Expected Idiosyncratic Skewness

Transcription

1 Expected Idiosyncratic Skewness BrianBoyer,ToddMitton,andKeithVorkink 1 Brigham Young University December 7, We appreciate the helpful comments of Andrew Ang, Steven Thorley, and seminar participants at Brigham Young University. We acknowledge financial support from the Harold F. and Madelyn Ruth Silver Fund, Intel Corporation, and our Ford research fellowships. We thank Greg Adams for research support. All authors are from the Marriott School of Management, Brigham Young University, Provo, UT Boyer: , bhb@byu.edu. Mitton: , tm@byu.edu. Vorkink: , keith_vorkink@byu.edu.

2 ABSTRACT A number of recent theories predict a pricing premium for stocks with high idiosyncratic skewness. In this paper, we empirically investigate the relation between idiosyncratic skewness and expected returns. Because lagged skewness alone does not adequately forecast skewness, we estimate a model of expected skewness that employs other predictive variables including, importantly, idiosyncratic volatility. Consistent with these recent theories, we find that the Fama-French alpha of the low-expected-skewness quintile exceeds the alpha of the high-expected-skewness quintile by 1.00% per month. We also find that the coefficients on predicted skewness in Fama-McBeth cross sectional regressions are significant and in the expected direction. We also observe that both the cross-sectional distribution of skewness and the cross-sectional pricing of skewness vary over time and appear to follow episodic behavior. Last, we find that expected skewness helps resolve the idiosyncratic volatility puzzle of Ang et al. (2006).

3 I. Introduction Alonghistoryoffinance research investigates the impact of return skewness on investor decision-making. Arditti (1967) and Scott and Horvath (1980) show that investors demonstrate a preference for positive skewness in return distributions. Building on these results, Kraus and Litzenberger (1976) and Harvey and Siddique (2000) show that an asset s coskewness with the market portfolio should be priced under conditions of no arbitrage. Inherent in these findings is the assumption that although fully diversified investors may care about coskewness, a stock s idiosyncratic skewness should be irrelevant. However, others have noted that because diversification erodes skewness exposure, investors may remain underdiversified in order to capture return skewness, and thus idiosyncratic skewness may be relevant [see Simkowitz and Beedles (1978), Conine and Tamarkin (1981)]. Several recent theories although starting from different sets of assumptions all concur that idiosyncratic skewness can be a priced component of stock returns. Mitton and Vorkink (2007), in a model incorporating heterogeneous investor preference for skewness, predict a pricing premium for stocks with idiosyncratic skewness. Barberis and Huang (2007) show that when investors have cumulative prospect theory preferences, stocks with greater idiosyncratic skewness may command a pricing premium. Brunnermeier and Parker (2005), and Brunnermeier, Gollier, and Parker (2007) solve an endogenous probabilities model that produces qualitatively similar asset pricing implications for skewness as Mitton and Vorkink (2007) and Barberis and Huang (2007). In this paper, we empirically investigate the pricing implications of idiosyncratic skewness. Despite the theoretical basis for the pricing effects of skewness preference, empirically testing the relationship between idiosyncratic skewness and returns is not a straightforward exercise. The primary obstacle is that ex ante skewness is difficult to measure. As opposed to variances and covariances, idiosyncratic skewness is not stable over time [see Harvey and Siddique (1999)], so variables other than lagged skewness may be required in order to effectively measure expected skewness. We follow the approach of Chen, Hong, and Stein (2001) in using a number of firm-level variables to predict idiosyncratic skewness. We find that, although lagged skewness is an important predictor of future skewness, other firm characteristics are more 1

4 important predictors of idiosyncratic skewness, including idiosyncratic volatility, a measure that has generated considerable interest in asset pricing. Other important predictive variables in our model include momentum and turnover [see Hong and Stein (2003), and Chen, Hong, and Stein (2001)], firm size, and industry designation. 1 Our predictive model also produces interesting variation in expected skewness both across stocks as well as over time. The time series variation in both realized and expected skewness appears to follow episodic behavior similar to observations on idiosyncratic volatility [see Campbell, Lettau, Malkiel, and Xu (2001) and Brandt, Brav, and Graham (2005)]. Using our model, we find that expected idiosyncratic skewness has significant cross sectional pricing effects. skewness. We first sort firms into quintiles based on their level of predicted We find that the average returns for the low-expected-skewness quintile exceed the average returns of the high-expected-skewness quintile by 0.67% per month. After adjusting for risk, the differences in returns are even more pronounced. We find that the Fama-French alpha of the low-expected-skewness quintile exceeds the Fama-French alpha of the high expected-skewness quintile by 1.00% per month. We then confirm the pricing effects of idiosyncratic skewness by estimating Fama-MacBeth regressions. negative relation between predicted idiosyncratic skewness and average returns. We find a strong This result is statistically significant, robust in a number of alternative specifications, and tends to have a stronger effect on returns than idiosyncratic volatility. We compare the time series variation in expected skewness to the pricing of skewness in the cross section and find that periods in which the cross-sectional estimate of expected skewness is most significant and negative correspond to periods of high dispersion in skewness as well as periods where our model of predicted skewness fits the data well (high R 2 ). interpret these relationships as supportive of speculative episodes in the pricing of skewness [see Kapadia (2006)]. Having documented pricing implications of idiosyncratic skewness, we then turn to the question of whether skewness can help explain the phenomenon that stocks with high idiosyncratic volatility have low expected returns. We The negative relationship between idiosyncratic 1 The inclusion of industry dummy variables in our skewness prediction regressions is similar in spirit to the analysis of Zhang (2005), who computes firm-specific skewness based on the cross-sectional skewness of firms in the same industry. 2

5 volatility and future returns, which we refer to as the IV puzzle, is documented in U.S. data in Ang et al. (2006), and shown to be present in international markets in Ang et al. (2007). The IV puzzle has generated substantial interest among researchers 2. A negative relationship between idiosyncratic volatility and expected returns is puzzling because standard theory does not account for the relationship. volatility should not be priced at all. If investors fully diversify, then idiosyncratic On the other hand, if investors do not fully diversify, then idiosyncratic volatility should be positively related to expected returns [see Merton (1987), and Malkiel and Xu (2006)]. However, our finding that idiosyncratic volatility is a strong predictor of idiosyncratic skewness suggests one reason why investors may be attracted to stocks with high idiosyncratic volatility. Investors may accept lower average returns on stocks which have experienced high idiosyncratic volatility in the past, not because they seek higher volatility, but because these stocks offer high idiosyncratic skewness in the future. We conduct tests to assess the impact of skewness preference on the IV puzzle. Using our model of predicted skewness, we study the relationship between idiosyncratic volatility and expected returns after controlling for forecasted skewness. We construct portfolios with wide dispersion in idiosyncratic volatility but low dispersion in idiosyncratic skewness, using the conditional sorting methodology of Ang et al. (2007). After controlling for idiosyncratic skewness, we find a much smaller and statistically insignificant difference between the average returns of the high-idiosyncratic-volatility portfolio and the average returns of the low-idiosyncratic-volatility portfolio. In addition, the difference between the Fama French (1993) alphas of the high-idiosyncratic-volatility portfolio and the low-idiosyncratic-volatility portfolio is reduced by 0.82% per month after controlling for expected idiosyncratic skewness. In summary, we find that skewness preference appears to be of first-order importance in explaining the low average returns of stocks with high idiosyncratic volatility. The rest of the paper is organized as follows. Section II presents the motivation for our empirical tests focusing on the theoretical connections for skewness preference and expected returns. Section III reports the results of our estimation of a skewness prediction model. Section IV investigates the pricing implications of idiosyncratic skewness. Section V explores 2 See, for example, Bali and Cakici (2005), Huang et al. (2006), Boehme et al. (2005), Duan, Hu, and McLean (2006), Jiang, Xu, and Yao (2006), Fu (2005), and Kapadia (2006). 3

6 whether skewness preference explains the negative relationship between idiosyncratic volatility and expected returns. The last section concludes. II. Motivation To motivate our empirical investigation, we draw on a couple of seemingly disparate strands of asset pricing. First, speculative behavior on the part of investors, particularly individual investors, has led to a number of papers attempting to motivate this type of behavior from preferences. For example, Mitton and Vorkink (2007) develop a model of heterogenous preference for skewness. Essential to their model is the assumption that some investors ( lotto investors ) have a preference for positive skewness while others ( traditional investors ) are mean-variance optimizers seeking to maximize the Sharpe ratio of their portfolios. Stock returns have exogenous first, second, and third moments. Lotto investors place a higher value on stocks with positive idiosyncratic skewness and choose to hold underdiversified portfolios to increase their exposure to positive skewness. In doing so, they entice traditional investors away from their preferred portfolio position for prices at which E [r i ] <r f + β i (E [r m ] r f ) (1) where r i is the return for stock i with positive idiosyncratic skewness, r m is the return on the market portfolio held by traditional investors, and β i is the regression slope of r i on r m. By tilting away from stocks with high idiosyncratic skewness, traditional investors increase the Sharpe ratio of their portfolios. In equilibrium, both lotto and traditional investors choose portfolio weights such that the marginal cost of holding additional shares is equal to the marginal benefit. That is, a unique pricing kernel exists given by the portfolio held by traditional investors. Stocks with positive idiosyncratic skewness have negative alphas measured against the market portfolio as indicated by equation (1). Thus in Mitton and Vorkink (2007) investors hold underdiversified portfolios in equilibrium, and total skewness (including idiosyncratic skewness) is priced. Empirically, Mitton and Vorkink (2007) find 4

7 their model s predictions help to explain the portfolio behavior of a dataset consisting of household accounts from a large discount brokerage. Alternatively, Barberis and Huang (2007) develop a model with cumulative prospecting investors where, similar to Mitton and Vorkink (2007), holdings are heterogenous across agents in equilibrium. In contrast to Mitton and Vorkink (2007), agents have identical preferences, but the equilibrium obtained is one where a set of agents hold highly underdiversified portfolios. Cumulative prospect utility theory suggests that, through a weighting function, agents overweight the probabilities in the extremes. Barberis and Huang (2007) show that when securities have return distributions that are skewed, agents maximize cumulative prospect utility functions, and equilibria result where idiosyncratic features of a stock s return, including skewness, are priced. In addition, Brunnermeier and Parker (2005) develop a structural asset pricing model where agents optimize over beliefs of outcomes, as opposed to taking probabilities as primitives. Agents derive felicity, analogous to utility, from increasing their beliefs about probabilities above the true levels on high positive payoff states as long as the costs of doing so do not offset this additional utility. This optimizing behavior of agents leads to, among other predictions, a strong preference for securities with skewed distributions (see Proposition II). A preference for skewed assets is also highlighted in Brunnermeier, Gollier, and Parker (2007). Given the multiple theoretical predictions of a negative relationship between idiosyncratic skewness and expected returns, an empirical investigation of the relationship seems in order. However, testing the relationship is complicated by the fact that skewness can be greatly influenced by small-probability events, and thus measures of skewness are typically not stable over time. Asset pricing tests often use lagged predictors of second moments (beta and volatility) as proxies for expected levels based on the widely supported assumption that second moments are quite persistent, but this approach appears problematic for third moments. For example, Harvey and Siddique (1999) estimate models of time-varying skewness and find that a lagged measure of skewness is, at best, a weak predictor of skewness (the autoregressive parameter for monthly skewness is about -0.4). The challenge is thus to find an appropriate way to measure expected skewness. 5

8 Our econometric approach follows Chen, Hong, and Stein (2001) by constructing measures of expected skewness for a given firm in each period using past returns and firm characteristics. Standard time series models are then used to predict skewness. A few alternatives to our approach exist. Zhang (2005) measures the expected skewness for a stock based on the recent cross-sectional return skewness of a group of stocks to which the stock belongs, such as an industry grouping. One challenge to measuring skewness cross sectionally is that little theory or guidance exists to find the appropriate peer group of stocks to construct the measure of skewness. An alternative approach could be to construct conditional skewness estimates using a distribution of returns that parameterizes skewness as well as the time-varying nature of skewness [see, for example Harvey and Siddique (1999)]. While this approach has an econometric appeal of more precise estimates, the difficulty of applying a model to the cross section of firms makes this approach particularly challenging given the current distribution alternatives. While our approach of using past returns and firm characteristics to predict skewness is not without criticism, we believe the approach is quite robust to alternative timeseries measures of skewness. One previous paper that analyzes predictors of skewness is Chen, Hong, and Stein (2001). Motivated by the model of Hong and Stein (2003), they estimate predictive regressions on firm skewness across a wide cross-section of stocks. 3 Chen, Hong, and Stein (2001) find, among other results, that lagged idiosyncratic volatility is a strong positive predictor of firm skewness [see also Kapadia (2006)]. Idiosyncratic volatility could be a strong predictor of skewness for a number of reasons. First, idiosyncratic volatility is positively related to corporate growth options [see Cao, Simin, and Zhao (2006) and Barinov (2006)], and the presence of growth options implies greater skewness in returns [see, e.g., Andrés-Alonso et al. (2006)]. Second, higher idiosyncratic volatility may be related to technological revolutions [see Pastor and Veronesi (2007)], and these revolutions may lead to industry shake-outs [Jovanovic and MacDonald (1994)] which in turn imply greater skewness in returns as a few winners emerge and other firms fail. Third, from a mechanical standpoint, limited liability of equity implies that greater volatility leads to greater skewness [see, e.g., Conine and Tamarkin (1981)], and 3 Chen, Hong, and Stein (2001) actually estimate models of negative skewness motivated by the crash model of Hong and Stein (2003). 6

9 volatility may be a better predictor of skewness simply because it is more persistent over time. In the next section we use idiosyncratic volatility, as well as other variables suggested by the literature, to estimate a model of expected skewness. III. A Skewness Prediction Model Three important issues arise as we consider how to model investor perceptions of future skewness. The first is choosing the horizon over which investors are hoping to experience an extreme positive outcome. This choice is of course, subjective, but our prior is a horizon in the range of several years rather than only a month as assumed in studies by Ang et al. (2007) and others. Second, the relation between expected skewness and firm-specific characteristics may vary over time. Finally, for our asset pricing tests the estimates of perceived expected skewness must use information available to investors at the time the estimate is made. Let the investment horizon over which investors are hoping to experience an extreme positive outcome be T months. Let is it and iv it be historical estimates of idiosycratic volatility and skewness for stock i using daily returns observed from the first day of month t (T +1) through the end of month t. Attheendofmontht an investor can determine how firm-specific characteristics are related to skewness in returns by estimating a cross sectional regression of the form is i,t = β 0 + β 1 is i,t T + β 2 iv i,t T + γx i,t T + ε it (2) where X i,t T is a vector of firm-specific variables including firm characteristics and a full set of industry dummies observable at the end of month t T. Equation (2) is similar to the panel estimations conducted in Chen, Hong, and Stein (2001) with the execption that we estimate the model separately each month. The investor can then use the regression parameters from equation (2), denoted byb, along with information observable as of time t to estimate expected skewness for each firm, E t [is i,t+t ], E t [is i,t+t ]=ˆβ 0 + ˆβ 1 is it + ˆβ 2 iv it + ˆγX it. (3) 7

10 We estimate the parameters of equation (2) for each month t in the sample, and use each month t s set of parameters to estimate E t [is i,t+t ] as in equation (3). This approach not only allows the relation between firm-specific variables and skewness to vary across time, but also provides feasible estimates of expected skewness that investors can use each month to forecast skewness. The main results of our paper use this approach to estimate expected skewness using horizons of T =60months. This implies an investor needs 10 years of prior data to estimate the parameters of equation (2) and generate an estimate of expected skewness as in(3). Since we use CRSP data from January 1925 through December 2005, we obtain estimates of expected skewness from January 1935 through December These estimates of expected skewness are based on parameters from monthly cross sectional regressions as in equation (2). For example, to obtain estimates of expected skewness as of January 1935, we first estimate a cross sectional regression as in (2) with explanatory variables observable as of the beginning of January 1930, and the dependent variable equal to historical firm skewness estimated from daily data from January 1930 through the end of December We then take these parameters and apply them to firm-specific variables as of Janaury 1935 to estimate expected skewness. 4 Figure 1 plots the cross-sectional distribution of is i,t+1 over the period January 1930 through December of is i,t+1 each month. In particular, we plot the 10, 25, 50, 75 and 90th percentiles All of the reported percentiles show time variation, but the 90th percentile exhibits the greatest variation, and particularly large movements during the early 1930s, the mid 1980s, and the mid 1990s. Increases in the upper tail of the cross section of is i,t+1 also occur in the 1960s and 1970s. occur during These periods of increased dispersion in skewness periods of high iv i,t+1 [see for example, Campbell, Lettau, Malkiel, and Xu (2001) and Brandt, Brav, and Graham (2005)]. Because is i,t+1 is raw skewness scaled by iv 3/2 i,t, raw skewness increases disproportionately in these periods (relative to variance) and US equity markets appear to experience episodes of high skewness and high dispersion of skewness described in Brandt, Brav, and Graham (2005) as speculative periods. Table I provides summary statistics for the variables used to estimate equation (2). Panel A of Table I reports descriptive statistics of the predictive variables and Panel B of Table I 4 In unreported results we estimate panel regression versions of equation (2) and find similar results. 8

11 reports the correlations of the variables. for idiosyncratic skewness and idiosyncratic volatility. The first three rows of the table report statistics We measure these variables over a five-year horizon (set equal to 60 months),but we also include a one-month iv measure for comparison with the results in Ang et al. (2006) in Section V. The last two rows report statistics for other instruments used in the predictive regressions including mom, the firm s cumulative return over months t 12 through t 2, andturn, the average daily turnover of the firmoverthepriormonth. Theinclusionofmom is motivated by Chen, Hong, and Stein (2001), who find that past returns are negatively correlated with forecasted skewness. The use of turn is motivated by the model in Hong and Stein (2003), which predicts that negative skewness is most pronounced during periods of heavy trading volume. In addition to the variables reported in Table I, we also include a number of dummy variables in our skewness prediction model. Unlike Chen, Hong, and Stein (2001), who use only NYSE data in their empirical tests, we include firms from NYSE and NASDAQ in our dataset. Because of the institutional differences (such as in turnover measurement) between the two exchanges, we include a NASDAQ dummy variable in our model. To control for firm size, we include dummy variables for small and medium-size firms, where firms are grouped into three equally sized categories of small, medium, and large based on market capitalization. 5 Finally, the use of industry dummies in the model is similar in spirit to the approach of Zhang (2005). Industry classifications are based on each firm s primary SIC code and are as defined on Ken French s website. We view our estimating equation as a parsimonious model for skewness prediction, and we omit other potential variables that have been mentioned in the literature. For example, in robustness checks, Chen, Hong, and Stein (2001) also use the book-to-market ratio and analyst coverage to predict skewness. In unreported tests we include these variables in our model, and we find that although they add some explanatory power, they also require the dropping 5 In alternative specifications (not reported) we include a continuous variable for market capitalization, the natural log of the firm s market capitalization. We find that inclusion of log size leads to generally similar results, but that allowing for a nonlinear size/skewness relationship, as in our dummy variables, improves the model fit and is less impacted by small firm outliers. 9

12 of a large number of observations. Given that our intent is to conduct cross-sectional asset pricing tests, we opt to use the full set of data with a more limited set of variables. 6 Table II reports the results of estimates of equation (2) represented by rows. In the tablewereporttheaveragecoefficient of the monthly cross-sectional regressions. Below the average coefficients, as a measure of statistical significance, we report the percentage of monthly regressions in which the coefficient has the same sign as the average coefficient and is significant at the 5% level. In Panel A, we use five years of data to estimate idiosyncratic skewness. The first regression uses only lagged is to predict is, while the second regression uses only lagged iv. Both lagged iv and lagged is positively predict is, andbothvariables are significant in 100% of the monthly regressions. Similar results hold in the third regression which includes lagged iv and is simultaneously. These regressions confirm the results of Harvey and Siddique (1999) and Chen, Hong, and Stein (2001) in that lagged iv is a stronger predictor of is than lagged is. The adjusted-r 2 for the iv regression is more than double the adjusted-r 2 for the is regression. In addition, variations in lagged iv lead to much greater variations in bis than variations in lagged is. Specifically, using the coefficients of the third regression, which includes both lagged iv and lagged is as predictors, we determine the impact on our forecast of skewness from a one-standard-deviation shock to each predictor. Using these coefficients and the standard deviations in Table I, we find that a one-standarddeviation shock in lagged iv will lead to variation in predicted skewness over twice as large as the variation resulting from a one-standard-deviation shock in lagged is. The fourth regression in Panel A includes lagged iv and is as well as our other predictive variables, mom, turn, and the dummy variables for NASDAQ, small, and medium-size firms. The signs of the coefficients in these regressions are consistent with Chen, Hong, and Stein (2001). Higher values for momentum and turnover are both associated with lower predicted skewness. The negative average NASDAQ dummy coefficient may seem puzzling given the casual observation of the type of firm listing on NASDAQ as opposed to the NYSE. However, turnover as measured on NASDAQ can be double that of similar trading activity as measured on the NYSE/AMEX exchanges. Hence, we interpret the negative coefficient on the NASDAQ 6 Though not included as a predictive variable by Chen, Hong, and Stein (2001), we also estimated models (not reported) that included a measure of leverage. The incremental explanatory power of leverage was relatively small, so we again opted to omit leverage as a variable and retain the larger set of data. 10

13 dummy variable as a mitigating parameter for the alternative turnover measuring conventions. Perhaps of note, our empirical models always predict higher average expected skewness for NASDAQ listed stocks relative to NYSE/AMEX listed stocks. The R 2 of the skewness predictive regression increases somewhat when the additional variables in the fourth row are included. All of the variables have fairly consistent statistical significance in the monthly regressions, although turn is significant in just 13% of the regressions. The inclusion of these additional variables reduces the predictive power of lagged iv more than lagged is, butthe estimated impact of a shock to lagged iv is still more than 60% greater than the estimated impact of lagged is. The fifth regression in Panel A reports results of regressions including only the industry dummies as explanatory variables and shows that the industry dummies alone have some ability to predict is. Including the industry dummies along with the other predictive variables (in the sixth row) results in the highest R 2 of any of the regressions. The regression in this row is the base model that is used our subsequent tests. The last row of Panel A reports results including only iv measured over a one-month horizon. Results using a one-month horizon for iv instead of a five-year horizon are included for comparison with Ang et al. (2006) as discussed in Section V. Panel B of Table II reports robustness checks of the predictive regressions reported in Panel A. The first row of Panel B repeats the regressions using older stock market data, from January 1930 through December Relative to the results in Panel A, the results using the older data generally have smaller estimated coefficients and less-frequent statistical significance (in part due to the smaller number of observations per year). However, the signs and relative magnitudes of the coefficients are quite similar to the results in Panel A. The last two rows of Panel B repeat the regressions from Panel A but use skewness measures with horizons of six months and two years instead of five-year skewness measures. The results are fairly similar in sign and magnitude to the results for the five-year skewness measures. We do observe that the coefficient on the NASDAQ dummy increases in value and decreases in average significance in these regressions relative to those of Panel A. We include a time series plot of the industry dummies to provide some insight into the behavior of these coefficients, found in Figure 2. We include industry dummies for Utilities, 11

14 Telecom, and Textiles stocks. The plots in Figure 2 show substantial time-series variation for each of the industries and similar variation exists for industries not included in the Figure. Much of the variation in these dummy plots appears to conform to expectations. For example, the sign on the Utility industry dummy is almost always negative from the 1940s forward, consistent with the belief that this heavily regulated industry offers comparatively little upside potential. The large positive spike in the Utility dummy during the late 1930s and early 1940s corresponds to the Natural Gas Act of 1938 which forced the splitting up of the utilities industry from a few dominant firms into many smaller companies. High positive coefficients on Telecom stocks during the late 1950s, 1970s, and early 1990s also conforms to periods where these types of companies generated positive attention and abnormally large returns. We include the Textiles industry dummy as it represents the industry with the largest time series variation as evidenced by the large swings in the coefficient s value from the 1970s through In sum, the results of this section suggest that a simple model including lagged idiosyncratic volatility and skewness, as well as momentum, turnover, size, and industry dummies, can help investors to forecast skewness. In the next section we assess whether idiosyncratic skewness, as predicted by this model, can help explain the cross-section of expected returns. IV. Expected Skewness and Average Returns To study the relationship between predicted skewness and expected returns, we first assess how average returns vary across stocks with differing levels of expected skewness. We then turn to the impact of predicted skewness in cross-sectional regressions using the methodology of Fama and MacBeth (1973). A. Portfolios Sorted on Predicted Skewness To construct measures of expected skewness we follow take estimated coefficients from regressions on equation (2) and use these coefficients as described in equation (3). We are able to construct predicted values of skewness, bis, every month so that we can form and rebalance portfolios monthly, analogous to the typical approach for iv in the asset pricing literature. 12

15 InTable III,wesortfirms into portfolios based on their levels of predicted skewness. Table III presents descriptive statistics for five quintiles, where the first quintile represents firms with the lowest predicted skewness, and the fifth quintile represents firms with the highest predicted skewness. The first column of Table III reports mean returns of the various portfolios. If predicted skewness is an important determinant of returns, we would expect to see differences in mean returns across quintiles. The first column shows that mean returns decline monotonically from the first quintile to the fifth quintile. Mean returns are substantially lower in the fifth quintile (0.52%) compared to the first quintile (1.19%), a difference of 0.67% per month. The largest decline in mean returns appears between the fourth and fifth quintiles. While predicted skewness appears to be an important determinant of returns, it is worth noting that in our model lagged skewness alone is not a good predictor of returns. In other tests (not reported) we repeat the analysisoftableiiibut sortfirms into portfolio based on predicted skewness using only our measure of lagged skewness. In this sorting, mean returns areonlyslightlylowerinthefifth quintile than in the first quintile, a difference of 0.04% per month. This result is in line with our assumption that additional variables are required to estimate expected skewness in an economically meaningful way. The other columns in Table III report other descriptive statistics of the predicted-skewness quintiles, showing, for example, that firms with higher predicted skewness tend to be smaller than firms with lower predicted skewness, consistent with the results of Table II. We also find a positive relationship between iv and predicted skewness, again consistent with the predictive regressions of Table II. We do find that the time-series measures of portfolio skewness, reported in the third column, increase almost monotonically with predicted skewness across the quintiles. This is somewhat surprising given that portfolio skewness has many coskewness determinants which do not confound the forecasted idiosyncratic skewness (reported in the fourth column) of the portfolio constituents. The last two columns of Table III indicate that firms with higher predicted idiosyncratic skewness have lower alphas relative to the CAPM and relative to the Fama-French three-factor model. The difference in the Fama-French alphas is particularly pronounced, with the first quintile having an alpha of 0.14% per month and the fifth quintile having an alpha of -0.86% per month, a difference of 1.00% per month. The 13

16 difference in the alphas of the first and fifth quintile is highly statistically significant. Table III indicates that predicted skewness is an important determinant of returns, with significantly lower returns associated with stocks with higher expected skewness. B. Fama-MacBeth Regressions To further assess the pricing effects of idiosyncratic skewness, we conduct cross-sectional regressions following the approach of Fama and MacBeth (1973). Each month we run the following cross-sectional regression: r p,t = γ 0 + γ 1 is t + γ 2 is t + γ 3 Z t + ε p,t (4) and report these results in Table IV. We include standard factor loadings, firm characteristics, and controls for our skewness predicting instruments as noted in equation (2) and Table II. We include loadings to a four-factor model that adds a momentum factor to the standard Fama-French (1993) three-factor model. We also include all firm characteristics associated with these factors (size, book-to-market, momentum) following Daniel and Titman s (1997) characteristic argument. Other factors included are a coskewness factor based on Harvey and Siddique (2000), a one-month idiosyncratic volatility measure following Ang, et al. (2006, 2007), and our predicted skewness measure. We include both momentum returns and lagged turnover to control for the previously established connections between expected returns and these variables given that these are instruments in our skewness prediction model of equation (2). We also include loadings on the Pastor and Stambaugh (2003) liquidity factor to capture potential liquidity pricing effects correlated with our turnover measure (similar to the inclusion of the loading on the momentum factor). To reduce measurement error problems we sort stocks each month into 100 portfolios based on predicted skewness and then construct value-weighted portfolio returns for the portfolios. Weconstructloadingsandcharacteristicsfortheseportfoliossimilarly. Wesetasourbase data the period from January 1988 through December 2005 as this allows our predicted skewness measure to include turnover on all stocks, particularly NASDAQ stocks which only begin to report turnover on a widespread basis in We run a cross-sectional regression every 14

17 month and report the time series averages of the γ coefficients along with t-statistics based on Newey and West (1987) constructed standard errors. The first two columns focus on the cross-sectional pricing of bis and iv. Predicted is, (bis), is negative and significant in both cases while iv is negative but not significant in models where both are included as possible pricing factors. The last two columns include all of our other pricing factors and these columns show a reduction in the average γ 1 on bis although theaverageremainsbothnegativeandsignificant in the presence of these other factors. The γ 2 coefficient is negative and significant in these larger models. The coefficient on the HML loading is negative and significant, perhaps related to our decision to exclude book-tomarket as an instrument in our skewness prediction model. 7 Results of Table IV indicate that predicted skewness helps to explain the cross-sectional variation in expected returns beyond common instruments. In contrast, coefficients on coskewness loadings are negative as predicted, but t-statistics on these coefficients are quite small. B.1. Robustness Checks We conduct a number of robustness checks to Table IV s results and report these in Table V. We vary the portfolio number, the data horizon, and the skewness measures to assess the stability of the skewness pricing results to these choices. In general, we find that predicted skewness remains negatively priced and significant to these checks. We conduct cross-sectional tests with both fewer (50) and greater (200 and all individual stocks) numbers of portfolios relative to our base case. In the first three columns of Table V we see that the magnitude of the γ 1 coefficient generally declines with the number of portfolios. The significance of γ 1 remains relatively strong along variations in the number of portfolios although the case where we use individual stocks in the cross-sectional regressions, the p-value on γ 1 is just above In contrast, the significance of iv increases quite dramatically with the number of portfolios, changing from a t-statistic of for the case of 50 portfolios to a t-statistic of for the individual stock cross-sectional case. One possible explanation for this change is that the iv pricing effectismostdominant inthetailoftheiv distribution, and 7 Chen, Hong, and Stein (2001) find that book-to-market is positively related to subsequent skewness, and consequently high book-to-market values would predict low average returns. Therefore the negative and significant coefficient on the HML loading may not seem suprising. 15

18 grouping stocks along any dimension (in our case predicted skewness) begins to erode this effect away. The average returns based on iv sorted portfolios [see Table VI Panel B of Ang et al. (2007) or our Table VI] drop dramatically for the highest iv stocks. We also extend the data back to 1935 and conduct tests using 100 portfolios. We have to drop some of our pricing factors in this case due to data limitations. Using this longer data sample, we continue to observe a γ 1 that is negative and significant at standard levels. 8 We finally construct bis using six-month and two-year skewness measures, and find that the γ 1 coefficient continues to be negative and significant as reported in the final two columns of Table V. C. Time Series Effects Given the time series features of predicted and realized skewness shown in Figures 1 and 2, a natural question may be whether our cross-sectional pricing results exhibit any related temporal variations. Our interest is also motivated by Kapadia (2006), who observes strong time series effects in the pricing of a cross-sectional skewness-based factor. Figure 3 plots the rolling twelve-month lagged average bis cross-sectional coefficient γ 1 from equation (4). The time series plot exhibits similar time series behavior as seen in Figures 1 and 2, including large negative swings in 1950s, 1960s, 1970s and 1990s corresponding to the speculative episodes occurring in cross-sectional skewness distributions in Figures 1 and 2. The figure also includes a plot of the adjusted-r 2 from the skewness prediction model of the bis for that month. This series appears to be negatively correlated to the γ 1 plot. These observations from the figures are suggestive of the possibility that skewness may be more strongly priced in speculative periods when expected skewness is high and more easily forecasted. In an attempt to formalize these relationships we conduct the following regression: γ 1,t = δ 0 + δ 1ˆμ skew,t + δ 2ˆσ skew,t + δ 3 R 2 pred,t + ε t (5) 8 We change our predictive skewness model during the period for this longer data set. During this period returns on NASDAQ stocks were available but turnover was not. Consequently, we exclude turnover as a predictor of skewness (all stocks not just NASDAQ) so that the forecasting model is consistent cross-sectionally. In months both before and after this period we are able to and do include turnover as a predictor of skewness. 16

19 where ˆσ skew,t is the month t cross-sectional standard deviation of forward looking skewness, ˆμ skew,t is the average cross-sectional skewness for month t, andrpred,t 2 is the adjusted-r2 from the predictive skewness model used to construct the month t predicted skewness measures. Results of this regression are in Table VI and show what Figure 3 suggests; the negative correlation between skewness and returns is strongest when dispersion in skewness is high, when average skewness is high and when skewness is forecastable. The statistical significance of all three of these explanatory variables is high. Clearly these tests are preliminary in nature and more work is required to pin down the temporal variations in skewness pricing, but our reported results in Table VI and from estimation of equation (5) strongly suggest the likelihood of time variation in skewness pricing. V. Skewness, Volatility, and Expected Returns Having established the pricing effects of idiosyncratic skewness, as an application we now assess whether idiosyncratic skewness can help explain the negative relationship between idiosyncratic volatility and expected returns as documented by Ang et al. (2006). Perhaps because a negative relationship between risk and return is difficult to reconcile with standard assumptions regarding investor utility, explanations for the IV puzzle tend to focus on market imperfections that could induce such a relationship, such as short-sale constraints or a lack of information disclosure [see Boehme et al. (2005), Duan, Hu, and McLean (2006), Jiang, Xu, and Yao (2006)]. In contrast, an explanation for the IV puzzle based on skewness preference does not focus on the existence of market imperfections. If investors have a preference for positive skewness, they will accept lower average returns on stocks that offer skewed returns. Ang et al. (2007) address the potential impact of idiosyncratic skewness on the negative relationship between idiosyncratic volatility and returns. They find that idiosyncratic skewness has a strong negative correlation with expected returns. In terms of statistical significance, they show that the negative relationship between lagged idiosyncratic skewness and returns is even stronger than the negative relationship between lagged idiosyncratic volatility and returns. At the same time, these authors also show that the IV puzzle persists even after controlling for lagged idiosyncratic skewness. However, because other variables are also 17

20 important predictors of skewness, a test like the one in Ang et al. (2007), which uses only lagged skewness as a measure of skewness, may underestimate the true impact of skewness preference on the IV puzzle. 9 To study the impact of predicted skewness on the IV puzzle, we begin with the key results of Ang et al. (2006), and then assess how those results change when we control for predicted skewness. A. Idiosyncratic Volatility and Expected Returns We replicate the results of Ang et al. (2006) with data consisting of security returns on firms in the CRSP database, beginning in January 1960 and ending in December We construct iv as the standard deviation of residuals from a Fama and French (1993) three-factor model regression using the prior month s daily returns, as detailed in equation (??). In Table VII, we provide summary statistics for the idiosyncratic volatility quintiles and find the portfolio characteristics to be similar to Ang et al. (2006). The first two and last two columns of Table VII are directly comparable to Panel B of Table VI of Ang et al. (2006). These columns indicate that average returns, return standard deviations, as well as pricing errors of the CAPM and Fama-French three-factor model are comparable to Ang et al. (2006). particular, the first column of Table VII illustrates the poor average performance of the high iv quintile portfolio. Whereas the low iv quintile has mean returns of 1.21% per month, the high iv quintile has mean returns of 0.12% per month. In addition, the last two columns show large pricing errors of the CAPM and Fama-French model. The difference in the Fama-French alphas between the first and fifth quintiles is 1.30% per month. In Table VII shows, similar to Ang et al. (2006), that the IV puzzle is most pronounced in the high iv quintile, where mean returns and alphas are disproportionately low relative to the other four quintiles. We also include two skewness variables in Table VII. The first skewness variable, in the third column of Table VII, is the time-series skewness estimate of the respective quintile portfolio. This statistic shows that quintiles of higher idiosyncratic volatility exhibit more positive levels of skewness in their returns. The other skewness variable, firm skewness (in the 9 In a similar vein, Fu (2005) argues that lagged idiosyncratic volatility is not a good measure of expected idiosyncratic volatility, and that the IV puzzle does not persist when using more sophisticated estimates of expected volatility. 18

21 fourth column), is the average firm idiosyncratic skewness (is) estimated from equation (??) using daily returns over the prior five years in an otherwise similar fashion to iv construction. 10 The pattern of increasing skewness in quintiles with higher volatility again illustrates a strong contemporaneous relationship between iv and is. These two measures of skewness suggest that, among possible alternative functions, lagged idiosyncratic volatility forecasts skewness of returns. 11 Consistent with the results for the mean returns, the largest increase in skewness occurs between the fourth and fifth quintiles. We also report firm size (market capitalization) in Table VII. The relationship between size, is, andiv is consistent with skewness-preferring investors taking speculative positions in small firms with highly skewed returns and accepting lower average returns in their positions for a chance of hitting an investment home run. B. Conditional Sorting We next directly address the question of whether predicted skewness can help explain the IV puzzle. Following the methodology in Ang et al. (2006, 2007) we conduct a double-sorting exercise to control for expected skewness. skewness predictions from estimations of equation (2). We first sort firms into five bis quintiles based on Then for each bis quintile, we sort firms into one of five iv portfolios based on lagged one-month iv measures. We then value weight each of the iv portfolios across each of the five bis quintiles, thereby controlling for the effect of expected skewness. The results of this exercise are reported in Table VIII. Table VIII suggests that skewness preference may play a substantial role in the differing returns of the iv portfolios. The spread of average returns across the conditionally sorted iv portfolios is much smaller than the unconditionally sorted quintiles. The return premium between the first and fifth portfolio is only 0.37% per month and is no longer significant, down from the 1.09% spread found in Table VII. The subsequent columns suggest that our model of expected skewness is doing a reasonable job. In particular, we find relatively small 10 We use one-month iv estimates to be consistent with the results of Ang et al (2006, 2007). We use fiveyear measures of is consistent with earlier work in our paper, but contrasting the six-month measures used in Chen, Hong, and Stein (2001). Our results are qualitatively robust to using one-month measures or five-year measures for both is and iv. 11 Portfolio skewness is comprised of idiosyncratic skewness and other coskewness components, so without further analysis we are not able to conclude that lagged idiosyncratic firm volatility is able to forecast idiosyncratic skewness. 19

22 variations in portfolio skewness across the five iv quintile portfolios. The average firm skewness measures (subsequent to portfolio formation) also show relatively small differences relative to the variation found in the unconditionally sorted iv portfolios in Table VII. In fact, skewness is now slightly lower in the high iv quintile relative to the low iv quintile. However, we still find a wide spread in idiosyncratic volatility across the portfolios. The spread is comparable to that of the portfolios unconditionally sorted on idiosyncratic volatility reported in Table VII. Finally, we control for risk to see if material pricing differences are found across the quintile portfolios. The high iv portfolio has a CAPM intercept estimate of -0.54% which is still significant with a t-statistic of 2.5. However, this point estimate is less than half the magnitude of the estimate for the unconditionally sorted high iv quintile in Table VII. Similarly, the high iv portfolio results in Table VIII indicate a Fama-French intercept of -0.37% on the high iv portfolio, which is 0.82% per month smaller than the intercept for the unconditionally sorted high iv quintile in Table VII. The evidence indicates that idiosyncratic volatility generates much smaller differences in pricing errors relative to the CAPM and Fama-French models after controlling for idiosyncratic skewness. In the finalrowoftable VIII,wereportresults from constructing a difference-in-differences portfolio, UCMC (unconditional minus conditional), to better assess the improvements associated with forecasting skewness. In particular, we long the zero investment unconditionally iv sorted long/short portfolio. This portfolio is constructed by taking a long position in the unconditionally sorted low iv quintile portfolio and shorting the unconditionally sorted high iv quintile portfolio; these portfolios are reported in Table VII. We then short the conditionally sorted 1-5 portfolio represented in the sixth row of Table VIII, or the portfolios that control for bis. The resulting portfolio is long one zero-investment portfolio and short another zero-investment portfolio, and should allow us to formally test the apparent improvements in explaining the IV puzzle found in the first fiverowsoftableviii. Theresultsarestrongly supportive. The CAPM alpha on the UCMC portfolio is 0.77% with a t-statistic of 3.2 and the Fama-French three-factor model has an intercept of 0.82% with a t-statistic of 3.5. As a whole, the results in Table VIII suggest that idiosyncratic skewness is of first-order importance in explaining the low average returns of stocks with high idiosyncratic volatility. 20