Variance Risk: A Bird’s Eye View∗

Fabian Hollstein† and Chardin Wese Simen‡

January 11, 2018

Abstract

Prior research documents a significant variance risk premium (VRP)

for the S&P 500 index but only for few equities. Using high-frequency

data, we show that these results are affected by measurement errors

in the realized variance estimates. We decompose the index VRP into

factors related to the VRP of equities and the correlation risk pre-

mium. The former mostly drives the variations in the index VRP while

the latter mainly captures the level of the index VRP. The two factors

predict excess stock returns in the time-series and cross-section, but at

different horizons. Together, they improve the return predictability.

JEL classification: G11, G12

Keywords: Correlation Swaps, Realized Variance, Return Predictability, Variance Swaps

∗We are grateful to Arndt Claußen, Maik Dierkes, Binh Nguyen, Marcel Prokopczuk, Roméo Tédon- gap and seminar participants at Leibniz University Hannover for helpful comments and discussions. Part of this research project was completed when Chardin Wese Simen visited Leibniz University Hannover. †School of Economics and Management, Leibniz University Hannover, Koenigsworther Platz 1, 30167 Hannover, Germany. Contact: [email protected]. ‡ICMA Centre, Henley Business School, University of Reading, Reading, RG6 6BA, UK. Contact: [email protected]. 1 Introduction

Carr & Wu(2009) and Driessen et al.(2009) document a significantly negative av-

erage variance swap payoff (VSP) for the U.S. stock market index.1 However, they find

significant VSPs only for a small number of individual equities that make up the stock

index. This finding appears surprising at first glance since the stock market index is

simply a portfolio of the constituent stocks and raises the question: where does the index

VSP come from?

This paper provides a bird’s eye view of the pricing of S&P 500 variance risk. We

make four contributions to the literature. First, we investigate whether the findings of the

aforementioned studies might be driven by statistical issues. In particular, we formulate

and evaluate explanations based on (i) the power of the statistical test, (ii) measurement

errors in the variance swap rate and (iii) measurement errors in the realized variance

estimate. We find evidence in support of the third explanation. Indeed, when realized

variance is computed using daily data, as is done in Carr & Wu(2009), Buraschi et al.

(2014b) and Dew-Becker et al.(2017), the average 1-month VSP of individual equities

is positive and statistically indistinguishable from zero (t-stat = 0.82). However, when

realized variance is accurately measured using high-frequency data, the average VSP

of individual equities becomes negative and statistically significant (t-stat = −1.99).

Additionally, the proportion of stocks for which the null of an insignificant VSP can be rejected rises from around 20 % to nearly 40 % when we compute realized variance using high-frequency, as opposed to noisy daily, data. We thus provide evidence of an upward bias in the realized variance estimates based on daily data. This bias is more pronounced

1Throughout this paper, we refer to the variance swap payoff as the difference between the realized variance, computed ex-post, and the risk-neutral expectation of future variance. We also analyze the variance risk premium, defined as the spread between the physical and risk-neutral expectations of future variance.

1 for firms with low market capitalizations, high leverage and illiquid stocks.

Second, we decompose the index VSP into (i) a factor that depends on the variance swap payoffs of the individual equities that make up the index and (ii) a factor that is a function of the correlation swap payoff. Our decomposition enables us to quantify the contribution of each of the two factors to the (i) level and (ii) variance of the index

VSP. Empirically, the two factors have distinct dynamics, with the first factor being less persistent and more volatile than the second factor. The factors are only weakly correlated, suggesting that they capture fundamentally different information. The factor related to the correlation swap payoff accounts for most of the average level of the index

VSP (70.4 %) while the factor linked to the VSP of individual stocks captures the lion’s share of the variations (64 %) in the index VSP.

Third, we go beyond the index variance swap payoff and study the index variance risk premium (VRP). We decompose the index VRP, which we model as in Bollerslev et al.(2009), into an individual VRP factor and a correlation risk premium (CRP) factor.

The CRP factor accounts for most of the level of the index VRP, whereas the individual

VRP factor is the primary force behind fluctuations in the index VRP. We explore the implications of the uncovered two-factor structure for the predictive power of the index

VRP. In particular, we seek to clarify the contribution of the Individual VRP and the

CRP factors to the predictability of S&P 500 excess returns over various horizons. Our analyses reveal that both factors are relevant for the predictability of S&P 500 excess returns, but at different horizons. The individual VRP factor is a highly significant predictor of excess returns over short horizons of up to 6 months whereas the predictive power of the CRP factor manifests itself also at longer horizons.

Fourth, we explore the cross-section of excess stock returns. Stocks that have a high

2 exposure to the index VRP earn a significantly higher future risk-adjusted return than

those that have a low exposure to the index VRP. This positive risk-adjusted return is

highly significant at horizons of up to 12 months. Analyzing the two factors that make up

the index VRP, we find that they both predict the cross-section of excess stock returns.

However, they operate at different horizons. Dependent sort analyses reveal that the

Individual VRP factor is a highly significant predictor for horizons of up to 12 months,

whereas the CRP factor’s predictive ability is also discernible at horizons such as 24

months. These results suggest that analyzing the index VRP alone may not reveal the

full scale of predictability in the cross-section of excess stock returns.

We perform several robustness checks. To begin with, we consider alternative sam-

pling frequencies for the computation of the realized variance estimates and reach similar

conclusions. Additionally, we consider alternative models to forecast realized variance

and obtain similar findings. Analyzing the term-structure of the VRP, we show that

the CRP factor accounts for most of the level of the index VRP of various maturities.

However, the contribution of the CRP factor to the variations of the index VRP rises

with the maturity of the variance swap.

Our study adds to the literature on measurement errors in the variance and correlation

swap payoff estimates.2 Du & Kapadia(2013) establish that the fixed leg of variance

swaps computed as in Carr & Wu(2009) is biased in the presence of jumps. 3 Faria

& Kosowski(2016) compare the synthetic correlation swap rates to those available in

the over-the-counter (OTC) market and show that the two are equal on average. We

differ from these studies in that we explore the role of measurement errors in the floating

2Bekaert & Hoerova(2014) analyze the impact of measurement errors in the index variance risk premium. We complement this work by also analyzing realized payoffs. Furthermore, we study correlation swaps and variance swaps related to the index as well as individual stocks. 3See also the works by Andersen et al.(2015), Martin(2017) and the references therein.

3 (rather than fixed) leg of variance and correlation swaps. We leverage developments in

the literature on high-frequency financial econometrics (Barndorff-Nielsen & Shephard,

2002; Andersen et al., 2003) to obtain more accurate estimates of the realized variance.

Cleaner realized variance estimates lead to more significant VSPs for individual equities

and affect the profitability of popular trading strategies.4 For instance, the average payoff

to a strategy that sells 1-month index variance swaps each month rises by 80 %, from

0.85 % to 1.53 % when realized variance is accurately measured. Relatedly, the average

payoff of a strategy that sells the 1-month correlation swap each month increases by 28 %,

from 10.74 % to 13.75 %, when realized variance is accurately measured. These findings

suggest that there is a large bias in the estimates reported in the literature.

Our paper belongs to the growing literature that dissects the index variance risk pre-

mium. Todorov(2010) and Bollerslev & Todorov(2011) decompose the index VRP into

components associated with (i) smooth and (ii) discontinuous movements. Bollerslev

et al.(2015) show that the compensation for the discontinuous movements plays an im-

portant role in the predictive power of the index VRP for aggregate excess stock returns.

Kilic & Shaliastovich(2015) and Feunou et al.(2017) decompose the index VRP into up-

side and downside parts and study their implications for the predictability of aggregate

excess returns. We decompose the index variance risk premium into factors linked to

the variance risk premium of stocks and the correlation risk premium. Our findings are

relevant for the literature on theoretical models of variance risk. For example, Driessen

et al.(2013) propose a model with correlation risk only. The model counterfactually

predicts that the variations in the index VRP are solely driven by the CRP factor. In

4The finding of significant average VSPs in stocks is relevant for studies that use implied variance to directly forecast the realized variance of stocks: it might be possible to improve the forecast accuracy by accounting for the variance risk premium in the spirit of Prokopczuk & Wese Simen(2014) and Kourtis et al.(2016).

4 the data, the CRP factor accounts for merely 28 % of the variations in the index VRP. A

realistic model should therefore feature two factors that are only weakly correlated and

at the same time match the decomposition results presented in this paper.

The remainder of this paper is organized as follows. Section 2 presents the data and

empirical methodology. Section 3 analyzes the variance swap payoffs. Section 4 focuses on the variance risk premia. Section 5 discusses the implications of our findings for theory and practice. Section 6 discusses additional results. Finally, Section 7 concludes.

2 Data and Methodology

This section begins with an overview of the dataset. It then introduces our method-

ology.

2.1 Data

We obtain options data related to the S&P 500 index and its constituents from IvyDB

OptionMetrics. The sample period ranges from January 1996 to August 2015.5 We use

the Volatility Surface provided by OptionMetrics. It contains implied volatilities for

different (i) constant time-to-maturity horizons and (ii) levels of delta, where delta is

defined as the sensitivity of the option price to a small change in the underlying asset

price.6

We process the options dataset as follows. First, we only retain options that are out-

of-the-money (OTM). Essentially, this means that we only keep put and call options with

5The starting date of the sample period is forced upon us as the OptionMetrics dataset is available from 1996 onwards. The end of our sample period reflects the data available at the time we started the study. 6IvyDB uses a kernel smoothing algorithm to compute the standardized options only “if there ex- ists enough option price data on that date to accurately interpolate the required values”. For further information, we refer the interested reader to the IvyDB technical document.

5 deltas that are higher than −0.5 and lower than 0.5, respectively. Second, we discard

observations related to options of time-to-maturity longer than 12 months. Third, we

download the time-series of discount rates available from OptionMetrics. We implement

a cubic spline interpolation to obtain the discount rate of same time-to-maturity as the

option. We match this time-series with our panel data so that for each option price, there

is a discount rate of corresponding maturity.

We also obtain the daily time-series of the S&P500 index, the prices and returns of all

individual equities that make up the index. The data come from the Center for Research

in Security Prices (CRSP).

2.2 Methodology

Our analysis revolves around the variance swap. A trader with a long position in

the variance swap pays a fixed rate called the variance swap rate (VSR). In return, she receives the realized variance (RV ) of the underlying asset computed over the maturity of the variance swap. Thus, the payoff to a long variance swap with a notional amount of $1 can be computed as follows:

VSP i,t+τ = RV i,t+τ − VSRi,t (1)

where VSP i,t+τ is the realized payoff on day t + τ of the variance swap on the underlying asset i. τ is the time to maturity of the contract, expressed in calendar days. RV i,t+τ is

the realized variance of the underlying asset over the life of the variance swap, i.e. for

the period starting at t and ending at t + τ. Notice that the realized variance can only be computed ex-post, i.e. at t + τ. VSRi,t is the variance swap rate, which is fixed at

inception, i.e. on day t. Because, by design no money changes hands at the inception

6 of the variance swap, no-arbitrage implies that the variance swap rate is the risk-neutral

Q 2 7 expectation of future variance (Et (σi,t+τ )). Thus, we can write:

Q 2 VSP i,t+τ = RV i,t+τ − Et (σi,t+τ ) (2)

Carr & Wu(2009), Buraschi et al.(2014b) and Dew-Becker et al.(2017), among

others, define the realized variance as follows:

t+τ 365 X RV = r2 (3) i,t+τ τ i,j j=t

where ri,j is the (log) return on security i on day j. Throughout this paper, the factor

365 τ annualizes the variance estimate.

For the computation of the variance swap rate, we build on the work of Bakshi et al.

(2003), who derive the following result:

365 τ 2 rft 2
EQ(σ ) = e 365 QUAD − µ (4) t i,t+τ τ t t with  h K i Z ∞ 2 1 − ln Z St  St 
St 2 1 + ln K QUADt = 2 Ct(τ, K)dK + 2 Pt(τ, K)dK (5) St K 0 K τ rft rf τ e 365 µ = e t 365 − 1 − QUAD (6) t 2 t

where rft denotes the (annualized) discount rate (of same maturity as that of the option)

on day t. Ct(τ, K) and Pt(τ, K) denote the price, on day t, of the call and put options with time to maturity τ and strike price K, respectively. All options relate to the underlying asset i. 7In this paper, we often refer to the risk-neutral expectation of future variance as the option implied variance.

7 Our construction of the variance swap rate broadly follows Chang et al.(2012). We begin by computing ex-dividend stock prices. From the Volatility Surface, we interpolate implied volatilities for different levels of moneyness using a cubic spline, where moneyness is defined as the ratio of the strike price over the underlying price. To be more precise, we interpolate the implied volatilities for 1,000 equally spaced moneyness levels between 0.3% and 300%. In practice, it is likely that the moneyness range available in the market does not completely span the interval starting at 0.3 % and ending at 300 %, raising the issue of extrapolation. Similar to Jiang & Tian(2005), we perform a nearest neighbourhood extrapolation. We then use the fine grid of interpolated/extrapolated implied volatilities to compute the Black & Scholes(1973) prices of the out-of-the-money options. Equipped with these prices, we implement the trapezoidal rule to numerically compute the risk- neutral expectation of future variance (see Equations (4)–(6)). We repeat the steps above for each security and observation day.

3 Variance Swap Payoffs

For our main empirical analysis, we study the S&P 500 index variance swap payoff of maturity 1 month.8 In order to mitigate concerns related to overlapping observation biases, we sample the last observation of each month.

3.1 Descriptive Analysis

Panel A of Table1 reports the average value of the floating and fixed legs of the variance swap linked to the S&P 500 index. We observe that the average realized variance is 3.75 %, whereas the average variance swap rate is 4.61 %. Thus, the variance swap

8As a robustness check, we consider alternative maturities for the variance swap. See Section 6.3 for further details.

8 payoff is negative on average (−0.85 %). The Newey & West(1987) corrected t-ratio (t-

stat=−2.14) indicates that we can reject the null hypothesis that the average index VSP

is equal to zero. The positive skewness and the large kurtosis estimates reveal that, while

a strategy that sells variance swaps is generally profitable, it is also prone to crashes.

These findings are consistent with those of earlier works, e.g. Carr & Wu(2009).

We now turn our attention to the payoffs of the variance swaps related to the stocks

that make up the S&P 500 index. Following Driessen et al.(2009), we compute the equal-

weighted average of the fixed and floating legs of the variance swaps of all constituent

stocks. We focus on stocks with at least 30 monthly observations of variance swap payoffs

during the sample period. In total, 796 stocks fulfill this screening criterion.9 Panel A of

Table1 documents that the average variance of individual firms is generally higher than

that of the S&P 500 index, reflecting the importance of diversification. For instance, the

average realized variance of the constituent stocks (16.38 %) is four times larger than that

of the S&P 500 index (3.75 %). A similar pattern emerges also for the option implied

variance (15.49 % vs. 4.61 %).

We notice that the average realized variance of individual equities, which is based on

daily data (see Equation (3)), is higher than the corresponding implied variance, implying

a positive average variance swap payoff (0.89 %). The positive average VSP of equities is

in contrast to the negative estimate observed for the S&P 500 index. The t-ratio of 0.82 associated with the mean VSP of individual stocks reveals that we cannot reject the null hypothesis that the mean is equal to zero. This result extends the finding of Driessen et al.(2009), who reach a similar conclusion for the stocks included in the S&P 100 index, to the constituents of the S&P 500 index.

One possible explanation for the average VSP of stocks that is statistically indistin-

9This number exceeds 500 due to index additions and deletions during our sample period.

9 guishable from zero is that some stocks have significantly positive variance swap payoffs

while others have significantly negative variance swap payoffs. If this is the case, the

payoffs might offset each other, making it hard to reject the null hypothesis that the

average variance swap payoff of stocks is equal to zero. We thus ask the question: how

often can we reject the null of a zero average variance swap payoff for individual stocks?

The results presented in Panel B of Table1 show that we cannot reject the null

hypothesis of a zero average VSP for 78.1 % of the individual stocks. We conclude, that

for an overwhelming number of individual equities, the average variance swap payoff is

not significantly different from zero. This finding echoes that of Carr & Wu(2009), who

document that 30 out of the 35 individual stocks with the most active option markets

are associated with an insignificant variance swap payoff.10

Summarizing, our analysis reveals that there is a significantly negative VSP at the index level but no evidence of a significant variance swap payoff for the average stock.

Taken together, these results raise the question: where does the index variance swap payoff come from?

3.2 Variance Swap Payoffs: Index vs. Individual Equities

This section sheds light on the gap between the average variance swap payoff of the index and that of the constituent equities. We begin by presenting and evaluating explanations based on statistical biases. Afterwards, we derive a formal relationship between the variance swap payoff of the index on the one hand and that of the constituent equities on the other.

10A legitimate concern may be that Carr & Wu(2009) analyze daily observations, whereas we focus on the monthly frequency. To shed light on this, we repeat our analysis at the daily (rather than monthly) frequency and obtain qualitatively similar results. Note that these results must be interpreted cautiously because of the large amount of overlapping observations. These results are available upon request.

10 3.2.a Statistical Biases

Power One may argue that the statistical test used in the preceding analysis has low power. This issue would affect the statistical inference by making it difficult to find a significant variance swap payoff for individual equities. There are reasons to be skeptical of this line of argument. First, it is not obvious why this problem might occur only when testing the null that the average VSP of stocks is equal to zero. Second, we use a sample period that covers nearly 20 years. Presumably, a long sample period should make the statistical test more powerful; yet, we reach similar conclusions as earlier studies.

Measurement Errors in the Variance Swap Rate A possible explanation for the results of Carr & Wu(2009) and Driessen et al.(2009) might be that their variance swap rate is biased. This could occur because the Britten-Jones & Neuberger(2000) implied variance formula used by the authors holds only approximately if the underlying return process contains jumps. Our analysis is robust to these concerns because our computation of the variance swap rate relies on the Bakshi et al.(2003) implied variance, which Du &

Kapadia(2013) show to be empirically robust to jumps.

One might also argue that option contracts, especially those on small equities, are illiquid and this may bias the estimates of the average variance swap payoff. There are several reasons to doubt this explanation. Carr & Wu(2009) study the 35 most active individual equities and find that the null that the average variance swap payoff is equal to zero can only be rejected for 5 out of the 35 stocks. Driessen et al.(2009) report broadly similar results for the firms in the S&P 100 index. As a whole, these studies show that by focusing on liquid equity derivatives, one cannot bridge the gap between the average

VSP of the index and that of the constituents.

11 Relatedly, the illiquidity argument suggests that individual equity option prices might be “relatively” high because of the illiquidity premium. An upshot of this is that we are quite likely to find a significantly negative average variance swap payoff in most equities.

This argument does not square well with the empirical evidence of Panel B of Table2, that indicates that the VSP of 78.1 % of the constituent stocks is statistically indistinguishable from zero.

Measurement Errors in Realized Variance Another possibility could be that the realized variance estimates are contaminated by measurement errors. This might occur because the realized variance estimates analyzed so far hinge on low-frequency (daily) data. However, recent works by Barndorff-Nielsen & Shephard(2002) and Andersen et al.(2003), among others, document the importance of high-frequency data for the accurate measurement of realized variance.

We repeat the previous analysis using high-frequency data. We obtain the data from

Thomson Reuters Tick History (TRTH). In order to process the final data set, we follow the data-cleaning steps outlined in Barndorff-Nielsen et al.(2009). That is, first, we use only data with a time stamp during the exchange trading hours, i.e., between 9:30AM and 4:00PM. Second, we remove recording errors in prices. To be more specific, we filter out prices that differ by more than 10 mean absolute deviations from a rolling centered median of 50 observations. Afterwards, we assign prices to every 5-minute interval using the nearest previous entry that occurred at most one day before. Finally, we follow

Bollerslev et al.(2016a) and supplement the TRTH data with data on stock splits and distributions from CRSP to adjust the TRTH overnight returns. We follow Bollerslev

12 et al.(2016a) and sample observations at the 75-minute frequency. 11 Equipped with this data, we use the following estimator for realized variance:

t+τ m 365 X X RV = r2 (7) i,t+τ τ i,j,k j=t k=0

ri,j,k is the return on security i on day j and at time interval k. Note that the case when k = 0 corresponds to the overnight return. Overall, we observe m + 1 intraday returns on each trading day (including the overnight return). All other variables are as previously defined.

Table2 repeats the analysis of Table1 using realized variance based on high-frequency

(rather than daily) data. The two tables highlight several differences. Panel A reveals that the average realized variance of the S&P 500 index estimated with intraday data (3.08 %) is lower than that based on the daily return series (3.75 %). This difference matters for the average index VSP. The average variance swap payoff based on intraday data (−1.53 %) is larger in magnitude than that based on the low-frequency data. Furthermore, the t-ratio associated with the mean variance swap payoff is clearly higher (t-ratio = −5.23) when realized variance is estimated with high-frequency data than in the benchmark case

(t-ratio = −2.14). This finding is consistent with an argument based on measurement errors.

Turning to the individual stocks, we find that the average realized variance based on intraday data (14.25 %) is lower than that based on the noisy daily data (16.38 %). This,

11Clearly, the choice of sampling frequency involves a trade-off. At first glance, one might think that pushing the frequency to the highest level may lead to more accurate estimates of the realized variance in the spirit of the theory of quadratic variation (Barndorff-Nielsen & Shephard, 2002). However, this needs not be the case. A too high sampling frequency may result in a host of microstructure issues, such as infrequent trading observations, i.e. too many zero returns and the Bid–Ask bounce, that could bias the realized variance estimates. Bollerslev et al.(2014) use a 75-minute frequency. We follow their lead. We also consider alternative frequencies and obtain qualitatively similar results. See Section 6.1 for further details.

13 in turn, leads to a negative average variance swap payoff based on high-frequency data

(−1.23 %). The negative estimate is in contrast to the positive average variance swap payoff (0.89 %) based on the daily data (see Panel A of Table1). Moreover, the t-ratio

associated with the average VSP computed with intraday data is statistically significant

(t-ratio = −1.99), whereas this is not the case for the initial analysis based on daily data.

We also investigate the proportion of stocks for which we can reject the null hypothesis

of an insignificant average VSP. Panel B of Table2 shows that we can reject this null

hypothesis for 41.3 % of individual stocks. This is twice as large as the share of stocks

for which the null can be rejected when realized variance is based on the daily data (see

Panel B of Table1).

One may wonder: what are the characteristics of the stocks most affected by mea-

surement errors in the realized variance estimate? To shed light on this, we perform a

Fama & MacBeth(1973) cross-sectional regression of the difference between the noisy

realized variance estimate (based on daily data) and the cleaner estimate (based on high-

frequency data) on several stock characteristics.12 As explanatory variable, we use the

logarithm of the market value of equity, the Book-to-Market ratio constructed as in Fama

& French(1992), the leverage ratio computed as the ratio of the book value of assets over

the market value of equity (Fama & French, 1992) and the Bid–Ask spread of the stock

associated with each firm. To be more specific, Bid–Ask is the average of the daily pro-

portional Bid–Ask spreads observed over the month. Table A.1 of the Online Appendix

reveals that the upward bias in the noisy realized variance estimate is high for firms with

low market capitalizations, high leverage ratios and illiquid stocks.

Summarizing, we show that our understanding of the VSP of individual equities is

12To be more precise, at the end of each month, we perform a cross-sectional regression of the difference between the noisy and clean realized variance estimates on the characteristics. We then perform the statistical inference on the time-series of the parameter estimates.

14 affected by errors in the measurement of the realized variance. The use of high-frequency data helps alleviate these measurement errors. Accordingly, the rest of this paper proceeds with estimates of realized variance based on high-frequency data.

3.2.b Dissecting the Market Variance Swap Payoffs

We now ask the question: what is the contribution of the VSP of individual equities to the index VSP? To tackle this question, we start with the identity linking the variance of the equity index returns to the sum of the weighted average of the variance of individual stock returns and covariance terms related to the index constituents:

N X 2 X p p RV I,t+τ = ωi,tRV i,t+τ + ωi,tωj,tρt+τ RV i,t+τ RV j,t+τ (8) i=1 i,j6=i

where RV I,t+τ is the realized variance of the index I at time t + τ. ωi,t is the market capitalization weight of asset i at time t. RV i,t+τ is the realized variance of asset i, computed at time t + τ. ρt+τ is the equicorrelation at time t + τ. To be more precise,

ρt+τ is the correlation that, if used instead of all the pairwise correlations, yields the same variance for the index (Skintzi & Refenes, 2005). In practice, we follow previous works, e.g. Driessen et al.(2009) and Buraschi et al.(2014b), and extract this term as a residual. That is, given the market capitalization weights, the realized variance of the index and that of individual stocks, we can re-arrange Equation (8) to derive the formula for the equicorrelation:13

PN 2 RVI,t+τ − i=1 ωi,tRV i,t+τ ρt+τ = P p p (9) i,j6=i ωi,tωj,t RVi,t+τ RVj,t+τ

13Throughout this paper, we use the weights computed at time t. Driessen et al.(2009) show that time- variations in the weights of the index constituents only marginally affect the estimate of equicorrelation.

15 A similar expression holds under the risk-neutral measure (Driessen et al., 2009):

N X X q q Q 2 2 Q 2 Q Q 2 Q 2 Et (σI,t+τ ) = ωi,tEt (σi,t+τ ) + ωi,tωj,tEt (ρt+τ ) E (σi,t+τ ) E (σj,t+τ ) (10) i=1 i,j6=i

Q 2 where Et (σI,t+τ ) is the time-t risk-neutral expectation of the future variance of the index.

Q 2 Q Et (σi,t+τ ) is the time-t risk-neutral expectation of the future variance of asset i. Et (ρt+τ )

is the time-t risk-neutral expectation of the future equicorrelation. We invert Equation

(10) to express the risk-neutral expected correlation as a function of observable quantities

(Driessen et al., 2009, 2013):

EQ(σ2 ) − PN ω2 EQ(σ2 ) Q t I,t+τ i=1 i,t t i,t+τ Et (ρt+τ ) = q q (11) P Q 2 Q 2 i,j6=i ωi,tωj,t Et (σi,t+τ ) Et (σj,t+τ )

Using Equations (2), (8) and (10), it is straightforward to show that:

Q 2 VSP I,t+τ = RV I,t+τ − Et (σI,t+τ ) N X 2 X  p p = ωi,tVSP i,t+τ + ωi,tωj,t ρt+τ RV i,t+τ RV j,t+τ i=1 i,j6=i q q  Q Q 2 Q 2 −Et (ρt+τ ) Et (σi,t+τ ) Et (σj,t+τ ) N X 2 = ωi,tVSP i,t+τ i=1 | {z } P ure V SP t+τ  q q  X Q p p Q 2 Q 2 + ωi,tωj,tEt (ρt+τ ) RV i,t+τ RV j,t+τ − Et (σi,t+τ ) Et (σj,t+τ ) i,j6=i | {z } Cross−VSP t+τ X p p Q
+ ωi,tωj,t RV i,t+τ RV j,t+τ ρt+τ − Et (ρt+τ ) i,j6=i | {z } CSP t+τ

= P ure V SP t+τ + Cross − VSP t+τ +CSP t+τ (12) | {z } Individual V SP t+τ

VSP I,t+τ = Individual V SP t+τ + CSP t+τ (13)

16 Equation (13) reveals that the variance swap payoff of the equity index can be de-

composed into 2 factors. The first factor (Individual VSP) is a function of the VSP

of individual equities.14 The second factor (CSP) is a function of the correlation swap

payoff.15 We next discuss the implications of this decomposition.

The Level of the Index Variance Swap Payoff Equation (13) shows that, if the

variance swap payoff of each stock equals 0, then the Individual VSP factor disappears.

An upshot of this is that the index variance swap payoff is equal to the CSP factor.16

Conversely, if the correlation swap payoff is equal to zero, the Individual VSP factor will

be equal to the index VSP.

However, if (i) the variance swap payoffs for individual equities are generally different

from zero (as the previous analysis shows using high-frequency data) and (ii) correlation

swap payoffs are on average negative, as Panel D of Table A.2 shows, it is interesting to

evaluate the contribution of each factor to the level and variance of the index VSP. We

can study the effect of the 2 factors on the level of the index VSP since Equation (13)

14This factor comprises two terms: the Pure VSP and the Cross-VSP. The Pure VSP provides a pure exposure to the variance swap payoffs of individual equities. To be more precise, it is the sum of the variance swap payoffs of individual equities multiplied by the squared value of the market capitalization weights. The Cross-VSP depends on the interaction between the swaps of different stocks scaled by the market capitalization weights and the risk-neutral expectation of the equity correlation. Empirically, we find that the Pure VSP and the Cross-VSP series are strongly related with a correlation coefficient equal to 87 %, suggesting that they contain similar information. This finding motivates us to merge the two terms under the label Individual VSP. 15The correlation swap is an over-the-counter derivative. The agent who takes a long position in the correlation swap (with notional of $1) pays the correlation swap rate and receives the realized correlation. Consistent with the works of Buraschi et al.(2014a) and Faria & Kosowski(2016), among others, Table A.2 documents significantly negative CSP. This is true, irrespective of whether realized equicorrelation is computed using 75-minute data or noisy daily data. Interestingly, Panels B and D of the same table show that measurement errors in the realized variance estimates translate to an upward bias of 3.01 % in the monthly average CSP (based on daily data). 16Driessen et al.(2013) develop a model where the average index VSP is completely determined by the correlation swap payoff. Our decomposition highlights that the correlation swap payoff is weighted by a P p p term corresponding to i,j6=i ωi,tωj,t RV i,t+τ RV j,t+τ . Absent this term, there will be a mismatch between the average index variance swap payoff and the average correlation swap payoff. For instance, the authors find an average correlation swap payoff of −6.87 %, more than 6 times the magnitude of the variance swap payoff of the S&P500 index (−1.05 %). Analyzing the (daily) time-series of 1-month correlation swap payoffs, where realized correlation is based on daily data as in Driessen et al.(2009), we find an average correlation swap payoff of −7.72 % (see Panel A of Table A.2 of the online Appendix).

17 implies that the level of the index VSP equals the sum of the levels of the two factors on

the right hand-side of Equation (13).

The Variance of the Index Variance Swap Payoff It is useful to recall that:

V ar(VSP I,t+τ ) = Cov(VSP I,t+τ ,VSP I,t+τ )

V ar(VSP I,t+τ ) = Cov(Individual V SP t+τ + CSP t+τ ,VSP I,t+τ ) (14)

It is straightforward to show that:

Cov(Individual V SP ,VSP ) Cov(CSP ,VSP ) 1 = t+τ I,t+τ + t+τ I,t+τ (15) V ar(VSP I,t+τ ) V ar(VSP I,t+τ )

where V ar(·) and Cov(·) are the variance and covariance operators, respectively.

Equation (15) shows that we can decompose the variance of the index VSP into two parts. The contribution of each factor corresponds to the slope estimate of a regression of the factor on a constant and the index VSP.

Results Panel A of Table3 summarizes the results of the decomposition. We find that the Individual VSP and CSP factors, yield average values of −0.45 % and −1.07 %, respectively, leading to a level of the index VSP equal to −1.53 %.17 The negative average values show that both factors make a positive contribution to the level of the index VSP.

Additionally, the estimates suggest that the bigger contribution to the level of the index

VSP stems from the CSP factor, which accounts for 70.4 %.

17Notice that the sign of the CSP factor is completely determined by that of the correlation swap payoff. This is because the weights and volatility terms are always positive. Thus, the negative average value of the CSP factor indicates that the correlation swap payoff is negative in general (see Table A.2 of the online Appendix). This finding is consistent with the works of Driessen et al.(2009), Buraschi et al.(2014a) and Faria & Kosowski(2016).

18 Turning to the variance decomposition results, the column entitled “Shrvar” reveals that both factors make a positive contribution to the variance of the index VSP, as evidenced by the positive entries. The bigger contributor is the Individual VSP factor, accounting for 64 % of the variability.

Taken together, the empirical analysis reveals that the CSP factor captures a large share of the level of the index VSP. This finding is consistent with the insight of Driessen et al.(2013). When we analyze the variance of the index VSP, we find that the Individual

VSP (rather than the CSP) factor is the main driving force.

4 Variance Risk Premia

Bollerslev et al.(2009) establish the predictive power of the index VRP, which we define as the spread between the physical and risk-neutral expectations of future vari- ance, for future aggregate excess stock returns. Motivated by this, we pose the following questions: do the decomposition results extend to the index VRP? If so, what are the implications for the predictability results? Which of the two factors matters for the predictability of aggregate excess stock returns and at what horizon? What does the two-factor structure imply for the cross-section of future excess stock returns?

4.1 Dissecting the Variance Risk Premium

Framework We define the variance risk premium as:

P 2 Q 2 VRP i,t = Et (σi,t+τ ) − Et (σi,t+τ ) (16)

19 P where VRP i,t is the variance risk premium of asset i at time t. Et (RV i,t+τ ) is the physical expectation at time t of the future variance. All other variables are as previously defined.

In order to decompose the variance risk premium of the aggregate index into its factors, it is useful to note the following:

N q q P X 2 P X P P P Et (RV I,t+τ ) = ωi,tEt (RV i,t+τ ) + ωi,tωj,tEt (ρt+τ ) Et (RV i,t+τ ) Et (RV j,t+τ )(17) i=1 i,j6=i

Using Equations (10), (16) and (17), we can thus decompose the index variance risk premium as follows:

P Q 2 VRP I,t = Et (RV I,t+τ ) − Et (σI,t+τ ) N X 2 = ωi,tVRP i,t i=1 | {z } P ure V RP t q q q q  X Q P P Q 2 Q 2 + ωi,tωj,tEt (ρt+τ ) Et (RV i,t+τ ) Et (RV j,t+τ ) − Et (σi,t+τ ) Et (σj,t+τ ) i,j6=i | {z } Cross−VRP t q q X P P P Q
+ ωi,tωj,t Et (RV i,t+τ ) Et (RV j,t+τ ) Et (ρt+τ ) − Et (ρt+τ ) i,j6=i | {z } CRP t

= P ure V RP t + Cross − VRP t +CRP t (18) | {z } Individual V RP t

VRP I,t = Individual V RP t + CRP t (19)

Two observations are in order. First, the variance risk premium can be computed in real-time since it is a conditional expectation of the future variance swap payoff. Hence, the ensuing analysis is cast in an ex-ante setting. Second, the variance risk premium involves the physical expectation of the future variance, which is not directly observable.

This forces us to specify a model to generate the physical expectation of the future

20 variance. Similar to Bollerslev et al.(2009), we assume a random walk model, i.e. the physical expectation for the future variance is equal to its most recent realization.

This assumption does not require any estimation, making it very easy to implement for all assets we study. Relatedly, it bypasses sampling errors associated with the vari- ance forecasting regressions. Additionally, this model ensures that we do not lose too many observations to estimate the parameters of the forecasting models.18 Moreover, an investor in real-time would have known and therefore considered this model. This is not necessarily the case for recently proposed variance forecasting models, e.g. Bollerslev et al.(2016b). Finally, it delivers positive forecasts of variance. This is not always guar- anteed for variance forecasting models that need to be estimated; a problem that could be acute for individual equities.

Figure1 summarizes the dynamics of the index VRP as well as its two factors. The index VRP is generally negative and takes large values during bad economic times, as defined by the National Bureau of Economic Research (NBER). Turning to the two factors, we notice that they exhibit distinct dynamics. The individual VRP factor is more volatile than the CRP factor, which tends to be rather sticky. Overall, the correlation between the two factors is low and negative (−24 %), suggesting that they are likely driven by different economic forces.

Results Panel B of Table3 reports that the level of the index VRP is significantly negative (−1.34 %, t-stat = −6.57) and is comparable to that of the index VSP (−1.53 %) documented in Panel A of the same table. An implication of this observation is that, on average, 87.58 % of the level of the index VSP was expected by an investor who uses the

18For instance, in the robustness check, we consider alternative variance forecasting models. In order to do so, we effectively lose the first year and a half of data, thus reducing our sample size available to study the variance risk premium. See Section 6.2 for further details.

21 random walk model to form her expectation about future realized variance.

The decomposition results reveal that the CRP factor accounts for 91.1 % of the level

of the index VRP. Turning to the variance decomposition results, “Shrvar” shows that the

Individual VRP factor makes the larger contribution (72.1 %) to the variations of the

index VRP.

Overall, these findings are consistent with the insights gleaned by analyzing the vari-

ance swap payoffs. The CRP factor is important for the level of the index VRP, whereas

the Individual VRP factor matters for the variations of the index VRP.

4.2 Implications for Return Predictability

In this section, we examine the implications of the two-factor structure for the pre-

dictability of excess returns. We begin by studying the predictability of aggregate market

excess returns. Afterwards, we explore the implications for the cross-section of future ex-

cess stock returns.

4.2.a Time-Series Evidence

We estimate the following regression:

erI,t→t+h = α + βV RP I,t + I,t+h (20)

where erI,t→t+h is the (annualized) excess return of the stock market index for the period starting at t and ending at t + h (h is expressed in months). Throughout this paper, the excess return of an asset is the difference between the total return of the asset and the risk-free rate. We obtain the U.S. Treasury bill rate data from Kenneth French’s website.

We always examine forecasting horizons of 1, 3, 6, 9, 12, 18 and 24 months. α is the

22 intercept. β is the slope of the regression model. I,t+h is the residual of the regression.

Panel A of Table4 reports the results of this analysis. We present the Newey & West

(1987) standard errors computed with h lags in parentheses. We also report Hodrick

(1992) standard errors in square brackets. The results indicate that the slope parameter is significantly negative (at the 5 % significance level) over horizons of up to 6 months.

This is true for both types of standard errors. Thus, the index VRP is a short-term predictor of S&P 500 excess returns. The adjusted R2 enables us to gauge the strength of predictability for different forecasting horizons. We can see a hump-shaped pattern, with the adjusted R2 peaking at the quarterly horizon (adjusted R2 = 9 %). Overall, these

findings confirm and update the results of Bollerslev et al.(2009) to the more recent period.19

The decomposition results show that we can express the index VRP as the sum of two factors. To understand which of the two factors is relevant for the predictability of excess returns, we directly use them in the following excess return forecasting regression:

erI,t→t+h = α + φIndividual V RP t + γCRP t + I,t+h (21)

where α, φ and γ are parameters to be estimated.

Panel B of Table4 shows that the slope estimates linked to the two factors are negative

and highly significant, suggesting they are both informative about risk premia. However,

the factors operate at different horizons. Figure2 illustrates this finding. We can see

that, using the 5 % significance level, the Newey–West corrected t-statistic associated

with the loading on the Individual VRP factor is significantly negative for horizons up

19In comparing our results to those of Bollerslev et al.(2009), it is worth remembering that the authors’ definition of the variance risk premium is the opposite of ours. Thus, their positive slope estimate is consistent with our negative slope parameter.

23 to 9 months, while the slope on the CRP factor remains significant for longer horizons.

Note also that the t-statistic of the loading on the Individual VRP is larger (smaller) in magnitude than that of the CRP factor for horizons up to (beyond) 6 months. Viewed as a whole, these patterns indicate that the CRP factor is a long-term predictor of aggregate excess stock returns whereas the individual VRP factor acts as a short-term predictor.20

It is worth noting that if one starts with the model in Equation (21) and imposes the restriction that φ = γ, then this model reduces to that in Equation (20). If the restriction is rejected by the data, then a forecaster who considers the two factors directly could achieve a higher forecasting power. Figure3 shows that the adjusted R2 of the unconstrained model is generally higher than that of the constrained model, suggesting that the restriction is not supported by the data. We also implement a Wald test to formally test the aforementioned restriction. The last two rows of Panel B of Table4 show that the null hypothesis is rejected for almost all horizons at the 5 % significance level.

One may wonder whether the predictability results are robust to the inclusion of well- known predictors of aggregate excess stock returns. We consider the following variables.

CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 index over the 12-month trailing sum of dividends paid by

S&P 500 firms. log(P/E) is the logarithm of the S&P 500 price index over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the

1-month U.S. T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the U.S. Treasury 10-year yield and the 3-month T-bill

20This is also true when considering the more conservative Hodrick(1992) standard errors. The Individual VRP factor predicts at horizons up to 3 months and the CRP factor at horizons up to 9 months.

24 rate. These variables are also used in Bollerslev et al.(2009). We obtain the data on these predictors from Amit Goyal’s website.21 Tables A.3–A.9 of the online appendix confirm that the predictability results are robust to the inclusion of these variables. This is true for both the constrained and unconstrained regressions.

4.2.b Cross-Sectional Evidence

We next focus on the predictive power of the index VRP for the cross-section of ex- cess stock returns. We obtain data on all stocks traded at the New York Stock Exchange

(NYSE), the American Stock Exchange (AMEX) and the National Association of Secu- rities Dealers Automated Quotations (NASDAQ) that are classified as ordinary common shares (CRSP share codes 10 or 11). We expunge closed-end funds and REITS (SIC codes 6720–6730 and 6798) from our dataset. We also discard stock-month observations with a price below $1. We account for delisting returns following Shumway(1997) and

Shumway & Warther(1999).

Our testing framework follows Kelly & Jiang(2014). At the end of a given month, we use a window containing the most recent 60-month period of data to estimate the following model:

eri,t→t+h = αi + βiVRP I,t + i,t+h (22)

where eri,t→t+h is the excess return on stock i for the period starting at the end of month t and ending in month t + h. αi and βi are the intercept and slope parameters, respectively. Note that these parameters are firm-specific and possibly time-varying, since we re-estimate the model on a rolling window basis. i,t+h is the residual associated with

21The data is available at the following address: http://www.hec.unil.ch/agoyal/.

25 firm i at t + h.

We estimate the model in Equation (22) for each firm, thus obtaining slope estimates

for all stocks. Next, we form quintile portfolios by sorting stocks in ascending order of

their slope estimates. As a result, portfolios 1 (P1) and 5 (P5) contain the stocks with

the lowest and highest 20 % slope estimates, respectively. We create a long-short strategy

that buys stocks in P5 and sells those in P1. We hold the P5−P1 position for the next

h months.

Panel A of Table5 reports the (annualized) value-weighted returns of the strategy

for various horizons.22 Based on the 5 % significance level, we conclude that the strategy

yields significantly positive raw returns for horizons of 6, 9, 12 and 18 months. The

positive average raw return is consistent with the work of Han & Zhou(2011), who

document a negative relationship between future excess stock returns and the spread

between the risk-neutral and physical expectations of the index variance.

Economically, stocks with the highest exposure to the index VRP outperform stocks

with the lowest exposure by 9.34 % per year (t-stat = 3.11) at the 12-month horizon. We investigate whether the performance of the long-short portfolio simply reflects passive exposure to well-known equity risk factors by computing the risk-adjusted return, i.e. the alpha, relative to the Carhart(1997) 4-factor model. 23 We find significant risk-adjusted

returns for the aforementioned horizons. It is worth noting that these risk-adjusted

returns are often higher than their unadjusted counterparts, indicating that the strategy

generally has a negative exposure to the risk factors with a positive price of risk.

Implicitly, the analysis based on Equation (22) suggests that ranking firms by their

22As a robustness check, we implement the overlapping portfolio approach of Jegadeesh & Titman (1993). Tables A.10 and A.11 of the online appendix summarize these results. 23The risk factors are available from Ken French’s website: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Research.

26 exposure to the index VRP is equivalent to sorting based on the exposure to the Individual

VRP or CRP factor. To test this assumption, we follow an alternative approach. At the end of each month, we use the previous 60-month period of data to estimate the following regression model:

ri,t→t+h = αi + φiIndividual V RP t + γiCRP t + i,t+h (23)

where αi, φi and γi are the intercept and slope parameters, respectively.

We compute the rank of firms sorted in ascending order of their φ-parameter. We repeat the same exercise for firms sorted in ascending order of their γ-parameter. We average the ranks of each firm across the two sorts. We then take a long position in the firms with the highest 20 % average score and a short position in the firms with the lowest 20 % average score. Panel B of Table5 reveals that this strategy yields significant risk-adjusted returns for horizons starting from 3 months. Generally, these risk-adjusted returns are larger than those obtained when sorting by exposure to the index VRP (see

Panel A of Table5). For instance, at the 3-month horizon, a strategy based on the average score improves the average risk-adjusted return by around 4.39 % per year.

Which of the individual VRP and the CRP factors matters for the cross-sectional return predictability? To answer this question, we perform a dependent double-sort analysis. By double-sorting, we aim to study the relationship between one factor and future excess stock returns while controlling for the other factor. The results of Panel B of Table6 ensue when we first sort stocks into quintile portfolios in ascending order of their sensitivity with respect to the CRP factor (γi,t). For each of these portfolios, we sort the stocks into quintile portfolios in ascending order of their sensitivity with respect to the Individual VRP factor (φi,t). We then compute the P5−P1 spread of each CRP

27 quintile portfolio and report the average of this spread across all CRP quintiles. After controlling for the CRP factor, stocks with a high exposure to the individual VRP factor deliver, on average, higher risk-adjusted returns than stocks with a low exposure to the individual VRP. The results are highly significant for holding horizons of 6 to 12 months.

Panel C of Table6 presents the results when we sort stocks first by their exposure to the individual VRP factor and then by their exposure to the CRP factor. Controlling for the exposure to the Individual VRP factor, the risk-adjusted returns of stocks with high exposure to the CRP factor are generally higher than those of stocks with low exposure to the CRP factor. This result is strongest, in terms of statistical significance, for horizons starting from 12 months onwards. In light of this, we conclude that the Individual VRP is a short-term predictor of the cross-section of excess stock returns whereas the CRP factor is a long-term predictor.

5 Is the View Worth It...

5.1 For Practitioners?

Our empirical analyses carry implications for the trading of variance and correlation risk. The term sheets of variance swap contracts and exchange-traded variance assets specify the formula used to compute the realized variance. Typically, this computation is based on daily data. Our results show that this framework could introduce large biases in the estimates of the swap payoffs. For instance, a hedge fund that sells 1-month index variance swaps each month realizes an average payoff of 0.85 % when realized variance is computed using daily data. However, if high-frequency data are used, the average payoff to such strategy rises by 80 %, from 0.85 % to 1.53 %. Similarly, the payoff of a

28 short position in the 1-month correlation swap increases by 28 %, from 10.74 % to 13.75 %, when realized variance is accurately computed. These differences could have an important economic impact on the profit (and loss) of market participants. As trading exchanges introduce more variance futures, it would be desirable to improve on the computation of the payoffs of these assets.24

5.2 For Academics?

Our work also carries implications for financial modeling. For instance, Bollerslev et al.

(2009) propose a model featuring time-varying macroeconomic volatility-of-volatility. In their model, the index variance risk premium reflects only this source of risk. The model provides a framework to rationalize the predictive power of the index VRP for index excess returns. Taken to the extreme, the 1-factor structure of that model implies that both the

Individual VRP and the CRP factors are positively and perfectly correlated (since they depend on the same source of risk). Clearly, this is at odds with our empirical results of a weak and negative correlation between these factors. This result is not surprising since the model is not designed to explain the joint dynamics of the index VRP and the CRP.

We thus focus specifically on models designed to jointly explain the index variance risk premium and the correlation risk premium. Driessen et al.(2013) propose a reduced- form model that features only a correlation risk premium. In their model, there is no risk premium for bearing the variance risk of individual stocks. Thus, the index variance risk premium arises solely via the CRP channel. Their model yields two empirically testable predictions. First, the CRP factor captures the level of the index VRP. This is consistent with our empirical evidence (see Table3). Second, the variations in the index VRP stem

24For instance, the Chicago Board of Options Exchange introduced S&P 500 Variance Futures on December 10, 2012. For further details on the computation of the payoff of this contract at expiration, we refer the interest reader to the following address: http : //cfe.cboe.com/products/spec_va.aspx.

29 solely from the CRP factor. This prediction is not borne out by the data. Indeed, Table

3 shows the CRP factor accounts for merely 34 % of the variations in the index VRP. Our results thus indicate that a successful model needs the 2-factor structure we uncover in this study.

Buraschi et al.(2014b) propose a two-tree Lucas economy where disagreement about beliefs is priced in equilibrium. More specifically, their model features unknown firm growth rates and heterogeneous beliefs about the state of the economy. The authors show that disagreement about stock fundamentals is important for the volatility risk premia of stocks, whereas disagreement about the economy-wide signal is the main driving force behind the correlation risk premium. Thus, this model potentially provides a structural explanation for the two-factor structure we document in this study. It could be that the Individual VRP factor is driven by the cross-sectional average of disagreement about each stock, whereas the CRP factor is driven by the disagreement about the state of the economy.

The authors propose a bottom-up measure of the economy-wide disagreement measure by aggregating the disagreement proxies of all firms. This is interesting because as a result, the variable that drives the Individual VRP factor should be the same that drives the CRP factor. However, we find a weak and negative correlation between these two factors. This is difficult to reconcile with the framework of Buraschi et al.(2014b). Thus, we conclude that while the authors’ model could potentially rationalize the findings that presented in this study, more work needs to be done to obtain valid empirical proxies for disagreement of beliefs that differentiate between the firm and the state of the aggregate economy.

30 6 Additional Analyses

In this section, we conduct several additional checks. First, we consider different sam- pling frequencies. Second, we assess the sensitivity of our results to alternative variance forecasting models. Third, we extend our decomposition to variance swaps of horizons longer than 1 month.

6.1 Alternative Sampling Frequencies

The computation of realized variance and its physical expectation exploits returns sampled at a frequency of 75 minutes. Naturally, one may wonder whether the results hold for higher sampling frequencies.

We repeat the main analyses using a sampling frequency of 30 minutes. Tables A.12 to A.15 of the online appendix point to conclusions similar to our baseline estimates.

In particular, the CRP factor accounts for most of the average index VRP, while the individual VRP factor captures the time-variations of the index VRP. The predictability analysis also confirms that both factors are useful and manifest themselves at different horizons. The individual VRP factor is a short-term predictor of the market excess return whereas the predictive power of the CRP factor is also discernible at longer horizons.

Turning to the cross-sectional results, we find similar results to our baseline findings.

These conclusions also extend to sampling frequencies of 5 and 15 minutes. These tables are not presented for brevity.

6.2 The Physical Expectation of Variance

As previously discussed, any study on the variance risk premium involves taking a stance on a model to generate the physical expectation of variance. While our baseline

31 analysis uses the simple random walk model suggested by Bollerslev et al.(2009), one

might wonder whether our conclusions hold if the variance risk premium is computed

using alternative approaches.

To shed light on this, we make the following assumptions. First, we assume that the

P physical expectation of the future variance of the index (Et (RV I,t+τ )) is generated by an

AR(1) process as in Della Corte et al.(2016). This enables us to easily compute the vari-

ance risk premium of the index (see Equation (16)). Second, we assume that the physical

P expectation of the future correlation (Et (ρt+τ )) is also generated by an AR(1) process.

Third, we model the sum of the cross-product of the weights and the square roots of

P p P p P the physical expectation of future variances ( i,j6=i ωi,tωj,t Et (RV i,t+τ ) Et (RV j,t+τ ))

as an AR(1) process. The last two assumptions enable us to directly compute the CRP

factor. Since we are able to compute the VRP of the index and the CRP factor, we

re-arrange Equation (19) to extract the Individual VRP factor.

This approach has several advantages. First, it bypasses the need to estimate a vari-

ance forecasting model for each stock in the index, facilitating our analysis. Furthermore,

it enables us to sidestep the question of what is the best variance forecasting model for

individual equities and whether specific models work better for stocks with particular

characteristics.25

Table7 presents some results on the alternative estimates of the index VRP. 26 It

shows that the index VRP is negative (−1.51 %) and highly significant (t-stat = −6.24).

The decomposition results confirm that the CRP factor captures well the level of the

25While there is a large body of works on how to forecast the variance of the equity index, relatively little attention has been dedicated to the task of forecasting the variance of individual equities. In future work, it would be very interesting to conduct a comprehensive analysis of variance forecasting models for all individual equities that make up the index. 26To implement the model, we use an expanding window. We require at least 300 available daily observations to estimate the model parameters. We use daily observations in order to obtain more precise parameter estimates.

32 index VRP. The individual VRP factor is the main force behind the variations in the index VRP. These findings are very similar to those in Table3. Tables8–10 present predictability results that are generally consistent with our benchmark findings.

It is worth mentioning that this alternative approach relies itself on forecasting mod- els to generate the explanatory variable(s) used to explore the predictability of aggregate excess stock returns. This might introduce sampling error in the analysis. In order to investigate the impact of the generated regressor problem, we follow the steps of Bekaert

& Hoerova(2014). We estimate each AR(1) model as before. We save the parameter estimates as well as their asymptotic covariance matrix. We draw 500 alternative coeffi- cients from the distribution of these parameters, which we use to generate the relevant return forecasting variables. We do this at the end of each calendar month and thus obtain alternative time-series of the return forecasting variable(s). Finally, we estimate the excess return forecasting model using each of the alternative time-series of predictive variables. Our untabulated results are consistent with the benchmark findings, indicating that sampling error does not materially affect our predictability results. This is consistent with the conclusion of Bekaert & Hoerova(2014).

As an additional check, we consider the HAR model of Corsi(2009), rather than an AR(1) process. Tables A.16–A.19 of the online appendix report results that are qualitatively similar to those based on the simpler AR(1) model.

6.3 Long-Term Variance Risk Premium Decomposition

Up to this point, we have analyzed the 1-month variance swaps. However, one may wonder if the results of our decomposition extend to variance swaps expiring in 3, 6 and

12 months.

33 Table 11 shows the decomposition of the index VRP for these maturities.27 Several observations are in order. First, the term-structure of the average of the index VRP is downward-sloping. This is evidenced by significantly negative average variance swap payoffs of −1.67 %, −1.84 % and −2.06 % for horizons of 3, 6 and 12 months, respectively.

Second, variance risk premia of longer maturities are less skewed than those of shorter maturities. Third, consistent with our baseline results, the CRP factor accounts for a large share of the average index VRP. Fourth, the contribution of the CRP factor to the overall variability of the index VRP rises from 27.9 % at the 1-month horizon to 48.3 % at the 12-month horizon.

7 Conclusion

We analyze the pricing of variance risk related to the index and its constituents. We show that the conclusion of insignificant VSPs for stocks reported in earlier works is affected by measurement errors in the realized variance estimates. Using high-frequency data to accurately compute the realized variance of stocks, we find significantly negative

VSPs for a substantial share of individual equities.

We decompose the index variance risk premium into two factors and assess their im- portance. The Individual VRP factor drives the variations in the 1-month index VRP, while the CRP factor captures the level of the index VRP. Both components are infor- mative about (i) future aggregate excess stock returns and (ii) the cross-section of future excess stock returns. The Individual VRP factor is important at short horizons while the

CRP factor is more prominent at longer horizons.

27In the interest of brevity, we only present the results for the VRP. The (untabulated) results for the VSP are qualitatively similar. These results are available upon request.

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37 Figure 1: Dissecting the Index VRP

This figure plots the time series of the 1-month index variance risk premium (dashed, blue line), the individual VRP factor (dash-dotted, orange line) and the

CRP factor (solid, green line). The shaded areas indicate business cycle contractions as identified by the NBER.

0.4

0.2

0 Index VRP

-0.2

0.4

38 0.2

0 Individual VRP Factor

-0.2

0.4

0.2

0 CRP Factor

-0.2

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Date Figure 2: Term-Structure of t-ratios

This figure displays the Newey & West(1987) corrected t-ratios (using h lags) of the slope parameters associated with the regression of S&P 500 h-month excess returns on a constant and the lagged Individual

VRP (dash-dotted, orange line) and CRP (solid, green line) factors. The dashed, blue line shows the t-statistic of the regression of h-month S&P 500 excess returns on a constant and the lagged index VRP.

The shaded area indicates the non-rejection region for the 5% significance level.

1 Index VRP Individual VRP Factor CRP Factor 0

-1

-2

-3 T-Statistics -4

-5

-6

-7

2 4 6 8 10 12 14 16 18 20 22 24 Horizon in months

39 Figure 3: Regression R2s: Constrained vs. Unconstrained Regressions

This figure plots the adjusted R2s of the constrained excess return predictability regression (solid line) and unconstrained excess return predictability regression (dashed line) for various horizons, ranging from

1 to 24 months. The constrained model involves regressing the h-month excess return on the S&P 500 index on a constant and the lagged index variance risk premium. The unconstrained model entails regressing the h-month excess return on the index on a constant and the (lagged) two factors that make up the index variance risk premium, namely the CRP and Individual VRP factors.

Constrained Unconstrained

0.1

0.08 2 0.06 Adjusted R

0.04

0.02

0 2 4 6 8 10 12 14 16 18 20 22 24 Horizon in months

40 Table 1: Variance Swap Payoffs: Daily Return Data

This table reports summary statistics on 1-month variance swap payoffs. Panel A presents the results linked to the S&P 500 index, as well as the equal-weighted average of the constituent stocks. RV and IV are the average (annualized) realized and Bakshi et al.(2003) option-implied variance, respectively. We use daily return data to compute the realized variance. VSP denotes the average variance swap payoff, defined as the spread between RV and IV . t-stat is the Newey & West(1987) corrected t-statistic (with 4 lags). Std Dev, Skew, Kurt and AR(1) denote the standard deviation, skewness, kurtosis and first-order autocorrelation of the variance swap payoff, respectively. Additionally, Median, q0.05 and q0.95 are the median, 5 % and 95 % quantiles of the variance swap payoff, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. The rows in Panel B relate to stocks with insignificant, significantly negative and significantly positive variance swap payoffs (at the 5% significance level), respectively. Share indicates the fraction of firms for which the variance swap payoff satisfies the condition [name in row].

Panel A: Market Variance Swap Payoff

RV IV V SP t-stat Std Dev Skew Kurt AR(1) Median q0.05 q0.95 Index VSP 0.0375 0.0461 -0.0085** -2.14 0.048 7.39 81.35 0.38 -0.0119 -0.0572 0.0308 Avg. Stocks 0.1638 0.1549 0.0089 0.82 0.113 5.87 45.35 0.48 -0.0138 -0.0701 0.1349

Panel B. Stock Variance Swap Payoff

Share RV IV V SP Std Dev Skew Kurt AR(1) Median q0.05 q0.95 = 0 not rejected 0.781 0.1930 0.1711 0.0219 0.295 4.25 36.01 0.13 -0.0232 -0.1509 0.3029 > 0 rejected 0.197 0.1005 0.1291 -0.0286 0.093 1.43 15.01 0.08 -0.0325 -0.1447 0.1070 < 0 rejected 0.021 0.3105 0.2421 0.0684 0.293 2.95 16.28 0.08 -0.0020 -0.1809 0.5535

41 Table 2: Variance Swap Payoffs: 75-Minute Data

This table reports summary statistics on 1-month variance swap payoffs. Panel A presents the results linked to the S&P 500 index, as well as the equal-weighted average of the constituent stocks. RV and IV report the average (annualized) realized and Bakshi et al.(2003) option-implied variance, respectively. We use high-frequency return data sampled at the 75-minute frequency to compute the realized variance. VSP shows the average variance swap payoff, defined as the spread between RV and IV . t-stat is the Newey & West(1987) corrected t-statistic (with 4 lags). Std Dev, Skew, Kurt and AR(1) denote the standard deviation, skewness, kurtosis and first-order autocorrelation of the variance swap payoff, respectively. Additionally, Median, q0.05 and q0.95 are the median, 5 % and 95 % quantiles of the variance swap payoff, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. The rows in Panel B relate to stocks with insignificant, significantly negative and significantly positive variance swap payoffs (at the 5% significance level), respectively. Share indicates the fraction of firms for which the variance swap payoff satisfies the condition [name in row].

Panel A: Market Variance Swap Payoff

RV IV V SP t-stat Std Dev Skew Kurt AR(1) Median q0.05 q0.95 Index VSP 0.0308 0.0461 -0.0153*** -5.23 0.041 7.27 90.38 0.21 -0.0148 -0.0585 0.0148 Avg. Stocks 0.1425 0.1549 -0.0123** -1.99 0.083 6.98 78.55 0.25 -0.0195 -0.0871 0.0829

Panel B: Stock Variance Swap Payoff

Share RV IV V SP Std Dev Skew Kurt AR(1) Median q0.05 q0.95 = 0 not rejected 0.587 0.1702 0.1709 -0.0007 0.186 4.07 37.27 0.12 -0.0226 -0.1471 0.2116 > 0 rejected 0.406 0.1214 0.1541 -0.0327 0.101 1.71 19.43 0.11 -0.0339 -0.1647 0.1030 < 0 rejected 0.008 0.2790 0.2112 0.0679 0.320 2.50 13.90 0.04 -0.0034 -0.2524 0.6843

42 Table 3: Decomposition Results

This table presents the results of the decomposition of the 1-month index variance swap payoff (Panel A) and variance risk premium (Panel B) into two factors. The variance swap payoff is the difference between the (annualized) realized and Bakshi et al.(2003) option-implied variance. We use high-frequency return data sampled at the 75-minute frequency to compute the realized variance. The variance risk premium is the difference between the physical expectation of the future variance, assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Mean is the average value. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. t-stat reports the

Newey–West corrected t-ratio (with 4 lags) associated with the mean. Shrmean reports the fraction of the mean of the (i) variance swap payoff (Panel A) or (ii) variance risk premium (Panel B) of the S&P

500 index associated with the variable [name in row]. Std Dev is the standard deviation. Shrvar reports the share of the variance of the (i) variance swap payoff (Panel A) or (ii) variance risk premium (Panel B) of the S&P 500 index captured by the variable [name in row]. Skew, Kurt and AR(1) denote the skewness, kurtosis and first-order autocorrelation, respectively. Median, q0.05 and q0.95 relate to the median, 5 % and 95 % quantiles of the distribution of the variable [name in row].

Panel A: Variance Swap Payoff

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VSP -0.0153*** -5.23 1.000 0.041 1.000 7.27 90.38 0.21 -0.0148 -0.0585 0.0148 Individual VSP -0.0045** -2.02 0.296 0.031 0.640 6.85 82.33 0.22 -0.0050 -0.0334 0.0274 CSP -0.0107*** -6.51 0.704 0.018 0.360 -1.76 36.12 0.41 -0.0084 -0.0315 0.0013

Panel B: Variance Risk Premium

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0134*** -6.57 1.000 0.027 1.000 5.88 70.23 0.12 -0.0116 -0.0443 0.0077 Individual VRP -0.0012 -0.59 0.089 0.027 0.721 8.61 106.8 0.15 -0.0034 -0.0212 0.0231 CRP -0.0122*** -8.34 0.911 0.014 0.279 -2.83 16.86 0.54 -0.0095 -0.0328 0.0002

43 Table 4: Predictability of S&P 500 Excess Returns

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of h-month on a constant and the lagged forecasting variable(s). Panel A considers the forecasting power of the market index variance risk premium. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Panel B considers the two factors of the index variance risk premium, namely the CRP and Individual VRP factors. We consider forecasting horizons of 1, 3, 6, 9, 12, 18 and 24 months. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with h lags) and Hodrick(1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2. Wald presents the results of a Wald test of the null hypothesis that the two slope parameters are equal. p − value reports the corresponding Newey & West(1987) corrected p-value.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0090 0.0070 0.0265 0.0407 0.0440 0.0451 0.0427 (s.e. NW) (0.035) (0.029) (0.031) (0.034) (0.036) (0.039) (0.039) [s.e. Hod] [0.036] [0.035] [0.037] [0.036] [0.036] [0.036] [0.036] Index VRP -4.6318 -3.6217 -2.0895 -1.1207 -0.8444 -0.6107 -0.5519 (s.e. NW) (1.141)*** (0.756)*** (0.684)*** (0.589)* (0.504)* (0.473) (0.362) [s.e. Hod] [1.279]*** [0.890]*** [0.763]*** [0.623]* [0.541] [0.443] [0.363] Adj. R2 0.052 0.090 0.051 0.018 0.012 0.007 0.008

Panel B: CRP and Individual VRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0269 -0.0275 -0.0070 0.0144 0.0241 0.0269 0.0260 (s.e. NW) (0.040) (0.034) (0.034) (0.036) (0.038) (0.041) (0.043) [s.e. Hod] [0.041] [0.034] [0.035] [0.035] [0.035] [0.034] [0.035] Individual VRP -4.4152 -3.2044 -1.6849 -0.8061 -0.6099 -0.4036 -0.3685 (s.e. NW) (1.050)*** (0.475)*** (0.370)*** (0.363)** (0.348)* (0.323) (0.237) [s.e. Hod] [1.122]*** [0.872]*** [0.859]* [0.683] [0.560] [0.430] [0.343] CRP -6.1231 -6.4953 -4.8750 -3.2931 -2.4716 -2.0752 -1.8724 (s.e. NW) (2.670)** (1.581)*** (0.963)*** (0.845)*** (0.788)*** (0.732)*** (0.862)** [s.e. Hod] [2.779]** [1.596]*** [1.635]*** [1.522]** [1.551] [1.469] [1.347] Adj. R2 0.049 0.104 0.077 0.040 0.026 0.024 0.024 Wald 0.320 4.038** 8.039*** 7.025*** 4.904** 4.120** 3.009* p-value (0.572) (0.044) (0.005) (0.008) (0.027) (0.042) (0.083)

44 Table 5: Cross-Sectional Predictability (I)

This table analyzes the predictability of the cross-section of excess stock returns. We use the past 5 years of data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. In Panel A, we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. For Panel B, we proceed as follows. We regress stock returns jointly on the lagged CRP and Individual VRP factors. We rank stocks in ascending order of their loadings on the CRP factor. We also rank stocks in ascending order of their loadings on the Individual VRP factor. We average the ranks across the two sorts. We then take long positions in the firms with the highest 20 % average score and short positions in stocks with the lowest 20 % average score. We report the average (annualized) value-weighted returns and Carhart (1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0377 0.0002 0.0728** 0.0752** 0.0934*** 0.0554** 0.0231 (s.e.) (0.048) (0.039) (0.037) (0.037) (0.030) (0.027) (0.015) 4-factor alpha 0.0556 0.0249 0.1043*** 0.0878*** 0.0998*** 0.0461** 0.0236 (s.e.) (0.041) (0.027) (0.033) (0.033) (0.023) (0.022) (0.015)

Panel B: Individual VRP & CRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0420 0.0359 0.0719* 0.0831** 0.0852*** 0.0687*** 0.0350** (s.e.) (0.036) (0.026) (0.039) (0.035) (0.029) (0.021) (0.017) 4-factor alpha 0.0582 0.0688** 0.1168*** 0.1001*** 0.0932*** 0.0596*** 0.0420*** (s.e.) (0.036) (0.029) (0.037) (0.031) (0.020) (0.017) (0.011)

45 Table 6: Cross-Sectional Predictability (II)

This table analyzes the predictability of the cross-section of excess stock returns. We use the past 5 years of data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. In Panel A, we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. Panels B and C are based on double- sorts. We regress stock returns jointly on the lagged CRP and Individual VRP factors. Panel B first sorts stocks into 5 portfolios based on the loadings on the CRP factor. It then sorts each of the portfolios into 5 portfolios based on the stock loadings on the individual VRP. Panel C sorts stocks first on the loading on the individual VRP factor and then on the CRP factor. We report the average (annualized) value-weighted returns and Carhart(1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0377 0.0002 0.0728** 0.0752** 0.0934*** 0.0554** 0.0231 (s.e.) (0.048) (0.039) (0.037) (0.037) (0.030) (0.027) (0.015) 4-factor alpha 0.0556 0.0249 0.1043*** 0.0878*** 0.0998*** 0.0461** 0.0236 (s.e.) (0.041) (0.027) (0.033) (0.033) (0.023) (0.022) (0.015)

Panel B: Individual VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0220 0.0116 0.0303 0.0529** 0.0474*** 0.0340** 0.0197 (s.e.) (0.046) (0.033) (0.021) (0.021) (0.016) (0.013) (0.013) 4-factor alpha 0.0347 0.0234 0.0571*** 0.0553*** 0.0398*** 0.0165 0.0259** (s.e.) (0.035) (0.024) (0.015) (0.017) (0.010) (0.012) (0.013)

Panel C: CRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0109 0.0309 0.0257 0.0324 0.0534** 0.0368* 0.0464*** (s.e.) (0.028) (0.038) (0.033) (0.024) (0.024) (0.021) (0.011) 4-factor alpha 0.0088 0.0493 0.0557 0.0507** 0.0695*** 0.0542*** 0.0385*** (s.e.) (0.036) (0.043) (0.034) (0.025) (0.020) (0.014) (0.009)

46 Table 7: Decomposition: Alternative VRP

This table presents the results of the decomposition of the 1-month index VRP into the Individual VRP factor and the CRP factor. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming an AR(1) process (Della Corte et al., 2016), and the Bakshi et al.(2003) option-implied variance. Mean is the average of the variable [name in row]. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. t-stat reports the Newey–West corrected t-ratio (with 4 lags) associated with the mean. Shrmean reports the fraction of the level of the S&P 500 index VRP captured by the variable [name in row]. Std Dev is the standard deviation of the variable [name in row]. Shrvar reports the share of the variance of the S&P 500 index VRP captured by the variable [name in row]. Skew, Kurt and AR(1) denote the skewness, kurtosis and first-order autocorrelation of the variable [name in row], respectively. Median, q0.05 and q0.95 relate to the median, 5 % and 95 % quantiles of the distribution of the variable [name in row].

0.05 0.95 Instrument Mean T -stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0151*** -6.24 1.000 0.026 1.000 0.28 22.59 0.28 -0.0100 -0.0536 0.0039 Individual VRP -0.0004 -0.19 0.025 0.027 0.641 6.97 89.23 0.11 0.0023 -0.0234 0.0114 CRP -0.0147*** -8.41 0.975 0.016 0.359 -3.71 23.97 0.59 -0.0113 -0.0378 -0.0008

47 Table 8: Predictability of S&P 500 Excess Returns: Alternative VRP

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of h-month on a constant and the lagged forecasting variable(s) on the S&P 500. Panel A considers the forecasting power of the market index variance risk premium. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming an AR(1) process (Della Corte et al., 2016), and the Bakshi et al. (2003) option-implied variance. Panel B considers the two factors of the index variance risk premium, namely the CRP and Individual VRP factors. We consider forecasting horizons of 1, 3, 6, 9, 12, 18 and 24 months. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with h lags) and Hodrick(1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2. Wald presents the results of a Wald test of the null hypothesis that the two slope parameters are equal. p − value reports the corresponding Newey & West(1987) corrected p-value.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0207 -0.0122 0.0047 0.0196 0.0245 0.0262 0.0219 (s.e. NW) (0.034) (0.032) (0.033) (0.035) (0.036) (0.039) (0.040) [s.e. Hod] [0.032] [0.034] [0.035] [0.034] [0.034] [0.034] [0.036] Index VRP -4.4221 -3.8736 -2.5714 -1.5802 -1.1834 -0.9583 -1.0226 (s.e. NW) (1.371)*** (0.745)*** (0.735)*** (0.621)** (0.527)** (0.458)** (0.395)** [s.e. Hod] [1.494]*** [0.898]*** [0.666]*** [0.584]*** [0.585]** [0.526]* [0.442]** Adj. R2 0.040 0.090 0.069 0.035 0.024 0.022 0.034

Panel B: CRP and Individual VRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0195 -0.0269 -0.0252 -0.0139 -0.0052 -0.0043 -0.0105 (s.e. NW) (0.043) (0.041) (0.039) (0.041) (0.043) (0.046) (0.048) [s.e. Hod] [0.043] [0.040] [0.040] [0.037] [0.036] [0.035] [0.037] Individual VRP -4.4367 -3.6974 -2.2143 -1.1885 -0.8426 -0.6206 -0.6845 (s.e. NW) (1.248)*** (0.539)*** (0.547)*** (0.512)** (0.479)* (0.430) (0.493) [s.e. Hod] [1.351]*** [0.858]*** [0.690]*** [0.577]** [0.546] [0.474] [0.403]* CRP -4.3389 -4.8821 -4.6154 -3.8520 -3.1853 -2.9998 -3.1522 (s.e. NW) (2.755) (2.167)** (1.173)*** (1.068)*** (1.119)*** (0.926)*** (0.948)*** [s.e. Hod] [2.796] [2.093]** [1.979]** [1.647]** [1.523]** [1.350]** [1.181]*** Adj. R2 0.035 0.088 0.087 0.070 0.059 0.075 0.109 Wald 0.001 0.253 2.292 3.327* 2.538 4.190** 3.912** p-value (0.979) (0.615) (0.130) (0.068) (0.111) (0.041) (0.048)

48 Table 9: Cross-Sectional Predictability: Alternative VRP (I)

This table analyzes the predictability of the cross-section of excess stock returns. We use the past 5 years of data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. In Panel A, we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming an AR(1) process (Della Corte et al., 2016), and the Bakshi et al.(2003) option-implied variance. We then sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. For Panel B, we proceed as follows. We regress stock returns jointly on the lagged CRP and Individual VRP factors. We rank stocks in ascending order of their loadings on the CRP factor. We also rank stocks in ascending order of their loadings on the Individual VRP factor. We average the ranks across the two sorts. We then take long positions in the firms with the highest 20 % average score and short positions in stocks with the lowest 20 % average score. We report the average (annualized) value-weighted returns and Carhart (1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0285 -0.0022 0.0605 0.0644* 0.0779** 0.0689** 0.0282* (s.e.) (0.047) (0.043) (0.042) (0.038) (0.035) (0.028) (0.017) 4-factor alpha 0.0382 0.0245 0.0951** 0.0863** 0.0991*** 0.0843*** 0.0342** (s.e.) (0.047) (0.037) (0.044) (0.035) (0.022) (0.021) (0.015)

Panel B: Individual VRP & CRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0423 0.0258 0.0631 0.0537* 0.0575** 0.0545* 0.0331* (s.e.) (0.033) (0.033) (0.040) (0.031) (0.029) (0.029) (0.017) 4-factor alpha 0.0449 0.0649 0.1024** 0.0729** 0.0773*** 0.0722*** 0.0363** (s.e.) (0.043) (0.042) (0.045) (0.031) (0.020) (0.019) (0.016)

49 Table 10: Cross-Sectional Predictability: Alternative VRP (II)

This table analyzes the predictability of the cross-section of excess stock returns. We use the past 5 years of data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. In Panel A, we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming an AR(1) process (Della Corte et al., 2016), and the Bakshi et al.(2003) option-implied variance. We then sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. Panels B and C are based on double-sorts. We regress stock returns jointly on the lagged CRP and Individual VRP factors. Panel B first sorts stocks into 5 portfolios based on the loadings on the CRP factor. It then sorts each of the portfolios into 5 portfolios based on the stock loadings on the individual VRP. Panel C sorts stocks first on the loading on the individual VRP factor and then on the CRP factor. We report the average (annualized) value-weighted returns and Carhart(1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0285 -0.0022 0.0605 0.0644* 0.0779** 0.0689** 0.0282* (s.e.) (0.047) (0.043) (0.042) (0.038) (0.035) (0.028) (0.017) 4-factor alpha 0.0382 0.0245 0.0951** 0.0863** 0.0991*** 0.0843*** 0.0342** (s.e.) (0.047) (0.037) (0.044) (0.035) (0.022) (0.021) (0.015)

Panel B: Individual VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0256 0.0072 0.0446** 0.0482*** 0.0422*** 0.0257* 0.0265*** (s.e.) (0.039) (0.026) (0.019) (0.015) (0.015) (0.013) (0.009) 4-factor alpha 0.0318 0.0319* 0.0691*** 0.0475*** 0.0357*** 0.0358*** 0.0191* (s.e.) (0.040) (0.019) (0.016) (0.014) (0.011) (0.012) (0.011)

Panel C: CRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0450 0.0371 0.0211 0.0308 0.0426* 0.0570*** 0.0429*** (s.e.) (0.038) (0.036) (0.028) (0.024) (0.023) (0.018) (0.009) 4-factor alpha 0.0370 0.0510 0.0418* 0.0529*** 0.0741*** 0.0704*** 0.0250*** (s.e.) (0.042) (0.037) (0.022) (0.016) (0.013) (0.012) (0.007)

50 Table 11: Decomposition: Longer Maturities

This table presents the results of the decomposition of the 3-month (Panel A), 6-month (Panel B) and 12-month (Panel C) index VRP into the Individual VRP factor and the CRP factor. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al. (2003) option-implied variance. Mean is the average of the variable [name in row]. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. t-stat reports the Newey–West corrected t-ratio

(with max(4, h) lags) associated with the mean. Shrmean reports the fraction of the level of the S&P 500 index VRP captured by the variable [name in row]. Std Dev is the standard deviation of the

variable [name in row]. Shrvar reports the share of the variance of the S&P 500 index VRP captured by the variable [name in row]. Skew, Kurt and AR(1) denote the skewness, kurtosis and first-order autocorrelation of the variable [name in row], respectively. Median, q0.05 and q0.95 relate to the median, 5 % and 95 % quantiles of the distribution of the variable [name in row].

Panel A: 3-Month Horizon

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0167*** -8.55 1.000 0.022 1.000 3.11 37.29 0.47 -0.0155 -0.0476 0.0047 Individual VRP -0.0027* -1.70 0.159 0.018 0.464 4.55 43.36 0.50 -0.0041 -0.0217 0.0198 CRP -0.0140*** -8.72 0.841 0.013 0.536 -2.47 13.86 0.74 -0.0100 -0.0405 -0.0032

Panel B: 6-Month Horizon

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0184*** -7.67 1.000 0.020 1.000 -0.01 9.22 0.72 -0.0170 -0.0475 0.0081 Individual VRP -0.0021 -0.95 0.116 0.018 0.432 1.25 9.84 0.77 -0.0042 -0.0208 0.0271 CRP -0.0163*** -8.33 0.884 0.013 0.568 -2.07 8.15 0.84 -0.0123 -0.0456 -0.0052

Panel C: 12-Month Horizon

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0206*** -6.20 1.000 0.022 1.000 0.26 8.43 0.83 -0.0183 -0.0535 0.0023 Individual VRP -0.0022 -0.63 0.109 0.023 0.517 1.76 12.85 0.89 -0.0043 -0.0290 0.0300 CRP -0.0183*** -6.58 0.891 0.014 0.483 -1.83 6.52 0.89 -0.0131 -0.0460 -0.0059

51 Appendix to

“Variance Risk: A Bird’s Eye View”

Not Intended for Publication!

Will be Provided as Online Appendix

52 Table A.1: Measurement Error in Realized Variance

This table presents the results of cross-sectional Fama & MacBeth(1973) regressions. The dependent variable in the first five columns is the absolute value of the difference between the realized variance based on daily data and the realized variance based on 75-minute data estimates. For the remaining columns, the dependent variable is simply the spread between the two realized variance estimates. We consider several characteristics as potential explanatory variables. “Market Cap” is the natural logarithm of the market value of equity. “Book-to-Market” denotes the Book-to-Market ratio of the corresponding firm. “Leverage” is the leverage ratio, defined as the ratio of the book value of assets over the market value of equity. Finally, “Bid–Ask” is the monthly average of the daily proportional Bid–Ask spreads of the stock of the corresponding firm. We present the Newey & West(1987) corrected standard errors ( s.e.) using 4 lags in parentheses. Adj. R2 denotes the adjusted R2 of the regression. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. 53 |RV daily − RVhigh−frequency| RV daily − RVhigh−frequency (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) Intercept 0.2251*** 0.0244*** 0.0360*** 0.0163*** 0.0950*** 0.0843** 0.0015 -0.0059 -0.0010 0.0246 (s.e.) (0.036) (0.004) (0.009) (0.005) (0.028) (0.033) (0.004) (0.006) (0.004) (0.024) Market Cap -0.0113*** -0.0047*** -0.0044** -0.0021* (s.e.) (0.002) (0.001) (0.002) (0.001) Book-to-Market 0.0341*** 0.0177* 0.0162 0.0073 (s.e.) (0.013) (0.009) (0.012) (0.008) Leverage 0.0153 0.0062 0.0327* 0.0246* (s.e.) (0.022) (0.017) (0.018) (0.014) Bid–Ask 25.893*** 7.8984*** 13.986** 0.5319 (s.e.) (6.989) (2.506) (6.881) (2.536) Adj. R2 0.02 0.02 0.01 0.03 0.05 0.00 0.01 0.01 0.01 0.03 Table A.2: Correlation Swap Payoffs

This tables presents the summary statistics of the 1-month correlation swap payoff (CSP) and correlation risk premium (CRP). CSP is computed by taking the difference between the realized equicorrelation, denoted by RC (see Equation (9)), and the implied equicorrelation (IC), computed as in Equation (10). CRP is the difference between the physical expectation of the future equicorrelation (EP(RC)) and IC. Panels A and B use daily return data to compute the realized variance estimates (see Equation (3)), whereas Panels C and D exploit high-frequency data sampled at the 75-minute frequency (see Equation (7)). Panels A and C focus on the daily, and thus overlapping, correlation swaps. Panels B and D relate to the monthly (non-overlapping) correlation swaps, with observations sampled at the end of each month. RC and IC denote the realized and implied equicorrelation estimates, respectively. Mean denotes the sample average. t-stat is the Newey & West(1987) corrected t-statistic (with 4 lags when using monthly observations and 21 lags when analyzing daily observations). Std Dev, Skew, Kurt and AR(1) denote the standard deviation, skewness, kurtosis and first-order autocorrelation, respectively. Additionally, Median, q0.05 and q0.95 are the median, 5 % and 95 % quantiles of the variance swap payoff, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Daily Return Data (Daily Observations)

Mean t-stat Std Dev Skew Kurt AR(1) Median q0.05 q0.95 RC 0.3211*** 36.64 0.142 0.88 3.59 0.98 0.2897 0.1375 0.6113 IC 0.3983*** 48.01 0.135 0.58 3.34 0.96 0.3866 0.1955 0.6423 CSP -0.0772*** -11.77 0.123 0.20 4.97 0.92 -0.0739 -0.2733 0.1211 EP(RC) 0.3230*** 36.72 0.142 0.84 3.41 0.98 0.2921 0.1408 0.6061 CRP -0.0754*** -14.07 0.102 -0.15 3.26 0.91 -0.0720 -0.2518 0.0889

Panel B: Daily Return Data (Monthly Observations)

Mean t-stat Std Dev Skew Kurt AR(1) Median q0.05 q0.95 RC 0.3129*** 20.52 0.139 0.82 3.37 0.55 0.2810 0.1436 0.5912 IC 0.4203*** 24.35 0.140 0.50 3.17 0.74 0.4075 0.2136 0.6556 CSP -0.1074*** -10.04 0.127 -0.06 4.01 0.18 -0.1022 -0.3302 0.1030 EP(RC) 0.3234*** 20.89 0.140 0.85 3.47 0.57 0.2850 0.1423 0.6051 CRP -0.0970*** -10.09 0.102 -0.11 3.22 0.25 -0.0927 -0.2656 0.0642

Panel C: 75-Minute Return Data (Daily Observations)

Mean t-stat Std Dev Skew Kurt AR(1) Median q0.05 q0.95 RC 0.2824*** 38.46 0.115 0.90 3.47 0.99 0.2576 0.1389 0.5099 IC 0.3983*** 48.02 0.135 0.58 3.34 0.96 0.3866 0.1955 0.6423 CSP -0.1159*** -21.52 0.099 -0.46 6.29 0.91 -0.1085 -0.2745 0.0211 EP(RC) 0.2826*** 38.41 0.115 0.93 3.61 0.99 0.2576 0.1400 0.5051 CRP -0.1158*** -24.43 0.090 -0.65 4.59 0.90 -0.1067 -0.2652 0.0146

Panel D: 75-Minute Return Data (Monthly Observations)

Mean t-stat Std Dev Skew Kurt AR(1) Median q0.05 q0.95 RC 0.2828*** 20.67 0.113 0.91 3.57 0.73 0.2597 0.1428 0.5081 IC 0.4203*** 24.35 0.140 0.49 3.16 0.74 0.4075 0.2137 0.6556 CSP -0.1375*** -13.35 0.105 -0.62 5.50 0.37 -0.1299 -0.2849 0.0192 EP(RC) 0.2828*** 20.65 0.113 0.95 3.75 0.73 0.2575 0.1420 0.4968 CRP -0.1375*** -14.75 0.092 -0.58 3.94 0.34 -0.1309 -0.2868 -0.0003

54 Table A.3: Predictability of S&P 500 Excess Returns: 1-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 1 month on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 1 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant -0.0090 -0.0269 0.0650 0.1429 1.2383 0.2965 0.0674 0.0515 1.6897 1.6337 2.6462 2.6009 (s.e. NW) (0.035) (0.040) (0.040) (0.128) (0.947) (0.388) (0.032)** (0.066) (0.826)** (0.837)* (0.792)*** (0.793)*** [s.e. Hod] [0.036] [0.041] [0.038]* [0.115] [0.864] [0.335] [0.033]** [0.066] [0.798]** [0.802]** [0.816]*** [0.815]*** VRP -4.6318 -4.5823 -4.2077 (s.e. NW) (1.141)*** (1.083)*** (1.250)*** 55 [s.e. Hod] [1.279]*** [1.217]*** [1.328]*** Individual VRP -4.4152 -4.2969 -3.8208 (s.e. NW) (1.050)*** (1.000)*** (1.171)*** [s.e. Hod] [1.122]*** [1.040]*** [1.163]*** CRP -6.1231 -6.6838 -7.0805 (s.e. NW) (2.670)** (2.959)** (2.841)** [s.e. Hod] [2.779]** [3.092]** [2.884]** CAY 1.6096 2.1706 1.7641 2.0209 1.5751 (s.e. NW) (1.353) (1.287)* (1.339) (1.333) (1.380) [s.e. Hod] [1.399] [1.355] [1.413] [1.403] [1.440] DFSP -8.9592 -20.286 -21.323 (s.e. NW) (13.69) (14.93) (14.80) [s.e. Hod] [11.96] [14.11] [13.92] log(P/D) -29.436 -41.316 -40.586 -64.044 -62.630 (s.e. NW) (23.26) (20.50)** (20.67)* (19.98)*** (20.14)*** [s.e. Hod] [21.30] [19.81]** [19.90]** [20.15]*** [20.21]*** log(P/E) -7.7205 4.9924 3.2842 (s.e. NW) (12.71) (11.63) (11.68) [s.e. Hod] [10.94] [11.23] [11.30] RREL 120.885 150.977 162.144 117.621 130.354 (s.e. NW) (69.80)* (66.86)** (70.42)** (75.08) (78.46)* [s.e. Hod] [63.07]* [61.39]** [64.78]** [69.14]* [72.56]* TMSP 0.0619 0.1789 0.8829 (s.e. NW) (2.671) (2.752) (2.798) [s.e. Hod] [2.561] [2.586] [2.618] Adj. R2 0.052 0.049 0.000 0.001 0.010 -0.001 0.013 -0.004 0.087 0.086 0.087 0.088 Table A.4: Predictability of S&P 500 Excess Returns: 3-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 3 months on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 3 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant 0.0070 -0.0275 0.0674 0.1004 1.3494 0.1867 0.0708 0.0443 1.8246 1.7261 2.4550 2.3869 (s.e. NW) (0.029) (0.034) (0.035)* (0.112) (0.822) (0.354) (0.029)** (0.058) (0.539)*** (0.556)*** (0.662)*** (0.627)*** [s.e. Hod] [0.035] [0.034] [0.038]* [0.116] [0.821] [0.297] [0.033]** [0.067] [0.714]** [0.711]** [0.826]*** [0.823]*** VRP -3.6217 -3.5638 -3.1790 (s.e. NW) (0.756)*** (0.645)*** (0.828)*** 56 [s.e. Hod] [0.890]*** [0.844]*** [0.853]*** Individual VRP -3.2044 -3.0627 -2.5972 (s.e. NW) (0.475)*** (0.459)*** (0.518)*** [s.e. Hod] [0.872]*** [0.804]*** [0.799]*** CRP -6.4953 -7.2545 -7.4997 (s.e. NW) (1.581)*** (1.558)*** (1.531)*** [s.e. Hod] [1.596]*** [1.566]*** [1.537]*** CAY 1.7333 2.3989 1.6851 2.1223 1.4517 (s.e. NW) (1.119) (1.039)** (1.023) (1.043)** (1.013) [s.e. Hod] [1.402] [1.339]* [1.343] [1.385] [1.366] DFSP -4.5299 -16.815 -18.374 (s.e. NW) (12.48) (9.262)* (9.293)** [s.e. Hod] [12.05] [13.36] [13.33] log(P/D) -32.131 -44.194 -42.911 -62.664 -60.538 (s.e. NW) (20.06) (13.45)*** (13.79)*** (16.06)*** (15.46)*** [s.e. Hod] [20.28] [17.71]** [17.68]** [19.74]*** [19.71]*** log(P/E) -4.1729 8.6413 6.0722 (s.e. NW) (11.63) (8.113) (8.408) [s.e. Hod] [9.710] [9.790] [9.860] RREL 128.616 161.047 180.658 147.463 166.614 (s.e. NW) (58.48)** (52.59)*** (53.10)*** (55.86)*** (54.88)*** [s.e. Hod] [61.44]** [57.86]*** [58.47]*** [64.62]** [65.53]** TMSP 0.4560 0.4489 1.5077 (s.e. NW) (2.512) (2.250) (2.185) [s.e. Hod] [2.586] [2.573] [2.607] Adj. R2 0.090 0.104 0.008 0.000 0.042 -0.002 0.051 -0.004 0.230 0.253 0.239 0.269 Table A.5: Predictability of S&P 500 Excess Returns: 6-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 6 months on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 6 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant 0.0265 -0.0070 0.0680 0.0425 1.4885 0.1389 0.0700 0.0264 1.9519 1.8629 2.0946 2.0351 (s.e. NW) (0.031) (0.034) (0.033)** (0.084) (0.667)** (0.345) (0.029)** (0.058) (0.447)*** (0.440)*** (0.607)*** (0.552)*** [s.e. Hod] [0.037] [0.035] [0.037]* [0.100] [0.756]* [0.259] [0.033]** [0.067] [0.724]*** [0.704]*** [0.851]** [0.845]** VRP -2.0895 -2.0076 -1.7733 (s.e. NW) (0.684)*** (0.555)*** (0.626)*** 57 [s.e. Hod] [0.763]*** [0.696]*** [0.602]*** Individual VRP -1.6849 -1.5499 -1.2525 (s.e. NW) (0.370)*** (0.297)*** (0.381)*** [s.e. Hod] [0.859]* [0.762]** [0.641]* CRP -4.8750 -5.3901 -5.6459 (s.e. NW) (0.963)*** (0.856)*** (0.871)*** [s.e. Hod] [1.635]*** [1.592]*** [1.549]*** CAY 2.0785 2.8989 2.2324 2.7436 2.1325 (s.e. NW) (1.160)* (0.994)*** (0.939)** (0.930)*** (0.858)** [s.e. Hod] [1.372] [1.366]** [1.366] [1.380]** [1.372] DFSP 1.2027 -8.2633 -9.6721 (s.e. NW) (9.431) (6.694) (6.089) [s.e. Hod] [10.29] [11.84] [11.40] log(P/D) -35.578 -46.771 -45.627 -53.871 -52.011 (s.e. NW) (16.22)** (11.16)*** (10.84)*** (13.68)*** (12.53)*** [s.e. Hod] [18.72]* [17.89]*** [17.45]*** [19.99]*** [19.79]*** log(P/E) -2.6756 6.3345 4.0494 (s.e. NW) (11.32) (5.292) (5.022) [s.e. Hod] [8.507] [8.893] [8.758] RREL 126.015 163.866 181.842 169.247 186.408 (s.e. NW) (58.78)** (55.53)*** (55.96)*** (60.76)*** (59.18)*** [s.e. Hod] [63.79]** [60.38]*** [61.17]*** [67.20]** [68.57]*** TMSP 1.1917 1.2191 2.1635 (s.e. NW) (2.386) (1.813) (1.706) [s.e. Hod] [2.592] [2.607] [2.595] Adj. R2 0.051 0.077 0.025 -0.004 0.096 -0.002 0.090 0.000 0.334 0.369 0.338 0.382 Table A.6: Predictability of S&P 500 Excess Returns: 9-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 9 months on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 9 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant 0.0407 0.0144 0.0679 0.0247 1.5626 0.1743 0.0713 0.0074 1.9968 1.9274 1.9170 1.8675 (s.e. NW) (0.034) (0.036) (0.034)** (0.073) (0.567)*** (0.324) (0.031)** (0.063) (0.389)*** (0.380)*** (0.559)*** (0.516)*** [s.e. Hod] [0.036] [0.035] [0.037]* [0.086] [0.712]** [0.239] [0.032]** [0.067] [0.699]*** [0.683]*** [0.851]** [0.848]** VRP -1.1207 -1.0399 -0.8878 (s.e. NW) (0.589)* (0.476)** (0.489)* 58 [s.e. Hod] [0.623]* [0.549]* [0.459]* Individual VRP -0.8061 -0.6815 -0.4497 (s.e. NW) (0.363)** (0.240)*** (0.265)* [s.e. Hod] [0.683] [0.575] [0.464] CRP -3.2931 -3.6921 -4.1450 (s.e. NW) (0.845)*** (0.960)*** (0.906)*** [s.e. Hod] [1.522]** [1.507]** [1.437]*** CAY 2.0861 2.9930 2.4611 3.0117 2.4938 (s.e. NW) (1.332) (1.023)*** (0.975)** (0.903)*** (0.840)*** [s.e. Hod] [1.363] [1.332]** [1.333]* [1.305]** [1.303]* DFSP 3.1119 -4.8277 -6.0241 (s.e. NW) (7.394) (6.604) (5.898) [s.e. Hod] [8.669] [10.07] [9.827] log(P/D) -37.371 -47.543 -46.656 -48.267 -46.721 (s.e. NW) (13.84)*** (9.782)*** (9.380)*** (12.35)*** (11.30)*** [s.e. Hod] [17.65]** [17.33]*** [16.96]*** [20.16]** [19.92]** log(P/E) -3.7563 3.3600 1.4480 (s.e. NW) (10.57) (5.391) (4.602) [s.e. Hod] [7.873] [9.001] [8.675] RREL 124.159 163.662 177.744 176.270 190.694 (s.e. NW) (65.73)* (61.22)*** (62.02)*** (70.81)** (69.34)*** [s.e. Hod] [65.09]* [63.01]*** [64.52]*** [70.69]** [72.44]*** TMSP 2.0573 2.3259 3.1193 (s.e. NW) (2.220) (1.612) (1.450)** [s.e. Hod] [2.526] [2.595] [2.572] Adj. R2 0.018 0.040 0.034 0.000 0.152 0.001 0.125 0.013 0.433 0.464 0.448 0.493 Table A.7: Predictability of S&P 500 Excess Returns: 12-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 12 months on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 12 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant 0.0440 0.0241 0.0665 0.0100 1.5893 0.1838 0.0692 -0.0161 1.9738 1.9250 1.6877 1.6487 (s.e. NW) (0.036) (0.038) (0.034)* (0.066) (0.509)*** (0.285) (0.033)** (0.068) (0.345)*** (0.346)*** (0.529)*** (0.504)*** [s.e. Hod] [0.036] [0.035] [0.037]* [0.078] [0.682]** [0.226] [0.033]** [0.066] [0.669]*** [0.664]*** [0.839]** [0.837]* VRP -0.8444 -0.7775 -0.7160 (s.e. NW) (0.504)* (0.401)* (0.372)* 59 [s.e. Hod] [0.541] [0.480] [0.395]* Individual VRP -0.6099 -0.5231 -0.3607 (s.e. NW) (0.348)* (0.210)** (0.186)* [s.e. Hod] [0.560] [0.474] [0.374] CRP -2.4716 -2.6608 -3.3532 (s.e. NW) (0.788)*** (0.774)*** (0.690)*** [s.e. Hod] [1.551] [1.500]* [1.459]** CAY 2.1499 2.9945 2.6057 3.2963 2.8696 (s.e. NW) (1.519) (1.100)*** (1.064)** (1.002)*** (0.938)*** [s.e. Hod] [1.438] [1.409]** [1.388]* [1.384]** [1.362]** DFSP 4.5483 -1.4263 -2.4130 (s.e. NW) (5.392) (6.357) (5.653) [s.e. Hod] [7.578] [8.877] [8.754] log(P/D) -38.031 -46.972 -46.352 -41.138 -39.919 (s.e. NW) (12.46)*** (8.721)*** (8.558)*** (11.67)*** (10.95)*** [s.e. Hod] [16.92]** [16.65]*** [16.51]*** [20.01]** [19.80]** log(P/E) -4.0685 -0.3903 -1.9215 (s.e. NW) (9.209) (5.607) (4.752) [s.e. Hod] [7.473] [8.862] [8.524] RREL 109.051 148.102 158.090 166.154 177.812 (s.e. NW) (62.99)* (55.46)*** (55.75)*** (66.70)** (64.85)*** [s.e. Hod] [64.51]* [62.21]** [63.55]** [69.64]** [71.34]** TMSP 3.0332 3.4806 4.1201 (s.e. NW) (1.999) (1.719)** (1.555)*** [s.e. Hod] [2.408] [2.503] [2.485]* Adj. R2 0.012 0.026 0.043 0.008 0.203 0.004 0.124 0.045 0.500 0.520 0.543 0.581 Table A.8: Predictability of S&P 500 Excess Returns: 18-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 18 months on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 18 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant 0.0451 0.0269 0.0630 -0.0074 1.6282 0.1102 0.0605 -0.0498 1.8714 1.8419 1.6049 1.5784 (s.e. NW) (0.038) (0.041) (0.034)* (0.068) (0.452)*** (0.259) (0.037) (0.073) (0.327)*** (0.341)*** (0.403)*** (0.392)*** [s.e. Hod] [0.036] [0.034] [0.036]* [0.070] [0.651]** [0.209] [0.033]* [0.066] [0.649]*** [0.648]*** [0.806]** [0.806]* VRP -0.6107 -0.5885 -0.4848 (s.e. NW) (0.473) (0.351)* (0.345) 60 [s.e. Hod] [0.443] [0.391] [0.343] Individual VRP -0.4036 -0.4333 -0.2078 (s.e. NW) (0.323) (0.198)** (0.160) [s.e. Hod] [0.430] [0.369] [0.309] CRP -2.0752 -1.7526 -2.5522 (s.e. NW) (0.732)*** (0.823)** (0.687)*** [s.e. Hod] [1.469] [1.428] [1.397]* CAY 2.3249 2.8670 2.6152 3.2431 2.8822 (s.e. NW) (1.785) (1.276)** (1.254)** (1.079)*** (1.017)*** [s.e. Hod] [1.484] [1.482]* [1.451]* [1.407]** [1.365]** DFSP 6.0634 -3.4274 -4.2558 (s.e. NW) (4.026) (4.051) (3.817) [s.e. Hod] [6.625] [6.523] [6.516] log(P/D) -39.020 -44.658 -44.290 -39.201 -38.363 (s.e. NW) (11.06)*** (8.218)*** (8.463)*** (8.206)*** (7.872)*** [s.e. Hod] [16.17]** [16.17]*** [16.12]*** [19.01]** [18.89]** log(P/E) -1.7945 -0.5130 -1.6702 (s.e. NW) (8.148) (3.891) (3.420) [s.e. Hod] [6.873] [7.643] [7.444] RREL 54.626 91.411 97.569 110.048 119.092 (s.e. NW) (40.58) (32.16)*** (31.49)*** (38.59)*** (35.85)*** [s.e. Hod] [58.13] [56.08] [57.13]* [63.18]* [64.60]* TMSP 4.3902 4.3557 4.8471 (s.e. NW) (1.877)** (2.002)** (1.795)*** [s.e. Hod] [2.178]** [2.415]* [2.424]** Adj. R2 0.007 0.024 0.064 0.026 0.299 -0.002 0.041 0.140 0.508 0.517 0.613 0.645 Table A.9: Predictability of S&P 500 Excess Returns: 24-Month Horizon

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of 24 months on a constant and the lagged forecasting variable(s). VRP is the index variance risk premium, defined as the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Individual VRP and CRP are the individual variance risk premium and the correlation risk premium factors. CAY is the consumption-to-wealth ratio. DFSP is the default spread, defined as the difference between BAA and AAA-rated corporate bond yields. log(P/D) is the logarithm of the level of the S&P 500 over the 12-month trailing sum of dividends paid by S&P 500 firms. log(P/E) is the logarithm of the price over the 12-month trailing sum of earnings. RREL is the stochastically detrended risk-free rate, i.e. the 1-month T-bill rate minus its 12-month trailing average. TMSP is the term spread, defined as the difference between the 10-year yield and the 3-month T-bill rate. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with 24 lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) Constant 0.0427 0.0260 0.0579 -0.0268 1.6204 0.0788 0.0510 -0.0724 1.7316 1.7174 1.4599 1.4466 (s.e. NW) (0.038) (0.043) (0.034)* (0.073) (0.380)*** (0.259) (0.039) (0.066) (0.307)*** (0.321)*** (0.372)*** (0.358)*** [s.e. Hod] [0.036] [0.035] [0.036] [0.068] [0.620]*** [0.183] [0.034] [0.067] [0.612]*** [0.615]*** [0.789]* [0.789]* VRP -0.5519 -0.5740 -0.5271 (s.e. NW) (0.362) (0.235)** (0.201)*** 61 [s.e. Hod] [0.363] [0.297]* [0.273]* Individual VRP -0.3685 -0.4991 -0.3129 (s.e. NW) (0.237) (0.140)*** (0.151)** [s.e. Hod] [0.343] [0.277]* [0.233] CRP -1.8724 -1.1374 -2.0997 (s.e. NW) (0.862)** (0.748) (0.618)*** [s.e. Hod] [1.347] [1.277] [1.278] CAY 2.4442 2.6485 2.5265 3.1039 2.8023 (s.e. NW) (1.888) (1.321)** (1.272)** (1.038)*** (0.951)*** [s.e. Hod] [1.491] [1.495]* [1.444]* [1.382]** [1.316]** DFSP 7.6028 -2.3346 -3.1124 (s.e. NW) (4.222)* (4.050) (3.701) [s.e. Hod] [6.075] [5.093] [5.192] log(P/D) -38.882 -41.498 -41.321 -33.969 -33.522 (s.e. NW) (9.196)*** (7.517)*** (7.788)*** (7.708)*** (7.241)*** [s.e. Hod] [15.42]** [15.28]*** [15.32]*** [18.07]* [18.07]* log(P/E) -0.8998 -3.6005 -4.4048 (s.e. NW) (7.920) (3.541) (3.079) [s.e. Hod] [5.989] [5.662] [5.570] RREL 5.1838 38.719 41.699 53.915 60.689 (s.e. NW) (27.82) (25.96) (26.44) (25.94)** (25.31)** [s.e. Hod] [45.64] [45.09] [46.29] [50.21] [51.89] TMSP 5.2721 4.7052 5.0830 (s.e. NW) (1.724)*** (1.832)** (1.695)*** [s.e. Hod] [2.087]** [2.368]** [2.385]** Adj. R2 0.008 0.024 0.080 0.056 0.380 -0.004 -0.004 0.261 0.508 0.509 0.668 0.692 Table A.10: Cross-Sectional Predictability: Overlapping Portfolios (I)

This table analyzes the predictability of the cross-section of excess stock returns. We use the past 5 years of data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. In Panel A, we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. For Panel B, we proceed as follows. We regress stock returns jointly on the lagged CRP and Individual VRP factors. We rank stocks in ascending order of their loadings on the CRP factor. We also rank stocks in ascending order of their loadings on the Individual VRP factor. We average the ranks across the two sorts. We then take long positions in the firms with the highest 20 % average score and short positions in stocks with the lowest 20 % average score. We construct overlapping portfolios as in Jegadeesh & Titman(1993) and report the average (annualized) value-weighted returns and Carhart(1997) 4-factor alpha of the long- short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0377 0.0037 0.0739* 0.0899** 0.1158*** 0.0872* 0.0476 (s.e.) (0.048) (0.040) (0.041) (0.043) (0.040) (0.047) (0.037) 4-factor alpha 0.0556 0.0294 0.0866* 0.0887** 0.1143*** 0.0771** 0.0493 (s.e.) (0.041) (0.032) (0.046) (0.043) (0.036) (0.038) (0.031)

Panel B: Individual VRP & CRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0420 0.0382 0.0709* 0.0972** 0.1082*** 0.0897*** 0.0651* (s.e.) (0.036) (0.028) (0.041) (0.040) (0.036) (0.034) (0.037) 4-factor alpha 0.0582 0.0662** 0.0917* 0.1057** 0.1145*** 0.0824*** 0.0654** (s.e.) (0.036) (0.033) (0.050) (0.041) (0.032) (0.028) (0.026)

62 Table A.11: Cross-Sectional Predictability: Overlapping Portfolios (II)

This table analyzes the predictability of the cross-section of excess stock returns. We use the past 5 years of data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. In Panel A, we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 75-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. Panels B and C are based on double-sorts. We regress stock returns jointly on the lagged CRP and Individual VRP factors. Panel B first sorts stocks into 5 portfolios based on the loadings on the CRP factor. It then sorts each of the portfolios into 5 portfolios based on the stock loadings on the individual VRP. Panel C sorts stocks first on the loading on the individual VRP factor and then on the CRP factor. We construct overlapping portfolios as in Jegadeesh & Titman(1993) and report the average (annualized) value-weighted returns and Carhart(1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0377 0.0037 0.0739* 0.0899** 0.1158*** 0.0872* 0.0476 (s.e.) (0.048) (0.040) (0.041) (0.043) (0.040) (0.047) (0.037) 4-factor alpha 0.0556 0.0294 0.0866* 0.0887** 0.1143*** 0.0771** 0.0493 (s.e.) (0.041) (0.032) (0.046) (0.043) (0.036) (0.038) (0.031)

Panel B: Individual VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0220 0.0136 0.0273 0.0606** 0.0531** 0.0398** 0.0192 (s.e.) (0.046) (0.033) (0.026) (0.025) (0.021) (0.019) (0.016) 4-factor alpha 0.0347 0.0354 0.0455* 0.0652*** 0.0591** 0.0351* 0.0181 (s.e.) (0.035) (0.025) (0.024) (0.022) (0.025) (0.020) (0.017)

Panel C: CRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0109 0.0302 0.0241 0.0395 0.0684** 0.0615 0.0666** (s.e.) (0.028) (0.040) (0.036) (0.031) (0.033) (0.039) (0.033) 4-factor alpha 0.0088 0.0359 0.0353 0.0470 0.0731** 0.0620* 0.0742** (s.e.) (0.036) (0.048) (0.043) (0.038) (0.034) (0.034) (0.030)

63 Table A.12: Decomposition Results: 30-Min

This table presents the results of the decomposition of the 1-month index variance swap payoff (Panel A) and variance risk premium (Panel B) into two factors. The variance swap payoff is the difference between the (annualized) realized and Bakshi et al.(2003) option-implied variance. We use high-frequency return data sampled at the 30-minute frequency to compute the realized variance. The variance risk premium is the difference between the physical expectation of the future variance, assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Mean is the average value. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. t-stat reports the

Newey–West corrected t-ratio (with 4 lags) associated with the mean. Shrmean reports the fraction of the mean of the (i) variance swap payoff (Panel A) or (ii) variance risk premium (Panel B) of the S&P

500 index associated with the factor [name in row]. Std Dev is the standard deviation. Shrvar reports the share of the variance of the (i) variance swap payoff (Panel A) or (ii) variance risk premium (Panel B) of the S&P 500 index associated with the factor [name in row]. Skew, Kurt and AR(1) denote the skewness, kurtosis and first-order autocorrelation, respectively. Median, q0.05 and q0.95 relate to the median, 5,% and 95 % of the distribution of the variable [name in row].

Panel A: Variance Swap Payoff

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VSP -0.0148*** -4.74 1.000 0.042 1.000 7.95 99.97 0.25 -0.0141 -0.0564 0.0137 Individual VSP -0.0033 -1.34 0.223 0.032 0.653 7.55 90.69 0.26 -0.0047 -0.0331 0.0263 CSP -0.0115*** -6.64 0.777 0.018 0.347 -2.57 40.24 0.45 -0.0086 -0.0335 0.0006

Panel B: Variance Risk Premium

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0129*** -5.62 1.000 0.029 1.000 6.33 77.91 0.20 -0.0118 -0.0430 0.0122 Individual VRP 0.0001 0.05 -0.010 0.030 0.764 9.10 114.6 0.21 -0.0030 -0.0206 0.0332 CRP -0.0130*** -8.37 1.010 0.014 0.236 -3.32 19.34 0.55 -0.0097 -0.0336 -0.0013

64 Table A.13: Predictability of S&P 500 Excess Returns: 30-Min

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of h-month on a constant and the lagged forecasting variable(s). Panel A considers the forecasting power of the market index variance risk premium. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 30-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. Panel B considers the two factors of the index variance risk premium, namely the CRP and Individual VRP factors. We consider forecasting horizons of 1, 3, 6, 9, 12, 18 and 24 months. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with h lags) and Hodrick(1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2. Wald presents the results of a Wald test of the null hypothesis that the two slope parameters are equal. (p − value) reports the corresponding Newey & West(1987) corrected p-value.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0069 0.0067 0.0277 0.0416 0.0455 0.0466 0.0441 (s.e.) (NW) (0.036) (0.029) (0.031) (0.034) (0.036) (0.038) (0.038) [s.e.] (Hod) [0.037] [0.037] [0.038] [0.037] [0.036] [0.036] [0.037] Slope -4.6413 -3.7742 -2.0632 -1.0921 -0.7562 -0.5112 -0.4631 (s.e.) (NW) (1.221)*** (0.839)*** (0.686)*** (0.598)* (0.494) (0.430) (0.336) [s.e.] (Hod) [1.408]*** [1.069]*** [0.845]** [0.687] [0.581] [0.469] [0.393] Adj. R2 0.059 0.111 0.056 0.020 0.010 0.005 0.005

Panel B: CRP and Individual VRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0319 -0.0384 -0.0121 0.0092 0.0218 0.0254 0.0238 (s.e.) (NW) (0.041) (0.034) (0.036) (0.038) (0.039) (0.042) (0.044) [s.e.] (Hod) [0.042] [0.034] [0.036] [0.035] [0.035] [0.035] [0.036] Slope VRP -4.4725 -3.4691 -1.7959 -0.8786 -0.6025 -0.3790 -0.3426 (s.e.) (NW) (1.052)*** (0.448)*** (0.333)*** (0.321)*** (0.305)** (0.261) (0.186)* [s.e.] (Hod) [1.201]*** [0.971]*** [0.884]** [0.710] [0.584] [0.455] [0.373] Slope CRP -6.5558 -7.2353 -5.1072 -3.5428 -2.5330 -2.0887 -1.9431 (s.e.) (NW) (2.586)** (1.149)*** (0.909)*** (0.823)*** (0.803)*** (0.729)*** (0.925)** [s.e.] (Hod) [2.766]** [1.339]*** [1.556]*** [1.472]** [1.521]* [1.449] [1.351] Adj. R2 0.058 0.134 0.089 0.049 0.029 0.025 0.028 Wald 0.454 8.030*** 9.839*** 8.415*** 4.955** 4.522** 2.768* (p-value) (0.501) (0.005) (0.002) (0.004) (0.026) (0.033) (0.096)

65 Table A.14: Cross-Sectional Predictability (I): 30-Min

This table summarizes the output of the sorting analysis. We use the past 60 observations of monthly data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. Panel A obtains when we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 30-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. To obtain Panel B, we proceed as follows. We regress stock returns jointly on the lagged CRP and Individual VRP factors. We rank stocks in ascending order of their loadings on the CRP factor. We also rank stocks in ascending order of their loadings on the Individual VRP factor. Next, we average the ranks across the two sorts. We then take long positions in the firms with the highest 20 % average score and short positions in stocks with the lowest 20 % average scores. We report the average (annualized) value-weighted returns and Carhart(1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0365 0.0202 0.0742** 0.0846** 0.0948*** 0.0585** 0.0210 (s.e.) (0.048) (0.048) (0.035) (0.038) (0.030) (0.026) (0.015) 4-factor alpha 0.0537 0.0381 0.1039*** 0.0975*** 0.1006*** 0.0519** 0.0303** (s.e.) (0.043) (0.030) (0.030) (0.034) (0.022) (0.022) (0.015)

Panel B: Individual VRP & CRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0656* 0.0484 0.0835** 0.0952** 0.0899*** 0.0748*** 0.0289* (s.e.) (0.036) (0.030) (0.039) (0.037) (0.032) (0.023) (0.016) 4-factor alpha 0.0805** 0.0807*** 0.1266*** 0.1170*** 0.1008*** 0.0668*** 0.0501*** (s.e.) (0.037) (0.031) (0.037) (0.033) (0.023) (0.019) (0.012)

66 Table A.15: Cross-Sectional Predictability (II): 30-Min

This table summarizes the output of the sorting analysis. We use the past 60 observations of monthly data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. Panel A obtains when we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 30-minute data and assuming a random walk model (Bollerslev et al., 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. Panels B and C are based on double-sorts. We regress stock returns jointly on the lagged CRP and Individual VRP factors. Panel B first sorts stocks into 5 portfolios based on the loadings on the CRP factor. It then sorts each of the portfolios into 5 portfolios based on the stock loadings on the individual VRP. Panel C sorts stocks first on the loading on the individual VRP factor and then on the CRP factor. We report the average (annualized) value-weighted returns and Carhart(1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0365 0.0202 0.0742** 0.0846** 0.0948*** 0.0585** 0.0210 (s.e.) (0.048) (0.048) (0.035) (0.038) (0.030) (0.026) (0.015) 4-factor alpha 0.0537 0.0381 0.1039*** 0.0975*** 0.1006*** 0.0519** 0.0303** (s.e.) (0.043) (0.030) (0.030) (0.034) (0.022) (0.022) (0.015)

Panel B: Individual VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0219 0.0097 0.0396* 0.0639*** 0.0474*** 0.0344*** 0.0134 (s.e.) (0.045) (0.036) (0.023) (0.022) (0.017) (0.012) (0.013) 4-factor alpha 0.0340 0.0167 0.0591*** 0.0636*** 0.0358*** 0.0193* 0.0327** (s.e.) (0.034) (0.025) (0.017) (0.019) (0.010) (0.012) (0.015)

Panel C: CRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0384 0.0413 0.0429 0.0345 0.0493** 0.0327 0.0355*** (s.e.) (0.029) (0.039) (0.031) (0.023) (0.021) (0.020) (0.011) 4-factor alpha 0.0368 0.0623 0.0682** 0.0555** 0.0666*** 0.0549*** 0.0339*** (s.e.) (0.036) (0.046) (0.034) (0.024) (0.021) (0.011) (0.009)

67 Table A.16: Decomposition Results: HAR

This table presents the results of the decomposition of the 1-month index variance swap payoff (Panel A) and variance risk premium (Panel B) into two factors. The variance swap payoff is the difference between the (annualized) realized and Bakshi et al.(2003) option-implied variance. We use high-frequency return data sampled at the 30-minute frequency to compute the realized variance. The variance risk premium is the difference between the physical expectation of the future variance, assuming a HAR model (Corsi, 2009), and the Bakshi et al.(2003) option-implied variance. Mean is the average value. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. t-stat reports the Newey–West

corrected t-ratio (with 4 lags) associated with the mean. Shrmean reports the fraction of the mean of the (i) variance swap payoff (Panel A) or (ii) variance risk premium (Panel B) of the S&P 500 index

associated with the factor [name in row]. Std Dev is the standard deviation. Shrvar reports the share of the variance of the (i) variance swap payoff (Panel A) or (ii) variance risk premium (Panel B) of the S&P 500 index associated with the factor [name in row]. Skew, Kurt and AR(1) denote the skewness, kurtosis and first-order autocorrelation, respectively. Median, q0.05 and q0.95 relate to the median, 5,% and 95 % of the distribution of the variable [name in row].

0.05 0.95 Mean t-stat Shrmean Std Dev Shrvar Skew Kurt AR(1) Median q q Index VRP -0.0165*** -6.81 1.000 0.022 1.000 -1.98 10.04 0.48 -0.0102 -0.0539 0.0036 Individual VRP -0.0022 -1.61 0.131 0.017 0.405 2.91 36.25 0.15 0.0010 -0.0252 0.0098 CRP -0.0143*** -8.37 0.869 0.015 0.595 -3.35 18.30 0.62 -0.0110 -0.0353 -0.0008

68 Table A.17: Predictability of S&P 500 Excess Returns: HAR

This table summarizes the results of the regression of S&P 500 (annualized) excess returns measured over a horizon of h-month on a constant and the lagged forecasting variable(s). Panel A considers the forecasting power of the market index variance risk premium. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 30-minute data and assuming a HAR model (Corsi, 2009), and the Bakshi et al.(2003) option-implied variance. Panel B considers the two factors of the index variance risk premium, namely the CRP and Individual VRP factors. We consider forecasting horizons of 1, 3, 6, 9, 12, 18 and 24 months. The entries in parentheses and square brackets indicate Newey & West(1987) corrected standard errors (with h lags) and Hodrick (1992) corrected standard errors, respectively. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively. Adj. R2 reports the adjusted R2. Wald presents the results of a Wald test of the null hypothesis that the two slope parameters are equal. (p − value) reports the corresponding Newey & West(1987) corrected p-value.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0208 -0.0113 -0.0066 0.0078 0.0137 0.0160 0.0100 (s.e.) (NW) (0.039) (0.035) (0.036) (0.037) (0.039) (0.042) (0.044) [s.e.] (Hod) [0.038] [0.035] [0.034] [0.033] [0.033] [0.033] [0.035] Slope -3.8713 -3.5336 -3.0134 -2.1134 -1.6774 -1.4359 -1.5773 (s.e.) (NW) (2.070)* (1.141)*** (0.823)*** (0.753)*** (0.696)** (0.569)** (0.622)** [s.e.] (Hod) [2.155]* [1.522]** [1.005]*** [0.867]** [0.856]* [0.776]* [0.682]** Adj. R2 0.020 0.052 0.068 0.047 0.037 0.038 0.062

Panel B: CRP and Individual VRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Constant -0.0101 -0.0149 -0.0185 -0.0097 -0.0013 -0.0008 -0.0067 (s.e.) (NW) (0.044) (0.044) (0.039) (0.040) (0.043) (0.045) (0.048) [s.e.] (Hod) [0.043] [0.042] [0.039] [0.036] [0.036] [0.035] [0.036] Slope VRP -4.5416 -3.3126 -2.2635 -1.0270 -0.7479 -0.3919 -0.5486 (s.e.) (NW) (2.573)* (1.236)*** (1.045)** (1.013) (0.914) (0.911) (0.998) [s.e.] (Hod) [2.677]* [1.657]** [0.990]** [0.798] [0.771] [0.695] [0.629] Slope CRP -3.0215 -3.8137 -3.9637 -3.4942 -2.8612 -2.7629 -2.8927 (s.e.) (NW) (2.872) (2.904) (1.176)*** (0.949)*** (0.996)*** (0.868)*** (0.915)*** [s.e.] (Hod) [2.703] [2.425] [2.020]* [1.615]** [1.471]* [1.260]** [1.101]*** Adj. R2 0.017 0.048 0.070 0.062 0.052 0.068 0.099 Wald 0.133 0.039 1.101 2.619 2.040 3.892** 2.886* (p-value) (0.716) (0.844) (0.294) (0.106) (0.153) (0.049) (0.089)

69 Table A.18: Cross-Sectional Predictability (I): HAR

This table summarizes the output of the sorting analysis. We use the past 60 observations of monthly data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. Panel A obtains when we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 30-minute data and assuming a HAR model (Corsi, 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. To obtain Panel B, we proceed as follows. We regress stock returns jointly on the lagged CRP and Individual VRP factors. We rank stocks in ascending order of their loadings on the CRP factor. We also rank stocks in ascending order of their loadings on the Individual VRP factor. Next, we average the ranks across the two sorts. We then take long positions in the firms with the highest 20 % average score and short positions in stocks with the lowest 20 % average scores. We report the average (annualized) value-weighted returns and Carhart (1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0387 0.0283 0.0569 0.0529 0.0604* 0.0569* 0.0162 (s.e.) (0.034) (0.039) (0.044) (0.039) (0.035) (0.031) (0.019) 4-factor alpha 0.0441 0.0687 0.1015** 0.0882** 0.0960*** 0.0776*** 0.0389*** (s.e.) (0.047) (0.048) (0.046) (0.035) (0.022) (0.022) (0.009)

Panel B: Individual VRP & CRP Factors

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0562* 0.0229 0.0521 0.0465 0.0561* 0.0371 0.0203 (s.e.) (0.031) (0.031) (0.041) (0.031) (0.030) (0.030) (0.017) 4-factor alpha 0.0554 0.0565 0.1023** 0.0772*** 0.0810*** 0.0603*** 0.0342*** (s.e.) (0.042) (0.038) (0.042) (0.030) (0.021) (0.018) (0.011)

70 Table A.19: Cross-Sectional Predictability (II): HAR

This table summarizes the output of the sorting analysis. We use the past 60 observations of monthly data to estimate a regression of the h-month excess return of a stock on a constant and the lagged forecasting variable(s). We do this for each stock and observation month. Panel A obtains when we use the index VRP as forecasting variable. The index variance risk premium is the difference between the physical expectation of the future variance, computed based on 30-minute data and assuming a HAR model (Corsi, 2009), and the Bakshi et al.(2003) option-implied variance. We sort all stocks, into quintile portfolios, in ascending order of the slope estimates. We simultaneously take long and short positions in the quintiles with the highest and lowest slope estimates, respectively. We hold this position over the next h months and present the results of this strategy. Panels B and C are based on double-sorts. We regress stock returns jointly on the lagged CRP and Individual VRP factors. Panel B first sorts stocks into 5 portfolios based on the loadings on the CRP factor. It then sorts each of the portfolios into 5 portfolios based on the stock loadings on the individual VRP. Panel C sorts stocks first on the loading on the individual VRP factor and then on the CRP factor. We report the average (annualized) value-weighted returns and Carhart(1997) 4-factor α of the long-short portfolio. We present Newey & West(1987) corrected standard errors with h lags in parentheses. *, ** and *** indicate significance at the 10%, 5%, and 1% level, respectively.

Panel A: Index VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0387 0.0283 0.0569 0.0529 0.0604* 0.0569* 0.0162 (s.e.) (0.034) (0.039) (0.044) (0.039) (0.035) (0.031) (0.019) 4-factor alpha 0.0441 0.0687 0.1015** 0.0882** 0.0960*** 0.0776*** 0.0389*** (s.e.) (0.047) (0.048) (0.046) (0.035) (0.022) (0.022) (0.009)

Panel B: Individual VRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0243 -0.0030 0.0520* 0.0499*** 0.0373** 0.0222 0.0231** (s.e.) (0.032) (0.021) (0.027) (0.018) (0.018) (0.015) (0.010) 4-factor alpha 0.0339 0.0235 0.0904*** 0.0597*** 0.0421*** 0.0349*** 0.0262*** (s.e.) (0.037) (0.020) (0.021) (0.016) (0.013) (0.012) (0.009)

Panel C: CRP

Horizon (in Months) 1 3 6 9 12 18 24

Return 0.0433 0.0360 0.0224 0.0210 0.0323 0.0515** 0.0375*** (s.e.) (0.046) (0.037) (0.031) (0.023) (0.020) (0.021) (0.010) 4-factor alpha 0.0273 0.0480 0.0410* 0.0448*** 0.0665*** 0.0712*** 0.0369*** (s.e.) (0.047) (0.037) (0.023) (0.016) (0.012) (0.013) (0.008)

71