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1 Aggregate volatility risk: International evidence Stanley Peterburgsky Brooklyn College, 2900 Bedford Ave, New York, NY Abstract Using a procedure analogous to that of Ang et al. (2006), this paper documents that aggregate volatility risk does not appear to be priced in European financial markets. Specifically, based on the period (for which European data is available), the price of aggregate volatility risk is not statistically different from zero. Additionally, aggregate volatility loadings do not appear to predict future returns, and alphas from time-series regressions of excess returns on a long/short aggregate volatility portfolio with respect to the CAPM, the Fama-French 3-factor model, and the Fama-French 5-factor model are not statistically different from zero. Analysis based on high-frequency data support these results. Consequently, contrary to what has been reported in some studies that examine U.S. data, whether aggregate volatility risk is priced is an open question. JEL classification: G12, Keywords: aggregate volatility, VSTOXX This version: 1/14/2018 I thank Geert Bekaert, Robert Hodrick, and seminar participants at Brooklyn College for very helpful comments. All errors are mine.

2 I. Introduction One of the main objectives of asset pricing is to uncover the full set of risk factors for which investors require a premium. Over the last several decades various risk factors have been proposed. Most of these have been either motivated by, or retroactively explained in the context of, Merton s (1973) ICAPM model, which shows that individuals who make lifetime consumption decisions are subject to not only the market risk of the original CAPM, but also to additional risk factors that arise due to conditional relationships between stock returns and unanticipated changes in state variables that affect future returns. Examples include the wellknown SMB (small market capitalization minus big market capitalization) and HML (high bookto-market ratio minus low book-to-market ratio) factors of Fama and French (1993) as well as more recently proposed RMW (robust profitability minus weak profitability) and CMA (conservative investment policy minus aggressive investment policy) factors of Fama and French (2015). Chen (2002) develops a model of stock returns in which market volatility is timevarying, 1 and demonstrates that stocks that perform poorly when market volatility rises earn a risk premium. The reason for the risk premium is that investors prefer to hedge against a possible rise in market volatility, and require higher compensation (even after controlling for market beta) to hold stocks that do not provide a hedge. Motivated by Chen s (2002) theoretical framework, a number of empirical papers examine the relationship between stocks sensitivity to changes in market volatility (also known as aggregate volatility risk, variance risk) and average returns. Most studies in the option pricing literature find that aggregate volatility risk carries a negative risk premium (e.g., Bakshi and Kapadia (2003), Arisoy, et al. (2007), Carr and Wu (2009), Da and Schaumburg (2011)). However, these studies do not control for non-market risk 1 The empirical observation that the stock market volatility is time-varying goes back at least four decades. See Black (1976), Merton (1980), Christie (1982), Poterba and Summers (1986), and Schwert (1989) for early studies of stochastic market volatility. 2

3 factors, such as SMB and HML. Failure to control for known risk factors can cause redundant factors to appear non-redundant. Ang et al. (2006) approach the issue of aggregate volatility risk from a different angle. Using U.S. stock price data, they document a number of cross-sectional relationships between sensitivity to changes in aggregate volatility and expected returns. First, they find that portfolios of stocks sorted on sensitivity to changes in aggregate volatility predict future returns, with lower aggregate volatility loadings associated with higher future returns. Second, they show that stocks with lower aggregate volatility loadings generate higher alphas with respect to the CAPM and the Fama-French 3-factor model than stocks with higher loadings. Finally, using a Fama and MacBeth (1973) procedure, they document that aggregate volatility risk carries a negative premium. However, Peterburgsky (2017) finds that these relationships disappeared in the period. Consequently, whether aggregate volatility risk is important to investors after taking into account known risk factors is unclear. In this paper, I use European stock price data to shed light on issues raised above. I find that, consistent with Peterburgsky (2017), the relationships between sensitivity to changes in aggregate volatility and expected returns documented by Ang et al. (2006) are not present in the European data. Specifically, aggregate volatility betas do not predict future returns. Alphas from time-series regressions of long/short high-minus-low aggregate volatility beta portfolio returns with respect to the CAPM, the Fama-French 3-factor model, and the Fama-French 5-factor model are not statistically different from zero. Finally, the price of aggregate volatility risk is not statistically different from zero. These findings are supported by results based on high-frequency intraday data. 3

4 The rest of this paper is organized as follows. In Section II, I discuss the data, explain the portfolio construction methodology, and report summary statistics. In Section III, I present my main findings on the (lack of) relationships between sensitivity to changes in aggregate volatility and expected returns in European markets. In Section IV, I examine the robustness of my results to alternative specifications of the portfolio construction methodology and measures of change in aggregate volatility. In Section V, I use high-frequency intraday data to re-examine the pricing of aggregate volatility risk. Finally, I summarize my research and offer concluding remarks in Section VI. II. Data, portfolio construction, and summary statistics This paper uses data from a number of sources to assess the importance of aggregate volatility risk for asset pricing in European financial markets. Daily stock returns, prices, number of shares outstanding (for computing market capitalization) and book equity (for computing book-to-market ratios) are from Bloomberg. 2 The daily VSTOXX index 3 level is from 4 European factor returns for SMB, HML, RMW, and CMA, as well as European risk-free rates, are from Ken French s web site ( Finally, high- 2 The syntax for Bloomberg formulas is as follows: =BDH(SecurityID&" Equity","CHG_PCT_1D","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y", "CshAdjNormal=Y", "CURRENCY=EUR") =BDH(SecurityID&" Equity","PX_LAST","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y", "CshAdjNormal=Y", "CURRENCY=EUR") =BDH(SecurityID&" Equity","EQY_SH_OUT","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y", "CshAdjNormal=Y", "CURRENCY=EUR") =BDH(SecurityID&" Equity","TOT_COMMON_EQY","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y", "CshAdjNormal=Y", "CURRENCY=EUR") 3 The VSTOXX index, also known as EURO STOXX 50 Volatility, represents the implied volatility of a synthetic option contract on the EURO STOXX 50 index with a maturity of one month. 4 STOXX is a subsidiary of Deutsche Börse Group. 4

5 frequency market volatility data is from Heber, et al. (2009). All prices and returns are in Euros unless otherwise noted. The companies used to analyze the importance of aggregate volatility risk are the members of STOXX Europe 600, 5 and account for approximately 88% of the total European stock market value (Plagge (2017)). Since Bloomberg data for STOXX Europe 600 components goes back to only 2002, I focus the period. Panel A of Table 1 shows the distribution of the number of firms across countries and over time, while Panel B presents a market value distribution. In the discussion that follows, I restrict my sample to firm-months that have at least 14 trading days. The VSTOXX index is the European version of the American VIX index. It is the most widely used measure of expected volatility in European markets. A graph of the index level over the period appears in Figure 1. The mean level is 24.76, while the median is As is the case for the VIX, the VSTOXX is highly positively skewed. The most pronounced spike in the index since its debut in 1999 occurred during the global financial crisis of Analogous to the procedure used in Ang, et al. (2006), I begin by forming portfolios of stocks by sorting them into quintiles based on the coefficient d from the following regression run each month, R(t) RF(t) = a + b[rm(t)-rf(t)] + dδvol(t) + e(t) (1) 5 The STOXX Europe 600 Index is a subset of the STOXX Global 1800 Index. With 600 component stocks, the index represents (mostly) large capitalization companies across 17 countries of the European region: Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Ireland, Italy, Luxembourg, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and the United Kingdom. In prior years, the index included Greek and Icelandic companies. 5

6 where R(t) is the daily return on the stock, RF(t) is the daily risk-free rate, RM(t) is the daily return on the STOXX Europe 600 index (henceforth, the market portfolio), ΔVol(t) is the daily change in the VSTOXX index, and a, b, and d are regression coefficients. Portfolio 1 is the quintile with the lowest loadings on the change in VSTOXX, while portfolio 5 is the quintile with the highest loadings. Table 2 reports summary statistics on firms in each of the five portfolios. The stocks in the 5 th quintile were somewhat smaller than those in the other quintiles, while the median bookto-market ratio decreases monotonically from quintile 1 to quintile 5 (although the mean does not). The table also shows that, as in Ang et al. (2006), loadings on the change in VSTOXX are highly unstable, with an average of more than 70% of the firms transitioning from one quintile to another each month. III. Results Ang et al. (2006) find, based on a U.S. sample, that stocks with low sensitivities to changes in aggregate volatility in one month have high average returns in the subsequent month. However, Peterburgsky (2017) documents that this relationship disappeared during the period. I follow Ang et al. s procedure to investigate whether low sensitivity to changes in aggregate volatility predicts high average returns in the European sample. I begin by sorting stocks into five portfolios based on the coefficient d from regression (1). Table 3 presents the equally- and value-weighted mean monthly total returns on the five portfolios in the subsequent month. The table shows that the spread between the returns on the 5 th and the 1 st portfolios was not statistically significant (and had the wrong sign) during (t = 1.30 for equally-weighted and t = 1.17 for value-weighted). The table also illustrates that (by construction) the average pre-formation loadings on the change in aggregate volatility 6

7 are monotonically increasing from portfolio 1 to portfolio 5. A similar pattern emerges in the post-formation loadings, although the variance is much smaller (as should be expected). Ang et al. (2006) also report that the alpha from a CAPM time-series regression for the long/short (high sensitivity to changes in aggregate volatility minus low sensitivity to changes in aggregate volatility) zero-cost portfolio is negative and statistically significant, as is the alpha from a Fama-French 3-factor time-series regression. However, Peterburgsky (2017) documents that in the more recent data, the alpha for the long/short portfolio is not significant in either specification. Consistent with Peterburgsky (2017), I find that the CAPM, Fama-French 3-factor, as well as Fama-French 5-factor long/short portfolio alphas are all insignificant (and have the wrong sign) in the European sample. Regression results appear in the middle columns of Table 3 (CAPM t = 1.29, three-factor t = 1.08, and five-factor t = 0.22). One of Ang et al. s (2006) most important contributions to the asset pricing literature is their analysis of the price of aggregate volatility risk. They estimate the price of risk based on the U.S. sample by running Fama-MacBeth regressions, 6 and find that the risk premium was negative and statistically significant. However, here too Peterburgsky (2017) documents that this result no longer holds when more recent U.S. data is examined. My analysis reveals that aggregate volatility risk is not priced in European markets. A battery of robustness tests confirms this conclusion. For the main specification, I follow a procedure similar to that of Ang et al. in constructing 25 portfolios by sorting stocks on their market betas and on their sensitivity to changes in aggregate volatility. Specifically, I first sort the members of STOXX Europe 600 into 6 The Fama-MacBeth procedure requires a set of time-series regression to estimate the sensitivities of test assets to various factors in stage one, and a set of cross-sectional regressions to estimate the period-byperiod prices of risks in stage two. After the risk premia have been estimated, a t-test can be conducted to determine whether each factor is priced. 7

8 one of 5 groups based on market betas from equation (1). Next, within each group, I sort the stocks into one of 5 portfolios based on sensitivity to changes in aggregate volatility. The 25 b d portfolios serve as test assets in the Fama-MacBeth procedure. In stage one of the procedure, the following time-series regression of monthly portfolio excess returns on factor returns gives the factor loadings for each portfolio: R(t) RF(t) = β0 + β f + e(t) (2) where β is a vector of factor loadings and f is a vector of factor returns. In stage two, the following cross-sectional regressions yield month-by-month risk premia: R(t) RF(t) = λ0 + λ β + e(t) (3) where λ is a vector of risk premia and β is a vector of factor loadings from stage one. Finally, a t-test determines whether the risk premia are statistically significant. Table 4 presents the results from six Fama-MacBeth regressions. In the first model, the only factor is assumed to be the excess return on the market portfolio. In the second model, ΔVol is added as an explanatory variable for asset excess returns. Models III and IV further add the Fama-French European SMB and HML factors, without and including the ΔVol, respectively. Models V and VI additionally add the Fama-French European RMW and CMA factors, without and including the ΔVol, respectively. The table indicates that, based on the European sample, aggregate volatility was not a priced factor in any of the three specifications that include the ΔVol variable: the CAPM plus volatility specification (labeled II ) yields t = 0.36 on ΔVol, the three-factor plus volatility specification (labeled IV ) yields t = 0.65, and the five-factor plus volatility specification (labeled VI ) yields t =

9 Estimating the price of risk of any factor is notoriously difficult. Ang et al. s (2006) point out that their regressions suggest a negative price of risk for both the SMB and the HML U.S.- based factors, which contradicts the results of most studies that examine much longer periods. This illustrates that 15 years of data is often insufficient to accurately estimate average risk prices. Therefore, it should not be surprising that questions related to the pricing of aggregate volatility risk are difficult to answer as well. IV. Robusness of results To determine whether my findings on factor risk premia are robust to alternative specifications, in this section I repeat the Fama-MacBeth analysis five times after making one of the following modifications: (1) using the unanticipated change in VSTOXX from an AR(1)/GARCH(1, 1)/EGARCH(1, 1) model (both in daily and monthly regressions) rather than the actual change, (2) excluding the global financial crisis period, (3) truncating/winsorizing the sample, (4) using an alternative parameter estimation window, and (5) using alternative test assets. IV.A Unanticipated change in VSTOXX Since stock market volatility is mean-reverting, a portion of the daily or monthly change in the VSTOXX can be predicted using volatility forecasting models such as AR(1), Bollerslev s (1986) and Taylor s (1987) GARCH, or Nelson s (1991) EGARCH. Therefore, only the remaining, unpredictable portion could theoretically be priced. In this subsection, I use the unanticipated change in the VSTOXX index instead of actual change, both in daily (preformation) and monthly (Fama-MacBeth) regressions. The unanticipated change in VSTOXX is 9

10 defined as the difference between the actual change and the predicted change from an AR(1) model. The model for daily regressions is, R(t) RF(t) = a + b[rm(t)-rf(t)] + duδvol (t) + e(t) (4) where UΔVol is the daily unanticipated change in VSTOXX, and the other variables are as defined previously. The correlation between the unanticipated change in VSTOXX and actual change is.99 at daily frequency, and.96 at monthly frequency. Table 5 indicates that the price of risk of unanticipated change in aggregate volatility during the sample period was not statistically significant, and had the wrong sign for all specifications (CAPM plus volatility t = 0.70, threefactor plus volatility t = 0.83, and five-factor plus volatility t = 0.84). Results from GARCH(1, 1) and EGARCH(1, 1) methodologies are similar. IV.B Excluding global financial crisis period One could argue that the period was unusually volatile. If so, aggregate volatility risk could still be priced (have a negative risk premium), since stocks with low loadings would only be expected to produce high returns on average, to compensate investors for low returns in volatile periods. 7 If the period was indeed unusually volatile, the apparent lack of relationship between aggregate volatility loadings and returns would be fully consistent with a risk-based story. 8 I believe the period was not unusually volatile. While it is true that it featured the global financial crisis during which the VSTOXX index topped 80, the fifteen-year period as a whole was likely not particularly risky, at least not compared to the preceding fifteen-year 7 This is the same argument as for any other risk factor in a single-factor or multifactor risk model. For example, in the CAPM (which carries a positive risk premium), stocks with high market betas are only expected to produce high returns on average, not when the market experiences a sharp decline. 8 I thank an anonymous referee for a related paper for proposing this line of reasoning. 10

11 period. Although the VSTOXX data doesn t go back much further than my sample period, the VIX (a U.S. analogue) had a more extreme spike during the global stock markets crash of October 1987, reaching a record of 150, than at any time during the financial crisis. Since the VSTOXX and the VIX are highly correlated, it stands to reason that Europe also experienced a more volatile environment during the fifteen year period preceding Nevertheless, to address the concern that was an unusually volatile time for European financial markets, in this subsection I exclude the global financial crisis period. For the purposes of this analysis, I define the beginning of the financial crisis period as September 15 th, 2008, the date on which Lehman Brothers filed for bankruptcy, and the end of the financial crisis period as October 20 th, 2009, the date on which the VSTOXX closed below the sample mean of for the first time in more than a year. The choice of the exact start and end dates do not affect the results (e.g., January 2008 December 2009). Here, the first-stage Fama-McBeth regressions use the entire period, while the cross-sectional regressions are omitted for the 14 month from September 2008 through October Table 6 indicates that the price of aggregate volatility risk when the financial crisis period is removed from the sample was not statistically significant for all specifications (CAPM plus volatility t = -0.18, three-factor plus volatility t = -0.11, and five-factor plus volatility t = -0.07). IV.C Truncating the sample The base case estimation is highly conditional, allowing for the possibility that factor loadings change very rapidly (i.e., from one month to the next). This flexibility is advantageous. On the other hand, since very few observations are used to estimate the betas, there is potential for estimated (sample) betas to not properly reflect true (population) betas. Furthermore, there is 11

12 a legitimate concern that firm-month observations with the most noisy betas will drive the results. In order to test whether outliers are indeed impacting the base case findings, I repeat the Fama-MacBeth analysis using a truncated sample. Specifically, I remove the top 10 and bottom 10 companies by market beta month-by-month. From the remaining 580 companies, I remove the top 15 and bottom 15 by aggregate volatility beta. Using the remaining 550 companies, I proceed as in the base case analysis. 9 Table 7 indicates that the price of aggregate volatility risk based on the truncated sample was not statistically significant during , and had the wrong sign for all specifications (CAPM plus volatility t = 0.19, three-factor plus volatility t = 0.26, and five-factor plus volatility t = 0.12). Similar results are obtained for a winsorized, rather than truncated, sample. IV.D Alternative parameter estimation window An alternative way to address the concern that one month is too brief an estimation window for producing reliable factor loadings is to simply use a longer window. In this subsection, I use a 3-month estimation period instead. Panel A of Table 8 hints that the longer estimation window produces more accurate factor loadings. For example, firms in the extremelow quintile during months 1-3 are 58.5% likely to remain in the same quintile during months 2-4, while the extreme-high quintile firms have a 57.5% probability of remaining in their original quintile. By comparison, the analogous figures for the 1-month estimation window are only 25.9% and 24.3%, respectively. 10 Panel B shows the frequency of the two approaches agreeing 9 In the base case, the 600 firms are divided evenly among 25 portfolios, resulting in 24 firms per portfolio. In the truncated sample, the 550 firms are likewise divided evenly among 25 portfolios, resulting in 22 firms per portfolio. 10 Although we would expect a lower likelihood of switching quintiles when using three month of data, of which two overlap, nevertheless, the magnitude of the increase in factor loading stability is noteworthy. For example, the 58.5% and 57.5% mentioned in the text are 38.5% and 37.5% higher than the 20% we would 12

13 as well as disagreeing as to which quintile a given firm should be assigned to. The results indicate that the two approaches agree between 27.0% and 51.1% of the time, depending on the quintile in question. The not insignificant probabilities far from the main diagonal suggest that the alternative estimation analysis is worthwhile. Table 9 reports the price of risk for several risk factors using a number of models specifications. The table indicates that the price of aggregate volatility risk during the sample period was not statistically significant and had the wrong sign for all specifications (CAPM plus volatility t = 0.14, three-factor plus volatility t = 0.23, and five-factor plus volatility t = 0.47). IV.E Alternative test assets In this subsection, I examine the pricing of aggregate volatility risk using an alternative set of test assets, namely the Fama-French 25 (5 5) European portfolios sorted on size and bookto-market (from Table 10 reports the price of risk for several risk factors using a number of models specifications. The table indicates that, based on the Fama-French 25 European portfolio returns, the price of aggregate volatility risk had a wrong sign for the three-factor plus volatility specification (t = 2.79), and was indistinguishable from 0 for the CAPM plus volatility (t = -0.96) and the fivefactor plus volatility (t = -0.95) specifications. expect from pure noise. On the other hand, the 25.9% and 24.3% for the 1-month estimation are merely 5.9% and 4.3% higher than would be expected from pure noise. 13

14 V. Evidence from high-frequency data Chen and Ghysels (2012) argue that volatility forecasting models that use intraday data outperform traditional GARCH-class models. To determine which variables play an important role in high-frequency models, Bekaert and Hoerova (2014) examine 31 competing specifications with various combinations of terms that capture the VIX, 1-month, 1-week, and 1- day historical volatilities, and 1-month, 1-week, and 1-day historical returns. Of all the models, the one that appears to most reliably forecast volatility is model 8. With estimated coefficients included, Bekaert and Hoerova s model 8 is, 2 (22) VIX t 22 (22) (5) (1) RVt RVt RVt RVt 22 (5) 12 where RV 22 (22) t RVt j 1 j 1 is the (monthly) realized variance, based on 5-minute returns, over the 22-day period ending on day t, RV 22 is the (monthly) realized variance over 5 (5) t RVt j 1 5 j 1 the 5-day period ending on day t, and 1 (1) RVt 22 RVt j 1 22RVt is the (monthly) realized j 1 variance on day t. I use an analogous model, with VSTOXX taking the place of VIX, to estimate conditional volatility. 11 This conditional volatility will serve as an input in a regression analogous to (1) above. It should be noted that the frequency of data is not the only difference between my analysis in this section and the previous ones. Another important distinction is that the 11 The coefficients in Bekaert and Hoerova s model 8 are based on U.S. data. Realized variance in European markets is greater than in the U.S., at least over the period I examine. Therefore, the constant term in the model should be somewhat larger if the model is to be successfully applied to European data for the purpose of volatility forecasting. However, since I am interested in the change in realized variance rather than in the realized variance itself, the constant term is irrelevant for this study. 14

15 VSTOXX, the main volatility variable in previous sections, is a function of both the expected future volatility and the volatility premium. On the other hand, equation (5) attempts to model only the expected future volatility. Before running a regression analogous to (1), I convert the high-frequency measure of volatility from monthly, variance-based to annual, standard deviation-based: CSD 12 RV (6) (22) (22) t t where (22) CSD t is the (annual) conditional standard deviation over the 22-day period ending on day t. Analogous to the procedure used above, I form portfolios of stocks by sorting them into quintiles based on the coefficient d from the following regression run each month, R(t) RF(t) = a + b[rm(t)-rf(t)] + dδcsd(t) + e(t) (7) where R(t) is the daily return on the stock, RF(t) is the daily risk-free rate, RM(t) is the daily return on the STOXX Europe 600 index, ΔCSD(t) is the daily change in conditional standard deviation, and a, b, and d are regression coefficients. Portfolio 1 is the quintile with the lowest loadings on the change in conditional standard deviation, while portfolio 5 is the quintile with the highest loadings. To determine whether aggregate volatility is a priced risk factor based on high-frequency European data, I follow a procedure analogous to that above. That is, I construct 25 portfolios by sorting stocks on their market betas and on their sensitivity to changes in conditional standard deviation. Specifically, I first sort stocks into one of 5 groups based on market betas from equation (7). Next, within each group, I sort stocks into one of 5 portfolios based on sensitivity to changes in conditional standard deviation. The 25 b d portfolios serve as test assets in the Fama- 15

16 MacBeth procedure. In stage one of the procedure, the following time-series regression of monthly portfolio excess returns on factor returns gives the factor loadings for each portfolio: R(t) RF(t) = β0 + β f + e(t) (8) where β is a vector of factor loadings and f is a vector of factor returns. In stage two, the following cross-sectional regressions yield month-by-month risk premia: R(t) RF(t) = λ0 + λ β + e(t) (9) where λ is a vector of risk premia and β is a vector of factor loadings from stage one. Finally, a t-test determines whether the risk premia are statistically significant. Table 11 presents the results from six Fama-MacBeth regressions, and is analogous to Table 4. In the first model, the only factor is assumed to be the excess return on the market portfolio. In the second model, ΔCSD is added as an explanatory variable for asset excess returns. Models III and IV further add the Fama-French European SMB and HML factors, without and including the ΔCSD, respectively. Models V and VI additionally add the Fama- French European RMW and CMA factors, without and including the ΔCSD, respectively. The table shows that the coefficients on conditional volatility change in all three regressions that include ΔCSD have the wrong sign (CAPM plus volatility t = 2.29, three-factor plus volatility t = 2.34, and five-factor plus volatility t = 2.34). Together, these results support the view that it is far too early to make any definitive statements about the pricing of aggregate volatility risk. VI. Conclusion I investigate whether the relationships between sensitivity to changes in aggregate volatility and expected return on stocks documented by Ang et al. (2006) for the U.S. data are 16

17 also present in the European data. I find that they are not. Specifically, aggregate volatility betas do not predict future returns. Alphas from time-series regressions of long/short high-minus-low aggregate volatility beta portfolio returns with respect to the CAPM, the Fama-French 3-factor model, and the Fama-French 5-factor model are not statistically different from zero. Finally, the price of aggregate volatility risk is not statistically different from zero. Analysis based on highfrequency data support the view that it is far too early to draw any definitive conclusions about the pricing of aggregate volatility risk. This is, of course, not to say that investors don t care about aggregate volatility. Almost every equilibrium model of asset returns (e.g., CAPM) assumes that investors dislike volatility and would be willing to pay a premium to reduce their exposure to it. What remains unclear, as I have argued in this paper, is whether aggregate volatility is priced after taking into account known risk factors such as market risk, SMB and HML. Contrary to some of the research that has focused on U.S. data, it appears that more time will be needed to better understand aggregate volatility risk. 17

18 References Anderson, Robert M., Stephen W. Bianchi, Lisa R. Goldberg, 2013, In Search of a Statistically Valid Volatility Risk Factor, Coleman Fung Risk Management Research Center Working Papers (UC Berkeley). Ang, A., R. J. Hodrick, Y. Xing, and X. Zhang The Cross-Section of Volatility and Expected Returns, Journal of Finance 61, Arisoy, Yakup E., Aslihan Salih, and Levent Akdeniz, 2007, Is volatility risk priced in the securities market? Evidence from S&P 500 index options, Journal of Futures Markets 27, Bakshi, G. and N. Kapadia, 2003, Delta-hedged gains and the negative market volatility risk premium, Review of Financial Studies 16, Bekaert, G. and M. Hoerova, 2014, The VIX, the variance premium and stock market volatility, Journal of Econometrics 183, Black, F., 1976, Studies of stock price volatility changes. In: Proceedings of the 1976 Meeting of the Business and Economic Statistics Section, American Statistical Association, Washington, D.C. Bollerslev, Tim, 1986, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 31, Carr, P. and L. Wu Variance Risk Premiums, Review of Financial Studies 22, Chen, Joseph, 2002, Intertemporal CAPM and the cross-section of stock returns, working paper, University of Southern California. Chen, X. and E. Ghysels, 2012, News good or bad and its impact on volatility predictions over multiple horizons, Review of Financial Studies 24, Christie, A., 1982, The stochastic behavior of common stock variances: Value, leverage and interest rates effects. Journal of Financial Economics, 10, Da, Z. and E. Schaumburg, 2011, The pricing of volatility risk across asset classes. Unpublished working paper, University of Notre Dame and Federal Reserve Bank of New York. Delisle, R. J., Doran, J. S. & Peterson, D. R Asymmetric pricing of implied systematic volatility in the cross-section of expected returns, Journal of Futures Markets 31,

19 Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stock returns, Journal of Finance 47, Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33, Fama, Eugene F., and Kenneth R. French, 2015, A five-factor asset pricing model, Journal of Financial Economics 116, Fama, Eugene F., and James D. MacBeth, 1973, Risk return, and equilibrium: Empirical tests, Journal of Political Economy 71, Heber, G., A. Lunde, N. Shephard and K. Sheppard, 2009, "Oxford-Man Institute's realized library", Oxford-Man Institute, University of Oxford. Merton, Robert C., 1973, An intertemporal capital asset pricing model, Econometrica 41, Merton, Robert C., 1980, On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics 8, Nelson, Daniel B., 1991, Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica 59, Peterburgsky, Stanley, 2017, Is aggregate volatility a priced risk factor?, SSRN working paper. Plagge, Jan-Carl, 2017, Gaining access to the European equity market: STOXX Europe 600, accessed at %20Equity%20Market%20-%20Feb% pdf on January 11, Poterba, James M. and Lawrence H. Summers, 1986, The persistence of volatility and stock market fluctuations, American Economic Review 76, Schwert, G., 1989, Why does stock market volatility change over time?, Journal of Finance 44, Taylor, Stephen J., 1987, Forecasting the volatility of currency exchange rates, International Journal of Forecasting 3,

20 Table 1. Distribution of STOXX Europe 600 firms by country/year This table contains the distribution of STOXX Europe 600 firms across countries and over time. Panel A shows the number of firms. Panel B shows the percentage of market value. Panel A: Number of companies by country Austria Belgium Czech Republic Denmark Finland France Germany Ireland Italy Luxembourg 1 Netherlands Norway Portugal Spain Sweden Switzerland United Kingdom Other Panel B: Percentage of market value by country Austria 0.09% 0.26% 0.46% 0.76% 0.96% 1.21% 1.18% 0.75% 0.94% 0.96% 0.67% 0.73% 0.54% 0.44% 0.40% Belgium 1.49% 1.74% 1.85% 2.54% 1.84% 1.91% 2.04% 2.07% 2.42% 2.32% 2.42% 2.55% 2.50% 2.85% 3.23% Czech Republic 0.17% 0.19% 0.15% Denmark 1.20% 1.26% 1.30% 1.31% 1.45% 1.35% 1.53% 1.68% 1.67% 1.89% 1.84% 1.97% 2.08% 2.41% 2.89% Finland 2.45% 2.04% 1.75% 1.52% 1.82% 1.66% 2.22% 1.84% 1.56% 1.69% 1.31% 1.27% 1.38% 1.40% 1.42% France 16.81% 16.70% 15.77% 14.59% 15.98% 17.71% 18.56% 19.12% 17.98% 16.89% 16.15% 16.53% 16.59% 16.25% 16.57% Germany 11.68% 9.47% 11.15% 10.49% 11.20% 10.78% 13.16% 13.52% 11.75% 12.29% 11.98% 12.99% 13.81% 13.37% 13.39% Ireland 1.28% 1.13% 1.21% 1.43% 1.43% 1.53% 1.12% 0.57% 0.60% 0.55% 0.65% 0.66% 0.70% 0.79% 0.96% Italy 7.34% 8.01% 7.27% 7.84% 7.44% 7.11% 6.54% 6.32% 5.92% 4.80% 4.61% 4.38% 4.25% 4.16% 4.29% Luxembourg 0.09% Netherlands 7.95% 8.40% 7.79% 7.30% 5.72% 6.37% 5.56% 4.65% 5.58% 5.47% 5.56% 5.45% 5.63% 5.80% 5.38% Norway 0.70% 0.83% 0.82% 1.02% 1.21% 1.56% 2.24% 1.51% 2.00% 2.12% 2.30% 2.25% 1.92% 1.61% 1.35% Portugal 0.56% 0.59% 0.57% 0.58% 0.54% 0.55% 0.76% 0.75% 0.87% 0.73% 0.50% 0.43% 0.41% 0.31% 0.32% Spain 4.53% 5.19% 5.81% 6.45% 6.33% 6.21% 6.16% 8.08% 6.83% 5.45% 5.66% 5.30% 5.69% 5.98% 5.59% Sweden 3.17% 2.79% 3.40% 3.61% 3.61% 3.80% 3.42% 3.73% 4.21% 5.21% 4.86% 5.07% 5.14% 5.00% 5.01% Switzerland 9.62% 9.95% 9.22% 9.38% 10.20% 9.72% 9.14% 11.66% 10.81% 11.46% 11.87% 11.45% 11.41% 11.87% 12.20% United Kingdom 30.32% 31.02% 30.67% 30.11% 29.09% 27.30% 24.62% 22.88% 26.04% 27.72% 29.45% 28.82% 27.58% 27.25% 26.77% Other 0.72% 0.63% 0.96% 1.07% 1.18% 1.23% 1.74% 0.87% 0.81% 0.46% 0.17% 0.15% 0.20% 0.33% 0.07% 20

21 Table 2. Summary statistics This table contains summary statistics on firm size (in Millions), book-to-market ratio, and transition probability for each of the 5 aggregate volatility sensitivity quintiles. Transition probability refers to the likelihood of a firm moving from one quintile in month t to another quintile in month t+1 (or staying in the same quintile). Transition Matrix Rank Average Size Median Size Average B/M Median B/M To Rank 1 To Rank 2 To Rank 3 To Rank 4 To Rank 5 1 (Lowest) % 20.4% 18.3% 16.8% 18.6% % 21.3% 20.9% 19.8% 17.7% % 20.9% 20.9% 21.5% 18.4% % 19.6% 21.1% 21.7% 20.6% 5 (Highest) % 17.9% 18.8% 20.3% 24.3% 21

22 Table 3. dδvol portfolio statistics This table contains data for each of the 5 portfolios sorted on dδvol from regression (1), as well as the portfolio that is long high dδvol stocks and short low dδvol stocks. The first column ranks the portfolios, with 1 corresponding to the lowest dδvol, and 5 corresponding to the highest. The second column shows the equally-weighted average monthly total returns in the post-formation month, calculated month-by-month and averaged over months, while the third column shows the value-weighted average monthly total returns, also calculated month-by-month and averaged over months. The following three columns report Jensen s alpha with respect to the CAPM, the Fama-French 3-factor model, and the Fama-French 5-factor model, using time-series regressions. The last two columns report the average value-weighted pre-formation and post-formation dδvol coefficients (multiplied by 100). Newey-West robust t statistics appear below the estimated coefficients. Factor Loadings CAPM FF-3 FF-5 Pre-formation Post-formation Rank EW Mean VW Mean Alpha Alpha Alpha 100 d ΔVol 100 d ΔVol 1 (Lowest) [-1.59] [-1.17] [1.13] [0.04] [0.67] [1.69] [0.63] [1.31] [2.48] [0.44] [1.02] [2.43] 5 (Highest) [-0.51] [-0.23] [1.31] [1.30] [1.17] [1.29] [1.08] [0.22] 22

23 Table 4. Price of aggregate volatility risk: base case This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well as prices of other risks. The 25 b d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the first-stage regression is a time-series regression of excess returns on the 25 b d portfolios on returns on the risk factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated coefficients. I II III IV V VI RM-RF [-0.2] [-0.23] [-0.23] [-0.24] [-0.33] [-0.49] SMB [0.24] [0.33] [-0.28] [-0.2] HML [0.19] [0.18] [0.8] [0.83] RMW [0.78] [0.74] CMA [-0.63] [-0.64] ΔVol [0.36] [0.65] [0.72] Alpha [1.63] [1.49] [1.33] [1.4] [1.08] [1.24]

24 Table 5. Price of aggregate volatility risk: unanticipated change in volatility This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well as prices of other risks. Here, I define the change in aggregate volatility as the unanticipated change in VSTOXX based on an AR(1) model. The 25 b d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to changes in aggregate volatility) from regression (4) in the text. For the Fama-MacBeth model, the first-stage regression is a time-series regression of excess returns on the 25 b d portfolios on returns on the risk factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the unanticipated change in aggregate volatility. t statistics appear below the estimated coefficients. I II III IV V VI RM-RF [-0.27] [-0.42] [-0.31] [-0.54] [-1.5] [-1.69] SMB [0.51] [0.59] [-0.19] [0.09] HML [0.34] [0.48] [0.34] [0.33] RMW [-0.02] [-0.36] CMA [-1.11] [-0.65] UΔVol [0.7] [0.83] [0.84] Alpha [1.75] [1.73] [1.49] [1.64] [2.1] [2.1] 24

25 Table 6. Price of aggregate volatility risk: excluding financial crisis period This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well as prices of other risks. Here, I exclude the financial crisis period (September 2008 October 2009) in the crosssectional regressions. The 25 b d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the first-stage regression is a time-series regression of excess returns on the 25 b d portfolios on returns on the risk factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated coefficients. I II III IV V VI RM-RF [-0.47] [-0.45] [-0.4] [-0.4] [-0.49] [-0.48] SMB [0.32] [0.31] [0.01] [0.01] HML [-0.02] [-0.02] [0.35] [0.35] RMW [0.52] [0.52] CMA [-0.47] [-0.47] ΔVol [-0.18] [-0.11] [-0.07] Alpha [2.13] [1.77] [1.69] [1.53] [1.56] [1.37] 25

26 Table 7. Price of aggregate volatility risk: truncated sample This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well as prices of other risks. Here, I truncate the sample by removing the extreme 10 firms by market beta (both high and low) and the extreme 15 remaining firms by aggregate volatility beta (both high and low), month-by-month. This leaves 550 of the original 600 firms per month. The 25 b d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the first-stage regression is a time-series regression of excess returns on the 25 b d portfolios on returns on the risk factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated coefficients. I II III IV V VI RM-RF [-0.42] [-0.43] [-0.16] [-0.16] [-0.61] [-0.61] SMB [-0.2] [-0.23] [0] [0.01] HML [-0.49] [-0.51] [-0.16] [-0.17] RMW [0.14] [0.12] CMA [-0.41] [-0.38] ΔVol [0.19] [0.26] [0.12] Alpha [2.04] [2.07] [1.03] [1.07] [1.38] [1.41] 26

27 Table 8. Transition matrix and quintile comparison: 3-month versus 1-month Panel A of this table presents the transition probability for each of the 5 aggregate volatility sensitivity quintiles when estimating sensitivities using 3 months of data versus when using 1 month of data. Transition probability refers to the likelihood of a firm moving from one quintile in month t to another quintile in month t+1 (or staying in the same quintile). For the 3-month estimation, a firm s quintile assignment for month t is determined by data in months t-2 through t, while for the 1-month estimation it is determined by data in month t alone (as in Table 2). Panel B presents a comparison between quintile assignments using 3-month estimation versus 1-month estimation. Panel A: Transition Matrix comparison Transition Matrix using 3-month estimation Transition Matrix using 1-month estimation (from Table 1) Rank To Rank 1 To Rank 2 To Rank 3 To Rank 4 To Rank 5 To Rank 1 To Rank 2 To Rank 3 To Rank 4 To Rank 5 1 (Lowest) 58.5% 23.4% 10.1% 5.0% 2.9% 25.9% 20.4% 18.3% 16.8% 18.6% % 35.6% 23.7% 12.3% 5.3% 20.3% 21.3% 20.9% 19.8% 17.7% 3 9.9% 23.2% 32.5% 23.7% 10.6% 18.3% 20.9% 20.9% 21.5% 18.4% 4 5.0% 12.2% 23.6% 35.8% 23.4% 17.0% 19.6% 21.1% 21.7% 20.6% 5 (Highest) 3.1% 5.5% 10.3% 23.5% 57.5% 18.7% 17.9% 18.8% 20.3% 24.3% Panel B: Quintile comparison 1-month estimation Rank 1 (Lowest) (Highest) 1 (Lowest) 51.1% 23.5% 12.9% 7.7% 4.8% % 30.2% 23.2% 14.8% 7.9% 3-month % 23.5% 27.0% 23.2% 13.5% estimation 4 7.5% 14.8% 23.4% 30.0% 24.3% 5 (Highest) 4.8% 8.1% 13.4% 24.5% 49.2% 27

28 Table 9. Price of aggregate volatility risk: 3-month estimation windows This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well as prices of other risks. Here, I use 3-month (rolling) estimation windows to estimate market betas and aggregate volatility betas, rather than 1-month. The 25 b d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the firststage regression is a time-series regression of excess returns on the 25 b d portfolios on returns on the risk factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated coefficients. I II III IV V VI RM-RF [-0.53] [-0.53] [-0.6] [-0.62] [-1.46] [-1.61] SMB [0.26] [0.29] [0.04] [0.02] HML [0.11] [0.15] [-1] [-0.88] RMW [-0.72] [-0.65] CMA [-0.7] [-0.7] ΔVol [0.14] [0.23] [0.47] Alpha [2.28] [2.11] [1.95] [1.81] [2.36] [2.47] 28