Ex-day returns for stock distributions: An anchoring explanation

Transcription

1 Ex-day returns for stock distributions: An anchoring explanation Abstract We propose an anchoring argument to explain the abnormal returns on the ex-days of stock splits, stock dividends, and reverse stock splits. We develop an anchoring model and find supporting evidence that stocks paying stock dividends or splitting their shares have positive ex-day abnormal returns, and that stocks reversely splitting their shares have negative ex-day abnormal returns. Cross-sectionally, the ex-day abnormal return is positively associated with the split factor and this positive association decreases with the split factor. In particular, when the split factor is 0.5 or 1, which is common and more visible, the anchoring bias is less severe. In addition, we document that the magnitude of the abnormal returns on the ex-days of stock distributions is greater among stocks that are largely traded by unsophisticated investors and stocks whose ex-day signal is noisier. (137 words) Key words: anchoring, stock dividend, stock split, ex-day return

2 1 Introduction Stock distributions, including stock dividends, stock splits, and reverse splits, do not directly affect future cash flows of a firm. In a frictionless and efficient market, these corporate actions are just cosmetic accounting changes and should not change firm value. However, previous studies have suggested that these noneconomic events are not trivial as the theory would lead us to believe. Grinblatt et al. (1984) examine 1,740 ex-day events for stock dividends and stock splits during the period from 1967 to 1976, and document significantly positive abnormal returns on the ex-day. Furthermore, they show that the ex-day effect differs between stock dividends and splits, with stock dividends exhibiting greater abnormal returns on the ex-day. The ex-day effect of stock distributions is puzzling. Woolridge (1983) cites an opinion from Committee on Accounting Procedure in the 1960s: Many recipients of stock dividends look upon them as distributions of corporate earnings and usually in an amount equivalent [emphasis added] to the fair value of the additional shares received. Furthermore, it is to be presumed that such views of recipients are materially strengthened in those instances, which are by far the most numerous, where the issuances are so small [emphasis added] in comparison with the shares previously outstanding that they do not have any apparent effect upon the share market price and, consequently, the market value of the shares previously held remains substantially unchanged. The above excerpt, in short, tells that investors fail to fully understand the substance of a distribution event, and incline to rely on past price in evaluating stock value on the ex-day. Such propensity is particularly strong when the split factor is small. While Woolridge (1983) argue that price discreteness, transaction costs, and tax are partially responsible for the documented anomaly, we interpret the phenomenon from another angle. We contend that past stock price actually acts as 2

3 an anchor in investors valuation process, and deters sufficient price adjustment after the distribution event takes effect. Motivated by the statement, we propose to develop a unified anchoring framework and explain the ex-day abnormal returns of stock dividends, stock splits, and reverse stock splits from a behavioral finance perspective. The anchoring effect refers to people s tendency to generate an estimate biased toward an anchor, which is an arbitrary value considered before the estimation (Jacowitz and Kahneman, 1995). Tversky and Kahneman (1974) advocate that, in making estimations, people often use the anchor as a starting point and directional adjust to yield the final estimate. Epley and Gilovich (2006) argue that as an adjustment is laborious and painful, people tend to stop the adjustment once the estimate enters into a plausible range, causing the adjustment to be insufficient. Based on the anchoring argument, we develop a model proposing that in estimating stock value on the ex-day, investors use the cum-day price observed one day before the ex-day as an anchor and insufficiently adjust for the split factor. For stock dividend payments and stock splits, the cum-day price is an upside anchor, making the ex-day price upward biased and bringing positive abnormal returns. For reverse stock splits, in contrast, the cum-day price acts as a downside anchor on the ex-day and induces negative abnormal returns. Overall, the model predicts a positively association between the split factor and the ex-day abnormal returns. Further, the model predicts the magnitude of the ex-day abnormal return to be affected by the degree to which investors are subject to the anchoring bias. The more investors are affected by the anchoring bias, the higher the ex-day abnormal return is for stock dividends and stock splits. We utilize a large stock distribution sample spanning from year 1926 to 2009 to test predictions generated by the anchoring model. Consistent with our model 3

4 predictions, we find that the abnormal return on the ex-day of stock dividends is on average positive, with a mean of 1.09% for 8,219 observations. The average abnormal return for stock splits is 0.76% for 13,698 observations. In addition, the ex-day abnormal return for reverse stock splits is on average negative with a mean of -1.79% for 1,238 observations. Cross-sectionally, the ex-day return is positively associated with the split factor. In addition, the propensity of the anchoring bias is affected by the split factor and the percentage of institutional investors in the shareholder base. Specifically, the split factor for stock dividends is often much lower than stock splits. According to the argument made by Committee on Accounting Procedures, as the stock dividend payment is less apparent than stock splits, investors are more likely to naively rely on the cum-day price as the prior and insufficiently adjust for the split factor, resulting in stronger anchoring bias and higher ex-day abnormal returns for such splits. Certain split factors like 0.5 and 1 are more apparent and common and thus in these cases, investors are less likely to suffer from the anchoring bias. In addition, we use the percentage of the institutional ownership to proxy for the aggregate propensity of the anchoring bias with lower institutional ownership representing higher propensity of the anchoring bias. We find that consistent with the model, as the institutional ownership is higher, the ex-day return is lower. Furthermore, consistent with the anchoring model, we find evidence that as the ex-day market is noisier, the anchoring bias in the wrong prior is incorporated into the ex-day return more and the ex-day abnormal return is higher. Our paper contributes to the literature in several aspects. First, we add to the behavioral finance literature by providing an additional piece of evidence on the influence of the anchoring bias on the stock market. Our empirical results show that the effect of the bias is prominent even in recent years. The anchoring effect has been well recognized as a deep-rooted and robust psychological bias, and appears to 4

5 be a fundamental cognitive bias related to many other behavioral biases, including the prospect theory, the disposition effect, the under-reaction to news, the money illusion, etc. Yet the application of anchoring in finance is at its infancy. By investigating the impact of the anchoring bias on stock valuation both theoretically and empirically, our study provides new evidence on the role played by the anchoring bias in the financial markets. Second, we provide an alternative explanation for the abnormal returns observed on the ex-day of stock distributions. While previous studies mainly address the issue from microstructure-related arguments (see, for example, Grinblatt et al. (1984), Lamoureux and Poon (1987), Maloney and Mulherin (1992), Conrad and Conroy (1994), and Bali and Hite (1998)), we propose to understand the phenomenon based on a simple anchoring argument. The argument is general enough to generate a unified framework to explain the ex-day abnormal returns for three types of stock distribution events, including stock dividends, stock splits, and reverse stock splits. We find strong empirical evidence supporting the conjecture that the anchoring effect is a driving force behind the ex-day anomaly. Third, we examine the cross-sectional difference in the magnitude of the exday abnormal returns, which lacks sufficient investigation in the literature. The findings in this paper could shed lights on trading strategies driven by distribution events. Given the well-documented positive abnormal returns on the ex-day of stock dividends and splits, some funds have already implemented trading strategies that go long on stocks which are about to go ex-distribution, expecting the ex-day stock price to adjust toward the cum-day level. By showing that the ex-day abnormal returns differ across stocks with different attributes, our study suggests that these funds could further increase their profits by focusing on distribution events of stocks that have a larger split factor, are heavily traded by unsophisticated investors, or have noisier ex-day signals. 5

6 The rest of the paper is organized as follows. In section 2, we go over related literature on stock distributions and the anchoring effect. In section 3, we develop an anchoring model to relate the split factor and the propensity of the anchoring bias to ex-day returns. Testable hypotheses are developed. In section 4, we provide descriptive statistics for ex-day returns and form portfolios to examine the impact of related variables on the ex-day returns. In section 5, we run cross-sectional regressions to examine the relation between ex-day abnormal returns and a set of variables of interest. Robustness tests are performed. Section 6 concludes the paper. 2 Literature review While a stock distribution event appears to be simply a superficial change of stock price and the number of shares outstanding, the empirical finance literature is bounded with evidence on the positive abnormal returns on both the announcement and ex-day of a stock dividend or split (See, for example, Fama et al. (1969), Woolridge (1983), Grinblatt et al. (1984), Eades et al. (1984), Ohlson et al. (1985), Dravid (1987), Lamoureux and Poon (1987), Lakonishok and Lev (1987), Brennan and Copeland (1988), Asquith et al. (1989), Maloney and Mulherin (1992), and Conrad and Conroy (1994)). Woolridge and Chambers (1983), among others, further show that in contrast to a stock dividend or split, the abnormal return on both the announcement and ex-day of a reverse stock split is negative. The announcement-day anomaly has been explored intensively. Two major arguments have been developed to explain the abnormal returns on the announcement day of stock distributions: signaling and optimal trading range. The first argument asserts that managers convey private information on future earnings through stock distribution events, and investors revise their belief on firm values accordingly. (see, for example, Brennan and Copeland (1988), Asquith et al. (1989), Mcnichols and 6

7 Dravid (1990), and Han (1995)). However, inconsistent with the signaling hypothesis, Byun and Rozeff (2003) fail to find long-run abnormal returns after stock splits. The optimal trading range argument argues that the distribution events realign per-share prices to a preferred trading range and improve liquidity and marketability. Empirical evidence on this argument is inconclusive. While some studies offer evidence suggesting that stock splits increase liquidity through either lowering the stock price or changing the the tick-to-price ratio ( Baker and Gallagher (1980), Angel (1997), Anshuman and Kalay (1998), Anshuman and Kalay (2002), and Brennan and Hughes (1991), among others), others find contradicting evidence and conclude that splits actually increase trading costs and attract uninformed trades (see, for example, Copeland (1979), Conroy et al. (1990), and Easley et al. (2001)). Although the ex-day anomaly for stock distributions has drawn equal attention in the finance literature, its source has been exploited less adequately compared to that of the announcement-day anomaly. As any information contained in the distribution event has been revealed on the announcement day, information-based arguments are not relevant in explaining abnormal returns on the ex-day. In addition, as stock distribution events are not taxable, tax argument is not likely to be responsible for the documented anomaly. Existing studies mainly attempt to explain the ex-day anomaly for stock distributions based on microstructure arguments (see, for example, Grinblatt et al. (1984), Lamoureux and Poon (1987), Maloney and Mulherin (1992), Conrad and Conroy (1994), and Bali and Hite (1998)). These studies conclude that the bid-ask bounce, price pressure, change in order flows, and price discreteness could contribute to the abnormal returns around the ex-day of stock distributions. In this study we focus on the ex-day anomaly, and provide a potential explanation based on an anchoring argument. 7

8 2.1 The anchoring effect The anchoring effect is first discussed by Tversky and Kahneman (1974). They report experiments where subjects are asked to estimate the percentage of African countries in the United Nation. They show that the estimations given by subjects are biased towards random numbers that the subjects have just observed, though the numbers are simply generated by a wheel of fortune. The anchoring effect has been documented to be a robust and reliable cognitive bias. Chapman and Bornstein (1996) and Englich and Mussweiler (2006) show that the outcomes of lawsuits are influenced by sentencing demands, in that the plaintiff gets more if he or she requests more. Northcraft and Neale (1987) find that in realestate pricing, experienced professionals are unconsciously influenced by arbitrary posted prices. Dodonova and Khoroshilov (2004) show that in on-line auctions, the bid price is anchored by the non-binding buy-now price. Galinsky and Mussweiler (2001) emphasize the anchoring role of the first price in negotiations, and Green et al. (1998) and Simonson (2004) identify the anchoring effect in determining purchase prices. Wansink et al. (1998) deal with the anchoring effect in purchase quantity decisions. Investigating the anchoring effect in the financial markets is not easy, as little is known about how investors update their reference points. Shefrin and Statman (1985) find that investors relay on their historical purchase price as a reference point in evaluating their portfolios. Shafir et al. (1997) postulate that anchoring on the nominal evaluation gives rise to the money illusion. George and Hwang (2004) show that investors use the 52-week high as a reference point in assessing the potential impact of news on stock price. Campbell and Sharpe (2009) find that consensus forecasts of monthly economic releases are biased toward the values of previous months data releases, and that the anchoring bias is anticipated by 8

9 market participants. Cen et al. (2010) examine the forecast errors induced by crosssectionally anchoring to the industry median, and such errors forecast future stock returns. Chang and Ren (2010) show that in issuing new shares with an existing class of shares cross-listed in another segmented market as a reference, participants anchor on the reference price and this anchoring effect contributes to IPO underpricing. Stock distributions provide another unique setting where the anchoring effect is most likely to present and where the anchor could be identified with less difficulty. The ex-day provides a clear cutoff and, as implied by the statement made by Committee on Accounting Procedure, stock price before the ex-day is a readily available anchor for investors. In addition, as any information on the distribution event is revealed on the announcement day and as the events examined in this study are all non-taxable, our investigation is free from the influence of management signaling or tax effect. 3 Model and hypotheses 3.1 The prior The true value of the security on the ex-day for stock dividend payments, stock splits, and reverse stock splits, is represented by θ, which is assumed to follow a normal distribution: θ N ( θ, σ 2 θ ). The historical cum-day closing price on the last trading day before the ex-day is denoted as P Cum, which acts as a biased prior about the value of the security on the ex-day. In a perfect market, the expected value of the ex-day price θ should be: θ = P Cum (1 + f), (1) where f is the adjustment factor as defined in CRSP. For stock dividend and stock 9

10 split events, f (0, + ). E.g., if 50% of stock dividend is paid, f is 0.5. On the exday, when there s no new information, the ex-day price should equal to P Cum /1.5. In a 2-for-1 split, the split factor f is 1 such that the ex-day stock price should be half of the cum-day price. For reverse splits, however, f ( 1, 0). E.g., after a 1-for-2 reverse split, the split factor is 0.5 such that the ex-day price should be twice as high as the cum-day price. However, participants may suffer from the anchoring bias when they refer to the cum-day price to get the prior θ. We contend that investors use the cum-day price at face value as the anchor and insufficiently adjust for the split factor. As a result, the anchored prior, denoted as θ A, is: θ A = P Cum 1 + γf = θ 1 + f 1 + γf, (2) where γ [0, 1]. γ measures the propensity of the anchoring bias, with a lower γ representing a less sufficient adjustment, and therefore a greater propensity of the anchoring bias. At one extreme when γ = 1, investors sufficiently adjust for the split factor and there is no anchoring effect such that θ A = θ. At the other extreme when γ = 0, investors completely ignore the effect of the stock distribution event and naively use the historical cum-day price P Cum at face value as the prior. We further define: y = θ A θ θ = 1 + f 1 + γf 1 = 1 γ 1/f + γ, (3) where y is the percentage of the anchoring effect manifested in the prior. As f ( 1, + ) and γ [0, 1], only when f > 0, we can have θ A >= θ and y > 0 regardless of the value of γ. When f < 0, it follows that θ A <= θ and y < 0. It can be found that the first derivative of y respect to f is positive, while the second derivative is negative. The absolute percentage difference between θ A and θ, γ, is 10

11 negatively associated with γ, and therefore positively associated with the propensity of the anchoring bias. Figure 1 illustrates the relation between the anchoring effect y and two parameters, the split factor f and the propensity of the anchoring bias γ. The vertical line represents y, the magnitude of the anchoring effect in the prior, and the horizontal line represents the split factor f. The solid line is plotted for γ = 0.8 (mild anchoring) and the dashed line is plotted for γ = 0.2 (strong anchoring). In both scenarios, y is positively associated with f, and this positive relation is decreasing with f. Throughout the range of f, y is monotonically increasing in f. The magnitude of the anchoring effect y, however, is larger in the strong anchoring scenario (γ = 0.2). It is worth noting that in the above analysis, we assume that γ, the propensity of the anchoring bias, is independent of the split factor f. This assumption is invalid sometimes in reality. For instance, when the split factor equals to 1, which implies a common and easily processed 2-for-1 split, investors are more likely to understand the substance of the event as they are more familiar with such a split. As a result, the anchoring effect is weakened. In contrast, when the split factor is just 0.05, which implies a 5% stock dividend paid, investors are more likely to overlook the stock distribution event as they are less acquainted with such an event and as the consequence of such a split is not obvious. In such a case, investors tend to suffer from a stronger anchoring bias and the ex-day abnormal returns are likely to be high, as described in the statement made by Committee on Accounting Procedure. Therefore, the relation between y and f is no longer monotonically positive due to the dependence of γ on f. For illustrative purpose, we set γ as a function of the split factor f: γ = 0.5 abs(f) when f is neither 0.5 nor 1. We adopt this functional form to make γ fall 11

12 within the rang of [0,1] as the majority of the observations in our sample has a split factor less than 2. Note that in this case, the first derivative of y respect to f is still positive, and the second derivative is still negative. When f equals to 0.5 or 1, however, the associated γ should arguably be closer to unit. We take the average of the original γ and 1 and set γ = (0.5 abs(f) + 1)/2. We plot γ as a function of f in Panel A of Figure 2 and find a rough V-shape relation between γ and f. We then insert this function into Eq. (3) and re-plot the magnitude of the anchoring bias y as a function of f in Panel B. We find that y is positively associated with f when f is small. As f grows after a certain point, the magnitude of the anchoring bias y turns to be negatively associated with f. Furthermore, the function is discontinuous when the split factor equals to 0.5 or The signal s In the previous subsection, we model the percentage of the anchoring bias incorporated into the prior formed at the market close of the cum-day. Between the market closes on the cum-day and on the ex-day, a noisy signal s is observed. It could be a piece of macroeconomic, industry-level, or firm specific news. The signal could also convey information on changes in microstructure-related factors, including bid-ask bounce, price pressure, order flows, or price discreteness. The signal s is modeled as: s = θ + ɛ, (4) where ɛ is a white noise and ɛ N (0, σ 2 ɛ ). ɛ is assumed to be orthogonal to θ, and could be either positive or negative. 12

13 3.3 The estimate and ex-day return Given the anchored prior θ A formed at the close of the cum-day and the new signal s observed on the ex-day, the rational expectation of θ is: E[θ θ A, s] = σ2 ɛ σ2 θ σθ 2 + θ A + σ2 ɛ σθ 2 + s. (5) σ2 ɛ The ex-day price is set based on this expectation. The ex-day return, Ret, is the percentage difference between E[θ θ A, s] and the comparable cum-day price θ: Ret E[θ θ A, s] θ 1. = σ2 ɛ σ 2 θ + σ2 ɛ 1 + f 1 + γf + σ2 θ σ 2 θ + σ2 ɛ θ + ɛ θ 1. (6) The expected return is therefore: E[Ret] = σ2 ɛ σ 2 θ + σ2 ɛ ( 1 + f 1 + γf 1) = σ2 ɛ σ 2 θ + σ2 ɛ 1 γ 1/f + γ = σ2 ɛ σθ 2 + y (7) σ2 ɛ Thus, the expected ex-day return is a fraction of y in Eq. (3). The fraction is σɛ 2, σθ 2+σ2 ɛ ranging between zero and one. When the ex-day signal s is noisier with a higher variance, and /or the true value θ is less uncertain with a lower variance, the value of the fraction will be higher and closer to unit, implying that a larger proportion of the anchoring bias in the prior is transmitted into the ex-day returns. The noise contained in the signal, or ɛ, also has influence on the ex-day return. When ɛ is realized to be positive, the ex-day return will be higher. When ɛ is realized to be negative, the ex-day return will be lower. 3.4 Testable hypotheses In Eq. (7), γ [0, 1]. For stock dividend and stock split events, f > 0 such that E[Ret] > 0 for all possible values of γ. For reverse stock splits, f ( 1, 0) such 13

14 that 1/f (, 0) and E[Ret] < 0 for all possible values of γ. Accordingly, we have the following testable hypothesis: Hypothesis 1 After a stock dividend or a stock split, the ex-day return is expected to be positive. After a reverse stock split, the ex-day return is expected to be negative. Cross-sectionally, in Eq. (3), the first derivative of y respect to f is positive, while the second derivative of y respect to f is negative. In Eq. (7), the expected ex-day return is a proportion of y. As a result, the ex-day return is expected to be positively associated with f and negatively associated with f 2. This hypothesis applies to the scenario when γ is independent of f as well as the scenario when γ is roughly a v-shape function of f. Accordingly, we derivate the following hypothesis: Hypothesis 2 Ceteris paribus, the ex-day return is expected to be positively associated with the split factor and negatively associated with the split factor square. Special cases are identified for events with a split factor of 0.5 (50% stock dividend) or 1 (2-for-1 split). These split factors are more commonly used and investors are more familiar with such splits. Consequently, investors are less likely to suffer from the anchoring bias and the anchoring effect is expected to be weaker. Hypothesis 3 Ceteris paribus, when the split factor is 0.5 or 1, the magnitude of the ex-day return is less than other scenarios. When γ is small, the propensity of the anchoring bias is stronger, and the expected ex-day return in Eq. (7) is greater in magnitude. We use the percentage of institutional investors in the shareholder base as reported in 13/F to proxy for the propensity of the anchoring bias, with lower institutional ownership implying a lower γ. We derive the following testable hypothesis: Hypothesis 4 Ceteris paribus, when the institutional ownership is higher, the ex- 14

15 day return is smaller in magnitude. When there is less uncertainty about the true value θ (σ 2 θ is small) or when the signal s is noisier (σ 2 ɛ is large), the anchoring bias in the prior θ A is transmitted more into the ex-day return, increasing the magnitude of the ex-day return. Accordingly, we develop the following hypothesis: Hypothesis 5 Ceteris paribus, the magnitude of the ex-day return is negatively associated with σ 2 θ and positively associated with σ2 ɛ. 4 Ex-day returns, split factors, institutional ownership, and variances 4.1 Data In the empirical analysis, we include all available observations from year 1926 to We collect stock distribution events including stock dividends, stock splits, and reverse stock splits from CRSP daily event database. We require a distribution event to have a CRSP event code of 5523 to be included in our sample. As such events are not taxable, tax-related arguments are not relevant in our study. Some stock distributions are mixed with cash dividend payments or other events. To avoid possible contamination from other concurring events, we eliminate a stock distribution event if the stock has another event concurring on the same ex-distribution date according to CRSP daily event database. Following Graham et al. (2003), we eliminate REITs and ADRs. The daily stock price and return data are collected from CRSP daily return database. The monthly stock returns are collected from CRSP monthly return database. Institutional holding data are obtained from Thomson-Reuters institutional holdings (13F) database, which starts distributing institutional holding in- 15

16 formation in year The institutional holding data are matched to the number of shares outstanding from CRSP daily event database to get the institutional ownership measure. 4.2 Descriptive statistics Table 1 shows the descriptive statistics of ex-day returns organized by the type of events. We calculate the raw returns on the ex-day and report the statistics for all available observations in Panel A. Statistics for abnormal returns on the ex-day are reported in Panel B. We follow Graham et al. (2003) and Bell and Jenkinson (2002) to define the abnormal return as the ex-day raw return minus the expected return from the market model. Beta in the market model is estimated using monthly data from the 60 months proceeding the month in which the ex-distribution takes place. Stocks with fewer than 24 monthly return observations are eliminated. In panel C, we report statistics for the trimmed samples where abnormal returns in the top and bottom 2.5 percentiles are set to missing. A rough comparison across the three panels in Table 1 reveals that for the full sample, the magnitude of the abnormal returns in Panel B is closer to that of the raw returns in Panel A. However, the magnitude of the abnormal returns for the trimmed sample in Panel C is on average lower than that for the full sample. All three panels show that, consistent with Hypothesis 1, reverse stock splits have significantly negative ex-day returns, and stock splits and stock dividends have significantly positive ex-day returns. Furthermore, the ex-day returns for stock dividends are on average higher than those for stock splits. In Panel C, 23,155 events are included in the trimmed sample. For 1,238 reverse stock splits, 61.6% observations show negative ex-day abnormal returns, with a mean of -1.79% and median of -1.61%. For 13,698 stock splits, 58.9% observations exhibit positive ex-day abnormal returns, with a mean of 0.76% and a median of 0.50%. For 8,219 stock dividends, 63.9% observations 16

17 show positive ex-day abnormal returns, with a mean of 1.09% and median of 0.80%. Hereafter, we focus the analysis on abnormal returns for the trimmed sample. Table 2 shows the sample distribution of ex-day abnormal returns for the trimmed sample, organized by decades from year 1926 to In Panel A, we show the statistics for pure reverse split events. CRSP records 182 reverse split events with an average abnormal return of -2.26% in the 1980s, 587 events with an average abnormal return of -1.93% in the 1990s, and 387 events with an average abnormal return of -1.26% in the 2000s. In all three decades, the ex-day abnormal return is significantly negative for the reverse split events. Panel B shows the statistics for pure stock split events. We find 1,038 split events in the 1960s, 1,618 events in the 1970s, 4,333 events in the 1980s, 4,113 in the 1990s, and 2,012 events in the 2000s. The average ex-day abnormal returns in all five decades are significantly positive, ranging from 0.35% in the 1960s to 1.05% in the 1980s. Panel C shows the results for pure stock dividend events. Stock dividends seem to be popular in the 1970s but lose their popularity in the 21 th century. We find 1,238 stock dividend payments in the 1960s, 1,939 events in the 1970s, 1,733 events in the 1980s, 1,406 in the 1990s, and 757 events in the 2000s. The average ex-day abnormal return is significantly positive in all five decades from the 1960s to the 2000s. It varies from 54% in the 1960s to 1.62% in the 1990s, and is as high as 1.53% in the 2000s. Hypothesis 1 is strongly supported from the sub-period analysis. Note that the ex-day abnormal return of the stock dividend events tend to be higher than that of split events. This is consistent with the conjecture that investors are more likely to suffer from the anchoring bias when the split factor is less noticeable, resulting in higher ex-day abnormal returns. 17

18 4.3 The impact of the split factor According to Hypothesis 2, a larger split factor will be associated with a higher ex-day return. In addition, Hypothesis 3 predicts that when the split factor is 0.5 (3-for-2 split or 50% of stock dividend) or 1 (2-for-1 split), the ex-day return will be lower than it is in other cases. We rank all the events by the split factor and form portfolios to calculate the average returns. In the original sample, the split factor reported by CRSP varies from to 99. We trim the sample by setting the split factors in the top and bottom 2.5 percentiles to missing. In the trimmed sample, the split factor varies from to 2. The results are reported in Panel A of Table 3. The relation between the ex-day abnormal return and the split factor is not monotonic. However, for events with a split factor below 0.25, the ex-day mean return increases with the split factor, supporting Hypothesis 2. We find that observations with a split factor that equals to 0.5 or 1 constitute almost half of the sample. As investors are more familiar with these two types of splits, they are less likely to subject to the anchoring bias. Consequently, stock splits with a split factor of 0.5 or 1 have lower ex-day returns compared to other split events, supporting Hypothesis 3. Figure 3 exhibits a scatter plot with ex-day abnormal return on the y-axis and the split factor f on the x-axis. To show the trend more clearly, we plot the curve of the fitted cubic function estimated using all available observations in the trimmed sample. It is shown that the abnormal return is not monotonically increasing in the adjustment factor. The shape of the curve is quite close to that of the figure in Panel B of Figure 2, which illustrates the relation between y and f when γ is allowed to change with f. A stock split often has a split factor greater than 0.5, while the split factor of a 18

19 stock dividend payment is usually less than 0.5. This raises a potential concern that the relation between the ex-day abnormal return and the split factor could simply reflects the fact that investors respond to different distribution events differently. To address this concern, we proceed to examine whether investors response to a distribution event is affected by size of the split factor or by the type of the event. Within each split factor category, we compare the ex-day abnormal returns between stock splits and stock dividends. The results are shown in Panel B of Table 3. There are totally 1,946 reported splits that are actually stock dividends with split factors below 0.5, and totally 716 reported stock dividends that are actually splits with split factors above 0.5. The last column in panel B shows that the difference in abnormal returns between splits and stock dividends is indistinguishable from zero for all the eight split factor categories. The results suggest that investors can see through the name of splits or stock dividends and confirm that investors response to a distribution event is affected by the split factor rather than the type of the event. 4.4 The impact of the institutional ownership Institutional investors, as a group of sophisticated investors, are less likely to subject to psychological biases. Thus, we rely on the percentage of institutional holding in the shareholder base to measure the propensity of the anchoring bias. According to Hypothesis 4, the percentage of institutional ownership is negatively associated with the magnitude of the ex-day abnormal returns. For firms without records in 13/F filing, we set the institutional ownership to zero. For observations with calculated institutional ownership higher than 100%, we set the institutional ownership to missing. It is noted that for the 1,238 reverse splits with available abnormal return data, only 11 firms have nonzero and non-missing institutional ownership. Thus, this test lacks power for the reverse split subsample. Consequently, in investi- 19

20 gating the relation between ex-day abnormal return and institutional ownership, we focus on stock dividends and splits only. We divide observations into groups based on institutional ownership, and calculate the average ex-day abnormal returns for each group. The results are reported in Table 4. For the 10,853 observations with nonzero institutional ownership data, the ex-day abnormal return decreases monotonically with institutional ownership. Such result supports the conjecture that the anchoring bias is less prominent among stocks that are largely held by sophisticated investors, and that the magnitude of the ex-day abnormal returns is lower among such stocks. The result is also consistent with the sentiment story proposed by Baker and Wurgler (2006) arguing that overoptimistic investors, who tend to be retail investors, push up the ex-day prices and generate positive returns. We thus perform multi-variate regression analysis in section 5 to further investigate the relation between ex-day abnormal return and institutional ownership with market sentiment controlled. 4.5 The impact of variance According to Hypothesis 5, when the valuation uncertainty is higher ( σθ 2 is lower) or when the ex-day signal s is noisier (σɛ 2 is higher), the anchoring bias in the prior will be transmitted into the ex-day return with a higher proportion, as shown in Eq. (7). In the empirical investigation, we rely on the variance of daily stock price over a six-month-period ending before the ex-distribution month to proxy for valuation uncertainty, or σθ 2. For stocks with fewer than 60 daily observations, we set σ2 θ to missing. To capture the noise contained in the ex-day signal, we first calculate the market beta for each stock using return data over the past six months. We then derive the expected ex-day return based on the market model and use the squared 20

21 fitted error from the model to measure the variance of the ex-day signal s, or σ 2 ɛ. To examine the relation between valuation uncertainty and ex-day returns, we divide all the observations into two groups based on σ 2 θ and compare the average ex-day abnormal returns of the two groups. The results are shown on the left-hand side of Panel A in Table 5. We find that the ex-day abnormal return is higher for the low σ 2 θ group than for the high σ2 θ group. The return difference between the two groups is 0.18%, which is significantly different from zero at the 1% level. The results imply that the ex-day abnormal return is negatively affected by the information uncertainty, which is consistent with Hypothesis 5. We repeat the test based on groups sorted by σ 2 ɛ to investigate the influence of the noise contained in the ex-day signal on ex-day returns. The results are shown on the right-hand side of Panel A in Table 5. The mean (median) ex-day abnormal return is 0.77% (0.55%) for the low σ 2 ɛ group and 0.89% (0.70%) for the high σ 2 ɛ group. Again, the difference in the ex-day abnormal returns between the two groups is significantly different from zero. The evidence that the ex-day abnormal return is high when the ex-day signal contains more noise renders support to Hypothesis 5. We further perform an independent two-way sort and divide the observations into four groups based on σ 2 θ and σ2 ɛ. 1 The average ex-day abnormal returns of the 2*2 groups are calculated and reported in Panel B of Table 5. The results show that σ 2 ɛ and σ 2 θ interacts with each other in affecting ex-day abnormal returns. The influence of σ 2 ɛ on return is stronger when σ 2 θ is low, and the impact of σ2 θ is stronger when σ 2 ɛ is high. In particular, the ex-day abnormal return is highest when σ 2 θ is low and σ 2 ɛ is high, with an average of 1.04%. Overall, the results are consistent with our conjecture that when valuation uncertainty is low and the ex-day market is noisier, investors are more likely to rely on the cum-day price in valuation and 1 Untabulated results show that a dependent two-way sort produces qualitatively similar results. 21

22 thus incorporate more anchoring bias into the ex-day stock price. 5 Cross-sectional regressions 5.1 The baseline regression In this section, we perform cross-sectional regressions to examine the relation between the ex-day abnormal return and the split factor, the institutional ownership, the information uncertainty, and the noisiness of the ex-day signal. The following regression is performed: Ret = α + β 1 f + β 2 f 2 + β 3 D0.5 + β 4 D1 + β 5 IO + β 6 σ 2 θ + β 7 σ 2 ɛ + ε, (8) where Ret is the abnormal return on the ex-distribution day of all observations in the trimmed sample, and f is the split factor. f and f 2 are included to test Hypothesis 2. As illustrated in Figures 1 and 2, the coefficient on f is expected to be positive, while the coefficient on f 2 is expected to be negative. D0.5 is a dummy variable that equals to 1 when f is 0.5, and zero otherwise. D1 is similarly defined for a split factor of 1. D0.5 and D1 are included to test Hypothesis 3, and both are expected to be negatively associated with ex-day abnormal returns. IO represents the institutional ownership, which is an inverse measure for γ, or the propensity of the anchoring bias in the aggregate shareholder base. Hypothesis 4 predicts the coefficient on IO to be negative. Finally, σθ 2 and σ2 ɛ are included to test Hypothesis 5. According to Eq. (7), the coefficients on σθ 2 and σ2 ɛ should be negative and positive, respectively. To reduce the influence of extreme values, σθ 2 and σ2 ɛ are trimmed at the upper and bottom 2.5 percentiles. The results of the cross-sectional regressions are reported in Table 6. In columns (1)-(5), we show the results without controlling for the fixed effects. The two-sided t-statistics are calculated based on white-errors. In column (1), the ex-day abnormal 22

23 return is regressed on the split factor f only. The coefficient on f is significantly positive at the 1% level, confirming that the ex-day abnormal return increases with the split factor. The f variable alone explains 0.07% of the cross-sectional variation in the ex-day abnormal returns. In column (2), where f 2 is added into the regression, the coefficient on f remains significantly positive, and the coefficient on f 2 is significantly negative, which is consistent with Hypothesis 2. The adjusted R-square increases significantly to 1.78%, indicating that the second-order effect of f is strong. In column (3), we find that the coefficients on D0.5 and D1 are both significantly negative at the 1% level, supporting Hypothesis 3. All together, the four split factor related variables, including f, f 2, D0.5 and D1, explain 3.39% of the cross-sectional variation in the ex-day abnormal returns. The institutional ownership variable is included in column (4). The coefficient on IO is significantly negative, indicating that for stocks that are largely traded by sophisticated investors, the anchoring effect is weakened and the associated ex-day abnormal return is reduced. The evidence supports Hypothesis 4. In column (5), the coefficient on σ 2 ɛ is significantly positive, confirming that when the ex-day market is noisier, the ex-day return is higher. The coefficient on σθ 2, however, is indistinguishable from zero. Overall, model (8) explains 3.61% of the cross-sectional variation in the ex-day abnormal returns for 18,996 observations. In columns (6) to (8), we further control year-fixed effects or/and industry-fixed effects in the regressions. We follow Kenneth R. French to classify observations into 48 industries by SIC codes 2. Pseudo R-squares are reported. We find that the control of year and industry fixed effects does not affect the results qualitatively. Furthermore, we conduct sub-period analysis to examine the impact of the anchoring bias on ex-day abnormal returns in different periods. We re-estimate model (8) for three sub-periods and report the results in Table 7. The coefficients on f, 2 From http : //mba.tuck.dartmouth.edu/pages/faculty/ken.french/data Library/changes ind.html 23

24 f 2, D0.5, D1 and IO are all significantly different from zero at the 1% level with predicted signs in each sub-period, supporting Hypotheses 2, 3, and 4. The coefficient on σ 2 θ is significantly positive in the post-2000 sub-period, but is insignificant in the other two sub-periods. The coefficient on σ 2 ɛ is insignificant in most cases. Hypothesis 5 is partially supported. The explanatory power of model (8) is strongest in the most recent decade, implying that the anchoring effect does not fade out as the stock market becomes more developed. 5.2 Multivariate regression To further examine the robustness of the baseline results reported in Table 6, we add multiple control variables into Eq. (8). Size is the logarithm of market capitalization, calculated by multiplying the cum-day closing price from CRSP daily return database by the number of shares outstanding recorded in CRSP daily event database. In addition, we include the announcement day raw return (Ret Ann ) to control for the possibility that investors are simply under-reacting to the announcement event (Eades et al., 1984). The cum-day raw return (Ret Cum ) and the logarithm of cum-day trading volume (Vol Cum ) are controlled as well. The cum-day percentage bid-ask spread (BA Cum ) is calculated using the difference between the closing ask and the closing bid, scaled by the closing price. BA Cum is used to capture the liquidity of individual stocks. The inclusion of the cum-day bid-ask spread into the regression, however, would result in a loss of almost half of the sample observations. Thus, we present results for both including and excluding the bid-ask spread in model (8). Kumar (2009) argues that stocks with low price, high idiosyncratic volatility, and high idiosyncratic skewness assemble lotteries. He finds that such lottery-type stocks are attractive to retail investors, especially poor, young, and less-educated people, and are avoided by sophisticated investors. The unexperienced and unsophisticated 24

25 investors are most likely to subject to various psychological biases, including the anchoring bias. We thus expect the ex-day anomaly to be stronger among stocks that possess the lottery features. Following Kumar (2009), we calculate idiosyncratic volatility (IV ) and idiosyncratic skewness (Skew) based on daily stock returns over a six-month-period before the event month. Stocks with fewer than 60 return observations are eliminated. Idiosyncratic volatility is the variance of the residual obtained by fitting a four-factor model to the daily stock return time-series. The idiosyncratic skewness is the skewness of the residual obtained by fitting a two-factor model to the daily stock returns time series, where the two factors are the excess market returns and the squared excess market returns. 3 To control for the possibility that the ex-day abnormal return is purely driven by periodic optimistic sentiment in the stock market, we include the monthly sentiment index developed by Baker and Wurgler (2006). 4 Following Eades et al. (1984) we include a dummy variable (DMonday) that equals to one if the ex-day is Monday and zero otherwise to control for a possible weekday effect. Table 8 reports the results of regressions where the aforementioned variables are controlled. The coefficients on f and f 2 remain significantly positive and negative, as predicted. The coefficients on D0.5 and D1 are both significantly negative, confirming that the ex-day abnormal return is lower for 3-for-2 and 2-for-1 splits. The significantly negative coefficient on IO is consistent with Hypothesis 4. And the significantly positive coefficient on σ 2 ɛ supports the conjecture that the ex-day anomaly is weakened when the market is noisier on the ex-day. The coefficient on σθ 2, however, is insignificant. In general, the inclusion of the control variables does 3 We do not include stock price in the regression as the correlation coefficient between the logarithm of firm size and stock price is as high as We thus focus on idiosyncratic volatility and skewness only in capturing the lottery features of stocks. 4 From jwurgler/ 25

26 not alter our main findings qualitatively and the results in 8 render support to all the five hypotheses laid out in Section 3.4. We then examine the coefficients on the control variables. We find no evidence that the ex-day return is just a continuation of the announcement-day return as the coefficient on Ret Ann is insignificant in all cases. The coefficients on Ret Cum and Vol Cum are significantly negative and positive in certain cases. The coefficient on the cum-day bid-ask spread is only marginally significant in column (11) with a t-ratio of It turns indistinguishable from zero in all other regressions. In unreported tests, we follow Graham et al. (2003) to add in dummies for periods when the tick size is 1/8, 1/16 and decimal, but none of these dummies have significant coefficients in the regressions. We conjecture that the microstructure-related arguments may not be adequate in explaining the ex-day abnormal returns, and that our anchoring argument is not capturing the liquidity effect in addressing the ex-day anomaly. The coefficient on IV is significantly positive in most cases, indicating that the ex-day abnormal return is high for stocks with high idiosyncratic volatility. This is consistent with our argument that the ex-day anomaly is stronger among stocks that are more difficult to arbitrage and that are particularly attractive to unsophisticated retail investors. The coefficient on Skew, however, is significantly negative in most cases, indicating that stocks with positively skewed returns have lower ex-day abnormal returns. Sent, the sentiment index, has a significantly positive coefficient in column (8), indicating that the optimistic sentiment contributes to the ex-day abnormal returns. The coefficient on Sent turns insignificant when we control for the year and industry-fixed effects. Consistent with findings in previous studies, the coefficient on DMonday is negative, indicating that if the ex-day happens to be Monday, the ex-day abnormal return would be lower. 26

27 6 Conclusion In this paper, we develop an anchoring model to explain the ex-day returns for stock dividends, stock splits, and reverse stock splits. The announcement effect of stock splits has attracted much attention over the past years, but the ex-day anomaly for stock distributions, in particular for stock dividends, has been investigated less frequently. Our model focuses on the relation between the split factor and the ex-day returns, and is general enough to explain the ex-day effect for all the three types of stock distribution events. Our anchoring model argues that in estimating the ex-day stock valuation, investors tend to use the cum-day price at face value as the reference and fail to sufficiently adjust for the split factor. As a result, for stock dividends and splits, the cum-day price is an upward biased reference for the ex-day valuation, incorporating an upward bias in the ex-day valuation and inducing positive ex-day abnormal returns. For reverse stock splits, on the opposite, the cum-day price is a downward biased reference for the ex-day valuation, resulting in negative ex-day abnormal returns. The model also predicts a positive relation between the ex-day return and the split factor. The positive relation between ex-day return and the split factor is expected to decrease with the split factor as investors are most likely to overlook a distribution event when its effect is not obvious. Therefore, for distribution events with a very small split factor, the propensity of the anchoring bias is particularly strong, resulting in exceptionally high ex-day abnormal returns. This argument generated by our anchoring model helps to explain why stock dividends experience higher ex-day abnormal returns than stock splits. In addition, as investors are more familiar with split events with a split factor of 0.5 or 1, they are more likely to notice and understand the substance of such split events. As a result, investors propensity 27