Master Thesis Finance THE ATTRACTIVENESS OF AN INVESTMENT STRATEGY BASED ON SKEWNESS: SELLING LOTTERY TICKETS IN FINANCIAL MARKETS

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1 ) Master Thesis Finance THE ATTRACTIVENESS OF AN INVESTMENT STRATEGY BASED ON SKEWNESS: SELLING LOTTERY TICKETS IN FINANCIAL MARKETS Iris van den Wildenberg ANR: Master Finance Supervisor: Dr. Rik Frehen Second reader: Dr. David Hollanders

2 I. INTRODUCTION Motivated by the fact that people irrationally spend countless dollars on purchasing lottery tickets in the hope of hitting the big jackpot, several models in the asset pricing literature investigated the impact of lottery-like stocks on investor decision making. These stocks offering a small chance of a very large pay-off are characterized by return distributions having a fatter and/or longer right tail, hereby being positively skewed. Recent behavioral theories concur that likewise lottery tickets, lottery-like stocks within financial markets are very popular among a large group of investors (Mitton and Vorkink, 2007; Barberis and Huang, 2008). Applying the market model of supply and demand consequently implies higher prices and several empirical studies indeed confirm that investors are paying a premium and therefore get lower returns for this type of stocks (Zhang, 2005; Boyer, Mitton and Vorkink, 2010). But if a significant part of investors is indeed willing to give up returns in order to be exposed to a greater possibility of outsized gains, does this not directly imply a very promising trading strategy within reach for the other group of traditional, often institutional, investors? Just like casinos, charities and other lottery organizations can make lots of money by selling lottery tickets, it seems that investors can easily realize gains by taking the other side of the trade, collecting premiums which lottery-preferring investors are willing to pay. Although the existence of such a prospering trading strategy in which one takes a long position in low expected skewness stocks and short position in high expected skewness stocks seems to be the obvious consequence of the equilibrium results of Boyer et al. (2010), we argue that making money in the financial markets by selling lottery-like stocks is not as straightforward as presented above for two reasons. The first argument starts with the observation that the autocorrelation between lottery-like return properties is very small. Lottery-like stocks are characterized by positive idiosyncratic skewness, and autocorrelation of skewness is generally demonstrated to be very low. This implies that it is very likely that an investor buying stocks today in pursuit of skewness exposure, does not end up having the exposure hoped for during the next months. Our own results show an extremely low autocorrelation coefficient of if we take the idiosyncratic skewness measure based on 1 month of daily returns, especially when comparing the result with the reported autocorrelation of for example volatility (0.689). Considering the less noisy idiosyncratic skewness measure based on 60 months of daily data, the non-overlapping autocorrelation (lagged 60 months) is still as low as As a consequence, keeping exposure to the lowest idiosyncratic skewness portfolio over time requires rebalancing very often, making a skewness-based trading strategy expensive. The more the investor rebalances to keep exposure to the desired long-short portfolio, the higher the transaction costs, like the bid-ask spread. Moreover, since small, start-up companies are often the ones having higher upside 1

3 potential, most lottery-like stocks are small and illiquid. Adding up the fact that the highest bid-ask spreads are typically seen in the universe of these illiquid shares, it can be argued that especially for this desired long-short portfolio rebalancing is costly. Thus, transaction costs will reduce the potential premium to be earned by the strategy, making us doubt whether selling lottery-like stocks indeed yields a positive net return in practice. Secondly, especially in the universe of illiquid shares, there may be only limited availability for immediate trading and some stocks may not be available to sell short, making the long-short strategy practically challenging. Hence, our research question is: Is selling lottery-like stocks a lucrative strategy for the rational mean-variance investor? The sample of this study is, for the sake of comparison, in line with that of Boyer et al. (2010) and consists of all ordinary common shares listed on the NYSE, AMEX and NASDAQ from the beginning of February 1978 through December 2005, enabling us to perform idiosyncratic skewness predictions for January 1988 through November Consistent with recent theories, our results show that returns of the lowest expected idiosyncratic skewness quintile are significantly higher than those of the highest skewness quintile. However, after subtraction of transaction costs and adjusting for the Fama-French (1993) risk factors and Pastor-Stambaugh (2003) liquidity factor, an investment strategy in which one goes long high expected skewness stocks and goes short low expected skewness stocks yields no significant net return. Rebalancing costs for lottery-like stocks are so large that they completely offset the positive returns attainable by the skewness based long-short strategy. Although we find that selling lottery-like stocks is not a profitable investment strategy, it turns out that investors can make a riskadjusted return, net of transaction costs, of about 2% on a yearly basis by the long-only strategy in which they buy the lowest expected skewness quintile. Therefore, it can still be optimal for traditional mean-variance investors to deviate from their optimal Markowitz (1952) portfolio, hereby collecting excess return. The results of this study are relevant to several parties. First, our research builds on the academic results of Boyer et al. (2010) by taking a position on the other side of the deal and trying to translate their results into a practically feasible and profitable trading strategy. Pursuing this, we highlight the trade-off between desired low skewness exposure and rebalancing costs in the implementation of the strategy. The finding that selling lottery-like stocks is not attractive for investors when taking transaction costs into account, provides empirical support for the reasoning in which the skewness premium is considered as irrational overpricing. Thus, this study questions previous theoretical models that regard the skewness premium as a market equilibrium price. Furthermore, the net returns of the long-only strategy give a possible explanation for the evidence that investors deviate from the theoretically optimal mean-variance portfolio developed by Markowitz (1952). Finally, our research 2

4 project provides guidance to portfolio managers by giving insight in a new investment strategy to make sustainable above-market returns. The rest of this paper is structured in the following way. Section II provides a review of the relevant literature, leading to the hypotheses tested in this study. Subsequently, the methodology, sample selection and empirical results are discussed in section III, and the conclusion of this study is presented in section IV. 3

5 II. LITERATURE REVIEW AND HYPOTHESES DEVELOPMENT One of the most fundamental assumptions that conventional economics and finance make is that people are rational wealth maximizers who seek to increase their own well-being. According to conventional economics, emotions and other extraneous factors do not influence people when it comes to making economic choices. Applying this traditional view to investors, Markowitz (1952) concludes that all investors should be mean-variance optimizers seeking to maximize the Sharpe ratio of their portfolios. However, reality shows that investors actions frequently deviate from this theoretically optimal behavior. Lottery-like assets, just like normal lottery tickets, are popular among investors and both theoretical and empirical research shows that lottery-like return properties, characterized by positive expected skewness of returns, are priced. Skewness pricing is explained in the asset pricing literature by both rational risk-based theories and by theories involving irrational or nonstandard investor behavior. The first group of theories extends the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965) to three moments and suggests that an asset s co-skewness with the market portfolio should be priced (Kraus and Litzenberger, 1976; Harvey and Siddique, 2000). In line with these studies, Simaan (1993) has shown that co-skewness with the market is a priced factor and generalizes the Security Market Line of the CAPM to a flat surface called the Security Market Plane, which includes the third moment risk premium as an additional dimension. In these extended CAPM models, the ultimate determinant of required risk premiums is the timing of losses the covariance with bad times as financial crises and economic recessions. Therefore, the main aspect of skewness here is not asymmetry in the asset s stand-alone returns, but rather the contribution to the portfolio s skewness. All else being equal, investors prefer assets that earn high returns during volatile markets, thus making their portfolios more positively skewed, and may be willing to pay a premium for this characteristic. Selling lottery-like assets that pay off in bad times is an investment that incurs rare but large losses in bad times, resulting in expected co-skewness with the market, and therefore justifies a high risk premium. Because the above risk-based models are set in the context of fully diversified investors, they imply that a stock s idiosyncratic skewness should be irrelevant. However, others have noted that because diversification erodes skewness exposure, some investors may remain underdiversified in order to capture return skewness, and thus idiosyncratic skewness may be relevant (Simkowitz and Beedles, 1978; Conine and Tamarkin, 1981). Ilmanen (2012) even argues that demand for lottery-like stocks is mainly represented by the idiosyncratic part of skewness. In his paper, he points out that positive 4

6 skewness can reflect both a truncated/thinner left tail or a thicker/longer right tail. A thinner left tail represents a reduced likelihood of large losses, hereby limiting the downside, especially in financial crises, and lowering systematic risk. An enhanced right tail represents the demand for lottery tickets and emphasizes idiosyncratic opportunities to enhance the upside. Therefore, he states that idiosyncratic skewness is the key aspect if we are looking for a measure to quantify lottery-like characteristics. In line with the theoretical structure employed by Ilmanen (2012), we argue that the literature explaining the pricing of lottery-like assets using firm-specific skewness, as opposed to co-skewness, can be divided into at least two theories: lottery preferences and investor overweighting of lowprobability events. Mitton and Vorkink s (2007) model of heterogeneous preference for skewness represents the first of these theories. Their model suggests that although some investors are indeed mean-variance optimizers seeking to maximize the Sharpe ratio of their portfolios, another meaningful segment of investors has a direct preference for positive skewness. This last group of lotto investors places a higher utility on positively skewed assets and is willing to sacrifice their Sharpe ratio in return for higher skewness. Recognizing the trade-off between skewness and diversification, these investors with a strong desire for upside potential may deliberately avoid diversification in favour of a small chance of a very large pay-off. Conversely, the traditional mean-variance investor is only willing to give up diversification, hereby increasing the volatility of its portfolio, if they are compensated for this in the form of additional return. When buying positively skewed assets from the traditional investor, the lotto investor passes that renumeration to the traditional mean-variance investor by paying a premium for those assets, hereby giving up return. In this way, in the model of Mitton and Vorkink (2007), supply and demand by those two groups of investors together determine the risk premium paid for lotterylike assets in market equilibrium. The other argumentation assumes, in contrast to the model of Mitton and Vorkink (2007), identical investor preferences, but considers the combination of irrational investor behavior and limits to arbitrage as the cause of persisting overpriced lottery-like assets. This line of reasoning is based on the prominent prospect theory of Kahneman and Tversky (1979). Prospect theory calculates values and is inconsistent with expected utility theory, applied in the literature discussed before, for two fundamental reasons. Firstly, whereas utility is dependent on final wealth, value is defined in terms of gains and losses (deviations from current wealth). Secondly, whilst utility is necessarily linear in the probabilities, value is not. Barberis and Huang (2008) are the first ones linking cumulative prospect theory to skewness properties of assets and, based on this theory, argue that positively expected future skewness even idiosyncratic skewness is priced. They state that, next to the sophisticated, 5

7 traditional investors maximizing their Sharpe ratio, there is a large group of less sophisticated prospect theory traders in the economy whose decisions are led by behavioral factors. Subsequently, higher prices and lower returns for lottery-like assets are explained by these traders adjusting their tangency portfolios based on irrational tendencies belonging to prospect theory. The figures below show the prospect theory value function and probability weighting function, facilitating the explanation of patterns of risk seeking and risk aversion. Figure 1: Value Function The graph plots the value function proposed by Tversky and Kahneman (1992) as part of cumulative prospect theory. The x-axis shows deviations from a reference point and the y- axis represents subjective value. The solid line shows that that the value function is normally concave for gains (implying risk aversion) and convex for losses (risk seeking). The dotted line emphasizes that the function is generally steeper for losses than for gains (loss aversion). - Source: Ilmanen (2012) - Figure 2: Probability-Weighting Function The graph plots the probability weighting function proposed by Tversky and Kahneman (1992) as part of cumulative prospect theory. The dotted 45 degree benchmark represents linear probability weighting used in expected utility theory, while the solid line corresponds to the estimated weighting function employed by behavioral investors. - Source: Ilmanen (2012) - 6

8 As the concave line in the winning part of the value function (Figure 1) indicates a tendency toward risk aversion when gambles involve only gains, the shape of the value function would imply a negative rather than a positive premium for lottery-like assets. As such, the overpricing of lottery-like assets cannot be explained by the shape of the value function. Rather, key in the prospect theory explanation for overpriced lottery-like assets is the overweighting of low-probability events (Figure 2). In this way, behavioral investors are attracted to lottery-like assets because they offer exceptionally large pay-offs with low probabilities, that investors overweight. Such overweighting of small probabilities can be strong enough to reverse the sign of risk appetite in the value function. Hence, assets offering a small chance of outsized returns can encourage risk seeking, despite a general propensity toward risk aversion when considering gains. Thus, this line of thinking considers the predicted skewness premium as mispricing resulting from irrational behavior by a group of less sophisticated investors. As these less sophisticated, individual investors, who are more likely to make buying and selling decisions based on behavioral and irrational factors, play a more important role for stocks of firms with lower market capitalizations, skewness overpricing is predicted to hold primarily within this subset of stocks. However, mispricing due to probability weighting can only subsist for stocks where the forces of arbitrage are likely to be weak for example, for stocks with low market capitalizations, illiquid stocks and stocks with low institutional ownership (Barberis, Mukherjee and Wang, 2014). Therefore, overpricing of lottery-like stocks can persist since it is very costly to take the position on the other side of the trade, as a consequence of illiquidity and low institutional ownership within this group of stocks. In summary, recent theories concur that lottery-like return properties are priced components of stock returns. Despite this extensive theoretical basis for the pricing effects of skewness, empirical tests of the relation between lottery-like stock properties and returns are far more scarce. This is caused by the fact that ex-ante skewness is difficult to measure. The primary obstacle is that, opposed to variances, skewness properties are not stable over time, and using lagged skewness does not effectively measure expected skewness (Harvey and Siddique, 1999). Zhang (2005) finds a negative relation between total skewness and average returns using cross-sectional returns of a group of comparable stocks as an attempt to estimate skewness. Boyer et al. (2010) contribute to this empirical evidence by developing a predictive regression model as a better measure for idiosyncratic skewness. Using their model of expected skewness, they find that the Fama-French (1993) alpha for the lowest expected skewness quintile exceeds the alpha of the highest expected skewness quintile by 1.00% per month. Moreover, estimating Fama-MacBeth (1973) regressions, they find a negative relation between their expected skewness measure and average returns. 7

9 Thus, Boyer et al. (2010) conclude that skewness is priced and provide us with the most detailed and sound empirical study about the idiosyncratic skewness premium. Having in mind their evidence that, on average, the return for the lowest skewness quintile is higher than the return of stocks in the highest skewness quintile, this study takes the point of view from the traditional, often large institutional, investor to determine whether the pricing of skewness can be translated into a profitable trading strategy. As a starting position for our research question, we will use Boyer et al. s (2010) model to predict skewness and to describe the statistics of the portfolios sorted on predicted skewness. Based on the existing literature and the results found by Boyer et al. (2010), our first hypothesis is stated as follows: H1: The difference in returns between the lowest and highest expected idiosyncratic skewness portfolios is, on average, positive. After testing the above hypothesis, we will distinguish our study from that of Boyer et al. (2010) by checking whether selecting a portfolio based on their measure predicting skewness over a horizon of 60 months, also results in real exposure to skewness when taking into account that investors do not keep these stocks in their portfolio for this full period. Since the autocorrelation of skewness is in general low and lagged skewness is demonstrated to be only a very weak predictor of expected skewness (Harvey and Siddique, 1999), investors need to rebalance often to keep exposure to the desired portfolio. In other words, as skewness is non-persistent, it is difficult to get exposure to skewness. When rebalancing monthly, this implies that part of the portfolio is already traded and exchanged for other stocks after one month. Thus, some stocks get only one month time (or at least a way shorter period than 60 months) to expose the investor to the forecasted and desired skewness. Since the portfolios are sorted on their expected skewness over a period of 60 months, we are skeptic about the realized exposure to skewness and our second hypothesis is the following: H2: Realized exposure to idiosyncratic skewness is not significantly higher in the high expected skewness portfolio than in the low expected skewness portfolio. The very low autocorrelation of skewness and resulting need to rebalance frequently in order to keep exposure to the desired long-short portfolio, also makes us doubt whether traditional investors can really profit from the pricing of idiosyncratic skewness in practice. Frequent readjustment of the portfolio, necessary to prevent divergence from the desired long-short portfolio, entails considerable transaction costs, especially among small and illiquid stocks. These costs that investors incur when selling and buying securities include brokers commissions and bid-ask spreads, of which the last one is by far the largest component (Damodaran, 2006). For our argumentation, it is also important to realize that small and illiquid stocks usually have the highest upside potential and therefore often 8

10 belong to the highest idiosyncratic skewness portfolio. Adding up the fact that the highest bid-ask spreads are typically seen in the universe of these illiquid shares (Stoll, 1978; Jegadeesh and Subrahmanyam, 1993), it can be argued that buying and selling stocks belonging to the highest skewness quintiles will be most expensive in terms of transaction costs. Since, in order to track our desired skewness portfolio, we very frequently need to buy and sell exactly this group of stocks, we expect transaction costs to be large enough to offset the estimated gains from maintaining a long low skewness and short high skewness portfolio. Relating this information to our study, we arrive at the last hypothesis: H3: The difference in returns between the lowest and highest expected idiosyncratic skewness portfolios is, net of transaction costs, not significantly positive anymore. Testing the above hypotheses will help us to determine whether selling lottery-like stocks can be a profitable investment strategy. 9

11 III. METHODOLOGY, SAMPLE SELECTION AND RESULTS Skewness prediction model In order to test our hypotheses, we aim to make five portfolios each month based on expected skewness. In purpose of this, daily data are used to arrive at a monthly rebalanced portfolio and its accompanying monthly returns and skewness characteristics, hereby following the idiosyncratic skewness prediction model developed by Boyer et al. (2010). This model of expected skewness incorporates past returns, trading volumes and the firm characteristics size, industry and the exchange on which stocks are traded. In this paragraph we explain their model and motivate its composition. Suppose that the investment horizon over which investors are hoping to experience an extremely positive outcome is T months, that S(t) denotes the set of trading days from the first day of month t-t+1 through the end of month t, and that N(t) indicates the number of days in this set. Let ε i,d be the regression residual using the Fama-French (1993) three-factor model on day d for firm i, where the regression coefficients that define this residual are estimated using daily data for days in S(t). In addition, let iv i,t and is i,t denote historical estimates of idiosyncratic volatility and skewness from firm i using daily data for all days in S(t), and let k be the number of factors in the regression. Iv i,t and is i,t1 can then be defined as follows: iv i,t = ( 1 1 ε N(t) 1 d S(t) i,d 2 2 ) (1) is i,t = 1 N(t) 2 3 d S(t) ε i,d iv i,t 3 (2) In pursuit of our monthly rebalanced portfolio based on expected skewness, rather than the measure of historical skewness defined in equation (2), we ultimately need a quantification of expected skewness over a horizon of T months for firm i at the end of month t, E t[is i,t+t]. To simulate investors best estimates of expected skewness, only information available to investors at the end of month t can be used. The first step in modelling investor perceptions of expected skewness is to separately run insample cross-sectional regressions at the end of each month t, is i,t = β 0,t + β 1,t is i,t T + β 2,t iv i,t T + λ t X i,t T + ε i,t, (3) where X i,t-t is a vector of additional firm-specific variables observable at the end of month t-t. 1 In order to avoid noisy results, the variables iv i,t and isi,t are set to a missing value at the end of month t if the number of observable returns in the set S(t) is less than

12 Time subscripts on regression parameters are included to emphasize that these parameters are estimated using information completely observable at the end of month t. By separately running these cross-sectional regressions each month, the parameters from Equation (3) can be estimated. Subsequently, these estimated monthly parameters, together with information observable at the end of each month t, can now be used to predict skewness for each firm T months ahead, E t [is i,t+t ] = β 0,t + β 1,t is i,t + β 2,t iv i,t + λ t X i,t. (4) This approach generates estimates of expected skewness each month, while allowing the relation between firm-specific variables and skewness to vary across time. In this study, we follow Boyer et al. (2010) by using a horizon of T = 60 months in the above model. Boyer et al. (2010) argue that investors typically focus on a stock s long-run upside potential (e.g. earlystage investments), as opposed to high returns over the short run, and therefore choose to focus on a multiple-year horizon. Since the choice of horizon is subjective, in absence of a better reasoning we use the same time frame in this study for comparability. This implies that an investor needs 10 years of prior data to estimate the parameters of Equation (3) and to generate an estimate of expected skewness as in Equation (4). The firm-specific variables, X i,t-t, that are used in the cross-sectional regressions defined in Equation (3) of the model include momentum (mom i,t-t), turnover (turn i,t-t), and dummy variables which are further 2 defined below. Mom i,t-t, defined as the cumulative return for firm i over months t T 12 through t T 1, is included based on Chen, Hong and Stein s (2001) finding that past returns are negatively correlated with forecasted skewness. The inclusion of turn i,t-t, defined as the average daily turnover of the firm over month t-t, is justified by the model of Hong and Stein (2003), which predicts that negative skewness is most pronounced during periods of heavy trading volume. For this computation, daily turnover for day d is defined as volume for day d divided by shares outstanding reported on day d. In addition to mom and turn, three sets of firm-specific dummy variables are used as independent variables in the cross-sectional regressions of this skewness prediction model developed by Boyer et al. (2010). First, to control for size, dummy variables for small- and medium-sized firms are included, where firms are grouped into three equally sized categories of small, medium, and large based on market capitalization. Second, industry dummy variables are included in the regressions, with industry 2 The variable momi,t-t, is set to missing at the end of month t T if a security does not have a complete set of 11 monthly returns in CRSP from t T 11 through month t T 1. 11

13 classifications based on each company s primary two-digit SIC code. 3 Third, a NASDAQ dummy is included to control for the unique institutional features of the NASDAQ exchange, such as differences in turnover measurement. 4 Sample selection The sample of this study is, for the sake of comparison, in line with that of Boyer et al. (2010) and consists of all ordinary common shares 5 listed on the NYSE, AMEX and NASDAQ from the beginning of February 1978 through December Since, as explained above, we need 5 years of previous daily returns for our estimates of is i,t and another 5 years to estimate our skewness prediction parameters, this sample enables us to perform idiosyncratic skewness predictions from January 1988 onwards. Stock returns in our regressions are measured in month t+1 (one month after expected skewness is estimated), so this sample will enable us to use expected skewness estimates from January 1988 until November The starting period is motivated by the fact that turnover data for NASDAQ stocks are only available on a widespread basis from January 1983 onwards. Since turnover data are used as an input to estimate our skewness prediction model parameters, our model only enables us to begin predicting skewness in January 1988 (assuming our horizon to estimate our expected skewness model is 5 years, or T=60 months). Thus, skewness estimates are estimated from January 1988 through November 2005 for a sample of 17,153 different shares. Daily and monthly stock returns and firm characteristics used to predict skewness are retrieved from the Center for Research and Security Prices (CRSP) database. Market capitalization data to determine size dummies and value-weighted results are originating from COMPUSTAT. Moreover, CAPM and Fama-French risk factors are downloaded from the Fama-French online data library and the Pastor- Stambaugh factors are retrieved from Pastor s online liquidity factor database. Portfolios sorted on predicted skewness Having obtained the idiosyncratic skewness predictions by the model described above, the next step is to make monthly portfolios sorted on predicted skewness in order to test the asset pricing implications of expected skewness. Stocks are sorted into quintile portfolios based on our constructed estimates of E t[is i,t+t], as outlined in equations (3) and (4) using a horizon of T=60 months, at the end of each month from January 1988 through November Table 1 provides information about the returns of the five quintiles, where the first quintile represents firms with the lowest predicted 3 Ken French s 17-industry classification, based on the SIC-codes, is used to generate dummies for 16 out of the 17 industries defined on his website. 4 Stocks traded on the NASDAQ are identified by CRSP exchange code 3. 5 This implies that predicted skewness estimates are only estimated for shares with CRSP share code 10 or

14 idiosyncratic skewness, and the fifth quintile represents firms with the highest predicted idiosyncratic skewness. Table 1 - Returns of Portfolios by Level of Predicted Skewness The table reports return characteristics over month t+1 for the five quintiles, where the first quintile represents stocks with the lowest predicted skewness, and the fifth quintile represents stocks with the highest predicted skewness. Column 1 reports the time-series average of the value-weighted portfolio returns over month t+1. Columns 2 and 3 report the time-series standard deviation and skewness of the portfolio returns. Columns 4, 5 and 6 report estimated alphas and accompanying Newey-West t-statistics (in parentheses) of the regressions of the monthly returns, in excess of the risk-free rate, on the CAPM, Fama French (1993) and extended Fama-French risk factors. The CAPM model controls for the market risk factor, the Fama-French three-factor model adds the size and value risk factors and the extended Fama-French model controls, next to these three conventionally used risk factors, for the Pastor-Stambaugh traded liquidity factor. The last row of this table shows for each statistic the difference in (risk-adjusted) returns between portfolio 1 and 5 and the corresponding Newey-West t-statistics of the paired-sample t-tests (in parentheses). (1) Mean return (2) Standard Deviation (3) Skewness (4) CAPM Alpha (5) FF Alpha (6) Extended FF Alpha 1 (Low) *** (2.88) * (-1.68) (-0.31) (-0.38) 5 (High) ** (-2.01) * ** (1.92) (2.46) *, **, *** Significant at the 0.10 level, 0.05 level and 0.01 level, respectively *** (3.63) ** (-2.12) (-1.46) ** (-1.99) *** (-3.28) *** (3.76) *** (3.58) ** (-2.04) (-1.62) ** (-2.16) *** (-3.34) *** (3.80) Column 1 of Table 1 reports the time-series average of the realized, value-weighted portfolio returns over month t+1. 6 Columns 2 and 3 provide additional information about the distribution of these returns and report the time-series standard deviation and skewness of the portfolio returns. As can be seen, the higher the skewness quintile, the higher the time-series volatility of portfolio returns in month t+1. Moreover, realized returns in month t+1 of the highest expected idiosyncratic skewness quintile are indeed positively skewed over time, whereas total skewness for all other quintiles is negative. The first column shows that except for an increase between the second and third quintile, mean returns decline from the first to the fifth quintile and that the largest part of the decline in returns occurs between the fourth and fifth quintile. Since macro-economic factors and other market 6 Market capitalization data, to compute the weights, are obtained from Compustat and calculated by the amount of outstanding shares of i multiplied by the last non-missing closing price of share i in month t. 13

15 circumstances influencing the returns in month t are the same for each quintile, the monthly returns of the quintiles are related. Therefore, we perform a matched-pair t-test for the return that is the result from a long position in quintile 1 and a short position in quintile 5 each month. The last row of column 1 shows the average difference in return between portfolio 1 and 5 and the corresponding Newey- West t-statistic 7 of the paired-sample t-test. This test provides us with the conclusion that mean returns are substantially higher in the first quintile (1.22%) compared to the fifth quintile (0.48%), a difference of 0.75% per month (t-statistic=1.92). In other words, the investment strategy in which one goes long the value-weighted first quintile portfolio and goes short the value-weighted fifth quintile portfolio, implemented by monthly rebalancing, makes an average return of 0.75% per month. Although the p-value is just above the 5% significance level (0.056), the result of the 1-5 portfolio is economically significant. Especially when converting the result to a yearly return and taking into account that an investor does not need to bring in own initial capital for this strategy, as it concerns a long-short portfolio, the magnitude of the attainable return is considered to be large. However, these results do not provide us with a complete analysis and conclusion about our first hypothesis. For this, we need to adjust the difference in return between the first and fifth predicted skewness quintile for risk. Therefore, we regress the monthly returns of the skewness portfolios, in excess of the risk-free rate, successively on the risk factors of the CAPM, Fama-French and our extended Fama-French model. The CAPM model controls for the market risk factor, whereas the Fama- French three-factor model adds the size and value risk factors. Finally, the extended Fama-French model combines the three conventionally used Fama-French factors with the Pastor-Stambaugh (traded) liquidity factor in order to control for the correlation between expected skewness and liquidity, thus making sure we only pick up the effect caused by skewness differences. Alphas (constants) of the regressions can be considered the risk-adjusted returns of the portfolios and are, together with the accompanying Newey West t-statistics, reported in columns 4, 5 and 6 of Table 1 for each of the quintiles. As can be seen in these columns, the risk-adjusted returns of all three models are significantly positive for the lowest skewness quintile. More salient are the alphas of the highest skewness quintile; the risk-adjusted returns for these lottery-like stocks are significantly negative in all risk models and have by far the largest absolute magnitude. Most importantly, the last row of columns 4, 5 and 6 shows the differences in returns between portfolio 1 and 5 after adjusting for risk. The regular and extended Fama-French alphas of the portfolio are particularly pronounced, and show that the attainable result of our long-short portfolio gets even larger and also becomes highly significant 7 Reported t-statistics in this paper are Newey-West statistics, with a lag length of 4 months, of the pairedsample t-tests. Hereby, empirical tests are controlled for both heteroscedasticity and autocorrelation problems, and we take into account that returns of the different quintiles are related by market circumstances. 14

16 after adjusting for the corresponding risk factors. In summary, the last three columns of Figure 1 indicate that expected skewness is negatively related to returns, even after controlling for standard measures of risk. As our Fama-French adjusted monthly result of 1.19% for the 1-5 portfolio is highly significant and even larger than the monthly return of 1.00% reported by Boyer et al. (2010), we can confirm their result that the difference in returns between the lowest and highest expected idiosyncratic skewness portfolios is, on average, positive. Thus, we can accept our first hypothesis. Realized idiosyncratic skewness exposure On the basis of our remaining hypotheses lies the assumption that autocorrelation of skewness, the third moment of a distribution, is very low. Therefore, Table 2 shows the autocorrelation coefficients of both the idiosyncratic skewness and the idiosyncratic volatility measures defined in equation (1) and (2) of this study. Table 2 Autocorrelation of Skewness and Volatility The table shows the autocorrelation coefficients of both the idiosyncratic skewness (column 1) and the idiosyncratic volatility (column 2) measures defined in equation (1) and (2) of this study. The first row reports the autocorrelation coefficients with a lag of 1 month for the skewness and volatility measures, based on a horizon of 60 months of previous daily returns. Row 2 reports the (non-overlapping) 60 months lagged autocorrelation coefficients of skewness and volatility, again based on a horizon of 60 months of daily returns. Moreover, row 3 reports the autocorrelation coefficients with a lag of 1 month, but uses only 1 month of daily returns as the horizon to calculate our volatility and skewness measures. Lag = 1 month, T = 60 months Lag = 60 months, T = 60 months Lag = 1 month, T= 1 month Autocorrelation coefficient idiosyncratic skewness Autocorrelation coefficient idiosyncratic volatility The first row reports the autocorrelation coefficients with a lag of 1 month for our skewness and volatility measures, based on a horizon of 60 months of previous daily returns (similar to the horizon used in the rest of this study). High correlation coefficients for both skewness and volatility can be explained by 59 out of 60 months of overlapping data used to calculate the measures and hence do not give us much insight. Therefore, row 2 reports the (non-overlapping) 60 months lagged autocorrelation coefficients of skewness and volatility, again based on a horizon of 60 months of daily returns. As can be seen in the table, the autocorrelation coefficient of idiosyncratic skewness is quite low (0.169), about half as large as the autocorrelation of idiosyncratic volatility. Moreover, row 3 reports the autocorrelation coefficients with a lag of 1 month, but uses only 1 month of daily returns 15

17 as the horizon to calculate our volatility and skewness measure. The skewness autocorrelation coefficient in this row is extremely small, especially when compared with that of volatility. However, it should be noted that this measure is noisier and less reliable, since only 1 month of previous returns is used in the definition. Having obtained our own evidence of the low autocorrelation of skewness, we continue with testing our next hypothesis and report additional descriptive statistics of the predicted skewness quintiles in Table 3. Column 1 reports the averages, across time, of the value-weighted cross-sectional averages of ln(size i,t) within each portfolio, where ln(size i,t) is the natural log of market capitalization for firm i at the end of month t. Consistent with our expectations about the correlation between size and skewness, we find that firms with higher expected idiosyncratic skewness tend to be smaller than firms with lower predicted skewness. Column 2 reports the averages, across time, of the value-weighted cross-sectional averages of iv i,t(1m) within each quintile. 8 This implies that our idiosyncratic volatility measure of equation (1) is now estimated with a horizon of T=1 month and thus, S(t) now only denotes the set of trading days within month t. Average values of iv i,t(1m) increase monotonically from the first quintile to the fifth quintile, indicating a positive relation between volatility, iv i,t(1m), and expected skewness, E t[is i,t+60]. Table 3 - Descriptive Statistics of Portfolios by Level of Predicted Skewness The table reports descriptive statistics for the five quintiles, where the first quintile represents stocks with the lowest predicted skewness, and the fifth quintile represents stocks with the highest predicted skewness. Columns 1, 2, 3 and 4 report averages, across time, of the value-weighted cross-sectional averages of ln(size i,t), iv i,t(1m), is i,t+1(1m) and is i,t+60(60m) within each portfolio. The last row of this table shows for each statistic the difference between portfolio 1 and 5 and the corresponding Newey-West t-statistics of the paired-sample t-tests (in parentheses). (1) Size (2) Idiosyncratic volatility (in %) (3) Realized firm skewness after 1 month (4) Realized firm skewness after 60 months 1 (Low) (High) *** (25.45) *** (-19.78) *** (-9.35) *** (-10.97) *, **, *** Significant at the 0.10 level, 0.05 level and 0.01 level, respectively. 8 In order to avoid noisy results, the realized volatility measure ivi,t(1m) is set to a missing value at the end of month t if the number of observable returns in the month is less than

18 Columns 3 and 4 report the averages, across time, of the value-weighted cross-sectional averages of is i,t+1(1m) and is i,t+60(60m), both concerning the realized exposure to skewness. 9 In order to test the realized skewness exposure, this study uses two different measures. The realized skewness measure used in column 3 is based on monthly rebalancing to track our desired skewness portfolio. Updating our portfolio on a monthly basis is required since the autocorrelation of skewness is low (as demonstrated in Table 2). However, this means we select our portfolio based on skewness forecasts over a horizon of 60 months and by contrast, keep some of these stocks only for 1 month in our portfolio. In order to investigate whether our quintiles also significantly differ in realized skewness exposure based on a 1 month holding period, we adjust our idiosyncratic skewness measure of equation (2) in such a way that S(t) now only denotes the set of trading days within month t+1. Column 3 indicates that is i,t+1(1m) of the highest skewness quintile is significantly higher than is i,t+1(1m) of the lowest skewness quintile, despite the mismatch between the forecasting horizon used for our classification and the realized holding period. However, the economic magnitude of this ex-post skewness measure is only small for each portfolio, as a period of 1 month is rather short to realize tail observations. The other measure used to test realized skewness in this study, is is i,t+60(60m). On the basis of this measure lies the assumption that we only rebalance after 60 months, and therefore hold stocks as long as the skewness prediction horizon upon which the portfolio ranking is based. In spite of this match between the forecasting measure used and the holding period, the combination of the low autocorrelation of our skewness measure and the low frequency of rebalancing will probably cause tracking error relative to our desired skewness exposure. Column 4 indicates that is i,t+60(60m) is significantly higher for quintile 5 than for quintile 1. The magnitude of the difference in skewness is larger than the difference reported in column 3, as the period to realize tail observations is 60 times as long as in the previous measure. Thus, columns 3 and 4 both indicate that realized exposure is significantly higher in the high expected skewness portfolio than in the low expected skewness portfolio, and therefore reject our second hypothesis. Transaction costs imposed by the strategy As noted earlier, low autocorrelation of skewness entails the need to frequently rebalance in order to prevent divergence from our investor s desired skewness exposure. The transaction costs that rebalancing imposes to the investor are important for our study, since these are one of the key determinants of net returns. Therefore, the next step is to approximate the transaction costs imposed 9 The variable isi,t+1(1m) is set to a missing value if the number of observable returns in month t+1 is less than 15, and isi,t+60(60m) is set to a missing value if the number of observable daily returns from the beginning of month t+1 through the end of month t+60 is less than

19 by the strategy in which the investor goes long the lowest expected skewness quintile and goes short the highest expected skewness quintile. Transaction costs include both brokers commissions and bidask spreads. The fees brokers charge for executing buy and sell orders are by far the smallest component of the two (Damodaran, 2006). Moreover, trading fees charged vary widely according to the level of service provided, the amount of money invested and the fee structure adopted by the broker. Therefore, this study disregards brokerage fees and uses bid-ask spreads as a conservative approximation for transaction costs. The bid-ask spread is the difference between the price at which an investor can buy an asset (the dealer s ask price) and the price at which investors can sell the same asset at the same point in time (the dealer s bid price), and compensates the dealer for the risk of holding inventory and the cost of processing orders. The spread is considered to be lost by the investor as a direct consequence of trading, since immediately after buying an asset for the higher ask price, it is only worth the lower bid price. Column 1 of table 4 reports for each of the quintiles the average value-weighted bid-ask spread of stocks as a percentage of the last ask price. Table 4: Transaction Costs per Quintile This table reports information about transaction costs, calculated with the help of bid-ask spreads, for each expected skewness quintile. Column 1 reports the average value-weighted bid-ask spread of stocks within each quintile as a percentage of the last ask price. Column 2 shows the average value-weighted transaction costs per month, based on monthly rebalancing, as a percentage of the value of the portfolio. The last row of this table shows for both statistics the difference between portfolio 1 and 5 and the corresponding Newey-West t-statistics of the paired-sample t-tests (in parentheses). Average bid-ask spread (%) Average monthly rebalancing costs (%) 1 (Low) (High) *** (-7.15) *** (-10.38) *, **, *** Significant at the 0.10 level, 0.05 level and 0.01 level, respectively. As can be seen, the average transaction costs per transaction gradually increase from the lower to the higher skewness quintiles. Whereas the average value-weighted bid-ask spread in the first quintile is only less than 1% of the last ask price, the same calculation for the highest skewness quintile indicates a percentage spread of almost 9%. The reported t-statistic of the matched-pair t-test in the last row indicates that this difference is highly significant. Thus, this result confirms our expectation that, 18

20 because of the high presence of small and illiquid stocks in the portfolio of lottery-like stocks, the average transaction costs per transaction are largest for the highest skewness quintile. However, what matters for our calculation of the net returns of the proposed long-short strategy are not only the average costs per transaction, but also the amount of transactions needed per quintile to keep exposure to the desired portfolio. That is, we need the total transaction costs of rebalancing. In order to calculate the monthly amount of transactions needed for each quintile to prevent tracking error, we need the absolute values of the differences in weight (with respect to the month before) for all stocks within the quintiles. Moreover, as investors incur transaction costs both when buying and selling stocks, we choose to recognize half of the spread as a cost at the moment a stock is bought and count the other half of the spread when a stock is sold. Therefore, we use the following approximation to compute the total transaction costs in percentages of the last ask price for quintile q on date t: Percentage trading costs q,t = n i=1 in which W i,t q denotes the weight of stock i within quintile q on date t. i [ (W q q i,t W i,t 1 ) 0.5 ask price i,t bid price i,t 100%], (5) ask price i,t We repeat this computation for all portfolios and report for each quintile the average value-weighted rebalancing costs per month based on monthly rebalancing, as a percentage of the value-weighted average of the portfolio value, in column 2 of Table 4. We observe that, just like the average transaction costs per transaction, the total costs of monthly rebalancing increase monotonically from the first to the fifth quintile. Furthermore, note that the average bid-ask spread per transaction in quintile 5 is almost three times as large as the average spread of stocks in quintile 4, while the jump in total transaction costs is much smaller (only 50% higher). As the total transaction costs are determined by the average bid-ask spread and the amount of transactions needed to prevent tracking error, this implies that less transactions are needed to keep exposure to the highest skewness portfolio. Since a similar pattern can be observed for stocks in the first portfolio, we conclude that stocks in the two extreme quintiles are relatively stable with respect to the quintile they belong to, possibly explaining the ex-post exposure to idiosyncratic skewness of our long-short strategy, despite low autocorrelation coefficients. Having obtained insight in the magnitude of monthly transaction costs per quintile, the next step is to calculate the average net return of our skewness based long-short strategy. For each month in our sample period, we subtract the monthly transaction cost approximations for both quintile 1 and quintile 5 from the corresponding monthly return that can be attained by our long-short portfolio. The time-series average of these monthly net returns is reported in the first column of Table 5. 19