asset pricing models.doc

8/19/2019 asset pricing models.doc

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BACHELOR THESIS

AssetPricing: Theory andEvidence

Filip Hájek

Charles University in Prague

Faculty of Social Sciences

Institute of Economic Studies

ay !""#


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CharlesUniversityinPrague

acultyo!socialsciencesInstituteo!Econo"icStudies

BACHELORTHESIS

#Asset Pricing: Theoryand Evidence$

Author:ili%H&'e(

Tutor: Ph)r* To"&+ ,aro+

ield o! study: Econo"ic theory

Acade"icyear: -../0-..1


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Pronounce"ent

I confirm that I have made this $achelor%s thesis independently and that I have used onlythe sources indicated in the thesis&

'''''''''''

Eisenstadt( ay !)& !""# Filip Hájek


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Ac(no2ledge"ent:

I *ould like to thank all *ho have helped me *ith my thesis and supported me

during my studies&


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BACHELOR3S THESISPROPOSAL

Su4"ission deadline: Spring semester !""+,!""#

Thesis author: Filip Hájek Thesis advisor: -omá. /aro.

Title: ASSETS 5ALUATIO6:THEOR7A6) E5I)E6CE

State"ento! o4'ectives:

Ever since 0illiam Sharpe and /ohn 1intner came up *ith the capital asset

pricing model 2C3P4 there has $een an academic de$ate over its validity& 5et over #"years later( the model is still *idely used in academia as *ell as practice&

-he model *as the first apparently successful attempt to sho* ho* to assess the

risk of the cash flo* from a potential investment project and ho* to estimate the project%scost of capital( *hich is the e6pected rate of return that investors *ill demand if they are

to invest in the project& In this model( a stock%s risk is summari7ed $y its $eta *ith themarket portfolio of all invested *ealth&

8ver the years C3P has come under heavy research a$out the validity of theassumption as *ell as on the validity of the predictions and the thesis *ill try to highlight

the most important papers that has shaped this rich academic de$ate&

1ately research has also focused on $ringing alternatives to Sharpe and 1intner%sC3P& -he market9derived capital9pricing model 2CP4 is $ased on financial marketinstruments( namely $ond yields and options prices and *ill $e introduced and used for

empirical estimates&-he purpose of the thesis is to $ring comprehensive summary of the C3P

de$ate and the CP and compare models empirically&

Thesis outline:

:& Capital 3sset Pricing odela& C3P description and theoretical frame*ork

$& C3P de$ate!& arket9derived Pricing odel+& Empirical evidence#& Conclusion

Preliminary literature:

;realey(


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A4stra(t

;akalá ská práce na tma 8ceGován aktiv = -eorie a Pra6e se skládá 7e dvou

v7ájemn doplGujcch se Jást& K prvn teoretick Jásti je p edstaven C3P a

nejd4leLitj. práce 7 akademick de$aty o jeho platnosti& Aále je vysvtlen teoretickM

7áklad konkurenJnho modelu CP( kterM Jerpá informace 7 opJnch a dluhopisovMch

trh4& Následuje kapitola vnujc se kritice tohoto postupu&

Nava7ujc Jást je empirická& /sou 7de odhadnuty ceny aktiv Jty spoleJnost

o$chodovanMch na americkMch $ur7ách a to dvma alternativnmi 7p4so$y& Nejprve

pomoc odhadu koeficientu $eta danMch spoleJnost prost ednictvm metody nejmen.ch

Otverc(QdodateOnRQpomocQmetodikyQCPQmodelu&

A4stract

;achelor thesis on the topic 3sset Pricing= -heory and Evidence consists of t*o

complementary parts& In the first part( C3P is introduced and the most important

articles of the ongoing academic de$ate of its validity are follo*ed& Frame*ork of rival

model arket9derived 3sset Pricing odel 2CP4 $ased on the information from

$ond and option markets is also introduced in this chapter and its critiue is provided&

Follo*ing part is empirical& -here are estimations of asset prices of four

companies traded on the US stock e6changes employing $oth models& Firstly $eta

estimation via ordinary least suare method for C3P is done and su$seuently asset

prices computed( secondly estimates employing CP frame*ork are performed&


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Content

8* Introduction**************************************************************************************************8

-* Asset valuation"odels********************************************************************************-

-*8* Ca%italasset %ricing"odel9CAP;*************************************************-

!&:&:& C3P&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& !

!&:&:&: arko*it7 portfolio theory&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& !

!&:&:&!& -o$in%s separation theorem&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

!&:&:&+& Sharpe9 1intner%s C3P &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

!&:&:& ;lack%s version of the model &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& T

-*-* CAP )e4ate**************************************************************************************8.

!&!&:& ethodology &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&:"

!&!&!& Early tests &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&::

!&!&!&: ;lack( /ensen and Scholes%s study&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&:!!&!&!&!& Fama and ac;eth%s study&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&:!

!&!&+& Challenges &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&:#

!&!&+&:& ;an7 study on si7e effect&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&:#

!&!&+&!& Fama French = Is $eta dead V &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&:)

!&!&


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!&+& Euity return risk &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&!

!&+&& Critiue&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&!T

/* Beta esti"ation and PCcalculation*************************************************/.

/*8* CAP****************************************************************************************************/.

+&:&:& Aata&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&+"

+&:&!& arket portfolio&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&+:

+&:&+&


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1. Introduction

In the last four decades asset pricing models and Capital asset pricing model in

particular have $een dominating financial te6t$ooks& ;ased on the ingenious

o$servations( reasona$le assumptions and due to the seductive simplicity( *e are today

not surprised to see that C3P has still not found too many competitors&

Nonetheless *ithout the possi$ility of testing it against other models the research

focused on testing its validity and the validity of its assumptions& C3P came under

heavy testing( *hich conseuently led to the de$ate that had $rought together advocates

and opponents of the model& Auring last four decades the de$ate gre* $ut the crucialuestion is still not settled&

In recent years nonetheless ne* innovative models have emerged& Some of them

are $ased on information from derivative markets such as arket9derived asset pricing

model 2CP4 *hich utili7es information from $ond and option markets& -hese models

try to $ypass the pro$lems of C3P via totally different approaches in calculations of

asset prices and thus challenge the prevailing models&

-he aim of this thesis is to follo* in the second chapter the $irth of C3P as *ell

as to map the long and very interesting de$ate that has started( and further( to reveal the

theoretical frame*ork of CP& Su$seuently( in the third chapter this theoretical $ase

*ill $e used for estimates of asset prices of four companies operating on the US market

employing $oth models& 8$servations from the practical application of CP are

concentrated in the su$chapter !&+&& Critiue( *hich for the sake of structure is presented

in the second i&e& theoretical chapter&

-his thesis is follo*ing the previous research on the Institute of economic studies

at Charles University in Prague done $y


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2. Asset valuation models

2.1. Capital asset pricing model (CAPM)

2.1.1. CAPM

-he C3P $uilds on the ground$reaking paper of Harry arko*it7 2:B!4 in

*hich he descri$es ho* to select a portfolio using the mean9variance rule 2E9K rule4 and

a latter paper of -o$in 2:BT4 *hich introduces the idea of linear efficient set and also

implies t*o steps in selection of desira$le portfolio( later kno*n as searationεtheorem&

In order to derive a C3P *e have to present their findings&

2.1.1.1 Markowitz portfolio theory

arko*it7 is one of the first to attack until then prevailing theory of only

ma6imi7ing the e6pected return that *as set among others $y 0illiams 2:B+B4& Diven

0illiams% theoretical frame*ork( investor *ould invest in an asset that $ears the highest

e6pected return *ithout taking into account a risk associated *ith it( assuming that “the

la# o% lar&e numbers #ill insure that the actual yield to the ort%olio #ill be almost the

same as the e'ected yields! 20illiams :B+B( pp& )T9)B4& arko*it7 ackno*ledges the

need of using an e6pected returns and allo*ing the returns for risk :

and a trade9off

$et*een them& “Thereεisεaεrateεatε#hichεinvestorεcanε&ainεe'ectedεreturnεbyεtakin&εon

variance, or reducevarianceby&ivin&ue'ectedreturn(!2arko*it7 :B!( pp& B4

arko*it7 $uilds his theory on the follo*ing assumptions 2/ensen :BT+4=

2i4 Investors select portfolio at time t9: that produces random return < at t&

2ii4 -he uantities of all assets are given and short sales are restricted 2 "≥i ) 4 for

all i&

2iii4 Investors are risk averse 2i&e& they minimi7e portfolio return variance given

e6pected return 4

:

Follo*ing the *ork of /&


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2iv4 Investors are not saturated 2i&e& they ma6imi7e portfolio e6pected return given

variance4

2v4 3ll assets are perfectly divisi$le and perfectly liuid( i&e& all assets are

marketa$le and there are no transaction costs

2vi4 -here are no ta6es

In order to solve the pro$lem of efficient portfolio selection arko*it7 considers

e6pected return as a random varia$le( that can $e descri$ed $y t*o moments= mean 2 iµ 4

and variance 2var W!σ 4&

Using the pro$a$ility theory *e can derive for the e6pected return and variance of

the portfolio the follo*ing formulas&

For e6pected return of portfolio=

∑=

=n

i

ii ' E :

µ 2!&:4

and for variance of the return of portfolio

∑∑= =

=n

i

n

+

+ii+ ' ': :

! σσ 2!&!4

*here E is e6pected return of portfolio(!

σ is variance of the portfolio( i ' is the

percentage of the investor%s asset *hich are allocated to the i th

security( + ' is the

percentage of the investor%s asset *hich are allocated to the + th security( and iµ is the

e6pected return of ith

security& i+σ is covariance $et*een iεand +εsecurity&th th

Covariance lies in the very heart of the portfolio selection theory and therefore

needs to $e defined properly& Covariance is defined as follo*s=

+ii+i+ σσρσ = 2!&+4


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*here i+σ is covariance $et*een ith

and +th

security( iσ is standard deviation of ith

security

2 i

σ var 4( +

σ is standard deviation of j th security and i+

ρ is a correlation coefficient

$et*een securities iεand +& Correlation coefficient varies $et*een 9: and X:& -he value of

X: implies perfect correlation i&e& fully align movement( 9: implies perfect negative

correlation i&e& securities move inversely of each other and " implies independent

movement of $oth securities&

Aue to the imperfect correlation $et*een securities( diversification is rational and

desira$le( $ecause portfolio of t*o imperfectly correlated securities *ill achieve

*eighted average of e6pected returns together *ith lo*er than *eighted variance of t*o

securities( *ith *eights $eing the portions of *ealth invested& 3s to the selection of

reasona$le iµ and i+σ arko*it7 suggests “asεtoεtentative iµ and i+σ is to use the

observed iµ σ , i+ %orsomeeriodo% the ast(!2arko*it7:B!(pp&B:4

-he E9K rule restricts the desira$le portfolios to the ones on the efficiency frontier

sho*n on Figure :& ean9variance efficient portfolio lie a$ove 2;4 in the Figure: and

investors *ill select portfolio from this frontier according to their aversion to the risk( in

here represented $y the indifference curves& -he optimal portfolio *ill lie on the point

2C4 *here efficient set of portfolios touches *ith the indifference curves of investor(

giving him the highest possi$le utility *ith given restrains&

-i&ure.: Marko#it/ e%%iciency set

0ource: 0hare, Ale'ander.112,( .3.

;

I+ I! I:

Standard deviation of portfolio

E6pected


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2.1.1.2. Toin!s separation theorem

-he ne6t ground $reaking *ork that marks the evolution of C3P is /& -o$in%s

paper 4iquidityεre%erenceεasεbehaviorεto#ardsεriskε2:BT4 in *hich the author discusses

the liuidity preference in respect to the mean and variance&

He allo*s investors to choose $et*een holding their *ealth in cash or in consoles(

*hich are in the a$sence of inflation risk9free in nominal sense( and further sho*s that

given these t*o assets( different investors( *ith regard to their risk a*areness( choose a

different optimum point of investment on the investment opportunity line 2"(C4 in Figure

!( i&e& different portions in the t*o assets&

-i&ure5: Tobin6sCaitalMarket 4ine

0ource:Reilly, 7ro#n.118,(593

3dding risk9free $orro*ing 2cash4 and lending greatly simplifies arko*it7%s

efficient set&

Consider a portfolio that invests the proportion 'εof portfolio funds in a risk9free

security and .'εin portfolio &

& % R ' 'R R 4:2 −+= 2!

than the e6pected return is simplified 2due to var 2 < 4 W " $y definition4f

m %

R ' 'R ER 4:2 −+=2!&4

arket

portfoclio 2C4

1ending

< W 2"4fC

E6pected

return of

portfolio

Standard deviation of portfolio


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as *ell as the variance

4var24:24var2!

m R ' R −= 2!&)4

-hese euations imply that efficient portfolios can $e o$tained $y varying 6& -hey

then plot a straight line 2C14 in Figure !& -he line starts at < 26 W :4( runs to the pointf

C 26 W "( i&e& all funds invested in C4 and continues on for portfolios that involve

$orro*ing at the risk9free rate 26 Y "4& -he key result is that *ith unrestricted risk9free

$orro*ing and lending( all efficient portfolios are com$ination of the single tangency

portfolio 2C4 and risk9free asset 2"4 *ith either lending at the risk9free rate 2points $elo*

C along the line from < 4 or risk9free $orro*ing 2points a$ove C along the line from < 4&f f

ore importantly *hen ela$orating on the arket portfolio 2C4 -o$in proves

“that the roortionate comosition o% the noncash assets is indeendent o% their

a&&re&ate share o% the investment balance( This %act makes it ossible to describe

investor6s decisions as i% there #ere a sin&le noncash asset, a comosite %ormed by

combinin& the multitude o% actual noncash assets in %i'ed roortions(“ 2-o$in :BT( pp& T#4 *hich restricts arko*it7%s efficient portfolio set into linear com$ination of risk9

free rate and one optimal portfolio as seen in Figure ! creating linear efficient set 2"(C4&

-his implies that process of choosing optimal investment “;canεbeεbrokenεdo#n

into t#o hases: %irst, the choice o% a unique otimum combination o% risky assets< and

second, a searate choiceconcernin& theallocationo% %undsbet#eensuchacombination

and a sin&le riskless asset! 2Sharpe :B)#( pp!)4( *hich has later $ecome kno*n as

-o$in%s separation theorem&

-hese t*o papers *ere the theoretical cornerstones necessary for the $irth of

Sharpe91inter%s C3P&


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2.1.1.". #harpe$ %intner!s CAPM

In :B)# Sharpe and in :B) 1inter have pu$lished papers concerning the prices!

of assets under conditions of risk& In their *ork they tried to esta$lish a theory that *ould

ans*er the crucial uestion= ho* to precisely evaluate the relationship $et*een higher

risk 2variance4 and e6pected return and ho* to distinguish the part of the risk that market

values( $ecause Sharpe o$serves that “throu&hεdiversi%ication,εsomeεo%εtheεriskεinherent

in an assetcan be avoided so thatits total risk is obviously not the relevant in%luence on

itsrice(! 2Sharpe:B)#(pp!)4

Sharpe and 1intner e6tended assumptions of arko*it7% model *ith the

follo*ing

2i4 3ll investors can $orro* and lend an unlimited amount at an e6ogenously

given risk9free rate of interest < and there are no restrictions on short sales of f

any asset&+

2ii4 Perfect capital markets&

2iii4 3ll investors have identical su$jective estimates of the means( variances( and

covariances of return among all assets&

-i&ure3:0ecurity Market 4ine

0ource:o#n&rahbasedonReilly, 7ro#n.118, ( 593

and Elton,=ruber.11., ( 51>,3>?

!Similar unpu$lished *ork of /& -reynor = >-o*ards a theory of market value of risky asset? 2:B):4 comes

to the similar conclusions&+-his assumption is ho*ever not ne*( $ut *e can see a clear analogy in -o$in%s *ork&

:&"

arket

portfolio 2C4

1ending


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0ith complete agreement a$out distri$utions of returns( all investors form the

same tangency portfolio C *ith < & In order for the market to clear assets in the marketf

portfolio 2C4 must $e priced so its *eight in given time is its total market value divided

$y the total value of all risky assets and such portfolio includes all assets&

In such situation *e can derive the familiar Sharpe91intner C3P risk9return

relation

i % m % i R ER R ER β42 −+= 2!&4

*here iβ W4var2

4(cov2

m

mi

R

R Rcan $e ver$ally interpreted as the covariance risk of asset iεin

portfolio C( measured relative to the risk of the portfolio( *hich is just an average of the

covariance risks of all assets i&e& the measure of asset%s contri$ution to overall portfolio

risk&&

-herefore covariance is the right measure of risk as /ensen notes= > Althou&h

investors take the variance @or equivalently the standard deviation o% their ort%olio

returns as an aroriate measure o% risk, there results imly that the aroriatemeasure o% the risk o% any individual asset is its covariance #ith the market ort%olio,

4(cov2 mi R R , andnotitso#nvariance, 42 R!

iσ (! 2/ensen :BT+4

2.1.1.&. 'lack!s version of the model

3ssumption that any investor may $orro* or lend any amounts he *ants at the

risk less rate of interest *as felt as the most restrictive one and there *as a feeling that

the model *ould $e changed su$stantially if this assumption *ould $e dropped& 3lso

research *as sho*ing that model using risk9free rate doesn%t hold very good the

empirical findings 2;lack( /ensen( and Scholes :B!4( $ut rather the t*o factor model

2!&T4 holds&

In the very same year ;lack pu$lished a paper in *hich he proves that in

euili$rium the portfolios of all investors consist of a linear com$ination of t*o $asic

portfolios& -his paper is one of the first contri$utions into the C3P de$ate on *hich *e


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*ill focus in the follo*ing su$9chapter( $ut for the sake of clarity and structure *e *ill

present it here&

0hile this point *as recogni7ed earlier ( ;lack sho*ed that under a$sence of #

risk9free $orro*ing and lending 2$ut *ithout restrictions on short selling4 “everyεe%%icient

ort%olio may be #ritten as a #ei&hted combination o% the market ort%olio m and the

minimum variance /eroβ ort%olio/(!2;lack:B!( pp!4

;lack demonstrates that in euili$rium the e6pected return on any asset *ill $e

given $y

mi / ii ER ER ER ββ +−= 4:2 2!&T4

*here / ER is the e6pected return on the 7ero $eta portfolio( other defined as previously&

-herefore the e6pected return on all assets is still a linear function of their systematic risk

and the euation is similar to the Sharpe91intner( only *ith the e6ception that / ER plays

the role of < &f

#Sharpe 2:B"4 chap& # in /ensen


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2.2. CAPM Debate

-he Sharpe91intner asset pricing model has received since its introduction in mid

)"%s *idespread attention and has $een heavily tested during the last #" years& -his

section *ill try to descri$e this long academic de$ate a$out the model%s validity& -here

has $een large amount of papers pu$lished that are connected to the topic( ho*ever in this

study only the major papers are used&

Included are studies that support C3P 2;lack( /ensen and ScholesZ Fama and

ac;eth4( studies that challenge it 2;an7Z Fama and French4 and studies that challengethese challenges 2;lackZ 3mihud( Christensen and endelsonZ /agannathan and 0ang4

2.2.1. Methodolo(y

Euation 2!&4 has three testa$le implications 2Fama and ac;eth :B+4=)

2i4 -he relationship $et*een the e6pected return on a security and its risk in any

efficient portfolio mεis linear and up*ard sloping

2ii4 iβ is a complete measure of the risk of security iεin the efficient portfolio m(

i&e& no other measure of the risk of iεappears in the euation 2!&4

2iii4 In a market of risk averse investors( higher risk should $e associated *ith

higher e6pected returnZ i&e& ">− % m ER ER

Euation 2!&4 is in terms of e6pected returns( $ut its implications must $e tested

*ith data on period9$y9period security and portfolio returns& In order to test the

implications the follo*ing stochastic euation is e6amined&

+

B

+ + r εγ βγ γ +++= ∑ =!:" 2!&B4

-he significance of the papers is su$jective( and author apologi7es if the reader is missing some *orks(

that are $elieved to $e important as *ell& Such is the nature of su$jective9ness&) 3s for the ;lack%s version of the model( third implication must $e modified to ">− / m ER ER &


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::

*here rεis an estimate of the e6pected e6cess return on portfolio 2 % ER ER − 4( β is an

estimate of $eta for portfolio ( :γ is the market risk premium for $earing one unit of $eta

risk 2 % m ER ER − 4Z "γ is the 7ero $eta rate( the e6pected return on an asset *hich has a

$eta of 7eroZ + for j W !(&&(j are non9$eta e6planatory varia$les assumed to $e relevant

for asset pricing and ε is a random distur$ance term in the regression euation&

In such a regression euation *e are trying to determine *hich e6planatory

varia$les are significant& -o do so *e use the econometric theory frame*ork and *e

e6amine the t9statistics of the respectedγ coefficients looking for significance&

-he estimated coefficients "γ and :γ o$tained from the regression are then

compared to < 2Sharpe91intner version4 or return on 7ero9$eta security 2;lack version4f

and % m R R − and / m ER R − respectively& m R is an average return on the market inde6(

*hich is usually taken as a pro6y for market return( over the period&

2.2.2. )arly tests

-he first test of C3P *as pu$lished $y Aouglas 2:B)B4& Aouglas regressed the

returns on a large cross9sectional sample of common stocks on their variance and on their

covariance *ith an inde6 constructed for the sample& He came to the same results as

1inter 2unpu$lished4& 3ccording to their findings for seven separate five9year periods

from :B!) to :B)"( the average reali7ed return *as significantly positively related to the

variance of the security%s return over the time $ut not to the covariance as the model

predicted &T

iller and Scholes revie*ed the theory and Aouglas91intner evidence and cameB

to the conclusion that these results are mainly due to omitted varia$le $ias(

3rithmetic averages of annual( uarterly( or monthly returns have generally $een used for $oth % R and

m R ( $ut a num$er of authors have also used average continuously compounded rates as *ell& 2/ensen

:BT+4T

In 1intner%s test the coefficient on the residual variance *as positive and as significant as the iβ term(2$oth having t9values greater then )4B


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:!

heteroscedasticity and errors in varia$le& -herefore nothing can $e said a$out the validity

of the results *ith certainty&

2.2.2.1 'lack* +ensen and #choles!s study

;lack( /ensen and Scholes 2;/S4 2:B!4 present some additional tests of Sharpe9

1intner 2S14 asset pricing model& 0hen testing the original S1 model( they conclude that

risk9free rate doesn%t hold to the empirical findings&

-hey find that the average returns on these portfolios are not consistent *ith

euation 2!&4 especially in the post*ar period :B#)9))& -heir estimates of the e6pected

returns on portfolios of stocks at lo* levels of iβ are consistently higher than predicted

$y euation 2!&4( and their estimates of the e6pected returns on portfolios of stocks at

high levels of iβ are consistently lo*er the predicted $y the same euation&

3ccording to their findings >t*o9factor model? 2!&T4 does a $etter jo$ in

e6plaining the risk9return relationship and further *ork of ;lack also provides theoretical

$acking of these results&

2.2.2.2. ,ama and Mac'eth!s study

Fama and ac;eth 2F4 in their study Risk,εReturn,εandεEquilibrium:εEmirical

Tests 2:B!4follo*ingthe;/S *ork( haveusedthe follo*ingregression

iiiii ur εσγ βγ βγ γ ++++= 42+!

!:" 2!&:"4

*here ir is an e6pected return on security i, "γ is an e6pected return on a 7ero9$eta

portfolio( :γ is an e6pected market premium(!

iβ is the average of the!β for all

individual securities in portfolio i( 42uiσ is the average of the residual standard

deviations and iε is a random distur$ance term in the regression euation&

-heir tests *ere carried out on the !" portfolios constructed from all securities on

N5SE in the period /anuary :B+9/une :B)T& -he portfolios *ere constructed in a *ay

that *ould minimi7e the measurement error $ias pro$lem in cross9sectional estimates of

coefficient t t t !:" [([([ γ γ γ and t +[γ in euation 2!&:"4&


21/53

:+

Specifically F first estimated $eta of each stock during one period and then

formed portfolios on the $asis of these estimated $etas& Ne6t( they re9estimated the $eta

of each portfolio $y using the returns in the su$seuent period& -his ensured that the

estimated $etas for each portfolio *ere largely un$iased and free from error& Finally(

these portfolio $etas *ere plotted against returns in an even later period&2;realey and

ayers :BTT4

-he F tests are $ased on the e6amination of time series of coefficients in the

euation 2!&:"4( estimated from the cross9sectional regression for each month& -he focus

*as put on tests of three major testa$le implications of euili$rium properties of the t*o9

parameter model=

2H:4 -he risk return relationship should $e linear 9 i&e& "42 ! =γ E

2H!4 No measure of risk in additional to β should $e systematically related to

e6pected returns 9 i&e& "42 + =γ E

2H+4 -he e6pected risk9return tradeoff should $e positive 9

i&e& "42 : >−= om ER ER E γ

3part from these the test for the implication of Sharpe91inter model *as

done

2H#4 % R E =42 "γ

In addition they provide tests of the proposition that each of the period9$y9period

coefficient are eual to 7ero( "[[[[ +!:" ==== t t t t γ γ γ γ and the implication of capital

market efficiency that each of the coefficient must $ehave as a fair game through time& If

the latter reuirement *ouldn%t hold( there *ould $e a trading rules $ased on the past

values of the +γ %s *hich *ould yield a$ove normal profits \ thus a violation of the

efficient markets hypothesis& 2/ensen :B!( pp& !!4:"

From the e6amination of the average values and tεstatistics for the estimated

coefficientsγ from the euation 2!&:"4 for the entire period and for various su$9periods

:"Citation from Fama( Eugene F& :B" >Efficient Capital arkets= 3


22/53

:#

F conclude that on the level of significance= ?oneεcannotεre+ectεtheεhyothesisεD.

3 that the ricin& o% securities is in line #ith the imlications o% the t#oarameter

model %or e'ected returns( And &iven a t#oarameter ricin& model, the behavior o%

returns throu&h time is consistent #ith an e%%icient caital market ? 2F9 :B+( pp& )!4

since the average values of !γ and +γ are generally small and insignificantly different

from 7ero&

3s for the H#( Fama and ac;eth confirm the ;/S results( that 42 "γ E does not

eual Rε& -herefore the S1 hypothesis is inconsistent *ith the data( on the other hand is %

not contradictory to the ;lack%s version of the C3P&

-he F tests of the period9$y9period values of the +γ indicate( that *hile *e

cannot reject the H: and H!( *e can reject them in each month& -herefore *hile there are

no 2systematic4 nonlinearities and effects of non9portfolio risk on security returns( such

effects appear in a random fashion from period to period& -hey conclude that even though

adding these t*o varia$les to the regression raises its average adjusted coefficient of

determination for the entire period( >theεkno#led&eεo%εtheseεe%%ectsεisεo%εnoεhelεtoεthe

investor since the coe%%icients themselves behave as a %air &ame throu&h time( Thus, the

investor can aarently do no better to actas i% the t#o%actor model, su&&ested by 7B0,

is valid &? 2/ensen:BT+(pp&!#4

2.2.". Challen(es

In "%s the C3P passed its first major empirical tests& Ho*ever in :BT: came

one of the first studies suggesting that C3P might $e missing something( only to $e

follo*ed decade later $y study of Fama and French 2:BB!4 that *as *arning that C3Pmight $e missing everything&

2.2.".1. 'anz study on size effect

In :BT: ;an7 tests *hether the si7e of the firm 2market value *as used as a

pro6y4 cannot e6plain the residual variation in the average returns across assets that is not

e6plained $y C3P%s $eta&

-he empirical tests are $ased on the regression euation


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:

immiiir εφφφγ βγ γ +−++= ],4^2!:" 2!&::4

*here rεis an e6pected return on security i,i "γ is an e6pected return on a 7ero9$eta

portfolio( :γ is an e6pected market premium( iφ is a market value of security i( mφ is a

market value( and iε is a random distur$ance term in the regression euation&

;an7 uses similar procedure to the portfolio9grouping of F 2:B!4 & He creates::

! portfolios containing similar num$er of securities( first to one of five on the $asis of

the market value of the stock( then the securities in each of those five are in turn assigned

to one of five portfolios on the $asis of their $eta&2;an7 :BT:( pp&4

For the entire period( :B+)9:B( ;an7 o$tains the follo*ing estimates 2and t 9

statistics4= ^ % R−"γ ] W "&""# 2!&)4( ^ 42: % m R R −−γ ] W 9"&"""B! 29:&"4( and !γ W

9"&"""! 29!&B!4&

;ecause the t 9statistic for !γ is large in a$solute value the si7e effect is

statistically significant& -he negativity of the estimated value implies that the shares of

firms *ith large market values have had smaller returns( on average( than similar small

firms&He finds that the >avera&eεreturnεonεsmallεstocksεareεtooεhi&hε&ivenεtheir β

estimates, and avera&e returnson lar&estocksaretoo lo#&? 2Famaand French :BB!(pp&

#!4 His estimates of "γ are consistent *ith Fama and ac;eth 2:B+4 findings( i&e&

contradicting the Sharpe91intner model&

-o assess the importance of these results( ;an7 tried to specify the >si7e

premium?& He constructs t*o portfolios( each *ith !" assets& 8ne portfolio containing

only stocks of small firms( the other containing only large firms& -he portfolios are

constructed in such a *ay that they have same $eta& Auring the :B+)9:B he concludes

that “theεavera&eεe'cessεreturnεD%romεholdin&εveryεsmallε%irmsεlon&εandεveryεlar&eε%irms

short is,on avera&e.(F5 ercent er month or .1(9 ercenton anannual basis(! 2;an7

:BT:( pp& :9:)4

-hus( the C3P seems to $e missing a significant factor= firm si7e& -his

o$servation has $ecome kno*n as the si/eεe%%ect &

::

In length descri$ed in ;lack( Fisher and Scholes( yron& 2:B#4 :!εTheεe%%ectsεo%εdividendεyieldεand dividendolicyon commonstockricesand returns!( /ournalofFinancial Economics:(ay( :9!!&


24/53

:)

2.2.".2. ,ama - ,rench Is eta dead /

In their paper “TheεCross0ectionεo%εE'ectedε0tockεReturns!ε2:BB!4 Fama and

French estimated the relations in the euation 2!&B4 for the period from /uly :B)+ to

Aecem$er :BB"& Using this cross9sectional regression they confirm that si7e( earnings9

price( de$t9euity( and $ook9to market ratios add to the e6planation of e6pected returns

provided $y market $eta&

-hey grouped stocks for firms listed on the N5SE( 3E_ and N3SA3` into :"

si7e classes and then into :" $eta classes( for a total of :"" portfolios& 3naly7ing firstly

the $eta and market si7e they o$tained the estimates of :γ W 9"&+ *ith t 9statistic of 9:&!:

and !γ W 9"&: *ith t 9statistic of 9+: respectively as seen in the ta$le :&

3s for the si7e itself( the average slope from monthly regression of returns on si7e

is 9"&: ( *ith t 9statistic of 9!&T& -his negative relation persist no matter *hat varia$les

are in the regression euation and more importantly the t 9statistic is al*ays close to or

more than ! in a$solute terms and thus si7e effect is ro$ust to the inclusion of other

varia$les in the :B)+9:BB" returns&

;eta is doing uite poorly in respect to si7e( ho*ever *hen left alone as the only

e6planatory varia$le it does even *orse& -he average slope from the regressions of

returns is "&: per month *ith t 9statistic of mere ")& Fama and French conclude= “Gn

contrast to the consistent e'lanatory o#er o% si/e, the re&ression sho# that market

β does not hel e'lain avera&e stock returns %or .1?3.11>! and “#e can also reort

that β sho#s no o#er to e'lain avera&e returns in re&ression that use various

combinations o% β #ith si/e, booktomarket equity, levera&e, and EHP(! 2Fama and

French :BB!( pp& #+T4

3s for the other e6planatory varia$les( FF find that the com$ination of si7e and $ook9to9market euity seems to a$sor$ the roles of leverage and E,P in average stock

returns&

-he conseuences of their findings are severe& If market $eta fails to e6plain

e6pected returns( then the market portfolio is not efficient( and the evidence on the si7e of

market premium is irrelevant& -he punch line is simple= if $eta is dead( so is the C3P&


25/53

:

Table.(:Avera&e 0loes@tstatistics%rom MonthbyMonthRe&ressiono% 0tockReturns

on β ,0i/e,7ooktoMarket Equity,4evera&e, andEHP: Buly.1?3 to "ecember.11>

0ource: [email protected], (231

2.2.&. 0esponses

-he Fama and French 2:BB!4 study has itself $een challenged& 8$viously most

claims have $een to*ards F9F findings that $eta has no role for e6plaining cross9sectional

variation in returns( that si7e has an important role as *ell as $ook9to9market euity&

3nother line of arguments against their findings stems from the disagreement on*hether value9*eighted portfolio of all stocks listed in N5SE and 3E_ is a reasona$le

pro6y for the return on the market portfolio of all assets( and *hether assuming that $etas

are constant over time is reasona$le&

2.2.&.1. 'lack

8ne of the first *ho stood up against FF study and pointed out that they are in

some aspects contradicting their o*n finding and even that some of them are mostly due

to data mining *as ;lack 2:BB+4& 3s a clear e6ample he chose the >si7e effect?&


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:T

In :BT: ;en7 found that si7e matters and favors small companies& Fama and

French e6tended the time up to :BB" and found no si7e effect at all in the period of

2:BT:9:BB"4 *hether or not they control for $eta& Ho*ever that did not stop them from

concluding that si7e is one of the varia$les that >captures? the cross9sectional variation in

average stock return&

3ccording to ;lack such anomalies as >si7e effect? might $e just as *ell just a

result of data mining& Since there are literally thousands of researchers looking for profit

opportunities in security and all of them looking at roughly the same data( “onceεinεa

#hile, +ust by chance, a strate&y #ill seem to have #orked consistently in the ast( The

researcher #ho %inds it #rites it u, and #e have a ne# anomaly( 7ut it &enerallyvanishes as soon as it6s discovered! 2;lack :BB+( pp& B4 *hich seems to $e the case(

since after the paper *as pu$lished small firms had tame and inconsistent performance&

-he lack of theoretical $acking of their findings also points out to the possi$ility

of data mining as is the e6ample of ratio $ook value to the market value of the firm&

3ccording to ;lack “theεastεsuccessεo%εthisεratioεmayεbeεdueεmoreεtoεtheεmarket

ine%%iciencies than “riced %actors! o% the kind that -ama and -rench %avor! 2;lack

:BB+( pp& :"4 3nnouncements of the death of $eta thus to him seemed premature&

2.2.&.2.Amihud* Christensen and Mendelson

3mihud( Christensen and endelson 23C4 took a different approach and tried

to use more efficient statistical methods& Using the same data as FF they “obtainedεa

si&ni%icantly ositive coe%%icient o% avera&e return on β ( An additional imrovement in

statistical si&ni%icance #as obtained #hen =40 #as alied to account %or

heteroskedasticity over time and across ort%olios as #ell as crossort%olio

correlations(! 23mihud( Christensen and endelson :BB!( pp& :)4 -he results *ere also

ro$ust to the data used&

3nother source of possi$le invalidity of FF findings stems from the data used and

the *ay portfolios are selected& FF used the portfolio selection method esta$lished $y

F( *here stocks are selected only in they had return data for the year of study and for

the preceding seven years& -he deletion of stocks that *ere delisted during the year of

study leads to >survivorship $ias? 9 the tendency for failed companies to $e e6cluded


27/53

:B

from performance studies due to the fact that they no longer e6ist or due to the merge

and acuisition&

If( for e6ample( high9risk stocks are more likely to $e delisted due to $ankruptcy(

e6cluding them from the sample makes the average return of the surviving stocks higher

than the average return of all stocks in that risk group( *hen accounting for the loss due

to $ankruptcy& In such a case( survivorship $ias may create the appearance of a positive

risk9return relationship *here none e6ist&

3C develop a method that tries to eliminate this $ias $y adjusting the returns

and employing D1S methodology& -his approach also implies that return9 β relationship

*as positive and significant&

2.2.&.". +a(annathan and an(

/agannathan and 0ang argue that poor empirical support of the C3P might $e

due to the inappropriate pro6y for the aggregate *ealth portfolio of all agents in the

economy& ost empirical studies of the C3P assume that the return on a $road stock

market inde6( like N5SE and 3E_( might $e a reasona$le pro6y for the true market

portfolio&

-he study of Aia79Dimene7( Presscott( Fit7gerald and 3lvare7 2:BB!4 points out

that( >almostεt#othirdsεo%εnon&overnmentεtan&ibleεassetsεareεo#nedεbyεtheεhousehold

sector, and only onethird o% the cororate assets are %inanced by equity(? Furthermore(

intangi$le assets( such as human capital( are omitted in stock market inde6es& 3n

evidence that strongly undermines the *ell spread $elieve&

/agannathan and 0ang 2:BB+4 try to tackle this pro$lem $y $roadening the market

inde6 and including human capital in their measure of *ealth&:!

Since human capital is

uno$serva$le they pro6y it *ith the gro*th in per capita income&

-heir version of C3P then degenerate into

labour

l

v#

i ER βαβαα !:" ++= 2!&:!4

:!

Follo*ing ayers 2:B+4& ayers *as the first to point out the importance of including return to humancapital in measuring the return to the aggregate *ealth portfolio& 2taken from /agannathan and 0ang( :BB+4


28/53

!"

*here ERεis the e6pected return on portfolio p(iv#

β is the risk of portfolio p relative to

the value9*eighted portfolio of all stocks traded on N5SE and 3E_( and labour

l

β is the

risk of the portfolio p relative to *ealth due to the human capital&:+

0ith human capital included in this *ay /agannathan and 0ang sho* that the

C3P model is a$le to e6plain !T percent of the cross9sectional variation in average

returns in the :"" portfolios e6amined $y Fama and French 2:BB!4& -his is a very

significant raise from :&+ that can $e done *ithβ alone&

2.2.. Conclusion

-he Sharpe91intner C3P has gone over the course of #" years under heavy

research a$out its validity *hich *e have follo*ed in this su$9chapter& -his line of de$ate

ho*ever also led to parallel research&


29/53

!:

6& -he $ehavioral C3P $y Statman and Shefrin 2:BB#4 incorporating the noise

tradersJ $eliefs

of *hich *e have ela$orated on the ;lack%s rela6ation of riskless $orro*ing assumption

and some others are dealt *ith in @eller 2!""+4& 0e have not $een follo*ing this line of

C3P de$ate for the reasons *e have already mentioned( $ut it is important to highlight

the conseuences of this de$ate&

Even though many su$seuent models are either the variants or e6tensions of the

original theory of portfolio selection of arko*it7 and they are often not empirically

ro$ust they have enriched the C3P de$ate& -hey ho*ever provide a very valua$le

$enchmark( so *e can see if the assets are correctly valued and economically justified in

terms of their e'εanteεfundamentals as *ell as their e'εostε performance&

No matter the length of the de$ate( it has never lost on intensity as *ell as uality&

8ver the course of forty years *e have learned a$out C3P greatly ho*ever de$ate%s

outcome is still uncertain&


30/53

!!

2.". Market$derived asset pricin( model 3MCPM4

Diven the fierce academic de$ate a$out the usefulness of C3P in reality(

research has $een done in the field in order to come up *ith a ne* model of asset pricing

that *ould overcome the pro$lems $edeviled in the C3P formula&

8ne of the ne*ly introduced models is the market9derived asset pricing model

2CP4 $y cNulty( 5eh( Schul7e and 1u$atkin 2!""!4 that *ill $e introduced in this

section&

Instead of focusing on dra*ing information on the historical stock9to9marketcorrelation( it tries to derive the information from the options market and yields on

government and corporate $onds& Such approach has an advantage over $eta since it

ena$les investor to use e6 ante data and also give the possi$ility to derive different rates

according to the time period of the investment( compared to a >fit9for9all? $eta& Ho*ever

it also has some $ottlenecks( *hich *ill $e discussed later&

2.".1. Assumptions aout the risk

CP is trying to define the risk premiums that investors value( and give them a

proper price tag& In order to come up *ith the risk premium that investors reuire for

their investment( CP $reaks the overall risk do*n into three parts&

2i4 National confiscation risk

9 *hich measures the risk that an investor *ill lose the value of investment

due to the national policy

2ii4 Corporate default risk

9 *hich measures the risk that a company *ill default as a result of

mismanagement independent of macroeconomic considerations

2iii4 Euity returns risk

9 *hich reflects the risk that euity investor has $ecause of the residual

claim on the company%s earnings is secondary to de$t holders% claim in

$ankruptcy


31/53

!+

2.".2. 5ational confiscation and corporate default risk

Information needed for valuing $oth these risks are o$tained from the $ond

market&

;y definition $ond is certificate issued $y a company or government that

promises repayment of $orro*ed money at a set rate of interest on a particular date& 0hat

interests us is the rate of interest that government or corporation has to pay to investors

and its logic& Even though U&S& -reasury ;ills are often referred to as a risk9free

investment they $ear an interest( in order to compensate investor for the risk of default(

confiscation or runa*ay inflation& In the case of the corporation( part of the $ond yield is

due to the risk associated *ith the mismanagement of the company that is $eyond the

po*ers of managers&

Kaluing these t*o types of risk is fairly simple& Compensation for the national

confiscation risk is reflected in the government $ond rate for a given period of time& -he

corporate default risk is company credit spread 9 the premium that a corporation has to

pay to its investors on the top of government $onds&

Chart.(:Term0tructure o% 7ondIields

0ource:###(bonda&e(comquotes%rom 51(3(5>>2

3s seen on the e6ample of 0al art Stores Inc& in the Chart : national

confiscation risk constitute for the majority of the overall risk compensated $y the yield

0.0%

1.0%

2.0%

3.0%

4.0%

2005 2006 2007 2008 2009 2010 2011

al Mart #tores Inc. Term structure of 'ond 6ields

governmemt bond rate company credit spread


32/53

!#

on corporate $ond and overall rate is rising in time& For companies that have no $onds

outstanding( yields can $e determined $y looking at the de$t of companies *ith

euivalent credit ratings( as done in our case &:#

2.".". 'lack$#choles ,ormula

Kaluation of the euity return risk( *hich derives most of the information from

options market( is $ased on the ;lack9Scholes Formula that *as presented in the paper

“The ricin& o% otions and cororate liabilities! 2:B:4& In order to present its

computation $asics of the formula have to $e provided&

-he essential idea $ehind the pricing formula is to set up a package of investment

in the stock and loan that *ill e6actly replicate the payoffs from the option& 8nce *e are

a$le to price the stock and the loan( *e can price the option&

Even though valuating of such a package seems tedious( ;lack and Scholes came

up *ith the follo*ing formula that calculates the price of put option=

4242 :! d 0J d J )e P rT −−−= − 2!&:+4

*here

t t

rt ) 0 d σ

σ !:4ln2

: ++

= 2!&:#4

and

t d d σ−= :! 2!&:4

and

∫ ∞−

−=

't

dt e ' J !

!

:

!

:42

π2!&:)4

and finally

42lnvar :!

−= t t 0 0 σ 2!&:4

P is the put option price( S is the stock price( N2'4 is the cumulative normal distri$ution

function( _ is the strike price( r is the annuali7ed risk9free interest rate 2as a decimal4( t is

:#

In this case uote for the 8ntario Prov Cda $ond *as used for year !""T and Citigroup Inc $ond for year !":"& ;oth companies have the same rating as 0al art Stores& Inc&


33/53

!

the time to e6piry 2in years4 and σ is the annuali7ed standard deviation of stock returns

as a decimal&

2.".&. )7uity return risk

Euity returns risk is the risk associated *ith the position of shareholders& In case

the company goes $ankrupt( they *ill stay *ith their claims at the very end and therefore

reuire for such risk a premium& In order to calculate itZ CP uses the follo*ing four

step calculation& 8nce such premium is calculated it can $e added to the previous t*o and

overall risk premium is achieved&

I* Calculationo! the!or2ard 4rea(


34/53

!)

*here AIK is current dividend( " P is current share price and &εis the e6pected rate of

dividend gro*th( *hich authors of CP assume to $e 7ero&:

8nce the minimal

reuired capital gain rate is o$tained( the price of the stock *hich needs to $e reached at

the end of the period in order to persuade the investor to invest in stock rather than $ond

can $e calculated using the standard future value formula 2;realey and yers !"""( pp&

#+4

t i P -* 4:2" += 2!&!"4

*here FK is future value( iεis minimal capital gain rate derived earlier and tεis the length

in years of the investor%s holding period&

II* Esti"ation o! the stoc(3s !uture volatility

Diven the calculated minimal price investor reuires( our focus shifts to the

likelihood that such price *ill $e achieved and *hat are the costs of ensuring such

outcome& 8ptions market and option valuation formula are used in order to derive such

an information&

Prices of options reflect truly the markets level of uncertainty a$out the a$ility of

the company to deliver the e6pected cash flo*s& High degree of uncertainty is tied closely

*ith the high volatility of the stock option( until the actual cash flo* proves other*ise& In

order to o$tain this information from the market ;lack9Scholes option valuation formula

is employed&

-aken the uotes for stock option for given period of time from the market *e

have all the necessary information reuired to calculate varia$ility& Such estimate reflects

the true value in case of the option market efficiency *hich *e assume to hold&

III* Calculatethecost o!do2nsideinsurance

:-his assumption is legitimi7ed $y reference to the research done in the field( *hich sho*s that the

current dividend is a good predictor of a company%s long9term dividend stream( due pro$a$ly to the factthat companies make a conscious effort not to surprise the markets *ith une6pected changes in their financial distri$ution policies& -his assumption seems to $e uite du$ious& It *ould $e natural to e6pect thatnot the dividend $ut div,profit ratio should $e constant and therefore the dividend *ould at least gro* *ith

inflation& Ho*ever( since the authors don%t document this part of the research( *e don%t have the means toverify their claim&


35/53

!

0hen *e have the estimate of volatility for a given time period( *e com$ine it

*ith the calculated for*ard $reakeven price to determine the price investors *ould $e

prepared to pay in order to insure against the chances that their shares *ill fall $elo* the

for*ard $reakeven price& -his is the premium that reflects the e6tra risk of euity over

de$t& Ho*ever it is nothing else than the price of the put option \ the right to sell in

particular time the shares an investor holds \ for given stock& Such option protects the

investor from the possi$ility of a drop in stocks price under a given strike price&

-his calculation can $e done very simply $y( again( employment of ;lack9Scholes

Formula in this case only *ith the e6ception that the unkno*n varia$le is not varia$ility

2*hich *e have just calculated4( $ut rather the price of the option&

I5* )erive the annuali>ede?cesse@uityreturn

In the final stage the price of an option needs to $e e6pressed in terms of annual

premium& -his can $e achieved in a very straightfor*ard *ay& insurance? in order to secure minimal price of stock in

particular time in the future&

In order to transform this ratio into annuali7ed premium( *e employ the standard

annuity formula 2;realey and yers !"""( pp& #!4

Presentvalueo%annuity

+−

t r r r

C 4:2

::2!&!:4

*here rεis the opportunity cost of investing the money for the period of one year( tεis the

num$er of years investor *ill receive annuity payment and C is the coupon& In our case

*e plug the ratio as a PK( $ond rate as rεand time period as t & In this *ay *e calculate the

annual e6cess euity return&

8nce the last part of the risk is calculated it can $e simply added to the t*o

preceding risks and the overall cost of euity is achieved&


36/53

!T

2.".. Criti7ue

Even though CP is one of the first models that attempts to derive information

a$out the asset pricing from the derivatives market( rather than from analysis of e6 post

stock performance( it is not invinci$le and posses some $ottlenecks&

It is o$vious that CP does not stand on such a strong theoretical $asis as

C3P( $ut rather is just a set of formulas put together in order to come to a desira$le

outcome& Its authors are focusing on three risks and are providing a reasona$le *ay of

pricing them( ho*ever( it is impossi$le to test *hether or not there are some not9so9

o$vious risks that the authors are unconsciously omitting and therefore not mentioning&

-he lack of testa$ility of such a model is crucial and the testimony of the authors

that CP provides more >reasona$le? and >realistic? outcomes than other models is

not forceful enough to make us thro* a*ay C3P( on the contrary& -his comes *ith a

great contrast to C3P( *hich has come during the last couple decades through serious

testing( as *e have seen in the previous su$9chapter&

8ther critiue falls on the actual calculation& Even though the $ond and

derivatives market *itnessed unprecedented $oom in the last decades it is still very

difficult to gather enough data for calculating CP& Even for large multi9nationalcompanies it is difficult to find data on $onds and options that *ill cover a large enough

time spectrum( *hich *ould then ena$le us to derive different rates according to the time

structure of the projects& For small companies and start9ups it is merely impossi$le&

-ake $onds first& 3uthors encourage the user of this formula( in case he or she is

not a$le to find $ond uotes on a given company( to use information from another

company in the same field or $etter *ith euivalent credit rating& If *e follo*ed their

advice I $elieve *e *ould $e committing $ias&

1ooking at the $ond market( *e see very clearly that the $ond rates for companies

*ith a same rating are not the same and thus such assumptions is some*hat du$ious&

Aifferences in rates of companies *ith the same credit rating are not a*fully large( $ut

they tend to rise *ith longer maturity( not to mention the fact that *e are currently

e6periencing very lo* rates *hich also might cause small spread& Ho*ever( if *e do not

follo* their advice *e are not a$le to complete the calculation at all&

It gets trickier *ith options& Even though CP is presented as a model that *ill

ena$le users to come up *ith time specific rates( looking at the data availa$le on the


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market *e see that it is only a partial truth& `uotes can $e found only for less than t*o

years ahead and therefore *e are not a$le to calculate rates for follo*ing periods&

-his is not the end of the pitfalls& 8ption market is used in the calculation to

derive e6pected volatility of the stock in a given period in the future& -his lays on the

assumption that options are priced $ased on the ;lack9Scholes Formula& ;ut if that *ould

hold( then *e *ould get the same volatility from all uotes for given time period

regardless of the fact that *e take into account put or call options& -his is sadly not the

case( and the varia$ility is( in this case( truly varia$le& Should *e take the most traded one

or rather the one that is closest to our for*ard $reak9even priceV 0e might argue that not

the most traded one( since it might $e due to the fact that investors see it as $adly pricedand therefore trade it& 3uthors use the latter case and do not e6plain *hy&

Even though CP is very appealing in its nature( one has to $ear these

$ottlenecks in mind& Until $etter theoretical frame*ork is found or proper testing proves

the usefulness of this model( its results should $e taken *ith high caution&


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". Asset price estimations

-he follo*ing section *ill focus on estimating of asset prices $ased on the

empirical data o$tained from the market&:)

3s for the C3P the e6 post stock data *ill

$e analy7ed in order to estimate $eta of the stocks and su$seuently its asset price& For

PC spot uotes( $ond uotes and uotes on options *ill $e used to derive the asset

prices&

In this section four stocks *ill $e analy7ed in order to provide data on *hich *e

*ill try to comment the suita$ility of the t*o models for asset pricing& -hese stocks

include=

• 0al art Inc& \ the *orld%s $iggest retailer *ith more than !" $illion dollars in

sales in the past :! months

• e;ay Inc& \ recently esta$lished leader in the e9commerce sector kno*n for its

online auction site

• yriad Denetics Inc& 9 $iopharmaceutical company focused on the development

of novel therapeutic products and the development and marketing of predictive

medicine products

• 3merican Electric Po*er Company( Inc& 23EP4 9 registered pu$lic utility holding

company& It is one of the $iggest company in the 3merican utility sector

-hese stocks have $een chosen $ecause they represent four very different

industries as *ell as giving a possi$ility to compare the firms from >ne*? and >old?

economy& 3lso they serve as good e6amples of $ottlenecks connected *ith CP&

3.1. CAPM

".1.1. 8ata

For the estimation of $eta using Capital asset pricing model and more importantly

for the linear regression analysis it is very important to decide *hat data *ill $e used for

the computation&

:)Aata used in this section are availa$le on reuest at hajekfbm$o6&fsv&cuni&c7


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+:


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+!

Ho*ever U&S& -9$ills do not provide uotes for every month and therefore *e

have to use :9month commercial paper rate 2CP


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++

Estimation of $eta is done using the ordinary least suare 281S4 method& Such

method provides the $est linear un$iased estimate of $eta given the fulfillment of the

81S assumptions& -he follo*ing assumptions must $e checked in order to use 81S

method and most importantly to get meaningful results= 2Dujarati !""+( pp& ))94

Ø Re&ression modelis linearinarameters

Ø *alues o% indeendentvariablesarenonrandom

Ø The residualsarenormallydistributed

Ø The e'ectedvalueo%residuals is equalto/ero

Ø The residualsareindeendent%romoneanother Ø omoskedasticity, i(e( equal variance o%residuum

Ø ero covariance bet#eenresiduumandindeendent variables

Ø Jumbero%observations is &reaterthannumbero%estimatedarameters

Ø *ariance o%indeendent variable is%initeandositive number

Ø Jone'istence o% seci%icvariance,orerrorin re&ression model

Ø Jone'istence o% multicolinearity amon&indeendent variables

".1..1. 5ormality

For the test of normality /arue9;era test:T

*as used& -his test pointed to the

normality pro$lem *ith 5DN as can $e seen in -a$le !& -his might $e happening due

to the limited num$er of o$servations&

Under the null hypothesis of a normal distri$ution( the /arue9;era statistic is

distri$uted as *ith ! degrees of freedom& -he reported Pro$a$ility is the pro$a$ility that a

/arue9;era statistic e6ceeds 2in a$solute value4 the o$served value under the null 9 asmall pro$a$ility value leads to the rejection of the null hypothesis of a normal

distri$ution&

:T/arue( C& &( ;era( 3& @&= 3 -est for Normality of 8$servations and


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+#

Table5(:Bar&ue7erastatistic results

Co"%any

,ar@ue


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+

".1..". 9omoscedasticity

-est of the assumption that the variance around the regression line is eual for all

values of the predictor varia$le i&e& homoscedasticity holds( *as carried out *ith 0hite%s

test&!"

Homoscedasticity *as detected in all samples and since 0hite%s test does not

assume normality these findings are valid for all four stocks&

Table2(:hite6sstatistic results

Co"%any hite3sstatistic P( Econometrica #T( :BT"( pp& T:9T:T&!:

-he star sign 24 $y the P9value in the ta$le indicates that the estimated coefficient is significant at : level( the dou$le star then on &


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+)

statistically significant at level of significance& Kalues seem to support the simple

assumption that ne* and I- or $iotechnology companies *ith short sales records should

$e more risky than companies *ith more sta$le cash flo*& ;oth e;ay and yriad

Denetics are aggressive stocks since their $eta is higher than :& 3EP and 0al art are

defensive as the very opposite is true&

TableF(:N40 Results

Co"%anyAl%ha

esti"ate

Standard

Errort


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+

;erndt notes that >over the last )" or so years in the United States the average

market risk premium has $een a$out T per year? 2;erndt :BB:( pp& +#4&!!

Since

market risk premium varies hugely in the last :" years as Chart ! uncovers( it is

appropriate to use ;erndt%s figure& 3s for the risk free asset( average from :9month

commercial paper rate for last :! months *ill $e used&!+

Chart5( : Market [email protected]>>3

Market Premium 3#-P


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+T

Table8: Results

Co"%any E?%ected return

3merican Electric Po*er 7.18%

e;ay 11.55%

yriad Denetics 15.96%

0al art 6.19%

0ource:o#ncalculations basedondata

3.2. MCPM

".2.1. 8ata

CP is a for*ard looking model that gathers information from the $ond and

option market& -imeframe is given $y the availa$ility of data rather than $y selecting past

period that is evaluated& For the estimation of asset prices using CP the follo*ing

data need to $e used= company $onds( option uotes and risk free rate&

".2.1.1. 'onds

For the purposes of the estimation it is necessary to o$tain information on the cost

of de$t for given companies& 3s *as mentioned in previous chapter it happens very often

that such information is not availa$le and *e need to derive it from the $onds of same

grade& Such is also the case here( *here $ond rates are o$tained in this *ay for e;ay and

yriad Denetics& Conseuences of such selection might $e( as *as said $efore( severe

and our estimates might $e $iased& -his has to $e taken into account *hen judging the

results&

Aata are taken from the $ond market uotes availa$le on :) ay !""#&th !#

".2.1.2. :ptions

Even though CP is presented as a model that is a$le to predict asset prices for

different time periods( in reality it is limited $y the availa$ility of data& Since option

markets do not provide uotes for far future 2in our case availa$le dates are at furthest for

/anuary !"")4& Aue to the lack of data $eyond this point( calculations are made only for

!#Source= ***&$ondpage&com


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+B

/anuary !"" and /anuary !"") *ith the e6ception of yriad Denetics& arket does not

provide uotes for yriad Denetics in the period $eyond Novem$er !""#( and therefore

rate only for this date is calculated and used&

Aata as such are taken from on9line uotes and are $ased on the uotes from :)th

ay !""#&!

".2.1.". 0isk free rate


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#"

3.3. Comparison

Since *e have calculated the same outcomes \ e6pected returns \ using t*o

different models *e *ould naturally *ant to compare them& Unfortunately in the case of

C3P and CP is not straightfor*ard& 3uthors of CP put as an undisputa$le

advantage of their model the possi$ility to derive different rates according to the time

period of the investment( compared to a >fit9for9all? $eta( ho*ever this hinders our a$ility

to make a comparison&

C3P uses only the historical data to achieve a risk premium and changes slo*ly

over the time& CP derives different a risk premium according to the length of the

investment $oth of them come essentially to the same result \ asset price& -he fact that

*e do not have a chance to compare the models head9to9head does not make either one of

them less helpful& In my opinion $oth models should $e used and the managers% final

decision should $e $ased on the information that $oth models are giving& Since they $oth

have their uniue point of vie* and focus on slightly different aspects com$ining them

seems to $e reasona$le&

In our case the only thing *e might conclude from the comparison of the t*o

results *e have is the fact that CP gives higher e6pected rates and therefore higher

cost of capital vis99vis C3P& 8ther conclusions are unfortunately impossi$le&


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#:

&. Conclusion

-he aim of this thesis has $een to map the introduction of C3P and to follo*

one line of the de$ate it started( as *ell as to introduce a challenging model $ased on

information from derivative markets& 0e have e6plained thoroughly $oth models and

pointed out the difficulties that may arise *hen interpreting the results as *ell as during

the calculations themselves& Employing $oth models *e have su$seuently derived asset

prices for four companies traded on the US market&

-he results of this thesis are ho*ever not as straight for*ard as *e *ould have

*ished them to $e( due to the fact that even though $oth models are meant to give ans*er to the same uestion( their results are not compara$le& 0e are missing a tool that *ould

ena$le us to decide *hich model is providing us *ith more valua$le and accurate

predictions of capital assets prices&

Final judgment has to $e given $y the practitioners around the *orld( *ho use not

only asset pricing models $ut also their shre*dness( intuition and e6perience& 0ithout

such insight *e are not a$le to ans*er this uestion( $ut that does not mean that *e

cannot take sides& In my o*n vie* C3P stands on firmer foundations and its

predictions a$out the risk return relation seem to $e valid in the long run&

3s for the CP( the current $oom of derivative markets around the glo$e

makes me $elieve that it *as *orth revealing its frame*ork& CP( despite the fact that

has a lot of $ottlenecks( as *as pointed out in one su$9chapter( provides us *ith a very

interesting alternative to the main stream asset pricing models& Further research in this

field should $e encouraged( *ith respect to the critiue that *as provided&

3midst the critiue of $oth models *e should ho*ever reali7e that even though

today *e have uite enough empirical evidence to comforta$ly reject most of the *orks

done during the last forty years( such rejection should not mislead us to discard the

portfolio theory as such& odern investment theory should $e perceived as the

indispensa$le $uilding $lock for every academician and practitioner to start off( $ut not to

$e engrossed( *ith&

3s Sharpe 2:BT#4 concludes( ehileεtheεrelativeεimortanceεo%εvariousε%actors

chan&esover time, asdo there%erences o%investors, #e need notcomletelyabandon a

valuable%rame#ork#ithin #hich #ecanaroach investmentdecisionmethodically(e


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#!

have develoed a use%ul set o% tools and should certainly continue to develo them(

Mean#hile, #e can use the tools #e have, as lon& as #e use them intelli&ently,

cautiously,andhumbly(


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References

A"ihudD7*DChristensenDB*,*DendelsonDH*98-;:„-urtherEvidenceεontheRisk

ReturnRelationshi!(StandfordUniversity(D R** 98=8;: „TheεRelationshiεbet#eenεreturnεandεmarketεvalueεo%εcommon

stocks!(/ournalofFinancialEconomics B( pp& +\:T

BerndtD E* R* 988;: „TheεPracticeεo%εeconometrics:εclassicεandεcontemorary!(

3ddison90esley(


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##

,ensenD*C*98F-;: „Theε-oundationsandεCurrent0tateo%CaitalMarketTheory“ (

in Studies in the -heory of Capital arkets ,εPraeger Pu$lishers

,ogannathanD R*D eierD I* 9-..-;: „"oε#eεneedεCAPMε%orεcaitalεbud&etin&S“ (

National ;ureau of Economic


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Internet resources=

***&$ondpage&com

***&economagic&com

***&moneycentral&msn&com

***&nyse&com

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