New univariate and multivariate tests of the S&P 500 comovement effect

Transcription

1 New univariate and multivariate tests of the S&P 500 comovement effect Yixin Liao Jerry Coakley and Neil Kellard Essex Finance Centre and Essex Business School Draft not for quotation! Abstract This paper employs new univariate and multivariate tests to examine the S&P 500 increase in betas or comovement effect when stocks are added to the index. It provides closed form solutions for a theoretical 3-factor model in which sentiment affects not just market returns but all factor returns that factor loadings from univariate regressions are also relevant. The model is tested using three- and fourfactor regressions for the sample of stocks added to the S&P 500 over the period. The changes in beta from the monthly data 3- and 4-factor models exceed those from univariate regressions. Finally the tiny impact of momentum on comovement in the 4- relative to the 3-factor model indicates that momentum cannot explain these increases. The conclusion is that comovement remains a puzzle! Key words: beta; index inclusion; univariate regressions; 3-factor model; 4-factor model We thank Dr. George Dotsis at the Department of Economics of the National and Kapodestrian University of Athens for providing data.

2 . Introduction In a path-breaking study Vijh (994) found that the daily and weekly betas S&P 500 additions averaged 0.2 and 0.3 respectively using the CRSP value-weighted returns for market returns over the period. He attributed most of this to the price pressure or excess volatility caused by index trading strategies. 2 The Barberies Shleifer and Wurgler [hereafter BSW] (2005) seminal study framed the debate in terms of changes in return comovement of added and deleted stocks while the prior debate had focused mainly on price change. 3 They find increasing comovement or betas increases after index addition events and decreasing comovement following deletions. Since S&P additions convey no new information about fundamentals these comovement findings are a puzzle. Virtually all of the extant comovement studies either rely on the BSW (2005) stylised model or no model at all as in the case of three-factor tests of comovement. This paper motivates and develops a novel stylised three-factor model and provides empirical evidence that the puzzle persists in the 3-factor model framework. BSW (2005) were the first to propose a non-fundamental explanation for excess comovement relative to that justified by stock fundamentals. Building on the Barberis and Shleifer (2003) concept of style investment they propose the category rationale for excess comovement. The basic idea is that some investors treating the S&P 500 as a category are noise traders with correlated sentiment. Index changes then induce correlated demand shocks for added stocks that impact on their prices and returns. This category or index common factor in returns is unrelated to stock s cash Vijh (994) described S&P stock betas as being overstated and did not use the comovement concept. 2 See Wurgler (20) for an interesting discussion of increasing index linked investment. 3 See for example Harris and Gurel (986) Shleifer (986) and Wurgler and Zhuravskaya (2002).

3 flows or discount rates and is what explains excess comovement for S&P inclusions and deletions. 4 Third BSW (2005) extended the methodology for examining comovement from univariate CAPM to bivariate regressions with both the S&P 500 and non-s&p 500 (rest of the market) indexes. Several studies of other country indexes have supported the BSW findings. Greenwood and Sosner (2007) found similar findings for the Nikkei 225 index. 5 Claessens and Yafeh (20) in a comprehensive study of 40 stock markets find support for the BSW univariate findings in the vast majority of their sample indices. Since most researchers on comovement employ the BSW (2005) univariate and bivariate regression approach they implicitly rely on the BSW stylised theoretical model. One notable exception is Chen Singal and Whitelaw (205) who develop a new univariate model that departs from the BSW model. They criticise the BSW bivariate methodology on the basis that the collinearity between the S&P and non- S&P 500 indices leads to parameter instability. Their model shows that the bivariate provides little information about comovement. Instead it shows that univariate regressions of the added stock on the non-s&p and S&P indices provide the relevant information on comovement. They compare the beta change of event-stocks with that of momentum-matched-stocks and do not find significant differences after Dimson adjustment for nonsynchronous trading. They interpret this as evidence of momentum effects being able to explain most of the changes in comovement. The principal contribution of this paper is the development of a stylised theoretical model that builds upon both the BSW (2005) and on the Chen et al. (205) 4 BSW actually propose three sentiment- or friction-based views of comovement where the category view is the one that contrasts most sharply with the fundamental view. Note that Greenwood (2008) and Claessens and Yafeh (20) combine the category and habitat views into a demand-driven view of comovement. 5 See also Greenwood (2008) who studied changes in the Nikkei 225 index weights. 2

4 models to provide an understanding of both the univariate regressions and the threefactor model. The latter is motivated by the fact that Kasch and Sarkar (204) criticise the asset pricing basis of the BSW methodology and instead argue for employing recent asset pricing regressions that condition returns on other common factors. We provide closed form solutions estimators of excess comovement in the three-factor model as well as in the univariate regression case. Importantly the model implies that sentiment can impact on changes in the loadings on the SML HML (and momentum) for added stocks precisely as it does in the case of market (S&P 500) factor. Whereas Kasch and Sarkar (204) interpret significant loadings on the SMB and HML factors as evidence against excess comovement (without a theoretical model) our model has the opposite implication. The paper s second contribution is that it provides novel supportive evidence on comovement when it re-examines beta changes after the addition and deletion events using data from 988 up to 204. The univariate and bivariate comovement results are broadly consistent with BSW (2005). The beta change results at the weekly frequency employing the Fama-French three-factor are statistically insignificant similar to those found by Kasch and Sarkar (204). However the monthly results from these models are novel and revealing. First the magnitude of the monthly comovement increases for added stocks from the Fama and French 3-factor and Carhart 4-factor models (0.23 and 0.20 respectively) exceeds the 0.7 increase from the univariate regression. 6 Note that these increase condition on the effects of size book-to-market and for the 4-factor model momentum also. Secondly the fourfactor model loading on momentum is significantly negative at Moreover the very small decrease (0.03) in the beta change in this relative to the 3-factor model 6 Surprisingly Kasch and Sarkar (204) do not consider monthly beta changes. 3

5 provides scant support for the Chen et al. (204) assertion that momentum can account for a large part of the apparent comovement puzzle. The remainder of the paper is organized as follows. Section two reviews the existing literature. In section three we outline a new theoretical model to help to understand the results from univariate and the 3-factor model regressions. Section four describes our methodology. Section five data and analyses the empirical results. A final section concludes. 2. Fundamentals versus other factors Following the BSW (2005) study there appeared to be a consensus in the literature on comovement. Behavioural finance proponents suggest that irrational investors frictions style investing and investor habitats are able temporarily to delink prices from n fundamentals and that these can explain comovement following index additions and deletions. However debate around the reasons for beta increases for S&P additions has recently resurfaced. Proponents of classical theory claims that betas increase because of changes to actual or expected fundamentals prior to index inclusions. 2. Competing models The modern theory of comovement has centred on employing the CAPM as the regression framework for testing for beta changes. The basic idea is that investors focus on style category (also denoted by basket in the case of index stocks) or index to simplify investment decisions. Often these concepts are not formally modelled but Barberis and Shleifer (2003) construct a style investment model. In this model noise traders can influence security prices due to limits to arbitrage. While Vijh (994) 4

6 assume the effect of the price pressure is on individual stocks Barberis and Shleifer (2003) consider the question based on style investing. The implication is that assets in the same group commove due to the same factors including price pressure where relevant. They assume that one group of investors makes decisions based on a momentum strategy as suggested by Jegadeesh and Titman (200) and the other makes decision based on fundamentals. The Barberis and Shleifer (2003) model has several interesting implications. First the covariance between a stock and a style increases after that asset is added to that style. The implication is that a stock should move more closely with the S&P 500 after it is added to the index which is similar to the prediction of the Vijh (994) classical model. Second stocks that are in the same style move more closely than their fundamentals do. Third stocks that are in different styles move less closely than their fundamentals do. The latter two predictions show that changes in comovement are not caused by fundamentals alone. BSW (2005) build on Barberis and Shleifer (2003) in constructing a model excess comovement based on notions of category and habitat and univariate and bivariate regressions. Both are based on the CAPM as the bivariate simply employs a broader (than just the S&P 500) definition of the market index. They also recognise that market frictions such as slow information diffusion may play a role. The BSW model assumes that risky assets are divided into different categories and noise traders invest funds in or withdraw funds from the categories depending on their sentiment. It also assumes that returns on assets depend on market-wide group-specific and idiosyncratic fundamental shocks and on noise trader sentiment. This model has two implications. First in their univariate model the beta coefficient of the group to which the asset is added increases after the asset is reclassified. Second in their 5

7 bivariate model the beta coefficient of the group to which the asset is added increases while that of the group the asset leaves decreases after the asset is reclassified and the absolute magnitudes of increases and decreases are equal. These two implications are similar to what Wurgler (20) refers to as the index inclusion and detachment effects of index-linked investment. Chen et al. (205) build a model that shares some of the BSW (2005) assumptions but also introduces new assumptions. They seek to show that the BSW bivariate model does not contain any information about excess comovement and that its regression coefficients are very sensitive to time variation in other characteristsics of the return process. They employ different assumptions and specifications. They do not specify what is included by common fundamental shocks. Further they assume that non-fundamental shocks are group-specific and correlations of non-fundamental shock across groups are zero. By contrast BSW (2005) assume that non-fundamental shocks such as sentiment are correlated across groups. The Chen et al. (205) assumption is not plausible because noise traders move funds from a group to the other one (see Barberis and Shleifer. 2003). The implication is that the demand of one group increases when that of the other group decreases. Chen et al. (205) also other unrealistic assumptions such as loadings of unity on the non-fundamental group shock and fundamental shocks. Finally they propose that univariate regressions of stock returns on S&P and non-s&p returns are more informative than the BSW bivariate regressions about comovement. 2.2 Contrasting results Vijh (994) regresses stock returns on the CRSP value-weighted index return to estimate the beta for pre- and post-event windows and then averages the difference 6

8 between pre and post-event betas calculated using data from 975 to 989. The t- statistic of the average difference is estimated to check its significance. Daily and weekly data are used for the test and a significant increase in beta is found at both frequencies for the whole sample. This finding is consistent with the prediction of Vijh s (994) model. However the beta changes vary over the sample and the magnitude of change in beta and its significance for daily data is higher than for weekly data. This indicates as Vijh (994) suggested that the effect of the price pressure disappears in the long term. BSW (2005) estimate changes in the S&P beta using univariate regressions and changes in the S&P and non-s&p beta employing bivariate regressions. Their empirical tests use data from 976 to 2000 at the daily weekly (455 additions n both cases) and monthly (on 324 additions) frequencies. 7 Their univariate regression results are similar to those of Vijh (994) in showing small to moderate beta increases for added stocks. However they also produced three novel results. First their bivariate results offered the strongest support for excess comovement. For instance over their full sample period they found that the mean daily S&P betas of added stocks increased by 0.30 while the corresponding non-s&p betas fell by Second they establish that their monthly subsample produces stronger results with S&P betas and non-s&p beta increases of 0.39 and -0.4 respectively. This they attribute to the S&P 500 index become more and more popular within the investor community in the later years of their sample. Finally using Dimson forward and lagged betas they establish that slow information 7 The addition numbers are smaller at the monthly frequency as this entails a three-year postimplementation window from

9 diffusion accounts for around one third of their univariate beta increases and up to two thirds of their larger bivariate increases The BSW (2005) results have been replicated in many studies and for a range of stock indices across the world. The most wide ranging study is that of Claessens and Yafeh (202). They employ data on forty developed and emerging markets over a 0 year sample span and find beta increases for stock added to a major index in most markets. Their test results are very similar to BSW. They support the BSW categoryhabitat views and they also find that information-related factors play a role. Some studies have focused on other major stock market indices. For instance Greenwood and Sosner (2007) find support for changes in the Nikkei 225 index. Greenwood (2008) finds a strong positive relation between overweighting and the comovement of a stock with other stocks in the Nikkei 225 index. 8 Changes in beta coefficients have been found not only in relation to indexing style. For example Green and Hwang (2008) find that stocks have higher beta coefficients with low-priced stocks and lower coefficients with high-priced stocks after splits. Boyer (20) finds that value index coefficients increase while growth index coefficients decrease after stocks are reclassified from a growth to a value index in the USA. The empirical results discussed above provide evidence that changes in the coefficients cannot be explained fully by fundamentals. As changes in constituents of the S&P 500 do not represent changes in fundamentals Barberis et al. (2005) claims that the changes in beta after changes in the S&P 500 constitutent stocks is a reflection of the fact that sentiment-based category or habit theories may explain the 8 Note that is a cross-sectional study and thus a cleaner test of comovement than the time series studies. 8

10 changes in comovement. Further Boyer (20) claims that this indicates changes in coefficients are not caused only by fundamentals that value index coefficients change after stocks are reclassified from growth index to value index because the reclassification from growth index to value index is not based on fundamentals. More recent supportive evidence is provided by Claessens and Yafeh (202) and Kumar et al. (203). Nonetheless not only is their evidence supporting behavioural finance theories but researchers also find evidences that support classical theories. For example Kasch and Sarkar 9 (204) claim empirical results discussed above is explained by changes in loadings on common factors in returns including size bookto-market ratio and momentum. The CAPM Fama and French (993) 3-factor model and Carhart (994) 4-factor model are used for detecting the reasons for the changes in comovement. Kasch and Sarkar (204) find the average change in the beta is insignificant while loading on size book-to-market ratio and momentum change after addition events of the S&P 500 through the Fama-French-3-factor-model and the Carhart-4-factor-model when the CRSP-value weighted index return is used as a proxy of market return. However the daily average change in the market beta is significant and positive in the 3-factor and 4-factor models with the S&P 500 index return which supports the view that changes in comovement is explained by not only fundamentals. 3. A stylised three-factor model A stylised model is required to interpret and understand the implications of the empirical results reported in the extant literature for excess comovement. Chen et al. (205) build a model to investigate the features of the univariate and bivariate 9 They drop the first two months after the month of inclusion announcement while post-inclusion time interval in Barberis et al. (2005) is from the first month after the month of inclusion announcement. 9

11 regressions in BSW (2005). They also develop new univariate regressions that they claim can shed new light on comovement. Kasch and Sarkar (204) employ three- and four-factor models instead of the CAPM to produce new empirical results on comovement without developing a model. Our model is similar to that of Chen et al. (205) and shares some of their predictions. However it also yields novel predictions and has implications for the three-factor model that are not considered by Chen et al. (205). 3. Setup and assumptions As Barberis and Shleifer (2003) suggested small and large cap value and growth and momentum portfolios are all investing styles that can be influenced by nonfundamental factors. As a result it seems entirely plausible to assume that returns on the size value and momentum effects are influenced by fundamental and nonfundamental factors. Let be the overall return on an S&P 500 index stock and let and denote the part of returns due to being added to the S&P 500 index the small-minus-big (SMB) the high-minus-low (HML) portfolio and momentum (MOM) portfolios respectively. = = + + = + + = + + = + + 0

12 where denotes the fundamental common factors denotes non-fundamental group-specific factors and denotes idiosyncratic fundamental factors. It is assumed that the returns on stocks that added to the S&P 500 index are affected by the fundamental common factors non-index index SMB HML and MOM groupspecific non-fundamental factors. We further assume that the SMB HML and MOM portfolios are influenced not only by fundamentals but also group specific nonfundamentals just like the market portfolio. Assumptions about the SMB and HML portfolios are not taken into account by Chen et al. (205) because they focus on univariate and bivariate models only. By contrast we focus not only on the univariate model but also on the 3-factor model. Moreover our assumptions about non-fundamental factors depart from those in BSW (2005). They assume that the sensitivity to the group to which a stock is added is unity but that it decreases to zero when the stock leaves that group. This is invalid for the SMB HML and MOM portfolios because these sensitivities vary in a range when the size or value ratio or momentum of a company changes. For example the sensitivity to the SMB portfolio can be negative for a large cap stock. We assume that group-specific non-fundamental factors are uncorrelated that fundamental factors are not correlated with non-fundamental factors and that idiosyncratic fundamental factors are not correlated with common fundamental factors. These are expressed as follows: =0 = =0 =0

13 These assumptions are shared with Chen et al. (205). BSW (2005) also assume that non-fundamental factors uncorrelated. Behavioural finance theories imply that excess comovement is driven by changes in the sensitivity to group-specific non-fundamental shocks. Specifically the sensitivity to non-index group-specific non-fundamental shocks goes to zero while that to index group-specific non-fundamental factors increases from zero after a stock is added to the S&P 500 index. Moreover sensitivities to SMB HML and MOM group-specific non-fundamental factors decrease after a stock is added to the index because empirical results conclude that such companies have larger cap and lower value ratios and exhibit prior momentum. We use underbars and overbars to denote stock values before and after they are added to the index. We further assume sensitivities to fundamentals do not change when the stock is added to the index but may change during sub-periods. This assumption is shared with Chen et al. (205). >0 =0 =0 >0 > 0 > 0 > Measures of excess comovement Excess comovement is defined as the fraction of the group return variance that can be explained by the non-fundamental shocks. It can be expressed as 2

14 "% #$ "% " % #&$' "#($) % $ " % &$' " ($) % % and " #$*$ "%. $*$ calculated for windows prior to and following index additions. The focus is on the CAPM and for reasons of algebraic tractability on the three-factor model. We also establish univariate regressions of stock return on the small-minus-big high-minus-low and momentum return are informative. We do not discuss the bivariate model or the univariate regression of stock returns on the group it exits from because they are discussed in Chen et al. (205). Consider the four regressions: = = = = = The probability limits of these regressions are: = 2 = 2+ 3 = = = =

15 = = = =+: ; = =+ : ; = =+ : ; 0 Following Chen et al (205 we assume also that sensitivities to the common factor the variances of the non-fundamental factors variances of the fundamental factors and correlations between returns on groups are constant over time. = =? =?? =@A@h@C@@ 3 = 3 3 >0 D = D D 2 = 2 2 =@A@h@C@@ regressions: Then we get the following estimators of changes in comovement for univariate = 3 >0 = 3 <0 0 See appendix for definitions of parameters 4

16 .. = 3 <0 = 3 <0 These estimators demonstrate that univariate regressions are informative about excess comovement. They predict that the CAPM beta increases when a stock is added to the market index. Moreover sensitivities to the SMB HML and MOM returns all decrease after the stock is added to the index. These are consistent with the empirical findings in Kasch and Sarkar (204). However the post-addition falls in the slopes of the SMB and HML returns cannot be interpreted in our model as evidence that comovement is driven by fundamentals only. Instead our univariate regressions confirm that changes in sensitivities to group-specific non-fundamental factors can impact on these decreasing slopes in line with the Chen et al. (205) model. The estimators of changes in comovement for the three-factor model are: =+ 45 ( ) 7 (.. ) 8 ( )9>0 =+ 45 ( ) 7 ( ) 8 (.. )9 =+ 45 (.. ) 7 ( ) 8 ( )9 +>0 5 >0 7 >0 8 >0 It is apparent that the three-factor model indicates that the beta should increase for a stock that is added to the index. However this is not always the case for loadings on the SMB and HML returns. The model implies that the change in loadings on the SMB and HML returns depends on relative changes in beta and loadings estimated 5

17 by univariate regressions on the SMB return and HML return. These changes are all driven by the change in sensitivities to group-specific non-fundamental risk and the variances of the group-specific non-fundamental factors " % #H "% mentioned by Chen et H al. (205). The loading estimated by the three-factor model on the SMB return decreases only when the absolute value of changes in beta and the loading on the SMB return are sufficiently high relative to the absolute value of changes in loading on the HML return. The HML loading in the three-factor model decreases only when the sum of the absolute values of the change in beta and the loading estimated through univariate regressions on the HML return is sufficiently high to offset the absolute value of the change in the loading on the SMB return. In sum the solutions clearly indicate that three-factor model loadings on the SMB and HML returns can increase or decrease when a stock is added to the index. Interestingly both increases and decreases can be evidence of excess comovement. Chen et al. (205) argue that the bivariate regression is not informative about excess comovement based on implausible assumptions. The variances of residuals are zero. The returns on stocks and portfolios have unit sensitivities to fundamental factors. Whether or not the sensitivity of a stock to each group-specific nonfundamental factor is zero or one depends on whether the stock belongs to that group. This paper seeks to establish whether the three-factor model is informative under these assumptions which are not general. Under the latter we get these estimators of the beta and the loadings on SMB and HML prior to and after additions: = % % % " #($) " #&$' " I "% #$ "% #($) " % I J" % #$ "% #&$' " % I J" % #($) "% #&$' " % I J" % #$ "% #&$' "% #($) = 6

18 = =0 = =0 The above estimators indicate that the three-factor model is still informative even under the Chen et al. (205) assumptions. Changes in the beta and the SMB and HML loadings all depend on the variances of the group-specific non-fundamental factor. Interestingly the change in beta does not depend on the variance of the index group-specific non-fundamental factor but on that of the SMB and HML groupspecific non-fundamental factors. Further the loadings on SMB and HML decrease to zero after additions while the beta increases to one from a negative value. The implication is that the stock should be neutral after it is added to the index when portfolios are well diversified and when the importance of the fundamental and the non-fundamental factors are equal. However these assumptions are too strong. In the empirical section we seek to provide evidence of that the three-factor model is still informative about excess comovement under these restrictions. 7

19 4. Methodology This section discusses the methodology used for detecting changes in beta around S&P 500 addition and deletion events and for empirical tests of implications from the theoretical model. 4. CAPM and bivariate models Following BSW (2005) univariate and bivariate models are used for tests of beta coefficient changes relative to the S&P 500 index and the S&P and non-s&p 500 indices respectively. Following equations give the univariate and bivariate models respectively where K denotes the added (deleted) stock return; K &M 0NN and K OP are S&P-500 and non-s&p-500 returns respectively. K =+ + K &M 0NN +- K =+ + OP0NN K OP0NN + OP K OP +- The beta coefficients for both the models are estimated using pre- and post-addition windows. The average change in all added (deleted) firm coefficients is estimated and t-statistics are calculated for a significance test. As BSW (2005) suggest the magnitude and significance of the average coefficient change should be smaller at lower frequencies (daily and weekly) because noise trader sentiment should disappear in the long-term when monthly data are employed with three-year windows. The preand post- windows are one year for daily and weekly data. We also check whether higher coefficient magnitude and statistical significance can be found using more recent data. 8

20 4.2 Univariate and multivariate models Our theoretical shows that the univariate regression loadings on both the SMB the HML factors as well as that on the market index feed into the three-factor model estimate of excess comovement. We also include the momentum factor as it has already been used by follow Kasch and Sarkar (204) and we expect it to play a similar role to the other factors in our theoretical framework. Thus we regress the added stock return on the SMB HML and MOM returns respectively. = = = We follow Kasch and Sarkar (204) in undertaking empirical tests for added stocks using the Fama-French three- and four-factor models as follows. = =+ Q Like most recent studies we follow BSW in defining window length and in excluding the month in which the event is announced and implemented due to noise. Note however that Kach and Sarkar (204) define the post-window as starting from the third month after the addition month. We leave the formal development of a theoretical 4-factor model to future work! 9

21 5. Data and Empirical Results 5. Data Data over the January 988-June 204 period are used for examining the beta change after changes in the S&P 500 index constituents. The list of the event firms is from the Compustat North America database. There are 572 inclusion and 44 deletion events over the period. Following BSW (2005) addition events are excluded if the firm results from restructuring or spinning off a firm already in the index or if the firm is involved in a merger or takeover around the event. We also exclude the event firm if the firm s sample in the pre- or post-event window is less than 30 at the daily and weekly frequencies. For the monthly test the event is excluded if we do not have data for the full 36-month post-event window. These criteria yield a daily sample of 55 events a weekly sample 509 and a monthly sample of 390 events. Deletion events are excluded if bankruptcy merger and takeover of the firm happen around the event or the data are not available. The final sample includes 44 events with daily 40 with weekly and 90 with monthly data. The prices of the event stocks the level of the S&P index and the level of the value-weighted CRSP NYSE AMEX and Nasdaq index are from the CRSP database. Log returns on the stock the S&P 500 and the value-weighted CRSP market index are calculated. The non-s&p 500 return index uses the formula K OP0NN = RK STUM VWP XYZ[[\]^ VWP _`ab&c\]^ VWP _`ab&c\]^k OP0NN d VWP _`ab&c\]^evwp XYZ[[\]^. 2 Total capitalisation of the S&P 500 index ghi OP0NNe and of the value-weighted CRSP market index 2 The formula is inferred from the identity K jakl = ghi mi500l ghi jakl K mi500l + ghi jakl ghi mi500l ghi jakl K oomi500l. 20

22 ghi STUMe are from the CRSP database. The Fama-French 3 factors and momentum factor are from the Fama-French Data Library. 5.2 Empirical Results We first analyse the traditional CAPM and bivariate results and then implement regression tests implied by our theoretical model CAPM and bivariate results Table reports the results from estimating the CAPM regression given in equation (). It gives the average change in the slope coefficient across all events in the sample and the average change in the K. [Table around here] The results support the BSW (2005) prediction that the beta coefficients increase after stocks are added to the S&P 500 index and that they decrease after stocks are deleted. The average increase in the daily and monthly betas over the full sample period is 0.7 in both cases while that in the weekly slope coefficient is 0.0. These increases in both the slope coefficient and the K are all highly significant. Our findings for the subsample show average increases in the daily and weekly beta coefficients and K but not for monthly data. Both the average beta and K changes are insignificant at the monthly frequency in line with BSW (2005). However the corresponding results for the 65 added stocks in are both economically and statistically significant suggesting that the BSW view needs to be amended in the light of the additional data up to The results for the subsample also confirm that comovement increases after stocks are added to the S&P 500 index but surprisingly the results are 2

23 now strongest at the monthly frequency followed by those at the daily frequency while those at the weekly frequency are insignificant. The average change in the daily slope coefficient is which is just one third of that for the subsample. 3 Chen et al. (205) find that the average change in the daily slope coefficient over the period is However these findings cannot be interpreted as evidence of the decreasing importance of index investing as Chen et al. (205) suggest. Chang Hong and Liskovich (205) find that shorting of index members has increased over the years and that mutual funds with large stocks in their portfolios supply liquidity for index trackers which should imply smaller beta chnages. 4 The bivariate regression results show that average changes in the S&P beta and the non-s&p beta confirm the BSW (2005) prediction that the S&P betas increase while the non-s&p 500 betas decrease after additions to the S&P 500. Table shows the average change in the daily S&P 500 coefficient across all addition events for the whole sample is in the weekly coefficient and 0.29 in the monthly coefficient. Correspondingly the mean decrease in the daily non-s&p 500 coefficient is in the weekly data and in the monthly data. The results from the bivariate model are statistically stronger than that from the univariate CAPM model in line with the BSW (2005) results. Finally our bivariate results like theirs show no evidence of sentiment reversion in the long run. 3 Barberis et al. (2005) reports monthly results up to 998 because data are not available. We divide the monthly sample into three subsamples for comparison. 4 Note that this study refers to the Russell 000 and 2000 indices. 22

24 New univariate factor and three- and four-factor loadings Table 2 reports the average change in loadings estimated through univariate regressions on the SMB HML and MOM portfolios and in K respectively from 988 to 20 at the daily weekly and monthly frequencies. [Table 2 around here] It shows that average change in the loading on the MOM portfolio is highest and most significant over all periods. The average change in the MOM loading over the full sample using daily data is the SMB loading decreases only by 0.43 but the HML loading is statistically insignificant. These findings are consistent with Kasch and Sarkar (204) although they use 3- and 4-factor model to estimate their loadings. These decreasing loadings on the SMB HML and MOM factors support the implication of addition events from our theoretical model. Weaker results are found when weekly data are used and only the loadings on the MOM show significant decreases for the whole sample and the two sub-samples. The SMB and the HML loadings are everywhere insignificant. In contrast the magnitude and significance of changes in these loadings increase when monthly data are employed. In particular the change in the MOM loading is now and highly significant. This is consistent with the finding that winner and loser performance persists for several years. However loadings on the SMB and the HML increase significantly especially in the recent sub-period. All of the loadings are largest for all portfolios at the monthly frequency which suggests no sign of sentiment disappearing at lower frequency. The change in the SMB loading has the largest (absolute) magnitude in the recent sub-period. 23

25 Table 3 reports changes estimated through the 3- and 4-factor models in beta and loadings on the SMB and HML and MOM. [Table 3 around here] The changes in beta estimated by means of the 3- and 4-factor models are smaller in magnitude and not always as significant as in the univariate results with the important exception of results at the monthly frequency. The changes in beta from the daily data 3- and 4-factor models are 0.08 and 0.03 respectively. These changes are highly significant but those estimated at the weekly frequency are insignificant as in Kasch and Sarwar (204). However the changes in beta from the monthly data 3- and 4-factor models are both larger and statistically significant at the % level at and 0.96 respectively for the full sample. The novel finding here is that of these changes in beta are larger in magnitude than those in the univariate regressions despite being conditioned on the SMB HML and MOM portfolios. The results for the sub-periods are also statistically significant for all but the sub-period and even then the 3-factor beta change is significant at the 0% level. In this respect they run contrary to the Kasch and Sarwar (204) and Chen et al. (205) premise that excess comovement can be explained by traditional risk loadings. Moreover the very small decrease (0.03) in the beta change in this relative to the 3-factor model provides scant support for the Chen et al. (204) assertion that momentum can account for a large part of the apparent comovement puzzle. 5 5 Nott that the Chen et al. (205) view is based on univariate results only. 24

26 The change in the loading on SMB is highly significant at the daily frequency for the the 3- and 4-factor models consistent with the Kasch and Sarkar (204) empirical findings. Note however the significant change in loading on the SMB can be driven by non-fundamental factors but not just by fundamental factors as our theoretical model suggests. The HML results are significant in the 3- the 4-factor models for the full sample and the sub-period at the monthly frequency. Note that the change in the HML loading also can be attributed to non-fundamentals in our theoretical model. The loading on the MOM in the 4-factor model decreases significantly by and 0.42 at the daily weekly and monthly frequencies respectively. The daily and weekly MOM results for the sub-period (and most other sub-periods) are both significant but those at the monthly frequency are significant only for the sub-period. Overall the results are consistent with our theoretical 3-factor model. This indicates that not only changes in beta but also those in the SMB HML and by extension MOM loadings provide evidence on excess comovement. 6. Conclusions This paper builds upon BSW (2005) and Chen et al. (205) to develop a theoretical 3- factor model in which sentiment plays a central role. Since both the BSW (2005) and Chen et al. (205) models focus mainly on univariate regressions a major difference is that sentiment affects all factor returns and not just market returns in our model. The paper provides closed form solutions for all the factor loadings including that on the market index. These suggests that univariate regressions on SMB HML (and by extension) MOM portfolios are informative about comovement. The models are tested 25

27 using and three- and four-factor models for the sample of stocks added to the S&P 500 over the period. The univariate results produce negative loadings on the SMB HML and MOM portfolios as predicted by our theoretical model. Regressions using the Fama and French 3- and Carhart (994) 4-factor models produce novel results. Notably the changes in beta at the monthly frequency (both around 0.2) are larger than those from the corresponding univariate regression and are also statistically significant at the % level. This is a novel result as since these increases condition on the effects of size book-to-market and for the 4-factor model momentum critics of the comovement effect assert that they should be smaller or insignificant. Moreover the very small decrease (0.03) in the beta change in the 4-factor relative to the 3-factor model provides scant support for the Chen et al. (204) assertion that momentum can account for a large part of comovement. The conclusion is that comovement remains a puzzle! 26

28 References Baffes J Does comovement among exchange rates imply market inefficieny?. Economics Letters Barberis N. & Shleifer A Style investing. Journal of Financial Economics Barberis N. Shleifer A. & Wurgler J Comovement. Journal of financial economics Boyer B. H. 20. Style-related comovement: fundamentals or labels?. The Journal of Finance Carhart M. 997 On Persistence in Mutual Fund Performance Journal of Finance Chelley-Steelley P. L. & Steeley J. M Changes in the comovement of European equity markets. Economic Inquiry Chen H. Singal V. & Whitelaw R. F Comovement revisited. NBER Working Paper 228. De Long J. B. Shleifer A. Summers L. H. & Waldmann R. J Noise trader risk in financial markets. Journal of Political Economy Fama E. F. & French K. R Common risk factors in the returns on stocks and bonds. Journal of Financial Economics Fama E. F. & French K. R Mutilfactor explanations of asset pricing anomalies. The Journal of Finance Green T. C. & Hwang B.-H Price-based return comovement. Journal of Financial Economics Greenwood R Excess comovement of stock returns:evidence from crosssectional variation in Nikkei 225 weights. The Review of Financial Studies Harris L. Gurel E Price and volume effects associated with changes in the S&P 500: new evidence for the existence of price pressure. Journal of Finance Jegadeesh N. & Titman S Profitability of momentum strategies: an evaluation of alternative explanations. Journal of finance Kasch M. & Sarkar A Is there and S&P 500 index effect? (March 204). FIRS 203. Available at SSRN: or Shleifer A Do demand curves for stocks slope down? Journal of Finance

29 Vijh A. M S&P 500 trading strategies and stock betas. The Review of Financial Studies Wahal S. & Yavuz M. D Style investingcomovement and return predictability. Journal of Financial Economics Wurgler J. 20. On the Economic Consequences of Index-Linked Investing Challenges to Business in the Twenty-First Century: The Way Forward edited by W.T. Allen R. KhuranaJ. Lorsch and G. Rosenfeld American Academy of Arts and Sciences. Wurgler J. Zhuravskaya K Does arbitrage flatten demand curves for stocks? Journal of Business ARTICLE IN PRESS 28

30 Table Average beta and p q changes This table present the average beta and K change after addition and deletion events from 988 to 204 through univariate and bivariate modles. The results are displayed at daily weekly and monthly frequencies. Table average r change and p q change Sample N Univariate Bivariate Panel A: daily returns r tttt(s.e.) p q (s.e.) r tttt uvwxx (s.e.) r tttt yzyuvwxx (s.e.) Additions *** (0.0220) *** (0.0072) *** (0.0304) *** (0.0339) *** (0.0345) *** (0.0255) Deletions *** (0.057) 0.057*** (0.0084) *** (0.09) *** (0.002) 0.44*** (0.0396) *** (0.0466) *** (0.62) *** (0.0476) *** (0.0484) *** (0.62) Panel B: weekly returns Additions *** (0.0349) *** (0.0546) (0.047) Deletions (0.0765) *** (0.0090) 0.036*** (0.05) 0.057*** (0.040) *** (0.04) *** (0.066) *** (0.089) ** (0.0987) *** (0.580) *** (0.0606) (0.0856) ** (0.0859) *** (0.52) Panel C: monthly returns Additions *** (0.0489) (0.0733) *** (0.744) ** (0.0594) Deletions ** (0.069) *** (0.08) (0.08) 0.209*** (0.0320) 0.087*** (0.065) * (0.024) *** (0.045) * (0.649) 0.692*** (0.29) 0.305** (0.3472) (0.2356) -0.24* (0.0856) * (0.464) (0.977) (0.055) (0.2073) *** ** and * denote significant differences from zero at the % 5% and 0% levels in one-sided test. 29

31 Table 2 Univariate regressions for SMB HML and MOM portfolios This table presents the average loadings on SMB HML and MOM portfolios and K change after addition and events from 988 to 204 through univariate modles. The results are displayed at daily weekly and monthly frequencies. Sample N SMB HML MOM Panel A: daily returns r ttttt(s.e.) q p q (s.e.) r ttttt(s.e.) { p q (s.e.) r ttttt(s.e.) p q (s.e.) Additions *** (0.042) 0.046*** (0.003) (0.0508) 0.025*** (0.0049) *** (0.074) 0.067*** (0.0056) ** (0.0587) *** (0.0604) 0.023*** (0.0036) * (0.005) (0.057) (0.0853) *** (0.0068) *** (0.007) -0.55*** (0.65) *** (0.0893) (0.0052) *** (0.00) Panel B: weekly returns Additions (0.06) (0.0902) (0.088) (0.0047) (0.0065) (0.0067) (0.073) (0.0947) (0.23) 0.072*** (0.0065) 0.03* (0.0092) 0.025*** (0.009) *** (0.0748) *** (0.84) *** (0.089) (0.0067) (0.006) (0.02) Panel C: monthly returns Additions * (0.067) * (0.076) (0.827) *** (0.0938) 0.052*** (0.0057) (0.0068) * (0.072) 0.097** (0.009) *** (0.0879) *** (0.23) (0.86) 0.603*** (0.473) * (0.0065) (0.0099) ** (0.094) 0.023* (0.0092) *** ( ) *** (0.355) *** (0.2077) *** (0.089) *** ** and * denote significant differences from zero at the % 5% and 0% levels in one-sided test *** (0.0066) (0.0079) *** (0.079) *** (0.007) 30

32 Table 3 three- and four-factor models This table presents average changes in beta and loadings estimated through three- and four-factor model on the SMB the HML and the MOM from The test is undertaken at daily weekly and monthly frequencies. Sample N Three-factor model Four-factor model Panel A: daily returns r tttt(s.e.) r ttttttttt(s.e.) }~ r tttttttt(s.e.) ~ r tttt(s.e.) r ttttttttt(s.e.) }~ r tttttttt(s.e.) ~ Additions *** *** * 0.033*** *** (0.0275) (0.0363) (0.0488) (0.0263) (0.0349) (0.0495) ** (0.046) *** (0.0287) -0.38*** (0.0586) *** (0.0407) (0.0703) -0.9** (0.0674) 0.307*** (0.0442) *** (0.0272) *** (0.0585) *** (0.036) (0.0759) (0.0626) r ttttttttt(s.e.) ~z~ *** (0.043) *** (0.0686) *** (0.052) Panel B: weekly returns Additions (0.044) (0.0664) (0.0479) -0.43** (0.065) *** (0.0955) (0.0755) -0.94*** (0.0773) * (0.208) ** (0.0946) (0.0428) (0.0677) (0.0509) (0.0642) -0.85* (0.004) -0.0 (0.078) (0.0847) (0.304) (0.063) -0.26** (0.063) (0.005) *** (0.074) Panel C: monthly returns Additions *** (0.0566) ** (0.098) (0.0678) (0.9) 0.85** (0.0865) 0.377*** (0.423) 0.964*** (0.0563) 0.75* (0.0926) (0.0686) (0.36) 0.985** (0.0889) ** (0.474) ** (0.0595) * (0.252) 3

33 *** (0.836) * (0.07) -0.5 (0.79) (0.093) (0.203) (0.287) *** (0.747) 0.7 (0.0729) (0.2205) (0.0925) *** ** and * denote significant differences from zero at the % 5% and 0% levels in one-sided test (0.2205) 0.06 (0.284) (0.354) (0.0659) 32

34 Appendix: proofs Assume the driving processes for returns before the stock is added to the S&P 500 index are: = = + + = + + = + + = + + D 3 () 2 and after the stock is added to the S&P 500 index: = = + + = + + = + + = + + D 3 () 2 >0 < 0 < 0 < 0 Univariate regressions 33

35 We run four univariate regressions = = = = the probability limit of the slope coefficients are = ( ) ( ) = ( ) ( ). = ( ) ( ) = ( ) ( ) estimators before and after the stock is added to the S&P 500 index = = 2 = =

36 . = = = = similarly = = = = = =

37 = = 2+ 3 Following Chen et al (205) we assume also that sensitivities to the common factor the variances of the non-fundamental factors variances of the fundamental factors and correlations between returns on groups are constant over time. = =? =?? =@A@h@C@@ 3 = 3 3 >0 D = D D 2 = 2 2 =@A@h@C@@ then = 3 >0 = 3 <0.. = 3 <0 = 3 <0 Three-factor regression The three-factor regression is =

38 The probability limits of slope coefficients are =(ƒ ƒ) e (ƒ ) = = +2?????? 5 =? 7 =??? 8 =??? = =? 7 =??? 8 =??? = =? 7 =??? 8 =??? again assuming parameters except sensitivity to the group-specific non-fundamental factor are fixed across the 2 sub-periods =+45 ( ) 7 (.. ) 8 ( )9>0 =+45 ( ) 7 ( ) 8 (.. )9 37