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Segmented Cross-Sectional Analyses
of Stock Splits
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MSc Finance – Asset Management
Master Thesis
University of Amsterdam, Amsterdam Business School
Segmented Cross-Sectional Analyses of Stock Splits
Michael van Amstel
July 2017
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Abstract
Public firms conduct a stock split to decrease its value of a single stock, which subsequently leads to a proportionate increase in the total number of shares outstanding. This research examines the short-run effects of a stock split in both the announcement period as well as the period of the ex-date. Asia is compared to the United States and the three tested variables are the return, trading volume and relative bid-ask spread. Event studies test different segments of the sample in order to compare the behavior of stocks with specific price adjustment factors and post-split prices. Setting aside the abnormal relative bid-ask spread in Asia and the abnormal return in the US in the date period, all three abnormal values significantly increased in the announcement and ex-date period of a stock split. Stock splits with a high adjustment factor tend to generate higher abnormal returns and abnormal trading volumes while stocks with a higher post-split price are associated with wider relative bid-ask spreads. The best suggestion for a trading strategy between the declaration and ex-date is investing in stocks with the highest post-split price and highest adjustment factor.
Acknowledgements
Writing this thesis marks the end of my MSc Finance at the University of Amsterdam. I want to thank the university for providing me the education and knowledge. I offer my sincerest gratitude to my supervisor, D. Güler, who supported me throughout writing my thesis and put remarkably much time and effort in supervising me on a regular basis. Finally, I want to thank my parents for providing me unfailing support during my academic career. This accomplishment would have not been possible without them. Thank you.
Statement of Originality
This document is written by student Michael van Amstel who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents (University of Amsterdam, 2017).
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Table of Contents
Abstract ... 1 1. Introduction... 3 2. Theoretical framework... 5 2.1 Overview... 6 2.2 Literature review... 7 2.3 Motivations... 102.4 Trading volume and bid-ask spread... 13
2.5 Hypotheses and added value... 14
3. Data... 15 3.1 United States... 15 3.2 Asia... 17 3.3 Data preparation... 18 3.4 Segmentation... 19 3.5 Summary statistics... 21 4. Methodology... 23 4.1 Analysis... 23
4.2 Trading strategy analysis... 27
4.3 Correlations... 28
5. Results... 28
5.1 Abnormal returns... 30
5.2 Abnormal trading volume... 32
5.3 Abnormal relative bid-ask spread... 33
5.4 Overall... 35
5.5 Cross-sectional comparison... 35
5.5.1 Return – Announcement period... 36
5.5.2 Return – Ex-date period... 37
5.5.3 Trading volume – Announcement period... 39
5.5.4 Trading volume – Ex-date period... 39
5.5.5 Relative bid-ask spread – Announcement period... 40
5.5.6 Relative bid-ask spread – Ex-date period... 40
5.6 Overview... 41
5.7 Comparability... 41
5.7 Trading strategy analysis... 44
5.7 Correlations... 46 6. Robustness checks... 47 6.1 Sign test... 47 6.2 Momentum analysis... 50 7. Conclusion... 53 Bibliography... 56 Appendix I... 59 Appendix II... 63
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1. Introduction
Suppose someone asks you to change $50 for five bills of $10. You are probably okay with it as you still have a total of $50 in your wallet. Does it change anything when someone asks you to do this for an Asian currency? The person who asks you to change a fifty-dollar bill into smaller denominations can be compared with a public firm that wishes to perform a stock split. The focus of this research is not on the question why one wishes to split its cash (equivalent) but instead what the effects are for the one who gets his/her assets split. Stocks are, however, somewhat more complex than a banknote. The prices fluctuate, trading volumes differ among days and bid-ask spreads get wider or narrower. All these aspects may be differently affected by a stock split and whether a stock split happens in Asia or in the United States may have effect as well. “Stock splits are one of the least understood phenomena in equity markets” (Easley et al., 2001) and “Stock splits are a puzzling corporate phenomenon.” (Ikenberry et al., 1996) indicate that the exact specifics of a stock split are unknown.
This study analyzes the effects of a stock split in the period when a stock split is announced and when the stock is actually being split, the ex-date. The examined traits in this research that are possibly affected by a stock split are the return, trading volume and bid-ask spread. These effects of a stock split may differ among specific split ratios. A stock split that performs a 10-for-1 split can have more effect than a 2-for-1 split for example. Another characteristic of a stock split that may have influence on the effects is the price to which a stock is being split, the post-split price. Hence, the sample is segmented into groups of price adjustment factors and post-split prices.
Rationales behind stock splits are frequently related to the concern about the amount that an investor has to pay for a stock. Reducing this amount is the essence of a stock split. Asia is a region where the standard of living is particularly lower than in the United States. The performance of a stock split may, for this reason, have different effects in both regions. Besides the fact that the GDP per capita is considerably lower in Asia, the markets in Asia are different regarding the operations as well as the regulatory condition compared to the United States. Accordingly, the analysis compares the effects of a stock split in the United States to those in Asia as well. The research question is what the effects of a stock split are on the abnormal return, abnormal trading volume and abnormal relative bid-ask spread, and if these differ amongst regions, specific stock split ratios or post-split prices.
4 The literature up to now has researched all aspects of this study but never compared the effects across regions, dates of interest, split ratios or post-split prices. The combination of all the examined elements makes this research profoundly unique and instructive. The contribution of this study to the scientific community is the insight in the effects of a stock split on the return, trading volume and bid-ask spread and by adding to the research of the stock-split puzzle. The results of this study may be informative to groups like investors and the general public via pension funds that can profit from knowledge about the characteristics of stock splits.
This research considers all distributions and corporate actions that are classified as stock split in the period of 2000 till and including 2016. The two analyzed 11-day event windows are the five days on each side of the announcement and ex-date. The Asian sample consists of stock splits of stocks in major indices from India, Indonesia, Malaysia and Thailand. The sample of the United States considers all stock splits of firms that have been in the S&P 500 during the examined period. The methodology to analyze the data is by performing event studies. A specified period before the announcement of the stock split is used to predict the normal values in the event window. Subsequently, the observed value is tested against the predicted value, which results in an abnormal value that may be significant at a certain level. The analysis is particularly comprehensive in the sense that the sample is segmented in groups of adjustment factors and post-split prices and that these are all separately examined. Additionally, the cumulative average abnormal values of the different groups are tested against each other in order to discuss possible significant different characteristics of stock splits. Specifically, the observed values are tested against values of other groups instead of predicted values. Investors apply specific trading strategies to generate returns. One such interesting trading strategy for investors may be the period between the announcement of a stock split and the ex-date of the stock split. This study examines if the above-mentioned trading strategy is profitable, compared to the index, and how the returns of different segments vary in this period.
The results found in this study show a significant positive abnormal return of 4.25 percent and 1.31 percent in the announcement period in Asia and the United States, respectively. Additionally, an even higher abnormal return is observed in the period of the ex-date in Asia. The abnormal trading volumes are again significant and positive in both regions. The values are higher in the announcement period compared to the period of the ex-date and are higher in Asia than in the United States. The abnormal bid-ask spreads as a percentage of the stock price in Asia and the United States are differently affected by a stock split. The observed value in Asia is lower
5 than the value that is predicted by the mean-adjusted model while the abnormal value in the United States is significant and positive. The effects have the same direction in both periods but are larger in the period of the ex-date.
The average abnormal values are cumulated and compared across the different groups in which the two samples are segmented. In this way, statements about significant differences between groups can be made. Evident is that groups with a relatively high adjustment factor are associated with higher abnormal returns and abnormal trading volumes in the announcement period as well as the period of the ex-date. Such a clear and distinctive characteristic in the adjustment factor groups is not visible for the abnormal relative bid-ask spread. On the other hand, groups that contain stock splits with a high post-split price are associated with wider abnormal relative bid-ask spreads. This implies that the adjustment factor is seemingly more related to the return and trading volume, while the post-split price is more representative to be an indication for the relative bid-ask spread.
The combination of the analysis of three variables, two regions, two event periods with daily data over a period of 11 days, two segmentations that are divided into numerous subgroups, plus the extensive comparison between different groups makes this study exceptionally comprehensive. Additionally, the exhaustive findings are robust in a sense that they are not driven by momentum. The remainder of this study continues as follows. Section 2 presents the theoretical framework in which this research operates. The understanding of a stock split, previous literature, motivations for a stock split and the hypotheses are discussed. Section 3 describes the data and variables, and explains the segmentation and summary statistics. Section 4 presents the methodology that is used in this study. The next section analyzes and interprets the effects of stock splits. Robustness checks are described in section 6 and the final section is the conclusion of the research.
2. Theoretical framework
On June 9th 2014 Apple split its stock with a 7-for-1 ratio. The price of a stock went from just over $645 and opened on the day of the split at a price of $92.70. What does this split imply for an investor who is holding the stock? What happens to his total position of Apple in his portfolio? Why did Apple conduct a 7-for-1 split and not a 2-for-1 split like they did in the past? Easley et al. even state that “stock splits remain one of the most popular and least understood phenomena in equity markets” (2001). Ford et al. mention that stock splits should be merely a
6 cosmetic event, but the announcements of splits are followed by significant market reaction (2012). These questions led to the interest in the phenomenon of a stock split and in how it performs in different circumstances. The different circumstances that are researched are regions, dates of interest, split ratios and post-split prices. A deeper understanding of stock splits and the position of this research is described in this section and is organized according to the following structure. First, some basic terminology and a specification of what a stock split is and how it works. Subsequently, previous research is discussed in a chronological order to get a better understanding of the position of this research. Applicable for all discussed research is that none of them analyzes the different effects of a split across ratios, post-split prices, regions and dates of interest. Afterwards, different rationales that firms could have for a stock split are pointed out. Finally, the effect a stock split has on the bid-ask spread and trading volume are mentioned and as a final point the hypotheses and added value are stated.
2.1 Overview
The terms stock split and stock dividend are often used interchangeably. Both are essentially the same thing, except that a stock split is usually expressed as a ratio while a stock dividend is generally expressed as a percentage. The rationale is that both increase the number of shares outstanding while the cash flow and ownership are unaffected, and thus the total stockholders’ equity (Conroy & Harris, 1999, Ford et al., 2012, Marshall, 2017 and Murray, 1985). Dividend payout is the distribution of earnings to the existing shareholders of the firm in a particular proportion of the number of shares each currently owns. A dividend payout can be done in two ways, using cash (cash dividend) or in the form of stock (stock dividend) (Lee & Lee, 2013). A difference between stock dividend and a stock split is that the par value1 is typically reduced proportionately to the split factor (Baker & Gallagher, 1980), while this is not the case for the issuance of stock dividend. Regarding the accounting of both procedures, the consequences of a stock split are negligible because there is no accounting entry required for a split. Conversely, stock dividend requires that an amount of retained earnings is transferred to the paid-in capital account2 (Angel, 1997, Grinblatt et al., 1984, Lakonishok & Lev, 1987, Lamoureux & Poon, 1987 and Marshall, 2017).
Two important dates to every stock split are the declaration date and the ex-date. The former is the date on which the board of directors authorizes to conduct a stock split. The ex-date is the
1 “An arbitrary value assigned to a share of stock when the corporation is organized. Sometimes used to refer to the stated value
or face amount of a security.” (Marshall, 2017)
2 “If the stock dividend percentage is more than 20 to 25 percent, only the par value or stated value of the additional common 2 “If the stock dividend percentage is more than 20 to 25 percent, only the par value or stated value of the additional common
7 date that the stock trades while it has already been split and thus implies that an investor’s shares will split as long as he bought them before the ex-date (Berk & DeMarzo, 2011, Lee & Lee, 2013 and Marshall, 2017).
2.2 Literature review
One of the most cited papers of Fama in the first years of his career is written in 1969 together with Fisher, Jensen and Roll. The focus of their paper is assessing the (speed of) price adjustment of common stock to information that is imbedded in a stock split. The empirical work of Fama et al. shows that the months before the announcement date of a stock split have highest returns. The explanation they give for this phenomenon is that the increase in earnings in the pre-split period is uncertain to be maintained at their increased level. Investors wish to observe indications that the increased earnings are durable and a stock split might be exactly such a sign. This explanation falls under the theory that stock splits have information content that affects the share prices. The information that comes with a stock split is incorporated in the price of the stock by the end of the split month. In the context of their research, this indicates that prices adjust immediately after the announcement date, which leads to their conclusion that the stock market is “efficient”. Regarding using a split to increase one’s returns, is by them perceived as not possible without insider trading. Analyzing the returns of splitting stocks from the day after the announcement until eight weeks later, results in the finding that split shares do not outperform nonsplit shares after the announcement. The methodology applied in their research is the residual approach by which averages of estimated regression residuals in the months around the split dates are tested for significance (Fama et al., 1969). Fama et al. use monthly data while daily data is used in this research. Due to this fact, the setup of the research differs in a sense that this research has a narrowed-down focus regarding the event of a stock split.
Bar-Yosef and Brown start off with five possible motives behind splitting a stock: “(1) increased marketability of firms’ shares, (2) conveyance of information regarding superior investment opportunities, (3) increased ownership base to avoid mergers, (4) increased product sales, and (5) improved employer/employee relations”. The first argument involves that splitting a stock would bring the price into more favorable trading ranges. Bar-Yosef and Brown have critique on the methodology of Fama et al. in which they assume that the systematic risk of split securities does not change around the time of a split. Previous literature3 found evidence that stock splits are often followed by increased total cash dividends. There is some uncertainty regarding future cash
3 Bar-Yosef and Brown (1977), Bellemore and Blucher (1959), Copeland (1979), Easley et al. (2001), Fama et al. (1969),
8 dividends and earnings before the split declaration and this may result in increased systematic risk around the split date. Bar-Yosef and Brown avoid this problem of fluctuating systematic risk by using moving betas and show that betas become higher in the period of a split. The moving betas approach is “a practice of deriving risk estimates from series of data symmetrical to the residual estimation period” (Charest, 1978). An intuitive way to see the symmetrical moving betas approach used by Bar-Yosef and Brown is a symmetrical rolling window around the variable that is estimated. The rolling window used for estimating the variable is of equal size on both sides of the variable and the estimation window moves together with the variable. Stock splits between 1945 and 1965, with a common share distribution greater than or equal to 25 percent, are obtained to test the hypothesis. Their hypothesis states that cumulative average residuals are lower when using the moving betas methodology compared to using the constant beta method. The conclusion of the paper is that Fama et al. were too optimistic about the benefit of a split for investors while underestimating the risks. The suggestion is that one should use the moving betas methodology when analyzing information content of financial events (Bar-Yosef & Brown, 1977).
The research that is written by Charest in 1978 examines stocks that are listed on the NYSE and performed a split between 1947 and 1967. He focuses on the proposals, approvals and realizations of stock splits and takes a closer look at some methodological issues in the process of estimating abnormal returns. Charest uses the Fama-MacBeth approach to estimate the residuals4 (abnormal returns) of stock splitting firms. The results are similar to those of Fama et al. and this is possibly due to the partly overlapping test period. However, one of the main results of Charest is that a strategy that involves buying a stock just after the split declaration date and holding it for three months, yields an excess return of 1.5 percent. Charest argues that it seems like investors are waiting for confirmation in the following three months of a split declaration that the split is actually going to be realized. Though, shareholders practically never withhold the realization of a split and the excess return thus seems inconsistent. The anomaly of abnormal excess returns in the subsequent three months of a declaration is primarily observed in a particular short sub-period. Any statements about market inefficiency are thus very cautious and have weak evidence. Buying the stock after the declaration date implies at the end of the announcement month of a stock split. Again, the research uses monthly data and this results in a form of imprecision in the analysis. Similar to Fama et al., he states that the adjustments of market prices to a stock split are consistent with market efficiency. Altogether, statements about market inefficiency are very
4 Charest explains ‘residuals’ as “[…] measures of the average percentage abnormal return experienced by the sampled stocks over
9 sensitive to the approach that is used to calculate the residuals and the adopted investment strategy5 (Charest, 1978).
Grinblatt et al. start off with the statement that a stock split is only a cosmetic accounting change without a direct cost or benefit (Schultz, 2000). The papers of Fama et al., Bar-Yosef and Brown, and Charest are subsequently mentioned as empirical research with evidence that is interpreted as the existence of a stock split announcement effect. These two statements are conflicting and hence two considerable remarks regarding previous research are made. Firstly, they have not controlled for contamination by other information releases that might influence the split announcement returns. Contamination is defined as other announcements in a three-day period around the stock split announcement day. The second remark is that their findings are even more limited in combination with the usage of monthly data. The stock price increase after the split announcement is most likely combined with other information releases that affect the return in the same month. Grinblatt et al. examine both the effect of stock dividend and stock split announcements, where the implications regarding the latter are discussed in this research. Stocks that are listed on the American or New York Stock Exchange with an initial split announcement between 1967 and 1976 are part of the sample that is used. The mean-adjusted returns methodology for event studies is then applied to analyze the sample. This methodology considers that the predicted return equals a constant variable, where the difference between the observed return and the predicted return equals the abnormal return (Brown & Warner, 1980). An essential point is to test day 0 as well as day 1 because the announcement is often after the close of trading on day 0 and thus would any effect be reflected in the return of day 1. Testing both days is taken literally because the mean two-day return of day 0 and 1 is compared to the benchmark to test for significance. Further, their research shows a phenomenon by which there are large price increases 40 days prior to the announcement. They call this the selection bias. The results of the empirical analysis indicate that there are significant positive excess returns on the announcement day of a stock split. The average daily stock return of pure stock splits on day 0 is 1.96% and 1.33% on day 1. These values together are significantly higher than the two-day return of the benchmark, which equals 0.16%. Additionally, they find high excess returns around the ex-date of stock splits. The average daily stock returns on days 0 to 3 are respectively 0.69%, 0.52%, 0.38% and 0.20%, which are significantly greater than the benchmark returns. A possible explanation why there are positive ex-date returns that is mentioned by Grinblatt et al. is that some investors view it as costly to get their stock split. A stock split might result in odd lots, which is an order amount
5 Charest analyzes two investment strategies, with equal weights and with variable weights. Investments are monthly and five
10 for a security that is not the usual amount that is traded. The positive return could be an indication that it is costly for an investor to hold a stock just before the ex-date. By way of contrast, the positive abnormal returns on days 1 to 3 after the ex-date are not justified by this argument (Grinblatt et al., 1984).
The first three papers that have been discussed only select stock splits that perform at least a 5-for-4 (25 percent) split while Grinblatt et al. choose to include all splits of ten percent or more. Ikenberry et al. merely analyze two-for-one splits and find a five-day announcement period return of 3.38 percent (1996). The various motives for splitting a stock that have been mentioned can be categorized such that one could argue that the split ratio might influence the effect of a split. Additionally, the post-split price could be a more accurate explanatory variable in assessing possible different magnitudes of effects of a stock split.
2.3 Motivations
Grinblatt et al.6 (1984) and Lamoureux and Poon (1987) state that there are no apparent costs involved in splitting a stock, and thus do not agree with Baker and Gallagher (1980), Bar-Yosef and Brown (1977), Fama et al. (1969), Lakonishok and Lev (1987) and Brennan and Copeland (1988) who argue that it is not costless to conduct a split. The latter research specifies the cost as the fixed cost element of brokerage commissions that increase the per-share trading costs of low-priced stocks (1988). In the late nineties, Angel states that the costs of splitting are more than one million dollars for a major stock (1997). In either case, there is at least some effort a firm has to put in to perform a stock split. Given this insight, a firm most certainly sees a particular rationale behind the performance of a stock split. The perception of the most prevalent reasoning behind a stock split varies across literature. There are three common hypotheses of why firms decide to split their stock.
A motivation for a stock split, that is often referred to as an inducement for firms, is marketability. The intention of the organization is to bring the stock price back to its preferred trading range and accordingly improving the liquidity of the stock (Baker & Gallagher, 1980, Ford et al., 2012, Grinblatt et al., 1984, Lakonishok & Lev, 1987 and Marshall, 2017). A round lot is a common trading unit of 100 shares or multiples of 100 shares (Lee & Lee, 2013). The lower price per share that follows from a stock split attracts a group of investors who find it costly to trade round lots on high stock prices, which improves the liquidity of the stock (Dennis, 2003
6 Grinblatt et al. argue that there are no apparent costs to stock splits but on the other hand analyze why managers with
11 and Ford et al., 2012). Criticism on this argument is that small investors can gather to economize on odd lots and that odd-lot trades are not a substantial fraction of trading activity (Copeland, 1979).
A characteristic of the stock market is the existence of asymmetric information between investors and managers. Communicating favorable information to the public in a trustworthy way may be done via signaling (Berk & DeMarzo, 2011 and Brennan & Copeland, 1988). Such a signal could be a stock split. This is immediately a second reason why firms could decide to conduct a stock split (Dennis, 2003, Grinblatt et al., 1984 and Lamoureux & Poon, 1987). A signal that a firm could give via a stock split is that it forecasts anticipated dividend increases (Bar-Yosef & Brown, 1977, Copeland, 1979, Fama et al., 1969, Grinblatt et al., 1984 and Lakonishok & Lev, 1987). An augmentation of this signaling hypothesis is the interpretation that an anticipated split is viewed as a more credible signal compared to splits that come as a surprise. This favorable signal is due to the better pre-split performance of the firm (Hwang et al., 2008). Grinblatt et al. find evidence that is consistent with this “trading range hypothesis”.
A third reason is that firms intent to increase the number of shareholders in the firm and consequently seek a broadening of the ownership base (Lamoureux & Poon, 1987 and Schultz, 2000). The rationale of executives is that they hope that a lower price per share results in more company’s stock purchases of individual investors. An increasing amount of individual investors lowers the percentage of shares that institutional investors own. The rationale is that by increasing the ownership base of the firm, the management makes it more difficult for any group of investors to initiate action against them (Dennis, 2003). A wider investor base of the firm gains political clout with regulators (Angel, 1997). The moment when institutional investors are claiming a large proportion of the volume of trading, a stock split could be seen as the solution for the company7 (Baker & Gallagher, 1980). An enhancement of this rationale is the reasoning of Schultz, who states that the increase in the number of small shareholders who own the stock is caused by a wider minimum bid-ask spread after a stock split. This on its turn leads to more incentive for brokers to promote the stock (2000). Brennan and Hughes confirm the negative relation between the number of analysts making forecasts and the share price (1991). The augmentation of Angel is that the explicit costs, that are higher after the split, are eventually
7 There is contradicting evidence on the broadening of the ownership base. Lamoureux and Poon (1987) and Mukherji et al.
(1997) find that the number of shareholders increases after a split. This same research of Mukherji et al., however, finds that the proportion of institutional ownership remains the same (1997).
12 beneficial for investors. The costs attract liquidity providers and limit orders8 and lead to enhanced liquidity for investors (1997).
Alternatively, there are other reasons that are seen as the driving force behind a stock split. (1) One such prime mover is mentioned9 by Grinblatt et al. and describes that firms draw attention with a stock split. Increased attention generates subsequently that market analysts reassess the firm’s future cash flows. However, a stock split is not the sole corporate action to call attention to a firm. Another remark is that a reverse-split draws attention to the firm as well, but is associated with an unfavorable market reaction (Lamoureux & Poon, 1987). These flaws in the hypothesis make it somewhat weaker (1984). (2) A motivation that is closely linked to the signaling hypothesis is the so-called “retained earnings hypothesis”. This implies that stock dividends of 20 or 25 percent or less10 result in a decrease in retained earnings and an increase in the firm’s capital account. This could result in a restricted ability to pay cash dividends in the future. These restrictions are expected to be binding for firms that foresee insufficient earnings. Therefore, lower-valued firms do not mimic the signals of firms that don’t see the restrictions as binding (Conroy & Harris, 1999, Dennis, 2003, Grinblatt et al., 1984, Ikenberry et al. 1996). Lamoureux and Poon do not associate apparent costs for the performance of a stock split but do acknowledge that the signaling argument requires some costs to false signaling. Without any costs it would be impossible to distinguish an overvalued splitting stock from an undervalued splitting stock (1987). Brennan and Copeland have a different approach of interpreting the costs of a stock split, namely that the fixed cost component of brokerage commissions increases the per-share trading costs of low-priced stocks (1996). This signaling argument is only relevant for the previous-mentioned percentages and thus a weaker one. (3) Conversely, Ford et al. find evidence that supports the signaling hypothesis in the form of a relation between market reaction to split announcements and information asymmetry. This information asymmetry is measured in terms of analyst coverage. More analyst coverage is associated with reduced information asymmetry between managers and investors. Subsequently, this leads to a reduced market reaction to the announcements of corporate actions (and thus splits as well) (2012). This finding is in favor of the argument that stock splits are used to communicate private information to the public11. (4) Further, providing the stock with a higher tax-option value is a motivation that is occasionally given for splitting a stock. The contention is that stock returns are more volatile after a stock split
8 “[…] liquidity-providing limit orders are used more frequently for stocks with higher relative ticks.” (Angel, 1997) 9 Grinblatt et al. find evidence that is consistent with this “attention hypothesis”.
10 All ‘stock dividends’ exceeding 25 percent are treated as splits and do not affect retained earnings.
11 Ford et al. argue that “because analysts improve the information environment, they help reduce the value of information
13 (Dravid, 1987 and Ohlson & Penman, 1985), which results in a higher tax-option value (Constantinides, 1984). This hypothesis is weak since there is contradicting evidence in previous literature12. (5) Finally, a reason that is closely related to the marketability of a stock is the fact that small investors have greater diversification opportunities with the same amount of money when a stock is split (Baker & Gallagher, 1980 and Copeland, 1979).
2.4 Trading volume and bid-ask spread
Various reasons for splitting a stock are related to the improvement of the market in terms of liquidity. Gray et al. describe market liquidity as “the ability of the market to absorb investor trading demand quickly and in size” (2003) and can be augmented with the ability that trades happen at reasonable prices (Lee & Lee, 2013). There is to a lesser extent literature that examines what happens to the bid-ask spread of a stock after a stock split. Dennis finds that the relative bid-ask spread of the Nasdaq-100 Index Tracking Stock widens after the two-for-one split in 2000. He argues that the effects of the split are only due to the liquidity effect because there can be no signaling with an index stock split (2003). Empirical studies that focus on the trading volume of a stock find that post-split liquidity is lower than before the split, based on decreased dollar trading volume13. Copeland (1979) and Murray (1985) both use representative bid-ask quotes14 as a measure of liquidity and only consider stock splits that have a ratio of 1.25-for-1 or greater. Copeland finds that volume increases less than proportionately after stock splits and that the bid-ask spreads as a percentage of the value of the stock after a stock split are significantly wider. The conclusion is that there is a permanent decrease in relative liquidity when a firm splits its stock, which is in contrast to the theory that stock splits are driven by the desire for more liquid markets (1979). The research of Murray is an extension of Copeland’s analysis. In the short term, the proportional post-split trading volume declined but this result is not significant. Contradicting to Copeland, Murray finds that the percentage15 bid-ask spread does not change after a stock split (1985). Conroy et al. investigate the relationship between stock splits and shareholder liquidity, measured as the bid-ask spread. The approach of research of Conroy et al. is somewhat more advanced because the lowest ask and highest bid that would have been available to an investor are employed. The finding is that percentage spreads increase after splits, from 0.951 to 1.229 percent. For splits greater than 1.5-for-1, less than 10 percent of the sample shows a decrease in percentage spread at the ex-date of the split (1990). Lin et al. link the liquidity of a stock to the cost of equity capital and its liquidity risk and find that these two decrease following
12 Lamoureux and Poon (1987) vs. Dhatt et al. (1997).
13 Copeland (1979), Lamoureux and Poon (1987) and Murray (1985). 14 These are the most commonly quoted bid- and ask prices by dealers.
14 a stock split. Additionally, firms that perform a split with a higher split factor experience greater liquidity improvements (2009). Research from Conroy et al. (1990), Easley et al. (2001), Gray et al. (2003) and Schultz (2000) find similar results in the sense that percentage bid-ask spreads increase and that proportional trading volume decreases. Lakonishok and Lev find that the volume decreases after a stock split and argue that the pre-split volume is remarkably high compared to the post-split volume of a stock. This would mean that stock splits have no permanent impact on trading volume (1987). Altogether, this implies that the consensus from these analyses is that stock splits lower the market liquidity of a stock. The lower post-split prices effectively increase the bid-ask spreads. Since brokers earn on this spread, this motivates them to promote split stocks, which is consistent with the findings in the above-discussed literature (Ford et al., 2012).
2.5 Hypotheses and added value
A stock split does not have any influence on the fundamentals of a firm and hence no abnormal return on the announcement- or ex-date is expected. All possible explanations that are given for performing a stock split would be viewed as positive information content regarding the firm. The alternative hypothesis for the effect on the return of a stock is that the abnormal return is positive. However, abnormal returns may be notably negative in certain groups. The alternative hypothesis is tested for significance by a two-sided test, to be able to give a useful interpretation of these possibly negative results. The null hypothesis regarding trading volume is that a stock split does not have an effect on the trading volume of a stock. The relative bid-ask spread null hypothesis is that it remains unaffected by a stock split. Applicable to the hypothesis of trading volume as well as that of the relative bid-ask spread is that the findings of previously discussed literature and the motivation of firms to split a stock are not aligned with each other. This results in two alternative hypotheses that are tested for significance using a two-sided test. The explicit hypotheses are as follows.
𝐻!: 𝐴𝑏𝑛𝑜𝑟𝑚𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 = 0 𝐻!: 𝐴𝑏𝑛𝑜𝑟𝑚𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 ≠ 0 𝐻!: 𝐴𝑏𝑛𝑜𝑟𝑚𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒 = 0 𝐻!: 𝐴𝑏𝑛𝑜𝑟𝑚𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒 ≠ 0 𝐻!: 𝐴𝑏𝑛𝑜𝑟𝑚𝑎𝑙 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐵𝑖𝑑 − 𝐴𝑠𝑘 𝑆𝑝𝑟𝑒𝑎𝑑 = 0 𝐻!: 𝐴𝑏𝑛𝑜𝑟𝑚𝑎𝑙 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐵𝑖𝑑 − 𝐴𝑠𝑘 𝑆𝑝𝑟𝑒𝑎𝑑 ≠ 0
15 The added value of this study in the current literature is significant. The cross-sectional analysis of stock splits between the United States and Asia focuses on the return, volume and bid-ask spread of a stock. Besides the cross-sectional comparison over a recent period that has not been studied before, a deeper understanding is developed using segmentation. Each region is segmented into groups of post-split prices and adjustment factors. The event study on all groups for both countries and all three variables give insight in the effect of a stock split. Additionally, this research tests the trading strategy that invests in the period between the announcement date and ex-date of a stock split. The research question regarding this analysis is whether this trading strategy generates abnormal returns and how these abnormal returns vary among different segments. Trading strategies may be more relevant and informative to investors and is therefore a valuable contribution to the analysis of stock splits.
3. Data
This study performs an event study to analyze the effects of stock splits on the stock price, volume and bid-ask spread. The data for this study is retrieved from two sources to identify stock splits that were performed in the period 2000 until and including 2016. Broadly, The Center for Research in Security Prices (CRSP) database is used for data from the United States and Datastream is consulted for data on Asia. The stocks splits that are considered in this research are the ones that are labeled as stock split by CRSP and Datastream. Further, this study assumes that the examined period are not contaminated by other information releases. This section first introduces the dataset from the United States and subsequently that of Asia. Then the data preparation and analysis are discussed next. Subsequently, the segmentation of the samples is discussed and summary statistics finalize the data part.
3.1 United States
Data from the United States (US) consists of firms from the S&P 500 constituent list and are further examined in this research. The S&P 500 is an index where the weightings are determined by the Dow Jones Indices. The index contains 500 leading companies in terms of market capitalization. It is a market-cap-weighted16 stock market index and was the first one in the US. The index focuses on the large-cap sector and captures around 80% of the total available market capitalization in the US and thus can be seen as a representation of the market as well (S&P Dow Jones Indices, 2017).
16 A market-capitalization-weighted index gives stocks with a higher market capitalization a higher weight than stocks with a lower
16 Only firms that were part of the S&P 500 in the period of 2000 to 2016, not necessarily the whole period, are selected. Firms that split their stock in the period when they were not a constituent of the S&P 500, are not removed from the dataset. This implies that the dataset that represents the United States consists of firms that have been part of the S&P 500 between 2000 and 2016 and conducted a split in these 17 years. The Compustat – Capital IQ database is used to determine which firms meet the requirements of being a constituent of the S&P 500.
The CRSP Stock Events – Distribution database includes distribution events in which this research only obtains the events classified as stock splits. The stock splits have a specific code17 that falls in the subsequent categories: (1) splits and stock dividends, (2) same issue of common stock, (3) split and (4) normal non-taxable (CRSP, 2017). Reverse splits18 are included in this list and thus have to be deleted. Because reverse stock splits have a different effect, the analysis only focuses on non-reverse splits (Dravid, 1987, Han, 1995, Lamoureux & Poon, 1987 and Peterson & Peterson, 1992). The database further provides the price adjustment factor19, declaration- and ex-date. The price adjustment factor is equal to the number of additional shares per old share issued. In the following formula, S is the number of shares outstanding and subscript t is a date after or on the ex-date, such that t-1 is a date before the split (CRSP, 2017).
𝑃𝑟𝑖𝑐𝑒 𝑎𝑑𝑗𝑢𝑠𝑡𝑚𝑒𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 = (!!!!!!!)
!!!! =
!!
!!!! − 1 (1)
An example shows that when a firm performs a 10-for-1 split, the price adjustment factor has a value20 of 9. Consequently, there is a clear distinction between the stock split ratio, which has a value of 10, and the price adjustment factor, which has a value of 9.
This study obtains data from CRSP, which uses PERMNO21 as an identifier, and data from Compustat, which uses GVKEY22 as an identifier. The CRSP/Compustat Merged Database - Linking Table is used to combine these two datasets.
17 The code for stock splits is 5523.
18 “A reverse stock split is a reduction in the number of shares outstanding, with each share increasing in value to keep the total
value of the firm unchanged.” (Lee & Lee, 2013)
19 ‘Factor to Adjust Price’ is described as a variable that “[..] is used to adjust stock prices after a distribution so that a comparison
can be made on an equivalent basis between prices before and after the distribution.” (CRSP) ‘Price adjustment factor’ and ‘adjustment factor’ are used interchangeably.
20 𝑆!= 10 and 𝑆!!!= 1 result in !"!!
! =
!"
!− 1 = 9
21 “A unique permanent identification number assigned by CRSP to each security.” (CRSP, 2017)
22 “A unique permanent number assigned by Compustat, that can be used to identify a Compustat record in different updates if
17 3.2 Asia
Data from Asia is retrieved in a different manner from Datastream. Each Asian country has a main index that represents the market in that country. The index that is used for every country is shown in Table A-1 in Appendix II. The constituent index variable of these indexes results in a list of all firms that are part of the index. A static request on the constituent index variable and ‘Corporate Action – Last 30 events’ as data type provides the last 30 corporate actions for events from 1 January 1988, for an individual stock (Datastream, 2017). The result is a list of mergers, takeovers, reverse split, consolidation, etc., where subsequently only the stock splits are preserved. Datastream provides the fundamentals like the announcement date, event date and number of old- and new shares. The price adjustment factor can be calculated conform the previous-mentioned formula using the number of old- and new shares.
The result of the static requests is a list of Asian firms that conducted a stock split in the period between 2000 and 2016. Subsequently, for the estimation- and event window a time series request on the splitting stocks gives the necessary data. The data includes the price, volume and bid- and ask price on a daily basis. The trading volume in Datastream is in thousands and thus has to be multiplied by one thousand to be comparable with the US data. The values are raw and thus not adjusted for corporate action events and represent the actual value as recorded on that day (CRSP, 2017 and Datastream 2017). Table I shows of which countries ‘Asia’ consists in this study.
While constructing the sample that represents Asia, more countries than the four in Table I have been considered. The countries that qualify themselves for the Asian sample have the criterion of
Country Frequency Percentage
India 5 6.33 Indonesia 61 77.22 Malaysia 6 7.59 Thailand 7 8.86 Total 79 100 Table I
Constituents Asia
The sample consists of stock splits that were conducted between 2000 and 2016. The table shows of which countries ‘Asia’ consists in this research. The frequency and corresponding fraction are given for all four countries. Only non-reverse stock splits and countries with a Gross Domestic Product per capita lower than USD 10,000 are considered.
18 having a Gross Domestic Product per capita23 that is lower than USD 10,000. Table A-1 in Appendix II shows all countries that have been examined and the corresponding constituent index. Some countries had splits where the announcement date was missing. Incomplete data is excluded from the sample, as the desire is that the number of stock splits is equal in the announcement and ex-date analysis. Many of the examined countries did not have any split in the sample period or only had a reverse split. Out of the 79 examined stock splits in Asia, 61 are in Indonesia and this accounts for 77.22 percent of the total. India has, with five stock splits, the least percentage of stock splits in this sample. Malaysia and Thailand complete the list with respectively six and seven stock splits. Seeing that all four countries have different currencies, a conversion has to be made. The data from Asia is eventually going to be compared with that from the United States. Therefore, all values are converted to US Dollars using the exchange rates24 from The Federal Reserve. The Federal Reserve does not publish the exchange rate from/to Indonesian rupiah. Hence, this exchange rate is retrieved from the International Monetary Fund.
3.3 Data preparation
The analyses comprise both the effect on the declaration date (declaration analysis) as well as on the ex-date (ex-date analysis). In the dataset where we examine the effect on the declaration date, all observations with an empty declaration date are removed, and vice versa for the ex-date. Some firms split their stock more than once. Important in these cases is that each split, instead of each firm, gets its own unique number so that no splits are lost in the process. Additionally, a couple of stock prices were negative due to some inaccuracy in the CRSP database. These numbers somehow had a negative sign in front of the correct price and are thus corrected by multiplying them with -1. The trading volume is adjusted from the ex-date and further such that the results remain sensible. For example, a stock that splits with a 2-for-1 ratio is expected to trade twice as many shares on a single day because the expected total trading value is expected to remain the same. The trading volume on the ex-date and the five consecutive days is hence divided by one plus the price adjustment factor. The same remark would be true for the bid-ask spread but since the bid-ask spread is divided by the price, it is made relative. In this way it is ensured that the results are sensible when put in a table.
23 Data from the World Bank; data on the GDP per capita from 2015 in USD at 21 April 2017. 24 The exchange rates are as of 21 April 2017.
19 3.4 Segmentation
The stock splits are divided in different groups. A group either consists of a specific range of price adjustment factors or post-split prices. The post-split price of a stock is the price of the stock after the split, which is on- or after the ex-date. Here, the two samples are not divided into bins of equal size because the price adjustment factors are clustered25 around several particular values and the segmentation in Table II shows this intuitive view on the sample. The distribution of splits in Asia is shown in Panel A of Table II. The stock splits are divided in six groups of post-split prices that have different ranges. The group with the lowest post-split prices has prices that are lower or equal than USD 0.015 and consists of nine splits. The group that contains most stock splits has post-split prices that range from USD 0.03 to USD 0.1 and consists of 23 splits. The smallest group regarding size has six stock splits with the highest post-split prices and is higher than USD 0.75. In the analysis of price adjustment factors, the 79 splits are divided in five groups with different adjustment factors. There are 16 splits with a price adjustment factor of one and the group with a price adjustment factor of nine has the same number of splits in it. The largest fraction of stock splits in Asia has an adjustment factor that is between and including 4 and 7. There are nine splits that have an adjustment factor higher than ten. Panel B of Table II shows how the 540 stock splits in the United States are divided in groups of post-split price and price adjustment factor. The sample from the US is larger, which results in more and larger groups. There are eight groups with different post-split prices and seven with different adjustment factors. Regarding the post-split prices, two groups have more than one hundred splits: the group that has a range of USD 10 to USD 30 and the group with range USD 40 to USD 50. The smallest group has a post-split price that is higher than USD 100 and consists of less than two percent of the US sample. The most common price adjustment factor has a value that is between and including 1 and 1.1 and counts 398 stock splits. Three groups have an adjustment factor lower than one, where 0.5 is the largest group and contains most splits. Five of the five hundred and forty stock splits in the sample of the US have a price adjustment factor that is larger than three.
25 Dividing the clustered sample into bins implies that observations with the same value end up in different bins. This precludes
20
Panel A. Distribution of Splits in Asia
Panel B. Distribution of Splits in the United States
r > 3 5 540 1 < r < 1.1 398 r = 2 85 r = 3 8 r = 0.25 6 r = 0.3333 7 r = 0.5 31 100 < P 10 540
Range Adjustment Factor
50 < P < 60 84 60 < P < 70 34 70 < P < 100 32 30 < P < 35 68 35 < P < 40 73 40 < P < 50 119 79
Range Post-split Price
10 < P < 30 120 16 13 25 16 9 r = 1 2 < r < 3 4 < r < 7 r = 9 10 < r 0.1 < P < 0.2 0.2 < P < 0.75 0.75 < P
Range Adjustment Factor
14 13 6
79
Range Post-split Price
P < 0.015 0.015 < P < 0.03 0.03 < P < 0.1 9 14 23 Table II
Sample distribution: Post-Split Prices and Price Adjustment Factors
Both panels give insight in the groups of post-split prices and price adjustment factors and how they are divided. Panel A gives a representation of Asia and Panel B of the United States. The upper part of both panels shows the different groups of post-split prices and its corresponding range. The stock splits in Asia are divided in six groups of post-split prices. The lower part of both panels shows the range of each group regarding the price adjustment factor. Asia has five different groups of adjustment factors and the United States has seven groups. All post-split prices are in US Dollars.
21 3.5 Summary statistics
Summary statistics are provided in Table III with Asia in Panel A and the United States in Panel B. Corresponding to Table II, the sample of Asia has 79 observations and that of the United States 540. Noteworthy is that all statistics, except the number of observations, regarding the price adjustment factor are lower in the US. Setting aside the maximum, all price adjustment factor statistics are at least four times larger in Asia. The split with the highest split ratio in Asia was a 100-for-1 split while the highest in the US was 50-for-1.
All post-split prices are in US Dollars and it is obvious that all these values are exceptionally lower in Asia. The highest post-split price in Asia is more than ten US Dollars lower than the lowest post-split price in the US. An Asian stock is usually split to a price that ranges between USD 0.07 and USD 0.28 while this is approximately USD 40 to USD 45 in the US. The two average percentage bid-ask spreads are rather similar to each other. Both medians are lower than the mean but the median in Asia is however lower than that of the US, which implies that the skew26 of Asia is more positive than that of the US. The positive skew indicates that the asymmetric tail extends towards positive values, the right-hand side of the standard normal distribution (Lee & Lee, 2013). Both samples have a minimum percentage bid-ask spread of zero. A bid- and ask price that are equal to each other result in a percentage bid-ask spread of zero. Remarkable is that the maximum relative bid-ask spread is larger in the US than in Asia.
The mean of the unadjusted volume in Asia is exceptionally higher compared to the US. Conversely, the median of the unadjusted volume in Asia is lower than in the US. The maximum unadjusted volume in Asia is around 12 times larger than in the US and also the standard deviation is slightly higher than in the US. The value for the minimum is zero in both samples, which implies that there are days when no stocks are traded. The unadjusted price has a different order of magnitude in both regions. The average value in Asia is less than two US Dollars while it is USD 75 in the United States. Noteworthy is the fact that the minimum unadjusted price in the US is higher than the mean and median of the Asian sample. The maximum stock price observed in the US is with USD 3,540 particularly high. As a result of all the higher values in the US, the standard deviation has a higher value as well.
Considerable is that the number of observations of the percentage bid-ask spread, unadjusted volume and unadjusted price are equal in the sample of the US while they are not in Asia. There
22 is incomplete data on the variables ‘percentage bid-ask spread’ and ‘unadjusted volume’. Consequently, the value of these variables is lower than the number of observations of the unadjusted price. The fact that the unadjusted price has higher values than the post-split price seems consistent with the purpose of a stock split, reducing the stock price, since most data is pre-split data. Regarding the statistics of the latter three variables, there is chosen to calculate these using both the estimation- as well as the event window relative to the announcement date. Applicable to all variables in Table III is that the median is lower than the mean. This implies that all variables have a positive skew and thus a longer right tail.
Table III
Summary Statistics: Asia and the United States
Each panel shows a table with the summary statistics for its region. Panel A contains all statistics for Asia and Panel B for the United States. The second column shows the number of observations for the variable in the first column. Column three to six show, respectively, the mean, median, minimum and maximum. The last column shows the standard deviation. The post-split price and unadjusted price are in US Dollars using the exchange rate of 21 April 2017. The percentage bid-ask spread is the difference between the bid- and ask price divided by the price of the stock. The unadjusted volume is the total number of shares sold expressed in units of one share. The last three variables are calculated using both the estimation- and event window relative to the announcement date of the stock splits.
Panel A.
N Mean Median Min Max Std. Dev.
Price-Adjustment Factor 79 7.4937 4 1 99 12.6996
Post-Split Price ($) 79 0.2862 0.0773 0.0098 3.3983 0.6476
Percentage Bid-Ask Spread 13,788 0.0333 0.0097 0 0.4331 0.0568
Unadjusted Volume 13,448 7,434,282 301,500 0 1.24e+9 2.98e+7
Unadjusted Price ($) 19,771 1.6744 0.3498 0.0112 29.2793 3.7916
Panel B.
N Mean Median Min Max Std. Dev.
Price-Adjustment Factor 540 1,1269 1 0.25 49 2.1763
Post-Split Price ($) 540 44.5474 40.82 13.42 171.75 20.38
Percentage Bid-Ask Spread 136,616 0.0314 0.0245 0 0.7656 0.0245
Unadjusted Volume 136,616 1,742,596 856,566 0 1.03e+8 3.39e+6
Unadjusted Price ($) 136,616 74.9147 62.08 4.8125 3,540 137.6844
Asia
23
4. Methodology
The main methodology used in this study to analyze the effects of a stock split is the event study. An event study can be described as “a research methodology designed the measure the impact of an event of interest on stock returns”. It primarily measures the reaction of a specific economic variable to an event ex-post (Lee & Lee, 2013). A comparison among stock splits with varying event dates is possible because the event days are consistently set as day 0. The splits are in a sense aligned to be able to employ an analysis. An event study has an estimation window, [𝑇!, 𝑇!], and an event window, which runs from 𝑡! to 𝑡!. The figure below shows that both event
windows are preceded by an estimation window that ends ten days before the first event window.
4.1 Analysis
The declaration date is set to day zero in the declaration analysis. The same is done for the ex-date; such that day zero is the day of the event. The day before the event day is -1 and the day after is day 1 and so forth. In assessing the market reaction of a stock split, the focus of this research is on the short run or announcement period reaction, which is the period that covers a few days following the announcement of the stock splits (Hwang et al., 2008). Accordingly the event window of -5 to 5 is created. The estimation window ranges from day -252 till -10, and is always27 calculated from the declaration date. The estimation window in the ex-date analysis does not include the period in which the declaration of the split falls because this avoids a possible bias28. The above figure is a graphical representation of the timeline where the second event window is around the date of a stock split. The time between the announcement and the ex-date differ among stocks and thus does not result in a set amount of days between the two events and its event windows. The datasets range from 1999 to January 2017 because a stock that is split in January 2000 has an estimation window that begins approximately a year prior to the split.
27 No matter if we analyze the effect on the declaration- or ex-date, the estimation window is -252 to -10 relative to the declaration
date in both cases.
28 An estimation window that includes the declaration date of a stock split is not representative for a period in which the normal
returns are estimated.
-252 -10 -5 +5 -5 +5 𝑇! 𝑇! 𝑡! 𝑡! 𝑡′! 𝑡′!
Event 1 Event 2 (t=0) (t’=0) Estimation window Event window 1 Event window 2
24 Likewise, a stock split at the end of December 2016 has an event window that contains a few days in January 2017.
In the event study analysis of a variable, the total value can be divided into two elements, the normal value and the abnormal value. The subscripts i and t indicate the stock that is being split and the day in the event window, respectively.
𝑉!,! = 𝑁𝑉!,! + 𝐴𝑉!,! (2)
The normal value is the value that would have been expected if the event did not take place. The abnormal value is the actual ex post value of the stock over the event window minus the normal value (Bun, 2016)29. First a normal value is predicted for the days in the event window, based on a specific model that differs among the variables but is explained in a later part. The observed value minus the normal value results in the abnormal value. The abnormal value could be either abnormal return, abnormal volume or abnormal relative bid-ask spread. The average of all abnormal values on a single day in the event window are computed to be able to analyze the effect. The average abnormal return (AAR) is for example calculated as follows.
𝐴𝐴𝑅!= !! !!!!𝐴𝑅!,! (3)
In the formula, N is the number of stock splits and AR is the abnormal return across stock splits (i = 1,…, N) at day t. The sum of all abnormal returns on day t is calculated and then divided by the number of abnormal returns. Subsequently the sample standard deviation of the abnormal returns is calculated as follows (Stock & Watson, 2012).
𝑠! = !!!! !!!!(𝐴𝑅!,!− 𝐴𝐴𝑅!)! (4)
Considering the predicted values in the event window, these are computed according to a specific formula. The expected normal return is based on a simple linear regression in the estimation window. An Ordinary Least Square (OLS) regression of an individual stock’s return on the return of the index is performed to compute the beta of the firm. The method to estimate the return is called the market model and controls for the relationship between stock returns and market returns. On an event day, the computed beta together with the return on the index is used to calculate the predicted return. The regression equation of the market model is given below. 𝐸 𝑅!,! = â! + 𝛽! 𝑅!,! (5)
The estimated constant is â! and the estimated beta is 𝛽!. The constant, plus the beta that is multiplied with the return of the market index 𝑅!,!, results in the predicted normal return on that day. This procedure is repeated for all splits and on every day in the event window.
25 The predicted value for the trading volume is computed in two ways. First, a regression of trading volume on the return of the index is performed to compute the trading volume beta. The market model relationship with the index return is however to a lesser extent expected. Therefore, a second method is applied to calculate the predicted trading volume in the event window using a different approach. This second method is simply the average of the trading volume in the estimation window and is called the mean-adjusted model. The predicted value for the bid-ask spread is computed using the mean-adjusted model as well and thus takes the average bid-ask spread in the estimation window (De Jong & De Goeij, 2011). The following matrix gives a representation of the dataset in the event window period when there are 𝑁 stock splits.
𝐴𝑉!,!! ⋯ 𝐴𝑉!,!! ⋮ ⋯ ⋮ 𝐴𝑉!,!! ⋯ 𝐴𝑉!,!! 𝐴𝑉!,! ⋯ 𝐴𝑉!,! 𝐴𝑉!,! ⋯ 𝐴𝑉!,! ⋮ ⋯ ⋮ 𝐴𝑉!,!! ⋯ 𝐴𝑉!,!!
Each column in the matrix is a time series of abnormal values for stock split i. 𝐴𝑉! is the abnormal value for the first stock split in the sample and there are as many columns as number of splits in the sample, which is denoted by 𝑁. The second subscript indicates the day relative to the event date, such that -1 is the day before the event date and 0 is the event date itself. Since the event window is 11 days, the values of 𝑡! and 𝑡! equal -5 and 5, respectively. The values for 𝑇! and 𝑇! are -252 and -10, respectively, as is indicated on the timeline. Each row is a cross section of abnormal values for time period t. The average abnormal return, but also the average abnormal volume and average abnormal bid-ask spread are tested for significance on every day in the event window. The following formula gives the test statistic for testing the abnormal return.
𝑇𝑆! = 𝑁!!"! !
! ≈ 𝑁(0,1) (6) The null hypothesis in this test is 𝐻!: 𝐸 𝐴𝑅!" = 0. The 𝐴𝐴𝑅! is the average abnormal return on
day t, 𝑁 is the number of abnormal returns and 𝑠! is the sample standard deviation of abnormal
returns in period t. Significance is tested assuming independent and identically distributed returns and using the usual method for a t-statistic. This equation does not use 𝑁 − 1 because De Jong and De Goeij state that there is strong evidence that stock returns do not satisfy the normality assumption, which is used to derive the distribution of this test (2011). Whereas 𝑇𝑆! tests
26 zero, testing a longer period is informative as well. The first step to do this is to aggregate the abnormal returns over the event period.
𝐶𝐴𝑅! = 𝐴𝑅!,!!+ ⋯ + 𝐴𝑅!,!! = 𝐴𝑅!"
!!
!!!! (7) The cumulative abnormal return follows from adding all individual abnormal returns of firm i over the event period. Secondly, the average of all cumulative abnormal returns is calculated to obtain the cumulative average abnormal return (𝐶𝐴𝐴𝑅) but can also be calculated by aggregating the average abnormal returns over the event period.
𝐶𝐴𝐴𝑅 = !! ! 𝐶𝐴𝑅!
!!! = !!!!! !𝐴𝐴𝑅! (8)
The 𝐶𝐴𝐴𝑅 has the following sample standard deviation.
𝑠 = !!!! !!!!(𝐶𝐴𝑅! − 𝐶𝐴𝐴𝑅)! (9)
The 𝐶𝐴𝐴𝑅 is subsequently tested for significance using the following t-test.
𝑇𝑆! = 𝑁!""#! ≈ 𝑁(0,1) (10)
The null hypothesis in this test is 𝐻!: 𝐸 𝐶𝐴𝑅! = 0. Dividing the 𝐶𝐴𝐴𝑅 by the sample standard
deviation 𝑠 and multiplying it with 𝑁 results in the t-statistic. This test statistic gives the opportunity to test the significance of cumulative abnormal returns over the complete pre-split and post-split period. In both test statistics, the central limit theorem preserves approximate normality for the t-statistic when the abnormal returns are independent. Additionally, it is assumed that the values are uncorrelated30 and that the standard deviation is the same in the event period as in the estimation period (De Jong & De Goeij, 2011).
The alternative hypothesis for the abnormal return is two-sided and has a critical value of 1.96 when a 95 percent confidence interval is used. The test statistic for the abnormal trading volume and abnormal relative bid-ask spread are similar to that of the abnormal return. Abnormal return is replaced by abnormal volume and abnormal relative bid-ask spread, respectively. The hypotheses for the trading volume and relative bid-ask spread are both two-sided. This implies that the critical value for the t-statistic equals a value lower than or equal to -1.96 or higher than or equal to 1.96. A value that is lower (higher) than the positive (negative) critical value does not reject the null hypothesis. An outcome of the t-test that lies in the critical area results in a rejection of the null hypothesis. This would then imply that stock splits do have an effect on the tested variable.
