The Pricing of Market Risks in Equity Options: Evidence

The Pricing of Market Risks in Equity Options: Evidence From Individual Variance Risk Premiums

∗

Jian Du†

First Draft: September 2011 Current Version: March 2012

∗

I thank Nikunj Kapadia, Hossein Kazemi, Bing Liang, Joe Zhang and seminar participants at Singapore Management University for helpful comments and suggestions. All remaining errors are my own. † Isenberg School of Management, University of Massachusetts, Amherst, MA 01003. E-mail address: [email protected].

Abstract

We examine the pricing of market level risks in equity options by studying the modelfree individual variance risk premiums (VRP) constructed from options and stocks data. We document significant empirical findings that a positive market equity risk premium, a negative market variance risk premium, and a negative market jump risk premium are priced in the cross-section of individual VRPs. The results are robust after controlling for firm-level variables, including firm size, book-to-market ratio, momentum, stock illiquidity, idiosyncratic volatility and options trading volume. We propose a non-parametric way to decompose the individual VRPs into three components: the common unexplained component, the systematic exposure component, and the idiosyncratic component. We find that the (conditional) common unexplained component is significantly negative and time varying, and it strongly predicts future 6-month to 2-year stock market return and future 3-month to 2-year growth of economic activities. Our findings support a missing risk factor story and suggest that the cross-section of individual VRPs can be understood as risk premiums rather than mispricing.

Keywords: equity options, variance risk premium, variance risk, jump risk

JEL classification: G1, G12, G13

1

Introduction

Options enable investors to trade on their view about future variance of the underlying asset. The profit or return of any delta-neutral option trading/hedging strategy to a large extent depends on the difference between the “variance price” paid by the investors and the variance realization. Understanding the gap between the expected variance implied from option prices and the realized variance, the so-called “variance risk premium” (VRP hereafter), is important in understanding the risk and return of option-based variance trading and the unexpected exposure of hedging with options; more generally, it helps us understand the pricing of options, investors’ risk preference, and asset pricing structure. For index options, it is a well-studied empirical fact that the implied variance is consistently higher than the realized variance; a negatively priced market variance risk has been proposed to explain this observation.1 At the individual firms level, the deviation between the option implied variance and the realized variance, which we term as “individual VRP”, receives less attention and is much less understood. A few existing literatures provide two distinctive explanations for the cross-section of individual VRPs. One takes the risk premium framework and suggests that a systematic variance risk factor is priced in individual VRP.2 The other explanation, represented by Goyal and Saretto (2009), suggests that options are mispriced as traders overreact to previous events and overestimate future variance after large negative stock returns.3 So far, no literature explains the cross-section of individual VRPs completely, which is the objective of this paper. How do we understand the cross-section of individual VRPs? Are they compensations for exposures to systematic risks, as suggested by the traditional asset pricing framework, or are they caused by market inefficiency and investors’ irrational behaviors? This paper provides 1

See Bakshi and Kapadia (2003a) and Carr and Wu (2008), among others. Bakshi and Kapadia (2003b) suggest that individual options price a negative market volatility premium based on analysis on how market volatility impacts the VRP of 25 individual equity options in the time series; Carr and Wu (2008) similarly conclude that a systematic variance risk factor is priced in individual VRP based on analysis on the model-free VRP of 40 individual and index options. 3 Goyal and Saretto (2009)’s explanation is based on the finding that options-based variance trading strategies constructed by sorting firms on the difference between realized volatility and implied volatility produce significant returns that cannot be explained by usual risk factor models. 2

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answers to the above questions. With a standard asset pricing framework, we empirically show that, three systematic risk factors, including the market equity risk, aggregate variance risk and aggregate jump risk, are all significantly priced in individual VRP and combine to capture about half of its cross-sectional variation. We also find evidence that there is a missing risk factor embedded in the average unexplained individual VRPs, and thus conclude that the individual VRPs can be completely understood from a rational perspective. Current literature, including Bakshi and Kapadia (2003b) and Carr and Wu (2008), supports that a negative aggregate variance risk premium is priced in individual VRP. Under Merton (1973)’s ICAPM framework, a negative systematic variance risk is priced due to option investors demanding a premium for bearing positive market volatility shocks, which are considered as unfavorable shocks to the investment opportunity. In this paper, the first question we try to explore is whether other risk factors that impact investment opportunity, such as market equity risk and jump risk, are also priced in individual equity option, and if so, how much of the cross-sectional variation of individual VRPs can be explained by these priced systematic factors? We are the first paper to show that, besides market variance risk, both market equity risk and market jump risk are significantly priced in individual VRPs. Why do we consider market equity risk and market jump risk as the relevant candidates in explaining the cross-section of individual VRPs? First, it’s a well documented fact that there is a significant negative correlation between market return and market variance. The stochastic variation of variance, which causes the variance risk premium, could either come from its correlation with return variation, or come from an independent variance risk factor, or come from both. Thus, the correlation between variation of return and variation of variance provides a channel that market equity return risk premium enters the the risk premium of variance. With the assumption that the individual stocks’ return and variance variations (which are also correlated) are driven by market level return and variance shocks (and idiosyncratic shocks), the individual VRP should contain components that are caused by the stochasticity of both market return and variance shocks.

2

VRP in our definition can be considered as the delta-hedged options return, so it may seem counter-intuitive that equity risk should be priced in VRP. However, as pointed out by Duarte and Jones (2007), since equity shock and volatility shock are significantly correlated, the regular delta hedging is not able to completely hedge away the underlying equity risk. Therefore, both market equity risk premium and variance risk premium play a role in determining delta-hedged equity option returns, which includes individual VRP. Carr and Wu (2008) use time series regression to test if market equity risk premium can explain the individual VRP, but they do not study whether equity risk premium is priced cross-sectionally in individual VRP. Second, option pricing literature such as Pan (2002) finds that a large market jump risk premium significantly improves the model fit for index options prices. A strand of literature on index option pricing also advocates the “stochastic volatility correlated-jump” model, where both the return and variance process have jump components that are correlated.4 In addition, Bollerslev and Todorov (2011) find that a large component of index options’ variance risk premium is actually caused by fear of market tail/jump risk. Motivated by these index options studies, we test whether the market jump risk is also priced in individual options by studying its contribution in explaining cross-sectional VRP. The results help us understand whether the jump risk priced in index options is really systematic and it provides implications for the modeling of individual options. Our main results about whether market risks are priced in VRP are based on Fama and MacBeth (1973) two-step regressions approach, complemented with results from sorting portfolios by systematic risk exposures (β). Based on the model-free (monthly) VRP constructed for more than 4,000 firms for the period 1996-2009, we find robust and significant results that a positive market equity risk premium and a negative market jump risk premium are priced in individual VRP. We also confirm with previous literature that a negative market variance risk premium is priced significantly. The result holds true for both full-sample and the subsample of S&P 500 component firms, which are large-cap firms and usually have more liquid options. Among the three factors, the aggregate variance risk alone can explain the most of 4

See Eraker, Johannes, and Polson (2003) and Broadie, Chernov, and Johannes (2007).

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the VRP cross-sectional variations (about 25%). The aggregate jump risk alone explains the second most (more than 19%), market equity risk alone can explain about 16%, and the three factors combined can explain 36% of the VRP cross-sectional variations for full sample and 45% for the S&P 500 component firms. The pricing significance of the market-level risk factors is robust after controlling for firm-level variables, including size, book-to-market ratio, momentum (previous 6-month stock return), illiquidity, idiosyncratic volatility and options trading volume. We also compare the performance of our three-factor model (equity risk, variance risk, and jump risk) with other standard pricing kernels that have been proposed in the equity literature. The alternative pricing models include CAPM, the polynomials of market excess return model proposed by Dittmar (1999) and Harvey and Siddique (2000), the market excess return and market variance two-factor model tested by Ang, Hodrick, Xing, and Zhang (2006), and Fama-French three-factor model. Our model explains a higher percentage of cross-sectional individual VRPs than all of these alternative models. The second contribution of this paper is that we use a non-parametric method to decompose the individual VRP into three components: the common unexplained component, systematic exposure component and idiosyncratic component for each firm in each month. By decomposition, we are able to extract the time series of each component conditionally. We find that after accounting for the three market level risks, there is a significant negative and time varying common VRP that is left unexplained. To further explore whether the common unexplained component is due to market inefficiency or some uncovered market level variable, we study whether it contains any information about future market by testing its predictability of future equity index excess return and future growth rate of macroeconomic variables. We find that the common unexplained component of individual VRPs significantly predicts future 6-month to 2-year stock market return and future 3-month to 2-year economic variables’ growth (including industrial production, capacity utilization, total retail sales excluding food services, civilian unemployment rate, and total non-farm payroll). The common unexplained component of individual VRP is also found to be significantly related to stock market liquidity, 4

options implied market skewness and kurtosis, however, only 10% of its time series variation can be explained by these market level variables. Our study is also related to some other literature. First, it bears implication on the pricing mechanism of individual options by identifying the common systematic risk factors priced in both index options and individual options. This contributes to the literature documenting the connection and difference between pricing structures of index options and individual options, such as Bakshi, Kapadia, and Madan (2003) and Duan and Wei (2008). Second, there is literature that provides alternative view on the pricing of options. For example, Bollen and Whaley (2004) and Gˆ arleanu, Pedersen, and Poteshman (2009) argue that the option-pricing puzzle can be explained by the demand-pressure effects. Our paper follows a different framework and our results imply that individual option prices can be explained by market risk factors rather than traders’ behaviors and mispricing. Third, some recent literature studies the information contents in individual VRP. Han and Zhou (2010) document the empirical finding that high VRP stocks outperform low VRP stocks by over 1% in the next month. Wang, Zhou, and Zhou (2009) find that individual VRPs have a significant explanatory power for credit default swap spreads. Our studies of the cross-sectional variation and the components of individual VRPs provide a way to identify the underlying source of the information in individual VRP. Lastly, literature has not agreed on the existence and sign of individual VRP, and our paper contributes to this area.

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Differing from most literature, our paper defines the individual

VRP as the return of variance trading, which is more intuitive and has better distributional properties; we find that under 20% of all firms have significant negative VRP and 35% of all firms have significant positive VRP. We also document that the cross-sectional distributional properties of VRP are different when we define it differently. The rest of this paper is organized as follows. Section 2 provides the theoretically frame5 Bakshi and Kapadia (2003b) find that on average the delta hedging strategy of 25 individual equity options loses a statistically significant 0.03% of underlying asset value. Carr and Wu (2008) report that estimates of mean log variance risk premia are significantly negative for 21 out of 35 individual stocks. Driessen, Maenhout, and Vilkov (2009) tests the significance of VRPs on the constituents of S&P 100 index and report that the null of an insignificant VRP cannot be rejected for the majority of these firms. Cao and Han (2010) study the average delta-hedged option returns of 5000 firms over 11 years and find that the returns are significantly negative for most stocks.

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work. Section 3 describes the data and the constructions of variance measures and VRP. Section 4 investigates whether standard market level risk factors are priced in individual VRPs. Section 5 provides the empirical results of the decomposition and document the properties and information of the unexplained VRP and implied market price of risk. Section 6 concludes.

2

Framework and Decomposition of Individual Variance Risk Premiums

In this section we describe the concept of variance swap rate and the definition of variance risk premium; with the assumption of no arbitrage and the existence of a pricing kernel that prices all traded assets, we provide the representation for the determinants of variance risk premium in a pricing kernel framework. We also describe the theoretical framework for the Fama-Macbeth empirical analysis and the decomposition of individual variance risk premium. A return variance swap is an over-the-counter financial derivative that enables investors to trade the return variance of the underlying asset. It has zero net market value on initiation date. At maturity, the payoff to the long side of the swap is equal to the difference between the realized variance over the life of the contract and a constant called the variance swap rate: RVt+1 − SWt , where RVt+1 denotes the realized variance of the underlying asset over the life of the contract and SWt denotes the fixed leg of the contract (i.e. the variance swap rate or the price of the variance). Let IVt denote the risk-neutral expectation of next-period variance at time t (IVt = EQ [RVt+1 ], which can also be called “implied variance”), and we have the relation IVt = SWt (1 + Rf ) (where Rf denotes the net risk-free return). Let Mt denote the stochastic discount factor, then for firm i we have the pricing relation:

IVi,t = SWi,t (1 + Rf ) = (1 + Rf )Et [Mt+1 RVi,t+1 ], IVi,t = (1 + Rf ) (Et [Mt+1 ]Et [RVi,t+1 ] − Covt [Mt+1 , RVi,t+1 ]) ,

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(1)

For the pricing relation of risk-free rate, we have:

Et [Mt+1 (1 + Rf )] = 1,

(2)

Combining equation (1) and equation (2) yields the following relation:

Et [RVi,t+1 ] − IVi,t = (1 + Rf )Covt [Mt+1 , RVi,t+1 ].

(3)

Here we define the individual variance risk premium (VRP) as the return of taking one unit long position in the variance of an individual stock:

VRPi,t =

Et [RVi,t+1 ] − IVi,t . IVi,t

(4)

Most of the literature define variance risk premium simply as the difference between the physical expected variance and risk neutral expected variance (i.e. Et [RVi,t+1 ] − IVi,t ), which can be viewed as the difference between expected payoff and the price in the context of variance swap.6 In contrast, our definition of VRP measures the expected return of a long position in variance swap. Since our focus is the cross-sectional variation of VRP, we propose that the return measure - the difference between RV and IV scaled by IV - has better distributional property than the difference between RV and IV (the profit); this is also consistent with the way standard asset pricing literature conducts empirical tests on stocks, where stock returns rather than the payoffs are used. It’s also worth noting that variance swap is not the only tool to trade a stock’s return variance. As documented by Derman, Demeterfi, Kamal, and Zou (1999), delta hedging a specific static option portfolio provides investors constant exposure to the underlying asset’s variance and thus replicates the position in a variance swap.7 As a result, the variance risk 6 See, for example, Carr and Wu (2008), Bollerslev, Tauchen, and Zhou (2009), and Drechsler and Yaron (2010). 7 This static option portfolio comprises of out-of-the-money options of all strikes and it replicates the Log contract, which pays the logarithm of stock return at maturity. As documented by Britten-Jones and Neuberger (2000) and Derman,R Demeterfi, Kamal, and Zou (1999), the option portfolio can be expressed as: R 1 C(S0 ; K, T )dK + K≤S0 K12 P (S0 ; K, T )dK, where K denotes the strike and K12 denotes the weight of K≥S0 K 2

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premium defined in this paper can be considered either as the return of a long position in variance swap, or as the return of delta hedging the replicating option portfolio. Similarly, the return of any option trading strategy whose equity risk is hedged away is determined by the variance risk premium of the underlying asset and the strategy’s specific variance exposure (vega), and this is the underlying rationale that existing literature implies the properties of variance risk premium by studying the delta-hedged at-the-money options return or staddle returns (for example, see Coval and Shumway (2001), Bakshi and Kapadia (2003a), and Cao and Han (2010)). Assuming that the stochastic discount factor Mt is a linear combination of s state variables and can be expressed in the following form,

Mt = λ0 +

s X

λj fj,t ,

(5)

j=1

we can substitute the stochastic discount factor into equation (3) and obtain the following relation,

Et [RVi,t+1 ] − IVi,t = (1 + Rf )Covt [λ0 +

s X

λj fj,t+1 , RVi,t+1 ]

j=1

(6)

Expressed in terms of individual VRPs, Et [RVi,t+1 ] − IVi,t IVi,t

= (1 + Rf )Covt [λ0 +

s X

λj fj,t+1 ,

j=1

VRPi,t = (1 + Rf ) = (1 + Rf )

s X j=1 s X

RVi,t+1 ], IVi,t

λj Covt [fj,t+1 , VRPi,t+1 ] λj Vart [fj,t+1 ]βij,t ,

(7)

j=1

where βij,t = Covt [fj,t+1 , VRPi,t+1 ]/Vart [fj,t+1 ] and it measures exposure of stock i’s VRP to each option.

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the jth state variable. Empirically, equation (7) can be tested using Fama and MacBeth (1973) two-step regressions. After we estimate VRPi,t from stocks and options data, as the first step we estimate βbij,t from time series regressions of VRPi on market-level risk factors fj s. Then in the second step, we do a cross-sectional regression of VRPi on βbij,t at each time t to estimate the market price of risk. The second step can be expressed as:

VRPi,t = α bt +

s X

θbj,t βbij,t + i,t , for i=1, 2, 3, ..., n,

(8)

j=1

where θbj,t denotes the estimates of market price of risk j at time t and n denotes the number of firms. In the tables that provide the empirical results of Fama-Macbeth regressions, we report the time series average of α bt and θbj,t and the associated Newey-West t values. We use equation (8) as the basis to interpret and decompose the individual VRP. Based on the equation, individual VRP can be considered as the summation of three components: • αt , which can be seen as the average/common unexplained portion of all individual VRPs; we denote it as UVRP; • the systematic component

Pb b θj,t βij,t , which is the market price of risks multiplied by the

exposure of firm i’s VRP to systematic risk factors; • the idiosyncratic VRP component i,t , which is the portion that cannot be explained by market risk factors and is not in the common αt . By the decomposition of individual VRP, we are also able to measure the magnitude of VRP explained by each systematic risk factor. Some recent literature find that individual VRPs contain substantial information about the underlying firms, such as predicting the crosssectional stock returns (Han and Zhou (2010)) and explaining cross-sectional CDS variations (Wang, Zhou, and Zhou (2009)). The decomposition we propose can be utilized to identify whether the source of VRP’s information content is its systematic component or idiosyncratic 9

component; we show an application of the decomposition in the end of this paper.

3

Data and Construction

In this section, we describe the stock and options data and the approach to construct the physical and risk-neutral expectations of integrated variance, which are then used to compute the individual VRPs. And we document the cross-sectional properties of VRP and its relation with firm-level variables.

3.1

Data Descriptions

The data sample spans from January 1996 to October 2009. The options related data (including quotes, implied volatility, and greeks) and the zero coupon yield data are from Ivy DB database of OptionMetrics. The equity price data, including daily closing price, daily and monthly return and discrete dividend distribution data for individual stocks and S&P 500 index, are from CRSP. The company characteristics data and index component data are from COMPUSTAT. The Fama-French three factors and the momentum data are downloaded from WRDS. The VIX (risk-neutral expected variance index of S&P 500) is downloaded from CBOE Website. We construct 30-day implied variance from option prices and construct expected 30-day physical variance from equity price data. We only use end-of-month measures of all variables and our analysis are based on monthly interval. We use the following procedures to clean the individual options data (in the following order): (1) only keep options with maturity between 7 days and 180 days; (2) only keep quotes with strictly positive bid and strictly positive bid-ask spreads; (3) delete options quotes with zero open interest; (4) delete options quotes with missing implied volatility or with implied volatility less than 1%; (5) delete options quotes with missing delta; (6) delete option quotes that we do not find matched closing price data in CRSP; (6) delete options with intrinsic values higher than the mid-quotes; (7) if two calls or puts with different strikes have identical mid-quotes,

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i.e. c(St ; T, K1 ) = c(St ; T, K2 ) or p(St ; T, K1 ) = p(St ; T, K2 ), we discard the further away from the money quote; (8) require at least two quotes that correspond to two different maturities (between one week and 180 days) at each available strike on each date (for interpolation); (9) retain only out-of-the-money (OTM) options; (10) require at least three available OTM strikes for each security on each date and require quotes for both OTM call and OTM put are available.

3.2

Measures of Physical and Risk-neutral Expected Variances

Following Carr and Wu (2008), we use the following formula to estimate the realized variance of each security or index at month t: n

RVt =

252 X 2 Si ln , n Si−1

(9)

i=1

where Si is the security’s closing price on the i’th trading day of each month and n denotes the number of trading days for that month. To obtain an estimation of the physical expectation of RVt+1 , we use a variation of the model advocated by Andersen, Bollerslev, and Diebold (2006) and also used in Bollerslev, Tauchen, and Zhou (2009). They forecast the one-month-ahead variance as a linear function of the current daily, weekly, and monthly realized variances. The advantage of this model is its simplicity and empirical good performance. In addition to the regressors used in their papers, we add in the past 15-day, 45-day, 60-day and 75-day realized variances. We fit an OLS regression using full sample daily observations for each firm and use the model forecasted one-month ahead realized variance as the physical expectation of realized variance of each firm for each month:

\ Et [RV bi + i,t+1 ] = α

n X

i d β i,j RVt (j),

j=1

where the dependent variable is the future 30-day realized variance and RVt (j) refers to the

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realized variance at time t computed from previous j days stock returns in the manner of equation (9); RVt (j) includes 1-day, 7-day, 15-day, 30-day, 45-day, 60-day and 75-day realized variances (RVt (1) refers to the square of one-day log-return at time t). We follow the Bakshi, Kapadia, and Madan (2003) (BKM (2003)) method to construct the model-free 30-day risk neutral expected variance. In the literature, there are two ways to compute the model-free risk-neutral variance from observed option prices: one proposed by Britten-Jones and Neuberger (2000) and the other documented by Carr and Madan (1998) and BKM (2003). As documented by Du and Kapadia (2011), these two approached are very close and measure the risk-neutral expected quadratic variation accurately under different model assumptions, and the method documented by BKM (2003) is more accurate for short-term options when the underlying asset process has jumps. Denoting C(St ; K, T ) and P (St ; K, T ) as the call and put of strike K and remaining time to expiration T , BKM (2003) demonstrate
2 IV = EQ e−rT (ln ST /S0 )2 − EQ e−rT (ln ST /S0 ) , where

Q

E (ln ST /S0 )

2

= e

rT

EQ (ln ST /S0 ) = erT

 Z 2(1 + ln(S0 /K)) 2(1 − ln(K/S0 )) C(S0 ; K, T )dK + P (S0 ; K, T )dK , K2 K2 K<S0 K>S0 Z  Z 1 1 − 1 − erT C(S ; K, T )dK + P (S ; K, T )dK , 0 0 2 2 K>S0 K K<S0 K

Z

and r is the constant risk-free rate. When price quotes for 30-day options are not available, we linearly interpolate the implied volatilities of two adjacent maturities to obtain the 30-day implied volatility. To construct the model-free options implied variance (IV), it requires a continuum of strikes. Following Jiang and Tian (2005) and Carr and Wu (2008), we interpolate the implied volatility across the range of observed strikes using a cubic spline, and assume the smile to be flat beyond the observed range of strikes. Using the interpolated curve, we compute 1001 option prices over a range of zero to three times the current stock price and then construct the model-free implied variance with the above formulas. To increase data availability and reduce estimation error, we take the average of daily implied variance estimations of the last week in each month as the end-of-month implied variance estimation. 12

3.3

Properties of VRP

In this section we report the summary statistics of VRP and document its relation with firmlevel variables. After construction of expected physical variance and risk neutral variances, we compute \ [ i,t = Et [RV measures of firm i VRP as: VRP i,t+1 ]/IVi,t − 1. We retain firms with at least 12 available end-of-month VRP estimation in the whole sample period and it leaves us 4,372 firms with 253,933 end-of-month firm-month observations in total. Table 1 provides the summary statistics. Panel A of Table 1 reports the summary statistics of realized variance, implied variance, stock return, firm size (SIZE) and firm book-to-market ratio (B/M), along with return, realized variance and implied variance for S&P 500 index. For individual firms, we first compute the time series average of each variable, and then report the summary statistics of the cross-sectional distribution across all firms. For variables of S&P 500 index, we report the time series summary statistics. The average of implied variance and realized variance for individual firms are very close, respectively 0.4225 and 0.4251. In contrast, the mean of realized variance for S&P 500 index is 0.0630, much higher than the mean of its implied variance 0.0432; this is consistent with a large negative market variance risk premium, as discussed by many literature including Bakshi and Kapadia (2003a) and Carr and Wu (2008). The implied variance of individual firms has smaller variations than the expected physical variance, and both variance measures show positive skewness and excess kurtosis cross-sectionally. Panel B of Table 1 reports the fit statistics including t-statistics of intercept and predictors, adjusted R2 , and root mean squared error of OLS regressions for realized variance prediction. We regress the future 30-day realized variance on the predictors using full-sample daily data for each firm following equation (10). The statistics reported in the table are the cross-sectional average of regression statistics for all firms. We compare different models’ performance by adding in more explanatory variables (the realized variance computed from previous returns). By construction, the explanatory variables are highly correlated, so the t statistics may not be reliable. By comparing the adjusted R2 , we choose the model that has the most predictors, in

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which we include 1-day log return square and realized variances constructed from past 7-day, 15-day, 30-day, 45-day, 60-day, and 75-day daily returns. On average, there are about 2210 observations for each firm used in the forecast regression. Panel C of Table 1 reports the cross-sectional percentiles of VRP means and the associated Newey-West (1987) t-values. In addition, we report the cross-sectional percentiles for individual variance risk premium defined in two alterative ways: the difference between expected physical variance and risk-neutral variance (E[RV] − IV), and the difference between natural logarithm of the two variances (ln E[RV] − ln IV, i.e. Log VRP).8 The median values of E[RV] and IV are very close (0.3307 and 0.3342), and E[RV] shows a greater variation and a wider range of values. More than 25% of all firms have significant negative E[RV] − IV and less than 15% firms have significant positive E[RV] − IV estimation. However, based on the t-values of VRP and Log VRP, under 20% of all firms have significant negative variance risk premium and 35% of all firms have significant positive variance risk premium. The histograms of E[RV] − IV, VRP and Log VRP (time series means) for all firms are shown in Figure 1, from which we can more clearly observe the distributional difference when the individual variance risk premium are defined differently. VRP and Log VRP are visibly more normally-distributed than E[RV] − IV. E[RV] − IV is more negatively skewed and VRP and Log VRP are slightly more positively skewed. We also study the relation between VRP and firm-level variables and report the results in Table 2. The results are from panel regressions with all available firm-month observations and the t-statistics are based on standard errors clustered by firms. Based on univariate regressions results, we observe that more negative VRPs are associated with smaller firm size and higher book-to-market ratio, which are both firm-level risk factors(Fama and French (1992)). Firms that experience worse returns in the past month have significantly more negative VRP. VRP is positively related to realized variance over previous one-month and negatively related to realized variance over previous one-year. More negative VRP is also related to positive historical skewness and low historical kurtosis. And interestingly, the idiosyncratic 8

Log VRP is the logarithm of one plus VRP defined in return term.

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variance defined relative to Fama-French 3-factor model is negatively related to individual VRP in the multivariate regression.9 We also find that more illiquid stocks tend to have more negative VRP, but the relation disappears after controlling for other effects.10

4

Pricing of Market Risk Factors in Individual VRPs

In this section, we answer the question whether market level risk factors are priced in individual VRPs, and study how much of the cross-sectional variation of individual VRPs can be explained by market level risk factors. We use both portfolio sortings and Fama-Macbeth two-step regressions to conduct the empirical analysis. We focus on the three market level risk factors including market equity risk, market variance risk and market jump risk, and use different proxies of aggregate variance risk and jump risk for robustness. In the Fama-Macbeth regression, we also compare the performance of the equity, variance, jump three-factor model with other standard pricing kernels.

4.1

Sorting Portfolios

We first attempt to find evidence of priced risk factors from portfolio sortings. By identifying a monotonic pattern in mean VRP of portfolios constructed by ranking firm’s exposure to risk factors, we may be able to imply the significance and sign of market price of risk factors. Essentially, we first regress monthly VRP of each individual firm on market risk factors to estimate its exposure to these factors (β) , expressed in the following equation,

e VRPi,t = αi + β1 RM,t + β2 VM,t + β3 JM,t + i , t = 1, 2, ..., T, 9

(10)

Following Ang, Hodrick, Xing, and Zhang (2006), we regress daily returns of individual stock of each month on the daily Fama-French three factors and use the mean of sum square errors as the estimate of idiosyncratic variance for that month. PD 10 1 Following Amihud (2002), we define stock i’s illiquidity of each month as ILLIQ = D d=1 |Ri,d |/Vi,d Pi,d , where D refers to the number of trading days in each month, and Ri,d , Vi,d and Pi,d respectively denote the daily return, daily trading volume, closing price of stock i on dth trading day of that month.

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e denote the market equity risk, V RM M denotes the market variance risk factor, and JM denotes

the market jump risk factor. And then we sort firms into portfolios based on the ranking of the βs and study the mean VRP of these portfolios.

4.1.1

Proxies for Risk Factors

e and use the logarithm We use the monthly excess return of CRSP value-weighted index for RM

of monthly realized variance of CRSP value-weighted return as a proxy for market variance risk. We mainly use two measures as proxies for jump risk. The first one is the jump/tail risk measure proposed by Du and Kapadia (2011), which is constructed from S&P 500 index options and is essentially to the first order determined by the implied third moment of the market distribution and in their framework related to the risk-neutral expected aggregate jump intensity. Besides DK(2011)’s jump/tail risk measure (JDK ), we also use the fear measure proposed by Bollerslev and Todorov (2011) (JBT ).11 Both of these measures are positive and higher values represent greater jump/tail risk. As a robustness check, we also use the monthly innovations in BT(2010)’s fear measure (∆JBT ) and DK(2011)’s jump/tail risk measure (∆JDK ) as jump proxies.12 As a further robustness check for the risk proxies, we construct the variance risk and tail risk mimicking portfolios following Ang, Hodrick, Xing, and Zhang (2006). We regress the daily innovation in VIX and daily innovation in the two jump risk measures respectively on the daily excess returns of 25 basis Fama-French size and book-to-market portfolios. The regression is specified as below, ∆Vt (or ∆Jt ) = c + b0 Xt + ut ,

(11)

where ∆Vt and ∆Jt denote the daily innovation in VIX and the jump risk proxies respectively, and Xt represents the excess returns on the 25 Fama-French size and book-to-market base 11 Bollerslev and Todorov (2011) fear measure basically measures the variance risk premium contributed by fear of negative jump risk. We thank Viktor Todorov for sharing the data. 12 We define the monthly innovation as the difference between the end-of-month measure and the beginningof-month measure divided by number of trading days in that month.

16

portfolios. The coefficient b can be interpreted as weights in a zero-cost portfolio. We then use bb0 Xt as the monthly returns on the variance/jump risk mimicking portfolios (∆VIXmim , mim and ∆J mim ). By constructing the mimicking portfolio, we obtain the risk factors ∆JDK BT

proxies that can be traded and whose data are available for the whole period. Figure 2 shows the time series plots of the market risks factors for the sample period January 1996-October 2009. Panel A plots the monthly value-weighted excess return of all CRSP stocks, which is a proxy for market equity risk. Panel B plots the monthly log realized variance of CRSP value-weighted return and the return of mimicking portfolio for innovation in VIX, both of which are proxies for market variance risk. Panel C plots the jump/tail risk measure proposed by Du and Kapadia (2011) (JDK ) and the return of mimicking portfolio for innovation in JDK .Panel D plots the jump/tail risk measure proposed by Bollerslev and Todorov (2011) (JBT ) and the return of mimicking portfolio for innovation in JBT , which are used later in the Fama-Macbeth analysis for robustness check.

4.1.2

Results of Sorting Portfolios

In this section, all the results reported are based on the following market risk factor proxies: the monthly value-weighted excess return of all CRSP stocks (equity risk), the monthly log realized variance of CRSP value-weighted return (variance risk), and the jump/tail risk measure proposed by Du and Kapadia (2011) (JDK , jump risk). We estimate βs in two different ways: (1) a constant β using full-sample univariate regressions (Table 3); β for each firm and each market risk is estimated by the following univariate regression using full sample monthly observations: VRPi,t = αi + βfM,t + i , t = 1, 2, ..., T , where fM refers to the market equity risk, variance risk, or jump risk; (2) a rolling β using previous 60-month multivariate regressions (Table 4); βs for each firm in month T are estimated simultaneously by the following multivariate regression using previous 60-month observations: e VRPi,t = αi + β1 RM,t + β2 VM,t + β3 JM,t + i , t = T − 59, T − 58, ..., T − 1, T . By running the

multivariate regression to estimate β simultaneously, we are able to measure the comovement

17

of VRP with each risk factor controlling for other risk exposures. After we have the estimation of β, for each month of our sample period, we divide all firms into deciles based on the crosssectional ranking of firms’ βs, take the average of firms’ VRP in each decile, and finally report the time series average of each decile’s mean VRP.13 We also report each decile portfolio’s alphas with respect to CAPM, Fama-French 3-factor model, and Carhart 4-factor model. The mean VRP, alphas and the Newey-West (1987) t-values of the long decile 10 and short decile 1 zero-cost strategy is also reported. e . There Panel A of Table 3 reports the results of sorting by β on market equity risk RM

is an increasing pattern of VRP from decile 1 to decile 10 sorted by market equity β. The portfolio VRPs increases from 0.1150 of decile 1 to 0.5371 of decile 10. The 10-1 portfolio has significantly positive mean VRP and alphas. Panel B of Table 3 reports the results of sorting by β on market variance risk VM . We can observe a strictly decreasing pattern of VRPs when we move from decile 1 to decile 10. The strategy that longs decile 10 portfolio and shorts decile 1 portfolio yields a mean of -0.6304 and is highly significant. The monotonic pattern and significance of 10-1 strategy remain after we compute the alphas using different models. The Newey-West (1987) t-values are all between -6 and -5. Another interesting observation from Panel B is that the firms with the largest and lowest variance betas seem to be firms of smaller sizes. Especially, the firms that have the most positive exposure to market variance risk are the smallest firms. Panel C of Table 3 reports the results of sorting by β on market jump risk JM . We can also observe a general decreasing pattern in the portfolio VRPs moving from decile 1 to decile 10. The mean VRP of 10-1 portfolio is about -0.4691 and the Newey-West t-values for both mean and alphas are between -5 and -4. Similar to the case of variance βs, we also observe an inverse U-shape pattern for the decile portfolios’ sizes: the firms that have the highest and lowest exposure to jump risk seem to be small firms. Since firm’s risk exposures vary over time, full-sample constant βs may not be able to capture the cross-sectional variation in risk exposures at different time points. Also, due to the 13

To remove the influence of outliers and estimation errors on the reported results, we delete the top 1% and bottom 1% of cross-sectional VRP for each month. The results are similar if we keep the top 1% and bottom 1%.

18

correlation between risk factors, β computed from univariate regressions may not distinguish the exposures to different risk factors clearly. As a robustness check, Table 4 reports the VRP means and alphas of decile portfolios sorted by βs computed from previous 60-month observations using multivariate regressions. We require that there are at least 36 observations (out of 60) in the regression for β to be included in the sample.14 The results of panel A are different from Panel A of Table 3; there is not a monotonic pattern of VRP from decile 1 to decile 10 sorted by market equity β, and instead, there appears to be an interesting U-shape pattern: the portfolio VRPs decreases from 0.3580 of decile 1 to 0.1176 of decile 6 and then starts to increase. The pattern for portfolios sorted by market variance risk β and portfolios sorted by market jump risk β are very similar to those in Table 3: they both show an overall decreasing pattern. Firms that are most positively exposed to variance risk or jump risk are the smaller firms. And in Table 4, the firms that are retained in the sample are larger firms as the mean firm size in each portfolio is larger than that of Table 3. It appears that book-tomarket ratio decreases with firms’ variance risk exposure and increases with firms’ jump risk exposure, but the pattern is not strong. To check whether the market risk exposure are proxies for firm-level variables, we report double-sorting results of each risk factor controlling for other risk factor exposures and firmlevel variables in Panel D of Table 4. The significance and risk premium sign of variance risk and jump risk remain the same controlling for β of the other two risk factors and controlling for firm size, book-to-market ratio, past returns (momentum) and idiosyncratic volatility. We still do not observe any support for a priced market equity risk premium from double sorting results. And interestingly, the 10-1 portfolio average VRP and alphas for variance risk controlling for jump risk β is much smaller than the result of single sorting and the 10-1 portfolio average VRP and alphas for jump risk controlling for variance risk β is slightly higher. In conclusion, the results of portfolio sortings show that market variance risk and market 14

As a result of the data requirement, the sample of Table 4 is much smaller than that of Table 3: 1996-1998 (the first three years) data are used only for estimating beta but not used in the formed portfolios, and firms with low availability of time series VRP (most of which are smaller firms that have illiquid options) are also excluded by design.

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jump risk are significantly priced in individual VRP, and the market price of both risks are negative. These results hold robustly no matter we use full-sample univariate betas or 60month multivariate betas, and the results are robust controlling for firm-level variables, such as size, book-to-market ratio, momentum and idiosyncratic volatility. The pricing of market equity risk in VRP is mixed from portfolio sorting results. From full-sample univariate beta, a positive market price of equity risk appears to be significantly priced in VRP. But based on 60-month multivariate beta, the pattern does not exist.

4.2

Fama-Macbeth Regressions

The advantage of portfolio sorting is that it provides a visible picture of the trend of portfolio means. However, it may not capture the true pattern, as usually we need to control for other risk factor exposures. Multiple sortings may alleviate the problem, but since we have more than two factors to study, multiple sorting is as efficient as Fama-Macbeth regression, which considers β of all risk factors simultaneously. In this subsection, we use Fama-Macbeth regressions to study the pricing of systematic risk factors and how much the standard pricing kernels can explain the cross-sectional variation of individual VRP. We use the model (mainly equation (7)) in Section 2 as the framework. We estimate βs using previous 60-month multivariate regressions following equation (10) and require a minimum of 36 observations for β to be included in the cross-sectional regression. While focusing on studying equity risk, variance risk and jump risk, we test the following standard pricing kernels that have been proposed by the equity literature in explaining crosssectional stock returns (Mt is the pricing kernel defined in Section 2): e . The only risk factor is market equity risk. • CAPM: Mt = λ0 + λ1 RM,t e e )2 + • The polynomials of market excess return model: Mt = λ0 + λ1 RM,t + λ2 (RM,t e )3 . The model is advocated by Dittmar (1999) and similar model without the λ3 (RM,t

cubic market return term is used by Harvey and Siddique (2000).

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e +λ V • Market excess return combined with market variance factor: Mt = λ0 + λ1 RM,t 1 M,t .

This pricing kernel has been tested by Ang, Hodrick, Xing, and Zhang (2006). • Market excess return combined with market variance factor and market jump risk factor: e Mt = λ0 + λ1 RM,t + λ1 VM,t + λ1 JM,t , which is this paper’s focus.15 e • Fama-French 3-factor model: Mt = λ0 + λ1 RM,t + λ2 HMLt + λ3 SMBt . We include the

model mainly for comparison and robustness check. Table 5 reports the results for Fama-Macbeth two-step regressions. Panel A reports the results for full-sample.16 From Panel A, we can see that the market equity risk is consistently positively priced in VRP across for all models. The market excess return square term and the market excess return cubic term are respectively negatively and positively priced. Market variance risk is significantly negatively priced for both measures we use: the logarithm of CRSP realized variance and the mimicking portfolio for VIX innovations. The estimate and significance of market price of risks are consistent across all models. The average market price of variance risk is between -1.34 and -1.18 when we use the logarithm of CRSP realized variance and is between -13.19 and -11.73 when we use the variance risk mimicking portfolio. And the result that market jump risk is negatively priced is also consistent across all models with all proxies used. The models that have the highest adjusted R2 are the models that incorporate both variance risk and jump risk. Market variance risk alone can explain the most of the cross-sectional variation, which is 25.32%. Market equity risk alone can explain 16.56% of the cross-sectional variation, and adding in the variance risk factor can increase the adjusted R2 to 32.82%. Further adding jump risk factor in the model increases the adjusted R2 , but not by a significant margin. The mimicking portfolios for variance risk and jump risk don’t perform as well as the models with the raw risk factor proxies, as their adjusted R2 are generally lower. We also observe that for the models that explain the highest percentage of cross-sectional 15

Cremers, Halling, and Weinbaum (2011) explore the pricing of aggregate variance risk and jump risk in cross-sectional stock returns. They find support for a priced aggregate variance risk and do not find evidence that aggregate jump risk is priced. 16 As we still require at least 36 observations available for estimating βs, the full sample in Table 5 has only 112,042 firm-month observations and the number of firms reduce to about 2,298 from more than 4,000 firms.

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variation, the intercepts are significantly negative. Panel B of Table 5 reports the results for the sample comprised of only S&P 500 component firms, which are generally the largest firms with liquid options and cover about half of the fullsample observations in Panel A. The pricing results are very similar to that of Panel A. Still, market variance risk explains the most of the cross-sectional variation of individual VRP (about 34%). In unreported results, we find that jump risk is the second best factor in explaining the cross-sectional variation. Different jump proxies explain about 25.5%-29.85% of the crosssectional variation of the S&P 500 sample. There are two minor differences from Panel A: firstly, the adjusted R2 is generally higher (as expected); the models with equity risk, variance risk and jump risk can explain more than 45% of the cross-sectional variation; secondly, the intercept of all models are significantly negative. Fama-French 3-factor model has significantly lower adjusted R2 than the equity, variance and jump risk model, and the HML factor is not significant in the cross-sectional regression.17 Panel C reports the Fama-Macbeth regression results with firm-level control variables for the sample of S&P 500 component firms. The control variables include firm size, book-tomarket ratio, previous returns (momentum), idiosyncratic volatility, the Amihud (2002) illiquidity measure and the monthly total option trading volume. Consistent with the results in Table 2, most of these firm-level control variables are significant, however, they do not subsume the effects of market risk exposures. In conclusion, from Fama-Macbeth regressions results, we find significant evidence that in individual VRP, market equity risk is positively priced, market variance risk is negatively priced, and market jump risk is negatively priced. The results are robust for different proxies for the variance risk and jump risk, are robust when we consider the sample of only S&P 500 component firms, and are robust after controlling for many firm-level variables. The market variance risk is the most important among these risk factors in explaining cross-sectional VRP, followed by market jump risk and then market equity risk. Other standard pricing kernels do 17

To study whether the pricing of jump risk is solely caused by the impact of crisis period, we repeat the Fama-Macbeth regressions for the subsample that excludes the NBER-recession period, and the significance of jump risk is unchanged.

22

not perform as well as the equity-variance-jump three factor model.

5

Decomposition of Individual VRP

The results of last section show that there is still a considerate portion of VRP that cannot be explained by the proposed risk factors, and we observe a significant negative alpha in the second step of Fama-Macbeth regression. How do we interpret the unexplained portion of VRP? Is it related to investors’ sentiment or market inefficiency such as traders’ irrational behaviors, or is there any systematic factor that we have missed in our analysis? In this section, we extract the time series of unexplained common individual VRP from the Fama-Macbeth regression and study whether it contain information about the future state of financial market. As pointed out by Cochrane (2001), any variable that forecasts asset returns (“changes in the investment opportunity set”) or that forecasts macroeconomic variables could be a candidate state variable in the ICAPM framework. If we find that the unexplained average individual VRP is related to future stock market return or macroeconomic variables, then it provides more support for a missing factor explanation or at least it requires a systematic story to interpret the unexplained VRP.

5.1

The Unexplained Component of VRP

For the empirical analysis in this section, we use the S&P 500 components firms only, so the data sample is the same as that of Panel B in Table 5. Throughout this section, we use valueweighted CRSP excess return and the logarithm of its monthly realized variance as the proxies for market equity risk and market variance risk, respectively. As we mentioned in Section 2, Fama-Macbeth regression naturally provides us a framework to interpret the individual VRP. Using the notations defined in Section 2, individual VRP for firm i can be expressed as:

VRPi,t = α bt + θbE,t βbiE,t + θbV,t βbiV,t + θbJ,t βbiJ,t + i,t

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(12)

where θbE,t , θbV,t and θbJ,t respectively denote the price of market equity risk, variance risk and jump risk at time t, and βbiE,t , βbiV,t and βbiJ,t respectively denote the risk exposure of firm i to market equity risk, variance risk and jump risk. Since we do a cross-sectional regression each month, empirically we are able to construct the time series of implied market price of different risk factors. In addition, the construction of these implied market risk premiums does not require any information beyond time t since we only use time t information to estimate individual VRP and β, so these implied market risk premiums (and unexplained VRP) can be considered conditional variables. We call the time t estimates of market price on these risk factors the “implied market equity risk premium” (IMERPt ), “implied market variance risk premium” (IMVRPt ) and “implied market jump risk premium” (IMJRPt ), respectively. So we can express the VRP of firm i as:

VRPi,t = UVRPt + IMERPt βbiE,t + IMVRPt βbiV,t + IMJRPt βbiV,t + i,t .

(13)

Figure 3 shows the time series plots of unexplained common individual VRP (UVRP), IMVRP, and IMJRP when we use different jump risk proxies. Panel A1-A3 are from the model where we use innovation in BT(2010) fear measure to represent jump risk. Panel B1-B3 are from the model where we use DK(2011) jump/tail risk measure as the jump risk proxy. Panel C1-C3 are from the model where we use innovation in DK(2011) jump/tail risk measure as the jump risk proxy. Visually, the time series pattern of the same quantity implied from models with different jump proxies are very similar. The correlations between UVRP constructed from the three models are above 0.86, the correlations between IMVRP are above 0.95, and the lowest correlations between IMJRP is 0.45 (still significant), which is between UJVRPs from the model using innovation in BT(2010) and the model using DK(2011) jump/tail risk measure. Next, we study whether the common unexplained individual VRP (UVRP) is related to some other market level factors that we have not included in our analysis. For the following analysis, we focus on the model where we use innovations of DK(2001) jump/tail risk measure

24

as the proxy, since it has the highest adjusted R2 in explaining VRP. In unreported results, we do all the following analysis (including predictions) using other jump risk proxies and the main results still hold. We compute the correlation of UVRP and the implied market risk premiums with other market level variables including: the value-weighted CRSP excess return e ), small-minus-big portfolio return (SMB), high-minus-low portfolio return (HML), the (RM M −IVM ), Pastor-Stambaugh variance risk premium of S&P 500 index (VRPM , defined as RVIV M

traded liquidity measure (LIQP S ), term spread defined as the difference between ten-year Tbond and three-month T-bill yields (TERM), default spread defined as the difference between Moody’s BAA and AAA corporate bond spreads (DEF), logarithm of price-dividend ratio of S&P 500 (log(P/D)), VIX square (IVM ), implied skewness (ISKEWM ) and implied kurtosis (IKURTM ) constructed from S&P 500 index options following BKM(2003), and the investor sentiment measure used and described in Baker and Wurgler (2007). Based on Table 6, both UVRP and IMVRP are positively correlated with implied skewness and negatively correlated with implied kurtosis, meaning that the unexplained VRP and the implied market price of risk are more negative when implied market return distribution is more negatively skewed and has fatter tails. The implied market price of market variance risk is positively correlated with both variance risk premium and the implied variance of S&P 500 index (VRPM and IVM ). In addition, we regress the UVRP, IMVRP, and IMJRP on market level factors to check how much of their variation can be explained by these factors. Only about 12.8% of UVRP’s time series variation can be explained by the considered market level factors in total, and nearly 50% of IMVRP and 40% of IMJRP can be explained. Both UVRP and IMVRP are significantly positively related to the variance risk premium of S&P 500 index and the investor sentiment measure. An interesting observation is that IMVRP is very significantly related to investor sentiment measure with the latter explaining about 10% adjusted R2 of the former. On the other hand, IMJRP is significantly negatively related to the variance risk premium of S&P 500 index.

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5.2

Predictability of UVRP

We then study whether UVRP contains any information about the future excess return of stock market and growth of economic activities. Following the prediction setup of Bollerslev, Tauchen, and Zhou (2009), who find that the variance spread of S&P 500 index significantly predicts medium horizon future stock market excess return (more than the traditionally used predictors), we specify the predictive regression as, f Rt+j − rt+j

= α + β1 · U V RPt + β2 · (IVM,t − RVM,t−1 ) + Γ0 · Z¯t + t ,

(14)

IVM,t −RVM,t−1 denotes the variance spread of S&P 500 index used in Bollerslev, Tauchen, and Zhou (2009), where they use VIX square as IV and use past month realized variance computed from intraday high-frequency data of S&P 500 index as RV.18 Rt+j denotes the log return of f the S&P 500 index from the end of month t to the end of month t+j. rt+j denotes the risk-free

return for the same horizon. We use five different values for j: 1 month, 3 months, 6 months, 1 year and 2 years. Z¯t denotes the commonly used predictors, in our table including log(P/D), CAY, TERM, and DEF. log(P/D) denotes the logarithm of price-dividend ratio for S&P 500 downloaded from Standard & Poor’s Website. CAY is as defined in Lettau and Ludvigson (2001) and downloaded from Lettau and Ludvigson’s Website. Data needed to calculate the term spread and the default spread are from the Website of the Federal Reserve Bank of St. Louis. Table 7 shows the predictive regressions results. The significance inferences are based on Hodrick (1992) t-values.19 We can observe that UVRP strongly predicts future 6-month to 2-year market excess return, and the significance does not change when we add the variance spread of S&P 500 index in the regression. Consistent with Bollerslev, Tauchen, and Zhou (2009), the variance spread of S&P 500 index is the most significant predictor for short to medium term return horizons. The sign of coefficient on UVRP is negative, indicating that 18

The realized variance data is from Hao Zhou’s Website. Ang and Bekaert (2007) show that inference in predictability regressions is critically dependent on the choice of standard errors, and they recommend Hodrick (1992) 1b standard errors under the null of no predictability. 19

26

the more negative the current UVRP, the higher the market excess return in the future. For 6-month to 2-year horizons, UVRP is the most significant predictor, even more significant than the S&P 500 variance spread. We follow the setup below for the economic activities prediction using UVRP,

ln EVt+j /EVt = α + β1 · U V RPt + β2 T ERMt + β3 DEFt + t ,

(15)

where EVt+j denotes the value of economic variable at time t + j, and TERM and DEF are as specified above (they are documented by the literature to predict the economic activities). We use five different values for j: 1 month, 3 months, 6 months, 1 year and 2 years. The macroeconomic variables considered include: (1)IPSA, industrial production, (2)TCU, capacity utilization, (3)NAPM, ISM Manufacturing: PMI Composite Index, (4)HOUST, housing starts, (5)RETAIL, total retail sales excluding food services, (6)UNRATE, civilian unemployment rate, and (7)PAYEM, total non-farm payroll. The economic activities variables are downloaded from Federal Reserve Board and Federal Reserve Bank of St. Louis Website. The first five of these variables relate to business activity, and the remaining two measure the level of employment. For control variables, we use term spread and credit spread as in Bakshi, Panayotov, and Skoulakis (2011). The inferences are all based on Hodrick (1992) standard errors. Table 8 reports the prediction results. The strongest result is observed in the unemployment rate (UNRATE) prediction: UVRP is significant for all horizons from 1-month to 2-year. UVRP also strongly predicts future 3-month to 2-year growth rate of IPSA, TCU, and PAYEM, both univariately and controlling for TERM and DEF, and it does not show evident predictability power for future growth rate of NAPM and HOUST. UVRP is also significant in predicting future 6-month to 1-year RETAIL growth rate. Being consistent with the market return prediction results, the relation between UVRP and future economic performance is negative: more negative UVRP is followed by greater growth rate of business activities, greater growth rate in payroll, and lower unemployment rate. The significance of UVRP tends to be

27

higher for longer horizons, and exceeds the significance of TERM and DEF for some economic variables. Before identifying the exact information in UVRP, we can consider it as an “overpricing” measure of overall individual options market controlling for market risk exposures. As more negative VRP indicates the relative expensiveness of individual equity options, the prediction results using UVRP show that the overall extra expensiveness of individual equity options that cannot be explained by risk factors, is associated with higher future market excess returns and better future economic activities. Based on these results, we conjecture that we need a macrorational explanation for the predictability of the unexplained average VRP, and it would be more difficult to reconcile the predictability results with the market inefficiency explanation involving factors such as traders’s behaviors and demand pressure.

5.3

Information in Different VRP Components

In this section, as an application of the individual VRP decomposition, we check the source of the cross-sectional predictability of VRP, which has been documented by Han and Zhou (2010).20 Specifically, we use the equity-variance-jump three factor model to decompose the individual VRP of each firm into the unexplained part αt , the exposure to systematic risks: E · IMERP , β bV · IMVRPt , and βbJ · IMJRPt , and the idiosyncratic component i,t . Then we βbi,t t i,t i,t

sort the stocks in our sample into quintiles based on the VRP and their components, respectively. We compute the next month value-weighted stock return and model-adjusted alphas for each quintile portfolio and also the long quintile 5 and short quintile 1 strategy (5-1). Table 9 reports the results. There are a few observations that are worth noting. Firstly, we confirm with Han and Zhou (2010) that there is a monotonic pattern of quintile portfolio returns and alphas sorted by individual VRP, and the 5-1 strategy yields a significant return of -0.88% for the next month. Secondly, there is also a monotonic decreasing pattern of quintile portfolio re20 Han and Zhou (2010) use E[RV] − IV to define the variance risk premium and sort stocks, and we use the variance risk premium defined in return terms to sort stocks and construct portfolios.

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V ·IMVRP ) turns and alphas if we sort the portfolios by the market variance risk component (βbi,t t

in VRP or by the idiosyncratic component i,t of VRP. The magnitude of the 5-1 performance sorted by market variance risk component (-0.89% return per month) is comparable to that sorted by the individual VRP. Thirdly, the components of VRP that are compensated for market equity risk and market jump risk do not seem to predict the cross-sectional stock returns. There are a few possible explanations for the cross-sectional predictability of the idiosyncratic component of VRP. Since the idiosyncratic component is defined relative to our model, it still could carry information for any systematic factor that is not specified in the model. Alternatively, the idiosyncratic component could be related to option traders’ private information about the underlying firms. More detailed tests are needed to understand this issue and we leave it to future research.

6

Conclusion

In conclusion, we find that market equity risk, market variance risk and market jump risk are priced in the cross-section of individual VRPs. The priced equity risk premium is positive and the priced variance and jump risk premiums are both negative. Market variance risk serves as the best risk factor in explaining the cross-sectional variation of individual VRP and the three factors together explain about 46% of the cross-sectional variation of individual VRP for S&P 500 component firms. Our results provide support that these three risk factors are systematic factors that impact the pricing of individual equity options. We also test the performance of standard pricing kernels used to explain cross-sectional stock returns, and none of these models is able to explain the cross-sectional variation of individual VRP as much as the equity-variance-jump three factor model. The results of market risk factors being priced are robust after controlling for firm-level variables, including firm size, book-to-market ratio, previous 1-month and 6-month returns, idiosyncratic volatility, stock illiquidity and monthly total option trading volume. A large portion of the cross-sectional variation of individual VRP cannot be explained by

29

market level risk factors and we also observe a significant negative alpha in the second step of Fama-Macbeth regression. We then decompose the individual VRP into three components: the common unexplained component, the systematic exposure component, and the idiosyncratic component. We find that the common unexplained component is significantly negative and time varying and strongly predicts future 6-month to 2-year stock market return and future 3month to 2-year economic activities. The predictability power of the unexplained average VRP calls for a systematic story (such as a missing risk factor) to further explain the cross-sectional variation of individual VRP. Our results tend to suggest that the cross-section of individual VRPs can be understood as risk premiums rather than mispricing. In our example of applying the individual VRP decomposition, we find evidence that the idiosyncratic components contain information about the cross-sectional variation of the underlying stocks’ future returns; we leave the detailed investigation to future research.

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References Amihud, Y., 2002, “Illiquidity and stock returns: cross-section and time-series effects,” Journal of Financial Markets, 5(1), 31–56. Andersen, T. G., T. Bollerslev, and F. X. Diebold, 2006, “Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility,” Review of Economics and Statistics, 89(4), 701720. Ang, A., R. J. Hodrick, Y. Xing, and X. Zhang, 2006, “The Cross-Section of Volatility and Expected Returns,” The Journal of Finance, 61, 259299. Baker, M., and J. Wurgler, 2007, “Investor Sentiment in the Stock Market,” The Journal of Economic Perspectives, 21(2), 129–151. Bakshi, G., and N. Kapadia, 2003a, “Delta-Hedged Gains and the Negative Market Volatility Risk Premium,” The Review of Financial Studies, 16(2), 527–566. , 2003b, “Volatility Risk Premiums Embedded in Individual Equity Options: Some New Insights,” Journal of Derivatives, 11(1), 4554. Bakshi, G., N. Kapadia, and D. Madan, 2003, “Stock Returns Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options,” The Review of Financial Studies, 16(1), 101–143. Bakshi, G., G. Panayotov, and G. Skoulakis, 2011, “Improving the predictability of real economic activity and asset returns with forward variances inferred from option portfolios,” Journal of Financial Economics, 100(3), 475–495. Bollen, N. P. B., and R. E. Whaley, 2004, “Does net buying pressure affect the shape of implied volatility functions?,” Journal of Finance, 59, 711754. Bollerslev, T., G. Tauchen, and H. Zhou, 2009, “Expected Stock Returns and Variance Risk Premia,” Review of Financial Studies, 22, 44634492.

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Bollerslev, T., and V. Todorov, 2011, “Tails, fears and risk premia,” The Journal of Finance, forthcoming. Britten-Jones, M., and A. Neuberger, 2000, “Option Prices, Implied Price Processes, and Stochastic Volatility,” The Journal of Finance, 55(2), 839–866. Broadie, M., M. Chernov, and M. Johannes, 2007, “Model Specification and Risk Premia: Evidence from Futures Options,” The Journal of Finance, 62(3), 1453–1490. Cao, J., and B. Han, 2010, “Cross-section of Stock Option Returns and Stock Volatility Risk Premium,” working paper. Carr, P., and D. Madan, 1998, “Towards a Theory of Volatility Trading,” Volatility, Risk Publication, pp. 417–427. Carr, P., and L. Wu, 2008, “Variance Risk Premiums,” Review of Financial Studies. Cochrane, J. H., 2001, Asset Pricing. Princeton University Press. Coval, J. D., and T. Shumway, 2001, “Expected Option Returns,” The Journal of Finance, 56(3), 983–1009. Cremers, M., M. Halling, and D. Weinbaum, 2011, “In Search of Aggregate Jump and Volatility Risk in the Cross-Section of Stock Returns,” working paper. Derman, E., K. Demeterfi, M. Kamal, and J. Zou, 1999, “More than you ever wanted to know of volatility swaps,” Quantitative Strategies Research Note, Goldman Sachs. Dittmar, R. F., 1999, “Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross Section of Equity Returns,” Journal of Finance, 57(1), 369–403. Drechsler, I., and A. Yaron, 2010, “Whats Vol Got to Do with It,” The Review of Financial Studies, 24(1). Driessen, J., P. Maenhout, and G. Vilkov, 2009, “The price of correlation risk: Evidence from equity options,” Journal of Finance, 64, 13771406. 32

Du, J., and N. Kapadia, 2011, “The Tail in the Volatility Index,” working paper. Duan, J.-C., and J. Wei, 2008, “Systematic Risk and the Price Structure of Individual Equity Options,” The Review of Financial Studies, 22, 19812006. Duarte, J., and C. S. Jones, 2007, “The price of market volatility risk,” working paper. Eraker, B., M. Johannes, and N. Polson, 2003, “The Impact of Jumps in Volatility and Returns,” The Journal of Finance, 58(3), 1269–1300. Fama, E. F., and K. R. French, 1992, “The Cross-Section of Expected Stock Returns.,” Journal of Finance, 47(2), 427 – 465. Fama, E. F., and J. D. MacBeth, 1973, “Risk, Return, and Equilibrium: Empirical Tests,” Journal of Political Economy, 81(3), 607–636. Gˆarleanu, N., L. H. Pedersen, and A. M. Poteshman, 2009, “Demand-Based Option Pricing,” Review of Financial Studies, 22(10), 4259–4299. Goyal, A., and A. Saretto, 2009, “Cross-section of option returns and volatility,” Journal of Financial Economics, 94(2009), 310326. Han, B., and Y. Zhou, 2010, “Variance Risk Premium and Cross-Section of Stock Returns,” working paper. Harvey, C. R., and A. Siddique, 2000, “Conditional Skewness in Asset Pricing Tests,” The Journal of Finance, 55(3), 1263–1295. Hodrick, R., 1992, “Dividend yields and expected stock returns: alternative procedures for inference and measurement,” Review of Financial Studies, 5(3), 357–386. Jiang, G., and Y. Tian, 2005, “The Model-Free Implied Volatility and Its Information Content,” Review of Financial Studies, 18(4), 1305–1342. Merton, R., 1973, “An Intertemporal Capital Asset Pricing Model,” Econometrica, 41, 867– 887. 33

Pan, J., 2002, “The jump-risk premia implicit in options: evidence from an integrated timeseries study,” Journal of Financial Economics, 63(1), 3–50. Wang, H., H. Zhou, and Y. Zhou, 2009, “Credit Default Swap Spreads and Variance Risk Premia,” working paper.

34

Table 1. Summary Statistics The sample period is January 1996- October 2009. Panel A reports the summary statistics of realized variance, implied variance, and firm characteristics. For individual firms, we first compute the time series average of each variable, and then report the summary statistics of the cross-sectional distribution. For market variables, we report the time series summary statistics. Ri , IVi , RVi , SIZE, and B/M respectively denote the monthly return, the end-of-month risk-neutral expected variance, monthly realized variance, market capitalization in $ millions, and book-to-market ratio of individual firms. RM , IVM , and RVM respectively denote the monthly return, the end-of-month risk-neutral expected variance, and monthly realized variance of S&P 500 index. The realized variance of each month for Pn 2 Si ln both individual securities and S&P 500 index is calculated as 252 i=1 n Si−1 , where Si is the underlying security’s closing price on the ith trading day of each month and n denotes the number of trading days of that month. The return and variance measures shown are scaled up by a factor of 100. There are 253,933 firm-month observations in total. Panel B reports the fit (t-statistics of intercept and explanatory variables, adjusted R2 , and root mean squared error) of OLS regressions for realized variance prediction. RV(n) denotes the historical realized variance computed with past n days return. Panel C reports the (cross-sectional) percentiles of the VRP means and Newey-West (1987) t-values. VRP is defined as: (E[RV] − IV)/IV, and Log VRP is defined as: ln E[RV] − ln IV.

Mean Std.Dev. Skew Kurt Min P5 Median P95 Max

Const. 24.07 20.96 17.83 16.07 14.97 14

RV(1) 0.6 0.69 0.71 0.71 0.7 0.69

Ri 1.21 3.02 2.33 47.17 -14.24 -3.14 1.11 5.69 64.78

RV(7) 6.11 0.72 0.97 0.95 0.93 0.91

Panel A. Summary Statistics IVi RVi SIZE B/M RM 42.25 42.51 3779.38 0.36 0.43 32.19 37.21 9598.20 0.29 4.68 2.05 2.96 6.30 0.54 -0.70 6.05 15.56 48.90 0.63 0.96 0.15 0.06 113.76 -0.17 -16.94 9.95 8.84 214.67 -0.17 -8.17 33.42 32.11 1085.53 0.33 0.92 108.54 114.14 15231.81 0.89 7.52 291.34 451.12 108049.00 1.61 9.67

IVM 4.32 6.93 5.72 41.70 0.43 0.61 2.45 11.72 64.82

Panel B. Fit of Realized Variance Prediction RV(15) RV(30) RV(45) RV(60) RV(75) Adj.R2 14.01% 5.55 18.36% 0.72 5.35 21.70% 0.95 0.62 3.6 23.48% 0.98 0.81 0.47 2.28 24.65% 1.02 0.87 0.7 -0.6 2.57 25.74%

Panel C. Percentiles of VRP Mean and 1% 5% 25% 50% E[RV] Mean 0.0473 0.0835 0.1850 0.3307 0.0548 0.0995 0.2041 0.3342 IV Mean (E[RV] − IV) Mean -0.4093 -0.1672 -0.0423 -0.0098 -5.37 -3.47 -1.67 -0.49 (E[RV] − IV) t value VRP Mean -0.3839 -0.2337 -0.0312 0.1051 VRP t value -8.41 -3.79 -0.42 1.13 Log VRP Mean -0.5820 -0.3590 -0.1080 0.0206 Log VRP t -7.39 -4.11 -1.30 0.26

35

t Value 75% 0.5429 0.5296 0.0348 1.12 0.3033 2.51 0.1716 1.89

95% 1.1677 1.0854 0.2729 4.78 0.8932 4.94 0.4986 5.15

RVM 6.30 6.03 3.47 16.23 1.15 1.51 5.08 19.01 46.23

RMSE 0.6627 0.6473 0.6330 0.6232 0.6161 0.6113

99% 1.8315 1.5826 0.6378 11.97 2.1519 7.60 0.9478 9.47

DF 2210 2209 2208 2207 2206 2205

36

ILLIQ

VarIDIO

KTt−12,t

SKt−12,t

KTt−1,t

SKt−1,t

RVt−12,t

RVt−1,t

Rt−6,t

Rt−1,t

B/M

LSIZE

Const.

Model 1 -0.0449*** (-4.02) 0.0062*** (4.31)

-0.0458*** (-8.80)

Model 2 0.0198*** (6.87)

0.0844*** (8.32)

Model 3 0.0012 (0.66)

0.0238*** (3.78)

Model 4 -0.0057*** (-2.66)

-0.0379*** (-5.19)

Model 5 0.0146*** (6.85)

-0.0064*** (-6.44)

Model 6 0.0029* (1.67)

0.0117*** (29.78)

Model 7 -0.0110*** (-6.38)

-0.0053*** (-3.93)

Model 8 0.0017 (1.00)

0.0014*** (8.97)

Model 9 -0.0065*** (-3.45)

1.4416 (1.02)

Model 10 0.0012 (0.60)

-57.0186* (-1.91)

Model 11 0.0049** (2.19)

-2.6325*** (-4.51) -39.4466 (-1.36)

Model 12 -0.0041 (-0.26) 0.0025 (1.30) -0.0367*** (-6.71) 0.1120*** (12.25) 0.0151*** (4.19) 0.0483*** (5.59) -0.0766*** (-8.51) -0.0085*** (-9.16) 0.0101*** (23.67)

Table 2. VRP and Firms Related Variables This tables reports the panel regression results of individual VRP regressed on firm related variables. LSIZE denotes the logarithm of market capitalization in $ millions. B/M denotes book-to-market ratio. Rt−i,t denotes the return over the previous i months. RVt−i,t , SKt−i,t and KTt−1,t respectively denote the historical realized variance, skewness, and kurtosis computed with daily log-return data over previous i months. VARIDIO denotes the monthly idiosyncratic variance defined relative to Fama-French 3-factor models as described in Ang, Hodrick, Xing, and Zhang (2006). ILLIQ denotes the stock illiquidity measure proposed in Amihud (2002). t-statistics are based on standard error clustered by firms.

Table 3. Portfolios Sorted by Full Period Univariate Beta The table reports the average VRP and alphas (with respect to CAPM, Fama-French 3-factor model, and Carhart 4-factor model) of decile portfolios sorted by firms’ βs with respect to market equity risk, market variance risk and market jump risk. β for each firm and each market risk is estimated by the following univariate regression using full sample monthly observations: VRPi,t = αi + βfM,t + i , t = 1, 2, ..., T , where fM refers to the market equity risk (value-weighted excess return of all CRSP stocks), variance risk (monthly realized variance of CRSP value-weighted return), or jump risk (the jump/tail risk measure proposed by Du and Kapadia (2011)). Panel A, B and C respectively report the results of sorting by β of market equity risk, β of market variance risk and β of market jump risk. The average β, size and book-to-market ratio (B/M) of each decile portfolio are also reported. 10-1 refers to the average return of a zero-cost strategy that longs decile 10 portfolio and shorts decile 1 portfolio. Newey-West (1987) t-statistics of 10-1 portfolios are also reported. The reported portfolio means and alphas are scaled up by a factor of 100. Panel A. Market Equity Risk Beta CAPM Alpha 3-factor Alpha 4-factor Alpha Est. t-value Est. t-value Est. t-value 11.74 11.65 11.40 5.20 5.05 4.97 7.54 7.41 7.33 7.59 7.37 7.25 8.46 8.31 8.20 10.34 10.17 10.00 16.76 16.53 16.30 18.71 18.34 18.09 25.76 25.56 25.28 52.58 52.12 51.53 40.84 [6.77***] 40.47 [6.72***] 40.13 [6.71***]

Rank 1 2 3 4 5 6 7 8 9 10 10-1

Mean Est. t-value 11.50 5.21 7.71 7.85 8.77 10.74 17.29 19.31 26.51 53.71 42.22 [6.24***]

Rank 1 2 3 4 5 6 7 8 9 10 10-1

Mean Est. t-value 64.77 30.62 20.30 14.32 12.75 9.25 5.55 3.89 2.87 1.73 -63.04 [-5.51***]

Panel B. CAPM Alpha Est. t-value 63.78 30.02 19.80 13.89 12.36 8.92 5.28 3.68 2.71 1.73 -62.05 [-5.65***]

Rank 1 2 3 4 5 6 7 8 9 10 10-1

Mean Est. t-value 56.30 33.87 22.64 15.85 13.18 8.95 7.05 3.93 4.32 9.39 -46.91 [-4.61***]

Panel C. Market Jump Risk Beta CAPM Alpha 3-factor Alpha 4-factor Alpha Est. t-value Est. t-value Est. t-value 55.44 54.56 54.05 33.22 32.63 32.17 22.13 21.57 21.25 15.36 15.05 14.82 12.79 12.54 12.34 8.61 8.38 8.23 6.78 6.70 6.60 3.69 3.60 3.55 4.20 4.32 4.31 9.30 9.59 9.51 -46.15 [-4.65***] -44.97 [-4.49***] -44.55 [-4.46***]

Market Variance Risk Beta 3-factor Alpha 4-factor Alpha Est. t-value Est. t-value 63.10 62.36 29.61 29.12 19.42 19.08 13.73 13.51 12.15 11.98 8.73 8.57 5.15 5.04 3.67 3.68 2.65 2.69 1.82 1.96 -61.29 [-5.61***] -60.40 [-5.56***]

37

β -1.14 -0.08 0.38 0.72 0.98 1.24 1.52 1.85 2.31 3.85

SIZE 4508.28 4272.14 7183.10 9819.25 10106.91 8794.29 8449.83 9298.68 9318.97 7989.05

B/M 0.41 0.42 0.37 0.36 0.37 0.37 0.38 0.39 0.37 0.38

β -0.5655 -0.3355 -0.2638 -0.2185 -0.1839 -0.1524 -0.1230 -0.0939 -0.0591 0.0081

SIZE 6314.55 9099.91 9019.07 10222.13 9163.07 9959.34 9060.14 7071.01 5629.13 4033.64

B/M 0.40 0.40 0.36 0.37 0.35 0.39 0.37 0.40 0.39 0.40

β -142.8453 -49.9159 -31.4868 -24.0527 -19.4435 -15.5636 -11.5501 -7.4027 -2.2222 11.7732

SIZE 4225.52 8127.09 9162.67 10556.06 10960.65 10486.13 9515.39 7093.16 6091.84 4655.14

B/M 0.38 0.39 0.38 0.36 0.37 0.37 0.37 0.38 0.42 0.42

Table 4. Portfolios Sorted By Previous 60-Month Multivariate Beta The table reports the average VRP and alphas (with respect to CAPM, Fama-French 3-factor model, and Carhart 4-factor model) of decile portfolios sorted by firms’ βs with respect to market equity risk (value-weighted excess return of all CRSP stocks), variance risk (monthly realized variance of CRSP value-weighted return) and jump risk (the jump/tail risk measure proposed by Du and Kapadia (2011)). βs for each firm in month T are estimated by the following multivariate regression using previous 60e month observations: VRPi,t = αi + β1 RM,t + β2 VM,t + β3 JM,t + i , t = T − 59, T − 58, ..., T . Panel A, B and C respectively report the results of sorting by β1 , β2 and β3 . The average β, size and book-to-market ratio (B/M) of each decile are also reported. 10-1 refers to the average return of a strategy that longs decile 10 portfolio and shorts decile 1 portfolio. Newey-West (1987) t-statistics of 10-1 portfolios are also reported. The reported portfolio means and alphas are scaled up by a factor of 100. Panel D reports the 5-1 portfolio return, alphas and Newey-West (1987) t-statistics from double-sorting controlling for β of other risk factors. In the double-sorting, we first sort firms into quintiles by ranking controlling variable and then again sort each quintile portfolio into quintiles by ranking β of interest. SIZE denotes the market capitalization in $ millions. B/M denotes book-to-market ratio. Rt−i,t denotes the return over the previous i months. VARIDIO denotes the monthly idiosyncratic variance defined relative to Fama-French 3-factor models as described in Ang, Hodrick, Xing, and Zhang (2006).

Rank 1 2 3 4 5 6 7 8 9 10 10-1

Mean Est. t-value 35.80 18.45 15.22 12.97 12.99 11.76 12.74 14.25 17.64 29.10 -6.10 [-1.22]

Panel A. Market Equity Risk Beta CAPM Alpha 3-factor Alpha 4-factor Alpha Est. t-value Est. t-value Est. t-value 35.79 35.60 35.99 18.42 18.18 18.38 15.17 14.98 15.18 12.85 12.53 12.73 12.92 12.74 12.96 11.61 11.42 11.58 12.57 12.39 12.54 14.15 13.77 13.90 17.53 17.38 17.46 28.95 28.95 29.11 -6.39 [-1.29] -6.13 [-1.18] -6.37 [-1.23]

Rank 1 2 3 4 5 6 7 8 9 10 10-1

Mean Est. t-value 61.58 26.58 17.59 13.49 10.18 9.05 7.45 6.14 5.21 7.74 -52.88 [-3.58***]

Panel B. CAPM Alpha Est. t-value 61.43 26.49 17.51 13.36 10.12 8.92 7.34 6.08 5.16 7.70 -52.69 [-3.64***]

Market Variance Risk Beta 3-factor Alpha 4-factor Alpha Est. t-value Est. t-value 61.37 60.69 26.38 26.08 17.40 17.12 13.20 12.99 10.00 9.84 8.92 8.84 7.28 7.17 6.15 6.12 5.31 5.32 8.08 8.16 -52.13 [-3.58***] -51.36 [-3.59***]

Rank 1 2 3 4 5 6 7 8 9 10 10-1

Mean Est. t-value 42.84 23.56 18.44 14.27 12.33 11.89 11.24 11.11 13.02 23.06 -19.08 [-3.21***]

Panel C. Market Jump Risk Beta CAPM Alpha 3-factor Alpha 4-factor Alpha Est. t-value Est. t-value Est. t-value 42.73 42.92 42.71 23.47 23.40 23.20 18.35 18.26 18.15 14.13 13.81 13.62 12.26 12.16 11.98 11.75 11.57 11.41 11.11 11.18 10.99 11.05 11.00 10.83 12.96 12.88 12.69 23.00 22.91 22.71 -19.01 [-3.26***] -19.17 [-3.27***] -19.20 [-3.24***]

38

β -2.53 -1.16 -0.62 -0.23 0.10 0.41 0.74 1.11 1.60 2.62

SIZE 7982.93 10103.88 11836.91 12140.40 13552.49 14104.77 13346.68 12191.40 11881.76 12418.13

B/M 0.41 0.41 0.40 0.39 0.39 0.38 0.38 0.38 0.40 0.39

β -0.4894 -0.2856 -0.2162 -0.1664 -0.1286 -0.0919 -0.0564 -0.0183 0.0294 0.1193

SIZE 9136.37 11114.08 13158.14 12378.71 12952.78 12345.32 12156.94 12853.40 12796.57 9102.64

B/M 0.46 0.42 0.39 0.38 0.37 0.38 0.39 0.39 0.38 0.37

β -132.7422 -77.4502 -58.1828 -44.3478 -32.8937 -21.7443 -10.0250 2.9005 21.4728 69.8688

SIZE 10206.84 13102.16 13936.96 13350.29 13203.56 14276.58 14010.86 10284.44 9372.32 8204.72

B/M 0.38 0.37 0.39 0.39 0.39 0.40 0.40 0.39 0.40 0.43

Panel D. Double Sorting

Control Variance β Jump β Size B/M Rt−1,t Rt−6,t VarIDIO

Mean Est. t-value 0.03 [0.01] -0.92 [-0.27] -4.17 [-1.11] -3.83 [-1.11] -3.34 [-0.94] -3.35 [-0.92] -3.38 [-0.97]

Control Equity β Jump β Size B/M Rt−1,t Rt−6,t VarIDIO

Mean Est. t-value -34.95 [-3.32***] -38.74 [-3.43***] -37.46 [-3.51***] -35.59 [-3.52***] -36.90 [-3.51***] -37.47 [-3.57***] -36.64 [-3.47***]

Control Equity β Variance β Size B/M Rt−1,t Rt−6,t VarIDIO

Mean Est. t-value -13.61 [-3.67***] -20.31 [-3.73***] -15.15 [-3.45***] -15.59 [-3.50***] -15.07 [-3.49***] -15.77 [-3.62***] -14.60 [-3.36***]

Market Equity Risk CAPM Alpha Est. t-value -0.08 [-0.03] -1.04 [-0.33] -4.28 [-1.14] -3.93 [-1.16] -3.44 [-1.00] -3.46 [-0.96] -3.48 [-1.02]

Beta 3-factor Alpha Est. t-value -0.12 [-0.04] -1.01 [-0.30] -4.36 [-1.11] -4.03 [-1.13] -3.53 [-0.97] -3.49 [-0.91] -3.60 [-1.01]

Market Variance Risk Beta CAPM Alpha 3-factor Alpha Est. t-value Est. t-value -34.86 [-3.41***] -34.56 [-3.36***] -38.65 [-3.52***] -38.14 [-3.46***] -37.38 [-3.58***] -37.10 [-3.54***] -35.51 [-3.60***] -35.18 [-3.54***] -36.83 [-3.58***] -36.45 [-3.52***] -37.40 [-3.65***] -37.09 [-3.60***] -36.56 [-3.54***] -36.21 [-3.48***] Market Jump Risk CAPM Alpha Est. t-value -13.55 [-4.10***] -20.26 [-3.89***] -15.11 [-3.59***] -15.54 [-3.65***] -15.03 [-3.63***] -15.73 [-3.77***] -14.56 [-3.49***]

39

Beta 3-factor Alpha Est. t-value -13.63 [-4.11***] -20.11 [-3.88***] -15.31 [-3.65***] -15.87 [-3.72***] -15.28 [-3.70***] -15.91 [-3.81***] -14.88 [-3.59***]

4-factor Alpha Est. t-value -0.02 [-0.01] -0.81 [-0.24] -4.15 [-1.07] -3.85 [-1.09] -3.37 [-0.94] -3.31 [-0.87] -3.44 [-0.97]

4-factor Alpha Est. t-value -34.00 [-3.36***] -37.58 [-3.45***] -36.57 [-3.54***] -34.68 [-3.54***] -35.91 [-3.52***] -36.55 [-3.60***] -35.69 [-3.48***]

4-factor Alpha Est. t-value -13.72 [-4.11***] -19.99 [-3.85***] -15.29 [-3.62***] -15.82 [-3.69***] -15.29 [-3.67***] -15.93 [-3.78***] -14.90 [-3.56***]

40

Adj.R2 Obs

SMB

HML

mim ∆JDK

mim ∆JBT

∆VIXmim

∆JDK

JDK

∆JBT

JBT

ln RVm

e 3 (Rm )

e 2 (Rm )

e Rm

Const.

16.56% 112042

Model 1 -0.0218 (-0.77) 15.2033*** (5.75)

31.04% 112042

Model 2 -0.0394* (-1.74) 8.4983*** (6.57) -44.9592** (-2.60) 444.2612*** (3.60)

25.32% 112042

-1.0164*** (-2.92)

Model 3 -0.0353 (-1.18)

32.82% 110236

-1.2650*** (-4.15)

33.72% 110236

-1.2390*** (-4.15) -0.0151*** (-3.52)

35.37% 112042

-0.0121*** (-5.81)

-1.3352*** (-4.58)

35.44% 112042

-0.0024*** (-2.96)

-1.1813*** (-3.98)

36.07% 112042

-0.0021*** (-4.72)

-1.2561*** (-4.09)

Panel A. Fama-Macbeth Regressions: Full Sample Model 4 Model 5 Model 6 Model 7 Model 8 -0.0683** -0.0442 -0.0653** -0.0504 -0.0704*** (-2.61) (-1.45) (-2.60) (-1.58) (-2.71) 6.7475*** 5.2813*** 4.7472*** 6.0765*** 6.1275*** (6.34) (6.90) (5.66) (6.10) (5.61)

22.40% 112042

-13.1896*** (-6.30)

Model 9 0.0028 (0.10) 13.3880*** (5.72)

28.05% 112042

-11.7322*** (-5.84) -0.0020*** (-3.71)

Model 10 -0.0084 (-0.33) 12.0809*** (5.28)

29.29% 112042

-0.0062*** (-5.47)

-11.9039*** (-6.49)

Model 11 -0.0282 (-1.05) 11.6137*** (6.07)

-0.2720 (-0.30) 2.7734* (1.98) 27.90% 112042

Model 12 -0.0276 (-0.88) 11.2618*** (5.65)

Table 5. Fama-Macbeth Tests of Market-Level Factors in Explaining Individual VRP This table reports the Fama-Macbeth results of testing standard market-level factors in explaining individual VRP. The model being tested include the market excess return factor (CAPM), the polynomials of market excess return (Harvey and Siddique (2000) and Dittmar (1999)), market excess return combined with market variance factor (Ang, Hodrick, Xing, and Zhang (2006)), market excess return combined with market variance and tail risk factors, and the Fama-French 3-factor model. Panel A reports the results for full-sample. Panel B reports the results for a subsample comprised of only S&P 500 component firms. Panel C reports the Fama-Macbeth regression results with firm-level e denotes the monthly excess return of CRSP value-weighted index. ln RVM control variables for the sample of S&P 500 component firms. RM denotes the logarithm of monthly realized variance of CRSP value-weighted return (by summing the squared daily log-return over each month). JBT denotes the fear measure proposed by Bollerslev and Todorov (2011). ∆JBT denotes the monthly innovation of BT(2010)’s fear measure. JDK denotes the jump/tail risk measure proposed by Du and Kapadia (2011). ∆JDK denotes the monthly innovation of DK(2011)’s jump/tail mim mim risk measure. ∆VIXmim , ∆JBT and ∆JDK denote the returns of portfolios that mimick the VIX innovation, BT(2010)’s fear measure innovation, and DK(2011)’s jump/tail risk measure innovation, respectively. SMB and HML respectively denote the small-minus-big and high-minus-low Fama-French factors. LSIZE denotes the logarithm of market capitalization in $ millions. B/M denotes book-to-market ratio. Rt−i,t denotes the return over the previous i months. VARIDIO denotes the monthly idiosyncratic variance defined relative to Fama-French 3-factor models. ILLIQ denotes the stock illiquidity measure proposed in Amihud (2002). Volume denotes the monthly total option trading e volume for each firm. t-statistics are based on Newey-West (1987) standard errors. RM , ∆VIXmim , SMB, and HML are scaled up by a factor of 100 (in the first step to estimate β).

41

Adj.R2 Obs

SMB

HML

mim ∆JDK

mim ∆JBT

∆VIXmim

∆JDK

JDK

∆JBT

JBT

ln RVm

3 Rm

2 Rm

Rm

Const.

26.05% 57459

Model 1 -0.1223*** (-4.88) 20.6191*** (6.51)

42.15% 57459

Model 2 -0.1303*** (-5.25) 13.6169*** (8.43) -57.5012*** (-3.40) 848.8583*** (5.45)

34.02% 57459

-1.0761*** (-3.16)

42.87% 57459

-1.4215*** (-5.13)

43.45% 56919

-1.3995*** (-4.95) -0.0159*** (-3.92)

45.26% 56919

-0.0162*** (-7.14)

-1.5014*** (-5.68)

44.95% 57459

-0.0030*** (-4.00)

-1.3706*** (-4.80)

45.52% 57459

-0.0029*** (-4.97)

-1.4273*** (-5.10)

32.88% 57459

-19.3397*** (-7.17)

Panel B. Fama-Macbeth Regressions: S&P 500 Components Firms Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 -0.0650** -0.1275*** -0.1025*** -0.1206*** -0.1112*** -0.1318*** -0.1073*** (-2.06) (-4.58) (-3.47) (-4.56) (-3.77) (-4.88) (-4.74) 10.3879*** 8.8338*** 8.1824*** 9.3524*** 9.6722*** 19.2522*** (10.54) (8.83) (7.39) (9.09) (9.18) (6.79)

36.95% 57459

-17.7231*** (-7.15) -0.0030*** (-5.79)

Model 10 -0.1111*** (-4.90) 17.7388*** (6.65)

38.06% 57459

-0.0087*** (-6.37)

-17.2019*** (-7.44)

Model 11 -0.1270*** (-4.60) 16.5740*** (7.26)

-0.5454 (-0.48) 3.3927*** (2.89) 37.86% 57459

Model 12 -0.1141*** (-4.20) 14.5468*** (6.74)

42

Adj.R2 Obs

Volume

ILLQ

VarIDIO

Rt−6,t

B/M

LSIZE

SMB

HML

mim ∆JDK

mim ∆JBT

∆VIXmim

∆JDK

JDK

∆JBT

JBT

ln RVm

3 Rm

2 Rm

Rm

Const.

0.0637*** (6.46) 0.1058** (2.37) 0.1118*** (3.36) 214.9201*** (5.20) -2766.1662 (-1.59) -0.0004*** (-5.42) 29.92% 57402

Model 1 -0.7776*** (-7.06) 20.3838*** (6.47)

0.0558*** (5.17) 0.0547** (2.00) 0.0768*** (3.14) 224.7554*** (4.80) 1276.7988 (1.10) -0.0003*** (-6.23) 45.38% 57402

Model 2 -0.7088*** (-5.70) 13.5453*** (8.32) -57.7844*** (-3.45) 863.1857*** (5.57)

0.0743*** (6.44) 0.0591 (1.52) 0.0369 (1.17) 246.4488*** (5.11) -73.2887 (-0.07) -0.0003*** (-5.99) 37.93% 57402

-1.0599*** (-3.11)

0.0564*** (6.03) 0.0635** (2.11) 0.0854*** (3.17) 222.7797*** (4.71) -547.4825 (-0.55) -0.0003*** (-6.60) 46.18% 57402

-1.4022*** (-5.02)

0.0569*** (5.94) 0.0681** (2.26) 0.0696*** (2.94) 214.4390*** (4.88) -703.3476 (-0.65) -0.0003*** (-6.07) 46.59% 56863

-1.3759*** (-4.82) -0.0158*** (-3.91)

0.0497*** (6.03) 0.0565* (1.97) 0.0931*** (3.47) 219.4617*** (4.62) -1122.5855 (-1.16) -0.0003*** (-6.54) 48.41% 56863

-0.0161*** (-7.35)

-1.4807*** (-5.57)

0.0558*** (5.83) 0.0682** (2.35) 0.0727*** (3.12) 217.0514*** (5.04) -888.8381 (-0.75) -0.0003*** (-6.33) 47.98% 57402

-0.0031*** (-4.28)

-1.3570*** (-4.75)

0.0526*** (5.35) 0.0533* (1.75) 0.0835*** (3.38) 217.7981*** (4.75) -762.5751 (-0.77) -0.0003*** (-6.71) 48.55% 57402

-0.0030*** (-5.48)

-1.4132*** (-5.04)

0.0631*** (7.02) 0.1148*** (3.11) 0.0962*** (3.28) 224.2890*** (5.43) -2432.4804 (-1.52) -0.0004*** (-5.94) 36.40% 57402

-19.0904*** (-7.11)

0.0511*** (4.71) 0.1166*** (4.18) 0.0836*** (3.14) 216.4584*** (5.57) -2580.6234* (-1.71) -0.0003*** (-6.11) 40.17% 57402

-17.4740*** (-7.00) -0.0031*** (-5.78)

Panel C. Fama-Macbeth Regressions With Control Variables: S&P 500 Components Firms Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 -0.8280*** -0.7150*** -0.6915*** -0.6383*** -0.6913*** -0.6774*** -0.7639*** -0.6539*** (-6.60) (-6.67) (-6.20) (-6.50) (-6.39) (-6.19) (-7.81) (-6.02) 10.1843*** 8.6271*** 8.0770*** 9.2119*** 9.5416*** 18.9889*** 17.4889*** (10.52) (8.74) (7.42) (9.10) (9.21) (6.74) (6.56)

0.0576*** (6.65) 0.0954*** (3.16) 0.0975*** (3.56) 256.8291*** (5.13) -1944.4020 (-1.58) -0.0003*** (-5.90) 41.25% 57402

-0.0086*** (-6.28)

-16.8813*** (-7.31)

Model 11 -0.7443*** (-7.87) 16.2450*** (7.17)

0.0528*** (5.67) 0.0905*** (3.71) 0.1025*** (3.37) 221.9526*** (4.87) -1087.3369 (-1.17) -0.0004*** (-6.18) 41.33% 57402

-0.6000 (-0.52) 3.5608*** (3.02)

Model 12 -0.6699*** (-6.72) 14.3698*** (6.61)

43

IMJRP

IMVRP

UVRP

Const. -0.0463 (-0.45) -0.9930* (-1.83) -0.0113*** (-3.7)

UVRP IMVRP IVJRP e RM SMB HML VRPM LIQP S TERM DEF log(P/D) IVM ISKEWM IKURTM SENTIMENT

UVRP 1.00

e RM -0.8715* (-1.79) -6.1791** (-1.99) -0.0135 (-0.89)

IMVRP 0.55 1.00

e RM -0.07 -0.28 -0.30 1.00

SMB 0.2294 (0.58) -0.7797 (-0.28) -0.0273*** (-3.12)

IVJRP -0.24 -0.09 1.00 IVM 0.11 0.61 0.42 -0.43 -0.08 -0.16 0.05 -0.38 0.31 0.75 -0.34 1.00

ISKEWM -0.1391* (-1.69) -0.3696 (-0.71) 0.0004 (0.26)

log(P/D) 0.24 0.05 -0.49 -0.02 0.04 0.14 -0.14 0.16 -0.35 -0.70 1.00

Panel B. Regression Results VRPM LIQP S TERM DEF 0.2336** 0.8121* 2.2520 -4.4785 (2.46) (1.8) (1.61) (-1.03) 0.9027* -0.4097 -8.8131 97.1900*** (1.67) (-0.14) (-1.09) (3.29) -0.0077** -0.0052 0.0293 0.3714*** (-2.46) (-0.53) (1.3) (2.62)

Panel A. Correlation Matrix HML VRPM LIQP S TERM DEF 0.09 0.13 0.16 0.06 -0.01 -0.02 0.12 -0.13 0.03 0.37 -0.11 -0.27 -0.30 0.20 0.33 -0.21 0.50 0.39 0.01 -0.11 -0.41 0.16 0.12 0.09 0.00 1.00 -0.20 0.11 -0.06 -0.13 1.00 0.16 0.10 0.29 1.00 -0.01 -0.21 1.00 0.42 1.00

HML 0.1090 (0.21) -5.3737 (-1.63) -0.0321*** (-2.63)

SMB 0.02 -0.02 -0.21 0.27 1.00

IKURTM -0.0232** (-2.32) -0.1694*** (-2.66) 0.0002 (0.93)

ISKEWM 0.20 0.37 -0.15 -0.19 0.09 0.10 0.02 -0.01 0.11 -0.04 0.18 0.13 1.00

38.9%

50.6%

Adj.R2 12.8%

SENTIMENT 0.27 0.34 -0.28 -0.23 -0.01 0.21 -0.04 0.10 -0.38 -0.30 0.56 -0.04 0.18 -0.19 1.00

SENTIMENT 0.0526** (2.21) 0.7868*** (5.53) -0.0009** (-2.25)

IKURTM -0.23 -0.43 0.11 0.25 -0.02 -0.05 0.07 0.05 -0.05 0.03 -0.18 -0.17 -0.93 1.00

Table 6. UVRP, IMVRP, IMJRP and Market Level Variables This table reports the relation between UVRP, IMVRP, IMJRP and other market level variables. Panel A reports the correlation matrix. Panel B reports the regression results of UVRP, IMVPR or IMJRP regressed on other market level factors. The market level variables include: the e value-weighted CRSP excess return (RM ), small-minus-big portfolio return (SMB), high-minus-low portfolio return (HML), the variance risk M −IVM ), Pastor-Stambaugh traded liquidity measure (LIQP S ), term spread defined as the premium of S&P 500 index (VRPM , defined as RVIV M difference between ten-year T-bond and three-month T-bill yields (TERM), default spread defined as the difference between Moody’s BAA and AAA corporate bond yields (DEF), logrithm of price-dividend ratio of S&P 500 (log(P/D)), VIX square (IVM ), implied skewness (ISKEWM ) and implied kurtosis (IKURTM ) constructed from S&P 500 index options following BKM(2003), and the investor sentiment measure used and described in Baker and Wurgler (2007) (SENTIMENT). t-statistics are based on White standard errors.

Table 7. Return Prediction With UVRP This table shows the results from predictive regressions of excess stock returns on UVRP and other f common predictors used in the literature. The predictive regression is specified as, Rt+j − rt+j = 0 ¯ α + β1 · U V RPt + β2 · (IVM,t − RVM,t−1 ) + Γ · Zt + t . IVM,t − RVM,t−1 denotes the variance spread of S&P 500 index where we use VIX square as IV and use past month realized variance computed from intraday high-frequency data as RV. Rt+j denotes the log return of the S&P 500 from the end of f month t to the end of month t + j. rt+j denotes the risk-free return for the same horizon. We use five different values for j: 1 month, 3 months, 6 months, 1 year and 2 years. Z¯t denotes the commonly used predictors, in our table including log(P/D), CAY, TERM, and DEF. log(P/D) denotes the logarithm of price-dividend ratio for S&P 500. CAY is as defined in Lettau and Ludvigson (2001). TERM is defined as the difference between ten-year T-bond and three-month T-bill yields, and DEF is defined as the difference between Moody’s BAA and AAA corporate bond yields. Hodrick (1992) t-values are reported under coefficient estimates.

Const. UVRP IVM,t − RVM,t−1 log(P/D) CAY TERM DEF Adj.R2 Obs

One-month 0.1520 0.0968 (1.32) (0.85) -0.0294 -0.0267 (-1.38) (-1.25) 0.4243*** (4.12) -0.0361 -0.0267 (-1.48) (-1.10) -0.3469 -0.6715* (-0.89) (-1.76) 0.2315 0.2017 (0.69) (0.60) -1.6282 -1.3360 (-1.39) (-1.15) 3.10% 8.37% 130 130

Three-month 0.3650 0.2086 (1.17) (0.66) -0.0754 -0.0680 (-1.61) (-1.46) 1.2032*** (6.52) -0.0881 -0.0613 (-1.33) (-0.91) -0.5840 -1.5047 (-0.51) (-1.30) 0.2966 0.2119 (0.29) (0.21) -2.8997 -2.0710 (-0.91) (-0.65) 5.80% 18.88% 130 130

Six-month 0.3258 0.1429 (0.55) (0.23) -0.2580*** -0.2493*** (-3.48) (-3.37) 1.4075*** (4.90) -0.0944 -0.0630 (-0.75) (-0.49) -0.3440 -1.4211 (-0.16) (-0.64) 0.7335 0.6344 (0.38) (0.33) -0.0653 0.9041 (-0.01) (0.14) 14.44% 21.49% 130 130

44

One-year 0.5806 0.4606 (0.70) (0.54) -0.4817*** -0.4759*** (-5.21) (-5.12) 0.9233** (2.29) -0.1746 -0.1540 (-0.95) (-0.82) 1.1918 0.4852 (0.35) (0.14) 2.6608 2.5957 (0.71) (0.69) 1.5253 2.1613 (0.18) (0.25) 31.82% 32.83% 130 130

Two-year 1.3677 1.3554 (1.16) (1.16) -0.3919*** -0.3910*** (-3.04) (-3.03) 0.2248 (0.25) -0.3849 -0.3845 (-1.30) (-1.30) 1.9698 1.8216 (0.47) (0.45) 13.6290* 13.5905* (1.95) (1.95) -7.4592 -6.9788 (-0.63) (-0.56) 66.17% 65.92% 120 120

Table 8. Economic Activities Prediction With UVRP This table shows the results from predictive regressions of future macroeconomic variable growth rate on UVRP and control variables. The predictive regression is specified as, ln EVt+j /EVt = α + β1 · U V RPt + β2 T ERMt + β3 DEFt + t , where EVt+j denotes the value of economic variable at time t + j, TERM denotes the difference between ten-year T-bond and three-month T-bill yields, and DEF denotes the difference between Moody’s BAA and AAA corporate bond yields. We use five different values for j: 1 month, 3 months, 6 months, 1 year and 2 years. The macroeconomic variables considered include: (1)IPSA, industrial production, (2)TCU, capacity utilization, (3)NAPM, ISM Manufacturing: PMI Composite Index, (4)HOUST, housing starts, (5)RETAIL, total retail sales excluding food services, (6)UNRATE, civilian unemployment rate, and (7)PAYEM, total non-farm payroll. Hodrick (1992) t-values are reported under coefficient estimates.

Const. UVRP TERM DEF Adj.R2 Obs

Const. UVRP TERM DEF Adj.R2 Obs

Const. UVRP TERM DEF Adj.R2 Obs

One-month -0.0001 0.0054*** (-0.10) (5.42) -0.0020 -0.0026 (-0.94) (-1.17) 0.0926** (2.37) -0.6374*** (-7.91) -0.54% 16.50% 130 130

Three-month -0.0009 0.0119*** (-0.55) (4.36) -0.0123** -0.0138** (-2.32) (-2.61) 0.2835** (2.52) -1.6015*** (-7.72) 1.15% 25.14% 130 130

Pane A. IPSA Six-month -0.0037 0.0113** (-1.09) (2.09) -0.0401*** -0.0427*** (-3.56) (-3.79) 0.5031** (2.28) -2.1376*** (-5.71) 5.74% 19.08% 130 130

One-year -0.0098 -0.0101 (-1.49) (-0.98) -0.0963*** -0.1012*** (-5.03) (-5.25) 1.0946** (2.55) -1.6074** (-2.58) 12.72% 19.75% 129 129

Two-year -0.0116 0.0111 (-1.02) (0.36) -0.0861*** -0.0991*** (-3.01) (-3.37) 3.3017*** (4.07) -7.5045** (-2.25) 6.01% 46.06% 117 117

One-month -0.0015*** 0.0029*** (-2.71) (2.85) -0.0031 -0.0040 (-1.30) (-1.64) 0.1615*** (4.18) -0.6429*** (-7.66) -0.22% 17.56% 130 130

Three-month -0.0052*** 0.0042 (-3.09) (1.53) -0.0158*** -0.0183*** (-2.84) (-3.27) 0.4893*** (4.33) -1.5918*** (-7.42) 2.19% 27.22% 130 130

Pane B. TCU Six-month -0.0119*** -0.0045 (-3.43) (-0.83) -0.0467*** -0.0512*** (-4.14) (-4.52) 0.9064*** (4.00) -2.0494*** (-5.27) 7.20% 24.15% 130 130

One-year -0.0248*** -0.0421*** (-3.76) (-4.14) -0.1078*** -0.1157*** (-5.55) (-5.90) 1.8271*** (4.08) -1.1520* (-1.83) 13.86% 32.70% 129 129

Two-year -0.0377*** -0.0737** (-3.23) (-2.29) -0.0915*** -0.1156*** (-3.03) (-3.75) 4.1081*** (4.88) -2.7243 (-0.81) 5.28% 60.89% 117 117

One-month 0.0033 -0.0132** (0.98) (-2.12) 0.0150 0.0135 (0.94) (0.85) 0.3592 (1.42) 0.9658* (1.87) -0.36% 2.31% 130 130

Three-month 0.0007 -0.0591*** (0.07) (-3.33) -0.0082 -0.0113 (-0.21) (-0.29) 0.8488 (1.11) 4.1874*** (2.75) -0.75% 10.28% 130 130

Pane C. NAPM Six-month -0.0037 -0.1352*** (-0.19) (-4.00) -0.0557 -0.0583 (-0.79) (-0.83) 0.9850 (0.65) 10.5449*** (3.37) -0.22% 21.97% 130 130

One-year -0.0133 -0.2524*** (-0.37) (-4.30) -0.1270 -0.1289 (-1.04) (-1.04) 1.2978 (0.45) 19.9374*** (3.70) 0.80% 41.89% 129 129

Two-year 0.0148 -0.3026 (0.22) (-1.66) 0.2686* 0.2139 (1.69) (1.29) 3.3284 (0.58) 27.4866 (1.33) 8.41% 38.89% 117 117

45

Const. UVRP TERM DEF Adj.R2 Obs

Const. UVRP TERM DEF Adj.R2 Obs

Const. UVRP TERM DEF Adj.R2 Obs

Const. UVRP TERM DEF Adj.R2 Obs

One-month -0.0084 0.0028 (-1.16) (0.19) 0.0001 -0.0033 (0.00) (-0.10) 0.6842* (1.77) -2.0556 (-1.45) -0.78% 0.07% 130 130

Three-month -0.0274 -0.0135 (-1.44) (-0.31) -0.0224 -0.0311 (-0.28) (-0.39) 1.7674 (1.60) -3.9483 (-1.01) -0.62% 3.89% 130 130

Pane D. HOUST Six-month -0.0526 -0.0563 (-1.46) (-0.72) -0.0554 -0.0718 (-0.42) (-0.54) 3.4339 (1.65) -4.8531 (-0.71) -0.35% 6.00% 130 130

One-year -0.1031 -0.1739 (-1.59) (-1.29) -0.0603 -0.0933 (-0.31) (-0.47) 7.5744* (1.77) -4.8628 (-0.42) -0.57% 14.15% 129 129

Two-year -0.1562 -0.0221 (-1.60) (-0.08) 0.4071 0.3337 (1.30) (1.11) 18.6880** (2.04) -43.0925 (-1.34) 2.95% 34.38% 117 117

One-month 0.0022** 0.0067*** (2.06) (2.83) -0.0033 -0.0035 (-0.81) (-0.85) 0.0285 (0.34) -0.4593* (-1.79) -0.60% 0.86% 130 130

Three-month 0.0062** 0.0157** (2.24) (2.43) -0.0111 -0.0117 (-0.98) (-1.00) 0.1074 (0.41) -1.0300 (-1.45) 0.08% 4.49% 130 130

Pane E. RETAIL Six-month 0.0103* 0.0217* (1.79) (1.98) -0.0390* -0.0405* (-1.94) (-1.96) 0.2782 (0.48) -1.4638 (-1.14) 3.91% 7.53% 130 130

One-year 0.0180 0.0191 (1.56) (1.04) -0.0862** -0.0898** (-2.32) (-2.33) 0.8070 (0.68) -1.3080 (-0.60) 8.94% 11.73% 129 129

Two-year 0.0466** 0.1035** (2.20) (2.22) -0.0513 -0.0631 (-0.92) (-1.08) 3.8936* (1.73) -11.9681** (-2.01) 1.00% 44.27% 117 117

One-month 0.0105*** -0.0126** (3.53) (-2.27) 0.0303** 0.0313** (2.47) (2.51) -0.1375 (-0.64) 2.3190*** (4.94) 2.67% 18.17% 130 130

Three-month 0.0268*** -0.0343** (3.12) (-2.07) 0.0576* 0.0615* (1.79) (1.86) -0.6131 (-0.96) 6.5206*** (4.96) 2.28% 32.35% 130 130

Pane F. UNRATE Six-month 0.0579*** -0.0389 (3.81) (-1.32) 0.1484*** 0.1579*** (3.06) (3.15) -1.6616 (-1.32) 11.3696*** (4.78) 5.65% 33.54% 130 130

One-year 0.1283*** 0.0386 (4.59) (0.75) 0.3801*** 0.4056*** (4.87) (5.01) -5.3422** (-2.32) 16.2074*** (3.93) 12.44% 33.05% 129 129

Two-year 0.2262*** 0.0405 (4.31) (0.28) 0.4912*** 0.5559*** (4.07) (4.30) -18.1715*** (-4.81) 47.5962*** (3.72) 7.96% 56.13% 117 117

One-month 0.0000 0.0032*** (0.02) (12.93) -0.0010 -0.0010 (-1.65) (-1.62) -0.0136 (-1.29) -0.2697*** (-12.25) 0.04% 63.52% 130 130

Three-month -0.0002 0.0088*** (-0.63) (12.32) -0.0042*** -0.0042*** (-2.75) (-2.72) -0.0180 (-0.57) -0.7934*** (-12.08) 1.07% 65.64% 130 130

Pane G. PAYEM Six-month -0.0009 0.0146*** (-1.33) (11.09) -0.0111*** -0.0114*** (-4.16) (-4.18) 0.0155 (0.24) -1.4386*** (-11.59) 2.76% 56.25% 130 130

One-year -0.0035** 0.0168*** (-2.51) (6.99) -0.0317*** -0.0330*** (-7.84) (-7.87) 0.2102* (1.68) -2.1657*** (-10.02) 8.00% 39.55% 129 129

Two-year -0.0063** 0.0380*** (-2.49) (5.40) -0.0586*** -0.0602*** (-9.72) (-9.31) 1.4489*** (6.25) -6.8344*** (-9.11) 11.59% 47.74% 117 117

46

Table 9. Cross-sectional Prediction With VRP Components This table documents the value-weighted return and model-adjusted alphas of stock portfolios sorted V E · · IMERPt , βbi,t by individual VRP and the different components of individual VRP respectively. βbi,t J b IMVRPt , and βi,t · IMJRPt respectively denote the components of VRP that are compensated for market equity risk exposure, market variance risk exposure and market jump risk exposure. Idio VRP (i,t ) denotes the idiosyncratic variance risk premium component. The return, alphas, and associated Newey-West t-values of the long-short strategy (5-1) are also reported.

Rank 1 2 3 4 5 5-1 1 2 3 4 5 5-1 1 2 3 4 5 5-1 1 2 3 4 5 5-1

Mean/Alpha Mean

CAPM

3-factor

4-factor

Est. 0.77 0.52 0.38 0.04 -0.10 -0.88 0.76 0.50 0.36 0.03 -0.12 -0.88 0.73 0.49 0.40 0.10 -0.07 -0.80 0.73 0.48 0.40 0.11 -0.04 -0.78

VRP t-value

[-2.71***]

[-2.79***]

[-2.94***]

[-2.90***]

E βbi,t · IMERPt Est. t-value 0.32 0.25 0.14 0.28 0.40 0.08 [0.25] 0.30 0.23 0.13 0.27 0.38 0.08 [0.25] 0.36 0.23 0.18 0.38 0.40 0.03 [0.11] 0.35 0.23 0.19 0.39 0.41 0.05 [0.18]

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V βbi,t · IMVRPt Est. t-value 0.64 0.55 0.37 0.18 -0.25 -0.89 [-2.28**] 0.62 0.54 0.35 0.17 -0.26 -0.88 [-2.21**] 0.72 0.57 0.37 0.23 -0.23 -0.95 [-3.19***] 0.72 0.58 0.37 0.23 -0.22 -0.94 [-3.27***]

J βbi,t · IMJRPt Est. t-value 0.32 0.32 0.32 0.36 0.15 -0.17 [-0.63] 0.30 0.31 0.31 0.34 0.13 -0.17 [-0.63] 0.29 0.40 0.39 0.37 0.14 -0.16 [-0.59] 0.29 0.41 0.39 0.38 0.16 -0.13 [-0.47]

i,t (Idio VRP) Est. t-value 0.54 0.48 0.34 0.30 -0.18 -0.72 [-2.46**] 0.52 0.46 0.32 0.28 -0.20 -0.72 [-2.56**] 0.50 0.47 0.35 0.34 -0.11 -0.61 [-2.37**] 0.52 0.48 0.34 0.35 -0.10 -0.61 [-2.43**]

Panel B

0.20

Density

0.10

6 4 0

0.00

2

Density

8

10

Panel A

−1

0

1

2

−20

0 (RV−IV) t−value

Panel C

Panel D

20

10

20

10

20

Density

1.0

0.10

1.5

0.20

10

0.0

0.00

0.5

Density

−10

(RV−IV) Mean

2.0

−2

−1

0

1

2

−20

−10

0

VRP Mean

VRP t−value

Panel E

Panel F

0.00

0.10

Density

2.0 1.0 0.0

Density

0.20

−2

−2

−1

0

1

2

−20

Log VRP Mean

−10

0 Log VRP t−value

Figure 1. Histogram of Cross-sectional Variance Risk Premiums: Mean and t Values The figure shows the cross-sectional distribution of the means (time series) and Newey-West (1987) t-values of E[RV] − IV, individual variance risk premiums (VRP, defined as (E[RV] − IV)/IV), and Log VRP (defined as ln E[RV] − ln IV).

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0.00 −0.15

Rm

Panel A

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

∆VIX Mimick

20 −10

0

−3

10

ln RVm ∆VIX Mimick

−5

ln RVm

−1

Panel B

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

∆JDK Mimick

0.005

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

−0.010

0.00

JDK

JDK ∆JDK Mimick

0.02

0.04

Panel C

0.004

∆JBT Mimick

−0.002

0.04

JBT ∆JBT Mimick

−0.02

JBT

Panel D

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Figure 2. Time Series Plots of Market Risk Factors The figure shows the time series plots of the market risks factors for the sample period January 1996October 2009. Panel A plots the monthly value-weighted excess return of all CRSP stocks, which is a proxy for market equity risk. Panel B plots the monthly log realized variance of CRSP value-weighted return and the return of mimicking portfolio for innovation in VIX, both of which are proxies for market variance risk. Panel C plots the jump/tail risk measure proposed by Du and Kapadia (2011) (JDK ) and the return of mimicking portfolio for innovation in JDK . Panel D plots the jump/tail risk measure proposed by Bollerslev and Todorov (2011) (JBT ) and the return of mimicking portfolio for innovation in JBT .

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Panel A3

Panel A2

1999

2001

2003

2005

2007

2009

−0.02

IMJRP

−4

−0.08

−2

IMVRP

0.0 −0.4

UVRP

0

0.02

Panel A1

1999

2001

2003

2007

2009

1999

2001

2003

2005

2007

2009

2007

2009

2007

2009

Panel B3

Panel B2

−0.01

−4

2001

2003

2005

2007

2009

1999

2001

2007

2009

1999

2001

2003

2005

2007

2009

2003

2005

0.02

1

IMJRP

0

−0.02

−4

1999

2001

Panel C3

−2

IMVRP

0.0 −0.4

UVRP

2005

Panel C2

0.4

Panel C1

2003

0.00

1999

0.01

IMJRP

0

IMVRP

−2

0.2 −0.2 −0.6

UVRP

2

Panel B1

2005

1999

2001

2003

2005

2007

2009

1999

2001

2003

2005

Figure 3. Time Series of UVRP, Implied Market Price of Variance Risk and Jump Risk The figure shows the time series plots of unexplained common individual VRP (UVRP), implied market variance risk premium (IMVRP), and implied market jump risk premium (IMJRP) when we use different jump risk proxies. Panel A1-A3 are for the model where we use innovation in BT(2010) fear measure to represent jump risk. Panel B1-B3 are for the model where we use DK(2011) jump/tail risk measure as the jump risk proxy. Panel C1-C3 are for the model where we use innovation in DK(2011) jump/tail risk measure as the jump risk proxy.

50