Lottery Preferences and the Idiosyncratic Volatility Puzzle* Doina C. Chichernea University of Denver

Transcription

1 Lottery Preferences and the Idiosyncratic Volatility Puzzle* Doina C. Chichernea University of Denver Haimanot Kassa Miami University and the U.S. Securities and Exchange Commission Steve L. Slezak University of Cincinnati This Draft: March 27, 2018 Abstract We investigate the empirical implications of investors heterogeneous preferences for skewness with respect to the idiosyncratic volatility (IVOL) puzzle, that is, the negative correlation between IVOL and mean returns. We show that the IVOL puzzle is stronger 1) within stocks held primarily by agents with a preference for lottery-like payoffs and 2) during economic downturns, when the demand for lottery-like payoffs is high. These results support recent theories that suggest lottery preferences could be a significant source of the IVOL puzzle. Keywords: Idiosyncratic volatility; skewness; lottery preferences; economic conditions. JEL classification: G11; G12 *Chichernea, Reiman School of Finance, Daniels College of Business, University of Denver, 2101 S. University Blvd., Denver, CO Tel ; doina.chichernea@du.edu. Kassa, Department of Finance, Farmer School of Business, Miami University, 800 E. High St., FSB 2058, Oxford, OH Tel ; kassah@miamioh.edu, and the U.S. Securities and Exchange Commission, 100 F Street N.E. Washington D.C Slezak, Department of Finance and Real Estate, Lindner College of Business, University of Cincinnati, 408 Carl H. Lindner Hall, PO Box , Cincinnati, OH Tel ; steve.slezak@uc.edu. We are thankful for the helpful comments and suggestions from Tom Boulton, Alexander Borisov, Kelly Brunarski, Colin Campbell, Michael Ferguson, Mark Griffiths, Hui Guo, Yvette Harman, Tyler Henery, Bochen Li, David Manzler, Terry Nixon, Chen Xue, Steve Wyatt, and Kim Yong. We benefited from discussions with seminar participants at Miami University, the University of Cincinnati, the Eastern Finance Association, the Midwest Finance Association, and the Southwestern Finance Association. We are responsible for any errors. The Securities and Exchange Commission disclaims responsibility for any private publication or statement of any SEC employee or Commissioner. This article expresses the author's views and does not necessarily reflect those of the Commission, the Commissioners, or other members of the staff.

2 Lottery Preferences and the Idiosyncratic Volatility Puzzle Abstract We investigate the empirical implications of investors heterogeneous preferences for skewness with respect to the idiosyncratic volatility (IVOL) puzzle, that is, the negative correlation between IVOL and mean returns. We show that the IVOL puzzle is stronger 1) within stocks held primarily by agents with a preference for lottery-like payoffs and 2) during economic downturns, when the demand for lottery-like payoffs is high. These results support recent theories that suggest lottery preferences could be a significant source of the IVOL puzzle. Keywords: Idiosyncratic volatility; skewness; lottery preferences; economic conditions. JEL classification: G11; G12

3 1. Introduction The negative empirical relation between idiosyncratic volatility (IVOL) and subsequent stock returns has been considered a puzzle since it was first documented by Ang, Hodrick, Xing and Zhang (2006). The subsequent literature suggests a link between the so-called IVOL puzzle and investor preferences for lottery-like features (i.e., skewness; see, e.g., Boyer, Mitton and Vorkink, 2010; Bali, Cakici and Whitelaw, 2011). If so, then the IVOL puzzle should be more pronounced within the universe of stocks held by agents who prefer lottery-like payoffs. Further, since the literature (e.g., Kumar, 2009) shows that the demand for lottery-like characteristics varies countercyclically over time, the IVOL puzzle should also be more pronounced during economic downturns, when the demand for lottery-like features increases. Our paper provides evidence that supports both hypotheses. A negative relation between positive skewness and subsequent returns is implied by heterogeneous investors preferences for skewness. In models in which the mean and variance are sufficient to characterize either security returns or investors preferences (e.g., the capital asset pricing model, or CAPM), all investors optimally hold mean variance-efficient portfolios. If, in addition to caring about mean and variance, some investors also derive utility from positive skewness, then these investors will, at prices implied by mean variance efficiency, create excess demand for securities with positive skewness. This has two implications: 1) the prices of lottery-like assets will be bid up to clear the market, resulting in lower subsequent returns relative to securities without lottery-like features, and 2) since the market-clearing price is a compromise between the valuations of each type of investor, the amount prices are bid up in equilibrium such that mean variance investors hold proportionally less and mean variance skewness investors hold proportionally more lottery-like securities. A third implication concerns the magnitudes of these effects. The greater the skewness benefit a security offers, the greater the reduction in the expected return and the greater the decrease in expected return, the greater the disparity between the proportional holdings of lottery-like securities between mean variance and mean variance skewness investors. 1

4 While it is intuitive that (at least some) investors might have a preference for positive skewness, it is not obvious why investors would have a preference for IVOL; thus, the above argument for a negative equilibrium relation between skewness and future returns does not apply for IVOL. However, the literature highlights a couple of reasons why IVOL and skewness could be connected empirically. First, it is possible that IVOL is an inherent feature of firms that have lottery-type returns. The literature presents at least two conceptual reasons for such a connection: 1) IVOL is positively related to corporate growth options, which induce greater skewness in returns (Andres-Alonso, Azofra-Palenzuela and La Fuente- Herrero, 2006; Cao, Simin and Zhao, 2008) and 2) a higher IVOL is related to technological revolutions, which also induce greater skewness in returns, since a few winners emerge and other firms fail (Jovanovic & MacDonald, 1994; Pastor & Veronesi, 2009). Second, the IVOL measures used in the literature are typically based on models in which investors are well diversified and hold portfolios on a mean variance frontier (e.g., multifactor CAPM), in which there is no (negative) risk premium for skewness. Such models will misspecify expected returns for lottery-like securities, which will likely result in upwardly biased estimates of the true IVOL for these securities. Thus, securities with greater skewness will have higher estimated IVOL. Finally, stocks with exceptionally high volatility are likely to have positive skewness by default, given the limited liability nature of equity (Conine and Tamarkin, 1981). Regardless of the exact channel, if there is a connection between IVOL and lottery-like features, then perhaps the IVOL puzzle is not symptomatic of a puzzle at all but is simply a proxy for the rational pricing of skewness instead. 1 To differentiate, it would be nice to be able to pit the alternative implications of each against each other. However, since the IVOL puzzle has no obvious theoretical underpinning, it provides no obvious alternative testable implications. However, the equilibrium skewness return relation based on heterogeneous investor preferences does have alternative testable implications suggested by evidence in the literature. First, as discussed above, the negative skewness 1 It is also consistent with behavioral bias. For example, Barberis and Huang (2008) argue that, under cumulative prospect theory, investors overweight small chances of large gains (and hence have lottery preferences). Therefore, such investors prefer (and bid up) positively skewed stocks, causing them to be overpriced and earn low subsequent returns. 2

5 return relation should be stronger among the set of securities held in greater proportion by those investors who have a preference for skewness. Thus, following Kumar (2009) and Han and Kumar (2013), who argue that lottery preferences are particularly pronounced for retail investors compared to institutional investors, the IVOL puzzle should be more pronounced for those securities held more by retail investors (rather than institutional investors). Second, following Kumar (2009), who shows that investor preference for skewness increases during economic downturns, the IVOL puzzle should be more pronounced in down markets. We start our empirical analysis by first showing that the negative skewness return relation and the negative IVOL return relation (i.e., the IVOL puzzle) are present in our sample from 1980 to Specifically, we control for market return (MKTRF), size (SMB), value (HML), momentum (UMD), and Pastor Stambaugh (2003) illiquidity (ILLIQ) and find an average five-factor alpha of -0.97% per month for the high-minus-low quintile IVOL hedged portfolio and -0.22% per month for the high-minus-low quintile skewness hedged portfolio in our sample. Our analysis then proceeds by showing that this IVOL puzzle varies by type of investor (i.e., institutional vs. retail), by type of firm (i.e., level of skewness), by type of market conditions (i.e., good vs. bad), and by the interaction of these three. In so doing, we attempt to empirically shed light on the economic underpinnings of the IVOL puzzle. 2 We identify institutional versus retail ownership using Thomson Reuters 13F institutional holdings data and define a stock s institutional ownership ratio (IOR) as the ratio of shares held by institutions to that stock s total shares outstanding. We find that low-ior stocks (i.e., stocks held primarily by retail investors) do indeed have higher skewness and higher IVOL than stocks held primarily by institutions (i.e., high-ior stocks). In our sample, the average skewness of the high-retail ownership quintile portfolio is approximately 50% higher than the average skewness of the low-retail ownership 2 It is important to note that the negative relation between IVOL and future returns is not accepted in the literature. For example, while Ang et al. (2006, 2009) document the IVOL puzzle (negative relation), Fu (2009) documents a positive relation between IVOL and future stock returns. Peterson and Smedema (2011) reconcile this conflicting evidence by showing that both of these results are robust, since the former refers to realized IVOL and the latter refers to expected IVOL. On the other hand, some studies show that this conflicting evidence is the result of a bias in the way expected IVOL is estimated (e.g., Fink, Fink, and He, 2012; Guo, Kassa, and Ferguson, 2014). 3

6 quintile portfolio (0.265 vs per month, respectively). Similarly, the average IVOL of the high-retail ownership quintile portfolio is approximately 67% higher than the average IVOL of the low-retail ownership quintile portfolio (2.5% vs. 1.5% per month, respectively). These results confirm that retail investors prefer stocks with lottery-like payoffs, consistent with the results of Kumar (2009) and Han and Kumar (2013). We then investigate whether the IVOL puzzle is related to skewness by performing a dependent bivariate sort, first by skewness (SKEW) and then by IVOL. 3 We find that the magnitude and statistical significance of the IVOL puzzle monotonically increases as we move from low to high skewness. Specifically, we find a risk-adjusted high-minus-low IVOL return of only -38 basis points (bps) per month (t-stat. = -2.16) in the lowest skewness quintile but -181 bps per month (t-stat. = -6.57) in the highest SKEW quintile. Thus, the IVOL puzzle is concentrated in stocks with high skewness (i.e., lotterylike features). Next, we use the same bivariate sort methodology to investigate whether the IVOL puzzle varies by the type of investor (retail vs. institutional) owning the stock. We again find that the economic and statistical significance of the IVOL puzzle increases monotonically as we move from the low retail (high institutional) ownership ratio to the high retail (low institutional) ownership ratio quintiles. Specifically, the risk-adjusted returns of the high-minus-low IVOL hedge portfolio is -38 bps per month for the lowretail (high institutional) ownership quintile (t-stat. = -2.49) but -166 bps per month in the high-retail (low institutional) ownership quintile (t-stat. = -6.58). Thus, the IVOL puzzle is prominent in stocks held primarily by retail investors but almost non-existent in stocks held mostly by institutional investors. To further investigate the relation between the IVOL puzzle and a preference for skewness, we combine IOR and SKEW to create an index that is inversely related to the degree to which the stocks have 3 The methodology is well established in the literature. For example, Stambaugh, Yu and Yuan (2015) use a bivariate sort to show that the IVOL return relation is negative within overpriced stocks and positive within underpriced stocks. 4

7 skewness and high retail ownership. 4 We then perform a bivariate dependent sort, first by this index and then by IVOL, and find that the IVOL puzzle varies greatly by this index. Specifically, we find that the average raw return for the high-minus-low IVOL hedge portfolio is -2.06% per month, with a t-statistic of -5.25, in the lowest index quintile (i.e., stocks with high skewness and high retail ownership) but it is only -41 bps per month, with a t-statistic of -1.60, in the highest index quintile (i.e., stocks with both low skewness and low retail ownership). The results are qualitatively similar for five-factor alphas. We confirm these results using Fama MacBeth (1973) cross-sectional regression by interacting IVOL with a lottery indicator. Regardless of our proxy for lottery-like features, we find a negative coefficient for both IVOL and the interaction term, thus confirming that the negative relation between IVOL and returns is more pronounced for lottery-like stocks. We then examine how the IVOL puzzle varies with market conditions. Kumar (2009) shows that retail investors preference for U.S. state lotteries varies countercyclically with economic conditions. Therefore, if retail investors preference for stocks with lottery-like payoffs is (at least partially) responsible for the IVOL puzzle, then the IVOL puzzle should vary systematically across expansionary and contractionary periods. Using several commonly accepted state variables to identify market conditions, we show that (ex post) high-ivol stocks underperform low-ivol stocks more during bad economic times than during good economic times. 5 For example, using the NBER recession dummy to identify expansions versus recessions, we find that the average return associated with the IVOL discount is -1.90% per month during recessions and about half of that (or -0.90%) during expansions. 4 We create the index by first independently ranking firms into quintiles by institutional ownership and skewness, with the lowest (highest) rank of one (five) being high (low) skewness or high (low) retail ownership. The index is simply the sum of the independent quintile ranks. Thus, a low (high) index value corresponds to stocks with high (low) skewness and high (low) retail ownership. 5 The variables considered for our main results are the economic recession indicator NBER from the National Bureau of Economic Research (NBER), the overall stock market return in excess of the risk free rate (MKTRF), the default premium (DEF), the term premium (TERM), the Chicago Fed National Activity Index (CFNAI), the Chicago Board Options Exchange (CBOE) Standard & Poor s (S&P) 100 market volatility (VXO), and the economic uncertainty measure UNC defined by Jurado, Ludvigson, and Ng (2015). We also consider additional macroeconomic variables for robustness checks (see Section 5.2). 5

8 Finally, we combine the three features of our study stock characteristics (i.e., skewness), investor characteristics (retail vs. institutional), and market conditions (good vs. bad) and investigate whether the IVOL puzzle is most pronounced for stocks with high skewness and low institutional ownership during bad economic conditions. We first separate the sample into two by market conditions (good vs. bad) and then we perform bivariate dependent sorts, first by the rank-based index formed from a combination of skewness and institutional ownership ratio and then by IVOL. Regardless of the macroeconomic variable used to identify economic conditions, the IVOL puzzle and its variation with our rank-based index is more pronounced under bad economic conditions than under good economic conditions. We also find that most of the IVOL puzzle is in stocks with low index values (i.e., high skewness and high retail ownership) and under bad market conditions. Taken together, all of our results confirm our main hypothesis, that the IVOL puzzle varies by the type of investor (retail vs. institutional), the type of firm (high vs. low skewness), and economic conditions (expansionary vs. recessionary). This supports the idea that one of drivers of the IVOL puzzle is a time-varying (i.e., economic state-dependent) retail investor preference for lottery-like payoffs. The connection between lottery preferences and the IVOL puzzle has already received some attention in the literature. Ang, Hodrick, Xing and Zhang (2009) address the potential impact of idiosyncratic skewness on the negative relation between IVOL and returns and show a significant connection between the two. Boyer et al. (2010) find that expected skewness helps explain why stocks with high IVOL have low expected returns. Hou and Loh (2016) evaluate the various alternative explanations for the IVOL puzzle and show that lottery preference-based explanations capture a good portion of the puzzle. Using maximum daily returns over the past month (MAX) as a proxy for lotterylike payoffs, Bali et al. (2011) and Annaert, De Ceuster and Verstegen (2013) show that controlling for MAX reverses the IVOL effect shown by Ang et al. (2006, 2009). More recently, Cheon and Lee (2017) find that the IVOL puzzle is stronger in the high MAX quintile for international data. Overall, the empirical evidence points to the conclusion that lottery preferences can explain a significant portion of the IVOL puzzle. 6

9 The research discussed above attributes the IVOL puzzle to IVOL potentially being a proxy for skewness. If researchers could precisely estimate investors expectations of IVOL and skewness, then the contribution of each to prices and subsequent returns could be precisely determined. The inherent statistical and theoretical connection between skewness and IVOL, however, makes it almost impossible to disentangle their effects. 6 Thus, as done herein, identifying and testing alternative falsifiable hypotheses implied by the lottery preference argument provide useful information. Moreover, our results contribute to the IVOL literature in general by documenting novel empirical regularities in the negative relation between IVOL and returns. Specifically, while the prior literature focuses on the cross-sectional relation between IVOL and returns, we show systematic variation in the magnitude of the IVOL puzzle over time, conditional on economic conditions. The paper also contributes to the literature on the effects of preferences for lottery-like features by documenting that differences in lottery preferences across investors (retail vs. institutional) varies over time (i.e., across contractions versus expansions) and systematically affects the cross section of stock returns. The remainder of the paper is organized as follows. Section 2 discusses the background and related literature and outlines our hypotheses. Section 3 describes the data, the sample, and the general methodology employed in the analysis. Section 4 reports our main empirical results. Section 5 discusses the interpretation of these results, along with a battery of robustness checks. Section 6 concludes the paper. 2. Background and Hypothesis Development 2.1. Lottery Preferences and IVOL: Theoretical and Empirical Evidence Models that incorporate investors heterogeneous preferences for skewness typically consider the equilibrium with two types of agents, one type with mean variance preferences and the other with mean variance skewness preferences. Maximizing their expected utility, each type has a set of optimal price- 6 Measuring the market s view of the skewness of future returns using historical data presents several difficulties (Boyer et al., 2010). An alternative approach to measuring skewness from the historical distribution of returns is to examine the ability of skewness implied from option prices to predict future returns (Conrad, Dittmar, and Ghysels, 2013; Boyer and Vorkink, 2014; Bali and Murray, 2013). 7

10 contingent demands for each asset and the set of equilibrium asset prices are those that clear the market. Prominent examples of such models include those of Kraus and Litzenberger (1976), Conine and Tamarkin (1981), Harvey and Siddique (2000), Brunnermeier, Gollier, and Parker (2007), and Mitton and Vorkink (2007). These models all have common implications with respect to the cross section of returns and differences in optimal demand by type of investor and security (i.e., amount of skewness). First, with respect to the cross section of returns, equilibrium prices are such that securities with greater skewness will, ceteris paribus, have higher prices and hence lower subsequent returns. This implication has broad empirical support. Boyer et al. (2010) document a negative correlation between expected idiosyncratic skewness and stock returns. Similarly, Bali et al. (2011) show that stocks with a recent extreme positive return, which can be thought of as a crude measure of skewness, have low future returns. Conrad, Dittmar and Ghysels (2013) show a negative relation between risk-neutral skewness and future stock returns. Chang, Christoffersen, and Jacobs (2013) show that stocks with high exposure to innovations in riskneutral market skewness have low returns. If high IVOL is an inherent feature of lottery-type stocks, then IVOL will be negatively related to returns, since it either proxies for actual skewness or is associated with perceived skewness (e.g., Boyer et al., 2010; Hou & Loh, 2016). 7 Second, investors with greater preference for skewness will, in equilibrium, hold portfolios that are not mean variance efficient (i.e., appear to be underdiversified). In particular, they hold portfolios that overweight securities with skewness, while mean variance investors underweight such securities. This implication also has empirical support in the literature. Using proprietary data from a discount brokerage, Mitton and Vorkink (2007) and Kumar (2009) show that underdiversified investors hold highly skewed stocks. If this is indeed the case, when IVOL is calculated relative to common factors, this apparent underdiversification will directly result in higher IVOL and lower returns for stocks with lottery-like 7 For example, Kumar (2009) argues that stocks with greater IVOL are more likely to be perceived as lotteries because the level of IVOL could influence the estimates of idiosyncratic skewness. When volatility is high, investors might believe that the extreme return events observed in the past are more likely to be realized again. In contrast, if a stock with low price and high skewness has low IVOL, extreme return events observed in the past might be perceived as outliers and recurrence of the events is likely to be assigned a considerably lower probability. 8

11 features. Furthermore, if preferences for skewness are heterogeneous across agents, the negative skewness return relation will be more pronounced within the universe of stocks held primarily by investors with high preferences for skewness (since they are likely to pay a higher price for their desired characteristics). If lottery preferences are at least partially responsible for the IVOL puzzle, the puzzle should be more pronounced within the universe of stocks held by these investors Retail Investors and a Preference for Lottery-Like Features Recent literature indicates that certain types of investors are drawn toward stocks with speculative features such as high skewness and high volatility (e.g., Kumar, 2009; Dorn & Huberman, 2010; Han & Kumar, 2013). This behavior is consistent with a preference for lottery-like features, as captured by positive skewness in the theoretical models presented in Section 2.1. Kumar (2009) provides empirical evidence that such preferences are more prominent for retail investors, as opposed to institutional investors. Kumar (2009) also shows that individual investors preference for lottery-type stocks is stronger among socioeconomic groups that are known to exhibit strong preferences for state lotteries. Specifically, individual investors invest disproportionately more (relative to institutional investors) in stocks that have lottery-like features; within the individual investor category, socioeconomic factors (e.g., wealth, age, education, gender, marital status, ethnicity, and religious affiliation) influence people s attitudes toward lottery playing and gambling and, in turn, their preferences toward investing in stocks with lottery features. If retail (individual) investors exhibit a stronger preference for stocks with lottery features (compared to institutions who exhibit a relative aversion for those stocks), the intuition of the models outlined in Section 2.1 implies that the IVOL puzzle (and the negative skewness return relation) should be stronger for stocks that are primarily held by retail investors. Furthermore, investigating the relation between skewness, we find that IVOL and returns only within the subset of stocks held primarily by retail investors can help assess how important lottery preferences are as an explanation for the IVOL puzzle. If skewness is responsible for the IVOL puzzle, then, to the extent that the demand for lottery features is 9

12 most pronounced within the subset held primarily by retail investors, skewness should be able to explain a larger portion of the IVOL puzzle in this subset Time Series Variation of the IVOL Puzzle Another dimension of investor heterogeneity concerns how general economic conditions affect investors propensity to gamble (i.e., demand for high-skewness stocks). If economic conditions influence an individual s gambling preferences, the aggregate demand for lottery-type stocks should vary systematically with those conditions. Kumar (2009) shows that, similar to the demand for state lotteries, the aggregate demand for lottery-type stocks is higher during economic downturns. If lottery preferences induce the IVOL puzzle, then this predictable time variation in the demand for lottery-type stocks should result in predictable variations in the IVOL puzzle. Specifically, the IVOL puzzle should be more pronounced during bad economic conditions than during good economic conditions, especially within the universe of stocks held by retail investors. Thus far, the literature has presented scarce empirical evidence on systematic time variations in the IVOL puzzle. One notable exception is Kapadia (2006), who shows substantial time series covariation between high-ivol stocks; this paper proposes that market wide measures of skewness are related to a systematic variable that also drives common time series variation. Another exception is Stambaugh et al. (2015), who propose that IVOL captures arbitrage risk and that the negative IVOL return relation exists because high-ivol overpriced stocks are corrected less than high-ivol underpriced stocks due to shortselling constraints. Our study investigates time series variation in the IVOL puzzle and shows that it varies over time with investor skewness preferences. 3. Data and Methodology The data in this paper come from the Center for Research in Security Prices (CRSP), Compustat, and Thomson Reuters 13F databases. We start with the universe of all common stocks covered by the CRSP (i.e., SHRCD = 10 or 11) that are traded on the New York Stock Exchange, American Stock Exchange, or 10

13 NASDAQ. We compute institutional holdings based on 13F filings for all firms covered by Thomson Reuters. 8 Given that data on institutional ownership (Thomson 13F) start in 1980, our monthly sample is from 1980 to We exclude penny stocks (i.e., stocks with a lagged price under a $1). 9 Next, we describe the construction of our main proxies for IVOL, skewness, and institutional ownership. Control variables such as beta, size, book to market, and illiquidity are constructed following the standard asset pricing literature (see the Appendix for a description of each variable). We construct monthly IVOL following the standard methodology of Ang et al. (2006). Every month, we regress the daily excess returns of each stock on the daily Fama French (1996) three factors, as follows: 10 R i,t r f,t = β 0 + β 1 MKTRF t + β 2 HML t + β 3 SMB t + ε i,t where R i,t is the daily stock return, r f,t is the daily risk-free rate, and MKTRF t, HML t, and SMB t are the daily excess market return, value, and size factors, respectively. Monthly IVOL is calculated as the standard deviation of the daily residuals from this regression (i.e., IVOL = var(ε i,t ).) Similar to Amaya, Christoffersen, Jacobs, and Vasquez (2015), we calculate the sample skewness for stock i every month using daily data as Skew i = skew(r i,t r f,t ). 11 A stock month is included in the sample only if there are at least 15 trading days within the month for that stock. The institutional ownership ratio is calculated based on quarterly Thomson 13F data as the ratio of total shares held by institutions to total shares outstanding for every stock. We transform quarterly data for institutional holdings to a monthly frequency by assuming that institutional ownership is constant every month during a particular quarter. We report our results in terms of the IOR, which we interpret to be a proxy that is inversely correlated with retail ownership. Thus, low-ior stocks are considered more likely to be held by retail investors who seek lottery-like payoffs. 8 In order to not limit our sample, we consider zero institutional holdings for stocks that do not appear in the Thomson Reuters database. Our results are qualitatively similar and even stronger if we restrict our sample to only stocks with an IOR greater than 0% (see Section 5.2 for robustness). 9 The results are qualitatively similar with different restrictions. See Section 5.2 for robustness checks. 10 The results are qualitatively similar when we compute IVOL using a four-factor model (that includes the momentum factor) or a five-factor model (that includes illiquidity). 11 Amaya et al. (2015) use intraday data to compute weekly realized skewness. 11

14 We retrieve macroeconomic variables to characterize economic conditions from various sources. Specifically, we use the NBER recession dummy (NBER) from the St. Louis Fed Database and the default and term premiums (DEF and TERM, respectively) from Professor Amit Goyal s website. 12 DEF is the difference between the return on a portfolio of bonds with at least 10 years to maturity and the return on the long-term government bond series (e.g., Fama & French, 1989) and TERM is the difference between the long-term government bond return and the one-month T-bill return. We also use the Chicago Fed National Activity Index (CFNAI) from the Federal Reserve Bank of Chicago. Following Bali, Brown, and Tang (2017), we also consider the economic uncertainty indexes (UNC) developed by Jurado, Ludvigson and Ng (2015). We obtain the one-, two-, and 12-month-ahead economic uncertainty indexes from Professor Sydney Ludvingson s website and use the three-month index for most of our tests. 13 Finally, we use the S&P 100 market volatility index, VXO, from the CBOE. The variable definitions are available in the Appendix. Additional macroeconomic variables used in robustness checks generate qualitatively similar results (see Section 5.2). [Insert Table 1 here] Our final sample contains approximately 2 million firm month observations covering the period from 1980 to Table 1 presents descriptive statistics. Panel A shows that the average firm in our sample has a monthly average return of 1.07%, with a standard deviation of 14.78%. Monthly stock returns are, on average, positively skewed, with a mean skewness of The monthly average IVOL is 2.70%, which is comparable to previous results in the literature (e.g., Hou and Loh, 2016). Approximately 35.2% of the shares of an average firm are owned by institutions. 14 From the correlation matrix in Panel B, we observe that IVOL is negatively correlated with returns, institutional ownership, and size and positively correlated with skewness, beta, and illiquidity. In other words, high-ivol stocks typically have low returns, high skewness, a low institutional ownership ratio, and small firm size. 12 We thank Professor Goyal for making the data available at 13 The one-, three-, and 12-month indexes are highly correlated. The results using the one- and 12-month indexes are qualitatively similar and are available upon request. 14 It should be noted that the IOR has increased substantially in recent years. This relatively low level of institutional ownership reflects the fact that we are averaging over the whole sample period, from 1980 to

15 Panel C of Table 1 shows summary statistics for the macroeconomic variables. The average value of the NBER dummy is ; that is, only 12.6% of our sample is from during a recession. The average market risk premium for our sample period is 0.65% per month, with a very high monthly standard deviation of 4.46%. The macroeconomic variables considered (CFNAI, UNC, VXO) present a relatively high degree of autocorrelation up to three lags, while the return factors (MKTRF, DEF, TERM) are less autocorrelated. Since we argue that our results are driven by extreme bad states, we identify bad economic conditions as those in the worst quintile for each macroeconomic indicator. 4. Empirical Results We organize our results as follows. Section 4.1 documents the presence of the IVOL puzzle in our sample. Section 4.2 shows that investors have a general preference for skewness (i.e., high-skewness stocks have higher current period prices and thus lower future returns compared to low-skewness stocks). Section 4.3 shows that this general preference for skewness is heterogeneous across investor types. Specifically, retail investors prefer lottery-like (i.e., high-skewness) stocks more than institutions do. Section 4.4 shows that the IVOL puzzle is concentrated in high-retail ownership and high-skewness stocks and is more pronounced during poor economic conditions, when retail lottery preferences are stronger Presence of the IVOL Puzzle in Our Sample We start our empirical analysis by confirming that high-ivol stocks are associated with low returns in our sample. We form quintile portfolios using the previous month s IVOL values and we report valueweighted returns and five-factor alphas that control for market (MKTRF), size (SMB), value (HML), momentum (UMD), and Pastor Stambaugh (2003) illiquidity (ILLIQ). 15 [Insert Table 2 here] 15 We also find qualitatively similar results for CAPM alphas, Fama French (1993) three-factor alphas, and Carhart (1997) four-factor alphas. To mitigate the influence of extremely large firms, we use the previous month s log size instead of size when forming value-weighted portfolios. The results are qualitatively similar for value-weighted portfolios using the previous month s size or equal-weighted portfolios. 13

16 Table 2 shows that the average raw and risk-adjusted returns monotonically decrease when we move from low to high IVOL quintiles. The last column, H - L, shows the raw and risk-adjusted return differences between the highest- and lowest-ivol portfolios are, respectively, -1.02% and -0.97% per month. Our results are qualitatively similar to Ang, et al. (2006) Table VI that report a monthly raw return of -1.06% for their sample period (July 1963 to December 2000) High-Skewness Stocks Have Lower Returns After confirming the existence of the IVOL puzzle in our sample, we show that high-skewness stocks have higher valuations and lower subsequent returns. This result empirically establishes that (at least some) investors prefer stocks with lottery-like payoffs (i.e., high skewness). [Insert Table 3 here] Table 3 reports univariate sort results by skewness. The average raw return for the lowest (highest) skewness quintile portfolio is 1.12% (0.89%) per month, which generates a statistically significant monthly return for the skewness hedge portfolio of -0.23% per month (t-stat. = -2.26). We find comparable results for five-factor alphas (the high-minus-low skewness hedge portfolio generates a statistically significant five-factor alpha of -0.22% month, t-stat. = -2.56). Taken together, these results confirm that investors prefer high-skewness stocks (i.e., the current prices of high-skewness stocks are high relative to low-skewness stocks and thus skewness and future returns are negatively correlated) Retail Investors Have Stronger Preferences for Lottery-Like Payoffs We start by investigating whether lottery preferences are heterogeneous across investors and whether retail investors, as a group, manifest higher affinity for lottery-like payoffs compared to institutional investors (see the discussion in Section 2.2). We sort stocks into quintiles based on the previous month s IOR and report the average characteristics of these portfolios (as well as the differences between extremes) in Table 4. [Insert Table 4 here] 14

17 The results in Table 4 show a monotonic relation between skewness and the IOR. The stocks in the lowest IOR quintile portfolio (or, equivalently, the portfolio with the highest proportion of retail investors) have much greater skewness and IVOL than those in the highest IOR quintile portfolio. The average difference in skewness between the two extreme IOR quintile portfolios is and strongly statistically significant, with a t-statistic of These results support the argument that lottery preferences are heterogeneous across the two groups of investors and that retail investors have a higher preference for high-skewness stocks. Similarly, the relation between IVOL and IOR is also nearly monotonic. The average IVOL value of stocks held primarily by retail investors (2.5% per month) is approximately 67% higher than the average IVOL value of stocks held primarily by institutional investors (1.5% per month). This result supports the argument that investors with greater demand for skewness (i.e., retail investors) will, in equilibrium, hold less diversified portfolios than investors with a lower demand for skewness (i.e., investors with a skewness preference hold mean variance skewness-efficient portfolios that are mean variance inefficient). Consistent with Mitton and Vorkink (2007) and Kumar (2009), our results show that underdiversified investors seek lottery-like payoffs and hold highly skewed stocks. Table 4 also indicates sufficient variation in the IOR within our data set. For a typical stock in the lowest IOR quintile, only 4.5% of the shares are held by institutions, whereas 71.4% of the shares are held by institutions in the highest IOR quintile. This result supports the validity of this particular measure as a proxy for heterogeneous lottery-type preferences. 16 Taken together, the results in Table 4 show that highly skewed firms are held primarily by retail investors, whereas little-skewed stocks are held primarily by institutions, consistent with the hypothesis that investors have heterogeneous preferences for lottery-like payoff stocks. Additionally, the IVOL of a 16 Insufficient variation in the affinity for lottery-type preferences would make it unlikely for us to observe any empirical difference in the pricing of lottery characteristics. 15

18 firm is negatively correlated with the percentage of institutional holdings of the firm s shares. 17 We note some caveats to interpreting the IOR as a good (inversely related) proxy for a stock s skewness characteristics. One could argue that certain types of ownership influence a firm s corporate policies (e.g., via monitoring) and, hence, the characteristics of the firm s return distribution. If so, for a given ownership ratio percentage, institutions are likely to have a greater effect on corporate decisions than retail investors (since institutional ownership is likely more concentrated than retail ownership, which is typically diffuse). Another caveat relates to insider holdings, which may be substantial but are not likely to reflect skewness preferences or characteristics. While we acknowledge these caveats, they are not likely to introduce a bias, so we do not specifically control for them in using our IOR proxy Combined Effect of Lottery Characteristics, Retail Ownership, and Macroeconomic Conditions on the IVOL Puzzle This section presents our main empirical findings, which show the extent to which heterogeneous preferences for lottery-like payoffs are associated with the IVOL puzzle. Section investigates 1) whether the IVOL puzzle is stronger among stocks with lottery-like characteristics and 2) whether the IVOL puzzle is stronger among stocks held primarily by retail investors. Section investigates whether the IVOL puzzle varies with general macroeconomic conditions Lottery-like Characteristics, Retail Investors, and the IVOL Puzzle We perform two bivariate dependent sorts to examine the relation between the IVOL puzzle and either skewness or retail ownership. Specifically, we sort first by skewness (our measure of lottery-like characteristics) and then by IVOL. Then, we independently sort first by IOR and then by IVOL. If the IVOL puzzle is driven by retail investors preference for lottery characteristics, then the puzzle (i.e., a negative relation between IVOL and subsequent returns) should be most pronounced among stocks with 17 Untabulated results show that the negative correlation between skewness and future returns is more pronounced among stocks held primarily held by retail investors, supporting the notion that retail investors preferences for lottery stocks affect current stock prices and future returns. 16

19 lottery-like characteristics (i.e., high skewness) and stocks held primarily by retail investors (low IOR). The results are presented in Table 5. [Insert Table 5 here] Panel A of Table 5 shows that the low returns associated with high-ivol stocks become more pronounced as we move from the lowest to the highest skewness portfolio quintiles. The H - L column documents the five-factor risk-adjusted returns for the IVOL hedge portfolio for each skewness quintile. Moving from the low to high skewness quintiles, the risk-adjusted returns, respectively, are -0.38%, -0.65%, -0.81%, -1.11%, and -1.81% per month, showing an increase of over 370% in the IVOL puzzle from the low to the high skewness quintiles. A similar pattern exists for unadjusted returns (not tabulated here for brevity). An obvious caveat concerning the aforementioned results is the high correlation between skewness and IVOL in the data. To the extent that these are mechanically correlated, the conclusions drawn based on double sorts on these characteristics could be misleading. An alternative approach to test our hypothesis is to use retail ownership as a proxy for skewness demand rather than the skewness characteristics of the securities themselves. Given that retail investors are more likely to have lottery preferences (see Section 4.3), we expect the IVOL puzzle to be most pronounced among stocks that are primarily held by retail investors. The advantage of this approach is that, for the double sort, it relies on an investor characteristic (IOR) and a stock characteristic (IVOL) that are likely to be less mechanically correlated than two stock characteristics (skewness and IVOL) are. The results are presented in Panel B of Table 5. The IVOL puzzle generates an economically large -1.65% five-factor alpha per month for the lowest IOR quintile, but an alpha of only -38 bps per month for the highest IOR quintile. Further, the magnitude of risk-adjusted returns decreases as the IOR increases. We find similar results for average raw returns, CAPM alphas, Fama French three-factor alphas, and Carhart four-factor alphas (untabulated). Overall, Panel B confirms that the IVOL puzzle is related to something correlated with retail investor demand. 17

20 To investigate and disentangle the connection between retail investors preference for skewness and the IVOL puzzle, we reinvestigate the results above using a combined rank index computed from IOR and SKEW. Specifically, every month, we rank stocks, independently, into groups using the IOR and skewness. We then add the IOR and skewness ranks to form an index so that a low index value is associated with a low IOR (i.e., retail investors) and high skewness. Conversely, a high index value is associated with high institutional ownership and low skewness. Since a low index value identifies a stock with the most lottery-like features, we expect the magnitude of the IVOL puzzle to decrease as we move from a low to a high index value. The results in Table 6 show that this is indeed the case. [Insert Table 6 here] Panel A of Table 6 shows that the magnitude of the IVOL puzzle for raw returns monotonically decreases as we move from low to high index quintiles. The average IVOL puzzle in the lowest index quintile (i.e., low-ior and high-skew group) is an economically large -2.06% per month and statistically significant, with a t-statistic of In contrast, the IVOL puzzle in the highest index quintile (i.e., high-ior and low-skew group) is only -41 bps and statistically insignificant, with a t- statistic of The results in Panel B show the same overall pattern when we look at average riskadjusted returns (five-factor alphas). Table 7 examines our double-sort results using Fama MacBeth (1973) regressions that include, in addition to standard variables (e.g., BETA), IVOL, SKEW, IOR, and interaction terms. [Insert Table 7 here] Model (1) presents the baseline regression that shows after controlling for beta, firm size, the book to market, momentum, and Amihud s (2002) illiquidity measure that IVOL and returns are negatively related (with high statistical significance). Model (2), which adds skewness to the baseline regression, continues to find a negative relation. Model (3) next adds an interaction term, which is the product of IVOL and a high skewness indicator that equals one for stocks in the highest skewness quintile and zero otherwise. IVOL and returns continue to be negatively related and the coefficient of the interaction term is negative (albeit not statistically significant), again indicating that the IVOL puzzle is 18

21 stronger for high-skewness stocks. For Models (4) and (5), we substitute IOR for skewness and estimate models analogous to Models (2) and (3). The results for Models (4) and (5) again show a negative relation between IVOL and subsequent returns. Importantly, the interaction term between IVOL and the dummy variable for high-retail investor holdings (i.e., a dummy equal to one for stocks in the lowest IOR quintile and zero otherwise) is negative and highly statistically significant. This result confirms that the IVOL puzzle is more pronounced for stocks with high retail holdings. Finally, we repeat this analysis using the rank-based index that combines skewness and the IOR. Model (6) is the baseline regression for this specification, while Model (7) adds the interaction variable equal to the product of IVOL and the lottery/retail index. Model (7) shows that the IVOL puzzle is more pronounced for stocks with high retail ownership and high skewness. Taken together, the results in Tables 5 to 7 provide clear cross-sectional evidence of a connection between retail investors preferences for lottery-like stocks and the IVOL puzzle. Specifically, the IVOL puzzle is most pronounced among stocks with high skewness that are held by retail investors. The next section examines how these results vary systematically over time with market conditions The IVOL Puzzle and Macroeconomic Conditions If the IVOL puzzle is related to the pricing of heterogeneous preferences for skewness, it should vary over time with the degree of heterogeneity. Following Kumar (2009), who shows that individuals (i.e., retail investors) have greater (less) demand for lottery-like features (e.g., skewness) during poor (strong) economic conditions, we investigate whether the IVOL puzzle varies with economic conditions (especially within the universe of lottery-like stocks held by retail investors). To capture variations in economic conditions, we use a variety of commonly accepted macroeconomic variables to characterize the economic condition of each sample period (see the Appendix for the definitions of the variables). We use each variable to assign each sample period to one of two categories, either a good or bad economic condition period. The bad economic condition is meant to capture an extreme condition and is based on the highest/lowest quintile associated with the poorest 19

22 state for a specific macroeconomic variable; the good economic condition is assigned for all the other four quintiles. 18 With periods thus assigned to good and bad economic conditions, we run two types of tests: 1) we form cross-sectional quintile portfolios sorted by IVOL and calculate the magnitude of the IVOL puzzle conditional on good or bad economic conditions and 2) we run Fama MacBeth (1973) regressions and test whether the coefficient of IVOL differs between the two different economic conditions. 19 The results are presented in Table 8. [Insert Table 8 here] Panel A of Table 8 shows that the magnitude of the IVOL puzzle is more pronounced during bad economic conditions relative to good economic conditions for every macroeconomic variable considered. In fact, the results indicate that the IVOL puzzle does not exist during good economic conditions when we use excess market returns (MKTRF) and the S&P 100 index option volatility (VXO) to identify economic conditions. This finding indicates that the unconditional results presented for the overall sample (in Table 2) are driven entirely by the months with bad economic conditions. The last row of Panel A shows that, for most of the macroeconomic variables considered, IVOL is associated with statistically significantly more negative subsequent returns in bad relative to good economic states. Panel B investigates the same issue using Fama MacBeth cross-sectional regressions that control for beta, size, book to market, momentum, and illiquidity, in addition to IVOL. For all of the macroeconomic indicators except DEF, the IVOL coefficient is statistically significantly lower during bad than good economic conditions. These results are consistent with preferences for skewness varying with economic conditions, generating the IVOL puzzle. A further implication of the lottery explanation is that the time series variation of the IVOL puzzle is especially pronounced within the universe of stocks held by retail investors. To quantify these effects, we use the IOR/skewness rankings index first used for Table 6. Using this index, we repeat the 18 The results are qualitatively similar (albeit weaker) when we split periods into the worst and best halves. These results are available upon request. 19 Since one could argue that the magnitude of IVOL differs between good and bad economic times, we standardize the IVOL variable so that the coefficients are comparable across the two states. 20