Event Study On Stock Split

De Almeida Ricardo Pedro, De Wolf David, Nehme Leila

December 11, 2015

Abstract

The purpose of this research is to analyze the effect of stock split on company specific returns. This paper

examines the event study methodology using five different hypothesis testing and three different models. The

data is retrieved from Datastream and models are applied following the Brown and Warner (1985) model,

Charles Corrado and Terry Zivney models. The study is conducted on a portfolio of fifty randomly selected

stocks and market returns from the S&P 500 index. The data range is from January, 4 1988 till October, 20 2015

reported daily. Noting that the use of daily data exhibit some issues like non-normality and non-synchronous

trading.

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Contents

1 Introduction 3

2 Literature Review 3

3 Methodologies 4

3.1 Sample construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2 Model construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2.1 Mean Adjusted Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2.2 Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2.3 OLS Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3 Test statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3.1 The Mean Excess Returns Test (T1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3.2 Mean Standardized Excess Returns(T2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3.3 Nonparametric Test on Mean Ranked Excess Return (T3) . . . . . . . . . . . . . . . . . . . 7

3.3.4 Mean Sign Excess Returns (T4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Empirical Results 8

4.1 Properties of daily abnormal returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 t-tests Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.3 Longer Event Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Conclusion 11

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1 Introduction

When a company decides to declare a stock split, the number of shares increases with the ratio of new stocks

that has been issued. In other words, when the management agrees on a 2/1 conversion ratio, all shareholders

will receive two new stocks in exchange for their one old stock. In this case, the total amount of outstanding

shares doubles, without affecting the market capitalization of the company itself.

The underlying reasons to opt for a stock split can be company specific but has mostly to do with shares being

too expensive and illiquid. This can be devoted to a strong growth path or due to other events that caused the

stock to grow beyond their original trading proportions. In this case, stock split decision takes places to insure

that purchasing power of private investors still exist. At some smaller markets, like the Belgian stock market,

private investors are still an important driver behind the system. Institutional parties carry of course more

power, money wise, but according to a study of the VFB the small investor still carries an important percentage

in the BEL20 with them.

Over this paper, abnormal return studies around stock splits are examined. To come to a result, five tests are

conducted under the null hypothesis that state that there is no abnormal performance around the event day.

The five tests that have been specified in this papers are the mean excess returns, mean standardized excess

returns, non parametric test on mean ranked excess returns and a less powerful sign test as well as a longer

event period test. In summary, the paper consists of three main sections and is closed by a general overview

about the empirical findings.

2 Literature Review

Since the late 70’s, academics and practitioners have been busy implementing event studies to determine whether

a certain event caused a significant shock in the share price or not. This could either be caused by a dividend

announcement, announcement of a merger or acquisition, stock split, terrorist attack such as 9/11 or any other

plausible social, political or economical occurrence that could affect stock prices in a positive or negative way.

By that time, there was very little evidence on the central issues of corporate finance. Since then, different

approaches have been used in the academic literature to improve on the way events are observed and analyzed.

Brown and Warner were one of the first researchers who published their empiricism in the Elsevier Science

Journal of Economics. Actually they were the first to link event studies to the Efficient Market Hypothesis

(1970) and the non normal distribution of daily (excess) stock returns, which was described later on by Fama in

1976. Although non normality could cause serious problems, Billingsley (1979) has proven that the cross

sectional sample mean excess returns converges to normality when relying on the assumptions of the Central

Limit Theorem. This means that excess returns in the cross section of the securities need to be independent

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identically distributed (iid) from the finite variance distribution. When this is the case, the distribution of the

sample mean excess returns converges back to normality when the number of securities increase.

Another problem that could occur while studying event studies is the non synchronous of trading. When trading

data are missing, OLS estimators are biased and inconsistent. The problem becomes worse when the frequency

increases. In other words, estimators will be more biased when daily observations are used than when for

example monthly observations are used. When looking to variance estimation, the variance can exhibit serial

correlation (Ruback, 1982), cross sectional dependence of the security - specific excess returns (Warner 1980,

Beaver 1981, Dent & Collins 1981) and the stationarity of the variance.

3 Methodologies

3.1 Sample construction

In order to construct our sample, we use Brown and Warner (1984) methodology.We used 250 samples of 50

different randomly picked securities. Each time, a random security is selected with replacement, while assuming

that they occur with an equal probability on each trading day from January, 4 1988 till October, 20 2015. We

define the hypothetical event day by equalizing the event day to zero. Additionally, an estimation window is added

to the set up that starts at 244 days before the hypothetical event and ends at 6 days before the event. Finally,

to isolate the event, we use the resisting 11 days (-5 days before, 0 at event and +5 after the event) to create an

event window. Crucial in this case is not to have missing observations in and around our event window.

Brown and Warner (1984) proposed to avoid missing observations over the last 20 days (-14 until + 5) in

the selected window. To make sure a sample can be used in order to come up with representative results, both

researchers proposed that a sample should at least consist of 30 observations.

3.2 Model construction

First we calculate individual stock returns from prices. Each daily stock return is calculated from the previous

day with a non-missing price to the current day.

3.2.1 Mean Adjusted Model

For every stock, the daily excess return in the event period is calculated as follow:

Ai,t = Ri,t − Ri (1)

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Where:

R =1

239

−6∑t=−244

Ri,t (2)

The observed arithmetic return for a security i at time t is subtracted by the simple average daily return of a

security i over the -244 and -6 estimation period.

3.2.2 Market Model

Ai,t = Ri,t −Rm,t (3)

Where Rm is the return on the equally weighted index of S&P 500 index.

The excess return of a stock i at day t is equal to its observed arithmetic return subtracted by the market return

at the time t.

3.2.3 OLS Market Model

This model is similar to the market model in addition we include an intercept term and a slope term that is

multiplying the market return at time t.

Ai,t = Ri,t − αi − βiRm,t (4)

where estimated alpha and beta are OLS values from the estimation period.

3.3 Test statistics

We have conducted four different two tailed t-tests at a significance level of 5%.

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3.3.1 The Mean Excess Returns Test (T1)

Brown and Warner (1984) assessed the excess return for each sample to test the significance of the each event

period. The null hypothesis states that the mean day “0” excess return (e.g. the simple average of the market

model excess return) is equal to zero. In other words, the test statistic could be seen as a ratio of the day “0”

mean excess return in relation to its estimated standard deviation, estimated from a time series of mean returns.

The test statistic for any event at time t (in this case t = 0) is described as follows:

T1 =A0

S(At)(5)

Where:

At =1

Nt

Nt∑i=1

Ai,t (6)

S(At) =

√√√√√[ −6∑t=−244

(At−=

A)2]

238(7)

=

A =1

239

t=−6∑t=−244

At (8)

with Nt the number of sample securities whose excess returns are available at time t.A test statistic like the one that has been described above (t-test) is widely used to determine whether anhypothetical event can be described as significant or not. If At are independent identically distributed (iid) andnormal distributed, the t-test statistics, which has been described earlier, can be seen as Student - t distributedunder the null hypothesis. Because the sample, which has been described by Brown and Warner exceeded 200degrees of freedom, one can conclude that the test statistic is assumed to be unit normal.

3.3.2 Mean Standardized Excess Returns(T2)

Each excess return at time t is divided by their estimated standard deviation to yield a standardized excess

return.

T2 =1√N

N∑i=1

A′

it (9)

Where:

A′

it =Ait

S(Ai)(10)

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S(Ai) =

√√√√ 1

238

−6∑t=−244

A2it (11)

Standardized excess return test formula at day 0 is:

T2depends on cross-sectional independence of security excess return. According to Brown and Warner (1985),

T2has more power than T1when the cross-sectional independence assumption is valid.

3.3.3 Nonparametric Test on Mean Ranked Excess Return (T3)

T3test is identical to T1. In order to apply the nonparametric test on mean ranked excess return, we should first

respectively rank the securities’ excess returns for each model. The ranking method transforms the distribution

of security excess returns into a uniform distribution. This implies that under the null hypothesis, a day 0 excess

return rank is a drawing from a uniform distribution In the formula (12), Kitis the rank of the excess return

Aitin security i’s time series of 250 excess returns:

Kit = rank(Ait) (12)

t = −244, ........,+5

where Ait ≥ Aij implies Kit ≥ Kijand 250 ≥ Kit ≥ 1. Noting that the average rank is one-half plus half the

number of observed returns (125.5).

The Rank test statistic will be as follow:

T3 =

1N

N∑i=1

(K0 − 125.5)

S(K)(13)

Where:

S(K) =

√√√√ 1

250

+5∑t=−244

(1

N

N∑i=1

(Kit − 125.5)

)2

(14)

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3.3.4 Mean Sign Excess Returns (T4)

A sign test is commonly used to examine statistical significance without requiring a symmetrical distribution of

excess return.

The sign of the excess return of each stock and for each day is represented byGit:

Git = sign(Ait −median(Ai)) (15)

t = −244, ........,+5

T4 =1√N

N∑i=1

Gi0

S(G)(16)

Where:

S(G) =

√√√√ 1

250

+5∑t=−244

(1√Nt

Nt∑i=1

Git

)2

(17)

Where Ntis equal to the number of non missing returns of the N stocks on the day of the event.

The aim of the sign test is to transform the stock excess returns into sign values (+1, -1 or 0), noting that the

probability to get a value of +1 is equal to the probability to get a value of -1. The studies conducted by Brown

and Warner (1980) (1985), Beny, Gallinger and Handerson (1990) prove a misspecification of the sign test when

the excess return median is assumed to be equal to 0, which is not the case this study.

4 Empirical Results

4.1 Properties of daily abnormal returns

Table 1 shows the properties of the various event study performance measures at the day of the event.

Referring to Table 1, we can see that the mean studentized ranges of the three measures are far the value of 6.85

for the 0.99 fractal of the studentized range of sample drawn from a normal population size of 200, especially the

studentized range value of the OLS measure that is 7.17. Mean values of skewness and kurtosis coefficient are

also distant from the value of the 0.99; except for the kurtosis of the Market model that is slightly closer to the

value of 0.99. This implies that daily excess returns are not normally distributed. In addition, Table 1 shows

that the three performance measure have almost same standard deviation values. This suggests that the

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Model Mean Median Standard Deviation Skewness Kurtosis Studentized Range

Mean Adjusted 0.92% 0.56% 2.58% 0.7 4.29 6.34Market 0.72% 0.53% 2.16% 0.06 3.65 6.17

OLS 0.94% 0.62% 2.55% 0.62 4.14 7.17

The table displays summary statistics for fifty randomly selected stocks at the day of the event (t=0) and for thethree different performance measure.

Variable 0.95 0.99

Skewness 0.06 0.13Kurtosis 3.52 3.87

Studentized Range (N=200) 6.15 6.85

Table 1: Summary Statistics for each performance measure

Models T1 T2 T3 T4

Mean Adjusted 2.89 3.22 -2.94 1.63Market 2.53 3.34 -3.44 2.91

OLS 2.35 3.30 -3.43 2.54

This Table shows the four different t-statistics for each performance measure at 5% significance level at time 0.

Table 2: t-statistics (T1, T2, T3and T4)

alternative measures of excess returns will exhibit similar ability to detect abnormal performance when it is

present. (Brown and Warner, 1985).

4.2 t-tests Results

Table 2 displays the t-statistics of the mean excess returns, the Mean Standardized Excess Returns, the Mean

Ranked Excess Returns as well as the Mean Sign Excess Returns for the three models.

At a 5% significance level, the t-statistic is equal to the absolute value of 1.965. Therefore, referring to Table 2,

we reject the null hypothesis of no abnormal performance or that the mean excess return is equal to 0 for each of

the three tests for the three models. How come we reject the null hypothesis, we can state that there is no

abnormal performance on the day of the event. However we cannot conclude that abnormal performance has

occurred. If we assume that abnormal performance has taken place, then we can state that the market is

efficient due to its reaction to the stock split event.

For both the market and OLS models, we reject the null hypothesis; stating that there is no abnormal

performance. However for the mean adjusted model we fail to reject the null hypothesis that the mean excess

return is equal to 0. Despite the t-statistics that we have computed, this test could be misspecified if incorrect

assumption about the date is imposed. In a research about specification and power of the sign test conducted by

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Cumulative average abnormal returns of the 50 randomly selected stocks from 10 days before the estimated dayof stock split event until 5 days after.

Figure 1: Cumulative Average Abnormal Returns

Corrado and Zevney in 1992, the statisticians were successful enough to prove that the power of the sign test

weakened when detecting ±1/2− percent and ±1− percent abnormal performance. Therefore, an assumption of

misinterpretations of the results could be concluded.

4.3 Longer Event Period

In order to be more specified, we have run a Longer event Period test statistics for the 11 equiprobable dates:

+5∑−5

At(+5∑−5

S2(At)

)1/2(18)

Table 3 shows the properties of the various event study performance measures at 11 days around the event.

Table 4 shows the results of the longer event period test with a sample of N=50. The null hypothesis is that the

cumulative main daily excess return over (-5, +5) is equal to 0.

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Model Mean Median Standard Deviation Skewness Kurtosis Studentized Range

Mean Adjusted 0.06% -0.00% 2.45% 1.31 12.25 12.44Market 0.11% 0.04% 2.06% 1.43 12.68 11.27

OLS 0.05% 0.01% 1.94% 1.23 12.55 11.74

The table displays summary statistics for fifty randomly selected stocks at the day of the event (-5, +5) and forthe three different performance measure.

Table 3: Summary Statistics for the longer period

Model Longer Event test

Mean Adjusted 1.44Market 3.30

OLS 1.57

This Table shows the t-statistics for a longer event period for each model at a 5% significance level at time (-5,+5).

Table 4: Longer Event Period t-statistic

Failing to reject the null hypothesis occurs in most of the models. That implies that the t-statistics show

misspecification of the results and fail to incorporate autocorrelation of the mean daily excess return. Plus, some

studies like “The Case of Event Study” of Brown and Warner (1985) have already demonstrated that the power

of the test decreases when the abnormal performance take place in a longer interval. While computing a longer

period interval around the event, the rejection rate decreases substantially achieving a higher failure of rejection

rate.

5 Conclusion

The paper study abnormal returns around stock split event. The study was conducted by calculating excess

mean return following different models and a series of computations of t-tests at different time intervals around

the event under the null hypothesis that no abnormal performance took place; in other words, that the mean

excess returns are equal to zero. Calculations were made following the methods used by important figures such

as Brown, Warner, Fama, Fisher, Corrado and many more.

Three models were tackled in order to calculate mean excess returns: Mean Adjusted Model, Market Model and

the OLS Market Model; thanks to literature review that took place.

Finally, a computation of five two-tailed test was held at a significance level of 5% at the day of the event as well

as for 11 days around the event for each of the models mentioned above in order to test for rejection or failure of

rejection of no abnormal performance. The five tests calculates were the Mean Excess Returns, Mean

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Standardized Excess Returns, Mean Rank Excess Returns, Mean Sign Excess Returns and a Longer Event

Period Test.

Rejection of the null hypothesis was observed in the majority of tests results. Mean Sign Excess Return test

failed to reject non occurrence of abnormal performance due to its misspecification and lack of power comparing

to the other tests as mentioned beforehand. Running a test for a longer period around the event fail to reject the

null hypothesis too for the Mean Adjusted And OLS Market Models this is because a longer interval of

estimated event period tend to affect the power of the test negatively as mentioned in some studies that took

place in the past. However, concluding the existence of abnormal returns at the day of the event is inaccurate,

because it would render the market to be efficient.

References

[1] Brown, S. J., & Warner, J. B. (1985). Using daily stock returns: The case of event studies. Journal of financial

economics, 14(1), 3-31.

[2] Corrado, C. J. (1989). A nonparametric test for abnormal security-price performance in event studies. Journal

of financial economics, 23(2), 385-395.

[3] Corrado, C. J., & Zivney, T. L. (1992). The specification and power of the sign test in event study hypothesis

tests using daily stock returns. Journal of Financial and Quantitative analysis, 27(03), 465-478.

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