ANALYST EXPECTATIONS AND STOCK RETURNS: A TALE OF TWO TAILS Pedro Bordalo, Nicola Gennaioli, Rafael La Porta, and Andrei Shleifer First Draft,

Transcription

1 ANALYST EXPECTATIONS AND STOCK RETURNS: A TALE OF TWO TAILS Pedro Bordalo, Nicola Gennaioli, Rafael La Porta, and Andrei Shleifer First Draft, November 2016 Preliminary and Incomplete 1

2 I. Introduction La Porta (1996) shows that expectations of stock market analysts about long term earnings growth of the companies they cover have strong predictive power for these companies future stock returns. Companies whose earnings growth the analysts are most optimistic about earn relatively poor stock returns, while companies whose earnings growth analysts are most pessimistic about earn relatively high stock returns. La Porta (1996) interprets his findings as evidence that analysts extrapolate past earnings growth into the future and make systematic errors of excessive optimism for stocks with rapidly growing earnings, and conversely for stocks with deteriorating earnings. In this paper, we revisit and extend La Porta s evidence with 20 additional years of data, and then suggest a new theoretical interpretation based on Gennaioli and Shleifer s (GS, 2010) model of Kahneman and Tversky s (1972) representativeness heuristic and on its application to stereotypes by Bordalo, Coffman, Gennaioli and Shleifer (BCGS, 2016). Figure 1 presents an updated version of a bar-graph from La Porta (1996), which shows average between 1981 and 2014 one year stock returns on equally weighted portfolios of stocks sorted by analyst earnings growth forecasts. We use the notation LTG for long term growth forecasts of earnings per share throughout, so the LLTG portfolio is the 10% of stocks with most pessimistic long term earnings growth forecasts, while the HLTG portfolio is the 10% of stocks with the most optimistic long term earnings growth forecasts. Consistent with La Porta (1996), the LLTG portfolio earns an average return of 17% in the year after formation, while the HLTG portfolio earns only 12%. Over this period, betting against extreme analyst optimism as reflected in LTG has been on average a good idea. Geometric differences in returns look even more dramatic. Figure 2 presents the outcomes of two investment strategies from 1981 to The first invests in a HLTG portfolio, and then 2

3 at the end of the year sells it and reinvests the proceeds in the new HLTG portfolio again based on new LTG-based classification, doing so 33 times. The second strategy does the same with the LLTG portfolio. At the end of 34 years, an investor of an initial $1 in the rolling HLTG portfolio has $7.50 while one with an initial $1 in a rolling LLTG portfolio has $ The average annual geometric mean is 15.1% for the LLTG portfolio vs. 6.1% for the HLTG one. This evidence raises a number of questions. What is the past performance of HLTG and LLTG companies? What is their future earnings growth, and how does it relate to expectations and their revisions? How do low returns materialize? Can we, more broadly, understand the coevolution of earnings, expectations, and returns for HLTG and LLTG stocks? After describing our data sources in Section 2, in Section 3 we provide empirical answers to these questions. We show that HLTG stocks exhibit fast past earnings growth, but that going forward their earnings growth slows down. On average, they substantially underperform earnings growth expectations, and these expectations are systematically downgraded. The opposite, but much less extreme, dynamics obtain for LLTG stocks. These patterns, as illustrated in Figures 1 and 2, also obtain in stock prices and returns. HLTG stocks have done very well in the past in terms of both returns and earnings growth, but their average returns going forward are lower than those of LLTG stocks. Centrally to our theory, we also show that while on average stocks with extremely optimistic analyst earnings growth forecasts disappoint in terms of both earnings growth and returns, a small minority of them exhibit both very fast earnings growth and spectacular future returns. There are a few stocks like Google in the data, but not nearly as many as analysts expect. The evidence suggests that a useful way to think about this analyst expectations anomaly is through the lens of Kahneman and Tversky s (1972) representativeness heuristic. As previously modeled by Gennaioli and Shleifer (2010) and Bordalo et al (2016 a,b) representativeness captures 3

4 the idea that perceptions and evaluations are comparative and relative rather than absolute. A trait X is representative of a group A when it occurs more frequently in group A than in a reference group B. Decision makers who judge by representativeness rather than Bayesian likelihood overestimate the share of members of group A who have trait X. Observed beliefs are distorted but depend tightly on true distributions, exaggerating differences between groups. This logic leads to a fairly basic insight. If two distributions differ in the tails (such as two normal distributions with different means), representativeness causes beliefs to overweight these tail differences. For example, people over-estimate the share of the elderly among Florida residents because the elderly are more common among the Floridians than in the US as a whole. This is so even though there are a lot fewer elderly than middle aged people in both Florida and the rest of the US. A representative Floridian is elderly, though a rationally expected Floridian is middle aged. This is known in psychology as the kernel of truth hypothesis: when people put too much weight on the tails, they get direction of the comparison between the two means correctly, but exaggerate its magnitude. There are more elderly in Florida, but not as many as people think. Belief formation through representativeness suggests a new way of thinking about the analyst expectations anomaly. At the most basic level, we argue that investors form expectations about and form valuations of stocks by comparison with other stocks. In doing so, they put too much weight on the attributes of a stock that are most distinctive from those of other stocks, and exaggerate the frequency of such attributes in the portfolio. To be more specific, good news about an already growing company is representative of it being the next Google rather than a volatile stock with a good realization, since repeated good news are much more likely among Googles than among regular volatile stocks, which would have some bad news as well. An analyst observing a string of extremely good earnings might then overweight the chances that the company is another 4

5 Google, rather than just volatile, in estimating its future growth prospects. Of course, a few stocks that look like Googles indeed end up being like Google, and in fact some of the high earnings growth companies end up being remarkable performers, but not as many as the analysts believe. Section 4 shows that a model relying on this basic insight is consistent with the evidence we present, particularly with the critical fact that among the stocks with extremely high earnings growth forecasts, a small minority exhibits extraordinarily high future earnings growth and returns while the majority disappoints. Similarly, stocks with a poor record of earnings growth would be seen as representing poor long run performance, rather than just occasional bad luck, leading to undervaluation. These stocks would, on average positively surprise both analysts and investors. Our paper is related to several strands of research in finance. Empirical research on crosssectional stock return predictability and anomalies started with DeBondt and Thaler (1985, 1987), and has evolved into a huge literature. Although this work is often framed in terms of concepts such as extrapolation (e.g., Cutler, Poterba and Summers 1991, Lakonishok et al 1994, Dechow and Sloan 1997), most studies in this area do not use expectations data. There are some older studies in finance that do use expectations data include Dominquez (1986) and Frankel and Froot (1987, 1988). There is also an enormous literature on analyst expectations, which often shows that they are on average too optimistic (Easterwood and Nutt 1999, Michaely and Womack 1999, Dechow, Hutton, and Sloan 2000). More recently, the use of survey expectations data not just by analysts but also by investors has been making a comeback into finance (e.g., Ben David, Graham and Harvey 2013, Greenwood and Shleifer 2014, Gennaioli, Ma, and Shleifer 2015, etc.) Our paper is also related to the theoretical literature on excessive optimism, over-reaction, and bubbles, e.g., DeLong et al. (1990), Barberis, Shleifer, and Vishny (1998), and more recently Barberis et al. (2015) and Glaeser and Nathanson (2015). The approach we take here is 5

6 conceptually new, and relates to our work on representativeness and stereotypes (Gennaioli and Shleifer 2010, Bordalo et al. 2016a, b). We use a portable model (Rabin 2013) of belief formation based on representativeness, which has been used to shed light on social phenomena ranging from stereotyping to credit cycles, and apply it to stock returns. We will argue that this approach has several advantages relative to the more mechanical models of over-reaction and extrapolation. We should stress from the start the relatively narrow scope of this paper. We do not try to explain the many anomalies in the cross-section of stock returns with one model. The range of anomalies has been growing and we are far from certain that one model is appropriate for all or even many of them. Instead, we focus on one which has been previously tightly linked to expectations. We put together additional evidence on expectations focusing on revisions and on the full distribution of outcomes -- and propose a model that can account for this evidence. Nor do we join the debate on whether the analyst expectations anomaly can be explained by so-called risk factors (Fama and French 1992), or requires a behavioral explanation. We feel it is instead more productive to develop explicit models of expectations formation that confront the evidence. Finally, we present a very stylized toy model to illustrate the fundamental features of belief formation by representativeness. We leave more general treatments to future work. II. Data and Summary Statistics II.A. Data We gather data on analysts expectations from IBES, stock prices and returns from CRSP, and accounting information from CRSP/COMPUSTAT. Below we describe the measures used in the paper and, in parentheses, provide their mnemonics in the primary datasets. 6

7 From the IBES Unadjusted US Summary Statistics file we obtain mean analysts forecasts for both the level of earnings per share and the expected long-run earnings per share growth rate (LTG) for the period December 1981 when LTG first becomes available through December of IBES defines LTG as the expected annual increase in operating earnings over the company s next full business cycle, a period ranging from three to five years. From the IBES Unadjusted US Actual file we get data on the earnings per share for the most recent fiscal year (fy0a). We use CRSP daily data on stock splits (cfacshr) to adjust IBES earnings per share figures. On December of each year between 1981 and 2014, we form deciles based on stocks that report earnings in US dollars by their LTG. The CRSP sample includes all domestic common stocks listed on a major US stock exchange (i.e. NYSE, AMEX, and NASDAQ) except for closed-end funds and REITs. Our CRSP sample starts in 1978 and ends in We present results for both buy-and-hold annual returns and daily cumulative-abnormal returns for various earnings announcement windows. For each calendar year, we compute annual stock returns by compounding monthly returns. We focus on equally-weighted returns for LTG portfolios. If a stock is delisted, CRSP makes an effort to establish its price after delisting. Whenever a post-delisting price exists, we use it in the computations for returns. When CRSP is unable to determine the value of a stock after delisting, we assume that the investor was able to trade at the last quoted price. After a stock disappears from the sample, we replace its return until the end of the calendar year with the return of the equallyweighted market portfolio. Given that IBES surveys analysts around the middle of the month (on Thursday of the third week of the month), LTG is in the information set when we form portfolios. Daily cumulative abnormal returns are defined relative to CRSP s equally-weighted index. We also gather data on market capitalization in December of year t as well as the pre-formation 3-year 7

8 return ending on December of year t. Finally, also for descriptive purposes, we rank stocks into deciles based on market capitalization using breakpoints for NYSE stocks. We get from the CRSP/COMPUSTAT merged file on assets (at), sales (sale), net income (ni), book equity, common shares used to calculate earnings per share (cshpri), adjustment factor for stock splits (adjex_f), and Wall Street Journal dates for quarterly earnings' releases (rdq). Our CRSP/COMPUSTAT data covers the period 1978 through We use both annual and quarterly accounting data. We define book equity as the sum of stockholders equity (depending on data availability seq, ceq+pfd, or at-lt) plus deferred taxes (depending on data availability txditc or txdb+itcb) minus preferred equity (depending on data availability pstkr, pstkl, or pstk). We define operating margin as the difference between sales and cost of goods sold (cogs) divided by assets and return on equity as net income divided by book equity. Finally, we compute the annual growth rate in sales per share in the most recent 3 fiscal years. When merging IBES with CRSP/COMPUSTAT, we follow the literature and assume that data for fiscal periods ending after June becomes available during the next calendar year. II.B. Summary Statistics Table 1 reports the means of some of the variables for LTG decile portfolios. The number of stocks with IBES data on LTG varies by year, ranging from 1,222 in 1981 to 3,845 in On average, each LTG portfolio contains 246 stocks. The expected growth rate in earnings per share ranges from 4% for the lowest LTG decile (LLTG) to 38% for the highest decile (HLTG), obviously an enormous difference. LLTG stocks are larger than HLTG stocks in terms of both total assets (7,982 MM vs. 1,061 MM) and market capitalization (3,699 MM vs. 1,617 MM). 8

9 However, differences in size are not extreme: the average size decile is 5.1 for LLTG and 3.6 for HLTG. The results on profitability show that LLTG stocks have lower operating margins than HLTG stocks but higher return on equity (4% vs -6%). In fact, 36% of HLTG firms have negative eps while the same is true for only 16% of LLTG stocks. The high incidence of negative eps companies in the HLTG portfolio underscores the importance of the definition of LTG in terms of annual earnings growth over a full business cycle. Current negative earnings thus do not appear to undermine analyst enthusiasm for medium term prospects of these companies. A look at history suggests why. In the three years prior to portfolio formation, LLTG stocks underperformed HLTG stocks in terms of annual growth in sales (3% vs. 19%), annual growth in eps (5% vs. 33%), and annual returns (7% vs. 24%). At least at this basic level, La Porta s (1996) narrative of extrapolation of past performance into the medium term future in forming LTG is broadly consistent with the evidence. This narrative sees the differential stock returns as evidence of error and disappointment. In the next section, we flesh out this broad picture, but also present some novel evidence on the time patterns of earnings, expectations, and returns for HLTG and LLTG stocks. III. Empirical Findings. In this section, we present the results on earnings, expectations, and stock returns of portfolios sorted by LTG. We present many of the findings comparing HLTG and LLTG portfolios graphically; most of these graphs have more detailed counterparts for all LTG decile portfolios and, generally speaking, the results are monotonic across portfolios. 9

10 III.A. Earnings Figure 3 presents the evolution of average per share earnings of HLTG and LLTG portfolios in the three years before, and 4 years after, portfolio formation. We normalize year t-3 average earnings of both portfolios to 1. Figure 3 shows that the earnings of LLTG firms decline sharply between t-3 and portfolio formation, while earnings of HLTG firms increase sharply. But the past does not repeat itself after portfolio formation. The data show that earnings per share of LLTG firms actually reverse their decline and begin growing, while the earnings growth of HLTG firms slows down considerably. In fact, earning per share actually decline a bit for HLTG firms in the first year after portfolio formation. HLTG firms exhibit high growth and remain more profitable on average than LLTG firms 4 years after portfolio formation, but the difference in growth rates is nowhere near the difference in LTG exhibited in Table 1. Figure 4 presents another perspective on this evidence, focusing on the distribution of growth rates of HLTG and LLTG firms. Because so many firms in the initial year have negative earnings, we normalize changes in earnings between the formation year 0 and year +5 by year zero book value per share. Figure 4 plots the distribution of this measure of earnings growth of HLTG and LLTG firms. We estimate kernel densities of the two distributions and present them in Figure 4. Two findings stand out. First, HLTG firms indeed have a slightly higher growth rate of earnings than LLTG stocks, as we saw in a somewhat different format in Figure 3. But second, and critically, the distribution of earnings changes is MUCH fatter-tailed for HLTG than for LLTG stocks. There are hardly any extreme performers in terms of earnings changes among LLTG stocks. In contrast, there are significant minorities of HLTG stocks whose earnings either collapse or grow extremely rapidly. Quantitatively, 69% of LLTG firms fall between the 25 th and the 75 th percentile of the distribution of these firms growth rates. The corresponding number for HLTG 10

11 firms is only 27%. In this sense HLTG firms contain a disproportionate number of great performers on the right tail, but also volatile firms that after doing well display very poor earnings growth post formation. The fat tails of the distribution of earnings changes of HLTG stocks, as compared to LLTG stocks, will form the foundation of our theoretical model. III.B. Expectations Figure 5 presents perhaps the most basic picture of earnings growth expectations and actual earnings for LLTG (left panel) and HLTG (right panel) stocks. Each panel presents two lines. The yellow line is average earnings forecasts if earnings grew at the LTG rate expected by the analysts. The brown line is the corresponding earnings realizations. Even the picture for LLTG stocks reflects small analyst over-optimism, established by many previous studies and usually explained by distorted analyst incentives (e.g., Dechow et al. 2000, Easterwood and Nutt 1999, Michaely and Womack 1999). But for LLTG stocks over-optimism is relatively minor. Earnings for LLTG stocks stagnate around 0.09 between t-3 and t+1 and rise slowly during the post-formation period to reach 0.11 in t+3. Analysts do not anticipate the period of stagnation for LLTG stocks and raise their forecasts for earnings per share fairly steadily from 0.10 to 0.12 between t-3 and the formation year. Based on year-0 LTG, earnings for LLTG stocks are projected to reach 0.16 in year t+3. As a result forecast errors rise steadily from in t-3 to in t+3. Analysts are consistently overoptimistic about LLTG stocks but forecasts don t deviate much from realized values. In contrast, the right panel of Figure 5 shows the extreme over-optimism of analysts with respect to HLTG stocks. By year 5, actual earnings are a small fraction of what analysts forecast. Earnings per share for HLTG stocks exhibit truly explosive growth during the pre-formation 11

12 period, rising from 0.01 in year -3 to 0.12 in year 0. Analysts forecasts for earnings closely track realized values rising from 0.06 in year -3 to 0.16 in year 0. Following the formation period, however, the two series diverge dramatically. Earnings drop to 0.08 in year +2 before returning to growth and rising to 0.21 in year +5. However, based on LTG forecasts in the formation year, earnings per share were expected to reach 0.70 in year +5. Although the sign of the errors made by analysts in forecasting earnings for HLTG and LLTG stocks is the same, only the errors for HLTG stocks are large in economic terms. Figure 6 presents another take on this disappointment by showing the evolution of LTG for the two portfolios. The figure shows that, over time, and particularly in the first year after portfolio formation, LTG of HLTG firms is revised sharply downwards, whereas LTG of LLTG firms is revised moderately up. Four years after portfolio formation, earnings of HLTG firms are still expected to grow faster than those of LLTG firms, but the spread in expected growth rates of earnings has narrowed considerably. Once again, this evidence is consistent with excessive initial optimism about HLTG firms that fades post formation, leading to systematic downward revision of expectations. The opposite occurs for LLTG firms. III.C. Returns The results on earnings growth and expectations would be interesting but their relevance rises if they are also reflected in the returns data. As we showed in Figures 1 and 2 in the introduction, however, the story of HLTG and LLTG stocks is one of returns and not just expectations vs realizations. In this subsection, we flesh out the returns story. 12

13 Figure 7 presents the time series of cumulative returns on HLTG portfolios relative to the formation year 0. It confirms the basic fact that HLTG stocks outperform LLTG stocks in returns and not just fundamentals prior to the formation period, but then underperform following formation. With the benefit of hindsight, an investment in year -3 of $0.47 in the HLTG portfolio would have turned into a $1 by the end of December of the formation year. In contrast, an investor would have needed to invest $0.77 in the LLTG portfolio in year -3 to have $1 in December of year 0. A year later, that $1 would have turned into $1.22 if invested in HLTG stocks vs $1.28 if invested in LLTG stocks. Three years after the formation year, that $1 had grown to $1.44 if invested in HLTG stocks vs $1.61 if invested in LLTG stocks. But again, the averages hide the picture of distributions, shown in Table 2 and in Figure 8. Just as we saw with earnings, HLTG stocks have a much fatter-tailed distribution of returns than LLTW stocks. At the portfolio return level, the standard deviation of the return on the LLTG portfolio is 18.5%, compared to 36.3% for the HLTG portfolio. Figure 8 shows the frequency distribution of pooled returns for LLTG stocks (on the left) and HLTG stocks (on the right). Figure 8 makes it clear that the HLTG portfolio includes many stocks with Google-like returns. For example, 9.3% of the HLTG earn annual returns in excess of 100%. The analogous figure for LLTG stocks is only 3.5%. Yet, HLTG stocks earn lower average returns than LLTG stocks because they have a fatter left tail of disappointing firms. Figures 9 and 10 complete our summary of basic facts by focusing on returns around earnings announcements. Figure 9, for every year before and after portfolio formation, presents event returns for HLTG and LLTG portfolios in narrow windows around the 4 annual earnings announcements. That is, for every stock in the portfolio, we compute the 12-day cumulative return during the four quarterly earnings announcement days, following the methodology of La Porta et 13

14 al. (1997). Figure 9 shows clearly that HLTG stocks positively surprise investors with their earnings announcements in the years prior to portfolio formation, but then severely disappoint them afterwards, especially in year. Recall that year 1 is when HLTG stocks really perform poorly. The converse holds for LLTG stocks, but in a much milder form: they disappoint investors prior to portfolio formation, but then their earnings positively surprise investors after portfolio formation. The total returns reported in Figure 8 at least to some extent accrue on earnings announcement dates (i.e. 35.6% of the annual spread between HLTG and LLTG stocks accrue during the 12 days in our earnings window). Figure 10 digs deeper into this evidence by focusing on how returns vary with earnings surprises using quarterly data. We follow the methodology of Skinner and Sloan (2002) and measure returns during the window that starts on the second day following the earnings announcement for the previous fiscal quarter ( FQi-1 ) and ends one day after the earnings announcement for the current fiscal period (FQi). To illustrate consider the example of Apple Inc. It announced earnings for the 3/31/2015 fiscal period on 4/27/2015 ( FQi-1 ) and for the 6/30/2015 fiscal period on 7/21/2015 ( FQi ). Accordingly, we compute returns for the period that starts on 4/29/2015 and ends on 7/22/2015. Next, we define earnings surprises as (epsi-e[epsi tq])/ptq, where epsi is the realized value of earnings for fiscal period i, E[epsi tq] denotes analysts expectations for eps in fiscal period i made right before the end the fiscal quarter ( tq ), and p is the stock price at time tq. Returning to the example of Apple Inc., IBES June monthly survey of analysts took place on 6/18/2015 (tq). Therefore, the eps surprise associated with Apple s 7/21/2015 eps announcement is based on forecasts and stock prices measured on 6/18/2015. The graph below puts these events on a time line: 14

15 Figure 10 shows that HLTG stocks outperform LLTG stocks when news is good. For example, for earnings news in the 9 th decile, returns are 5.8% for HLTG vs. 1.4% for LLTG stocks. In fact, the spread between the LLTG and HLTG portfolio is negative for earnings surprises at or above the median and ranges from.4% for surprises in the tenth decile to -5.2% for surprises in the eighth decile. Critically, returns for the LLTG portfolio exceed those of the HLTG portfolio when earnings news is bad, i.e. for earnings surprises below the median. For example, LLTG drop by 8% during periods with the worst earnings surprises while HLTG stocks drop by 14.4% during those periods. This pattern of returns to earnings surprises is reminiscent of Skinner and Sloan s (2002) torpedo effect. Overall, the evidence provides the key motivations for how one might want to model the analyst expectations puzzle. At a broad level, the data indicate that analysts react to news. Optimism about HLTG firms follows the observation of fast earnings growth, and is reversed after earnings growth slows down. Likewise, pessimism about LLTG firms follows the observation of declining earnings, and is reversed post formation as earnings start to recover (see Figure 3). These patterns suggest that analysts are trying to learn the earnings generating capacity of firms, and do so in light of the firm s observed performance. On the other hand, the data cast serious doubt on the notion that analysts learning process is rational. In a Bayesian model, expectations formation would rely on the true posterior distribution of firm types. As a result, analysts would be unlikely 15

16 to make the large mistakes we see in Figure 5. These large mistakes are made on average in a large portfolio of different firms, indicating that the true posterior distribution of firm types is different from the one that analysts have in mind. We can also narrow down the set of departures from full rationality that might help explain the LTG puzzle. In particular, the data seem inconsistent with the possibility that analysis may be subject to forms of inattention to information (Sims 2003). As Figures 3 and 5 show, analysts react swiftly to information, and in fact over-react in the direction of the news. Two noteworthy models that could generate over-reaction to news are the mechanical extrapolation of adaptive expectations models and the representativeness-based model of Barberis, Shleifer and Vishny (BSV, 1998). Under mechanical extrapolation, LTG is formed as a distributed lag of past earnings growth rates. As such, this model has a hard time accounting for the fast reversal in expectations and returns of Figures 6 and 8. 1 The BSV approach allows for subtler learning dynamics. In this model, the true process driving a firm s earnings is a random walk, but analysts perform Bayesian updating across two incorrect models. In one of these models earnings permanently trend, in the other they always mean revert. The logic of over-reaction to news then goes as follows. After some periods of fast earnings growth, the analysts attaches a high probability that the firm is of the trending type. Because no firm is in reality trending (earnings of all firms follow a random walk), the stock becomes over-valued. This model is motivated with representativeness: positive earnings growth 1 One could of course consider a version of adaptive expectations in which the coefficient on the last observed growth performance is very large. This model would imply very high volatility in LTG expectations. In particular, it would imply, similarly to BSV, that all firms experiencing high growth would be overvalued, which is difficult to reconcile with the fact that HLTG firms can earn high returns. It would also imply that even temporary low growth would imply a dramatic downgrading of firms s earnings growth prospects. 16

17 is representative of a trending firm. As a result, after seeing earnings growth, the probability that the trend continues is exaggerated. Still, we think this approach can be improved. First, it cannot explain why a sizable share of HLTG firms earn very high returns. In the BSV world, firms in the HLTG bucket are all overvalued, and would all deliver disappointing returns. In the data, even when considering firms that have performed well in the past, analysts keep in mind the possibility that these firms may have been lucky. This moderates expectations in the HLTG group so that the truly outstanding firms in the group exceed expectations. Methodologically, the BSV approach relies on Bayesian updating of wrong models. Here we start with a more basic and portable formulation of representativeness, in which beliefs are based on reality rather than on incorrect models. Our approach is based on the formal model of representativeness developed in GS (2010) and applied to the study of stereotypes in BCGS (2016). In our model, as in BSV, past performance brings representative firm types to mind. Unlike in BSV, however, these representative types are selected from the true distribution of possible outcomes. Thus the most representative HLTG firm is the one most common in this group relative to the overall population. In Figure 4, the far right tail of Googles in HLTG is most representative of this group, even if unlikely in absolute terms. Second, in the BCGS model updating is non-bayesian because representative firms are overweighed at the expense of others. Over-reaction comes from the exaggeration of the true incidence of tail events, but its size is limited by the scarcity of Googles in reality (and by the fat left tail in Figure 4). As a result, reality itself moderates the analysts over-reaction, allowing the few real Googles to positively surprise analysts. In the next section, we pursue this approach. 17

18 IV. A model of learning with representativeness IV.A. Representativeness and beliefs Our model of learning builds on research on heuristics and biases in judgments and decision-making. Kahneman and Tversky (1972) argue that individuals often make probabilistic assessments using the representativeness heuristic, estimating a trait as likely in a group whenever it is merely representative of that group. KT define representativeness as follows: an attribute is representative of a class if it is very diagnostic; that is, the relative frequency of this attribute is much higher in that class than in the relevant reference class (TK 1983). Starting with KT (1972), much experimental evidence has accumulated to support the role of representativeness. Gennaioli and Shleifer (2010) build a model in which decision makers overweight events that are representative in the sense of KT s definition, and show this can account for a number of puzzles in human probability judgment. BCGS (2016) develop a representativeness-based model of stereotypical thinking that accounts for a variety of evidence on social stereotypes. Our model is most closely related to Bordalo, Gennaioli and Shleifer s (2016) model of expectations formation, called diagnostic expectations, which we review below. A decision maker assesses the distribution of a trait in a group. According to GS (2010), the representativeness of the trait for is:, where is a relevant comparison group. As in KT, a trait is more representative if it is relatively more frequent in than in. The representative heuristic then implies that decision makers overweigh the representative types in their assessment of. The notion is that decision makers 18

19 have limited working memory, and representative types are easier to recall and thus dominate judgment (GS 2010). This implies, in particular, that beliefs about depend on context. The setup above is extremely broad, and can be applied to prediction problems (as in BCGS 2016 and BGS 2016) as well as inference problems. Here we focus on an inference, or learning, setting. To illustrate, consider a doctor who assesses the health status of a patient,,. Taking a medical test provides some new information about the health status. How does the doctor update his prior in light of a positive result? If the test is informative, namely if a positive result increases the probability of being sick relative to baseline ( ), then being sick is representative of testing positive: Pr Pr Pr Pr According to representativeness, a positive test brings to mind the sick type and the doctor increases its probability excessively, relative to the Bayesian benchmark. 2 Thus, decision makers inflate the likelihood of diagnostic types, i.e. those whose objective probability rises the most in relative to the reference context. This type of the base rate neglect fallacy, verbally described in TK s (1974), plays an important role in the analysis below. Crucially, in this learning setting context is provided by lagged expectations, held before new information arrived. IV.B. Learning under representativeness 2 In the evidence collected by Casscells et al. (1978), physicians often inflated the probability of diseases that are very rare, committing a form of base rate neglect. 19

20 We now apply this framework to the problem of investors learning about firms future performance. We first present the definition of representativeness and of diagnostic expectations with minimal assumptions on the learning environment. We then work out the model in a particularly simple setting that captures key features of the evidence on fundamentals reviewed in Section III.A. In Section V, we show how, in this simple setting, our model helps account for the evidence on expectations and returns presented in Sections III.B and III.C. Firms are indexed by 0,1, and are characterized by their type, namely the data generating process of their earnings growth. Investors know the unconditional distribution over types, Pr for 0,1, but learn the type of specific firms over time. At each time period 0, firm reports earnings,, which we assume to be positive throughout. Earnings growth at time is then,,, 1. At each time 1, analysts use firm s history of earnings growth,,,,, to infer its type. Going forward, we identify firms by their history and omit the index. Figure 11 below presents the time line of the model.,,,,, 1 update types Pr, forecasts Pr, and prices. Returns accrued. 1 Figure 11. Timeline. Given a firm s history, beliefs about its performance going forward can then be decomposed into beliefs about the firm type, Pr, together with a conditional expectation about future performance conditional on firm type, Pr, namely: 20

21 Pr Pr Pr Expectations of future earnings growth are given by Pr, and these determine prices in a standard Gordon growth framework, reviewed in section V. 3 In the rational benchmark, beliefs about firm type are updated in light of new information according to Bayes rule. For example, after history, rational beliefs are: Pr Pr Pr Pr Pr In our model, representativeness distorts the updating of beliefs about firm types relative to the rational benchmark above. Just like the doctor assesses the distribution of health status conditioning on test results, analysts assess distribution of firm types conditioning on its performance history. Pursuing the analogy, analysts form beliefs about a group of firms :, in comparison to their lagged beliefs about those firms, prior to new information, :,. We therefore have: Definition 1. At time, the representativeness of type for a firm with history is: ; Pr Pr Denote by ; the vector of representativeness of all types for history. 3 It is useful to clarify our focus on expectations of the long term growth in earnings. In the Gordon model, prices are determined, largely, by the steady state earnings growth in the long run. In the short run, earnings may deviate from their long term behavior for example, they can be negative but this has a minor impact on prices. The key assumption in our approach is that earnings growth in the short run is informative about the firm type, which regulates earnings growth in the long run. 21

22 Under Definition 1, the more representative types given history are those whose objective likelihood rises the most when the latest growth data, is observed. We can then define: Definition 2. Given history, diagnostic beliefs about the firm s type are given by: Pr Pr Pr where, ;, and is a function that is weakly increasing in its first argument, and weakly decreasing (symmetrically) in each of the other arguments. As in BCGS (2016) and BGS (2016), probabilistic assessments overweight the most representative types relative to the others. In BGS (2016) we used a continuous weighing scheme, where,, which was most tractable in the context of normal distributions. In the model developed below, it is more tractable to use rank based discounting defined in BCGS (2016). We thus set, where 1 measures the role of representativeness, and is the rank order of, in the vector. In particular, the most representative type has 1 and receives the highest weight. For 1, beliefs are rational. As increases, the most representative type absorbs more and more of the probability weight, eventually to the exclusion of all others. IV.C. The Google, the Volatile, and the Boring We now develop our model of learning under representativeness in a stylized setting that is as simple as possible yet captures the key features of the evidence on earnings growth of LTGsorted portfolios described in Section III.A. Our choice of setting is motivated by two key features 22

23 of the data on earnings growth. First, Figure 3 shows that prior to formation, stocks that end up in the HLTG portfolio have higher earnings growth on average than stocks that end up in the LLTG portfolio. Second, Figure 4 shows that, while the HLTG portfolio has higher expected growth than the LLTG portfolio, it also fatter right and left tails of earnings growth. Thus, high growth prior to formation is informative of higher growth going forward, but also of higher variance in growth: while some HLTG firms do extremely well, some do worse than the LLTG firms. The departures of from rationality in our model concern precisely the associated signal extraction problem. To capture these features of the fundamentals, we make two simplifying assumptions. First, we take firms earnings growth to be i.i.d., as it significantly simplifies the inference problem, and the application of representativeness to it, while providing a good approximation to the evidence. Second, we assume that earnings growth can take only three values, high, medium, and low, denoted, which is the minimal support compatible with the fact that the HLTG portfolio has higher left and right tails. Table 3 then presents the simplest distributions of earnings growth that entails a signal extraction problem in this three state i.i.d. setting. There are three firm types. A small measure 0 of googles has earnings growth on the right tail. A measure of volatile firms has volatile earnings growth, concentrated on both tails. The remaining 1 firms are boring, as they often experience mediocre, sometimes poor, but never exciting, growth. (googles) (volatiles) 1 0 (boring) 1 0 Table 3. Distributions of earnings growth 23

24 The group of volatile firms serves two purposes: it makes signal extraction difficult, as high performance can indicate a google or a volatile firm, and poor performance can indicate a volatile firm or a boring firm. Second, it modulates the variance in growth rates of the LTG portfolios. To this end, we assume that volatile firms have a higher likelihood of low earnings growth, and lower average growth, than boring firms. Formally: A.1 i) 1. ii) 1 but not too large. Assumption A.1 simply says that the firms that are worst on average (volatile ones) are also relatively more likely, and indeed more representative, after a low earning growth realization. To see how it maps to the data, suppose analysts form LTG portfolios given the first observation of earnings growth, i.e. at time 1. A.1 then implies that the firms for which, have the highest (rational) growth expectations, while the firms for which, have the lowest expectations. Thus, as in Figure 3, we can define the High LTG portfolio as the group HLTG :,, and the Low LTG portfolio as the group LLTG :,. 4 The HLTG portfolio includes googles and high performing volatile firms, while the LLTG portfolio includes low performing volatile and boring firms. Then, the HLTG portfolio has fatter tails as in Figure 4 in the following circumstances: 4 In the current setting (where there are few firm types and a single piece of news) it makes sense to define portfolios in terms of expectation thresholds. However, this leads to portfolios with different numbers of firms. We could adopt a definition closer to the empirical one, in terms of deciles of growth expectations, by looking at firms with more data (and possibly with more types). Our results depend only on the gradient of expectations across portfolios, and not on the details of portfolio formation or size. 24

25 Lemma 1 Under A.1 ( 1 ), there is a threshold 0 such such that the HLTG portfolio has higher probability of extreme performances and than the LLTG portfolio if and only if googles are sufficiently uncommon, i.e.,. It is natural for HLTG to have a fatter right tail of earnings growth, given that it includes all googles and no boring firms. However, because googles are rare, most HLTG firms are volatile, leading this portfolio to have a fat left tail of earnings growth. In particular, HLTG has a fatter left tail than LLTG if its share of volatiles is sufficiently high, namely if is sufficiently small. The simple setting outlined above is designed to explore the properties of LTG portfolios formed after the first observation of earnings growth. In particular, in most cases firm type can be inferred with certainty after a few observations. This can be seen as a tractable approximation to a more realistic Kalman-filter setting in which firm types can change over time, thus requiring constant learning. V. Mapping the Model to the Data V.A. Representativeness and the Features of Expectations We now describe how expectations are formed in our model with representativeness. We show how the model accounts for three features of the evidence on expectations of long term growth, described in Section III.B: first, Figure 5 shows that earnings growth forecasts are excessively optimistic for HLTG firms, and much less so for LLTG firms. Second, Figure 6 shows that these expectations exhibit systematic reversals, so that LTG displays a boom-bust pattern for HLTG firms and he reverse bust-boom pattern for LLTG firms. Moreover, according to Figure 6 25

26 expectations of earnings growth in the HLTG portfolio are much more volatile than expectations of earnings growth in the LLTG portfolio. We start by deriving the implications of representativeness for the formation of expectations about firm s growth in the HLTG and LLTG portfolios. From Definition 1, the representativeness of different firm types of the high growth portfolio is,,, 0 and,. The most representative firm type of the high growth portfolio is a google, for which exceptional performance is more common. The next most representative firms is a volatile one, which occasionally experiences high growth. Following Definition 2, in assessing expected future growth, the stereotypical thinker overweights more representative firm types. Under the rank-based smooth discounting we adopt, earnings growth expectations are given by:,, 9 As increases, the perceived share of googles in HLTG increases, so expectations of future growth are inflated and tend to,,. The representativeness of different types in the LLTG portfolio is given by, 0,,, and,. After observing low growth, the most representative firm type is a volatile firm because it is relatively more likely to produce bad performance (by A.1, 1 ). The next most representative firms is a boring one, which is also occasionally produces low earnings growth. The expectation of future growth is then: 26

27 1 1, 1 1, 11 As increases, the perceived share of volatile firms in the LLTG portfolio rises, so expectations tend to,, which is lower than,. In sum, we have: Proposition 1. (Overreaction of Expectations) Suppose that 1. Then, expectations of firms in the HLTG portfolio improve pre-formation and are too optimistic at formation, while expectations of firms in the LLTG portfolio deteriorate pre-formation and are too pessimistic at formation. Formally:,,,,,. 12 Representativeness causes expectations to move too much in the direction of the information received, illustrating the kernel of truth logic nested in diagnostic expectations (see also BGS 2016). This result maps to Figure 5 panel b), which shows that growth expectations for HLTG firms are wildly optimistic on formation. Figure 5 panel a) shows that expectations for LLTG firms are much less optimistic, but they are not pessimistic as the model predicts; this could be attributed to the well-documented overall optimism in analysts forecasts discussed in Section 3. We now consider how expectations are updated after a second piece of news. Observing low performance from a firm in the HLTG portfolio identifies it as a volatile firm. Thus, expectations are not distorted:,,,,. In contrast, it is easy to check that googles are more representative than volatiles for firms that experience a second period of high performance. Thus, after a path,, expected earnings growth becomes: 27

28 ,,. 13 Similarly, observing high performance from a firm in the LLTG portfolio identifies it as volatile, so that,,. Intermediate growth identifies the firm as boring, so that,, 1. Finally, volatile is again more representative than boring for firms that experience a second period of low performance. Thus, after a path,, expectations are: 1,, We can now establish the following result. Proposition 2. (Mean Reversion in Expectations) Earnings growth expectations systematically revert post-formation, for both HLTG and LLTG firms. Formally:,,,,,,. Under representativeness, expectations of future earnings growth systematically adjust after formation. Intuitively, most firms in the HLTG portfolio are initially overvalued (given that googles are very rare). Further earnings announcements reduces the share of firms in the HLTG portfolio that are initially (and excessively) mistaken for googles, reducing the average expectation of growth for the portfolio. Similarly, more information reduces the share of firms that get mistaken for volatiles in the LLTG portfolio, thus increasing its average expectation of growth. Together with Proposition 1, this result helps account for the boom-bust pattern of expectations for the HLTG portfolio, and bust-boom pattern of the LLTG portfolio, documented in Figure 6. 28

29 One important feature of our representativeness based model is that the correction in the initial over-optimism is very fast. Even a single piece of disconfirming evidence (low earnings growth) generates a drastic adjustment. This is because that low growth is not representative of googles, and suffices to debunk the inflated stereotype. The speed at which reversals occur is evident in Figure 6, and may help distinguish representativeness, which is based on the actual fundamentals in Table 3, from other mechanical extrapolation mechanisms. For example, in an adaptive expectations framework where forecasted growth is a weighted average of recently observed growth, one observation of low growth lowers expectations commensurably with its weight. A high weight in the latest observation is required to obtain a fast reversal but would also suggest all firms in HLTG are maximally overvalued upon formation of the portfolio, which is difficult to reconcile with the high returns of some HLTG firms going forward (see Figure 9 and Section V.B. below). Figure 5, and Proposition 2, set up the stage for analysts surprise about HLTG and LLTG firms performance in period 2. Figure 6 shows expectations about HLTG firms are more volatile, in that they react more to positive news prior to formation, as well as to disappointing news after formation. The model predicts higher volatility of HLTG expectations around formation, as in Figure 6, but also in a stronger sense of a displaying a wider spread in the cross section of HLTG firms after formation: HLTG firms with good (bad) news exhibit stronger increase (decrease) in expectations than the corresponding LLTG firms. Formally: Proposition 3. (Volatility of Expectations) There is threshold 1 such that, for, HLTG firms exhibit more extreme revisions of expectations than LLTG firms. Formally:,,,,,,, 29

30 ,,,,,,. The logic of this result is straightforward. HLTG expectations are volatile because they update the probability of an infrequent but very extreme outcome, namely the possibility that the firm is a google. By contrast, LLTG expectations are stable because they update the probability of two firms, the boring and volatile ones, that deliver quite similar mediocre average earnings growth. The result thus arises from the assumptions on fundamentals, and obtains even in the rational case 1. In fact, by inflating expectations for HLTG firms, representativeness weakens the subsequent upward revision for HLTG firms with positive news, and strengthens the downward revisions for HLTG firms with negative news. The constraint on the magnitude of ensures that expectations are not mechanical extrapolators; decision makers understand that high performance firms may just have gotten lucky, and experience large upward revisions in expectations when further news is consistent with them being a new google. The converse holds for LLTG firms. 5 V.B. Representativeness and Returns To develop the implications of the model for stock returns, we use the Gordon pricing formula, in which the price of a stock is the present value of its expected future earnings discounted at the required rate of return of. The price of the stock of a firm with history is then the weighted average Pr, 5 In Proposition 4, the upward revision of LLTG firms happens when intermediate growth is observed (as opposed to high growth), as that is indicative of boring firms, which have higher average growth rates. 30

31 where, is the value of stock of a firm of type at time,,,. For example, the price for firms that have had an realization (which we denote by HLTG) is then:, Because of this linear relationship between price and earnings growth expectations, the behavior of the return of HLTG stock prices mirrors the behavior of HLTG expectations. Define the return on stocks at formation ( 1 and post-formation ( 2) as,,,,. We then have that: Proposition 4. (Predictable Returns) The return of HLTG stocks displays a boom bust pattern around the formation period 1.,,,. 16 Conversely, LLTG stock returns display a bust boom pattern,,,,. That is, the HLTG portfolio exhibits abnormally high returns up to portfolio formation and abnormally low returns after portfolio formation, and conversely for the LLTG portfolio, just as we saw in Table 1 and Figures 1, 2, and 7. Leading up to formation, good news renders googles representative for HLTG firms. Market participants inflate these firms earnings growth prospects and their prices rise excessively, generating a high return. This is the first inequality (a return higher than upon formation arises also in the rational model, as in the second inequality, as is objectively good news). After formation, however, expectations revert back on average. The inflated fundamentals give way to a less extreme assessment. On average, then, investors are 31

32 disappointed and returns are abnormally low (the third inequality). This boom-bust cycle is a product of representativeness. In the rational model, expected returns should always equal. Finally, the last property we documented about returns is that the distribution of post formation returns for HLTG stocks has fatter tails than the distribution of returns for LLTG stocks. Our model yields this property by the same logic of Proposition 3, namely the idea that for moderate degrees of stereotypical thinking, expectations of earnings growth about HLTG firms exhibit larger surprises on the left as well as the right tail relative to LLTG firms. This is again a distinctive implication of representativeness, and in particular of the fact that investors do not assume that every HLTG firm is a google. Investors inflate the probability that any HLTG firm is a google, but they are aware that the firms is quite likely to be volatile. This effect moderates the extent of over-pricing and allow the few googles in the portfolio to indeed earn very high returns going forward. 32

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35 Table 1 Descriptive Statistics for Portfolios Formed on LTG. We form decile portfolios based on analysts' expected growth in earnings per share (LTG) in December of each year between 1981 and The table reports time-series means of the variables described below for equally-weighted LTG portfolios. Unless otherwise noted, accounting variables correspond to the most recently available fiscal year, where we follow the standard assumption that data for fiscal periods ending after June become available during the next calendar year. Assets is book value of total assets (in millions). Market capitalization is the value of common stock on the last trading day of year t (in millions). Size decile refers to deciles of market capitalization with breakpoints computed using only NYSE stocks. Operating margin equals the difference between sales and cost of goods sold divided by assets. Return on equity is net income divided by the book value of equity. Percent eps positive is the fraction of firms with positive earnings. Growth in sales is the annual geometric average of the growth rate in sales per share in the most recently available 3 fiscal years. Growth in eps is the annual geometric average of the growth rate in earnings per share in the most recently available 3 fiscal years, excluding firms with negative earnings in period t-3. (eps t -eps t-3 )/book equity t-3 is the increase in earnings per share between year t-3 and t divided by book equity in year t-3, excluding firms with negative book equity in year t-3. Previous 3-year return is the (annualized) compounded return for the 3-year period ending on December of year t. The variable is set to missing if any of the monthly observations during that period is missing. Observations is the number of observations in a year. All variables are capped at the 1% and 99% levels. LTG decile Expected growth in eps (LTG) 4% 9% 11% 12% 14% 15% 17% 20% 25% 38% Assets (MM) 7,993 10,828 9,617 8,020 5,214 3,658 2,472 1,938 1,356 1,082 Market capitalization (MM) 3,699 4,620 5,048 4,531 3,819 3,196 2,669 2,285 1,641 1,617 Size decile Operating margin 19% 24% 29% 33% 37% 40% 41% 42% 42% 37% Return on equity 5% 8% 10% 10% 10% 9% 9% 7% 3% 6% Percent eps positive 84% 87% 89% 90% 89% 87% 87% 84% 77% 64% Growth in sales (t 3, t) 3% 5% 6% 8% 9% 10% 12% 14% 16% 19% Growth in eps (t 3, t)t 5% 8% 9% 11% 12% 14% 18% 22% 27% 33% (eps t eps t 3 )/book equity t 3 1% 3% 4% 5% 6% 7% 9% 12% 17% 18% Previous 3 year return 7% 8% 9% 10% 11% 12% 14% 17% 21% 24% Observations

36 Table 2 Return Distribution for LTG portfolios In December of each year between 1981 and 2014, we form decile portfolios based on ranked analysts' expected growth in earnings per share and compute average one-year raw portfolio returns. For each LTG portfolio return, we report the value of its: (a) mean, (b) minimum, (c) 5 th percentile, (d) median, (e) 95 th percentile, (f) maximum, and (g) the standard deviation. LTG Mean Min 5 th pctl Median 95 th pctl Max Std Dev % 32.1% 13.5% 19.3% 43.3% 47.0% 18.5% % 37.1% 28.7% 21.6% 41.4% 51.2% 19.6% % 43.0% 21.5% 18.1% 45.1% 68.1% 21.2% % 39.1% 21.9% 16.2% 50.5% 52.9% 19.7% % 37.2% 17.0% 18.5% 47.7% 50.3% 18.7% % 38.9% 16.5% 16.7% 60.8% 65.6% 21.0% % 41.3% 20.2% 14.2% 50.4% 71.0% 22.5% % 42.9% 34.1% 12.6% 55.2% 73.6% 24.2% % 46.9% 37.5% 9.0% 69.2% 86.2% 28.8% % 54.2% 48.9% 5.4% 86.9% 119.9% 36.3% 36

37 Figure 1. Annual Returns for Portfolios Formed on LTG (January 1981-December 2015). In December of each year between 1981 and 2014, we form decile portfolios based on ranked analysts' expected growth in earnings per share and report the average one-year return over the subsequent calendar year for equally-weighted portfolios. 37

38 Figure 2 Compounded Returns for Portfolios Formed on LTG (January 1981-December 2015). In December of each year between 1981 and 2014, we form decile portfolios based on ranked analysts' expected growth in earnings per share and report the average one-year return over the subsequent calendar year for equally-weighted portfolios. We show the cumulative compounded return for the lowest (Lo LTG) and highest (Hi LTG) decile portfolios. 38

39 Figure 3 Evolution of EPS In December of each year t between 1984 and 2011, we form decile portfolios based on ranked analysts' expected growth in earnings per share. We report the mean value of earnings per share for the highest (Hi) and lowest (Lo) LTG deciles for each year between t-3 and t+4. We normalize earnings per share in year t-3 to equal 1. 39

40 Figure 4. Kernel Density Estimates of Changes in Earnings per Share of Stocks in LTG Portfolios. In December of each year t between 1981 and 2010, we form decile portfolios based on ranked analysts' expected growth in earnings per share (LTG). For each stock, we then compute the five-year change in earnings per share divided by book equity in year t, excluding stocks with negative book equity in year t. Next, we cap the values of that variable at the 1% and 99% levels before separately estimating the kernel density function for stocks in the highest (Hi) and lowest (Lo) LTG deciles. The graph shows the estimated density kernel for five-year post-formation changes in earnings in per share (scaled by book equity). The vertical lines indicate the means of each distribution. 40

41 Figure 5. Realized vs. Expected EPS for LTG Portfolios. In December of each year t between 1984 and 2010, we form decile portfolios based on ranked analysts' expected growth in eps. We plot two series. First, for each year between t-3 and t+5, we plot the mean value of eps. Second, we plot the mean: (a) one-year-ahead eps forecast made in years t-2 and t-1; and (b) eps forecasts made in year t for eps in years t through t+5 based on increasing the lagged value of eps in year t by LTG (i.e. eps_ltm*(1+ltg) i+1, where i ranges from 0 through 5 and eps_ltm is the lagged value of eps in year t). We exclude from the sample all observations with negative values for lagged eps. 41

42 Figure 6. Evolution of LTG. In December of each year t between 1984 and 2011, we form decile portfolios based on ranked analysts' expected growth in earnings per share (LTG) and report the mean value of LTG on December of years t-3, t-2, t-1, t, t+1, t+2, t+3, and t+4 for the highest (Hi) and lowest (Lo) LTG deciles. We include in the sample stocks with LTG forecasts in year t-3. Values for t+1, t+2, t+3, and t+4 are based on stocks with IBES coverage for those periods. 42

43 Figure 7 Cumulative returns for LTG Portfolios. Time series evolution of returns. We form decile portfolios based on ranked analysts' expected growth in earnings per share (LTG) in December of years 1981, 1984,, 2008, and We compute cumulative returns for holding periods between year t-3 and year T, where T ranges between t-3 and t+3. We rebalance portfolios annually and normalize cumulative returns to equal 1 in December of year t. 43

44 Figure 8 Returns as lottery tickets In December of each year between 1981 and 2014, we form decile portfolios based on ranked analysts' expected growth in earnings per share. The graph on the left (right) shows the pooled frequency distribution of stocks in the lowest (highest) LTG decile. The vertical red line indicates the sample average return (14.19%). We cap returns at 100%. 44

45 Figure 9. Three-day Returns on Earnings Announcements for LTG Portfolios. In December of each year t between 1981 and 2011, we form decile portfolios based on ranked analysts' expected growth in earnings per share. Next, for each stock, we compute the 3-day market-adjusted return centered on earnings announcements in years t-3, t-2,, t+4. Next, we compute the annual return that accrues over earnings announcements by compounding all 3-day stock returns in each year. We report the equally-weighted average annual return during earnings announcements for the highest (Hi) and lowest (Lo) LTG deciles. Excess returns are defined relative to the equally-weighted CRSP market portfolio. 45

46 Figure 10. Cumulative Abnormal Returns for LTG Portfolios. In December of each year t between 1983 and 2014, we form decile portfolios based on ranked analysts' expected growth in earnings per share. Then, for each stock and each quarter q during the first postformation year (t+1), we compute the cumulative abnormal return (CAR) during the period that starts 2 days after the earnings announcement for the quarter preceding q and ends one day after the earnings announcement for quarter q; abnormal returns are defined relative to the equally-weighted CRSP market portfolio. Next, we define earnings surprises (SUEs) as (eps q -E[eps q ])/p, where eps q is the realized value of earnings per share in quarter q, E[eps q ] denotes the mean analysts forecast for eps q made at the end of quarter q, and p is the stock price also at the end of the quarter q. Each quarter, we rank SUEs into deciles. Finally, for each SUE and LTG decile, we compute the equally-weighted CAR for each quarter and plot its time-series mean value against the median SUE for both the highest (Hi) and lowest (Lo) LTG deciles. 46