Cointegration between Fama-French Factors

Transcription

1 1 Conegraon beween Fama-French Facors Absrac Conegraon has many applcaons n fnance and oher felds of scence researchng me seres and her nerdependences. The analyss s a useful mehod o analyse non-conegraon me seres, whch n hs sudy are Fama-French facors. If he prevous relaonshps can be formed o a saonary lnear combnaon, he prevous seres are conegraed. FF-facors are facors, whch affec he share's yeld expecaons n he long erm.e. each facor has a premum of dfferen sze over he rsk-free neres. In hs sudy we use conegraon o fnd dependences beween dfferen facors. We also sudy he balances and dynamcs beween prces, afer whch we creae an error correcon model based on 4 hedge porfolos. Keywords: conegraon, hree-facor model, Johansen procedure, hedgng 1. Inroducon The conegraon analyss became an mporan par of economercs shorly afer was publshed (Engle, Granger 1987). I has radonally been appled broadly o a wde varey of me seres, when one has desred o nvesgae nerrelaonshps of hose seres. The mehod s hus useful n many secors of scence, bu n parcular has araced neres n he fnancng, where probably has been appled o sudy he relaonshps beween dfferen local markes. The dea s smply o fnd common sochasc rends beween he seres. The mehod s nended o examne nonsaonary me seres, as are usually he prces of secures. Compared wh he radonal correlaon analyss, conegraon allows creang a model o forecas he me seres. Conegraon also demands nvesgaed me seres o be reverng owards he mean. When seres dverge, hey may be srongly correlang, hough no necessarly conegraed. Referrng o he prevous, he correlaon analyss may f beer he shor-erm perspecve, whle he conegraon can be used n boh shor-erm and long-erm dynamc sudes (Alexander 1999). In hs sudy, he examnaon focuses analysng he Fama-French hree-facor model (Fama, French 1996) wh he conegraon of he marke porfolo. The purpose s o examne he prce balances and he dynamcs of he earnngs. Moreover, we also mus pay aenon o he long-erm rends of he seres I s possble o apply an adjusmen erm when he seres drfs are dfferen, whch erm mus be used beween he FF-facors, as hese facors are parly dvergng from each oher. The sarng assumpon s ha he markes and he FF-facors are alone non-saonary, bu ha s possble o generae a saonary lnear combnaon beween hose seres. Non-saonary means n hs sudy a random walk process, whle he assumpon beween he facors s ha hey don behave compleely random walk. Thus ogeher seres can wander anywhere, bu no alone. Prevous examnaon responds que well wh he conegraon beween he ndex and he sngle sock (e.g. Alexander 1999). Frs, he FF facors average yelds above he rsk-free neres rae are compared o Russell's syle nvesmen ndces for comparable yelds over he rsk-free neres. Then he dfferences

2 2 beween he ndces and facors are analysed. In hs case he dfferences are he sze of he share and he rao beween he book value and marke value (BE/ME). Then an ADF es (Dckey-Fuller 1979) decdes wheher he seleced ndces are non-saonary, oherwse he conegraon analyss canno be appled. Then he conegraon beween marke porfolo and he ndces s analysed wh he CVAR model (conegraed vecor auoregressve model), and when conegraon s found he error correcon model ECM s used o make predcons. Now he mulple-me seres conegraon revew mus be done wh he Johansen procedure (Johansen 1988; Johansen, Juselus 1990). Fnally, 4-syle nvesmen hedge porfolos are bul for dfferen ranng perods based on ECM's forecas properes. Afer ha hese sraeges are compared wh he marke porfolo and ndex reurns. 2. Fama-French hree-facor model The Fama-French hree-facor model (Fama, French 1993, 1996) s currenly he bes and mos wdely used model for "anomales" (Cochrane 1999), whch CAPM (Sharpe 1964; Lnner 1965) canno explan o. These "anomales" coun he share sze and book-o-marke BE/ME-value s effec on he long-erm expeced reurns. The shares seem o have a srong value premum a (hgh- BE/ME-value), whch has been observed n emprcal sudes (Rosenberg, Red, Lansen 1985), whch n urn demonsraes ha he value shares reurns are sgnfcanly hgher han he growh shares. In addon, small socks have been found o produce hgher reurns han large socks n he long-erm perod. In he hree-facor model, expeced reurns depend on he marke rsk b and he rsk-free neres rae R f and n addon, share sze and BE/ME-value also affec expeced reurns. The model follows he equaon E( R ) R b ( E( R ) R ) s E(SMB) h E(HML), (1) f M f Where E (R ) s he expeced reurn on he chosen shares and E (R M ) s he expeced reurn on he whole marke porfolo. SMB s he dfference beween he expeced reurns on small and large shares. Correspondngly HML s he dfference beween expeced reurns on hgh-be/me and low- BE/ME. Facor weghs b, s and h can be deermned from porfolo componens n a smple lnear regresson. In our sudy, he marke s dvded no 9 porfolos B/L, B/M, B/H, M/L, M/M, M/H, S/L, S/M, S/H, where he frs characer ells he porfolo s share sze bg B, md M and small S and he second characer ells BE/ME value hgh H, medum M and low L. One facor s always dvded no hree pars, where a sngle componen represens 33% of he number of shares. SMB and HMB conss of he followng equaons (2) and (3) SMB (S/L S/M S/H)/3 (B/L B/M B/H)/3 (2) HML (S/H M/H B/H)/3 (S/L M/L B/L)/3. (3) The prevous wo FF facors are very low correlaed 0.13 (Davs, Fama, French 2000). In he same arcle, he prevous 9 porfolos yelds were suded n he U.S. marke n he me nerval , n whch 339 shares of NYSE were used over he perod ,

3 3 snce n he year of 1953 he number of NYSE shares had been doubled. In he year of 1996 he number of shares was 4,562 n NYSE, AMEX and Nasdaq sock markes. Moreover he model has been esed n oher major capal markes. The FF-facor model s reurn dfferences (Davs, Fama, French 2000) have been repored n Table 1. As he rsk-free neres rae R f he Uned Saes 1- monh reasure bll s used. SockSymbol BE/ME Sze(mllons) Exra reurns (R -R f )(%)/annum Volaly()(%) B/L B/M B/H M/L M/M M/H S/L S/M S/H Russell 3000 marke porfolo Russell 1000 growh large cap growh Russell 1000 value large cap value Russell 2000 growh small cap growh Russell 2000 value small cap value Table 1. Fama-French facors versus Russell ndces and her exra reurns/annum and volales. In laer conegraon analyses, he facors suded are Russell's fve ndces (Russell 2006) n The marke porfolo s descrbed by he Russell 3000 Index, whch covers approxmaely 98% of he U.S. sock marke's value. As a large cap porfolo s used he Russell 1000 growh ndex, whch corresponds o he B/L-porfolo and he oher large cap porfolo s he Russell 1000 value ndex, whch n urn corresponds o he B/H porfolo. Sngle shares o he prevous ndces have been seleced n he Russell 1000 ndex based on he BE/ME values and he Russell 1000 ndex conans he 1,000 larges shares. In he same ways as he small cap porfolo s used he Russell 2000 growh ndex, whch corresponds o he S/L-porfolo and as he oher small cap porfolo s he Russell 2000 value ndex, whch n urn corresponds o he S/H porfolo. Sngle shares o prevous ndces have been seleced n he Russell 2000 ndex based on he BE/ME values and he Russell 2000 ndex conanng he 2,000 smalles shares. The porfolos, whch Fama and French formulaed, correspond que well o Russell's ndces (Table 1). Value shares, whch have a hgh BE/ME value, have clearly hgher average reurn han growh shares, bu he sze mpac s no so ha clear. However, has been observed ha n he longer erm he small-value shares produce hgher reurns han larger shares. Ths s clearly seen n (Table 1) and (Fgure 1), where value shares have gven beer reurns beween he years On he oher hand he growh-shares show conradcory resuls, because large growh shares have surprsngly reurned more durng he years han smaller, hough n he years he suaon (Davs, Fama, French 2000) has been he oppose. The resuls are neverheless que conradcory comparng o he CAPM model, because durng he years

4 he hgher volaly growh socks have gven he lowes reurns compared o oher Russell ndces. Fgure1. Normalzed prces of Russell ndces. 3 Conegraon Conegraon beween me seres s one of he mos mporan economerc ools, whch has been used wdely, snce he Engle-Granger wo-sep mehod appeared (Engle, Granger 1987, he Nobel Commee 2003). Conegraon can be used o oban mporan nformaon of he me seres long-erm srucure, whch hen can be used o mprove he economc decson-makng. The prevous conegraon eher exss or no (on/off-prncple). A good resul can be acheved only by usng carefully sascal analyss, leavng sll a small probably o fal. Two or more nonsaonary me seres, whch are negraed o a degree of I(n), can represen lnear combnaons, where hey are saonary. Thus hese seres are conegraed I(0). In hs revew seres are negraed o he degree I(1) (non-saonary) or hey are no negraed I(0) (saonary). If he seres x and y are negraed o a degree I(1), bu her lnear combnaon y a (4) bx s I(0), he seres x and y are conegraed and he error erm s n he form z ( ) y a bx ~ I(0) (5) beng saonary, so ha a and b exs and a s a possble drf vecor. Now he vecor z s called a conegraon vecor, whch properes are esed laer n hs sudy. If here exs a number of n seres, here canno be more han (n-1) conegraon vecors z. If here exs only wo me-seres,

5 5 here can hus be only one vecor, because oherwse he orgnal seres should be saonary (Alexander 1999). One of he mos mporan feaures of he conegraed seres s her common sochasc rend (Sock, Wason, 1988). The seres x and y are hus lnked o each oher n a long-erm perod. These seres may be separaed n he shor erm, bu n long erm hey follow su, whch s called "long-run equlbrum". If he seres dverge whou lm and no correcon erm s used, hese seres do no have a common balance relaonshp and hus conegraon does no exs. A sochasc rend can be presened for wo seres as follows x y, (6) x y x, (7) y where x and y are averages of he seres x and y, whch depend on prevous averages and her relaed error erms. x and y are dsances from he averages. Now ha x and y are conegraed, hey can be presened as a lnear combnaon b 1 y b2x ( b1 y b2 x ) b1 y b2 x, (8) where c b b ) ( 1 y 2 x mus be saonary. The coeffcens b 1 and b 2 can be solved wh e.g. lnear regresson. The seres x and y may now be presened n he form x y, (9) x b b x c 2 x y, (10) 1 b1 because hey have a common sochasc rend and hen hey are hus conegraed. In he followng chapers , he prevous heory s explaned based on he Johansen procedure and laer s used o creae a smple hedgng sraegy. 3.1 Johansen procedure Usng he Johansen procedure n conegraon analyss can be appled more han jus wo me-seres (Johansen 1988; Johansen, Juselus 1990), herefore he procedure has become a man ool n conegraon analyss. We use also n hs sudy, because he Engle-Granger mehod canno be used for he fve me-seres suaon. The Johansen procedure s based on fndng a sochasc marx egenvalues, whch wll also help o reduce he correlaon relaed problems. The bgges dfference from he Engle-Granger mehod s o focus maxmum saonary nsead of he mnmum varance prncple. Furhermore, he es s more versale and sophscaed compared o he Engle-Granger mehod, bu correspondngly more complex (Alexander 1999). Conegraon can also be found dvergenng from seres, f he rend correcve erm s used, whch s requred when nvesgang FF-facors, because he reurns dffer n he long erm.

6 6 Frs we mus creae an n-degree and p-dmensonal long-erm VAR model (vecor auoregressve model) (11), from whch o creae a conegraon basc model. y y... y D 1 1 n n (11) y s now he process vecor ( p 1) a he me, n whch a leas wo componens mus be nonsaonary. ( p 1) s he error-erm vecor, where he errors are ndependen of each oher. ( p p) s he process y - coeffcen marx a he me, whle D s he vecor of non-sochasc varables, such as he dummy varables for whch he s he coeffcen marx. n urn s he unlmed drf, whch akes no accoun he dfferen szed drfs from he observed me-seres. Non-sochasc effecs vecors don appear n our case, so now D 0. The nex CVAR model (conegraed vecor auoregressve model) s bul on equaon (11) y y y y n n1 n, (12) where ( ), where 1,..., n -1 (13) 1... ) (14) ( 1 n and y y y 1. Apar from he VAR model he revew s now focused on he marx rank() degree, whch wll ell he mos essenal nformaon of he seres long-erm relaonshps. Now he erm y -n mus be saonary I(0). If marx degree s rank () p, he marx s hen a full degree marx and all he componens of y are saonary, when he nal assumpons of nonsaonary canno be revsed. If also rank( ) 0, s null marx and consequenly he model s no longer a CVAR model. When he marx degree s 1 rank( ) (1 p), here exs conegraon vecors and he marx can be represened n he form = T, where and are full degree marces. descrbes a long-erm adjusmen speed and n urn descrbes he conegraon vecors. The conegraon hypohess H 0 (r) s of he form T H 0 ( r) :, (15) n whch case he process y s saonary and accordng o he nal assumpons of he seres y a leas wo of prevous seres are non-saonary. The prevous esmaon of he model begns wh maxmum lkelhood procedure, whch nally begns by fndng he Gaussan errors n he mulvarae conegraon model (Johansen 1988, 1991; Johansen, Juselus 1990). The lkelhood funcon parameers,, n-1 and mus be defned usng regresson wh he y and y -n ulzng he erms y -1,,y -n+1. Ths gves he resduals R 0 and R n, whch can be used o deermne he cross momen marx resdual s S j

7 7 T 1 T Sj T RR j,, j 0, n, (16) where T s he me marx from null o me T. The cenralzed lkelhood funcon s 1 T R 0 R, (17) where represens he error. The regresson equaon (17) can be used o esmae as a funcon of n ˆ S, (18) T 1 0 n( Snn) afer whch can be deermned by solvng egenvalues from he equaon (19). 1 S nn Sn0S00 S0n 0 (19) Egenvalue problem (19) has now p soluons, 1 ˆ... ˆ 1 p 0. Correspondng egenvecors are found as Vˆ ( vˆ,..., vˆ ) and hey can be represened n a normalzed form 1 p ˆ ( vˆ,..., vˆ ), (20) 1 r where βˆ s he maxmum lkelhood esmae and s also gven by r rank( ). The maxmum lkelhood funcon has he form L r max S00 1 ˆ ) 1 (. (21) Nex he lkelhood rao es s carred ou for he hypoheses (15) n he equaon (11) VAR model (Johansen, Juselus 1990). There exs wo dfferen LR-ess, of whch he frs one s he Trace sasc (22) and he second s he max sasc (23). The ess solve he relevan roos or egenvalues, whereby he rank of he marx can be decded. LR race T p r 1 ln( 1 ˆ ) (22) LR ln( 1 ˆ max T r 1) (23) In he LR-es he null hypohess s appled H 0 : r 1 r , whch gves he sysem p-r un roos, whch are he sarng pon for fndng sysem rank. The roos are found sep by sep, where frs s assumed ha here exs p un roos. If he null hypohess H 0 has o be rejeced, he answer s 1 0, afer whch he hypohess H 0:... p 0 s appled. If hs agan s rejeced, 2 3 he resul s 2 0, afer whch he process s repeaed all he way o p, unless he un roo can be

8 8 found. If he hypohess s fnally acceped, he amoun of conegraon vecors are found by usng he un roos. The las descrbed rank of he marx s he mos mporan and dffcul par he of Johansen procedure. If he rank s esmaed oo low, he conegraon may be unnoced. A oo large degree of rank can lead o dscoverng a conegraon, hough does no really exs. 3.2 Resuls Before conegraon can be esed, he orgnal me seres logarhms saonary mus be esed. To realze saonary, he coeffcens of he me-seres erm (equaon 24) mus have a smaller absolue value han 1. In saonary seres he shocks are emporary and hey wll always reurn slowly o her average level. Non-saonary seres ypcally do no have a long-erm equlbrum, lke socks and ndces n general. If he seres s non-saonary, here s always a un roo. For roos exploraon, here are many ess. However, n hs sudy we use he mos wdely known augmened Dckey-Fuller es (ADF-es) (Dckey-Fuller 1979). The esed me seres y s now gven by y a y y... y p n, (24) where a s a consan (drf), s he coeffcen of lag change y and n s he lag degree n he auoregressve process. Nex s esed he marke porfolo s and FF facor s saonary, whch are now solely creaed from he Russell ndces. As null hypohess H 0 s appled 1, hus he seres s non-saonary and here exss a un roo. As alernave hypohess s appled H 1, hus he seres s saonary. Prevous es resuls have been gahered n Table 2. wh lags = 1-5 and wll be noced from hem ha all porfolos are non-saonary H 0 (p>0.05) for each fve lags. lag = 1 lag = 2 lag = 3 lag = 4 lag = 5 Porfolo name Dckey- Dckey- Dckey- Dckey- Dckeyp-value p-value p-value p-value Fuller Fuller Fuller Fuller Fuller p-value marke porfolo large cap growh large cap value small cap growh small cap value Table 2. Sascal sgnfcance of non-saonary p-value. Complee he nex (logarhmc) error analyss for ndces usng he equaon (4), where a and b can be defned wh a lnear regresson (for example OLS regresson). In hs suaon he error erm does no need o be noced. Analysng errors, marx rank s assumed o be r p 5. The me-seres error correlaons (-resdual) and her sandard devaons are lsed n Table 3. Beween some errors, here are sgnfcan correlaons, as for example beween he marke and large cap growh porfolos, where he correlaon s as hgh as 0.967, whle he correlaon beween he large cap value and he small cap growh s only The normaly of errors has furher been esed wh he Shenon-Bowman es (Shenon, Bowman 1977; Doornk, Hansen 1994), where

9 9 normaly or auocorrelaon n errors do no occur, when lags s rased o sx. Auocorrelaon, however, can be found, f he used lag s oo small. -resdual marke porfolo Large cap growh large cap value small cap growh small cap value marke porfolo 1 large cap growh large cap value small cap growh small cap value sandard devaons of he resduals Table 3. Correlaon marx and sandard devaons of he resduals. The nex sep s o es he exsence of he long-erm conegraon beween marke porfolo and FF facors (Hansen, Juselus 1995). Frs all he fve me-seres mus undergo he race sasc esmaon accordng o he equaon (22). In Table 4., here are he prevous es p-values of he null hypohess H 0. In Table 4. can be found hree dfferen un roos, whch are λ 3 λ 4 λ 5 0, 1 0 and 0 2. The marx rank s enavely 2, so here are also 2 conegraon vecors. Hypohess r 0 r 1 r 2 r 3 r 4 p-value Table 4. Johansen Trace sascs p-values. The marx rank r can be furher verfed by usng he companon marx A egenvalues (Hansen, Juselus 1995), 1 I p A 0 0 I 0 p 0 where I p s a p-dmensonal deny marx and s defned n he equaon (11). 30 egenvalues of he marx A are descrbed n he Pcure 2. un crcle. The egenvalues mus be locaed a he un crcle or nsde, unless he marx rank canno be 2. In our suaon all he egenvalues are, however, a he un crcle or he nsde, so he marx rank s 2. Thereby has also 2 conegraon vecors, whch reflec he marke rsk for each facor and marke porfolo. 2 I n1 0 0 p n (25)

10 10 Roos of he Companon Marx 1.0 Rank(PI)= Fgure 2. A scaer plo of he egenvalues of he companon marx. Marx has now go an esmae ˆ and so have all he oher CVAR equaon (12) parameers been esmaed. Thus, he marx has been resolved, afer whch can be buld error correcon model ECM for forecas. The prevous revew has also been represened separaely n Table 5. beween all sngle me-seres. In combnaon of he wo me-seres, here canno be found conegraons excep of a par of large cap value and small cap value. lag = 6 marke porfolo large cap growh large cap value small cap growh small cap value marke porfolo - large cap growh 0 - large cap value small cap growh small cap value Table 5. Rank of conegraon marx beween sngle me-seres. Fnally, usng he error correcon model, here has been creaed smple hedgng sraegy, where he arge nvesmens are he 4 FF-facors based on he Russell ndces. The hedge porfolo weghs are updaed every hree monhs, so ha he facor, whch he ECM model predcs o grow, he hghes reurn receves a porfolo wegh 1 and he oher facors weghs are 0. The weghs are presened every quarer n Appendx 1. Four dfferen ranng perods are used o es he predcons. The used ranng perods are 5-year, 7-year, 10-year and all prevous daa, and he predcons are for he perod 1991Q1-2004Q3. Also n he nal scenaro used perod 1980Q1 1990Q4 has a marx, whch rank s 2, so s raonal o use wo conegraon vecors, a every sage. All 4 ranng perods cumulave nomnal reurns are llusraed n Fgure 3. The suded 14 year perod conegraon hedgng sraegy has won a marke porfolo (Russell 3000 ndex). Reurns are also compared apar from he marke porfolo o he bes ndex small cap value reurns. In he Fgure 3. cases b), c) and d), where a 7 years or longer ranng perod has been used for a 3 monh predcon pror o s sarng pon, he hedgng sraegy was able o wn even he hgh reurn small cap value ndex (Russell 2000 value). On he oher hand a 5-year ranng sraegy was unable o

11 11 wn he bes ndex, hough won he marke porfolo. In Table 6., he marke porfolo, facors and hedge porfolos reurns over rsk-free neres, are shown ogeher as has been done n Table 1. as also her mos mporan descrpve sascs for he 14-year perod. Fgure 3. Cumulave nomnal reurns of 5,7,10 years and all prevous daa hedge porfolos. Exra reurns(r -R f )(%) Volaly()(%) Sharpe rao Alpha(%) marke porfolo large cap growh large cap value small cap growh small cap value hedge porfolo (5 year) hedge porfolo (7 year) hedge porfolo (10 year) hedge porfolo (all prev. daa) Table 6. Porfolos descrpve numbers/annum(%) 1991Q1-2004Q3. Accordng o Fgure 1. and 3. he dfferences beween he ndces have been greaes around he urn of he mllennum, when he IT bubble has made growh socks unreasonably expensve. Precsely n ha perod of hedgng sraegy has performed bes. In Fgure 3. sraeges b), c) and d) has exploed really well he unusual hgh reurns of he growh socks n he lae 1990s, and afer

12 12 ha managed o jump off he rde changng o value socks, when he growh socks have become dsproporonaely expensve. Ths s very undersandable, because he value socks were rsng hen que moderaely and he conegraon expecs he seres o be mean reversng. When he gap grew oo hgh, he model smply jumped ou of he growh shares. Conegraon has been suded much beween he marke locaon facors and he properes of sngle sock he ndex (e.g. Alexander 1999), bu conegraon can also be examned beween he FF-facors, whch n hs sudy was able o wn, n 3 ou of 4 cases, he ndvdual FF facors and marke porfolo durng he IT bubble perod. The clam of exsng conegraon n he effcen markes s unclear. I has been presened dfferen vews for and agans conegraon n effcen markes. For example (Granger 1986) and (Balle, Bollerslev 1989) have argued ha predcably of conegraon would mean neffcen markes. On he oher hand, (Dwyer, Wallace 1992) and (Ferre, hall 2002) have argued ha neffcency and conegraon s no he same hng. In our emprcal research, seems o be hsorcally unque, ha he facors had very large devaons a he me of he IT bubble. Those prevous feaures mgh refer o a marke neffcency. Also opnons on small cap value premum have been dvdng he scens opnons. For example, s argued wheher he value premum s from an anomaly or somehng oher rse (such as an exreme loss a very bad mes). If hs s a pure anomaly, should dsappear. For example Table 1. shows clearly a resrucurng o he small cap growh facor afer he 80s, afer whch he shares of hs facor have gven a lo weaker reurns han he oher facors. 4. Concluson In hs sudy we evaluae conegraon beween shares prces of he Fama-French facors (Fama, French 1993, 1996) and he marke n he perod The FF facors were used by he Russell syle nvesmen ndces. Conegraon s especally useful for he revew of non-saonary daa, whch he syle nvesmen ndces are. The prevous analyss wll be a good addon, when one waned o go furher han he correlaon analyss and possbly creae some knd of model for predcons. Inally he FF facors hsorcal reurns were compared o he correspondng syle ndces reurns. The resuls were oherwse smlar excep for he small cap growh porfolo, whch gave gven lower reurns n our sudy, so ha he small cap growh porfolo premum was sgnfcanly decreased snce he 80s. The man neres cenered on conegraon beween mulple me seres usng Johansen procedure (Johansen 1988; Johansen, Juselus 1990). Frs an error analyss was carred ou, whch was found ha he errors were somewha correlaed. On he oher hand neher auocorrelaon nor normaly was seen a suffcenly long lags. For he fve me-seres suded he conegraon was rank 2, whch also means ha here are wo conegraon vecors. Fnally an error correcon model (ECM) was creaed beween he marke porfolo and he ndces, whch were successfully used o creae he syle nvesng based hedge porfolos, whch were used successfully whn four dfferen long ranng perods from hree monhs me clps. In parcular hedge sraeges were que successful, as durng he IT bubble hey bea n all cases he marke porfolo and n hree cases of four he bes succeeded ndex. Consequenly was found ha he componens can form a saonary enrey and her behavour s nerdependen on each oher.

13 13 Moreover all ndvdual ndces and marke porfolo was esed beween he wo combnaons of conegraons, whch ddn occur excep one excepon. Thus he ECM model canno be creaed hs suaon. Crcsm, however, has been heard on he exsence of conegraon n effecve markes (Granger 1986; Balle, Bollerslev 1989). Bu some researchers do no see a conradcon beween conegraon and effcen markes (Dwyer, Wallace, 1992; Ferrer, hall 2002). Also small cap value share fuure premum has rased dscusson, as should dsappear f s an anomaly. For nsance he small cap growh premum has radcally dmnshed afer he 80s. References Alexander C. 1999, Opmal hedgng usng conegraon. Phlosophcal Transacons of he Royal Socey, Seres A 357: Balle R., Bollerslev T. 1989, Common sochasc rends n a sysem of exchange raes. Journal of Fnance, 44, Cohcrane J. 1999, New facs n fnance. Economc Perspecves XXIII (3) Thrd quarer 1999 (Federal Reserve Bank of Chcago), also NBER workng paper 7169 Davs J., Eugene F., French K. 2000, Characerscs, Covarances, and Average Reurns: 1929 o The Journal of Fnance, Vol. 55, No Dckey D., Fuller W. 1979, Dsrbuon of he esmaes for auoregressve me seres wh a un roo. J. Am. Sascal Assoc. 74, Doornk J., Hansen H. 1994, An omnbus es for unvarae and mulvarae normaly. Workng paper, Nuffeld college, Oxford. Dwyer G., Wallace M. 1992, Conegraon and marke effency. Journal of Inernaonal Money and Fnance, 11, Engle R., Granger C. 1987, Conegraon and error correcon: represenaon, esmaon, and esng. Economerca 55, Fama E., French K. 1993, Common rsk facors n he reurns on socks and bonds. Journal of Fnancal Economcs 33, Fama E., French K. 1996, Mulfacor explanaons of asse prcng anomales. Journal of Fnance 51,

14 14 Ferre M., Hall S. 2002, Foregn exchange marke effency and conegraon. Appled Fnancal Economcs 12, Granger C. 1986, Developmens n he sudy of conegraed varables. Oxford Bullen of Economcs and Sascs 48, Hansen H., Juselus K. 1995, CATS n RATS, Conegraon Analyss of Tme Seres. Esma: Illnos, USA. Johansen S. 1988, Sascal analyss of conegraon vecors. J. Econ. Dyn. Conrol 12, Johansen S. 1991, Esmaon and hypohess esng of conegraon n Gaussan auoregressve models. Economerca, 59, Johansen S., Juselus K. 1990, Maxmum lkelhood esmaon and nference on conegraon wh applcaons o he demand for money. Oxford Bull. Econ. Sas. 52, Lnner J. 1965, The valuaon of rsk asses and he selecon of rsky nvesmens n sock porfolos and capal budges. Revew of Economcs and Sascs 47, Nobel Commee 2003, Advanced Informaon: Tme Seres Economercs: Conegraon and Auoregressve Condonal Heeroskedascy. Rosenberg B., Red K., Lansen R. 1985, Persuasve evdence of marke neffcency. Journal of Porfolo Managemen 11,9-17 Russell Invesmen Group, 2006, Russell U.S. Equy Index Defnons. hp:// Sharpe W. 1964, Capal Asse Prces: A Theory of Marke Equlbrum Under Condons of Rsk. Journal of Fnance 19, Shenon L., Bowman K. 1977, Abvarae model for he dsrbuon of b 1 and b 2. Journal of Amercan sascal assocaon 72, Sock, J. M. Wason M. 1988, Varable Trends n Economc Tme Seres. Journal of Economc Perspecves, Vol 2, No. 3,

15 15 Appendx 1. Porfolos weghs n every quarer. ranng perod 5 year 7 year 10 year all daa A B C D A B C D A B C D A B C D 1991Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q

16 Q Q Q Q Q Q Q Q Q Q Q A = large cap value B = large cap growh C = small cap value D = small cap growh Table A1. Porfolos weghs 1991Q1-2004Q3.