Condtonal Beta Captal Asset Prcng Model (CAPM) and Duraton Dependence Tests By Davd E. Allen 1 and Imbarne Bujang 1 1 School of Accountng, Fnance and Economcs, Edth Cowan Unversty School of Accountng, Fnance and Economcs & FEMARC Workng Paper Seres Edth Cowan Unversty September 2009 Workng Paper 0907 Correspondence author: Davd E. Allen School of Accountng, Fnance and Economcs Faculty of Busness and Law Edth Cowan Unversty Joondalup, WA 6027 Australa Phone: +618 6304 5471 Fax: +618 6304 5271 Emal: d.allen@ecu.edu.au
ABSTRACT Ths paper uses a sample of 50 companes contnuously lsted on Man Board of Bursa Malaysa from January 1994 untl December 2001 and uses duraton dependence tests whlst applyng two asset prcng models based on the CAPM; the two Factor Model developed by Fama and French (F&F)(1998) and Ferson, Sarkssan and Smn s (FSS) (2008) condtonal beta model appled to estmate the condtonal beta of CAPM as to generate the postve and negatve abnormal returns. The fndngs suggest that both the Log Logstc and Webull hazard models seem to support the exstence of negatve duraton dependence for both postve and negatve runs of abnormal returns, consstent wth the presence of bubbles theory as predcted by McQueen and Thorley (1994). The negatve runs of abnormal returns for both the F&F and FSS models show that more than 80% of the sample seems to support the exstence of negatve duraton dependence usng both hazard models. Meanwhle the postve runs show that not more than 80% of the sample rejects the null hypothess based on LR tests of the absence of duraton dependence. Therefore, the analyss and results suggest that the negatve runs of abnormal returns are consstent for both hazard models. Keywords: Duraton dependence, Two factor models; Ratonal bubbles; Log logstc and Webull hazard models
1. Introducton The ssue of whether stock markets are ratonal s a central one. (In ths paper, the concern s whether the Malaysan stock market s ratonal?). One way of approachng ths ssue s by means of the concept of ratonal bubbles. Bubbles can be defned as the phenomena of stock prces movng consstently away from fundamental values. A recent study conducted by Chan, McQueen and Thorley (1998) used the ratonal bubble concept to assess whether the US market as represented by the Standard and Poor 500 Index, s ratonal through a technque called a duraton dependence test. However, they found that there s no duraton dependence n ths market. Harman and Zuehlke (2004) dsagreed and asserted that there s sgnfcant duraton dependence evdent n runs of both postve and negatve abnormal returns that are nconsstent wth the model of ratonal bubbles as proposed by McQueen and Thorley (1994). Watanapalachakul and Islam (2007) also nvestgated the exstence of ratonal bubbles n the Tha stock market. Smlar to Chan et al. (1998), they employed the duraton dependence test but they used two hazard models namely () the Log-Logstc and () the Webull Hazard Model. In addton, they also analysed ratonal bubbles usng annual data. Ths s n contrast to the prevous studes (McQueen and Thorley, 1994; Chan et al., 1998), whch used one-short perod of analyss. Based on ther study, Watanapalachakul and Islam (2007) contended that t was easer to detect the exact sze of bubbles n each partcular year usng a yearly based analyss. In another study conducted by Harman and Zuehlke (2004) further support was provded for Watanapalachakul and Islam (2007). They employed the generalzed Welbull hazard model on value weghted portfolos of NYSE stocks from 1927 through 1997. Ths model provdes much more flexblty whch allows changes n the drecton of duraton dependence. Ther fndngs suggest that duraton dependence s not monotonc. Bascally, the dea s of duraton elastcty beng ntally postve. However, the sgn can be negatve as the duraton of the run of abnormal returns ncreases. The presence of sgnfcant duraton dependence n runs of negatve abnormal return returns s nconsstent wth McQueen and Thorley (1994). McQueen and Thorley (1994) suggested that the duraton dependence pattern s ndcatve of the exstence of bubbles snce t cannot be the result of asymmetrc or leptokurtc nnovatons n fundamentals alone. Though a study by Harman 1
and Zuehlke (2004) agreed wth Mudholkar, Srvasta, and Kolla (1996) who reported negatve duraton over much of the range of data they examned, and t s evdence n support of both postve and negatve runs of abnormal returns. Another study conducted by Harman and Zuehlke (2004) appled the duraton dependence test to both equally weghted and value weghed portfolos of NYSE traded securtes and found that both portfolos produced evdence of negatve and postve duraton dependence, whch once agan was not supported by the model proposed by McQueen and Thorley (1994). One potental explanaton for ths s a falure of the assumpton of market effcency. The ntal research on the exstence of ratonal bubbles n equty securtes on the Malaysan stock market was conducted by Chan et al. (1998), usng monthly and weekly Kuala Lumpur Composte Index returns from January 1977 through Aprl 1994 employng duraton dependence tests usng a Log-Logstc Hazard Model. They found that there was no evdence of the exstence of ratonal bubbles n the Malaysan stock market between the years 1977 untl 1999. However, n ths paper, the focus wll be on a sample of Man Board companes from the Kuala Lumpur Stock Exchange (Bursa Malaysa). The performance of the South East Asan economes has been relatvely erratc throughout the 1990s and untl 2007, most sgnfcantly durng the 1997 Asan fnancal crss. Do ratonal bubbles n the equty premum exst n the Malaysan stock market? The present study attempts to detect ratonal bubbles n the equty premum usng Duraton Dependence Tests. In these tests two models are employed: (1) the Log Logstc Hazard Model and (2) the Webull Hazard Model. There are a few relevant papers publshed about the Malaysan stock market, and n one of the latest papers, publshed by Mokhtar at. el. (2006), t was revealed that bubbles dd not exst durng the economc crss of 1997. However, the detecton of ratonal bubbles n ths study lmts ther analyss to a study of sectoral ndces data rather than ndvdual stocks, as opposed to the use of ndvdual stocks that we adopt for forecastng the equty premum. Furthermore, ths s the frst study of Malaysan data to employ a condtonal beta verson of CAPM. We apply two models whch are; (1) the two Factor model developed by Fama and French (1998) and an asset prcng model wth tme varyng 2
coeffcents as developed by Ferson, Sarkssan and Smn (2008) to estmate the abnormal returns and utlse them n duraton dependence tests. Ths paper contans 4 sectons whch nclude an ntroducton, a data and methodology secton, an explanaton of the estmaton of the hazard model and a further secton presentng the results followed by a concludng secton. 2. Data and Methodology We adopt the Duraton dependence test usng the Log Logstc Hazard Model and Webull Hazard Model that are more wdely accepted (Fung 2001; Harman and Zuehlke 2004), because of ther robustness n testng for ratonal bubbles. Furthermore, both hazard models are chosen because of ther flexblty n applyng the hazard rate dstrbuton. The paper uses monthly aggregate returns on the Malaysan Stock Exchange for the perod 1994 to 2001 and collected the data for dvdend prce ratos and dvdend yelds from Professor Kenneth French s personal webste (data for Malaysa). These wll be analysed usng regressons applyng the two factor model of Fama & French (1998) and a tme varyng coeffcent model by Ferson, Sarkssan and Smn (2008). These models are used to assess the relatonshp between the equty premum and dvdend yelds. The dvdend yeld has been chosen as an ndependent varable as the ntal regresson result shows sgnfcantly ablty to explan the equty premum. The model can be llustrated as follows: The two Factor Model By Fama and French (1998) R t - R ft = α + β 0 (Rm Rf) + β 1 (HB- LB /M) t + u t ( 1 ) β* = β 0 + β 1 (HB- LB /M) t Where, HB and LB are defned as hgh and low book value to market value respectvely. Asset Prcng Tme Varyng coeffcents model by Ferson, Sarkssan and Smn (FSS) (2008) R t = α 0 + α 1 Z t-1 + β 0 (Rm t ) + β 1 (Rm t Z t-1 ) + u t ( 2 ) β*= β 0 + β 1 (Z t-1 ) 3
Where the Z t-1 s the lagged predctor varable. For the purposes of ths paper, one month lagged dvdend yelds are used to estmate condtonal betas. The regresson models are used to forecast aggregate returns (the equty premum) and to measure realsed excess returns. These excess returns are then subjected to duraton dependence tests based on hazard models and the Log Logstc and Webull dstrbutons. The FF model and FSS model assume that the both the two factor and condtonal CAPM mply that α 0 =0 and α 1 =0. Where, f the model holds, then E(u t )=E(u t [R m -R f ) t ])=0 and E(u t ) = E(u t [Rm t Z t-1 ])=0 respectvely. Once the betas have been estmated, then equaton (1 and 2) are used to calculate the forecast stock returns and can then be appled to dentfy abnormal returns for both posttve and negatve seres for the duraton dependence tests. Thus the sample perod adopted for ths paper s shown n table 1: Table 1: Perod of Estmatng coeffcent and forecast for duraton Dependence Test Model Benchmark Total Perod Estmaton perod Two Factor Model * (Model 1) Fama and French (1998) Condtonal beta Based on Asset prcng Tme varyng* (Model2) Ferson, Sarkssan &Smn (2008) 50 companes of Man Board KLSE 50 companes of Man Board KLSE January 1994 untl October 2001 January 1994 untl October 2001 January 1994 untl November 1997 January 1994 untl November 1997 Forecast Perod December 1997 untl October 2001 December 1997 untl October 2001 Usng the LR test, both versons of the CAPM models (F&F and FSS) are tested based on the followng hypothess: H 0 : H 1 : There s no ratonal bubble. Excess returns do not exst. There s no duraton dependence; there s a constant hazard rate. There are ratonal bubbles. Excess returns do exst. There s postve or negatve duraton dependence; there are decreasng or ncreasng hazard rates. 4
3. Log Logstc and Webull Hazards Models The Log Logstc Hazard Model s defned as estmated by McQueen and Thorley (1994) and Harman and Zuehlke (2004): N ln L(, β ) = { J ln[ g( t )] + ( 1 J ) ln[ 1 G( t )]} = 1 α (3) where α s the shape parameter of the lognormal dstrbuton, β s the duraton elastcty of the hazard functon, J s a duraton of the process or tme to ext from a state, g s the dscrete functon for duraton, and G s the correspondng dstrbuton functon. The dscrete densty and dstrbuton functons for duraton are related as: = k = t ( ) g( k) G t 1 (4) However, f the law of condtonal probablty s appled (Harman and Zuehlke, 2004), the densty for completed duraton s: k 1 ( k) = h( k) [ h( m) ] 1 m= 0 g (5) For postve nteger k, where h(0) s defned as zero. In addton, McQueen and Thorley (1994) used the logstc dstrbuton functon ψ evaluated at a lnear transformaton of log duraton as: h(k) = ψ [α + β ln (k)] = {1+ exp [-α -β ln (k)]} -1 (6) The Webull hazard Model (Mudholkar, Srvastava and Kola, 1996) s defned as S(t) = exp (-αt bt+1 ) (7) where S(t) s the probablty of survval n the data up to at least tme (t), The correspondng hazard functon s: 5
h(t) = α (β + 1)t β (8) where α s the shape parameter of the Webull dstrbuton, and β s the duraton elastcty of the hazard functon. The fundamental assumpton of the Webull Hazard model s a lnear relatonshp between the log of the hazard functon and the log of duraton, where: ln [h(t)] = ln[α (β + 1] + β ln(t) (9) Further nformaton regardng Webull Hazard Model can be found n Harman and Zuehlke (2004). 4. Fndngs The man objectve of the paper s to llustrate whether the forecastng of returns usng CAPM as n the Fama and French (1998) model or the condtonal verson of the Ferson, Sarkssan and Smn model (2008) as a flter, are approprate n evaluatng the exstence of duraton dependence n excess returns on the Malaysan Stock Market. Based on the LR test for the absence of duraton dependence, there s evdence of negatve duraton dependence n Malaysa stock market for the perod for December 1997 untl October 2001. Table 2 presents a summary of number of companes that both rejected and faled to reject the null hypothess. Table 2 Results summary: percentage of sample of 50 companes falng to detect the exstence of duraton dependence by falure to reject the null hypothess of no duraton dependence. Bubble Model Log Logstc Hazard Model Webull Hazard Model CAPM Fama and French Ferson, Sarkssan Fama and French Ferson, Sarkssan Model Model (1997) and Smn (2006) Model (1997) and Smn (2006) Abnormal Returns Postve Negatve Postve Negatve Postve Negatve Postve Negatve H 0 = β 0 15 9 12 7 14 8 10 5 % Faled to detect 30 18 24 14 28 16 20 10 Notes: H 0 = the null hypothess for the absence of duraton dependence 6
Table 2 shows that n the case of negatve runs of abnormal returns for both the F&F and FSS model t seems more than 80% (82%, 86%, 84% and 90%) of the sample seem to support the exstence of negatve duraton dependence usng both hazard models. Meanwhle the postve runs suggest not more than 80% (70%, 76%, 72%% and 80%) of the sample reject the null hypothess at a 95% confdence level based on LR tests for the absence of duraton dependence whch ndcates that there s both negatve duraton dependence and postve runs of abnormal returns for the perod December 1997 untl October 2001 (forecast perod). Therefore, based on ths analyss, there are postve and negatve runs of abnormal return prevalent for both hazard models. Table 3 Pared Sample t-test on the number of postve, negatve and total runs of abnormal returns for both F&F Model and FSS Model Statstc Postve F&F Postve FSS Negatve F&F Negatve FSS Total F&F Total FSS Model Model Model Model Model Model Mean 13.66 12.42 13.72 12.66 27.38 25.08 Standard Devaton 1.184 2.081 1.819 2.309 3.568 3.948 Pared t- test 4.128 3.409 4.195 (0.0001)*** Correlaton 0.412 (0.003)*** (0.001)*** 0.453 (0.001)*** (0.0001)*** 0.472 (0.001)*** Notes: fgures n the parentheses are the t-statstcs and Correlaton p-values. * denotes sgnfcance at the 10% level, ** denotes sgnfcance at the 5% level and *** denotes sgnfcance at the 1% level. The purpose of usng Table 3 s to compare whether the estmaton of the run lengths of postve and negatve abnormal returns for both F&F and FSS models are sgnfcantly dfferent. The result suggests that the number of runs for both the condtonal beta models by F&F and FSS shows no dfference at all as the pared sample t-test proved that there were no dfferences n the mean of runs of postve abnormal return for all types of runs (Postve, negatve and Total ). Lastly, the analyss of the hazard rate dctates that the Log Logstc results suggest that the seres of abnormal return runs were decreasng whle the Webull Hazard rate found that the runs seres were decreasng then ncreasng. 5. Concluson The tests for duraton dependence examne a partcular nonlnear form of return predctablty that s consstent wth the presence of bubbles allowng predctablty 7
assocated wth tme varaton of the equty returns fltered through a prcng model. In ths paper two factor model developed by Fama and French(1998) and the condtonal model of Ferson, Sarkssan and Smn (2008) were used to generate return predctons that were then compared wth actual returns to derve excess returns. These were then subjected to duraton dependence tests. Both models suggest the exstence of duraton dependence n excess returns though tests on the means of the results of the two approaches suggest they are sgnfcantly dfferent. Ths paper featured novel tests of the mplcatons and estmaton of the run lengths of postve and negatve abnormal returns usng two models developed by F&F (1998) and FSS (2008). The overall results suggest that both postve and negatve runs show negatve duraton dependence as the LR test for absence of duraton dependence was rejected whch s consstent wth the evdence of McQueen and Thorley (1994). 8
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