Sources of Over-Performance in Equity Markets: Mean Reversion, Common Trends and Herding

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1 The Universiy of Reading THE BUSINESS SCHOOL FOR FINANCIAL MARKETS Sources of Over-Performance in Equiy Markes: Mean Reversion, Common Trends and Herding ISMA Cenre Discussion Papers in Finance Firs version: May 2003 This version: Ocober 2003 Carol Alexander ISMA Cenre, Universiy of Reading, UK Anca Dimiriu ISMA Cenre, Universiy of Reading, UK Copyrigh 2003 Carol Alexander and Anca Dimiriu. All righs reserved. The Universiy of Reading ISMA Cenre Whieknighs PO Box 242 Reading RG6 6BA UK Tel: +44 (0) Fax: +44 (0) Web: Direcor: Professor Brian Sco-Quinn, ISMA Chair in Invesmen Banking The ISMA Cenre is suppored by he Inernaional Securiies Marke Associaion

2 Absrac In he field of opimisaion models for passive invesmens, we propose a general porfolio consrucion model based on principal componen analysis. The porfolio is designed o replicae he firs principal componen of a group of socks, insead of a radiional benchmark, hus capuring only he common rend in he sock reurns. The main advanage of his approach is ha he reducion of he noise presen in sock reurns faciliaes he replicaion ask considerably and he opimal porfolio srucure is very sable. We analyse he porfolio performance over differen ime horizons and in differen inernaional equiy markes. The sraegy over-performs boh equally weighed and price weighed benchmarks, even afer ransacion coss. A marke premium, a value premium associaed wih mean reversion in sock reurns, and a volailiy premium which give he sraegy characerisics of a benchmark enhancer, all explain he over-performance, bu have ime-varying conribuions o i. A behavioural explanaion for he mean reversion mechanism leads o he conclusion ha he porfolio performance is influenced by he exen of invesors herding owards he common rend in sock reurns. Auhor Conacs: Prof. Carol Alexander Chair of Risk Managemen and Direcor of Research ISMA Cenre, School of Business, Universiy of Reading, Reading RG6 6BA Anca Dimiriu (corresponding auhor) ISMA Cenre, School of Business Universiy of Reading, Reading RG6 6BA Tel +44 (0) Fax +44 (0) JEL classificaion: C32, C51, G11, G23 Keywords: common rends, mean reversion, herding, principal componen analysis, abnormal reurns, value sraegies, behavioural finance The auhors would like o hank Glen Larsen for his valuable commens on he behavioural implicaions of heir resuls. Also, he auhors graefully acknowledge he commens of Myron Scholes, Lionel Marellini and he paricipans a Quan 03 conference, which have helped improving his paper. All errors remain our responsibiliy. This discussion paper is a preliminary version designed o generae ideas and consrucive commen. The conens of he paper are presened o he reader in good faih, and neiher he auhor, he ISMA Cenre, nor he Universiy, will be held responsible for any losses, financial or oherwise, resuling from acions aken on he basis of is conen. Any persons reading he paper are deemed o have acceped his.

3 Inroducion Comparisons beween he wo main equiy invesmen syles acive and passive have a long hisory, being much influenced by boh academic research and he invesmen managemen indusry. 1 The ineres in replicaing marke performance hrough a passive sraegy, mos frequenly in he form of indexaion, is subsaniaed by he principles of efficien markes and modern porfolio heory, where he only way ha invesors can bea he marke over he long erm is by aking greaer risks (Fama, 1970). Addiionally, acive managemen has been shown o ofen under-perform is passive alernaive (Jensen, 1968; Elon, Gruber, Das, Hlavka, 1993; Carhar, 1997) due o ransacion coss and adminisraion fees, mosly in bull, bu also in bear markes. For example, he S&P acive/passive scorecard for he las quarer of 2002 shows ha he majoriy of acive funds have failed o bea heir relevan index even in he bear marke of he las few years. As a consequence of hese rends, he passive invesmen indusry has winessed a remarkable growh during he las en years, wih a huge number of funds pegging heir holdings o broad marke indexes such as SP500. Currenly, i is esimaed ha more han $1.4 rillion are invesed in index funds in he US alone (Blake, 2002). Tradiionally, indexaion has argeed price weighed and value weighed indexes, which are easy o replicae wih porfolios comprising he enire se of socks and mirroring he benchmark weighs. Such porfolios are self-adjusing o changes in sock prices and do no require any rebalancing, provided here are no changes in he index composiion or in he number of shares in each issue. Despie he self-replicaion advanage, holding all he socks in he benchmark may no always be desirable or possible. 2 More involved sraegies are also required for racking equally weighed indexes, since frequen rebalancing is required in order o mainain equal dollar amouns in each sock. Larsen and Resnick (1998) provide a horough empirical invesigaion of he relaionship beween he indexed porfolio s composiion and he racking performance. Their resuls show ha value weighed indexes are easier o replicae han equally weighed indexes, and capialisaion dominaes oher sraificaion crieria such as indusry classificaion. Given he disadvanages of direc replicaion, recen research has focused on developing opimisaion models for passive invesmens. Convenionally, racking sraegies using fewer socks are consruced on basic capialisaion or sraificaion consideraions. Opimisaion echniques have also been 1 As a consequence, he very conceps of acive and passive invesmen syles have evolved. Now, hey can only be discriminaed based on heir invesmen objecive, all oher feaures, e.g. amoun of research involved, porfolio opimisaion echniques, frequency of rades, being similar. The acive managemen is seeking o over-perform he marke, usually hrough sock selecion or marke iming, while passive managemen is aiming o replicae he marke performance. Also, sraegies such as enhanced index racking, which exend a passive syle ino acive managemen, have been developed. Copyrigh 2003 Carol Alexander and Anca Dimiriu 3

4 developed using objecive funcions based on he correlaion of he porfolio reurns wih he benchmark, he mean deviaion of he racking porfolio reurns from he benchmark, he variance of his deviaion (ofen referred o as racking error ) or he ransacion coss. Some examples are given in Rudd (1980), Meade and Salkin (1989), Adcock and Meade (1994), Connor and Leland (1995), Alexander (1999), Larsen and Resnick (1998 and 2001). The presen paper conribues o his line of research by proposing a general porfolio consrucion model based on principal componen analysis. The model idenifies, of all possible combinaions of socks wih uni norm weighs, he porfolio ha capures he larges amoun of he oal join variaion of he sock reurns. Such a propery makes i he opimal porfolio for capuring he common rend in a sysem of socks whils filering ou a significan amoun of noise. In finance, he use of saisical echniques o model asse reurns has been exensive, especially in he conex of facor models. Going back o Feeney and Heser (1967) and Lessard (1973), or in more recen years, Schneeweiss and Mahes (1995) and Chan, Karceski and Lakonishok (1998), principal componen and facor analysis have been used o examine he exisence of common movemens in sock reurns. They are seen as alernaives o fundamenal approaches which relae he facors influencing financial asse reurns o macroeconomic measures such as inflaion, ineres raes and marke indices, or o company specifics such as size, book o marke raio or dividend yield. A grea deal of saisical facor analysis has been performed for esing he arbirage pricing model (Ross, 1976). In his conex, hisorical reurns are used o esimae orhogonal saisical facors and heir relaionship wih he original variables. The consrucion of mimicking porfolios for he saisical facors has been formalised by Huberman, Kandel and Sambaugh (1987). Furhermore, alernaives o sandard principal componen analysis have been developed, e.g. asympoic PCA (Chamberlain and Roschild, 1983, Connor and Korajczyk, 1986 and 1988) or independen componen analysis (Common, 1994). A common finding in he lieraure is ha he firs principal componen of a group of socks capures he marke facor (Chan, Karceski and Lakonishok, 1998; Connor and Korajczyk, 1988). 3 This assessmen is based on wo observaions. Firs, provided ha sock reurns are reasonably correlaed, hey will have similar loadings on he firs principal componen, so a shock o his facor will generae a common rend in he sysem. Secondly, he R 2 from a simple regression of an equally weighed 2 This happens mainly because of difficulies in purchasing odd los o exacly mach he marke weighs, or he increased ransacion coss/marke impac relaed o rading less liquid socks. 3 To noe, hroughou he paper we use he erm marke o denoe he specific universe of socks argeed by he passive invesmen sraegy, which can be anyhing beween a selecion of socks and he rue marke porfolio, comprising all asses. Copyrigh 2003 Carol Alexander and Anca Dimiriu 4

5 porfolio of all socks on heir firs principal componen is usually found o be high, above 0.8, he firs principal componen explaining o a large exen he reurns on an equally weighed porfolio. This resul can be exrapolaed also o oher ype of indexes, such as price weighed, provided ha he reurns of price weighed and equally weighed indexes represening he same universe are generally highly correlaed. Our porfolio consrucion model is based precisely on he resemblance of he firs principal componen of he sock reurns o he marke facor proxied by a radiional index. The sandard approach o consrucing facor mimicking porfolios uses he facor loadings in he sock selecion process (e.g. Fama and French, 1993). The socks are ranked according o heir loading on a paricular facor, hen a self-financed porfolio is se up wih long posiions on he socks wih he highes loadings on ha facor and shor posiions on he socks wih he smalles loadings. Mos frequenly, here is no porfolio opimisaion, equal dollar amouns being invesed in each sock. An alernaive proposed by Fung and Hsieh (1997) for facor mimicking porfolios considers, in he sock selecion sage, only he socks ha are highly correlaed solely o he principal componen for which he replica is consruced. Having seleced he socks, heir porfolio weighs are opimised as o deliver he maximal correlaion of he mimicking porfolio reurns wih he corresponding principal componen. In hese wo mehods, principal componen analysis is used as a sock selecion echnique and he porfolio consrucion is a separae sage, based eiher on a sandard opimisaion, or on an arbirary mehod such as equal weighing. In his paper we propose a differen approach in which a porfolio replicaing he firs principal componen is consruced direcly from he normalised eigenvecors of he covariance marix of sock reurns. Such a porfolio, by consrucion, capures he larges proporion of he variaion in he sock reurns and filers ou a significan amoun of noise. Therefore, i is naurally suied for a passive invesmen framework, requiring a fully invesed porfolio of all socks, bu involving a very small amoun of rebalancing rades because i capures only he major common rend in sock reurns. This procedure involves a single opimisaion, he one producing he principal componens. Moreover, here is no arbirary choice of he porfolio consrucion model, such as equal weighing of socks. In order o invesigae he porfolio performance, we use a group of socks included in he Dow Jones Indusrial Average (DJIA) a he end of year To suppor he feaures of he sraegy observed in he DJIA case, we also consruc random subses of socks from he SP100, FTSE100 and CAC40 universes. The performance is analysed boh before and afer ransacion coss: we examine he reurns volailiy and correlaion (uncondiional over he enire sample and also shor-erm ime series Copyrigh 2003 Carol Alexander and Anca Dimiriu 5

6 esimaes), and he higher order momens of reurns disribuions, boh from an overall perspecive and condiional on marke circumsances. Even if a benchmark does no ener he porfolio consrucion model, we follow convenion o use boh price weighed and equally weighed indexes as benchmarks for he porfolio performance. Unsurprisingly, our resuls indicae ha he firs principal componen capures he marke facor, being highly correlaed wih he benchmark reurns. Moreover, he facor weighs prove o be very sable in ime, so ransacions coss are minimal. However, wha does come as some surprise is ha, ou of sample, he porfolio replicaing he firs principal componen, while being highly correlaed wih is benchmarks, significanly over-performs boh of hem. We demonsrae ha one cause of he overperformance is a mean reversion in reurns for he group of socks which are over-weighed by he porfolio. We show ha hese are precisely hose socks ha have had higher volailiy and have also been highly correlaed as a group during he porfolio calibraion period. Subsequenly, we observe wo behavioural mechanisms which could explain he mean reversion for hese socks: he aenion capuring effec documened by Odean (1999) and he over-reacion based models of De Long, Shleifer, Summers and Waldmann (1990a), Lakonishok, Shleifer and Vishny (1994) and Shleifer and Vishny (1997). Separaely, our resuls show ha he abnormal reurn 4 is relaed o a behavioural measure of he invesors herding owards he marke facor, driving he mean reversion in sock reurns. A decomposiion of he sraegy s over-performance ino a marke premium, a value premium and a volailiy premium reveals a ime-varying srucure. Throughou mos of he period sudied, he value componen dominaed he oher wo, bu during he volaile periods of he las years he sraegy earned a significan volailiy premium. The remainder of he paper is organised as follows: secion one inroduces he saisical model for he firs principal componen porfolio, secion wo describes he DJIA daa and he performance esing mehodology, secion hree reviews he empirical properies of he firs principal componen, secion four analyses he ou-of-sample performance of he firs principal componen porfolio, secion five repors he resuls of applying he sraegy o oher inernaional equiy markes, and finally, secion six summarises and concludes. 4 We define he abnormal reurn as he difference beween he facor mimicking porfolio reurns and he reurns of a price weighed benchmark, reconsruced from he same socks as he porfolio. Copyrigh 2003 Carol Alexander and Anca Dimiriu 6

7 I. The common rend replicaion model Principal componen analysis (PCA), inroduced by Hoelling (1933) in connecion o he analysis of daa in psychology, was recommended as an imporan ool in he mulivariae analysis of economic daa more han half a cenury ago (Tinner, 1946). This echnique is now a sandard procedure for an orhogonal ransform of variables, reducing dimensionaliy and he amoun of noise in he daa. Given a se of k saionary random variables, X 1, X 2,...X k, PCA deermines linear combinaions of he original variables, called principal componens and denoed by P 1, P 2,... P k, so ha (1) hey explain, successively, he maximum amoun of variance possible and (2) hey are orhogonal. By convenion, he firs principal componen is he linear combinaion of X 1, X 2,...X k ha explains he mos variaion. Each subsequen principal componen accouns for as much as possible from he remaining variaion and is uncorrelaed wih he previous principal componens. The i h principal componen, where i = 1,..., k, may be wrien: P i = w 1i X 1 + w 2i X w ki X k (1) Thus, if we denoe by Σ he covariance marix of X, hen: var(p i ) = w i Σ w i ; cov(p i, P j ) = w i Σ w j, where w i = [w 1i w 2i...w ki ] and i is sandard o impose he resricion of uni lengh for hese vecors, i.e. w i w i = 1. 5 Noe ha hese are, in fac, he eigenvecors of Σ. The specral decomposiion of he covariance marix is Σ = WΛW, where Λ is a diagonal marix of eigenvalues (ordered by convenion so ha λ λ >... > λ 0 ) and W is an orhogonal marix of eigenvecors (which have also been 1 > 2 k > ordered according o he size of he corresponding eigenvalue). The principal componens defined as P = XW observe he condiions above. Noe ha he variance of each principal componen is equal o he corresponding eigenvalue, so he oal variabiliy of he sysem is he sum of all eigenvalues. To reproduce he oal variaion of a sysem of k variables, one needs exacly k principal componens. However, when he firs few principal componens ogeher accoun for a large par of he oal 5 Eigenvecors are no unique, and so i is sandard o impose he orhonormal consrain. A more naural consrain in a porfolio consrucion framework would be o have he sum of he eigenvecors, raher han he sum of heir squares, equal o one. However, his does no ensure a balanced porfolio srucure, which is essenial for indexing. In order o avoid large exposures o individual socks, we keep he uni lengh consrain for he eigenvecors, and hen normalise hem o sum up o one. Copyrigh 2003 Carol Alexander and Anca Dimiriu 7

8 variabiliy, he dimensionaliy and much of he noise in he original daa can be significanly reduced. Since he principal componens define a k-dimensional space in erms of orhogonal coordinaes, he disances defined in he principal componens space depend on he amoun of correlaion in he original variables. The higher he correlaion in he original sysem, he beer a principal componen can accoun for he original join variaion and he larger he iner-poin disances will be in ha dimension. The elemens of he firs eigenvecor are he facor loadings on he firs principal componen in he represenaion of he variables in erms of principal componens. In a highly correlaed sysem, hese elemens will be of similar size and sign. Consequenly, when porfolio weighs are direcly proporional o he elemens of he firs eigenvecor, as in (2) below, he more correlaed he socks, he more evenly balanced he porfolio. When applied o large sock universes, previous research has shown ha he firs principal componen is capuring he marke facor, explaining a very high proporion from he reurns of an equally weighed porfolio of all socks. Moivaed by hese resuls, we propose a porfolio consrucion model which is based on replicaing he firs principal componen of a se of sock reurns. 6 For a porfolio of k socks, he porfolio weigh of sock i is defined as: w i k = w 1 / w 1 (2) i j= 1 j where w i1 is he i h elemen from he firs column in he eigenvecors marix ordered as above. In he PCA framework he firs eigenvecor is obained, independenly of he ohers, by maximising he variance of he corresponding linear combinaion of socks, under he consrain of uni norm. Therefore he porfolio based on he sock weighs deermined as in (2) is, of all possible combinaions of k socks wih uni norm, he porfolio ha accouns for he larges par of he oal join variaion of he k socks. This propery ensures ha i is he opimal porfolio for capuring he common rend in a sysem of socks. Considering ha he model maximises he variance of he porfolio under some consrain, i 6 We noe ha, ofen, he original saionary variables are sandardised o have zero mean and uni variance before he principal componen analysis ha is, ha he eigenvecors of he correlaion marix are used o consruc he principal componens, raher han he eigenvecors of he covariance marix. This ensures ha he variable wih he highes volailiy does no dominae he firs principal componen. However, in a realisic porfolio consrucion seing, he assumpion of equal volailiies for all asses is no feasible. Such an assumpion would resul in he porfolio model being consruced solely on he correlaion srucure of he asses, raher han he complee covariance srucure of he daa. Therefore, for he purpose of our model, we do no sandardise he sock reurns. Copyrigh 2003 Carol Alexander and Anca Dimiriu 8

9 will over-weigh, relaive o benchmark, he socks ha were boh highly correlaed and had higher han average volailiy over he esimaion period. The common rend replicaion model is differen from he radiional approaches o porfolio opimisaion (Markowiz, 1952; Chan, Karceski and Lakonishok, 1999; Jagannahan and Ma, 2002) in more han one respec. Firsly, i is maximising and no minimising porfolio variance, and his migh appear counerinuiive a a firs glance. However, when combined wih uni norm consrain on he facor loadings, he resul is a balanced porfolio wih a sable srucure which also explains mos of he join variance in he sysem of socks. 7 Secondly, i is no aiming a sock selecion, bu raher a diversifying over he enire universe of socks. All socks will be represened in he porfolio replicaing he common rend and he porfolio will be fairly evenly balanced if here is a high level of correlaion in he sock reurns. Finally, despie being a passive invesmen model, he benchmark does no ener ino he mehodology anywhere. This eliminaes he problems associaed o using an inappropriae benchmark in he porfolio consrucion, bu also limis he relevance of radiional indexing performance measures such as racking error, so cauion is needed when inerpreing such resuls. II. Daa, benchmarks and porfolio ou-of-sample performance measuremen In order o examine he properies of he porfolio replicaing he firs principal componen, we use a main daa se comprising daily closing prices on he 25 of he socks currenly included in he DJIA which have a hisory available for he period Jan-80 o Dec-02. Four ou of he five socks which are currenly in he DJIA, bu which do no have a hisory going back o Jan-80, are echnology socks. Therefore, our porfolio has a lower loading on echnology han he curren DJIA and he laer canno be considered he relevan benchmark because of a echnology bias. Also, he sock selecion mehodology may raise he concern of performance biases such as survivorship and look-ahead, because we are selecing he socks which had a hisory of a leas 23 years of daa available. We deal wih all hese poenial biases by creaing benchmarks from exacly he same socks as our porfolio, so ha he benchmarks are affeced by he same biases as he porfolio. 8 Subsequenly, we analyse all performance on a relaive basis. 7 The uni norm consrain ensures a balanced porfolio srucure, wihou large exposures o individual socks. This consrain can also be inerpreed as a Bayesian approach o limiing he effec of ouliers in he hisoric sock reurns a large weigh on an individual sock resuls when he sock has a very high in-sample volailiy, bu his could simply be due o measuremen errors or single ouliers (Jagannahan and Ma, 2002). 8 The alernaive would be o include in our porfolio a ime he socks ha were in he benchmark a ime. However, his would necessiae a complex dynamic back-es procedure and he underlying principle, ha he socks in he benchmark are he same as he socks in he porfolio, is he same. Acually, over he enire daa sample he price weighed benchmark had a Copyrigh 2003 Carol Alexander and Anca Dimiriu 9

10 Despie he fac ha a benchmark does no formally ener he porfolio consrucion model, i is needed o evaluae is performance. By resricing he informaion used in he benchmark consrucion o he informaion used in he porfolio consrucion (i.e. he hisory of sock prices, no capialisaion figures) here are wo alernaive benchmarks: a price weighed benchmark (PW) and an equally weighed benchmark (EW). The firs implies no rading as long as he universe of socks does no change and i is self-adjusing o price changes. Therefore, PW is a naural choice as benchmark for a passive invesmen sraegy. However, he reurns differenial beween EW and PW will also ener our performance analysis, as a proxi for a value porfolio: by consrucion, PW places more weigh on growh socks, so heir reurns difference can be inerpreed as a value premium. For he purpose of performing principal componen analysis, we are paricularly ineresed in he average correlaion of he sock reurns, as his has a srong influence on he effeciveness of principal componen analysis. We find ha he average correlaion of he daily sock reurns from he DJIA se is in he range of 0.3 o 0.4, occasionally going o as low as 0.2. The highes average correlaion in sock reurns occurs in down, volaile markes, such as 1987, 1990, or , his being a common finding for sock markes. Regarding he general marke condiions during he sample period, i is worh menioning ha, in 10 ou of he 23 years, he socks in DJIA had average reurns above 20%. By conras, in only 5 years ou of 23, he average reurn was negaive, which, however, was he case for he las 3 years in he sample. The average volailiy sayed in he range of 20%-30%, increasing significanly in he las par of he daa sample. The year 1987 sands ou from he sample, in erms of reurns correlaion, volailiy, excess kurosis and negaive skewness, because of he Ocober crash. For he ou-of-sample reurns analysis, he porfolio opimisaion and rebalancing procedure is as follows: a each rebalancing momen, he sock weighs are deermined from he eigenvecors of he covariance marix of he sock reurns esimaed from he mos recen 250 observaions prior o he momen of he porfolio consrucion. 9 For he ou-of-sample performance assessmen, he porfolio cumulaive reurn of 203% and he acual DJIA reurned 215%. Therefore, he survivorship bias due o he difference in echnology loadings is in fac negaive. 9 The rolling sample PCA raises he issue of consisen idenificaion of he facor loadings because he choice of he sign of he eigenvecors is arbirary. Choosing a paricular normalisaion is no relevan if he esimaion of he principal componens is performed over he enire daa sample. However, when he opimisaion is performed over a rolling sample, in order o have consisen principal componen esimaes from successive esimaions, one needs o ensure ha he same normalisaion is used hroughou he enire daa sample. To his end, following Chan, Karceski and Lakonishok (1998), we impose an addiional Copyrigh 2003 Carol Alexander and Anca Dimiriu 10

11 consruced in he previous sep is lef unmanaged for he nex 10-rading days, and hen rebalanced based on he new sock weighs from principal componen analysis. We have used a 10-day rebalancing period having in mind an insiuional invesor, for which his rading frequency is sandard. However, he rebalancing frequency could be easily reduced, wihou affecing he porfolio performance because he porfolio weighs are sable over ime. In order o accoun for ransacion coss we assume an amoun of 20 basis poins on each rade value o cover he bid-ask spread and he brokerage commissions. 10 In addiion o DJIA socks, we use several ses of daily closing prices of socks included a he end of year 2002 in he CAC40, FTSE100 and SP100 indexes. The lengh of he daa sample ranges from 1,600 daily observaions for SP100 (Apr-96 o Jun-02) o 2,100 daily observaions for FTSE100 (Jul-94 o Dec-02). III. Empirical properies of PCA This secion examines he empirical resuls of esimaing he firs principal componen on a rolling sample of 250 observaions on daily reurns o DJIA socks. Firs, we show ha he in-sample properies of he firs principal componen jusify he use of he PC1 porfolio o capure a marke facor. Subsequenly we examine he size and he sabiliy of he facor loadings on he firs principal componen, as his will deermine he srucure of he PC1 porfolio and he associaed ransacions coss. The price weighed benchmark (PW) and he PC1 porfolio have very similar informaion raios, as shown in Figure 1. Each poin in Figure 1 represens he informaion raio over he las 250 observaions. The main excepions are he periods and , during which he informaion raios of he PC1 porfolio are significanly higher. Addiionally, heir reurns are also highly correlaed. The correlaion coefficien ranges from 0.7 o Lower correlaion occurs beween 1992 and 1996, bu mos of he ime i is sill above 0.9. A sandard regression of he benchmark reurns on he firs principal componen, esimaed over he enire sample, has an R 2 of 0.8. Therefore, we can safely conclude ha he firs principal componen largely capures he marke facor. 11 resricion on he principal componen analysis, i.e. he firs principal componen needs o be posiively correlaed wih he price-weighed porfolio of all socks. 10 We do no accoun for poenial ax implicaions for individual invesors, assuming ha he sraegy is primarily designed for insiuional invesors. 11 The erm marke refers o he specific universe of socks included in our analysis. As his universe increases, i will converge o he rue marke porfolio. Copyrigh 2003 Carol Alexander and Anca Dimiriu 11

12 The amoun of oal variaion explained by he firs principal componen in our sysem urns ou o be in he range of 30-40%, in line wih previous research on his issue (Chan, Karceski and Lakonishok, 1998; Connor and Korajczyk, 1988). This is direcly relaed o he amoun of correlaion in he original sysem of reurns. Figure 2 repors he proporion of variance explained by he firs principal componen and he average correlaion of reurns, boh based on a rolling sample of he las 250 observaions. Clearly, he average correlaion in he original daa is he single mos imporan deerminan of he proporion of variance explained by he firs principal componen. 12 The lowes average correlaion (and, consequenly, amoun of variaion explained by he firs principal componen) occurs beween 1992 and 1997, and again in 1999 and These imes were relaively calm periods for he developed sock markes, and correlaions are generally higher during more volaile periods. Of cenral ineres o our analysis are he eigenvecors of he covariance marix of sock reurns, as hese will deermine he sock weighs in he porfolio replicaing he firs principal componen. The eigenvecor corresponding o he firs principal componen comprises he sensiiviy of each sock o changes in he firs principal componen, he so-called facor loading. If he sock reurns were perfecly correlaed, he firs principal componen would capure he enire variaion of he sysem and he facor loadings would all be equal. More generally, in a highly bu no perfecly correlaed sysem, he facor loadings on he firs principal componen will be similar bu no idenical so ha a change in he firs principal componen generaes a nearly parallel shif in he original variables. In his case we can associae he firs principal componen wih he exisence of a common rend in sock reurns. In he DJIA case, he facor loadings on he firs principal componen are largely in he same range: during periods of high average correlaion (e.g. afer he 1987 crash) he facor loadings are high and very similar bu more recenly hey end o be lower and less similar. This observaion is jusified by Figure 3, which plos he sandard deviaion of he facor loadings. The similariy of he facor loadings is an imporan feaure of he model, as i allows he consrucion of balanced porfolios, wihou exreme exposures o individual socks. In consequence we observe ha even hough here are no shor-sale resricions imposed on he model, shor posiions occur very rarely. 12 A ghos feaure caused by he Ocober-87 crash can be idenified in boh of hem: he correlaion and he percenage of variance explained remain very high for as long as he Ocober crash says in he esimaion sample and drop immediaely afer excluding ha observaion from he sample. This is an arefac of he euqal weighing in reurns and would no be eviden if exponenial weighing of he covariance marix were applied. Copyrigh 2003 Carol Alexander and Anca Dimiriu 12

13 The dispersion of he facor loadings has been used as a measure of herding behaviour in recen research in behavioural finance (Hwang and Salmon, 2001), and we shall reurn o he implicaions of his in he nex secion when we analyse he ou-of-sample performance of he model. Apar from he cross-secional variabiliy of he facor loadings, a very aracive feaure is heir low ime variabiliy. The facor loadings are very sable in ime, which, in a porfolio consrucion seing, is ranslaed ino a reduced amoun of re-balancing rades and low ransacion coss. IV. Ou-of-sample performance of he saisical facor equiy porfolio To recap, he porfolio replicaing he firs principal componen (PC1) is consruced from he 25 socks ha were boh included in DJIA a he end of 2002 and had a hisory ha goes back as far as Jan The benchmarks for he performance assessmen are a price weighed porfolio (PW) and an equally weighed porfolio (EW) boh wih all 25 socks. For he ou-of-sample analysis, PC1 is firs se up in Jan 81, based on he principal componen analysis performed on he 250 observaions preceding he porfolio consrucion momen and furher rebalanced every 10-rading days. In beween rebalancing, he number of socks in each porfolio is kep consan and he ou of sample porfolio performance recorded. The performance saisics for ou-of-sample daily reurns series generaed by he PC1 porfolio and he wo benchmark porfolios are repored in Table I. In erms of annual reurns, he PC1 porfolio overperforms PW by an average of 5% per year, wih only 1% exra volailiy. EW is also over-performed by he PC1 porfolio wih an annual average of 2.70%. The superior performance resuls in an informaion raios of 0.75 for PC1, compared wih 0.5 for PW and 0.63 for EW. The PC1 porfolio reurns appear o be marginally closer o normaliy han he reurns on he benchmark porfolios, bu all hree porfolios have heavy ailed and negaively skewed reurns disribuions. There are oher srong similariies beween he hree porfolios: when ou-of-sample reurns are analysed period by period all porfolios are affeced by he main marke crises during he period in observaion: Oc-87, he Gulf War, he Asian Crisis, he burs of he echnology bubble and Sep-01. They all have a srong January effec, which, however, is less eviden in he case of he PC1 porfolio. The correlaion beween he PC1 and he boh benchmark porfolios reurns is very high, indeed he PC1 porfolio and he benchmarks have very similar shor-erm volailiy and correlaion properies. The exponenially weighed moving average (EWMA) volailiies and correlaion for PC1 and PW wih a smoohing parameer of 0.96 are shown in Figure 4. The volailiy of he PC1 porfolio is slighly higher, Copyrigh 2003 Carol Alexander and Anca Dimiriu 13

14 especially during he las par of he sample, bu is closely following he benchmark volailiy. Wih very few excepions, he EWMA correlaion is high, saying above 0.8 mos of he ime and i is paricularly high during marke crises such as Oc-87 or Sep-01. Finally, he ransacion coss are almos negligible, amouning o an average of 0.24% per year for implemening he PC1 sraegy. As our arge is o explain he pure over-performance, i.e. he difference beween he PC1 porfolio reurn and he PW benchmark reurn, afer esablishing ha he overall profiabiliy of he sraegy does no disappear afer ransacion coss, we will perform he analysis of he porfolio reurns before ransacion coss. Considering he PC1 porfolio over-performance wih respec o he price weighed benchmark, he abnormal reurn can be hough of as being produced by a self-financed sraegy which, a each momen in ime, is long on he PC1 porfolio and shor on PW. In his case (see Table I) he 5.19% annual reurn is associaed wih an annual volailiy of 6.3%. Is informaion raio is 0.82, higher han hose of he benchmarks and PC1 porfolio. Moreover, he abnormal reurn is uncorrelaed wih he benchmark reurn and much closer o normaliy han he laer. The fac ha he over-performance of he PC1 porfolio is no caused by singular evens is eviden from Figure 5, which shows he cumulaive reurns difference beween he PC1 porfolio and he wo benchmarks. Any sraegy ha over-performs in he DJIA sock universe should lead one o quesion if here is any connecion wih he famous Dogs of he Dow or Fool s Four value sraegies. Indeed, we shall see ha he PC1 sraegy has a significan value il. Bu apar from his, here are no oher similariies. The Dogs of he Dow and Fool s Four sraegies pick a small number of socks from DJIA (en and four respecively) and rebalance as rarely as once a year. They are known o be a classic case of daa mining, ha is, an exensive search hrough a large number of rading sraegies for he ones which have hisorically over-performed he benchmark. Apar from he low diversificaion and increased volailiy, hese sraegies have been shown no o be robus o ou-of-sample ess and associaed ransacion coss (Hirschey, 2000; McQueen and Thorley, 1999). In conras, he PC1 porfolio model is no he resul of a blind search hrough he hisorical performance of differen rading rules. I is consruced on srong heoreical foundaions, o capure he common rend in sock reurns. Moreover, i is no being on few socks, having a similar indusry and sock diversificaion o is benchmarks. Finally, i is robus o ou-of-sample ess and inclusion of ransacion coss. 13 We have also analysed he performance of a porfolio comprising all 30 socks currenly included in DJIA, over he period Jan-91 o Dec-02. The resuls are very similar o he ones obained wih he 25-socks porfolio. For reasons of space, we have Copyrigh 2003 Carol Alexander and Anca Dimiriu 14

15 In order o explain he performance of he porfolio replicaing he firs principal componen, we shall consruc a simple model which is based on he relaionship beween he PC1 porfolio and he benchmarks. Given he high correlaion beween PW and EW, in order o avoid near mulicollineariy, we include in he model only he price weighed benchmark as a proxi for he marke, and, separaely, he reurns differenial beween EW and PW, as a proxi for a value facor. Over he enire daa sample here is a negaive bu no very significan correlaion beween he wo explanaory variables. This is o be expeced, given ha mos of he value over-performance has been documened in negaive marke circumsances. Thus, we esimae he following model on daily daa covering he period Jan-1981 o Feb-2003 (from which we have eliminaed he wo ouliers represening he marke crashes from Ocober 1987 and Sepember 2001), using ordinary leas squares: PC1_reurn = α + β * PW_reurn + β * (EW_reurn - PW_reurn) + ε 1 2 This very simple specificaion is no robus o heeroskedasiciy ess, he paern in he auocorrelaion of squared residuals indicaing a GARCH(1,1) as he alernaive, his being a common specificaion for sock marke index volailiy. Addiionally, he model does no pass specificaion error ess, indicaing a possible non-linear relaionship beween PC1 reurns and one of he explanaory variables. These specificaion ess improve when squared reurns on he price weighed benchmark are included as follows: PC1_reurn = α + β 1 * PW_reurn + β 2 * (EW_reurn - PW_reurn) + β 3 * PW_reurn 2 + ε ε I 2 N( 0, σ ) (3) σ 2 = ω + α ν ε β ν σ 2-1 The esimaion resuls for model (3) covering he sample are repored in Table II. 14 The very high R squared, above 0.98, comes as no surprise, considering he srong correlaion beween he porfolio reurn and is benchmarks. All coefficiens, excep for he mean regression inercep, are posiive and highly significan a 1% significance level. The variance regression model esimaes show an almos inegraed vanilla GARCH model for he variance of he PC1 porfolio, wih he persisence coefficien (0.96) and reacion coefficien (0.037) in he usual range for sock marke volailiy during no included hem in his paper. They are available by reques from he auhors. 14 This version of he model passes he auocorrelaion and ARCH ess, having slighly non-normal residuals, probably due o he presence of ouliers. The informaion crieria clearly favour his specificaion as compared o he iniial one. Copyrigh 2003 Carol Alexander and Anca Dimiriu 15

16 his sample period. The mean regression model provides a complee decomposiion of he PC1 porfolio performance ino risk facor premia, given ha he inercep erm is no saisically significan. The coefficien of he price weighed benchmark reurns, which measures he porfolio s sensiiviy o he marke facor, is above uniy (i.e. a 1.04). Therefore, par of he porfolio over-performance can be aribued o a higher loading on he marke risk facor. Also, he posiive relaionship beween he porfolio reurn and he value facor proxi indicaes ha par of he over-performance is due o a higher loading on value socks. Finally, he posiive and significan coefficien of he squared price weighed benchmark reurns can be inerpreed as he porfolio sensiiviy o a volailiy facor, proxied by he squared marke reurns. The porfolio premium from each facor is defined as he produc of he porfolio sensiiviy o ha facor imes he facor premium. If we combine he marke premium earned by he PC1 porfolio wih he volailiy premium, he reurns differenial beween he wo and he price weighed benchmark has a sraddle paern: he PC1 porfolio over-performs large negaive and large posiive benchmark reurns, and marginally under-performs small negaive marke reurns. This feaure of he sraegy is imporan, as i allows he invesor o reduce he porfolio exposure o negaive marke circumsances, while increasing he exposure o posiive ones. From his poin of view, he sraegy acs like a benchmark enhancer. Anoher imporan issue o invesigae is he evoluion of he individual conribuions o he PC1 porfolio over-performance. Over he enire daa period, he conribuion of he hree sources o he oal over-performance of he PC1 porfolio is he following: marke premium 11%, value premium 60% and volailiy premium 29%. However, his disribuion is far from being saionary, as shown by Figure 6. This figure shows he conribuions of he hree over-performance sources esimaed from model (3) on a rolling window of 500 daily observaions. Each premium is compued as he porfolio sensiiviy o he facor imes he annualised mean facor premium over he esimaion sample. The marke premium has he mos sable, bu also he smalles conribuion o he porfolio overperformance, becoming negaive from 2001 as he sock marke generally declined. The PC1 porfolio, having a high marke bea, will under-perform in down markes. The value premium accouns for he larges par of he porfolio over-performance during mos of he 80s and 90s. This finding is consisen wih abnormal reurns being generaed by a mean Copyrigh 2003 Carol Alexander and Anca Dimiriu 16

17 reversion mechanism, as we shall argue below. However, he value premium falls sharply, becoming negaive, around he ime of he Ocober 87 crash, afer he Gulf War, during he Asian crisis and again a he end of he sample. An explanaion for his change is ha he normal mean reversion cycle is broken around he ime of marke crises, because invesors behaviour changes significanly. Given he increased sock marke urbulence during he las few years, he volailiy premium increased markedly during 2001 and I accouns for 70% of he oal over-performance by he end of he sample. Having idenified he sources of porfolio over-performance, we now explain a mechanism hrough which his over-performance may be achieved. As shown in secion I, he sock weighs in he PC1 porfolio are chosen o maximise he porfolio variance, subjec o he consrain of uni norm for he facor loadings. Since porfolio variance increases wih boh individual asse variance and he covariance beween asses, he porfolio will over-weigh, relaive o he PW benchmark, socks ha have higher volailiy over he esimaion period and which are also highly correlaed as a group. Separaely, he PW benchmark is under-weighing socks ha have recenly declined. Now, if i does hold rue ha markes end o be more urbulen afer a large price fall han afer a similar price increase (i.e. he leverage effec ha is commonly idenified in sock markes, as in Black, 1976; Chrisie, 1982; French, Schwer and Sambaugh, 1987), hen he same group of socks will be impaced hrough he over-weighing of volaile, correlaed socks in he PC1 porfolio and he under-weighing of declining socks in he PW benchmark. These socks have had a volaile, declining period over he esimaion sample. From his perspecive, he over-performance of he PC1 porfolio mus be due o a mean reversion in sock reurns over he one-year esimaion period used for our porfolio. The porfolio over-weighs socks ha have recenly declined in price, relaive o he benchmark, so he relaive profi on he porfolio has o be he resul of a consequen rise in price of hese socks. The hypohesis ha mean reversion akes place over a period of one-year is suppored by he fac ha when he PC1 esimaion sample is reduced, he in-sample over-performance of he firs principal componen wih respec o he PW benchmark disappears. These resuls are in line wih he research on shor-erm momenum and long-erm reversals ha has frequenly been idenified in sock reurns. For example, De Bond and Thaler (1985), Lo and MacKinlay (1988), Poerba and Summers (1988) and Jagadeesh and Timan (1993) idenify posiive Copyrigh 2003 Carol Alexander and Anca Dimiriu 17

18 auocorrelaion in sock reurns a inervals of less han one year and negaive auocorrelaion a longer inervals. In behavioural finance, wo explanaions are usually proffered for long-erm reversals and shor-erm momenum in sock markes. The firs explanaion focuses on relaively volaile socks, which capure he aenion of noise raders for whom hey are he bes buy candidaes (Odean, 1999). 15 The rading behaviour of noise raders creaes an upward price pressure on hese volaile socks, forcing mean reversion when heir high volailiy was associaed wih a recen decline in price. The same explanaion is no applicable o a selling decision, creaing symmerically downward price pressure on volaile socks, because he range of choice in a selling decision is usually limied o he socks already held (Barber and Odean, 2002). Addiionally, we noe ha volaile socks which have recenly experienced a price decline, also qualify as value socks, so his explains he value premium previously observed. A second behavioural explanaion of he shor-erm momenum followed by mean reversion has been provided by De Long, Shleifer, Summers and Waldmann (1990a), Lakonishok, Shleifer and Vishny (1994) and Shleifer and Vishny (1997). This explanaion is based on invesors senimen, over-reacions and excessive opimism/pessimism. The occurrence of some bad news regarding one sock creaes an iniial excess volailiy and, according o hese models, some invesors will become pessimisic abou ha sock and sar selling. If here is posiive feedback in he marke, more selling will follow and he selling pressure will drive he price below is fundamenal level. However, he arbirageurs (someimes called smar money, or raional invesors) will no ake posiions agains he mispricing eiher because (1) he mispricing is oo small o jusify arbirage afer ransacion coss, or (2) here is no appropriae replica available for ha sock, so he fundamenal risk canno be hedged away, or (3) here is a noise rader risk arising from posiive feed-back, where he excessive invesors pessimism will drive he price even furher down over he shor erm. In he presence of posiive feedback, De Long, Shleifer, Summers and Waldmann (1990b) show ha he arbirageurs will iniially join he noise raders in selling, in order o close heir posiions when he mispricing has become even larger. This ype of invesor behaviour jusifies boh shor-erm momenum and longer-erm mean reversion. In addiion o he above explanaions which jusify he over-performance of he PC1 porfolio by a mean reversion in sock reurns, we also find ha here is a connecion beween he abnormal reurns 15 Noise raders are usually defined in he lieraure as no fully raional invesors, making invesmen decisions based on beliefs or senimens which are no fully jusified by fundamenal news, or which are subjec o a sysemaic biases. Copyrigh 2003 Carol Alexander and Anca Dimiriu 18

19 generaed by he PC1 porfolio and anoher behavioural phenomenon ha is well documened in sock markes invesors herding. We shall show ha he more inense he herding behaviour, as measured by a decrease in he cross secional sandard deviaion of he facor loadings, he higher he abnormal reurns generaed by he PC1 porfolio. The use of he cross secional disribuion of sock reurns as an indicaion of herding was firs inroduced by Chrisie and Huang (1995) in he form of he cross secional sandard deviaion of individual sock reurns during large price changes. Hwang and Salmon (2001) build on his idea bu insead advocae he use of a sandardised sandard deviaion of PCA facor loadings o measure he degree of herding. Their measure has he advanage of capuring inenional herding owards a given facor, such as he marke facor, raher han spurious herding during marke crises. Following Hwang and Salmon (2001), we assume ha he sandard deviaion of he facor loadings (shown in Figure 3) capures he inenional herding of he invesors owards he firs principal componen of he socks, or heir common rend. An inense herding of he invesors owards he common rend of he socks should reduce he differences in he individual socks loadings on he firs principal componen. Therefore, we inerpre a low sandard deviaion of he facor loadings as an indicaion of herding. From Figure 3 we see ha more inense herding appears o happen before 1993, and hen again before 1998, which suppors he findings in Hwang and Salmon (2001) ha his ype of herding occurs especially during quie periods for he marke. During he marke crises of he las five years, he herding behaviour appears o be significanly reduced. Given he inerpreaions of mean revering behaviour presened above, an inense herding owards he firs principal componen, indicaed by a sharp reducion in he sandard deviaion of he facor loadings, should enhance and speed up he mean reversion. Therefore he sandard deviaion of he facor loadings should be negaively relaed o he over-performance of he PC1 porfolio. Indeed, he correlaion beween he sandard deviaion of he facor loadings and he abnormal reurn, esimaed over all non-overlapping sub-samples of 120 observaions, is negaive (-0.33) and significan a 5%. From he hree differen sources of over-performance, his measure of herding is mosly relaed o he value premium. The smaller he sandard deviaion of he facor loadings, he more similar he PC1 porfolio will be o an equally-weighed porfolio, and he closer is over-performance will be o he way we have defined he value premium. From Figures 3 and 6 we see ha he highes value premium coincides wih he periods when he sandard deviaion of he facor loadings was he smalles. Copyrigh 2003 Carol Alexander and Anca Dimiriu 19

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