TESTING THE RELATION BETWEEN RISK AND RETURNS USING CAPM AND APT: THE CASE OF ATHENS STOCK EXCHANGE (ASE) Abstract

Transcription

1 TESTING THE RELATION BETWEEN RISK AND RETURNS USING CAPM AND APT: THE CASE OF ATHENS STOCK EXCHANGE (ASE) Therou, N 1., V. Aggelds and D. Madtnos TEI of Kavala, Greece Abstract Ths artcle nvestgates the determnants of stock returns n Athens Stock Exchange (ASE) usng both frameworks the classcal CAPM and the statstcal APT model. The analyss s conducted wth monthly data from the Greek stock market. Emprcal tests n ths study suggest that the relatonshp between β and return n the ASE, over the perod January December 2001, s weak. More analytcally, the Captal Asset Prcng Model (CAPM) has poor overall explanatory power, whereas the Arbtrage Prcng Theory (APT) model, whch allows multple sources of systematc rsks to be taken nto account, performs better than the CAPM, n all the tests consdered. Shares and portfolos n the ASE seem to be sgnfcantly nfluenced by a number of systematc forces and ther behavour can be explaned only through the combned explanatory power of several factors. Factor analyss replaces the arbtrary and controversal search for factors of the APT model by tral and error wth a real systematc and scentfc approach. Key words: CAPM, APT, rsk-return trade-off. 1 Contact person: Assocate Professor of Strategc Management, TEI of Kavala, School of Busness and Economcs, Department of Busness Admnstraton, Agos Loukas, Kavala, Greece, Tel. and Fax: , E-mal: ntherou@tekav.edu.gr 1

2 1. Introducton Fnance has evolved nto a hghly techncal subject snce the 1950s. Pror to that tme t was largely nsttutonally orented and a descrptve subject (Ryan, Scapens and Theobald, 2002). The two man precursors of the change n the scope of fnance were the treatment of rsk n the portfolo context by Markowtz (1952) and the mathematcal economc analyss by Modglan and Mller (1958) of captal structure. Markowtz s modern portfolo theory (MPT), as such, dd not provde a prcng model as ts scope s lmted to normatve prescrptons as to how an nvestor should operate n order that a rskavers wealth maxmser mght maxmse hs or her utlty. By dentfyng rsk wth the dsperson of returns t s possble to develop mean varance effcent sets and wth the ntroducton of rsk-free borrowng and lendng opportuntes a separaton theorem can be derved n mean-standard devaton space. That s, the composton of an nvestor s optmal rsky asset portfolo can be separated from hs or her rsk preferences. An mportant concept that drves from MPT s the dstncton between dversfable and non-dversfable rsk; That s, n large portfolos the contrbuton of an ndvdual securty to total portfolo rsk derves from the non-dversfable rsk of that securty, measured by the covarance of that securty s returns wth the exstng portfolo. Whle portfolo theory dealt wth the ndvdual nvestor s portfolo decson t provded the bass for an equlbrum asset prcng model-the captal asset prcng model (CAPM)-developed by Sharpe (1964), Lntner (1965) and Mossn (1966). The CAPM assumes that all nvestors maxmse the utlty of termnal wealth defned over the mean and varance of portfolo returns, and that all nvestors have uncondtonal homogeneous expectatons of means, varances, and covarance s. The captal market s assumed to be perfect. In the sngle perod analyss, a separaton theorem at the aggregate level wll arse (manfested by the captal market lne) n the presence of a rsk-free asset and the captal asset prcng model can be derved by the Lagrangean multplers technque. The lnear rsk return trade off, wth rsk measured by the beta coeffcent (whch reflects covarance or non-dversfable rsk) s perhaps one of the best known models n the fnance feld. Its message s smple; the only rsk that s prced at equlbrum n the market s that rsk whch cannot be dversfed away. The CAPM was developed n a relatvely restrcted theoretcal envronment. However, t dd provde strong emprcal mplcatons, that s, that systematc rsk and return are lnearly related n the captal market. In the last twenty years the feld of asset prcng, n both the theoretcal and emprcal domans, has advanced consderably, although wth a remarkable amount of controversy on the way. Ths was the man reason that nduced us to the decson of devotng ths paper on the emprcal testng of both CAPM and APT on ASE. To our knowledge, there s no such research undertaken for the ASE, up to now, probably because many authors consder t as an emergent market. So, the man argument of ths artcle s to dscover whch of the two models 2

3 (CAPM or APT) fts better,.e., has more explanatory power to explan the relatonshp between stock returns and rsk, n the Greek captal market (ASE). The rest of the paper conssts of fve sectons. Secton two ncludes a summary of the lterature revew on CAPM and APT. Secton three descrbes the research methodology used. Secton four proceeds to the statstcal analyss and gves the outcomng results. Fnally, secton fve ends the paper wth all mportant conclusons dervng from the precedng statstcal analyss and the comparson of the two models, CAPM and APT. 2. Lterature revew We can trace many of the ssues addressed n modern fnance back to the remarkable paper presented to the Imperal Academy of Scences n St Petersburg by Danel Bernoull (1738). Bernoull examnes the proposton that expected values are computed by multplyng each possble gan by the number of occurrences, and then dvdng the sum of these products by the total number of possble cases. Durng the eghteenth and nneteenth centures, Bernoull's concept of utlty was regarded as the provnce of mathematcans rather than economsts. Fnance was transformed wth the publcaton of the Markowtz (1952) artcle on Portfolo Selecton. The most mportant contrbuton made by Markowtz s hs dstncton between the varablty of returns of an ndvdual securty and ts contrbuton to the rskness of a portfolo. He notes that, n tryng to make varance small, an nvestor s not enough to nvest n many securtes. It s necessary to avod nvestng n securtes wth hgh covarances among themselves. Tobn (1958) takes Markowtz's analyss one step further by showng how to dentfy whch effcent portfolo should be held by an ndvdual nvestor. He consders how an nvestor should dvde hs or her funds between a safe lqud asset such as cash (or treasury blls) and a rsky asset (a bond or equty portfolo). He shows that the proportonate composton of the non-cash assets s ndependent of ther aggregate share of the nvestment balance. Ths fact makes t possble to descrbe the nvestor's decsons as f there was a sngle non-cash asset, a composte formed by combnng the multtude of actual non-cash assets n fxed proportons. After the publcaton of Markowtz's (1959) Portfolo Selecton book, Treynor (1961) started ntensve work on the theory of asset prcng. The ntenton of Treynor's paper s to lay the groundwork for a theory of market value whch ncorporates rsk. Shortly after Treynor began hs work on asset prcng, Sharpe also set out to determne the relatonshp between the prces of assets and ther rsk attrbutes. The paper publshed by Sharpe (1964) notes that through dversfcaton, some of the rsk nherent n an asset can be avoded so that ts total rsk s obvously not the relevant nfluence on ts prce; unfortunately lttle has been sad concernng the partcular rsk component whch s relevant. Sharpe ams to use the theory of portfolo selecton to construct a market 3

4 equlbrum theory of asset prces under condtons of rsk and notes that hs model sheds consderable lght on the relatonshp between the prce of an asset and the varous components of ts overall rsk. After the publcaton of the Sharpe (1964), Lntner (1965) and Mossn (1966) artcles, there was a wave of papers seekng to relax the strong assumptons that underpn the orgnal CAPM. The most frequently cted modfcaton s the one made by Black (1972), who shows how the model changes when rskless borrowng s not avalable; hs verson s known as the zero-beta CAPM. Another mportant varant s that of Brennan (1970), who proves that the structure of the orgnal CAPM s retaned when taxes are ntroduced nto the equlbrum. Also, Mayers (1972) shows that when the market portfolo ncludes non-traded assets, the model also remans dentcal n structure to the orgnal CAPM. Solnk (1974) and Black (1974) extended the model to encompass nternatonal nvestng. The captal asset prcng models of Sharpe-Lntner-Black (SLB) have been subjected to extensve emprcal testng n the past 30 years (Black, Jensen and Scholes, 1972; Blume and Frend, 1973; Fama and MacBeth, 1973; Basu, 1977; Reganum, 1981; Banz, 1981; Gbbons, 1982; Stambaugh, 1982 and Shanken, 1985). In general, the emprcal results have offered very lttle support of the CAPM, although most of them suggested the exstence of a sgnfcant lnear postve relaton between realsed return and systematc rsk as measured by β. However, the specal predcton of Sharpe-Lntner verson of the model, that a portfolo uncorrelated wth market has expected return equal to rsk free rate of nterest, has not done well, and the evdence have suggested that the average return on zero-beta portfolos are hgher than rsk free rate. Most of the early emprcal testng of CAPM has employed the methodology of two-stage examnaton, frst estmatng betas usng tme seres regresson, and then, runnng a cross secton regresson usng the estmated betas as explanatory varables to test the hypothess mpled by the CAPM. APT, founded upon the work of Ross (1976), ams to analyse the equlbrum relatonshp between assets rsk and expected return just as the CAPM does. The two key CAPM assumptons of perfectly compettve and effcent markets and homogeneous expectatons are mantaned. Moreover, n lne wth the CAPM, the APT assumes that portfolos are suffcently dversfed, so that the contrbuton to the total portfolo rsk of assets unque (unsystematc) rsk s approxmately zero. The frst dfference s, that the CAPM was essentally derved from a sngle-ndex (snglefactor) model,.e., from a process generatng asset returns whch was only a functon of returns unque to the asset (predctable and unpredctable) and returns on two factors, the market portfolo tself and the rsk less asset, or the zero-beta portfolo. The senstvty of the asset s returns to the markets was defned as the asset s beta, measurng systematc (market) rsk, whle the unsystematc 4

5 (unque) rsk of the asset (portfolo) tended to zero through dversfcaton. APT can then be seen as a mult-ndex (mult-factor) model,.e., one n whch the returns generatng process of the portfolo s a functon of several factors, generally excludng the market portfolo. The factors are not specfed a pror and ther choce depends on the queston at hand. Possble factors may nclude partcular sector-specfc nfluences, such as prce-dvdend ratos, leverage, and stock sze, as well as aggregate macroeconomc varables such as nflaton and nterest rate spreads. The second dfference between the CAPM and the APT has to do wth the equlbrum noton. In contrast to the CAPM s assumpton of an effcent market portfolo, whch every nvestor desres to hold, the APT reles on the absence of free arbtrage opportuntes. In partcular, two portfolos wth the same rsk cannot offer dfferent expected returns, because that would create an arbtrage opportunty wth a net nvestment of zero. An nvestor could then guarantee a rsk less postve expected return by short-sellng one portfolo and holdng an equal and opposte long poston n the other. As such, free arbtrage cannot persst; equlbrum n the APT specfes a lnear relatonshp between expected returns and the betas of the correspondng rsk factors. The shortsellng assumpton s crucal to the equlbrum, as t consttutes one sde of the arbtrage portfolo. Equally mportant s the requrement that the proceeds from short sellng are mmedately avalable. However, despte ts shortcomngs, especally the dffcultes assocated wth emprcal testng of ts valdty, the APT has caught on n fnancal practce as t allows for a more detaled and custommade approach to portfolo rsk management than the CAPM. Ths has become more relevant wth the wdespread use of dervatve nstruments and ther partcular types of rsk. Testng the APT s trcky. Arbtrage arguments can only be used to provde an approxmate factor prcng equaton for some unknown number of undentfed factors. Shanken (1982), however, argues that testng requres an exact prcng equaton, whch n turn requres addtonal assumptons. One way to produce an exact factor prcng equaton s to formulate a compettve equlbrum model, wth the entre attendant assumptons about nvestors' tastes added back nto the model. Moreover, Shanken contends that the role of the (unobservable) market portfolo n Connor's (1982) exact prcng equaton may preclude meanngful tests of the APT. In the same way Roll (1977) mantans that the nablty to observe the true market portfolo precludes tests of the multvarable CAPM. Lke many ssues n emprcal fnance, the contenton that testng the APT requres an exact prcng equaton s open to debate. Frst, Dybvg (1983) and Grnblatt and Ttman (1985) argue that, gven a reasonable specfcaton of the parameters of the economy, theoretcal devatons from exact factor prcng are lkely to be neglgble. Hence, they conclude that we may not need to rely on equlbrum-based dervatons of the APT. Dybvg and Ross (1985) and Shanken (1985) debate the ssue. Second, Roll and Ross (1980) put forth a seres of arguments to support the contenton that 5

6 the APT could be rejected wthout havng to rely on exact factor prcng. Ths explans why, from the very earlest APT tests, t was always consdered essental to check the cross-sectonal relaton between sample mean returns and factor betas by ncludng addtonal regressors that should not be there (accordng to the APT). The prme canddate whch should not be there s a beta aganst a market ndex. But own varance and other varables were also tred. Of course, one can never prove the absence of arbtrage, but one could certanly demonstrate ts exstence and hence reject the APT. The Arbtrage Prcng Theory of Ross (1976) provdes a theoretcal framework to determne the expected returns on stocks, but t does not specfy the number of factors nor ther dentty. Hence, the mplementaton of ths model follows two avenues: factors can be extracted by means of statstcal procedures, such as factor analyss or be pre-specfed usng manly macro-economc varables. 2.1 The emprcal testng of CAPM and APT n the ASE There has been lmted research on the behavour of stocks traded on the ASE. Papaoannou (1982; 1984) reports prce dependences on stock returns for a perod of at least sx days. Panas (1990) provdes evdence of weak-form effcency for ten large Greek frms. Koutmos, Negaks, and Theodossou (1993) fnd that an exponental generalsed ARCH model s an adequate representaton of volatlty n weekly Greek stock returns. Barkoulas and Travlos (1996) test whether Greek stock returns are charactersed by determnstc non-lnear structure (chaos). More recently, Dacoganns, Glezakos and Segredaks (1998) examned the effect of the Prce / Earnngs (P/E) rato and the Dvdend Yeld (DY) on expected returns of ASE common stocks for the perod He found that P/E s statstcally sgnfcant varable explanng the cross secton varaton of expected returns, whle the explanatory power of DY was documented rather weak. Karankas (2000) examned the role of sze, book to market rato and dvdend yelds on average stock returns n the ASE for the perod from January 1991 to March Followng Fama and MacBeth s (1973) cross sectonal regresson methodology, enhanced wth Shanken s adjustments for the Error n Varables (EIV) problem, he found that a statstcally sgnfcant postve relatonshp exsts between book to market rato, dvdend yelds and average stock returns. He also found that the market captalsaton varable ( sze effect ) does not seem to explan a sgnfcant part of the varaton n average returns. Therou et al. (2005) explore the ablty of CAPM, as well as the frm specfc factors, to explan the cross-sectonal relatonshp between stock returns and rsk by adoptng a methodology smlar to Fama and French (1992). The fndngs ndcate that n the Greek stock market there s not 6

7 a postve relaton between rsk, measured by β, and average stock returns. On the other hand, there s a sze effect on the cross-sectonal varaton n the average stock returns. Narchos and Alexaks (2000) examned whether t s possble to predct stock market returns wth the use of macroeconomc varables n the ASE, for the perod from January 1984 to December 1995, on a monthly base usng contegraton analyss and as explanatory varables some macroeconomcs factors. The macroeconomc factors used are, the nflaton rate measured by the Consumer Prce Index (CPI), the M3 measure of money supply, and the exchange rate of US Dollar/ Drachmae (Drachmae s the Greek currency pror to Euro). Wth the results of ther nvestgatons, they reject statstcally the Effcent Market Hypothess for the case of the Athens Stock Exchange; they noted the statstcal sgnfcance of the lagged returns whch suggest that the monthly returns n the ASE are postvely correlated. The above fndngs can not be explaned as a thn tradng effect or as non synchronous tradng effect because of the monthly tme nterval used n the nvestgaton. On the contrary, someone can reasonably assume that ether news s reflected wth some delay on stock market prces or that the Greek stock market s nfluenced by psychologcal factors.e. a perod of prce ncrease lead to optmsm and further prce ncrease, and a perod of prce decrease leads to pessmsm and further prce decrease. In addton they found that there s statstcal evdence that the lagged values of nflaton rate have explanatory power n a model were the stock return s the depended varable. 3. Research Methodology 3.1 Data Collecton Our data s daly closng prces of the common stocks traded n the Athens Stock Exchange. They are row prces n the sense that they do not nclude dvdends but are adjusted for captal splts and stock dvdends. The data was taken from Athens Stock Exchange data bank. The data set covers the 180 month perod from January 1987 to December 2001 and s dvded nto three non overlappng 60 month sub-perods for analyss (see table 1). Securtes are ncluded n a sub-perod sample f they have a complete prce relatve hstory (no mssng values) n that perod. The market return s obtaned from the ASE Composte (General) Share Prce Index. Tme seres of excess returns on the market and ndvdual securtes are taken over the three month Government Treasury Bll rate, whch s consdered to be the short term rsk-free nterest rate. Daly returns are calculated usng the logarthmc approxmaton. 7

8 Source : Sample Perod : Athens Stock Exchange January 1987 December 2001 nclusve. The entre perod s dvded nto three sub perods January 1987 December 1991 January 1992 December 1996 January 1997 December 2001 January 1987 December 2001 Selecton Crtera : The selecton crtera for the shares n the sub perods are : Shares wth no mssng values n all the sub perod Shares wth adjusted R 2 <=0 or F sgnfcant >0.05 of the frst pass regresson of the excess returns on the market rsk premum are excluded. Shares are grouped by alphabetc order nto groups of 30 ndvdual securtes. The alphabetcally last shares were not used snce complete groups of 30 were requred. Table 1: Sample perods and selecton crtera. Then daly returns are aggregated to compose the monthly returns, whch are the nput of our nvestgaton. To reduce the dmenson of the equaton system to feasble proportons, securtes n each sub-perod are allocated by alphabetc order nto groups of 30 ndvdual securtes (see Roll and Ross, 1980). It s mportant to note that portfolos (rather than ndvdual assets) are used for the reason of makng the analyss statstcally feasble. Ths s n contrast wth the reasonng of usng portfolos n tradtonal (unvarate) CAPM tests. In these tests, portfolos were formed to attenuate the problem of errors-n-varables (EIV), ntroduced by the well known two-stage testng approach (Campbell, Lo and MacKnlay, 1997). 3.2 Sample summary statstcs The data s fltered by keepng only the shares that have no mssng values n the sub perods. Ths procedure produces sample szes of 71, 145 and 217 for the three sub-perods, respectvely. Frstly, summary statstcs are produced to check out the null hypothess of the normal dstrbuton 2 of our data. Analytcally, we estmate the mean, standard devaton, skewness 3 and kurtoss 4, and then, based on those statstcs, we proceed to the normalty test of Kolmogorov & Smrnov 5 (per cent of Gausan dstrbuton), of all the shares ncluded n the sub perods under examnaton, usng SPSS. As table 2 (fnal column) shows the null hypothess of normalty cannot be rejected at the 5 per cent level of confdence n 21 per cent of the shares n the perod , 28 per cent n the perod , 22 per cent n the perod and 0 per cent n the perod The Normal Dstrbuton has skewness and kurtoss values equal to zero. It s fully descrbed by the frst two central moments, the mean and standard devaton. 3 Skewness measures the drecton and degree of asymmetry of a dstrbuton. A value of zero ndcates a symmetrcal dstrbuton. 4 Kurtoss measures the degree of peakedness and heavness of the tals of a dstrbuton. A normal dstrbuton has a kurtoss value equal to 0. 5 Kolmogorov-Smrnov (Lllefors): s a modfcaton of the Kolmogorov-Smrnov test that tests for normalty when means and varances are not known, but must be estmated from the data. The Kolmogorov-Smrnov test s based on the largest absolute dfference between the observed and the expected cumulatve dstrbutons. 8

9 The above results are n accord wth the fndngs of Mandelbrot (1963) and Fama (1970) for the US market. After the test of normalty, excess returns (for each ndvdual securty) and market premums are computed n each of the sub perods and we regress the excess returns on the market premum. Table 2 (column three) shows that 7 per cent of the shares of our sample n the perod and 13 per cent n the perod have negatve adjusted R squared 6, whch probably means that these specfc equatons could be non lnear. In order to make our model more sutable we elmnate all the shares that they produce negatve adjusted R squared. Table 2 (column four) shows that 8 per cent of the shares of our sample n the perod and 23 per cent n the perod have sgnfcance value of the F statstc 7 hgher than In order to make our model more effcent we also elmnate all the shares that they produce sgnfcance value of the F statstc hgher than Perod Number of Shares Negatve Adjusted R 2 F sgnfcant > 0.05 Durbn Watson 1.8<=.p. <= 2.2 Durbn Watson < 1.5 % Gausan dstrbuton % 8% 46.5% 0.0% 21% % 23% 45.8% 3.5% 28% % 0% 58.3% 1.3% 22% % 0% 80% 0.0% 0% Table 2: Regresson results of the sample n the sub perods The analyss of the resduals ncludes also the Durbn Watson (DW) statstc test 8. Table 2 (column fve) shows that 46.5, 45.8, 58.3, and 80 per cent of shares n the three sub perods and all the perod respectvely, have a Durbn Watson test between 1.8 and 2.2, and only 3.5 and 1.3 per cent of the shares, n the second and thrd sub perod respectvely (column sx) show a very strong seral correlaton (p<1.5). The above procedure of flterng produces sample szes of 65, 110 and 217 for the three sub-perods, respectvely. At ths pont shares are grouped by alphabetc order nto groups (portfolos) of 30 ndvdual securtes; the alphabetcally last shares were not used snce complete groups of 30 were requred. Ths procedure produce sample sze of 2, 3 and 7 groups of 30 shares n each sub perod respectvely. 6 The R-squared statstc, or the coeffcent of determnaton, s the percentage of total response varaton explaned by the ndependent varables. Adjusted R-squared s preferable to use f you have a lot of ndependent varables snce R- squared can always be made larger by addng more varables Adjusted R squared s the relatve predctve power of a model and s a descrptve measure between 0 and 1. The closer t s to one, the better your model s. By "better" we mean a greater ablty to predct. 7 The F statstc s the regresson mean square (MSR) dvded by the resdual mean square (MSE). If the sgnfcance value of the F statstc s small (smaller than say 0.05) then the ndependent varables do a good job explanng the varaton n the dependent varable. If the sgnfcance value of F s larger than say 0.05 then the ndependent varables do not explan the varaton n the dependent varable. 8 The Durbn Watson s a test for frst order seral correlaton and measures the lnear assocaton between adjacent resduals from a regresson model. If there s no seral correlaton, the DW statstc wll be around 2. The DW statstc wll fall f there s postve seral correlaton (n worst case, t wll be near zero). If there s a negatve correlaton, the statstc wll le somewhere between 2 and 4. Usually the lmt for non seral correlaton s consdered to be DW= (1.8; 2.2). A very strong postve seral correlaton s consdered at DW lower than

10 ) 3.3 Data analyss methodology After the data flterng process and the formaton of the groups (portfolos) the man part of our nvestgaton begns. It s focused on the testng and comparson of CAPM and APT. The test used for the CAPM and APT s a two-step test, whch s extensvely used n the lterature (see Roll and Ross, 1980; Chen, 1983; Lehmann and Modest, 1988; Cheng, 1995; Groenewold and Fraser, 1997). The frst step nvolves the use of tme seres to estmate the betas of the shares for the CAPM and a set of factor scores, through factor analyss, for the APT; the second step, then, regresses the sample mean excess returns on the beta (for the CAPM) and on the factor scores (for the APT). Test of CAPM The analyss proceeds n the followng stages: 1. For a group (or portfolo) of ndvdual securtes, (n our case a portfolo of 30 selected alphabetcally), we estmate the excess returns ( R t R f, t, ) of each securty from a tme seres of returns of ASE lsted stocks for each sub perod under examnaton. We, also estmate the market Premum ( R m t R f, t, ) for the same perod. We then regress the excess returns (dependent varable) on the market premum (ndependent varable). Such regresson s called frst pass regresson. The outputs of the regressons are the beta coeffcents of the ndvdual shares n the sub perod under examnaton. The formula used for the above estmaton s the followng: R, t R f, t = ( R m, t R f, t ) β where R, t s the average monthly returns of the securty (dependent varable), R, s the rsk free nterest rate and f t R m, t s the average monthly return of the market (ndependent varable). 2. Then a regresson of the average holdng perod excess returns of the securtes on the estmated betas are computed, ths cross sectonal regresson s called second pass regresson. The second pass regresson has the followng form: R, t = λ o + λ1β + ε, t where ) R, t s the average monthly excess returns of the securty (dependent varable) and β the estmated beta coeffcent of securty (ndependent varable). 3. Steps 1 through 2 are repeated for all groups n all the sub perods and the results are tabulated. (see table 5) Tests of APT The analyss proceeds n the followng stages: 1. For a group of ndvdual assets, (n our case a group of 30, selected alphabetcally), a sample product moment covarance matrx s computed from a tme seres of excess returns (of all Athens Stock Exchange lsted stocks for the sub perod under examnaton). 10

11 2. A prncpal component analyss s performed on the covarance matrx. Ths estmates the number of factors and the factors scores. The results of KMO and Bartlett tests are tabulated. To decde the number of factors to retan, both the scree test and the Kaser crteron were used. 3. Then a regresson of the average holdng perod excess returns of the securtes on the factor scores s computed, ths cross sectonal regresson s called second pass regresson. The ~ ~ ~ second pass regresson has the followng form: R = λ + λ b + λ b b, where λn R,t s the average monthly excess returns of the securty (dependent varable) and β j the factor scores (ndependent varable). 4. Steps 1 through 3 are repeated for all groups (portfolos) n all the sub perods and the results are tabulated (see table 6). Comparson of the models The comparson of the two models proceeds n the followng stages: 1. For each group (portfolo) of ndvdual shares a frst comparson s done usng as measures the adjusted R squared and the F statstcs of the cross sectonal regressons of the two models under examnaton. 2. A second comparson of the two models s done usng the Davdson and MacKnnon R = α + 1 α +, where R APT equaton. Ths equaton has the followng form RAPT ( ) RCAPM e and R CAPM are the expected excess returns generated by the APT and the CAPM respectvely, as ndependent varables and R, t the average monthly excess returns of the securty as dependent varable; α s a measure of the effectveness of the two models. 3. A thrd comparson s done usng the posteror odds rato usng the formula N ( ) k 0 k 1 2 ESS R = N where ESS s the error sum of squares, N s the number of ESS observatons, and k s the dmenson (.e., the number of ndependent varables) of the respectve models (k 0 =APT mode and k 1 =CAPM model). 4. A forth comparson s done usng the resdual analyss. In order to test the effcency of the CAPM a regresson s computed wth e (the resduals of the CAPM) as dependent varable and the factor scores of the APT as ndependent. Then an analogous regresson s computed of the APT resduals on the CAPM β to fnd out whether CAPM captures nformaton mssed by the APT model. 5. Steps 1 through 4 are repeated for all groups n all the sub perods and the results are tabulated (see tables 7, 8, 9, 10). n 11

12 4. Results In order to test CAPM and APT as descrbed above, the two-step methodology s adopted, whch s extensvely used n the lterature. In the frst step, a regresson s computed of the excess returns on market premum n order to estmate the betas of the shares for the CAPM and a factor analyss n order to produce a set of factor scores for the APT. In the second step, a cross sectonal regresson s computed of the average monthly excess returns on the estmated betas (for the CAPM) and on the factors scores (for the APT). 4.1 Test for the CAPM and the APT Tests for the CAPM The results of the test for the CAPM are dsplayed n Table 5 (Appendx I). The p-values for the t test of sgnfcance are dsplayed below the coeffcents n talcs. The beta s prced 9 at the 95 per cent level of confdence, n the perod portfolo 2, (p value 0.008), when the percentage of varance explaned, represented by the adjusted R 2, s 19.6 per cent 10. In the perod beta s prced n portfolo s 1,5,6,7, and n all shares (p value: 0.000, 0.040, and respectvely), wth adjusted R , 11.1, 10.1, 25.2 and 13 per cent respectvely. In the perod beta s prced n portfolo 1 and n all shares (p value 0.015, and respectvely), wth adjusted R and 12.3 per cent respectvely. As we notce n all cases where betas are prced they have a negatve sgn, somethng that does not support the theory and ts assumptons (rsk averson). Durng the perod β s not sgnfcant. In concluson, the above results suggest that the relatonshp between β and return s weak n the Greek stock market and s consstent wth the fndngs of Fama and French (1992), Chen (1983), Cheng (1995) and Groenewold and Fraser (1997) for the US, UK and Australan stock markets respectvely. The weak explanatory power dsplayed by β suggests that addtonal varables may be needed to explan the behavour of shares prces n the ASE. Tests for the APT The number of factors and factor scores n the APT model are determned through Prncpal Component Analyss (PCA), and Varmax rotaton n order to mnmse the number of varables that have hgh loadngs on a factor. The matrx X n our tests s the (60, 30) matrx of excess returns formed by the 60 share vectors (each vector has 30 components (shares), correspondng to the 60 9 By the term prced we mean that the specfc value s statstcally sgnfcant 10 We use the Adjusted R 2 as a measure of the total varance explaned by the models to adjust for the fact that a large number of exogenous varables can artfcally produce a hgh R 2 causng n our case bas toward the APT. 12

13 monthly observatons of excess returns). The Kaser-Meyer-Olkn (KMO) test value for al the tests are very hgh and Bartlett s test of sphercty s sgnfcant at 99 per cent level, ndcatng that the factor analyss s an approprate technque for our data. Table 3 reports the Kaser-Meyer-Olkn test 11 and the Bartlett s test of sphercty n all the sub perods and n all the formed portfolos. Bartlett's test tests the hypothess that the correlaton matrx s an dentty matrx (.e., a matrx contanng ones on the leadng dagonal and zeros elsewhere). In other words, t s a test provng that there s no shared varance n the matrx. The test produces a ch-square statstc. A large and hghly sgnfcant ch-square ndcates that the data s sutable snce the correlaton matrx s not adequately descrbed by an dentty matrx; a non-sgnfcant ch-square suggests that factor analyss (FA) s not approprate for the data set under examnaton. In our case both tests prove the approprateness of the adopted FA. Perods Portfolos Bartlett Bartlett Bartlett Bartlett KMO KMO KMO KMO Ch sq. Sg Ch sq. Sg Ch sq. Sg ch sq. Sg Portfolo Portfolo Portfolo Portfolo Portfolo Portfolo Portfolo Table 3: KMO and Barlet tests for all the formed portfolos To decde the number of factors to retan, both the scree test and the Kaser crteron 12, were used. Table 4 reports the number of the factors and the total varance explaned on all the cases under examnaton. As we could observe from ths table the number of factors changes from case to case but the total varance explaned by these factors n all the cases s greater than 70 per cent. Perods Portfolos Total Total Total Total Factors Factors Factors Factors Varance Varance Varance Varance Portfolo Portfolo Portfolo Portfolo Portfolo Portfolo Portfolo Table 4: Results of factor analyss of all the formed portfolos To test the model, we examne n the second step, accordng to Chen (1983), the results of the cross sectonal regresson of average excess returns of each securty for each sub-perod 11 KMO test descrbes values between 1 and 0.9 as marvellous; values between 0.8 and 0.9 as excellent; values between 0.7 and 0.8 as good; values between 0.6 and 0.7 as medocre, values between 0.5 and 0.6 as mserable and values below 0.5 as unacceptable; 12 The KMO crteron conssts n droppng the egenvalues less than one 13

14 (dependent varable) on the estmated factor scores ( b ~ ) (ndependent varables). The results of the regresson are shown n Table 6 (Appendx I). Sgnfcance levels (p-values) are reported n talcs. The APT s overall sgnfcant (F statstc) and outperforms the CAPM n every perod: curously, ts worst performance s durng the perod where most of the betas of the CAPM are prced. In fact, n ths perod no one factor s prced for the second, thrd, forth and ffth portfolo. Another result that we could observe s that n the frst portfolo, durng the perod , where the beta s prced n the CAPM, no one factor s prced. In the perod , only factor 7 of the frst portfolo s statstcally sgnfcant (p=0.024<0.05) and the adjusted R squared for ths portfolo s 37.6 per cent; n the second portfolo no one factor s prced and the adjusted R squared s 29.9 per cent. In the perod n the frst portfolo no one factor s statstcally sgnfcant and the adjusted R squared s 10.3 per cent; n the second portfolo all factors:1, 2, 3, 4, 5, 6 and 7, are prced wth adjusted R squared 47.7 per cent, whle n the thrd portfolo factors 1, 2, 4 and 5 are statstcally sgnfcant wth adjusted R squared 77.9 per cent, the best performance of the cases under examnaton. In the perod none of the factors s statstcally sgnfcant for the second, thrd, forth and ffth portfolo wth adjusted R squared 11.2, 25.3, 52.8 and 19.6 per cent respectvely; n the frst and sxth portfolos all the factors are statstcally sgnfcant wth adjusted R squared 47.7 and 32.8 per cent respectvely; n the seventh portfolo factors 1 and 4 are prced wth adjusted R squared 23.9 per cent. Fnally n the perod n the frst portfolo no one factor s prced whle n the second portfolo all the factors are statstcally sgnfcant, the adjusted R squared s 27.3 and 67.2 per cent respectvely. Observng the adjusted R squared of all the cases under examnaton we could say that shows a consderable mprovement compared wth the lack of explanatory power of the CAPM for the same cases. 4.2 Comparsons between CAPM and APT The next step s to assess whch one of the two competng models, CAPM or APT s supported by the data. Followng the approach used by Chen (1983), we use three methods, the Davdson and McKnnon equaton, the posteror odds rato and the resdual analyss. Davdson and McKnnon Equaton The CAPM could be consdered as a partcular case of the theoretcal APT wth k=1 (b k =β). However, when we consder the APT wth artfcal factors, ths s true f and only f there exsts a rotaton of the factors such that one of the factors s the market. The two models, CAPM and APT, are thus defned as non-nested. One method to dscrmnate among non-nested models was suggested by Davdson and McKnnon (1981). 14

15 ) Let R APT and R CAPM be the expected excess returns generated by the APT and the CAPM, and consder the followng equaton APT ( ) CAPM R = αr + 1 α R + e where α s a measure of the effectveness of the two methods. When s close to 1, the APT s the correct model relatve to the CAPM. The results of the regresson, reported n table 7 (Appendx I), are heavly n favour of the APT, wth the possble excepton of the perod (portfolo 7 and all shares), for whch the results n favour of the APT are less substantal. The Davdson and McKnnon (DM) equaton has been crtcsed because, even f the models are non-nested, there s stll a rsk of multcollnearty between the varables as the β of the CAPM could be strongly correlated wth APT factors. However, the method has been extensvely appled n the lterature (Chen, 1983; Groenewold and Fraser, 1997). Posteror Odds Rato Gven the assumpton that the resduals of the cross-sectonal regresson of the CAPM and the APT satsfy the IID (Independently and dentcally dstrbuton) multvarate normal assumpton (Campbell, Lo and MacKnlay, 1997), t s possble to calculate the posteror odds rato between the two models. In general, the formula for posteror odds n favour of model A (n our case APT) over model B (n our case CAPM) s gven by Zellner (1979): R ESS N ( k A k ) 2 B A 2 = N where ESS s the ESSB error sum of squares, N s the number of observatons, and k s the dmenson (.e., the number of ndependent varables) of respectve models. If the value produced by the above equaton s lower than 1 then model A has better performance than ths produced by model B. The posteror odds computed are overwhelmngly n favour of the APT. Table 8 (Appendx I) shows that the posteror odds computed are all less than one, and thus n favour of the APT. The posteror odds rato s n general a more formal method than the Davdson and MacKnnon equaton and has sounder theoretcal foundatons. Resdual Analyss The resduals from the CAPM are of nterest as they are used for performance measurement. If the CAPM s not mss-specfed, the expected return of an asset would be captured by β and the resdual e wll behave lke whte nose wth zero mean across tme. Thus, f expectatons n the market are ratonal, the realsed excess return can be wrtten as ratonal expected excess return and v s the error term. R + = E v where E s the market If the CAPM s not mss-specfed, R can also be wrtten as (Chen, 1983) R = E CAPM ) + e (. Thus e = [ E E ( CAPM )] + v return from the CAPM wth the market proxes., where (CAPM ) ) E s the expected excess 15

16 ) If the CAPM s correct then E = E (CAPM ) and v = e should behave lke whte nose and should not be prced by any other models. If e s prced by any other model, e contans nformaton that s not captured by (CAPM ) and the CAPM s mss-specfed. Therefore, a logcal method to ) E test the CAPM s to run a regresson wth e (the resduals of the CAPM) as dependent varable and the factor scores of the APT as ndependent. We then run an analogous regresson of the APTs resduals on the CAPM βs to check whether CAPM captures nformaton mssed by APT. The results, reported n Tables 9 and 10 (Appendx I) are clearly n favour of the APT. The CAPM fals to explan the varance of APT resduals n all the perods (table 9). On the other hand, the APT explans, n the perod , 34.4, 23 and 12.8 per cent respectvely of the varance unexplaned by the CAPM. In the perod the explanatory power of the APT has the best performance and explans 4.6, 32.7, 78.7 and 42 per cent of the varance unexplaned by the CAPM. As expected, the worst performance of the APT s n the perod , when the varance explaned s only 9, 5.7, 26.5, 47.5, 11.2, 15.5, 0.8 and 9.9 per cent ths s the perod when β s prced and has some explanatory power. Fnally n the perod APT explans, 7.9, 56.0 and 27.7 per cent of portfolos 1, 2, and all-shares respectvely of the varance unexplaned by the CAPM. However, care s needed when lookng at the results n tables 5, 6, 9 and 10. Tables 6, 10 and 5, 9 are strctly connected. Any factor not prced n Table 6 should also not be prced n table 10 and any factor not prced n table 5 should also not be prced n table 9. If a factor s not prced wth the orgnal data, but s prced n the regresson of e on the ~ β, the estmated λ may be spurously nduced by ~ β. Analysng tables 6 and 10, we see that n the perod factor 7 s prced n both regressons. Ths result strongly supports the ablty of the APT to explan nformaton not captured by the CAPM. In the perod we could observe that factors 1, 4 and 5 of thrd portfolo and factor 7 of the last portfolo are prced n both regressons but n the second portfolo factors 1, 2, 4, 5, 6, and 7are prced n the cross sectonal regresson but not n the resdual regresson. In the perod no one of the factors that s prced n the cross sectonal regresson s prced n the resdual regresson; ths fact confrms that ths perod s the worst for the APT. Fnally n the perod we could observe that n the second portfolo factors 1, 3, 5, and 6 are prced n both regresson but factors 2, 4 are prced only n the cross sectonal regresson and n the last portfolo (all-shares) no one of the factors that are prced n the cross sectonal regresson are prced n the resdual regresson. On the other hand, analysng tables 5 and 9 we notce that any of the factors prced on the orgnal CAPM regressons, n table 5 (Appendx I), are not prced by the regressons of the 16

17 resduals n table 9 (Appendx I). Ths means that CAPM s totally ncapable of explanng the varance, whch s not explaned by the APT model. 5. Conclusons Examnng the hstory of the Greek Stock Exchange (ASE), we observe that n the last 15 years a number of reforms have been ntroduced n order to ncrease the lqudty, effcency, and transparency of stock tradng. Lberaton of the captal market should further mprove the possbltes for the Greek stock market to respond more rapdly to new nformaton. Also we observed that the market captalsaton of the Greek stock Exchange as a percentage of Gross Domestc Product, from 1987 up to 1997, s the lowest comparng wth all other European stock exchanges. However, we observe that durng the years 1998 and 1999 the captalsaton rate ncreases tremendously whle n 2000 and 2001 t drops very much, n both cases wthout any specfc reason. These fndngs confrm the fndngs of Narchos and Alexaks (2000) that the Greek stock market s nfluenced by psychologcal factors. At the end of 2001 there were 345 lsted companes, whch represent the 74.5 per cent of GDP. Ths fact shows us that the ASE plays an mportant role n Greek economy the last few years. The analyss of the chosen sample shows that 21, 28 and 22 per cent of the shares, n the three sub perods are normally dstrbuted. Ths s an mportant fndng, suggestng that, more or less, 25 per cent of the returns dstrbuton of the ASE may be normal n any sub perod, n accord wth the fndngs of Mandelbrot (1963) and Fama (1970) for the US stock exchange, whch are wdely accepted n the modern fnancal theory. The relatonshps between β and return n the ASE n all the sub perods and all the formed portfolos s weak, and the Captal Asset Prcng Model dsplays poor explanatory power. It s however dffcult to assess the extent of ths dependence that s due solely on the specfcaton of the model tself. The apparent low nformatonal effcency of the ASE, the fact that there are few nsttutonal nvestors, and that prvate nvestors n Greece often regard the stock market more as a place to gamble than to nvest, could cause market rratonalty, undermnng the assumptons upon whch the CAPM s based. The Arbtrage Prcng Theory performs better, compared to the CAPM, n all the tests consdered. From the evdence gathered n ths study, the APT s a more powerful method that allows consderaton of the rsk borne on addtonal systematc state varables, other than the market portfolo. The percentage of varance explaned for the portfolos formed n the sub perods under examnaton s rangng from 10.3 to 77.9 per cent. Ths performance can be consdered a good result compared wth the results obtaned by Chen (1985) n the US stock market (results 17

18 rangng from 4 to 27.8 per cent n dfferent sub-perods from 1963 to 1978) and Cheng (1995) n the UK (11 per cent n the perod January 1965-December 1988). The study was orgnally desgned to compare CAPM and APT, but one of the man results obtaned, s the apprecaton of the wde range of potentaltes offered by a relatvely new tool used n testng the APT: factor analyss. If the dentfcaton of the number of factors and ther dentty s one of the most mportant drectons for future research (Chen 1983), factor analyss technque, s a powerful nstrument to replace the arbtrary and controversal search for factors by tral and error wth a real systematc approach. The overall concluson of the study s that even f the market return s an mportant element, the behavour of securtes returns n the ASE s complex and cannot be fully explaned by a sngle factor. Shares and portfolos are sgnfcantly nfluenced by a number of systematc forces and ther behavour can be explaned only through the combned explanatory power of several factors or macroeconomc varables. Consderng that the APT does not explan the overall varance, we can ask ourselves where the mssng nformaton s, and why the APT fals to explan fully the returns covarance s and means returns. There can be several possble explanatons (Cheng, 1995). Frst, rsk and expected return may not be statonary durng the perod n consderaton, whle one of the assumptons n the study of the APT s that rsk and expected returns are assumed not to change durng the perod. Second, the APT prcng relatonshp could hold only n some months of the year, and there s evdence of a January effect on the capablty of the APT to explan the return-rsk relatonshp (Gültekn and Gültekn, 1987). Thrd, and n our opnon more probable, there s the possblty of non-lnear prcng relatonshps. The assumpton of lnear relatonshps between the APT and factors or macroeconomc varables s a strong assumpton, whch s often overlooked. The lnear model s a smple model, deal to explan observed correlatons. If nstead the objectve s to predct mean returns, hgher-order factor models would provde more accurate predctons as mnor factors relatvely unmportant n explanng covarances, may be fundamental to explan mean returns. These, we thnk, may be mportant drectons for future research. 6. References Banz, R. (1981), The Relatonshp Between Return and Market Value of Common Stocks, Journal of Fnancal Economcs 9(1), pp Barkoulas, J. T. and N. G. Travlos (1996), Chaos n An Emergn Market? The Case of the Athens Stock Exchange, Appled Fnancal Economcs, 8(3), pp Basu, S. (1997), The Investment Performance of Common Stocks n Relaton to ther Prce to Earnngs Rato: A Test of the Effcent Markets Hypothess, Journal of Fnance, 33(3), pp Bernoull, D. (1738), Exposton of a New Theory on the Measurement of Rsk, (transl.) by L. Sommer (1954), Econometrca, 22(1), pp

19 Black, F., M. C. Jensen and M. Scholes (1972), The Captal Asset Prcng Model: Some Emprcal Tests, In M. C. Jensen (eds) (1972), Studes n the Theory of Captal Markets, New York: Praeger Publsher, pp Black, F. (1972), Captal Market Equlbrum wth Restrcted Borrowng, Journal of Busness 45(3), pp Black, F. (1974), Internatonal Captal Market Equlbrum wth Investment Barrers, Journal of Fnancal Economcs, 1(4), pp Blume, M. E. and I. Frend, (1973), A New Look at the Captal Asset Prcng Model, Journal of Fnance, 28(1), pp Brennan, M. J. (1970), Taxes, Market Valuaton and Corporate Fnancal Polcy, Natonal Tax Journal, 23(4), pp Campbell, J. Y., A. W. Lo and A. C. MacKnlay (1997), The Econometrcs of Fnancal Markets, Prnceton: Prnceton Unversty Press. Chen, N. F. (1983), Some Emprcal Tests of Arbtrage Prcng, Journal of Fnance, 38(5), pp Cheng, A. C. S. (1995), UK Stock Market and Economc Factors: A New Approach, Journal of Busness Fnance and Accountng, 22(1), pp Connor, G. (1982), A Factor Prcng Theory of Captal Assets, North Western Unversty: Unversty Press. Davdson, R. and J. McKnnon (1981), Several Tests for Model Specfcaton n the Presence of Alternatve Hypotheses, Econometrca, 49(3), pp Dacoganns, G. P., M. Glezakos and K. N. Segredaks (1998), Exploraton of the Impact of Prce-Earnngs (P/E) and Dvdend Yelds on Expected Returns of Common Stocks of the Athens Stock Exchange, (n Greek), Economc Revew Bank of Greece, 4(1), pp Dybvg, P. (1983), An Explct Bound on Indvdual Assets' Devatons from APT Prcng n a Fnte Economy, Journal of Fnancal Economcs, 12(4), pp Dybvg, P. and S. Ross (1985), Yes, the APT s Testable, Journal of Fnance, 40(4), pp Fama, E. F. (1970), Mult-Perod Consumpton-Investment Decsons, Amercan Economc Revew, 60(1), pp Fama, E. F. and J. D. MacBeth (1973), Rsk, Return and Equlbrum: Emprcal Tests, Journal of Poltcal Economy, 81(3), pp Fama, E. F. and K. French (1992), The Cross-Secton of Expected Returns, Journal of Fnance, 47(2), pp Gbbons, M. R. (1982), Multvarate Tests of Fnancal Models: A New Approach, Journal of Fnancal Economcs, 10(1), pp Grnblatt, M. and S. Ttman (1985), Approxmate Factor Structures: Interpretatons and Implcatons for Emprcal Tests, Journal of Fnance, 40(5), pp Groenewold, N. and P. Fraser, P (1997), Share Prces and Macroeconomc Factors, Journal of Busness Fnance and Accountng, 24(9/10), pp Karankas E. (2000), CAPM Regulartes for the Athens Stock Exchange, SPOUDAI, 50(1/2), pp Koutmos, G., C. Negaks and P. Theodossou (1993), Stochastc Behavour of the Athens Stock Exchange, Appled Fnancal Economcs, 3(2), pp Lehmann, B. N. and D. M. Modest (1988), The Emprcal Foundatons of the Arbtrage Prcng Theory, Journal of Fnancal Economcs, 21(2), pp Lntner, J. (1965), Securty Prces, Rsk and Maxmal Gans from Dversfcaton, Journal of Fnance, 20(4), pp Mandelbrot, B. (1963), The Varaton of Certan Speculatve Prces, Journal of Busness, 36, pp Markowtz, H. M. (1952), Portfolo Selecton, Journal of Fnance, 7(1), pp