Capital Asset Pricing Model and Arbitrage Pricing Theory

Transcription

1 Aarhus School of Business M. Sc. of Finance and Inernaional Business Dearmen of Finance Fuglesangs Allé 4 DK-821 Aarhus V Maser Thesis Caial Asse Pricing Model and Arbirage Pricing Theory An alicaion of marke equilibrium models o he Polish marke Auhors: Agnieszka Sawa Slawomir Sklinda Wrien under suervision of Paula Peare, Associae Professor Dearmen of Finance Aarhus, 23

2 INTRODUCTION...4 CHAPTER I MARKET EQUILIBRIUM MODELS -THEORY AND ASSUMPTIONS APPLICATION OF MARKET EQUILIBRIUM MODELS ARBITRAGE Arbirage mechanism and marke equilibrium Limis of Arbirage CAPITAL ASSET PRICING MODEL (CAPM) Sandard version Zero bea version of he CAPM model Assumions of he sandard Caial Asses Pricing Model Marke efficiency Decisions based on he mean-variance crieria Homogenous beliefs ARBITRAGE PRICING THEORY No arbirage ooruniies Facor Model Firm- secific risk APT relaion Mehodological concerns...23 CHAPTER II CAPITAL MARKET EQUILIBRIUM MODELS EMPIRICAL TESTS CAPM EMPIRICAL EVIDENCE Early CAPM ess Linner es (1968) Black, Jensen and Scholes es (1972) Fama and MacBeh es (1973) Roll s criique (1977) Laer ess of he CAPM model Banz es (1981) Fama and French es (1992) Kozicki s and Shen s es (22) EMPIRICAL STUDIES ON APT Invesigaion on variables influencing reurns Aroaches o APT model esimaion Saisical esimaion of beas and facors Porfolio mehod of facor esimaion Beas arbirary choice APT CONTRA CAPM EMPIRICAL EVIDENCES IN POLAND Tess of marke efficiency Mulifacor models on Warsaw Sock Exchange...5 CHAPTER III DATA DESCRIPTION DATA CHOICE Choice of he roxy for he marke orfolio Lengh of esimaion eriod Observaion frequency CHARACTERISTICS OF DATA USED FOR CAPM AND APT TESTS Characerisics of daa used for CAPM es Reurns on Shares Warsaw Marke Index (WIG) Risk Free Rae Variables used for APT es S&P Polish Zloy (PLN) Exchange Rae Inernaional Price of Gold

3 CHAPTER IV EMPIRICAL TEST OF CAPM CALCULATION PROCEDURE CAPM es mehodology Porfolio grouing Risk free rae variabiliy EMPIRICAL TEST OF THE CAPM Time-series regression Cross-secional regression...78 CHAPTER V APT ESTIMATION AND TESTS METHODOLOGY Esimaion rocedure Mehods of esing and Esimaion Facor Analysis overview Facor Analysis formal model FACTORS ESTIMATION- EMPIRICAL RESULTS Variables analyzed Suboimizaion Number of cases Samling adequacy Number of facors Kaiser rule Caell rule Variance crierion Facoring mehods Maximum Likelihood Facoring PCA versus PFA PCA resuls TIME-SERIES REGRESSION CROSS-SECTIONAL REGRESSION CHAPTER VI POSSIBLE REASONS FOR CAPM AND APT FAILURE BETA INSTABILITY INAPPROPRIATE PORTFOLIO GROUPING APT CASE MARKET INEFFICIENCY AND LIQUIDITY VALUE WEIGHTED INDEX AND CAPITAL DOMINANCE OF A FEW COMPANIES LOW SIGNIFICANCE OF THE MARKET AS A SOURCE OF CAPITAL SHORTCOMINGS OF APT FACTOR ANALYSIS DIVERSIFICATION OF THE FIRM-SPECIFIC RISK SMALL NUMBER OF VARIABLES SHORT ESTIMATION PERIOD REFERENCES: APPENDIXES

4 Inroducion Polish caial marke is very young. I was se u in 1812 however jus before he II. World War i was liquidaed for over fify years. Warsaw Sock Exchange was reacivaed in 1991 and since hen i has been develoing consanly. Sudies carried ou by Szyszka (23) revealed ha he efficiency of ha marke is imroving as well. If Polish caial marke was efficien enough, he marke equilibrium models could be assumed o work on i. Marke equilibrium models ha are Caial Asse Pricing model and model creaed on he basis of Arbirage Pricing Theory have alicaions in many fields of finance. They could suor in he decision making rocess of Polish cororae managers and invesors. Due o imlemenaion of APT or CAPM models, decisions concerning he choice of a orfolio ha mees cerain invesor crieria could be made. Moreover, invesors using hese models would be able o idenify overriced and underriced asses. Furhermore, asse ricing heories could be alied in budgeing rocess, as hey hel wih cos of equiy esimaion. Thus, he aim of his aer is o es he sandard versions of he marke equilibrium models on Warsaw Sock Exchange and herefore answer he quesion if sandard CAPM or APT could be useful for Polish invesors and cororae managers. Desie of he fac ha such sudies were conduced on many foreign markes, his research is he firs ha emirically analyses boh marke equilibrium models on Polish caial marke. However, he sudy conduced in his hesis faced a few imoran roblems. The sudy limiaions are associaed mainly wih characerisics of he WSE ha is sill develoing. The fac Polish caial marke is only welve years old resuled in a relaively small 4

5 number of firms analyzed and shor esimaion eriod. Those are resecively 1 comanies examined wihin hree-year eriod from 2 o 23. The research objecive deermined he srucure of his aer. Therefore, chaer one concerns heoreical issues associaed wih marke equilibrium models. I describes shorly heir alicaion and assumions required o creae hese models. Furhermore, i discusses hese issues wih relaion o Polish business environmen. The second chaer analyzes sudies concerning CAPM and APT emirical ess ha migh sugges he mehodology of develoing hese models. Chaer hree discusses he choice of daa emloyed in he models. Two nex secions discuss esing echniques ha are radiionally used when esimaing hese models, chose he mehod mos aroriae o Polish caial marke and hen describe ess resuls. Finally, he las chaer focuses on roblems faced while develoing and esing boh models. Moreover, i suggess imrovemens ha could be made in order o creae models delivering more reliable resuls. Based on he mehodology alied in his aer boh marke equilibrium models do no describe execed raes of reurns on he WSE. Neiher marke bea nor oher macroeconomic facors examined saisfacory exlain he reurns. According o obained resuls, sandard models should no be alied by Polish invesors and cororae managers in heir decision making rocess. However, similar research can be carried ou in a few years. I is likely ha Polish caial marke will be more efficien because of he increasing inegraion of he caial markes. Furhermore, i is believed ha marke economy in Poland will be much more liberalized and herefore marke efficiency will be bigger. Due o he fac ha he esimaion eriod would be longer, he esimaed resuls would be more reliable. 5

6 CHAPTER I Marke Equilibrium Models -Theory and assumions This chaer resens marke equilibrium models, heir alicaion and assumions discussing hem wih relaion o Warsaw Sock Exchange. 1.1 Alicaion of marke equilibrium models In sie of he fac ha CAPM and APT models esed in his sudy were imlemened for he firs ime over hiry years ago, hey are sill alied in finance. Firs of all, hey are used in a modern orfolio heory ha is an aem o undersand he marke as a whole. According o his heory invesmens are described saisically, in erms of heir execed reurn rae and heir execed volailiies. Marke equilibrium models hel o idenify acceable level of risk olerance, and hen find a orfolio wih he maximum execed reurn a ha level of risk. Second, alicaion of CAPM and APT is a ool on he idenificaion of unique ooruniies, when shares are eiher over- or undervalued. The caial marke equilibrium models enable esimae he execed value of he equiy in erms of reurns. Finally, hese models are of crucial imorance in caial budgeing rocesses. Alying ime value echniques involves esimaing a discouning rae. A very common aroach is weighed average cos of 6

7 caial, which is comosed of he cos of equiy and he cos of deb. The firs one can be esimaed by using caial marke models. As marke equilibrium models are useful in so many areas of finance, hey could suor Polish invesors and cororae managers in heir decision rocesses as well. 1.2 Arbirage Boh marke equilibrium models assume ha markes arbirageurs are able o ensure marke equilibrium and herefore o reven misricing. There is, however, he quesion ha needs o be answered, namely wheher he marke equilibrium is a fac or only a heory Arbirage mechanism and marke equilibrium The heory of arbirage rovides he answer. I says ha he mechanism of arbirage revens any deviaions from he equilibrium, as he acions underaken by invesors will immediaely increase he rice of undervalued asses and decrease he rice of he overvalued ones. The mechanism of arbirage was discussed for examle by Elon and Gruber (1998), Francis (2) and Grinbla and Timan (1998). An examle of he mechanism is resened below. Assume ha Comany 1 has overvalued socks (Asse 1) and socks of Comany 2 are undervalued (Asse2). Hence he marke is no in equilibrium and he asses mus be laced beyond he Securiy Marke Line. This siuaion is grahically resened on Diagram 1.1: 7

8 Diagram No 1.1: Pricing arbirage undervalued asse (green) and overvalued (red) E(R) R 1 R 1 R M R 2 R F. Asse 2. Asse 1 SML 1 β Source: M. Grinbla i and S. Timan: Financial Markes and Cororae Sraegy, Irvin McGraw-Hill Comanies, 1998: 12 If he above siuaion occurs, following he arbirage ooruniy he abnormal rofi could be earned wihou bearing risk. If asses are overvalued, an invesor can obain any oin on he Securiy Marke Line. I can be done hrough eiher urchasing /selling of individual shares or creaing (from oher accessible asses) a orfolio, which can be laced in any oin on he SML. In such case, selling shor he asse/orfolio laced on he SML (wih he same bea coefficien as Asse 1) and buying long Asse 1, an invesor can earn rofis on he higher execed reurn. The value of he gain is marked by he red line and equals R 1 - R 1. This invesor will receive he abnormal reurn ill he rice of he Asse 1 reaches is rue level R 1. This mechanism is analogical for undervalued asses. In his siuaion an invesor using he arbirage mechanism can receive he abnormal reurn as well. Now one would sell shor Asse 2 buying long orfolio on he SML wih he same bea coefficien as Asse 2. Such invesmen will generae abnormal reurns wihou any addiional risk. The remium is marked green and equals R 2 - R F. 8

9 However, roonens of behavioural finance noiced ha in a real business world rices can diverge from equilibrium Limis of Arbirage Firs of all, here are risks and coss associaed wih arbirage ha migh limi i. Furhermore, according o behavioural finance he deviaions from he fundamenal value of asses can be caused by raders ha are no fully raional and i migh be difficul for raional raders o undo he caial dislocaions made by less raional ones. According o he heory, a raional invesor, who is aking advanage of he arbirage rofi ooruniy ensures marke equilibrium. Following he behavioral finance heory, no all he arbirage ooruniies can generae rofis (Barberis and Thaler, 22). The invesmen sraegies may someimes be cosly and risky. Barberis and Thaler (22) believe ha raders erusing arbirage sraegy usually face fundamenal and noise-rader risk. Furhermore, hey need o ay he imlemenaion coss. Fundamenal risk migh no be fully hedged as securiies are no erfec subsiues and he sraegy of selling one asse and urchasing anoher can incur he fundamenal risk. The noise-rader risk is a risk ha irraional invesors can make he misriced asses even more deviae from equilibrium. This risk may force arbirageurs o liquidae heir osiions oo early as here is a searaion of brains and caial (Shleifer and Vishny, 21). I is an agency roblem, since orfolios managers do no oerae heir own money and reurn maximizaion insead of ensuring ricing equilibrium is of crucial imorance for hem. However, according o De Long e al. 9

10 (199) arbirageurs migh make he rice diverge from equilibrium, as well. If he marke is dominaed by osiive feedback raders, overriced asses migh be urchased by invesors making arbirage rofi, as hey would exec ha higher rice will be ushed u even higher. Moreover, exloiing arbirage is no cosless. For examle ransacion coss ha overwhelm oenial arbirage rofi, would deer raders from exloiing arbirage ooruniy. Furhermore, i migh be cosly o learn abou misricing. I was believed ha reurns redicabiliy is a sign of incorrecly se rices. However, Summers (1986) and Shiller (1984) resened ha he demand of he irraional raders migh be so srong ha he reurns can be unredicable. Neverheless, behavioral finance blames for misricing no only invesors bu cororae managers as well who are resonsible for asses issue. Theoreically, if asses are overriced managers would decide o issue more asses in order o sell hem a aracive rices. Then he oversuly should ush rices back o equilibrium. However, he issue incurs coss and managers canno be sure ha invesors overesimae heir shares. Therefore, hey migh no decide o issue equiy. Behavioral finance icks u he weaknesses of marke equilibrium models focusing mosly on invesor sychology and beliefs. To dae, here are no behavioral models ha migh be alied insead of CAPM or APT in all of heir alicaions. The reason for ha migh be sychological facors ha canno be quanified easily. I is ossible ha models combining boh aroaches will be develoed in he fuure. 1

11 1.3 Caial Asse Pricing Model (CAPM) CAPM model was imlemened almos 4 years ago by Share (1964), Linner (1965) and Mossin (1966) indeendenly from each oher. This model was he firs and, as is oulariy roves, successful aem of defining he risk of cash flows from an invesmen anamoun o he execed rae of reurn. The CAPM model is he simles version of he caial marke equilibrium models, and is also called one facor caial marke equilibrium model. Zero bea model 1, which is one of many derivaive of a sandard version will be resened furher in his secion Sandard version The CAPM model describes he relaionshi beween risk and execed reurn. Execed reurn of a securiy or a orfolio is defined by he sysemaic risk affecing he comany and equals he sum of he rae on a risk-free asse and risk remium. The caial marke equilibrium akes he following form: R i = r f + ( r m - r f )β i, Where R i is he execed reurn on he equiy (of a single comany or orfolio), r f is a risk free rae, r m defines he execed reurn on he marke orfolio and β measures he sensiiviy (risk) of execed reurn on equiy R i o he reurn on he marke orfolio r m. Formal derivaion o he model does no make many roblems and is available in he lieraure ( for examle: Elon and Gruber 1998, Francis 2, Bernd 1996, Haugen 1996). Alicaion of he CAPM model requires defining a risk free rae, execed marke risk remium ( r m - r f ) and calculaion of he bea coefficien. I is comued based on he hisorical values as he sloe coefficien in he regression of reurns on he equiy agains marke risk 1 well known as Black CAPM. 11

12 remium. The Caial Asse Pricing Model is he mos oular among American comanies when esimaing he cos of equiy in he caial budgeing rocess. A recen sudy conduced by Graham and Harvey (21) confirms he oulariy of CAPM. According o heir resuls over 73 ercen of US firms aly he model while esimaing he cos of equiy. However, here was no such sudy conduced on he Polish marke, hence no commens on he model s oulariy can be done Zero bea version of he CAPM model The sandard version of CAPM model may be considered unrealisic, as i is raher imossible for a single invesor o borrow or lend wihou any limiaion a he risk free rae. Black (1972) releases his assumion resening a version of he CAPM model where all asses are risky. The equilibrium can be achieved by subsiuion of any of he zero bea orfolios, which are laced on he coninuous secion R F C, insead of R F from he CAPM equaion. Since R F Z is unrealisic, he minimum variance orfolio wih bea coefficien ha equals zero is marked wih a sign Z and is laced on he crossing oin of he curve K and he line R F C and he corresonding execed reurn rae equals R z = R F. The zero bea coefficiens mean a lack of correlaion wih a marke orfolio. A grahical exlanaion of he siuaion is resened in Diagram No

13 Diagram No1.2 Zero bea orfolio (Z) on he efficien orfolio curve E(R) K M. R Z = R F.S. Z C σ Source: E. J. Elon and M.J. Gruber: Nowoczesna eoria orfelowa i analiza aierów warościowych, WIG-Press, Warszawa, 1998, Hence, he securiy marke line can be resened as follows: R i = R + β Z i ( RM RZ ) The orfolio Z has cerainly lower execed rae of reurn han he marke orfolio, since is reurn is reresened by he crossing oin of he angen o he curve K and he verical axis. As he execed marke reurn is a angen oin wih he curve K, i mus be laced above R z. The orfolio Z will no be an efficien orfolio as well, because i is laced below he lowes variance orfolio, so i is ossible o find a orfolio of he same variance bu higher execed reurn. Hence, orfolio Z, alhough laced on he lowes variance curve is no efficien (formal derivaion in Elon and Gruber 1998). Assuming ha oin S reresens he lowes variance orfolio, all invesors will hold orfolios laced on he efficien orfolios curve (SMK). Alhough orfolio Z is no efficien i is used in analysis because of is zero correlaion wih marke orfolio. Since a combinaion of minimum variance orfolios gives a minimum variance orfolio, he aim of he analysis is o cerae such a combinaion ha would be lacaed on he curve SMK. Invesors ha chose reurns beween R S and R M, will have a combinaion of zero bea orfolio and 13

14 marke orfolio. On he oher hand, invesors ha own a orfolio on he righ side from he marke orfolio M, will have a orfolio comosed by shor sale of Z orfolio and long urchase of marke orfolio. Since he orfolio Z is inefficien, he overall sum of shor and long osiions mus equalize. Tha is because in equilibrium all invesors will osses only marke orfolio Assumions of he sandard Caial Asses Pricing Model The realiy of financial evens is so comlex ha in order o build any model describing i in a lain way simlifying assumions need o be adoed. These assumions are o eliminae facors, which marginally affec he modelled even. The CAPM assumions can be groued in hree general condiions (Bailey 21): A. Markes are efficien. B. All invesors make heir decisions on he basis of he mean-variance crieria. C. Invesors have homogenous beliefs Marke efficiency This assumion is he leas real and is comosed of a number of inerrelaed facors: No ransacion coss, which means ha here are no coss associaed wih sale or urchase of an asse. This assumion is jusified by he fac ha ransacion coss migh change he reurn 14

15 rae from an insrumen deending on he invesor s osiion a he beginning of a considered eriod. Taking hese coss ino accoun significanly increase he comlexiy of he model. However, recognising heir relaively low value low significance can be assumed. No insiuional barriers in asses rade, which is direcly relaed o he unlimied shor sales ossibiliy. This assumion in racice is rarely fulfilled, since invesors are usually no able o sale shor any amoun of asses. Unlimied shor sale and long buys a he risk free rae. As in he revious assumion he firs ar of his assumion is raher imossible, however is second ar may be saisfied. Invesors may lend heir money a he risk free or even higher rae. Black (1972) relaxed his assumion and buil a well known and acceed Marke model. Asses are infiniely divisible, which means ha invesors can hold any amoun of he asse. This assumion was creaed because redicions of he model would be inaccurae, as he indivisible asses would require a grea amoun of iniial wealh of an invesor. All asses can be bough or sold a he observed marke rice. There is a marke for all kind of asses, even human caial. Individual invesors decisions abou heir osiion in any asses will no affec heir rice(hsu e al. 1974). This assumion imlicaes ha he marke is erfecly comeiive wih no mono- or oligooly. The rice is a resul of all acions, no a aricular invesor. Taxes are neural, hence all invesors ay he same ax from all forms of income: dividends, ineress and caial gains Decisions based on he mean-variance crieria Should an invesor be able o make his/her decision based only on hese wo variables, he reurns mus have normal disribuion. Risk 15

16 ha accomanies he invesmen can be defined by he disribuion of ossible reurns. As his disribuion is assumed o be normal, i can be described by wo measures: mean and variance (or is square roo called sandard deviaion, which is much more oular in finance). Since all normal disribuions are virually he same, hey differ from each oher only in heir means and variances. While all invesors refer higher reurns o lower reurns, ceeris aribus, i is rue ha hey do refer lower risk. I is described by he sandard deviaion of reurns. This leads o he conclusion ha if invesmen/orfolio/share risk is high, invesors would acce i only if hey would be rewarded by a high execed reurn. Consequenly, if he execed reurn is low i would be acceed only if he relaed risk is low as well. Commonly alied mean-variance analysis is herefore a rade off analysis beween he acceed risk and he required rae of reurn. Even if an invesmen bears no risk, invesors would sill exec nonzero reurn as an incenive o delay he curren consumion. Shorerm governmen bonds may be considered an examle of his kind of invesmen as heir defaul risk is virually zero. I is addiionally assumed ha all invesors make heir decisions only one eriod ahead and all of hem define his eriod in he same manner. This assumion fringe in is classificaion uon he nex grou, inroducing homogenous beliefs Homogenous beliefs This condiion saes ha all invesors have he same, homogenous believes abou he rimary daa, which are necessary o make orfolio decisions. These are mainly hree grous of daa characerisics: execed reurns, variance (or sandard deviaion) of he reurns and 16

17 he marix including correlaion coefficiens beween he reurns on all airs of shares. The execaions are a resul of all available informaion and herefore he discussed assumion refers direcly o efficien marke heory, which resumes ha each invesor has an access o he same informaion. In racice his assumion is unlikely o be fulfilled, since informaion is no disribued among all of he invesors o he same exend. Big financial insiuions have much beer access o he informaion han individuals. On he oher hand individuals sae oo small ercenage of marke layers o consider hem saisically insignifican. However, banks may have more informaion abou he comanies hey service han oher invesors, as banks remain in a close business conac wih heir cusomers. The roblem of lack of informaion among he small invesors is usually solved by observaion of he invesmen decisions made by bigger and beer informed insiuions. Alhough he laer have ime advanage, i mus be discouned by he rice of gaining informaion Arbirage Pricing Theory The classic APT was imlemened by Ross in 1976 as an alernaive o Caial Asse Pricing Model. I considers more han one facor influencing asses reurns. The heory deduces ha firm-secific risk is fairly unimoran o invesors holding well-diversified orfolios and i migh be reended ha firm-secific- risk is no resen. Thus, he risk of securiies can be described by facor bea coefficiens only. In equilibrium, where 17

18 arbirage ooruniy does no exis, asses reurns will saisfy an equaion relaing execed reurns of securiies o heir facor beas. This risk-execed reurn relaion was called APT and can be wrien formally: R i = R F + K j= 1 γ ik λ + ε k i The derivaion of APT requires only hree general condiions o be me: 1. No arbirage ooruniies. 2. Reurns can be described by a facor model. 3. There is large number of securiies, so ha i is ossible o form orfolios ha diversify he firm secific risk of individual socks No arbirage ooruniies The Arbirage Pricing Theory is based on he Law of One Price. The rule says ha all goods of he same risk should be sold a he same rice hus marke can reach equilibrium revening arbirage. There are furher requiremens similar o assumions concerning CAPM ha relae o marke efficiency, invesors homogenous beliefs and heir mean- variance invesmen crieria. These assumions ensure ha here is no arbirage ooruniy and marke is in equilibrium wha is crucial for APT. According o Jajuga and Jajuga (1999) here are eigh such requiremens for APT significance: No ransacional coss. Asses divisibiliy No axes on incomes generaed by caial marke Unlimied shor sale and long buys Invesors can borrow a he risk free rae No barriers on asses sale or urchase 18

19 Decisions based on he mean-variance crieria Individual invesors decisions abou heir osiion in any asses will no affec heir rice Above assumions are arly me by Warsaw Sock Exchange. Firs of all, here are ransacional coss in Polish marke bu if he ransacion size is large, coss are relaively small and hey can be negleced (Rubaszek, 22). Furhermore, asses divisibiliy condiion can be assumed as well. In a real business world, he smalles uni ha can be raded on a real marke is one share ha canno be divided ino ars and hen raded. Neverheless, i can be suosed ha marke aricians inves in he urchase of one exensive share. Assuming his siuaion shares could be seen as divisible. Moreover, he no axes condiion could be suosed. Taxes on caial gains are going o be inroduced in 24 or 25 bu here is no exlici regulaion of his issue righ now. Nowadays, only dividends and ineress on bank deosis are axed. There is a visible imac of imoran insiuional invesors on sock rices of he comanies wih he greaes caialisaion in Poland. However, firms characerized by smaller caialisaion are very sensiive o seculaive acions of individuals or a grou of noninsiuional invesors. As an examle, i migh be said ha he grou maniulaed he rice of Efek s socks making i few imes greaer. This inciden ook lace en years ago, bu i is a roof ha Polish marke does no mee he assumion ha individual invesor is no able o influence sock rices (Rubaszek, 22). Moreover, APT allows shor sale of securiies. Polish law has regulaed his issue on 21 s December Warsaw Sock Exchange oened a 19

20 secial inerne laform in order o collec invesors orders and encourage marke aricians o shor sale or urchase of securiies. However, here are only five brokerage houses dealing wih shor sale. These ransacions accouned for.1 or 1.18 ercen of all ransacions on WSE. According o Maciejewski and Mejszuowicz (23) he reason for his siuaion is ha he whole sysem is oo comlicaed. Furhermore, borrowers charge high fees wha deer lenders. Therefore, shor sale of securiies is no very oular among Polish invesors. This fac migh imly ha arbirage can be limied and hus making rices diverge from equilibrium. Furhermore, in real business world invesors ha borrow funds need o ay remium o he borrower as a rice of he loan. Thus, he assumion of borrowing a risk free rae is no fulfilled in realiy. The nex condiion requires securiies o be raded wihou any barriers. I is believed ha his requiremen is me on Polish marke as i is ensured by law (Dz. U. of 22 year. No 49, osiion. 447). The las issue refers o invesors. They are assumed o allocae heir funds aking ino accoun only execed reurns and securiies risks. Tha is rue ha he majoriy of invesors while making financial decisions focuses mosly on hese wo issues (Rubaszek, 22). However, i canno be assumed ha all invesors behave in he same manner. Having in mind he assumions resened above, he APT model can be imlemened. Neverheless, hese eigh assumions are usually no me in a real business world. APT roonens believe ha he basic advanage of he heory is he fac ha no all of he assumions need o be me (Haugen, 1996). 2

21 1.4.2 Facor Model APT begins wih he assumion on he reurn generaing rocess. If individuals believe ha he random reurns on he se of asses are exlained by K-facor linear model: R i = a i + K k= 1 γ ik I k + ε i where: i=1,, n Ri is random reurn on asse i a i is he execed reurn on he asse i γ ik facors coefficiens ε i are he mean zero asse secific disurbances assumed o uncorrelae wih he δ K and wih each oher Then, he securiy is differenly sensiive o each I k facor. However, all of I k facors have he same value for all securiies. Moreover, each I k variable have imac on more han one securiy. Reurns of all securiies deend on I k ha are changing consanly and coefficiens ha are secific for each securiy. Termsε i are assumed o reflec he random informaion ha is unrelaed o oher asses. Too srong deendence on ε i would sugges ha here are more han k common facors. Furhermore, n should be much bigger han number of facors k. γ ik According o Roll and Ross (198) he formula reflecs he naure of asses in differen saes of naure. 21

22 1.4.3 Firm- secific risk The assumion of diversified firm-secific risk is of umos imorance for APT, as i allows for relaing reurns o facor beas. Roll and Ross assumed ha he number of asses analysed is aroximae o infiniy and he orfolios are erfecly diversified. Moreover, all variances of residuals have weighs equal squared amoun invesed in asse, as residuals are uncorrelaed. Thus, for he erfecly diversified orfolios he residuals risk would be close o zero. If ε i were excluded from he model, he formula would say ha each asse i has reurns ha are an exac linear combinaion of he reurns on riskless asse and he reurns on k oher facors. Thus, he riskless reurn and each of he k-facors can be described as linear combinaion of k+1 ohers reurns. Any oher asse s reurn, since i is a lineal combinaion of he facors, mus be also a combinaion of he firs k+1 reurns. Therefore, he orfolios of he firs k+1 asses can be a erfec subsiue for all of he asses in he marke. Such subsiue should be riced equally. Thus, he APT sugges ha only limied number of risk comonens exiss. Therefore, if here are only a few sysemaic risk comonens, economic aggregaes (for examle GDP, inflaion rae, ineres raes ec.) could be execed o be such facors APT relaion In order o rack he reurn on he orfolio wih no firm-secific risk a racking orfolio wih weighs of K 1 γ ik on he risk free securiy, γ i1 j= 1 on facor orfolio 1 and γ i2 on facor orfolio 2,., finally γ ik on facor orfolio k can be consruced. Therefore, he execed reurn of he orfolio ha racks he invesmen would be: 22

23 R F + K j= 1 γ ik λ k where λ 1 λ k are risk remiums of facor orfolios. If he invesmen and his racking orfolio have he same execed reurn, here is no arbirage ooruniy. Thus, he APT equaion for all invesmens wih no- firm secific risk can be formulaed as follows: R i = R F + K j= 1 γ ik λ k This relaion should hold in he absence of arbirage ooruniies. On he lef-hand side an execed reurn on invesmen is resened and on he righ-hand side here is he execed reurn of a racking orfolio deending on he same facor coefficiens. If here is only one significan facor in APT model hen he asse ricing equaion can be resened as a sraigh line. Two-facor model s grahical resenaion will be a lane as here are hree oins necessary o describe a lane. These oins will be wo coefficiens and execed reurn. More han wo facor model resens a hyerlane Mehodological concerns The heory roonens believe ha here are wo mos imoran advanages of APT. The firs one is he liberal characer of is assumions comared o CAPM assumions. The second benefi is 23

24 ha he heory significance can be verified saisically. The second issue is discussed by he heory oonens. There were economiss rying o assess if he APT model is esable. The firs sudies on APT saisical ess show ha a researcher carrying ou a facor analysis may face mehodological difficulies. APT oonens usually believe ha he assumion of diversified firmsecific risk is weak APT oonens such as Schanken (1982) and Dhrymes e all (1985), researched how he heory works when limied number of asses is assumed or when he economy analysed is of he limied size. Shanken (1982) criicized he idea of APT esabiliy. The reurn-facors lineariy assumion was inoined as he misake in he heory formulaion. They concluded ha emloying he infinie number of asses is no enough o neglec he firm-secific risk. Furhermore, APT models are rone o maniulaion, as neiher he facors generaing reurns nor heir number were secified in he heory. One of he mos imoran roblems concerning he arbirage ricing heory esabiliy is he number of asses in he analysed orfolios. Dhrymes e al. (1984, 1985) saed ha for he number of asses ranging from 15-6 he number of significan facors increases from 3 o 6. Therefore, he number of asses analysed in grous is of umos imorance in he model esimaion. Furhermore, Dhrymes e al show ha he number of facors generaed by facor analysis deends on he number of observaions hroughou he ime and he numbers of analysed macroeconomic variables. However, Haugen (1996) believes ha he mos significan facors will be esimaed even on small samles. Weaker and herefore less imoran facors can surely disaear, unless he samle analysed is large enough. The less visible facors are no valuable for emirical 24

25 researchers, hus APT roonens believe ha he samle size does no maer. Furhermore, APT in he conrary o CAPM gives exlici redicions abou he orfolios efficiency. Haugen (1996) gives he following examle. I was assumed ha here are n facors and n orfolios and ha each of hem is a subsiue for one of he facors. According o Grinbla and Timan (1998) hese orfolios will be efficien orfolios only if hey were creaed according o APT. Thus, he emirical verificaion of he heory is easier han CAPM emirical ess. However, he roblem of APT esabiliy is sill unsolved. 25

26 CHAPTER II Caial marke equilibrium models emirical ess This chaer focuses on CAPM and APT emirical ess. Presened sudies migh be useful in esing hese models on Warsaw Sock Exchange as hey resen differen mehodologies of esimaion and ess. Furhermore, he emirical basis of researches includes hese ha were carried ou on Polish marke. These aers migh sugges, wheher hese models can be imlemened on he WSE. 2.1 CAPM emirical evidence In his secion he resuls of he mos significan CAPM ess will be chronologically resened and discussed. The lieraure can be divided ino wo ars: early CAPM ess, conduced in 7 s and laer ess (afer he Roll s criique) Early CAPM ess In he early CAPM ess he echnique of wo-sage regression analysis was commonly alied. The firs hase of his analysis was he imeseries regression, which was o esimae he bea coefficiens of each analysed comany. In he second hase he cross-secional regression was run, while he average rae of reurn was a deenden variable and a corresonding bea coefficien became an indeenden variable. The 26

27 aim of he regression was o esimae he Securiy Marke Line, which would allow o sae if is heoreical values were consisen wih he emirical findings. The mos significan emirical sudies on his oic were conduced by: Linner, quoed by Douglas (1968), Black, Jensen and Scholes (1972), Blume and Friend (1973) as well as Fama and MacBeh (1973) Linner es (1968) Based on he samle of 31 randomly chosen comanies Linner esimaed bea coefficiens regressing yearly reurns on shares agains yearly reurns on marke index. Years were he esimaion eriod for he equaion: R i, = α i + b i R M, + e i, where b i is he bea coefficien for he comany i. The second hase was he cross-secional regression: R i = a 1 + a 2 b i + a 3 S 2 ei + η i where 2 S ei is a variance of he residual e i. The obained resuls say in conradicion o he CAPM heory because of hree reasons. Firs of all, he coefficien a 1 should aroximaely equal he value of he risk free rae, bu i aeared o be higher of any value ossibly aken by R F in he examined eriod. Second, he coefficien a 2, which defines he rice of he acceed risk, alhough saisically significan, had much lower value han execed. 27

28 Finally, assuming ha he CAPM is rue a 3 as a coefficien of addiional indeenden variable should be saisically insignifican. Therefore, Linner s resuls are no coheren wih he heory, since a 3 is osiive and saisically significan. A resonse o he above analysis was a sudy conduced by Miller and Scholes (1972), who concenraed heir effors on a criique of he mehodology alied by Linner. There were hree lanes of he criique. Firs, he noaion of esed equaion was incorrec, since i did no include he model s reliance on he risk free rae in he roer manner. If he original model akes he form of: ( R R ) R i, = RF, + β i M, F, hen ( β i ) RF, + i RM R i, = 1 β, The siuaion ges more comlicaed if R F is no consan over ime. Second, heeroskedasiciy, which is ofen resen in he financial imeseries, is inerreed as an inconsan variance of he reurns over he esimaion eriod. Alhough Miller and Scholes found he heeroskedasiciy, hey decided ha i is no a cause of he CAPM rejecion. The las, and as i aeared he mos imoran reason for he CAPM failure was a bea esimaion error in he ime-series regression. The esimaed bea coefficien, biased wih he esimaion error, becomes an indeenden variable, which mus lead o he false esimaion of he arameer describing he variable. 28

29 Black, Jensen and Scholes es (1972) Black, Jensen and Scholes (BJS) overcame he roblem of bea coefficien s esimaion error, which was a cause of Linner s sudy failure. The mehodology of his sudy will be discussed in deails laer on, since based on a similar mehodology he es of he CAPM model on he Polish marke will be conduced. BJS used only he shares, which were lised on he NYSE wihin he eriod Based on he daa from he suberiod bea coefficiens of he individual shares were esimaed. These arameers were comued agains he unweighed marke index, comosed of all shares lised on NYSE. The nex sage was o sor he comanies according o heir bea value and dividing hem ino en orfolios, so ha he firs orfolio conains he decile of he comanies wih he highes bea coefficien and he las orfolio was creaed by he decile of comanies wih he lowes values of his coefficien. Nex, for each of he orfolios, series of welve monhly reurns realised in he nex year 1931, were calculaed. This rocess was reeaed shifing he sequence one year ahead, which means ha based on he eriod bea coefficiens were esimaed. Based on he esimaed bea coefficiens comanies were sored in orfolios for which welve monhly reurns were comued. Having he orfolios monhly reurns calculaed, BJS could esimae he orfolios bea coefficiens using he same index, which was used o bea coefficiens esimaion of individual comanies. The final version of he esed model ook he following form: ( R R ) R P, = RF, + β P M, F, 29

30 where R P is a reurn on he orfolio P. Using he esimaed bea arameers of each orfolio, in he cross-secional regression he Securiy Marke Line was esimaed: R P = a + a β 1 P where a is a risk free rae, if he one exiss. The value of his coefficien was.519, which is ercen yearly. Because i is saisically more han he average ineres rae of he governmen bonds in he sudied eriod, he resuls suor Black CAPM version. Black allows long buying of he governmen bonds a he risk free rae bu forbids heir shor sailing. The obained marke risk remium was.181, which is ercen yearly. The resuls srongly suor he zero bea version of CAPM model. The esimaed SML does no reveal any signs of curvilineariy and he deerminaion coefficien for he cross-secional regression equals almos uniy Fama and MacBeh es (1973) Alying a similar rocedure o BJS Fama and MacBeh (FM) formed 2 orfolios, for which he bea coefficiens were esimaed in he firs hase. The difference comes from he fac ha he bea coefficiens comued agains daa from he eriod were a basis o form he redicions of he raes of reurn in he eriod +1. Unlike BJS, Fama and MacBeh in he second hase reeaed he regression searaely for each monh in he eriod Due o he fac ha his echnique was emloyed, FM could analyse he changes in he arameer values over ime. The esimaed cross-secional equaion was as follows: 3

31 2 R P, = γ, + γ 1, β P γ 2, β P + γ 3, Se, + η P, Having he monhly regressions esimaed, he average values of esimaed arameers were calculaed in order o es he hyohesis regarding all four coefficiens. The resuls can be summarised in four oins. Firs, he average value of he inerce γ, should be equal o (for he sandard version of he CAPM) or greaer han (for he Black CAPM) he risk free rae. Second, he average value of γ 1, coefficien should be osiive. Third, he average value of γ 2, coefficien deermines if here are any signs of curvilineariy. According o he heory his coefficien should be saisically insignifican and eliminaing his variable should no decrease he value of he deerminaion coefficien. Finally, he residual variance should no be saisically significan while esimaing he average reurns on he orfolio. I is because invesors can eliminae his facor hrough diversificaion of heir orfolios. Hence, he average value of he coefficien γ 3, should no be saisically differen from zero. The resuls obained in he FM sudies are consisen wih CAPM heory. Similarly o he BJS research, he oucomes suor he Black s version of he CAPM model. However, one asec of he sudy should be criicised, namely FM did no use a weighed index, hence i can no be reaed as he reliable roxy of he marke orfolio. This argumen is resened by Roll (1977), who criicised he early ess of he CAPM model Roll s criique (1977) Roll criicised ess of he CAPM model of ha ime arguing ha hey are mahemaical auologies. The resened rove confirms ha even 31

32 if he index alied is he marke orfolio, bu some oher on he efficien orfolio, hen here always will be a linear relaionshi found beween he execed reurn on a share and is bea coefficien esimaed wih he resec o he efficien orfolio. Furhermore, he indices used are no marke orfolios. According o he definiion of he marke orfolio, i should consis of all asses available o he invesor. Marke indices do no include bonds, real esaes, gold and many oher invesmen ooruniies. Hence, he so far conduced ess may only verify he hyohesis of he aricular index efficiency, bu can no be considered a CAPM model ess. The conclusion is ha esing he CAPM model is no ossible because of he urely heoreical idea of he marke orfolio. Since he marke orfolio grouing all risky asses is non-exisen, i is imossible o calculae is reurn and herefore CAPM can no be a esable heory. However, Sambaugh (1982) roved ha he CAPM model es is no sensiive o he enrichmen of he roxy for he marke orfolio in addiional invesmen ooruniies. He buil a few versions of he CAPM model saring wih NYSE index as a roxy for he marke orfolio, nex exending i by he governmen and cororae bonds marke, hen adding he real esae marke and finally including even durable goods marke 2. Alying Lagrange mulilier ess o verify he hyohesis, Sambaugh could no rejec he Black CAPM version, concluding ha he Roll s criique was oo srong. Increasing he comosiion of he roxy for he marke orfolio did no influence Sambaugh s resuls. 2 Such as vehicle marke. 32

33 2.1.3 Laer ess of he CAPM model A brand new series of emirical aacks on he CAPM model consised in idenifying variables oher han bea coefficien ha could exlain he average execed reurns on shares Banz es (1981) One of he firs sudies of his ye was conduced by Banz (1981), who decided o es if he firm size can exlain his ar of variance in he reurns, which is no exlained by he bea coefficien. I was found ha in he eriod he average reurns on he comanies wih low caialisaion were saisically higher han he average reurns on he big comanies, afer adjusing for he risk in boh grous. This relaionshi is commonly known as a size effec. The rocedure of orfolios building alied in Banz (1981) es is similar o he BJS. All orfolios consis of comanies lised on NYSE and he cross-secional regression defines he relaionshi beween he average reurns, bea coefficien and relaive size of he orfolios. Since he coefficien of he relaive size variable is saisically significan even a he low levels of significance, Banz (1981) concludes ha he CAPM model is no fully secified hence fails. The negaive value of he coefficien should be inerreed as follows: he shares of he comanies wih higher caialisaion are characerised by lower, on average, raes of reurn han shares of he small comanies. To suor he resuls Banz conduced one more es. Two orfolios were creaed, each consising of 2 shares. The firs orfolio included shares of he comanies wih low caialisaion, whereas he second one included big firms. Boh were consruced in ha heir beas were equal. Based on he same eriod as in he revious sudy Banz found 33

34 ha he firs orfolio indicaes monhly on average 1.48 ercen higher reurn han he second one and he difference is saisically significan. This oucome is consisen wih reviously obained. The subsequen sudies suored doubs abou model missecificaion. Basu (1983) found ha he raios defining he firm size and E/P are inerrelaed, hence E/P should exlain he execed reurns as well. Addiionally, Bhandari (1988) roves ha he financial leverage raio is osiively correlaed wih execed rae of reurn. On he oher hand, he same year as he Banz s sudy was released, Chrisie and Herzel (1981) ublished a aer, in which hey indicaed ha he comanies decreasing heir caialisaion became more risky and since bea coefficien was measured based on he hisorical daa, i could no caure an increase in risk over he esimaion eriod, hence he bea was lower. Reiganum (1981) and Roll (1981) indicaed ha he bea coefficien of small comanies would be lower as i was an effec of hin rading Fama and French es (1992) A samle, on which Fama and French (FF) conduced heir es of he CAPM model, was consiued of comanies lised on NYSE, AMEX and NASDAQU in he eriod July 1963 December 199. FF creaed 1 orfolios firs soring he shares in en orfolios wih resec o heir caialisaion and hen, wihin each grou, shares were sored wih regard o heir bea coefficien value. Based on he cross-secional regression analysis of he equaion: RP = + γ 1β P + γ 2 γ ψ + η P P 34

35 where ψ P is an indeenden variable defining he firm size, hey came o he conclusion ha γ 2 coefficien is negaive (-.17) and saisically significan ( saisics = -3.41). The bea coefficien, however, is saisically insignifican and his conclusion will no change even afer exclusion of he size variable ψ P. FF include one more facor in heir analysis, book-o-marke equiy raio (BV/P) and conclude ha his variable exlain much beer he variance of average reurns han he size variable. Shares characerised by he high value of he BV/P raio generae higher reurns on average. Even hough his relaionshi does no necessarily have o be rue in he shor-erm, i is held in he long-erm. Unexeced migh be he fac ha FF alying he same mehodology as FM (1973) came o comleely inconsisen conclusions. This inconsisency is assigned o differen esimaion eriod. FF reeaed he es for he eriod alied in FM sudies and reached coheren resuls. Sudies ha are a resonse o he FF criique refer mosly o he daa used. Three years afer FF ublicaion Korhari, Shanken, and Sloan (KSS) (1995) wroe a aer, in which hey rove ha he resuls obained by FF deend mainly on he inerreaion of he saisical ess. KSS conclude ha bea coefficien from he esimaed form of equaion has a very high sandard error, which does no allow o rejec saisically a high range of he risk remiums. For examle he esimaed coefficien γ 1 wih a value of.24 ercen, has a sandard deviaion of.23 ercen, which means ha values of γ 1 may range from zero o.5. Amihud e al. (1992) share he view abou he saisical noise, concluding ha when alying more sohisicaed esimaion echniques he value of γ 1 coefficien would be osiive and saisically significan. The same resuls were obained by Black (1993), who suggesed ha he size effec migh have been relaed only o he 35

36 esimaion eriod alied by FF. He roved ha for he eriod he size variable did no affec he average rae of reurn and was saisically insignifican. Even if he size effec exiss, here is a remaining quesion if is significance is high enough, because of he relaively low value of he small comanies. Jagannahan and Wang (JW) (1993) sae ha in each of he grous esed by FF, 4 ercen of he bigges comanies were more han 9 ercen of he marke value of all comanies lised on he NYSE and AMEX. In his case, he CAPM model holds is emirical validiy. JW criicise ha he marke indices are used as a roxy for he marke orfolio. They indicae ha in he U.S.A. only one hird of non-governmenal asses is held by he indusrial secor and only 3 ercen of his amoun is financed by he caial markes. Furhermore, he inangible asses like human caial can no be refleced by marke indices. Finally, hey conclude ha bea coefficien of an individual comany is no a consan value over ime and he realiy can be much beer described by he CAPM model ha allow he coefficien o vary over ime. The KSS criique refers o he second variable as well roving ha he comanies wih high BV/P raio a he beginning of he esimaion eriod had much lower chances o survive, hence he lower chances of being included in Comusa, which was used in he survey. On he oher hand, he comanies ha managed o survive ogeher wih comanies added o he survey in he laer eriod indicaed on average higher reurns. Taking ino consideraion he above reasoning Breen and Korajczyk (1993) verify his hyohesis using he same sofware and daa as FF. Their conclusion is ha he BV/P variable should be definiely less significan. 36

37 Kozicki s and Shen s es (22) The auhors of his es quesion he conemorary mehodology alied while esing he CAPM model. Kozicki and Shen (KS) consider he hyoheses saed incorrecly by FM were he basis for many furher sudies. They argue ha insufficien evidence o rejec he null hyohesis was considered sufficien o rejec he model. KS sae ha his manner of esing leads o false rejecion of he model in a leas half of he sudies. They sugges o es he CAPM based on he saisical es in which he heory is rue under he null hyohesis. Alying his alernaive saisical es, he model can no be rejeced based on he daa used by FF. The invered formulaion of he null hyohesis, in which he bea coefficien equals zero is following: H : β i = I causes ha he es is a subjec o he error ye I and II, which means ha rejecs he model when i is rue and acces when CAPM is false. There are four mos oular reasons for he ye I error o occur: Low value of execed remium for he marke risk. High variance of he errors in execed remium for he marke risk var( η ) = σ m High variance of he error in he CAPM model. Small number of surveyed eriods. A high frequency of he error ye I occurrence reflecs he roblem of limied access o informaion, which causes a lack of sufficien 3 R M, = E R ) + η ( M, 37