Skewness of Firm Fundamentals, Firm Growth, and Cross-Sectional Stock Returns

Transcription

1 Skewness of Firm Fundamentals, Firm Growth, and Cross-Sectional Stock Returns Yuecheng Jia Shu Yan December 2015 Abstract We present a novel interpretation of the conditional sample skewness of firm fundamentals as a proxy of the expected growth rate of firm cash flows and therefore being positively related to the expected stock return. Empirically, we document significant evidence that the skewness of firm fundamentals positively predicts cross-sectional stock returns and future firm growth. Our findings cannot be explained by existing predictors and risk factors. The evidence further indicates that the alternative skewness measures of firm fundamentals are proxies of different factors driving firm cash flows. We would like to thank Yuzhao Zhang for helpful comments. All errors are our own. Both Jia and Yan are at Department of Finance, Spears School of Business, Oklahoma State University, Stillwater, OK 74078, USA. Phone: (Yan). Fax: Please address correspondence to yuecheng.jia@okstate.edu or yanshu@okstate.edu.

2 Skewness of Firm Fundamentals, Firm Growth, and Cross-Sectional Stock Returns Abstract We present a novel interpretation of the conditional sample skewness of firm fundamentals as a proxy of the expected growth rate of firm cash flows and therefore being positively related to the expected stock return. Empirically, we document significant evidence that the skewness of firm fundamentals positively predicts cross-sectional stock returns and future firm growth. Our findings cannot be explained by existing predictors and risk factors. The evidence further indicates that the alternative skewness measures of firm fundamentals are proxies of different factors driving firm cash flows. JEL Classification: G12 Keyword: Skewness, fundamental, growth, stock return, earnings, profitability

3 1 Introduction Fama and French (2006, 2008) point out that most stock return anomalies, no matter whether they are rational or irrational, are consistent with the basic stock valuation equation, which is a mathematical identity that relates firm cash flows and stock returns (e.g., Campbell and Shiller (1988) and Vuolteenaho (2002)). According to the equation, higher expected growth rate of future cash flows implies higher expected stock return if the book-to-market ratio is fixed. 1 Fama and French argue that the anomaly variables such as book-to-market ratio, net stock issuance, accruals, lagged stock return, asset growth, and profitability are proxies of expected cash flows. One notable feature of the aforementioned return predictors is that they are the first moments of firm fundamentals or stock returns. In this paper, we propose the conditional sample skewness of firm fundamentals as a new proxy of expected growth rate of cash flows. We demonstrate analytically and numerically that, for very general data-generating specifications, the conditional sample skewness of firm cash flows is positively correlated with the expected growth rate of the underlying process. 2 Based on the basic stock valuation equation, this interpretation of skewness of firm fundamentals implies our main testable hypothesis: The conditional sample skewness of firm fundamentals positively predicts future stock returns. Beyond the first moments, researchers have examined whether the second moments of firm fundamentals can predict stock returns and firm performance (e.g., Diether, Malloy, and Scherbina (2002), Johnson (2004), Dichev and Tang (2009), and Gow and Taylor (2009)). 3 To our knowledge, our paper is the first to examine the return predictability of the skewness 1 This conclusion can also be drawn from the q-theory of investment (e.g., Hou, Xue, and Zhang (2014)). The valuation equation is a tautology and can be interpreted in alternative ways. 2 In addition, we show that a high value of the conditional sample skewness implies high acceleration rate and high level of the underlying process. Both effects strengthen the positive relation between the conditional sample skewness of firm fundamentals and future stock returns. 3 There is also a literature on return predictability by stock volatility, the second moment of stock returns. See, for example, Goyal and Santa-Clara (2003), Ang, Hodrick, Xing, and Zhang (2006), and Jiang, Xu, and Yao (2009). 1

4 of firm fundamentals. Using the conditional sample skewness of firm fundamentals to infer the expected growth rate of firm cash flows has a couple of advantages over other econometric approaches. First it is very easy to calculate. Second, the approach does not make any assumptions about the underlying data-generating mechanism and is robust to many alternative model specifications. Our paper also differs from the previous studies on the return predictability of stock return skewness in several ways. 4 First, our analysis is preference-free as we do not require investors to prefer positive skewness in stock returns. Second, in contrast to the negative relation between the return skewness and future stock returns documented in the literature, our model dictates a positive relation between the skewness of firm fundamentals and future stock returns. Third, as we will see later, incorporating the return skewness as a control variable in the empirical analysis does not change our results for the skewness of firm fundamentals. To test our hypothesis, we use three measures of cash flows: gross profitability of Novy- Marx (2013) (GP), earnings per share (EP S), and analyst earnings forecasts (AF ). 5 We denote the skewness of these three measures by SK GP, SK EP S, and SK AF. Unlike the first two skewness measures which are time-series estimates, SK AF is the cross-sectional skewness of analysts forecasts, and does not conform to our argument that the conditional sample skewness is a proxy of expected growth rate of the underlying cash flow process. To see that the return predictability of SK AF is consistent with that of SK GP and SK EP S, we draw support from the literature of stock analysts. 6 Both theory and empirical evidence in the 4 The literature on stock return (co)skewness dates back to the seminal work of Kraus and Litzenberger (1976). Recent studies include Harvey and Siddique (2000), Dittmar (2002), Barone-Adesi, Gagliardini, and Urga (2004), Chung, Johnson, and Schill (2006), Mitton and Vorkink (2007), Boyer, Mitton, and Vorkink (2010), Engle (2011), Chang, Christoffersen, and Jacobs (2013), Conrad, Dittmar, and Ghysels (2013), and Chabi-Yo, Leisen, and Renault (2014). 5 We have also considered alternative cash flow measures including ROE and earnings surprises. The results for the alternative measures are very similar and available upon request. 6 The important and relevant articles in this literature include Scharfstein and Stein (1990), Avery and Chevalier (1999), Hong, Kubik, and Solomon (2000), Clement and Tse (2005), Clarke and Subramanian (2006), and Evgeniou, Fang, Hogarth, and Karelaia (2013). 2

5 literature indicate that less skilled analysts tend to herd while more skilled analysts tend to be bolder and are likely to deviate from the consensus. Because skewness is driven by outliers, SK AF is likely to pick up the forecasts of more skilled analysts, which are supposed to be more accurate. Consequently, SK AF should be positively correlated with expected future earnings and predict future stock returns. Strongly supporting our hypothesis, all three skewness measures are positively significant in predicting cross-sectional stock returns. For example, when stocks are sorted on SK GP into decile portfolios, the equal-weighted average next-quarter return increases from decile 1 to decile 10. The H-L spread between deciles 10 and 1 is 1.55% per quarter and statistically significant. Value-weighting stock returns and adjusting average returns by the conventional risk factors do not change the results. The predictability evidence is corroborated by the Fama-MacBeth regressions, even in the presence of other return predictors including the level of GP. To strengthen our argument, we further test whether the skewness measures positively predict future firm cash flows, which are measured by ROE and GP. The evidence overwhelmingly supports our argument. All three skewness measures can significantly forecast future ROE and GP for up to 4 quarters. If we sort stocks on SK GP into decile portfolios, the next-quarter H-L spread in ROE is 1.20% and statistically significant. The next-quarter H-L spread in GP is much higher at 3.76%. We find similar evidence for SK EP S but weaker evidence for SK AF. These results suggest that the alternative skewness measures are related to different aspects of firm cash flows. We further examine the differences across the alternative skewness measures by comparing their return predictability. Only for the sample of stocks with valid observations of SK AF, the predictability of SK EP S is subsumed by that of SK AF. We don t find SK GP and SK AF to dominate each other. For the larger sample of stocks with valid observations of SK GP and SK EP S, neither skewness measure dominates the other. These findings further support the view that the alternative skewness measures are proxies of different factors of firm cash 3

6 flows. An alternative motivation of our main hypothesis is the positive impact of cash flow skewness on firm growth option. In the literature, a firm s growth options are treated as options written on the firm s cash flow process (e.g., Berk, Green, and Naik (1999) and Carlson, Fisher, and Giammarino (2004)). A standard option pricing argument illustrates that everything else fixed, higher skewness of firm cash flows leads to higher value of firm growth option. Following the approach of Bernardo, Chowdhry, and Goyal (2007), this implies higher stock risk and therefore higher stock return. We also test this argument with two proxies of growth option: market asset-to-book asset ratio and Tobin s q. We find strong evidence supporting this argument as well. Studies of higher moments of firm fundamentals or macroeconomic variables are scarce. Colacito, Ghysels, and Meng (2013) show that the skewness of forecasts on the GDP growth rate made by professional forecasters can predict stock market return. In a separate study, we consider the skewness of aggregate stock market and find that it can predict stock market return. The current paper differs from these two studies in a couple of ways. First, we consider the skewness of firm fundamentals and individual stock returns. Second but more importantly, we do not make specific assumptions on the data-generating processes and investors utility functions as our approach is based on the basic stock valuation equation. Also related to our paper, Scherbina (2008) constructs a non-parametric skewness measure of analysts earnings forecasts as the difference of median and mean forecasts and finds that it negatively predicts future stock returns, opposite to the positive relation we find for SK AF. 7 In addition to the methods of measuring skewness of analyst forecasts, there are a couple of important differences between our papers. First, Scherbina s evidence is weak for large firms while our evidence is very strong for large firms. Second, she argues that her findings are caused by negative information withheld by analysts being incorporated in stock 7 The non-parametric skewness measure is not an accurate proxy of population skewness for many probability distributions. One such example is a bi-modal distribution where the non-parametric skewness measure can have the wrong sign. 4

7 prices with a significant delay while we argue that more skilled analysts are more likely to issue bolder (but more accurate) forecasts. The rest of the paper is organized as following. In Section 2, we show why the conditional sample skewness of firm fundamentals contains information on the firm cash flows and as a result positively predicts stock returns. We describe the data and econometric methodology in Section 3. Section 4 discusses the empirical evidence. Section 5 concludes. 2 Information Content of Conditional Sample Skewness of Firm Fundamentals In this section, we first demonstrate a novel approach to infer sampling properties of general time-series processes from the conditional sample skewness. Then we apply the results to firm fundamentals, and derive the relation between the conditional sample skewness of firm fundamentals and expected stock return. 2.1 Conditional Skewness of Small Samples To estimate the conditional skewness of a time-series process at time t using the past sample of size n, {x t n+1,..., x t }, the standard formula is 1 n m 3 ˆb = s = n i=1 (x t n+i x) 3 3 [ 1 n i=1 (x t n+i x) 2], (1) 3/2 n 1 where x is the sample mean, s is the sample standard deviation, and m 3 is the sample third central moment. Alternative formulas can be used but they do not affect our results. We show next that the estimated ˆb is informative about the order of the sample observations of the change of x. Define the change of x as x t = x t x t 1. For the ease of presentation, assume x t n = 0. Using x t = n i=1 x t n+i, we can express the first three 5

8 sample moments as x = 1 n s 2 = = 1 n n i=1 i j=1 x t n+j n (n i + 1) x t n+i, (2) i=1 1 n 1 ( n i x t n+j x i=1 j=1 ) 2 ( 1 n i j 1 = n 1 n x t n+j i=1 j=1 ( m 3 = 1 n i ) 3 x t n+j x n = 1 n i=1 j=1 ( n i i=1 j=1 j 1 n x t n+j n j=i+1 n j=i+1 ( 1 j 1 ) ) 2 x t n+j, (3) n ( 1 j 1 ) ) 3 x t n+j. (4) n In the sample mean x, earlier observations of x t n+i are clearly over-weighed than later observations. To see how the location of an observation affects its weight in s 2 and m 3, we consider two examples. For n = 3, simple calculations show s 2 = 2 3 ( ) x x 2 x 3 + x 2 3, m 3 = 1 9 ( x 3 x 2 ) ( 2 x x 2 x x 2 3). In this case, s 2 is symmetric with respect to x 2 and x 3 while m 3 is monotonically increasing in x 3 x 2. For n = 4, we can write s 2 = 1 4 ( 3 x x x x 2 x x 2 x x 3 x 4 ), m 3 = 3 8 ( x 4 x 2 ) ( x x x 2 x x 2 x x 3 x 4 ). In this case, s 2 is symmetric with respect to x 2 and x 4. m 3 is monotonically increasing in x 4 x 2 if the second part of m 3 is positive, which is the case when x 3 = 0. These two 6

9 examples suggest that the sign and magnitude of m 3 depend on the order of observations { x i } while s 2 is not. This seems intuitive from the construction of s 2 and m 3. The second moment is insensitive to whether an observation occurs early or late in the sample but the third moment may over-weighs/under-weighs an observation depending on its location in the sample. Taken together, a high value of ˆb seems to suggest relatively high (low) values for more recent (earlier) observations of x t. It is messy to extend the above examples to general settings without specifying the underlying data-generating process. In the following, we consider the class of AR(1) processes x t = ρx t 1 + u t, (5) where ρ 1 is a constant and u t is an iid standard white noise process. Note that x t is a random walk when ρ = 1. The initial value x 0 is set to be zero for simplicity. There is no constant term on the right-hand side although including one does not change the results. Instead of providing analytical proofs, we conduct the following numerical analysis. To be consistent with our later empirical work, we consider n = 8, 12, 16, and 20 and ρ = 0.9, 0.95, and 1. 8 To take into account of sampling errors, we use the Monte Carlo simulation method to examine the correlations between the conditional sample skewness and cross-sections of sample observations of x t. The steps are detailed as following. Step 1: For fixed n and ρ, independently generate N = 1, 000, 000 paths of x t according to equation (5). Denote the observations of the ith path by {x it } n t=0. Step 2: For the ith path, compute the sample skewness ˆb i. 8 The small sample sizes are appropriate when we consider low-frequency financial accounting data such as the quarterly earnings. Using larger sample sizes to estimate the conditional skewness is problematic if the underlying data-generating mechanism is time-varying and non-stable. The near-unit-root or unit-root specification for x is also reasonable as most financial accounting variables are highly persistent. Negative values may arise in the AR(1) process, which are undesirable since most accounting variables are nonnegative. This problem can be solved by taking logarithm. In most practical cases, the results are not affected by the log transformation. 7

10 Step 3: For each value of t = 2,..., n, compute the correlation of ˆb i and x it across the N sample paths and denote it by c(t). Figure 1 shows the plots of c(t) as a function of t for different values of n and ρ. Several patterns are important to note. First, for every (n, ρ) pair, the value of c(t) is negative during the first half of the sample but positive during the second half of the sample. Second, c(t) is monotonically increasing in t for the cases of n = 8, from less than 0.2 to over 0.2 when ρ = 1. For the cases of n = 12, 16, 20, c(t) is monotonically increasing except for the two ends of the sample. The minimum and maximum of c(t) still occur near the beginning and ending of the sample, respectively. Third, when n is fixed, the increasing pattern of c(t) becomes more significant as ρ increases to 1. Fourth, when ρ is fixed, the shape of c(t) becomes flatter as n increases. The minimum and maximum of c(t) are located further away from the first and last observations. These correlation patterns of c(t) are not sensitive to the iid assumption for u t as we have checked various heteroskedastic specifications for u t. We have also considered numerous alternative ARMA(p,q) specifications for x t and find qualitatively similar results. The numerical results confirm our conjecture that the conditional sample skewness ˆb is informative about the order of observations of x t at least for small sample size up to 20. A high value of ˆb suggests that the recent growth rates are likely high while the earlier growth rates are likely low. A low value of ˆb suggests the opposite. Moreover, ˆb is not only an indicator of the current growth rate of x t but also positively associated with the acceleration rate (change of growth rate) of x t. In addition to the growth rate of x t, it is also interesting to examine the correlations of ˆb with the levels of x t. To this end, we use the same Step 1 an Step 2 of the previous analysis. But in the new Step 3, we calculate the cross-sectional correlations of ˆb i with x it (denoted by C(t)) instead of x it. Figure 2 shows the plots of C(t). We make several comments. First, C(t) is mostly negative except when t is close to the sample end n. Second, C(t) is initially decreasing but becomes increasing as t increases, with the maximum attained at 8

11 t = n. The near-convexity of C(t) is consistent with the near-monotonicity of c(t) shown in Figure 1. Third, with fixed n, C(t) becomes smaller when ρ increases. Fourth, with fixed ρ, C(t) becomes larger when n increases. Overall, the most important finding is that a higher value of ˆb is more likely associated with a higher value of the last sample observation than past observations and the effect is stronger for lower ρ and higher n. Having seen how ˆb is related to the past growth rates of x, an important question is: What does ˆb tell us about the expected future growth rate of x. If x t is iid over time, the above results are not useful for prediction purpose because knowing ˆb and therefore the order of the past observations of x t does not provide additional information about future x t. In real financial data, however, x t is often non-iid, and ˆb can be informative about the expected growth of x. As an example, consider the following process x t = u t + ε t (6) u t = µ + θu t 1 + e t (7) where µ and 0 < θ < 1 are constants, and ε t and e t are iid standard white noise processes. In this model, u t is the expected growth rate of x and follows an AR(1) process which is unobserved. A high value of x t implies a high value of u t and consequently higher future growths of x due to the persistence of the growth rate process. Estimates and predictions of this type of models can be obtained using methods such as the Kalman filters. But there are some serious issues with the parametric approach. First, accurate estimates of such models require long time-series data, which are not available. Second, the models are not stable over time. This can happen, for example, when there are structural breaks in the underlying data-generating process. Third, the models are likely misspecified. Alternative ARMA specifications or regime-switching models can provide similar fit of the same data. Using the conditional sample skewness ˆb to imply the expected growth rate of x circum- 9

12 vents these problems. It doesn t need long time series to estimate. More importantly, it doesn t rely on any parametric models. It allows many different types of model specifications. As long as the growth rate of x is positively autocorrelated to a certain extent, a high value of ˆb implies that the future growth rate of x is also likely high. The same argument also implies potentially high acceleration and level of x when ˆb is high. 2.2 Skewness of Firm Fundamentals and Stock Returns We now apply the previous results for general time series to firm fundamentals. Let A t denote a measure of the fundamental cash flows of a firm at time t. We do not impose any parametric specifications on A t other than that it satisfies certain time-series properties so that, as shown in the last section, the conditional sample skewness is positively related to the growth rate, acceleration, and level of A t. Suppose that we have calculated ˆb using the historical data of size n 20, {A t n+1,..., A t }. As we will argue next, there are three channels through which ˆb positively predicts future stock returns. First, we have demonstrated that a high value of ˆb implies that the recent growth rates of A are likely high. If the growth rate of A is persistent, then the expected future growth rate of A will also be high. According to the basic stock valuation equation, higher expected growth rate of firm cash flows implies higher stock returns. Consequently, ˆb positively predicts future stock returns. Second, we have also shown previously that a high value of ˆb implies positive acceleration of A, in other words, increasing growth rate of A during the recent sample period. If the positive acceleration persists into the future, then the expected future growth rate of A will be even higher than that implied by the previous point. This second-order effect strengthens the positive return predictability of ˆb. Lastly, recall that a high value of ˆb also implies that the current level of A is potentially high. To see how this fact leads to higher future stock returns, we resort to the literature of firm growth options. In this literature, the firm value is decomposed into two parts: firm 10

13 fundamental cash flows and growth option. We follow the approach of Bernardo, Chowdhry, and Goyal (2007) by treating the growth option as a call option on A. It is well-known that the call option value is an increasing function of the underlying process. Hence a high value of ˆb likely causes the value of firm growth option to increase, which leads to more weight of the growth option in the firm total value. As demonstrated in Bernardo, Chowdhry, and Goyal (2007), the risk and therefore return of the growth option are higher than those of the firm cash flows. Combining the above results leads to the third channel that ˆb positively predicts future stock returns. The effects of all three channels are in the same direction regardless which of them is at work. We therefore conclude that the conditional sample skewness of firm fundamentals positively predicts future stock returns. 3 Data and Methodology In this section, we first show the definitions of skewness measures of firm fundamentals. We then describe the data. Finally, we discuss our econometric methods. 3.1 Definition of Skewness Measures In quarter t, we follow Gu and Wu (2003) to define skewness of GP and EP S as the standard skewness coefficient of lagged observations during the rolling window of quarters t n to t 1: SK GP,t = SK EP S,t = n (n 1)(n 2) n (n 1)(n 2) t 1 τ=t n τ=t n ( ) 3 GPτ µ GP, (8) s GP t 1 ( ) 3 EP Sτ µ EP S, (9) s EP S where µ GP (µ EP S ) and s GP (s EP S ) are, respectively, the sample average and standard deviation of GP (EP S) for the rolling window. In the benchmark case reported in the paper, we 11

14 fix n = 8. The results for n up to 20 are very similar and available upon request. Note that we don t use the quarter t GP and EP S in the definitions because they not reported until quarter t + 1. When examining whether the skewness of earnings skewness up to quarter t predicts the stock returns in quarter t + 1, using future information that is available in quarter t + 1 but not in quarter t biases the statistical inference. We in fact have conducted (unreported) empirical analysis without skipping quarter t and have found even stronger (but biased) results. To define the skewness of analysts forecasts, we follow Diether, Malloy, and Scherbina (2002) to use the most recent valid analysts annual earning forecasts by the end of quarter t. 9 We restrict our sample to stocks with at least three analyst forecasts in a quarter so that skewness is well defined. The skewness across analysts is then SK AF,t = N (N 1)(N 2) N ( ) 3 epsj µ eps, (10) where eps j is the earnings forecast of the jth analyst, µ eps and s eps are, respectively, the sample average and standard deviation of all forecasts, and N is the number of analysts. Note that we don t skip quarter t as for the first two skewness measures of firm fundamentals j=1 because analyst forecasts are known before quarter t + 1. s eps 3.2 Data Stock return and accounting data are obtained from CRSP and COMPUSTAT. The I/B/E/S provides analysts earnings forecasts. We consider all NYSE, AMEX and NASDAQ firms in the CRSP monthly stock return files up to December, 2013 except financial stocks (four digit SIC codes between 6000 and 6999) and stocks with share price less than $5. To be 9 Our measure is updated every quarter while Diether, Malloy, and Scherbina (2002) construct their dispersion in analysts forecasts at monthly frequency. We have also considered dropping the forecasts made in the last five days of quarter t to make sure all forecasts are observed. The results are almost identical to those reported in paper. 12

15 included in the sample of SK GP or SK EP S, we require a firm to have at least 16 quarters of gross profitability or earnings data during The construction of each observation of skewness measure needs observations of 8 consecutive quarters. We begin portfolio sorting and regression analysis for SK GP and SK EP S from Q1, 1973, and for SK AF from Q1, For each quarter, the accounting variables are defined as follows. GP : Following Novy-Marx (2013), gross profitability is quarterly revenues minus quarterly cost of goods sold scaled by quarterly asset total. EP S: Quarterly earnings per share before extraordinary items. M C: Market capitalization is the quarter-end shares outstanding multiplied by the stock price. BM: Book-to-market ratio is the ratio of quarterly book equity to quarter-end market capitalization. Quarterly book equity is constructed by following Hou, Xue, and Zhang (2014) (footnote 9), which is basically a quarterly version of book equity of Davis, Fama, and French (2000). ROE: Return on equity is defined as income before extraordinary items (IBQ) divided by 1-quarter-lagged book equity. M ABA: Market asset-to-book asset ratio is defined as [Total Asset Total Book Common Equity+Market Equity]/Total Assets. Tobin s q: It is defined as [Market Equity+Preferred Stock+Current Liabilities Current Assets Total+Long Term Debt]/Total Assets. Disp: Dispersion of analysts forecasts is defined in almost the same way as Diether, Malloy, and Scherbina (2002) using the detail history file of I/B/E/S. The only difference is that we use quarterly frequency instead of monthly frequency. If an analyst makes more than one forecast in a given quarter, only the most recent forecast is used in the calculation. 13

16 Firm size and book-to-market ratio are standard control variables in asset pricing studies. ROE is a popular measure of firm cash flows and has been shown to predict stock returns (e.g., Hou, Xue, and Zhang (2014)). We follow Cao, Simin, and Zhao (2008) to use MABA and Tobin s q as proxies of firm growth options. We have also considered the dispersion in analysts forecasts defined in Diether, Malloy, and Scherbina (2002) as a control variable when we examine the return predictability of SK AF. But controlling the dispersion does not change the results. So we do not include it in the paper. The variables related to stock returns are defined in the following. MOM: Momentum for month t is defined as the cumulative return between months t 6 and t 1. We follow the convention in the literature by skipping month t when MOM is used to predict returns in month t + 1. We have also used the cumulative return between months t 11 and t 1 and obtained similar results. Idvol: Idiosyncratic volatility is, following Jiang, Xu, and Yao (2009), the standard deviation of the residuals of the Fama and French (1993) 3-factor model using daily returns in the quarter. Idskew: Following Harvey and Siddique (2000) and Bali, Cakici, and Whitelaw (2011), it is defined as the skewness of the regression residuals of the market model augmented by the squared market excess return. We use daily returns in the quarter to estimate the regression. P rskew: It is predicted idiosyncratic skewness defined in Boyer, Mitton, and Vorkink (2010). We obtain the P rskew data from Brian Boyer s website. M AX: Following Bali, Cakici, and Whitelaw (2011), it is the average of the two highest daily returns in quarter t. Note that we use quarterly frequency instead of monthly frequency. 14

17 We use Iddvol as a control because a number of studies have documented that it predicts returns. The skewness measures of stock returns, Idskew, P rskew, and M AX are good controls to evaluate additional return explanatory power of skewness of firm fundamentals. We have also considered total return skewness of daily stock returns in the quarter and obtained similar results. We winsorize all the variables at 1% and 99% levels although the results do not change significantly without winsorizing or winsorizing at 0.5% and 99.5% levels. 3.3 Econometric Methods We rely on the portfolio sorts and cross-sectional regressions of Fama and MacBeth (1973) for our empirical investigation. For single portfolio sorts, we rank stocks on a skewness measure of firm fundamentals into decile portfolios and then consider both equally-weighted and value-weighted portfolio returns. If the skewness is positively related to stock returns, we expect an increasing pattern of portfolio returns from decile 1 to decile 10. For double portfolio sorts, we first rank stocks into quintiles by a control variable such as MC and then further sort stocks within each portfolio into quintiles by the skewness measure. If the control variable can explain the predictability of skewness, we expect the increasing pattern of returns in skewness to be much less significant in each quintile of the control variable. To compute t-statistics of average portfolio returns, we use the Newey-West adjusted standard errors because of the persistence in the portfolios. For the Fama-MacBeth regressions, we expect the estimated average coefficient of the skewness measure to be positive and significant. The cross-sectional regressions allow us to estimate the marginal effect of the skewness measure when controlling for other variables known to predict stock returns. In the most general specification, we include all the control variables in the regression. If the skewness measure captures information about expected stock returns beyond that in other variables, the coefficient of the skewness measure should be significant even in the presence of all the control variables. 15

18 We also use the Fama-MacBeth regression approach to compare the explanatory power of different skewness measures. To do so, we include two or three skewness measures in one regression. If the coefficient of one skewness measure is no longer significant in the presence of another skewness measure, it indicates that the later skewness measure dominates the first measure in the sense that it subsumes all the explanatory power of the first measure. 4 Empirical Evidence 4.1 Data Descriptions The data of skewness measures are unbalanced. There are 350,050, 384,402, and 162,782 firm-quarter observations for SK GP, SK EP S, and SK AF, respectively. The sample size for SK AF is less than half of those for the other two measures because not only the sample period for the I/B/E/S data is shorter but also many small firms do not have enough analyst coverage to compute skewness. Table 1 reports the average contemporaneous cross-sectional correlations of quarterly skewness measures and some control variables. Amongst the three skewness measures, SK GP and SK EP S are mildly correlated with correlation coefficient of The forward-looking measure, SK AF, is essentially uncorrelated with the other two time-series measures. The results indicate that different skewness measures have different information contents about firm cash flows. SK GP is mildly correlated with MOM and level of GP but uncorrelated with other controls. SK EP S seems to be slightly correlated with the control variables but all correlation coefficients are below In contrast, the forward-looking skewness, SK AF, is uncorrelated with all the control variables. 16

19 4.2 Single Portfolio Sorts Table 2 reports the average time-series returns and characteristics of the decile portfolios formed by sorting stocks on the three skewness measures. When sorted on SK GP as in panel A, the average equal-weighted quarterly return increases from decile 1 (2.99%) to decile 10 (4.54%). The average H-L spread between is 1.55% per quarter (or 6.20% per year) and highly significant (t = 5.67). To make sure that the significant H-L spread is not driven by higher stock risks, we estimate the risk-adjusted α using the four-factor model of Carhart (1997). The risk-adjusted H-L spread is even higher at 1.69% and significant. The results for value-weighted returns are very similar to those for equal-weighted returns, implying that the findings are not dominated by small stocks. Looking at the characteristics of the decile portfolios, low-sk GP stocks have low past return, GP, and ROE but slightly high book-to-market ratio and idiosyncratic volatility. One reason of these patterns in control variables is that low-sk GP stocks are past underperformers in terms of profitability. To make sure that the return predictability of SK GP is not a result of the firm characteristics, we will use double portfolio sorts and Fama-MacBeth regressions. The results of portfolios sorts on SK EP S in panel B are very close to those for SK GP. The unadjusted and adjusted H-L spreads for SK EP S are actually slightly higher than those for SK GP. The firm characteristics of the decile portfolios also exhibit similar patterns as those in panel A. Panel C shows the results for SK AF. The portfolio returns are generally increasing from decile 1 to decile 10. All average equal-weighted/value-weighted unadjusted/adjusted H-L spreads are higher than 1.26% per quarter and statistically significant. The numbers for SK AF are smaller and less significant than those for SK GP and SK EP S. Interestingly, we do not observe any significant patterns in firm characteristics. There are several reasons why the results for SK AF are different from those for SK GP and SK EP S. First, the sample period for SK AF is shorter. Second, the stocks for SK AF tend to be larger and better followed firms. 17

20 Third, as we see in Table 1, SK AF are uncorrelated with firm characteristics. Overall, in spite of the obvious differences across the three skewness measures, we find consistent positive predictive relation between skewness of firm fundamentals and future stock returns, confirming our hypothesis. The results are robust regardless whether the returns are equal-weighted or value-weighted, and unadjusted or risk-adjusted. 4.3 Double Portfolio Sorts We now investigate whether the predictability of the skewness measures are caused by firm characteristics. We use the double portfolio sort approach by first sorting stocks on firm characteristics and then sorting on the skewness measures. Table 3 reports the average equalweighted returns of double-sorted portfolios for the six characteristics reported in Table 2. The results for value-weighted returns are very similar and not shown for brevity. We have also considered a number of other control variables and their results are available upon requests. First, we consider the results for SK GP in panel A. When stocks are initially ranked by MC, the H-L spreads of skewness quintiles show a decreasing pattern from MC quintile 1 (2.51%) to MC quintile 5 (0.58%), suggesting that the predictability of SK GP is stronger for small stocks. In contrast, the predictability of SK GP is stronger for high momentum, GP, and idiosyncratic volatility stocks but there is no clear pattern for BM and ROE. No matter which firm characteristic is considered, all H-L spreads remain positive and most of them are statistically significant. The evidence indicates that the return predictive power of SK GP can not be explained the firm characteristics. The results for SK EP S in panel B are similar to those for SK GP with a few differences. The predictability of SK EP S is strong for low BM and high ROE stocks. The H-L spreads for GP quintiles exhibit a U-shape pattern. If any firm characteristic can explain SK EP S, it is ROE because only one of the five H-L spreads is significant. Some loss of statistical significance can be attributed to the higher standard errors due to smaller sample sizes. Close 18

21 inspection of the ROE quintiles reveals non-linear interactions among stock return, SK EP S, and ROE. We will get a clearer picture when we estimate Fama-MacBeth regressions with other control variables. Finally, we consider the results for SK AF in panel C. In all cases but one, the H-L spreads are positive, many of which are significant. The relations between H-L spreads and the firm characteristics are not generally linear. For example, the H-L spreads for BM quintiles are hump-shaped. In sum, no characteristic seems to be able to explain the predictive power of SK AF. 4.4 Fama-MacBeth Regressions We further examine the return predictability of the skewness measures with the Fama- MacBeth regressions, which allow us to control for other return predictors. The results are reported in Table 4. For each skewness measure, we estimate eight regressions. The first model uses a skewness as the only explanatory variable. Models (2)-(7) consider the six control variables, one at a time. Because of different sample sizes for different measures, we reestimate these models for each skewness measure. Model (8) includes the skewness measure and all six control variables. First, let us look at the results for SK GP in panel A. The average coefficient of SK GP in model (1) is positive and significant at the 1% level (0.24 and t = 6.18). Every control variables but MC is significant when it is used alone to forecast returns. In model (8) where all controls are incorporated, the average coefficient of SK GP is smaller in magnitude than that in model (1) but still significant at the 1% level (0.11 and t = 3.87). Interestingly, the average coefficient for MC is now significant at the 10% level and has the same negative sign as that documented in the literature. Next, as shown in panel B, the estimation results for SK EP S are very similar to those for SK GP. By itself, SK EP S positively predicts stock returns in model (1). Even when all controls are included in model (8), the average coefficient of SK EP S remains positive and 19

22 significant at the 5% level (0.09 and t = 2.37). Finally, panel C shows the results for SK AF. The average coefficient of SK AF in model (1) is 0.21 and significant at the 5% level (t = 2.60). Among the control variables, MOM, GP, and ROE are significant while MC, BM, and Idvol are insignificant in predicting stock returns by themselves. In model (8), the average coefficient of SK AF is even larger and more significant (0.28 and t = 3.31) than that in model (1). Among the control variables, GP and ROE are consistently significant return predictors in all three samples, confirming the evidence documented in Novy-Marx (2013) and Hou, Xue, and Zhang (2014). In sum, the results of Fama-MacBeth regressions are consistent with those of portfolio sorts. All three skewness measures of firm fundamentals positively predict stock returns. While in the presence of control variables some evidence is not as significant as in the portfolios sorts, the overall return predictability by the skewness measures cannot be explained by other predictors. 4.5 Skewness and Future Firm Profitability We now test the implication of our model that the skewness of firm fundamentals is positively related to future profitability or growth of firm cash flows. We proxy growth rate by ROE and GP. We look forward for four quarters. We use both portfolio sorts and Fama-MacBeth regressions to examine the issue. Table 5 reports the average equal-weighted future ROE and GP of decile portfolios formed by sorting stocks on the skewness measures. Value-weighted results are very similar and not reported for brevity. The results of panel A for SK GP indicate that high-skewness stocks have higher profitability in future four quarters. The H-L spreads of both ROE and GP are positive and significant at the 1% level for all four quarters. The H-L spreads decline gradually as horizon increases, suggesting mean reversion in firm performance. Panel B shows the results for SK EP S. The results are similar to those for SK GP as stocks with high values of SK EP S have higher profitability in terms of ROE and GP in the 20

23 futures. There are a couple of small differences. First, the H-L spreads in ROE in panel B are larger than those in panel A. Second, the H-L spreads in GP in panel B are smaller than those in panel A. These results are not surprising as the skewness of earnings should be more significant in predicting ROE than the skewness of GP while the opposite is true for the skewness of GP. The results for SK AF shown in panel C are in start contrast to those for SK GP and SK EP S. Although the H-L spreads in ROE and GP are all positive and significant at the 10% level, they are much smaller than those in panels A and B. Looking closely, the positive H-L spread are mostly driven by the high profitability of decile 10. There is no increasing pattern in ROE or GP from decile 1 to decile 10. The results of the portfolio sorts strongly support our argument for SK GP and SK EP S but the evidence for SK AF is weak. We reexamine the above evidence using the Fama-MacBeth regressions and report the regression estimates in Table 6. We only show the results where the dependent variable is the next-quarter ROE and GP as the estimates for long horizons up to four quarter are very similar. The regression results confirm what we have found with the portfolio sorts: the skewness measures positively predicts future profitability. The results are robust to inclusion of control variables. Interestingly, SK AF becomes more significant in predicting ROE and GP when the control variables are included. Overall the evidence in this section confirms our hypothesis that the skewness measures forecast future profitability. 4.6 Skewness and Future Growth Option We have argued for a second channel through which the skewness of firm fundamentals is positively related to expected stock return: skewness is positively related to the growth option. We test this argument using two measures of growth option, MABA and Tobin s q. Again we conduct both portfolio sorts and Fama-MacBeth regressions. Table 7 reports the average equal-weighted future MABA and Tobin s q up to four quarters of decile portfolios formed by sorting stocks on the skewness measures. The results 21

24 of portfolio sorts generally support our argument that higher skewness implies higher growth option. The evidence is very strong for SK GP and SK EP S as the H-L spreads are all positive and significant at the 1% level for all four future quarters and both growth option proxies. The evidence for SK AF is weaker but is still consistent with that for SK GP and SK EP S. The H-L spreads are much smaller than those for SK GP and SK EP S albeit still positive and marginally significant in most cases, particularly in the short run. In Table 8, we present the estimates of Fama-MacBeth regressions where the dependent variable is the next-quarter MABA or Tobin s q. The results for the two proxies of growth options are very similar. When a skewness measure is the only predictor, its estimated coefficient is positive. The estimates for SK GP and SK EP S are significant at the 1% level while the estimate for SK AF is significant at the 10% level. When all control variables including the lagged value of the growth option proxy are incorporated, the coefficients on the skewness measures are much smaller and less significant. The estimates for SK GP and SK EP S are still significant at the 10% level but the estimate for SK AF becomes insignificant. Taken together with the evidence of portfolio sorts, these results are consistent with our hypothesis that high skewness of firm fundamentals implies high growth option. But this effect seems less significant than that on the growth rate of cash flows shown earlier. 4.7 Controlling for Skewness of Stock Returns One concern about our main findings so far is whether the return predictability of the skewness of firm fundamentals is related to the return predictability of the skewness of stock returns. We address this concern by incorporating three popular return skewness measures (M ax, Idskew, and P rskew) in the Fama-MacBeth regressions of the fundamental skewness measures. Table 9 reports the estimation results of the Fama-MacBeth regressions. In models (1) (3), we only use one of the three return skewness measures. MAX and P rskew are significant but Idskew is insignificant in predicting returns. However, the sign of average coefficient for MAX changes signs for different samples. Model (4) use all three 22

25 measures, Max, Idskew, and P rskew. MAX and Idskew are significant in the sample of SK GP while P rskew is significant in the samples of SK EP S and SK AF. For our samples, the return skewness measures do not appear to consistently predict stock returns. We next combine the skewness of fundamentals with the return skewness measures. In model (4), we only include one skewness measure of fundamentals. The average coefficients of all three skewness measures of fundamentals are significant at the 1% level but among the skewness measures of returns only MAX is significant at the 1% level for the SK GP sample and P rskew is significant at the 10% level in the SK EP S and SK AF samples. Model (5) augments model (4) by including all the control variables that we used in Table 3. MAX and P rskew are still significant in some cases. Most importantly, the estimates for all three skewness measures of fundamentals are significant at the 1% level even when all the control variables are incorporated. The evidence of this section indicates that our findings for SK GP, SK EP S, and SK AF can not be explained by the skewness measures of stock returns. 4.8 Comparison of Alternative Skewness Measures Given the different constructions of the skewness measures, it is interesting to investigate their relative return predictive power. To do this, we estimate Fama-MacBeth regressions with multiple skewness measures as explanatory variables. The estimation results are reported in Table 10. In models (1) (3), we compare one skewness measure against another. Model (4) includes all three measures. There is no control variables in models (1) (4). We include all control variables in model (5) together with all three skewness measures. The estimated coefficients of the control variables are not shown for brevity. In model (1), the average coefficients of SK GP and SK EP S are both positive (0.19 and 0.13) and significant at the 5% level, indicating that the two time-series skewness measures do not dominate each other. In model (2), the average coefficient of SK AF is significant but the average coefficient of SK EP S is insignificant. This suggests that for the sample of 23

26 stocks with valid SK AF, the return predictability of SK EP S is subsumed by that of SK AF. In contrast, the estimates of model (3) show that both SK GP and SK AF are significant in positively predicting returns when they are considered together. The estimates of model (4) are consistent with those of the first three models as SK GP and SK AF are significant but SK EP S is no longer significant. Finally, when all the control variables are incorporated in model (5), the average coefficients of SK GP and SK AF remain significant at the 1% level and the average coefficient of SK EP S is insignificant. Despite the loss of significance in the joint regressions above, we should not dismiss the return predictability of SK EP S. One reason is that the sample of stocks with valid SK AF is much smaller than that for SK EP S. Furthermore, as seen in the estimates of model (1), for the larger sample of stocks with valid SK GP and SK EP S, the predictability of SK EP S is not subsumed by that of SK GP. In summary, the evidence suggests that the alternative skewness measures capture different factors driving the firm cash flows. 5 Conclusions We present a novel interpretation of sample skewness of a general time series as a proxy of the growth rate of the underlying process. Applying this interpretation to the firm cash flows together with the basic stock valuation equation, we posit a positive relation between the skewness of firm fundamentals and expected stock return. Using three skewness measures of firm fundamentals based on gross profitability, earnings, and analysts earnings forecasts, we find strong evidence supporting our hypothesis. The skewness measures positively predict not only cross-sectional stock returns but also future firm performance in terms of growth rate and growth option. The evidence is robust to control variables including the skewness of stock returns. The evidence also suggests that the alternative skewness measures capture different factors of the underlying firm cash flow process. 24