DOES IT PAY TO HAVE FAT TAILS? EXAMINING KURTOSIS AND THE CROSS-SECTION OF STOCK RETURNS

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1 DOES IT PAY TO HAVE FAT TAILS? EXAMINING KURTOSIS AND THE CROSS-SECTION OF STOCK RETURNS By Benjamin M. Blau 1, Abdullah Masud 2, and Ryan J. Whitby 3 Abstract: Xiong and Idzorek (2011) show that extremely large price movements, which are rare under a normal distribution, occur approximately 10 times more often than predicted by a normal distribution indicating that the normality assumption required in many asset pricing models is often violated. When examining whether the higher moments of return distributions are priced in stocks, prior research has demonstrated that skewness predicts negative returns. While some studies have investigated the return premium associated with skewness, fewer studies have focused on the relation between kurtosis and future returns. In this paper we attempt to determine whether there is a significant return premium associated with kurtosis. Using a broad sample of stocks 1980 to 2010, we find that the average stock exhibits excess kurtosis and that kurtosis is negatively related to firm size and positively related to volatility and turnover. Further, we show that stocks with high kurtosis have higher raw returns than stocks with low kurtosis. However, when we control for other risk factors that are more common in traditional asset pricing models, the return premium associated with kurtosis vanishes. These results indicate that kurtosis is not priced in stocks. 1 Benjamin M. Blau is an Assistant Professor in the Department of Economics and Finance at Utah State University, 3565 Old Main Hill, Logan, Utah Phone: Fax: Abdullah Masud is a Graduate Student in the Department of Economics and Finance at Utah State University, 3565 Old Main Hill, Logan, Utah Ryan J. Whitby is an Assistant Professor in the Department of Economics and Finance at Utah State University, 3565 Old Main Hill, Logan, Utah Phone: Fax: Electronic copy available at:

2 1. INTRODUCTION The presence of bubbles and subsequent reversals in securities markets is a reminder that stock returns tend to be leptokurtic, or have fat tails. That is, extreme events happen more regularly than predicted by a normal distribution. While many asset pricing models assume normality in the return distribution, most evidence suggests that this normality assumption is often violated (see Fama (1965) and Rosenberg (1974)). Almost all asset classes and portfolios have returns that are not normally distributed. Events that are very rare under a normal distribution seem to occur empirically approximately 10 times more often than predicted by a normal distribution (see Xiong and Idzorek (2011)). While the non-normality of return distributions is a generally accepted empirical result, there is still widespread resistance by both practitioners and academics to models that use non-normal distributions. This reluctance often stems from the fact that non-normal asset pricing models can increase estimation error, which leads to additional uncertainty. Nevertheless, understanding how the non-normality of asset returns affects the pricing of assets is important. While standard portfolio theory assumes that returns are normally distributed, the outcomes associated with fatter tails in the return distribution are similar to those of a distribution with a larger standard deviation. Namely, there is a greater probability of a particular return that is further from the mean in any given time period. If investors perceive that leptokurtic return distributions represent some level of risk that is unaccounted for, then, all else equal, investors may prefer less kurtosis to more kurtosis for a given expected return. Further, assets that increase the kurtosis of a portfolio may not only be less desirable, buy may also have higher expected returns. It is important to note, however, that for kurtosis to be priced, kurtosis must not covary perfectly with volatility or any other risk factor. The main objective of this 2 Electronic copy available at:

3 paper is the determine whether or not kurtosis helps explain the cross-section of returns, especially after accounting for the other known risk factors that influence the cross-section of the returns. In particular, we attempt to identify the presence of a return premium associated with kurtosis. In this study, we ask two simple questions: First, what stock-specific characteristics determine the level of kurtosis? Second, are investors compensated in the form of a return premium for holding stocks that exhibit excess kurtosis? We find that the average stock has a leptokurtic distribution. Additionally, both our univariate and multivariate tests demonstrate that the kurtosis of a stock is negatively related to size and positively related to volatility and turnover. As expected, smaller and more volatile stocks have distributions that are leptokurtic. Results are similar when we examine idiosyncratic kurtosis. In our second set of tests, we examine whether higher returns are associated with higher kurtosis. While stocks with higher kurtosis have higher raw returns, which is indicative of a significant kurtosis return premium, once we control for size, the return premium associated with kurtosis disappears. Thus, although the return distributions of stocks are, on average, leptokurtic, any return premium associated with kurtosis seems to be captured almost entirely by the common negative relation between firm size and future returns. Both theoretical and empirical research has examined the higher moments of return distributions. Harvey and Siddique (2000) develop an asset pricing model that incorporates conditional skewness while Barberis and Huang (2008) model the preferences for skewness and show theoretically that skewness leads to lower future returns. Similar results are found in Mitton and Vorkink (2007). Further, empirical results in Mitton and Vorkink show that retail investors under-diversify in order to obtain greater skewness in their portfolios. Boyer, Mitton, 3

4 and Vorkink (2010) find that expected idiosyncratic skewness and returns are negatively related. Similarly, Zhang (2005) also finds that skewness explains the cross-section of returns. 1 The consensus in the literature suggests that skewness is priced. While most of the prior research examining the non-normality of return distributions has focused on skewness, other papers have also considered kurtosis. Xiong and Idzorek (2011) examine the effect of skewness and kurtosis on optimized portfolio. Conrad, Dittmar, and Ghysels (2013) examine ex-ante skewness and kurtosis implied through option prices and find both are related to the cross-section of stock returns. Our findings extend this growing stream of literature by showing that kurtosis does not predict higher future returns. Any return premium associated with kurtosis is, in general, unrelated to the cross-section of returns after accounting for firm size. These findings have important implications for investors, fund managers, and analysts that use traditional asset pricing models or focus on mean-variance investment strategies. 2. DATA DESCRIPTION The data used in this analysis comes from two primary sources. From Compustat, we gather annual balance sheet information including assets, liabilities, and total equity. From the universe of stocks available on the Center for Research on Security Prices (CRSP), we obtain trading volume, prices, shares outstanding, etc., and aggregate this data to the yearly level. We also obtain daily returns and daily value-weighted market returns. Using these returns, we estimate a market model by regressing daily returns on value-weighted market returns in order to 1 In other studies that examine higher moments of return distributions, Patton (2004) finds that distributional asymmetries and skewness are advantageous to managers of firms that face short-sale constraints. Further, Harvey, Liechty, Liechty, and Muller (2010) use a bayesian framework to model portfolio selection incorporating higher moments while Ilmanen (2011) investigates the skewness of returns with respect to insurance and lotteries. 4

5 obtain daily residual returns. Using CRSP daily raw returns and residual returns from our market model, we estimate kurtosis in the following way. ( 1) n n n + ( rt r) Kurtosis = 4 ( n 1)( n 2)( n 3) i 1 σ 4 2 3( n 1) ( n 2)( 3) = n (1) r t is either the raw return or the residual return and σ is the standard deviation of either the raw return or the residual return, respectively. We denote the results from estimating equation (1) as total kurtosis when r t is defined as the raw return and idiosyncratic kurtosis when r t is defined as the residual return. We estimate equation (1) for the universe of stocks available on CRSP for each year during the period 1980 to The total number of unique stocks used in the analysis is 10,465. Table 1 reports statistics that describe our sample. Kurt is the estimate for total kurtosis while IdioKurt is the estimate of kurtosis that uses the daily residuals from the market model. Price is the CRSP closing price for each stock at the end of each year. Size is the market capitalization obtained from CRSP. B/M is the book-to-market ratio using the end-of-year balance sheet equity from Compustat and market capitalization from CRSP. Turn is the share turnover and is calculated by dividing the annual trading volume by shares outstanding. IdioVolt is the idiosyncratic volatility or the standard deviation of the daily residual returns (residuals from daily market models) for each stock during each year. Beta is obtained by estimating CAPM using CRSP returns. The summary statistics reported in Table 1 are equally weighted by stock. Panel A reports the results when examining all stock-year observations. We find that the average stock has a total kurtosis of and an idiosyncratic kurtosis of A normal distribution has kurtosis equal to 0. These results suggest that the average stock has a return distribution with excess kurtosis suggesting that the common normality assumption required in 5

6 Table 1 Summary Statistics The table reports statistics that describe our sample. Panel A reports the summary statistics for the entire time period ( ). Panels B through D report the statistics for each of the three decades during the sample time period. Kurt is the kurtosis of daily returns calculated for each stock during each year while IdioKurt is the idiosyncratic kurtosis of daily residual returns (residuals from daily market models). Price is the CRSP closing price. Size is the market capitalization from CRSP. B/M is the book-to-market ratio. Turn is the share turnover of the annual volume scaled by shares outstanding. IdioVolt is the idiosyncratic volatility or the standard deviation of the daily residual returns (residuals from daily market models) for each stock during each year. Beta is the CAPM beta during each year using CRSP returns. The summary statistics reported are equally-weighted for each stock. Panel A. All Stock-Year Observations (N = 10,465) Mean Median Standard Dev. Min Max [1] [2] [3] [4] [5] Kurt IdioKurt Price Size B/M Turn IdioVolt Beta ,199,575, ,279, Panel B. Stock-Year Observations from 1980 to 1989 (N = 3,022) Kurt IdioKurt Price Size B/M Turn IdioVolt Beta ,532, ,052, Panel C. Stock-Year Observations from 1990 to 1999 (N = 6,507) Kurt IdioKurt Price Size B/M Turn IdioVolt Beta ,100,532, ,805, Panel D. Stock-Year Observations from 2000 to 2010 (N = 6,877) Kurt IdioKurt ,517,823, ,948,101, ,602,142, ,042,478, ,4602,228, ,873,055, Price Size B/M Turn IdioVolt Beta ,267,998, ,071, ,553,057, , ,492,491,

7 most asset pricing models is violated. The average stock has a share price of $16.38, a market cap of nearly $1.2 billion, a B/M ratio of 0.44, turnover of 13.51, idiosyncratic volatility of 3.97%, and a CAPM beta of Panels B through D report the results for each of the three decades during the sample time period. In panel B, there are 3,022 unique stocks during the sub-period. Again, we find that the average stock has excess kurtosis as the mean Kurt is while the mean IdioKurt is Further, the mean price is $14.91, the mean market cap is $425 million, the mean B/M ratio is 0.57, the mean turnover is 5.96, the mean idiosyncratic volatility is 3.17%, and the mean beta is approximately equal to one. Panel C presents the statistics for 6,507 unique stocks during the period of 1990 to During this sub-period, we find markedly less kurtosis of and idiosyncratic kurtosis of in the average stock. Relative to the panel A, we find that the average stock has a higher price of $18.98, greater market cap, $1.1 billion, a higher B/M ratio of , more turnover, , more idiosyncratic volatility of 4.25%, and a lower beta of Panel D reports the summary statistics for 6,877 unique stocks during the period 2000 to The average stock has total kurtosis of , idiosyncratic kurtosis of , a share price of $18.36, market capitalization of $2.3 billion, a B/M ratio of 0.34, turnover of 17.42, idiosyncratic volatility of 3.99%, and a beta of EMPIRICAL RESULTS In Table 1, we find the presence of excess kurtosis, on average. Given that much of asset pricing theory rests on the assumption of normality in the return distribution, we explore which stock-specific characteristics are related to the level of kurtosis. The first set of tests attempts to identify the characteristics that determine the level of kurtosis. We conduct both univariate and multivariate tests when identifying the determinants of kurtosis. The second set of tests 7

8 examines the effect of kurtosis on the cross-section of returns, which is the main objective of this study. Using familiar methods for calculating future returns, we attempt to determine whether there is a return premium associated with kurtosis. 3.1 Determinants of Kurtosis Univariate Tests We begin by sorting our sample into five portfolios based on both total kurtosis and idiosyncratic kurtosis. We then examine different stock characteristics across each of these portfolios. Portfolios are re-sorted at the end of each year. We also examine the univariate correlation coefficients between our measures of kurtosis and several stock characteristics. Table 2 reports the results from this analysis. Panel A presents the results when sorting by total kurtosis while panel B shows the results when sorting by idiosyncratic kurtosis. We report share price (Price), market capitalization (Size), total assets (Assets), the book-to-market ratio (B/M), the debt-to-equity ratio (D/E), share turnover (Turn), idiosyncratic volatility (IdioVolt), and systematic risk (Beta). In columns [1] through [5] of panel A, we find that share prices, market cap, and total assets are decreasing monotonically across increasing kurtosis quintiles. Column [6] reports the difference between extreme quintiles and shows differences in prices, size, and assets are negative and significant. We also find that Pearson correlation and Spearman correlation coefficients are negative and significant. These results indicate that kurtosis is higher for lower priced stocks and stocks with less market capitalization and less assets. We also find that B/M ratios, idiosyncratic volatility, and beta are increasing monotonically across increasing kurtosis quintiles. Differences, reported in column [6], are positive and significant. However, we do not find a significant Pearson correlation coefficient between total kurtosis and the B/M ratio. We also find that share turnover is generally increasing across kurtosis portfolios, although not 8

9 Table 2 Determinants of Kurtosis Univariate Tests The table reports a variety of stock characteristics across quintiles sorted by kurtosis (panel A) and idiosyncratic kurtosis (panel B). Column [6] reports the difference between extreme quintiles while column [7] presents the Pearson correlation coefficients and column [8] reports the Spearman rank correlation coefficients. Price is the CRSP close price. Assets is the assets reported in Compustat (in millions). B/M is the book-to-market ratio. D/E is the debt-to-equity ratio. Turn is the share turnover. Idiovolt is the idiosyncratic volatility and Beta is the CAPM beta. The p-values are reported in parentheses in columns [6] through [8]. *,**,*** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. Panel A. Kurtosis Quintiles and Correlation Q I Price Size Assets B/M D/E Turn IdioVolt Beta (Low) Q II Q III Q IV Q V (High) Q V-Q I Pearson Correlation [1] [2] [3] [4] [5] [6] [7] [8] *** *** 4,086,646,920 2,442,737,900 1,871,065,820 1,458,545,650 1,135,650,440-2,950,996,480*** *** 11, , , , , *** *** ** (0.018) (0.631) ** (0.025) (0.903) *** *** *** *** *** *** Spearman Correlation *** *** *** *** *** *** *** *** 9

10 Panel B. Sort by Idiosyncratic Kurtosis Q I (Low) Q II Q III Q IV Price Size Assets B/M D/E Turn IdioVolt Beta Q V (High) Q V-Q I Pearson Correlation [1] [2] [3] [4] [5] [6] [7] [8] *** *** 3,547,477,120 2,394,210,780 1,983,734,110 1,612,644,490 1,456,377,000-2,091,100,120*** *** *** *** * (0.054) (0.929) *** (0.002) (0.866) *** *** *** *** *** *** Spearman Correlation *** *** *** *** *** *** *** *** 10

11 monotonically. The correlation coefficients are positive and significant. These findings suggest that kurtosis is higher for stocks with higher B/M ratios, higher share turnover, higher idiosyncratic volatility, and larger betas. We do not find a reliable pattern for D/E ratios across kurtosis quintiles. Panel B reports the results for idiosyncratic kurtosis sorts. Similar to our findings in panel A, share prices, market capitalization, and assets are negatively related to idiosyncratic kurtosis. We also find that B/M ratios, share turnover, idiosyncratic volatility, and beta are directly related to idiosyncratic kurtosis. The univariate results in Table 2 suggest that stocks with leptokurtic return distributions are related to specific stock characteristics like size, prices, volatility, and liquidity. 3.2 Determinants of Kurtosis Multivariate Tests We recognize the need to explore the determinants of kurtosis in a multivariate setting. For instance, the size of a particular stock is likely related to the level of risk and liquidity for a particular stock. Therefore, we estimate the following equation using pooled stock-year observations. Kurt i,t or IdioKurt i,t = β 0 + β 1 Ln(Price i,t ) + β 2 Ln(Size i,t ) + β 3 Ln(Assets i,t ) + β 4 B/M i,t + β 5 D/E i,t + β 6 Turn i,t + β 7 IdioVolt i,t + β 8 Beta i,t + ε i,t (2) The dependent variable is either total kurtosis for stock i during year t or idiosyncratic kurtosis for stock i during year t. The independent variables include the natural log of prices (Ln(Price i,t )), the natural log of size (Ln(Size i,t )), the natural log of assets (Ln(Assets i,t )), the bookto-market ratio (B/M i,t ), the debt-to-equity ratio (D/E i,t ), the share turnover (Turn i,t ), the idiosyncratic volatility (IdioVolt i,t ), and the CAPM beta (Beta i,t ). The results from estimating equation (2) are reported in Table 3. Panel A reports the results when the dependent variable is 11

12 defined as total kurtosis while panel B presents the results when the dependent variable is defined as idiosyncratic kurtosis. Because of the pooled nature of the data, we estimate an F- statistic to determine whether we need to control for fixed effects. The F-statistic, testing for the presence of fixed effects across years, is (p-value = 0.000) in panel A. Similarly the F- statistic is (p-value = 0.000) in panel B indicating the need to control for fixed effects. In columns [1] through [4], we report the results while including year fixed effects. We recognize the possibility that our results are affected by the presence of multicollinearity. In unreported tests, we estimate variance inflation factors. For each of our independent variables, the variance inflation factors are well below five suggesting that the significance of our tests is not inflated from multicollinearity. In Table 3, we also report the results using different combinations of independent variables to show how the magnitude of the coefficients changes when other variables are included. The p-values that are reported are obtained from White (1980) robust standard errors. In panel A column [1], we find that the estimates for Ln(Price), Ln(Size), and Ln(assets) are each negative and significant. These findings are consistent with our univariate tests in the previous subsection. Column [2] shows that both B/M and D/E produce estimates that are statistically close to zero. Column [3] shows that both Turn and IdioVolt produces reliably positive estimates. Column [4] reports the results of the full model. We find that, when including all of the independent variables, Ln(Price), IdioVolt, and Turn produce estimates that are positive and significant while Ln(Size), Ln(Assets), B/M, and Beta produce estimates that are negative and significant. The results in this panel suggest that smaller stocks, both in terms of market cap and total assets, have higher total kurtosis. Further, stocks with higher turnover, higher idiosyncratic volatility, and smaller betas also have higher kurtosis. The relation between 12

13 Table 3 Determinants of Kurtosis Multivariate Tests The table reports the results from estimating the following equation using pooled stock-year data. Kurt i,t or IdioKurt i,t = β 0 + β 1 Ln(Price i,t ) + β 2 Ln(Size i,t ) + β 3 Ln(Assets i,t ) + β 4 B/M i,t + β 5 D/E i,t + β 6 Turn i,t + β 7 IdioVolt i,t + β 8 Beta i,t + ε i,t The dependent variable is either Kurtosis for stock i during year t (panel A)or Idiosyncratic Kurtosis for stock i during year t (panel B). The independent variables include the natural log of prices (Ln(Price i,t )), the natural log of size (Ln(Size i,t )), the natural log of assets (Ln(Assets i,t )), the book-to-market ratio (B/M i,t ), the debt-to-equity ratio (D/E i,t ), the share turnover (Turn i,t ), the idiosyncratic volatility (IdioVolt i,t ), and the CAPM beta (Beta i,t ). An F- statistic reports the presence of fixed effects by year. Therefore in columns [1] through [4] we include year fixed effects. The p-values obtained from White (1980) robust standard errors are reported in parentheses. *,**,*** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. Panel A. Dependent Variable is Kurtosis i,t [1] [2] [3] [4] Intercept Ln(Price i,t ) Ln(Size i,t ) Ln(Assets i,t ) B/M i,t D/E i,t Turn i,t IdioVolt i,t Beta i,t *** *** *** *** *** (0.689) -5.4E-6 (0.971) (0.792) *** *** *** *** *** ** (0.038) ** (0.017) -2.9E-5 (0.832) *** *** *** Adj. R 2 Fixed Effects

14 Panel B. Dependent Variable is Idiosyncratic Kurtosis i,t [1] [2] [3] [4] Intercept *** *** (0.892) Ln(Price i,t ) *** Ln(Size i,t ) ** (0.017) Ln(Assets i,t ) *** B/M i,t D/E i,t Turn i,t IdioVolt i,t Beta i,t (0.882) -1.6E-5 (0.976) *** *** *** *** (0.388) *** (0.002) (0.118) -3.7E-5 (0.788) *** *** *** Adj. R 2 Fixed Effects

15 kurtosis and prices, B/M ratios, and D/E ratios depend on the variables that we include in the analysis. Panel B reports the results when defining our dependent variable as idiosyncratic kurtosis. For brevity, we only discuss the differences in these results with respect to the results in the previous panel. The most notable difference is that Ln(Size) produces an estimate that is reliably positive, instead of negative, as in column [1]. In addition, we do not find a significant estimate for B/M. However, according to our analysis in panel B, stocks with fewer assets, higher turnover, higher idiosyncratic volatility, and smaller betas exhibit higher levels of idiosyncratic kurtosis. 3.3 Kurtosis and the Cross-Section of Stock Returns The analysis in the two previous subsections was exploratory in nature and attempted to identify stock-specific characteristics that affect the level of kurtosis in the distribution of daily returns. The findings suggest that kurtosis is related to several other characteristics, such as market capitalization, B/M ratios, idiosyncratic volatility, etc., that have shown to have predictability in the cross-section of returns. The main objective of this study is to determine whether kurtosis is priced in the cross-section of stock returns. As mentioned previously, stocks with leptokurtic return distributions may present a certain level of risk that is not captured when examining the standard estimates of volatility and investors holding stocks with excess kurtosis may therefore command a significant return premium. Table 4 reports the results from sorting stocks into five portfolios based on our estimates of kurtosis and then reporting various measures of future returns across each of the kurtosis portfolios. In particular, we sort stocks into quintiles based on the level of kurtosis during a particular year. We then report returns during the next year across each of the portfolios. The 15

16 idea is to determine the profitability of a long-short portfolio, where an investor is potentially long in the portfolio of stocks with the highest kurtosis and short in the portfolio of stocks with the lowest kurtosis. We report future CRSP raw returns, along with three other measures of returns that are obtained from estimating the following equation. R i,t R f,t = α + β 1 (R m,t R f,t ) + β 2 HML t + β 3 SMB t + β 4 UMD t + ε i,t (3) The dependent variable is the difference between the returns for stock i on day t and the risk free rate on day t. The independent variables include the market risk premium (R m,t R f,t ), the highminus-low Fama-French factor (HML), the small-minus-big Fama-French factor (SMB), and the up-minus-down Carhart momentum factor (UMD). We estimate three variants of equation (3). First, we restrict β 2, β 3, and β 4 to equal zero such that equation (3) is the standard estimation of the Capital Asset Pricing Model (CAPM). When estimating the CAPM, we obtain the annual estimates of α and denote this measure of future returns as CAPM α. Second, we restrict β 4 to equal zero such that equation (3) is the standard estimation of the Fama-French 3-Factor model. As before, we obtain the annual estimate of α and denote this measure of future returns as FF3F α. Finally, we estimate the full version of equation (3) which is the standard estimation for the Fama-French-Carhart 4-Factor model. The alpha obtained from the full version of the model is denoted as the FF4F α. Table 4 reports CRSP raw returns, CAPM α, FF3F α, and the FF4F α in year t+1 across total kurtosis portfolios (panel A) and idiosyncratic kurtosis portfolios (panel B) that are sorted at the end of year t. The portfolios are re-sorted each year. Similar to Table 2, columns [1] through [5] report the results across each of the five portfolios. Column [6] reports the difference between extreme quintiles, which is the long-short return. Columns [7] and [8] report the Pearson Correlation and Spearman Correlation coefficients, respectively. 16

17 In the first row of panel A, we find that raw returns are increasing monotonically across increasing kurtosis portfolios. The difference between extreme portfolios in column [6] is (p-value = 0.000). These results suggest that stocks with high kurtosis have larger nextyear returns compared to stocks with low kurtosis. The return premium on this long-short strategy is nearly 4% per year. The correlation coefficients in columns [6] and [7] seem to confirm the assertion that kurtosis is priced in stocks. In the second row of panel A, we find that CAPM α does not increase monotonically across kurtosis portfolios. However, the CAPM α is significantly larger in quintile 5 than in quintile 1 as the difference in column [6] is (pvalue = 0.000). In economic terms, the alpha obtained in the portfolio with the highest kurtosis (column [5]) is 1.05%. Said differently, stocks with the highest kurtosis outperform the market by only 1.05%. Interestingly, stocks with the least kurtosis outperform the market by 54 basis points annually. These results seem to suggest that while our results are statistically significant, the return premium associated with kurtosis is economically modest. In the third and fourth row of panel A, we find that FF3Fα and the FF4Fα appear to be unrelated to kurtosis. In fact, the differences between extreme quintiles in both rows are statistically close to zero. These findings indicate that, after controlling for the Fama-French factors and the Carhart momentum factor, the kurtosis return premium documented in the first row of the panel disappears. Panel B shows the results when we sort stocks into portfolios based on idiosyncratic kurtosis. As before, we find that raw returns are increasing monotonically across idiosyncratic kurtosis quintiles. Similar to our findings in the second row of panel A, we find that the CAPM α is 1.04% in the portfolio of stocks with the highest kurtosis (column [5]). However, when examining the third and fourth rows of panel B, we find that any return premium associated with 17

18 Table 4 Kurtosis and the Cross-section of Stock Returns The table reports various measures of returns in year t+1 across quintiles sorted by Kurtosis during year t in panel A and Idiosyncratic Kurtosis during year t in panel B. We report CRSP raw returns (Raw Return), alphas obtained from a capital asset pricing model (CAPM α), alphas from a Fama-French three-factor model (FF3F α), and alphas from a Fama-French-Carhart four-factor model (FF4F α). In column [6], we report the difference between extreme quintiles along with corresponding p-values in parentheses that are obtained from t-tests testing for statistical significance. Columns [7] and [8] report the Pearson and Spearman Rank Correlation between Kurtosis and our future return measures. *,**,*** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. Panel A. Kurtosis Quintiles Q I Raw Return CAPM α FF3F α FF4F α (Low) Q II Q III Q IV Q V (High) Q V-Q I Pearson Correlation [1] [2] [3] [4] [5] [6] [7] [8] *** ** (0.025) *** *** (0.112) (0.908) (0.175) (0.942) Panel B. Idiosyncratic Kurtosis Quintiles Raw Return CAPM α FF3F α FF4F α *** *** (0.146) (0.208) (0.415) ** (0.019) (0.941) (0.920) Spearman Correlation (0.188) *** ** (0.011) (0.991) ** (0.015) * (0.091) (0.339) (0.204) 18

19 idiosyncratic kurtosis disappears when controlling for the HML, SMB, and UMD factors. Our findings suggest that forming long-short portfolios based on annual estimates of kurtosis does not provide excess returns for investors. Table 5 replicates the analysis in Table 4 for the different sub-periods described above. In particular, we test for a return premium associated with kurtosis in each of the three decades analyzed in this study. Panels A through C report the results when forming portfolios based on total kurtosis during the 1980s, the 1990s, and the 2000s, respectively. Panels D through E present the results when forming portfolios based on idiosyncratic kurtosis during each of the three decades. The conclusions that we are able to draw in the latter three panels are qualitatively similar to those that we draw in the first three panels. Therefore, for brevity, we only discuss our findings in panels A through C. Panel A shows that raw returns are neither increasing nor decreasing across increasing total kurtosis portfolios during the 1980s. While the correlation coefficients in columns [7] and [8] are positive and significant, the difference between extreme portfolios is not reliably different from zero. Similar results are found when examining the CAPM α in the second row of panel A. We note, however, that the difference between extreme portfolios is positive and marginally significant (difference = , p-value = 0.091). However, this difference only represents a 25 basis point return premium in excess of the market. We do not find a reliable relation between kurtosis and either the FF3F α or the FF4F α in the third and fourth rows of panel A. Similar results are found in the previous table. Panel B presents the results during the 1990s. Similar to our findings in the previous panel, we do not find a significant return premium associated with total kurtosis. In fact, column 19

20 Table 5 Kurtosis and the Cross-section of Stock Returns by Sub-Periods The table reports various measures of returns in year t+1 across quintiles sorted by Kurtosis during year t and Idiosyncratic Kurtosis during year t. We report CRSP raw returns (Raw Return), alphas obtained from a capital asset pricing model (CAPM α), alphas from a Fama-French three-factor model (FF3F α), and alphas from a Fama-French-Carhart four-factor model (FF4F α). In column [6], we report the difference between extreme quintiles along with corresponding p-values in parentheses that are obtained from t-tests testing for statistical significance. Columns [7] and [8] report the Pearson and Spearman Rank Correlation between Kurtosis and our future return measures. In Panels A through C we report the results using total kurtosis quintiles during each of the three decades. Panels D through F show the results using idiosyncratic kurtosis during each of the three decades. *,**,*** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. Panel A. Kurtosis Quintiles Q I Raw Return CAPM α FF3F α FF4F α (Low) Q II Q III Q IV Q V (High) Q V-Q I Pearson Correlation [1] [2] [3] [4] [5] [6] [7] [8] ** (0.305) (0.035) * ** (0.091) (0.035) ** (0.462) (0.037) (0.881) (0.427) Panel B. Kurtosis Quintiles Raw Return CAPM α FF3F α FF4F α (0.884) (0.399) (0.399) (0.436) (0.333) ** (0.029) (0.584) (0.648) Spearman Correlation *** *** (0.003) (0.496) ** (0.039) *** (0.119) (0.739) (0.231) 20

21 Panel C. Kurtosis Quintiles Q I (Low) Q II Q III Q IV Raw Return CAPM α FF3F α FF4F α Q V (High) Q V-Q I Pearson Correlation [1] [2] [3] [4] [5] [6] [7] [8] *** (0.154) *** *** (0.003) * (0.094) (0.802) (0.154) (0.976) Panel D. Idiosyncratic Kurtosis Quintiles Raw Return CAPM α FF3F α FF4F α (0.384) (0.214) (0.268) (0.785) ** (0.032) (0.241) ** (0.010) (0.363) Spearman Correlation *** *** *** *** (0.002) (0.284) (0.590) (0.169) *** (0.004) 21

22 Panel E. Idiosyncratic Kurtosis Quintiles Q I (Low) Q II Q III Q IV Raw Return CAPM α FF3F α FF4F α Q V (High) Q V-Q I Pearson Correlation [1] [2] [3] [4] [5] [6] [7] [8] (0.791) (0.139) ** (0.417) (0.026) (0.263) (0.529) (0.260) (0.587) Panel F. Idiosyncratic Kurtosis Quintiles Raw Return CAPM α FF3F α FF4F α *** *** (0.115) (0.192) (0.868) * (0.066) (0.887) (0.956) Spearman Correlation *** (0.119) (0.952) (0.416) (0.105) (0.858) (0.111) (0.505) 22

23 [6] suggests that the differences between extreme portfolios are all statistically close to zero. These results hold regardless of how future returns are defined. Panel C reports the analysis for the period 2000 to In the first row of panel C, we find a positive monotonic relation between total kurtosis and future raw returns. The difference between the extreme portfolios is 7.20% (p-value = 0.000). These findings suggest that a significant return premium is associated with kurtosis. In the second row of panel C, we also find a positive and monotonic relation between kurtosis and the CAPM α. The difference between extreme portfolios is again positive and significant. However, in the third and fourth rows of panel C, we do not find a positive relation between total kurtosis and future returns. In fact, the difference between extreme portfolios is negative and significant when examining FF3Fα. While the difference is negative in the fourth row, it is not statistically significant at the 0.10 level. These results suggest that the positive return premium associated with total kurtosis that is documented when focusing on raw returns disappears when controlling for other known factors that affect the cross-section of returns. We are able to draw similar conclusions when examining idiosyncratic kurtosis in panels D through F. Our results that suggest that kurtosis does not predict the cross-section of stock returns in Table 4 are robust to these different subperiods. 3.4 Kurtosis and the Cross-Section of Stock Returns: Additional Tests In this final section, we continue to explore the relation between kurtosis in the distribution of stock returns and future returns. Tables 4 and 5 provide some evidence that nextyear s raw returns are directly related to this year s level of kurtosis. However, when we control for other factors in the CAPM, the Fama-French 3-Factor Model, and the Fama-French-Carhart 4-Factor Model, the relation disappears. The purpose of this final section is to determine 23

24 whether stock-specific characteristics reduce the return premium associated with kurtosis. In particular, we estimate the following equation using pooled stock-year data. Returns i,t+1 = β 0 + β 1 Beta i,t + β 2 Ln(Size i,t ) + β 3 B/M i,t + β 4 Returns i,t-1 + β 5 Turn i,t + β 6 IdioVolt i,t + β 7 Kurt i,t or IdioKurt i,t +ε i,t (4) The dependent variable is the CRSP raw return for stock i during year t+1. The independent variables include the CAPM beta (Beta i,t ), the natural log of size (Ln(Size i,t )), the book-to-market ratio (B/M i,t ), the CRSP returns for stock i during year t-1 (Returns i,t-1 ). These stock-specific characteristics capture the potential explanatory power associated with the common risk factors described in equation (3). We also include the idiosyncratic volatility (IdioVolt i,t ), and share turnover (Turn i,t ) as additional independent variables. The variables of interest are the kurtosis of daily returns (Kurt i,t ) or the idiosyncratic kurtosis of daily returns (IdioKurt i,t ). Table 6 reports the results from estimating equation (4). As before, an F-statistic testing for the presence of year fixed effects is significantly large indicating the need to control for fixed effects. Therefore, we report our results while including year fixed effects. In panel A, we report the results when the variable of interest is total kurtosis. Focusing primarily on the stock-specific characteristics that correspond with the factors in the Fama- French-Carhart 4-Factor Model, we estimate separate regressions while including kurtosis and each of stock-specific characteristics that correspond with the 4 factors independently. For instance, column [1] shows the results when Kurt and Beta are included as the only two independent variables. Column [2] presents the results when Kurt and Ln(Size) are included, and so on. Column [5] reports the results when including all of the stock-specific characteristics that correspond with the 4 factors while column [6] presents the results from the full model. 24

25 Column [1] shows that both Beta and Kurt produce reliably positive estimates. The positive estimate for Beta supports the main implications of the CAPM discussed in a broad stream of literature. The positive estimate for Kurt suggests that, after controlling for Beta, kurtosis still has explanatory power with respect to the cross-section of returns. However, when we control for Ln(Size), both the estimates for Ln(Size) and Kurt are negative and significant. The negative estimate for Ln(Size) is consistent with the size effect discussed in Banz (1981) and Fama and French (1992). The negative estimate for Kurt suggests that, when controlling for market cap, the return premium associated with kurtosis disappears. In fact, when controlling for size, the kurtosis return premium becomes negative and significant. Given our results in Table 2 that show that market capitalization is negatively related to kurtosis, the results in column [2] of this table indicate that any observable direct relation between kurtosis and future raw returns is driven primarily by the size effect. Column [3] shows that, when controlling only for the B/M ratio, both B/M and Kurt produce positive estimates. The positive estimate for B/M supports the findings in Fama and French (1992) while the positive estimate for Kurt indicates that, when only holding B/M ratios constant, kurtosis produces a positive and significant return premium. Column [4] presents the results when including both last year s return and kurtosis. The estimate on past returns is negative and significant and supports the idea of return reversals discussed in Lehmann (1990) and Jegadeesh (1990). The positive estimate for Kurt suggests that, while controlling for past returns, kurtosis explains the cross-section of returns. In column [5], we report the results when including the stock-specific characteristics associated with the 4 factors described in equation (3) as well as kurtosis. A few of these results are noteworthy. First, Beta, Ln(Size), and Return i,t-1 produce estimates that are similar to those in columns [1], [2], and [4]. However, the estimate 25

26 Table 6 Kurtosis and the Cross-section of Stock Returns The table reports the results from estimating the following equation using pooled stock-year data. Returns i,t+1 = β 0 + β 1 Beta i,t + β 2 Ln(Size i,t ) + β 3 B/M i,t + β 4 Returns i,t-1 + β 5 Turn i,t + β 6 IdioVolt i,t + β 7 Kurt i,t or IdioKurt i,t +ε i,t The dependent variable is the CRSP raw return for stock i during year t+1. The independent variable include the CAPM beta (Beta i,t ), the natural log of size (Ln(Size i,t )), the book-to-market ratio (B/M i,t ), the CRSP returns for stock i during year t-1 (Returns i,t-1 ), the idiosyncratic volatility (IdioVolt i,t ), and the share turnover (Turn i,t ). The variables of interest is the kurtosis of daily returns (Kurt i,t ) or the idiosyncratic kurtosis of daily returns (IdioKurt i,t ). We report the results when using total kurtosis in panel A and idiosyncratic kurtosis in panel B. An F-statistic reports the presence of fixed effects by year. Therefore in columns [1] through [6] we include year fixed effects. Columns [7] and [8] report the results without controlling for year fixed effects. The p-values obtained from White (1980) robust standard errors are reported in parentheses. *,**,*** denote statistical significance at the 0.10, 0.05, and 0.01 levels, respectively. Panel A. Independent Variable of Interest is Kurtosis [1] [2] [3] [4] [5] [6] Intercept Beta i,t Ln(Size i,t ) B/M i,t Return i,t-1 IdioVolt i,t Turn i,t Kurt i,t *** ** (0.046) *** *** *** *** *** * (0.085) *** *** *** *** (0.005) *** *** *** (0.851) *** (0.329) *** (0.722) *** (0.987) *** *** *** (0.008) *** Adj. R 2 Fixed Effects

27 Panel B. Independent Variable of Interest is Idiosyncratic Kurtosis [1] [2] [3] [4] [5] [6] Intercept *** *** *** *** Beta i,t * (0.053) Ln(Size i,t ) B/M i,t Return i,t-1 IdioVolt i,t Turn i,t IdioKurt i,t *** *** (0.455) * (0.084) *** *** *** (0.002) *** *** *** (0.858) *** (0.855) *** (0.705) *** (0.993) *** *** ** (0.013) *** Adj. R 2 Fixed Effects

28 for B/M is statistically close to zero. Second, and perhaps more important, when we control for each of these variables, the estimate for Kurt is not reliably different from zero. Given our findings in column [2] of this table, these results suggest that when controlling for several stockspecific characteristics and, in particular the market capitalization of a particular stock, the return premium associated with kurtosis disappears. In column [6] we extend the model in column [5] by including both idiosyncratic volatility and share turnover. The positive estimate for idiosyncratic volatility supports the notion that riskier stocks have greater returns while the negative estimate for share turnover indicates that stocks with less liquidity have higher returns. When controlling for these variables, the estimate for Kurt is negative and significant instead of positive and significant. Again, these results suggest that after controlling for a variety of explanatory variables, stocks with high kurtosis underperform stocks with low kurtosis. Further, panel B reports the results when we include idiosyncratic kurtosis as the independent variable of interest. Again, we are able to draw the same conclusions as the estimates for idiosyncratic kurtosis in columns [2], [5], and [6] are non-positive. Combined with the results in Tables 4 and 5, our findings in Table 6 seems to suggest that any observable return premium associated with kurtosis is driven primarily by the size effect (Banz, 1981; Fama and French, 1992). 4. CONCLUSION While most traditional asset pricing models assume normality in the distribution of stock returns, much of the empirical evidence suggests that return distributions are non-normal. In fact, Xiong and Idzorek (2011) show that extremely large price movements, which are rare under a normal distribution, occur approximately 10 times more often than predicted by a normal 28

29 distribution. When examining whether the higher moments of return distributions are priced in stocks, prior research has demonstrated that skewness in the return distribution is negatively related to future returns (Mitton and Vorkink, 2007; Barberis and Huang, 2008; and Boyer, Mitton, and Vorkink, 2010). While the literature has investigated the return premium associated with skewness, fewer studies have focused the relation between kurtosis and future returns. In this paper we attempt to determine whether there is a significant return premium associated with kurtosis. If investors perceive that stocks with excess kurtosis represent risk that is unaccounted for in other traditional risk factors, then investors holding stocks with higher levels of kurtosis should command a significant return premium. This paper shows that the return distribution of the average stock has excess kurtosis. We show that this high level of kurtosis is robust across a thirty-year time period. We also show that kurtosis is negatively related to firm size and positively related to volatility and turnover. Because the literature has shown that firm size, volatility, and turnover predict future returns, it is important that we control for these stock-specific characteristics in a multivariate setting. We show that while stocks with high kurtosis have higher raw returns than stocks with low kurtosis, when controlling for various risk factors that are used in more common asset pricing models, we do not find a significant return premium associated with kurtosis. In other tests, we show that the direct relation between raw returns and kurtosis disappears once we control for firm size. These findings indicate that any return premium associated with kurtosis is entirely captured by the common return premium associated with firm size (Banz, 1981; Fama and French, 1992). 29

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