Volatility-of-Volatility Risk in Asset Pricing

Te-Feng Chen, Tarun Chordia, San-Lin Chung, and Ji-Chai Lin*

November 2017

Abstract

This paper develops a general equilibrium model in an endowment economy with time-varying

uncertainty and provides empirical support for an outcome of the model that the market volatility-

of-volatility (VOV) predicts market returns. A second outcome is an asset pricing model that

incorporates the market, the market volatility and VOV as a pricing factors. The differential long-

short quintile portfolio return amounts to 72 basis points per month when sorting on the VOV beta.

The risk premium on VOV is robust across different testing strategies and different test assets.

Market risk and volatility risk are not priced in constant beta models but consistent with theory,

the conditional factor loadings on both the market risk and volatility risk are priced. The pricing

impact of VOV strengthens during market crashes while that of market volatility obtains during

normal times.

JEL Classification: G14; G12;

Contacts

Chen Chordia Chung Lin

Voice: 852-3400-3856 1-404-727-1620 886-2-3366-1084 852-3400-8454

Fax: 852-2330-9845 1-404-727-5238 886-2-2366-0764 852-2330-9845

E-mail: tfchen@polyu.edu.hk Tarun.Chordia@emory.edu chungsl@ntu.edu.tw jclin@polyu.edu.hk

Address: HK Polytechnic

University

Kowloon, Hong Kong

Goizueta Business School

Emory University

Atlanta, GA 30322

National Taiwan

University

Taipei, Taiwan

HK Polytechnic

University

Kowloon, Hong Kong

*We thank seminar participants at Hong Kong Polytechnic University.

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Volatility-of-Volatility Risk in Asset Pricing

Abstract

This paper develops a general equilibrium model in an endowment economy with time-varying

uncertainty and provides empirical support for an outcome of the model that the market volatility-

of-volatility (VOV) predicts market returns. A second outcome is an asset pricing model that

incorporates the market, the market volatility and VOV as a pricing factors. The differential long-

short quintile portfolio return amounts to 72 basis points per month when sorting on the VOV beta.

The risk premium on VOV is robust across different testing strategies and different test assets.

Market risk and volatility risk are not priced in constant beta models but consistent with theory,

the conditional factor loadings on both the market risk and volatility risk are priced. The pricing

impact of VOV strengthens during market crashes while that of market volatility obtains during

normal times.

JEL classification: G12, G13, E4

1  

1. Introduction

Turmoil in financial markets while episodic has become more frequent, leading to higher

uncertainty in markets.1 In order to capture the dynamics of market uncertainty, Bollerslev,

Tauchen, and Zhou (2009) build a general equilibrium model for a representative investor with

Epstein and Zin (1989) preferences and introduce the volatility-of-volatility (VOV) as an additional

source of uncertainty. Their model shows that VOV drives the time-varying variance risk premium

and its ability to predict stock market returns.

This paper develops a macroeconomic model that synthesizes the seminal long-run risk model

of Bansal and Yaron (2004) and the variance-of-variance model of Bollerslev, Tauchen, and Zhou

(2009). We solve the macro-finance model explicitly and derive the equilibrium aggregate asset

prices. The model establishes that an increase in VOV reflects higher uncertainty about market

volatility, and raises the market risk premium, implying an immediate price decline and higher

future returns. Further, VOV drives the contemporaneous covariance between market returns and

market volatility (denoted CVRV). CVRV is time-varying and becomes more negative as VOV

increases. Further, VOV and CVRV contain similar information for predicting market returns.

We use aggregate asset prices to characterize the macroeconomic risks, and transform the

underlying macro-based model to a market-based three-factor asset pricing model. The market-

based model has distinct advantages: (i) Financial data provide useful information because asset

prices tell us how market participants value risks; and (ii) Markets disseminate financial data in a

timely fashion. In the asset pricing model, the expected return of security i is determined by three

sources of risks: (i) the covariance with market returns, ov , , , ; (ii) the covariance

                                                       1As Mr. Olivier Blanchard, IMF’s chief economist, points out in The Economist in January 2009, “Crises feed uncertainty. And uncertainty affects behaviour, which feeds the crisis.”

2  

with the market variance, ov , , , , where , ar , ; and, (iii) the

covariance with the variance of market variance, ov , , , , where ,

ar , . The first risk is the market risk of the classical capital asset pricing model (CAPM;

Sharpe, 1964; Lintner, 1965). The second risk corresponds to the volatility risk of Ang, Hodrick,

Xing, and Zhang (2006). The last risk, which is the main focus of our paper, is the volatility-of-

volatility, VOV, risk.

While the aggregate volatility risk is proxied by the volatility index, VIX, obtained from the

Chicago Board of Options Exchange (CBOE), we measure VOV using high frequency S&P 500

index option data. We first convert the tick-by-tick options data to equally spaced five-minute

observations and then use the model-free methodology (see Carr and Madan 1998, Britten-Jones

and Neuberger 2000, Bakshi, Kapadia, and Madan, 2003, and Jiang and Tian 2005) to estimate the

market variance implied by index option prices for each five-minute interval. For each day, we

estimate VOV by calculating the realized bipower variance from a series of five-minute model-free

implied market variance within the day. The bipower variation, introduced by Barndorff-Nielsen

and Shephard (2004), delivers a consistent estimator solely for the continuous component of VOV

while excluding the jump component.2 In other words, our empirical results are not affected by

the potential jump risk embedded in volatility as shown by Pan (2002), Eraker (2008), and

Drechsler and Yaron (2011).

Empirically, VOV strongly affects the market return-volatility covariance (CVRV), and,

moreover, VOV and, to a lesser extent, CVRV predict market returns. VOV predicts market returns

even after controlling for the variance risk premium and jump risk, as documented by Bollerslev,

                                                       2 Measures of realized jump based on the difference between realized variation and bipower variation have been proposed by Barndorff-Nielsen and Shephard (2004), Huang and Tauchen (2005), and Andersen, Bollerslev, and Diebold (2007).

3  

Tauchen, and Zhou (2009) and Drechsler and Yaron (2011), respectively. Thus, consistent with our

model, VOV is a state variable that drives the time-varying market risk premium. The next

question is whether VOV is a priced factor in the cross-section of asset returns.

We test the three-factor asset pricing model using stock returns on NYSE, AMEX, and

NASDAQ listed stocks over the period from 1996 to 2015. The main finding is that VOV is indeed

priced in the cross-section of stock returns with a return differential of -72 basis points per month

across quintile portfolios sorted on the VOV beta. The portfolio returns decrease with the VOV beta

because the market return decreases with VOV and so a stock whose return increases with VOV

i.e., a positive VOV beta, will provide a hedge against VOV risk and thus earn lower returns than a

stock with a negative VOV beta. The risk premium on VOV is robust in Fama-MacBeth (1973)

regressions when the test assets are 25 portfolios formed using independent quintile sorts on the

VOV beta and the VIX beta as well as when using individual stocks as test assets. The VOV premium

is robust in the presence of the Fama and French (1993) and the Carhart (1997) factors (SMB,

HML and UMD) as well as the market skewness and the market kurtosis factor of Chang,

Christoffersen, and Jacobs (2013) and the jump factor of Cremers, Halling and Weinbaum (2015).

Moreover, the VOV premium remains robust in the presence of the firm level characteristics

including firm size, the book-to-market ratio, past returns, the Amihud (2002) illiquidity, the

implied-realized volatility spread (IVOL-TVOL) and the call-put implied volatility spread (CIVOL-

PIVOL) of Bali and Hovakimian (2009) and Yan (2011).

The unconditional VIX beta and the MKT beta are not priced in our sample. However,

consistent with the theoretical model, the conditional component of the factor loadings of both,

VIX and the MKT, are priced. While Ang et al. (2006) and Adrian and Rosenberg (2008) show that

market volatility is a priced factor, Chang, Christoffersen, and Jacobs (2013) find mixed results.

4  

Interestingly, we find that all the three risks (MKT, VIX, VOV) in the asset pricing model are

significantly priced during normal times when we remove the observations with large market

declines from our sample. However, during crash periods (with extreme negative market returns),

the price of VOV risk becomes even more significant while the premiums on MKT and VIX have

the wrong sign. The positive premium on VIX during market turmoil combined with the negative

premium during normal periods, results in an overall VIX premium that is indistinguishable from

zero. The results imply that market turmoil strengthens investors’ demand for compensation for

VOV risk.

Our study is related to several strands of the literature. Bollerslev, Tauchen, and Zhou (2009)

use the variance risk premium as an indirect measure of VOV to predict stock market returns.

However, Drechsler and Yaron (2011) show that jump shocks can capture the size and predictive

power of the variance risk premium. Further, Drechsler (2013) shows that model uncertainty also

has a large impact on the variance risk premium, helping explain its power to predict stock returns.

Unlike these studies, we focus directly on VOV. We empirically construct a VOV measure that

excludes jumps to show that it is a determinant of the variance risk premium and that it can predict

stock market returns. Moreover, we provide new evidence that VOV risk is important for pricing

the cross-section of stock returns.

Our paper is also related to the pricing model with higher moments of the market return as

risk factors, as proposed by Chang, Christoffersen, and Jacobs (2013). They find that market

skewness is a priced risk factor in the cross section of stock returns. Both our paper and their work

extend Ang et al. (2006) to extract implied moments from index option prices. Our results are

robust to the inclusion of the market skewness factor as well as the jump factor of Cremers, Halling,

and Weinbaum (2015). In fact, we find that VOV risk subsumes the pricing power of the higher

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moment and jump betas documented in the literature. Our findings imply that, in the spirit of the

Merton (1973) ICAPM, the time-varying VOV seems better in capturing shifts in the investment

opportunity set than the time-varying market skewness and jumps. Finally, Baltussen, Van Bekkum,

and Van Der Grient (2013) show evidence that an ambiguity measure based on firm-level historical

volatility of option-implied volatility (vol-of-vol) is associated with future stock returns, which is

inconsistent with the rational pricing of uncertainty. Our study complements theirs as we show that

our VOV measure is in supportive of the rational pricing of VOV risk.

The remainder of the paper is organized as follows. The next section describes the economic

dynamics and develops our market-based three-factor model for the empirical implementation.

Section 3 constructs the measure of market volatility-of-volatility. Section 4 describes the data,

presents the summary statistics, and provides evidence for the VOV’s predictability on future stock

market returns. In section 5, we show empirical evidence on the pricing of VOV risk in the cross-

sectional stock returns and investigate the VOV risk-return tradeoff during market turmoil. Finally,

section 6 contains our concluding remarks.

2. An asset pricing model with VOV risk

This section develops an asset pricing model with volatility-of-volatility (VOV) risk starting

with a macroeconomic model that incorporates the seminal long-run risk model of Bansal and

Yaron (2004) and the variance-of-variance model of Bollerslev, Tauchen, and Zhou (2009). We

solve the macro-finance model explicitly and derive the equilibrium aggregate asset prices. Then,

we use the properties of aggregate asset prices to characterize the macroeconomic risks and

develop a market-based three-factor model for the cross-sectional asset prices.

2.1. Economic dynamics and the equilibrium market risk premium

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The setting of our model is a discrete-time endowment economy. The dynamics of the

consumption growth rate, , and the dividend growth rate, , , are governed by the

following processes:

, ,

, ,

, ,

, ,

, , ,

(1)

where , , , , , , , , , ~ 0,1 , represents the long-run consumption

growth, is the time-varying economic uncertainty, and is the economic volatility-of-

volatility, which is the conditional variance of the economic uncertainty. These features of the

long-run risk and the time-varying economic uncertainty have been proposed by Bansal and Yaron

(2004), while the additional feature of economic volatility-of-volatility has been introduced by

Bollerslev, Tauchen, and Zhou (2009).

The representative agent is assumed to have recursive preferences of Epstein and Zin (1989).

Thus, the logarithm of the Intertemporal Marginal Rate of Substitution (IMRS) is

log 1 , ,

where , is the return on consumption claim, is the time discount factor, is the risk-

aversion parameter, is the intertemporal elasticity of substitution (IES) parameter, and ≡

1 1 1/ . With 1, and 1, 0. Based on Campbell and Shiller’s (1988)

approximation, , , where is the logarithm of price–

consumption ratio, which in equilibrium is an affine function of the state variables, i.e.

.3 Substituting the equilibrium consumption return, , , into the

                                                       

3 The equilibrium solutions are: / 0, / 0, and 0.

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IMRS, the innovation in the pricing kernel is

, , , , , (2)

where 0, 1 0, 1 0, 1 0 .

The underlying parameters drive the prices for short-run risk ( ), long-run risk ( ), volatility

risk ( ), and volatility of volatility risk ( ). An analogous expression holds for the stock market

return, , , , , , , , where , is the log price–dividend

ratio, which in equilibrium is an affine function of the state variables, , , ,

, , .4 Since, 0, we have , 0, , 0, and , 0.

The innovation in market return can be expressed as

, , , , , , , , , ,

where , , , 0, , , , 0, and , , , 0 . The

conditional variance of market return is readily calculated as

, ≡ ar , , , , ,

and the process for innovations in market variance is , , , ,

, , , where , , and , , . Thus, innovations in market variance

are related to both the economic volatility shock and the economic volatility-of-volatility shock. It

follows that the market volatility-of-volatility (i.e., the conditional variance of market variance) is

, ≡ ar , , , , and the process for its innovations is ,

, , , , where , , . Note that, in our model, innovations to the

market volatility-of-volatility are solely determined by economic variance of variance shocks with

a scaling factor, , . Thus, the volatility of market VOV (i.e., the conditional variance of variance

                                                       4 The solutions are: ,

/

,, ,

. ,

,, and ,

. ,

,, where

, , and , , .

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of market variance), , ≡ ar , , , is constant in our model.

It is straightforward now to derive the equity premium on the market portfolio. In the

following proposition, the expected market return is determined by the covariance risks with

respect to three sources of risks in the pricing kernel.

Proposition 1: The risk premium for the market, , , , is governed by three

covariance risks as follows:

, , 0.5 ar , ov , ,

ov , , ov , , ov , ,

, , , .

(3)

The first two terms in (3)represent the long-run risk premium and the volatility risk premium,

which are the same as in Bansal and Yaron (2004), while the last one represents the volatility-of-

volatility risk premium, as in Bollerslev, Tauchen, and Zhou (2009).5

2.2. Market return-volatility covariance and future return predictability

The contemporaneous market return and market volatility tend to be negatively correlated,

which is commonly referred to as the leverage effect in the literature (e.g. Black, 1976; Christie,

1982; among others). This covariance could also be understood as the volatility risk for the

aggregate market under ICAPM (Ang et al., 2006). The model endogenously generates a time-

varying contemporaneous correlation between the market return and the market volatility as well

as a negative contemporaneous correlation between the market return and the volatility-of-

volatility. Straightforward calculations yield the following proposition.

Proposition 2: The market return is negatively correlated with both the market volatility and the

                                                       5 Since we do not assume the square root process for the volatility-of-volatility as Bollerslev, Tauchen, and Zhou (2009) do, the volatility risk in the resulting equity premium does not confound with the volatility-of-volatility risk.

9  

market volatility-of-volatility, as shown below:

, ≡ ov , , , , , , , 0,ov , , , , , 0.

(4)

The contemporaneous covariance between the market return and the market volatility is time-

varying and driven by the volatility-of-volatility. By contrast, the contemporaneous covariance

between the market return and the volatility-of-volatility is constant.

In the absence of the time-varying economic volatility-of-volatility i.e., when is constant

and =0, the second term of , is zero, resulting in a less negative value. Thus, the

economic volatility-of-volatility amplifies the impact of volatility on returns. Moreover, driven by

, the volatility risk for the aggregate market is time-varying and provides information about

future market returns since the time-varying market risk premium from (3) is driven by as

well. Further, the market volatility-of-volatility ( , ), which varies directly with the economic

volatility-of-volatility ( ), possesses similar information about future market returns. The

following proposition summarizes market return predictability of , through the channel

of the underlying economic volatility-of-volatility ( ).

Proposition 3. With the presence of time-varying economic volatility-of-volatility ( ), the

projection coefficient for the predictive regression of market returns on , is negative, i.e.,

, , , , with 0.

The projection coefficient for predictive regression of market returns on , is positive, i.e.,

, , , ,with 0.

Specifically, the projection coefficients are:

, , , ,

, ,0,

, , , ,

,

,

,0.

(5)

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The negative association between the future market return and the volatility risk for the

market and a positive association between the future market return and the market volatility-of-

volatility are consistent with the feedback effect through the time-varying risk premium

documented in the literature (see, e.g. Campbell and Hentschel 1992; Bekaert and Wu 2000; Wu

2001; Bollerslev, Sizova, and Tauchen, 2012; among others). When increases, , also

increases (and , decreases), which leads investors to demand a higher risk premium,

resulting in a negative contemporaneous market return due to a higher discount rate pushing the

price lower, and leading to a higher future return to compensate for the increased risk.

Furthermore, Bollerslev, Tauchen, and Zhou (2009) show that the variance risk premium

(VRP), which is the proxy for the economic volatility-of-volatility, , in their equilibrium model,

can predict the future market return. However, Drechsler and Yaron (2011), challenge the role of

in Bollerslev, Tauchen, and Zhou, since the jump risk premium could be an alternative state

variable that contributes to both the variance risk premium and the market return predictability.

Thus, instead of using VRP, an indirect measure used by Bollerslev, Tauchen, and Zhou to proxy

for , we will use the direct measure - market volatility-of-volatility ( , . Further, we will

isolate the jump component of , from the continuous component and use only the latter.

2.3. A market-based three-factor model for pricing individual stocks

Based on the sources of risk in (1), we assume that the innovation in stock return i is6

, , , , , , , , , .

Given the pricing kernel in (2), the expected stock return can be written as

, , 0.5 ar ,

, , , . (6)

The expected stock return is determined by three sources of economic risks: economic long-run

                                                       6 We have not included the risk of consumption growth as the empirical support for it is weak.

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risk ( , ), economic volatility risk ( , , and economic volatility-of-volatility risk ( , ). The

following proposition provides the risk premium in terms of market risks.

Proposition 4: The risk premium for stock i, , , , is governed by

, , 0.5 ar ,

ov , , , ov , , ,

ov , , , ,

(7)

where ,, ,

,, , ,

,. (8)

The proof of the above proposition uses the properties of the aggregate asset prices to

characterize the macroeconomic risks in (6). In equilibrium, the market volatility-of-volatility risk,

which is the return covariance with respect to the variance of market variance, is solely determined

by the economic volatility-of-volatility risk ( , ), i.e., ov , , , , , . Also,

the return sensitivities with respect to the market variance and with respect to the market return

provide additional information for the economic volatility risk and the long-run risk; that is,

ov , , , , , , , , and ov , , , , ,

, , , , . Substituting out the economic risks in (6), results in the market-based

three-factor model for individual stock excess returns in (7).

Thus, the expected return on stock i is determined by three sources of risks related to aggregate

asset prices. The first term measures the market risk of classical capital asset pricing model (CAPM;

Sharpe, 1964; Lintner, 1965). The second term corresponds to the aggregate volatility risk in Ang

et al. (2006). The last term, which is the main focus of this paper, measures the aggregate volatility

of volatility risk. The resulting three risk premiums in our market-based model, , , and ,

are related to the three economic risk premiums through a linear transformation.

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The market-based model developed in this paper has several advantages. Financial data

provides useful information because asset prices tell us how market participants value risks.

Moreover, financial data conveys information to investors in a timely fashion. Further, the market-

based risk factors are easier to estimate. For instance, , is easier to estimate than . Also, the

empirical design of our model is compatible with a large literature of multi-factor models for

explaining cross-sectional stock returns (see, for instance, Fama and French, 1993, 2015; Ang et

al., 2006; Maio and Santa-Clara, 2012; Hou, Xue and Zhang, 2015; among others).

3. Estimating the variance of market variance

To test the proposed market-based three-factor model, we need three risk factors – the market

return, the market variance, and the variance of market variance. For the first two factors, we use

the CRSP value-weighted market index and the CBOE VIX index, respectively. The VIX index

has been used as a proxy for the market variance by Ang et al., (2006), Bollerslev, Tauchen, and

Zhou (2009), and Chang, Christoffersen, and Jacobs (2013) among others. We estimate the

variance of market variance by calculating the realized bipower variation from five-minute model-

free implied market variances, using the high-frequency S&P 500 index option data as follows.

First, we extract the model-free implied variance, using the spanning methodology proposed

by Carr and Madan (2001), Bakshi and Madan (2000), Bakshi, Kapadia, and Madan (2003), and

Jiang and Tian (2005). Bakshi, Kapadia, and Madan (2003) show that the price of a -maturity

return variance contract, _

≡ ℚ e , Log , which is the discounted conditional

expectation of the square of market return under the risk-neutral measure, and can be spanned by

a collection of out-of-money call options and out-of-money put options, i.e.,

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_ 2 1 log ⁄;

2 1 log ⁄; ,

(9)

where ; and ; are the prices of European calls and puts at time written on the

underlying stock with strike price and expiration date at τ. The conditional variance of

market return can be calculated by e ,_

, where satisfies the risk-

neutral valuation relationship, which is related to the first four risk-neutral moments of market

returns as described in equation (39) of Bakshi, Kapadia, and Madan (2003).

Next, we use the model-free realized bipower variance, introduced by Barndorff-Nielsen and

Shephard (2004), to estimate the variance of market variance. Define the intraday stock return as

, ≡ log ⁄ log ⁄ , 1, . . . , , where M is number of five minute intervals

per trading day. Barndorff-Nielsen and Shephard (2004) provide two measures of realized

variation; the first one is the realized variance, ∑ , ,, and the second one is

the bipower variance, ∑ , , . Andersen, Bollerslev, and

Diebold (2002) show that the realized variance converges to the integrated variance plus the jump

contributions, i.e. →

∑ , , where is the number of

return jumps in day t+1 and , is the squared jump size. Moreover, Barndorff-Nielsen and

Shephard (2004) show that →

. In other words, the bipower

variance provides a consistent estimator of the realized variance but solely for the diffusion part.

Our measure for the variance of market variance is estimated from five-minute model-free

implied variances. The intraday model-free implied variances, ⁄ , 1, . . . , , are

calculated using equation (9). Since the process of market variance is a (semi-)martingale, we

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apply the bipower variance formula on the changes in annualized model-free implied variances

and obtain a measure for the variance of market variance:

Δ2 1 , , (10)

where Δ , ≡ ⁄ ⁄ . Similarly, we estimate diffusion part of

the covariance between the market return and the market variance, CVRV, using the bipower

covariation formula of Barndorff-Nielsen and Shephard (2008):

14

Δ Δ (11)

Based on these measures, our empirical results will not be affected by volatility jumps (or the

return jumps embedded in the volatility).

4. Data and descriptive statistics

4.1. Data

To calculate implied volatility, we use the tick-by-tick quote data for the S&P 500 index (SPX)

options from CBOE’s Market Data Report (MDR) tapes over the time period from January 1996

to December 2015. The underlying SPX prices are also available on the tapes. Daily data for equity

options and S&P 500 index options are obtained from OptionMetrics. We use the Zero Curve file,

which contains the zero-coupon interest rate curve and the Index Dividend file, which contains the

dividend yield, from OptionMetrics. Daily and monthly stock return data are from the CRSP while

intraday transactions data are from TAQ. Financial statement data are from the COMPUSTAT.

Fama and French (1993) factors including their momentum factor, UMD, are obtained from the

15  

online data library of Professor Ken French.7 The VIX index is obtained from the CBOE website.8

While we use the ‘new’ VIX index to calculate the market variance risk premium as proposed by

Bollerslev, Tauchen, and Zhou (2009), we also use the ‘old’ VIX, which is based on the S&P 100

options and Black–Scholes implied volatilities, as our volatility factor, following Ang et al. (2006).

We use index option prices from the Option Price file to replicate the market skewness factor and

the market kurtosis factor of Chang, Kristoffersen, and Jacobs (2013).

We follow the literature (see Jiang and Tian 2005; Chang, Kristoffersen, and Jacobs 2013;

among others) to filter out index option prices that violate arbitrage bounds. We eliminate all

observations for which the ask price is lower than the bid price, the bid price is equal to zero, or

the average of the bid and ask price is less than 3/8. We also eliminate in-the-money options (e.g.

put options with K/S>1.03 and call options with K/S<1.03) because prior studies suggest that they

are less liquid. We use the daily SPX low and high prices, downloaded from Yahoo Finance,9 to

filter out the MDR data that are outside the [low, high] interval.

To compute the market volatility-of-volatility, we first partition the tick-by-tick S&P500 index

options data into five-minute intervals. For each maturity within each interval, we linearly

interpolate implied volatilities for a fine grid of one thousand moneyness levels (K/S) between

0.01% and 300%. For moneyness levels below or above the available moneyness level in the

market, we use the implied volatility of the lowest or highest available strike price. The model-

free implied variance is estimated from (9), using out-of-money call and out-of-money put prices.

Linearly interpolated maturities are used to obtain the estimate at a fixed 30-day horizon. Each day,

the market volatility-of-volatility (VOV) is calculated by using the bipower variance formula of

equation (10) with the 81 within-day five-minute annualized 30-day model-free implied variance

                                                       7 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ 8 http://www.cboe.com/micro/vix/historical.aspx 9 http://finance.yahoo.com/q/hp?s=^GSPC+Historical+Prices

16  

estimates covering the normal CBOE trading hours from 8:30 a.m. to 3:15 p.m. Central Time. We

estimate the daily bipower covariance, CVRV, between the market return and the market variance

using the intraday five-minute logarithmic return multiplied by 22 and the five-minute implied

variance for S&P 500 index.

Following Bollerslev, Tauchen, and Zhou (2009), we define the market variance risk premium

( , ) as the difference between the ex-ante implied variance ( , ) and the ex-post realized

variance ( , ), i.e. , ≡ , , . We focus on the 30-day variance risk premium.

Market implied variance ( , ) is the squared ‘new’ VIX index divided by 12. The sum of the

SPX five-minute squared logarithmic returns across the past 22 trading days is used to calculate

the one-month market realized variance ( , ).

To implement our empirical model, we construct innovations in market moments. Following

Ang et al. (2006), the innovation in market volatility (ΔVIX) is its first order difference, i.e.

. Chang, Kristoffersen, and Jacobs (2013) indicate that while the first

difference is appropriate for VIX, an ARMA(1,1) model is needed to remove the time-series

dependence in the skewness and kurtosis factors. Following their approach, the innovations in

market volatility-of-volatility (ΔVOV) is computed as the ARMA(1,1) model residual of the market

volatility-of-volatility. Similarly, an ARMA(1,1) model is used to obtain the innovations in the

bipower covariance, ΔCVRV.   

4.2. Descriptive statistics

Figure 1 plots the daily S&P 500 logarithmic return (rSPX) and changes in market volatility

(ΔVIX) over the time period from January 1996 to December 2015. There are clear spikes on the

graph—the Asian financial crisis in 1997, the LTCM crisis in 1998, September 11, 2001, the

WorldCom and Enron bankruptcies in 2001 and 2002, the subprime loan crisis in 2007, the recent

17  

financial crisis in 2008, the flash crash in 2010, the battle over the fiscal cliff in 2011 and 2013,

and the recent sovereign debt crisis in Europe.

Table 1 reports descriptive statistics for the daily factors used in this paper, including the four

factors (MKT, SMB, HML, and UMD) of Fama and French (1993) and Carhart (1997), the market

variance risk premium (VRP), the volatility index (VIX; Ang et al., 2006), our measure of variance

of market variance (VOV), the market skewness factor (SKEW) and market kurtosis factor (KURT)

of Chang, Kristoffersen, and Jacobs (2013), the zero beta straddle factor of Coval and Shumway

(2001), and the options return volatility factor (VOL) and the options return jump factor (JUMP)

of Cremers, Halling, and Weinbaum (2015).

In our sample, the mean of 30-day market variance risk premium (VRP) is 0.155%, which is

slightly smaller than 0.183% in Bollerslev, Tauchen, and Zhou (2009) sample. The mean VOV is

0.047%, and its standard deviation is 0.497%. The mean SKEW is -1.864 and the mean KURT is

10.971, suggesting that the risk-neutral distribution of the market return is asymmetric and has fat

tails. The mean delta-neutral straddle return (STR) is -0.346%; the mean delta-neutral, gamma-

neutral, vega positive straddle return (VOL) is -0.039%; the mean delta-neutral, vega-neutral,

gamma positive straddle return (JUMP) is -0.209%. These sample estimates are consistent with

the findings of Cremers, Halling, and Weinbaum (2015) that the STR documented by Coval and

Shumway (2001) is in large part due to the jump component (JUMP).

Panel B reports the Spearman correlations between the daily factors, where the non-return

based state variables are measured by their innovations. As expected, MKT is strongly negatively

correlated with ΔVIX (-0.786), but much less with ΔVOV (-0.054). However, the correlation of

MKT and ΔVOV increases to -0.160 on days when the S&P 500 index declines by more than 1.91%,

(the 5th percentile value of daily rSPX), while the correlation of MKT and ΔVIX increases to -0.581.

VRP is positively correlated with ΔVOV (0.140) and JUMP (0.090), consistent with Bollerslev,

18  

Tauchen, and Zhou (2009) and Drechsler and Yaron (2011), respectively. ΔKURT and ΔSKEW are

highly correlated at -0.888, which is comparable to the -0.83 reported by Chang, Kristoffersen,

and Jacobs (2013). Further, STR has a much higher correlation with JUMP (0.929) than that with

VOL (0.263), which is similar to the findings reported by Cremers, Halling, and Weinbaum (2015).

Interestingly, ΔVOV has low correlations with ΔVIX (0.065), ΔSKEW (-0.006), ΔKURT (-0.016),

STR (0.065), VOL (0.055), and JUMP (0.055), which suggests that the pricing power of VOV, if

any, is unlikely to be due to these market moments proposed by previous studies.

While ΔVOV has a low correlation with ΔVIX, VOV determines the market volatility risk.

According Proposition 2, VOV dictates the extent to which the contemporaneous market return and

market volatility covary. Indeed, the results of the following two simple regressions show that the

daily bipower covariance, CVRV, between the market return and the market variance (using the

intraday 5-min logarithmic return multiplied by 22 and the 5-min implied variance for S&P 500

index) is significantly and negatively related to VOV:

0.05310.8

0.45245.656

, 0.293 

0.0020.6

0.39351.42

, 0.344 

The regression results suggest that the correlation between market returns and the market

variance becomes more negative as VOV increases. Figure 2 depicts VOV and -CVRV (i.e., the

negative of CVRV) over time. The two time-series strongly co-move together. Furthermore, the

results of the following two simple regressions show that the market variance risk premium (VRP)

is significantly and positively related to VOV and significantly and negatively related to CVRV.

0.17317.3

0.145 4.33

, 0.069

0.16316.5

0.277 6.48

, 0.146

19  

The results are consistent with our model in which an increase in VOV makes the market more

uncertain, and leads investors to demand a higher risk premium, resulting in a lower stock price

and a negative contemporaneous market return. If VOV is indeed a determinant of the equity risk

premium, the lower stock price now paves the way for a higher future return, which means that

future stock returns are predictable by VOV. We examine this prediction in the next subsection.

4.3. VOV’s predictability of future market returns

Table 2 reports the results of the one-period return predictability regression of daily S&P 500

logarithmic returns (rSPX) in excess of the risk-free rate (rf) on the lagged bipower covariance

(CVRV), market volatility (VIX), variance of market variance (VOV), variance risk premium (VRP),

market skewness (SKEW), market kurtosis (KURT), and jump risk (JUMP). We use robust Newey-

West (1987) t-statistics with the optimal lags to account for autocorrelations.

Panel A presents the regressions at a daily frequency. Consistent with Proposition 3, in

columns [1] through [3], we find that VOV positively predicts one-period ahead daily market return

in all of the specifications that include the aforementioned control variables (except for CVRV).

Consistent with Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011), VRP

significantly predicts future stock market returns. Nevertheless, VOV contains information about

future stock market returns beyond that in VRP.

Moreover, column [4] reports that CVRV negatively predicts one-period ahead daily market

returns. That is, future market returns will be higher when the market return and the market

variance are more negatively correlated (i.e., when VOV is higher). Thus, CVRV also impacts future

market returns and this impact is driven by VOV. In column [5], we find that VOV and CVRV

become less significant when both are in the regression model, suggesting that they contain similar

information about future market returns. Since VOV is still marginally significant and CVRV is

20  

insignificant in the regression, it suggests that VOV is the main driving force for the market return

predictability of the covariance of the market return and the market variance.

However, the market variance itself, as measured by VIX, is insignificant in all the

specifications. Thus, while VIX is inversely correlated with the contemporaneous market return, it

contains little information about future market returns.

In Panel B of Table 2, we use the monthly S&P 500 logarithmic returns (rSPX) in excess of

the risk-free rate (rf) as the dependent variable, and the independent variables are sampled at the

end of the previous month. We similarly find that VOV positively predicts one-period ahead

monthly market returns in all of the specifications, except when CVRV is included as one of the

independent variables. Overall, the test results support our proposition that VOV is an important

determinant of the market risk premium, and a state variable that affects the aggregate asset prices.

5. VOV risk in the cross-sectional stock returns

This section examines how VOV risk affects cross-sectional stock returns. Following Ang et

al. (2006), we use daily returns to estimate betas each month t for each firm i listed on the NYSE,

AMEX, and NASDAQ with more than 17 daily observations in a given month.

, , , , , , . (12)

We then use the pre-ranking betas obtained from the above regression in the previous month to

form the test assets. Specifically, we independently sort the pre-ranking betas, , and , ,

each into quintile portfolios to obtain 25 (5x5) test portfolios. After portfolio formation, we

calculate the value-weighted daily and monthly stock returns for each portfolio. Using these 25

portfolio test assets, we test our proposition that, holding other things constant, the expected return

on portfolio p is determined by its three systematic risks implied by our proposed model,

21  

, , , , (13)

where , , , , and , are the post-ranking betas obtained from the following time

series regression estimated over the entire sample period,

, , , , , , . (14)

In addition, since our theoretical model (see Proposition 4) results in factor loadings that are related

to time-varying volatility and time-varying volatility-of-volatility, we follow Avramov and

Chordia (2006) to estimate a conditional beta model as follows,

, , , , , , , ,

, , , , ,

, , , , , ,

(15)

Thus, the expected return on portfolio p is determined by

, , , , , , ,

, , , , , . (16)

The above specification facilitates the identification of two time-varying systematic risks,

, , , , , , , and , , , , , ,

which not only nest the constant betas of , and , examined by Ang et al. (2006) but

also incorporate the time-varying betas of , , , , , , and , , implied by our

theoretical model.

5.1.Portfolios results

We first use the , and , sorted portfolios to test how portfolio returns vary with

their post-ranking VIX betas and VOV betas and report the results in Table 3. Specifically, we sort

stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile

5), and also independently sort stocks into quintile portfolios based on , . For each of the

quintile portfolios sorted on , , we average across the five , portfolios that intersect

22  

with the , portfolio, resulting in , quintile portfolios controlling for , . A similar

approach yields , quintile portfolios controlling for , .

We expect the portfolio returns to decrease with VOV beta for the following reason.

Proposition 2 shows that the market return decreases with VOV and so a stock whose return

increases with VOV i.e., a positive , will provide a hedge against the VOV risk, thus earning

a lower return than a stock with a negative , . Panel A of Table 3 reports that controlling for

, , the quintile 5 stocks (whose returns co-move more positively with VOV beta) have lower

average returns than the quintile 1 stocks with a negative VOV beta, by 72 basis points per month

(t-statistic = -2.25). Controlling for Fama-French (1993) and Carhart (1997) four factor model, the

“5-1” long-short portfolio yields a significant alpha of -0.86 percent per month with a t-statistic of

-2.79. The test results suggest that VOV risk is priced independent of volatility risk and the widely

used factors –market, size, B/M, and momentum factors.

The post-ranking beta, , , lines up with the pre-ranking beta, , and increases from

quintile 1 to the quintile 5 portfolio. The , -sorted “5-1” long-short portfolio has a significant

positive post-ranking , of 0.18 (t=2.26), an insignificant , of 0.01 (t=0.23) and an

insignificant , of 0.01 (t=0.23). Furthermore, it is clear that the post ranking betas are time-

varying. Consistent with Proposition 4, we find that , varies with the market variance and

the VOV and , varies with VOV.10 In terms of the firm characteristics, the quintile 5 firms

are the larger more liquid firms and have lower past month returns as compared to the quintile 1

firms.

Controlling for , , we also find that the return difference between stocks with high

volatility risk and stocks with low volatility risk is significantly negative, at -0.42 percent per

                                                       10  Proposition 4 actually shows that , varies with the economic volatility and the economic volatility of volatility but these are in turn related to market volatility and VOV.

23  

month with a t-statistic of -1.77. The , -sorted “5-1” long-short portfolio has significant

positive post-ranking , of 0.10 (t=3.97) as well as significant , of 0.24 (t=4.49) and

significant , of 0.03 (t=1.89). The negative impact of , on the cross-section of stock

returns is also consistent with our model and with the results in Ang et al. (2006). Proposition 2

shows that the market return and variance are negatively correlated. Thus, a stock whose return is

positively correlated with the market variance will provide a hedge and earn negative returns.

Note that the , -sorted portfolios are accompanied by dispersed post-ranking factor

loadings with respect to not only the VIX factor but also the MKT and the VOV factors. Further,

unlike in the case of the , -sorted portfolios, the betas are no longer time-varying. In terms of

the firm characteristics, the quintile 5 firms are smaller than the quintile 1 firms.

5.2. Controlling for firm characteristics, high moment risks, and jump risk

We now check whether our results are robust to other competing systematic risks and also

firm characteristics. Table 4 presents results on , -sorted portfolios, controlling for market

capitalization (Size), book-to-market ratio (B/M), past 11-month returns (RET_2_12), past one-

month return (RET_1) and Amihud’s illiquidity (ILLIQ). We also control for other competing

systematic risks including the Chang, Christoffersen, and Jacobs (2013) market skewness risk

( , ) and market kurtosis risk ( , ), the Coval and Shumway (2001) volatility risk with

respect to the zero beta straddle factor ( , ), the Cremers, Halling, and Weinbaum (2015)

volatility risk ( , ) and jump risk ( , ), and the Kelly and Jiang (2014) return tail risk

( , ). As before, we independently sort stocks into quintile portfolios based on , and on

the other factors or characteristics. We then average the returns across the , portfolios.

Panel A of Table 4 presents the post-ranking portfolio alphas for , quintile portfolios

controlling for the firm characteristics and other potential sources of systematic risk. The Fama-

24  

French (1993) and Carhart (1997) four factor alpha of the “5-1” long–short portfolio remains

significant at -0.40 percent (t=-1.94) after controlling for Size; at -0.68 percent (t=-2.48) after

controlling for B/M; at -0.54 percent (t= -2.08) after controlling for RET_2_12; at -0.61 percent

(t=-2.16) after controlling for RET_1; and at -0.42 percent (t=-1.98) after controlling for ILLIQ.

Thus, the lower returns on higher , stocks are not driven by the well-known firm

characteristics that impact the cross-section of stock returns.

Panel A further shows that the four-factor alpha of the “5-1” long–short portfolio sorted by

, also remains significant after controlling for the competing systematic risks. Specifically,

the alpha is -0.63 percent (t=-2.34) after controlling for , ; -0.58 percent (t=-2.14) after

controlling for , ; -0.67 percent (t=-2.35) after controlling for , ; -0.71 percent (t=-2.40)

after controlling for , ; -0.78 percent (t=-2.75) after controlling for , ; and -0.70 percent

(t=-2.37) after controlling for , . Again, the results imply that the lower returns on higher

, stocks are not likely due to the existing high-moment risks, jump risk, or tail risk.

Table 4 also presents results for the quintile portfolios sorted on each of the competing

systematic risks. Panel C controls for , , while Panel B does not. Panel B shows that, in our

sample period, only , and , are significantly priced, with their four factor “5-1”

alphas at -0.60 (t=-2.50) and at -0.55 (t=-1.65), respectively. Controlling for , , as shown in

Panel C, the four factor “5-1” alpha for , remains significantly negative at -0.45 (t = -2.13),

but the four factor “5-1” alpha for , is no longer significant. This indicates that a substantial

part of the jump risk premium is due to VOV risk.

In summary, the return differential associated with , cannot be explained by either firm

characteristics or other competing systematic risks. On the other hand, while , and ,

are the only two other competing systematic risks that carry significant return differentials in our

sample period, and moreover , subsumes the pricing effect of , .

25  

5.3. Cross-Sectional Regressions

We now run Fama-MacBeth (1973) regressions to estimate the risk premium on VOV risk

while controlling for the other risk factors including the Fama-French and Carhart factors (SMB,

HML, and UMD), the Chang, Christoffersen, and Jacobs (2013) market skewness factor (ΔSKEW)

and market kurtosis factor (ΔKURT), and the Cremers, Halling, and Weinbaum (2015) jump factor

(JUMP). Our main set of test assets are the 25 portfolios formed on intersection of , and

, quintile portfolios. Since , and , are the only two other competing

systematic risks that carry significant return differentials in our sample period as shown in Table

4, we also form 125 (5x5x5) portfolios by sorting on , , β , , and , and yet another

set of 25 portfolios by sorting on , and , . For each portfolio, we use the post-

formation daily value-weighted returns to obtain the post-formation factor loadings. We then run

the cross-sectional regressions to estimate the premiums. Robust Newey and West (1987) standard

errors with six lags that account for autocorrelations are used.

Panel A of Table 5 presents the constant beta results while Panel B presents results for the

time-varying betas. In column [1] of Panel A, the estimated price for VOV beta, , is -3.35 (t

= -2.51), implying that investors are willing to accept a lower return to the tune of 3.35% on their

investments for a unit increase in VOV beta. To apply this price of VOV risk to the “5-1” hedge

portfolio in Table 3, a VOV beta difference of 0.18 between quintile portfolio 5 and quintile

portfolio 1 contributes 0.60 percent lower (i.e., -3.35×0.18=-0.60) per month to the hedge

portfolio’s expected return, which accounts for about 83% of the hedge portfolio’s return of -

0.72% per month. Column [2] shows that adding other risks to the regressions does not materially

change the risk premium on VOV. In fact, none of the other risk factors are significant. In column

[3] where we use the 125 portfolios independently sorted on , , , , and , , we

26  

find that the estimate of is significantly negative at -1.60 with a t-statistic of -6.51, but the

estimates of and are insignificant with t-statistics of 0.30 and -0.54, respectively.

Further, in column [4] where we use the 25 portfolios independently sorted on , and , ,

we similarly find that the estimate of remains significantly negative at -2.45 with a t-statistic

of -1.98, but the estimate of is insignificant with a t-statistic of 0.77. Thus, VOV risk is

robust in the presence of other risk factors and with different test assets.

In contrast, while the estimated price of VIX risk, , is significant at -7.09 (t =-1.84) in

column [1], it becomes insignificant when Fama-French-Carhart factors and other higher moment

factors are included as shown in columns [2], [3], and [4]. This suggests that VOV risk has stronger

pricing power than VIX risk.

In Panel B, we estimate the time-varying beta model of equation (15), using VIX and the

VOV as conditioning variables. The risk premium estimates are obtained for the expected return in

equation (16), which gives the estimates of premiums, , , , , , , , , and

. As with the constant beta results in Panel A, we find that is significant across all

different combinations of risk factors. In addition, as reported in column [6], , , , , and

, are significant suggesting that betas are indeed time-varying as suggested by Proposition

4. Even with all the other factors in columns [8], , , and , are still marginally

significant consistent with the time-varying factor loadings as per our theoretical model.

Overall, VOV risk is a surprisingly robust pricing factor!

5.4. Firm-level Fama-MacBeth regressions

Thus far, we have used various portfolio approaches to show that VOV risk is priced. In this

section, we further examine whether the pricing of VOV risk is robust to the firm-level analysis.

Using individual stocks as test assets may avoid potentially spurious results that could arise when

27  

the test portfolios exhibit a factor structure (Lewellen, Nagel, and Shanken, 2010). Furthermore,

Lo and MacKinlay (1990) argue that portfolio sorts could lead to data snooping biases. In addition,

a stock-level analysis allows us to control for important firm characteristics.

Specifically, we run the following cross-sectional regression:

, , , , , , , , , , , ,

, , , , , , , ,

, , , , (17)

where the dependent variable is the monthly individual excess stock return. We use the 25

portfolios formed on intersection of , quintile portfolios and , quintile portfolios and,

following the methodology of Fama and French (1992), we assign each of the 25 portfolio-level

post-ranking beta estimates to the individual stocks within the portfolio at that time. Small stocks

ranked in the first decile based on NYSE breakpoints are excluded from the sample. For stock i

that belongs to portfolio p at time t, , , , , , , , , , , , , , , , , , , and

, , are the post-ranking betas from the time-series regression (15). Thus, not only do we

estimate time-varying betas, the individual stock betas can also vary over time because the

portfolio compositions can change each month. We also report the time-constant betas results using

the post-ranking betas , , , , , , and , , from the time-series regression (14).

One again, we use Newey and West (1987) standard errors with six lags.

, denotes a set of firm characteristic variables that consist of Size, B/M, RET_1,

RET_2_12, and ILLIQ. We also check whether our results are robust to existing well-known

volatility spreads that affect cross-sectional stock returns. We construct the implied-realized

volatility spread (IVOL-TVOL), which, as described in Bali and Hovakimian (2009), is defined as

the average of implied volatilities of at-the-money call and put options minus the total volatility

calculated using daily returns in the previous month; the call-put implied volatility spread (CIVOL-

28  

PIVOL), which, as described in Bali and Hovakimian (2009) and Yan (2011), is defined as the at-

the-money call option implied volatility minus the at-the-money put option implied volatility.

Since we extract the volatility data from OptionMetrics Volatility Surface file as in Yan (2011), we

choose the 30-day maturity put and call options with deltas equal to -0.5 and 0.5, respectively.

Thus, , denotes a set of volatility characteristic variables that include IVOL-

TVOL, and CIVOL-PIVOL.

Table 6 presents the results from the firm-level Fama-MacBeth regressions. Panel A presents

the constant beta results while Panel B presents the conditional beta results. Column [1] of Panel

A shows that while is insignificant, is significantly negative at -3.378 (t=-5.10) and

the risk premium of VOV at the firm level is close to that estimated from the portfolio approach,

as reported in Table 5. Controlling for IVOL-TVOL and CIVOL-PIVOL, the estimate of

remains significantly negative at -3.10, with a t-statistic of -4.19, as shown in column [2]. In Panel

B, we find that remains significantly negative at -1.11(t=-1.89) controlling for firm

characteristics as shown in column [3] and at -1.24 (t=-1.98) further controlling for volatility

characteristics as shown in column [4]. In addition, consistent with previous findings, , , and

, are significant with t-statistics of 4.08 and -4.40, as reported in column [3], respectively

and 2.60 and -2.87, as reported in column [4], respectively. Thus, once again, consistent with our

theoretical model, we find evidence for time variation in the betas.

The firm characteristics - illiquidity, firm size, the past one month return, the realized

volatility spread, and the call-put implied volatility spread are all significant with signs that are

consistent with the prior literature.

Overall, the firm-level evidence confirms our results that VOV is a priced risk factor.

29  

5.5. Market crash and volatility risk

In contrast to the results in Ang et al. (2006), Tables 5 and 6 show that the risk premium on

the volatility (VIX) betas is essentially zero. To better understand why VOV risk is priced but

volatility risk is not, we investigate the performance of the 25 portfolios independently sorted on

, and , during the normal periods and crash periods. Table 7 presents the Fama–

MacBeth (1973) factor premiums during the market crash period and the normal period for our

time-constant and time-varying beta models. In Panel A, Crash is equal to one if the monthly

market factor (MKT) is below -13.10%, its time-series mean minus three times standard deviation;

and zero otherwise; Normal is defined by 1-Crash. The results are similar when we define Crash

differently in Panels B (MKT is lower than the time-series mean minus twice the standard deviation)

and C (MKT is lower than the 5th percentile value).

During normal times, when we remove the observations on the days with extreme market

downturns, Panel A of Table 7 shows that is significantly positive at 1.97 with a t-statistic

of 1.70, is significantly negative at -7.35 with a t-statistic of -1.88, and is significantly

negative at -3.29 with a t-statistic of -2.47. When we consider the time-varying beta, as shown in

column [2], , , , , , and , are significant, but and become

insignificant.

In contrast, during crash periods with extreme market downturns, as shown in columns [3]

and [4], becomes even more negative while becomes significantly negative and

becomes significantly positive. Thus, while large market declines reverse the risk-return

tradeoff for the market risk and volatility risk, the market turmoil strengthens investors’ demand

for compensation for VOV risk. This implies that volatility risk in asset pricing, as proposed by

30  

Ang et al. (2006), has a negative premium during normal times but when combined with the

positive premium during crash periods, the overall premium is zero in Tables 5 and 6.

5.6. Time-varying volatility risk

In this section, we examine the risk premiums of the time-constant and the time-varying

volatility risk. Table 8 reports the performance of portfolios sorted on the pre-ranking , and

, , using the time-varying beta model of

, , , , , , ,

, , , , , . (18)

We first sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the

highest (quintile 5), and within each quintile we further sort stocks into quintile portfolios based

on , , . Portfolios are rebalanced monthly and are value weighted. In Panel A, each of the

quintile portfolios sorted on , , is then averaged over the five portfolios intersected with the

quintile portfolios sorted on , , resulting in , , quintile portfolios controlling for , .

In Panel B, similar approach yields , quintile portfolios controlling for , , . The results

show that, stocks with high , , (quintile 5) have a lower average return than stocks with more

negative , , (quintile 1) by 0.42 percent per month with a t-statistic of -2.35. Controlling for

Fama-French (1993) and Carhart (1997) four factor model, the “5-1” long-short portfolio yields a

significant alpha of -0.57 percent per month with a t-statistic of -2.87. In contrast, the “5-1” long-

short portfolio sorted on , only produces insignificant differentials in stock returns, at 0.10

percent per month with a t-statistic of 0.54. The results confirm the previous findings that while

the constant volatility risk premium has limited explanatory power, the time-varying volatility risk

(conditional on VOV) is priced in the cross-section of stock returns.

6. Conclusion

31  

This paper extends Bollerslev, Tauchen, and Zhou (2009) to study the asset pricing

implications of the volatility-of-volatility (VOV), and contributes to the literature along two

dimensions. First, we establish that VOV is an important determinant of the market risk premium

by showing that VOV dictates the negative covariance between the contemporaneous market return

and market volatility and that it predicts future market returns.

Second, we develop a market-based three-factor model in which the market (MKT) risk,

market volatility (VIX) risk, and market VOV risk determine the cross-section of asset returns. We

find that VOV risk is robustly priced and that stocks with more negative VOV betas have

significantly higher future stock returns, even after we account for the exposures to the Fama-

French (1993) and Carhart(1997) factors, the market skewness and kurtosis factors, the jump risk

factor, as well as firm characteristics.

While, the unconditional VIX beta and the MKT beta are not priced in our sample, consistent

with the theoretical model, the conditional component of the factor loadings of both, VIX and the

MKT, are priced. All the three risks (MKT, VIX, VOV) in the asset pricing model are significantly

priced during normal times when we remove the observations with large market declines from our

sample. However, during crash periods, the price of VOV risk becomes even more significant while

the premiums on MKT and VIX have the wrong sign. The positive premium on VIX during market

turmoil combined with the negative premium during normal periods, results in an overall VIX

premium that is indistinguishable from zero. The results imply that market turmoil strengthens

investors’ demand for compensation for VOV risk.

32  

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36  

Table 1: Properties of the daily factors

We report summary statistics and Spearman correlations for the daily factors, including the four factors (MKT, SMB, HML, and UMD) of Fama and French (1993) and Carhart (1997), the market variance risk premium (VRP), the VIX index, our measure of variance of market variance (VOV), the bipower covariance (multiplied by 22) between the intraday five-minute logarithmic return and the five-minute implied variance for S&P 500 index (CVRV), the market skewness factor (SKEW) and market kurtosis factor (KURT) of Chang, Christoffersen, and Jacobs (2013), the zero beta straddle factor of Coval and Shumway (2001), and the options return volatility factor (VOL) and the options return jump factor (JUMP) of Cremers, Halling, and Weinbaum (2015). ΔVIX is the first difference of VIX; ΔCVRV, ΔVOV, ΔSKEW, and ΔKURT are the residuals from fitting an ARMA(1,1) regression using VOV, SKEW, and KURT, respectively. The sample period is from January 1996 to December 2015.

Panel A: Summary statistics

MKT SMB HML UMD VRP VIX CVRV VOV SKEW KURT STR VOL JUMP

(%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%)

Mean 0.030 0.007 0.009 0.026 0.155 21.523 -0.074 0.047 -1.864 10.971 -0.346 -0.039 -0.209

Median 0.070 0.030 0.010 0.070 0.130 20.200 -0.016 0.002 -1.781 9.600 -1.320 -0.079 -0.757

Std.Dev. 1.233 0.627 0.647 0.958 0.197 9.175 0.415 0.497 0.641 5.538 6.212 1.528 4.406

Panel B: Spearman correlation

MKT SMB HML UMD VRP ΔVIX ΔCVRV ΔVOV ΔSKEW ΔKURT STR VOL JUMP

MKT 1.000

SMB 0.138 1.000

HML -0.223 -0.112 1.000

UMD -0.032 0.045 -0.135 1.000

VRP -0.229 -0.060 0.016 0.051 1.000

ΔVIX -0.786 -0.053 0.162 0.007 0.186 1.000

ΔCVRV 0.262 0.038 -0.058 0.036 -0.204 -0.239 1.000

ΔVOV -0.054 -0.008 -0.024 0.034 0.140 0.065 -0.208 1.000

ΔSKEW -0.253 -0.046 0.022 0.021 0.073 0.266 -0.041 -0.006 1.000

ΔKURT 0.324 0.044 -0.043 -0.005 -0.093 -0.321 0.070 -0.016 -0.888 1.000

STR -0.243 -0.085 0.033 -0.038 0.111 0.531 -0.140 0.065 0.128 -0.143 1.000

VOL -0.275 -0.094 -0.022 0.000 0.125 0.376 -0.115 0.055 0.098 -0.111 0.263 1.000

JUMP -0.171 -0.062 0.032 -0.035 0.090 0.421 -0.109 0.055 0.101 -0.112 0.929 -0.034 1.000

37  

Table 2: CVRV, VOV, and future stock returns

This table reports the estimates of the one-period return predictability regression using the logarithmic returns on the S&P 500 index (rSPX) in excess of the risk-free rate (rf) on the lagged bipower covariance (multiplied by 22), between the intraday five-minute logarithmic return and the five-minute implied variance for S&P 500 index (CVRV), market volatility (VIX), variance of market variance (VOV), variance risk premium (VRP), market skewness (SKEW), market kurtosis (KURT), and jump risk (JUMP). The dependent variable in Panel A is the daily rSPX-rf multiplied by 22. In Panel B, the dependent variable is the monthly rSPX-rf and the independent variables are sampled at the end of the month. Robust Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1996 to December 2015.

Panel A: Dependent variable= daily rSPX-rf (t)

[1] [2] [3] [4] [5]

Intercept -0.457 (-0.29) -0.729 (-0.46) 0.843 (0.36) 2.807 (0.99) -0.229 (-0.16)

VIX (t-1) 0.024 (0.29) -0.059 (-0.63) -0.062 (-0.61) -0.145 (-1.12) 0.012 (0.16)

VOV (t-1) 5.009 (2.10) 4.770 (2.05) 4.767 (2.05) 4.739 (1.66)

CVRV (t-1) -4.864 (-1.91) -0.674 (-0.28)

VRP (t-1)

13.136 (3.75) 13.307 (3.83) 14.847 (3.72)

SKEW (t-1) 2.076 (1.56) 2.300 (1.74)

KURT (t-1) 0.213 (1.43) 0.201 (1.33)

JUMP (t-1) 0.074 (0.70) 0.064 (0.62)

rSPX-rf (t-1) -0.061 (-3.29) -0.039 (-2.14) -0.037 (-2.05) -0.033 (-1.77) -0.061 (-3.26)

Adj. R2 0.012

0.019 0.019 0.016

0.012

Panel B: Dependent variable= monthly rSPX-rf (t)

[1] [2] [3] [4] [5]

Intercept 0.058 (0.06) -0.087 (-0.12) 0.418 (0.33) 0.825 (0.67) 0.029 (0.04)

VIX (t-1) 0.006 (0.13) -0.022 (-0.70) -0.029 (-0.87) -0.048 (-1.39) 0.008 (0.20)

VOV (t-1) 1.671 (2.73) 1.142 (2.02) 1.141 (1.92) 1.753 (1.52)

CVRV (t-1) -0.478 (-2.25) 0.050 (0.10)

VRP (t-1)

4.747 (4.14) 4.591 (3.83) 5.061 (4.97)

SKEW (t-1) -0.230 (-0.19) -0.098 (-0.08)

KURT (t-1) -0.069 (-0.56) -0.059 (-0.48)

JUMP (t-1) 0.014 (0.29) 0.006 (0.12)

rSPX-rf (t-1) 0.104 (1.39) 0.069 (1.02) 0.080 (1.17) 0.100 (1.35) 0.101 (1.34)

Adj. R2 0.008

0.059 0.050 0.052

0.003

38  

Table 3: Two-way sorted portfolios on , and ,

At the end of each month, we run the following regression for each stock using daily returns: , , , , , , .

We sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile 5), and independently sort stocks into quintile portfolios based on , . Portfolios are rebalanced monthly and are value weighted. In Panel A, each of the quintile portfolios sorted on , is then averaged over the five portfolios intersected with the quintile portfolios sorted on , , resulting in , quintile portfolios controlling for , . In Panel B, similar approach yields , quintile portfolios controlling for , . The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. Using the post-formation daily portfolio returns, we estimate the post-formation factor loadings for each portfolio separately from the following time-series regressions,

, , , , , , ,

, , , , , , , , , , where , , , , , , , , , , , , , , and , , = , . Using the post-formation monthly portfolio returns, we compute the excess returns and the risk-adjusted returns of each portfolio with respect to Fama-French and Carhart four factors (MKT, SMB, HML, and UMD). Robust Newey–West (1987) t-statistics are in parentheses. The sample period is from January 1996 to December 2015.

Portfolios ranking

1 2 3 4 5 5-1 t-stat

Panel A: Performance of , sorted portfolio, controlling for ,

Excess return 0.92 0.71 0.50 0.51 0.20 -0.72 (-2.25)

α-CAPM 0.16 0.10 -0.06 -0.11 -0.58 -0.74 (-2.16)

α-FF3 0.14 0.08 -0.08 -0.12 -0.58 -0.72 (-2.33)

α-FFC4 0.32 0.12 -0.08 -0.12 -0.54 -0.86 (-2.79)

Post-ranking factor loadings

, 1.34 1.09 1.01 1.07 1.35 0.01 (0.23)

, 0.07 0.02 0.00 0.01 0.08 0.01 (0.23)

, -0.03 -0.01 0.02 0.02 0.15 0.18 (2.26)

Post-ranking time-varying factor loadings

, 1.17 0.98 0.96 1.10 1.50 0.32 (4.93)

, , 0.18 0.10 0.14 0.11 -0.06 -0.24 (-1.95)

, , 2.22 1.86 4.43 -0.89 -12.52 -14.74 (-2.00)

, 0.03 0.00 0.00 0.03 0.12 0.09 (3.67)

, , 1.08 0.74 1.94 -0.29 -5.09 -6.17 (-1.90)

, 0.05 0.03 0.00 0.00 0.08 0.03 (2.08)

Pre-formation firm characteristics

Size($b) 1.46 3.39 4.03 3.75 1.81 0.35 (4.01)

B/M 1.08 0.88 0.84 0.82 1.01 -0.07 (-1.38)

RET_2_12 12.46 14.80 15.06 14.61 12.34 -0.12 (-0.10)

RET_1 2.10 1.06 0.64 0.28 1.03 -1.07 (-5.68)

ILLIQ(106) 7.81 3.31 2.96 3.31 7.54 -0.27 (-1.65)

   

39  

Table 3 (continued.)

Portfolios ranking

1 2 3 4 5 5-1 t-stat

Panel B: Performance of , sorted portfolio, controlling for ,

Excess return 0.68 0.65 0.68 0.57 0.26 -0.42 (-1.77)

α-CAPM -0.02 0.07 0.11 -0.08 -0.57 -0.55 (-2.59)

α-FF3 0.00 0.06 0.08 -0.12 -0.60 -0.59 (-2.78)

α-FFC4 0.11 0.12 0.10 -0.11 -0.53 -0.64 (-2.58)

Post-ranking factor loadings

, 1.23 1.01 1.01 1.13 1.47 0.24 (4.49)

, 0.03 -0.01 0.00 0.03 0.13 0.10 (3.97)

, 0.05 0.03 0.00 0.00 0.08 0.03 (1.89)

Post-ranking time varying factor loadings

, 1.27 1.10 0.99 1.08 1.28 0.02 (0.26)

, , 0.26 0.00 0.06 -0.05 0.20 -0.07 (-0.42)

, , -10.89 -5.52 -1.86 7.28 6.08 16.97 (1.53)

, 0.08 0.02 0.00 0.01 0.07 -0.01 (-0.40)

, , -4.73 -2.64 -0.61 3.18 3.17 7.90 (1.64)

, -0.02 -0.01 0.02 0.02 0.15 0.17 (2.21)

Pre-formation firm characteristics

Size($b) 1.87 4.03 4.13 3.07 1.32 -0.55 (-4.33)

B/M 1.03 0.88 0.81 0.83 1.07 0.04 (0.94)

RET_2_12 11.44 15.03 15.47 14.96 12.38 0.94 (0.57)

RET_1 1.30 0.71 0.72 0.80 1.57 0.27 (1.14)

ILLIQ(106) 8.25 3.12 2.60 2.96 8.00 -0.25 (-1.41)

40  

Table 4: Two-way sorted portfolios on , and control variables

This table shows performance of portfolios sorted on , , with controlling market capitalization (Size), book-to-market ratio (B/M), past 11-month returns (RET_2_12), past one-month return (RET_1), and Amihud’s illiquidity (ILLIQ), Chang, Christoffersen, and Jacobs’s (2013) market skewness risk ( , ) and market kurtosis risk ( , ), Coval and Shumway’s (2001) volatility risk with respect to the zero beta straddle factor ( , ), Cremers, Halling, and Weinbaum’s (2015) volatility risk ( , ) and jump risk ( , ), and Kelly and Jiang’s (2014) return tail risk ( , ), respectively. We sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile 5), and independently sort stocks into quintile portfolios based on each control variable. Portfolios are rebalanced monthly and are value weighted. Panel A presents the results for , quintile portfolios controlling for the quintile portfolios sorted on each control variable. The results for the quintile portfolios sorted on each of the competing systematic risks without and with controlling for , quintile portfolios are presented in Panel B and in Panel C, respectively. The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. Using the post-formation monthly portfolio returns, we compute the risk-adjusted returns of each portfolio with respect to Fama-French and Carhart four factors (MKT, SMB, HML, and UMD). Robust Newey–West (1987) t-statistics are in parentheses. The sample period is from January 1996 to December 2015.

Portfolios ranking

Control variables 1 2 3 4 5 5-1 t-stat

Panel A: Performance of , sorted portfolio, controlling for each control variable

Size 0.24 0.29 0.28 0.21 -0.16 -0.40 (-1.94)

B/M 0.42 0.14 0.01 0.03 -0.26 -0.68 (-2.48)

RET_2_12 0.17 0.08 0.12 -0.03 -0.37 -0.54 (-2.08)

RET_1 0.15 0.11 -0.05 0.02 -0.47 -0.61 (-2.16)

ILLIQ 0.10 0.25 0.23 0.13 -0.32 -0.42 (-1.98)

, 0.22 0.20 0.00 0.00 -0.41 -0.63 (-2.34)

, 0.15 0.09 0.06 -0.06 -0.43 -0.58 (-2.14)

, 0.26 0.18 -0.05 -0.03 -0.41 -0.67 (-2.35)

, 0.18 0.15 -0.05 -0.12 -0.53 -0.71 (-2.40)

, 0.26 0.18 -0.07 0.00 -0.52 -0.78 (-2.75)

, 0.39 0.28 0.09 -0.06 -0.31 -0.70 (-2.37)

41  

Table 4 (continued.)

Portfolios ranking

Control variables 1 2 3 4 5 5-1 t-stat

Panel B: Performance of each control variable sorted portfolio

, 0.28 0.01 0.06 -0.08 -0.31 -0.60 (-2.50)

, -0.24 0.02 0.05 0.01 0.04 0.28 (1.11)

, 0.06 0.15 0.15 -0.22 -0.28 -0.34 (-1.01)

, -0.27 -0.03 0.12 -0.02 -0.07 0.20 (0.76)

, 0.13 0.17 0.01 -0.06 -0.41 -0.55 (-1.65)

, -0.15 -0.01 0.17 0.18 0.23 0.38 (1.44)

Panel C: Performance of each control variable sorted portfolio, controlling for ,

, 0.20 0.01 0.08 -0.04 -0.25 -0.45 (-2.13)

, -0.12 -0.08 -0.01 -0.01 0.04 0.16 (0.71)

, 0.10 0.12 0.11 -0.18 -0.19 -0.29 (-0.96)

, -0.23 -0.10 0.06 -0.05 -0.04 0.19 (0.90)

, 0.10 0.13 -0.02 -0.01 -0.35 -0.44 (-1.53)

, -0.15 -0.04 0.15 0.26 0.17 0.32 (1.30)

Table 5: Fama-MacBeth regressions

This table presents the Fama–MacBeth (1973) factor premiums for the volatility-of-volatility factor (ΔVOV), with controlling for the market factor (MKT), the volatility factor (ΔVIX), Fama-French and Carhart factors (SMB, HML, and UMD), Chang, Christoffersen, and Jacobs’s (2013) market skewness factor (ΔSKEW) and market kurtosis factor (ΔKURT), and Cremers, Halling, and Weinbaum’s (2015) jump factor (JUMP). We estimate the first stage return betas using the daily full-sample post-formation value-weighted returns. Then, we regress the cross-sectional monthly portfolio returns on daily return betas from the first stage, using Fama–MacBeth (1973) cross-sectional regression. In Panel A, the testing model is , , , where , , , , and , are the post-ranking betas from the time-series regression of , , , , , , . In Panel B, the testing model is , , , , , , , , , , , , , where the post-ranking betas are from , , , , , , , , , where , , , , , , , , and , , , , , Three sets of test portfolios are considered. In columns [1], [2], and [5] through [8], the test portfolios are the 25 portfolios independently sorted on , and , . Column [3] uses the 125 portfolios independently sorted on , , , , and , . Column [4] uses the 25 portfolios independently sorted on , and , . Portfolios are rebalanced monthly and are value weighted. Robust Newey–West (1987) t-statistics that account for autocorrelations are in parentheses. The sample period is from January 1996 to December 2015. 

Fama-MacBeth cross-sectional regressions

Panel A: Constant beta models Panel B: Time-varying beta models

Test portfolios , × , , × , , × , , × , , × , , × , , × , , × ,

× ,

[1] [2] [3] [4] [5] [6] [7] [8]

Intercept -1.23 (-1.08) 0.30 (0.23) -1.95 (-1.87) 2.42 (1.58) 1.04 (1.39) -0.62 (-0.54) 2.36 (2.19) 0.25 (0.18)

, 1.84 (1.60) 0.33 (0.24) 2.95 (2.59) -1.92 (-1.16) -0.36 (-0.45) 1.22 (1.02) -1.72 (-1.51) 0.34 (0.23)

, -7.09 (-1.84) -4.19 (-0.96) 0.44 (0.22) 5.14 (1.12) 1.31 (0.41) -4.47 (-1.12) 3.06 (0.92) -3.17 (-0.71)

, -3.35 (-2.51) -3.01 (-2.84) -1.60 (-6.51) -2.45 (-1.98) -3.27 (-2.54) -2.72 (-2.59) -3.23 (-2.87) -2.53 (-2.77)

, , 0.90 (1.82) 0.61 (1.24)

, , 0.13 (1.85) 0.14 (1.72)

, , -0.31 (-1.87) -0.32 (-1.73)

, -0.47 (-0.58) -1.36 (-1.74) 1.69 (1.40) -0.34 (-0.54) -0.41 (-0.50)

, -0.14 (-0.26) -0.39 (-0.72) 1.38 (1.54) 0.16 (0.33) -0.05 (-0.08)

, -0.85 (-0.95) -0.19 (-0.16) 0.83 (0.87) -2.29 (-2.13) -0.35 (-0.31)

, 1.79 (1.09) 0.03 (0.30) 2.19 (1.29) 1.71 (1.06) 1.88 (1.12)

, -8.09 (-0.72) -0.15 (-0.54) -16.57 (-1.31) -15.52 (-1.22) -8.56 (-0.75)

, -1.78 (-0.10) 4.24 (0.40) 11.82 (0.77) 13.46 (0.79) -7.12 (-0.37)

Adj. R2 0.70 0.67 0.39 0.61 0.55 0.72 0.63 0.66

 

Table 6: Firm-level Fama-MacBeth regressions

This table reports the results for the firm-level Fama-MacBeth regressions. We run the following cross-sectional regression:

, , , , , , , , , , , , , ,

, , , , , , , , , ,

where the dependent variable is the monthly individual stock returns; We use the 25 portfolios formed on intersection of , quintile portfolios and , quintile portfolios and, following the methodology of Fama and French (1992), we assign each of the 25 portfolio-level post-ranking beta estimates to the individual stocks within the portfolio at that time. Small stocks ranked in the first decile based on NYSE breakpoints are excluded. In Panel A, for stock i that belongs to portfolio p at time t, , , , , , , and , , are the post-ranking betas from the time-series regression of , , , , , , ; In Panel B,

, , , , , , , , , , , , , , , , , , and , , are from regression of , ,

, , , , , , , , , , , .

, consists of Size, B/M, RET_2_12, RET_1, and ILLIQ; , includes the implied-realized volatility spread (IVOL-TVOL) and the call-put implied volatility spread (CIVOL-PIVOL). Robust Newey and West (1987) t-statistics that account for autocorrelations are reported in parentheses. The sample period is from January 1996 to December 2015.

Fama-MacBeth regressions: individual stocks

Panel A: Constant beta models Panel B: Time-varying beta models

[1] [2] [3] [4]

Intercept -1.09 (-0.96) 0.56 (0.44) -0.01 (-0.01) 1.45 (1.15)

Log(Size($b)) -0.27 (-4.95) -0.19 (-3.02) -0.27 (-4.92) -0.19 (-2.98)

Log(B/M) 0.11 (0.96) 0.10 (0.77) 0.11 (0.95) 0.10 (0.78)

RET_2_12 0.34 (0.99) 0.20 (0.56) 0.35 (1.02) 0.21 (0.57)

RET_1 -1.70 (-2.80) -1.52 (-2.12) -1.68 (-2.77) -1.52 (-2.12)

ILLIQ(106) 0.49 (2.46) 2.91 (3.08) 0.50 (2.51) 2.93 (3.09)

, 1.96 (1.65) 0.37 (0.28) 0.93 (0.77) -0.47 (-0.35)

, 0.08 (0.02) 4.25 (0.91) 2.57 (0.66) 5.81 (1.28)

, -3.00 (-4.33) -3.10 (-4.19) -1.11 (-1.89) -1.24 (-1.98)

, , 0.23 (0.50) -0.37 (-0.73)

, , 0.20 (4.08) 0.16 (2.60)

, , -0.49 (-4.40) -0.40 (-2.87)

IVOL-TVOL 0.76 (3.72) 0.76 (3.71)

CIVOL-PIVOL 4.05 (6.63) 4.01 (6.66)

Adj. R2 0.07 0.08 0.07 0.08

No. obs 570,718 381,187 570,718 381,187

44  

Table 7: Market crash and factor premiums

This table presents the Fama–MacBeth (1973) factor premiums during the market crash period and the normal period for the market factor (MKT), the volatility factor (ΔVIX), and the volatility-of-volatility factor (ΔVOV). The test portfolios are the 25 portfolios independently sorted on , and , and the portfolios are rebalanced monthly and value weighted. Crash is equal to one if the monthly market factor (MKT) is below -13.10%, its time-series mean minus three times standard deviation; and zero otherwise; Normal is defined by 1-Crash. Using the post-formation daily portfolio returns, we compute the risk exposures with respect to our constant beta model and time-varying beta model. Then we regress the cross-sectional monthly portfolio returns on betas from the first stage, using Fama–MacBeth (1973) cross-sectional regression. The constant beta model is , ,

, where the post-ranking betas are from the time-series regression of , ,

, , , , . The time-varying beta model is

, , , , , , , , , , , , , where the post-ranking betas are from , , , , , , , , , where

, , , , , , , , and , , , , , . We report the estimates of factor premiums using the Normal periods in columns [1] and [2] and using the Crash periods in columns [3] and [4]. Robust Newey and West (1987) t-statistics that account for autocorrelations are reported in parentheses. The sample period is from January 1996 to December 2015.

Normal periods Crash periods

[1] [2] [3] [4]

Panel A: Crash(t) =1 if MKT(t) <-13.10% (Mean-3*SD of MKT)

Intercept -1.22 (-1.07) -0.66 (-0.56) -2.68 (-6.04) 3.79 (1.16)

, 1.97 (1.70) 1.40 (1.17) -14.02 (-80.10) -20.63 (-5.45)

, -7.35 (-1.88) -4.92 (-1.21) 23.62 (2.74) 48.87 (7.35)

, -3.29 (-2.47) -2.70 (-2.57) -10.43 (-1.96) -5.34 (-4.20)

, , 0.94 (1.87) -3.84 (-4.68)

, , 0.13 (1.83) 0.41 (0.53)

, , -0.31 (-1.84) -0.98 (-0.62)

Adj. R2 0.69 0.70 0.27 0.18

   

45  

Table 7 (continued.)

 

Normal periods Crash periods

[1] [2] [3] [4]

Panel B: Crash(t) =1 if MKT(t)< -8.54% (Mean- 2*SD of MKT)

Intercept -1.59 (-1.36) -1.16 (-0.98) 9.23 (1.70) 15.02 (3.01)

, 2.60 (2.17) 2.16 (1.81) -20.18 (-4.03) -25.98 (-5.80)

, -8.67 (-2.17) -6.14 (-1.54) 38.59 (1.66) 43.80 (1.90)

, -2.53 (-2.39) -2.31 (-2.47) -26.94 (-3.10) -14.41 (-2.16)

, , 1.12 (2.16) -5.57 (-2.29)

, , 0.14 (1.71) 0.04 (0.06)

, , -0.31 (-1.69) -0.40 (-0.30)

Adj. R2 0.54 0.53 0.74 0.76

Panel C: Crash(t) =1 if MKT(t)< -7.88 (5th percentile of MKT)

Intercept -1.37 (-1.20) -1.03 (-0.86) 1.43 (0.22) 7.14 (1.04)

, 2.55 (2.17) 2.19 (1.83) -11.55 (-1.82) -17.27 (-2.56)

, -8.28 (-2.06) -5.91 (-1.43) 15.42 (0.65) 22.88 (1.02)

, -2.41 (-2.28) -2.35 (-2.43) -21.15 (-2.85) -9.71 (-1.73)

, , 1.07 (2.12) -2.35 (-0.92)

, , 0.13 (1.58) 0.20 (0.40)

, , -0.29 (-1.54) -0.74 (-0.71)

Adj. R2 0.53 0.52 0.74 0.79

46  

Table 8: Performance of portfolios formed on , and , ,

At the end of each month, we run the following regression for each stock using daily returns:

, , , , , , ,

, , , , , . We first sort stocks into quintile portfolios based on , , from the lowest (quintile 1) to the highest (quintile 5), and within each quintile we further sort stocks into quintile portfolios based on , , . Portfolios are rebalanced monthly and are value weighted. In Panel A, each of the quintile portfolios sorted on , , is then averaged over the five portfolios intersected with the quintile portfolios sorted on , , resulting in , , quintile portfolios controlling for , . In Panel B, similar approach yields , quintile portfolios controlling for , , . The column “5-1” refers to the hedge portfolio that longs portfolio 5 and shorts portfolio 1. Using the post-formation monthly portfolio returns, we compute the excess returns and the risk-adjusted returns of each portfolio with respect to Fama-French and Carhart four factors (MKT, SMB, HML, and UMD). Robust Newey–West (1987) t-statistics are in parentheses. The sample period is from January 1996 to December 2015.

Portfolios ranking

1 2 3 4 5 5-1 t-stat

Panel A: Performance of , , sorted portfolio, controlling for ,

Excess return 0.68 0.71 0.65 0.49 0.26 -0.42 (-2.35)

α-CAPM -0.03 0.08 0.04 -0.18 -0.56 -0.53 (-3.09)

α-FF3 -0.03 0.07 0.03 -0.18 -0.58 -0.56 (-3.01)

α-FFC4 0.05 0.11 0.06 -0.17 -0.51 -0.57 (-2.87)

Panel B: Performance of , sorted portfolio, controlling for , ,

Excess return 0.39 0.66 0.70 0.56 0.49 0.10 (0.54)

α-CAPM -0.44 0.05 0.16 -0.07 -0.34 0.09 (0.48)

α-FF3 -0.41 0.02 0.14 -0.09 -0.36 0.05 (0.27)

α-FFC4 -0.22 0.04 0.14 -0.13 -0.29 -0.07 (-0.33)

 

   

47  

Figure 1. Daily market returns (rSPX) and market volatility (VIX). We plot daily market returns (rSPX) and changes in market volatility (ΔVIX) over the time period from January 1996 through December 2015.

48  

Figure 2 Daily market covariance between return and variance (CVRV), and market volatility-of-volatility (VOV). We plot daily minus market covariance between return and variance (-CVRV) and market volatility-of-volatility (VOV) over the time period from January 1996 through December 2010. We partition the tick-by-tick S&P500 index options data into five-minute intervals, and then we estimate the model-free implied variance for each interval. For each day, we use the bipower variation formula on the five-minute based annualized 30-day model-free implied variance estimates within the day, resulting in our daily measure of market volatility-of-volatility (VOV). We estimate the covariance between the market return and the market variance (CVRV) using the bipower covariation formula on the five-minute based S&P500 index returns and the five-minute based annualized 30-day model-free implied variance within the day.