Disagreement, Underreaction, and Stock Returns Ling Cen University of Toronto ling.cen@rotman.utoronto.ca K. C. John Wei HKUST johnwei@ust.hk Liyan Yang University of Toronto liyan.yang@rotman.utoronto.ca Keywords: Disagreement, short-sale constraints, underreaction, cross section of stock returns. This Draft: October 2013 * We appreciate the helpful comments and suggestions from Snehal Banerjee, James Choi, Ming Dong, Bing Han, Kewei Hou, Wei Xiong, and Jialin Yu. We also acknowledge financial support from the Research Grants Council of the Hong Kong Special Administrative Region, China (GRF 644212), the Social Sciences and Humanities Research Council of Canada (SSHRC), and jointly from the Institute of New Economic Thinking (INET) and the Centre for International Governance Innovation (CIGI). All remaining errors are our own.
Disagreement, Underreaction, and Stock Returns Abstract We explore the analyst earnings forecasts data to study the interactive effect between disagreement and underreact to earnings news on asset prices. We find that (1) changes in the mean of forecasted earnings as an underreaction measure positively predict future returns, that (2) changes in the standard deviation of forecasted earnings as a disagreement measure negatively predict future returns, and more importantly, that (3) changes in the standard deviation predict future returns significantly only when changes in the mean are negative. Our results are robust both in the standard crosssectional return setting and in the event-study setting around earnings announcements. Our evidence suggests that the analyst dispersion measure in Diether, Malloy, and Scherbina (2002) is mainly an underreaction measure instead of a disagreement measure. JEL Classification: G02, G12, G14 Keywords: Disagreement, short-sale constraints, underreaction, cross-section of stock returns.
1. Introduction Investors disagreement on public information affects asset prices. Miller (1977) suggests that, when investors hold diverse opinions, negative information held by pessimistic investors will not be fully reflected in stock prices in the presence of short-sale constraints and, therefore, stocks are likely to be overpriced. 1 Any attempt to test the pricing impact of disagreement would certainly require a reasonable measure for differences of opinions. One of the most popularly used measures is the dispersion of analyst forecast earnings proposed by Diether, Malloy, and Scherbina (2002) (DMS, hereafter). 2 This measure is computed as the standard deviation of analyst forecast earnings scaled by the absolute mean of analyst forecast earnings for each stock at the end of each month. Consistent with Miller s prediction, DMS show that analyst forecast dispersion indeed has a significant predictive power for future returns in the cross section. According to DMS (2002), the numerator in their measure i.e., the standard deviation of forecasted earnings captures the disagreement among analysts (and also investors), while the denominator i.e., the absolute mean of forecasted earnings is simply a normalization scalar to remove the common factor pertaining to the number of shares outstanding in the numerator so that this measure is comparable across stocks. Their argument that the scalar should not predict future returns relies on an implicit assumption that the mean of analyst forecasts is, on average, an unbiased estimator for future returns. However, this assumption contrasts with a vast literature on investors underreaction of public earnings news. For example, Bernard (1993) examines the role of underreaction of stock prices to announced earnings (i.e., the post earnings announcement drift (PEAD) effect first discovered by Ball and Brown (1968)), which is formalized theoretically by 1 Chen, Hong, and Stein (2002), Hong and Stein (2003), and Cen, Lu, and Yang (2013) extend Miller s (1977) model to study the breadth-return relation and stock market crashes, respectively. Harrison and Krep (1978), Harris and Raviv (1993), and Scheinkman and Xiong (2003) develop dynamic models that build on heterogeneous beliefs and short-sale constraints and derive the same conclusions. Hong and Stein (2007) provide an excellent review of the literature on disagreement. 2 This measure of disagreement has been used in more than 30 papers in top-tier finance and accounting journals. 1
Barberis, Shleifer, and Vishny (1998). Recent studies mainly focus on the economic rationales underlying the underreaction to earnings news, including investors overconfidence in the precision of their private signals (Daniel, Hirshleifer, and Subrahmanyam (1998)), investors failure to condition their estimates on past and current prices (Hong and Stein (1999)), investors limited attention to earnings news (Hirshleifer and Teoh (2003), DellaVigna and Pollet (2009), and Hirshleifer, Lim, and Teoh (2011)), and analysts anchoring their forecast earnings per share to the industry median (Cen, Hillary, and Wei (2013)). Therefore, there is a good reason for us to believe that an underreaction component would arise in the denominator of analyst forecast dispersion measure in DMS (2002) and that this component may confound the results when this measure is used to examine return predictability in the literature. Our doubt above is echoed by two surprising empirical observations that we document in Section 2. First, although the scalar (i.e., the absolute mean of forecast earnings per share) itself is not directly comparable across stocks, we find that most empirical patterns documented by DMS can be maintained when we sort stocks by the scalar of the DMS analyst forecast dispersion measure. This result is surprising since the cross-sectional variation in the disagreement component in the numerator is completely eliminated in this exercise. Second, we find that the economic and statistical significance of the hedging portfolio returns based on the analyst forecast dispersion measure are sensitive to the choice of scalars. The tensions in both theories and empirics mentioned above motivate us to ask the following research questions in this paper: What information contained in the denominator of the analyst forecast dispersion measure (i.e., the mean of analyst forecast earnings) could predict future stock returns? Does the disagreement component contained in the numerator (i.e., the standard deviation of analyst forecast earnings) predict stock returns independently? Do the numerator and the denominator of the analyst forecast dispersion measure interact with each other in predicting future stock returns? 2
We make an effort to answer these questions both theoretically and empirically in this paper. In essence, our story is a combination of underreaction and disagreement. First, investors underreaction to public earnings information implies a positive correlation between the updates of earnings estimates and future stock returns: Since investors react insufficiently to a good (bad) earnings news and the stock price goes up (down) too little, subsequent returns will continue to be high (low) on average till the fundamentals are fully reflected. Second, in the presence of short-sale constraints, the mechanism described by Miller (1977) and DMS (2002) would imply that the standard deviation of analyst forecast earnings, as a proxy for the difference in opinions, negatively predicts future stock returns: When the standard deviation of analyst forecast earnings is surprisingly high, investors beliefs are likely to be disperse, and so short-sale constraints cause pessimistic views not to be fully reflected in stock prices, raising the stock prices and predicting low future returns. Lastly, and most importantly, the disagreement component in the standard deviation and the underreaction component in the mean of analyst forecast earnings interact in predicting future returns. We argue that the short-sale constraints are more likely to be binding in the down market when investors underreact to bad news for the following three reasons: First, when investors underreact to bad news in the down market and gradually adjust their estimates downwards, they are more likely to sell or short sell stocks; second, as the stock price continues to drop, the short selling is likely to be restricted by the short-sale rules (e.g., the tick-test rule ) before 2007; third, even if the short selling is allowed, its costs can be much higher in the down market when the demand of short selling increases. Therefore, the predictive power of the standard deviation of analyst forecast earnings is particularly strong when the mean of analyst forecast earnings is decreasing. In Section 3, we formalize all the above predictions in a parsimonious model which incorporates investors 3
underreaction to earnings news into a disagreement model in the spirit of Miller (1977). We then make an effort to empirically test these predictions in Sections 4 and 5. The main difficulty in conducting such an empirical analysis to test our model predictions is that neither the standard deviation (numerator) nor the mean (the denominator) of analyst forecast earnings are comparable in the cross section. To overcome this difficulty, we decompose the analyst forecast dispersion measure in a way similar to the decomposition of the book-to-market ratio employed by Daniel and Titman (2006) and Fama and French (2008). Specifically, we decompose the natural logarithm of the analyst forecast dispersion measure at period t into three components: the natural logarithm of the lagged analyst forecast dispersion measure at period t-k, the logdifference in the standard deviations of forecast earnings between period t-k and period t ( the change in standard deviation hereafter), and the log-difference in the means of forecast earnings between period t-k and period t ( the change in mean hereafter). We then increase k to a value such that the first component (i.e., the natural logarithm of the lagged analyst forecast dispersion measure at period t-k) becomes an insignificant predictor for future stock returns. This design has three advantages. First, when k is large enough, the predictive power of the DMS analyst forecast dispersion measure for future stock returns can only be attributed to the change in standard deviations and the change in means of analyst forecast earnings. Second, because these two predictors are constructed in the form of growth rates, they are comparable across stocks. Finally, it is much easier to interpret the economic meanings of the two growth rates obtained from decomposition than it is to interpret the economic meanings of the numerator and the denominator of the analyst forecast dispersion measure. More specifically, the change in standard deviation represents the change in investors disagreement in the numerator of the analyst dispersion measure, while the change in mean captures investors (under)reaction to earnings news in the denominator. 4
Our empirical results, obtained using the U.S. data from 1983 to 2009, are consistent with our theoretical predictions. First, we show that the change in standard deviations predicts future stock returns negatively, and its predictive power for future stock returns is only statistically significant among small-sized firms (which have the most short-sale constraints). On the contrary, the change in means predicts future stock returns positively, and its predictive power is statistically significant among both small-sized and medium-sized firms. Second, our results from portfolio sorts and the Fama and MacBeth (1973) regressions suggest that the predictive power of the change in standard deviations for future stock returns is only statistically significant when the mean of forecast earnings is decreasing (i.e., when the short-sale constraints are more likely to be binding). In contrast, the return predictability of the change in means is not affected by whether the change in standard deviations is positive or negative. Our results are robust irrespective of whether portfolios in all subgroups are equal-weighted or value-weighted, and whether or not we control for common risk factors or firm characteristics. As a result of the above decomposition analysis, we find that a trading strategy based on the change in means generates a much higher average hedging portfolio return than those based on the change in standard deviations. For example, for equal-weighted portfolios in the group of small-sized firms, the average monthly hedging portfolio return based on the change in means is 0.96%, which is more than twice as large as the monthly hedging portfolio return (i.e., 0.36%), based on the change in standard deviations. In addition to the traditional setting for the stock returns in the cross section, our results are also robust under an event study where the source of information shocks that trigger disagreement and underrection among investors can be identified. In our event study, we focus on 138,874 quarterly earnings announcements and examine the impact of the change in the mean and the standard deviation of forecasted earnings around the earnings announcements on the stock returns 5
in the post earnings announcement period. Our results emerging from this event study are consistent with those based on the cross-sectional stock returns. There are two main messages in this paper. First, after qualitatively and quantitatively decomposing the return predictability of the DMS analyst forecast dispersion measure into two components, we find that the predictive power of the DMS measure for future stock returns is mainly contributed by the underreaction component contained in the denominator, instead of the disagreement component contained in the numerator. This message contributes to the literature by providing a clear guidance for the use and interpretation of the DMS analyst forecast dispersion measure in future studies. Second and more importantly, using a unified theoretical framework to study the interactive pricing effects on investor disagreement and underreaction to earnings news, we show that disagreement strongly predicts future returns only in a down market where short-sale constraints are more likely to be binding. This message contributes to the literature by reminding us that the effect of disagreement on future stock returns depends on the level of short-sale constraints and, henceforth, is conditional on the direction of the market movement. We organize the remainder of our paper as follows: Section 2 discusses our empirical motivation and decomposition approach. Section 3 presents a theoretical framework formalizing the pricing implications of investor disagreement and the underreaction to earnings information. We then propose testable implications generated by our theoretical framework. Section 4 describes the sample characteristics of our data. We present our empirical results in Section 5 and conclude the paper in Section 6. 2. Empirical Motivation 2.1. A Surprising Observation in DMS (2002) We start our analysis by replicating DMS s (2002) results for the extended sample period from 1983 to 2009. The results are reported in Panel A of Table 1. The patterns of monthly hedging 6
portfolio returns based on the analyst forecast dispersion measure (i.e., the trading strategy of purchasing stocks in the lowest dispersion group and short selling stocks in the highest dispersion group simultaneously) are almost identical to those reported by DMS (2002). First, the analyst forecast dispersion measure could predict cross-sectional stock returns in the full sample. If investors long stocks in the lowest dispersion quintile in each size group and short stocks in the highest dispersion quintile in each size group, they can generate an average monthly hedging portfolio return of 57 basis points, which is statistically significant at the 1% level. Second, the hedging portfolio returns in size quintiles are monotonically decreasing as firm size increases. The hedging portfolio returns based on the analyst forecast dispersion measure are not statistically significant for the largest and second largest size quintiles. [Insert Table 1 Here] To give us a rough sense of the return predictability of the numerator and the denominator of the analyst forecast dispersion measure, we replace the sorting variable by the standard deviation of analyst forecast earnings (the numerator) or the absolute mean of forecast earnings (the denominator). 3 Our results from the numerator and the denominator of the dispersion measure are reported in Panels A and B of Table A1 in Appendix C, respectively. We find that, although the numerator still has strong predictive power for future returns in the smallest size quintile, the denominator by itself can almost replicate both the pattern and the magnitude of hedging portfolio returns based on the analyst forecast dispersion measure. 4 3 Because the numerator and the denominator of the analyst forecast dispersion measure are not comparable across firms, it is important to point out that the purpose of this test is not to examine whether these two components can predict returns in the cross section independently. Instead, the focus of this test is to see whether one component still has the predictive power for future returns when the (information in) other component is distorted. 4 Cen, Hillary, and Wei (2003) find that a trading strategy similar to Panel B of Table A1 by replacing the absolute mean of forecast earnings per share (FEPS) with a firm s FEPS relative to its industry median can generate a mean riskadjusted return (alpha) of 0.71% per month, or 8.52% per year, across size groups based on the Carhart four-factor model. 7
In DMS (2002), the disagreement theory is mainly associated with the information in the numerator, the standard deviation of analyst forecasts. The absolute level of mean forecasts in the analyst forecast dispersion measure is just a scalar to ensure that the common component pertaining to the number of shares outstanding in both the numerator and the denominator is cancelled out so that the analyst forecast dispersion measure can be comparable across different stocks. If this is true, any scalars that are of a per share basis should be able to serve the same purpose without affecting the return predictability of the analyst forecast dispersion measure, i.e., the replacements of the scalar in the original analyst forecast dispersion measure should not significantly weaken the return predictability of the revised measure. In this spirit, we construct two alternative analyst forecast dispersion measures: DISP-A is constructed as the standard deviation of forecasted earnings scaled by the book value of equity per share and DISP-B is constructed as the standard deviation of forecasted earnings scaled by the stock price per share. The information contained in both scalars is ex ante, i.e., the book value of equity per share is corresponding to the information for the most recent fiscal year end before the portfolio construction and the stock price per share is the close price of the stock at the month end before the portfolio construction. The results of portfolio sorts based on these two alternative analyst forecast dispersion measures are reported in Panels B and C of Table 1. Our results suggest that the results in DMS (2002) are sensitive to the choice of scalars. For example, when we replace the original scalar by the book value of equity per share, the magnitude of the monthly hedging portfolio returns based on the this analyst forecast dispersion measure (DISP- A) drops to 0.40% in the full sample, which is 30% lower than the 0.57% monthly hedging portfolio returns based on the measure in DMS (2002). When we replace the original scalar by the stock price per share, the monthly hedging portfolio returns based on this analyst forecast dispersion (DISP-B) is merely 0.06% in the full sample, which is statistically indifferent from zero. It is worth mentioning 8
that, although the hedging portfolio returns in the smallest size decile are also sensitive to scalar replacements, they remain statistically significant at the 5% level with both alternative measures. While it is questionable whether the hedging portfolio returns in the smallest size decile under alternative analyst forecast dispersion measures can survive the transactional costs, this result at least hints that the disagreement component contained in the numerator seems to have a strong and persistent predictive power of future returns in the smallest decile group. Both results above are surprising. They hint that the results of DMS (2002) seem to largely depend on the information contained in the scalar of the analyst forecast dispersion measure and, therefore, has led us to pose those research questions laid out in our introduction. 2.2. Decomposition of the Analyst Forecast Dispersion Measure We decompose the analyst forecast dispersion measure in a way similar to the decomposition of the book-to-market ratio by Daniel and Titman (2006) and Fama and French (2008). Our decomposition has two purposes in mind. First, the decomposition allows us to single out components that can be compared across stocks. Second, and more importantly, the decomposition allows us to isolate and identify meaningful economic components that can predict future returns. The decomposition is formally shown as follows: ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( )] [ ( ) ( )] ( ) ( ) ( ) ( ) 9
In this equation, Disp t, Std t, and Mean t represent the analyst forecast dispersion measure, the standard deviation, and absolute mean of analyst forecast earnings at month t, respectively. adj t is the cumulative adjustment factor for stock splits and dividend distributions in month t, which makes the standard deviation and the mean of analyst forecast earnings comparable over time. Chg_Std(t-k,t) and Chg_Mean(t-k,t) are the change in standard deviation and the change in absolute mean (both in logarithm) of analyst forecast earnings from month t-k to month t: Chg_Std(t-k,t) log(adj.std t ) log(adj.std t-k ) and Chg_Mean(t-k,t) log(adj. Mean t ) log(adj. Mean t-k ). 5 One can easily see that the analyst forecast dispersion measure at month t can be decomposed into three components: the lagged analyst forecast dispersion measure at month t-k (i.e., log(disp t-k )), the change in standard deviations of analyst forecast earnings from month t-k to month t (i.e., Chg_Std(t-k,t)), and the change in the consensus forecasts from month t-k to month t (i.e., Chg_Mean(t-k,t)). As alluded to above, such decomposition overcomes two weaknesses of directly comparing means and standard deviations of analyst forecast earnings. First, the standard deviation and the mean of analyst forecast earnings, although economically meaningful, are not comparable across stocks. Second, these two components may be highly correlated and, hence, may capture common information such as the stock price level. In equation (1), Chg_Std(t-k,t) and Chg_Mean(t-k,t) are essentially constructed as growth rates of the standard deviation and the mean of analyst forecast earnings from month t-k and month t, respectively. Therefore, we can compare them across stocks meaningfully. Further, while the means and the standard deviations of forecast earnings may be highly correlated, as will be discussed in Section 4, this is not the case for Chg_Std(t-k,t) and Chg_Mean(t-k,t). 5 Under this specification, Chg_Mean(t-k,t) may be meaningless if the mean of forecast earnings at time t or t-k is negative. However, after we screen the data according to the criteria stated in Section 4.1, only 5% of observations in our sample have negative forecast earnings. Further, we do a robustness check based on a subsample that excludes all firms with negative earnings. Our results remain qualitatively and quantitatively unchanged. 10
Our main interest in this paper is to understand how Chg_Std(t-k,t) and Chg_Mean(t-k,t) independently or jointly determine future stock returns. To isolate the predictive power for stock returns from the remaining component in equation (1), log(disp t-k ), in our empirical analysis, we gradually increase k until log(disp t-k ) loses its return predictability. The results are shown in Appendix C of Table A2 and suggest that the predictive power of log(disp t-k ) for future stock returns is indeed declining as k increases. When k is equal to or larger than 5, log(disp t-k ) loses its return predictability in all size quintiles. 6 Therefore, we decompose log(disp t ) into log(disp t-5 ), Chg_Std(t-5,t), and Chg_Mean(t-5,t) in our paper. Obviously, in this decomposition, the predictive power of log(disp t ) for future returns must come mainly from Chg_Std(t-5,t) and Chg_Mean(t-5,t). Under this decomposition, there is no need to include log(disp t-5 ) in our analysis, and we can solely focus on the dynamics between Chg_Std(t-5,t) and Chg_Mean(t-5,t) in predicting future returns. 3. Theoretical Framework and Hypothesis Development 3.1. Theoretical Framework In this section, we develop a parsimonious model of return predictability of analyst forecasts to formalize our idea. The economy lasts for one period and has two tradable assets: one bond with a risk-free rate of zero, 7 and one stock that pays a dividend of at the end of the period. 8 We assume that: ( ) where parameter is a constant representing the unconditional mean of the firm s cash flow, the random variable ( ) (with ) represents the component that analysts can forecast, 6 DMS (2002) also find that the return predictability of Disp t-k decreases when k increases and loses its predictability six months after portfolio formation. 7 The assumption of a zero risk-free rate is not essential for our results. 8 Throughout the paper, a tilde (~) always signifies a random variable. 11
and the random variable ( ) is the residual uncertainty of the firm s cash flow. 9 The stock has a total supply of one unit and its price is denoted by. We employ analyst forecasts to generate heterogeneous beliefs among investors whose trading, in turn, affects prices. In this way, analyst forecasts are reflected in prices and generate their return predictability in our economy. Specifically, we assume that there are two (types of) analysts an optimistic analyst and a pessimist analyst whose forecast reports are signals about. The optimistic analyst reports an upward-biased signal of: ( ) while the pessimist analyst reports a downward-biased signal of: ( ) where captures disagreement among analysts. On average, the signals of analysts give the true information. 10 We assume that is uniformly distributed over [0,1]; that is, [ ]. In addition, we assume that all the underlying random variables i.e., (the mean of analyst forecasts), (the analyst disagreement), and (the residual cash flow uncertainty), are mutually independent. We deliberately assume that and are independent of each other to make sure that the interdependence between their return predictability is endogenously generated by the market equilibrium. Our later empirical analysis also provides evidence consistent with this assumption. There is a continuum [0,1] of traders who have constant-absolute-risk-aversion (CARA) utility with a risk tolerance normalized to be 1. They are further categorized into three classes according to their beliefs and investment constraints. The first class is buyers, of a mass of. 9 The assumption that and have a non-zero mean is without loss of generality, since their means can be absorbed by the constant. Also, the assumption that has a unit variance does not affect our results. 10 The literature suggests that analyst forecasts may be biased in aggregate (e.g., Hong and Kacperczyk, 2010). Our results would hold under this alternative assumption, as long as analyst forecasts contain information, which has received much supportive evidence from the empirical research (e.g., Bradley et al., 2013). 12
They face short-sale constraints and each of them is a dogmatic believer of one or the other analyst. Thus, buyers as a group inherit the disagreement in analyst forecasts. Their trading will generate the return predictability of the disagreement. We can think of buyers as mutual funds. If buyer i believes in analyst, the CARA-normal structure implies that his/her demand is (see Appendix A for the proof): ( ) We assume that half of the buyers believe in, while the other half believe in. Thus, the aggregate demand of buyers is: ( ) ( ) The second class of traders is underreactors, of a mass of. They completely ignore analyst forecasts, underreact to information, and are not subject to short-sale constraints. Thus, their total demand is: ( ) ( ) ( ) can be understood as the effective mass of underreactors. The behavior of underreactors could be motivated by their limited ability to process information in the market (e.g., Hirshleifer and Teoh, 2003; DellaVigna and Pollet, 2009; Hirshleifer, Lim, and Teoh, 2011). We can think of underreactors as individual investors. The third class of traders is arbitrageurs, of a mass of. These traders correctly aggregate all analyst signals and do not face short-sale constraints. We can think of them as hedge funds. The total demand of arbitrageurs is: ( ) ( ) ( ) The first two classes of traders are essential for our results. We incorporate arbitrageurs simply to demonstrate the robustness of our results. 13
In equilibrium, the market clears. That is, ( ) ( ) ( ) ( ) which states that the total stock demand from the three classes of traders is equal to the total unit supply. In Appendix A, we use the demand functions and the market clearing condition to compute the equilibrium price as follows: ( ) ( ) ( ) ( ) ( ) { ( ) ( ) The price is equal to the expected dividend adjusted by a term representing the equity premium. The three different regions of ( ) reflect the tightness of short-sale constraints of buyers. By equation (10), the price reacts positively to the information, but not to a full extent in the sense that. This is because underreactors do not respond to information fully, which generates the positive return predictability of (i.e., the mean of analyst forecasts). 11 The price also positively reacts to disagreement in the middle region where short-sale constraints are binding for some buyers, but not for others. Because the dividend is independent of, this sensitivity of prices to generates a negative return predictability of (i.e., the standard deviation of analyst forecast earnings). This mechanism is effectively the Miller (1977) hypothesis. The realized stock return per share is defined as: ( ) Then, the regression coefficients of and used to forecast return are, respectively, 11 Note that when there are no underreactors i.e., when (and hence ), regression coefficient of used to forecast returns will degenerate to 0 in equation (12). 14. In this case, the
( ) ( ) ( ) ( ) ( ) We label the above two s as the earnings news beta and the disagreement beta, respectively. In Appendix A, we show that and. The following proposition summarizes our discussion above. Proposition 1. The standard deviation of analyst forecast earnings negatively predicts future returns, while the mean of analyst forecast earnings positively predicts future returns. That is, and. We next examine how the mean and the standard deviation of analyst forecast earnings interact to predict future stock returns. We define conditional disagreement betas as: ( ) ( ) ( ) ( ) ( ) where and are the regression coefficients of used to forecast returns when information is surprisingly high (i.e., above its unconditional mean) and low (i.e., below its unconditional mean), respectively. 12 Note that the Miller hypothesis works through the combination of disagreement and short-sale constraints. When the market is on the rise, that is, when is higher than the conditional mean, buyers are likely to purchase the stock, and hence, the short-sale constraints are likely to be irrelevant. As a result, the analyst disagreement is unable to predict future returns (i.e., ). When the market is on the decline, buyers are likely to short the stock and the short-sale constraints are likely to be binding. Hence, the analyst disagreement will predict future returns in a down market (i.e., ). Recall that the unconditional mean of is 0.5. We can similarly define conditional earnings news betas as: 12 In our parsimonious static model, we compare the realized value of a random variable with its unconditional mean to capture the concept of surprises. Our later empirical analysis is conducted in a dynamic context and news is measured as changes in variables over time. 15
( ) ( ) ( ) ( ) ( ) which are the regression coefficients of used to forecast returns when disagreement is surprisingly high and low (i.e., the realized is above and below its unconditional mean, 0.5), respectively. However, the return predictability of is mainly driven by the underreaction of the second group of investors (i.e., by a positive ), and hence, the value of does not affect the power of in predicting future returns, and as a result, and. In Appendix A, we formally prove the following proposition. Proposition 2. When the mean of analyst forecast earnings is positive, the standard deviation of analyst forecast earnings does not predict future returns. But when the mean of analyst forecast earnings is negative, the standard deviation of analyst forecast earnings negatively predicts future returns. That is, and. Independent of the standard deviation of analyst forecast earnings, the mean of analyst forecast earnings always positively predicts future returns. That is, and. 3.2. Hypothesis Development Based on the decomposition approach discussed in Section 2.2, the change in standard deviation Chg_Std(t-5,t) in equation (1) relates to investor disagreement (i.e., variable in our theoretical framework in Section 3.1), while the change in mean Chg_Mean(t-5,t) captures investors underreaction to earnings news (i.e., variable in our model). Therefore, we can translate Propositions 1 and 2 into the following two hypotheses with clear empirical predictions: Hypothesis 1 (H1): Chg_Std(t-5,t) negatively predicts future stock returns, while Chg_Mean(t-5,t) positively predicts future stock returns. Hypothesis 2 (H2): The predictive power of Chg_Mean(t-5,t) for future returns does not depend on the sign of Chg_Std(t-5,t). That is, Chg_Mean(t-5,t) always predicts future returns irrespective of 16
whether the standard deviation of analyst forecast earnings is increasing or decreasing. However, the predictive power of Chg_Std(t-5,t) for future returns depends on the sign of Chg_Mean(t-5,t). That is, Chg_Std(t-5,t) significantly predicts future stock returns only when Chg_Mean(t-5,t) is negative. 4. Sample Characteristics 4.1. Sample Selection Our basic sample consists of all NYSE-, Amex-, and Nasdaq-listed common stocks in the intersection of: (a) the Center for Research in Security Prices (CRSP) stock file, (b) the merged Compustat annual industrial file, and (c) the unadjusted summary historical file of the Institutional Brokers Estimate System (I/B/E/S) for the period from January 1983 to December 2009. 13 To be included in the sample for a given month (i.e., month t), a stock must satisfy the following five criteria: First, the mean and standard deviation of analyst forecasts of the one-year-ahead (FY1) earnings per share of the stock in this and the previous five months i.e., from month t-5 to month t must be available from the I/B/E/S unadjusted summary historical file. Second, the prices, returns, and total market capitalizations of the stock for the period from month t-6 to month t+1 must be available from CRSP. Third, there must be sufficient data from CRSP and Compustat to compute the Fama and French (1992, 1993) book-to-market ratio of the stock as of December of the previous year. Fourth, the stock must be priced above five dollars at the end of the current month (i.e., month t), and the book value of stockholders equity for the stock must be positive in the Compustat records. Finally, as in the work of Fama and French (1992, 1993), the stock must not be a certificate, an American depositary receipt (ADR), shares of beneficial interest (SBIs), a unit trust, a closed-end fund, a real estate investment trust (REIT), or a financial firm. This screening 13 Although I/B/E/S provides data starting from 1976, we restrict our sample period to January 1983 to December 2009 for two reasons: first, before January 1983, the coverage of stocks by I/B/E/S was limited, which would reduce the power of our tests; second, the data sample of DMS (2002) also starts from 1983, so we adopt the same starting years to make our analysis comparable with their analysis. 17
process yields 852,387 stock-month observations from June 1983 to December 2009 or an average of 2,630 stocks per month. 14 4.2. Summary Statistics Table 2 provides summary statistics describing our sample. All of the independent variables, as mentioned above, are either lagged by one month or computed based on public information that were already available to investors at the end of month t and can be used to execute our trading strategies. Panel A of Table 2 reports the time-series averages of the cross-sectional means, medians, standard deviations, and other summary statistics. The mean and median of firm size (Size) are $2.05 billion and $0.51 billion, respectively, which are much larger than the corresponding values for all CRSP stocks (untabulated). This is not surprising because in arriving at our final sample we have eliminated all small stocks in the CRSP sample that are priced below five dollars. Because the return anomaly based on the analyst forecast dispersion measure is stronger for small stocks, according to DMS (2002), our sample selection criteria are actually biased against finding significant results. [Insert Table 2 Here] The average change in mean forecast earnings from month t-5 to month t (i.e., Chg_Mean(- 5,0)) is 0.018. While the distribution of Chg_Mean(-5,0) is negatively skewed (skewness= 0.475), we find that Chg_Mean(-5,0) is positive in 70% of all observations, which suggests that forecast earnings are revised upwards in a five-month period for most stocks. The average change in standard deviation of analyst forecast earnings from month t-5 to month t (i.e., Chg_Std(-5,0)) is also positive (0.036), but it is more symmetrically distributed than Chg_Mean(-5,0). 14 Following previous studies (e.g., Jegadeesh and Titman (1993)), we exclude stocks priced below five dollars because such stocks not only have small analyst followings, they also incur large transaction costs due to their poor market liquidity (thin trading and large bid-ask spreads), which could distort the feasibility of any trading strategy. We also exclude stocks with a negative book value of stockholders equity simply to ensure that financially distressed firms do not drive our results. Including these observations leads to (untabulated) results that are economically and statistically more significant. 18
Panel B of Table 2 reports the univariate correlations among these variables. Not surprisingly, because Chg_Std(-5,0) affects the numerator of the analyst forecast dispersion measure (Disp), it is positively correlated with Disp. Similarly, because Chg_Mean(-5,0) affects the denominator of Disp, the correlation between them is negative. However, the correlation between Chg_Std(-5,0) and Chg_Mean(-5,0) is very low, suggesting no clear relation between the dynamics of the mean and the standard deviation of analyst forecast earnings. Further, their correlations with other independent variables, such as Size (MV), past returns (i.e., Ret(-1,0) and Ret(-7,-1)), earnings-to-price ratios (E/P), book-to-market ratios (BM), and accounting accruals (Accrual), are also quite low. This suggests that multicollinearity is not a major concern when we include Chg_Std(-5,0) and Chg_Mean(-5,0) with other independent variables in the predictive regressions of future stock returns. 5. Empirical Results 5.1. Portfolio Sorts in the Full Sample At the beginning of each month, we sort stocks into three equal groups based on market capitalization (MV) at the end of the previous month. Stocks in each size group are then sorted into three subgroups based on the change in the mean forecasts, Chg_Mean(-5,0). Stocks in each subgroup are further sorted into three groups based on the change in the standard deviation of forecast earnings, Chg_Std(-5,0). Panels A and B of Table 3 report the time-series averages of onemonth-ahead stock returns of equal-weighted and value-weighted portfolios based on 3 3 3 (=27) subgroups, respectively. [Insert Table 3 Here] The results show several interesting patterns. First, consistent with H1, we find that both Chg_Mean(-5,0) and Chg_Std(-5,0) can predict future stock returns. The return predictability of Chg_Mean(-5,0) and Chg_Std(-5,0) diminishes as firm size increases. This finding is consistent with those of DMS (2002) and confirms the notion that both investors short-sale constraints and 19
underreaction to new information are likely to be more severe among smaller stocks. Second, our results suggest that the predictive power of Chg_Mean(-5,0) is much stronger than that of Chg_Std(- 5,0). To illustrate this point, let us take for example the equal-weighted portfolios of small-sized firms where both Chg_Mean(-5,0) and Chg_Std(-5,0) can predict future stock returns. The hedging portfolios based on Chg_Mean(-5,0) generate a monthly return of 0.96% on average, 15 which is more than 2.6 times larger than the return (i.e., 0.36% on average) generated by the hedging portfolios based on Chg_Std(-5,0). 16 Such differences in return predictability are even more pronounced for the value-weighted portfolios and for the medium-sized firms. The most striking observation in Table 3 is that, while the predictive power of Chg_Mean(- 5,0) does not depend on the sign of Chg_Std(-5,0), the predictive power of Chg_Std(-5,0) is only statistically significant at the 5% level for the subgroups of firms with the smallest Chg_Mean(-5,0). For example, the hedging portfolio based on Chg_Std(-5,0) for the equal-weighted portfolios of small-sized firms with the smallest Chg_Mean(-5,0) generates an average monthly return of 0.56%, which is statistically significant at the 1% level. In contrast, the hedging portfolio returns based on Chg_Std(-5,0) for the equal-weighted portfolios of small-sized firms with medium and the largest Chg_Mean(-5,0) are both much lower and neither one is statistically significant at the 10% level. Again, we find that this pattern is more pronounced for the value-weighted portfolios and for the medium-sized firms. To further investigate the characteristics of stocks with the smallest Chg_Mean(-5,0), we report the average values of Chg_Mean(-5,0) and Chg_Std(-5,0) for each of the 27 (3 3 3) subgroups in Panels C and D of Table 3, respectively. We find that the average values of Chg_Mean(-5,0) for the subgroups of stocks with the smallest Chg_Mean(-5,0) are all negative. This provides a useful hint for 15 The corresponding trading strategy is to long the stocks with the largest Chg_Mean(-5,0) and to short stocks with the smallest Chg_Mean(-5,0) simultaneously. This strategy is denoted by CM(H-L). 16 The corresponding trading strategy is to long stocks with the smallest Chg_Std(-5,0) and to long stocks with the largest Chg_Std(-5,0) simultaneously. This strategy is denoted by CS(L-H). 20
a further test of H2 to see if Chg_Std(-5,0) only exhibits return predictability when the mean forecasts are decreasing, that is, when most investors are considering selling or short-selling stocks. To ensure that our results are not driven by common risk factors, we consider Alpha instead of raw portfolio returns in Table 4. Alpha is the intercept from a time-series regression of returns based on the Fama and French (1993) three-factor model, plus a Carhart (1997) momentum factor described as follows: ( ) ( ) where R p is the monthly return on portfolio p, MKT is the monthly return on the market portfolio, and R f is the monthly risk-free rate (proxied by the one-month Treasury bill rate). MKT R f, SMB, and HML are returns on the market, size, and book-to-market factors, respectively, as constructed by Fama and French (1993). UMD is Carhart s (1997) momentum factor. Factor returns and the risk-free rates are obtained from Kenneth French s website. 17 We estimate the intercept (Alpha) for each of the 27 (3 3 3) equal-weighted and value-weighted portfolios and the results are reported in Panels A and B of Table 4, respectively. [Insert Table 4 Here] The results in Table 4 confirm our discussions based on raw portfolio returns. We find that the portfolios with either significantly positive or negative alphas mostly come from portfolios of small-sized firms. The result is consistent with the argument that small-sized firms are more vulnerable than large-sized firms to investor behavioral bias that leads to asset mispricing. Once again, we find that the results from the trading strategy based on Chg_Mean(-5,0) are stronger than those from the trading strategy based on Chg_Std(-5,0). In addition, we observe that the hedging portfolios from the trading strategy based on Chg_Std(-5,0) for small- and medium-sized firms 17 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/. 21
generate statistically significant alphas (i.e., statistically significant at the 5% level) only for the subgroups with the smallest Chg_Mean(-5,0). However, the profitability of a trading strategy based on Chg_Mean(-5,0) for firms in small- and medium-sized groups is relatively more pervasive across all Chg_Std(-5,0) subgroups. 5.2. Sample Partitioned Based on Chg_Mean(-5,0) To test H2, we could partition each cross section into two groups conditional on whether Chg_Mean(-5,0) is positive or negative, and repeat our portfolio sorts within the group of stocks with positive or the negative Chg_Mean(-5,0). This test strategy is simple enough. However, it triggers potential issues that may affect portfolio properties when we perform the sorting. Specifically, because the economic environment varies constantly, the number of firms with negative Chg_Mean(- 5,0) and the number of firms with positive Chg_Mean(-5,0) will be different in each month. In the extreme case of an economic boom (recession), most firms experience a positive (negative) Chg_Mean(-5,0) with very few experiencing a negative (positive) Chg_Mean(-5,0), so that the portfolios obtained from our 3 3 3 sorts would not be properly diversified. As a compromise, we compute the mean of Chg_Mean(-5,0) for each month and partition the time series into two groups depending on whether the monthly average Chg_Mean(-5,0) is positive or negative. The argument is essentially similar: When the monthly average Chg_Mean(-5,0) is positive (negative), most investors are more likely to buy (sell or short-sell), and the short-sale constraints are less (more) binding. In this partition, we find that a positive average Chg_Mean(-5,0) is seen in 183 out of 324 months in total in our sample, while a negative average Chg_Mean(-5,0) is seen in the remaining 141 months. We present the raw returns of portfolio sorts for the 183 months with positive average Chg_Mean(-5,0) vs. the 141 months with negative average Chg_Mean(-5,0) in Table 5. Not surprisingly, higher raw returns in general are seen in the months with positive average Chg_Mean(- 22
5,0) than in the months with negative average Chg_Mean(-5,0). What is striking, however, is that the return spreads from the trading strategy based on Chg_Std(-5,0) diminish in the months with positive average Chg_Mean(-5,0). Even with the small-sized firms, the average hedging portfolio return is only 0.22%, and none of the other hedging portfolio returns in the months with positive average Chg_Mean(-5,0) are statistically significant at the 5% level. However, in the months with negative average Chg_Mean(-5,0), the hedging portfolio returns from the trading strategy based on Chg_Std(- 5,0) are highly significant. The hedging portfolios based on Chg_Std(-5,0) for small-sized firms can generate an average monthly return of 0.54%, which is statistically significant at the 5% level. The difference between hedging portfolio returns based on Chg_Std(-5,0) for small-sized firms in the months with positive average Chg_Mean(-5,0) and those based on Chg_Std(-5,0) for smalls-sized firms in the months with negative average Chg_Mean(-5,0) (i.e., 0.22% vs. 0.54%) is also statistically significant at the 5% level. [Insert Table 5 Here] While the trading strategy based on Chg_Std(-5,0) exhibits a statistically different predictive power for future stock returns in months with positive Chg_Mean(-5,0) than in months with negative Chg_Mean(-5,0), this is not the case for the trading strategy based on Chg_Mean(-5,0). Specifically, we find that Chg_Mean(-5,0) can predict stock returns equally well in all months. For small-sized firms, the average hedging portfolio returns based on Chg_Mean(-5,0) are 0.85% and 1.10% for the months with positive and negative average Chg_Mean(-5,0), respectively, and their difference is not statistically significant. For the medium-sized firms, the hedging portfolio returns based on Chg_Mean(-5,0) are slightly higher in the months with positive average Chg_Mean(-5,0), than in the months with negative average Chg_Mean(-5,0). Overall, Chg_Mean(-5,0) does not predict future stock returns in a statistically different manner between the positive and negative average Chg_Mean(-5,0) months. 23
In Table 6, we compute the alphas of portfolio returns in the 183 months with positive Chg_Mean(-5,0) and the 141 months with negative Chg_Mean(-5,0) examined in Table 5. Our results again confirm the prediction in H2. We find that (1) the hedging portfolio returns based on Chg_Std(-5,0) can only generate statistically significant Alphas among small-sized firms when the monthly means of Chg_Mean(-5,0) are negative; and (2) the hedging portfolio returns based on Chg_Mean(-5,0) generate statistically significant Alphas among small- and medium-sized firms no matter whether the monthly means of Chg_Std(-5,0) are positive or negative. [Insert Table 6 Here] 5.3. Fama and MacBeth (1973) Regressions The above portfolio sorts provide evidence supporting H1 and H2. However, this technique has four potential limitations. First, considerable variations in each sorting variable might still exist across firms within each of the 27 (3 3 3) subgroups. Second, other than the sorting variables, risk factors, and firm characteristics that we have controlled for in the computation of portfolio alphas, it is difficult for us to consider other risks and firm characteristics at the same time (i.e., we usually cannot go beyond three-way sorts). Third, with portfolio sorts, we are not able to precisely identify the quantitative impacts of Chg_Mean(-5,0) and Chg_Std(-5,0) on the future stock returns. Most importantly, as we mentioned in the last subsection, it is difficult to maintain diversified portfolios when we partition firms into the cross section according to the sign of Chg_Mean(-5,0). These limitations, however, can be easily addressed in the regression framework. Therefore, we perform the Fama and MacBeth (1973) regression tests at the individual stock level to further test H1 and H2. In Fama-MacBeth regressions, the dependent variable is measured in the unit of raw returns because all controls of firm characteristics can be addressed by including relevant explanatory 24