The Behavior of Beta in the 19 th Century

Transcription

1 The Behavor of Beta n the 9 th Century Lord Mensah Unversty of Ghana Busness School, Department of Fnance, Legon, Accra-Ghana Correspondence: Lord Mensah, Unversty of Ghana Busness School, Department of Fnance, Legon, P. O. Box 78, Legon, Accra-Ghana. E-mal: lordmensah@ug.edu.gh Receved: August 30, 03 Accepted: September 6, 03 Onlne Publshed: September 5, 03 do:0.5430/afr.vn4p34 URL: Abstract Ths paper uses completely new data to study the varatons n beta when t devates from the constancy assumpton presumed by the market model. The concentraton of the lterature on beta s based on post 96 data. Ths makes the 9 th century Brussels Stock Exchange (BSE) data a very good out-sample dataset to test beta varatons. Varous models proposed n the lterature to capture the varatons n beta were studed. Blume s correlaton technques reveal that beta s not stable at the ndvdual stocks level and that the stablty can be farly mproved by portfolo formatons. Usng root mean square error (RMSE) crteron, t was shown that the market model betas are weak n predctng future betas. The predctablty can be mproved by adustng betas wth the Blume and Vascek mean reverson technques. Further results from ths study reveal that few stocks have lead or lag relatonshp wth the market ndex. Small szed stocks were detected to be more prone to outlers. In effect betas n the 9 th century exhbts smlar pattern as betas n the post 96 era. Keywords: Fnancal hstory, Hstorcal beta estmaton, Brussels stock exchange, Robust betas Many thanks to the Unversty of Antwerp for makng the 9 th century Brussels Stock Exchange data avalable to me and also for sponsorng the research. I wll lke to thank all the partcpants of the Frst World Fnance Conference n Vana do Castelo, Portugal for ther comments and suggestons.. Introducton Beta stablty, bas and robustness evaluatons have become the centre stage of research n fnance snce the development of the Captal Asset Prcng Model (CAPM) by Sharpe (964). Beta, one of the parameters of the tme seres regresson, plays a key role n CAPM applcatons. Beta s usually estmated wth the standard market model (MM). The MM s a statstcal model that relates the return of any stock to the return of the market ndex. Beta estmated wth the MM has been tested not to be stable over tme. The fnancal lterature has ndcated that the stablty can be mproved by Blume s and Vascek s autoregressve adustment methods (Blume (97), Vascek (973)). The obectves of these adustment methods are based on the tendency of betas from successve perods to be mean revertng. Whle Vascek consders the accuracy of the hstorcal betas, Blume depend on how far the hstorcal betas devate from ther average value. The lterature has also shown that betas estmated by the market model are based. That s when stocks are ranked based on ther market captal (sze); beta s based downwards for small sze stocks and upwards for large sze stocks. Scholes and Wllams (977), Dmson (979), Fowler and Rorke (983) and Marsh (979) developed methods to correct the bas. The method adopted by Marsh (979) s MM estmaton wth returns calculated on trade to trade bass. Dmsom (979) adopted the aggregate coeffcent method whch computes the lead and the lag betas n a multple regresson of the stock return on a number of lead and lag market returns. The sum of the estmated beta coeffcents from the multple regressons s the Dmson s estmate of beta. On the other hand, Scholes and Wllams (977) lead and lag beta coeffcents are computed from unvarates regressons. The lterature also hghlghts the mpact of outlers (extreme observatons) on the beta estmates. It proposes varous technques of mnmzng the mpact of outlers on beta estmates. The weghted least square estmaton technque s one of the methods used n mnmzng the mpact of outlers on beta estmates. Based on the assumpton that outlers have lower probablty of occurrng n the future, less weght s placed on them (Martn et al. (003)). An alternatve technque s to get rd of the extreme observatons and perform the regresson wth fewer observatons. To the best of my knowledge the fnancal lterature has not traced the stablty, bas and robustness of beta usng data beyond 96 n any country. Luoma et al. (994) generated dfferent artfcal market returns data and Publshed by Scedu Press 34 ISSN E-ISSN

2 documented that dfferent beta estmaton technques behave dfferently on dfferent markets. Ths shows that there s no beta adustment technque whch has been accepted unversally and performs well ndependent of the data. The obectve of ths study s to ntroduce a new set of data from the 9 th Century Brussels Stock Exchange (BSE) to test the performance of the alternatve technques of estmatng beta. By so dong one would know the beta estmaton technque needed to estmate beta on ths data. Indvdual and portfolo betas from the varous estmaton technques wll be compared based on ther predctve accuracy. The root mean square error (RMSE) and the mean absolute error (MAE) crteron adopted by Blume (97) and Klemkosky et al. (975) are used as an error predctor of the varous technques. The 9 th century data provde a very good platform to determne f betas before 94 exhbt smlar pattern as betas after 94. Ths paper s organzed as follows: the next secton provdes a summary of related lterature on statonarty, based and robustness of beta. In secton 3, varous models of estmatng beta are dscussed. Secton 4 provdes a detaled descrpton of the data and descrptve statstcs of beta when estmated wth the least square regressons of the market model. In secton 5, Blume s correlaton technque of detectng the stablty of beta s studed. Betas are adusted wth the Blume and Vascek s autoregressve technques n secton 6. The predctve accuracy of the adustment technques based on the root mean square error crteron s tested n the same secton. The modfed Debold and Marano test proposed by Harvey et al. (997) s used to test for equal predctve accuracy of the models. Secton 7 determnes how the stocks n each perod lead or lag the market ndex usng the Dmson s model. Betas estmated wth the teratve reweghted least square (outler resstant) technques are compared to the benchmark market models betas n secton 8. The fnal secton presents the concluson.. Related Papers There s substantal evdence n the fnancal lterature that has establshed that securty betas are not stable over tme. Partcularly, prevous studes have ndcated that portfolo betas are more stable as compared wth ndvdual securty betas. Blume (97 and 975) poneered the research on beta stablty focusng on the cross-sectonal correlaton between beta estmates for successve perods. Hs study s based on the assumpton that beta s statonary wthn each perod. He concluded that beta nstablty can be attenuated by the formaton of portfolos. Subsequent publcatons on beta stablty assume that beta varatons exhbt stochastc behavor. They formally propose a model to capture the varatons n beta overtme and conduct a test to confrm the model. For nstance, Fabozz and Francs (978) found strong evdence of beta followng the Hldreth-Houck random coeffcent model. Colln et al. (987) utlzed Ohlsen and Rosenberg s model to analyze the varatons n beta for both ndvdual and randomly formed portfolos. Ther fndngs were n support of the mxed sequental and purely random varatons of the beta. It was clear from ther results that ndvdual securtes and portfolo betas exhbt the mxed sequental and purely random varatons. Faff et al. (99) used Australan data to establsh evdence n beta nonstatonarty. Detals of ther result ndcates a relatonshp between beta nonstatonarty, the market ndex as a proxy for the general market movement, the magntude of the betas (degree of rskness) and the market captal. Further nvestgaton by Gregory-Allen et al. (994) confrm that when the varance of the beta estmate s takng nto consderaton, there s no evdence that portfolo betas are more stable than ndvdual securty betas (as clamed by Blume (97)). The focus of the above lterature s on the stablty of beta, but the ssue of beta bas s worth consderaton. The bas n beta estmates s traced n the lterature to be due to nfrequent tradng. Ibbotson et al. (997) documented that beta estmates for small stocks are severely downwards based that s they exhbt severe nfrequent tradng. They recommend the ncluson of lagged market returns n the estmaton of beta when securtes are nfrequently traded. On the other hand Bartholdy and Rdng (994) found on the New Zealand Market, the MM betas are less based, more effcent and consstent compared to the Dmson s, Scholes and Wllams betas (corrected for bas). In effect, the lterature has shown that nonsynchronous tradng bases betas estmated wth the MM. When stocks are ranked based on ther market captal (sze), beta s based downward for small sze stocks and upward for large sze stocks. Scholes and Wllams (977), Dmson (979) and Fowler and Rorke (983) developed methods to correct the bas. Dmson (979) adopted the aggregate coeffcent method, whch computes the lead and the lag betas n a multple regresson of the stock return on a number of lead/lag market returns. The sum of the estmated beta coeffcents from the multple regressons s Dmson s estmate of beta. In the MM least square estmaton, t s assumed that the return resduals follow a normal dstrbuton, and that nfluental ponts are rare. Practcally ths assumpton fals most of the tme. The falure s connected to the occurrence of a small fracton of exceptonally large or small returns (outler or nfluental ponts) observaton. Chatteree and Jacques (994) studed the effect of these ponts on the beta parameter. They establshed that the market model beta estmaton method sheld the outlers and aggregate ther nfluence on the beta due to the square of ther errors. They proposed that the effect of these observatons on beta can be reduced by applyng least square method that can detect and assgn zero weghts to the outlyng observatons n the estmaton process. Chan and Publshed by Scedu Press 35 ISSN E-ISSN

3 Lakonshok (99) emphaszed the robust method (outler detectng methods) of estmatng beta are good for stocks whch are susceptble to outlers (mostly due to stock splts, dvdend cuttngs, and ntal publc offerngs (IPO s)). Stocks whch exhbt ths behavor are mostly small szed stocks. Martn and Smn (003) confrmed the outler resstance beta better predcts future beta than the market model beta. They also reveal that, small szed stocks betas are more vulnerable to outlers. 3. Varous models for estmatng beta Frstly, beta wll be estmated from the market model (MM) whch s bascally the statstcal model that relates the return of any gven stock to the return of the market ndex. Thus for returns of a stock n perod p we wrte: Where (Note ) E p 0 and Var p and the market ndex return respectvely. R p R, () mp p, p, and R and mp pecular to stock. The MM assumes that the estmated parameter R are the perod p returns on the stock are the estmated parameters of the market model s constant over tme but emprcal evdence confrm otherwse. One has to establsh a model to capture the varablty of the beta. Beta coeffcents estmated n successve perods have been shown to be mean revertng by Blume (97). Blume uses ths autoregressve tendency to mprove the accuracy of beta forecasts. The estmaton procedure adopted by Blume s obtaned from cross-sectonal regresson of the betas of perod p on the betas of perod p. The betas of perod p are then substtuted for perod p betas to predct perod p beta coeffcents., p a b for, p,,..., N, () where N s the number of stocks n the perod. Vascek (973) also utlzed the autoregressve tendency of beta n a successve perod to mprove the betas n a perod based on pror nformaton on betas. Vascek (973) appled the Bayesan correcton method by utlzng the cross-sectonal nformaton of the prevous perod betas:, p, p, p var var, p, p for,,..., N, (3) where s the mean of the posteror dstrbuton of beta for stock whch serves as the beta forecast., p s the varance of the market model regresson coeffcents., p s the cross-sectonal mean of betas n perod, p p, and var, p s the varance n the cross-secton of betas. Blume and the Vascek Bayessan adustment technques above control the stablty of beta. Unbased beta can be obtaned by adoptng the based-correcton procedures proposed by Dmson (979). The Dmson s technque nvolves estmatng a multple regresson of the form R 3 p 3 R, (4) m, p 3 3 The Dmson beta estmate s then gven by. Here R p s the return seres of stock and R mp s dm the return seres of the market ndex. The error term p s assumed to follow the assumptons of the classcal lnear regresson model. More lag and lead terms wll be requred for very thnly traded stocks to correct for bas. Fowler and Rorke (983) presented analytcal evdence that the Dmson and Scholes-Wllams methods do not suffcently controls thn tradng bas n beta estmaton. They proposed a varant verson of Scholes and Wllams model whch yelds betas whch are consstent and unbased. We use only three months lag and lead because beta bas has been documented as not prevalent n monthly returns data (see Cohen, Hawawn, Maer, Schwartz and Whtcomb (983)). The least square approach adopted for the market model mnmzes the sum of square of resduals wth respect to the model parameters and : p Publshed by Scedu Press 36 ISSN E-ISSN

4 mn, R R. Squarng the resduals magnfes the effect of the outlers on the estmated parameters. For nstance, expected returns are lkely to be shfted towards the outlers whereas covarance matrx wll be nflated towards outlers. To reduce the nfluence of outlers, the statstcs lterature has emphaszed on the use of teratve reweghted least-square (IRLS) method. Ths method estmates beta by teratvely mnmzng a weghted functon whch depends on standardzed return resduals: R Rmp mn W, (5), wth weght functon W and the standard devaton of the return resdual. Frst, estmate regresson parameters from least square regressons and use these parameters as ntal nput for the teraton. A weght functon s appled to the standardzed resdual. The bsquare weght functon defne as: R s B Rmp for s T wth the standardzed resdual W s T s and T, a tunng 0 for s T constant s consdered n ths case. Small values of T ntroduce more resstance to outlers, but at the expense of effcency when return resduals are Gaussan. Statstcally, the value of T s chosen n order for the method to be 95- percent effcent when errors are normal and t wll stll provde protecton aganst outlers. For the bsquare weght functon the value of T s set to Wth the ntal nput from the frst least square regressons, standardzed the return resdual and use the weght functon defne above to transform them. Estmate the parameters ' 0 ' 0 by the weghted least square: b W W R where R mp wth a vector of ones wth the 0 same sze as R mp (market ndex returns). W s an ntal standardzed weght dagonal matrx, b the estmated parameter. The new set of parameters serves as nput for the next teraton. The above procedure s repeated untl the parameter of nterest (beta) converges. Wth the above weght functon, return par observatons whose absolute standardzed resdual exceeds the tunng constant s assgned zero weght (outler). 4. The Data and Beta Coeffcent Descrptve Statstcs The data consdered for ths study was taken from the Brussels Stock Exchange (BSE). The data set was constructed at the Unversty of Antwerp n Belgum (Studecentrum voor Ondernemng en Beurs (SCOB)). The data for our study start from 83 and ends at 94, ust before the outbreak of World War I. Durng the World War I perod, the BSE was closed and ths can be regarded as a natural breakng pont of the long tme seres of the stock returns. The BSE was consdered one of the bggest markets n the world at that tme, because Belgum was one of the frst natons on the European contnent to become ndustralzed (see Van der Wee (996) and Neymarck (9)). On the ndustral output per head ladder, Belgum stood second after Brtan n 860, and thrd n 93, after the UK and the USA (see Baroch (98)). Durng ths perod, hghly developed bankng system coupled wth lberal stock market regulatons attracted a great deal of domestc and foregn captal n Belgum. In confrmaton, Van Neuwerburgh, Buelens and Cuyvers (006) document that the development of the fnancal sector, accompaned wth the stock market-based fnancng of frms, played an mportant role n the economc growth of 9th century Belgum. The database contans monthly common stocks return from 83 to 94 for every stock lsted on the BSE as of that tme. The returns have been adusted for dvdends and stocks splt. Consderng the envronment the BSE was operatng, the perod studed can be dvded nto two: thus the perod of hgh regulatory envronment ( ) and the perod of deregulaton and expanson (867-9). Nowadays, the BSE s not recognzed as one of the top stock exchange but t s ranked among the largest stock markets n the world durng the 9 th and the frst half of the 0 th centures (Annaert et al. (004)). Therefore, studyng the return patterns on ths market can be compared to the current UK market. The stocks on the BSE are well dversfed across ndustres such as transportaton, mnng and extracton, fnancals, utltes and ndustrals. The BSE as of that tme lsted Belgum and foregn owned stocks. Ths study consders only Belgum owned stocks. The data sample s dvded nto ffteen fve-year non-overlappng sub perods p for,..., 5. Stocks wth complete return data for a perod of fve years are consdered n each sub perod. The removal of stocks that dd not trade fully wthn the fve-year estmaton perods does not ntroduce survvorshp bas snce there s no sgnfcant dfference n the beta coeffcent descrptve statstcs when stocks wth at least 4 observatons are added to the sample (not reported). For the proxy of the market portfolo we consder value-weghted market ndex constructed by the studecentrum. The number of stocks n each perod and the statstcal characterstcs of the betas estmated n each perod are presented n Table. The Table dsplays the ffteen perods studed startng from January 837 to December 9. We gnore the frst fve years snce the perod can only boast of only one lsted stock. The number Publshed by Scedu Press 37 ISSN E-ISSN mp

5 Accountng and Fnance Research Vol., No. 4; 03 of stocks wth full return data n each perod ranges from to 44 as shown n column. It can be seen n Table that the number of stocks that have fve years of data does not exceed 00 before 877. The beta coeffcent of each stock n each perod s estmated by smply regressng the monthly returns of the stock on the correspondng value weghted ndex usng model above. Cross-sectonal statstcs of the betas n each perod are computed and the result s dsplayed. Equal and value weghted cross-sectonal average of betas are dsplayed n column 3 and 4 respectvely. We compute the value-weghted mean beta by consderng ndvdual stock relatve to market captalzaton at the begnnng of the perod. One can easly see from the weghted average that small sze stocks contrbute relatvely hgh beta values n almost every perod. Table. Beta Coeffcent Descrptve Statstcs for the 5 estmated perods Note: Ths table dsplays cross-sectonal descrptve statstcs of betas estmated by market model () of returns on ther market counterparts over the ffteen fve year sub perods between January 83 and December 9. The number of stocks n the varous perods ranges from to 44 for stocks wth full return data wthn a perod. The table also reports the percentage of beta sample n a perod that s less than zero. The maxmum and mnmum beta of each perod s also recorded. The last column reports the average coeffcent of determnaton R. Equally weghted (EQ) and value weghted (VW) mean betas are also dsplayed n the table. The value weghted mean s the average of the betas weghted by ther market captal at the begnnng the perod. The fgures n the last column show the average coeffcent of determnaton R n percentages, whch s a measure of explanatory power of the market model. Surprsngly, the lterature reports negatve betas on dfferent markets after World War I. For example, Altman, Jacqullat and Levasseur (974), Dmson and Marsh (983) recorded a large number of negatve betas n ther weekly returns nterval estmaton of beta on the French and the Amercan markets. Table. Average Beta and Average Coeffcent of Determnaton of the szed based sub-samples Note: In these table stocks n each perod was sorted n descendng order based on ther market captal. The frst 30% of stocks n the perods are classfed as large stocks, the next 40% as medum stocks and last 30% as small stocks. Cross-sectonal statstcs of the betas of the sub-samples were computed and ther equally weghted averages of betas and coeffcent of determnaton and Small stocks respectvely. Publshed by Scedu Press R are dsplayed. The L, M and S subscrpts on betas ndcates Large, Medum 38 ISSN E-ISSN

6 Average Betas large stocks small stocks /837-/84 /84-/846 /847-/85 /85-/856 /857-/86 /86-/866 /867-/87 /87-/877 /877-/88 /88-/886 /887-/89 /89-/896 /897-/90 /90-/906 /907-/9 Fgure. Graph of the average beta of each perod for large stocks and small stock Note: The average beta estmates for large and small stocks are plotted aganst ther correspondng perods of estmaton. The damond ponts ndcate the averages of large stock and the square ponts for small stocks. Ths depcts the extreme values of beta estmates recorded for the small stocks relatve to the benchmark beta of one. It s also mportant to note the very hgh and low betas n the 9 th century BSE compared to the post-96 market betas recorded n the lterature. The possble explanaton for these extreme values of the beta and low coeffcents of determnaton mght be lnked to the nfrequent tradng effect and the nfluence of extreme observatons (outlers) n the returns seres of the stocks. In order to capture the effect of sze (market captalzaton) on the average beta and R values, we computed the same cross-sectonal statstcs for betas n each perod. However, the stocks n each perod are sub-dvded nto three mutually exclusve sze-based sub-samples. The composton of the sub-samples s as follows: the sample of stocks n each perod s sorted n descendng order based on ther market captalzaton (sze) at the begnnng of the perod. The frst 30 percent of stocks are classfed as the large stocks portfolo, next 40 percent as medum stocks portfolo, and the last 30 percent as the small stocks portfolo. Table reports the average beta and the R values for each R recorded for large stocks are strkng because on the sub-sample n each perod. Generally, the hgh values of UK market, Dmson (979) recorded a smlar pattern wth 5 years of monthly data. Ths mght be attrbuted to the value-weghted ndex used n the computaton, snce the ndex wll have more explanatory power for large-szed stocks. Fgure depcts the cross-sectonal average beta of each perod for large stocks and small stocks. It s clear from Fgure that small stocks record comparatvely low betas n the perods before 867 and hgh betas thereafter. The result n the perods before 867 corroborates the asserton of Dmson (979), Scholes and Wllams (977) and Beer (997). They sad that f tradng frequency s hghly correlated wth the market captalzaton of the stock, betas of small stocks (nfrequently traded stocks) are lower when estmated wth the market model. On the contrary, Ibbotson et al. (997), based on NYSE data between 96 and 994, found that small stock returns due to nfrequent tradng show a hgh degree of autocorrelaton and that they are capable of recordng hgh betas. On the 9 th century BSE, we confrm ther result n the perods after 867. The only anomaly s the perod between 88 and 886, where the small stock average beta les slghtly below the average beta of large stocks. 5. Beta Stablty Here, we adopt the Blume (97) and Altman et al. (974) correlaton method of nvestgatng the stablty of beta estmates. Blume (975) shows that the beta coeffcent between two successve perods s statonary f (Note ) E E Var Var corr, where are the betas n perod p. Betas p p, p, p p p n perod p are used to rank stocks exstng n perods p and p n ascendng order. In perod p equally weghted portfolos of s,,3,... stocks are formed as follows: the frst portfolo conssts of stocks wth s smallest beta estmates. The next portfolo conssts of stocks wth the next smallest beta estmates. Ths process of portfolo formaton s repeated untl the number of stocks left s less than s. For each s, the betas of all portfolos n perod p are also computed. We compute the correlaton and Spearman rank order correlaton of the betas between each two adacent perods. Table 3 reports the weghted average correlaton across all the sub-perods p Publshed by Scedu Press 39 ISSN E-ISSN

7 Accountng and Fnance Research Vol., No. 4; 03 studed. The weghted average correlaton takes nto account the number of stocks or portfolos n each adacent ten-year perod. The weghted average correlaton across the successve perods ranges from 0.54 for ndvdual stocks to 0.95 for portfolos of 0 stocks. These values ndcate that the beta of ndvdual stocks have, on average, less nformaton about ther future beta than the portfolo beta. Blume (97) found a smlar result on the US market. Due to the lmted number of stocks, we were not able form up to 50-szed portfolos on the BSE. Blume fnds 0.6(0.67) and 0.9(0.93) mean correlaton (rank correlaton) for - and 0-szed portfolos usng 84-month estmaton perods respectvely. Furthermore, based on 5-week estmaton perods, Levy ( 97) records 0.44 and 0.8 mean correlaton for - and 0-szed portfolos on the same exchange. On the UK market between 955 and 979, Dmson and Marsh (983) used value-weghted market ndex, monthly returns nterval measurement and sxty-month estmaton perods to obtan an average correlaton of 0.56 and 0.9 for - or 0-szed portfolos respectvely. The correlatons n Table 3 show that the betas from the 9th century BSE are stable compared to the betas n post-96 US and UK markets. Table 3. Weghted Average of Correlaton and Spearman Rank Order Correlaton across successve Perods Note: Ths table shows the weghted average correlaton and Spearman s Rank correlaton of betas of ndvdual stock and portfolos n successve perods across the ffteen perods studed. For a stock to be ncluded n ths analyss t has data for completely two consecutve perods. Betas n a perod (estmaton perod) are used to rank the betas n that perod and the next adacent perod (predcton perod) n ascendng order. Portfolos are formed wth ther consttuent as follows: The frst portfolo s the frst s stocks for s (,,4,7,0,0). The second portfolo contans the followng s stocks and so on untl avalable stocks s less than s. Assumng equal amount s nvested n each stock then the portfolo beta wll ust be the mean of the betas of stocks ncluded n the portfolo. We computed the weghted average correlatons weghted by the number of portfolos n each ten year perod. 6. Blume and Vascek stablty adustment technques Indvdual stock betas estmated from the MM are noted as unstable n the prevous secton (also n Blume (97), Collns, Ledolter and Rayburn (987), Faff and John (99), Gregory-Allen et al. (994), Esenbess, Kauermann and Semmler (007)). There s a tendency for a hgh beta estmate to overstate ts true value and vce versa. Therefore, we use the Blume (97) autoregressve adustment model to mprove the stablty of beta estmates for both ndvdual stocks and portfolos. Table 4. Measurement of Regresson Tendency of Estmated Beta Coffcents for Indvdual Stocks Publshed by Scedu Press 40 ISSN E-ISSN

8 Table 4 presents regresson tendences mpled between adacent perods, where a and b are the constant term and slope coeffcents, respectvely. The values of the coeffcents n the perods between 837 and 867 are strkng. It s not consstent wth the Blume asserton that all the coeffcents le between zero and one. The last two Columns show the t-statstcs of the test of a hypothess of the slope coeffcent equal to zero or one. The t-statstcs of the slope coeffcents show that the null hypothess of the slope coeffcent equal to zero s reected n the adacent perods after 87. In addton, the null hypothess of the slope coeffcent equal to one s reected n all the adacent perods (except the frst two perods). The R values also show that the betas n the perods after 87 have more explanatons for ther pror betas than those before 87. As can be seen n Table 4, the coeffcents change over tme, but there are extreme coeffcents outsde the nterval between zero and one n the frst two perods. The extreme values may be attrbuted to the number of stocks n a perod as we record less than 50 stocks n our frst two fve-year perods. A result not reported shows that ncreasng the length of the estmaton perod (such as seven years n Blume (97)) mproves the R values and the t-statstcs but at the expense of losng more stocks, snce fewer stocks have complete returns' data for longer perods. Wth the regresson tendences, suppose we want to forecast the beta for any stock or portfolo n the perod We compute ts beta n The forecast of the beta s obtaned by substtutng t for β p- n equaton () wth the coeffcents n the frst row of Table.4. β p s then computed from the equaton and used as the forecast. The adustment process s repeated for stocks n the subsequent 3 adacent perods usng ther respectve coeffcents. We also ntroduce the Vascek adustment model (3) to adust betas n successve adacent perods. We test the predctve performance of the varous adusted betas by usng the root mean square error (Note 3) (RMSE) crteron. The RMSE tests the performance of the autoregressve methods based on varaton and unbasedness of ther beta forecast. The adustment method s repeated for all adacent perods on equally weghted portfolos of sze, 4, 7, 0 and 0. In order to compare the predctve performance of the MM betas and the autoregressve-adusted betas, we compute the RMSE of betas estmated wth MM n adacent perods. Table 5 dsplays the average RMSE of the adusted betas and MM betas across the 5 perods studed. It s clear from the Table that the average RMSE of the adusted betas s lower than that of the market model betas. Table 5 also shows that the predctve performance mproves as the number of stocks n a portfolo ncreases for both adusted and the MM betas. For ndvdual stocks, the Bayesan adustment technque proposed by Vascek s superor to Blume s adustment as reflected n the small average RMSE. Blume (97) and Klemkosky and Martn (975) recorded smlar patterns of the predctve performance on the NYSE market. Ther adusted betas mean square errors were smaller than ther MM betas. For portfolos of sze 7 or more, one cannot see much dfference between the Blume adustment method and the Bayesan approach. Table 5. Predctve Performance of Blume and Vascek (Bayesan) procedures of estmatng Beta Note: Average RMSE and MAE across the ffteen perods studed were used to compare the predctve performance of the varous adusted betas and the market model (MM) betas n successve perods. The average RMSE and MAE across the successve adacent perods for the varous equally weghted portfolo formatons are dsplayed. The concluson s that on the 9 th century BSE, the predctve accuracy of betas estmated by the MM can be mproved by adustng betas usng ether the Blume or Bayesan adustment methods and a portfolo wth a szeable number of stocks. The relablty of the concluson above can be confrmed by performng an addtonal test on the root mean squared error values. The possble method s to test whether the dfferences n the values of the RMSE s are statstcally sgnfcant. Harvey et al. (997) presented a modfed Debold and Marano test statstc that wll be used for ths BL purpose. Therefore, suppose we want to compare the forecasts of Blume (BL) and Vascek (VA) models. and VA are the forecastng errors from the Blume and Vascek models, respectvely. In our case, we consder the root mean square error functon, BL f root mean square error of the Blume adusted betas. The test s based on BL VA the loss dfferental functon d f f for,..., H. The null hypothess of expected equal predctve Publshed by Scedu Press 4 ISSN E-ISSN

9 performance s H 0 : Ed 0 and the alternatve hypothess of the Blume model predctng worse than the Vascek model s H : 0. H where d H d a E d The Modfed Debold-Marano (MDM) statstc s: H hh h h d MDM. ~ t, H 0,, H 0 cov d, d t a student s t dstrbuton,, h s the horzon of forecast and, (0,) wth H degrees of freedom and v s the sgnfcant level usually set at 5%. The test compares the Debold-Marano test statstcs to crtcal values from the student s t dstrbuton. We reect the null hypothess of equal predctve accuracy when the test statstc s greater than the crtcal value at level. In order to apply ths test, betas estmated wth the Market, Blume and Vascek models n all perods are pooled together to form three seres of length H. Then we perform the test on the three seres.table 6 reports the modfed Debold-Marano test statstcs between the varous models under study. Betas estmated from the varous models are consdered across the entre perod studed. Table 6. Modfed Debold-Marano test statstcs (p-value n parentheses). H 0 :E(d )=0 and H a :E(d )>0 H Once agan the perod studed s dvded nto two sub-perods based on the envronment n whch the BSE operated, a perod of strct regulaton and a perod of deregulaton and expanson. The values n the frst row of the Table reveal that we can confdently reect the null hypothess of one-step ahead equal predctve accuracy of the Blume and MM betas. For nstance, n the overall perod of our sample the null hypothess can be reected at the 5 percent level. Durng the strctly regulated perod, we fnd that Blume-adusted beta sgnfcantly outperforms the market model beta. Between the Vascek adusted betas and the MM betas, we can reect the null hypothess of equal predctve accuracy for the deregulated and expanson perod. The sgnfcance level of the reecton becomes weak n the strctly regulated perod. The equalty n the predctve performance of the Vascek and the MM betas s strongly reected n the entre perod. The values from the bottom row of Table 6 show that there s no sgnfcant dfference between the Vascek betas and the Blume betas n terms of ther one-step ahead forecast. Therefore, we cannot reect the null hypothess of equal predctve accuracy between the two models. 7. Beta Bas Consderng the perod of the study and the tradng frequency of the market, we mght expect that some stocks may not trade every month for economc reasons or because of regulatory condtons. These stocks may systematcally lead or lag the market movement, producng based betas when beta s estmated wth the MM. In order to expose the presence of possble lead or lag effects, we test the sgnfcance of the coeffcent of the returns on the lagged or lead market ndex. The Dmson bas adustment equaton wth maxmum lag or lead of three months s consdered, that s 3,...,3. We use only three months lag and lead because beta bas has been documented as not prevalent n monthly returns data (see Cohen, Hawawn, Maer, Schwartz and Whtcomb (983)). For each stock, the estmates of the parameters ndcate the lagged, matched and lead beta coeffcents. We test the hypothess H 0 : 0 aganst the alternatve H : 0 for each stock. Table 7 reports the cross-sectonal average of the lag and lead betas n each perod. The numbers n parentheses are the percentage of stocks that reect the null hypothess. Publshed by Scedu Press 4 ISSN E-ISSN

10 Accountng and Fnance Research Vol., No. 4; 03 0 are not sgnfcantly dfferent from zero for the maorty of the stocks. Ths shows that the explanatory power of the model for 0 s approxmately zero for Evdence from ths Table ndcates that beta coeffcents for most of the stocks. Unsurprsngly, there are some stocks wth lead and lag coeffcents that are statstcally sgnfcant, but ther numbers does not exceed the coeffcents correspondng to the match. Ths ndcates that there s no severe tmng problem n the 9th century data. As most of the lead and lagged coeffcents are sgnfcantly equal to zero, we can nterpret ths as evdence of the market model (MM) producng statstcally relable beta estmates n relaton to the other models, whch ncorporate the lagged and lead market ndexes. These results can be compared to the results from the post-world War I markets. For example, Hawawn and Mchel (979) found a smlar pattern of results on the Belgum stock exchange by usng weekly nterval returns data between 963 and 976. The result also follows Cohen et al. (983) hypothess that there s a strong relatonshp between beta estmates and the length of the nterval over whch returns are measured. They establshed that beta bas mostly shows up n the short length nterval (daly) of returns, and the bas dsappears when the dfference of the nterval s lengthened (monthly). Smlarly, on the New Zealand market, Bartholdy and Rdng (994) used monthly data to establsh that betas estmated from MM are less based. On the contrary, Ibbotson et al. (997) reports that lagged coeffcents should be consdered when estmatng beta. 8. Impact of Outlyng observatons on Beta The extreme (maxmum/mnmum) beta estmates recorded by some stocks n Table for the fve-year perods studed mght be due to the nfluence of outlers or unexpected movement by the stock or the market returns. The lterature shows that outlers have a tendency to reduce or ncrease the magntude of the beta when t s estmated wth the MM (Chatteree and Jacques (994)). In such cases, reducng the mpact of outlers n the estmaton of the beta can sgnfcantly change the value of beta. We apply the IRLS (outler resstant), whch mnmzes a weghted sum of squares of resduals. The weghts gven to each return par observaton depends on the dstance between the observaton and the ftted lne (Martn and Smn (003)). Table 8 reports how the presence of outlers affects the beta value. The number of stocks n each fve-year perod s grouped nto two. As explaned n prevous secton, stocks wth dentfed outlers less than or equal to 4 are grouped nto Category A and those wth dentfed outlers greater than 4 are grouped n category B. We compute the average beta of each category. Table 7. Dmson Aggregate Coeffcent (AC) beta Adustment Publshed by Scedu Press 43 ISSN E-ISSN

11 In each perod, we compare the cross-sectonal average betas of the MM and IRLS for each category. For example, n the frst perod out of the stocks, three fall n Category A wth an average market model (MM) beta estmate of.9 and IRLS beta of In Category A, the dfference between the average MM beta and the average IRLS beta s Lookng across perods, except for perods to 3, the rest of the perods have more stocks n Category A than Category B. In each perod, the dfference between the MM betas and the IRLS betas for Category B s greater than the dfference n Category A (last column). Ths mples that the more the outler observatons n the return seres, the hgher the market model overestmates beta. The MM beta s always greater than the IRLS beta n Category A (across perods n Fgure ). It confrms the result by Chatteree and Jacques (994) that the weghted least squares estmaton reduces the MM betas by a certan percentage. We apply the modfed Debold-Marano test to compare one-step ahead predctve accuracy of the MM and IRLS estmated betas. Table 8. The modfed Debold Marano test statstcs proposed by Harvey et al. (997) are employed to test the null hypothess of equal predctve accuracy aganst the alternatve of IRLS betas forecastng better than the MM betas. A pooled sample of betas wthn the perod studed s consdered. The modfed Debold- Marano's test statstc between the two models s.37 wth p-value of 0.09 for one-step forecasts n the perod of our study (shown n Table 9). Ths shows that we cannot reect the null hypothess of equal predctve accuracy at the 5 percent sgnfcance level n the overall perod. The null hypothess can be reected only at the 0 percent level. From the deregulaton and expanson perod, the null hypothess of equal predctve accuracy s not reected. From Table 9, we conclude Publshed by Scedu Press 44 ISSN E-ISSN

12 that on the 9 th century BSE, the IRLS method can help to curb the nfluence of outlers on estmated betas, but t does not sgnfcantly outperform the standard MM n terms of ther ablty to predct one-step ahead n the perod of deregulaton and expanson. Fgure. Plot of average MM/IRLS betas for stocks wth outler observatons less than 4 9. Conclusons Ths study evaluates the relatve performance of dfferent methods of estmatng beta based on ther ablty to predct subsequent beta on the 9 th century BSE. The analyss of the dfferent beta technques reveals that beta estmated wth the market model s not stable. Specfcally, the study reveals that for ndvdual stocks, the market model beta s weak n ts ablty to predct the future beta. The predctablty can be mproved by groupng 0 or more stocks to form a portfolo or adustng betas wth the Vascek and Blume autoregressve technques. The study also shows no sgnfcant dfference between the Blume and Vascek adusted betas n terms of ther predctve accuracy. Applyng the Dmson method, correctng nonsynchronous tradng effect reveals that returns of few stocks have a sgnfcant relatonshp wth the lead and lag market returns. There s no sgnfcant dfference n the predctve accuracy of the betas estmated wth the IRLS method and the market model n the deregulaton and expanson perod. In the next chapter, we study the ablty of beta to explan returns n the cross-secton of stocks, whch s the prmary mplcaton of the CAPM. References Altman, E. I., Jacqulla, B. & Levasseur, M. (974). Comparatve Analyss of Rsk measures: France and the Unted States, Journal of Fnance, Annaert J., Buelens F., De Ceuster J. K. (004). Equty return estmaton for 9th century Belgum. Workng Paper, Unversty of Antwerp Bartholdy J., Rdng A. (994). Thn Tradng and the Estmaton of Betas: The effcacy of Alternatve Technques. The Journal of Fnancal Research, Beer F. M. (997). Estmaton of Rsk on the Brussels Stock Exchange: Methodologcal ssues and emprcal results, Global Fnance Journal, Blume, M. E. (975). Betas and ther Regresson Tendences. Journal of Fnance, Blume, M. E. (97). On the Assessment of Rsk, The Journal of Fnance, Chan L.K.C. & Lakonshok J. (99). Robust Measurement of Beta rsk. Journal of Fnancal and Quanttatve Analyss, Chatteree S. & Jacques W. E. (994). An Outler-Resstant Approach to Rsk Estmaton, Fnancal Analyst Journal, Cohen, Kalman J., Gabrel A. Hawawn, Steven F. Maer, Robert A. Schwartz, & Davd K. Whtcomb. (983). Estmatng and adustng for the ntervallng-effect bas n beta, Management Scence 9, Collns D. W., Ledolter J., & Rayburn J. (987). Some Further Evdence on the Stochastc Propertes of Systematc Rsk, Journal of Busness, Publshed by Scedu Press 45 ISSN E-ISSN

13 Dmson, E. (979). Rsk measurement when shares are subected to nfrequent tradng, Journal of Fnancal Economcs, Dmson, E. and Marsh, P. R., (983), The Stablty of UK Rsk Measures the problem of thn tradng, Journal of Fnance, Esenbess M., Kauermann G. & Semmler W. (007). Estmatng Beta-Coeffcent of German Stock Data: A Non-parametrc Approach, The European Journal of Fnance, Faff R. W., John H.H. (99). Tme Statonarty of Systematc Rsk: Some Australan Evdence, Journal of Busness Fnance and Accountng,9() Fabozz, J. F. & Francs C. J. (978). Beta Random Coeffcent, Journal of Fnancal and Quanttatve Analyss, Fowler, D.J. & Rorke, C.H. (983). Rsk Measurement when shares are Subected to nfrequent tradng, Journal of Fnancal Economcs, Gregory-Allen R, Impson M. C. & Karafath I. (994). An emprcal nvestgaton of Beta Stablty: Portfolo verses ndvdual Securtes, Journal of Busness and Accountng, Harvey D., Leybourne S. & Newbold P. (997). Testng the Equalty of predcton mean squared errors, Internatonal Journal of Forecastng 3, Hawawn, G. A. & Mchel, P. A. (979). An assessment of rsk n the thnner markets: the Belgan case, Journal of Economcs and Busness Hawawn, G. A, Maer S. F., Cohen, K. J. & Whtcomb, D. K. (983). Estmaton and Adustng for the ntervallng-effect bas n Beta, Management scence, Ibbotson, R. G., Kaplan, D. P & Peterson, J. D. (997). Estmates of Small stocks Betas are much too small, The ournal of portfolo management, Klemkosky, C. R. & Martn, J. D. (975). The Adustment of Beta Forecasts, Journal of Fnance, Levy, R. A. (97). On the short-term statonarty of beta coeffcents, Fnancal Analysts Journal, Luoma M., Martkanen T., Perttunen J. & Pynnonen S. (994). Dfferent Beta Estmaton Technques n Infrequently traded and Ineffcent Stock Markets, Internatonal Journal of Management Scence (Omega), Martn R.D. & Smn T.T. (003). Outler-Resstant Estmates of Betas, AIMR, Neymarck, A. (9). Les fnances contemporanes. L'epargne françase et les valeurs moblères (Felx Alcan, Pars). Sharpe, W. F. (964). Captal Asset Prces: A theory of Market Equlbrum Under condtons of Rsk, Journal of Fnance. Scholes, M. & Wllams, J. (977). Estmatng Betas from Non-synchronous data, Journal of Fnancal Economcs, Sunder S. (980). Statonarty of Market Rsk: Random Coeffcent Test for Indvdual Stocks, The Journal of Fnance, Van der Wee, Herman. (996). The ndustral revoluton n belgum, n M. Tech, and Roy Porter, eds.: The ndustral revoluton n natonal context. Europe and the USA (Cambrdge Unversty Press, Cambrdge). Van Neuwerburgh, Stn, Frans Buelens, & Ludo Cuyvers. (006). Stock market development and economc growth n belgum, Exploratons n Economc Hstory 43, Vascek, O. A. (973). A note on usng the cross-sectonal nformaton n Bayesan estmaton of securty betas, Amercan Economc Revew, Notes Note. Ex s the expected value of x, Var( x ) s the varance of x Note. corr( x, y ) s the correlaton between x and y. for t t Note 3. The root mean square error was calculated by N,..., N Publshed by Scedu Press 46 ISSN E-ISSN