Have we solved the idiosyncratic volatility puzzle?*

Transcription

1 Have we solved the idiosyncratic volatility puzzle?* Kewei Hou Ohio State University Roger K. Loh Singapore Management University This Draft: June 2014 Abstract We propose a simple methodology to evaluate a large number of potential explanations for the negative relation between idiosyncratic volatility and subsequent stock returns (the idiosyncratic volatility puzzle). We find that surprisingly many existing explanations explain less than 10% of the puzzle. On the other hand, explanations based on investors lottery preferences, short-term return reversal, and earnings surprises show greater promise in explaining the puzzle. Together they account for 60-85% of the negative idiosyncratic volatility-return relation. Our methodology can be applied to evaluate competing explanations for a broad range of topics in asset pricing and corporate finance. Keywords: Idiosyncratic volatility; Cross-section of stock returns; Lottery preferences; Market frictions; Earnings surprises JEL Classification Codes: G12; G14 * We thank Jack Bao, Geert Bekaert, Hank Bessembinder, Steve Dimmock, Fangjian Fu, Dong Hong, Chuan-Yang Hwang, Jung-Min Kim, Sehoon Kim, Bob Kimmel, William Leon, Angie Low, René Stulz, Avanidhar Subrahmanyam, Mitch Warachka, and seminar participants at Hong Kong Polytechnic University, Nanyang Technological University, National University of Singapore, Ohio State University, Peking University, Seoul National University, Singapore Management University, SUNY Buffalo, University of Central Florida, University of Delaware, University of Exeter, and University of Hong Kong for their comments and suggestions. We also thank Karl Diether, and Keith Vorkink for sharing data. Address Correspondence: Kewei Hou, Ohio State University, Department of Finance, Fisher College of Business, 820 Fisher Hall, 2100 Neil Avenue, Columbus, OH 43210, USA. Tel: (614) , hou.28@osu.edu. Roger Loh, Singapore Management University, Lee Kong Chian School of Business. 50 Stamford Rd, #04-01, Singapore , Singapore. Tel: (65) , rogerloh@smu.edu.sg.

2 Have we solved the idiosyncratic volatility puzzle? 1. Introduction Ang, Hodrick, Xing, and Zhang (2006), in a highly influential paper, document a negative relation between idiosyncratic volatility and subsequent stock returns. 1 To the extent that realized idiosyncratic volatility proxies for expected idiosyncratic volatility, this result is very puzzling because traditional asset pricing theories either predict no relation between expected idiosyncratic volatility and expected returns under the assumptions that markets are complete and frictionless and investors are well-diversified, or predict a positive relation under the assumptions that markets are incomplete and investors face sizeable frictions and hold poorly-diversified portfolios (see, e.g., Merton (1987) and Hirshleifer (1988)). Consequently, many papers have been written trying to explain the puzzle, with each paper proposing a different economic mechanism linking idiosyncratic volatility to subsequent stock returns. 2 However, to date there has been no comprehensive examination about which explanations best explain the puzzle. Further complicating this matter is the fact that existing studies typically differ in terms of empirical methodology and sample construction, thus making direct comparisons of their results difficult. Motivated by these concerns, this paper provides a simple unified framework to evaluate a large number of candidate explanations of the puzzle. Most studies in this literature typically 1 Ang, Hodrick, Xing, and Zhang (2009) show that the relation also exists in international markets. 2 The long list of candidate explanations includes those based on expected idiosyncratic skewness (Boyer, Mitton, and Vorkink (2010)), coskewness (Chabi-Yo and Yang (2009)), maximum daily return (Bali, Cakici, and Whitelaw (2011)), retail trading proportion (Han and Kumar (2013)), one-month return reversal (Fu (2009) and Huang, Liu, Rhee, and Zhang (2009)), illiquidity (Bali and Cakici (2008) and Han and Lesmond (2011)), uncertainty (Johnson (2004)), average variance beta (Chen and Petkova (2012)), and earnings surprises (Jiang, Xu, and Yao (2009) and Wong (2011)). In addition, several papers show that the idiosyncratic volatility puzzle is stronger among stocks that are short-constrained (Boehme, Danielsen, Kumar, and Sorescu (2009) and George and Hwang (2011)), in financial distress (Avramov, Chordia, Jostova, and Philipov (2013)), have low investor attention (George and Hwang (2011)), have prices greater than five dollars (George and Hwang (2011)), and in non-january months (George and Hwang (2011) and Doran, Jiang, and Peterson (2012)). 1

3 promote a new explanation of the puzzle while controlling for a limited number of existing explanations. We believe that our paper provides the most comprehensive examination of existing explanations to date. More importantly, our methodology allows us to quantify the fraction of the puzzle that is explained by each candidate explanation, either by itself or after controlling for other competing explanations. To summarize our methodology, we start from Fama and MacBeth (1973) cross-sectional regressions of individual month t stock returns on month t 1 idiosyncratic volatility. We find, as many papers do, that the estimated regression coefficient, which we denote as γ t, is on average negative and highly statistically significant. Next, we decompose the γ t coefficient into one or more components, each related to a candidate explanation of the puzzle (e.g., skewness), and a residual component. The ratio of the component related to a particular candidate explanation to the original γ t coefficient then measures the fraction of the idiosyncratic volatility puzzle that is captured by that explanation, and the ratio of the residual component to γ t measures the fraction of the puzzle left unexplained by all candidate explanations considered. Our decomposition methodology ensures that the components related to the candidate explanations and the residual component add up to γ t. This makes for intuitive interpretation and easy comparisons when we pit existing explanations against one another. 3 To guide our analysis, we break up existing explanations into three groups. The first group of explanations attributes the idiosyncratic puzzle to lottery preferences of investors (they propose different proxies for the lottery feature of a stock, e.g., skewness, coskewness, expected idiosyncratic skewness, maximum daily return, and retail trading proportion). The second group of explanations appeals to various forms of market frictions (e.g., one-month return reversal, the 3 Our methodology can be easily used to compare competing explanations of a broad range of topics in empirical asset pricing and corporate finance. 2

4 Amihud illiquidity measure, and the zero-return measure) to try to explain the puzzle. Explanations that do not fall naturally into the first two groups (e.g., uncertainty, average variance beta, and pre- and post-formation earnings surprises) are then included in the third group. Using the sample of CRSP common stocks from , we find that surprisingly many existing explanations, when evaluated alone, explain less than 10% of the idiosyncratic volatility puzzle. This holds true for the explanations based on skewness, coskewness, illiquidity, uncertainty, and average variance beta. For example, skewness and analyst dispersion (a proxy for uncertainty) can only explain 6.8% and 6.4%, respectively, of the puzzle. Or consider the Amihud illiquidity measure. Despite being highly correlated with idiosyncratic volatility, it also fails to capture a significant fraction of the puzzle. On the other hand, explanations based on expected idiosyncratic skewness, maximum daily return, retail trading proportion, one-month return reversal, pre- and post-formation earnings surprises show promise in explaining the puzzle. In particular, post-formation earnings surprises alone can explain 33.9% of the puzzle, followed by retail trading proportion at 24.4%, one-month return reversal at 24.1%, pre-formation earnings surprises at 11.4%, and expected idiosyncratic skewness at 10.5%. For the maximum daily return variable proposed by Bali et al. (2011), it turns out that it can explain the entire puzzle. The problem, however, is that this variable is close to being perfectly collinear with idiosyncratic volatility (correlation of about 0.90). When we break this mechanical relation using alternative measures of maximum return, the explained fraction drops to 24-55%, which is still impressive. Finally, we include all explanations that on their own can explain more than 10% of the puzzle in a multivariate framework, so that we can evaluate the marginal contribution of each 3

5 explanation. We are also interested in the total fraction of the puzzle they can collectively explain. We find that after controlling for competing explanations, maximum daily return (using less collinear versions) explains 29-61% of the puzzle and expected idiosyncratic skewness explains 3-5% of the puzzle, depending on the specification. Together, these two lottery preference-based explanations capture a good 34-64% of the puzzle. On the other hand, the onemonth return reversal, a market friction-based explanation, explains 8-25% and earnings surprises (pre- and post-formation) explain 5-17% of the puzzle in the multivariate analysis. Collectively, the above explanations account for 60-85% of the puzzle in the overall sample. In addition, among subsamples of stocks with prices greater than five dollars, low analyst coverage, poor credit ratings, high short-sale constraints, high leverage, or for non-january months (which have been shown by previous studies to exhibit a stronger idiosyncratic volatility puzzle), existing explanations account for a smaller fraction (52-73%) of the puzzle. We therefore conclude that a significant portion of the idiosyncratic volatility puzzle remains unexplained. The rest of the paper is organized as follows. Section 2 describes the data and methodology and gives an overview of the various explanations that have been proposed for the idiosyncratic volatility puzzle. Section 3 evaluates the explanations one at a time, and Section 4 investigates multiple explanations at the same time. Section 5 considers a number of robustness tests including using idiosyncratic volatility-sorted portfolios in the cross-sectional regressions, and finally, Section 6 concludes. 4

6 2. Data and methodology 2.1. Stock return and idiosyncratic volatility data We start our sample from the standard CRSP common stock (share codes of 10 or 11) universe from July 1963 to December Monthly returns are adjusted for delisting following Shumway (1997). To be included in the analysis, we require a firm to have non-missing values for size and book-to-market equity (B/M), where Size is the most recent June-end market cap of the firm and B/M is computed according to Fama and French (2006). We apply a price screen of one dollar to remove penny stocks but also adjust this screen in subsample robustness tests. We compute idiosyncratic volatility (IVOL) following Ang et al. (2006) as the standard deviation of the residuals from a regression of daily stock returns in month t 1 on the Fama and French (1993) factors. We require at least ten daily returns to compute IVOL, although our results are unaffected if we require at least 15 daily returns or do not impose any minimum observation restriction. The estimates for IVOL start in July 1963, and month t 1 estimates of IVOL are matched to month t returns from August 1963 to December Candidate variables related to lottery preferences of investors A battery of candidate variables is constructed as potential explanations of the idiosyncratic volatility puzzle. The first group of explanations concerns lottery preferences of investors. Barberis and Huang (2008) argue that under cumulative prospect theory, investors overweigh small chances of large gains (hence the lottery preferences). As a result, they prefer positively-skewed stocks, causing them to be overpriced, which would then earn low subsequent returns. Several papers attribute the idiosyncratic volatility puzzle to idiosyncratic volatility being correlated with skewness. We measure skewness (denoted Skew) using the daily returns in 5

7 month t 1. In addition to the raw skewness measure, we also compute alternative measures of skewness. Chabi-Yo and Yang (2009) develop a model showing that the effect of idiosyncratic volatility on stock returns is related to a stock s coskewness with the market portfolio. We measure coskewness (Coskew) as the regression coefficient of squared daily individual stock returns on market returns. 4 Boyer et al. (2010) use the forecasts from a regression model to proxy for expected idiosyncratic skewness and show that it helps explain the idiosyncratic volatility puzzle. 5 We obtain the estimates of expected idiosyncratic skewness (E(Idioskew)) from the authors for We then extend their sample period by constructing the measure for and We also consider the maximum daily return (Maxret) and the retail trading proportion (RTP) of a stock, which are proposed by Bali et al. (2011) and Han and Kumar (2013), respectively, as indicators for stocks that are preferred by lottery-seeking retail investors. Maxret is measured using daily returns in month t 1. RTP is measured as the fraction of the dollar trading volume in month t 1 that comes from trades less than or equal to $5000. It is computed using the Institute for the Study of Securities Markets (ISSM) database for and the Trades and Quotes (TaQ) database for We have also calculated Harvey and Siddique (2000) s measure of coskewness by regressing daily individual stock returns on squared market returns. The results are similar to those based on Chabi-Yo and Yang (2009) s coskewness measure. 5 Expected idiosyncratic skewness is estimated by regressing idiosyncratic skewness on lagged idiosyncratic skewness, idiosyncratic volatility, momentum, turnover, dummy variables for small firms and medium-sized firms, two-digit SIC dummies, and a Nasdaq dummy. Boyer et al. (2010) measure idiosyncratic skewness using the residuals from a regression of the past five years of daily returns on the Fama-French (1993) factors. 6 For the period, we drop turnover from the forecast model since turnover data for Nasdaq stocks is not available for this early sample period. 7 Following Han and Kumar (2013), we exclude the post-decimalization years because it is more difficult in those years to identify retail trades using dollar screens due to greater incidence of order-splitting by institutions. 6

8 2.3. Candidate variables related to market frictions The second group of explanations attributes the idiosyncratic volatility puzzle to market frictions. Fu (2009) and Huang et al. (2009) argue that once we control for the one-month return reversal effect, which is likely driven by the bid-ask bounce and other microstructure biases, the negative idiosyncratic volatility-return relation is no longer significant. 8 We measure the onemonth reversal effect using the month t 1 return (Lagret). Illiquidity can also affect the idiosyncratic volatility-return relation. We examine two measures of illiquidity. The Amihud (2002) measure (Amihud) is computed as the month t 1 average of daily absolute return divided by daily dollar trading volume. We also include the fraction of trading days in month t 1 with a zero return (Zeroret) as another proxy for illiquidity (Han and Lesmond (2011)) Candidate variables related to other explanations The third group of explanations consists of those that do not fall naturally into the lottery preference or market friction category. First, idiosyncratic volatility could proxy for the fundamental uncertainty surrounding a stock. Johnson (2004) argues that uncertainty is negatively related to future stock returns because stock is a call option on a levered firm s underlying assets. We measure uncertainty using analyst dispersion (Dispersion) which is the standard deviation of analysts FY1 forecasts scaled by the absolute value of the mean consensus forecast for month t 1. Analysts forecasts are obtained from the I/B/E/S Summary Estimates unadjusted file. 8 A recent paper by Chen, Jiang, Xu, and Yao (2012) challenges the robustness of this result. 7

9 Chen and Petkova (2012) argue that a stock s exposure to the average variance component of the market variance explains the idiosyncratic volatility puzzle. We replicate their measure of average variance beta (AvgVarβ) for the sample period of and include it in the analysis. 9 We also examine SUE (the most recent quarter s standardized unexpected earnings reported as of month t 1) and NextSUE (next quarter s SUE). Wong (2011) shows that high idiosyncratic volatility stocks suffer negative earnings surprises both before and after portfolio formation, which could explain the poor return performance of those stocks. 10 Standardized unexpected earnings are measured as the Compustat quarterly earnings before extraordinary items (item IBQ) minus the earnings four quarter ago, divided by the standard deviation of the difference over the last eight quarters Decomposition methodology Our decomposition methodology is based on firm-level Fama-MacBeth cross-sectional regressions, which are commonly used in the literature to study the relation between idiosyncratic volatility and returns. For each month t, we regress the cross-section of individual stock returns on their month t 1 IVOL as follows: R it = α t + γ t IVOL it 1 + ε it. (1) For our baseline sample, the average γ t coefficient ( 100 and reported in percent) equals percent with a t-statistic of 4.11 (hence the idiosyncratic volatility puzzle). For simplicity, we use raw returns in the regressions. Our results are robust to using returns adjusted 9 For each month t, average variance beta is estimated by regressing a stock s returns over the past 60 months (24-month minimum) on changes in the average variance (AV) of the market portfolio, controlling for changes in the average correlation (AC) of the market portfolio and the Fama-French (1993) factors. AV is the average of the individual stock daily return variances. AC is the average pairwise correlation between stocks. 10 See also Jiang et al. (2009). 8

10 for size, value, and momentum factors/characteristics or to using three-month buy-and-hold returns as the dependent variable. Next, we regress IVOL it 1 on a candidate explanatory variable (Candidate it 1 ): IVOL it 1 = a t 1 + δ t 1 Candidate it 1 + μ it 1. (2) This regression allows us to assess the relation between idiosyncratic volatility and the candidate variable as any candidate variable that can potentially explain the puzzle must be correlated with idiosyncratic volatility. 11 We then use the regression coefficient estimates to decompose IVOL it 1 into two orthogonal components: δ t 1 Candidate it 1 is the component of IVOL it 1 that is related to the candidate variable and (a t 1 + μ it 1 ) is the residual component that is unrelated to the candidate variable. The final step is to use linearity of covariances to decompose the estimated γ t coefficient from Equation 1: γ t = Cov[R it, IVOL it 1 ] Var[IVOL it 1 ] = Cov[R it, (δ t 1 Candidate it 1 + a t 1 + μ it 1 )] Var[IVOL it 1 ] = Cov[R it, δ t 1 Candidate it 1 ] Var[IVOL it 1 ] + Cov[R it, (a t 1 + μ it 1 )] Var[IVOL it 1 ] = γ t C + γ t R. (3) The ratio of γ C t /γ t then measures the fraction of the idiosyncratic-return relation (the idiosyncratic volatility puzzle) explained by the candidate variable, and γ R t /γ t measures the fraction of the puzzle left unexplained by the candidate variable. While the means and variances of the two ratios are unattainable in closed-form, we can approximate them using the means, 11 However, as we will demonstrate later, a high correlation in and of itself does not guarantee that the candidate variable will explain a large fraction of the puzzle. 9

11 variances, and covariances of γ t C, γ t R, and γ t using the multivariate delta method based on Taylor series expansions (see, e.g., Casella and Berger (2002)): and C ) E γ t C E(γ t γ t E(γ t ), Var γ t C Var γ t R C ) E(γ t γ t E(γ t ) R ) E(γ t γ t E(γ t ) 2 2 R E γ t E(γ t γ t E(γ t ), (4) R ) Var(γ t C ) (E(γ C t )) 2 + Var(γ t) (E(γ t )) 2 2 Cov(γ t C, γ t ) E(γ C, (5) t )E(γ t ) Var(γ t R ) (E(γ R t )) 2 + Var(γ t) (E(γ t )) 2 2 Cov(γ t R, γ t ) E(γ R, (6) t )E(γ t ) and the corresponding estimated means and variances of the ratios are based on the respective time series of γ t C, γ t R, and γ t estimates (over T months): and E γ t C γ C t, γ t γ t Var γ t C 1 C γ t T γ t γ t E γ R t γ R t, (7) γ t γ t 2 s 2 C γt C γ 2 + t 2 s γt 2 γ 2 ρ γ C t,γ t s γt Cs γt, (8) C t γ t γ t Var γ t R 1 R γ t T γ t γ t 2 s 2 γt R R γ 2 + t 2 s γt 2 γ 2 ρ γ R t,γ t s γt Rs γt. (9) R t γ t γ t Our decomposition methodology is different from the conventional approach to evaluate a candidate variable, which usually involves including the candidate variable as a control in the regression of returns on idiosyncratic volatility: R it = α t + γ t R IVOL it 1 + γ t C Candidate it 1 + ε it. (10) In this regression, if the average coefficient on IVOL is zero, researchers typically conclude that the candidate variable explains the idiosyncratic volatility puzzle. However, if the average coefficient on IVOL is not zero, this conventional approach does not allow one to easily quantify the fraction of the puzzle that is explained by a candidate variable, especially when it is 10

12 evaluated against other competing candidate variables. The important advantage of our decomposition methodology is that by requiring that γ t C and γ t R add up to the original γ t coefficient, we can make a direct statement about the fraction of the idiosyncratic volatility puzzle that is explained by the candidate variable. In addition, our methodology can easily accommodate multiple candidate variables at the same time so we can objectively quantify the marginal contribution of each variable in a horse race. It is important to point out that a candidate variable that is highly correlated with idiosyncratic volatility may not necessarily explain a large fraction of the puzzle in our decomposition methodology. This is because the part of idiosyncratic volatility that is related to the candidate variable may not be the part that is responsible for the negative relation between idiosyncratic volatility and returns. In the Appendix, we show that γ t C from our decomposition methodology (Equation 3) is related to the coefficients from the conventional approach (Equation 10) in the following way: γ t C = γ t C δ t 1 + γ t R Var[δ t 1 Candidate it 1 ] Var[IVOL it 1 ]. 12 This suggests that γ t C not only depends on the fraction of the variation of idiosyncratic volatility explained by the candidate variable Var[δ t 1Candidate it 1 ], but also on the component of the candidate variable Var[IVOL it 1 ] that is uncorrelated with idiosyncratic volatility but correlated with future returns (captured by γ t C in Equation 10). Consequently, a candidate variable that is highly correlated with idiosyncratic volatility could actually have a small or even negative contribution to the puzzle if the component of the candidate variable that is uncorrelated with idiosyncratic volatility predicts returns positively. Empirically, we show in Section 3 that this is indeed the case for a number of candidate variables we investigate. The bottom line is that our decomposition methodology is not simply picking up candidate variables based solely on their correlations with idiosyncratic 12 We also consider the general case of multiple candidate variables in the Appendix. 11

13 volatility. Rather, we attribute a high explanatory power to a variable for capturing a significant fraction of the negative relation between idiosyncratic volatility and returns. 3. Evaluating candidate explanations one at a time 3.1. Sample descriptive statistics Panel A of Table 1 reports the descriptive statistics of our sample. There are more than two million firm-month observations in our baseline sample. The average return is 1.1% per month with a standard deviation of 15.8%. The average IVOL estimated using daily returns is 2.7%. The average market beta (estimated with three years of past monthly returns), size (June market cap), B/M ratio (computed following Fama and French (2006)), and momentum (buyand-hold return from month t 12 to t 2) are 1.141, $1.424 billion, 0.895, and 16.9%, respectively. [Insert Table 1 here] The rest of Panel A reports the descriptive statistics for the three groups (lottery preference, market friction, and others) of candidate variables. Among the lottery preference variables, the average skewness is 0.260, suggesting that stock returns are on average positively skewed. The average maximum daily return is 7.2%. The average retail trading proportion is 16.1%, indicating that retail investors typically do not account for a large fraction of the trading volume of a stock. Among the market friction variables, last month s return has an average value of 1.6% (higher than the month t average return of 1.1% due to the one-dollar price screen we impose at the end of month t 1). The average value of the zero-return proportion, an illiquidity proxy, is 20.8%, which indicates that on average about one-fifth of the trading days in a month have zero returns. Among the other candidate variables, analyst dispersion has an average value 12

14 of 20.2% and the average pre- and post-formation earnings surprises are 17.0% and 16.1%, respectively. Panel B of Table 1 reports the time-series averages of cross-sectional correlations between the variables. The average correlation between month t 1 IVOL and month t returns is 3.3%, which is consistent with the negative idiosyncratic volatility-return relation documented in the literature. The second column of Panel B shows that IVOL is positively correlated with skewness, coskewness, expected idiosyncratic skewness, maximum daily return, retail trading proportion, last month s return, Amihud s illiquidity measure, zero-return proportion, analyst dispersion, and average variance beta, and negatively correlated with pre- and post-formation earnings surprises. These correlations are generally consistent with the various explanations that have been proposed for the idiosyncratic volatility puzzle. For example, the average correlation between IVOL and skewness is 0.20, which is consistent with the lottery preference explanation that IVOL predicts returns because of its correlation with skewness. Or consider pre- and postformation earnings surprises. The average correlations between them and IVOL are 0.10 and 0.09, respectively. These correlations are in line with Wong (2011) s conclusion that the poor earnings performance of high idiosyncratic stocks is responsible for their low returns. Among all the candidate variables, the one that has the highest correlation with IVOL is maximum daily return (average correlation of 0.88), suggesting that it is well placed to explain the idiosyncratic volatility puzzle. 13 However, collinearity could be a concern for this variable. 13 The high correlation between the two variables is not surprising considering that price range has been used as a volatility estimator in the literature (see, e.g., Alizadeh, Brandt, and Diebold (2002) and Brandt and Diebold (2006)). 13

15 3.2. The idiosyncratic volatility puzzle To set the stage, Table 2 reports the results of monthly Fama-MacBeth cross-sectional regressions of month t individual stock returns on month t 1 IVOL and different candidate variables. 14 [Insert Table 2 here] Model 1 regresses returns on IVOL alone. The sample period is August 1963 to December 2012 with an average of 3,651 stocks per month. The average coefficient on IVOL is percent (t= 4.11) and its magnitude and statistical significance are in line with past findings in the literature. 15 Models 2-6 add the lottery preference-based candidate variables one at a time to Model 1. For each model, the number of observations and sample period may differ from that of Model 1 due to data availability of the candidate variable examined. The results from Models 2-6 show that in all but one case, the coefficient on IVOL remains negative and statistically significant. Only when maximum daily return (Maxret) is included in the regression does the coefficient on IVOL become positive (and statistically insignificant), consistent with the results in Bali et al. (2011). Models 7-9 and in Table 2 investigate the candidate variables related to market frictions and other explanations, respectively. The results show that the coefficient on IVOL is always negative and statistically significant, irrespective of the candidate variable included in the regressions. 14 Asparouhova, Bessembinder, and Kalcheva (2013) show that microstructure noise introduces an upward bias to stock returns, which could potentially bias the inferences from Fama-MacBeth regressions. In unreported tests, we follow their paper by using (one plus) month t 1 return as the weight in the Fama-MacBeth regressions and find that our results are robust to this noise-adjustment procedure. 15 The average coefficients from all of our cross-sectional regressions are multiplied by 100 and reported in percent. 14

16 The main takeaway from Table 2 is that the negative idiosyncratic volatility-return relation remains significant after controlling for almost all of the candidate explanatory variables (except for maximum daily return). But the question remains: Even if these candidate variables cannot completely explain away the idiosyncratic volatility puzzle, can they at least explain part of it? If so, what fraction of the puzzle can the candidate variables capture? We investigate this next using the decomposition methodology described in Section Candidate variables related to lottery preferences of investors We first examine the candidate variables related to lottery preferences of investors. We start off with a detailed account of the decomposition analysis using skewness (Skew) in Panel A of Table 3. Stage 1 reports monthly Fama-MacBeth cross-sectional regressions of month t returns on month t 1 IVOL using firm-month observations with non-missing IVOL and Skew. This is to ensure that the sample is kept constant when we add Skew to the analysis later. The average coefficient on IVOL is percent with a t-statistic of In Stage 2, we add Skew to the cross-sectional regressions. This is identical to Model 2 in Table 2. The average coefficient on Skew is percent with a t-statistic of 4.07, which is consistent with Barberis and Huang (2008) s argument that investors overprice positivelyskewed stocks and as a result the future returns of those stocks are low. Controlling for Skew, however, we see that the average coefficient on IVOL is still significantly negative ( percent, t= 3.88). In Stage 3, we regress IVOL each month on Skew to study the relation between the two variables. The average coefficient on Skew is percent with a t-statistic of 34.37, suggesting that part of IVOL is indeed related to the skewness of a stock (a unit change in Skew is associated with a 0.376% change in IVOL). However, the adjusted R-squared shows that only 4.4% of the 15

17 variation in IVOL can be explained by Skew. The Stage 3 estimated coefficients allow us to separate IVOL each month into two components: the first one (δ t 1 Skew it 1 ) is the component of IVOL that is related to Skew and the second (a t 1 + μ it 1 ) is the residual component that is unrelated to Skew. In Stage 4, we follow Equation 3 and use the above two components of IVOL to decompose the Stage 1 IVOL coefficient (γ t ) into a component that is related to Skew (γ t Skew ) and a residual component (γ t R ). The time-series averages of γt Skew and γ t R are percent and percent, respectively. Since by construction the two coefficients sum up to the Stage 1 coefficient of percent, we can readily calculate the fraction of the Stage 1 coefficient attributable to Skew as =6.8% (t-statistic=4.07, calculated based on Equation 8), and the fraction attributable to the residual component is =93.2% (t=56.19). We therefore conclude that skewness can only explain a very small fraction of the idiosyncratic volatility puzzle. [Insert Table 3 here] We also examine the other skewness variables in Panel A of Table 3. The coskewness measure of Chabi-Yo and Yang (2009) (Coskew) explains only 3.3% (t=1.80) of the puzzle, and the expected idiosyncratic skewness measure (E(Idioskew)) of Boyer et al. (2010) captures 10.5% (t=1.69) of the puzzle. On the other hand, the portion of the puzzle that is explained by the maximum daily return measure (Maxret) of Bali et al. (2011) is 102.2% (t=11.83). 16 It thus appears that Maxret explains the idiosyncratic volatility puzzle entirely. However, given the near perfect collinearity between Maxret and IVOL (a correlation of 0.88, Table 1 Panel A), one might be concerned that 16 The reason this fraction is above 100% is because the adding-up constraint in Stage 4 requires the Maxret component and the residual component must add up to the Stage 1 coefficient on IVOL. 16

18 this finding is mechanical. To mitigate these concerns, we examine a number of alternative measures of maximum daily return that are less collinear with IVOL but at the same time retain the ability to capture the lottery feature of a stock. The first measure we consider is the maximum daily return for the three-month period ending in month t 1 (Maxret(3mth)), as it is possible that lottery-seeking investors will use daily returns beyond the past one month to determine the lottery feature of a stock. The correlation between Maxret(3mth) and IVOL is lower at We find that Maxret(3mth) explains 54.5% (t=11.34) of the puzzle, which is still very impressive. We also examine LagMaxret(3mth), which is the maximum daily return for the three-month period ending in month t 2. This variable is even less correlated with IVOL (correlation of 0.48), which suggests that collinearity is not as serious of an issue. LagMaxret(3mth) is able to explain 24.0% (t=3.54) of the puzzle. The last column of Panel A shows that the RTP measure of Han and Kumar (2013) explains 24.4% (t=4.37) of the idiosyncratic volatility puzzle. However, we note that this result is obtained over a relatively short sample period ( ) compared to the rest of the candidate variables. Overall, the results from Table 3 Panel A suggest that among all the lottery preferencebased variables we examine, the expected idiosyncratic skewness measure of Boyer et al. (2010), the maximum daily return measure of Bali et al. (2011), and the RTP measure of Han and Kumar (2013) are the most promising candidates in that they each explain a sizable (more than 10%) fraction of the idiosyncratic volatility puzzle, whereas the rest of the candidate variables fail to capture economically significant fractions of the puzzle. 17

19 3.4. Candidate variables related to market frictions Panel B of Table 3 examines the candidate variables related to market frictions. We first consider the one-month return reversal effect (Fu (2009) and Huang et al. (2009)), which is likely driven by the bid-ask bounce and other microstructure effects. We see that the month t 1 return (Lagret) explains about a quarter (24.1%, t=3.72) of the idiosyncratic volatility puzzle, with the residual component capturing the remaining 75.9% (t=11.75) of the puzzle. The rest of Panel B examines the impact of illiquidity on the puzzle using the Amihud (2002) measure (Amihud) and the zero-return proportion (Zeroret) of Han and Lesmond (2011). The results show that neither of them can explain a significant fraction of the puzzle. In fact, the fractions explained are both negative at 8.1% (t= 1.07) and 0.9% (t= 0.42) for Amihud and Zeroret, respectively. Intuitively, the reason that these illiquidity proxies have negative contributions to the idiosyncratic volatility puzzle is because they are both positively correlated with IVOL but their return predictability (after controlling for idiosyncratic volatility) is also positive, which is in the opposite direction of the idiosyncratic volatility puzzle. The low explanatory power of these illiquidity candidate variables despite their high correlations with idiosyncratic volatility shows that our decomposition methodology does not necessarily attribute a large explained fraction to a candidate variable just because it has a high correlation with idiosyncratic volatility. In sum, with the exception of the one-month return reversal effect, the other market friction variables we examine have very little success in explaining the idiosyncratic volatility puzzle. 18

20 3.5. Candidate variables related to other explanations Panel C of Table 3 examines candidate variables that cannot be grouped into the lottery preference or market friction category. We first look at analyst dispersion (Dispersion) and find that it can only explain a small fraction (6.4%, t=1.93) of the idiosyncratic volatility puzzle. The next candidate variable we investigate is the average variance beta (AvgVarβ) of Chen and Petkova (2012). This variable shows little promise in explaining the puzzle, with the explained fraction equal to 1.0% (t=2.07). Finally, we examine whether pre- and post-formation earnings surprises (SUE and NextSUE, respectively) can explain the idiosyncratic volatility puzzle. We find that SUE captures 11.4% (t=5.33) of the puzzle whereas NextSUE captures 33.9% (t=4.18) of the puzzle. 4. Evaluating multiple candidate explanations at the same time 4.1. The most promising candidate variables After investigating each of the candidate variables in isolation, we now turn to multivariate analysis and evaluate against each other the variables that account for at least 10% of the puzzle. 17 We want to know the marginal contribution of each variable after controlling for competing variables. In addition, we are interested in the total fraction of the puzzle these candidate variables can collectively explain. The candidate variables that can explain 10% or more of the puzzle include the expected idiosyncratic skewness measure (E(Idioskew)) of Boyer et al. (2010), different variants of the maximum daily return measure (Maxret) of Bali et al. (2011), past one month s return (Lagret), 17 Choosing 10% as the cutoff is admittedly arbitrary. That said, our inferences are robust to using higher or lower levels of cutoff or removing the cutoff (and thereby using the entire set of candidate variables). 19

21 and pre- and post-formation earnings surprises (SUE and NextSUE, respectively). 18 We put these variables through our decomposition methodology and the linear adding-up constraint ensures that their contributions plus that of the residual component add up to 100% of the puzzle. [Insert Table 4 here] Table 4 reports the multivariate results. The sample period is In Model 1, we measure maximum daily return using Maxret(3mth) to avoid the near perfect collinearity between IVOL and the original Maxret variable. We see that the five candidate variables (E(Idioskew), Maxret(3mth), Lagret, SUE, and NextSUE) collectively explain 85.1% of the idiosyncratic volatility puzzle and the residual component accounts for the remaining 14.9% of the puzzle. The best performing variable is Maxret(3mth) which alone captures 60.9% (t=12.72) of the puzzle. In fact, the explained fraction is even higher than in the univariate analysis (54.5%, t=11.34). In contrast, the contributions of the other candidate variables are all significantly lower than their univariate counterparts. Take NextSUE for example. In the univariate analysis, it is able to explain 33.9% (t=4.18) of the puzzle. After controlling for Maxret(3mth) and other candidate variables, the explained fraction shrinks considerably to 9.6% (t=4.18). Similarly, the explained fraction goes down from 10.5% (t=1.69) in the univariate analysis to a statistically and economically insignificant 3.3% (t=0.80) in the multivariate analysis for E(Idioskew), from 24.1% (t=3.72) to 8.1% (t=2.53) for Lagret, and from 11.4% (t=5.33) to 3.2% (t=3.90) for SUE. Model 2 replaces Maxret(3mth) with LagMaxret(3mth), which has an even lower correlation with IVOL (0.48 vs. 0.70). The total fraction of the puzzle explained by the five candidate variables declines to 75.8%, with LagMaxret(3mth), E(Idioskew), Lagret, NextSUE, and SUE each contributing 31.0% (t=8.39), 3.3% (t=0.55), 24.7% (t=3.68), 12.2% (t=4.26), and 18 RTP also meets the cutoff, but because of its short sample period ( ) we exclude it from our main multivariate analysis but perform robustness checks using it for the period. 20

22 4.5% (t=4.12), respectively. In Model 3, we drop NextSUE from Model 2 to focus on candidate variables that are strictly ex ante (measured prior to the returns being predicted). The total explained fraction declines further to 60.1%, with LagMaxret(3mth), E(Idioskew), Lagret, and SUE contributing 28.7% (t=9.74), 5.0% (t=1.00), 21.2% (t=4.12), and 5.2% (t=4.83), respectively. 19 The results from Models 1-3 can also be seen in Panel A of Figure 1 where we plot the marginal contribution of each candidate variable using pie charts. [Insert Figure 1 here] The main takeaway from Table 4 is that a significant portion of the idiosyncratic volatility puzzle remains unexplained by the candidate variables that have the most success in the univariate analysis. In this multivariate setting, the best performing candidate variable is the maximum daily return measure, which is related to the lottery preferences of investors Subsample analysis In this subsection, we repeat the multivariate decomposition analysis in Table 4 for subsamples of stocks that have been shown by previous studies to exhibit a stronger idiosyncratic volatility puzzle. These include stocks that have prices greater than five dollars, low analyst coverage, poor credit ratings, high short-sale constraints, high leverage, or for non-january months (see, e.g., George and Hwang (2011), Avramov et al. (2013), Boehme et al. (2009), Johnson (2004), and Doran, Jiang, and Peterson (2012)). The five dollar price screen is imposed at the end of the previous month (t 1). We define low analyst coverage as having three or fewer analysts issuing FY1 earnings forecasts in month t 1 (firms with no I/B/E/S coverage are excluded). Poor credit rating is defined as belonging to 19 If we restrict the sample period to and include RTP in the multivariate analysis, the total explained fraction varies from 65% to 80% depending on the model used. The marginal contribution of RTP ranges from 10% to 13%. 21

23 the lowest credit rating tercile using the S&P long-term issuer rating reported in Compustat (SPLTICRM); short-constrained is defined as belonging to the highest tercile of short interest in month t 1 using the data in Cohen, Diether, and Malloy (2007) provided by Karl Diether; and high leverage is defined as belonging to the highest tercile of leverage where leverage is measured as the Compustat long-term debt (LTDEBT) over total assets (AT) from the previous fiscal year end. [Insert Table 5 here] Table 5 reports the results of the multivariate decomposition analysis on the six subsamples. 20 Overall, they show that a smaller fraction of the idiosyncratic volatility puzzle is explained by the candidate variables that are most successful in the univariate analysis. Specifically, the total fraction of the puzzle that is explained by the best candidate variables (across the three models) is 57-78% for stocks with prices greater than five dollars, 46-64% for low coverage stocks, 36-62% for poor rating stocks, 57-72% for short-constrained stocks, 53-79% for high leverage stocks, and 61-81% for non-january months, compared with 60-85% for the overall sample reported in Table 4. We plot in Panel B of Figure 1 the average fraction explained by each candidate variable across the six subsamples. The first pie chart (Model 1) shows that across subsamples, the best candidate variables explain a total of 72.5% of the idiosyncratic volatility puzzle while the residual component captures the remaining 27.5%. Comparing across the different candidate variables in Model 1, we see that the two lottery preference-based variables (Maxret(3mth) and E(Idioskew)) combine to explain 57.3% of the puzzle (49.8% for Maxret(3mth) and 7.5% for 20 To conserve space, we only report the Stage 4 results in Table 5. The results confirm the findings in past research that the idiosyncratic volatility puzzle is stronger for these subsamples. For example, when stocks with prices less than five dollars are removed, the average regression coefficient of returns on IVOL is percent (t= 4.57), compared to the percent (t= 4.06) coefficient reported in Table 4. 22

24 E(Idioskew)) across the six subsamples, followed by the two earnings surprise proxies (SUE and NextSUE) which together explain 12.6%, and the market friction-related Lagret which explains 2.7% of the puzzle. The second and third pie charts show that, when we replace Maxret(3mth) with LagMaxret(3mth) (Model 2) and also drop the ex post variable NextSUE (Model 3), the total explained fraction declines to 64.7% and 51.6%, respectively, thus leaving 35.3% and 48.4% of the puzzle unexplained. The two lottery preference-based variables (LagMaxret(3mth) and E(Idioskew)) continue to dominate the other candidate variables. They combine to explain 36-38% of the puzzle in those two models. The contributions of Lagret and SUE are 10.5% and 4.0%, respectively, in Model 2 and 10.6% and 4.9%, respectively, in Model 3. Finally, NextSUE explains 12.2% of the puzzle in Model Additional robustness tests In this section, we show that our decomposition methodology is robust to using idiosyncratic volatility-sorted portfolios to control for sampling uncertainty at the individual stock level and to allowing for non-linear relations between idiosyncratic volatility, candidate variables, and returns. We also show that the methodology can be used to evaluate explanations for other asset pricing anomalies Portfolio-level analysis Thus far, our decomposition analysis has been based on cross-sectional regressions estimated at the individual stock level. The advantage of using individual stocks, as opposed to portfolios, is that it is robust to data mining and loss of information concerns. However, this does 23

25 raise the question about sampling uncertainty at the individual stock level as both idiosyncratic volatility and many of the candidate variables we investigate are generated regressors. To see the effect of sampling uncertainty, let us assume that both idiosyncratic volatility and the candidate variable are measured with error, e.g. IVOL = IVOL + u, and Candıdate = Candidate + v, where the true variables (IVOL and Candidate) are unobservable and u and v are mean zero measurement errors. It can be shown (in proofs available upon request) that while the sampling error in idiosyncratic volatility has no effect on the mean or standard error of the fraction of the idiosyncratic volatility puzzle explained by the candidate variable (i.e. the γ t C / γ t ratio), the sampling error in the candidate variable does lead to a downward bias in the mean as well as the standard error of the ratio by a factor of Var(Candidate)/[Var(Candidate) + Var(v)] although the t-statistic for the ratio is still unbiased. To address this concern about sampling uncertainty, we follow the literature and perform robustness checks by using idiosyncratic volatility-sorted portfolios instead of individual stocks in the decomposition analysis. The motivation for using portfolios is that if the errors in an estimated variable are not perfectly correlated across stocks, we can improve the precision of the estimates by grouping stocks into portfolios because the errors will tend to offset each other. The disadvantage of aggregating stocks into portfolios, as pointed out by Ang, Liu, and Schwarz (2010), is that it loses information by reducing the cross-sectional variation in the estimated variable, which could lead to large efficiency losses. At the beginning of each month t, we sort individual stocks into 200 portfolios based on their month t 1 IVOL. 21 Value-weighted returns are calculated for these idiosyncratic volatilitysorted portfolios for month t and the portfolios are rebalanced at the beginning of month t+1. We 21 Dividing stocks into 200 portfolios seems like an appropriate compromise between concerns about loss of information versus sampling uncertainty, as it allows us to retain much of the variation in the candidate variables but at the same time offer insight into how the measurement errors in the candidate variables affect our results. 24

26 then use the portfolio returns to compute portfolio-level IVOL and the return-based candidate variables (Skew, Coskew, different Maxret measures, Lagret, Amihud, and AvgVarβ). The other candidate variables (E(Idioskew), RTP, Zeroret, Dispersion, SUE, and NextSUE) are computed as value-weighted averages of the firm-level numbers within each portfolio. [Insert Table 6 here] Table 6 presents the portfolio-level decomposition analysis. Panel A reports the univariate results and Panel B reports the multivariate results. Panel A first shows that the idiosyncratic volatility puzzle remains significant among idiosyncratic volatility-sorted portfolios. The average Fama-MacBeth regression coefficient of portfolio returns on IVOL ranges from percent (t= 5.50) to percent (t= 6.11) depending on the candidate variable examined. More importantly, the results in Panel A show that the candidate variables that are most successful in the firm-level analysis also tend to perform well in the portfolio-level analysis. Specifically, among the eight candidate variables that capture at least 10% of the puzzle in the firm-level univariate analysis (E(Idioskew), Maxret, Maxret(3mth), LagMaxret(3mth), RTP, Lagret, SUE, and NextSUE), six of them (E(Idioskew), Maxret, Maxret(3mth), LagMaxret(3mth), RTP, and NextSUE) continue to explain more than 10% of the puzzle in the portfolio-level univariate analysis whereas the explained fractions of the other two (Lagret and SUE) are only slightly below 10%. 22 In addition, Coskew and Amihud, which fail to capture significant fractions of the puzzle in the firm-level analysis, now each account for more than 10% of the puzzle in the portfolio-level analysis. 22 For E(Idioskew), Maxret(3mth), LagMaxret(3mth), and RTP, their contributions in the portfolio-level analysis are higher than those in the firm-level analysis. The opposite is true for Maxret, Lagret, SUE, and NextSUE. In particular, Maxret, which completely explains the puzzle in the firm-level analysis, sees its contribution drop to 57.3%. 25

27 In Panel B, we put all the candidate variables that explain at least 10% of the puzzle in Panel A through our multivariate decomposition framework. The list includes the two maximum daily return measures that are less collinear with IVOL (Maxret(3mth) and LagMaxret(3mth)), two skewness measures (E(Idioskew) and Coskew), Amihud (2002) s illiquidity measure (Amihud), and post-formation earnings surprise (NextSUE). 23 The results in Panel B show that these candidate variables collectively explain 71-76% of puzzle (depending on the combination of the variables used), thus leaving 24-29% of the puzzle unexplained. The total explained fraction is comparable to that from the firm-level analysis (60-85%) albeit in a tighter range. Furthermore, when we compare across different candidate variables, we see that, again consistent with the firm-level analysis, the lottery preference-based variables (Maxret(3mth), LagMaxret(3mth), E(Idioskew), and Coskew) dominate the rest of the candidate variables (Amihud and NextSUE) in the multivariate analysis. As before, the maximum daily return measures take the lion s share in terms of explanatory power. In sum, the analysis in Table 6 confirms that our main results are robust to using idiosyncratic volatility-sorted portfolios to mitigate the sampling uncertainty at the individual stock level. It also shows that our decomposition methodology can be easily applied to characteristic-sorted portfolios in addition to individual stocks Non-linear relations To be consistent with existing literature, we adopt the simple linear specifications in Equations 1 and 2 to study to what extent different candidate variables can explain the idiosyncratic volatility puzzle. However, our decomposition methodology can easily 23 We again exclude RTP from the multivariate analysis due to its short sample period ( ). However, our results are robust to including it for the sample period. 26

28 accommodate non-linear relations between idiosyncratic volatility and candidate variables and between idiosyncratic volatility and returns. To study potential non-linear relations between idiosyncratic volatility and the candidate variables, we include both the level and the squared term of a candidate variable in the decomposition analysis. Panel A of Table 7 shows that adding the squared term leaves the contribution largely unaffected for most of the candidate variables. Specifically, among the eight candidate variables that capture at least 10% of the puzzle under the univariate linear specification (E(Idioskew), Maxret, Maxret(3mth), LagMaxret(3mth), RTP, Lagret, SUE, and NextSUE), all eight of them (with their squared terms) continue to explain more than 10% of the puzzle with the combined fraction explained by both the level and squared terms equal to 10.9% (vs. 10.5% in the linear case) for E(Idioskew), 104.5% (vs %) for Maxret, 57.4% (vs. 54.5%) for Maxret(3mth), 24.2% (vs. 24.0%) for LagMaxret(3mth), 28.9% (vs. 24.4%) for RTP, 16.4% (vs. 24.1%) for Lagret, 13.7% (vs. 11.4%) for SUE, and 39.9% (vs. 33.9%) for NextSUE. 24 Panel B of Table 7 shows that, together these candidate variables (excluding Maxret due to its near perfect collinearity with IVOL, and RTP due to its short sample period) and their squared terms explain 58-79% of the idiosyncratic volatility puzzle (with Maxret(3mth) and its squared term contributing the most), which is similar to the fraction of the puzzle explained under the multivariate linear specification (60-85%). [Insert Table 7 here] Another possible source of non-linearity is that between idiosyncratic volatility and returns. Previous literature argues that much of the idiosyncratic volatility puzzle is driven by high idiosyncratic volatility stocks earning low returns (see e.g., Ang et al. (2006)). To 24 For the other candidate variables, they again fail to explain significant fractions of the puzzle even after taking into account possible non-linear relations with IVOL. 27

29 investigate this possibility, we define a dummy variable, HIGHIVOL, which equals one when a firm belongs to the top decile ranked by IVOL in month t 1 and zero otherwise. Panel C of Table 7 confirms that there is indeed a negative and significant relation between HIGHIVOL and subsequent stock returns with the average Fama-MacBeth regression coefficient on HIGHIVOL ranging from (t= 3.82) to (t= 6.61) percent. We then use our decomposition methodology to study the HIGHIVOL-return relation. We find that the candidate variables that are most successful in explaining the linear IVOL-return relation also capture significant fractions of the relation between HIGHIVOL and returns. Specifically, out of the eight candidate variables that capture at least 10% of the IVOL-return relation (E(Idioskew), Maxret, Maxret(3mth), LagMaxret(3mth), RTP, Lagret, SUE, and NextSUE), six of them again explain more than 10% of the HIGHIVOL-return relation in the univariate analysis (78.0% for Maxret, 41.6% for Maxret(3mth), 17.4% for LagMaxret(3mth), 19.8% for RTP, 20.4% for Lagret, and 26.9% NextSUE) whereas the contributions of the other two (E(Idioskew) and SUE) are only slightly below 10% (8.9% and 7.8%, respectively). In the multivariate analysis reported in Panel D of Table 7, Maxret(3mth) (or LagMaxret(3mth)), Lagret, and NextSUE combine to explain 40-72% of the negative relation between HIGHIVOL and returns, again with Maxret(3mth) contributing the most in terms of explanatory power. Overall, the analysis above shows that our results are robust to modifying the decomposition methodology to accommodate non-linearity in the relations between returns, idiosyncratic volatility, and different candidate variables Decomposing other anomalies In this paper, we treat the negative idiosyncratic volatility-return relation as a puzzle and use the candidate variables proposed in the literature (e.g., maximum daily return) to try to 28

30 explain the puzzle. In this subsection, we turn the tables and use idiosyncratic volatility as the candidate variable to explain the relations between returns and variables such as maximum daily return and pre-formation earnings surprise as the return predictability of those variables has also been cited in the literature as being anomalous relative to traditional asset pricing theories. For brevity, we focus our analysis on three anomaly variables maximum daily return (Maxret), one-month return reversal (Lagret), and pre-formation earnings surprise (SUE) and use IVOL as the only candidate variable, although the analysis can be easily extended to other anomaly variables or to include other candidate explanatory variables. [Insert Table 8 here] Table 8 shows that IVOL, when considered alone, explains 71.0% of the negative relation between Maxret and subsequent returns. This number, though smaller than the fraction of the IVOL puzzle explained by Maxret (102.2%), is still impressive and identifies idiosyncratic volatility as a major contributor to the maximum daily return puzzle. In contrast, IVOL can explain none ( 1.6%) of the one-month return reversal effect based on Lagret and only a small fraction (5.6%) of the post-earnings announcement drift based on SUE, compared to 24.1% and 11.4% of the IVOL puzzle explained by Lagret and SUE, respectively. In short, the above results show that our decomposition methodology can be used to evaluate explanations for other asset pricing anomalies. 6. Conclusion In this paper, we propose a simple methodology to examine a large number of explanations that have been proposed in the literature for the negative relation between idiosyncratic volatility and subsequent stock returns (the idiosyncratic volatility puzzle). The 29

31 main advantage of our approach is that it allows us to quantify the contribution of each explanation either by itself or when evaluated against competing explanations. We find that surprisingly many existing explanations explain less than 10% of the idiosyncratic volatility puzzle. On the other hand, maximum daily return (Bali et al. (2011)), expected idiosyncratic skewness (Boyer et al. (2010)), one-month return reversal (Fu (2009) and Huang et al. (2009)), and pre- and post-formation earnings surprises (Wong (2011) and Jiang et al. (2009)) show much greater success in explaining the puzzle, although together they still leave 15-40% of the puzzle unexplained. Our decomposition methodology is robust to subsample analysis, using portfolios instead of individual stocks, and potential non-linearity in the idiosyncratic volatility-return relation, and can be easily adapted to evaluate competing explanations for a broad range of topics in empirical asset pricing and corporate finance. 30

32 Appendix In this appendix, we demonstrate the relation between our decomposition methodology (Equation 3) and the conventional approach of regressing returns on idiosyncratic volatility and a candidate variable (Equation 10). Specifically, for each month t, we can substitute Equation 2 into Equation 10 and obtain: R it = α t + γ t R (a t 1 + μ it 1 + δ t 1 Candidate it 1 ) + γ t C Candidate it 1 + ε it = α t + γ t R (a t 1 + μ it 1 ) + (γ t C + δ t 1 γ t R )Candidate it 1 + ε it = α t + γ t R (a t 1 + μ it 1 ) + γ t C Candidate it 1 + ε it, (11) where γ t C, which equals γ t C + δ t 1 γ t R, is identical to the coefficient of regressing returns on the candidate variable alone because (a t 1 + μ it 1 ) and Candidate it 1 are uncorrelated by construction. We can then rewrite γ t C from Equation 3 as follows: γ t C = Cov[R it, δ t 1 Candidate it 1 ] Var[IVOL it 1 ] = Cov[R it, δ t 1 Candidate it 1 ] Var[δ t 1 Candidate it 1 ] = C γ t Var[δ t 1Candidate it 1 ] δ t 1 Var[IVOL it 1 ] Var[δ t 1Candidate it 1 ] Var[IVOL it 1 ] = γ t C + γ R δ t Var[δ t 1Candidate it 1 ]. (12) t 1 Var[IVOL it 1 ] To show the relation more generally for k candidate variables, we simplify the notation by denoting IVOL it 1 as V and R it as R, both are n 1 vectors where n is the number of firms in the month t cross-sectional regression. We also denote an n 1 vector of ones by ι and we can now rewrite Equation 1 as: R = ια + Vγ + ε. (13) 31

33 Next, we regress V on ι and the n k matrix of k candidate variables (measured contemporaneously with V in month t 1) denoted by C = (C 1 C k ), where C j is an n 1 vector: V = ιa + Cδ C + μ. (14) δ C is the k 1 vector of coefficients. In the last step, we decompose the idiosyncratic volatilityreturn relation γ into k components each related to a candidate variable and a residual component: γ = (v v) 1 V r = (v v) 1 (Cδ C + ιa + μ) r = (v v) 1 (ιa + μ) r + (v v) 1 (Cδ C ) r, (15) where v and r (both n 1 vectors) are demeaned versions of V and R, respectively. The first term in the last line of Equation 15 represents the unexplained component of the idiosyncratic volatility puzzle and the second term represents the combined contribution of all k candidate variables. The contribution of the jth candidate variable is then γ C j = (v v) 1 C j δ C j r. Now, take the conventional approach of regressing R on V and C: R = ια + Vγ R + Cγ C + ε. (16) We can rewrite Equation 16 by substituting in Equation 14 as follows: R = ια + (ιa + Cδ C + μ)γ R + Cγ C + ε = ια + (ιa + μ)γ R + C(γ C + δ C γ R ) + ε. (17) Because C and (ιa + μ) are uncorrelated by construction, the coefficient on the jth candidate variable γ C j + δ C j γ R should be identical to the slope coefficient when R is regressed on the regression residual of C j on the other k 1 candidate variables. Specifically, we define an n (k+1) matrix C = (ι C 1 C k ), an (k+1) (k+1) matrix J which is an identity matrix except that the 32

34 (j+1)th diagonal term is set to zero, and θ which is the (k+1) 1 vector of coefficients from regressing C j on CJ. Then we have: γ j C + δ j C γ R = C j CJθ C j CJθ 1 C j CJθ r γ C j + δ C j γ R = C j CJθ C j CJθ 1 C j r (CJθ) r C j r = C j CJθ C j CJθ γ C j + δ C j γ R + (CJθ) r. (18) We can then rewrite Equation 15 to give us the relation between γ C j (the contribution of the jth candidate variable to the idiosyncratic volatility puzzle) and γ C j (the coefficient on the jth candidate variable in Equation 16): γ C j = (v v) 1 C j δ C j r = (v v) 1 δ C j C j r = (v v) 1 δ C j C j CJθ C j CJθ γ C j + δ C j γ R + (CJθ) r. (19) When k = 1, the above relation collapses to (v v) 1 δ C j C j C j γ C j + δ C j γ R, which is the matrix form of Equation

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37 Table 1: Sample descriptive statistics Sample statistics from are reported. Panel A shows the distribution of firm characteristics and Panel B shows the timeseries averages of cross-sectional correlations. The sample consists of all CRSP common stocks with share prices greater than $1 at the end of the previous month. N is the total number of firm-month observations. IVOL is the stdev of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Beta is the regression coefficient of the past three years of monthly returns on market returns. Size and B/M are measured and aligned as in Fama and French (2006), and Momentum is the buy-and-hold month t 12 to t 2 return. Skew is the month t 1 skewness of raw daily returns. Coskew is the coskewness measure in Chabi-Yo and Yang (2009). Maxret is the maximum daily return in month t 1. E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). RTP is the retail trading proportion computed from ISSM and TaQ. Lagret is the month t 1 return. Amihud is the illiquidity measure in Amihud (2002). Zeroret is the fraction of trading days in month t 1 with a zero return. Dispersion is the dispersion in analysts FY1 forecasts. AvgVarβ is a stock s exposure to the average variance component of the market variance as in Chen and Petkova (2012). SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. Panel A: Distribution of firm characteristics Variable Mean Std Dev N 1st Pctl 10th Pctl 25th Pctl 50th Pctl 75th Pctl 90th Pctl 99th Pctl Return IVOL Size ($m) B/M Momentum Beta Lottery preference variables Skew Coskew E(Idioskew) Maxret RTP Market friction variables Lagret Amihud Zeroret Other variables Dispersion AvgVarβ SUE NextSUE

38 Table 1 (Cont d) Panel B: Time-series averages of cross-sectional correlations between firm characteristics Variable Ret IVOL Beta Size BM Mom Skew Coskw Eiskw Maxret RTP Lagret Amihud Zeroret Disp AVβ SUE IVOL Beta Size B/M Momentum Skew Coskew E(Idioskew) Maxret RTP Lagret Amihud Zeroret Dispersion AvgVarβ SUE NextSUE

39 Table 2: The negative relation between idiosyncratic volatility and returns Firm-level Fama-MacBeth cross-sectional regressions are estimated each month from August 1963 to December Stocks with prices less than $1 at the end of the previous month are excluded. Time-series averages of the coefficients ( 100) and the associated time-series t-statistics (in parentheses) are reported. Idiosyncratic volatility (IVOL) is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Skew is the month t 1 skewness of raw daily returns. Coskew is the coskewness measure in Chabi-Yo and Yang (2009). Maxret is the maximum daily return in month t 1. E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). RTP is the retail trading ( $5000 trades) proportion computed from ISSM and TaQ. Lagret is the month t 1 return. Amihud is the illiquidity measure in Amihud (2002). Zeroret is the fraction of trading days in month t 1 with a zero return. Dispersion is the dispersion in analysts FY1 forecasts. AvgVarβ is a stock s exposure to the average variance component of the market variance as in Chen and Petkova (2012). SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Variable Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 Model 11 Model 12 Model 13 Intercept 1.574*** 1.590*** 1.564*** 1.627*** 1.498*** 1.860*** 1.489*** 1.591*** 1.521*** 1.519*** 1.579*** 1.513*** 1.277*** (8.62) (8.66) (8.70) (7.87) (8.33) (6.21) (8.44) (8.84) (7.19) (6.63) (8.10) (7.28) (6.10) IVOL *** *** *** *** *** *** *** *** *** *** *** *** (-4.11) (-3.88) (-4.03) (-6.75) (1.42) (-5.45) (-2.67) (-4.74) (-4.25) (-3.06) (-4.99) (-4.30) (-2.66) Skew *** (-4.07) Coskew *** (-2.78) E(Idioskew) (1.19) Maxret *** (-7.96) RTP (0.05) Lagret *** (-10.33) Amihud 0.040*** (3.46) Zeroret (1.17) Dispersion ** (-2.24) AvgVarβ * (-1.74) SUE 0.130*** (12.57) NextSUE 0.370*** (28.67) Avg Adj R #firms/mth Startdate Enddate

40 Table 3: Decomposing the idiosyncratic volatility puzzle: Univariate analysis Using firm-level Fama-MacBeth cross-sectional regressions, the negative relation between idiosyncratic volatility and returns is decomposed into a component that is related to a candidate variable and a residual component. Stage 1 regresses month t returns on month t 1 IVOL (R it = α t + γ t IVOL it 1 + ε it ). Stage 2 adds a candidate variable (Candidate it 1 ) to the regression. Stage 3 regresses IVOL on the candidate variable (IVOL it 1 = a t 1 + δ t 1 Candidate it 1 + μ it 1 ) to decompose IVOL it 1 into two orthogonal components: δ t 1 Candidate it 1 and (a t 1 + μ it 1 ). In Stage 4, the γ t coefficient from Stage 1 is decomposed as: γ t = Cov[R it,ivol it 1 ] Cov R it,(a t 1 +μ it 1 ) Var[IVOL it 1 ] 39 Var[IVOL it 1 ] = Cov[R it, δ t 1 Candidate it 1 ] + Var[IVOL it 1 ] = γ t C + γ t R. The time-series average of γ t C divided by the time-series average of γ t then measures the fraction of the negative idiosyncratic volatility-return relation explained by the candidate variable, and the average γ t R divided by the average γ t measures the fraction of the relation left unexplained by the candidate variable, with the standard error of the ratios being determined using the multivariate delta method. Stocks with prices less than $1 at the end of the previous month are excluded from the analysis. IVOL is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Panel A examines lottery preference-based candidate variables, where Skew is the month t 1 skewness of raw daily returns, Coskew is the coskewness measure in Chabi-Yo and Yang (2009), E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010), RTP is the retail trading proportion computed from ISSM and TaQ, Maxret is the maximum daily return in month t 1, Maxret(3mth) (LagMaxret(3mth))is the maximum daily return for the three-month period ending month t 1 (t 2). Panel B examines market friction-based candidate variables, where Lagret is the month t 1 return, Amihud is the illiquidity measure in Amihud (2002), and Zeroret is the fraction of trading days in month t 1 with a zero return. Panel C considers all other candidate variables, where Dispersion is the dispersion in analysts FY1 earnings forecasts, AvgVarβ is a stock s exposure to the average variance component of the market variance in Chen and Petkova (2012), and SUE and NextSUE are the standardized unexpected earnings from the previous quarter and following quarter respectively. Time-series averages of estimated coefficients ( 100) are reported with t-statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Panel A: Various lottery preference variables Stage Description Variable Skew Coskew E(Idioskew) Maxret Maxret(3mth) LagMaxret(3mth) RTP 1 Regress returns on IVOL 2 Add candidate variable 3 4 IVOL on candidate variable Decompose Stage 1 IVOL coefficient Intercept 1.588*** (8.63) 1.574*** (8.62) 1.617*** (8.43) 1.574*** (8.62) 1.562*** (8.58) 1.564*** (8.60) 1.876*** (6.37) IVOL *** (-4.20) ***(-4.11) ***(-5.35) ***(-4.11) ***(-4.04) ***(-3.99) ***(-4.95) Intercept 1.590*** (8.66) 1.564*** (8.70) 1.627*** (7.87) 1.498*** (8.33) 1.601*** (8.83) 1.592*** (9.03) 1.860*** (6.21) IVOL *** (-3.88) ***(-4.03) ***(-6.75) (1.42) ***(-3.77) ***(-5.06) ***(-5.45) Candidate *** (-4.07) *** (-2.78) (1.19) *** (-7.96) *** (-2.91) (-0.58) (0.05) Intercept 2.415*** (90.36) 2.495*** (87.56) 1.260*** (61.15) 0.775*** (80.94) 1.022*** (61.89) 1.471*** (69.75) 2.108*** (66.69) Candidate 0.376*** (34.37) 0.612*** (9.51) 1.542*** (41.72) *** (242.64) *** (107.37) *** (76.67) 5.077*** (46.00) Adj R 2 4.4% 4.0% 18.6% 78.1% 49.0% 23.7% 22.7% Candidate %*** (4.07) 3.3%* (1.80) 10.5%* (1.69) 102%*** (11.83) 54.5%*** (11.34) 24.0%*** (3.54) 24.4%*** (4.37) Residual %*** (56.19) 96.7%*** (52.81) 89.5%*** (14.36) -2.2% (-0.25) 45.5%*** (9.46) 76.0%*** (11.21) 75.6%*** (13.55) Total *** (-4.20) *** (-4.11) *** (-5.35) *** (-4.11) *** (-4.04) *** (-3.99) *** (-4.95) 100% 100% 100% 100% 100% 100% 100% Sample period 1963 to to to to to to to 2001 Avg # firms/mth

41 Table 3 (Cont d) Panel B: Market friction variables Panel C: Other variables Stage Description Variable Lagret Amihud Zeroret Dispersion AvgVarβ SUE NextSUE 1 Regress returns on IVOL 2 Add candidate variable 3 IVOL on candidate variable 4 Decompose Stage 1 IVOL coefficient Intercept 1.574*** (8.62) 1.564*** (8.52) 1.574*** (8.62) 1.517*** (6.60) 1.586*** (8.13) 1.612*** (7.72) 1.527*** (7.30) IVOL ***(-4.11) ***(-4.02) ***(-4.11) ***(-3.12) ***(-5.00) ***(-4.75) ***(-3.94) Intercept 1.489*** (8.44) 1.591*** (8.84) 1.521*** (7.19) 1.519*** (6.63) 1.579*** (8.10) 1.513*** (7.28) 1.277*** (6.10) IVOL ***(-2.67) ***(-4.74) ***(-4.25) ***(-3.06) ***(-4.99) ***(-4.30) ***(-2.66) Candidate *** (-10.33) 0.040*** (3.46) (1.17) ** (-2.24) * (-1.74) 0.130*** (12.57) 0.370*** (28.67) Intercept 2.369*** (84.89) 2.396*** (86.01) 2.455*** (89.06) 2.194*** (71.15) 2.506*** (91.45) 2.622*** (88.39) 2.615*** (88.80) Candidate 2.365*** (22.00) 0.040*** (19.30) 0.805*** (10.64) 0.210*** (17.01) 0.009*** (6.07) *** (-34.07) *** (-35.87) Adj R 2 8.3% 11.2% 1.0% 1.8% 0.5% 1.2% 1.0% Candidate %*** (3.72) -8.1% (-1.07) -0.9% (-0.42) 6.4%* (1.93) 1.0%** (2.07) 11.4%*** (5.33) 33.9%*** (4.18) Residual %*** (11.75) 108%*** (14.20) 101%*** (45.97) 93.6%*** (28.17) 99.0%*** (198.20) 88.6%*** (41.28) 66.1%*** (8.14) Total *** (-4.11) *** (-4.02) *** (-4.11) *** (-3.12) *** (-5.00) *** (-4.75) *** (-3.94) 100% 100% 100% 100% 100% 100% 100% Sample period 1963 to to to to to to to 2012 Avg # firms/mth

42 Table 4: Decomposing the idiosyncratic volatility puzzle: Multivariate analysis Using firm-level Fama-MacBeth cross-sectional regressions, the negative relation between month t 1 idiosyncratic volatility and month t returns is decomposed into a number of components each related to a candidate variable and a residual component. The candidate variables considered here are those that explain more than 10% of the puzzle in the univariate analysis. Stocks with prices less than $1 at the end of the previous month are excluded. IVOL is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Maxret(3mth) (LagMaxret(3mth)) is the maximum daily return for the three-month period ending in month t 1 (t 2). E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). Lagret is the month t 1 return. SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. The standard errors of the fractions of the puzzle explained in Stage 4 are determined using the multivariate delta method. Time-series averages of estimated coefficients ( 100) are reported with t- statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Stage Description Variable Model 1 Model 2 Model 3 1 Regress returns on Intercept 1.569*** (7.68) 1.570*** (7.69) 1.646*** (8.10) IVOL IVOL *** (-4.06) *** (-4.07) *** (-4.93) 2 Add candidate Intercept 1.227*** (5.80) 1.237*** (5.93) 1.525*** (7.37) variables IVOL (-0.98) (-1.54) *** (-3.48) 3 IVOL on candidate variables 4 Decompose Stage 1 IVOL coefficient Maxret(3mth) *** (-4.68) LagMaxret(3mth) *** (-4.14) *** (-4.23) E(IdioSkew) 0.249** (2.15) 0.265** (2.34) (1.32) Lagret *** (-12.75) *** (-13.30) *** (-11.01) NextSUE 0.382*** (30.82) 0.382*** (30.77) SUE 0.025*** (2.90) 0.026*** (2.90) 0.146*** (14.19) Intercept 0.618*** (42.35) 0.807*** (42.73) 0.822*** (42.58) Maxret(3mth) *** (82.07) LagMaxret(3mth) 9.037*** (59.95) 8.421*** (57.77) E(IdioSkew) 0.660*** (41.79) 0.990*** (41.84) 1.025*** (42.94) Lagret 0.780*** (11.55) 2.088*** (24.27) 2.247*** (26.08) NextSUE *** (-25.28) *** (-27.26) SUE *** (-23.48) *** (-23.38) *** (-27.26) Adj R % 39.2% 37.6% Maxret(3mth) %*** (12.72) LagMaxret(3mth) %*** (8.39) 28.7%*** (9.74) E(IdioSkew) % (0.80) 3.3% (0.55) 5.0% (1.00) Lagret %** (2.53) 24.7%*** (3.68) 21.2%*** (4.12) NextSUE %*** (4.18) 12.2%*** (4.26) SUE %*** (3.90) 4.5%*** (4.12) 5.2%*** (4.83) Residual %*** (2.77) 24.2%*** (3.90) 39.9%*** (9.78) Total *** (-4.06) *** (-4.07) *** (-4.93) 100% 100% 100% Sample 1971 to to to 2012 Avg # firms

43 Table 5: Decomposing the idiosyncratic volatility puzzle: Subsample analysis Using firm-level Fama-MacBeth cross-sectional regressions, the negative relation between month t 1 idiosyncratic volatility and month t returns is decomposed into a number of components each related to a candidate variable and a residual component. The candidate variables considered here are those that explain more than 10% of the puzzle in the full sample univariate analysis. We perform the multivariate decomposition analysis on subsamples of stocks that have prices greater than five dollars (Panel A), low analyst coverage (1-3 analysts, Panel B), poor credit ratings (the lowest rating tercile, Panel C), high short-sale constraints (the highest short interest tercile, Panel D), high leverage (the highest leverage tercile, Panel E), and for non-january months (Panel F). IVOL is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Maxret(3mth) (LagMaxret(3mth)) is the maximum daily return for the three-month period ending in month t 1 (t 2). E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). Lagret is the month t 1 return. SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. The standard errors of the fractions of the puzzle explained are determined using the multivariate delta method. Time-series averages of estimated coefficients ( 100) are reported with t-statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Variable Model 1 Model 2 Model 3 Coeff. Fraction t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Maxret(3mth) %*** (12.46) Panel A: Prices $5 LagMaxret(3mth) %*** (8.25) %*** (7.71) E(Idioskew) % (-0.05) % (-0.28) % (-0.38) Lagret %*** (3.08) %*** (4.08) %*** (3.93) NextSUE %*** (4.56) %*** (4.62) SUE %*** (3.71) %*** (3.93) %*** (4.01) Residual %*** (4.32) %*** (4.45) %*** (7.94) Total *** 100% (-4.57) *** 100% (-4.57) *** 100% (-4.45) Sample 1971 to to to 2012 Avg # firms Panel B: Low analyst coverage Maxret(3mth) %*** (7.46) LagMaxret(3mth) %*** (4.12) %*** (4.66) E(Idioskew) %*** (3.04) %*** (3.26) %*** (3.21) Lagret % (-0.88) % (0.87) % (1.35) NextSUE %*** (3.50) %*** (3.61) SUE %** (2.45) %*** (2.64) %*** (3.67) Residual %*** (5.37) %*** (6.95) %*** (8.99) Total *** 100% (-4.09) *** 100% (-4.09) *** 100% (-4.50) Sample 1982 to to to 2012 Avg # firms Panel C: Poor credit ratings Maxret(3mth) %*** (3.94) LagMaxret(3mth) %*** (3.22) %*** (4.00) E(Idioskew) %* (1.96) %** (2.23) %*** (3.30) Lagret % (0.27) % (0.04) % (0.57) NextSUE %*** (3.46) %*** (3.48) SUE % (1.34) %* (1.93) %*** (3.12) Residual %*** (3.71) %*** (4.51) %*** (10.22) Total *** 100% (-3.48) *** 100% (-3.46) *** 100% (-5.13) Sample 1986 to to to 2012 Avg # firms

44 Table 5 (Cont d) Variable Model 1 Model 2 Model 3 Coeff. Fraction t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Panel D: High short-sale constraints Maxret(3mth) %*** (7.99) LagMaxret(3mth) %*** (5.62) %*** (5.95) E(Idioskew) %*** (4.98) %*** (5.27) %*** (5.85) Lagret % (-0.55) % (1.02) % (1.35) NextSUE %*** (4.01) %*** (4.04) SUE %** (2.45) %*** (2.89) %*** (4.08) Residual %*** (4.17) %*** (4.21) %*** (7.24) Total *** 100% (-4.05) *** 100% (-4.05) *** 100% (-4.77) Sample 1988 to to to 2005 Avg # firms Maxret(3mth) %*** (11.12) Panel E: High leverage LagMaxret(3mth) %*** (7.74) %*** (8.76) E(Idioskew) % (-0.12) % (-0.27) % (0.26) Lagret %*** (2.65) %*** (3.54) %*** (3.99) NextSUE %*** (3.93) %*** (4.07) SUE %*** (3.76) %*** (4.04) %*** (5.27) Residual %*** (3.81) %*** (5.31) %*** (9.41) Total *** 100% (-4.14) *** 100% (-4.18) *** 100% (-5.37) Sample 1971 to to to 2012 Avg # firms Panel F: Non-January months Maxret(3mth) %*** (21.86) LagMaxret(3mth) %*** (14.05) %*** (15.28) E(Idioskew) %*** (5.88) %*** (6.09) %*** (6.86) Lagret %** (2.52) %*** (4.68) %*** (4.72) NextSUE %*** (6.87) %*** (7.12) SUE %*** (5.65) %*** (6.09) %*** (6.66) Residual %*** (6.32) %*** (7.97) %*** (13.89) Total *** 100% (-7.16) *** 100% (-7.17) *** 100% (-8.17) Sample 1971 to to to 2012 Avg # firms

45 Table 6: Decomposing the idiosyncratic volatility puzzle: Portfolio-level analysis Using portfolio-level Fama-MacBeth cross-sectional regressions, the negative relation between month t 1 idiosyncratic volatility and month t returns is decomposed into a number of components each related to a candidate variable and a residual component. Panel A (Panel B) performs univariate (multivariate) analysis using 200 idiosyncratic volatility-sorted portfolios. At the beginning of each month t, we sort individual stocks into 200 portfolios based on their month t 1 IVOL. Value-weighted returns are calculated for these idiosyncratic volatility-sorted portfolios for month t and the portfolios are rebalanced at the beginning of month t+1. We then use the portfolio returns to compute portfolio-level IVOL and return-based candidate variables (Skew, Coskew, different Maxret measures, Lagret, Amihud, and AvgVarβ). The other candidate variables (E(Idioskew), RTP, Zeroret, Dispersion, SUE, and NextSUE) are computed as value-weighted averages of the firm-level numbers within each portfolio. IVOL is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Skew is the month t 1 skewness of raw daily returns. Coskew is the coskewness measure in Chabi-Yo and Yang (2009). Maxret is the maximum daily return in month t 1. Maxret(3mth) (LagMaxret(3mth))is the maximum daily return for the three-month period ending in month t 1 (t 2). E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). RTP is the retail trading ( $5000 trades) proportion computed from ISSM and TaQ. Lagret is the month t 1 return. Amihud is the illiquidity measure in Amihud (2002). Zeroret is the fraction of trading days in month t 1 with a zero return. Dispersion is the dispersion in analysts FY1 forecasts. AvgVarβ is a stock s exposure to the average variance component of the market variance as in Chen and Petkova (2012). SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. Stocks with prices less than $1 at the end of the previous month are excluded from the analysis. The standard errors of the fractions of the puzzle explained are determined using the multivariate delta method. Time-series averages of estimated coefficients ( 100) are reported with t-statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Panel A: Univariate analysis Candidate IVOL Coeff. Candidate Component Residual Component Coeff. t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Skew *** (-5.52) %** (2.43) %*** (59.25) Coskew *** (-5.50) %*** (4.22) %*** (29.92) E(Idioskew) *** (-6.62) %*** (5.49) %*** (14.62) Maxret *** (-5.50) %*** (11.71) %*** (8.74) Maxret(3mth) *** (-5.50) %*** (15.34) %*** (9.26) LagMaxret(3mth) *** (-5.50) %*** (15.57) %*** (9.88) RTP *** (-6.11) %*** (11.19) %*** (11.06) Lagret *** (-5.50) %*** (2.84) %*** (26.18) Amihud *** (-5.54) %*** (6.33) %*** (21.99) Zeroret *** (-5.50) %* (1.67) %*** (41.63) Dispersion *** (-5.47) %*** (4.83) %*** (48.22) AvgVarβ *** (-6.50) %*** (4.45) %*** (94.08) SUE *** (-6.35) %*** (4.94) %*** (53.24) NextSUE *** (-6.48) %*** (7.84) %*** (28.68) Panel B: Multivariate analysis Model 1 Model 2 Model 3 Variable Coeff. Fraction t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Maxret(3mth) %*** (17.23) LagMaxret(3mth) %*** (16.47) %*** (17.18) E(Idioskew) %*** (3.41) %*** (2.98) %*** (2.80) Coskew %*** (3.59) %*** (4.06) %*** (3.89) Amihud % (-0.04) % (-0.77) % (-0.69) NextSUE %*** (6.71) %*** (6.32) Residual %*** (7.72) %*** (7.62) %*** (8.90) Total *** 100% (-6.62) *** 100% (-6.62) *** 100% (-6.65) Sample 1971 to to to

46 Table 7: Decomposing the idiosyncratic volatility puzzle: Non-linear relations Using firm-level Fama-MacBeth cross-sectional regressions with non-linear terms, the negative relation between month t 1 idiosyncratic volatility and month t returns is decomposed into a number of components each related to a candidate variable and a residual component. Panel A examines together the level and the squared terms of each candidate variable. Panel B performs multivariate analysis using the candidate variables whose level and squared terms together explain more than 10% of the puzzle in Panel A. Panel C reports the results of univariate decomposition analysis using a dummy variable HIGHIVOL where HIGHIVOL=1 when IVOL is in the highest decile in month t 1 and 0 otherwise. Panel D reports the results of multivariate decomposition analysis using the candidate variables that explain more than 10% of the HIGHIVOL-return relation in Panel C. IVOL is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Skew is the month t 1 skewness of raw daily returns. Coskew is the coskewness measure in Chabi-Yo and Yang (2009). Maxret is the maximum daily return in month t 1. Maxret(3mth) (LagMaxret(3mth))is the maximum daily return for the threemonth period ending in month t 1 (t 2). E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). RTP is the retail trading ( $5000 trades) proportion computed from ISSM and TaQ. Lagret is the month t 1 return. Amihud is the illiquidity measure in Amihud (2002). Zeroret is the fraction of trading days in month t 1 with a zero return. Dispersion is the dispersion in analysts FY1 forecasts. AvgVarβ is a stock s exposure to the average variance component of the market variance as in Chen and Petkova (2012). SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. The standard errors of the fractions of the puzzle explained are determined using the multivariate delta method. Time-series averages of estimated coefficients ( 100) are reported with t-statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Panel A: Level and squared terms of each candidate variable IVOL Coeff. Candidate Component Squared Candidate Component Residual Component Candidate Coeff. t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Skew *** (-4.20) %*** (4.06) %*** (3.68) %*** (41.78) Coskew *** (-4.11) % (1.61) %* (1.85) %*** (29.38) E(Idioskew) *** (-5.35) % (1.03) % (1.11) %*** (14.16) Maxret *** (-4.11) %*** (11.80) %*** (-6.82) % (-0.50) Maxret(3mth) *** (-4.04) %*** (9.92) %*** (-4.34) %*** (8.12) LagMaxret(3mth) *** (-3.99) %*** (3.45) %*** (-3.94) %*** (8.51) RTP *** (-4.95) %*** (5.47) %*** (-5.28) %*** (12.42) Lagret *** (-4.11) % (1.55) %** (2.14) %*** (14.45) Amihud *** (-4.02) % (-0.99) % (0.89) %*** (11.12) Zeroret *** (-4.11) % (-0.36) % (-0.14) %*** (31.13) Dispersion *** (-3.12) %** (2.13) % (-1.29) %*** (19.55) AvgVarβ *** (-5.00) %** (2.27) %*** (4.10) %*** (77.14) SUE *** (-4.75) %*** (5.46) % (-1.19) %*** (34.36) NextSUE *** (-3.94) %*** (4.18) %*** (-3.11) %*** (6.30) 45

47 Table 7 (Cont d) Panel B: Level and squared terms of multiple candidate variables together Variable Model 1 Model 2 Model 3 Coeff. Fraction t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Maxret(3mth) %*** (12.52) Maxret(3mth) %*** (-5.67) Lagmaxret(3mth) %*** (8.03) %*** (9.63) Lagmaxret(3mth) %*** (-6.18) %*** (-7.64) E(Idioskew) % (0.89) % (0.65) % (1.13) E(Idioskew) % (-0.02) % (-0.02) % (0.10) Lagret % (0.22) %*** (3.29) %*** (3.41) Lagret %*** (3.65) %*** (3.34) %*** (4.19) NextSUE %*** (4.21) %*** (4.26) NextSUE % (-1.36) % (-1.29) SUE %*** (3.89) %*** (4.09) %*** (5.12) SUE % (-0.75) % (-1.27) % (-1.21) Residual %*** (5.76) %*** (7.50) %*** (10.45) Total *** 100% (-4.06) *** 100% (-4.07) *** 100% (-4.93) Sample Avg # firms 1971 to to to 2012 Panel C: Univariate analysis using HIGHIVOL instead of IVOL Candidate HIGHIVOL coef. Candidate Component Residual Component Coeff. t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Skew *** (-5.83) %*** (2.67) %*** (64.10) Coskew *** (-5.81) %*** (3.30) %*** (100.67) E(Idioskew) *** (-6.61) % (1.61) %*** (16.51) Maxret *** (-5.81) %*** (14.76) %*** (4.16) Maxret(3mth) *** (-5.66) %*** (7.05) %*** (9.90) LagMaxret(3mth) *** (-5.65) %*** (3.17) %*** (15.08) RTP *** (-4.16) %** (2.45) %*** (9.90) Lagret *** (-5.81) %*** (4.23) %*** (16.51) Amihud *** (-5.52) % (-1.04) %*** (19.69) Zeroret *** (-5.81) %* (-1.71) %*** (67.35) Dispersion *** (-3.28) %** (2.14) %*** (36.55) AvgVarβ *** (-4.97) %* (1.86) %*** (152.25) SUE *** (-5.24) %*** (6.02) %*** (71.11) NextSUE *** (-3.82) %*** (4.06) %*** (11.03) Panel D: Multivariate analysis using HIGHIVOL instead of IVOL Model 1 Model 2 Model 3 Variable Coeff. Fraction t-stat Coeff. Fraction t-stat Coeff. Fraction t-stat Maxret(3mth) %*** (6.70) LagMaxret(3mth) %*** (3.25) %*** (3.77) Lagret % (1.14) %*** (3.38) %*** (4.57) NextSUE %*** (3.93) %*** (3.94) Residual %*** (3.59) %*** (3.76) %*** (12.73) Total *** 100% (-3.76) *** 100% (-3.74) *** 100% (-5.65) Sample 1971 to to to 2012 Avg # firms

48 Table 8: Decomposing other anomalies Using firm-level Fama-MacBeth cross-sectional regressions, the relations between three anomaly variables (Maxret, Lagret, and SUE) and returns are decomposed into a component that is related to IVOL and a residual component. Stage 1 regresses month t returns on an anomaly variable. Stage 2 adds IVOL as the candidate variable to the regression. Stage 3 regresses the anomaly variable on IVOL to decompose the anomaly variable into two orthogonal components. In Stage 4, the coefficient on the anomaly variable from Stage 1 is decomposed into a component that is related to IVOL and a residual component. The time-series average of the IVOL component divided by the time-series average of the stage 1 coefficient on the anomaly variable then measures the fraction of the anomaly explained by IVOL, and the average residual component divided by the average stage 1 coefficient measures the fraction of the anomaly left unexplained by IVOL. The standard errors of the fractions explained are determined using the multivariate delta method. Stocks with prices less than $1 at the end of the previous month are excluded from the analysis. IVOL is the standard deviation of residuals from a regression of daily stock returns in month t 1 on the Fama-French (1993) factors. Maxret is the maximum daily return in month t 1, Lagret is the month t 1 return, and SUE is the standardized unexpected earnings from the previous quarter. Time-series averages of estimated coefficients ( 100) are reported with t- statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. Stage Description Variable Maxret Lagret SUE 1 Regress returns on anomaly variable 2 Add IVOL 3 Anomaly variable on IVOL 4 Decompose Stage 1 anomaly coefficient Intercept 1.554*** (7.82) 1.165*** (5.07) 1.090*** (4.13) Anomaly *** (-6.23) *** (-9.16) 0.135*** (10.91) Intercept 1.498*** (8.33) 1.489*** (8.44) 1.513*** (7.28) Anomaly *** (-7.96) *** (-10.33) 0.130*** (12.57) IVOL (1.42) *** (-2.67) *** (-4.30) Intercept *** (-26.19) *** (-15.44) *** (22.92) IVOL *** (203.51) *** (20.75) *** (-34.26) Adj R % 8.3% 1.2% IVOL %*** (10.66) -1.6% (-0.38) 5.6%* (1.93) Residual %*** (4.35) 102%*** (24.30) 94.4%*** (32.84) Total *** (-6.23) *** (-9.16) 0.135*** (10.91) 100% 100% 100% Sample period 1963 to to to 2012 Avg # firms/mth

49 Figure 1: Average fractions of the idiosyncratic volatility puzzle explained by various explanations The fractions of the idiosyncratic volatility puzzle explained by various explanations from Tables 4 are plotted in Panel A. Panel B plots the average fractions from the six subsamples in Table 5. Maxret(3mth) (LagMaxret(3mth)) is the maximum daily return for the three-month period ending in month t 1 (month t 2). E(Idioskew) is the expected idiosyncratic skewness measure in Boyer et al. (2010). Lagret is the month t 1 return. SUE and NextSUE are the standardized unexpected earnings from the previous quarter and the following quarter, respectively. Panel A: Overall sample from Table 4 Model 1 SUE 3.2% NextSUE 9.6% Lagret 8.1% E(Idioskew) 3.3% Residual 14.9% Unexplained Others Frictions Lottery Maxret(3mth) 60.9% Panel B: Average of six subsamples from Table 5 Model 1 Residual Maxret(3mth) 49.8% 27.5% Model 2 SUE 4.5% NextSUE 12.2% Model 2 Residual 24.2% Unexplained Others Residual 35.3% Frictions Lagret 24.7% LagMaxret(3mth) 31.0% Lottery Johnson Held E(Idioskew) 3.3% LagMaxret(3mth) 27.4% Model 3 Model 3 Residual 39.9% Unexplained Others SUE 5.2% Residual 48.4% Frictions Lagret 21.2% LagMaxret(3mth) 28.7% Lottery E(Idioskew) 5.0% LagMaxret(3mth) 25.1% SUE 2.8% NextSUE 9.8% Unexplained Others Frictions Lagret 2.7% E(Idioskew) 7.5% Lottery SUE 4.0% Unexplained Others NextSUE 12.2% Lottery Johnson Held Frictions Lagret 10.5% E(Idioskew) 10.6% Unexplained Others SUE 4.9% Lottery Frictions Lagret 10.6% E(Idioskew) 11.0% 48