Variance Risk Premium and Cross Section of Stock Returns

Transcription

1 Variance Risk Premium and Cross Section of Stock Returns Bing Han and Yi Zhou This Version: December 2011 Abstract We use equity option prices and high frequency stock prices to estimate stock s variance risk premium (V RP ), defined as the difference between the expected variances under the risk-neutral measure and under the empirical measure. Cross-sectionally, V RP is significantly related to stocks sensitivities to common risk factors. In particular, stocks whose returns tend to be low when systematic volatility increases have higher variance risk premium. We find that stock s expected returns increase with their variance risk premium. Stocks ranked in the top V RP quintile on average outperform those in the bottom quintile by 1.84% (resp. 1.44%) per month when portfolios are value-weighted (resp. equal-weighted). We reject explanations based on stock mispricing or informed trading in options. Our result is linked to another new finding: stocks with low beta with respect to change in the average idiosyncratic volatility or the market variance risk premium have high expected returns. JEL Classification: G12. We thank Jie Cao, Darwin Choi, John Griffin, Umit Gurun, Andrew Karolyi, Leonid Kogan, Shimon Kogan, Dick Roll, Yuliy Sannikov, Alessio Saretto, Clemens Sialm, Sheridan Titman, Malcolm Wardlaw, Amir Yaron, Stan Zin, as well as participants of the seminars at Chinese University of Hong Kong, Hong Kong University of Science and Technology, University of Texas at Austin, York University, and 21st Annual Derivatives Securities and Risk Management Conference, Risk Management Conference at University of Oklahoma, 2011 China International Conference in Finance, 18th Annual Meeting of the Multinational Finance Society for invaluable discussions. All remaining errors are our own. University of Texas at Austin McCombs School of Business, 1 University Station B6600, Austin, TX 78712, bhan@mail.utexas.edu, Tel: (512) University of Oklahoma, Michael F. Price College of Business, Finance Division, 307 West Brooks Street, Adams Hall 250, Norman, OK 73069, yi.zhou-2@ou.edu, Tel: (405)

2 Variance Risk Premium and Cross Section of Stock Returns Abstract We use equity option prices and high frequency stock prices to estimate stock s variance risk premium (V RP ), defined as the difference between the expected variances under the riskneutral measure and under the empirical measure. Cross-sectionally, V RP is significantly related to stocks sensitivities to common risk factors. In particular, stocks whose returns tend to be low when systematic volatility increases have higher variance risk premium. We find that stocks expected returns increase with their variance risk premium. Stocks ranked in the top V RP quintile on average outperform those in the bottom quintile by 1.84% (resp. 1.44%) per month when portfolios are value-weighted (resp. equal-weighted). We reject explanations based on stock mispricing or informed trading in options. Our result is linked to another new finding: stocks with low beta with respect to change in the average idiosyncratic volatility or the market variance risk premium have higher expected returns. JEL Classification: G12. Keywords: variance risk premium, model free implied variance, high frequency realized variance, cross-section of stock returns

3 1 Introduction It is well known that risks of stocks as measured by their return variances are time-varying. However, little is known about how the risk of variation in stock variance is priced. This paper provides an in-depth analysis of individual stock s variance risk premium (V RP ), defined as the difference between the risk neutral expectation and the objective expectation of stock return variation. We find that V RP is an important determinant of the cross-section of expected stock returns. We adopt a new approach to estimate variance risk premium based on information from options markets. Since options are sensitive to changes in variances, they are ideal instruments to study the variance risk premium. Early studies explicitly model the variance dynamics and make parametric assumptions about the variance risk premium. 1 We estimate the variance risk premium without making such assumptions. Our model-free estimate of the variance risk premium follows several recent studies including Carr and Wu (2008), Bollerslev, Tauchen, and Zhou (2009), Drechsler and Yaron (2011) and Todorov (2010). We infer the risk-neutral expectation of the future return variation from a collection of option prices without the use of a specific pricing model. This model-free implied variance incorporates the option information across all ranges of moneyness and thus is more informative than the implied variance from at-the-money option alone (e.g., Jiang and Tian (2005)). The objective expectation of stock return variation is computed from high-frequency intraday stock returns, which provides more accurate ex post observations on the actual return variation than the traditional sample variances based on daily or coarser frequency return observations (e.g., Andersen et al. (2001)). Due to limitation on the availability of sufficient options data, we only estimate the variance risk premium for about 500 stocks on average in each month. For our V RP sample, the average market cap of is 6.78 billion dollars, the average institutional ownership is 74.48%, and the average number of analyst coverage is Our results are not driven by small, 1 See, e.g., Bakshi, Cao, and Chen (1997), Heston (1993), Bates (2000), and Pan (2002). 1

4 neglected, illiquid stocks. We find that two-thirds of the individual stocks have significantly positive variance risk premium. The cross-sectional dispersion in individual stock variance risk premium is large. V RP varies systematically with stock characteristics. Variance risk premium is significantly higher for small stocks, value stocks, past loser stocks, stocks with high volatility and high analyst disagreement. V RP is also significantly related to sensitivities of stock returns with respect to common risk factors such as the Fama-French factors and several proxies of systematic volatility factors (including the average idiosyncratic volatility). In particular, stocks whose returns tend to be low when systematic volatility increases also have higher variance risk premium. We find that the risk exposures of stocks have more explanatory power for V RP than stock characteristics. The main finding of the paper is that V RP is significantly and positively related to the cross section of expected stock returns. The value-weighted (resp. equal-weighted) average return of a portfolio that is long the stocks ranked in the top V RP quintile and short the bottom V RP quintile is 1.84% (resp. 1.44%) per month. This finding is robust to variations in the measurement of the variance risk premium, to differences in sorting procedures. It is confirmed in Fama-MacBeth regressions, after controlling for all the stock characteristics that are related to both stock return and V RP. The magnitude of outperformance of the high V RP stocks is barely changed after we control for the Fama-French factors and the momentum factor. The higher average returns of high V RP stocks could reflect compensation for stocks exposures to systematic volatility risks. High V RP stocks tend to have lower returns when systematic volatility risks increase. They are riskier and command higher risk premium. On the other hand, low V RP stocks serve as useful hedges for systematic volatility risks, and therefore have lower expected return. Consistent with this hypothesis, we find that stocks whose returns have higher beta with respect to change in average idiosyncratic volatility 2

5 have lower expected returns. In addition, we find stocks that are more sensitive to the market variance risk premium have lower expected returns. This cross-sectional relation for individual stock returns complements the time-series relation between market return and market variance risk premium documented in Bollerslev, Tauchen, and Zhou (2009), and Drechsler and Yaron (2011). More importantly, controlling for stock s beta with respect to the systematic volatility risks reduces the V RP effect by about 22% to 30%. We reject behavioral explanations for why stocks with large (resp. small) variance risk premium have higher (resp. lower) expected returns. Our results are strong and significant in all subsamples of stocks regardless whether they face high limits to arbitrage. In fact, the outperformance of stocks with high variance risk premium is somewhat larger in magnitude and more significant among the biggest stocks and stocks in the S&P 500 index. In addition, most of the return difference between high V RP and low V RP stocks is due to the large average return of stocks with high variance risk premium. It is hard to argue that the high V RP stocks are undervalued and there are significant frictions that prevent arbitragers from buying these stocks to eliminate the mispricing. Previous studies have focused on the variance risk premium of the aggregate stock market (e.g., Bollerslev, Tauchen, and Zhou (2009), Todorov (2010) and Drechsler and Yaron (2011)). A growing body of research has studied variance risk premium at individual stock level. Carr and Wu (2008) and Driessen, Maenhout, and Vilkov (2009) find large variation in the individual stock variance risk premium across 35 stocks and 127 stocks respectively for the 1996 to 2003 sample period, but they do not study what determines such cross-sectional variation, or the relation between variance risk premium and expected stock return. Vedolin (2010) documents significant time-series variations in the volatility risk premia of individual stocks. Cross-sectionally, she links individual stock variance risk premium to belief disagreement among investors. Our paper is related to but differs from the literature that examines the relation between stock return and (idiosyncratic) volatility (see, e.g., Ang, Hodrick, Xing, and Zhang (2006), 3

6 Fu (2009) and Huang, Liu, Rhee, and Zhang (2009)). Variance risk premium is positively correlated with stock volatility. But controlling for volatility does not materially affect the positive relation between expected stock return and the variance risk premium. Our paper is also related to recent studies documenting that various variables computed from the options data can predict the underlying stock returns, including (Bali and Hovakimian (2009), Ang, Bali, and Cakici (2010), Cremers and Weinbaum (2010), and Xing, Zhang, and Zhao (2010). These variables are viewed as capturing information contained in the options that are not yet reflected in the stock prices. We find that the significant positive relation between expected stock return and variance risk premium is little changed after controlling for all of the option-related variables used in previous studies. Further, we conduct several tests and reject an information-based explanation of our main finding. The strong link between the variance risk premium and the expected stock return can not be explained by the idea that the variance risk premium proxies for some private information yet to be reflected in the stock prices. The reminder of this paper is organized as follows. Section 2 introduces the measurement of individual stock variance risk premium, the data and our sample of stocks. Section 3 reports summary statistics and cross-sectional determinants of variance risk premium. Section 4 documents a robust finding that expected stock return increases with stock s variance risk premium. Section 5 examines several potential explanations for the positive relation between stock variance risk premium and expected return, including risk compensation, mispricing and informed trading in options. Section 6 concludes. 4

7 2 Variance Risk Premium and Data 2.1 Measuring Variance Risk Premium Following Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011), we define the variance risk premium for a given stock i in a month t as the difference between the risk neutral expectation and the objective (physical) expectation of its return variation over the next month. We estimate he risk neutral expectation of return variation with the model free option-implied variance, following the methodology used in Britten-Jones and Beuberger (2000) and Jiang and Tian (2005). The estimation of the physical expected variance is based on high frequency return data. On the last trading day of each month t, we extract the implied volatilities for one-month call options from the standardized Volatility Surface provided by OptionMetrics 2, and translate these implied volatilities into call option prices using the Cox Ross Rubinstein binomial lattice model (Cox, Ross, and Rubinstein (1979)). Following Jiang and Tian (2005)), the one-month model-free implied variance is: IVt i C = 2 t(t i + T, K)/B(t, t + T ) max[0, St/B(t, i t + T ) K] dk, (1) 0 K 2 where S i t denotes the stock price of firm i at t, T = 1/12 year (i.e., 1 month), C i t(t + T, K) denotes the option price of a call option with time-to-maturity T and strike price K. B(t, t+ T ) denotes the present value of a zero-coupon bond that pays off one dollar at time t + T. The integral is evaluated numerically where we set the number of strike prices fine enough to ensure that the discretization has minimal impacts on the estimation of the model-free 2 OptionMetrics computes the implied volatility of a traded option from its price using a proprietary pricing algorithm that is based on the Cox-Ross-Rubinstein binomial tree model, which can accommodate underlying securities with either discrete dividend payments or a continuous dividend yield. Based on the implied volatilities of traded options, OptionMetrics computes a surface of standardized option implied volatilities using a kernel smoothing technique. 5

8 implied variance. The model-free implied variance is more informative than the implied variance from at-the-money options alone as it incorporates the option information across all moneyness (e.g., see Jiang and Tian (2005)). To compute the model-free estimate of the realized variance RV i t of each stock i in each month t, we extract from TAQ intraday equity trading data spaced by = 15 minutes interval. 3 Let p i j denote the logarithmic price of stock i at the end of the jth 15-minutes interval in the month t. The realized variance for month t is measured as: RV i t = 12 n j=1 [ ] p i j p i 2 j 1. (2) where n is the number of 15-minutes interval in month t. We multiply by 12 to get an annualized variance estimate that is comparable to the option implied variance. This model-free realized variance based on high-frequency intraday data is more accurate than the realized variance based on daily returns (e.g., Andersen, Bollerslev, Diebold, and Labys (2001)). Following Drechsler and Yaron (2011), we adopt a linear forecast model to estimate the expected variance under the physical measure with lagged risk-neutral expected variance and historical realized variance. For eacj stock i, we run the following regression RV i t+1 = α + βiv i t + γrv i t + ɛ i t+1, (3) and take the physical expected variance EV i t as the fitted value from the regression: EV i t RV i t+1 = α + βiv i t + γrv i t. (4) To summarize, our empirical estimate of individual stock s variance risk premium is the 3 All of our results are robust when we estimate the realized variance (and hence variance risk premium) using stock prices sampled every 30 minutes or every hour. 6

9 difference between risk-neutral expected variance inferred from the option prices and physical expected variance: V RP i t = IV i t EV i t. (5) Our measure of variance risk premium is the same as that in Drechsler and Yaron (2011) and Wang, Zhou, and Zhou (2010). Bollerslev, Tauchen, and Zhou (2009) measures the variance risk premium as the difference between model free implied variance inferred from the option prices and the realized variance based on high frequency return data. We verify that all of our results are robust to this measure of variance risk premium. Finally, Carr and Wu (2008) and Driessen, Maenhout, and Vilkov (2009) measure the variance risk premium as the difference between the realized variance and the variance swap rate. Their realized variance is estimated from daily stock returns. Since the variance swap rate is the risk-neutral expectation of realized variance, their measure of variance risk premium is essentially our measure multiplied by (-1). In a previous version of the paper, we compute the option implied volatilities ourselves rather than relying on the Volatility Surface provided by OptionMetrics. 4 Using the variance risk premium estimated this way does not change any of our results. Our focus is the difference in the average returns of stocks with high variance risk premium and those with low variance risk premium. Small variations in the variance risk premium estimates would not change the ranking of stocks by V RP. Therefore, our results stay the same regardless of whether we use the implied volatility surface provided by OptionMetrics or compute the implied volatilities and interpolate/extrapolate them on our own. We choose to estimate the variance risk premium based on the Volatility Surface provided by OptionMetrics so that our results can be readily verified and replicated by other researchers. Previous studies have linked the variance risk premium of the aggregate stock market 4 We apply the Cox Ross Rubinstein binomial lattice model to invert the prices of traded strikes into implied volatilities and then fit a smooth cubic splines function to the implied volatilities to interpolate/extrapolate to other strikes. For options with strike prices beyond the available range, we use the endpoint implied volatility to extrapolate their option value. 7

10 to uncertainty about economic fundamentals (e.g., Bollerslev, Tauchen, and Zhou (2009), Todorov (2010) and Drechsler and Yaron (2011)). In Bollerslev, Tauchen, and Zhou (2009), the market variance risk premium is proportional to the volatility of the aggregate consumption growth volatility. In Drechsler and Yaron (2011), variance premium can also reveal the level of the jump intensity in the state variables driving the pricing kernel. Todorov (2010) emphasizes that jump in volatility plays an important role in the variance risk-premium. Time variation in economic uncertainty and a preference for early resolution of uncertainty together generate a positive market variance premium that is time-varying and positively predicts aggregate stock market excess returns. We find that the average return difference between high and low VRP stocks can not be explained by the market variance risk premium. 2.2 Data Sources We use data from both the equity option and stock markets. For the January 1996 to October 2009 sample period, we obtain data on U.S. individual stock options from the Ivy DB database provided by OptionMetrics. For each month during our sample period, we will estimate the variance risk premium for all optionable stocks that we deem to have sufficient reliable option observations in that month. To ensure the quality of the variance risk premium estimate we derive from the options, we first apply several filters to the options data. We exclude all option observations that violate obvious no-arbitrage conditions such as S C max(0, S Ke rt ) for a call option C where S is the underlying stock price, and K is the option strike price, T is time to maturity of the option, and r is the riskfree rate. To avoid microstructure related bias, we only retain options that have positive bid quotes, with the bid price strictly smaller than the ask price, and the mid-point of bid and ask quotes is at least $1/8. We keep only the options whose price dates match the underlying security price dates. To ensure the reliability of the variance risk premium estimates, for each stock at the 8

11 end of a given month, we require that there are at least five traded call options on the stock with maturity between 15 days and 60 days that survive the data filters above. Among these options, we further require at least two are out of money, two in the money, one close to being at the money. 5 With these additional data filters, on average there are about 464 stocks in each month for which we estimate the variance risk premium. The set of such stocks increases from about 350 in the begining of our sample ( ) to about 670 towards the end of the sample ( ). Compared to the whole CRSP stock universe, the sample of stocks with traded options have larger market cap, more institutional ownership and analyst coverage. The set of stocks for which we estimate the variance risk premium is a subsample of all optionable stocks that have even larger market cap, higher institutional ownership, higher analyst coverage, and tilt more towards growth stocks. For the whole sample of optionable stocks, the average market cap of is 3.81 billion dollars, the average B/M ratio is 0.63, the average institutional ownership is 66.68%, and the average number of analyst coverage is For the V RP sample, the average market cap of is 6.78 billion dollars, the average B/M ratio is 0.45, the average institutional ownership is 74.48%, and the average number of analyst coverage is It is important to keep in mind the characteristics of our sample of V RP stocks. Our results are not driven by small, neglected, illiquid stocks. We obtain daily and monthly split-adjusted stock returns, stock prices, trading volume and shares outstanding from the Center for Research on Security Prices (CRSP). For each stock, we also compute the book-to-market ratio using the book value from COMPUSTAT. We obtain analyst coverage and earnings forecasts data from I/B/E/S, and quarterly institutional holding (13f filling) from Thomson Financial. Further, we obtain the monthly Fama-French factor returns and risk-free rates from Kenneth French s data library. 6 Finally, we use high-frequency stock trading information from TAQ database. 5 We obtain similar results when we require each stock in our sample to have three, or seven traded call options with maturity under 60 days. 6 The data library is available at 9

12 3 Individual Stock Variance Risk Premium 3.1 Summary Statistics Table 1 Panel A presents descriptive statistics on the variance risk premium, which is reported in annualized, percentage squared format. The average individual stock variance risk premium is 587 across all firms and all years. This is about twice as high as the average market variance risk premium for the same sample period. Figure 1 plots the time series of average individual stock V RP and market V RP. Each year from 1999 to 2009, the average individual stock variance risk premium is larger than the market variance risk premium. But at the beginning of our sample (from 1996 to 1998), the average individual stock variance risk premium is lower than the market variance risk premium. Table 1 Panel A and Figure 2 show significant cross-sectional dispersion in the individual stock variance risk premium. The 10 percentile of the cross-section of individual stock V RP is usually negative, especially in the first two years of the sample. The 90 percentile is always positive, usually between 1000 and 3000 (in percentage squared format). The average 10th percentile individual stock V RP is -224 and the average 90th percentile is 1,797. Table 1 Panel B reports the mean tests of individual stock variance risk premium. There are 3932 stocks in our sample period for which we have computed variance risk premium in at least one month. For each of these stocks, we use its monthly time-series of variance risk premium to investigate whether its mean V RP is zero. We find that the mean variance risk premium is positive for 3682 stocks. Among them, 3029 stock s mean V RP is significantly positive at 5% level. Only 206 stocks have significantly negative mean V RP. Our results shed new lights on the statistical and economic significance of individual stock variance risk premium. The common perception is that individual stock variance risk premium is usually insignificant. For example, Driessen, Maenhout and Vilkov (2009) find that the null of a zero variance risk premium is not rejected at the 5% confidence level for 98 stocks out of the 127 stocks in their sample. They find that on average realized individual 10

13 variance exceeds option-implied variance for stocks in the S&P 100 index. On the other hand, Carr and Wu (2009) find option-implied variance are higher than the average realized variance (i.e. positive variance risk premium) for most of the 35 individual stocks they study. But only seven generate significant variance risk premium. The sample period is from 1996 to 2003 in both Carr and Wu (2009) and Driessen, Maenhout and Vilkov (2009). In contrast, our sample period is twice as long, and our sample is more comprehensive, covering all stocks with sufficient number of traded options to compute reliable estimates of the variance risk premium. Table 1 Panel C reports the R 2 of time-series regression of each stock s variance risk premium on the market variance risk premium, which is estimated using S&P 500 index options data. 7 We find that the the regression R 2 is generally small. Across all stocks in our sample, the mean (median) regression R 2 is only 7.38% (3.38%). For three quarters of the stocks, the R 2 is below 10%. This indicates that the majority of the time-series variations in individual stock variance risk premium can not be explained by the market variance risk premium. Consistent with the significant premium for many of the individual stock volatility risks, we find common comovements among the individual stock return volatilities using principal component analysis. For each stock in our sample and in each month, stock volatility is measured as the standard deviation of daily stock returns in that month. The idiosyncratic volatility is the standard deviation of the residuals of the Fama-French 3-factors model estimated using the daily stock returns in that month. Principal component analysis show that the first common component explains about 35% of the variations in individual stock volatilities. This common component is well approximated by the equal-weighted average of individual stock idiosyncratic volatilities, with 0.9 correlation between the two. Table 1 Panel D and E report the time-series R 2 of regressing change in individual stock 7 Our estimates of the market variance risk premium closely match those obtained by Bollerslev, Tauchen, and Zhou (2009). We thank the authors for making the data on the market variance risk premium available at 11

14 return volatility on change in V IX (a proxy of market volatility), and on change in the average individual stock idiosyncratic volatility, respectively. Market volatility on average explains about 6.69% of the variation in individual stock return volatility. The average individual stock idiosyncratic volatility on average explains about 18.9% of the variation in individual stock return volatility. These results collaborate common comovements in the stock return volatilities. The average individual stock idiosyncratic volatility is a more important factor than the market volatility explaining the common comovements in the stock return volatilities. Table 1 Panel F reports some summary statistics on the regression specification (3) which estimates the expected variance under the physical probability measure based on lagged predictors (historical realized variance and risk-neutral expected variance implied from options). Across the 5,000 stocks for which this regression is run, the mean R 2 is 0.44, the median is 0.46, the 25 and 75 percentile of the R 2 is 0.3 and 0.59 respectively. Such goodness of fit is comparable to the 0.59 R 2 for the aggregate stock market index reported in Drechsler and Yaron (2011). The average regression coefficient on the historical realized variance and risk-neutral expected variance is 0.35 and 0.3. They are close to the corresponding coefficient estimates of 0.4 and 0.26 for the market index reported in Drechsler and Yaron (2011). 3.2 Determinants of Variance Risk Premium To better understand the individual stock variance risk premium, we study the cross-sectional determinants of VRP using Fama-MacBeth regressions. Table 2 examines the relation between stock s variance risk premium and the sensitivities of both stock returns and return volatilities with respect to several proxies of systematic volatility risk factors. We use a two-pass procedure. First, in each month, we estimate the beta s for each stock from timeseries regressions using a rolling window of past 60 month s observations. Then, we run 12

15 cross-sectional regressions of stock s V RP on the returns beta s and/or volatility beta s. Stock return betas are obtained by regressing a given stock s returns on the Fama-french three factors and one of the systematic volatility factors. The volatility beta s (β V ) are obtained by regressing a given stock s return volatility on one of the systematic volatility factors. We adopt three volatility risk factors. The first factor ( VIX) is the monthly change of the VIX index from the Chicago Board Options Exchange, which proxies for innovation in the systematic volatility risk. The second factor ( AvgIndVol) is the monthly change of the equal-weighted average idiosyncratic volatility of individual stocks. The third factor M arket V RP is the market variance risk premium. Bollerslev, Tauchen, and Zhou (2009), and Drechsler and Yaron (2011) both show that high market variance risk premium is associated with high expected market returns over the next few months. Table 2 Model 1 through 3 show that stocks whose return volatilities have higher exposures to the systematic volatility factors have significantly larger variance risk premium. This holds for all proxies of volatility factors we use. Model 4 through 6 show that stock s variance risk premium is also significantly related to the stock return s beta with respect to the Fama-French factors and the systematic volatility factors. Specifically, stocks with higher market beta and SMB beta tend to have larger variance risk premium. On the other hand, stocks whose returns load more heavily on the HML factor or the systematic volatility factors tend to have lower variance risk premium. Interestingly, for regressions Model 7 through 9 which include both the stock returns beta s and the volatility beta s as independent variables, the coefficients for the volatility beta s switch signs and become insignificant, while the stock return beta s are still highly significant. To summarize, Table 2 documents a robust finding: stock s variance risk premium is significantly related to sensitivities of stock returns with respect to common risk factors such as the Fama-French factors and several proxies of systematic volatility factors (including the average idiosyncratic volatility). In particular, stocks whose returns tend to be low when systematic volatility increases also have higher variance risk premium. 13

16 Table 3 documents systematic differences in the variance risk premium across stocks of different characteristics. Variance risk premium is significantly larger for small stocks, value stocks, past loser stocks (both over the last month and over the last one year), as well as for stocks with high volatility and high institutional ownership. These findings are new. We also find that stocks with high analyst forecast dispersion tend to have larger variance risk premium. This result is consistent with Vedolin (2010). Based on regression R 2, the risk exposures of stocks have more explanatory power for V RP than stock characteristics. The regression R 2 is 4% for Model 7 through 9 in Table 2 while it is 3% in Model 7 of Table 3. When we include both stock beta s and characteristics as regressors, their coefficients have the same sign and statistical significance as reported in Model 7 through 9 of Table 2 and Model 7 of Table 3, for all cases except two. First, the stock return volatility beta (β V ) with respect to the average idiosyncratic volatility is positive and significantly related to V RP. Second, institutional ownership loses it significance. Together, stock beta s and characteristics explain 6% of the cross-sectional variation in V RP. 4 Variance Risk Premium and Equity Returns This section examines the relation between variance risk premium and expected return of individual stocks. Our basic results are documented using the portfolio sorting approach. The portfolio sorting procedure does not impose a restrictive relation between variance risk premium and stock returns. In subsequent sections, we also conduct Fama-MacBeth crosssectional regressions for robustness checks and to incorporate multiple controls variables. 4.1 Basic Results Table 4 reports the average monthly returns of portfolios of stocks sorted by variance risk premium (V RP ). Each month, a stock s V RP is measured as the difference between the 14

17 model free implied variance inferred from individual stock option prices at the end of each month and the expected variance under the empirical measure estimated using realized returns sampled at 15-minutes interval during the trading days of the month. We sort stocks into ten (resp. five and three) portfolios by their variance risk premium in Panel A (resp. Panle B and Panel C). We report both the value-weighted and the equal-weighted average returns of V RP portfolios over the next month. We also report the average return of the portfolio that is long the top V RP stocks and short the bottom V RP stocks, as well as its CAPM alphas, FF-3 alphas and Carhart-4 alphas. Table 4 documents a strong finding: the average future stock return increases with variance risk premium. This result is also robust, as it holds for ten, five and three sorts on V RP, for both value-weighted and the equal-weighted portfolios. For example, Panel B shows that a value-weighted portfolio of stocks ranked in the top V RP quintile on average outperforms the bottom V RP quintile stocks by 1.84% per month over the first month after portfolio formation. This outperformance is statistically significant (with a t-statistic of 2.96) and economically significant as well. It can not be explained by the Fama-French or momentum factors. The portfolio that is long the decile ten stocks and short the decile one stocks has a CAPM alpha of 1.58% (t statistic = 2.88), Fama-French three-factor alpha of 1.78% (t statistic = 4.14), and Carhart four-factor alpha of 2% (t statistic = 4.14). The difference in average return of the high and low V RP stocks is larger (2.67%) when we use ten V RP sorts. Even for three V RP sorts, the top tercile stocks on average outperform the bottom tercile stocks by 1.68% per month, which is significant both statistically and economically. The average returns of equal-weighted portfolios sorted by V RP display a similar pattern as the value-weighted ones. The larger the variance risk premium, the higher the expected stock returns. The return difference between high V RP and low V RP stocks becomes somewhat smaller when portfolios are equal-weighted. For example, Table 4 Panel B shows that when equal-weighted, the top V RP quinitle on average outperforms the bottom VRP 15

18 quintile by 1.44% per month, compared to the 1.84% return difference for value-weighted portfolios. The comparison between value-weighted and equal-weighted portfolio returns suggest that high V RP stocks outperform low V RP stocks by a larger amount among bigger stocks. This will be supported by results in Table 5 and Table 10, to be discussed later. Figure 3 plots the monthly time series of the return to the equal-weighted long-short portfolio that buys (shorts) stocks ranked in the top (bottom) decile ranked by the variance risk premium. This portfolio has significantly positive average returns in various sub-periods (e.g., in each of the three periods of about five years in length, or when excluding the financial crisis period). 4.2 Robustness Checks We further document the robustness of the positive relation between expected stock return and the variance risk premium. First, we classify in each month all stocks for which we estimate the variance risk premium into two subsamples. One subsample consists of stocks belonging to the S&P500 index, and the rest are classified into the Out of S&P500 subsample. For each subsample in each month, we sort stocks into five portfolios by their V RP. Table 5 reports both the value-weighted and the equal-weighted average returns of these quintile portfolios over the next month. Table 5 shows that for both value-weighted and equal-weighted portfolios, regardless of whether the stocks belong to the S&P 500, high V RP stocks on average significantly outperform low V RP stocks. The results are actually stronger for S&P 500 subsample. For stocks within the S&P 500 index, the top V RP quintile outperforms the bottom quintile by 1.99% on value-weighted basis and by 1.82% on equal-weighted basis. By comparison, for non S&P 500 stocks, the top V RP quintile outperforms the bottom quintile by 1.44% when value-weighted and by 1.27% when equal-weighted. All of these outperformances are significant both statistically and economically. 16

19 Our results are also robust to variations in the measurement of variance risk premium. In previous tables, we estimate the realized variance of stock return in a month using high frequency stock prices data sampled every 15 minutes during all the trading days in that month. In Panel A (resp. Panel B) of Table 6, we sample stock prices data every 30 minutes (resp. 60 minutes) when computing the realized variance. With the variance risk premium estimates obtained this way, we still find that the top V RP decile outperforms the bottom decile by a significant 1.25% to 1.38% per month. For all reported results, the risk-neutral expected variances are inferred from the call option prices. In unreported tables, we have also verified that our results do not change materially when we estimate the risk-neutral expected variance from put options. 5 Explanations 5.1 Risk Compensations One explanation for higher expected return of high V RP stocks is risk compensation. We have shown that high V RP stocks tend to have lower returns when systematic volatility risks increase (see Table 2). Thus, high V RP stocks are riskier and command higher risk premium. On the other hand, low V RP stocks have high beta s with respect to systematic volatility risks proxied by change in V IX (market volatility), change in the average idiosyncratic volatility, or the market variance risk premium. Low V RP stocks serve as useful hedges for systematic volatility risks, and therefore have lower expected return. To examine the extent to which the positive relation between expected stock return and variance risk premium can be explained by stock s exposure to the systematic volatility risks, we form volatility beta (β E ) and V RP double sorted portfolios. At the end of each month and for each stock, β E is estimated from a four-factor model (Fama-French three factors plus a volatility factor) using a rolling window of past 60 month s observations. We first sort 17

20 stocks into five portfolios by their exposure (β E ) to a systematic volatility factor. Within each β E sort, we further rank stocks into five quintiles by V RP. Finally, we compute the equal-weighted average return of each V RP quintile across the five β E portfolios. These returns, reported in Table 7, show the relation between stock s V RP and next month s average return after controlling for stock s beta with respect to the systematic volatility risks. We find that after controlling for stock s beta with respect to the systematic volatility risks, an equal-weighted portfolio of stocks ranked in the top V RP quintile on average outperform an equal-weighted portfolio of stocks of bottom V RP quintile stocks by about 1.13%. This is 0.31% lower than the corresponding number without controlling for the exposure to systematic volatility risks. Recall Table 4 Panel B shows that the equal-weighted return spread between the top and the bottom V RP quintile in univariate sorts is 1.44%. Thus, controlling for stock s exposure to the systematic volatility risks explains about 22% of the V RP effect on stock returns. Table 8 further documents the importance of stock s exposure to the systematic volatility risks for the positive relation between V RP and expected stock return using Fama MacBeth cross-sectional regressions. In the univariate regression of next month s stock return on current estimate of variance risk premium, the coefficient for V RP is 3.96, with a t statistic of After controlling for stock s exposure to the systematic volatility risks, the V RP coefficient is reduced to about 2.8 to 2.9, with a t statistic of 2.3 to 2.4. This 30% decrease in the regression coefficient of V RP is in line with the above finding based on portfolio sorts. Table 8 shows that β E, stock return beta with respect to systematic volatility factors, have negative coefficients for all three proxies for systematic volatility risk: change in market volatility, change in the average idiosyncratic volatility of individual stocks, and the market variance risk premium. The result for market volatility beta is consistent with the finding in the literature that stocks with high beta with respect to market volatility risk have low average returns (see, e.g., Ang, Hodrick, Xing, and Zhang (2006)). 18

21 The results for beta with respect to the other two systematic volatility factors are new. We find that stocks whose returns have higher beta with respect to change in average idiosyncratic volatility have lower expected returns. In our sample, the effect of beta with respect to average idiosyncratic volatility is stronger and more significant than effect of market volatility beta. In addition, we find stocks that are more sensitive to the market variance risk premium have lower expected returns. This cross-sectional relation for individual stock returns complements the time-series relation between market return and market variance risk premium documented in Bollerslev, Tauchen, and Zhou (2009), and Drechsler and Yaron (2011). Both double sorts results in Table 7 and Fama MacBeth regressions results in Table 8 show that after controlling for stock s exposures to systematic volatility risks, stock s variance risk premium is still positively and significantly related to the expected stock return, although the V RP effect is weakened. One possible explanation of this finding is that our estimates of stock s beta s with respect to systematic volatility risks are more noisy than the information obtained from variance risk premium. Note that the beta s with respect to the systematic volatility factors are estimated using rolling window of 60 months historical data, while the variance risk premium is based on current month s data and incorporates forward looking information from the options. Further, our proxies of the systematic volatility risks are measured with noise as well. Alternatively, there could be other reasons for why large V RP stocks have higher expected return. In subsequent sections, we test alternative possible explanations for the positive relation between stock s V RP and future returns. For example, we control for several stock charactersitics (Table 9) or various proxies for private information in options (Table 11). In the presence of these control variables, the V RP coefficient is reduced from 3.96 to about 3.4, as opposed to about 2.8 V RP coefficient when we control for stock s beta with respect to systematic volatility factors. This evidence already suggests that stock s exposure to the systematic volatility risk is the most important explanation of the relation between V RP 19

22 and stock return. In fact, results of our tests in the subsequent sections call into doubt alternative explanations based on mispricing or private information in options. 5.2 Mispricing and Limits to Arbitrage This section examines the possibility that the relation between stock s variance risk premium and expected return reflects some type of mispricing. For example, Table 3 shows that individual stock s variance risk premium varies systematically with many stock characteristics (e.g., size, book-to-market ratio and past returns). It is well known that expected stock returns depend on these characteristics, and these findings are usually viewed as anomalies caused by some investors irrational trading behavior or mistaken beliefs. Table 9 reports the results of monthly Fama MacBeth regressions where we regress in each month the next month s stock return on lagged regressors including stock s variance risk premium and multiple stock characteristics. The coefficients for stock characteristics such as size, book-to-market ratio and past stock returns have expected signs. But after controlling for all the stock characteristics, the coefficient for V RP is still 3.46 with a t-statistics of The predictive power of variance risk premium for stock return is largely distinct from the effects of known stock characteristics. In particular, the high average future returns of stocks with large variance risk premium can not be explained by the relation between stock return and volatility. Ang, Hodrick, Xing, and Zhang (2006) and others find that high volatility stocks tend to have lower average returns. If this were true in our sample, then it can not explain why high V RP stocks, which tend to have high volatility, have higher (not lower) average return. It turns out that for our sample, there is a significant and positive relation between stock return and volatility. 8 However, controlling for stock s volatility only marginally reduces the V RP coefficient. The 8 It is known that the result of Ang, Hodrick, Xing, and Zhang (2006) is driven by very small stocks, which do not characterize stocks in our sample. The average market cap of our sample stocks is 6.78 billion dollars. 20

23 larger average return of high V RP stocks is not a mere reflection of the positive returnvolatility relation for our sample stocks. To further test the possibility of a behavioral explanation for our results, we use double sorts to examine how the relation between stock s V RP and next month s average return depends on several proxies of limits to arbitrage. Any behavioral explanations would imply that our results would mostly come from stocks facing high limits to arbitrage. We use several limits to arbitrage proxies related to transaction costs and information uncertainty, including size, stock price, bid-ask spread, and analyst dispersion. For each such proxy, we first sort stocks into five portfolios, and then within each sort, we further rank stocks into five quintiles by variance risk premium. Table 10 reports the equal-weighted average returns of these double-sorted portfolios over the next month, as well as the difference in the average returns of top V RP quintile and the bottom V RP quintile within each sort by the arbitrage cost measures. Contrary to the prediction of behavioral explanations, Table 10 shows that our results are strong and significant in all subsamples of stocks regardless whether they face high limits to arbitrage. In fact, the outperformances of stocks with high variance risk premium are actually somewhat larger in magnitude and more significant among largest stocks, which are easier to arbitrage. This is consistent with the earlier finding that our results are stronger for S&P 500 stocks (see Table 5). Asset pricing anomalies apply mostly to small, negelected stocks facing short sales constraints, high transaction costs and high uncertainty. However, stocks in our sample have large market cap, high institutional ownership and high analyst coverage. Table 4 shows that most of the return difference between high V RP and low V RP stocks comes from the large average return of stocks with high variance risk premium. Stocks ranked in the top V RP quintile have an average market cap of 3.32 billion dollars, average institutional ownership of 71.6%, and the average number of analyst coverage of 9.8. It is hard to argue that the high returns of such stocks are due to underpricing. Stocks ranked in the bottom V RP quintile 21

24 have an average market cap of 7.09 billion dollars and 71.45% institutional ownership. It is easy to short these stocks, and hard to argue that the low returns of low V RP stocks are due to overpricing. Taken together, the results above argue against behavioral explanations for why stocks with large (resp. small) variance risk premium have higher (resp. lower) expected returns. 5.3 Informed Trading in Options Another possible explanation for the predictive power of the variance risk premium for stock returns is informed trading in the equity options by investors with private information. It is possible that V RP, estimated using the equity option prices, contains some useful information about the underlying stocks not currently reflected in the stock prices. 9 is the line of argument in Bali and Hovakimian (2009), Xing, Zhang, and Zhao (2010), and Cremers and Weinbaum (2010) for the ability of other option-related variables to predict stock returns. This Table 11 reports the results of Fama-macBeth regressions where we control for all optionrelated variables that are known to predict stock returns. First, we control for CV OL, change of at-the-money call implied volatility, and P V OL, change of at-the-money put implied volatility. Second, we control for CV OL P V OL, the difference in implied volatilities between at-the-money calls and puts. 10 RV OL IV OL used by Bali and Hovakimian (2009). Third, we control for the volatility spread variable It is measured as the difference between realized volatility and at-the-money option implied volatility, where the realized volatility is calculated using daily returns over a month. Fourth, we control for the slope of 9 Several previous studies have provided evidence for informed trading in the equity options (see, e.g., Cao, Chen, and Griffin (2005) and Pan and Poteshman (2006). 10 Following Bali and Hovakimian (2009), for a given stock at the end of a each month, we average the implied volatilities of all call (resp. put) options with time-to-maturity between 30 days and three months, and with absolute values of the natural log of the ratio of the stock price to the exercise price being smaller than