Aggregate volatility risk: International evidence

Stanley Peterburgsky

Brooklyn College, 2900 Bedford Ave, New York, NY 11210

E-mail: phinance@hotmail.com

Abstract

Using a procedure analogous to that of Ang et al. (2006), this paper documents that aggregate

volatility risk does not appear to be priced in European financial markets. Specifically, based on

the 2002-2016 period (for which European data is available), the price of aggregate volatility risk

is not statistically different from zero. Additionally, aggregate volatility loadings do not appear

to predict future returns, and alphas from time-series regressions of excess returns on a

long/short aggregate volatility portfolio with respect to the CAPM, the Fama-French 3-factor

model, and the Fama-French 5-factor model are not statistically different from zero. Analysis

based on high-frequency data support these results. Consequently, contrary to what has been

reported in some studies that examine U.S. data, whether aggregate volatility risk is priced is an

open question.

JEL classification: G12, Keywords: aggregate volatility, VSTOXX

This version: 1/14/2018

I thank Geert Bekaert, Robert Hodrick, and seminar participants at Brooklyn College for very

helpful comments. All errors are mine.

2

I. Introduction

One of the main objectives of asset pricing is to uncover the full set of risk factors for

which investors require a premium. Over the last several decades various risk factors have been

proposed. Most of these have been either motivated by, or retroactively explained in the context

of, Merton’s (1973) ICAPM model, which shows that individuals who make lifetime

consumption decisions are subject to not only the market risk of the original CAPM, but also to

additional risk factors that arise due to conditional relationships between stock returns and

unanticipated changes in state variables that affect future returns. Examples include the well-

known SMB (small market capitalization minus big market capitalization) and HML (high book-

to-market ratio minus low book-to-market ratio) factors of Fama and French (1993) as well as

more recently proposed RMW (robust profitability minus weak profitability) and CMA

(conservative investment policy minus aggressive investment policy) factors of Fama and French

(2015). Chen (2002) develops a model of stock returns in which market volatility is time-

varying,1 and demonstrates that stocks that perform poorly when market volatility rises earn a

risk premium. The reason for the risk premium is that investors prefer to hedge against a possible

rise in market volatility, and require higher compensation (even after controlling for market beta)

to hold stocks that do not provide a hedge. Motivated by Chen’s (2002) theoretical framework, a

number of empirical papers examine the relationship between stocks’ sensitivity to changes in

market volatility (also known as aggregate volatility risk, variance risk) and average returns.

Most studies in the option pricing literature find that aggregate volatility risk carries a

negative risk premium (e.g., Bakshi and Kapadia (2003), Arisoy, et al. (2007), Carr and Wu

(2009), Da and Schaumburg (2011)). However, these studies do not control for non-market risk

1 The empirical observation that the stock market volatility is time-varying goes back at least four decades.

See Black (1976), Merton (1980), Christie (1982), Poterba and Summers (1986), and Schwert (1989) for

early studies of stochastic market volatility.

3

factors, such as SMB and HML. Failure to control for known risk factors can cause redundant

factors to appear non-redundant.

Ang et al. (2006) approach the issue of aggregate volatility risk from a different angle.

Using 1986-2000 U.S. stock price data, they document a number of cross-sectional relationships

between sensitivity to changes in aggregate volatility and expected returns. First, they find that

portfolios of stocks sorted on sensitivity to changes in aggregate volatility predict future returns,

with lower aggregate volatility loadings associated with higher future returns. Second, they show

that stocks with lower aggregate volatility loadings generate higher alphas with respect to the

CAPM and the Fama-French 3-factor model than stocks with higher loadings. Finally, using a

Fama and MacBeth (1973) procedure, they document that aggregate volatility risk carries a

negative premium. However, Peterburgsky (2017) finds that these relationships disappeared in

the 2001-2015 period. Consequently, whether aggregate volatility risk is important to investors

after taking into account known risk factors is unclear.

In this paper, I use European stock price data to shed light on issues raised above. I find

that, consistent with Peterburgsky (2017), the relationships between sensitivity to changes in

aggregate volatility and expected returns documented by Ang et al. (2006) are not present in the

European data. Specifically, aggregate volatility betas do not predict future returns. Alphas from

time-series regressions of long/short high-minus-low aggregate volatility beta portfolio returns

with respect to the CAPM, the Fama-French 3-factor model, and the Fama-French 5-factor

model are not statistically different from zero. Finally, the price of aggregate volatility risk is not

statistically different from zero. These findings are supported by results based on high-frequency

intraday data.

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The rest of this paper is organized as follows. In Section II, I discuss the data, explain the

portfolio construction methodology, and report summary statistics. In Section III, I present my

main findings on the (lack of) relationships between sensitivity to changes in aggregate volatility

and expected returns in European markets. In Section IV, I examine the robustness of my results

to alternative specifications of the portfolio construction methodology and measures of change in

aggregate volatility. In Section V, I use high-frequency intraday data to re-examine the pricing of

aggregate volatility risk. Finally, I summarize my research and offer concluding remarks in

Section VI.

II. Data, portfolio construction, and summary statistics

This paper uses data from a number of sources to assess the importance of aggregate

volatility risk for asset pricing in European financial markets. Daily stock returns, prices, number

of shares outstanding (for computing market capitalization) and book equity (for computing

book-to-market ratios) are from Bloomberg.2 The daily VSTOXX index3 level is from

www.stoxx.com.4 European factor returns for SMB, HML, RMW, and CMA, as well as

European risk-free rates, are from Ken French’s web site

(http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). Finally, high-

2 The syntax for Bloomberg formulas is as follows: =BDH(SecurityID&" Equity","CHG_PCT_1D","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y",

"CshAdjNormal=Y", "CURRENCY=EUR")

=BDH(SecurityID&" Equity","PX_LAST","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y",

"CshAdjNormal=Y", "CURRENCY=EUR")

=BDH(SecurityID&" Equity","EQY_SH_OUT","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y",

"CshAdjNormal=Y", "CURRENCY=EUR")

=BDH(SecurityID&"

Equity","TOT_COMMON_EQY","startdate","enddate","DIR=V","DTS=S","DAYS=W","CapChg=Y",

"CshAdjNormal=Y", "CURRENCY=EUR") 3 The VSTOXX index, also known as EURO STOXX 50 Volatility, represents the implied volatility of a

synthetic option contract on the EURO STOXX 50 index with a maturity of one month. 4 STOXX is a subsidiary of Deutsche Börse Group.

5

frequency market volatility data is from Heber, et al. (2009). All prices and returns are in Euros

unless otherwise noted.

The companies used to analyze the importance of aggregate volatility risk are the

members of STOXX Europe 600,5 and account for approximately 88% of the total European

stock market value (Plagge (2017)). Since Bloomberg data for STOXX Europe 600 components

goes back to only 2002, I focus the 2002-2016 period. Panel A of Table 1 shows the distribution

of the number of firms across countries and over time, while Panel B presents a market value

distribution. In the discussion that follows, I restrict my sample to firm-months that have at least

14 trading days.

The VSTOXX index is the European version of the American VIX index. It is the most

widely used measure of expected volatility in European markets. A graph of the index level over

the 2002-2016 period appears in Figure 1. The mean level is 24.76, while the median is 22.31.

As is the case for the VIX, the VSTOXX is highly positively skewed. The most pronounced

spike in the index since its debut in 1999 occurred during the global financial crisis of 2008-

2009.

Analogous to the procedure used in Ang, et al. (2006), I begin by forming portfolios of

stocks by sorting them into quintiles based on the coefficient d from the following regression run

each month,

R(t) – RF(t) = a + b[RM(t)-RF(t)] + dΔVol(t) + e(t) (1)

5 The STOXX Europe 600 Index is a subset of the STOXX Global 1800 Index. With 600 component

stocks, the index represents (mostly) large capitalization companies across 17 countries of the European

region: Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Ireland, Italy,

Luxembourg, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and the United Kingdom. In

prior years, the index included Greek and Icelandic companies.

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where R(t) is the daily return on the stock, RF(t) is the daily risk-free rate, RM(t) is the daily return

on the STOXX Europe 600 index (henceforth, the market portfolio), ΔVol(t) is the daily change

in the VSTOXX index, and a, b, and d are regression coefficients. Portfolio 1 is the quintile with

the lowest loadings on the change in VSTOXX, while portfolio 5 is the quintile with the highest

loadings. Table 2 reports summary statistics on firms in each of the five portfolios. The stocks in

the 5th quintile were somewhat smaller than those in the other quintiles, while the median book-

to-market ratio decreases monotonically from quintile 1 to quintile 5 (although the mean does

not). The table also shows that, as in Ang et al. (2006), loadings on the change in VSTOXX are

highly unstable, with an average of more than 70% of the firms transitioning from one quintile to

another each month.

III. Results

Ang et al. (2006) find, based on a 1986-2000 U.S. sample, that stocks with low

sensitivities to changes in aggregate volatility in one month have high average returns in the

subsequent month. However, Peterburgsky (2017) documents that this relationship disappeared

during the 2001-2015 period. I follow Ang et al.’s procedure to investigate whether low

sensitivity to changes in aggregate volatility predicts high average returns in the European

sample. I begin by sorting stocks into five portfolios based on the coefficient d from regression

(1). Table 3 presents the equally- and value-weighted mean monthly total returns on the five

portfolios in the subsequent month. The table shows that the spread between the returns on the

5th and the 1st portfolios was not statistically significant (and had the “wrong” sign) during 2002-

2016 (t = 1.30 for equally-weighted and t = 1.17 for value-weighted). The table also illustrates

that (by construction) the average pre-formation loadings on the change in aggregate volatility

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are monotonically increasing from portfolio 1 to portfolio 5. A similar pattern emerges in the

post-formation loadings, although the variance is much smaller (as should be expected).

Ang et al. (2006) also report that the alpha from a CAPM time-series regression for the

long/short (high sensitivity to changes in aggregate volatility minus low sensitivity to changes in

aggregate volatility) zero-cost portfolio is negative and statistically significant, as is the alpha

from a Fama-French 3-factor time-series regression. However, Peterburgsky (2017) documents

that in the more recent data, the alpha for the long/short portfolio is not significant in either

specification. Consistent with Peterburgsky (2017), I find that the CAPM, Fama-French 3-factor,

as well as Fama-French 5-factor long/short portfolio alphas are all insignificant (and have the

“wrong” sign) in the European sample. Regression results appear in the middle columns of Table

3 (CAPM t = 1.29, three-factor t = 1.08, and five-factor t = 0.22).

One of Ang et al.’s (2006) most important contributions to the asset pricing literature is

their analysis of the price of aggregate volatility risk. They estimate the price of risk based on the

1986-2000 U.S. sample by running Fama-MacBeth regressions,6 and find that the risk premium

was negative and statistically significant. However, here too Peterburgsky (2017) documents that

this result no longer holds when more recent U.S. data is examined. My analysis reveals that

aggregate volatility risk is not priced in European markets. A battery of robustness tests confirms

this conclusion.

For the main specification, I follow a procedure similar to that of Ang et al. in

constructing 25 portfolios by sorting stocks on their market betas and on their sensitivity to

changes in aggregate volatility. Specifically, I first sort the members of STOXX Europe 600 into

6 The Fama-MacBeth procedure requires a set of time-series regression to estimate the sensitivities of test

assets to various factors in stage one, and a set of cross-sectional regressions to estimate the period-by-

period prices of risks in stage two. After the risk premia have been estimated, a t-test can be conducted to

determine whether each factor is priced.

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one of 5 groups based on market betas from equation (1). Next, within each group, I sort the

stocks into one of 5 portfolios based on sensitivity to changes in aggregate volatility. The 25 b×d

portfolios serve as test assets in the Fama-MacBeth procedure. In stage one of the procedure, the

following time-series regression of monthly portfolio excess returns on factor returns gives the

factor loadings for each portfolio:

R(t) – RF(t) = β0 + β’f + e(t) (2)

where β’ is a vector of factor loadings and f is a vector of factor returns. In stage two, the

following cross-sectional regressions yield month-by-month risk premia:

R(t) – RF(t) = λ0 + λ’β + e(t) (3)

where λ’ is a vector of risk premia and β is a vector of factor loadings from stage one. Finally, a

t-test determines whether the risk premia are statistically significant.

Table 4 presents the results from six Fama-MacBeth regressions. In the first model, the

only factor is assumed to be the excess return on the market portfolio. In the second model, ΔVol

is added as an explanatory variable for asset excess returns. Models III and IV further add the

Fama-French European SMB and HML factors, without and including the ΔVol, respectively.

Models V and VI additionally add the Fama-French European RMW and CMA factors, without

and including the ΔVol, respectively. The table indicates that, based on the 2002-2016 European

sample, aggregate volatility was not a priced factor in any of the three specifications that include

the ΔVol variable: the CAPM plus volatility specification (labeled “II”) yields t = 0.36 on ΔVol,

the three-factor plus volatility specification (labeled “IV”) yields t = 0.65, and the five-factor

plus volatility specification (labeled “VI”) yields t = 0.72.

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Estimating the price of risk of any factor is notoriously difficult. Ang et al.’s (2006) point

out that their regressions suggest a negative price of risk for both the SMB and the HML U.S.-

based factors, which contradicts the results of most studies that examine much longer periods.

This illustrates that 15 years of data is often insufficient to accurately estimate average risk

prices. Therefore, it should not be surprising that questions related to the pricing of aggregate

volatility risk are difficult to answer as well.

IV. Robusness of results

To determine whether my findings on factor risk premia are robust to alternative

specifications, in this section I repeat the Fama-MacBeth analysis five times after making one of

the following modifications: (1) using the unanticipated change in VSTOXX from an

AR(1)/GARCH(1, 1)/EGARCH(1, 1) model (both in daily and monthly regressions) rather than

the actual change, (2) excluding the global financial crisis period, (3) truncating/winsorizing the

sample, (4) using an alternative parameter estimation window, and (5) using alternative test

assets.

IV.A Unanticipated change in VSTOXX

Since stock market volatility is mean-reverting, a portion of the daily or monthly change

in the VSTOXX can be predicted using volatility forecasting models such as AR(1), Bollerslev’s

(1986) and Taylor’s (1987) GARCH, or Nelson’s (1991) EGARCH. Therefore, only the

remaining, unpredictable portion could theoretically be priced. In this subsection, I use the

unanticipated change in the VSTOXX index instead of actual change, both in daily (pre-

formation) and monthly (Fama-MacBeth) regressions. The unanticipated change in VSTOXX is

10

defined as the difference between the actual change and the predicted change from an AR(1)

model. The model for daily regressions is,

R(t) – RF(t) = a + b[RM(t)-RF(t)] + dUΔVol (t) + e(t) (4)

where UΔVol is the daily unanticipated change in VSTOXX, and the other variables are as

defined previously. The correlation between the unanticipated change in VSTOXX and actual

change is .99 at daily frequency, and .96 at monthly frequency. Table 5 indicates that the price of

risk of unanticipated change in aggregate volatility during the sample period was not statistically

significant, and had the “wrong” sign for all specifications (CAPM plus volatility t = 0.70, three-

factor plus volatility t = 0.83, and five-factor plus volatility t = 0.84). Results from GARCH(1, 1)

and EGARCH(1, 1) methodologies are similar.

IV.B Excluding global financial crisis period

One could argue that the 2002-2016 period was unusually volatile. If so, aggregate

volatility risk could still be priced (have a negative risk premium), since stocks with low loadings

would only be expected to produce high returns on average, to compensate investors for low

returns in volatile periods.7 If the 2002-2016 period was indeed unusually volatile, the apparent

lack of relationship between aggregate volatility loadings and returns would be fully consistent

with a risk-based story.8

I believe the 2002-2016 period was not unusually volatile. While it is true that it featured

the global financial crisis during which the VSTOXX index topped 80, the fifteen-year period as

a whole was likely not particularly risky, at least not compared to the preceding fifteen-year

7 This is the same argument as for any other risk factor in a single-factor or multifactor risk model. For

example, in the CAPM (which carries a positive risk premium), stocks with high market betas are only

expected to produce high returns on average, not when the market experiences a sharp decline. 8 I thank an anonymous referee for a related paper for proposing this line of reasoning.

11

period. Although the VSTOXX data doesn’t go back much further than my sample period, the

VIX (a U.S. analogue) had a more extreme spike during the global stock markets crash of

October 1987, reaching a record of 150, than at any time during the 2008-2009 financial crisis.

Since the VSTOXX and the VIX are highly correlated, it stands to reason that Europe also

experienced a more volatile environment during the fifteen year period preceding 2002-2016.

Nevertheless, to address the concern that 2002-2016 was an unusually volatile time for European

financial markets, in this subsection I exclude the global financial crisis period. For the purposes

of this analysis, I define the beginning of the financial crisis period as September 15th, 2008, the

date on which Lehman Brothers filed for bankruptcy, and the end of the financial crisis period as

October 20th, 2009, the date on which the VSTOXX closed below the sample mean of 24.76 for

the first time in more than a year. The choice of the exact start and end dates do not affect the

results (e.g., January 2008 – December 2009).

Here, the first-stage Fama-McBeth regressions use the entire 2002-2016 period, while the

cross-sectional regressions are omitted for the 14 month from September 2008 through October

2009. Table 6 indicates that the price of aggregate volatility risk when the financial crisis period

is removed from the sample was not statistically significant for all specifications (CAPM plus

volatility t = -0.18, three-factor plus volatility t = -0.11, and five-factor plus volatility t = -0.07).

IV.C Truncating the sample

The base case estimation is highly conditional, allowing for the possibility that factor

loadings change very rapidly (i.e., from one month to the next). This flexibility is advantageous.

On the other hand, since very few observations are used to estimate the betas, there is potential

for estimated (sample) betas to not properly reflect true (population) betas. Furthermore, there is

12

a legitimate concern that firm-month observations with the most noisy betas will drive the

results. In order to test whether outliers are indeed impacting the base case findings, I repeat the

Fama-MacBeth analysis using a truncated sample. Specifically, I remove the top 10 and bottom

10 companies by market beta month-by-month. From the remaining 580 companies, I remove

the top 15 and bottom 15 by aggregate volatility beta. Using the remaining 550 companies, I

proceed as in the base case analysis.9 Table 7 indicates that the price of aggregate volatility risk

based on the truncated sample was not statistically significant during 2002-2016, and had the

“wrong” sign for all specifications (CAPM plus volatility t = 0.19, three-factor plus volatility t =

0.26, and five-factor plus volatility t = 0.12). Similar results are obtained for a winsorized, rather

than truncated, sample.

IV.D Alternative parameter estimation window

An alternative way to address the concern that one month is too brief an estimation

window for producing reliable factor loadings is to simply use a longer window. In this

subsection, I use a 3-month estimation period instead. Panel A of Table 8 hints that the longer

estimation window produces more accurate factor loadings. For example, firms in the extreme-

low quintile during months 1-3 are 58.5% likely to remain in the same quintile during months 2-

4, while the extreme-high quintile firms have a 57.5% probability of remaining in their original

quintile. By comparison, the analogous figures for the 1-month estimation window are only

25.9% and 24.3%, respectively.10 Panel B shows the frequency of the two approaches agreeing

9 In the base case, the 600 firms are divided evenly among 25 portfolios, resulting in 24 firms per portfolio.

In the truncated sample, the 550 firms are likewise divided evenly among 25 portfolios, resulting in 22

firms per portfolio. 10 Although we would expect a lower likelihood of switching quintiles when using three month of data, of

which two overlap, nevertheless, the magnitude of the increase in factor loading stability is noteworthy. For

example, the 58.5% and 57.5% mentioned in the text are 38.5% and 37.5% higher than the 20% we would

13

as well as disagreeing as to which quintile a given firm should be assigned to. The results

indicate that the two approaches agree between 27.0% and 51.1% of the time, depending on the

quintile in question. The not insignificant probabilities far from the main diagonal suggest that

the alternative estimation analysis is worthwhile.

Table 9 reports the price of risk for several risk factors using a number of models

specifications. The table indicates that the price of aggregate volatility risk during the sample

period was not statistically significant and had the “wrong” sign for all specifications (CAPM

plus volatility t = 0.14, three-factor plus volatility t = 0.23, and five-factor plus volatility t =

0.47).

IV.E Alternative test assets

In this subsection, I examine the pricing of aggregate volatility risk using an alternative

set of test assets, namely the Fama-French 25 (5×5) European portfolios sorted on size and book-

to-market (from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html).

Table 10 reports the price of risk for several risk factors using a number of models specifications.

The table indicates that, based on the Fama-French 25 European portfolio returns, the price of

aggregate volatility risk had a “wrong” sign for the three-factor plus volatility specification (t =

2.79), and was indistinguishable from 0 for the CAPM plus volatility (t = -0.96) and the five-

factor plus volatility (t = -0.95) specifications.

expect from pure noise. On the other hand, the 25.9% and 24.3% for the 1-month estimation are merely

5.9% and 4.3% higher than would be expected from pure noise.

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V. Evidence from high-frequency data

Chen and Ghysels (2012) argue that volatility forecasting models that use intraday data

outperform traditional GARCH-class models. To determine which variables play an important

role in high-frequency models, Bekaert and Hoerova (2014) examine 31 competing

specifications with various combinations of terms that capture the VIX, 1-month, 1-week, and 1-

day historical volatilities, and 1-month, 1-week, and 1-day historical returns. Of all the models,

the one that appears to most reliably forecast volatility is “model 8”. With estimated coefficients

included, Bekaert and Hoerova’s model 8 is,

2(22) (22) (5) (1)22

22 22 223.730 0.108 0.199 0.330 0.10712

tt t t t

VIXRV RV RV RV

(5)

where 22

(22)

1

1

t t j

j

RV RV

is the (monthly) realized variance, based on 5-minute returns, over the

22-day period ending on day t, 5

(5)

1

1

22

5t t j

j

RV RV

is the (monthly) realized variance over

the 5-day period ending on day t, and 1

(1)

1

1

22 22t t j t

j

RV RV RV

is the (monthly) realized

variance on day t. I use an analogous model, with VSTOXX taking the place of VIX, to estimate

conditional volatility.11 This conditional volatility will serve as an input in a regression

analogous to (1) above.

It should be noted that the frequency of data is not the only difference between my

analysis in this section and the previous ones. Another important distinction is that the

11 The coefficients in Bekaert and Hoerova’s model 8 are based on U.S. data. Realized variance in

European markets is greater than in the U.S., at least over the period I examine. Therefore, the constant

term in the model should be somewhat larger if the model is to be successfully applied to European data for

the purpose of volatility forecasting. However, since I am interested in the change in realized variance

rather than in the realized variance itself, the constant term is irrelevant for this study.

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VSTOXX, the main volatility variable in previous sections, is a function of both the expected

future volatility and the volatility premium. On the other hand, equation (5) attempts to model

only the expected future volatility.

Before running a regression analogous to (1), I convert the high-frequency measure of

volatility from monthly, variance-based to annual, standard deviation-based:

(22) (22)12t tCSD RV (6)

where (22)

tCSD is the (annual) conditional standard deviation over the 22-day period ending on

day t. Analogous to the procedure used above, I form portfolios of stocks by sorting them into

quintiles based on the coefficient d from the following regression run each month,

R(t) – RF(t) = a + b[RM(t)-RF(t)] + dΔCSD(t) + e(t) (7)

where R(t) is the daily return on the stock, RF(t) is the daily risk-free rate, RM(t) is the daily

return on the STOXX Europe 600 index, ΔCSD(t) is the daily change in conditional

standard deviation, and a, b, and d are regression coefficients. Portfolio 1 is the quintile

with the lowest loadings on the change in conditional standard deviation, while portfolio

5 is the quintile with the highest loadings.

To determine whether aggregate volatility is a priced risk factor based on high-frequency

European data, I follow a procedure analogous to that above. That is, I construct 25 portfolios by

sorting stocks on their market betas and on their sensitivity to changes in conditional standard

deviation. Specifically, I first sort stocks into one of 5 groups based on market betas from

equation (7). Next, within each group, I sort stocks into one of 5 portfolios based on sensitivity to

changes in conditional standard deviation. The 25 b×d portfolios serve as test assets in the Fama-

16

MacBeth procedure. In stage one of the procedure, the following time-series regression of

monthly portfolio excess returns on factor returns gives the factor loadings for each portfolio:

R(t) – RF(t) = β0 + β’f + e(t) (8)

where β’ is a vector of factor loadings and f is a vector of factor returns. In stage two, the

following cross-sectional regressions yield month-by-month risk premia:

R(t) – RF(t) = λ0 + λ’β + e(t) (9)

where λ’ is a vector of risk premia and β is a vector of factor loadings from stage one. Finally, a

t-test determines whether the risk premia are statistically significant.

Table 11 presents the results from six Fama-MacBeth regressions, and is analogous to

Table 4. In the first model, the only factor is assumed to be the excess return on the market

portfolio. In the second model, ΔCSD is added as an explanatory variable for asset excess

returns. Models III and IV further add the Fama-French European SMB and HML factors,

without and including the ΔCSD, respectively. Models V and VI additionally add the Fama-

French European RMW and CMA factors, without and including the ΔCSD, respectively. The

table shows that the coefficients on conditional volatility change in all three regressions that

include ΔCSD have the “wrong” sign (CAPM plus volatility t = 2.29, three-factor plus volatility t

= 2.34, and five-factor plus volatility t = 2.34). Together, these results support the view that it is

far too early to make any definitive statements about the pricing of aggregate volatility risk.

VI. Conclusion

I investigate whether the relationships between sensitivity to changes in aggregate

volatility and expected return on stocks documented by Ang et al. (2006) for the U.S. data are

17

also present in the European data. I find that they are not. Specifically, aggregate volatility betas

do not predict future returns. Alphas from time-series regressions of long/short high-minus-low

aggregate volatility beta portfolio returns with respect to the CAPM, the Fama-French 3-factor

model, and the Fama-French 5-factor model are not statistically different from zero. Finally, the

price of aggregate volatility risk is not statistically different from zero. Analysis based on high-

frequency data support the view that it is far too early to draw any definitive conclusions about

the pricing of aggregate volatility risk.

This is, of course, not to say that investors don’t care about aggregate volatility. Almost

every equilibrium model of asset returns (e.g., CAPM) assumes that investors dislike volatility

and would be willing to pay a premium to reduce their exposure to it. What remains unclear, as I

have argued in this paper, is whether aggregate volatility is priced after taking into account

known risk factors such as market risk, SMB and HML. Contrary to some of the research that

has focused on U.S. data, it appears that more time will be needed to better understand aggregate

volatility risk.

18

References

Anderson, Robert M., Stephen W. Bianchi, Lisa R. Goldberg, 2013, In Search of a

Statistically Valid Volatility Risk Factor, Coleman Fung Risk Management Research

Center Working Papers (UC Berkeley).

Ang, A ., R. J. Hodrick, Y. Xing, and X . Zhang. 2006. The Cross-Section of Volatility

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19

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20

Table 1. Distribution of STOXX Europe 600 firms by country/year This table contains the distribution of STOXX Europe 600 firms across countries and over time. Panel A shows the

number of firms. Panel B shows the percentage of market value.

Panel A: Number of companies by country

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Austria 2 4 5 7 11 11 11 11 12 11 10 9 7 7 6

Belgium 11 12 13 14 12 12 13 17 18 18 18 14 13 14 14

Czech Republic 2 2 2

Denmark 15 16 15 14 15 15 17 19 17 16 16 16 18 19 20

Finland 12 13 15 15 21 18 20 19 19 21 20 19 18 16 15

France 77 76 71 71 74 78 77 83 83 84 88 84 81 82 82

Germany 58 53 53 52 57 57 62 58 58 60 63 66 66 65 66

Ireland 12 12 13 13 15 14 13 10 9 9 8 9 9 9 8

Italy 49 47 45 43 42 41 33 35 34 32 32 30 31 31 32

Luxembourg 1

Netherlands 34 34 30 29 30 30 28 28 28 29 30 32 32 32 29

Norway 12 11 8 9 9 15 17 14 15 15 15 15 15 13 12

Portugal 9 8 7 7 7 7 9 10 10 9 6 5 6 4 4

Spain 32 38 39 41 41 38 36 36 32 33 31 28 27 29 30

Sweden 41 38 43 43 43 44 44 47 46 48 51 54 55 52 59

Switzerland 27 29 28 32 32 33 34 35 36 36 36 35 33 33 33

United Kingdom 200 199 202 198 179 174 171 166 172 170 172 181 184 187 186

Other 10 9 13 12 12 12 15 12 11 9 4 3 3 6 2

Panel B: Percentage of market value by country

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Austria 0.09% 0.26% 0.46% 0.76% 0.96% 1.21% 1.18% 0.75% 0.94% 0.96% 0.67% 0.73% 0.54% 0.44% 0.40%

Belgium 1.49% 1.74% 1.85% 2.54% 1.84% 1.91% 2.04% 2.07% 2.42% 2.32% 2.42% 2.55% 2.50% 2.85% 3.23%

Czech Republic 0.17% 0.19% 0.15%

Denmark 1.20% 1.26% 1.30% 1.31% 1.45% 1.35% 1.53% 1.68% 1.67% 1.89% 1.84% 1.97% 2.08% 2.41% 2.89%

Finland 2.45% 2.04% 1.75% 1.52% 1.82% 1.66% 2.22% 1.84% 1.56% 1.69% 1.31% 1.27% 1.38% 1.40% 1.42%

France 16.81% 16.70% 15.77% 14.59% 15.98% 17.71% 18.56% 19.12% 17.98% 16.89% 16.15% 16.53% 16.59% 16.25% 16.57%

Germany 11.68% 9.47% 11.15% 10.49% 11.20% 10.78% 13.16% 13.52% 11.75% 12.29% 11.98% 12.99% 13.81% 13.37% 13.39%

Ireland 1.28% 1.13% 1.21% 1.43% 1.43% 1.53% 1.12% 0.57% 0.60% 0.55% 0.65% 0.66% 0.70% 0.79% 0.96%

Italy 7.34% 8.01% 7.27% 7.84% 7.44% 7.11% 6.54% 6.32% 5.92% 4.80% 4.61% 4.38% 4.25% 4.16% 4.29%

Luxembourg 0.09%

Netherlands 7.95% 8.40% 7.79% 7.30% 5.72% 6.37% 5.56% 4.65% 5.58% 5.47% 5.56% 5.45% 5.63% 5.80% 5.38%

Norway 0.70% 0.83% 0.82% 1.02% 1.21% 1.56% 2.24% 1.51% 2.00% 2.12% 2.30% 2.25% 1.92% 1.61% 1.35%

Portugal 0.56% 0.59% 0.57% 0.58% 0.54% 0.55% 0.76% 0.75% 0.87% 0.73% 0.50% 0.43% 0.41% 0.31% 0.32%

Spain 4.53% 5.19% 5.81% 6.45% 6.33% 6.21% 6.16% 8.08% 6.83% 5.45% 5.66% 5.30% 5.69% 5.98% 5.59%

Sweden 3.17% 2.79% 3.40% 3.61% 3.61% 3.80% 3.42% 3.73% 4.21% 5.21% 4.86% 5.07% 5.14% 5.00% 5.01%

Switzerland 9.62% 9.95% 9.22% 9.38% 10.20% 9.72% 9.14% 11.66% 10.81% 11.46% 11.87% 11.45% 11.41% 11.87% 12.20%

United Kingdom 30.32% 31.02% 30.67% 30.11% 29.09% 27.30% 24.62% 22.88% 26.04% 27.72% 29.45% 28.82% 27.58% 27.25% 26.77%

Other 0.72% 0.63% 0.96% 1.07% 1.18% 1.23% 1.74% 0.87% 0.81% 0.46% 0.17% 0.15% 0.20% 0.33% 0.07%

21

Table 2. Summary statistics This table contains summary statistics on firm size (in € Millions), book-to-market ratio, and transition probability for

each of the 5 aggregate volatility sensitivity quintiles. Transition probability refers to the likelihood of a firm moving

from one quintile in month t to another quintile in month t+1 (or staying in the same quintile).

Rank

Average

Size

Median

Size

Average

B/M

Median

B/M To Rank 1 To Rank 2 To Rank 3 To Rank 4 To Rank 5

1 (Lowest) 9465 3786 0.99 0.66 25.9% 20.4% 18.3% 16.8% 18.6%

2 10733 4263 0.82 0.60 20.3% 21.3% 20.9% 19.8% 17.7%

3 10165 4222 0.90 0.58 18.3% 20.9% 20.9% 21.5% 18.4%

4 9140 3911 0.82 0.57 17.0% 19.6% 21.1% 21.7% 20.6%

5 (Highest) 7195 3242 0.78 0.54 18.7% 17.9% 18.8% 20.3% 24.3%

Transition Matrix

22

Table 3. dΔVol portfolio statistics This table contains data for each of the 5 portfolios sorted on dΔVol from regression (1), as well as the portfolio that is

long high dΔVol stocks and short low dΔVol stocks. The first column ranks the portfolios, with “1” corresponding to

the lowest dΔVol, and “5” corresponding to the highest. The second column shows the equally-weighted average

monthly total returns in the post-formation month, calculated month-by-month and averaged over months, while the

third column shows the value-weighted average monthly total returns, also calculated month-by-month and averaged

over months. The following three columns report Jensen’s alpha with respect to the CAPM, the Fama-French 3-factor

model, and the Fama-French 5-factor model, using time-series regressions. The last two columns report the average

value-weighted pre-formation and post-formation dΔVol coefficients (multiplied by 100). Newey-West robust t

statistics appear below the estimated coefficients.

Rank EW Mean VW Mean

CAPM

Alpha

FF-3

Alpha

FF-5

Alpha

Pre-formation

100×d ΔVol

Post-formation

100×d ΔVol

1 (Lowest) 0.0041 0.0018 -0.0043 -0.0030 0.0028 -0.875 -0.379

[-1.59] [-1.17] [1.13]

2 0.0073 0.0053 0.0001 0.0012 0.0034 -0.474 -0.331

[0.04] [0.67] [1.69]

3 0.0075 0.0061 0.0012 0.0025 0.0050 -0.266 -0.302

[0.63] [1.31] [2.48]

4 0.0071 0.0057 0.0009 0.0020 0.0048 -0.064 -0.267

[0.44] [1.02] [2.43]

5 (Highest) 0.0065 0.0046 -0.0012 -0.0005 0.0034 0.304 -0.257

[-0.51] [-0.23] [1.31]

5-1 0.0023 0.0027 0.0031 0.0024 0.0006

[1.30] [1.17] [1.29] [1.08] [0.22]

Factor Loadings

Table 4. Price of aggregate volatility risk: base case This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. The 25 b×d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to

changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the first-stage regression

is a time-series regression of excess returns on the 25 b×d portfolios on returns on the risk factors, and estimates factors

loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b×d portfolios on the

factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market

portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the change in

aggregate volatility. t statistics appear below the estimated coefficients.

I II III IV V VI

RM-RF -0.00118 -0.00141 -0.00163 -0.00169 -0.00265 -0.00399

[-0.2] [-0.23] [-0.23] [-0.24] [-0.33] [-0.49]

SMB 0.00104 0.00141 -0.00124 -0.00090

[0.24] [0.33] [-0.28] [-0.2]

HML 0.00068 0.00065 0.00298 0.00311

[0.19] [0.18] [0.8] [0.83]

RMW 0.00185 0.00177

[0.78] [0.74]

CMA -0.00196 -0.00199

[-0.63] [-0.64]

ΔVol 0.57112 0.79957 0.86995

[0.36] [0.65] [0.72]

Alpha 0.00486 0.00531 0.00549 0.00633 0.00490 0.00636

[1.63] [1.49] [1.33] [1.4] [1.08] [1.24]

24

Table 5. Price of aggregate volatility risk: unanticipated change in volatility This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. Here, I define the change in aggregate volatility as the unanticipated change in VSTOXX based

on an AR(1) model. The 25 b×d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to

changes in aggregate volatility) from regression (4) in the text. For the Fama-MacBeth model, the first-stage regression

is a time-series regression of excess returns on the 25 b×d portfolios on returns on the risk factors, and estimates factors

loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b×d portfolios on the

factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market

portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the

unanticipated change in aggregate volatility. t statistics appear below the estimated coefficients.

I II III IV V VI

RM-RF -0.00156 -0.00258 -0.00224 -0.00409 -0.01142 -0.01367

[-0.27] [-0.42] [-0.31] [-0.54] [-1.5] [-1.69]

SMB 0.00216 0.00254 -0.00074 0.00038

[0.51] [0.59] [-0.19] [0.09]

HML 0.00146 0.00209 0.00159 0.00155

[0.34] [0.48] [0.34] [0.33]

RMW -0.00005 -0.00085

[-0.02] [-0.36]

CMA -0.00352 -0.00227

[-1.11] [-0.65]

UΔVol 0.84907 1.02805 1.07481

[0.7] [0.83] [0.84]

Alpha 0.00514 0.00635 0.00625 0.00883 0.00891 0.01190

[1.75] [1.73] [1.49] [1.64] [2.1] [2.1]

25

Table 6. Price of aggregate volatility risk: excluding financial crisis period This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. Here, I exclude the financial crisis period (September 2008 – October 2009) in the cross-

sectional regressions. The 25 b×d portfolios are formed by sorting on b (market beta) and then on d (sensitivity to

changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the first-stage regression

is a time-series regression of excess returns on the 25 b×d portfolios on returns on the risk factors, and estimates factors

loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b×d portfolios on the

factor loadings from the first stage, and estimates risk premia. The factors include the excess return on the market

portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the change in

aggregate volatility. t statistics appear below the estimated coefficients.

I II III IV V VI

RM-RF -0.00248 -0.00236 -0.00240 -0.00240 -0.00375 -0.00370

[-0.47] [-0.45] [-0.4] [-0.4] [-0.49] [-0.48]

SMB 0.00137 0.00133 0.00006 0.00005

[0.32] [0.31] [0.01] [0.01]

HML -0.00006 -0.00006 0.00132 0.00131

[-0.02] [-0.02] [0.35] [0.35]

RMW 0.00123 0.00124

[0.52] [0.52]

CMA -0.00135 -0.00134

[-0.47] [-0.47]

ΔVol -0.25251 -0.11805 -0.07765

[-0.18] [-0.11] [-0.07]

Alpha 0.00617 0.00592 0.00643 0.00635 0.00644 0.00638

[2.13] [1.77] [1.69] [1.53] [1.56] [1.37]

26

Table 7. Price of aggregate volatility risk: truncated sample This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. Here, I truncate the sample by removing the extreme 10 firms by market beta (both high and

low) and the extreme 15 remaining firms by aggregate volatility beta (both high and low), month-by-month. This

leaves 550 of the original 600 firms per month. The 25 b×d portfolios are formed by sorting on b (market beta) and

then on d (sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model,

the first-stage regression is a time-series regression of excess returns on the 25 b×d portfolios on returns on the risk

factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on

the 25 b×d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the

excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French

(1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated coefficients.

I II III IV V VI

RM-RF -0.00244 -0.00252 -0.00119 -0.00120 -0.00541 -0.00541

[-0.42] [-0.43] [-0.16] [-0.16] [-0.61] [-0.61]

SMB -0.00080 -0.00088 0.00000 0.00003

[-0.2] [-0.23] [0] [0.01]

HML -0.00207 -0.00216 -0.00072 -0.00078

[-0.49] [-0.51] [-0.16] [-0.17]

RMW 0.00034 0.00030

[0.14] [0.12]

CMA -0.00133 -0.00124

[-0.41] [-0.38]

ΔVol 0.16939 0.22983 0.11175

[0.19] [0.26] [0.12]

Alpha 0.00605 0.00616 0.00484 0.00494 0.00707 0.00717

[2.04] [2.07] [1.03] [1.07] [1.38] [1.41]

27

Table 8. Transition matrix and quintile comparison: 3-month versus 1-month Panel A of this table presents the transition probability for each of the 5 aggregate volatility sensitivity quintiles when

estimating sensitivities using 3 months of data versus when using 1 month of data. Transition probability refers to the

likelihood of a firm moving from one quintile in month t to another quintile in month t+1 (or staying in the same

quintile). For the 3-month estimation, a firm’s quintile assignment for month t is determined by data in months t-2

through t, while for the 1-month estimation it is determined by data in month t alone (as in Table 2).

Panel B presents a comparison between quintile assignments using 3-month estimation versus 1-month estimation.

Panel A: Transition Matrix comparison

Rank To Rank 1 To Rank 2 To Rank 3 To Rank 4 To Rank 5 To Rank 1 To Rank 2 To Rank 3 To Rank 4 To Rank 5

1 (Lowest) 58.5% 23.4% 10.1% 5.0% 2.9% 25.9% 20.4% 18.3% 16.8% 18.6%

2 23.0% 35.6% 23.7% 12.3% 5.3% 20.3% 21.3% 20.9% 19.8% 17.7%

3 9.9% 23.2% 32.5% 23.7% 10.6% 18.3% 20.9% 20.9% 21.5% 18.4%

4 5.0% 12.2% 23.6% 35.8% 23.4% 17.0% 19.6% 21.1% 21.7% 20.6%

5 (Highest) 3.1% 5.5% 10.3% 23.5% 57.5% 18.7% 17.9% 18.8% 20.3% 24.3%

Panel B: Quintile comparison

Rank 1 (Lowest) 2 3 4 5 (Highest)

1 (Lowest) 51.1% 23.5% 12.9% 7.7% 4.8%

2 23.9% 30.2% 23.2% 14.8% 7.9%

3 12.8% 23.5% 27.0% 23.2% 13.5%

4 7.5% 14.8% 23.4% 30.0% 24.3%

5 (Highest) 4.8% 8.1% 13.4% 24.5% 49.2%

Transition Matrix using 3-month estimation Transition Matrix using 1-month estimation (from Table 1)

3-month

estimation

1-month estimation

28

Table 9. Price of aggregate volatility risk: 3-month estimation windows This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. Here, I use 3-month (rolling) estimation windows to estimate market betas and aggregate

volatility betas, rather than 1-month. The 25 b×d portfolios are formed by sorting on b (market beta) and then on d

(sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-MacBeth model, the first-

stage regression is a time-series regression of excess returns on the 25 b×d portfolios on returns on the risk factors, and

estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on the 25 b×d

portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the excess return on

the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French (1993, 2015), and the

change in aggregate volatility. t statistics appear below the estimated coefficients.

I II III IV V VI

RM-RF -0.00300 -0.00305 -0.00373 -0.00395 -0.00920 -0.01056

[-0.53] [-0.53] [-0.6] [-0.62] [-1.46] [-1.61]

SMB 0.00074 0.00083 0.00012 0.00005

[0.26] [0.29] [0.04] [0.02]

HML 0.00039 0.00054 -0.00398 -0.00342

[0.11] [0.15] [-1] [-0.88]

RMW -0.00141 -0.00128

[-0.72] [-0.65]

CMA -0.00194 -0.00198

[-0.7] [-0.7]

ΔVol 0.15456 0.25728 0.55518

[0.14] [0.23] [0.47]

Alpha 0.00628 0.00637 0.00702 0.00738 0.00813 0.00927

[2.28] [2.11] [1.95] [1.81] [2.36] [2.47]

29

Table 10. Price of aggregate volatility risk: alternative test assets This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. Here, I use alternative test assets. Namely, the 25 b×d portfolios are the Fama-French 25

European portfolios sorted on size and book-to-market (from

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html) rather than portfolios formed by sorting on b

(market beta) and then on d (sensitivity to changes in aggregate volatility) from regression (1) in the text. For the Fama-

MacBeth model, the first-stage regression is a time-series regression of excess returns on the 25 b×d portfolios on

returns on the risk factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of

excess returns on the 25 b×d portfolios on the factor loadings from the first stage, and estimates risk premia. The

factors include the excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in

Fama and French (1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated

coefficients.

I II III IV V VI

RM-RF 0.00297 0.00460 0.00334 0.00198 0.00209 0.00368

[0.45] [0.69] [0.5] [0.29] [0.26] [0.44]

SMB 0.00197 0.00238 0.00245 0.00241

[1.35] [1.63] [1.68] [1.65]

HML 0.00150 0.00125 0.00105 0.00106

[0.87] [0.73] [0.61] [0.62]

RMW 0.00673 0.00706

[3.44] [3.51]

CMA -0.00380 -0.00400

[-1.68] [-1.77]

ΔVol -1.18485 4.78253 -1.46671

[-0.96] [2.79] [-0.95]

Alpha 0.00573 0.00350 0.00422 0.00533 0.00566 0.00414

[1.06] [0.66] [0.79] [0.99] [0.82] [0.57]

30

Table 11. Price of aggregate volatility risk: evidence from high-frequency data This table contains the results of Fama-MacBeth regressions that estimate the price of aggregate volatility risk as well

as prices of other risks. Here, I define the change in aggregate volatility as the change in conditional standard deviation

based on Bekaert and Hoerova’s (2014) model 8. The 25 b×d portfolios are formed by sorting on b (market beta) and

then on d (sensitivity to changes in aggregate volatility) from regression (7) in the text. For the Fama-MacBeth model,

the first-stage regression is a time-series regression of excess returns on the 25 b×d portfolios on returns on the risk

factors, and estimates factors loadings. The second-stage regression is a cross-sectional regression of excess returns on

the 25 b×d portfolios on the factor loadings from the first stage, and estimates risk premia. The factors include the

excess return on the market portfolio, the SMB, HML, RMW, and CMA portfolios as defined in Fama and French

(1993, 2015), and the change in aggregate volatility. t statistics appear below the estimated coefficients.

I II III IV V VI

RM-RF -0.00050 -0.00202 -0.00025 -0.00285 -0.00661 -0.00960

[-0.09] [-0.35] [-0.04] [-0.45] [-0.98] [-1.43]

SMB -0.00096 -0.00195 -0.00015 -0.00167

[-0.25] [-0.51] [-0.04] [-0.4]

HML -0.00086 -0.00015 -0.00207 -0.00148

[-0.23] [-0.04] [-0.55] [-0.39]

RMW -0.00188 -0.00156

[-0.78] [-0.65]

CMA -0.00057 -0.00026

[-0.21] [-0.09]

ΔCSD 2.08525 2.11413 2.19305

[2.29] [2.34] [2.34]

Alpha 0.00438 0.00500 0.00391 0.00572 0.00715 0.00876

[1.47] [1.71] [1.06] [1.65] [2.01] [2.58]

31

Figure 1. VSTOXX Index This figure shows the time series of the VSTOXX volatility index during 2002-2016, with a mean value of 24.76. The

distribution percentiles are as follows: p5 = 13.81, p10 = 15.09, p25 = 17.89, median = 22.31, p75 = 28.00, p90 =

39.61, p95 = 46.70.