How Aggregate Volatility-of-Volatility Affects Stock Returns

How Aggregate Volatility-of-Volatility Affects Stock Returns Fabian, Hollstein,;Marcel, Prokopczuk, 2018-12-01 00:00:00 Abstract A stylized theoretical model with stochastic volatility suggests the existence of a trade-off between returns and volatility-of-volatility. Using the VVIX, a measure of the option-implied volatility of the volatility index, we confirm this prediction and detect that time-varying aggregate volatility-of-volatility commands an economically substantial and statistically significant negative risk premium. We find that a two-standard-deviation increase in aggregate volatility-of-volatility factor loadings is associated with a decrease in average annual returns of about 11%. These results are robust to controlling for aggregate volatility, jump risk, and several other characteristics and factor sensitivities, as well as various additional tests. Received September 05, 2016; accepted June 21, 2017 by Editor Raman Uppal Introduction In this paper, we study whether aggregate volatility-of-volatility influences stock returns.1 Aggregate volatility-of-volatility is important because it characterizes the distribution of stochastic volatility. Although there seems to be a consensus that stochastic volatility is important for asset pricing (e.g., Ang et al. 2006; Barndorff-Nielsen and Veraart 2012), volatility-of-volatility, thus far, has received considerably less attention. However, the volatility of stochastic volatility likely also carries important information about the future investment opportunity set. If an innovation to aggregate volatility-of-volatility predicts worsening future investment opportunities, investors seek to insure against this and demand stocks with low sensitivities to innovations in volatility-of-volatility. Using a simple stylized theoretical model with recursive preferences and rich consumption volatility dynamics in the spirit of Bollerslev, Tauchen, and Zhou (2009), we demonstrate the existence of a volatility-of-volatility-return trade-off in addition to the commonly employed risk-return and volatility-return trade-offs. Based on the model predictions, our main contribution is to show empirically that volatility-of-volatility is significantly priced in the cross-section of stock returns, earning a negative risk premium. We document the trade-off between volatility-of-volatility and returns using several steps. First, we show that market volatility-of-volatility, measured using the VVIX index, is distinct from market volatility. The shocks to the two measures are only moderately correlated (0.67), and the overall dynamics appear to be substantially distinct. Moreover, empirically, the VIX forecasts future market volatility, whereas the VVIX forecasts the future volatility of the VIX. Second, following the empirical framework of Ang, Chen, and Xing (2006) and Cremers, Halling, and Weinbaum (2015), we show that the single-sorted hedge portfolio on innovations in aggregate volatility-of-volatility yields an annual return of –4.91% and an alpha of –6.35% with respect to the factor model of Hou, Xue, and Zhang (2015b).2 Additionally, using the Fama and MacBeth (1973) cross-sectional regressions, we find that a two-standard-deviation increase in aggregate volatility-of-volatility factor loadings is associated with a significant decrease in average annual returns of about 11%. We show that these results hold both for value-weighted and equally weighted sorts and regressions and prevail after controlling for canonical characteristics and popular market risk factors, including, for example, beta, aggregate volatility, size, liquidity, and jump risk. Third, we show that our findings are robust to various additional tests. Performing multivariate sensitivity estimations to control for various aggregate factors, our conclusions are unchanged. Likewise, we find that our results are not driven by the 2008–2009 financial crisis. Lastly, we show that innovations in aggregate volatility-of-volatility can be hedged in the cross-section of stock returns when estimating the sensitivities from market- and delta-neutral straddle returns on the VIX or directly using delta-hedged options written on the volatility index. Our work is related to recent studies on uncertain volatility. Epstein and Ji (2013) formulate a multiple priors model with ambiguity about volatility. Our approach is similar in that we consider a distribution of expected volatility. However, we use a direct smooth measure to characterize this distribution. Huang and Shaliastovich (2014) show that there is a volatility-of-volatility risk premium in the cross-section of delta-hedged S&P 500 and VIX options. Unlike Huang and Shaliastovich (2014), we study the pricing of aggregate volatility-of-volatility in the cross-section of equity returns. Studying the cross-section of stock returns allows us to control for several other cross-sectional risk factors, something not easily possible when working with hedged options portfolios. Baltussen, Van Bekkum, and Van Der Grient (forthcoming) show that the volatility-of-volatility of individual stocks is significantly negatively priced in the cross-section of stock returns. As opposed to our study, Baltussen, Van Bekkum, and Van Der Grient (forthcoming) primarily use a measure of idiosyncratic volatility-of-volatility and do not find a significant effect using past sensitivities from aggregate factor specifications with high-minus-low idiosyncratic volatility-of-volatility portfolios or the volatility-of-volatility from at-the-money (ATM) S&P 500 options. We, in turn, examine model-free aggregate market volatility-of-volatility, represented by the VVIX, as a state variable focusing on systematic, instead of idiosyncratic, effects. Chen, Chung, and Lin (2014) measure volatility-of-volatility using high-frequency index option data and empirically find volatility-of-volatility to carry a significantly negative risk premium. Our study differs from theirs in two important aspects. First, we directly use the VVIX index provided by the Chicago Board Options Exchange (CBOE) instead of a high-frequency intraday realized variance measure of the VIX index. Consequently, we use a forward-looking volatility measure instead of past variation in forward-looking volatility. This is important because there appears to be a consensus in the literature that using implied, instead of historical, volatility estimates significantly improves prediction accuracy (e.g., Jiang and Tian 2005; Prokopczuk and Wese Simen 2014). Second, we present a model framework with recursive Epstein and Zin (1989) preferences and rich consumption volatility dynamics that rationalizes our empirical results. Our work also connects to the aggregate risk factor literature including Ang et al. (2006), Adrian and Rosenberg (2008), Han and Zhou (2012), Chang, Christoffersen, and Jacobs (2013), Lettau, Maggiori, and Weber (2014), and Cremers, Halling, and Weinbaum (2015). However, our focus is different. Whereas previous studies have tested whether aggregate volatility, its components, or higher moments are priced, we examine the implications of aggregate volatility-of-volatility for the stock market. Finally, our paper relates to the literature on long-run risks pioneered by Bansal and Yaron (2004). Our empirical model especially builds on Bollerslev, Tauchen, and Zhou (2009), Campbell et al. (2014), and Bali and Zhou (2016). We show that with the model assumptions commonly employed in the literature, especially those of more recent work, the expected return beta representation implies that volatility-of-volatility is priced in asset markets. 1. The Model We build on the results of Bollerslev, Tauchen, and Zhou (2009), Campbell et al. (2014), and Bali and Zhou (2016) and impose a stylized intertemporal asset pricing model with stochastic volatility to theoretically motivate the volatility-of-volatility-return trade-off. The representative agent has Epstein and Zin (1989) preferences with the value function Vt as Vt=[(1−δ)Ct1−γθ+δ(Et[Vt+11−γ])1θ]θ1−γ, (1) where Ct is the consumption at time t, and the preference factors of the representative agent are denoted by δ, the subjective discount factor, and γ, the coefficient of relative risk-aversion. As is commonly done, for convenience we define θ=(1−γ)/(1−1/ψ) ⁠, where ψ is the elasticity of intertemporal substitution. As shown by Epstein and Zin (1991), the corresponding stochastic discount factor (SDF) can be expressed as Mt+1=(δ(CtCt+1)1/ψ)θ(Wt−CtWt+1)1−θ. (2) Wt is the market value of the agent’s consumption stream. The logarithm of the SDF is then mt+1=θln⁡δ−θψgt+1+(θ−1)rt+1. (3) rt+1=ln⁡(Wt+1/(Wt−Ct)) is the log return on wealth and gt+1=Δct+1 is the log consumption growth. We follow Bollerslev, Tauchen, and Zhou (2009) and Bali and Zhou (2016), assuming the following joint dynamics for consumption growth and consumption growth volatility: gt+1=μg+σg,t2zt+1, (4) σg,t+12=aσ+ρσσg,t2+qtzv,t+1, (5) qt+1=aq+ρqqt+ϕqqtzq,t+1. (6) μg is the constant mean growth rate, σg,t2 denotes the conditional variance of consumption growth, qt represents the volatility uncertainty process, while zt+1, zv,t+1 ⁠, and zq,t+1 describe independent i.i.d. N(0, 1) processes. The parameters satisfy aσ>0 ⁠, aq > 0, |ρσ|<1, |ρq|<1 ⁠, and ϕq>0 ⁠. Let ωt denote the logarithm of the price-dividend ratio, or price-consumption or wealth-consumption ratio, of the asset that pays the consumption endowment. To find the equilibrium, we use the standard Campbell and Shiller (1988) approximation for the return on wealth, rt+1, rt+1=κ0+κ1ωt+1−ωt+gt+1 ⁠. Doing this, one can conjecture a solution for ωt as an affine function of the state variables σg,t2 and qt (Bollerslev, Tauchen, and Zhou 2009): ωt=A0+Aσσg,t2+Aqqt. (7) Note that, under the assumption that γ>1 and ψ>1 ⁠, hence θ<0 ⁠, it holds that both Aσ<0 and Aq < 0. Substituting the Campbell and Shiller (1988) approximation into Equation (3), one obtains a pricing kernel without reference to consumption growth (Campbell et al. 2014; Bali and Zhou 2016): mt+1=θln⁡δ+θψκ0−θψωt+θψκ1ωt+1−γrt+1. (8) Assuming a conditional joint log-normal distribution with time-varying volatility for the asset returns, the risk premium on any asset j is given by Et(rj,t+1)−rf,t+12Vart(rj,t+1)=−Covt(mt+1,rj,t+1). (9) Inserting the pricing kernel without reference to consumption growth in Equation (8) into Equation (9), one can obtain an ICAPM pricing relation of the following form: Et(rj,t+1)−rf,t+12Vart(rj,t+1)=γCovt(rt+1,rj,t+1)−θψκ1Covt(ωt+1,rj,t+1). (10) Inserting Equation (7) into Equation (10) yields: Et(rj,t+1)−rf,t+12Vart(rj,t+1)=γCovt(rt+1,rj,t+1)−θψκ1AσCovt(σg,t+12,rj,t+1)−θψκ1AqCovt(qt+1,rj,t+1). (11) Apart from the variance term Vart(rj,t+1) ⁠, there is the usual risk-return trade-off γ, a trade-off of returns and variance −θψκ1Aσ ⁠, and a volatility-of-volatility-return trade-off −θψκ1Aq can be detected from this formulation. This formulation may already directly serve as a motivation for why both shocks to volatility and volatility-of-volatility should be priced in the stock market. However, in this formulation, σg,t+12 and qt+1 represent the moments of the consumption processes. The model also makes direct predictions on the relation of the expected return on wealth and the consumption moments. From this, one can directly infer information on the variance of the return on aggregate wealth (Bollerslev, Tauchen, and Zhou 2009) as well as on volatility-of-volatility on aggregate wealth. We use the equations σr,t2≡Vart(rt+1)=σg,t2+κ12(Aσ2+Aq2ϕq2) and Qt≡Vart(σr,t+12)=qt(1+ϕq2κ14(Aσ2+Aq2ϕq2)2) ⁠, to obtain the following relation:3 Et(rj,t+1)−rf,t+12Vart(rj,t+1)=γCovt(rt+1,rj,t+1)−θψκ1AσCovt(σr,t+12,rj,t+1) + θψκ1[Aσκ12(Aσ2+Aq2ϕq2)−Aq][1+ϕq2κ14(Aσ2+Aq2ϕq2)]Covt(Qt+1,rj,t+1). (12) Equation (12) directly predicts a relation of current excess returns with variance and volatility-of-volatility of aggregate wealth. Note, however, that technically, Covt(σr,t+12,rj,t+1) and Covt(Qt+1,rj,t+1) denote the time-t covariances of asset returns with t + 1 expectations of variance and volatility-of-volatility over time t + 2. Hence, for these higher-order terms to be estimable and the model to be directly applicable, we have to assume that the current expectation of aggregate variance and aggregate volatility-of-volatility provide good proxies for next-period expectations.4 Further, note that Equation (12) implies a relation of the variance as well as, technically, the variance-of-variance of total wealth with stock returns. In our empirical tests, we use shocks to the empirically more robust measures of aggregate volatility and aggregate volatility-of-volatility, which are less strongly affected by outliers and measurement errors than variance measures due to the square-root transformation. Shocks to both volatility and squared volatility always have the same sign and are highly correlated by construction. Hence, for the purpose of motivating our empirical analysis, although not perfectly similar, the model delivers a reasonable description. Transforming Equation (12) to a conditional beta representation, replacing the conditional covariances with conditional betas, we get Et(rj,t+1)−rf,t+12Vart(rj,t+1)=Y˜·βj,tM+V˜·βj,tV+Z˜·βj,tQ. (13) Y˜, V˜ ⁠, and Z˜ denote the risk premiums, whereas βj,tM, βj,tV ⁠, and βj,tQ are the conditional market, variance, and volatility-of-volatility betas. From Equation (12), one can infer the model predictions on the market prices of risk. The market prices of risk are proportional to the derivative of the negative of the stochastic discount factor to shocks in the state variables. It can be easily shown that the market price of risk of shocks to aggregate wealth is positive and, given that under our assumptions the parameters satisfy ψ>1, θ<0 ⁠, and Aσ<0 as well as Aq < 0, the market price of shocks to variance is negative. The market price of risk on aggregate volatility-of-volatility is not clearly signed and depends on the relation of Aσκ12(Aσ2+Aq2ϕq2) and Aq. Because most of these parameters are difficult to interpret economically, the sign and size of the market price of risk on aggregate volatility-of-volatility remain an empirical question. 2. Data and Methodology 2.1 Data We base our study on all stocks traded on the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), and the National Association of Securities Dealers Automated Quotations (NASDAQ) that are classified as ordinary common shares (Center for Research in Security Prices (CRSP) share codes 10 or 11), excluding closed-end funds and REITs (SIC codes 6720–6730 and 6798), for the sample period between March 01, 2006 and September 30, 2016. We obtain data on the VIX and the VVIX from the CBOE. The VIX is constructed so that it represents the model-free 30-day implied volatility of the S&P 500 index. On February 24, 2006, the CBOE began trading options written on the VIX and recently the CBOE started reporting the VVIX, which represents the model-free 30-day implied volatility of the VIX.5,6 We also obtain 5-minute intraday high-frequency data on the VIX and the S&P 500 from the Thompson Reuters Tick History (TRTH) database. We obtain daily and monthly price data as well as data on dividend payments, trading volumes, firm age, and shares outstanding from CRSP. Following Amihud (2002) and Zhang (2006), we exclude “penny stocks” with prices below 1 dollar. Additionally, we require a market capitalization of at least 225 million dollars (D’Avolio 2002; Baltussen, Van Bekkum, and Van Der Grient forthcoming). These two thresholds serve to eliminate the most illiquid stocks that exhibit potential microstructure problems and may bias the results (Fama and French 2008). Furthermore, they ensure that only stocks with relatively low short-sale constraints are selected (D’Avolio 2002). On average, these selection criteria result in about 2,396 stocks included in our sample per month.7 We adjust for delisting returns following Shumway (1997) and Shumway and Warther (1999). We collect balance sheet and income statement data from the Compustat database. Data on option prices and Black and Scholes (1973) option sensitivities for options on the S&P 500 and the VIX are from the IvyDB OptionMetrics database.8 Data on the Fama and French (1993) and momentum factors as well as the risk-free (Treasury Bill) rate are from Kenneth French’s data library. We obtain data on the Pastor and Stambaugh (2003) liquidity factor from Robert Stambaugh’s homepage and data on the Hou, Xue, and Zhang (2015b) factors directly from Chen Xue.9 We provide detailed variable definitions of all variables used in the paper in Appendices A and B. 2.2 Empirical framework Our goal is to test whether stocks with different sensitivities to innovations in aggregate volatility-of-volatility have different returns. To examine that, we follow a large body in the asset pricing literature, and examine the contemporaneous relation between realized factor loadings and realized returns (e.g., Black, Jensen, and Scholes, 1972; Fama and French, 1992, among many others). Ang, Chen, and Xing (2006) argue that whereas pre-formation factor loadings reflect both actual variation in factor loadings and measurement error, post-formation factor loadings are almost exclusively affected by stock return covariations with risk factors. Additionally, they point out that if risk exposures, and hence factor loadings, are highly time-varying, pre-formation factor loadings might be poor predictors of ex post risk exposures, leaving the analysis with low power to detect relations between factor loadings and realized returns. Addressing these concerns, our research design follows Ang, Chen, and Xing (2006) and Cremers, Halling, and Weinbaum (2015) by estimating factor loadings for individual stocks using daily returns over rolling annual periods from the regression: rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tVdVIXτ+βj,tQdVVIXτ+ϵj,τ. (14) rj,τ is the daily return of asset j on day τ, rM,τ is the return of the market on that day, and rf,τ is the risk-free rate. dVIXτ and dVVIXτ are the daily innovation in the VIX (Ang et al., 2006) and VVIX indices. To mirror the specification implied by our theoretical model in Equation (13), the regression in Equation (14) only includes the market excess return, innovations in aggregate volatility, and innovations in aggregate volatility-of-volatility. Furthermore, Ang et al. (2006) argue that directly including additional factors in the regression in Equation (14) may add a lot of noise. Although we do not include additional possible cross-sectional risk factors when estimating the factor sensitivities to aggregate volatility-of-volatility using Equation (14), we are careful to ensure that we control for these factors when performing the time-series and cross-sectional asset pricing tests. As a robustness check and to account for possible model misspecification, we also consider multivariate joint factor loading estimations controlling for several aggregate risk factors previously documented in the literature: rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tVdVIXτ+βj,tQdVVIXτ+βj,tζζτ+ϵj,τ. (15) ζτ contains one or more market factors, such as the Fama and French (1993) and Hou, Xue, and Zhang (2015b) factors, the innovations in market skewness and kurtosis (dSkew, dKurt) as shown by Chang, Christoffersen, and Jacobs (2013), the Cremers, Halling, and Weinbaum (2015) straddle vol and jump factors, or innovations in the market variance risk premium (dVRP; Han and Zhou 2012). For the regressions in Equations (14) and (15), we use daily returns over rolling annual periods to estimate the sensitivities. For each period and stock, we require at least 100 nonmissing return observations in order to estimate the factor sensitivities.10 Turning the focus on the measurement of innovations in an economic variable, there generally exists a trade-off between a possible errors-in-variables problem using simple first differences, if that fails to completely filter out the expected movement, versus the danger of misspecifying a more complex equation for the expected movement in a variable (Chen, Roll, and Ross 1986). We choose to measure the innovations in the VVIX index using the daily first differences in the variable, similar to the way that Ang et al. (2006) obtain innovations in the VIX, because the VVIX is highly serially correlated with a first-order autocorrelation of 0.94 during our sample period. Therefore, the current value of the VVIX appears to be a relatively good proxy for tomorrow’s expectation making the first difference quite adequately capture its innovation. For robustness, we also consider measuring innovations in aggregate volatility-of-volatility by fitting an ARMA(1,1) model on the complete time-series of the VVIX index. This approach results in a measure of innovations of dVVIXτ=VVIXτ−0.9989VVIXτ−1+0.0839dVVIXτ−1 ⁠. The results of both approaches are qualitatively similar, which is further discussed in the next section.11 3. Empirical Results 3.1 Descriptive statistics In addition to various firm characteristics, we consider the impact of several aggregate state variables that have previously been examined in the literature. In Table 1 we report the sample correlations between daily innovations in aggregate volatility-of-volatility (dVVIX), innovations in aggregate volatility (dVIX; Ang et al. 2006), the Fama and French (1993), Carhart (1997), and Hou, Xue, and Zhang (2015b) factors, and also the factors on market skewness and kurtosis of Chang, Christoffersen, and Jacobs (2013), stochastic volatility and jump risk of Cremers, Halling, and Weinbaum (2015), and innovations in the market variance risk premium (Han and Zhou 2012). Table 1 Sample correlations of different aggregate factors dVVIX dVIX Straddle vol dSkew dKurt Jump dVRP MKT SMB HML Momentum SMBHXZ IA ROE dVVIXARMA dVoVIX * 0.67 −0.06 0.09 −0.16 0.46 −0.37 −0.55 −0.09 −0.12 0.02 −0.13 0.10 0.10 1.00 0.10 dVVIX * −0.05 0.15 −0.22 0.43 −0.77 −0.84 −0.07 −0.27 0.22 −0.18 0.21 0.18 0.66 0.07 dVIX * 0.03 −0.04 −0.69 −0.01 −0.02 −0.03 −0.03 0.05 −0.05 −0.01 −0.03 −0.06 −0.11 Straddle vol * −0.89 0.05 −0.11 −0.24 −0.01 −0.09 0.12 −0.07 0.08 0.06 0.08 −0.05 dSkew * −0.09 0.14 0.30 0.07 0.06 −0.09 0.11 −0.09 −0.07 −0.16 0.02 dKurt * −0.20 −0.27 −0.05 −0.06 −0.02 −0.07 0.06 0.07 0.46 0.19 Jump * 0.64 −0.02 0.22 −0.22 0.06 −0.18 −0.12 −0.37 0.01 dVRP * 0.17 0.40 −0.39 0.33 −0.23 −0.18 −0.55 −0.07 MKT * −0.10 0.03 0.78 −0.05 −0.08 −0.09 0.03 SMB * −0.59 0.36 0.19 −0.06 −0.12 −0.01 HML * −0.33 0.12 0.15 0.02 −0.02 Momentum * 0.04 −0.13 −0.14 0.01 SMBHXZ * 0.20 0.10 −0.02 IA * 0.11 0.00 ROE * 0.11 dVVIXARMA * dVoVIX dVVIX dVIX Straddle vol dSkew dKurt Jump dVRP MKT SMB HML Momentum SMBHXZ IA ROE dVVIXARMA dVoVIX * 0.67 −0.06 0.09 −0.16 0.46 −0.37 −0.55 −0.09 −0.12 0.02 −0.13 0.10 0.10 1.00 0.10 dVVIX * −0.05 0.15 −0.22 0.43 −0.77 −0.84 −0.07 −0.27 0.22 −0.18 0.21 0.18 0.66 0.07 dVIX * 0.03 −0.04 −0.69 −0.01 −0.02 −0.03 −0.03 0.05 −0.05 −0.01 −0.03 −0.06 −0.11 Straddle vol * −0.89 0.05 −0.11 −0.24 −0.01 −0.09 0.12 −0.07 0.08 0.06 0.08 −0.05 dSkew * −0.09 0.14 0.30 0.07 0.06 −0.09 0.11 −0.09 −0.07 −0.16 0.02 dKurt * −0.20 −0.27 −0.05 −0.06 −0.02 −0.07 0.06 0.07 0.46 0.19 Jump * 0.64 −0.02 0.22 −0.22 0.06 −0.18 −0.12 −0.37 0.01 dVRP * 0.17 0.40 −0.39 0.33 −0.23 −0.18 −0.55 −0.07 MKT * −0.10 0.03 0.78 −0.05 −0.08 −0.09 0.03 SMB * −0.59 0.36 0.19 −0.06 −0.12 −0.01 HML * −0.33 0.12 0.15 0.02 −0.02 Momentum * 0.04 −0.13 −0.14 0.01 SMBHXZ * 0.20 0.10 −0.02 IA * 0.11 0.00 ROE * 0.11 dVVIXARMA * dVoVIX This table presents the sample correlation coefficients of the aggregate factors dVVIX, dVIX, straddle vol, dSkew, dKurt, jump, dVRP, MKT, SMB, HML, momentum, SMBHXZ, IA, ROE, dVVIXARMA ⁠, and dVoVIX. Appendix A provides detailed variable definitions. Table 1 Sample correlations of different aggregate factors dVVIX dVIX Straddle vol dSkew dKurt Jump dVRP MKT SMB HML Momentum SMBHXZ IA ROE dVVIXARMA dVoVIX * 0.67 −0.06 0.09 −0.16 0.46 −0.37 −0.55 −0.09 −0.12 0.02 −0.13 0.10 0.10 1.00 0.10 dVVIX * −0.05 0.15 −0.22 0.43 −0.77 −0.84 −0.07 −0.27 0.22 −0.18 0.21 0.18 0.66 0.07 dVIX * 0.03 −0.04 −0.69 −0.01 −0.02 −0.03 −0.03 0.05 −0.05 −0.01 −0.03 −0.06 −0.11 Straddle vol * −0.89 0.05 −0.11 −0.24 −0.01 −0.09 0.12 −0.07 0.08 0.06 0.08 −0.05 dSkew * −0.09 0.14 0.30 0.07 0.06 −0.09 0.11 −0.09 −0.07 −0.16 0.02 dKurt * −0.20 −0.27 −0.05 −0.06 −0.02 −0.07 0.06 0.07 0.46 0.19 Jump * 0.64 −0.02 0.22 −0.22 0.06 −0.18 −0.12 −0.37 0.01 dVRP * 0.17 0.40 −0.39 0.33 −0.23 −0.18 −0.55 −0.07 MKT * −0.10 0.03 0.78 −0.05 −0.08 −0.09 0.03 SMB * −0.59 0.36 0.19 −0.06 −0.12 −0.01 HML * −0.33 0.12 0.15 0.02 −0.02 Momentum * 0.04 −0.13 −0.14 0.01 SMBHXZ * 0.20 0.10 −0.02 IA * 0.11 0.00 ROE * 0.11 dVVIXARMA * dVoVIX dVVIX dVIX Straddle vol dSkew dKurt Jump dVRP MKT SMB HML Momentum SMBHXZ IA ROE dVVIXARMA dVoVIX * 0.67 −0.06 0.09 −0.16 0.46 −0.37 −0.55 −0.09 −0.12 0.02 −0.13 0.10 0.10 1.00 0.10 dVVIX * −0.05 0.15 −0.22 0.43 −0.77 −0.84 −0.07 −0.27 0.22 −0.18 0.21 0.18 0.66 0.07 dVIX * 0.03 −0.04 −0.69 −0.01 −0.02 −0.03 −0.03 0.05 −0.05 −0.01 −0.03 −0.06 −0.11 Straddle vol * −0.89 0.05 −0.11 −0.24 −0.01 −0.09 0.12 −0.07 0.08 0.06 0.08 −0.05 dSkew * −0.09 0.14 0.30 0.07 0.06 −0.09 0.11 −0.09 −0.07 −0.16 0.02 dKurt * −0.20 −0.27 −0.05 −0.06 −0.02 −0.07 0.06 0.07 0.46 0.19 Jump * 0.64 −0.02 0.22 −0.22 0.06 −0.18 −0.12 −0.37 0.01 dVRP * 0.17 0.40 −0.39 0.33 −0.23 −0.18 −0.55 −0.07 MKT * −0.10 0.03 0.78 −0.05 −0.08 −0.09 0.03 SMB * −0.59 0.36 0.19 −0.06 −0.12 −0.01 HML * −0.33 0.12 0.15 0.02 −0.02 Momentum * 0.04 −0.13 −0.14 0.01 SMBHXZ * 0.20 0.10 −0.02 IA * 0.11 0.00 ROE * 0.11 dVVIXARMA * dVoVIX This table presents the sample correlation coefficients of the aggregate factors dVVIX, dVIX, straddle vol, dSkew, dKurt, jump, dVRP, MKT, SMB, HML, momentum, SMBHXZ, IA, ROE, dVVIXARMA ⁠, and dVoVIX. Appendix A provides detailed variable definitions. First, we note that whether innovations are measured as simple first differences (dVVIX) or as innovations in an ARMA(1,1) model (⁠ dVVIXARMA ⁠) in fact does not make a difference. The two innovation measures are (almost) perfectly correlated with a correlation of 1.00. There is a negative correlation of dVVIX with MKT of –0.55. The linear relation between dVVIX and dVRP, jump, and dVIX, to which it might be related by construction, is of similar magnitude.12 Correlations of dVVIX with other factors are negligible. Consequently, aggregate volatility-of-volatility appears to be distinct from other factors documented in the previous literature. In Table A1 of the Online Appendix, we also examine the Spearman rank correlations of the aggregate factor sensitivities. In general, these are very low. The rank correlation of sensitivities to aggregate volatility-of-volatility and aggregate volatility is even negative with –0.40, and substantially smaller in magnitude than the correlation of factor innovations. We provide additional summary statistics in Table 2. Panel A shows that the mean and the median innovations in the VVIX are close to zero. Measuring innovations in the VVIX using the first difference is shown to result in a factor with very low autocorrelation (–0.13), whereas using residuals from the fitted ARMA model reduces the first-order autocorrelation to practically zero. The remaining factors are mostly constructed as returns with means close to zero and negligible autocorrelations. Table 2 Summary statistics A. Market factors Variable Mean Median SD AR(1) P10 P90 dVVIX 0.00011 −0.00350 0.0471 −0.0838 −0.0471 0.0511 dVIX 0.00001 −0.00095 0.0192 −0.1284 −0.0170 0.0175 Straddle vol −0.00014 0.00109 0.0158 −0.0408 −0.0166 0.0156 dSkew −0.00228 −0.00351 0.0790 0.2265 −0.0969 0.0896 dKurt 0.00109 −0.00232 0.1205 0.1339 −0.1369 0.1454 Jump −0.00171 −0.00962 0.0557 −0.0953 −0.0447 0.0428 dVRP 0.00000 −0.00002 0.0086 −0.0379 −0.0039 0.0037 MKT 0.00033 0.00080 0.0129 −0.0841 −0.0130 0.0128 SMB 0.00004 0.00010 0.0059 −0.0430 −0.0069 0.0067 HML −0.00003 −0.00020 0.0069 −0.0251 −0.0063 0.0061 Momentum −0.00001 0.00050 0.0104 0.1214 −0.0102 0.0099 SMBHXZ 0.00013 0.00004 0.0062 0.0057 −0.0072 0.0068 IA 0.00017 0.00008 0.0042 0.0541 −0.0043 0.0049 ROE 0.00040 0.00038 0.0044 0.0191 −0.0043 0.0054 dVVIXARMA 0.00127 −0.00226 0.0498 0.0005 −0.0481 0.0540 dVoVIX 0.00022 0.00503 0.0421 0.1784 −0.0338 0.0295 A. Market factors Variable Mean Median SD AR(1) P10 P90 dVVIX 0.00011 −0.00350 0.0471 −0.0838 −0.0471 0.0511 dVIX 0.00001 −0.00095 0.0192 −0.1284 −0.0170 0.0175 Straddle vol −0.00014 0.00109 0.0158 −0.0408 −0.0166 0.0156 dSkew −0.00228 −0.00351 0.0790 0.2265 −0.0969 0.0896 dKurt 0.00109 −0.00232 0.1205 0.1339 −0.1369 0.1454 Jump −0.00171 −0.00962 0.0557 −0.0953 −0.0447 0.0428 dVRP 0.00000 −0.00002 0.0086 −0.0379 −0.0039 0.0037 MKT 0.00033 0.00080 0.0129 −0.0841 −0.0130 0.0128 SMB 0.00004 0.00010 0.0059 −0.0430 −0.0069 0.0067 HML −0.00003 −0.00020 0.0069 −0.0251 −0.0063 0.0061 Momentum −0.00001 0.00050 0.0104 0.1214 −0.0102 0.0099 SMBHXZ 0.00013 0.00004 0.0062 0.0057 −0.0072 0.0068 IA 0.00017 0.00008 0.0042 0.0541 −0.0043 0.0049 ROE 0.00040 0.00038 0.0044 0.0191 −0.0043 0.0054 dVVIXARMA 0.00127 −0.00226 0.0498 0.0005 −0.0481 0.0540 dVoVIX 0.00022 0.00503 0.0421 0.1784 −0.0338 0.0295 B. VVIX summary statistics Year Mean Median SD P10 P90 2006 0.8024 0.8073 0.1126 0.6601 0.9467 2007 0.8768 0.8618 0.1331 0.7194 1.0469 2008 0.8185 0.7741 0.1560 0.6763 1.1088 2009 0.7978 0.7913 0.0863 0.6954 0.9225 2010 0.8836 0.8622 0.1307 0.7538 1.0346 2011 0.9294 0.9146 0.1021 0.8182 1.0491 2012 0.9484 0.9384 0.0838 0.8416 1.0740 2013 0.8052 0.7960 0.0897 0.6967 0.9203 2014 0.8301 0.7991 0.1433 0.6771 0.9990 2015 0.9482 0.9041 0.1475 0.8002 1.1533 2016 0.9258 0.8981 0.1021 0.8260 1.0949 Total 0.8696 0.8523 0.1332 0.7100 1.0450 B. VVIX summary statistics Year Mean Median SD P10 P90 2006 0.8024 0.8073 0.1126 0.6601 0.9467 2007 0.8768 0.8618 0.1331 0.7194 1.0469 2008 0.8185 0.7741 0.1560 0.6763 1.1088 2009 0.7978 0.7913 0.0863 0.6954 0.9225 2010 0.8836 0.8622 0.1307 0.7538 1.0346 2011 0.9294 0.9146 0.1021 0.8182 1.0491 2012 0.9484 0.9384 0.0838 0.8416 1.0740 2013 0.8052 0.7960 0.0897 0.6967 0.9203 2014 0.8301 0.7991 0.1433 0.6771 0.9990 2015 0.9482 0.9041 0.1475 0.8002 1.1533 2016 0.9258 0.8981 0.1021 0.8260 1.0949 Total 0.8696 0.8523 0.1332 0.7100 1.0450 C. βQsummary statistics Year Mean Median SD P10 P90 2006–2007 0.0093 0.0074 0.0502 −0.0425 0.0650 2007–2008 0.0085 0.0072 0.0730 −0.0693 0.0904 2008–2009 0.0433 0.0367 0.1050 −0.0619 0.1585 2009–2010 0.0066 0.0053 0.0687 −0.0593 0.0751 2010–2011 0.0105 0.0093 0.0521 −0.0443 0.0675 2011–2012 0.0131 0.0118 0.0559 −0.0455 0.0766 2012–2013 0.0037 0.0018 0.0627 −0.0617 0.0729 2013–2014 −0.0054 −0.0040 0.0648 −0.0724 0.0581 2014–2015 −0.0057 −0.0043 0.0590 −0.0631 0.0489 2015–2016 −0.0060 −0.0073 0.0626 −0.0654 0.0559 Total 0.0078 0.0049 0.0684 −0.0589 0.0789 C. βQsummary statistics Year Mean Median SD P10 P90 2006–2007 0.0093 0.0074 0.0502 −0.0425 0.0650 2007–2008 0.0085 0.0072 0.0730 −0.0693 0.0904 2008–2009 0.0433 0.0367 0.1050 −0.0619 0.1585 2009–2010 0.0066 0.0053 0.0687 −0.0593 0.0751 2010–2011 0.0105 0.0093 0.0521 −0.0443 0.0675 2011–2012 0.0131 0.0118 0.0559 −0.0455 0.0766 2012–2013 0.0037 0.0018 0.0627 −0.0617 0.0729 2013–2014 −0.0054 −0.0040 0.0648 −0.0724 0.0581 2014–2015 −0.0057 −0.0043 0.0590 −0.0631 0.0489 2015–2016 −0.0060 −0.0073 0.0626 −0.0654 0.0559 Total 0.0078 0.0049 0.0684 −0.0589 0.0789 Panel A of this table presents summary statistics on the aggregate factors dVVIX, dVIX, straddle vol, dSkew, dKurt, jump, dVRP, MKT, SMB, HML, momentum, SMBHXZ, IA, ROE, dVVIXARMA ⁠, and dVoVIX. Appendix A provides detailed variable definitions. Panel B provides yearly summary statistics on the VVIX, and panel C shows yearly summary statistics on the individual stocks’ sensitivities to aggregate volatility-of-volatility, βQ ⁠, with the sensitivity estimation starting in the year denoted first in the first column. Mean, median, and SD refer to the sample average, median, and standard deviation of the factors, respectively. AR(1) indicates the first-order autocorrelation. P10 and P90 refer to the 10% and 90% percentiles, respectively. Table 2 Summary statistics A. Market factors Variable Mean Median SD AR(1) P10 P90 dVVIX 0.00011 −0.00350 0.0471 −0.0838 −0.0471 0.0511 dVIX 0.00001 −0.00095 0.0192 −0.1284 −0.0170 0.0175 Straddle vol −0.00014 0.00109 0.0158 −0.0408 −0.0166 0.0156 dSkew −0.00228 −0.00351 0.0790 0.2265 −0.0969 0.0896 dKurt 0.00109 −0.00232 0.1205 0.1339 −0.1369 0.1454 Jump −0.00171 −0.00962 0.0557 −0.0953 −0.0447 0.0428 dVRP 0.00000 −0.00002 0.0086 −0.0379 −0.0039 0.0037 MKT 0.00033 0.00080 0.0129 −0.0841 −0.0130 0.0128 SMB 0.00004 0.00010 0.0059 −0.0430 −0.0069 0.0067 HML −0.00003 −0.00020 0.0069 −0.0251 −0.0063 0.0061 Momentum −0.00001 0.00050 0.0104 0.1214 −0.0102 0.0099 SMBHXZ 0.00013 0.00004 0.0062 0.0057 −0.0072 0.0068 IA 0.00017 0.00008 0.0042 0.0541 −0.0043 0.0049 ROE 0.00040 0.00038 0.0044 0.0191 −0.0043 0.0054 dVVIXARMA 0.00127 −0.00226 0.0498 0.0005 −0.0481 0.0540 dVoVIX 0.00022 0.00503 0.0421 0.1784 −0.0338 0.0295 A. Market factors Variable Mean Median SD AR(1) P10 P90 dVVIX 0.00011 −0.00350 0.0471 −0.0838 −0.0471 0.0511 dVIX 0.00001 −0.00095 0.0192 −0.1284 −0.0170 0.0175 Straddle vol −0.00014 0.00109 0.0158 −0.0408 −0.0166 0.0156 dSkew −0.00228 −0.00351 0.0790 0.2265 −0.0969 0.0896 dKurt 0.00109 −0.00232 0.1205 0.1339 −0.1369 0.1454 Jump −0.00171 −0.00962 0.0557 −0.0953 −0.0447 0.0428 dVRP 0.00000 −0.00002 0.0086 −0.0379 −0.0039 0.0037 MKT 0.00033 0.00080 0.0129 −0.0841 −0.0130 0.0128 SMB 0.00004 0.00010 0.0059 −0.0430 −0.0069 0.0067 HML −0.00003 −0.00020 0.0069 −0.0251 −0.0063 0.0061 Momentum −0.00001 0.00050 0.0104 0.1214 −0.0102 0.0099 SMBHXZ 0.00013 0.00004 0.0062 0.0057 −0.0072 0.0068 IA 0.00017 0.00008 0.0042 0.0541 −0.0043 0.0049 ROE 0.00040 0.00038 0.0044 0.0191 −0.0043 0.0054 dVVIXARMA 0.00127 −0.00226 0.0498 0.0005 −0.0481 0.0540 dVoVIX 0.00022 0.00503 0.0421 0.1784 −0.0338 0.0295 B. VVIX summary statistics Year Mean Median SD P10 P90 2006 0.8024 0.8073 0.1126 0.6601 0.9467 2007 0.8768 0.8618 0.1331 0.7194 1.0469 2008 0.8185 0.7741 0.1560 0.6763 1.1088 2009 0.7978 0.7913 0.0863 0.6954 0.9225 2010 0.8836 0.8622 0.1307 0.7538 1.0346 2011 0.9294 0.9146 0.1021 0.8182 1.0491 2012 0.9484 0.9384 0.0838 0.8416 1.0740 2013 0.8052 0.7960 0.0897 0.6967 0.9203 2014 0.8301 0.7991 0.1433 0.6771 0.9990 2015 0.9482 0.9041 0.1475 0.8002 1.1533 2016 0.9258 0.8981 0.1021 0.8260 1.0949 Total 0.8696 0.8523 0.1332 0.7100 1.0450 B. VVIX summary statistics Year Mean Median SD P10 P90 2006 0.8024 0.8073 0.1126 0.6601 0.9467 2007 0.8768 0.8618 0.1331 0.7194 1.0469 2008 0.8185 0.7741 0.1560 0.6763 1.1088 2009 0.7978 0.7913 0.0863 0.6954 0.9225 2010 0.8836 0.8622 0.1307 0.7538 1.0346 2011 0.9294 0.9146 0.1021 0.8182 1.0491 2012 0.9484 0.9384 0.0838 0.8416 1.0740 2013 0.8052 0.7960 0.0897 0.6967 0.9203 2014 0.8301 0.7991 0.1433 0.6771 0.9990 2015 0.9482 0.9041 0.1475 0.8002 1.1533 2016 0.9258 0.8981 0.1021 0.8260 1.0949 Total 0.8696 0.8523 0.1332 0.7100 1.0450 C. βQsummary statistics Year Mean Median SD P10 P90 2006–2007 0.0093 0.0074 0.0502 −0.0425 0.0650 2007–2008 0.0085 0.0072 0.0730 −0.0693 0.0904 2008–2009 0.0433 0.0367 0.1050 −0.0619 0.1585 2009–2010 0.0066 0.0053 0.0687 −0.0593 0.0751 2010–2011 0.0105 0.0093 0.0521 −0.0443 0.0675 2011–2012 0.0131 0.0118 0.0559 −0.0455 0.0766 2012–2013 0.0037 0.0018 0.0627 −0.0617 0.0729 2013–2014 −0.0054 −0.0040 0.0648 −0.0724 0.0581 2014–2015 −0.0057 −0.0043 0.0590 −0.0631 0.0489 2015–2016 −0.0060 −0.0073 0.0626 −0.0654 0.0559 Total 0.0078 0.0049 0.0684 −0.0589 0.0789 C. βQsummary statistics Year Mean Median SD P10 P90 2006–2007 0.0093 0.0074 0.0502 −0.0425 0.0650 2007–2008 0.0085 0.0072 0.0730 −0.0693 0.0904 2008–2009 0.0433 0.0367 0.1050 −0.0619 0.1585 2009–2010 0.0066 0.0053 0.0687 −0.0593 0.0751 2010–2011 0.0105 0.0093 0.0521 −0.0443 0.0675 2011–2012 0.0131 0.0118 0.0559 −0.0455 0.0766 2012–2013 0.0037 0.0018 0.0627 −0.0617 0.0729 2013–2014 −0.0054 −0.0040 0.0648 −0.0724 0.0581 2014–2015 −0.0057 −0.0043 0.0590 −0.0631 0.0489 2015–2016 −0.0060 −0.0073 0.0626 −0.0654 0.0559 Total 0.0078 0.0049 0.0684 −0.0589 0.0789 Panel A of this table presents summary statistics on the aggregate factors dVVIX, dVIX, straddle vol, dSkew, dKurt, jump, dVRP, MKT, SMB, HML, momentum, SMBHXZ, IA, ROE, dVVIXARMA ⁠, and dVoVIX. Appendix A provides detailed variable definitions. Panel B provides yearly summary statistics on the VVIX, and panel C shows yearly summary statistics on the individual stocks’ sensitivities to aggregate volatility-of-volatility, βQ ⁠, with the sensitivity estimation starting in the year denoted first in the first column. Mean, median, and SD refer to the sample average, median, and standard deviation of the factors, respectively. AR(1) indicates the first-order autocorrelation. P10 and P90 refer to the 10% and 90% percentiles, respectively. Panels B and C of Table 2 present yearly summary statistics on the VVIX and βj,tQ factors of individual stocks, respectively. It is quite interesting to observe that the yearly average level of the VVIX is smallest in the crisis year 2009. Thus, it seems that investors largely agreed about the presumably high level of risk during that period. In the years 2011, 2012, 2015, and 2016 the average volatility-of-volatility is substantially higher, with values above 0.9. In 2013, there is a sharp decrease, almost returning to the 2009 level. The mean and median sensitivities of individual stocks to innovations in aggregate volatility-of-volatility, βj,tQ ⁠, presented in panel C, are mostly close to zero. The highest average βj,tQ and average cross-sectional standard deviation results for rolling annual estimation periods starting in 2008. The time-series of the VIX and VVIX are plotted in Figure 1. We find that the level of the VVIX is higher than that of the VIX throughout the entire sample period. The averages of the VVIX and VIX are 0.87 and 0.20, respectively. Both the S&P 500 and the VIX are directly tradable, for example, using futures on the respective indexes. Thus, the underlyings of the option-implied volatilities are comparable and our results indicate that the risk-neutral expectation of the variance of the returns on the VIX is substantially higher than that of the returns on the S&P 500. Furthermore, the VVIX itself is by far more volatile and less persistent compared to the VIX. The VVIX exhibits pronounced spikes that correspond with news which increase economic uncertainty. For these news, typically the spikes in the VIX are, by far, less pronounced or are not perceptible. These news events include, for example, the Bear Sterns Hedge Funds Collapse (August 2007), the Lehman Brothers bankruptcy (September 2008), the Freddie Mac and Fannie Mae crisis (May 2010), and the near-collapse of the Russian ruble (December 2014). Consequently, this stylized evidence provides additional insights about the notion that VVIX is a proxy for economic uncertainty, distinct from volatility. Figure 1 View largeDownload slide Time series of VIX and VVIX This figure plots the time-series of the VIX (dashed, gray), and the VVIX (solid, black) during the sample period February 24, 2006 until September 30, 2016. The shaded area indicates the time period marked as business-cycle contraction by the NBER. Figure 1 View largeDownload slide Time series of VIX and VVIX This figure plots the time-series of the VIX (dashed, gray), and the VVIX (solid, black) during the sample period February 24, 2006 until September 30, 2016. The shaded area indicates the time period marked as business-cycle contraction by the NBER. One may wonder whether the VVIX helps predict future volatility and volatility-of-volatility. To analyze this, we estimate predictive regressions for the realized volatility of the S&P 500 and the VIX. We use the end-of-month observations of the VIX and VVIX to predict the realized volatility, based on 5-minute returns, over the subsequent month. We present the results in Table 3. We find that, univariately, both the VIX and the VVIX significantly predict future realized variance. However, in a multivariate predictive regression, only the VIX significantly predicts the future realized volatility of the market index. On the other hand, we find that only the VVIX positively predicts the future realized volatility-of-volatility, in both a univariate and a multivariate framework. These results are consistent with those in Huang and Shaliastovich (2014) for a shorter sample period. Table 3 Predictability of the VIX and VVIX Volatility forecasting Volatility-of-volatility forecasting Constant −0.0190 −0.0173 −0.0185 0.3035*** 0.0215 0.0217 (SE) (0.040) (0.076) (0.040) (0.070) (0.046) (0.046) Slope VIX 0.8847*** 0.8850*** −0.0139 −0.1753* (SE) (0.047) (0.124) (0.084) (0.096) Slope VVIX 0.2068** −0.0007 0.3238*** 0.3649*** (SE) (0.094) (0.050) (0.054) (0.059) Adj. R2 0.583 0.053 0.580 −0.008 0.135 0.148 Volatility forecasting Volatility-of-volatility forecasting Constant −0.0190 −0.0173 −0.0185 0.3035*** 0.0215 0.0217 (SE) (0.040) (0.076) (0.040) (0.070) (0.046) (0.046) Slope VIX 0.8847*** 0.8850*** −0.0139 −0.1753* (SE) (0.047) (0.124) (0.084) (0.096) Slope VVIX 0.2068** −0.0007 0.3238*** 0.3649*** (SE) (0.094) (0.050) (0.054) (0.059) Adj. R2 0.583 0.053 0.580 −0.008 0.135 0.148 This table presents the results of monthly nonoverlapping predictive regressions for the realized variance of the S&P 500 and the VIX over the following month. Constant refers to the regression intercept, whereas slope VIX and slope VVIX denote the regression slope coefficients on the VIX and VVIX, respectively. Adj. R2 denotes the adjusted R2 of the regression. Robust Newey and West (1987) standard errors (SE) using four lags are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 3 Predictability of the VIX and VVIX Volatility forecasting Volatility-of-volatility forecasting Constant −0.0190 −0.0173 −0.0185 0.3035*** 0.0215 0.0217 (SE) (0.040) (0.076) (0.040) (0.070) (0.046) (0.046) Slope VIX 0.8847*** 0.8850*** −0.0139 −0.1753* (SE) (0.047) (0.124) (0.084) (0.096) Slope VVIX 0.2068** −0.0007 0.3238*** 0.3649*** (SE) (0.094) (0.050) (0.054) (0.059) Adj. R2 0.583 0.053 0.580 −0.008 0.135 0.148 Volatility forecasting Volatility-of-volatility forecasting Constant −0.0190 −0.0173 −0.0185 0.3035*** 0.0215 0.0217 (SE) (0.040) (0.076) (0.040) (0.070) (0.046) (0.046) Slope VIX 0.8847*** 0.8850*** −0.0139 −0.1753* (SE) (0.047) (0.124) (0.084) (0.096) Slope VVIX 0.2068** −0.0007 0.3238*** 0.3649*** (SE) (0.094) (0.050) (0.054) (0.059) Adj. R2 0.583 0.053 0.580 −0.008 0.135 0.148 This table presents the results of monthly nonoverlapping predictive regressions for the realized variance of the S&P 500 and the VIX over the following month. Constant refers to the regression intercept, whereas slope VIX and slope VVIX denote the regression slope coefficients on the VIX and VVIX, respectively. Adj. R2 denotes the adjusted R2 of the regression. Robust Newey and West (1987) standard errors (SE) using four lags are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Overall, the results indicate that aggregate volatility-of-volatility, measured by the VVIX, and aggregate volatility, measured by the VIX, have differential dynamics and are only weakly related. 3.2 Portfolio sorts To test whether aggregate volatility-of-volatility is priced in the stock market, we first perform portfolio sorts. At the beginning of each month, we sort the stocks in ascending order with respect to their sensitivities to innovations in volatility-of-volatility (⁠ βj,tQ ⁠) over the following year. We form quintile portfolios, so that quintile 1 contains the stocks with the lowest exposure to aggregate volatility-of-volatility, whereas quintile 5 contains those stocks with the highest volatility-of-volatility factor loadings. The hedge portfolio (5 minus 1) buys the quintile of stocks with the highest exposure and simultaneously sells the stocks in the quintile with the lowest exposure to aggregate volatility-of-volatility. The portfolio sorting approach maximizes the spread in the exposure to aggregate volatility-of-volatility and, thus, differences in average returns can be quite accurately attributed to differences in the sorting variable. Fama and French (2008) point out that by building value-weighted portfolios the hedge portfolio can be dominated by few big stocks, whereas for equally weighted portfolios the hedge portfolio can be dominated by micro caps. To address this issue, we analyze both value-weighted and equally weighted portfolio sorts. For the portfolio sorts, within each quintile, we weight the stocks by their relative market value at the beginning of the estimation period for βj,tQ ⁠. While our research design involves successive 12-month periods employing partly overlapping information, it introduces moving average effects. To account for that, in all analyses, we adjust the standard errors following Newey and West (1987) using 12 lags.13 Table 4 reports various summary statistics for the quintile portfolios sorted by contemporaneous volatility-of-volatility sensitivities. We find that the average annual raw return obeys a nearly strictly monotonically decreasing pattern, from 10.0% and 10.6% in quintiles 1 and 2 to 5.1% in quintile 5. Consequently, the better the stocks insure against positive shocks to aggregate volatility-of-volatility, the lower are their returns, on average. The difference in raw returns of –4.91% between quintiles 5 and 1 is statistically significant with a p-value of 2.7%. Looking at the line labeled CAPM alpha, which reports the results when controlling for systematic risk, we find a similar abnormal return for the 5 minus 1 portfolio of –4.87%, which is also significant at 5%. Controlling for the Carhart (1997) factors (four-factor) and additionally including the Pastor and Stambaugh (2003) factor (five-factor) leads to alphas of –4.91% and –4.62% per year, both significant at 5%. The alpha relative to the new factor model by Hou, Xue, and Zhang (2015b) amounts to –6.35%, which is also significant at 5%.14 Interestingly, the alpha relative to the model of Hou, Xue, and Zhang (2015b) is larger in magnitude than that for the classical four-factor model. This results because the hedge portfolio has a positive exposure on the investment and profitability factors, which earn a positive risk premium, on average, and also a positive exposure on the value factor, which earns a negative risk premium during our sample period. Once we (dis-)account for these exposures, the abnormal return gets even more negative. Table 4 Portfolios sorted by exposures to aggregate volatility-of-volatility Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1004 0.1062** 0.0987* 0.0826 0.0514 −0.0491** (0.156) (0.049) (0.069) (0.141) (0.460) (0.027) CAPM alpha −0.0106 0.0201 0.0123 −0.0064 −0.0593** −0.0487** (0.699) (0.113) (0.292) (0.629) (0.014) (0.021) 4-factor alpha −0.0072 0.0189*** 0.0147*** −0.0052* −0.0563*** −0.0491*** (0.504) (0.000) (0.000) (0.081) (0.000) (0.004) 5-factor alpha −0.0023 0.0194*** 0.0137*** −0.0049 −0.0485*** −0.0462** (0.840) (0.000) (0.000) (0.120) (0.000) (0.022) HXZ alpha 0.0103 0.0264*** 0.0153*** −0.0074 −0.0532*** −0.0635** (0.457) (0.000) (0.004) (0.305) (0.000) (0.015) Value-weighted Mean return 0.0967 0.0951** 0.0914** 0.0652 0.0433 −0.0533* (0.111) (0.047) (0.044) (0.198) (0.518) (0.089) CAPM alpha −0.0003 0.0193*** 0.0185*** −0.0140 −0.0629*** −0.0626* (0.985) (0.000) (0.000) (0.121) (0.007) (0.056) 4-factor alpha −0.0039 0.0161*** 0.0195*** −0.0091 −0.0507*** −0.0468* (0.755) (0.001) (0.000) (0.196) (0.004) (0.075) 5-factor alpha 0.0063 0.0194*** 0.0144*** −0.0126* −0.0508** −0.0572* (0.553) (0.000) (0.000) (0.058) (0.015) (0.057) HXZ alpha 0.0198 0.0204*** 0.0173*** −0.0148 −0.0657*** −0.0855** (0.199) (0.000) (0.000) (0.264) (0.009) (0.029) Factor loadings βM 1.0708*** 0.9253*** 0.9729*** 1.0989*** 1.3848*** 0.3140*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) βV 0.1441*** 0.0395*** −0.0041 −0.0401*** −0.0907*** −0.2349*** (0.000) (0.000) (0.150) (0.000) (0.000) (0.000) βQ −0.0599*** −0.0177*** 0.0062* 0.0313*** 0.0777*** 0.1376*** (0.000) (0.000) (0.057) (0.000) (0.000) (0.000) Return characteristics SD 0.0601 0.0474 0.0450 0.0504 0.0668 0.0311 Skewness −0.1632 −0.4981 −0.8644 −0.8132 −0.0727 0.0175 Kurtosis 4.0501 3.6484 3.9630 3.6487 3.8287 2.3774 Stock characteristics Mkt. share 0.2042 0.2467 0.2209 0.2034 0.1248 −0.0794 log(Size) 16.863 17.444 17.330 17.087 16.314 −0.5490 Book-to-market 0.5416 0.4850 0.4805 0.5196 0.6716 0.1299 Bid-ask spread 0.0008 0.0006 0.0006 0.0006 0.0009 0.0002 Age 33.491 41.099 38.442 35.074 28.564 −4.9267 Leverage 0.5567 0.5549 0.5622 0.5861 0.6015 0.0448 Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1004 0.1062** 0.0987* 0.0826 0.0514 −0.0491** (0.156) (0.049) (0.069) (0.141) (0.460) (0.027) CAPM alpha −0.0106 0.0201 0.0123 −0.0064 −0.0593** −0.0487** (0.699) (0.113) (0.292) (0.629) (0.014) (0.021) 4-factor alpha −0.0072 0.0189*** 0.0147*** −0.0052* −0.0563*** −0.0491*** (0.504) (0.000) (0.000) (0.081) (0.000) (0.004) 5-factor alpha −0.0023 0.0194*** 0.0137*** −0.0049 −0.0485*** −0.0462** (0.840) (0.000) (0.000) (0.120) (0.000) (0.022) HXZ alpha 0.0103 0.0264*** 0.0153*** −0.0074 −0.0532*** −0.0635** (0.457) (0.000) (0.004) (0.305) (0.000) (0.015) Value-weighted Mean return 0.0967 0.0951** 0.0914** 0.0652 0.0433 −0.0533* (0.111) (0.047) (0.044) (0.198) (0.518) (0.089) CAPM alpha −0.0003 0.0193*** 0.0185*** −0.0140 −0.0629*** −0.0626* (0.985) (0.000) (0.000) (0.121) (0.007) (0.056) 4-factor alpha −0.0039 0.0161*** 0.0195*** −0.0091 −0.0507*** −0.0468* (0.755) (0.001) (0.000) (0.196) (0.004) (0.075) 5-factor alpha 0.0063 0.0194*** 0.0144*** −0.0126* −0.0508** −0.0572* (0.553) (0.000) (0.000) (0.058) (0.015) (0.057) HXZ alpha 0.0198 0.0204*** 0.0173*** −0.0148 −0.0657*** −0.0855** (0.199) (0.000) (0.000) (0.264) (0.009) (0.029) Factor loadings βM 1.0708*** 0.9253*** 0.9729*** 1.0989*** 1.3848*** 0.3140*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) βV 0.1441*** 0.0395*** −0.0041 −0.0401*** −0.0907*** −0.2349*** (0.000) (0.000) (0.150) (0.000) (0.000) (0.000) βQ −0.0599*** −0.0177*** 0.0062* 0.0313*** 0.0777*** 0.1376*** (0.000) (0.000) (0.057) (0.000) (0.000) (0.000) Return characteristics SD 0.0601 0.0474 0.0450 0.0504 0.0668 0.0311 Skewness −0.1632 −0.4981 −0.8644 −0.8132 −0.0727 0.0175 Kurtosis 4.0501 3.6484 3.9630 3.6487 3.8287 2.3774 Stock characteristics Mkt. share 0.2042 0.2467 0.2209 0.2034 0.1248 −0.0794 log(Size) 16.863 17.444 17.330 17.087 16.314 −0.5490 Book-to-market 0.5416 0.4850 0.4805 0.5196 0.6716 0.1299 Bid-ask spread 0.0008 0.0006 0.0006 0.0006 0.0009 0.0002 Age 33.491 41.099 38.442 35.074 28.564 −4.9267 Leverage 0.5567 0.5549 0.5622 0.5861 0.6015 0.0448 At the beginning of each month, we form quintile portfolios based on the stocks’ sensitivities to innovations in aggregate volatility-of-volatility (⁠ βj,tQ ⁠) over the following year. To obtain the sensitivities, we regress daily excess stock returns on dVVIX, controlling for MKT and dVIX like in Equation (14). Stocks with the lowest βj,tQ are sorted into portfolio 1, those with the highest βj,tQ into portfolio 5. The column labeled 5 minus 1 refers to the hedge portfolio buying the quintile of stocks with the highest βj,tQ and simultaneously selling the stocks in the quintile with the lowest βj,tQ ⁠. Each month, we set up new 12-month portfolios. The row labeled Mean return is based on simple returns. CAPM alpha, four-factor alpha, and five-factor alpha refer to the alphas of the CAPM, the Carhart (1997) four-factor, and the five-factor models (including Pastor and Stambaugh 2003 liquidity), respectively. HXZ alpha denotes the alpha relative to the Hou, Xue, and Zhang (2015b) factor model. The segment Factor loadings denotes the value-weighted average annual factor loadings, where βM, βV ⁠, and βQ refer to the factor loadings on the market factor, dVIX, and dVVIX, respectively. The segment Stock characteristics presents average (value-weighted) portfolio characteristics with Mkt. share denoting the average market share of the portfolios. Appendix A provides detailed variable definitions. Robust Newey and West (1987)p-values using 12 lags are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 4 Portfolios sorted by exposures to aggregate volatility-of-volatility Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1004 0.1062** 0.0987* 0.0826 0.0514 −0.0491** (0.156) (0.049) (0.069) (0.141) (0.460) (0.027) CAPM alpha −0.0106 0.0201 0.0123 −0.0064 −0.0593** −0.0487** (0.699) (0.113) (0.292) (0.629) (0.014) (0.021) 4-factor alpha −0.0072 0.0189*** 0.0147*** −0.0052* −0.0563*** −0.0491*** (0.504) (0.000) (0.000) (0.081) (0.000) (0.004) 5-factor alpha −0.0023 0.0194*** 0.0137*** −0.0049 −0.0485*** −0.0462** (0.840) (0.000) (0.000) (0.120) (0.000) (0.022) HXZ alpha 0.0103 0.0264*** 0.0153*** −0.0074 −0.0532*** −0.0635** (0.457) (0.000) (0.004) (0.305) (0.000) (0.015) Value-weighted Mean return 0.0967 0.0951** 0.0914** 0.0652 0.0433 −0.0533* (0.111) (0.047) (0.044) (0.198) (0.518) (0.089) CAPM alpha −0.0003 0.0193*** 0.0185*** −0.0140 −0.0629*** −0.0626* (0.985) (0.000) (0.000) (0.121) (0.007) (0.056) 4-factor alpha −0.0039 0.0161*** 0.0195*** −0.0091 −0.0507*** −0.0468* (0.755) (0.001) (0.000) (0.196) (0.004) (0.075) 5-factor alpha 0.0063 0.0194*** 0.0144*** −0.0126* −0.0508** −0.0572* (0.553) (0.000) (0.000) (0.058) (0.015) (0.057) HXZ alpha 0.0198 0.0204*** 0.0173*** −0.0148 −0.0657*** −0.0855** (0.199) (0.000) (0.000) (0.264) (0.009) (0.029) Factor loadings βM 1.0708*** 0.9253*** 0.9729*** 1.0989*** 1.3848*** 0.3140*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) βV 0.1441*** 0.0395*** −0.0041 −0.0401*** −0.0907*** −0.2349*** (0.000) (0.000) (0.150) (0.000) (0.000) (0.000) βQ −0.0599*** −0.0177*** 0.0062* 0.0313*** 0.0777*** 0.1376*** (0.000) (0.000) (0.057) (0.000) (0.000) (0.000) Return characteristics SD 0.0601 0.0474 0.0450 0.0504 0.0668 0.0311 Skewness −0.1632 −0.4981 −0.8644 −0.8132 −0.0727 0.0175 Kurtosis 4.0501 3.6484 3.9630 3.6487 3.8287 2.3774 Stock characteristics Mkt. share 0.2042 0.2467 0.2209 0.2034 0.1248 −0.0794 log(Size) 16.863 17.444 17.330 17.087 16.314 −0.5490 Book-to-market 0.5416 0.4850 0.4805 0.5196 0.6716 0.1299 Bid-ask spread 0.0008 0.0006 0.0006 0.0006 0.0009 0.0002 Age 33.491 41.099 38.442 35.074 28.564 −4.9267 Leverage 0.5567 0.5549 0.5622 0.5861 0.6015 0.0448 Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1004 0.1062** 0.0987* 0.0826 0.0514 −0.0491** (0.156) (0.049) (0.069) (0.141) (0.460) (0.027) CAPM alpha −0.0106 0.0201 0.0123 −0.0064 −0.0593** −0.0487** (0.699) (0.113) (0.292) (0.629) (0.014) (0.021) 4-factor alpha −0.0072 0.0189*** 0.0147*** −0.0052* −0.0563*** −0.0491*** (0.504) (0.000) (0.000) (0.081) (0.000) (0.004) 5-factor alpha −0.0023 0.0194*** 0.0137*** −0.0049 −0.0485*** −0.0462** (0.840) (0.000) (0.000) (0.120) (0.000) (0.022) HXZ alpha 0.0103 0.0264*** 0.0153*** −0.0074 −0.0532*** −0.0635** (0.457) (0.000) (0.004) (0.305) (0.000) (0.015) Value-weighted Mean return 0.0967 0.0951** 0.0914** 0.0652 0.0433 −0.0533* (0.111) (0.047) (0.044) (0.198) (0.518) (0.089) CAPM alpha −0.0003 0.0193*** 0.0185*** −0.0140 −0.0629*** −0.0626* (0.985) (0.000) (0.000) (0.121) (0.007) (0.056) 4-factor alpha −0.0039 0.0161*** 0.0195*** −0.0091 −0.0507*** −0.0468* (0.755) (0.001) (0.000) (0.196) (0.004) (0.075) 5-factor alpha 0.0063 0.0194*** 0.0144*** −0.0126* −0.0508** −0.0572* (0.553) (0.000) (0.000) (0.058) (0.015) (0.057) HXZ alpha 0.0198 0.0204*** 0.0173*** −0.0148 −0.0657*** −0.0855** (0.199) (0.000) (0.000) (0.264) (0.009) (0.029) Factor loadings βM 1.0708*** 0.9253*** 0.9729*** 1.0989*** 1.3848*** 0.3140*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) βV 0.1441*** 0.0395*** −0.0041 −0.0401*** −0.0907*** −0.2349*** (0.000) (0.000) (0.150) (0.000) (0.000) (0.000) βQ −0.0599*** −0.0177*** 0.0062* 0.0313*** 0.0777*** 0.1376*** (0.000) (0.000) (0.057) (0.000) (0.000) (0.000) Return characteristics SD 0.0601 0.0474 0.0450 0.0504 0.0668 0.0311 Skewness −0.1632 −0.4981 −0.8644 −0.8132 −0.0727 0.0175 Kurtosis 4.0501 3.6484 3.9630 3.6487 3.8287 2.3774 Stock characteristics Mkt. share 0.2042 0.2467 0.2209 0.2034 0.1248 −0.0794 log(Size) 16.863 17.444 17.330 17.087 16.314 −0.5490 Book-to-market 0.5416 0.4850 0.4805 0.5196 0.6716 0.1299 Bid-ask spread 0.0008 0.0006 0.0006 0.0006 0.0009 0.0002 Age 33.491 41.099 38.442 35.074 28.564 −4.9267 Leverage 0.5567 0.5549 0.5622 0.5861 0.6015 0.0448 At the beginning of each month, we form quintile portfolios based on the stocks’ sensitivities to innovations in aggregate volatility-of-volatility (⁠ βj,tQ ⁠) over the following year. To obtain the sensitivities, we regress daily excess stock returns on dVVIX, controlling for MKT and dVIX like in Equation (14). Stocks with the lowest βj,tQ are sorted into portfolio 1, those with the highest βj,tQ into portfolio 5. The column labeled 5 minus 1 refers to the hedge portfolio buying the quintile of stocks with the highest βj,tQ and simultaneously selling the stocks in the quintile with the lowest βj,tQ ⁠. Each month, we set up new 12-month portfolios. The row labeled Mean return is based on simple returns. CAPM alpha, four-factor alpha, and five-factor alpha refer to the alphas of the CAPM, the Carhart (1997) four-factor, and the five-factor models (including Pastor and Stambaugh 2003 liquidity), respectively. HXZ alpha denotes the alpha relative to the Hou, Xue, and Zhang (2015b) factor model. The segment Factor loadings denotes the value-weighted average annual factor loadings, where βM, βV ⁠, and βQ refer to the factor loadings on the market factor, dVIX, and dVVIX, respectively. The segment Stock characteristics presents average (value-weighted) portfolio characteristics with Mkt. share denoting the average market share of the portfolios. Appendix A provides detailed variable definitions. Robust Newey and West (1987)p-values using 12 lags are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Consequently, accounting for systematic risk factors, the 5 minus 1 portfolio is expected to earn substantially higher returns than realized. This effect is mostly driven by the portfolio of the stocks with the highest sensitivities to innovations in aggregate volatility-of-volatility. For each factor model specification, the alpha of this portfolio is strongly significantly negative, whereas that of portfolio 1 is close to zero. Hence, investors are willing to pay a premium in order to hold stocks with high sensitivities to innovations in aggregate volatility-of-volatility, whereas the stocks with the lowest sensitivities do not earn significant positive abnormal returns. We obtain similar results for value-weighted returns. The return and the alpha on the 5 minus 1 portfolio are typically even larger than for the equally weighted approach, although the significance of the abnormal returns of the hedge portfolio is typically slightly weaker. In the remainder of Table 4, we examine value-weighted characteristics of the portfolios. The 5 minus 1 portfolio has a significantly positive exposure to the market excess return. The average market beta of the hedge portfolio of 0.31 would, following the logic of the CAPM, predict a substantially positive excess return, whereas the return that is realized is significantly negative. Interestingly, the hedge portfolio has a significantly negative exposure to aggregate volatility. Given the positive correlations of the innovations in the VIX and VVIX, one might expect the opposite. The exposure to aggregate volatility-of-volatility is monotonically increasing by construction. Furthermore, we find that the firms in portfolio 5, on average, are smaller than those in portfolio 1, have higher book-to-market ratios, are younger, and have slightly higher bid-ask spreads and leverage. Another potentially interesting issue relates to the time-series of excess returns of the 5 minus 1 hedge portfolio, presented in Figure 2. First, this analysis shows whether the cumulative returns are generated by a certain time period. Second, the plot also shows the periods in which the insurance strategy pays off. To obtain a time-series of monthly returns, we average the returns of all active portfolios during that month.15 We present the results for both value- and equally weighted portfolios in Figure 2. Over the entire sample period, the cumulative excess return is roughly –37% for the value-weighted and –32% for the equally weighted portfolios. The trend is in general negative but there are also periods for which the insurance strategy pays off and the 5 minus 1 portfolio yields large positive returns. These times roughly correspond with the first half of 2008, where the National Bureau of Economic Research (NBER) indicates a recession, as well as short time periods in the beginning of 2009, in the beginning of 2012, in the second half of 2014, and in the beginning of 2016. These time periods typically correspond with positive innovations in and correspondingly high levels of the VVIX index. Figure 2 View largeDownload slide Time series of cumulative hedge portfolio returns This figure plots the cumulative returns of the 5-1 hedge portfolio. We present both value-weighted (dashed) and equally weighted (dashed-dotted) returns. Monthly returns are obtained by averaging the returns across all active portfolios. The shaded area indicates the time period marked as business-cycle contraction by the NBER. Figure 2 View largeDownload slide Time series of cumulative hedge portfolio returns This figure plots the cumulative returns of the 5-1 hedge portfolio. We present both value-weighted (dashed) and equally weighted (dashed-dotted) returns. Monthly returns are obtained by averaging the returns across all active portfolios. The shaded area indicates the time period marked as business-cycle contraction by the NBER. 3.3 Double sorts Given the close relation of aggregate volatility-of-volatility and aggregate volatility and also the market variance risk premium (as shown by Bollerslev, Tauchen, and Zhou 2009), we aim to further dissect the effects of these variables. To do that, we estimate the factor sensitivities in a joint regression and perform a double-sort.16 At the beginning of each month we independently sort all stocks into quintiles, using NYSE breakpoints, on the characteristic we want to control for as well as on their VVIX sensitivities (Novy-Marx 2013).17 This results in a total of 25 portfolios. Thus, we obtain quintile portfolios on the exposure to aggregate volatility-of-volatility controlling for another characteristic without making assumptions on the parametric form of the relationships. For each quintile of both sorting variables, we obtain a 5 minus 1 hedge portfolio by buying the final portfolio 5 and selling portfolio 1 in the quintile. We consider both equally and value-weighted portfolios. For value-weighted portfolios, within each of the 25 portfolios we weight the stocks by their relative market value at the beginning of the estimation period for the exposures to aggregate volatility-of-volatility. Table 5 reports all 25 portfolios resulting from the double sorts as well as the respective 5 minus 1 portfolios. We report Hou, Xue, and Zhang (2015b) factor alphas and robust Newey and West (1987)p-values in parentheses.18 Panels A and B report the results for equally and value-weighted double sorts with respect to volatility-of-volatility and volatility betas. According to the model’s result in Equation (13), there are three risk factors in the market. Stock returns depend on the stocks’ βj,tM, βj,tV ⁠, and βj,tQ ⁠. Hence, given that the price of risk on both volatility and volatility-of-volatility appears to be negative, stocks with low βj,tV and low βj,tQ should earn high average returns. For each quintile of βj,tV ⁠, stocks with higher βj,tQ should earn lower returns. Similarly, for each quintile of βj,tQ ⁠, stocks with higher βj,tV should earn lower returns. This is precisely the pattern we find in panels A and B of Table 5. Portfolios with both low βj,tV and low βj,tQ have positive alphas. Both with increasing volatility-betas and volatility-of-volatility betas, the alphas decrease. The largest negative alpha results for the portfolio of stocks with the highest values for both βj,tV and βj,tQ ⁠. For all volatility-of-volatility quintiles, we detect significantly negative alphas for the 5 minus 1 portfolios. The alphas range between –13.7% and –12.0% for equally weighted returns and are even larger in magnitude for value-weighted returns. Similarly, for all volatility quintiles, beside quintile 5 for value-weighted returns, we find significantly negative alphas for the 5 minus 1 portfolios. The alphas range between –11.0% and –8.0% for equally weighted returns and –12.2% and –6.4% for value-weighted returns. Consequently, as also indicated by the negative loading of the 5 minus 1 portfolio on aggregate volatility in Table 4, when controlling for aggregate volatility, the trade-off of volatility-of-volatility and returns is even more pronounced than for the univariate sorts. Table 5 Dissecting the effects of aggregate risk and aggregate volatility-of-volatility A. Aggregate volatility, equally weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.0741* 0.0716*** 0.0578*** 0.0277** −0.0497*** −0.1238*** (0.053) (0.000) (0.000) (0.036) (0.000) (0.000) 2 0.0777*** 0.0684*** 0.0591*** 0.0068 −0.0566*** −0.1343*** (0.000) (0.000) (0.000) (0.268) (0.000) (0.000) 3 0.0653*** 0.0443*** 0.0288*** 0.0041 −0.0699*** −0.1352*** (0.000) (0.000) (0.000) (0.618) (0.000) (0.000) 4 0.0454*** 0.0327*** 0.0111 −0.0307** −0.0920*** −0.1374*** (0.000) (0.000) (0.165) (0.016) (0.000) (0.000) 5 −0.0075 −0.0339*** −0.0523*** −0.0810*** −0.1277*** −0.1202*** (0.510) (0.001) (0.000) (0.001) (0.001) (0.004) 5 minus 1 −0.0817* −0.1055*** −0.1101*** −0.1086*** −0.0781* (0.058) (0.000) (0.000) (0.001) (0.060) A. Aggregate volatility, equally weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.0741* 0.0716*** 0.0578*** 0.0277** −0.0497*** −0.1238*** (0.053) (0.000) (0.000) (0.036) (0.000) (0.000) 2 0.0777*** 0.0684*** 0.0591*** 0.0068 −0.0566*** −0.1343*** (0.000) (0.000) (0.000) (0.268) (0.000) (0.000) 3 0.0653*** 0.0443*** 0.0288*** 0.0041 −0.0699*** −0.1352*** (0.000) (0.000) (0.000) (0.618) (0.000) (0.000) 4 0.0454*** 0.0327*** 0.0111 −0.0307** −0.0920*** −0.1374*** (0.000) (0.000) (0.165) (0.016) (0.000) (0.000) 5 −0.0075 −0.0339*** −0.0523*** −0.0810*** −0.1277*** −0.1202*** (0.510) (0.001) (0.000) (0.001) (0.001) (0.004) 5 minus 1 −0.0817* −0.1055*** −0.1101*** −0.1086*** −0.0781* (0.058) (0.000) (0.000) (0.001) (0.060) B. Aggregate volatility, value-weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.1007*** 0.0664*** 0.0394*** 0.0135 −0.0803*** −0.1810*** (0.001) (0.000) (0.006) (0.335) (0.000) (0.000) 2 0.0794*** 0.0574*** 0.0278*** −0.0250*** −0.0547*** −0.1341*** (0.000) (0.000) (0.000) (0.000) (0.001) (0.000) 3 0.0594*** 0.0315*** 0.0241*** −0.0118 −0.0978*** −0.1573*** (0.000) (0.000) (0.000) (0.192) (0.000) (0.000) 4 0.0360*** 0.0032 −0.0017 −0.0507** −0.1186*** −0.1546*** (0.007) (0.737) (0.912) (0.016) (0.000) (0.000) 5 −0.0217 −0.0305 −0.0769*** −0.0915*** −0.1444*** −0.1227*** (0.128) (0.220) (0.000) (0.005) (0.000) (0.002) 5 minus 1 −0.1224*** −0.0969** −0.1163*** −0.1051** −0.0641 (0.003) (0.018) (0.000) (0.012) (0.145) B. Aggregate volatility, value-weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.1007*** 0.0664*** 0.0394*** 0.0135 −0.0803*** −0.1810*** (0.001) (0.000) (0.006) (0.335) (0.000) (0.000) 2 0.0794*** 0.0574*** 0.0278*** −0.0250*** −0.0547*** −0.1341*** (0.000) (0.000) (0.000) (0.000) (0.001) (0.000) 3 0.0594*** 0.0315*** 0.0241*** −0.0118 −0.0978*** −0.1573*** (0.000) (0.000) (0.000) (0.192) (0.000) (0.000) 4 0.0360*** 0.0032 −0.0017 −0.0507** −0.1186*** −0.1546*** (0.007) (0.737) (0.912) (0.016) (0.000) (0.000) 5 −0.0217 −0.0305 −0.0769*** −0.0915*** −0.1444*** −0.1227*** (0.128) (0.220) (0.000) (0.005) (0.000) (0.002) 5 minus 1 −0.1224*** −0.0969** −0.1163*** −0.1051** −0.0641 (0.003) (0.018) (0.000) (0.012) (0.145) C. Market variance risk premium, equally weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0270 0.0312*** 0.0505*** 0.0703*** 0.0993*** 0.1262*** (0.128) (0.009) (0.000) (0.000) (0.000) (0.000) 2 −0.0361*** 0.0181*** 0.0420*** 0.0665*** 0.0573*** 0.0934*** (0.000) (0.003) (0.000) (0.000) (0.000) (0.000) 3 −0.0301*** 0.0024 0.0314*** 0.0356*** 0.0530*** 0.0831*** (0.003) (0.739) (0.000) (0.000) (0.000) (0.000) 4 −0.0606*** −0.0313*** 0.0014 0.0201*** 0.0281*** 0.0886*** (0.000) (0.004) (0.871) (0.004) (0.000) (0.000) 5 −0.1294*** −0.0721*** −0.0568*** −0.0473*** −0.0410*** 0.0884*** (0.000) (0.000) (0.004) (0.000) (0.001) (0.004) 5 minus 1 −0.1025*** −0.1033*** −0.1073*** −0.1176*** −0.1403*** (0.007) (0.000) (0.000) (0.000) (0.000) C. Market variance risk premium, equally weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0270 0.0312*** 0.0505*** 0.0703*** 0.0993*** 0.1262*** (0.128) (0.009) (0.000) (0.000) (0.000) (0.000) 2 −0.0361*** 0.0181*** 0.0420*** 0.0665*** 0.0573*** 0.0934*** (0.000) (0.003) (0.000) (0.000) (0.000) (0.000) 3 −0.0301*** 0.0024 0.0314*** 0.0356*** 0.0530*** 0.0831*** (0.003) (0.739) (0.000) (0.000) (0.000) (0.000) 4 −0.0606*** −0.0313*** 0.0014 0.0201*** 0.0281*** 0.0886*** (0.000) (0.004) (0.871) (0.004) (0.000) (0.000) 5 −0.1294*** −0.0721*** −0.0568*** −0.0473*** −0.0410*** 0.0884*** (0.000) (0.000) (0.004) (0.000) (0.001) (0.004) 5 minus 1 −0.1025*** −0.1033*** −0.1073*** −0.1176*** −0.1403*** (0.007) (0.000) (0.000) (0.000) (0.000) D. Market variance risk premium, value-weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0770*** 0.0047 0.0455*** 0.0641*** 0.1059*** 0.1829*** (0.000) (0.766) (0.000) (0.000) (0.000) (0.000) 2 −0.0603*** −0.0133 0.0146** 0.0440*** 0.0822*** 0.1424*** (0.000) (0.113) (0.016) (0.000) (0.000) (0.000) 3 −0.0785*** −0.0050 0.0237** 0.0220*** 0.0461*** 0.1245*** (0.000) (0.673) (0.012) (0.002) (0.000) (0.000) 4 −0.0890*** −0.0446** −0.0089 −0.0071 0.0298*** 0.1188*** (0.000) (0.031) (0.601) (0.587) (0.003) (0.000) 5 −0.1386*** −0.0977*** −0.0731** −0.0604** −0.0799*** 0.0587* (0.000) (0.000) (0.014) (0.010) (0.000) (0.084) 5 minus 1 −0.0616* −0.1025*** −0.1186*** −0.1245*** −0.1859*** (0.081) (0.006) (0.002) (0.000) (0.000) D. Market variance risk premium, value-weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0770*** 0.0047 0.0455*** 0.0641*** 0.1059*** 0.1829*** (0.000) (0.766) (0.000) (0.000) (0.000) (0.000) 2 −0.0603*** −0.0133 0.0146** 0.0440*** 0.0822*** 0.1424*** (0.000) (0.113) (0.016) (0.000) (0.000) (0.000) 3 −0.0785*** −0.0050 0.0237** 0.0220*** 0.0461*** 0.1245*** (0.000) (0.673) (0.012) (0.002) (0.000) (0.000) 4 −0.0890*** −0.0446** −0.0089 −0.0071 0.0298*** 0.1188*** (0.000) (0.031) (0.601) (0.587) (0.003) (0.000) 5 −0.1386*** −0.0977*** −0.0731** −0.0604** −0.0799*** 0.0587* (0.000) (0.000) (0.014) (0.010) (0.000) (0.084) 5 minus 1 −0.0616* −0.1025*** −0.1186*** −0.1245*** −0.1859*** (0.081) (0.006) (0.002) (0.000) (0.000) This table reports Hou, Xue, and Zhang (2015b) factor alphas for double-sorted portfolios. We obtain the factor sensitivities in a joint multivariate estimation. At the beginning of each month, we independently sort stocks into quintiles based on their sensitivity to aggregate volatility (panels A and B) and the market variance risk premium (panels C and D) as well as their volatility-of-volatility sensitivities (⁠ βj,tQ ⁠) using NYSE breakpoints. We report the alphas of all 25 portfolios as well as the respective long minus short (5 minus 1) portfolios. Each month, we set up new 12-month portfolios. Robust Newey and West (1987)p-values using 12 lags are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 5 Dissecting the effects of aggregate risk and aggregate volatility-of-volatility A. Aggregate volatility, equally weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.0741* 0.0716*** 0.0578*** 0.0277** −0.0497*** −0.1238*** (0.053) (0.000) (0.000) (0.036) (0.000) (0.000) 2 0.0777*** 0.0684*** 0.0591*** 0.0068 −0.0566*** −0.1343*** (0.000) (0.000) (0.000) (0.268) (0.000) (0.000) 3 0.0653*** 0.0443*** 0.0288*** 0.0041 −0.0699*** −0.1352*** (0.000) (0.000) (0.000) (0.618) (0.000) (0.000) 4 0.0454*** 0.0327*** 0.0111 −0.0307** −0.0920*** −0.1374*** (0.000) (0.000) (0.165) (0.016) (0.000) (0.000) 5 −0.0075 −0.0339*** −0.0523*** −0.0810*** −0.1277*** −0.1202*** (0.510) (0.001) (0.000) (0.001) (0.001) (0.004) 5 minus 1 −0.0817* −0.1055*** −0.1101*** −0.1086*** −0.0781* (0.058) (0.000) (0.000) (0.001) (0.060) A. Aggregate volatility, equally weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.0741* 0.0716*** 0.0578*** 0.0277** −0.0497*** −0.1238*** (0.053) (0.000) (0.000) (0.036) (0.000) (0.000) 2 0.0777*** 0.0684*** 0.0591*** 0.0068 −0.0566*** −0.1343*** (0.000) (0.000) (0.000) (0.268) (0.000) (0.000) 3 0.0653*** 0.0443*** 0.0288*** 0.0041 −0.0699*** −0.1352*** (0.000) (0.000) (0.000) (0.618) (0.000) (0.000) 4 0.0454*** 0.0327*** 0.0111 −0.0307** −0.0920*** −0.1374*** (0.000) (0.000) (0.165) (0.016) (0.000) (0.000) 5 −0.0075 −0.0339*** −0.0523*** −0.0810*** −0.1277*** −0.1202*** (0.510) (0.001) (0.000) (0.001) (0.001) (0.004) 5 minus 1 −0.0817* −0.1055*** −0.1101*** −0.1086*** −0.0781* (0.058) (0.000) (0.000) (0.001) (0.060) B. Aggregate volatility, value-weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.1007*** 0.0664*** 0.0394*** 0.0135 −0.0803*** −0.1810*** (0.001) (0.000) (0.006) (0.335) (0.000) (0.000) 2 0.0794*** 0.0574*** 0.0278*** −0.0250*** −0.0547*** −0.1341*** (0.000) (0.000) (0.000) (0.000) (0.001) (0.000) 3 0.0594*** 0.0315*** 0.0241*** −0.0118 −0.0978*** −0.1573*** (0.000) (0.000) (0.000) (0.192) (0.000) (0.000) 4 0.0360*** 0.0032 −0.0017 −0.0507** −0.1186*** −0.1546*** (0.007) (0.737) (0.912) (0.016) (0.000) (0.000) 5 −0.0217 −0.0305 −0.0769*** −0.0915*** −0.1444*** −0.1227*** (0.128) (0.220) (0.000) (0.005) (0.000) (0.002) 5 minus 1 −0.1224*** −0.0969** −0.1163*** −0.1051** −0.0641 (0.003) (0.018) (0.000) (0.012) (0.145) B. Aggregate volatility, value-weighted dVIX 1 2 3 4 5 5 minus 1 dVVIX 1 0.1007*** 0.0664*** 0.0394*** 0.0135 −0.0803*** −0.1810*** (0.001) (0.000) (0.006) (0.335) (0.000) (0.000) 2 0.0794*** 0.0574*** 0.0278*** −0.0250*** −0.0547*** −0.1341*** (0.000) (0.000) (0.000) (0.000) (0.001) (0.000) 3 0.0594*** 0.0315*** 0.0241*** −0.0118 −0.0978*** −0.1573*** (0.000) (0.000) (0.000) (0.192) (0.000) (0.000) 4 0.0360*** 0.0032 −0.0017 −0.0507** −0.1186*** −0.1546*** (0.007) (0.737) (0.912) (0.016) (0.000) (0.000) 5 −0.0217 −0.0305 −0.0769*** −0.0915*** −0.1444*** −0.1227*** (0.128) (0.220) (0.000) (0.005) (0.000) (0.002) 5 minus 1 −0.1224*** −0.0969** −0.1163*** −0.1051** −0.0641 (0.003) (0.018) (0.000) (0.012) (0.145) C. Market variance risk premium, equally weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0270 0.0312*** 0.0505*** 0.0703*** 0.0993*** 0.1262*** (0.128) (0.009) (0.000) (0.000) (0.000) (0.000) 2 −0.0361*** 0.0181*** 0.0420*** 0.0665*** 0.0573*** 0.0934*** (0.000) (0.003) (0.000) (0.000) (0.000) (0.000) 3 −0.0301*** 0.0024 0.0314*** 0.0356*** 0.0530*** 0.0831*** (0.003) (0.739) (0.000) (0.000) (0.000) (0.000) 4 −0.0606*** −0.0313*** 0.0014 0.0201*** 0.0281*** 0.0886*** (0.000) (0.004) (0.871) (0.004) (0.000) (0.000) 5 −0.1294*** −0.0721*** −0.0568*** −0.0473*** −0.0410*** 0.0884*** (0.000) (0.000) (0.004) (0.000) (0.001) (0.004) 5 minus 1 −0.1025*** −0.1033*** −0.1073*** −0.1176*** −0.1403*** (0.007) (0.000) (0.000) (0.000) (0.000) C. Market variance risk premium, equally weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0270 0.0312*** 0.0505*** 0.0703*** 0.0993*** 0.1262*** (0.128) (0.009) (0.000) (0.000) (0.000) (0.000) 2 −0.0361*** 0.0181*** 0.0420*** 0.0665*** 0.0573*** 0.0934*** (0.000) (0.003) (0.000) (0.000) (0.000) (0.000) 3 −0.0301*** 0.0024 0.0314*** 0.0356*** 0.0530*** 0.0831*** (0.003) (0.739) (0.000) (0.000) (0.000) (0.000) 4 −0.0606*** −0.0313*** 0.0014 0.0201*** 0.0281*** 0.0886*** (0.000) (0.004) (0.871) (0.004) (0.000) (0.000) 5 −0.1294*** −0.0721*** −0.0568*** −0.0473*** −0.0410*** 0.0884*** (0.000) (0.000) (0.004) (0.000) (0.001) (0.004) 5 minus 1 −0.1025*** −0.1033*** −0.1073*** −0.1176*** −0.1403*** (0.007) (0.000) (0.000) (0.000) (0.000) D. Market variance risk premium, value-weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0770*** 0.0047 0.0455*** 0.0641*** 0.1059*** 0.1829*** (0.000) (0.766) (0.000) (0.000) (0.000) (0.000) 2 −0.0603*** −0.0133 0.0146** 0.0440*** 0.0822*** 0.1424*** (0.000) (0.113) (0.016) (0.000) (0.000) (0.000) 3 −0.0785*** −0.0050 0.0237** 0.0220*** 0.0461*** 0.1245*** (0.000) (0.673) (0.012) (0.002) (0.000) (0.000) 4 −0.0890*** −0.0446** −0.0089 −0.0071 0.0298*** 0.1188*** (0.000) (0.031) (0.601) (0.587) (0.003) (0.000) 5 −0.1386*** −0.0977*** −0.0731** −0.0604** −0.0799*** 0.0587* (0.000) (0.000) (0.014) (0.010) (0.000) (0.084) 5 minus 1 −0.0616* −0.1025*** −0.1186*** −0.1245*** −0.1859*** (0.081) (0.006) (0.002) (0.000) (0.000) D. Market variance risk premium, value-weighted dVRP 1 2 3 4 5 5 minus 1 dVVIX 1 −0.0770*** 0.0047 0.0455*** 0.0641*** 0.1059*** 0.1829*** (0.000) (0.766) (0.000) (0.000) (0.000) (0.000) 2 −0.0603*** −0.0133 0.0146** 0.0440*** 0.0822*** 0.1424*** (0.000) (0.113) (0.016) (0.000) (0.000) (0.000) 3 −0.0785*** −0.0050 0.0237** 0.0220*** 0.0461*** 0.1245*** (0.000) (0.673) (0.012) (0.002) (0.000) (0.000) 4 −0.0890*** −0.0446** −0.0089 −0.0071 0.0298*** 0.1188*** (0.000) (0.031) (0.601) (0.587) (0.003) (0.000) 5 −0.1386*** −0.0977*** −0.0731** −0.0604** −0.0799*** 0.0587* (0.000) (0.000) (0.014) (0.010) (0.000) (0.084) 5 minus 1 −0.0616* −0.1025*** −0.1186*** −0.1245*** −0.1859*** (0.081) (0.006) (0.002) (0.000) (0.000) This table reports Hou, Xue, and Zhang (2015b) factor alphas for double-sorted portfolios. We obtain the factor sensitivities in a joint multivariate estimation. At the beginning of each month, we independently sort stocks into quintiles based on their sensitivity to aggregate volatility (panels A and B) and the market variance risk premium (panels C and D) as well as their volatility-of-volatility sensitivities (⁠ βj,tQ ⁠) using NYSE breakpoints. We report the alphas of all 25 portfolios as well as the respective long minus short (5 minus 1) portfolios. Each month, we set up new 12-month portfolios. Robust Newey and West (1987)p-values using 12 lags are reported in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Panels C and D of Table 5 report the results of double sorts of volatility-of-volatility betas and betas with respect to the variance risk premium. For all quintiles of volatility-of-volatility betas, we find positive 5 minus 1 portfolio alphas on the variance risk premium. More importantly, for all variance risk premium beta quintiles, we detect a significantly negative hedge portfolio alpha for volatility-of-volatility. Bollerslev, Tauchen, and Zhou (2009) find that the variance risk premium is associated with macroeconomic volatility-of-volatility. In light of these results, one may argue that our result that both aggregate volatility-of-volatility and the market variance risk premium are priced is surprising. However, estimating the variance risk premium faces the problem of specifying an expectation of the variance under P ⁠. Empirically, innovations in the market variance risk premium are mostly driven by innovations in the risk-neutral expectation. dVIX and dVRP are highly correlated, with –77%. Hence, empirically, innovations in the market variance risk premium mostly capture innovations in aggregate volatility rather than consumption volatility-of-volatility. This suggests that it is important to directly use stock market volatility-of-volatility instead of simply following Bali and Zhou (2016) in proxying consumption volatility-of-volatility with the market variance risk premium. 3.4 Regression tests The portfolio sorts present strong evidence that sensitivities to innovations in aggregate volatility-of-volatility are related to stock returns. The double sorts indicate that this relation persists when controlling for, and hence cannot be explained by, aggregate volatility or the variance risk premium. Following on from that, in this section, we estimate Fama and MacBeth (1973) regressions that simultaneously control for different variables and test if a stock’s sensitivity to innovations in aggregate volatility-of-volatility contains information about stock returns beyond that of various other firm characteristics and factor sensitivities. We perform the regressions using individual stocks instead of stock portfolios. Lo and MacKinlay (1990) and Lewellen, Nagel, and Shanken (2010) argue against the use of portfolios in cross-sectional regressions, since the particular method, by which the portfolios are formed can severely influence the results. Furthermore, Ang, Liu, and Schwarz (2010) show that creating portfolios ignores important information on individual factor loadings and leads to higher asymptotic standard errors of risk premium estimates. Consequently, using individual stocks, we utilize this additional information and, at the same time, avoid the specification of breakpoints. Each month, we perform cross-sectional regressions of stock excess returns over the following year on the stocks’ sensitivities to innovations in aggregate volatility-of-volatility and several control variables, measured over the same period. We winsorize all regressors at the 1st and 99th percentile to restrict the effect of outliers (Fama and French, 2008; Baltussen, Van Bekkum, and Van Der Grient, forthcoming).19 For the regressions, we use ordinary least squares (equally weighted) or weighted least squares (value-weighted) with a diagonal weighting matrix, where the inverse of the firm’s market value at the beginning of the estimation period for βj,tQ is along the diagonal, with the following regression equation: rj,t−rf,t=λt0+λtMβj,tM+λtVβj,tV+λtQβj,tQ+λtζβj,tζ+ϵj,t. (16) rj,t is the annual return of stock j and rf,t is the risk-free rate during that period. βj,tM, βj,tV ⁠, and βj,tQ are the stock’s market beta, sensitivity to innovations in aggregate volatility and sensitivity to innovations in aggregate volatility-of-volatility over the evaluation period, respectively. The term βj,tζ denotes a vector collecting additional variables hypothesized to explain returns. λt0 denotes the regression intercept and λtM, λtV, λtQ ⁠, and λtζ are the prices of factor risk associated with the respective variables. ϵj,t is the idiosyncratic return component of stock j at time t. In the next step, we perform tests on the time-series averages λ0¯, λM¯, λV¯, λQ¯ ⁠, and λζ¯ of the estimated intercept and slope coefficients, λt0^, λtM^, λtV^, λtQ^ ⁠, and λtζ^ ⁠. We account for potential autocorrelation, heteroscedasticity, and errors-in-variables concerns, computing robust Newey and West (1987) (again using 12 lags) and Shanken (1992) adjusted p-values based on the time-series of coefficient estimates. Table 6 reports the results of the basic Fama and MacBeth (1973) regressions. We report the results of a regression of excess returns on βj,tM, βj,tV, βj,tQ ⁠, and various other canonical characteristics. In the basic regression specification suggested by our theoretical model (1), the price of risk on the market is essentially zero and the price of risk on aggregate volatility amounts to –0.2279, which is significant at 5%. The annual factor risk premium of aggregate volatility-of-volatility (⁠ λQ¯ ⁠) is –0.7753 with a p-value of 0.010.20 Consequently, a two-standard-deviation increase in a stock’s volatility-of-volatility-sensitivity is associated with a 10.61% decrease in average annual returns.21 Table 6 Fama-MacBeth regressions Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1005*** −0.4661*** 0.1202*** 0.1226*** 0.1077*** 0.1009*** 0.1002*** 0.0807** −0.3124*** (0.004) (0.000) (0.001) (0.000) (0.002) (0.003) (0.002) (0.020) (0.000) Beta 0.0000 0.0191 0.0018 0.0030 −0.0206 −0.0048 −0.0010 0.0015 −0.0008 (0.999) (0.767) (0.977) (0.962) (0.723) (0.939) (0.987) (0.981) (0.989) dVIX −0.2279** −0.2015** −0.2243** −0.2232** −0.2138** −0.2251** −0.2289** −0.2305** −0.1851** (0.017) (0.037) (0.020) (0.019) (0.013) (0.016) (0.015) (0.016) (0.033) dVVIX −0.7753** −0.6954** −0.7532** −0.7680** −0.7710*** −0.7736*** −0.8065*** −0.7680** −0.6886** (0.010) (0.025) (0.015) (0.012) (0.008) (0.009) (0.008) (0.012) (0.019) ln(Size) 0.0375*** 0.0304*** (0.000) (0.000) Book-to-market −0.0363*** −0.0350*** (0.000) (0.000) Bid-ask spread −18.085*** −4.3590*** (0.000) (0.004) Momentum −0.0510 −0.0698 (0.304) (0.157) Short-term reversal −0.0496 −0.0903 (0.324) (0.115) ln(Age) 0.0008 −0.0117** (0.879) (0.040) Leverage 0.0341 0.0164 (0.172) (0.559) Adj. R2 0.0928 0.1097 0.1047 0.1019 0.1057 0.0987 0.0982 0.0977 0.1435 Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1005*** −0.4661*** 0.1202*** 0.1226*** 0.1077*** 0.1009*** 0.1002*** 0.0807** −0.3124*** (0.004) (0.000) (0.001) (0.000) (0.002) (0.003) (0.002) (0.020) (0.000) Beta 0.0000 0.0191 0.0018 0.0030 −0.0206 −0.0048 −0.0010 0.0015 −0.0008 (0.999) (0.767) (0.977) (0.962) (0.723) (0.939) (0.987) (0.981) (0.989) dVIX −0.2279** −0.2015** −0.2243** −0.2232** −0.2138** −0.2251** −0.2289** −0.2305** −0.1851** (0.017) (0.037) (0.020) (0.019) (0.013) (0.016) (0.015) (0.016) (0.033) dVVIX −0.7753** −0.6954** −0.7532** −0.7680** −0.7710*** −0.7736*** −0.8065*** −0.7680** −0.6886** (0.010) (0.025) (0.015) (0.012) (0.008) (0.009) (0.008) (0.012) (0.019) ln(Size) 0.0375*** 0.0304*** (0.000) (0.000) Book-to-market −0.0363*** −0.0350*** (0.000) (0.000) Bid-ask spread −18.085*** −4.3590*** (0.000) (0.004) Momentum −0.0510 −0.0698 (0.304) (0.157) Short-term reversal −0.0496 −0.0903 (0.324) (0.115) ln(Age) 0.0008 −0.0117** (0.879) (0.040) Leverage 0.0341 0.0164 (0.172) (0.559) Adj. R2 0.0928 0.1097 0.1047 0.1019 0.1057 0.0987 0.0982 0.0977 0.1435 This table presents average coefficient estimates from monthly Fama and MacBeth (1973) regressions. Each month, we regress excess stock returns during the following year on a constant, the market betas, the sensitivities to innovations in aggregate volatility, and the sensitivities to innovations in aggregate volatility-of-volatility over the same time, as well as a series of stock characteristics, all also measured over the following year. Appendix A provides detailed variable definitions. We report robust Newey and West (1987),p-values using 12 lags, that also incorporate the Shanken (1992) errors-in-variables correction, in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 6 Fama-MacBeth regressions Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1005*** −0.4661*** 0.1202*** 0.1226*** 0.1077*** 0.1009*** 0.1002*** 0.0807** −0.3124*** (0.004) (0.000) (0.001) (0.000) (0.002) (0.003) (0.002) (0.020) (0.000) Beta 0.0000 0.0191 0.0018 0.0030 −0.0206 −0.0048 −0.0010 0.0015 −0.0008 (0.999) (0.767) (0.977) (0.962) (0.723) (0.939) (0.987) (0.981) (0.989) dVIX −0.2279** −0.2015** −0.2243** −0.2232** −0.2138** −0.2251** −0.2289** −0.2305** −0.1851** (0.017) (0.037) (0.020) (0.019) (0.013) (0.016) (0.015) (0.016) (0.033) dVVIX −0.7753** −0.6954** −0.7532** −0.7680** −0.7710*** −0.7736*** −0.8065*** −0.7680** −0.6886** (0.010) (0.025) (0.015) (0.012) (0.008) (0.009) (0.008) (0.012) (0.019) ln(Size) 0.0375*** 0.0304*** (0.000) (0.000) Book-to-market −0.0363*** −0.0350*** (0.000) (0.000) Bid-ask spread −18.085*** −4.3590*** (0.000) (0.004) Momentum −0.0510 −0.0698 (0.304) (0.157) Short-term reversal −0.0496 −0.0903 (0.324) (0.115) ln(Age) 0.0008 −0.0117** (0.879) (0.040) Leverage 0.0341 0.0164 (0.172) (0.559) Adj. R2 0.0928 0.1097 0.1047 0.1019 0.1057 0.0987 0.0982 0.0977 0.1435 Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1005*** −0.4661*** 0.1202*** 0.1226*** 0.1077*** 0.1009*** 0.1002*** 0.0807** −0.3124*** (0.004) (0.000) (0.001) (0.000) (0.002) (0.003) (0.002) (0.020) (0.000) Beta 0.0000 0.0191 0.0018 0.0030 −0.0206 −0.0048 −0.0010 0.0015 −0.0008 (0.999) (0.767) (0.977) (0.962) (0.723) (0.939) (0.987) (0.981) (0.989) dVIX −0.2279** −0.2015** −0.2243** −0.2232** −0.2138** −0.2251** −0.2289** −0.2305** −0.1851** (0.017) (0.037) (0.020) (0.019) (0.013) (0.016) (0.015) (0.016) (0.033) dVVIX −0.7753** −0.6954** −0.7532** −0.7680** −0.7710*** −0.7736*** −0.8065*** −0.7680** −0.6886** (0.010) (0.025) (0.015) (0.012) (0.008) (0.009) (0.008) (0.012) (0.019) ln(Size) 0.0375*** 0.0304*** (0.000) (0.000) Book-to-market −0.0363*** −0.0350*** (0.000) (0.000) Bid-ask spread −18.085*** −4.3590*** (0.000) (0.004) Momentum −0.0510 −0.0698 (0.304) (0.157) Short-term reversal −0.0496 −0.0903 (0.324) (0.115) ln(Age) 0.0008 −0.0117** (0.879) (0.040) Leverage 0.0341 0.0164 (0.172) (0.559) Adj. R2 0.0928 0.1097 0.1047 0.1019 0.1057 0.0987 0.0982 0.0977 0.1435 This table presents average coefficient estimates from monthly Fama and MacBeth (1973) regressions. Each month, we regress excess stock returns during the following year on a constant, the market betas, the sensitivities to innovations in aggregate volatility, and the sensitivities to innovations in aggregate volatility-of-volatility over the same time, as well as a series of stock characteristics, all also measured over the following year. Appendix A provides detailed variable definitions. We report robust Newey and West (1987),p-values using 12 lags, that also incorporate the Shanken (1992) errors-in-variables correction, in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adding additional control variables including size, book-to-market, bid-ask spread, momentum, short-term reversal, age, and leverage in models (2) to (8) does not change our main results. The risk premium estimate on volatility-of-volatility remains economically large, with values ranging from –0.81 to –0.70 and highly significant for every specification. Adding all canonical characteristics jointly yields a factor risk premium of aggregate volatility-of-volatility of –0.69, which is statistically significant at 5%. The risk premium estimate on the market excess return is not significantly different from zero and even negative for some specifications. This result is consistent with recent evidence that market beta is not priced in the cross-section of stock returns (e.g., Frazzini and Pedersen 2014). Many of our control variables yield insignificant risk premium estimates for our sample period or conditional risk premium estimates with different signs compared to the previous literature. Hence, our results are in line with the view that prominent return anomalies have attenuated recently (Chordia, Subrahmanyam, and Tong 2014). Adding characteristics univariately, we find a positive conditional risk premium on size and negative conditional risk premiums on the book-to-market ratio and the bid-ask spread. Hence, stocks of large firms and stocks with lower bid-ask spreads, conditional on all remaining variables being the same, earn higher returns than stocks of small firms and stocks with higher bid-ask spreads and value stocks earn lower returns than growth stocks during the recent sample period. On the other hand, momentum, short-term reversal, age, and leverage do not obtain significant prices of risk, conditional on our main model specification. Table A2 of the Online Appendix repeats the analysis for value-weighted regressions. The results are qualitatively similar. We obtain risk premium estimates on aggregate volatility-of-volatility of around –0.8, which are statistically significant at 5% throughout. Additionally, in Table A3 of the Online Appendix, we present the results when controlling for additional returns distributions and liquidity-related characteristics, like idiosyncratic volatility, coskewness, or Amihud illiquidity. For each of the model specifications, we obtain a significantly negative cross-sectional price of risk on aggregate volatility-of-volatility. Lastly, we examine whether the cross-sectional regression results are consistent with the model predictions of Equation (13). The first model prediction is that the cross-sectional risk premium on the market portfolio is positive and proportional to the coefficient of relative risk-aversion. For example, the coefficient of relative risk-aversion in the basic model specification (1) of Table 6 is approximately equal to zero.22 The coefficient of relative risk-aversion therefore is not really in a reasonable range (Friend and Blume 1975). For some specifications, the risk premium estimate on the market is even negative, implying a negative relative risk-aversion. However, the standard error of the risk premium estimate on the market is typically large; for example, it amounts to 6.26% in the basic model specification (1). Hence, the point estimate is ambiguous and has to be interpreted with caution. Next, we turn the focus on the price of risk on aggregate volatility, which is significantly negative throughout all the test specifications. This is in line with the model’s prediction. Finally, we find that the market price of risk on volatility-of-volatility, which could be positive or negative according to our theoretical model, is significantly negative. This result is consistent with Bali and Zhou (2016), who proxy consumption volatility-of-volatility with the variance risk premium. 4. Additional Analyses 4.1 Multivariate estimation In this section, we examine the robustness of our results to jointly estimating the sensitivities to innovations in aggregate volatility-of-volatility with those to other factors, as presented in Equation (15). Table 7 reports the results of Fama and MacBeth (1973) regressions using multivariate sensitivity estimation regressions. Incorporating the Fama and French (1993) factors in model (1) leaves the risk premium estimate of aggregate volatility-of-volatility strongly significant at 1%. Adding the other market factors like dSkew, dKurt, straddle vol, jump, or dVRP (models (2) to (7)) only marginally affects the risk premium estimate on aggregate volatility-of-volatility. It ranges from –0.76 to –0.71 and remains highly statistically significant. Adding the Hou, Xue, and Zhang (2015b) factors in a joint sensitivity estimation, the risk premium estimate on aggregate volatility-of-volatility gets even more negative, reaching a value of –0.90 and is statistically significant at 1%. Table 7 Fama-MacBeth regressions, multivariate estimation (1) (2) (3) (4) (5) (6) (7) (8) Constant 0.0835** 0.1020*** 0.0832** 0.0931** 0.0739* 0.1046*** 0.0821** 0.0406* (0.025) (0.002) (0.015) (0.011) (0.056) (0.002) (0.022) (0.098) MKT 0.0354 −0.0025 0.0361 −0.0010 0.0456 −0.0051 0.0369 0.0918* (0.560) (0.967) (0.540) (0.987) (0.461) (0.935) (0.541) (0.089) dVIX −0.2147** −0.2077** −0.2179** −0.2674*** (0.022) (0.020) (0.016) (0.001) dVVIX −0.7101*** −0.7539*** −0.7300*** −0.7801** −0.7615*** −0.7323** −0.7314*** −0.8947*** (0.007) (0.007) (0.003) (0.011) (0.005) (0.012) (0.005) (0.001) SMB −0.0186 −0.0174 −0.0198 −0.0190 (0.340) (0.349) (0.304) (0.329) HML −0.0599** −0.0596** −0.0590** −0.0608** (0.015) (0.014) (0.020) (0.012) dSkew 0.6213 0.4023 (0.167) (0.340) dKurt −1.3352* −0.9010 (0.073) (0.146) Straddle vol 0.2078*** 0.1192* (0.001) (0.071) Jump −1.0184*** −0.7976** (0.003) (0.038) dVRP 0.0500** 0.0517** (0.043) (0.037) SMBHXZ −0.0674** (0.010) IA 0.0109 (0.583) ROE 0.0713*** (0.001) Adj. R2 0.1359 0.1032 0.1442 0.0926 0.1392 0.0936 0.1362 0.1588 (1) (2) (3) (4) (5) (6) (7) (8) Constant 0.0835** 0.1020*** 0.0832** 0.0931** 0.0739* 0.1046*** 0.0821** 0.0406* (0.025) (0.002) (0.015) (0.011) (0.056) (0.002) (0.022) (0.098) MKT 0.0354 −0.0025 0.0361 −0.0010 0.0456 −0.0051 0.0369 0.0918* (0.560) (0.967) (0.540) (0.987) (0.461) (0.935) (0.541) (0.089) dVIX −0.2147** −0.2077** −0.2179** −0.2674*** (0.022) (0.020) (0.016) (0.001) dVVIX −0.7101*** −0.7539*** −0.7300*** −0.7801** −0.7615*** −0.7323** −0.7314*** −0.8947*** (0.007) (0.007) (0.003) (0.011) (0.005) (0.012) (0.005) (0.001) SMB −0.0186 −0.0174 −0.0198 −0.0190 (0.340) (0.349) (0.304) (0.329) HML −0.0599** −0.0596** −0.0590** −0.0608** (0.015) (0.014) (0.020) (0.012) dSkew 0.6213 0.4023 (0.167) (0.340) dKurt −1.3352* −0.9010 (0.073) (0.146) Straddle vol 0.2078*** 0.1192* (0.001) (0.071) Jump −1.0184*** −0.7976** (0.003) (0.038) dVRP 0.0500** 0.0517** (0.043) (0.037) SMBHXZ −0.0674** (0.010) IA 0.0109 (0.583) ROE 0.0713*** (0.001) Adj. R2 0.1359 0.1032 0.1442 0.0926 0.1392 0.0936 0.1362 0.1588 This table presents average coefficient estimates from monthly Fama and MacBeth (1973) regressions. Each month, we regress excess stock returns during the following year on a constant, the market betas, the sensitivities to innovations in aggregate volatility, the sensitivities to innovations in aggregate volatility-of-volatility over the same time, and/or a series of βj,tζ estimates obtained in a joint multivariate regression like in Equation (15), using daily data over the following year. Appendix A provides detailed variable definitions. We report robust Newey and West (1987),p-values using 12 lags, that also incorporate the Shanken (1992) errors-in-variables correction, in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 7 Fama-MacBeth regressions, multivariate estimation (1) (2) (3) (4) (5) (6) (7) (8) Constant 0.0835** 0.1020*** 0.0832** 0.0931** 0.0739* 0.1046*** 0.0821** 0.0406* (0.025) (0.002) (0.015) (0.011) (0.056) (0.002) (0.022) (0.098) MKT 0.0354 −0.0025 0.0361 −0.0010 0.0456 −0.0051 0.0369 0.0918* (0.560) (0.967) (0.540) (0.987) (0.461) (0.935) (0.541) (0.089) dVIX −0.2147** −0.2077** −0.2179** −0.2674*** (0.022) (0.020) (0.016) (0.001) dVVIX −0.7101*** −0.7539*** −0.7300*** −0.7801** −0.7615*** −0.7323** −0.7314*** −0.8947*** (0.007) (0.007) (0.003) (0.011) (0.005) (0.012) (0.005) (0.001) SMB −0.0186 −0.0174 −0.0198 −0.0190 (0.340) (0.349) (0.304) (0.329) HML −0.0599** −0.0596** −0.0590** −0.0608** (0.015) (0.014) (0.020) (0.012) dSkew 0.6213 0.4023 (0.167) (0.340) dKurt −1.3352* −0.9010 (0.073) (0.146) Straddle vol 0.2078*** 0.1192* (0.001) (0.071) Jump −1.0184*** −0.7976** (0.003) (0.038) dVRP 0.0500** 0.0517** (0.043) (0.037) SMBHXZ −0.0674** (0.010) IA 0.0109 (0.583) ROE 0.0713*** (0.001) Adj. R2 0.1359 0.1032 0.1442 0.0926 0.1392 0.0936 0.1362 0.1588 (1) (2) (3) (4) (5) (6) (7) (8) Constant 0.0835** 0.1020*** 0.0832** 0.0931** 0.0739* 0.1046*** 0.0821** 0.0406* (0.025) (0.002) (0.015) (0.011) (0.056) (0.002) (0.022) (0.098) MKT 0.0354 −0.0025 0.0361 −0.0010 0.0456 −0.0051 0.0369 0.0918* (0.560) (0.967) (0.540) (0.987) (0.461) (0.935) (0.541) (0.089) dVIX −0.2147** −0.2077** −0.2179** −0.2674*** (0.022) (0.020) (0.016) (0.001) dVVIX −0.7101*** −0.7539*** −0.7300*** −0.7801** −0.7615*** −0.7323** −0.7314*** −0.8947*** (0.007) (0.007) (0.003) (0.011) (0.005) (0.012) (0.005) (0.001) SMB −0.0186 −0.0174 −0.0198 −0.0190 (0.340) (0.349) (0.304) (0.329) HML −0.0599** −0.0596** −0.0590** −0.0608** (0.015) (0.014) (0.020) (0.012) dSkew 0.6213 0.4023 (0.167) (0.340) dKurt −1.3352* −0.9010 (0.073) (0.146) Straddle vol 0.2078*** 0.1192* (0.001) (0.071) Jump −1.0184*** −0.7976** (0.003) (0.038) dVRP 0.0500** 0.0517** (0.043) (0.037) SMBHXZ −0.0674** (0.010) IA 0.0109 (0.583) ROE 0.0713*** (0.001) Adj. R2 0.1359 0.1032 0.1442 0.0926 0.1392 0.0936 0.1362 0.1588 This table presents average coefficient estimates from monthly Fama and MacBeth (1973) regressions. Each month, we regress excess stock returns during the following year on a constant, the market betas, the sensitivities to innovations in aggregate volatility, the sensitivities to innovations in aggregate volatility-of-volatility over the same time, and/or a series of βj,tζ estimates obtained in a joint multivariate regression like in Equation (15), using daily data over the following year. Appendix A provides detailed variable definitions. We report robust Newey and West (1987),p-values using 12 lags, that also incorporate the Shanken (1992) errors-in-variables correction, in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. The price of risk on HML regularly is significantly negative, whereas that on MKT and SMB is never significantly different from zero. The prices of jump risk are significantly negative and those on straddle vol and dVRP are significantly positive. Regarding the new investment and profitability factors of Hou, Xue, and Zhang (2015b), ROE obtains a positive risk premium estimate, whereas that on IA is indistinguishable from zero. The results for value-weighted regressions, presented in Table A4 of the Online Appendix, are qualitatively similar. 4.2 Orthogonalized factor innovations To account for concerns related to high correlations among market excess returns, innovations in the VIX, and innovations in the VVIX, this section reports results for orthogonalized factor innovations. We perform the orthogonalization following Sims (1980). We orthogonalize innovations in the VIX with respect to the market excess return and innovations in the VVIX with respect to both the market excess return and the VIX. With this order of orthogonalization, it is least likely that we find a positive market price of risk on aggregate volatility-of-volatility, since the common variation with the market excess return and aggregate volatility is removed from the variable. The orthogonalization approach delivers two important insights. First, we can directly assess the market prices of risk on the factors which potentially deviate from the factor risk premiums of a cross-sectional regression when the factors are strongly correlated. Second, we can test whether there is a clear incremental contribution of the information contained in aggregate volatility-of-volatility in explaining the cross-sectional variation in equity returns compared to only using the information contained in aggregate volatility and the market excess return. The results are presented in Table 8. We find that, consistent with the results presented in Table 6, the market price of risk on the market excess return is not significantly different from zero. The market price of risk on aggregate volatility is statistically significant at 1% in any case and is slightly smaller in magnitude compared to the nonorthogonalized version of Table 6, with values ranging between –0.22 and –0.16. Hence, the information contained in aggregate volatility has a strong marginal contribution in explaining the cross-section of stock returns relative to the market excess return. Finally, the market price of risk on the orthogonalized innovations in aggregate volatility-of-volatility ranges between –0.41 and –0.35. The market price of risk of the orthogonalized factor innovations is only about half that of the nonorthogonalized innovations. Hence, as indicated by the correlation of innovations in volatility-of-volatility with those in volatility as well as the market excess return, there seems to be some interaction of these factors. However, the market price of risk on the orthogonalized aggregate volatility-of-volatility is still significant with p-values ranging between 3.7% and 8.8%. For model specification (1), a two-standard-deviation increase in factor loadings on orthogonalized volatility-of-volatility innovations is associated with a decrease in annual returns of about 5.2%. Consequently, we find compelling evidence that volatility-of-volatility is a priced risk factor in the cross-section of equity returns, over and above aggregate volatility. Table 8 Fama-MacBeth regressions, orthogonalized factors Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1038*** −0.4782*** 0.1230*** 0.1263*** 0.1103*** 0.1041*** 0.1016*** 0.0834** −0.3288*** (0.003) (0.000) (0.001) (0.000) (0.002) (0.002) (0.002) (0.019) (0.000) Beta −0.0073 0.0124 −0.0046 −0.0044 −0.0277 −0.0119 −0.0074 −0.0055 −0.0074 (0.910) (0.852) (0.943) (0.946) (0.642) (0.851) (0.907) (0.932) (0.903) dVIX −0.2076*** −0.1550*** −0.2037*** −0.1983*** −0.2168*** −0.2093*** −0.2098*** −0.2086*** −0.1605*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) dVVIX −0.3829* −0.3509* −0.3688* −0.3872* −0.3958** −0.3869* −0.4086** −0.3732* −0.3688* (0.056) (0.088) (0.070) (0.055) (0.042) (0.051) (0.037) (0.068) (0.060) ln(Size) 0.0385*** 0.0317*** (0.000) (0.000) Book-to-market −0.0365*** −0.0348*** (0.000) (0.000) Bid-ask spread −18.216*** −4.3741*** (0.000) (0.002) Momentum −0.0501 −0.0695 (0.312) (0.160) Short-term reversal −0.0478 −0.0891 (0.334) (0.117) ln(Age) 0.0012 −0.0118** (0.817) (0.043) Leverage 0.0349 0.0166 (0.150) (0.547) Adj. R2 0.0948 0.1125 0.1064 0.1040 0.1073 0.1006 0.1000 0.0997 0.1453 Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1038*** −0.4782*** 0.1230*** 0.1263*** 0.1103*** 0.1041*** 0.1016*** 0.0834** −0.3288*** (0.003) (0.000) (0.001) (0.000) (0.002) (0.002) (0.002) (0.019) (0.000) Beta −0.0073 0.0124 −0.0046 −0.0044 −0.0277 −0.0119 −0.0074 −0.0055 −0.0074 (0.910) (0.852) (0.943) (0.946) (0.642) (0.851) (0.907) (0.932) (0.903) dVIX −0.2076*** −0.1550*** −0.2037*** −0.1983*** −0.2168*** −0.2093*** −0.2098*** −0.2086*** −0.1605*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) dVVIX −0.3829* −0.3509* −0.3688* −0.3872* −0.3958** −0.3869* −0.4086** −0.3732* −0.3688* (0.056) (0.088) (0.070) (0.055) (0.042) (0.051) (0.037) (0.068) (0.060) ln(Size) 0.0385*** 0.0317*** (0.000) (0.000) Book-to-market −0.0365*** −0.0348*** (0.000) (0.000) Bid-ask spread −18.216*** −4.3741*** (0.000) (0.002) Momentum −0.0501 −0.0695 (0.312) (0.160) Short-term reversal −0.0478 −0.0891 (0.334) (0.117) ln(Age) 0.0012 −0.0118** (0.817) (0.043) Leverage 0.0349 0.0166 (0.150) (0.547) Adj. R2 0.0948 0.1125 0.1064 0.1040 0.1073 0.1006 0.1000 0.0997 0.1453 This table presents average coefficient estimates from monthly Fama and MacBeth (1973) regressions using orthogonalized factors. Each month, we regress excess stock returns during the following year on a constant, the market betas, the sensitivities to innovations in aggregate volatility, and the sensitivities to innovations in aggregate volatility-of-volatility over the same time, as well as a series of stock characteristics, all also measured over the following year. Appendix A provides detailed variable definitions. We report robust Newey and West (1987),p-values using 12 lags that also incorporate the Shanken (1992) errors-in-variables correction, in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 8 Fama-MacBeth regressions, orthogonalized factors Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1038*** −0.4782*** 0.1230*** 0.1263*** 0.1103*** 0.1041*** 0.1016*** 0.0834** −0.3288*** (0.003) (0.000) (0.001) (0.000) (0.002) (0.002) (0.002) (0.019) (0.000) Beta −0.0073 0.0124 −0.0046 −0.0044 −0.0277 −0.0119 −0.0074 −0.0055 −0.0074 (0.910) (0.852) (0.943) (0.946) (0.642) (0.851) (0.907) (0.932) (0.903) dVIX −0.2076*** −0.1550*** −0.2037*** −0.1983*** −0.2168*** −0.2093*** −0.2098*** −0.2086*** −0.1605*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) dVVIX −0.3829* −0.3509* −0.3688* −0.3872* −0.3958** −0.3869* −0.4086** −0.3732* −0.3688* (0.056) (0.088) (0.070) (0.055) (0.042) (0.051) (0.037) (0.068) (0.060) ln(Size) 0.0385*** 0.0317*** (0.000) (0.000) Book-to-market −0.0365*** −0.0348*** (0.000) (0.000) Bid-ask spread −18.216*** −4.3741*** (0.000) (0.002) Momentum −0.0501 −0.0695 (0.312) (0.160) Short-term reversal −0.0478 −0.0891 (0.334) (0.117) ln(Age) 0.0012 −0.0118** (0.817) (0.043) Leverage 0.0349 0.0166 (0.150) (0.547) Adj. R2 0.0948 0.1125 0.1064 0.1040 0.1073 0.1006 0.1000 0.0997 0.1453 Factor (1) (2) (3) (4) (5) (6) (7) (8) (9) Constant 0.1038*** −0.4782*** 0.1230*** 0.1263*** 0.1103*** 0.1041*** 0.1016*** 0.0834** −0.3288*** (0.003) (0.000) (0.001) (0.000) (0.002) (0.002) (0.002) (0.019) (0.000) Beta −0.0073 0.0124 −0.0046 −0.0044 −0.0277 −0.0119 −0.0074 −0.0055 −0.0074 (0.910) (0.852) (0.943) (0.946) (0.642) (0.851) (0.907) (0.932) (0.903) dVIX −0.2076*** −0.1550*** −0.2037*** −0.1983*** −0.2168*** −0.2093*** −0.2098*** −0.2086*** −0.1605*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) dVVIX −0.3829* −0.3509* −0.3688* −0.3872* −0.3958** −0.3869* −0.4086** −0.3732* −0.3688* (0.056) (0.088) (0.070) (0.055) (0.042) (0.051) (0.037) (0.068) (0.060) ln(Size) 0.0385*** 0.0317*** (0.000) (0.000) Book-to-market −0.0365*** −0.0348*** (0.000) (0.000) Bid-ask spread −18.216*** −4.3741*** (0.000) (0.002) Momentum −0.0501 −0.0695 (0.312) (0.160) Short-term reversal −0.0478 −0.0891 (0.334) (0.117) ln(Age) 0.0012 −0.0118** (0.817) (0.043) Leverage 0.0349 0.0166 (0.150) (0.547) Adj. R2 0.0948 0.1125 0.1064 0.1040 0.1073 0.1006 0.1000 0.0997 0.1453 This table presents average coefficient estimates from monthly Fama and MacBeth (1973) regressions using orthogonalized factors. Each month, we regress excess stock returns during the following year on a constant, the market betas, the sensitivities to innovations in aggregate volatility, and the sensitivities to innovations in aggregate volatility-of-volatility over the same time, as well as a series of stock characteristics, all also measured over the following year. Appendix A provides detailed variable definitions. We report robust Newey and West (1987),p-values using 12 lags that also incorporate the Shanken (1992) errors-in-variables correction, in parentheses. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. 4.3 Pre-formation factor loadings In Section 3, we demonstrate a strong negative relation between the contemporaneous sensitivities to innovations in aggregate volatility-of-volatility and stock returns. The previous analysis strongly indicates that volatility-of-volatility is a priced risk factor, but, for practical purposes, how this risk can be hedged is also of interest. Therefore, in this section, we examine whether one can create the same return spread using pre-formation factor loadings. First, we directly employ pre-formation factor loadings on innovations in aggregate volatility-of-volatility. Second, we try an alternative approach based on VIX-straddle returns. We start by examining the relation of average returns and past sensitivities to innovations in aggregate volatility-of-volatility. Ang, Chen, and Xing (2006) point out that pre-formation factor loadings, beside actual variation in exposures toward that factor, additionally reflect measurement error. This measurement error is likely to be especially high during highly volatile times – as is the case, in particular, in the first part of our sample period. Taken together, noisy estimates of our sorting variable and a relatively short sample increase the standard error estimates and yield a test with reduced statistical power to detect significant spreads in returns and alphas. Hence, while for these reasons we concentrate on the results generated using post-formation factor loadings, the results from pre-formation factor loadings might be weaker than those presented in the previous sections. To address the issue of potential measurement error, we estimate Equation (14) using an expanding window and weighted least squares (WLS) with an exponentially decreasing weighting scheme. The expanding window makes it possible that parameters are estimated more precisely, while the weighting ensures that we can capture time-variation in the factor loadings. As weights we use  exp(−|t−τ|h)∑τ=1t−1exp(−|t−τ|h) with h= log(2)100 ⁠, so that the half-life of the weights converges to 100 for large samples. τ denotes the time of one return observation.23 Table A5 of the Online Appendix reports the results when using pre-formation factor loadings on innovations in aggregate volatility-of-volatility. Sorting the stocks by past exposure to aggregate volatility-of-volatility does not produce a significant spread in average returns and in factor alphas. In many instances, the point estimates are even positive. The main explanation for this is provided by the realized ex post factor sensitivities of the portfolios. In the presence of higher measurement errors, the spread in ex post exposures to aggregate volatility-of-volatility is only 0.0017. This spread is substantially smaller than the 0.13 produced by sorting on contemporaneous factor loadings (see Table 4). Hence, the current factor loadings exhibit very weak predictability for future factor loadings in aggregate volatility-of-volatility. Thus, it seems that the exposure to innovations on aggregate volatility-of-volatility is very difficult to hedge. We examine whether this also holds for an alternative measure of volatility-of-volatility innovations. To that end, we directly use the VIX options market.24 Following Coval and Shumway (2001) and Cremers, Halling, and Weinbaum (2015), we create market-neutral straddle returns from at-the-money options on the VIX. Since these options portfolios are still sensitive to changes in the underlying, that is, aggregate volatility, we further use two straddles with different maturities to make the options position delta-neutral as well. That is, we first create zero-beta straddles using Black and Scholes (1973) option sensitivities by imposing the condition βCωC+βP(1−ωC)=0 ⁠, where βC and βP are the market betas of the at-the-money call and put options with the same strike and maturity. ωC is the portfolio weight of the call option. Following Cremers, Halling, and Weinbaum (2015), we use an at-the-money straddle maturing in the next month (t1) and one maturing in the month after (t2). In a second step, we obtain the final options position by imposing ΔS,t2+ΔS,t1ω1=0 ⁠, where ΔS,t is the Black and Scholes (1973) delta of one straddle.25 Going forward, we use the market and delta-neutral straddle returns instead of dVVIX to estimate the sensitivities to innovations in aggregate volatility-of-volatility using the expanding window WLS approach described above. We present the results in Table 9. First, we find that the ex post exposure spread in βQ (based on innovations to the VVIX) is more than twice as high as when using pre-formation factor loadings on dVVIX. Consistent with this, we also obtain significant spreads in returns and factor alphas. For equally weighted portfolios, the return of the hedge portfolio and the factor alphas are smaller in magnitude compared to Table 4 with values ranging between –2.0% and –2.5%. However, they are all highly statistically significant. The value-weighted average return amounts to –3.1%, which is not statistically significant. However, the value-weighted factor alphas are all statistically significant. Hence, once more, it seems that the value-weighted hedge portfolio loads positively on the systematic risk factors with a positive factor risk premium. Table 9 Pre-formation factor loadings, straddles Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1102 0.0962 0.1009* 0.0953 0.0851 −0.0251*** (0.144) (0.121) (0.095) (0.122) (0.229) (0.005) CAPM alpha 0.0005 0.0042 0.0116 0.0043 −0.0200 −0.0205*** (0.985) (0.758) (0.221) (0.752) (0.414) (0.005) 4-factor alpha −0.0035 0.0003 0.0083*** −0.0019 −0.0238*** −0.0203*** (0.686) (0.947) (0.001) (0.489) (0.003) (0.001) 5-factor alpha 0.0116 0.0003 0.0090*** 0.0014 −0.0129 −0.0245*** (0.112) (0.956) (0.001) (0.646) (0.267) (0.001) HXZ alpha 0.0068 0.0111* 0.0127*** 0.0056 −0.0181*** −0.0250*** (0.192) (0.053) (0.002) (0.193) (0.004) (0.000) Value-weighted Mean return 0.1203* 0.0864* 0.0815 0.0801 0.0892 −0.0311 (0.055) (0.097) (0.137) (0.154) (0.168) (0.156) CAPM alpha 0.0287 0.0067 0.0025 0.0001 −0.0070 −0.0357* (0.150) (0.267) (0.732) (0.992) (0.278) (0.080) 4-factor alpha 0.0332** 0.0129** −0.0009 −0.0088 −0.0073 −0.0405** (0.043) (0.015) (0.836) (0.215) (0.357) (0.046) 5-factor alpha 0.0578*** 0.0166*** −0.0076** −0.0150** −0.0021 −0.0598*** (0.000) (0.000) (0.013) (0.018) (0.863) (0.005) HXZ alpha 0.0439*** 0.0126*** 0.0158** 0.0033 −0.0084 −0.0523*** (0.003) (0.000) (0.016) (0.622) (0.285) (0.003) Ex post βQ 0.0074 0.0058 0.0059 0.0077 0.0112 0.0039 Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1102 0.0962 0.1009* 0.0953 0.0851 −0.0251*** (0.144) (0.121) (0.095) (0.122) (0.229) (0.005) CAPM alpha 0.0005 0.0042 0.0116 0.0043 −0.0200 −0.0205*** (0.985) (0.758) (0.221) (0.752) (0.414) (0.005) 4-factor alpha −0.0035 0.0003 0.0083*** −0.0019 −0.0238*** −0.0203*** (0.686) (0.947) (0.001) (0.489) (0.003) (0.001) 5-factor alpha 0.0116 0.0003 0.0090*** 0.0014 −0.0129 −0.0245*** (0.112) (0.956) (0.001) (0.646) (0.267) (0.001) HXZ alpha 0.0068 0.0111* 0.0127*** 0.0056 −0.0181*** −0.0250*** (0.192) (0.053) (0.002) (0.193) (0.004) (0.000) Value-weighted Mean return 0.1203* 0.0864* 0.0815 0.0801 0.0892 −0.0311 (0.055) (0.097) (0.137) (0.154) (0.168) (0.156) CAPM alpha 0.0287 0.0067 0.0025 0.0001 −0.0070 −0.0357* (0.150) (0.267) (0.732) (0.992) (0.278) (0.080) 4-factor alpha 0.0332** 0.0129** −0.0009 −0.0088 −0.0073 −0.0405** (0.043) (0.015) (0.836) (0.215) (0.357) (0.046) 5-factor alpha 0.0578*** 0.0166*** −0.0076** −0.0150** −0.0021 −0.0598*** (0.000) (0.000) (0.013) (0.018) (0.863) (0.005) HXZ alpha 0.0439*** 0.0126*** 0.0158** 0.0033 −0.0084 −0.0523*** (0.003) (0.000) (0.016) (0.622) (0.285) (0.003) Ex post βQ 0.0074 0.0058 0.0059 0.0077 0.0112 0.0039 At the beginning of each month, we form quintile portfolios based on the stocks’ past sensitivities to aggregate volatility-of-volatility (⁠ βj,tQ ⁠). To obtain the sensitivities, we regress daily excess stock returns on dVVIX, controlling for MKT and dVIX like in Equation (14) and using an expanding exponential weighting scheme. Stocks with the lowest βj,tQ are sorted into portfolio 1, those with the highest βj,tQ into portfolio 5. The column labeled 5 minus 1 refers to the hedge portfolio buying the quintile of stocks with the highest βj,tQ and simultaneously selling the stocks in the quintile with the lowest βj,tQ ⁠. Each month, we set up new 12-month portfolios. The row labeled Mean return is based on simple returns over the following year. CAPM alpha, four-factor alpha, and five-factor alpha refer to the alphas of the CAPM, the Carhart (1997) four-factor, and the five-factor models (including Pastor and Stambaugh 2003 liquidity), respectively. HXZ alpha denotes the alpha relative to the Hou, Xue, and Zhang (2015b) factor model. The row ex post βQ reports the average annual ex post sensitivities to aggregate volatility-of-volatility of the portfolios. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Table 9 Pre-formation factor loadings, straddles Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1102 0.0962 0.1009* 0.0953 0.0851 −0.0251*** (0.144) (0.121) (0.095) (0.122) (0.229) (0.005) CAPM alpha 0.0005 0.0042 0.0116 0.0043 −0.0200 −0.0205*** (0.985) (0.758) (0.221) (0.752) (0.414) (0.005) 4-factor alpha −0.0035 0.0003 0.0083*** −0.0019 −0.0238*** −0.0203*** (0.686) (0.947) (0.001) (0.489) (0.003) (0.001) 5-factor alpha 0.0116 0.0003 0.0090*** 0.0014 −0.0129 −0.0245*** (0.112) (0.956) (0.001) (0.646) (0.267) (0.001) HXZ alpha 0.0068 0.0111* 0.0127*** 0.0056 −0.0181*** −0.0250*** (0.192) (0.053) (0.002) (0.193) (0.004) (0.000) Value-weighted Mean return 0.1203* 0.0864* 0.0815 0.0801 0.0892 −0.0311 (0.055) (0.097) (0.137) (0.154) (0.168) (0.156) CAPM alpha 0.0287 0.0067 0.0025 0.0001 −0.0070 −0.0357* (0.150) (0.267) (0.732) (0.992) (0.278) (0.080) 4-factor alpha 0.0332** 0.0129** −0.0009 −0.0088 −0.0073 −0.0405** (0.043) (0.015) (0.836) (0.215) (0.357) (0.046) 5-factor alpha 0.0578*** 0.0166*** −0.0076** −0.0150** −0.0021 −0.0598*** (0.000) (0.000) (0.013) (0.018) (0.863) (0.005) HXZ alpha 0.0439*** 0.0126*** 0.0158** 0.0033 −0.0084 −0.0523*** (0.003) (0.000) (0.016) (0.622) (0.285) (0.003) Ex post βQ 0.0074 0.0058 0.0059 0.0077 0.0112 0.0039 Rank 1 2 3 4 5 5 minus 1 Equally weighted Mean return 0.1102 0.0962 0.1009* 0.0953 0.0851 −0.0251*** (0.144) (0.121) (0.095) (0.122) (0.229) (0.005) CAPM alpha 0.0005 0.0042 0.0116 0.0043 −0.0200 −0.0205*** (0.985) (0.758) (0.221) (0.752) (0.414) (0.005) 4-factor alpha −0.0035 0.0003 0.0083*** −0.0019 −0.0238*** −0.0203*** (0.686) (0.947) (0.001) (0.489) (0.003) (0.001) 5-factor alpha 0.0116 0.0003 0.0090*** 0.0014 −0.0129 −0.0245*** (0.112) (0.956) (0.001) (0.646) (0.267) (0.001) HXZ alpha 0.0068 0.0111* 0.0127*** 0.0056 −0.0181*** −0.0250*** (0.192) (0.053) (0.002) (0.193) (0.004) (0.000) Value-weighted Mean return 0.1203* 0.0864* 0.0815 0.0801 0.0892 −0.0311 (0.055) (0.097) (0.137) (0.154) (0.168) (0.156) CAPM alpha 0.0287 0.0067 0.0025 0.0001 −0.0070 −0.0357* (0.150) (0.267) (0.732) (0.992) (0.278) (0.080) 4-factor alpha 0.0332** 0.0129** −0.0009 −0.0088 −0.0073 −0.0405** (0.043) (0.015) (0.836) (0.215) (0.357) (0.046) 5-factor alpha 0.0578*** 0.0166*** −0.0076** −0.0150** −0.0021 −0.0598*** (0.000) (0.000) (0.013) (0.018) (0.863) (0.005) HXZ alpha 0.0439*** 0.0126*** 0.0158** 0.0033 −0.0084 −0.0523*** (0.003) (0.000) (0.016) (0.622) (0.285) (0.003) Ex post βQ 0.0074 0.0058 0.0059 0.0077 0.0112 0.0039 At the beginning of each month, we form quintile portfolios based on the stocks’ past sensitivities to aggregate volatility-of-volatility (⁠ βj,tQ ⁠). To obtain the sensitivities, we regress daily excess stock returns on dVVIX, controlling for MKT and dVIX like in Equation (14) and using an expanding exponential weighting scheme. Stocks with the lowest βj,tQ are sorted into portfolio 1, those with the highest βj,tQ into portfolio 5. The column labeled 5 minus 1 refers to the hedge portfolio buying the quintile of stocks with the highest βj,tQ and simultaneously selling the stocks in the quintile with the lowest βj,tQ ⁠. Each month, we set up new 12-month portfolios. The row labeled Mean return is based on simple returns over the following year. CAPM alpha, four-factor alpha, and five-factor alpha refer to the alphas of the CAPM, the Carhart (1997) four-factor, and the five-factor models (including Pastor and Stambaugh 2003 liquidity), respectively. HXZ alpha denotes the alpha relative to the Hou, Xue, and Zhang (2015b) factor model. The row ex post βQ reports the average annual ex post sensitivities to aggregate volatility-of-volatility of the portfolios. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively. Overall, we find that it is possible to hedge innovations in aggregate volatility-of-volatility. However, to form the portfolios, one should use straddle returns instead of innovations to the VVIX.26 4.4 Predicting future factor loadings Given the weak results when directly using pre-formation factor loadings on innovations the VVIX, we examine the cross-sectional predictability of several ex ante firm characteristics and factor sensitivities for future sensitivities to innovations in aggregate volatility-of-volatility by performing Fama and MacBeth (1973) regressions. Cross-sectional predictors of the stocks’ volatility-of-volatility sensitivities are measured during the 12 months directly prior to the 12-month estimation period for sensitivities to innovations in aggregate volatility-of-volatility. The results are reported in Table A6. Consistent with the results for the pre-formation factor loadings in Table A5 of the Online Appendix, we find that past volatility-of-volatility sensitivities do not significantly predict future relative volatility-of-volatility sensitivities.27 Hence, there appears to be large time-variation in factor loadings on aggregate volatility-of-volatility. The best predictors for future volatility-of-volatility factor sensitivities are past beta with a positive sign and the bid-ask-spread, momentum, short-term reversal, and idiosyncratic volatility with a negative sign. Further, consistent with the results of the previous section, we find that the sensitivities to market- and delta-neutral VIX straddle returns (StraddVIX) predict future factor loadings on aggregate volatility-of-volatility far better than past dVVIX sensitivities. The coefficient on StraddVIX is at least weakly significant. 4.5 Delta-neutral VIX option returns The preceding sections show that volatility-of-volatility is sigificantly priced in the stock market but not easy to hedge. In this section, we examine the VIX options market that likely delivers very good hedging opportunities. Since the vega, that is, the sensitivity of the options price with respect to the volatility of the unerlying, of all options is positive, these are natural instruments to hedge against increases in the volatility of the underlying. We follow Bakshi and Kapadia (2003) and Huang and Shaliastovich (2014), computing continuously rebalanced delta-hedged option returns. The delta-hedged VIX option position is approximately insensitive to innovations in aggregate volatility. Hence, it only delivers insurance against increases in aggregate volatility-of-volatility. The option gain can be computed as follows (Huang and Shaliastovich 2014): Πt,T=MT−Mt−∑τ=0N−1Δtτ(Fτ+1−Fτ)−∑τ=0N−1rf,τMtT−tN. (17) Πt,T is the option gain and Ft is the forward price of the option’s underlying. We divide Πt,T by the forward price Ft to obtain the delta-hedged return on the option. MT is the final payoff of the option (put or call) at time T and Mt is the option price at initiation t. Δtτ is the Black and Scholes (1973) delta of the option at time τ. N is the number of days between t and T that are trading days. All other variables are as previously defined. We use all options with volume and open interest greater than zero. For the strategy, we buy options with 30 days to maturity and hold them until expiration, rebalancing the delta-hedge at the end of each trading day. We present the results for delta-hedged option returns in Table A7 of the Online Appendix. Consistent with Huang and Shaliastovich (2014), who end their sample period in September 2012, we find negative delta-hedged option returns both for calls and puts for each moneyness bin, with moneyness computed as strike price over futures price at initiation (⁠ KFt ⁠), between 0.7 and 1.3. These returns are significantly negative for in-the-money calls and both in-the-money and out-of-the-money puts. This finding confirms our previous results of a negative risk premium on volatility-of-volatility. On average, investors on the options market are also willing to pay a premium in order to insure against increases in aggregate volatility-of-volatility. Worth noting are also the distributional characteristics of the delta-hedged option returns. The returns are highly positively skewed with skewness of 0.74 up to 1.68 and have high kurtosis between 8.36 and 9.13. The 5% quantile of option return lies between –4.8% and –2.0% for the different moneyness bins. The 95% quantile lies between 1.3% and 7.8%. Hence, the option return can become strongly positive. These distributional characteristics further indicate that the options on the VIX are adequate hedges for volatility-of-volatility risk.28 4.6 Realized measure of aggregate volatility-of-volatility Taking account of the literature on volatility estimation, we further test the robustness of our results employing an inferior measure of aggregate volatility-of-volatility. Instead of utilizing the forward-looking expected volatility-of-volatility, in this section, we rely on the realized volatility of the VIX. To perform the analysis, we use 5-minute high-frequency data on the VIX from the TRTH database and estimate each day the realized bipower volatility over the past month (Barndorff-Nielsen and Shephard 2004). Since the high-frequency data are available for a longer period, we extend the sample period to January 01, 2000 until September 30, 2016. We perform standard data cleaning operations following Barndorff-Nielsen et al. (2009). Following on from that, we obtain daily realized volatilities of the VIX and compute the innovations (dVoVIXt) again as simple first differences. We estimate factor sensitivities using Equation (14), replacing dVVIX by dVoVIX. We present the results of portfolio sorts based on dVoVIX in Table A8 of the Online Appendix. The return differential between stocks with high and those with low sensitivities to innovations in aggregate volatility-of-volatility is only slightly smaller than when using the VVIX in Table 4, with about –3.4%. The return differential is not significant with a p-value of 0.19. However, once controlling for systematic risk, we find a significant volatility-of-volatility-return trade-off with alphas relative to the CAPM, four-factor, and five-factor models of –6.6% up to –4.4% which are all significant. When using the Hou, Xue, and Zhang (2015b) factor model, the alpha becomes insignificant. When restricting the sample to the horizon in which data on the VVIX is available (2006 until 2016), we also do not find a significant return differential but a significant Hou, Xue, and Zhang (2015b) alpha. Even though the portfolio of stocks with high sensitivities to innovations in aggregate volatility-of-volatility has highly significantly negative alphas in most cases, these are not directly carried over to the hedge portfolio since the alpha of the portfolio with the lowest sensitivities is close to zero or also slightly negative, on average. Hence, overall the model prediction that investors strongly demand stocks which insure against increases in aggregate volatility-of-volatility is confirmed by these results. However, using a realized measure of volatility-of-volatility, the monotonicity in portfolios 1 to 4 vanishes. Consequently, when using this theoretically inferior measure, there is still some indication of a volatility-of-volatility-return trade-off, but this result is less significant and also less robust. Therefore, it is important to use the forward-looking information from options markets.29 5. Conclusion Using a simple stylized theoretical model with Epstein-Zin preferences and rich consumption volatility dynamics, we motivate a potential trade-off of volatility-of-volatility and stock returns. In our empirical study, we confirm this prediction. We find that aggregate volatility-of-volatility carries a considerable negative risk premium. We use both uni- and bivariate portfolio sorts and show that the quintile portfolio of stocks with the highest sensitivity toward innovations in aggregate volatility-of-volatility underperforms the quintile of stocks with the lowest exposure to aggregate volatility-of-volatility by about 5% per annum. Using regression tests, we estimate the cross-sectional price of factor risk of aggregate volatility-of-volatility to be both economically substantial and statistically highly significant. The estimated factor price on aggregate volatility-of-volatility cannot be explained by known risk factors. Our results are also consistent with any multifactor model, in which aggregate volatility-of-volatility is priced with a negative sign if investors relate a positive change in aggregate volatility-of-volatility to future unfavorable shifts in the investment opportunity set. Appendix A Variable Definitions: Main Control Variables Age (Zhang 2006) is the number of years up to time t since a firm first appeared in the CRSP database. In regressions, we take the natural logarithm to remove the extreme skewness in this variable. Aggregate volatility (Ang et al. 2006; “dVIX”), is the coefficient βj,tV obtained by the regression in Equation (14). Beta is the coefficient βj,tM obtained by the regression in Equation (14). Bid-ask spread is the stock’s average daily relative bid-ask spread over the examination period. Book-to-market (Fama and French 1992) is the weighted average of book equity divided by market equity over the examination period. Book and market equity are updated every 12 months for the beginning of the year. Book equity is defined as stockholders’ equity, plus balance sheet deferred taxes and investment tax credit, plus post-retirement benefit liabilities, minus the book value of preferred stock. Leverage (Bhandari 1988) is defined as the weighted average of one minus book equity (see “Book-to-market”) divided by total assets (Compustat: AT). Book equity and total assets are updated every 12 months for the beginning of the year. Market factors: “MKT” is the value-weighted excess return on all CRSP firms. “SMB”, and “HML” are the Fama and French (1993) size and value factors. “SMBHXZ,” “IA,” and “ROE” are the size, investment, and profitability factors of Hou, Xue, and Zhang (2015b). Market variance risk premium (Han and Zhou 2012, “dVRP”), is the coefficient βj,tdVRP in the regression rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tdVRPdVRPτ+ϵj,τ ⁠, using daily returns over the examination period (Ang, Chen, and Xing 2006), where the market variance risk premium is defined as the difference between the physical expected variance and the risk-neutral expected variance (VIX2) of the S&P 500 index over a 30-day horizon using daily return data. First, we compute the expected variance (⁠ EVτ ⁠) under the physical measure by regressing the annualized realized variance (⁠ RVτ˜+30 ⁠) on the lagged implied (⁠ VIXτ˜2 ⁠) and the lagged annualized historical (⁠ RVτ˜ ⁠) realized variance, using an expanding window of daily data that is available at time τ, starting with data from January 1, 1996 (⁠ τ˜ refers to those dates), RVτ˜+30=ατ+βτVIXτ˜2+γτRVτ˜+ϵτ˜+30 in a first step and then computing EVτ=ατ^+βτ^VIXτ2+γτ^RVτ ⁠. The market variance risk premium (⁠ VRPτ ⁠) is obtained as VRPτ=EVτ−VIXτ2 ⁠. dVRP is obtained as the first difference in VRP. Momentum (Jegadeesh and Titman 1993) is the cumulative stock return over the period from t – 12 until t – 1. Short-term reversal (Jegadeesh 1990) is the preceding month’s stock return (from t – 1 to t). Size (Banz 1981) is the average of firm’s market capitalization over the examination period. Market capitalization is computed as the product of the price times the number of shares outstanding. In regressions, we take the natural logarithm to remove the extreme skewness in this variable. Stochastic volatility (Cremers, Halling, and Weinbaum 2015; “straddle vol”), market skewness, and kurtosis (Chang, Christoffersen, and Jacobs 2013; “dSkew”, “dKurt”), and aggregate jump risk (Cremers, Halling, and Weinbaum 2015; “jump”) are the coefficients βj,tF in the regression rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tFFτ+ϵj,τ ⁠, using daily returns over the examination period (Ang, Chen, and Xing 2006), where F is one of the following: Innovations in implied market skewness and kurtosis (Chang, Christoffersen, and Jacobs 2013), which are defined as the difference of daily implied skewness (kurtosis) computed from S&P 500 index options using the formulas of Bakshi, Kapadia, and Madan (2003) and its expectation, which is obtained by fitting an ARMA(1,1) model on the complete time series of skewness (kurtosis) estimates. The resultant measure of innovations in market skewness then is dSkewτ=Skewτ−0.9993Skewτ−1+0.6647dSkewτ−1 and that of innovations in market kurtosis is dKurtτ=Kurtτ−0.9999Kurtτ−1+0.4971dKurtτ−1 ⁠. Market-neutral straddle returns (Cremers, Halling, and Weinbaum 2015) are computed by first constructing at-the-money zero beta straddles. Afterward, the straddle vol factor is the return of a gamma neutral and vega positive portfolio of the two straddles maturing in the next month and the month after next, while the jump factor is the return of a gamma positive and vega neutral portfolio using the same straddles. To construct the factors, we use Black and Scholes (1973) option sensitivities. B Variable Definitions: Additional Controls Amihud illiquidity (Amihud 2002) is the absolute value of the stock’s return divided by the daily dollar volume, averaged over the examination period. Specifically, it is Illiqt=1n∑τ=1n|rj,τ|Volume$τ ⁠, with the daily dollar volume (⁠ Volume$τ ⁠, in thousands dollars), calculated as the last trade price times shares traded on day τ, whereas the summation is taken over all n trading days during the examination period. Coskewness (Harvey and Siddique 2000) and Cokurtosis (Dittmar 2002) are the coefficients βj,tCS and βj,tCK in the regression rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tCS(rM,τ−rf,τ)2+βj,tCK(rM,τ−rf,τ)3+ϵj,τ ⁠, including the market excess return, the squared market excess return, and the cubed market excess return. The regression is estimated using daily returns over the examination period. Demand for lottery (Bali, Cakici, and Whitelaw 2011; “MAX”) is the average of the five highest daily returns during the examination period. Downside beta (Ang, Chen, and Xing 2006) is the coefficient βj,tD in the regression rj,τ−rf,τ=αj,t+βj,tD(rM,τ−rf,τ)+ϵj,τ ⁠, using daily returns over the examination period only when the market return is below the average daily market return over that year. Idiosyncratic volatility (Ang et al. 2006; “idio. volatility”) is the standard deviation of the residuals ϵj,τ in the Fama and French (1993) three-factor model rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tSSMBτ+βj,tHHMLτ+ϵj,τ ⁠, using daily returns over the examination period. SMBτ and HMLτ denote the returns on the Fama and French (1993) factors. Kurtosis is the stock’s scaled fourth moment, computed using daily returns over the examination period. Pastor-Stambough liquidity (Pastor and Stambaugh 2003, “PS liquidity”) is the coefficient βj,tL in the following regression rj,τ−rf,τ=αj,t+βj,tM(rM,τ−rf,τ)+βj,tSSMBτ+βj,tHHMLτ+βj,tLLτ+ϵj,τ ⁠, where Lτ is the liquidity factor provided by Lubos Pastor and rM,τ−rf,τ=MKTτ, SMBτ ⁠, and HMLτ are the Fama-French factors provided by Kenneth R. French. We run the regression using the monthly returns during the examination period. Skewness (Xu 2007) is the stock’s scaled third moment, computed using daily returns over the examination period. Turnover (Datar, Naik, and Radcliffe 1998) is the number of shares traded in one month divided by the total shares outstanding, averaged over all months in the examination period. Volatility (Zhang 2006) is the stock’s standard deviation, computed using daily returns over the examination period. Volume (Gervais, Kaniel, and Mingelgrin 2001) is the stock’s average daily dollar trading volume over the examination period. In regressions, we take the natural logarithm to remove the extreme skewness in this variable. Footnotes 1 Aggregate volatility-of-volatility denotes the volatility of the volatility of the aggregate stock market. 2 The alpha with respect to the classical Carhart (1997) four-factor model is –4.91%, which is also significantly different from zero. 3 We rearrange the two equations and plug them in into Equation (11) for σg,t+12 and qt+1 ⁠. 4 In our empirical study, we use daily observations. Our proxies for aggregate volatility and volatility-of-volatility denote expectations over the next 30 days. Hence, because of the large overlap of our empirical measures, this assumption appears reasonable. 5 For the period prior to 2007, we compute the VVIX ourselves using the VIX methodology and options on the VIX from OptionMetrics. 6 For reliable implied moments, option liquidity is an important issue. The average daily trading volume of VIX options was already greater than 8,000 in February 2006. It increased gradually to more than 500,000 contracts per day toward the end of the sample. For more information, refer to the CBOE homepage. 7 The minimum is 1,807 stocks eligible in February 2009, whereas the maximum is 2,794 stocks eligible in April 2006. The total number of firm-month observations is 280,333. 8 Options data have only been available up to April 30, 2016, when we revised this manuscript. Hence, all tests that include options data (e.g., jump, dSkew) are performed for the sample period until April 30, 2016. 9 The data from Chen Xue are monthly and range until December 2015. Therefore, we construct the factors ourselves following the description in Hou, Xue, and Zhang (2015a) to add the data for January until September 2016, as well as to obtain daily returns on the factors. 10 For the factor loading estimation regressions, potentially low explanatory power might be a concern. In the basic specification, we find the model in Equation (14) to exhibit an average R2 of 0.31 with median 0.29. Thus, we conclude that the factor loading regressions possess substantial explanatory power. In these regressions, the coefficient βj,tM is significantly different from zero (at 10%) in 96% of the cases, whereas the coefficients βj,tV and βj,tQ are significantly different from zero in about 19% and 13% of the cases, respectively. 11 One may argue that the innovation in the VVIX captures both the innovation in aggregate volatility-of-volatility and in the volatility-of-volatility risk premium. However, trying to estimate the risk premium as the difference between the expected volatility-of-volatility, obtained using VIX futures and the VVIX as described for the market variance risk premium in Appendix A, and the risk-neutral expectation, we find the innovations in the risk premium to be only weakly related to total innovations in the VVIX with a correlation of –0.12. Furthermore, when separating innovations in the VVIX from innovations in the risk premium and testing their significance in a joint cross-sectional regression, we find that only the “true” innovations in aggregate volatility-of-volatility are significantly priced. These results are available on request. 12 Bollerslev, Tauchen, and Zhou (2009) show that volatility-of-volatility is theoretically (positively) linked to the variance risk premium. However, the sign depends on the definition of the variance risk premium as risk-neutral minus physical expected variance. Although we define it the other way around, our results are fully consistent with their derivation. 13 Although only 11 lags are theoretically required, we follow Ang, Chen, and Xing (2006) and Cremers, Halling, and Weinbaum (2015) and include an additional lag for robustness. 14 In light of the results of Fabozzi, Huang, and Wang (2016), we decide to use the model of Hou, Xue, and Zhang (2015b) instead of that of Fama and French (2015). 15 Because of the overlapping approach with 12-month holding periods, typically, there are 12 active 5 minus 1 hedge portfolios. 16 For double sorts with aggregate volatility, we directly obtain the betas from Equation (14). When controlling for the variance risk premium, we use the same regression equation, replacing dVIXτ by dVRPτ ⁠. 17 While the double-sorting approach we use is unconditional, the results are qualitatively similar when performing a conditional double-sort, that is, first sorting on a certain characteristic and subsequently sorting on the VVIX sensitivities. These results are available on request. 18 The results for Carhart (1997) four-factor alphas are qualitatively similar. These results are available on request. 19 The results of nonwinsorized regression tests are qualitatively similar. These results are available on request. 20 Note that the correlation between factor loadings on dVVIX and dVIX amounts to only –40% (compared to the factor correlation of 67%), which makes it unlikely that problems due to multicollinearity are present. 21 This number is obtained as follows. We multiply the risk premium estimate with 2 times the average cross-sectional standard deviation of the sensitivities to innovations in aggregate volatility-of-volatility from Table 2 (0.0684). Plugging in yields −0.7753 *(2*0.0684)=−0.1061 ⁠. 22 The coefficient of relative risk-aversion of the representative investor can be approximated by dividing the cross-sectional price of risk by the variance of the market return, which amounts to 0.0019. 23 Simply using rolling 12-month windows of past factor loadings instead of the expanding window WLS approach yields similar, though somewhat weaker results. 24 We thank an anonymous referee for suggesting this. 25 To prevent very extreme portfolio positions, if one of the straddles has an absolute delta smaller than 10−5 ⁠, we directly use the return on that straddle. 26 Using the market- and delta-neutral straddle returns in a contemporaneous setup also implies a negative volatility-of-volatility-return trade-off. For example, for equally weighted returns, the average return of the hedge portfolio is –3.1% and the Hou, Xue, and Zhang (2015b) alpha is –2.7%. Both are significantly different from zero at 5%. 27 We find that the factor loadings on the VIX are slightly more predictable than those on the VVIX, but not very much. The average 12-month autocorrelation of VIX factor loadings 7.4%. 28 Huang and Shaliastovich (2014) further show that volatility-of-volatility can predict future delta-hedged option returns. 29 This result is similar to that of Ang et al. (2006), who only find a significant spread in the cross-section of returns when using an option-implied, as opposed to a simple historical or high-frequency, measure to proxy for innovations in aggregate volatility. References Adrian T. , Rosenberg J. . 2008 . Stock returns and volatility: Pricing the short-run and long-run components of market risk . Journal of Finance 63 : 2997 – 3030 . Google Scholar Crossref Search ADS Amihud Y. 2002 . 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