1. No category

Selectivity and Sample Bias in Dividend Drop-Off Studies
Michael D. McKenzie* and Graham Partington#
Abstract
There is a large literature that estimates dividend values from the dividend drop-off
ratio around company ex-dividend dates. This paper focuses on how choices relating
to the sample and method of measurement impact on the drop-off estimate. Specific
issues considered are the use of opening and closing ex-dividend prices, adjustments
for outliers, the impact of the market adjustment, thin trading, the size of the spread
and tick relative to the dividend, and ex-dividend event clustering. The results show
that conclusions regarding the drop-off ratio are sensitive to small changes in choices
about the sample and measurement. Thus, rather than focus on deriving a point
estimate of the drop-off ratio, it may be more sensible to talk in terms of a feasible
range.
Keywords: Dividend
Measurement
Drop-off,
Ex-Dividend,
Sample
Selection,
Dividend
Acknowledgements: The authors would like to thank Securities Industry Research
Centre of Asia-Pacific (SIRCA) for providing the data used in this paper.
*
Corresponding Author. The University of Liverpool and The University of Sydney. Email:
[email protected].
#
The Discipline of Finance, Faculty of Economics and Business, The University of Sydney 2006.
Email: [email protected].
Introduction
How much is a dollar of dividends is worth? Since Miller and Modigliani’s (1961)
paper, this question has been hotly debated. The answer is important in the
formulation of company financial policy, in valuation and in questions regarding the
cost of capital. Under the Australian imputation tax system, the value of dividends and
associated imputation tax credits has been a very contentious issue in regulatory
hearings. This is because the valuation directly affects the prices that regulated entities
are allowed to charge. Thus, the valuation of dividends is a matter of intense academic
and practical interest.
Following the work of Elton and Gruber (1970) ex-dividend studies have become the
method that is extensively used to empirically address dividend valuation. Such
studies provide an estimate of the drop-off ratio, which is the ratio of the ex-dividend
price drop to the dividend. However, there has been extensive debate about the values
produced by ex-dividend studies. Elton and Gruber estimated a drop-off ratio of 0.78,
but Kalay (1982) found that the drop-off ratio was not significantly different to one.
The considerable subsequent literature has produced a range of estimates and the
debate is ongoing (for example Elton Gruber and Blake, 2005, recently revisited this
issue and found support for the original Elton and Gruber, 1970, study).1
1
In addition to variation across studies, Eades, Hess and Kim (1994) showed that there was substantial
variation in the drop-off ratio through time. In Australia, Brown and Clarke (1993) also report
substantial yearly variation in drop-of ratios varying from 0.37 to 0.87, over the period 1974 to 1991.
Also in Australia, Beggs and Skeels (2006) examine the period from 1986 to 2004 and estimate dropoff ratios ranging from 0.43 to 0.78.
2
Variation in drop-off ratios across studies has meant that the debate over ex-dividend
valuation remains unresolved. One purpose of the current study is to demonstrate that
this debate is likely to remain unresolved because of the variation in drop-off
estimates that can arise due to modest changes in sample and/or choices about
methods of analysis. We show, for example, that switching between ex-dividend
prices observed at the beginning and end of the day reverses the conclusion about
whether or not the drop-off ratio is one, as does a simple scaling of the regression
equation used to estimate the drop-off ratio. Thus we caution against attaching too
much weight to any one ex-dividend based estimate of dividend value.
An important and potentially highly influential part of any study of ex-day dividend
drop-offs relates to the choice of the sample of data.2 On the one hand, it is desirable
to include as many observations as possible. For example, the seminal paper by Boyd
and Jagannathan (1994) discusses the need for a large sample to combat sampling
error, discreteness and parameter instability. They estimate their results using a
sample of 132,057 dividend events, with only minimal filtering to exclude dividends
that had more than one type of dividend being paid, or that were partly tax-paid.
2
A related literature has considered the choice of estimation technique and, in particular, how to deal
with the multicolinearity that exists between the size of the dividend and the franking credit (see
Bellamy and Gray (2004)). As the focus of this paper is on the data sample selection issue, we do not
specifically address the issue of estimation technique.
3
The need to assemble a large sample of data calls for an approach to sampling in
which all dividend events are included.3 This inclusive approach however, presents
the researcher with a number of unique challenges.
All theories of dividend policy ascribe a positive value to dividends. In theory,
therefore, we expect ex-dividend price movements to be strictly positive. In reality,
however, drop-off ratios estimated using unadjusted price data typically contain a
disproportionately high number of zero values. Where the price change is adjusted for
overnight movements in the market, these observations of zero drop-offs are masked.
Even if the stock price does not change, the market adjustment will ensure that some
price movement is recorded. This will generate a non-zero and possibly even negative
drop-off ratios.
Events other than the dividend, such as news releases, may also affect the price about
the ex-dividend event, reducing the signal (dividend value) to noise ratio. In order to
reduce the period over which such events may impact the stock price, it is possible to
use the closing cum-dividend price and the opening ex-dividend price, rather than the
more commonly used close to close prices. However, prices are not necessarily
observed at the close or open, or even on the cum-dividend or ex-dividend days. Thus,
stale cum or ex-prices introduce the possibility that the change in price observed on
the ex-dividend date is a mix of old information and the value of the dividend. Thus it
3
Further, the sample period itself is typically set as long as possible, which means the sampling of
dividend events over a long period of time. The use of a long data series however, is conditional on
stationarity in the underlying values of dividends and franking credits. The assumption must therefore
be made that the time series variation documented by Eades, Hess and Kim (1994) is not reflective of
true variation in the underlying value, but is a consequence of measurement problems
4
can be a challenge to separate the dividend valuation from the noise of extraneous
impacts on price.
In addition to stale prices, infrequently traded stocks may also be limited in their
ability to adjust to ex-dividend events due to wide bid-ask spreads. Specifically,
where the spread is large and the dividend relatively small, then the theoretical dropoff in the share price may be close to, or even less than, the spread in the stock.
Obviously, the ability of the change in market price to fully reflect the change in the
value of the stock is limited in such circumstances. The minimum tick size (see Bali
and Hite, 1998) is a further factor that may limit the ability of price changes to fully
account for the dividend.
One issue that has received relatively little attention in ex-dividend studies, is the
issue of the independence of the observations. Ex-dividend dates tend to be clustered
on certain days-of-the-week and at certain times of the year. As a consequence
observations will not be independent and the standard errors will be understated. One
solution is to form portfolios of stocks which are observed within a common time
period, but whether that period should be a day or longer is an open question.
Time clustering also has significance for the method of adjusting for overnight
movements of the market. Brown and Warner (1980) present evidence that the
adjustment for market movements should be firm specific where there is time
clustering in events. That is, they should reflect the sensitivity of the individual firm
to movements in the market. However, Australian ex-dividend studies, as in Beggs
and Skeels (2006), typically use the movement of the whole market to make the
5
adjustment. In effect, they assume that the sensitivity of stock returns to the market
(the equity beta) is one. Unfortunately, estimated betas for individual stocks are
frequently so unreliable that using firm specific adjustments is unlikely to be an
improvement and may make things worse. The alternative is to use industry betas,
which may be more reliably estimated.
One final issue relates to the presence of holidays and weekends in the dataset.
Where a dividend event is impacted by the occurrence of a holiday or weekend, this
means that the return is not measured across successive calendar days. So the return
likely incorporates a greater amount of news, which has accumulated over the longer
interval and the price movement observed is less indicative of the value of the
dividend. As such, a case may be made to exclude these observations and we
investigate what effect this has.4
The purpose of this paper is to empirically investigate the impact of the following on
the estimation of the dividend drop-off ratio:
•
the use of opening and closing ex-dividend prices,
•
adjustments for outliers,
•
the adjustment for market movements,
•
the effect of thin trading,
4
A case might also be made to exclude data from extremely volatile markets. For example, Beggs and
Skeels (2006) exclude “data from the extremely volatile month of October 1987” on the basis that
“price volatility in that month was the highest measured over the past 100 years” which “could distort
interpretation of results.” What constitutes an abnormally volatile period however, is not obvious.
Further, a similar argument could just as easily be made for other market metrics, such as trading
volume. Indeed, we may even extend this concept beyond market activity to the individual stock level.
Since objective criteria for such exclusions are difficult to establish, we do not consider such deletions
in the context of this paper.
6
•
the size of the spread and tick relative to the dividend
•
ex-dividend event clustering through time and by day of the week.
It is unknown what impact, if any, each of these issues may have on the estimation of
the dividend drop-off ratio. It is worth emphasising that the point of this paper is not
to suggest the optimal choice for data selection when undertaking dividend drop-off
studies. Rather, it is to help other researchers understand the likely consequences of
the choices that they make.
Data and Method
The sample of dividend event data used in this paper consists of all dividend events
for the ASX over the period July 1997 to February 2010. This data is provided by
SIRCA and consists of 13,080 unique observations. To ensure that the data is as close
as possible to that used in other dividend drop-off studies, we only consider those
dividends identified as interim or final. Further, as in Beggs and Skeels (2006) we
only consider dividends paid by ordinary class shares and trusts (SIRCA security type
1 or 6). Finally, we limit our sample to only those stocks we could match with the
Datastream database, which allows us to externally validate the price data. Thus, we
arrive at our final sample of 10,160 dividend events for 1,129 individual companies.
The empirical ex-dividend work typically uses regression estimates rather than
computing the drop-off ratio directly from raw data. The relationship between the
dividend price drop-off and the size of the dividend may simply be estimated as:
, − , = + + 7
(1)
where , is the cum-price for stock i, , is the ex-price, is the dividend, and are parameters to be estimated and is the residual. Beggs and Skeels (2006, p.6)
point out that dividend drop-off models, “are typically troubled by the presence of
heteroscedasticity”.5 To account for any potential variation in the variance, a white
heteroscedasticity consistent covariance matrix is specified for all regression
equations. If one takes the view however, that large residuals are simply outliers,
rather than the realization of large variance due to heteroscedasticity, then an
alternative adjustment may be appropriate. One possible solution is to simply exclude
the most influential observations (which may be determined using a measure such as
Cook’s D-statistic6).
However, this approach does suffer from the limitation of
reducing the sample size.
If the view is held that outliers are a feature of dividend drop-off data, their exclusion
may be viewed as a form of data mining, which leads to the omission of important
information about the inadequacy of the model. As Skeels (2009a, p. 13) puts it, ‘not
all influential observations are unreliable’ and ‘not all unreliable observations are
influential’. An alternative approach, is to use a robust regression method, such as
MM estimators as in Armitage, Hodgkinson and Partington (2006). This approach
offers the advantage of not excluding observations, but simply reweighting them to
lessen their influence. As such, the estimation results are still based on the complete
5
Note that Beggs and Skeels (2006, p.6) opt for the more complex method of specifying a functional
form for the variance of the errors and use a feasible generalised least squares estimator (FGLS), which
is asymptotically more efficient than the White adjustment. As the focus of this paper is not on the
econometrics associated with estimating drop-off ratios, we opt for the more general White adjustment.
6
Cook’s D-statistic is a useful technique to detect influential observations, however, it does have
limitations in the area of a group of observations being jointly significant. Further, it is entirely
arbitrary as to what percentage of observations are excluded.
8
sample of data, however the influence of the outlier observations is lessened resulting
in more reliable estimates.
Following Boyd and Jaganathan (1994), it has become common to express the
variables of Equation 1 relative to the cum-dividend price. After scaling by the cumdividend price Equation 1 becomes:
, − ,
=+
+ (2)
,
,
In estimating this equation we either specify a White heteroscedasticity consistent
covariance matrix for the errors, or a robust regression technique is applied.
Benchmark results
To provide a benchmark set of results, Equations 1 and 2 were applied to the sample
of raw data and the estimation results are presented in Table 1. Panel A presents the
results where the price change is measured from the cum-date close to the ex-date
open (close-to-open hereafter), and Panel B measures the drop-off price change from
the cum-date close to the ex-date close (close-to-close hereafter).
The results
highlight the variable nature of the dividend value estimates, which can lead to quite
different conclusions.
For example, consider the first row of panel A, for the raw data, using the close to
open returns. Equation (1) generated the higher coefficient estimate of 0.9018, which
is not significantly different from 1 at the 5% level, while Equation (2) generated the
lower coefficient estimate of 0.7858, which is significantly less than one. In the first
case we would conclude that dividends are fully valued and in the second case we
9
would conclude that they are priced at a discount to face value. In contrast, in the first
row of Panel B, using the close to close returns, it is Equation 1 that now provides the
lower value at 0.7980 (significantly less than one) with 0.8936 (not significantly
different from one for Equation (2). Thus, utilising the same ex-dividend events quite
different conclusions would be reached about the value of dividends depending on
which regression model was chosen, and using almost the same ex-dividend events
the conclusions would reverse depending on whether the price change was measured
close to close or close to open.
All slope coefficient estimates in Table1 are significantly different from zero at the
5% level and the constant terms tend to be mostly insignificant. This pattern is
repeated throughout subsequent results, although there is one insignificant slope
coefficient in Table 5. For the sake of brevity therefore, discussion of significance is
omitted in the analysis of subsequent results. Instead we focus on the variability of the
dividend valuation estimate, which is the central theme of this paper.
Outliers and partitioning the data
Outliers are a common feature of ex-dividend data and we investigate alternative
ways of handling this in Panel A and B of Table 1. These results provide the dividend
drop-off estimates generated using the MM robust regression model and the drop-off
ratio using a sample of data with the top 1% of influential outliers excluded (identified
using Cook’s D-statistic). The results reveal that the adjustment for outliers tends to
yield higher estimates of the drop-off ratio, but the close-to-open estimates are not
particularly sensitive to this adjustment. The close-to-close estimates differ rather
10
more. For example, the close-to-close MM robust regression estimate increases from
0.7980 for the raw data to 0.9987 for Equation (1) and in the case of Equation 2, the
robust regression estimate is greater than one (β = 1.0070).
The presence of noise in the data and measurement errors mean that it is common in
ex-dividend studies to partition the data into various samples, as a form of robustness
test. The most commonly used partitions segregate the data according to firm size,
dividend yield classes and, in Australian studies, by franking status. Such partitions
commonly show that the drop-off ratio increases with dividend yield classes, and we
also find this result using our data. For example, the drop-off ratio using close-toclose prices and Equation 1 for high dividend yield stocks is 0.88 (the top half of the
sample ranked by dividend yield), whereas for low dividend yield stocks it is 0.68 (the
bottom half of the sample) and 0.22 when only the bottom third of the sample is
considered. This is a consistent result in the literature from Elton and Gruber (1970)
onwards.
Larger firms also tend to have higher drop-off ratios (see Hathaway and Officer,
2004) and our data corroborates this finding.
For example, the drop-off ratio
estimated using close-to-close prices and Equation 1 for large stocks is 0.80 (the top
half of the sample ranked by size), whereas for small stocks it is 0.63 (the bottom half
of the sample) and 0.44 when only the bottom third of the sample is considered. This
result is most probably because the dividends of large firms tend to be larger and these
stocks are easier to trade (in particular the bid ask spread is lower). Further, stale
prices are less of a problem and dividend arbitrage is a more attractive proposition.
11
Partitioning the data between fully franked and unfranked dividends the drop-off ratio
is higher for fully franked dividends as in Bellamy (1994) and Truong and Partington
(2006). The drop-off ratio, measured using close-to-close prices and Equation 1, for
fully franked dividends is 0.9726, and for unfranked dividends it is 0.8623. However,
for partially franked dividends the estimate of 0.8548 is similar to that for unfranked
dividends.
Does the Market Adjustment Matter?
It is common in the literature to measure the price drop-off using an adjusted ex-date
= Px,i / (1+Rm,i) where
price to account for the overnight market movement, ie. ,
Rm,i is the overnight movement in the market for dividend event i. In this case, the
beta of each stock is assumed to be one and in the current study the market return is
measured using the Datastream Australia Market Price Index.
To assess the impact of this adjustment to the overnight return, Table 1 presents the
estimated results for Equation (1) and (2) where the market beta of one is used to
provide an adjusted ,
. The close-to-open drop-off ratios are similar to their raw data
counterparts as is the close-to-close estimate for Equation 2. However, the close to
close estimate for Equation 1 generated a somewhat higher dividend value.
We demonstrate later that there is time clustering in the ex-dividend events in the
sample. The work of Brown and Warner (1980) suggests that adjustment for market
movements should be firm specific where there is time clustering in events. That is,
they should reflect the sensitivity of the individual firm to movements in the market.
Unfortunately, estimated betas for individual firms are frequently so unreliable that
12
using firm specific adjustments is unlikely to be an improvement and may make
things worse. A reasonable compromise therefore, might be to measure the price
drop-off relative to the industry-beta adjusted return for the stock.7
The regression equations are re-estimated using the industry beta adjusted change in
prices and the results are presented in the final rows of Panel A and B of Table 1. The
drop-off ratio estimates are typically quite close to their benchmarks, except for the
close-to-close returns for Equation (1), where the beta estimate is slightly higher than
the benchmark.
In summary, the results of the data analysis so far suggest that dividend drop-off ratios
vary considerably. In particular, where the returns are measured from close-to-close,
the drop-off ratio varies from around 0.80 to 1.00 irrespective as to whether raw price
changes or scaled price changes are considered, whereas the close to open drop-off
ratios are more less variable. Further, robust regression and a sample truncated to
exclude outliers both produce higher drop-off ratio estimates. Finally, the adjustment
of the return for the overnight movement in the market has relatively little impact on
the drop-off ratio.
The remainder of this paper will focus on the data issues of thin trading, the size of
the spread and tick relative to the dividend and dividend event day clustering. While
we considered the impact of these issues across a number of different data
specifications and estimation options (the use of close-to-open returns, the use of
7
The industry grouping for each firm is established using Datastream’s own industry classifiers. The
industry beta for each stock is estimated using Datastream industry and market price index data for the
previous 60 months (see Brailsford, Faff and Oliver, 1997).
13
market and industry beta adjusted prices in the analysis, and also the use of robust
regressions and outlier reduced samples), the discussion of the regression results will
be limited solely to an analysis of the close-to-close returns data. This allows us to
present the basic tenor of our results without burdening the reader with a myriad of
tables. We specifically chose to focus on close-to-close returns as they are the most
commonly used in the literature. Further, we also choose not to detail our robustness
checks on firm size, franking status and dividend yield as these did not serve to alter
the results in any meaningful way.
The Impact of Stale Prices on the Drop-off Ratio
Stale prices are an important issue for any ex-dividend study, as they are likely to
confound the ex-dividend effect with other effects. In particular, stale prices can give
rise to apparent dividend drop-off values of zero. In order to gain a sense of the nature
of the problem, we first consider the extent to which cum and ex-date prices are stale.
For the cum-date data, only 26% (2,675 observations) of prices were generated at the
market close. A further 59% of the dividend events have a trade on the cum-date but
not at the close. Panel A of Table 2 presents descriptive statistics of the number of
minutes prior to the close that the last trade occurred for these 5,985 observations and
Figure 1 presents a histogram of the number of minutes prior to the close that the last
trade occurred. The median trade occurred over 8 minutes (08:10) prior to the close.
There is a great deal of variation in this data however, as the standard deviation is
more than an hour and a half (1:31:09). About fifteen percent (1,500 observations) of
the sample had no trade on the cum-date. The descriptive statistics of the number of
14
days prior to the cum-date on which the last trade is observed are presented in Panel B
of Table 2 and a plot of this data is presented in Figure 2. The median number of days
of the last trade prior to the cum-date was three, with a standard deviation of 50 days,
again highlighting the great degree of variation in the data.8
The dividend drop-off is commonly measured from the cum-date close to the ex-date
close, in which case the last trade of the ex-date is of interest. Of the 8,066 events that
had a trade on the ex-date, 24% (2,456 observations) of trades were generated at the
market close. A further 55% (5,610 observations) of the dividend events have a trade
on the ex-date but not at the close. Panel A of Table 2 presents descriptive statistics
of the number of minutes prior to the close that the last trade occurred for these 5,610
observations. Figure 3 presents a histogram of the number of minutes prior to the
close that the last trade occurred. Twenty-one percent (2,094 observations) of the
sample do not have a trade on the ex-date. The descriptive statistics of the number of
days after the ex-date on which a trade is observed are presented in Panel B of Table 2
and a visual representation of the data is provided in Figure 4.
The analysis in the previous section showed that the close-to-open returns generated
less variable drop-off estimates. It is interesting to note there is higher proportion of
prices observed at the open (50% or 5,050 observations) compared to the close.
However, this is based on a start time of 10:10am to account for the staggered
8
We note the extreme maximum values for both sets of summary statistics presented in Panel B of
Table 2. We allowed these observations to remain in the data as it is not obvious what rule should be
used to filter them out, ie. at what point do they become plausible. Further, as our goal is to highlight
the problems inherent to standard dividend drop off studies, we did not want to employ any additional
filters beyond those commonly used in the literature.
15
opening of stocks for trading between 10:00 and 10:10. A further 30% (3,016
observations) of the sample have a trade on the ex-date but after the open. The
descriptive statistics for the number of minutes after the open that the first trade
occurred are presented in Panel A of Table 2 and a visual representation of the data is
provided in Figure 5. Note that this figure shows the trade time of all stocks after the
10:00am open and so exhibits clustering in the first ten minutes due to the staggered
start of the trading day.
Over the entire sample, we observe 1,075 events which have no trade on either the ex
or the cum date, 1,444 events that have a trade on either the ex or the cum date and
finally, 7,641 events that had a trade on both the ex and the cum date.
To assess the impact of stale prices on the drop-off ratio estimate, the close-to-close
regression analysis is replicated excluding those events where stale prices are a
feature.
The estimated results are presented in Panel A of Table 3, where the
benchmark close-to-close results from Table 1 are reproduced in the first row of
results to aid the reader in interpreting the impact of stale prices. The first (second)
modified model excludes the dividend events where there is no trade on the cum (ex)day. The final row of results in Panel A, presents the results generated using only
data which had a trade on both the ex and the cum day. Excluding stale prices
generally gave a slightly lower estimate of the drop-off ratio with the biggest impact
for estimates from Equation 1.
Rather than excluding data from the analysis, an alternative approach would be to
preserve the data by adjusting the prices to account for the length of time the stock has
16
not traded. Thus a hypothetical cum (ex)-price is created by adjusting the last (first)
traded price using either a beta of one, or that stock’s industry beta, and the market
return over the period from the day of the last (first) traded price to the cum (ex)-date.
These adjusted price series are used to re-estimate the raw price regression analysis
and the results are presented in Panel B of Table 2. For Equation (1), updating stale
prices using the market beta, or industry beta, results in consistently higher drop-off
estimates. Where Equation (2) is used, the stale price adjusted estimates are generally
lower than the benchmark, but the effect is rather small.
The Impact of Spread Size on Drop-off Ratios
A potentially important source of measurement error in ex-dividend studies arises
from the bid-ask spread. The problem is that where the dividend is small, and/or the
bid ask spread is wide, movements in price due to the stock going ex-dividend will be
difficult to measure accurately. To provide some insights into this issue, Panel A of
Table 4 presents descriptive statistics for the dividend relative to the spread at the
open and the close on the ex-date. Since spreads can be affected by the open and
close a comparable set of metrics is also produced for half an hour after the open and
half an hour before the close. Table 4 shows that a large proportion of the data is
potentially affected by the problem of dividends inside the spread. In particular, the
data at the open and the close has a high percentage of observations where the
dividend to spread ratio is less than one. It is worth noting that the spread data has a
number of missing observations, in particular at the start of the day. The 3.30pm data
provides the most complete sample and to aid interpretation, Figure 6 presents a plot
of the dividend to spread ratio for the 3.30pm data.
17
To assess the impact of the spread on the estimation of the dividend drop-off ratio,
Table 5 presents the estimation results for Equation (1) and (2) using close-to-close
prices for all events where spread data is available at a given point during the day and
a restricted sample of data where the dividend to spread ratio is greater than one. For
Equation (1) excluding those observations with a large spread relative to the size of
the dividend generally serves to raise the drop-off ratio. While for Equation (2) the
effect is to reduce the drop-off ratio. It is also evident from Table 5 that the drop-off
ratio is very sensitive to the filtering applied to the data. The sample requiring spread
data for 10.00am gives a low drop-off at 0.57, while the sample requiring spread data
at 4.00pm, gives a high drop-off at 1.04.
The Impact of Tick Size on Drop-off Ratios
Boyd and Jaganathan (1994) discuss the difficulties that discreteness of stock prices
creates when estimating the expected price drop. In essence, discreteness introduces
noise to the event as prices are prevented from fully adjusting where the dividend is
not a multiple of the tick size. This resulting measurement error is greatest when the
dividend is small relative to the tick size. Bali and Hite (XXX) argue that the tick size
effect results in a downward bias in the drop-off ratio. Given that tick sizes in
Australia are typically one cent we would not expect tick size to be an important
issue, but this requires empirical verification.
Panel B of Table 4 summarises the dividend to tick size ratio for the sample of data
considered in this paper. Dividends are typically several multiples of the tick size, but
this is not always the case. As the table shows about 1% of observations have a
18
dividend that is less than the tick size. To assess the impact of these small dividend
events, the drop-off equations are estimated for close-to-close prices for a restricted
sample of data where the tick to dividend ratio is greater than one. The market and
industry beta analysis is also re-estimated. Panel A of Table 6 shows that there is little
effect when using Equation (2), but using Equation (1) there is a modest increase in
the drop-off ratio.
Panel B of Table 4 shows the results from restricting the sample to cases where the
ratio of dividend to tick size is a whole number. Again there is little effect when using
Equation (2) but there is a modest increase in the drop-off ration when using Equation
(1).
Are Dividend Days Clustered Across the Calendar Year?
The nature of the reporting cycle for listed companies means that there is potentially a
great deal of clustering of observations in a sample. This violates the assumption of
independent and identically distributed errors and so introduces a possible bias to the
estimation process.
A histogram of the number of dividend events for each day of the year is presented in
Figure 7. A clustering of dividend events is clearly evident in this plot. The standard
analysis assumes that the data are iid, and such clustering clearly violates this
assumption. The effect of such clustering is that events in the high frequency period
will have a disproportionately large impact on the regression results.
One possible solution to counter the effects of this clustering is to form portfolios of
stocks which are observed within a common time period. It is not clear however,
19
whether that period should be a day, or longer. As such, we form portfolios for
dividend events that occur simultaneously on a given day, week and month during the
sample period. The estimation results for this portfolio approach to estimation
dividend drop-off ratios is presented in Table 7 and the estimate for the daily data is
0.8959 and 0.9777 for Equations (1) and (2) respectively.
These estimates are
considerably higher compared to the unclustered benchmark results presented in
Table 1. Where a weekly timeframe is specified, the drop-off ratio estimates are
higher again, including 1.1166 for Equation 2..The weekly timeframe also gives a
notable improvement in fit relative to the benchmark regressions and nearly all of the
other regressions in the paper. A monthly time frame reduces the sample size to 152
and generates considerably lower drop-off ratios across the two equations
The main problem with adopting a portfolio type approach is the substantial loss of
data that results from having only one observation per day, week or month. An
alternative approach is to use weighted regression teachnique, whereby each day has
an equal contribution to the overall regression and the influence of each observation
on a given day is proportional to the number of events on that day. Table 8 presents
the estimation results using weighted regression analysis and the drop-off ratio
increases against the benchmark to 0.8411 for Equation (1) but decreases to 0.7174
for Equation (2).
Recall that the data sample considered in this paper consists of ordinary shares and
trust units. Figure 8 and 9 present a histogram of the number of dividend events for
each day of the year for SIRCA security type 1 and 6 respectively. While the trust
data has relatively fewer observations, the Figure 9 highlights that the clustering is far
20
more prevalent among trust data. As such, it may be useful to consider the drop-off
ratio of these two different assets classes individually. To this end, the latter part of
Table 8 presents the estimated dividend drop-off ratio for ordinary shares and trusts
using a weighted regression equation. The results reveal that the drop-off ratio for
trusts is considerably higher (0.8948 and 1.0510 for Equations (1) and (2)
respectively) compared to ordinary shares (0.7721 and 0.7405 for Equations (1) and
(2) respectively). The fir for the trust data is also dramatically better than for the other
regressions.
Are Dividend Days Clustered Across the Days-of-the-Week?
The final issue considered in this paper focuses on whether dividend events are
clustered on a particular day-of-the-week. For the sample of data considered in this
paper, we find that dividend ex-dates are approximately twice as likely to occur on a
Monday. Specifically, 3,504 ex-dividend events occurred on a Monday, whereas only
1,657 events occurred on a Tuesday, 1,347 on a Wednesday, 1,840 on a Thursday and
1,812 on a Friday.
Thus, Mondays are overrepresented in the sample and this may presented a problem
for the estimation of the dividend drop-off ratio. The problem with Mondays is that
the return is not measured across successive calendar days, in which case the return
potentially incorporates a greater amount of news that has accumulated over the
longer interval.
To test for the impact of the weekend return on the results, we re-estimate the drop-off
ratio using close-to-close prices for Mondays and all other days of the week and the
21
results are presented in Table 9. The drop-off ratio for Equation (1) is reasonably
consistent compared to the benchmark. Where Equation (2) is used to estimate the
drop-off ratio however, Mondays (0.9601) are associated with a noticeably higher
value compared to the equation estimated for the data sampled on the other days-ofthe-week (0.8715).
Conclusions
This paper shows that attempting to estimate the ‘true’ drop-off ratio is ultimately
futile because drop-off ratio estimates are so variable. In this paper they vary from
0.57 to 1.12, although most of the estimates lie between 0.75 and 1. Even the smallest
modification to the sample, or method of measurement, can change the estimate
substantially and can change the conclusion about whether or not the drop-off ratio is
significantly different to one. Thus, it is not sensible to talk about the drop-off ratio in
a point estimate sense but more meaningful to talk about a feasible range. Further, it is
important to acknowledge the limitations of ex-dividend studies and triangulate their
results with alternative approaches to estimating dividend values.
22
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25
Figure 1
Distance in time (hrs:min: sec) prior to the close that the last trade occurred on
the cum-date
120
100
No. Obs.
80
60
40
20
5:40:34
4:26:25
3:30:45
2:23:17
1:35:25
1:07:42
0:48:42
0:34:45
0:22:53
0:14:30
0:08:45
0:04:03
0:00:00
0
Time
Figure 2
The number of days prior to the cum-date on which the last trade is observed
600
500
No. Obs.
400
300
200
100
0
1
5
9
13
17
21
25
Days
26
29
33
37
41
47
5:47:25
4:49:56
3:56:37
3:07:52
2:20:59
1:45:57
1:20:43
1:01:57
0:48:28
0:35:58
0:25:57
0:18:01
0:12:01
0:07:18
0:03:23
100
90
80
70
60
50
40
30
20
10
0
0:00:00
No. Obs.
Figure 3
Distance in time (hrs:min: sec) prior to the close that the last trade occurred on
the ex-date
Time
Figure 4
The number of days after the ex-date on which the first trade is observed
800
700
500
400
300
200
100
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
>30
No. Obs.
600
Days
27
Figure 5
Distance in time (hrs:min: sec) after the open that the first trade occurred on the
ex-date
60
50
No. Obs.
40
30
20
10
5:38:50
4:48:23
3:59:12
2:57:29
2:19:02
1:49:26
1:27:01
1:09:35
0:54:27
0:41:27
0:31:19
0:22:05
0:15:24
0:10:01
0:05:22
0:00:00
0
Time
Figure 6
The dividend to spread ratio measured at the close of trade
240
Dividend / Spread
200
160
120
80
40
0
2500
5000
Observation
28
7500
10000
29
Date
11-Dec
21-Nov
1-Nov
12-Oct
22-Sep
2-Sep
13-Aug
24-Jul
4-Jul
14-Jun
25-May
5-May
14-Apr
25-Mar
5-Mar
14-Feb
24-Jan
2-Jan
No. Obs.
11-Dec
21-Nov
1-Nov
12-Oct
22-Sep
2-Sep
13-Aug
24-Jul
4-Jul
14-Jun
25-May
5-May
14-Apr
25-Mar
5-Mar
14-Feb
24-Jan
2-Jan
No. Obs.
Figure 7
The number of dividend events for each day of the year
180
160
140
120
100
80
60
40
20
0
Date
Figure 8
The number of ordinary share dividend events for each day of the year
180
160
140
120
100
80
60
40
20
0
30
Date
15-Dec
27-Nov
6-Nov
26-Sep
21-Oct
12-Sep
30-Aug
16-Aug
6-Aug
23-Jul
25-Jun
31-May
5-May
18-Apr
28-Mar
3-Mar
15-Mar
16-Feb
3-Feb
8-Jan
No. Obs.
Figure 9
The number of trust dividend events for each day of the year
180
160
140
120
100
80
60
40
20
0
Table 1
Benchmark Estimation Results
The following table presents the estimation results for Equations (1) and (2) applied to dividend ex-date price changes measured from the cum-date close to ex-date open
(Panel A) and ex-date close (Panel B). The raw data regression results are augmented with Robust Regression results and the results for a restricted sample that excludes the
top 1% most influential observations. Finally, estimation results are also presented where ex-prices are adjusted for the overnight market return.
Equation (1)
Equation (2)
2
R
No.
Obs
R2
No. Obs
α
β
α
β
Panel A: Returns measured close to open
Raw Data
-0.0056
0.9018
** 0.11
10030
-0.0006
0.7858
** 0.12
10030
(0.012)
(0.125)
(0.002)
(0.085)
Raw Data (Robust Regression MM)
-0.0111
0.9518
** 0.18
10030
-0.0022
0.8058
** 0.09
10030
(0.001)
(0.004)
(0.001)
(0.013)
Raw Data (top 1% outliers removed)
-0.0100
0.9170
** 0.40
9930
-0.0007
0.7857
** 0.11
9930
(0.001)
(0.018)
(0.001)
(0.023)
Market Beta Adjusted Prices
-0.0055
0.9042
** 0.11
10030
-0.0006
0.7868
** 0.12
10030
(0.012)
(0.124)
(0.002)
(0.085)
Industry Beta Adjusted Prices
-0.0054
0.9036
** 0.11
9858
-0.0001
0.7709
** 0.12
9858
(0.012)
(0.124)
(0.002)
(0.086)
Panel B: Returns measured close to close
Raw Data
0.0112
0.7980
** 0.08
10068
-0.0007
0.8935
** 0.14
10068
(0.007)
(0.076)
(0.002)
(0.089)
Raw Data (Robust Regression MM)
-0.0062
0.9987
** 0.19
10068
-0.0035
1.0070
** 0.11
10068
(0.001)
(0.005)
(0.001)
(0.013)
Raw Data (top 1% outliers removed)
-0.0007
0.8890
** 0.30
9968
-0.0013
0.9069
** 0.13
9968
(0.002)
(0.024)
(0.001)
(0.025)
Market Beta Adjusted Prices
0.0074
0.8604
** 0.09
10068
-0.0005
0.8897
** 0.14
10068
(0.007)
(0.073)
(0.002)
(0.089)
Industry Beta Adjusted Prices
0.0096
0.8361
** 0.08
9894
-0.0002
0.8819
** 0.14
9694
(0.007)
(0.072)
(0.002)
(0.091)
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
31
Table 2
Descriptive Statistics for Stale Prices in the Database
Mean
Median Mode
StDev
Max
Min
Panel A
Distance in time (hrs:min:sec) of the last trade prior to the close where
0:53:50 0:08:10 0:00:02 1:31:09 6:00:14 0:00:01
there is no trade at the close on the cum-date
Distance in time (hrs:min:sec) of the first trade after the open where
1:43:00 1:03:16 0:03:21 1:43:59 5:55:58 0:00:01
there is no trade at the open on the ex-date
Distance in time (hrs:min:sec) of the last trade prior to the close where
0:59:43 0:11:03 0:00:02 1:33:45 6:00:16 0:00:01
there is no trade at the close on the ex-date
Panel B
Number of days prior to the cum-date on which the last trade occurs
13
where there is no trade on the cum-date
3
2
50
775
2
Number of days after the ex-date on which the first trade occurs where
21
there is no trade on the ex-date
2
1
128
1836
1
32
Table 3
The Impact of Stale Prices on Drop-off Ratio Estimates
Panel A presents the estimation results for Equations (1) and (2) applied to a restricted sample of data that includes only those events with a trade on the cum-day, the ex-day
and both the cum and ex-dividend day respectively. The benchmark raw data close-to-close results are also reproduced from Table 1. Panel B presents the estimation results
where the stale prices are adjusted using either the market or the industry beta.
Equation (1)
Equation (2)
2
R
No.
Obs
R2
No. Obs
α
β
α
β
Panel A:
Raw Data (close to close prices)
0.0112
0.7980 **
0.08
10068
-0.0007
0.8935 **
0.14
10068
(0.007)
(0.076)
(0.002)
(0.089)
Raw Data (close to close prices)
0.0105
0.7429 **
0.12
8583
-0.0006
0.8848 **
0.13
8583
- Last trade is on cum-day
(0.007)
(0.083)
(0.003)
(0.107)
Raw Data (close to close prices)
0.0077
0.7738 **
0.13
7996
-0.0009
0.8947 **
0.14
7996
- First trade is on ex-day
(0.007)
(0.083)
(0.003)
(0.107)
Raw Data (close to close prices)
0.0091
0.7550 **
0.12
7575
-0.0008
0.8854 **
0.14
7575
- Trade on cum and ex-day
(0.008)
(0.087)
(0.003)
(0.115)
Panel B:
MktBeta Adj close to close
0.0088
0.8358 **
0.07
10047
0.0000
0.8867 **
0.14
10047
(0.007)
(0.074)
(0.003)
(0.102)
IndBeta Adj close to close
0.0100
0.8305 **
0.06
10018
-0.0002
0.8904 **
0.13
10018
(0.007)
(0.073)
(0.003)
(0.102)
Close to MktBeta Adj Close
0.0112
0.8417 **
0.07
10030
-0.0004
0.8766 **
0.13
10030
(0.007)
(0.078)
(0.002)
(0.089)
Close to IndBeta Adj Close
0.0120
0.8375 **
0.06
9985
-0.0005
0.8810 **
0.13
9985
(0.008)
(0.08)
(0.002)
(0.089)
MktBeta Adj close to MktBeta Adj close
0.0086
0.8796 **
0.06
10009
0.0003
0.8691 **
0.13
10009
(0.008)
(0.075)
(0.003)
(0.102)
IndBeta Adj close to IndBeta Adj close
0.0104
0.8701 **
0.05
9956
0.0000
0.8785 **
0.13
9956
(0.008)
(0.076)
(0.003)
(0.103)
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
33
Table 4
Descriptive Statistics for the Dividend to Spread Ratio and the Dividend to Tick Ratio
Panel A
Panel B
Div/Spread
(Open)
1.56
0.80
1.00
3.34
100.00
0.00
Div/Spread
(10.30am)
5.66
2.50
1.00
10.32
200.28
0.01
Div/Spread
(3.30pm)
6.12
2.50
1.00
12.12
264.00
0.01
Div/Spread
(Close)
4.27
1.53
2.00
9.09
236.00
0.00
Div / Tick
13.66
7.00
6.00
41.34
1012.70
0.00
2393
4017
1619
5612
2151
7679
2812
7056
135
10160
% of obs. where Div/Spread < 1 (Div/Tick < 1)
60%
29%
28%
Note: not all dividend events have bid/ask data and so the number of observations differs.
40%
1%
Mean
Median
Mode
Standard Deviation
Maximum
Minimum
Total obs where Div/Spread < 1 (Div/Tick < 1)
No. Obs with data
34
Table 5
Assessing the Impact of the Dividend to Spread Ratio on the Estimation Results
The following table presents the estimation results for Equations (1) and (2) applied to all dividend events for which spread data is available and a restricted sample of data
that only includes those observations that have a dividend to spread ratio of greater than one. The spreads are measured at the market open, 10.30am, 3.30pm and the market
close. Note that as the number of observations differs for each spread, the benchmark equation must be re-estimated in each case.
Equation (1)
Equation (2)
2
R
No.
Obs
R2
No. Obs
α
β
α
β
Raw Data (close to close) – 10am spread sample
0.0245
0.5696 **
0.07
3892
0.0035
0.6300 **
0.08
3892
(0.015)
(0.184)
(0.004)
(0.164)
Raw Data (close to close) – Div/Spread(10am) >1
0.0173
0.6330 **
0.09
1533
0.0071
0.5002
0.05
1533
(0.013)
(0.168)
(0.008)
(0.323)
Raw Data (close to close) – 10.30 am spread sample
-0.0163
0.9415 **
0.26
5525
-0.0049
0.9485 **
0.19
5525
(0.018)
(0.18)
(0.002)
(0.077)
Raw Data (close to close) – Div/Spread (10.30am) >1
-0.0209
0.9773 **
0.29
3881
-0.0042
0.9171 **
0.18
3881
(0.021)
(0.214)
(0.003)
(0.101)
Raw Data (close to close) – 3.30pm spread sample
-0.0114
0.9047 **
0.23
7559
-0.0011
0.8037 **
0.13
7559
(0.015)
(0.157)
(0.003)
(0.103)
Raw Data (close to close) – Div/Spread (3.30pm) >1
-0.0147
0.9380 **
0.27
5379
-0.0003
0.7647 **
0.14
5376
(0.018)
(0.185)
(0.003)
(0.13)
Raw Data (close to close) –4.00pm spread sample
-0.0167
0.9621 **
0.34
6889
0.0003
0.7485 **
0.13
6889
(0.014)
(0.142)
(0.003)
(0.108)
Raw Data (close to close) – Div/Spread (4.00pm) >1
-0.0243
1.0400 **
0.41
4096
-0.0001
0.7662 **
0.15
4096
(0.018)
(0.182)
(0.004)
(0.172)
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
35
Table 6
Assessing the Impact of the Dividend to Tick Ratio on the Estimation Results
Panel A of the following table presents the estimation results for Equations (1) and (2) applied to a restricted sample of data for which the dividend to tick ratio is greater than
one. Estimation results are also presented for these restricted sample regressions where ex-prices are adjusted for the overnight market return and the benchmark raw data
close-to-close results are also reproduced from Table 1. Panel B presents the estimation results for Equations (1) and (2) applied to a a restricted sample of data for which the
dividend to tick ratio is a whole number (WN).
α
Equation (1)
R2
β
Equation (2)
R2
β
No. Obs
α
-0.0007
(0.002)
-0.0006
(0.002)
-0.0003
(0.002)
-0.0001
(0.002)
0.8935
(0.089)
0.8972
(0.092)
0.8932
(0.091)
0.8852
(0.094)
-0.0022
(0.001)
-0.0020
(0.001)
-0.0016
(0.001)
No. Obs
Panel A
Raw Data (close to close)
Raw Data (close to close) - Div/Tick > 1
MktBeta (close to close) - Div/Tick > 1
IndBeta (close to close) - Div/Tick > 1
0.0112
(0.007)
0.0115
(0.007)
0.0085
(0.007)
0.0105
(0.007)
0.7980
(0.076)
0.8087
(0.078)
0.8615
(0.075)
0.8383
(0.074)
**
0.08
10068
**
0.08
9730
**
0.08
9730
**
0.07
9564
0.0106
(0.006)
0.0048
(0.006)
0.0069
(0.006)
0.7462 **
(0.027)
0.8331 **
(0.027)
0.8116 **
(0.028)
0.09
6691
0.12
6691
0.11
6572
**
0.14
10068
**
0.15
9730
**
0.14
9730
**
0.14
9564
0.9549 **
(0.027)
0.9476 **
(0.027)
0.9393 **
(0.027)
0.15
6691
0.15
6691
0.14
6572
Panel B
Raw Data (close to close) - Div/Tick = WN
MktBeta (close to close) - Div/Tick = WN
IndBeta (close to close) - Div/Tick = Wn
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
36
Table 7
Portfolio Based Regression Results to Account for Event Clustering
The following table presents the estimation results for Equations (1) and (2) applied to data in which portfolios are formed on each day, week and month in the sample period
respectively.
Equation (1)
Equation (2)
2
R
No.
Obs
R2
No. Obs
α
β
α
β
Raw Data (close to close)
0.0117
0.8950 **
0.09
2317
-0.0014
0.9777 **
0.17
2317
- daily clustering
(0.011)
(0.085)
(0.002)
(0.104)
Raw Data (close to close)
0.0045
0.9230 **
0.25
650
-0.0029
1.1166 **
0.20
650
- weekly clustering
(0.011)
(0.060)
(0.006)
(0.294)
Raw Data (close to close)
0.0419
0.5890 **
0.08
152
0.0044
0.7856 **
0.09
152
- monthly clustering
(0.025)
(0.185)
(0.008)
(0.341)
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
Table 8
The Impact of Event Clustering on the Estimation of Dividend Drop-off Ratios
The following table presents the results for Equations (1) and (2) estimated using a weighted regression technique applied to the entire dataset, and Ordinary Shares (Security
Type 1) and Trusts (Security Type 6) individually. The weights in the regression equation are set such that each day (week, month) has an equal contribution to the overall
regression. The benchmark raw data close-to-close results are also reproduced from Table 1.
Equation (1)
Equation (2)
2
R
No. Obs
R2
No. Obs
α
β
α
β
Raw Data (close to close)
0.0112
0.7980 **
0.08
10068
-0.0007
0.8935 **
0.14
10068
(0.007)
(0.076)
(0.002)
(0.089)
Raw Data (close to close) - All Data
0.0261
0.8411 **
0.11
10068
0.0029
0.7174 **
0.15
10068
- Weighted OLS
(0.016)
(0.079)
(0.003)
(0.138)
Raw Data (close to close) - Data Type = 1
0.0274
0.7721 **
0.03
9205
0.0027
0.7405 **
0.14
9205
- Weighted OLS
(0.023)
(0.259)
(0.004)
(0.147)
Raw Data (close to close) - Data Type = 6
-0.0005
0.8948 **
0.95
863
-0.0078
1.0510 **
0.70
863
- Weighted OLS
(0.005)
(0.049)
(0.003)
(0.037)
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
37
Table 9
The Impact of Day-of-the-Week Clustering on the Estimation of Dividend Drop-off Ratios
The following table presents the estimation results for Equations (1) and (2) for a sample of data that distinguishes between Mondays and all other days-of-the-week. The
benchmark raw data close-to-close results are also reproduced from Table 1.
Equation (1)
Equation (2)
2
R
No.
Obs
R2
No. Obs
α
β
α
β
Raw Data (close to close)
0.0112
0.7980 **
0.08
10068
-0.0007
0.8935 **
0.14
10068
(0.007)
(0.076)
(0.002)
(0.089)
Raw Data (close to close) – Excl. Mondays
0.0140
0.7957 **
0.07
6599
0.0003
0.8715 **
0.13
6599
(0.009)
(0.098)
(0.002)
(0.113)
Raw Data (close to close) – Mondays Only
0.0058
0.8015 **
0.08
3469
-0.0034
0.9607 **
0.15
3469
(0.010)
(0.117)
(0.002)
(0.083)
Note: standard errors in parentheses. ** = indicates significant at the 5% level.
38