Download Comovement Among Stocks with Similar Book-to

Comovement Among Stocks with Similar
Book-to-Market Ratios
Brian H. Boyer∗
Brigham Young University
This Version: February 14, 2006
abstract
This paper presents empirical evidence that style investor sentiment generates comovement among stocks with similar book-to-market ratios. Data on constituents of
the S&P/Barra Value and Growth indices are used to conduct empirical tests that
lead to five main results. (1) When a stock switches from one index to the other,
return comovement with the new index increases significantly. (2) Cross-sectionally,
there is evidence of a discontinuity in comovement at the book-to-market cutoff which
defines the two indices. (3) Turnover comovement with an index is found to increase
significantly when a stock switches to the index (4) The value and growth definitions
of S&P/Barra are a significant determinant of what value and growth style funds
hold in their portfolios. (5) Similar empirical patterns do not exist among the universe of stocks that would have been in the indices from 1981 to 1991. The indices
did not exist before 1992. In summary, the evidence of this paper suggests trading
behaviors unrelated to systematic risk generate comovement among stocks with similar book-to-market ratios. The results are further evidence that limits-to-arbitrage
are binding in U.S. equity markets.
∗
Email contact: [email protected]. I wish to thank Barra for providing data on index constituents.
Barra Inc. is an investment software and research company and the data has been provided as part
of a broad academic program to foster the study of investment technology. I also am grateful to
John Ammer, Joseph Chen, Michael Gibson, Lars Lefgren, Grant McQueen, Matt Pritsker, Brian
Reid, Tyler Shumway, Clemens Sialm, Wilbert Van Der Klaauw, Jonathan Wright, Kathy Yuan,
Lu Zheng, and seminar participants at Brigham Young University, the University of Michigan
Business School, the 2004 EFA meetings in Maastricht, and the 2005 WFA meetings in Portland
for helpful suggestions. I also thank Greg Adams for computer support and Brandon Bates for
excellent research assistance.
1
I.
Introduction
A book-to-market factor generates common variation among stock returns (Fama
and French (1993), (1995)). While asset loadings on common factors in returns are
frequently used to explain or understand differences in average returns, it is not well
understood why assets load on common factors in the first place. According to the
traditional view, common variation in returns is generated by common factors in cash
flows. An alternative view, contradictory to standard asset pricing theory, is that
shifting investor sentiment unrelated to fundamentals gives rise to common variation
in returns. Barberis and Shleifer (2003) develop a model in which the shifting sentiment of “style investors”, investors who allocate wealth across broad investment categories rather than individual assets, generates common factors in returns unrelated
to fundamentals. Provided there are no limits to arbitrage, sophisticated investors
should be able to arbitrage away such price effects. Real world arbitrage, however,
entails both costs and risk, and therefore, is in some sense limited (DeLong et al.
(1990), Shleifer and Vishney (1997)). Such limits to arbitrage may allow investor
sentiment unrelated to fundamentals to affect prices. Using constituent data for the
S&P/Barra Value and Growth indices, I present empirical evidence that shifting investor sentiment of style investors does generate common variation, or comovement,
among stocks with similar book-to-market (BE) ratios unrelated to fundamentals.
The evidence suggests the impact of style investor sentiment on comovement is both
statistically and economically significant, often accounting for 20 to 50 percent of
the common variation in returns orthogonal to the rest of the market.
Fama and French (1992), (1993), (1995) actually uncover a three-part puzzle.
First, value stocks (stocks with high BE ratios) earn a premium that cannot be
explained by the CAPM. Second, there is a common factor in returns related to BE
ratios. Third, the common factor in returns does not respond to common shocks in
earnings. The first two findings suggest BE ratios may proxy for systematic risk.
The third finding, however, hints that something other than a common factor in
cash flows drives the common variation in returns. Fama and French (1995) suggest
the results are caused by noisy measures of expected earnings. In contrast, I find
evidence that a considerable component of the common variation in stocks with
2
similar BE ratios is driven by style investor sentiment.
These empirical patterns have subsequently received considerable attention in
the literature.1 Some have used VAR methods developed by Campbell and Shiller
(1988a), (1988b) and Campbell (1991) to decompose stock returns into news about
cash flows and news about discount rates to better understand the BE factor in returns.2 In this paper, instead of relying on ad hoc VAR models to decompose returns,
I turn to methods used by others to show that trading unrelated to fundamentals
effects prices. Some of the most definitive evidence of this type comes from index
reclassifications. Harris and Gurel (1986), Shleifer (1986), and Lynch and Mendenhall (1997) find strong evidence for price effects when stocks are included in the S&P
500. Kaul, Mehrotra, and Morck (2000) and Greenwood (2005) find similar price
effects in the Toronto Stock Exchange TSE 300 and Nikkei 225 indices respectively.
Vijh (1994) and Barberis, Shleifer and Wurgler (2005) find that index inclusion in
the S&P 500 is associated with greater comovement with the S&P 500 index return.
The index classifications used in this paper are the S&P 500/Barra Value and
Growth indices. These index classifications are likely to be associated with what style
investors hold and trade since they define financial products by which investors can
acquire exposure to styles in a simple, low-cost manner. The Chicago Mercantile Exchange launched options and futures contracts on the S&P/Barra Value and Growth
indices in November of 1995, and Barclays issued exchange traded funds (iShares) on
both indices in May of 2000. Mutual funds are another method by which investors
access styles in a simple, low-cost manner, and the S&P/Barra indices serve as a
benchmark by which value and growth fund managers are evaluated. As protection
against poor performance fund managers may closely track their benchmark, implying the indices are associated with what style investors hold indirectly through fund
holdings.
1
See for example, Fama and French (1996), Vassalou (2003), Vassalou and Liew (2000), Daniel
and Titman (1997), Davis, Fama and French (2000), Campbell and Vuolteenaho (2004), and Petkova
and Zhang (2005).
2
See for example Bansal, Dittmar, and Lundblad (2005), Hansen, Heaton and Li (2005), and
Campbell, Polk, and Vuolteenhao (2005).
3
The methodology used to construct these indices is very simple. All stocks in the
S&P 500 are divided into two groups based on a single variable: the BE ratio. At
the end of every June and December, a cut-off value is determined and stocks with a
BE ratio above the cut-off are placed in the value index, while all others are placed
in the growth index. The indices are rebalanced twice a year, with the cut-off value
determined such that the total market capitalizations of the two indices are about
equal.
The simple index methodology gives the S&P/Barra indices three advantages
from an empirical standpoint relative to most other indices. First, a clean control
sample exists to test whether index inclusion is a credible instrument uncorrelated
with other effects such as fundamental risk. The indices were first created in 1992.
However, Barra has back-dated the constituent data to 1981, dividing all stocks in
the S&P 500 into two groups and rebalancing the groups at the end of June and
December as if the indices had existed over this period. Consequently, I have index
definitions over a time period in which investors were not using the arbitrary style
boundary of the S&P/Barra indexes to make trading decisions. If the results are
being driven by effects other than style investor sentiment, the same results should
be found in the data prior to 1992. However, the empirical patterns found in support
of the style investing hypothesis are not found in the data prior to 1992.
A second advantage of the index methodology is that it allows identification
of stocks for which index reclassification is not likely to be related to changes in
fundamentals. For example, BE ratios of stocks that switch to the growth index are
likely to decrease prior to switching. If BE ratios proxy for risk factor loadings, then
such stocks should be expected to comove more strongly with the growth index after
switching even if style investor sentiment has no effect on comovement. However,
because the indices are rebalanced twice a year so that they each have approximately
the same market capitalization, a stock may switch to the high (low) BE index even
though its own BE ratio actually decreased (increased) prior to the switch. For
example, if growth stocks do very well over a given period, then in order to keep the
total market capitalization of the two indices the same, some stocks in the growth
index must be switched to the value index even though their BE ratios actually
4
decreased prior to the switch. That is, even though a stock is more like a growth
stock in terms of its BE ratio, its label is changed from growth to value. For clarity
in exposition I refer to these stocks as “index balancers” since they switch indexes
to balance index market capitalizations.
A third advantage of the index methodology is that I can identify and compare
securities in different indices with similar fundamentals by comparing stocks in each
index whose BE ratios are very near the cut-off point. This approach is often referred to as a“regression discontinuity analysis” (Thistlewaite and Campbell (1960),
Trochim (1984)). Since the only variable used to define the indices is the BE ratio,
stocks with BE ratios near the cut-off point may be expected, on average, to be very
similar in terms of fundamentals. The unique methodology used to construct the
S&P/Barra indices makes this analysis possible. Such methods would be entirely
impossible using an index such as the S&P 500 for which membership is determined
using a variety of loose decision rules.3
The empirical tests of this paper use data on constituents of the S&P/Barra indices to determine if style investor sentiment generates comovement among returns
of stocks with similar BE ratios (the “investor sentiment hypothesis”). The alternative hypothesis is that comovement is entirely driven by fundamental cash flow
risk (the “fundamental risk hypothesis”). The tests lead to five main results. First,
when a stock switches from one index to the other, return comovement with the new
index increases significantly. For example, using daily data for all stocks that switch
to the growth index from 1992 though 2004, the average level of comovement with
the growth index increases by 17 percent after the reclassification event. For index
balancers switching to the growth index, the average level of comovement increases
by 68 percent after the reclassification event. Second, stocks with similar fundamentals in different indices differ significantly in their level of comovement with the two
indices. For example, using a regression discontinuity design with daily data, I find
that over the period from 1992 to 2004, stocks just above the cut-off in the value
index exhibit 33 percent higher comovement with the value index than stocks just
3
Complete details on the criteria for index additions and removals for the S&P 500 are available
at www.standardandpoors.com/indices, under Index Committee Policy.
5
below the cutoff in the growth index. Similarly, I find that stocks just below the
cut-off in the growth index exhibit 36 percent higher comovement with the growth
index than stocks just above the cutoff in the value index. Third, I find evidence that
turnover comovement with an index increases significantly when a stock switches to
the index over the period from 1992 through 2004. Fourth, the value and growth
definitions of S&P/Barra are a significant determinant of what value and growth
style funds hold in their portfolios over the period from 2002 though 2004. Fifth,
and perhaps most importantly, similar empirical patterns do not exist among the
universe of stocks that would have been in the indices from 1981 to 1991. The indices did not exist before 1992. In summary, the evidence of this paper suggests
trading behaviors unrelated to systematic risk generate comovement among stocks
with similar book-to-market ratios.
How do the results of this paper apply to stocks in general? This question is indeed important, while at the same time, difficult to answer. In the model of Barberis
and Shleifer (2003), noise trader risk creates limits-to-arbitrage that prevents sophisticated investors from arbitraging away the price effects of style investors. Given that
such limits-to-arbitrage exist among stocks within the S&P 500, it is perhaps extreme
to believe that no such limits-to-arbitrage exist among stocks that are not included
in these indices, or that such limits-to-arbitrage did not exist before the S&P/Barra
indices were created. Without an instrument to identify such effects however, it is
difficult to extrapolate the results to stocks in general in a manner that would be
convincing to all. In any event, this paper provides evidence that style investor sentiment does generate common variation related to BE ratios among a subset of very
liquid, heavily analyzed stocks, that make up on average about 71 percent of the
total market capitalization of the entire market from 2002 through 2004, contrary to
standard asset pricing theory.4
The paper proceeds as follows. In Section 2, I further discuss the S&P/Barra
index classification methodology and present summary statistics for the index constituent data. Section 3 presents empirical methodologies and results to test for
4
Other authors have found empirical evidence consistent with the style sentiment view of co-
movement. See for example Lee, Shleifer and Thaler (1991), Froot and Daborah (1999).
6
impacts of style investor sentiment on return comovement. Section 4 presents empirical methodologies and results to test for impacts of style investor sentiment on
trading behavior. Section 5 concludes the paper.
II.
The S&P/Barra Indices
Several sources of data are used in this paper. Data on constituents of the
S&P/Barra Value and Growth indices from May 1981 through March 2003 are obtained directly from Barra. These data are updated through December 2004 using
data on index changes from the Standard and Poors’ website.5 All data on stock returns, prices, shares outstanding, and volume are obtained from the CRSP database.
Mutual fund holdings data come from the CDA/Spectrum database which are collected from reports filed by mutual funds with the SEC and from voluntary reports
generated by the funds. Finally, book values are measured as Common Equity (item
60) from Compustat. Since this paper is the first to use the S&P/Barra constituent
data, this section provides a brief description of these data and presents summary
statistics.
The S&P/Barra Value and Growth indices are created by dividing all stocks in
the S&P 500 into two mutually exclusive groups according to a single variable, the
BE ratio. The value-weighted indices are periodically rebalanced to maintain equal
market capitalization in each index. In June and December of each year, Barra
determines a “cut-off” point that will divide all 500 stocks into two groups with
approximately equal market capitalization. Throughout the year the indices are also
adjusted as stocks move in and out of the S&P 500.6
The indices were first created in May of 1992. However, the simple index methodology has allowed Barra to back-date the constituent data to 1981, rebalancing the
indices every June and December over this period as if the indices existed. The sample from 1981 to 1992 is therefore used as a control sample. For clarity, the sample
from 1992 through 2004 is referred to as the “test sample.”
5
6
www.standardandpoors.com
When a stock enters the S&P 500 its book-to market value is compared to the cut-off point
established during the most recent rebalancing month and placed in the appropriate index.
7
Panel A of Figure 1 presents time series plots of the number of stocks in each
index, Panel B reports index performance measured as the log value of $1 invested
in each index at the beginning of the sample, and Panel C provides a time-series plot
of the equally weighted average monthly turnover for each index. Value-weighted
returns and equally-weighted turnover on each index were calculated by aggregating
data on individual stocks.7 Monthly turnover is calculated as the sum of daily
turnover (volume on day t divided by shares outstanding on day t − 1) during the
month and is adjusted for splits. The approximate date when the indices were first
created is marked by a vertical line in each panel.
The value index is always composed of more stocks since these firms are, on average, smaller in terms of market capitalization. Comparing Panels A and B of Figure
1, this difference appears to narrow (widen) as the value index outperforms (underperforms) the growth index. When the value index outperforms the growth index,
more stocks must be switched to the growth index to keep the market capitalization
of the two indices equal. Similarly, when the growth index outperforms the value
index, stocks must be switched from the growth index to the value index.
In Panel B, the value index is seen to generally outperform the growth index.
However, during much of the latter half of the late 1990’s the growth index greatly
outperforms the value index. From the beginning of 1997 through the end of 1999,
the average growth rates of the value and growth indices are 17% and 39% per year
respectively. The strong performance of growth relative to value does not carry over
past 1999 however. From 2000 through 2004 the value index beats the growth index
every year by an average of 9.6% on a simple-return basis. The poor performance of
the growth index over this period is especially stark during 2000, when the growth
index lost 19.7% and the value index gained 6.2%. In Panel C, average monthly
turnover of stocks in the growth index is generally higher than that for stocks in the
value index. In the late 1990’s however, average turnover for stocks in the growth
index increases dramatically.
The late 1990’s therefore represent a unique period in which the growth index
7
Monthly return data on the two indices can also be found on Barra’s web site and closely
matches the monthly return data created for this paper.
8
outperforms the value index followed by a considerable increase in growth index
turnover. From a rational pricing perspective, growth stocks experienced a sequence
of positive shocks to expected earnings growth in the late 1990’s which subsequently
reversed. The behavioral view is that a moderate increase in the prices of growth
stocks sparked investor enthusiasm for growth stocks as a style. This enthusiasm
consequently fed upon itself and was able to push prices away from fundamentals
for a period of time because arbitrage is somehow limited. Similar price patterns
are uncovered by Barberis and Shleifer (2003) in simulations of their model of style
investor sentiment. The abrupt increase in growth index turnover in 1999 certainly
suggests that strong growth performance fueled market interest in growth stocks.
Chan, Karceski, and Lakonishok (2000) and Chan and Lakonishock (2004) find that
earnings growth and sales growth of value stocks over this period were as high as
those of growth stocks and unusually good by historical standards. In other words,
the performance of growth stocks over value stocks during this period cannot be
easily linked to fundamentals. Hence, the high turnover and strong performance of
the growth index cause this period to be of special interest. If style investor sentiment
is driving stock prices in a manner unrelated to fundamentals, the effects may be
especially strong over this subperiod.
Figure 2 shows the return distribution of stocks that switch indices over the
five-month interval before the switch.8 This five-month interval is the time between
successive rebalancing months. A total of 399 stocks switch from the value to the
growth index over the period from 1992-2004 (Panel A). Of these, about 15 percent
drop in value before moving to the growth index. A total of 519 stocks switch from
the growth to the value index over this same period and of these, about 26 percent
increase in value before moving to the value index. A similar pattern is evident in
Panels C and D of Figure 2. Given that book-values are relatively constant, this
implies that some value index stocks switch to the growth index after their BE ratios
8
To be counted as a stock that switched to a new index, the stock must be in the S&P 500
index and the same sub-index (value or growth) throughout the entire five-month period before the
switch. The stock must also have a complete return history in CRSP over this same five-month
period.
9
have actually increased, and similarly, some growth index stocks move to the value
index after their BE ratios have actually decreased. Taking note of such stocks is
important to the analysis since it is unlikely these stocks switch indices because of
any change in their own fundamental risk characteristics. For example, if BE ratios
proxy for a loadings on a distress factor (Fama and French (1996)), it is unlikely a
stock that has increased in value since the previous rebalancing month is any more
distressed than before.
Stocks that switch to the growth index whose prior five month return was negative, and stocks that switch to the value index whose prior five month return is
positive are referred to throughout the paper as “index balancers.” The term is
motivated by the observation that these stocks apparently switch indexes merely to
balance out index market capitalizations. Index balancers are examined separately
in the empirical tests below.
Figure 3 provides time series plots of the cross sectional average BE ratio and
decile for each index. The cross sectional averages are calculated at the end of June
and December of each year, just after each index has been rebalanced. BE ratios are
constructed to be similar to those used by S&P/Barra when rebalancing the indices.
Equity value is defined as size (price time shares outstanding) at the end of May and
November. Book value is common equity reported in Compustat at the end of the
latest fiscal quarter at least six months prior to the end of June or December. Decile
cutoff points are calculated using all stocks in the CRSP database with a share code
of 10 or 11. Over the entire sample from 1981 to 2004, the average BE ratio for
stocks in the growth (value) index is about 0.28 (0.76), while the average decile is
about 2.83 (6.37). There is perhaps a slight downward trend in average BE ratios
and deciles across both the control and test samples. However, the change in average
BE ratios and deciles from any one rebalancing period to another is not large.
Panel A of Table I provides summary statistics on stocks in the two indices. The
left side of Panel A covers the period from June 1992 through December 2004, while
the right side covers the period from May 1981 through May 1992. To eliminate any
speculation that differences in empirical results across the two periods are caused by
10
the crash of October 1987, the crash is removed from the control sample.9 To compare
the summary statistics across the value and growth indices, pair-wise t−statistics,
calculated by GMM, are reported in parentheses.10 Average returns of the two
indexes are not significantly different in either the test sample or the control sample.
For example, in the test sample average monthly returns of the growth and value
indexes are 0.97 and 1.00, and the t-statistic for the difference in average returns is
-0.11. Correlation among stocks within the same index is found to be strong. To
measure the correlation of stocks within the same index, stocks in the index were
randomly split into two value-weighted portfolios. The complete indices were used
to calculate the cross-index correlations. For the test sample, the correlation of
growth (value) index stocks with other growth (value) index stocks is found to be
0.92 (0.95). The correlation between the growth and value indexes however, is only
about 0.76. In both the test and control samples, within-index correlation is found
to be statistically higher than cross-index correlation.
Panel B of Table I reports loadings on the Carhart (1997) four-factor model.11
These estimates along with their standard errors are obtained by jointly estimating
the regression of each index on the four factors and an intercept with GMM. Joint
estimation allows for the computation of pair-wise t-statistics for differences in estimated parameters across the two indexes. As expected, due to its construction, the
value index has a much higher loading on the HML factor than the growth index.
In summary, the value index outperforms the growth index in most years. The
late 1990’s represent a unique period in which the growth index outperforms the
value index followed by a considerable increase in growth index turnover. Further,
loadings on HML and average BE ratios differ considerably across groups, and stocks
within each group are significantly more correlated with each other than with stocks
across groups. I now explain the hypotheses and empirical tests.
9
Including the crash in general does not change the results, and in many instances, even further
weakens the results over the control sample.
10
I use the Newey West (1987) estimator and automatic lag selection technique of Newey and
West (1994) in both Panels A and B of Table I.
11
Factor portfolio returns are obtained from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.
11
III.
Empirical Tests for Return Comovement
The empirical tests explained in this section are designed to determine if style
investor sentiment generates comovement among returns of stocks with similar BE
ratios (the investor sentiment hypothesis). The alternative hypothesis is that comovement is entirely driven by fundamental cash flow risk (the fundamental risk
hypothesis).
A.
Index Rebalancing Test for Changes in Return Comovement
Throughout both the control and test samples, stocks often switch from one index
to the other. Given that index reclassification is not related to stock fundamentals,
and stocks in an index are more likely to be traded by style investors of a given
type, the effects of style-based trades on comovement can be identified by comparing
comovement before and after a stock switches to a new index. Similar tests have been
conducted by Vijh (1994)and Barberis, Shleifer and Wurgler (2005) using constituent
data for the S&P 500.
For each stock that switches to a new index, the following two regressions are
estimated
rit = αi + βiG rGt + βiV rV t + εit
t ∈ pre-rebalance window
∗
∗
rGt + βiV
rV t + ε∗it
rit = α∗i + βiG
t ∈ post-rebalance window
(1)
where rit is the return of some stock that switches to a new index, rGt is the return
on the growth index and rV t is the return on the value index. All stocks that switch
indices are excluded from the indices before estimating the regressions to avoid measuring any changes in comovement caused by changes in index composition. Hence,
the index returns are based on the same set of stocks both before and after the
rebalance month. Each stock is regressed on both indices simultaneously since a
substantial amount of the variation in either the value or growth index is caused by
market-wide shocks.12 The objective is to measure changes in style-related comove12
Tests on univariate measures of comovement were also conducted and similar results were found.
These results are discussed later in the text.
12
ment that is not related to broad market movements. Both daily and weekly data
are used in these tests. Weekly data is defined as the total return from Wednesday
close to Wednesday close. The pre- and post-rebalance windows using daily data are
the five-month intervals before and after each rebalancing month. For example, the
pre-rebalance window for December of year t covers July through November of the
same year. Since the rebalancing months are June and December, the pre- and postrebalance windows for consecutive rebalance months perfectly overlap. The pre- and
post-rebalance windows using weekly data are the eleven month intervals before and
after each rebalance month. For example, the pre-rebalance window for December
of year t covers January through November of the same year.
The estimated parameters of (1) are used to conduct the following test.
Test 1 (Index Rebalancing Test for Return Comovement) If style investor sentiment
impacts comovement, then on average, when a stock switches to a new index, its beta
with the new index increases. In addition, beta with the old index decreases.
For stocks that switch to the growth index:
P
∗
(T1-A) H0 : ∆β G = ni=1 (βiG
− βiG ) /n ≤ 0
Pn
∗
− βiV ) /n ≥ 0
(T1-B) H0 : ∆β V = i=1 (βiV
For stocks that switch to the value index:
Pn
∗
(T1-C) H0 : ∆β G = i=1 (βiG
− βiG ) /n ≥ 0
Pn
∗
− βiV ) /n ≤ 0
(T1-D) H0 : ∆β V = i=1 (βiV
where n is the number of stocks switching to a given index.
Significance of test results is measured in two ways. Since pre- and post-rebalance
windows overlap, regression coefficients are likely to be cross-sectionally correlated.
The pre- and post-rebalance windows overlap perfectly using the daily data in a
manner suitable for block-bootstrapping. Each block is defined as the five-month
interval between rebalance months. Dates within each block are sampled with replacement and the complete cross-section of stock returns observed on the chosen
dates, regardless of whether the return is part of a pre- or post-rebalance window, is
used to create a new block of data. Blocks are then stacked in sequential order to cre13
ate a new sample. This sampling technique preserves the cross-sectional dependence
among stocks whose pre- or post-rebalance windows overlap. The new data is used
to compute the test statistics of Test 1, and the procedure is repeated 1000 times.
Given that the null hypothesis for a test statistic is positive (negative), a p-value to
test the null hypothesis is estimated as the fraction of bootstrapped samples whose
estimated test statistics are positive (negative).
The pre- and post-rebalance windows do not overlap in a manner suitable for
the block bootstrap approach using the weekly data.13 Hence, I calculate consistent
standard errors in a simple manner, assuming residuals are i.i.d., that accounts for
the cross-sectional dependence in regression coefficients. See Appendix A for details.
Both techniques are used with the daily data and statistical inference is consistent
across both techniques, providing some confidence that the i.i.d. assumption does
not systematically impair the results using weekly data.
Stocks that switch indices are passed through a series of screens before being used
in Test 1. First, they must be in the S&P 500 index throughout the entire pre- and
post-rebalance windows. Second, they must be in the same index throughout the
entire pre- and post-rebalance windows. This is not a serious issue for the daily data
since the pre- and post-rebalance windows lie perfectly between rebalance months.
For the weekly data however, this second screen is important since pre- and postrebalance windows each overlap with other rebalance months. Third, the stock must
have a complete return history in CRSP over the pre- and post rebalance months.
Of course, a stock may comove with an index after switching because fundamentals have changed. I therefore conduct Test 1 using two data sets uniquely qualified
to settle this issue. First, I conduct Test 1 using index balancers only.14 Index
13
Any regression within a defined block must have data on every date within the block. This
causes some blocks to be defined too narrowly for sampling to be a reliable representation of the
true underlying distribution. For example, the pre-rebalance window for December covers January
through November, overlapping the month of June. However, for stocks that switch in June, no
data is used within that month. Hence, June would need to be a block in and of itself, providing
about only four weekly observations from which to sample.
14
Appendix B further investigates the statistical properties of regressions using index balancers
and shows that the conditioning information does not lead to any bias in the slope coefficients.
14
balancers are unlikely to have switched indices because of any change in their own
fundamental risk characteristics. In addition, I conduct Test 1 using the control
sample. If results are being driven by effects other than style investor sentiment, the
same results should be found in the data prior to 1992.
Results for Test 1 are given in Tables II and III. Table II provides results using
daily data and Table III provides results using weekly data. Both tables report the
average change in regression coefficients for stocks that switched to a new index, as
well as the average level of the coefficients over the pre-rebalance window. In almost
all cases, the average coefficient levels over the pre-rebalance window are significant
at the 1% level, and hence p-values or t-statistics are not reported for these measures
in the interest of clarity.
Table II provides evidence in favor of rejecting the null hypotheses of Test 1.
Using the full test sample (1992-2004), the estimated value of βG in Panel A for
stocks that switch to the growth index is found to increase on average by 0.054, or
17 percent relative to the average level of βG over the pre-rebalance window (0.311).
On the other hand, the average change in βG in Panel B for stocks that switch to
the value index is found to be -0.047, an 11 percent drop relative to the average
level of βG over the pre-rebalance window (0.417). Both results are significant at
the 10 percent level, providing some evidence in favor of rejecting Tests 1A and 1C.
Stronger results are obtained for index balancers alone. The estimated value of βG
for index balancers that switch to the growth index is found to increase on average
by 0.215, or 67 percent relative to the average level of βG over the pre-rebalance
window (0.314). This result is significant at the 1 percent level, and thus provides
strong evidence in favor of rejecting Test 1A. For index balancers that switch to the
value index, the average change in βG is -0.073 (-19 percent) and the average change
in βV is 0.090 (15 percent). Both of these results are significant at the 10 percent
level, supplying evidence in favor of rejecting Tests 1C and 1D.
The strongest evidence in favor of rejecting the null hypotheses of Test 1 in Table II is found over the period from 1998 through 2002. Over this period investor
enthusiasm for growth stocks in particular was apparently very strong as evidenced
by abnormally high growth index turnover (see Figure 1). In addition, this period
15
was unique in that growth stocks beat value stocks in most years, and other authors
have not been able to easily link differences in performance to differences in fundamentals.15 If style investor sentiment is generating comovement among stocks with
similar BE ratios, we may expect the effects to be especially strong over this period.
Using all switchers, the estimated value of βG for stocks that switch to the growth
index is found to increase on average by 0.146 (50 percent). The result is significant
at the 1 percent level, providing strong evidence in favor of rejecting Test 1A. Results
for stocks that switch to the value index are similar to those using the full test sample. However, using index balancers over this time period, all components of Test 1
are soundly rejected at the 5 percent level, and most are rejected at the one percent
level. The average value of βG for index balancers that switch to the growth index
is found to increase by 0.264 (86 percent) while the average change in βV is found
to be -0.206 (-34 percent). The former result is significant at the 1 percent level,
while the latter result is significant at the 5 percent level, providing strong evidence
in favor of rejecting Tests 1A and 1B. The average change in βG for index balancers
that switch to the value index is found to be -0.305 (-70 percent) while the average
change in βV is found to be 0.444 (73 percent). Both results are significant at the 1
percent level, providing strong evidence in favor of rejecting Tests 1C and 1D.
Interestingly, these results are not found over the control sample (1981-1991). In
fact, for stocks that would have switched to the growth index (had it existed), the
estimated value of βG is found to actually decrease, while the estimated value of βV
is found to increase. These results are driven mostly by index balancers. Since BE
ratios of index balancers that switched to the growth index likely increased prior to
the month in which they would have switched to the growth index, the results are
reasonable from the perspective that fundamental risk has some influence on return
comovement. For stocks that switch to the value index over the control sample,
the average change in βV is positive, and barely significant at the 10 percent level
according to the t-statistic calculated under the i.i.d. assumption. The bootstrapped
p-value however, does not indicate the positive change is significant. For index
balancers that switched to the value index over the control sample, the estimated
15
See Chan, Karceski, and Lakonishok (2000) and Chan and Lakonishock (2004).
16
value of βG is found to actually increase, while the estimated value of βV is found
to decrease. Since BE ratios of index balancers that switched to the value index
likely decreased prior to the month in which they would have switched to the value
index, the findings are again reasonable from the view that fundamental risk has
some influence on return comovement.
The results of Table III are in some sense even stronger than those of Table II.
For example, using the full test sample (1992-2004), the estimated value of βG in
Panel A for stocks that switch to the growth index is found to more than triple
in size. The average change in βG is 0.374 while the average level of βG over the
pre-rebalance window is only 0.160. The estimated average change in βV in Panel
B for stocks that switch to the growth index over the full test sample is found to
be -0.314 (36 percent). The estimated change in βG for stocks that switch to the
value index is found be -.297 (-57 percent), while the estimated change in βV is found
to be 0.223 (35 percent). All these results are significant at the one percent level,
providing strong evidence in favor of rejecting all null hypotheses of Test 1. Similar
results are found using index balancers over this time period, as well as all switchers
and index balancers over the period from 1998 through 2002. The results using index
balancers from 1998-2002 are only significant at the 10 percent level. However, since
only 15 (10) index balancers switch to the growth (value) index and pass the screens
to be included in the tests over this period, the low significance is perhaps not too
surprising.
The weekly data pose greater risk that any changes in comovement are simply
the result of changing fundamentals. Since estimation windows are longer and stocks
are required to remain in the same index for a longer period, more time is allowed for
stock fundamentals to change. The fundamental risk hypothesis can be distinguished
from the investor sentiment hypothesis again by looking at results using the control
sample. For stocks that switch from the value to the growth index, none of the
changes in comovement are found in the control sample that were identified in the
test sample. For stocks that switch from the growth index to the value index, the
estimated changes in βG and βV are significant and consistent with the results of
the test sample. These results suggest that using weekly data and all stocks that
17
switch from growth to value, some of the change in comovement can be attributed
to changes in fundamental risk. However, similar results are not found using index
balancers over the control sample, as no significant change in comovement is found
using index balancers that switch to either index during this period. Results over the
control sample therefore suggest that changes in comovement identified with index
balancers are not driven by changes in fundamental risk.
Similar tests were also conducted on univariate measures of comovement (correlation and univariate regression slope), and similar results were found. When a
stock switches to the growth (value) index, its correlation and beta with the growth
(value) index significantly increase. Again these results are not found using the control sample. A size factor was also added to the multivariate regressions.16 As stocks
switch among the indices, changes in comovement could be caused by changes in
loadings on cash flow factors related to size. However, when a size factor is included
in the multivariate regression tests, the results discussed above are actually found to
be somewhat stronger.
In summary, the index rebalancing tests provide considerable evidence in favor
of rejecting the null hypotheses of Test 1. When a stock switches to the growth
index, its slope coefficient in on the growth index significantly increases while the
slope coefficient on the value index decreases. Similarly, when a stock switches to
the value index, its slope coefficient on the value index increases while the slope
coefficient on the growth index decreases. For the daily data, the effect is especially
strong over the period from 1998-2002, while for the weekly data, the effect is strong
over the entire test sample. Similar results are not found over the control sample,
suggesting that the change in comovement observed over the test sample is caused
by style investor sentiment.
16
At the end of each month all stocks in the S&P 500 are split according to size (price times
shares outstanding) with the cut-off point being the median size of all stocks in the S&P 500. Value
weighted returns are calculated for the two portfolios throughout the rest of the following month
and the size factor is defined as the difference between the return on the small portfolio and the
large portfolio each day.
18
B.
Regression Discontinuity Analysis
If stocks were separated at random to the two indices, then fundamental risk
characteristics would be independent of index assignment. In such a completely
randomized experiment, the effect of style investor sentiment on comovement could
be identified by simply comparing average comovement across assets in each index.
Although index membership is not randomly determined and fundamental risk characteristics are not likely to be independent of index assignment, an approach similar
to the completely randomized experiment can still be used to identify index inclusion
effects using the S&P/Barra indices.
A unique characteristic of the S&P/Barra indices is that the mechanism by which
stocks are assigned to an index is perfectly known and determined by a single variable: stocks with a BE ratio above the cutoff go to the value index while all others
go to the growth index. Consequently, a comparison of comovement for observations
just above and just below the cutoff could be very similar to a completely randomized experiment. Since the cutoff point is defined exogenously by S&P/Barra, risk
characteristics should be expected to be similar for firms just above and just below
the cut-off. The exogenously specified cutoff point allows for identification. This
empirical approach is referred to in the statistics and econometrics literature as a
“regression discontinuity analysis”.17
Two necessary conditions must be satisfied for identification. First, experimental units must be separated into two groups, one that receives some treatment and
another that does not, according to a single sorting variable, zi , where group membership is completely defined by a cut-off point, z0 . Second, for some outcome variable,
yi , E[yi |zi] must be continuous at z0 under the null of no treatment effect. In this
paper the sorting variable is the BE ratio, the cut-off is the BE ratio that defines
which index a stock is included, and the treatment effect is index inclusion.
17
An early application used the fact that National Merit Awards are given to students whose
score on a test exceeds some threshold to estimate the effect of the award on a student’s future
academic ambitions and achievements (Thistlewaite and Campbell (1960). Others that have used
the design include Berk and Rauma (1983), Black (1999), Angrist and Lavy (1999), Guryan (2000),
Hahn et al. (1999) and Jacob and Lefgren (2002).
19
If the two necessary conditions for a regression discontinuity analysis mentioned
above are met, the treatment effect is identified by estimating
E[yi|yi ∈ A] − E[yi|yi ∈ B]
(2)
where A is the set of observations just above the cutoff and B is the set of observations
just below the cutoff. For instance, if many observations arbitrarily close to the cutoff point are observed, then the effect of treatment can be achieved by estimating
the difference in the mean outcome variable for these observations.
Although index membership is known with certainty, a drawback of the S&P/Barra
index data is that the actual book-values used to rebalance the S&P/Barra indices
as well as the cut-off point to define the indices are not observed. Consequently I
create my own cut-off following the methodology used by Barra to the best of my
ability. Barra’s web site18 states that it uses “Compustat data for book value . . .
Compustat updates its information as it receives 10-Q report data from companies
throughout the year.” From this information it is not clear which data or combination of data items from Compustat Barra uses to compute book value, nor when the
data is observed, since 10-Q reports are filed with the SEC at various times following
the end of the fiscal quarter.19 I measure book value as common equity reported in
Compustat at the end of the latest fiscal quarter at least six months prior to the
end of June or December. Barra’s web site also states that market values “used at
the time of rebalancing are the equity’s position at the close of trading one month
prior (i.e., November 30 and May 31).” I therefore define equity value as total market
capitalization at the end of May and November. On average across rebalancing periods, the calculated BE ratio correctly predicts index assignment about 94 percent
of the time in both the test and control samples. The concern however, is that in
estimating the treatment effect as in (2), an unobserved variable helps determine
index assignment, namely, the difference between observed BE and the actual BE
used by S&P/Barra, that could also be correlated with the outcome variable. In
this case, OLS estimates of the treatment effect on the outcome variable would be
biased.
18
19
www.barra.com
See for example, Alford, Jones, and Zmijewski (1994), and Fama and French (1992).
20
The solution to this endogeneity problem proposed by Hahn et al. (1999) and
Jacob and Lefgren (2002) is to estimate the effect of treatment using an instrumental
variables approach. The first step is to define a treatment dummy, Ti , where Ti = 1
if stock i is in the value index and zero otherwise, and to regress the treatment
variable on observable characteristics which predict treatment, including a dummy,
Di , where Di = 1 if the observed BE ratio is above the estimated cutoff and zero
otherwise. The second step is to use only the variation in treatment correlated
with observable characteristics in estimating the treatment effect on comovement.
It follows that point estimates of this effect should be unaffected by the correlation
between treatment and unobservable characteristics.
Formally the first stage is to estimate the regression
Ti = λ0 + δ1ln(BE i ) + δ2Di + δ3ln(ME i ) + ηi
(3)
where MEi is market capitalization of stock i (the same used to define the observed
BE ratio) and other variables are as defined above. The second stage is then to
estimate
yi = λ0 + λ1 ln(BEi )i + λ2 E[Ti] + λ3 ln(MEi ) + vi
(4)
where E[Ti] is taken from the first stage, and yi is one of the estimated slope coef∗
∗
or βiV
. The approach
ficients from (1) over the post-rebalance window, either βiG
amounts to modeling the baseline relationship between BE ratios and comovement.
Given that this relation is modeled correctly, λ2 measures the effect of treatment on
the outcome variable for observations arbitrarily close to the BE cutoff, conditional
on log size.
The IV estimates are obtained for each rebalancing period and the cross-sectional
regression results are aggregated using an approach similar to Fama and McBeth
(1973). First, mean parameter estimates across rebalancing periods are calculated.
The IV covariance matrix of the coefficients in (4) is also obtained for each rebalancing period. For some given coefficient, the variance of the mean across rebalancing
periods is estimated simply as the sum of the variances divided by n2 where n is the
total number of rebalancing periods.
The estimated coefficients are then used to conduct the following test:
21
Test 2 (Regression Discontinuity Analysis) If style investor sentiment impacts comovement, then stocks whose BE ratios are just above the cutoff which defines the
S&P/Barra indices should comove more with the value index than stocks whose BE
ratios are just below the cutoff. Similarly, stocks whose BE ratios are just below the
cutoff should comove more with the growth index than stocks whose BE ratios are
just above the cutoff.
(T2-A) H0 : λ2 ≥ 0
∗
using βiG
as the outcome variable
(T2-B) H0 : λ2 ≤ 0
∗
using βiV
as the outcome variable
Proper estimates of λ2 rely on knowing the functional form of the relationship
between the outcome variable and the variable that determines treatment. In this
paper, the relationship between comovement for stock i and ln(BEi) is assumed to
be linear. Any non-linearities between these variables can be played out through λ2 .
Most regression discontinuity designs validate the baseline model by examining the
relationship between the sorting variable and the outcome variable over a period in
which no treatment occurred (e.g., Jacob and Lefgren, (2002). With the S&P/Barra
data, I not only observe these variables over the control sample during which no
treatment occurred, but I also know exactly which variables would have received
treatment had treatment been given. The finding that the coefficient on the treatment dummy representing index inclusion is not significant over the control sample
is evidence that the results estimated over the test sample are not being driven by
non-linearities.
Test 2 is conducted using daily data and weekly data. Similar to Test 1, the
∗
∗
estimation window to obtain βiG
and βiV
using daily data is the five month period
following the rebalancing month in which BE and ME are observed. For weekly data,
it is the eleven month period following the rebalancing month. To be included in the
cross sectional regression for any given rebalancing month, a stock must be in the
same S&P/Barra index over the entire estimation window. Regression discontinuity
designs commonly use data within some narrow range around the cut-off point. Here
a tradeoff exists since using a narrower range helps eliminate the effects of any non22
linearities, while using a wider range provides more degrees of freedom to estimate
the baseline relationship. For each cross-sectional regression, I only exclude stocks
whose log BE ratio is more than two standard deviations away from the mean log BE
ratio across stocks for that rebalancing month. This eliminates the effects of large
outliers possibly caused by mismeasured book values, but also provides a sample
sufficient to estimate the baseline relationship between BE ratios and comovement.
Results for Test 2 are reported in Table IV. Panel A reports results using daily
data and Panel B reports results using weekly data. In Table IV, the average relation
between BE ratios and βiG is found to be significantly negative in all cases, as should
be expected. For example, over the full test sample in Panel A, the average estimated
value of λ1 when the dependent variable is βiG is found to be -2.83 with a t-statistic
of -2.58. Moreover, the average relation between BE ratios and βiV is found to be
significantly positive in all cases.
Panel A of Table IV offers substantial evidence in favor of rejecting the null
hypotheses of Test 2 using daily data. For example, over the full test sample, the
∗
is the dependent variable is found to be -0.358 and is
average value of λ2 when βiG
significant at the five percent level. That is, stocks whose BE ratios are just above
the cutoff point (T reati = 1) comove on average about 36 percent less with the
growth index than stocks whose BE ratios are just below the cutoff point. Further,
∗
is the dependent variable
the average value of λ2 over the full test sample when βiV
is found to be 0.326 and is also significant at the 5 percent level. That is, stocks
whose BE ratios are just above the cutoff point comove on average about 32 percent
more with the value index than stocks whose BE ratios are just below the cutoff
point. Similar results are found over the period from 1998 though 2002 in Panel A.
However, none of these results are significant over the control sample (1981-1991).
This evidence thus suggests that results over the test sample are not driven by a
non-linear relation between log BE ratios and comovement.
Panel B of Table IV also offers evidence in favor of rejecting the null hypotheses
of Test 2 using weekly data. Over the full test sample, the average value of λ2 when
∗
is the dependent variable is found to be -0.395. The average value of λ2 over
βiG
∗
is the dependent variable is found to be 0.334. Both
the full test sample when βiV
23
∗
results are significant at the 5 percent level. Results using βiG
are somewhat difficult
to interpret however, since λ2 is significant at the 10 percent level using the control
sample. Using weekly data, some non-linearity may exist between log BE ratios
∗
and comovement. However, the same cannot be said for results using βiV
as the
dependent variable. Over the control sample in Panel B, the estimate of λ2 is not
significant.
Finally, Table IV also provides estimates of mean coefficients of the first stage
regression. First, note that R2 measures for the first stage regressions are all in
the range of 0.75 to 0.83, indicating the observed variables used in the first stage
regression effectively predict the treatment dummy, Ti . Next, note that a substantial
amount of the variation in expected treatment is driven by whether the observed BE
ratio is above the estimated cutoff. For instance, over the test sample in Panel A, the
estimated value of δ2 is 0.702 and is significant at the one percent level. These results
help verify the usefulness of the instruments used to predict treatment, and provide
additional assurance that measures of λ2 are driven by variation in comovement near
the cutoff since this is where the vast majority of variation in expected treatment
occurs.
In summary, the results of Table IV suggest the null hypotheses of Test 2 should
be rejected. Over the test sample, stocks whose BE ratios are just below the cutoff
which defines the two S&P indices comove significantly more with the value index,
while stocks whose BE ratios are just above the cutoff comove significantly more
with the growth index. These same results are not found among stocks over the
control sample, suggesting the effects identified over the test sample are not driven
by non-linearities between BE ratios and comovement.
C.
Marginal Portfolio Tests
If two stocks with identical risk fundamentals could be found in different indexes,
the impact of style investing on comovement could be identified by simply comparing
comovement of the two stocks. Since finding such stocks is not possible, an alternative approach is to compare comovement of portfolios that are similar in terms
of fundamentals, but are classified in different indexes. Two portfolios are created
24
that are rebalanced at the end of every December and June. The first is a valueweighted portfolio of growth stocks that just switched to the growth index, or that
will switch to the value index during the next rebalancing period. The second is a
value-weighted portfolio composed of value stocks that just switched to the value
index or that will switch to the growth index the next rebalancing period. Similar
portfolios and tests are conducted by Barberis, Shleifer, and Wurgler (2005) using
index constituent data for the S&P 500. I refer to such portfolios as “marginal
portfolios” since they are composed of stocks on the margin of index boundaries.
Both portfolios are constructed using monthly returns, which are regressed on the
contemporaneous monthly growth and value index returns,
rit = αi + βiG rGt + βiV rV t + εit ,
(5)
where rit is either the total return on the growth marginal portfolio (i = G) or the
value marginal portfolio (i = V ), rGt is the return on the growth index, and rV t is
the return on the value index. Stocks included in marginal portfolios are excluded
from the index returns to prevent measuring the relation between identical stocks.
The two regressions are estimated jointly by GMM, which allows construction of a
variance covariance matrix to compare differences in estimated parameters across
regressions.20 The estimated coefficients of the two regressions are then used to
conduct the following test:
Test 3 (Comovement of Marginal Portfolio Returns) If style investor sentiment impacts comovement, then the growth marginal portfolio should comove more with the
growth index, and the value marginal portfolio should comove more with the value
index.
(T3-A) H0 : βGG ≤ βV G
(T3-B) H0 : βV V ≤ βGV
20
The covariance matrix is constructed using the Newey West (1987) estimator and automatic
lag selection technique of Newey and West (1994).
25
If the growth marginal portfolio and value marginal portfolios share similar fundamental risk characteristics, then any additional comovement of the growth marginal
portfolio with the growth index and any additional comovement of the value marginal
portfolio with the value index can be considered excess comovement above and beyond what can be explained given fundamentals. The objective of Test 3 is to determine if such excess comovement exists. To determine if differences in fundamental
risk can explain the results, the same tests are conducted over the control sample.
The results for Test 3 are reported in Table V. Over the full test sample, the
growth marginal portfolio exhibits much stronger comovement with the growth index
than with the value index. The slope coefficient of the growth marginal portfolio on
the growth index (βGG ) is 0.875 with a t-statistic of 3.65 while the slope coefficient
on the value index (βGV ) is only 0.235 with a t-statistic of 1.11. Similarly over the
full test sample, the value marginal portfolio exhibits much stronger comovement
with the value index than with the growth index. The slope coefficient of the value
marginal portfolio on the value index (βV V ) is 0.920 with a t-statistic of 9.61 while
the slope coefficient on the growth index (βV G ) is only 0.339 with a t-statistic of
3.97.
Table V also provides evidence in favor of rejecting the null hypotheses of Test
3. Comovement between the growth marginal portfolio and the growth index is
stronger than that between the value marginal portfolio and the growth index. The
difference between βGG and βV G , 0.537, is significant at the 5 percent level. Similarly,
comovement between the value marginal portfolio and the value index is stronger
than that between the growth marginal portfolio and the value index. The difference
in βV V and βGV , 0.685, is significant at the 1 percent level. Somewhat stronger
results are found over the period from 1998 though 2002. The difference between
βGG and βV G is 0.695 with a t-statistic of 2.14, while the difference between βV V and
βGV is 0.964 with a t-statistic of 3.10.
The results are much weaker over the control sample. The difference between
βGG and βV G is only 0.130 and the result is not significant at any reasonable level of
significance, while the difference between βV V and βGV , 0.174, is significant at the 10
percent level. Hence, results over the control sample suggest that some of the excess
26
comovement of the value marginal portfolio with the value index could be due to
fundamental risk, similar to the findings of the index balancing tests using weekly
data in Table III. The same cannot be said for results using the growth marginal
portfolio. Further, although the difference in βV V and βGV over the control sample
is marginally significant, it is significantly less than the difference between βV V and
βGV over the test sample. Assuming regression estimates across time are independent
(estimation windows do not overlap), the difference-in-difference estimate,
[(βV V ) − (βGV )]T EST − [(βV V ) − (βGV )]CON T ROL
(6)
is 0.512 with a standard error of 0.28. From the perspective of a one-tailed test, this
difference-in-difference is significant at the 5 percent level.
In summary, the evidence presented in Table V suggests excess comovement exists between the growth marginal portfolio and the growth index, as well as excess
comovement between the value marginal portfolio and the value index. Since similar
differences in comovement cannot be identified using the control sample, it is difficult
to justify differences in comovement by differences in fundamental risk characteristics
of the two portfolios.
IV.
Empirical Tests for Trading Activity
The previous section used index definitions as an instrument to understand the
source of comovement among returns. The evidence suggests style investor sentiment drives the comovement of stocks with similar BE ratios in a manner that is
both statistically and economically meaningful. For the style investing story to be
complete, similar patterns should be found in trading activity. I first use the same
index definitions to first determine if index reclassification is associated with changes
in turnover comovement. I then provide evidence that index reclassification effects
what mutual funds hold, and therefore trade over the test sample, but not the control
sample.
27
A.
Index Balancing Tests for Turnover
Index rebalancing tests can also be performed using some measure of trading
activity as the dependent variable. The literature on trading activity in financial
markets is voluminous and several measures have been proposed.21 In this paper I use
turnover as the measure of trading activity. Daily turnover is defined as total volume
on day t divided by total shares outstanding as of t − 1 adjusted for splits. Weekly
turnover is defined as the sum of daily turnover over the week, from Wednesday close
to Wednesday close. Index turnover is measured as equally-weighted stock turnover
across stocks in the index. Under the investor sentiment hypothesis, stock turnover
should be more correlated with index turnover after the stock is reclassified in the
index.22
For each stock that switches to a new index, the following two regressions are
estimated
τit = αi + γiG τGt + γiV τV t + εit
t ∈ pre-rebalance window
∗
∗
rit = α∗i + γiG
τGt + γiV
τV t + ε∗it
t ∈ post-rebalance window
(7)
where τit is the turnover of some stock that switches to a new index, τGt is the
turnover on the growth index and τV t is the turnover on the value index. All stocks
that switch indices are excluded from the turnover indices before estimating the
regressions. Hence, index turnover is based on the same set of stocks both before
and after the rebalance month. Similar to the return regressions, each stock is
regressed on both indices simultaneously to control for variation in market-wide
shocks to turnover. Again, the pre- and post-rebalance windows using daily data are
the five-month intervals before and after each rebalancing month, and the pre- and
post-rebalance windows using weekly data are the eleven month intervals before and
after each rebalance month.
The estimated parameters of (7) are used to conduct the following test.
21
22
For a review of the literature, see Karpoff (1987), and Lo and Wang (2000).
In the special case that portfolio weights of style investors are proportional to value-weights,
style-driven turnover is identical across all stocks within the style.
28
Test 4 (Index Rebalancing Test for Turnover Comovement) If style investor sentiment has a meaningful influence on how a stock is traded, then on average, when a
stock switches to a new index, its gamma with the new index increases. In addition,
gamma with the old index decreases.
For stocks that switch to the growth index:
P
∗
(T4-A) H0 : ∆γ G = ni=1 (γiG
− γiG ) /n ≤ 0
Pn
∗
− γiV ) /n ≥ 0
(T4-B) H0 : ∆γ V = i=1 (γiV
For stocks that switch to the value index:
Pn
∗
(T4-C) H0 : ∆γ G = i=1 (γiG
− γiG ) /n ≥ 0
Pn
∗
− γiV ) /n ≤ 0
(T4-D) H0 : ∆γ V = i=1 (γiV
where n is the number of stocks switching to a given index.
Standard errors of coefficient estimates for Test 4 are measured in exactly the
same way as standard errors for the coefficient estimates of Test 1, the index rebalancing test for changes in return comovement.
Unlike returns, turnover is highly persistent. For example, using daily data,
autocorrelations are about 0.82 and 0.88 for the value and growth indexes over the
test sample. A simple Dicky Fuller test rejects the null of a unit root in turnover for
either index over either the test or control sample using either weekly or daily data.23
However, time-series regressions also reject the null of no deterministic trend in index
turnover, particularly across the test sample. Other authors have found evidence of
some kind of non-stationarity in turnover. In their study of weekly turnover, Lo
and Wang (2000) investigate four different detrending methods and find substantial
differences in the time-series properties of the different detrended series. With no
prior as to which detrending method is correct, these authors choose to use raw
turnover rather than any detrended version, and estimate time series regressions
using 5-year sub-periods as a means of addressing issues of non-stationarity. I take
23
Under the null that the true process for turnover contains a unit root and an intercept, the
standard error of the estimated autocorrelation coefficient in a regression of turnover on an intercept
and lagged turnover can be derived using the standard OLS formulas. See Hamilton (1994) for
details.
29
a similar approach in this paper rather than filter the data through a specific trend
process that may not be convincing to others. Point estimates of the regression
coefficients in (7) are obtained using raw turnover within the rebalancing windows,
which are either five or eleven months, depending on whether daily or weekly data
is used. These estimation windows are much shorter than the 5-year estimation
intervals used by Lo and Wang (2000) in their study of weekly turnover. The implicit
assumption is that the mechanisms governing turnover are relatively stable over these
five, and eleven month periods. Finally, given the strong persistence in turnover, the
regression models of (7) are also estimated including the lagged turnover of both
indexes as explanatory variables. Similar results were obtained.
Results for Test 4 are presented in Table VI using daily data and Table VII using
weekly data. Table VI provides evidence in favor of rejecting the null hypotheses
of Test 4. For stocks that switch from the value index to the growth index, the
results are particularly strong for index balancers over the period from 1998 through
2002. The estimate of γG over this period increases by 0.194 (148 percent) while the
estimate of γV changes by -0.291 (-31 percent). Both results are significant at the 5
percent level, providing evidence in support of rejecting the null hypotheses of Test 4.
Over the full test sample, estimated average changes in γG and γV are not significant
for stocks that switch to the growth index. This finding is somewhat consistent with
the results of Table II. Evidence that investor sentiment impacts return comovement
using daily data is strongest over the period from 1998 though 2002. For stocks
that switch from the growth index to the value index, significant changes in turnover
comovement are found using both the full test sample and the sub-period from 19982002. For instance, looking at index balancers over the period from 1998-2002, the
estimate of γG over this period changes by -0.139 (-90 percent) while the estimate of
γV increases by 0.311 (51 percent). The former result is significant at the 10 percent
level, while the latter result is significant at the 5 percent level, providing evidence
in support of rejecting the null hypotheses of Test 4. Again, none of these results
are found over the control sample from 1981 though 1991, with the exception that
the average estimated change in γV is positive and significant at the 10 percent level
for stocks that switch to the value index. However, the average value of γV for index
30
balancers that switch to the value index over this period appears to significantly
decrease.
Table VII presents further evidence in favor of rejecting the null hypotheses of
Test 4. The results can be compared with those of Table III. For nearly every period
in both Panels A and B of Table III for which there is evidence in support of the
style investing story for return comovement, there is evidence in favor of rejecting
at least one of the null hypotheses of Test 4 for the corresponding period and panel
in Table VII. There is also some evidence in favor of rejecting the null hypotheses
of Test 4 over the control sample for stocks that switch from the growth index to
the value index, again suggesting the weekly data is picking up other effects over the
pre- and post rebalance window for stocks that switch to the value index. However,
this is not the case for stocks that switch to the growth index. For example, the
increase in γG for index balancers that switch to the growth index over the period
from 1998 to 2002 is found to increase by .253 (from -.114) while the estimated value
of γV is found to change by -0.601 (-54 percent). Similar results are not found over
the control period.
In summary, despite the difficulty in working with turnover data which seem
to exhibit some non-stationarity, Tables VI and VII provide evidence in favor of
rejecting the null hypotheses of Test 4. When a stock switches to a new index,
evidence exists that turnover comovement increases with the new index and decreases
with the old index.
B.
Mutual Fund Holdings
If the S&P/Barra index definitions are correlated with what style investors hold,
then when a stock switches to a new index, style investors aligned with the style
of the new index should buy the stock, while style investors aligned with the style
of the old index should sell the stock. Mutual funds are one plausible channel by
which style investing flows reach the market. In this section, empirical methods
are presented to test whether fund holdings are influenced by the S&P/Barra index
definitions. Does the change in fund holdings of stocks that are reclassified to a
new index depend on the fund style? These tests therefore compliment the index
31
balancing test on turnover comovement and provide additional evidence that stylebased trading behavior is behind the patterns in prices documented above.24
The data used in these tests are quarterly mutual fund holdings from the CDA/
Spectrum database, collected from reports filed by mutual funds with the SEC and
from voluntary reports generated by the funds. The data begins March 1984 and
ends December 2002. To measure a fund manager’s style, the method of Kacperczyk,
Sialm, and Zheng (2004) is used. Each stock traded on the major U.S. exchanges is
grouped into respective quintiles according to its size (ME) and BE ratio. Using the
quintile information, the value-weighted size score and value score for each mutual
fund in each quarter is computed. For example, a fund that invested only in stocks
in the largest ME quintile and the smallest BE quintile would have a size score of
5 and a value score of 1. Mutual funds in the CDA/Spectrum tapes classified as
“aggressive growth”, “growth”, or “growth and income” were used in the analysis.
These funds are primarily domestic-equity funds.25
Fund holdings are examined before and after each S&P/Barra index rebalancing
month (June and December of each year). In particular, fund holdings are examined
at the end of March and June (for the June rebalancing month) and at the end of
September and December (for the December rebalancing month). The set of stocks
that switch from one index to the other during each rebalancing month is determined.
The change in fund holdings of these stocks around index rebalancing months is then
measured. First, for each stock that switches in June (of any given year), the change
in fund holdings for each fund is measured as the difference in the fraction of total
shares outstanding held at the end of March and at the end of June. Likewise, for
each stock that switches in December, the change is measured as the difference in
the fraction of total shares outstanding held by each fund at the end of September
and at the end of December.
In the CDA/Spectrum data, if the fund holds less than 1 million shares of the
24
Of course, there are many other channels by which style-based flow could reach the market.
This paper focuses on mutual funds because of the availability of mutual fund holdings data.
25
Funds classified as “international”, “municipal”, “bond & preferred”, “balanced”, “metals”,
and “unclassified” were removed from the analysis.
32
stock, then holdings of that stock are not reported. To deal with this issue, for a
fund-stock pair to be included in the analysis, the fund must report holding some
shares of the stock prior to the rebalancing month. If no record of any holdings
exists for the fund-stock pair at the end of the rebalancing month, I assume the fund
owned zero shares of the stock at the end of the rebalancing month. This assumption
is made only when the fund reported holdings of some other kind in that quarter.
If the fund reported no holdings of any stock at the end of the rebalancing month,
then the fund is removed from the analysis for that rebalancing period.
For each fund, the mean change in stock holdings over each rebalancing period is
then calculated separately for stocks that switch to value and for stocks that switch
to growth, ∆HikV and ∆HikG . For example, ∆HikV is the mean change in holdings
for fund i over rebalancing period k for stocks that switched from the growth to the
value index.
After applying the above screens, the data include on average about 453 unique
funds each rebalancing month over the control sample and an average of about 1482
unique funds each rebalancing month over the test sample. During the period from
1984 through 1991, these mutual funds held an average of about 5% of the total
market capitalization of stocks that would have been reclassified to a new S&P/Barra
index had the indices existed, and during the period 1992-2002, these funds held an
average of about 14% of the total market capitalization of stocks that were reclassified
to a new index.
Summary statistics related to the styles and holdings of the funds used in the
analysis are reported in Table VIII. For each fund and each rebalancing period,
the mean change in holdings of stocks that switched to the growth/value index was
first calculated ∆HikG and ∆HikV . Time-series averages of cross-sectional summary
statistics were then calculated. Panel A is for the period from 1992 to 2002 while
panel B is for the period from 1984 to 1991. Fund attributes (value score, size score
and log fund size) are all calculated prior to each rebalancing period.
In panel A, there does not appear to be much difference in average styles of funds
that held stocks that switched to the growth index and funds that held stocks that
switched to the value index. For example, over the test sample, the mean value score
33
of funds that held stocks that switched to the growth index is 1.73 with a standard
deviation of 0.35, while the mean value score of funds that held stocks that switched
to the value index is also 1.73 with a standard deviation of 0.36.
For each rebalancing period the following two sets of cross-sectional regressions
are estimated:
∆HikG = θk0 + θk1 Vik + θk2 Sik + θk3 Mik + εikG
(8)
∆HikV
(9)
= φk0 + φk1 Vik + φk2 Sik + φk3 Mik + εikV ,
where Vik , Sik , and Mik are the value score, size score, and the log market capitalization of fund i’s holdings at the beginning of rebalancing period k. The results
are then aggregated across rebalancing periods by calculating the mean estimated
regression coefficients across rebalance periods k, separately for (8) and (9). The
main objective is to determine if on average, θk1 is negative and φk1 is positive. If
θk1 is negative, then around index rebalancing periods, growth-style funds tend to
increase holdings of stocks that switch to the growth index more than value funds.
Likewise, if φk1 is positive, then value style funds tend to increase their holdings of
stocks that switch to the value index more than growth funds around rebalancing
periods.
Standard errors for the mean regression coefficients are calculated using an approach similar to Fama and McBeth (1973). Since the number of rebalancing periods is small, instead of using the time series of estimated coefficients to estimate
the standard errors, the OLS covariance matrices of the coefficients in (8) and (9)
are obtained for each rebalancing period. For some given coefficient, the variance of
the mean across rebalancing periods is estimated simply as the sum of the variances
divided by n2 where n is the total number of rebalancing periods.
The relevant empirical tests are now defined.
Test 5 (Mutual Fund Holdings) If the S&P/Barra Value and Growth index definitions influence the holdings of mutual fund managers, then on average, growth
managers increase their holdings of stocks that switch to the growth index relative to
value managers, and value managers increase their holdings of stocks that switch to
34
the value index relative to growth managers.
(T5-A) H0 : θ̄1 ≥ 0
(T5-B) H0 : φ̄1 ≤ 0
The results for Test 5 are presented in Table IX. Panel A of Table IX gives results
for all stocks that switch. All estimated coefficients in this table are scaled by 100.
In both the test and control samples, the average level of θ1 is found to be negative
and significant, and the average level of φ1 is found to be positive and significant.
This implies that over both samples, funds with higher value scores were inclined to
increase their holdings of stocks that switched to the value index while funds with
lower value scores were inclined to increase their holdings of stocks that switched to
the growth index. Given that BE ratios of stocks that switch to the value (growth)
index are on average increasing (decreasing), the fact that the result is found over
both samples should not be surprising.
Panel B of Table IX however, gives results using only the index balancing stocks.
For the test sample, the average level of θ1 is again found to be negative and significant, and the mean level of φ1 is found to be positive and significant. Interestingly,
over the control sample, the mean levels of θ1 and φ1 are not found to be significant.
The results therefore suggest that index definitions did have an effect on mutual fund
holdings of the index balancers over the test sample.
The average level of θ1 in panel B over the period from 1992-2002 is found to
be -0.00033. Using the summary statistics from Table VIII, this implies that a onestandard deviation increase in value score of a single fund that held these stocks
before the rebalancing period (0.35) is associated with the fund selling an additional
0.01 percent (or buying 0.01 percent less) of the total shares outstanding of the stock
during the rebalancing period. Aggregated across many funds, this small relative
change in holdings becomes economically meaningful. The average level of φ1 in
Panel B over the test sample is found to be 0.00012. Again, using the summary
statistics from Table VIII, this implies that a one-standard deviation increase in
value score of a single fund that held these stocks before the rebalancing period
(0.36) is associated with the fund purchasing an additional 0.004 percent (or selling
35
0.004 percent less) of the total shares outstanding of the stock during the rebalancing
period. Again, when aggregated across many funds, this small relative change in
holdings can become economically meaningful.
In summary, the results suggest that index definitions do influence what mutual
funds hold. Funds with higher value scores tend to buy (or sell less) shares of index
balancers that switch to the value index while funds with lower value scores tend to
buy (or sell less) shares of index balancers that switch to the growth index. These
results are not found over the control sample, and thus provide further evidence that
the index definitions influence the trading behavior of style investors.
V.
Conclusion
This paper investigates the source of comovement among stocks with similar BE
ratios. The traditional explanation for comovement is that common factors exist in
dividend or earnings news. This paper examines whether style investor sentiment
may generate comovement among stocks with similar BE ratios.
The empirical tests of this paper lead to five main results. First, when a stock
switches from one index to the other, return comovement with the new index increases significantly. Second, stocks with similar fundamentals in different indices
differ significantly in their level of comovement with the two indices. Third, turnover
comovement with an index increases significantly when a stock switches to the index.
Fourth, the value and growth definitions of S&P/Barra are a significant determinant
of what value and growth style funds hold in their portfolios. Fifth, and perhaps
most importantly, these patterns are all identified over the period from 1992 though
2004. Similar empirical patterns do not exist among the universe of stocks that would
have been in the indices from 1981 to 1991. The indices did not exist before 1992.
In summary, using a wide variety of methodologies, the results of this paper strongly
suggest that style investor sentiment has an important effect on the comovement of
stocks with similar BE ratios.
An interesting avenue of future research would be to explore whose trades, institutions or individuals, are leading to such patterns in comovement, and how such patterns may change as relative institutional holdings increase. More importantly, one
36
of our most pressing concerns as economists is that capital be allocated to the most
promising investment opportunities. In capital markets, stock prices serve as a signal
to allocate capital. The finding that limits-to-arbitrage allow investor sentiment to
impact prices in a manner unrelated to fundamentals among liquid, highly-analyzed
stocks such as those in the S&P 500 should be somewhat disturbing. Further research should focus on understanding how to overcome these limits-to-arbitrage, so
that stock prices can more effectively function as signals to efficiently allocate capital.
37
Appendix A. Consistent Standard Errors for Index Balancing Tests
The pre- and post-rebalance windows do not overlap in a manner suitable for
the block bootstrap approach using the weekly data. Hence, I calculate consistent
standard errors in a simple manner, assuming residuals are i.i.d., that accounts for
the cross-sectional dependence in regression coefficients. For example, the variance
of the test statistic ∆β G , can be written
#
" n n
1 XX
∗
∗
∗
V ar ∆β G = 2
Cov(βiG
, βjG
) + Cov(βiG, βjG ) − 2Cov(βiG
, βjG ) . (10)
n i=1 j=1
Coefficients are assumed to be independent across regressions if estimation windows
do not overlap. Otherwise, the covariance matrix of coefficients across regressions is
estimated using a generalization of the standard OLS formula.
For illustration, let Xi be the n × k matrix of explanatory variables for regression
i, and Xj be the p × k matrix of explanatory variables for regression j. Assume some
observations of both regressions are observed on the same date. If the residuals of
each regression are i.i.d. across time and correlated cross-sectionally, the parameter
covariance matrix across regressions, Σ̂ij , can be consistently estimated as
Σ̂ij = (X0i Xi )−1 X0i σ̂ij Ωij Xj (X0j Xj )−1
(11)
where the matrix Ωij is an n × p matrix with element (a, b) equal to one if the ath
row of Xi and the bth row of Xj are observed on the same date and zero otherwise.
The contemporaneous covariance among residuals, σ̂ij is estimated as
σ̂ij = ˆ0i ˆj /T
(12)
where ˆi and ˆj are the vectors of overlapping residuals and T is the number of
observations which overlap.
Using this approach, I estimate the covariance matrix of estimated parameters
across regressions, obtain the elements of (10), and calculate the standard error of
∆β G as the square-root of the variance. Standard errors of other test statistics for
the index rebalancing tests are obtained in a similar manner.
38
Appendix B. Index Balancers and Estimated Regression Coefficients
Some results of this paper rely on estimates of conditional mean slope coefficients,
that is, mean slope coefficients of all stocks whose return over the time period is
positive or negative. This appendix shows that the conditioning information does
not bias the estimated mean slope coefficients in any manner. On the other hand,
estimates of conditional mean intercepts are probably biased. None of the tests in
this paper rely on estimates of the mean intercept. For clarity, I consider a regression
with an intercept and only a single explanatory variable. The same results can be
obtained if more explanatory variables are included in the regression at the cost of
extra math, and no extra added intuition.
Assume the objective is to estimate the parameters of the time series regression
yit = αi + βi xit + eit
(13)
for several time series i = 1, ..., M , and obtain an estimate of the conditional mean
slope coefficient
(β̄|i ∈ Ω) =
X
βi /N
(14)
i∈Ω
where Ω represents some subset of the time series used to estimate the regression and
N is the number of elements in the subset. For example, if yit represents stock returns
Ω could be the subset for which the total return over the period was abnormally large.
Plugging in the usual least-squares estimate for β into (14) and taking expectations gives
"
E
X
i∈Ω
#
"
X
X P yit xit − P xit P yit
1
tP
t
t
P
β̂i/N ≡ E
=
βi /N
2
2
N
t xit − (
t xit )
#
i∈Ω
(15)
i∈Ω
As long as xit is independent of the “true” error term, eit, β̂ is an unbiased estimate
of βi , implying the estimated conditional slope coefficient is an unbiased estimate of
the actual conditional mean slope.
The same cannot be said of the conditional mean intercept, which can be estimated as
"
#
! #
"
"
#
X X
X
X
X
1
1
E
α̂i /N ≡ E
yit − β̂i xit /T = E
(αi +
eit/T ) (16)
N
N
t
t
i∈Ω
i∈Ω
i∈Ω
39
where T is the number of time-series observations (assumed to be the same for all i).
P
Equation (16) shows the estimated conditional mean is biased, since E[ t eit /T |i ∈
Ω] is probably not equal to zero. For example, let yit represent stock returns and
assume Ω is the subset for which the total return over the period was abnormally
large. Such stocks are likely to have experienced abnormal positive shocks to the
P
“true” error terms, eit over the time period, causing E[ t eit/T |i ∈ Ω] 6= 0. Since
none of the tests in this paper reply on an estimate of the conditional mean intercept,
this is not an issue that impairs any of the paper’s results.
40
REFERENCES
Alford, A., J. Jones, and M. Zmijewski, 1994, Extensions and violations of the statuatory sec form 10-k filing date, Journal of Accounting and Economics 17, 229–254.
Angrist, Joshua D., and Victor Lavy, 1999, Using maimonides rule to estimate the
effect of class size on scholastic achievement, Quarterly Journal of Economics 114,
535–575.
Bansal, Ravi, Robert Dittmar, and Christian Lundblad, 2005, Consumption, dividends, and the cross section of equity returns, The Journal of Finance 60, 1639–
1672.
Barberis, N., and A. Shleifer, 2003, Style investing, Journal of Financial Economics
68, 161–199.
Barberis, Nicholas, Andrei Shleifer, and Jeffrey Wurgler, 2005, Comovement, Journal
of Financial Economics 75, 283–317.
Berk, R. A., and D. Rauma, 1983, Capitalizing on nonrandom assignment to treatments: A regression-discontinutiy evaluation of a crime-control program, Journal
of the American Statistical Association 78, 21–28.
Black, S., 1999, Do better schools matter? parental valuation of elementary eduaction, Quarterly Journal of Economics 114, 577–599.
Campbell, J., and R. Shiller, 1988a, The dividend-price ratio and expections of future
dividends and discount factors, Review of Financial Studies 1, 195–228.
, 1988b, Stock prices, earnings and expected dividends, Journal of Finance
43, 661–676.
Campbell, John Y., 1991, A variance decomposition for stock returns, The Economic
Journal 101, 157–179.
, Christopher Polk, and Tuomo Vuolteenaho, 2005, Growth or glamour?
fundamentals and systematic risk in stock returns, Working Paper.
41
Campbell, John Y., and Tuomo Vuolteenaho, 2004, Bad beta, good beta, American
Economic Review 94, 1249–1275.
Carhart, M., 1997, On persistence in mutual fund performance, Journal of Finance
52, 57–82.
Daniel, K., and S. Titman, 1997, Evidence on the characteristics of cross sectional
variation in stock returns, Journal of Finance 52, 1–33.
Davis, J., E. Fama, and K. French, 2000, Characteristics, covariances, and average
returns, Journal of Finance 55, 389–406.
DeLong, J., A. Shleifer, L. Summers, and R. Waldman, 1990, Noise trader risk in
financial markets, Journal of Political Economy 98, 703–738.
Fama, E., and K. French, 1992, The cross-section of expected stock returns, Journal
of Finance 57, 427–465.
, 1993, Common risk factors in the returns on stocks and bonds, Journal of
Financial Economics 33, 3–56.
, 1995, Size and book-to-market factors in earnings and returns, Journal of
Finance 50, 131–155.
Fama, Eugene F., and Kenneth R. French, 1996, Multifactor explanations of asset
pricing anomalies, The Journal of Finance 51, 55–84.
Fama, Eugene F., and James D. MacBeth, 1973, Risk return and equilibrium: Empirical tests, Journal of Financial Political Economy 71, 607–636.
Froot, K., and E. Daborah, 1999, How are stock prices affected by location of trade?,
Journal of Financial Economics 53, 189–216.
Greenwood, Robin, 2005, Short- and long-term demand curves for stocks: Theory
and evidence on the dynamics of arbitrage, Journal of Financial Economics 75,
607–649.
42
Guryan, J., 2000, Does money matter? regression discontinuity estimates from education finance reform in massachussetts, Working Paper, University of Chicago.
Hahn, J., P. Todd, and W. van der Klaauw, 1999, Evaluating the effect of an antidiscrimination law using a regression-discontinuity analysis, NBER Working Paper
No. 7131.
Hamilton, James D., 1994, Time Series Analysis (Princeton University Press: New
Jersey).
Hansen, Lars Peter, John C. Heaton, and Nan Li, 2005, Consumption strikes back?
measuring long-run risk, Working Paper.
Harris, Lawrence, and Eitan Gurel, 1986, Price and volume effects associated with
changes in the s&p 500 list: New evidence for the existence of price pressures, The
Journal of Finance 41, 815–829.
Jacob, B., and L. Lefgren, 2002, Remedial education and student achievement: a
regression-discontinuity analysis, NBER Working Paper No. 8918.
Kacperczyk, M., C. Sialm, and L. Zheng, 2004, On the industry concentration of
actively managed mutual funds, Journal of Finance Forthcoming.
Karpoff, J., 1987, The relation between price changes and trading volume: A survey,
Journal of Financial and Quantitative Analysis 22, 109–126.
Kaul, Aditya, Vikas Mehrotra, and Randall Morck, 2000, Demand curves for stocks
do slope down: New evidence from an index weights adjustment, The Journal of
Finance 55, 893–912.
Lee, C., A. Shleifer, and R. Thaler, 1991, Investor sentiment and the closed-end fund
puzzle, Journal of Finance 46, 75–110.
Lo, Andrew W., and Jiang Wang, 2000, Trading volume: Definitions, data analysis,
and implications of portfolio theory, The Review of Financial Studies 13, 257–200.
43
Lynch, Anthony W., and Richard R. Mendenhall, 1997, New evidence on stock price
effects associated with changes in the s&p 500, The Journal of Business 70, 351–
383.
Newey, W. K., and K. D. West, 1987, A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703–708.
, 1994, Automatic lag selection in covariance estimation, Review of Economic
Studies 61, 631–653.
Petkova, Ralitsa, and Lu Zhang, 2005, Is value riskier than growth?, Journal of
Financial Economics 78, 187–202.
Shleifer, Andrei, 1986, Do demand curves for stocks slope down?, The Journal of
Finance 41, 579–590.
Shleifer, Andrew, and Robert W. Vishny, 1997, The limit of arbitrage, Journal of
Finance 52, 35–55.
Thistlewaite, D., and D. Campbell, 1960, Regression discontinuity analysis: An
alternative to the ex-post facto experiment, Journal of Educational Psychology
51, 309–317.
Trochim, W., 1984, Research design for Program Evaluation: The Regression Discontinuity Analysis (Sage Publications: Beverly Hills, CA).
Vassalou, M., 2003, News related to gdp growth as a risk factor in equity returns,
Journal of Financial Economics 68, 47–73.
Vassalou, Maria, and Jimmy Liew, 2000, Can book-to-market, size, and momentum
be risk factors that predict economic growth?, Journal of Financial Economics
57, 221–245.
Vijh, A., 1994, S&p trading strategies and stock betas, Review of Financial Studies
7, 215–251.
44
Table I
Summary Statistics and Factor Loadings
In Panel A, summary statistics for stocks in the S&P/Barra value and growth indices are reported using end-of-month data. Mean # ref ers to the mean number of stocks in
each index across time. Turnover is the equally weighted cross-sectional mean monthly turnover across stocks in each index. Index returns are value-weighted.
Correlations among the indices are also reported. To calculate the correlation of an index with itself, stocks in the index were randomly split into two portfolios and
monthly value weighted returns were calculated. All stocks in each index were used when calculating the correlation of the value index with the growth index. Panel B
presents loadings of the S&P 500/Barra Value and Growth indices on the Carhart (1997) four-factor model. The factor loadings and their estimates were obtained by jointly
estimating the regressions
r Gt = β 0 + β 1 MKT t + β 2 SMB t + β 3 HML t
r Vt = β 0 + β 1 MKT t + β 2 SMB t + β 3 HML t
by GMM, where rGt is the return on the growth index, rVt is the return on the value index, and MKT, SMB, HML, and MOM are the market, size, book-to-market and
momentum factors. The t-statistics were calculated by GMM to compare estimated parameters across the two indices in each panel and period.
Panel A. Summary Statistics
June 1992- December 2004
Returns
Turnover
Return Correlation
(percent)
(percent)
Matrix
mean stdev mean
stdev Growth Value
Growth 155.58 0.97
4.66
13.53
5.75
0.92
.
Value
340.85 1.00
4.19
10.73
3.79
0.76
0.95
ρGG-ρGV ρVV-ρGV
t-stat difference
-(0.11) (1.54) (4.59) (4.34) (3.06) (3.56)
Mean #
Growth
t-stat
Value
t-stat
difference
t-stat difference
May 1981- May 1992 (Control)
Turnover
Returns
Return Correlation
(percent)
(percent)
Matrix
mean stdev mean stdev Growth Value
206.40 1.46
4.54
6.62
1.32
0.96
.
292.75 1.53
4.06
6.45
1.49
0.90
0.96
ρGG-ρGV ρVV-ρGV
-(0.42) (2.61) (1.55) -(2.81) (3.49) (3.63)
Mean #
Panel B. Factor Loadings
June 1992- December 2002
Inter. Rm-Rf SMB
HML MOM
May 1981- May 1992 (Control)
Inter. Rm-Rf SMB HML MOM
0.561 0.919 -0.318 -0.383 0.067
(6.14) (34.70) -(10.14) -(14.73) (3.17)
0.788 0.962 -0.245 -0.302
(8.44) (44.02) -(6.69) -(9.65)
0.271 1.050 -0.097
(4.05) (49.20) -(4.89)
0.400 -0.127
(15.05) -(8.67)
0.629 1.041 -0.186 0.338 -0.089
(13.20) (69.38) -(9.71) (10.74) -(4.51)
-0.289 0.132
-(2.14) (3.05)
0.783 -0.193
(30.13) -(9.17)
-0.160 0.079 0.059 0.640 -0.186
-(1.46) (2.59) (0.76) (20.48) -(6.35)
0.220
(3.73)
0.097
(3.30)
Table II
Change in Daily Return Comovement: Test 1
Daily returns of stocks (r it ) that switched among the S&P/Barra Growth and Value indices are regressed on the value-weighted average returns of the indices (r Gt and
r Vt ) before and after the switch in the following manner
r it = α + β G r Gt + β V r Vt +e it.
The measures of average index returns exclude the returns of stocks that switched. The indices are rebalanced every June and December. The pre-event window is the 5month period before the rebalancing month, and the post-event window is the 5-month period after the rebalancing month. Average changes in β G and β V across the preand post event windows and across all stocks that switched indices are in columns labeled “Δ”. OLS t-statistics and bootstrap p-values, both of which take into account
the overlapping estimation windows, are in parentheses. Columns labeled “Level” report average levels of the parameter values over the pre-event window, which are all
significant at the 1% level. Index balancers are stocks that switched to the growth index whose 5-month return was negative prior to the switch and stocks that switched to
the value index whose 5-month return was positive prior to the switch. Results over the period from 1981 through 1991 exclude the crash of October 1987. Significance
of the one-tailed tests described in the paper at the 1%, 5%, and 10% levels is indicated respectively by ***, **, and *.
βG
T1-A OLS t-statistic
Bootstrap p-value
βV
T1-B OLS t-statistic
Bootstrap p-value
βG
T1-C OLS t-statistic
Bootstrap p-value
βV
T1-D OLS t-statistic
Bootstrap p-value
Panel A. Stocks that Switch from the Value Index to the Growth Index
1992-2004
1998-2002
All Switchers
Index Balancers
All Switchers
Index Balancers
N=38
N=399
N=60
N=156
Δ
Level
Δ
Level
Δ
Level
Δ
Level
0.054
0.311
0.215
0.314
0.146
0.286
0.264
0.306
(1.62) *
(2.54) ***
(3.04) ***
(2.87) ***
(0.08) *
(0.01) ***
(0.01) ***
(0.01) ***
0.040
(1.13)
(0.85)
0.604
-0.097
-(1.07)
(0.17)
0.597
-0.022
-(0.45)
(0.39)
0.591
-0.206
-(2.00) **
(0.04) **
0.613
Panel B. Stocks that Switch from the Growth Index to the Value Index
1992-2004
1998-2002
All Switchers
Index Balancers
All Switchers
Index Balancers
N=519
N=136
N=207
N=32
Δ
Level
Δ
Level
Δ
Level
Δ
Level
-0.047
0.417
-0.073
0.368
-0.073
0.518
-0.305
0.433
-(1.48) *
-(1.51) *
-(1.47) *
-(4.45) ***
(0.07) *
(0.09) *
(0.12)
(0.00) ***
0.017
(0.49)
(0.32)
0.639
0.090
(1.67) **
(0.06) *
0.614
0.033
(0.62)
(0.32)
0.695
0.444
0.611
(5.33) ***
(0.00) ***
1981-1991 (Control)
All Switchers
Index Balancers
N=406
N=75
Δ
Level
Δ
Level
-0.090
0.452
-0.321
0.447
-(1.88)
-(2.56)
(0.96)
(1.00)
0.141
(2.67)
(0.99)
0.500
0.441
(3.02)
(1.00)
0.508
1981-1991 (Control)
All Switchers
Index Balancers
N=466
N=128
Δ
Level
Δ
Level
-0.053
0.436
0.160
0.298
-(1.28)
(2.39)
(0.11)
(0.99)
0.063
(1.36) *
(0.14)
0.551
-0.119
-(1.57)
(0.93)
0.657
Table III
Change in Weekly Return Comovement - Test 1
Weekly returns of stocks (r it ) that switched among the S&P/Barra Growth and Value indices are regressed on the value-weighted average returns of the indices (r Gt and
r Vt ) before and after the switch in the following manner
r it = α + β G r Gt + β V r Vt +e it.
The measures of average index returns exclude the returns of stocks that switched. The indices are rebalanced every June and December. The pre-event window is the 11month period before the rebalancing month, and the post-event window is the 11-month period after the rebalancing month. Average changes in β G and β V across the preand post event windows and across all stocks that switched indices are in columns labeled “Δ”. OLS t-statistics which take into account the overlapping estimation
windows are in parentheses. Columns labeled “Level” report average levels of the parameter values over the pre-event window. Index balancers are stocks that switched
to the growth index whose 5-month return was negative prior to the switch and stocks that switched to the value index whose 5-month return was positive prior to the
switch, consistent with Table 2. Results over the period from 1981 through 1991 exclude the crash of October 1987. Significance of the one-tailed tests described in the
paper at the 1%, 5%, and 10% levels is indicated respectively by ***, **, and *.
βG
T1-A OLS t-statistic
βV
T1-B OLS t-statistic
βG
T1-C OLS t-statistic
βV
T1-D OLS t-statistic
Panel A. Stocks that Switch from the Value Index to the Growth Index
1992-2004
1998-2002
All Switchers
Index Balancers
All Switchers
Index Balancers
N=161
N=25
N=69
N=15
Δ
Level
Δ
Level
Δ
Level
Δ
Level
0.374
0.160
0.349
0.160
0.348
0.136
0.249
0.063
(4.62) ***
(2.45) ***
(3.27) ***
(1.51) *
-0.314
0.886
-(3.92) ***
-0.148
-(0.98)
0.701
-0.368
0.806
-(3.66) ***
-0.255
-(1.51) *
0.741
5.999
Panel A. Stocks that Switch from the Growth Index to the Value Index
1992-2004
1998-2002
All Switchers
Index Balancers
All Switchers
Index Balancers
N=240
N=57
N=87
N=10
Δ
Level
Δ
Level
Δ
Level
Δ
Level
-0.297
0.523
-0.282
0.474
-0.276
0.602
-0.349
0.443
-(3.81) ***
-(2.75) ***
-(2.35) ***
-(2.02) **
0.223
0.636
(2.83) ***
0.295
0.587
(2.56) ***
0.184
(1.57) *
0.662
0.289
(1.43) *
0.818
1981-1991 (Control)
All Switchers
Index Balancers
N=116
N=16
Δ
Level
Δ
Level
0.033
0.532
-0.244
0.640
(0.32)
-(0.98)
0.142
(1.26)
0.472
0.356
(1.23)
0.351
1981-1991 (Control)
All Switchers
Index Balancers
N=139
N=35
Δ
Level
Δ
Level
-0.271
0.610
-0.137
0.636
-(2.76) ***
-(0.80)
0.192
(1.82) **
0.522
-0.025
-(0.14)
0.464
Table IV
Cross-Sectional Tests on Return Comovement - Test 2
This table presents coefficient estimates of the following system of equations:
Treat i = δ 0 + δ 1 ln(BE i ) + δ 2 D i + λ 3 ln(MEi) + v i
y i = λ 0 + λ 1 ln(BE i ) + λ 2 E [Treat i ] + λ 3 ln(MEi) + e i
where yi is either β * or β * , one of the coefficients in the regression of returns on the value and growth indices
G
V
as in Table 1. The regressions are estimated for each rebaalncing period, where right hand variables are observed at
the end of the rebalacnig month, and left hand variables are observed at the end of the post-rebalance window.
Average coefficient estiamtes across rebalance periods are presented along with t -statistics. None of the estimated
values of λ3 or δ3 were found to be significant, and are not reported in the interest of clarity. Also estimated, but not
reported are the regression intercepts. Significance of the one-tailed tests described in the paper at the 1%, 5%, and
10% levels is indicated respectively by ***, **, and *.
Dependent Variable is βG
Dependent Variable is βV
T2-A: λ2≥0
T2-B: λ2≤0
λ1
λ2
λ1
λ2
First Stage
δ1
δ2
R2
Panel A. Daily Data
1992-2004
-0.283 *** -0.358 **
0.253 ***
0.326 **
0.163 ***
-(2.58)
(2.40)
(1.98)
(5.92)
-(2.10)
0.702 ***
0.762
(18.80)
1998-2002
-0.288 *** -0.344 **
0.253 ***
0.321 **
0.167 ***
-(2.67)
(2.35)
(1.94)
(5.97)
-(2.07)
0.704 ***
0.761
(18.76)
1981-1991 (Control)
-0.422 *** -0.212
0.425 ***
0.167
0.188 ***
-(3.08)
(2.91)
(0.85)
(5.41)
-(1.16)
0.683 ***
0.754
(17.23)
Panel B. Weekly Data
1992-2004
-0.284 **
-0.395 **
0.233 **
0.334 **
0.141 ***
-(2.17)
-(1.92)
(1.81)
(1.66)
(5.61)
0.750 ***
0.812
(20.93)
1998-2002
-0.220 **
-0.440 **
0.217 **
0.243
0.126 ***
-(2.16)
-(2.26)
(2.15)
(1.26)
(6.96)
0.760 ***
0.828
(22.73)
1981-1991 (Control)
-0.429 *** -0.327 *
0.454 ***
0.259
0.156 ***
-(2.85)
(2.69)
(1.15)
(4.94)
-(1.64)
0.746 ***
(19.91)
0.823
Table V
Monthly Return Comovement of Marginal Portfolios - Test 3
Marginal portfolio returns are jointly regressed on the returns of the S&P/Barra Value and
Growth indices in the following manner,
r Gt = α G + β GG r Gt + β GV r Vt
r Vt = α V + β VG r Gt + β VV r Vt
where r Gt and r Vt are monthly returns on the marginal growth and value portfolios, and r Gt
and r Vt are returns on the S&P/Barra Value and Growth indices. The marginal growth
portfolio is composed of stocks that are in the growth index and either 1) just switched from
the value index or 2) will switch to the value index in the subsequent rebalancing month.
Similarly, the marginal value portfolio is composed of stocks that are in the value index and
either 1) just switched from the growth index or 2) will switch to the growth index in the
subsequent rebalancing month. Portfolio returns are value weighted and portfolios are
rebalanced at the end of each June and December, the months in which the S&P/Barra
Value and Growth indices are rebalanced. GMM t-statistics are in parentheses.
Significance of the one-tailed tests described in the paper at the 1%, 5%, and 10% levels is
indicated respectively by ***, **, and *.
1992-2004
Marginal Growth Portfolio
β GG
0.875 ***
(3.65)
β GV
0.235
(1.11)
T3-B
βVV - βGV
0.685 ***
(2.71)
1998-2002
Marginal Growth Portfolio
βGV
βGG
1.060 ***
0.019
(4.31)
(0.08)
Marginal Value Portfolio
β VG
0.339 ***
(3.97)
β VV
0.920 ***
(9.61)
T3-A
βGG - βVG
0.537 **
(1.86)
T3-A
βGG - βVG
0.695 **
(2.14)
Marginal Value Portfolio
βVG
βVV
0.365 ***
0.983 ***
(3.10)
(8.36)
T3-B
βVV - βGV
0.964 ***
(3.10)
1981-1991 (Control)
Marginal Growth Portfolio
Marginal Value Portfolio
βGV
βVG
βVV
βGG
0.498 ***
0.477 ***
0.368 ***
0.651 ***
(6.23)
(6.22)
(4.41)
(8.55)
T3-A
βGG - βVG
0.130
(0.98)
T3-B
βVV - βGV
0.174 *
(1.41)
Table VI
Change in Daily Turnover Comovement - Test 4
Daily turnover of stocks (τ it ) that switched among the S&P/Barra Growth and Value indices are regressed on the equal-weighted average turnover of the indices (τGt and
τ Vt ) before and after the switch in the following manner
τ it = α + γ G τ Gt + γ V τ Vt +e it.
Measures of average index turnover exclude the turnover of stocks that switched. The indices are rebalanced every June and December. The pre-event window is the 5month period before the rebalancing month, and the post-event window is the 5-month period after the rebalancing month. Average changes in γG and γV across the preand post event windows and across all stocks that switched indices are in columns labeled “Δ”. OLS t-statistics and bootstrap p-values, both of which take into account the
overlapping estimation windows, are in parentheses. Columns labeled “Level” report average levels of the parameter values over the pre-event window, which are all
highly significant at the 1% level. Index balancers are stocks that switched to the growth index whose 5-month return was negative prior to the switch and stocks that
switched to the value index whose 5-month return was positive prior to the switch. Results over the period from 1981 through 1991 exclude the crash of October 1987.
Significance of the one-tailed tests described in the paper at the 1%, 5%, and 10% levels is indicated respectively by ***, **, and *.
γG
T4-A OLS t-statistic
Bootstrap p-value
Panel A. Stocks that Switch from the Value Index to the Growth Index
1992-2004
1998-2002
All Switchers
Index Balancers
All Switchers
Index Balancers
N=399
N=60
N=156
N=38
Δ
Level
Δ
Level
Δ
Level
Δ
Level
0.047
0.296
0.017
0.211
0.063
0.242
0.194
0.131
(0.75)
(0.15)
(0.94)
(2.23) **
(0.26)
(0.43)
(0.23)
(0.02) **
γV
T4-B OLS t-statistic
Bootstrap p-value
0.004
(0.05)
(0.55)
0.775
-0.081
-(0.53)
(0.30)
0.861
0.085
(0.78)
(0.70)
0.746
-0.291
-(1.89) **
(0.06) *
0.952
γG
T4-C OLS t-statistic
Bootstrap p-value
Panel A. Stocks that Switch from the Growth Index to the Value Index
1992-2004
1998-2002
All Switchers
Index Balancers
All Switchers
Index Balancers
N=519
136
N=207
N=32
Δ
Level
Δ
Level
Δ
Level
Δ
Level
-0.205
0.487
-0.244
0.422
-0.177
0.464
-0.139
0.154
-(3.36) ***
-(2.71) ***
-(1.89) **
-(1.39) *
(0.00) ***
(0.01) ***
(0.05) **
(0.08) *
γV
T4-D OLS t-statistic
Bootstrap p-value
0.122
(1.43) *
(0.09) *
0.721
0.099
(0.83)
(0.27)
0.612
0.057
(0.39)
(0.35)
0.878
0.311
(1.99) **
(0.06) *
0.616
1981-1991 (Control)
All Switchers
Index Balancers
N=406
N=75
Δ
Level
Δ
Level
-0.100
0.603
0.014
0.368
-(1.35)
(0.08)
(0.92)
(0.46)
0.098
(1.28)
(0.89)
0.437
0.055
(0.27)
(0.60)
0.553
1981-1991 (Control)
All Switchers
Index Balancers
N=466
N=128
Δ
Level
Δ
Level
-0.041
0.516
0.289
0.286
-(0.63)
(2.68)
(0.27)
(1.00)
0.099
(1.40) *
(0.10) *
0.413
-0.352
-(3.05)
(1.00)
0.651
Table VII
Change in Weekly Turnover Comovement - Test 4
Weekly turnover of stocks (τ it ) that switched among the S&P/Barra Growth and Value indices are regressed on the equal-weighted average turnover of the indices (τGt and
τ Vt ) before and after the switch in the following manner
τ it = α i + γ iG τ Gt + γ iV τ Vt +e it.
Measures of average index turnover exclude the turnover of stocks that switched. The indices are rebalanced every June and December. The pre-event window is the 11month period before the rebalancing month, and the post-event window is the 11-month period after the rebalancing month. The average change in γiG and γ iV across the
pre- and post event windows and across all stocks that switched indices are in columns labeled “Δ”. OLS t-statistics which take into account the overlapping event
windows, are in parentheses. Columns labeled “Level” report average levels of the parameter values over the pre-event window. Index balancers are stocks that switched
to the growth index whose 5-month return was negative prior to the switch and stocks that switched to the value index whose 5-month return was positive prior to the
switch, consistent with Table 3. Significance of the one-tailed tests described in the paper at the 1%, 5%, and 10% levels is indicated respectively by ***, **, and *.
γG
T4-A OLS t-statistic
Panel A. Stocks that Switch from the Value Index to the Growth Index
1992-2004
1998-2004
All Switchers
Index Balancers
All Switchers
Index Balancers
N=161
N=25
N=69
N=15
Δ
Level
Δ
Level
Δ
Level
Δ
Level
0.116
0.320
0.269
-0.202
0.004
0.093
0.253
-0.114
(0.87)
(1.69) **
(0.04)
(1.48) *
γV
T4-B OLS t-statistic
-0.240
-(1.55) *
γG
T4-C OLS t-statistic
Panel A. Stocks that Switch from the Growth Index to the Value Index
1992-2004
1998-2004
All Switchers
Index Balancers
All Switchers
Index Balancers
N=240
N=57
N=87
N=10
Δ
Level
Δ
Level
Δ
Level
Δ
Level
-0.470
0.757
-0.291
0.490
-0.401
0.740
0.071
0.155
-(4.11) ***
-(2.06) **
-(2.48) ***
(0.44)
γV
T4-D OLS t-statistic
0.991
0.395
0.520
(2.83) ***
-0.461
-(2.17) **
0.111
(0.69)
1.203
0.611
-0.004
-(0.02)
0.186
(0.78)
0.883
0.597
-0.601
-(2.31) **
-0.315
-(1.34) *
1.108
0.678
1981-1991 (Control)
All Switchers
Index Balancers
N=116
N=16
Δ
Level
Δ
Level
0.208
0.396
-0.369
1.227
(0.99)
-(0.68)
-0.250
-(1.18)
0.875
-0.093
-(0.15)
0.318
1981-1991 (Control)
All Switchers
Index Balancers
N=139
N=35
Δ
Level
Δ
Level
-0.226
0.630
-0.427
0.541
-(1.88) **
-(2.26) **
0.171
(1.34) *
0.415
0.419
(2.00) **
0.325
Table VIII
Summary Statistics on Mutual Funds
This table presents time-series averages of cross-sectional summary statistics for the mutual fund panel data. The data includes only funds that
reported holding stocks that switched among the S&P/Barra Value and Growth indices before index rebalancing dates. Using data at the end
of the calendar quarter before each rebalancing date, cross sectional means, standard deviations, minimums and maximums were calculated
for each variable. Time-series averages of these statistics were then calculated across all rebalancing dates within the date range for each
panel. Value Score and Size Score are measures of fund style that range on a continuum from 1 to 5. Equity Holdings are the total market
capitalization of all equity held by the fund. Before and after each rebalancing period, fund holdings of each stock that swtiched to a new
index were calculated as a percent of total shares outstanding. The change in stock holdings was then calculated as the level of holdings after
the rebalancing period minus the level of holdings before the rebalancing period. For each fund and rebalancing period, the average change in
holdings was then calculated across all stocks that switched to a new index. Time series averages of cross sectional summary statistics on this
measure of holding changes are in the column labeled ΔH. The bottom line of each panel reports the average number of funds per rebalancing
date. For consistency with other tables, the period from 1981 through 1991 excludes the crash of October 1987.
Funds Holding Stocks that
Switch from Value to Growth
Mean
Stdev
Mean
Stdev
Funds Holding Stocks that
Switch from Growth to Value
Panel A. 1992-2002
ΔH
ΔH
Value Score Size Score Equity Holdings
Value Score Size Score Equity Holdings
(millions)
(percent)
(millions)
(percent)
1.73
4.99
859
0.01
1.73
4.99
799
0.02
0.35
0.05
2,906
0.10
0.36
0.05
2,724
0.10
Avg # Funds = 1058.2
Avg # Funds = 1205.3
Panel B. 1981-1991
ΔH
ΔH
Value Score Size Score Equity Holdings
Value Score Size Score Equity Holdings
(millions)
(percent)
(millions)
(percent)
2.41
4.98
254
0.02
2.42
4.98
240
0.03
0.48
0.07
523
0.10
0.48
0.06
502
0.13
Avg # Funds = 296.5
Avg # Funds = 337.1
Table IX
Index Definitions and Mutual Fund Holdings - Test 5
This table presents time-series averages of cross-sectional regression coefficients. For each rebalancing period, the following two crosssectional regressions were estimated:
ΔHiG = θ0 + θ1Vi + θ2Si + θ3Mi + εi
ΔHiV = φ0 + φ1Vi + φ2Si + φ3Mi + εi
where ΔHiG is the average change in the holdings of fund i of all stocks that switched from the S&P/Barra Value index to the Growth index,
ΔHiV is the average change in the holdings of fund i of all stocks that switched from the S&P/Barra Growth index to the value index, Vi is the
fund value score, Si is the fund size score, and Mi is the log market capitalization of total fund equity holdings. All right hand variables are
measured at the end of the calendar quarter before the rebalancing date. Before and after each rebalancing period, fund holdings of each stock
that switched to a new index were calculated as a percent of total shares outstanding. The change in stock holdings was then calculated as the
level of holdings after the rebalancing period minus the level of holdings before the rebalancing period. For each fund and rebalancing period,
the average change in holdings was then calculated across all stocks that switched to a new index. All coefiicients in this table are scaled by
100. Statistical significane at the 1%, 5% and 10% levels is indeicated respectively by ***, ** and *.
θ0
Value to Growth
T5-A θ1≥0
θ1
θ2
1992-2002
t -statistic
0.188 *
(1.67)
θ3
φ0
Panel A: All Stocks
-0.009 *** -0.065 *** 0.009 *** -0.175 **
-(3.47)
-(2.90)
(21.73)
-(2.45)
1984-1991
t -statistic
0.046
(0.21)
-0.015 *** -0.027
-(4.41)
-(0.63)
1992-2002
t -statistic
0.498
(1.42)
-0.033 *** -0.124 *
-(4.64)
-(1.77)
1984-1991
t -statistic
2.499 *
(1.90)
-0.008
-(1.12)
-0.521 ***
-(4.87)
Growth to Value
T5-A φ1≤0
φ1
φ2
φ3
0.019 *** 0.002
(8.73)
(0.16)
0.008 ***
(19.80)
0.008 *** -0.03369
(7.56)
-(0.20)
0.0141 *** -0.042
(3.10)
-(1.26)
0.0127 ***
(8.82)
Panel B: Index Balancers
0.011 *** 0.108
(7.97)
(0.27)
0.012 *** -0.050
(3.48)
-(0.63)
0.008 ***
(12.27)
-0.013
-(1.29)
0.0084 ***
(3.57)
0.0076 ***
(3.91)
0.6005
(1.31)
-0.14
-(1.54)
Figure 1. The S&P/Barra Value and Growth Indices
This figure plots the number of stocks in each of the S&P/Barra Value and Growth indices (Panel A), the performance of each index measured as the log value of
$1 invested in each index at the beginning of the period (Panel B), and average monthly turnover (Panel C). Each chart begins May 1981 and ends December
2004. The indices were first created in May 1992 (indicated by the solid vertical line in each panel). Data prior to May 1992 are based on index classifications if
the indices had existed.
Panel A. Number of Stocks
Panel B. Index Performance
Panel C. Average Turnover
Figure 2. Distribution of Returns Before Switching Indices
For stocks that switch among the S&P/Barra indices (in June or December of each year), this figure shows the distribution of the five-month return before the
switch. Panels A and B are for stocks that switch during the period 1992-2004. Panel A is for stocks that switch to the growth index while Panel B is for stocks
that switch to the value index. Panels C and D are for stocks that would have switched during the period 1981-1991 if the indices had existed over this time
period. Panel C is for stocks that switch to the growth index while Panel D is for stocks that switch to the value index.
Panel A. 1992 – 2004 Value to Growth
Panel C. 1981 – 1991 Value to Growth
Panel B. 1992-2004 Growth to Value
Panel D. 1981-1991 Growth to Value
Figure 3. Book-to-Market Ratios of the S&P/Barra Value and Growth Indices
This figure plots the cross-sectional average book-to-market ratio and decile for the S&P/Barra indexes after being rebalanced at the end of June and December
of each year. Each chart begins June 1981 and ends December 2004. The indices were first created in May 1992 (indicated by the sold vertical line in each
panel). Data prior to May 1992 is based on index classification had the indices existed over this period. The book-to-market ratios are constructed to be similar to
those use by S&P/Barra when rebalancing the indices. Equity value is defined as size (price time shares outstanding) at the end of May and November. Book
value is common equity reported in Compustat at the end of the latest fiscal quarter at least six months prior to the end of June or December.
Panel A. Average Book-to-Market Ratio
Panel B. Average Book-to-Market Decile