and a beta vector B* as long as

,*1* BLBandfLf == −

where L is a non-singular k × k matrix.


Chen Roll and Ross (1986)Chen, Roll and Ross (1986)Ch R ll d R (1986 “E i fChen, Roll and Ross (1986, “Economic forces and the stock market”, Journal of Business) find the following macroeconomic factors arethe following macroeconomic factors are relevant– Unanticipated changes in industrial production– Unanticipated changes in industrial production– Unanticipated changes in inflation– Unanticipated changes in risk premiums (asUnanticipated changes in risk premiums (as

measured by the return difference between low grade and high grade bonds

– Unanticipated changes in the term structure of interest rates


Why are those factors chosenWhy are those factors chosenStocks prices in a discounted cash flow frameworkStocks prices in a discounted cash flow framework

[ ]CEP =

– E[C] = expected dividends– r = discount rate

r

Relevant factors are those that influence either the discount rate or the expected dividends:

Discount rate:– Discount rate:• Change in risk premium• Changes in Term Structure

Dividends:– Dividends:• Changes in inflation rate• Changes in industrial production


Construction of the factorsConstruction of the factors

Industrial production growth:( ) ( ) ( )1ll PIPIMP

Inflation:( ) ( ) ( )1lnln −−= tPItPItMP

– Unexpected inflation rate( ) ( ) ( )[ ]1= ttIEtItIU

– Changes in expected inflation rate

( ) ( ) ( )[ ]1−−= ttIEtItIU

g p( ) ( ) ][ ( ) ][ 11 −−+= ttIEttIEtIED


Construction of the factors (2)Construction of the factors (2)Risk premium:

( ) ( )tBGLreturnbondunderandBaatUPR −= ""note that UPR(t) is negatively related to unexpected changes in aggregate risk aversion

( ) ( )tBGLreturnbondunderandBaatUPR =

changes in aggregate risk aversionTerm Structure:

( ) ( ) ( )1tBTtBGLtSTUAdditionally examined variables:

EWNY(t) t ll i ht d NYSE i d

( ) ( ) ( )1−−= tBTtBGLtSTU

– EWNY(t): return on equally weighted NYSE index– VWNY(t): return on value-weighted NYSE index


Empirical ResultsEmpirical Results

Model equations:eSTURPUIUIEDPMaR UTSUPRUIDEIMP ++++++= βββββ

Results (see the table):εβββββα ++++++= UTSUTSUPRUPRUIUIDEIDEIMPMP bbbbbR

Results (see the table):• bMP > 0

b 0• bDEI < 0• bUI < 0• bUPR > 0• bUTS<0


Results (Panel C of table 4)Results (Panel C of table 4)


Results (Panel D of table 4)Results (Panel D of table 4)


InterpretationInterpretation

bMP>0: stocks less sensitive to production risk are more valuable (therefore has a lower return)bUPR>0: stocks less sensitive to the return of low-grade bond returns (relative to that of high-grade g ( g gbonds) are more valuablebUI<0,bDEI<0: stocks doing well when inflation isbUI 0,bDEI 0: stocks doing well when inflation is unexpectedly high are more valuableb <0: stocks doing well when long-term realbUTS<0: stocks doing well when long-term real rate is unexpectedly low are more valuable.


Further remarksFurther remarks

Market return as a factor:– The macroeconomic factors don’t lose their

significance, even if market return is included as a factor.

– The influence of market return is not significantsignificant

– CAPM would suggest the oppositeTh f th lt t APT i t– Therefore the results support APT against CAPM


APT and portfolio managementAPT and portfolio management

S R ll R d S A R 1984 ThSee Roll, R. and S. A. Ross, 1984, The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning, Financial Analyst Journal, p. 14-26.The central focus of portfolio strategy is the choices of an appropriate pattern of t e c o ces o a app op ate patte ofactor sensitivitiesThis choice should depend on theThis choice should depend on the economic characteristics of the investors


ExampleExample

Two factors:inflation and industrial production.Is zero exposure to both factors desirable?– No hedge against

inflationinflation– No risk premium


Analyzing portfolio strategiesAnalyzing portfolio strategies

Analysis of the investors’ economic situation – What is the investor’s expenditure structure?

What is the investor’s income structure?– What is the investor s income structure?– E.g. Institutional investors: Expenditures are

l i fl d b f d i b t th iless influenced by food prices; but their travelling expenses are likely to be higher th th f i tthan those of an average investor.

Compare with the market portfolioAsset Management Youchang Wu 27

p p

Strategic portfolio decisionsStrategic portfolio decisions

If an institution does not need to hedge so much against inflation due to low sensitivity to food price ⇒ choose a low βI (relative to the average).βI ( g )If an institution does not need to hedge so much against production fluctuationsagainst production fluctuations⇒ choose a high βIP.By bearing such additional systematic risks theBy bearing such additional systematic risks the institution can earn a higher expected return


Tactical portfolio decisionsTactical portfolio decisions

If the firm’s idiosyncratic risk depends on energy cost => heavily invest in the stocks of energy sector.If the firm is unconcerned about inflation inIf the firm is unconcerned about inflation in agricultural price=> skew portfolio holdings out of this sector=> skew portfolio holdings out of this sectorCommon mistake: pension funds investing in the stock of the sponsoring firmstock of the sponsoring firm


APT and CAPMAPT and CAPMAPT ll f lti l i k f t b t iAPT allows for multiple risk factors but requires zero correlation among idiosyncratic returnsCAPM considers only one factor but does not requiredCAPM considers only one factor but does not required zero correlation among idiosyncratic returnsCAPM is derived under the assumption that investors phold mean-variance efficient portfolios, while APT is derived under the assumption of no arbitraged in expectationsexpectationsCAPM recommends investors to hold the market portfolio, while APT suggests investors to optimize p gg pexpoures to risk factors acording to their own profiles.


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