The Variance Risk Premium Around the World

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1035 November 2011 The Variance Risk Premium Around the World Juan M. Londono NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

The Variance Risk Premium around the World y Juan M. Londono (cid:3) November 28, 2011 Abstract This paper investigates the variance risk premium in an international setting. First, I provide new evidence on the basic stylized facts traditionally documented for the US. I show that while the variance premiums in several other countries are, on average, positive and display signi(cid:133)cant time variation, theydonotpredictlocalequityreturns. Then, IextendthedomesticmodelinBollerslev, Tauchen and Zhou (2009) to an international setting. In light of the qualitative implications of my model, I provide empirical evidence that the US variance premium outperforms that of all other countries in predicting local and foreign equity returns. JEL Classi(cid:133)cation: E44, F36, G12, G13, G15. Keywords: variance risk premium, economic uncertainty, interdependence, international integration, comovements, return predictability. This paper has bene(cid:133)ted signi(cid:133)cantly from the comments of Lieven Baele and Joost Driessen. I would y also like to thank the valuable comments from Geert Bekaert, Esther Eiling, Eric Engstrom, Rik Frehen, Bruno Gerard, Frank de Jong, Clara Vega, Hao Zhou, and participants at the UvT Department of (cid:133)nance Brownbag and the Basque Country University Department of Economics Seminars. All errors are mine. Theauthorisasta⁄economistintheDivisionofInternationalFinance,BoardofGovernorsoftheFederal (cid:3) Reserve System, Washington, D.C. 20551 U.S.A. E-mail juan-miguel.londono-yarce@frb.gov. The views in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

1 Introduction Traditional asset pricing models have mainly focused on characterizing the reward for equity risk. However, such models typically fail to capture the reward for bearing variance risk. The variance risk premium is formally de(cid:133)ned as the di⁄erence between the risk neutral and the physical expectation of the total return variation. It can be estimated using model-free measures as the di⁄erence between the option implied variance and the expected realized variance. The observed variance premium in the US is large and varies signi(cid:133)cantly over time. In order to generate a time-varying variance premium, standard asset pricing models have been adjusted in di⁄erent ways. One strand of the literature, and the one that will be followed in this paper, links the variance risk premium to macroeconomic uncertainty. This strand follows the intuition behind the long-run risk model in Bansal and Yaron (2004) (BY hereafter), and the idea that agents have a preference for an early resolution of uncertainty in Bansal et. al. (2005). Extending BY(cid:146)s model, Bollerslev, Tauchen and Zhou (2009) (BTZ hereafter) show that the variance premium predicts equity returns; an implication for which they (cid:133)nd empirical evidence for the US. An alternative strand of the literature relates the variance premium to agents(cid:146)attitudes towards non-normalities in the distribution of returns. In Bakshi and Madam (2006), for example, the variance risk premium is explained by the desire of risk averse agents to buy protection against extreme events. In a similar vein, Bekaert and Engstrom (2010), Todorov (2010), and Gabaix (2009), using di⁄erent methodologies, focus on the interplay between returns, risk aversion and extreme events to explain many asset pricing regularities, including the variance risk premium. Existing work, both theoretical and empirical, has predominantly focused on the US market. This paper adds to the literature by extending the variance premium analysis to an international setting. The contribution is threefold. First, I provide new evidence on the basic stylized facts related to the variance premium for a total of eight countries. I show that while the variance premiums display signi(cid:133)cant time variation in all countries analyzed, the local return predictability does not hold internationally. Then, I extend the domestic model in Bollerslev, Tauchen and Zhou (2009) to an international setting. My model links the variance premium to local and aggregate macroeconomic uncertainty and yields a qualitative explanation for the local predictability puzzle. Finally, I provide new empirical evidence to investigate the main qualitative implications of my model. The empirical evidence suggests that the US variance premium predicts the equity returns in the US as well as in any other country in the sample. In addition, the evidence also suggest that the US variance premium plays a key role in predicting the variance premium correlations as well as the equity return correlations across countries. The di⁄erent parts and contributions of the paper are discussed in more detail in turn. Inthe(cid:133)rstpart, Iinvestigatethemainstylizedfactsrelatedtothevariancepremiumpreviously documented for the US in an international setting. In particular, I investigate whether the timevarying and positive nature of the variance premium as well as its capacity to predict returns holds internationally. In order to do so, I collect data for the US, Germany, UK, Japan, Switzerland, The Netherlands, Belgium, and France for the sample period between 2000 and 2009. As it has become standard in the literature, the variance premiums for all countries are estimated using model-free measures of the expected variance of returns. Thus, the (squared of the) model-free implied volatil- 1

ity (IV) index for each equity market approximates the expectation of the total return variation undertheriskneutralmeasure(CarrandMadan,(1998),andBritten-JonesandNeuberger,(2000)) while the expectation under the physical measure is approximated by a conditional forecast of the actual realized variance. Thesingle-countryevidenceshowsthatthevariancepremiumsdisplaysigni(cid:133)canttimevariation and are, on average, positive for all countries in the sample. This international evidence is in line with previous (cid:133)ndings for the US.1 However, I show that the local variance premium can predict local equity returns only in the US and Belgium. For any other country analyzed, the evidencesuggeststhatthelocalvariancepremiumscannotpredictlocalequityreturns. This(cid:133)nding suggestsapuzzlethatcannotbesolvedbytheexistingdomesticmodelswherethevariancepremium implicitly explains the variation in the local equity premium.2 The strictly domestic nature of the existing models motivates the theoretical contribution of this paper. In the second part, I propose a model to investigate the role of the variance premium in explaining the interactions across international equity and option markets. The model is a two-country extension of that in BTZ and extends the intuition that agents have a preference for an early resolution of uncertainty to an international setting. The macroeconomic uncertainty is characterized in my model by the dynamics of the consumption growth volatility of each country and is allowed to be transmitted across countries given a unique representative agent endowed with recursive preferences. In such a setting, the shocks to macroeconomic uncertainty in any country characterize the variance premium in all countries. In particular, the variance premiums of the two countries reveal the volatility of volatility of consumption generated in both countries. Given that changes in the volatility of volatility also explain a portion of the total risk premiums of any country,themodelnotonlyimpliesthatvarianceriskispricedbutalsoprovidestheintuitionforthe potentialroleofthevariancepremiumofanycountryinpredictinglocalandforeignequityreturns. In other words, agents demand a reward for the existing local and foreign sources of risk (i.e., the volatility and the volatility of volatility of consumption). Although this uncertainty transmission mechanism is bidirectional, the model explicitly features a leader economy. The consumption process of this leader economy is entirely driven by local shocks. However, the shocks of the leader country consumption process can be partially transmitted to a second country, the follower. My model yields several qualitative implications for the interactions across international equity andoptionmarketsthatexplaintheinabilityofthevariancepremiumtopredictlocalequityreturns incountriesotherthantheUS.The(cid:133)rstmainimplicationofmymodelisthatthevariancepremium in each country is uniquely characterized by the volatility of volatility of consumption (VoV) of the two countries. The load of each country(cid:146)s VoV increases with the relative size of its economy and the degree of economic dependence among countries (leader-follower relation). As a consequence of having common components, the variance premiums are highly correlated across countries; and the cross-country variance premium correlation is mainly driven by the VoV generated in the leader country. Thus, the leader country variance premium plays the key role in predicting the variance 1See for instance Britten-Jones and Neuberger (2000), Jiang and Tian (2005), Bakshi and Madan (2006), Carr and Wu (2009), Bollerslev, Gibson and Zhou (2011), and BTZ, among others. 2BTZ, Zhou (2010), and Drechsler and Yaron (2011) (cid:133)nd empirical evidence for their respective model-implied return predictability. However, Bekaert and Engstrom (2010) (cid:133)nd weak evidence of return predictability. 2

premium correlations across countries. The second main implication of my model is that the VoV of the two countries also load on all countries(cid:146)equity premiums. Similar to the implication for the variancepremiums,theloadofVoVincreaseswiththerelativesizeofeacheconomyandtheimplied correlation of the consumption processes. This second implication links the variance premium to all countries(cid:146)equity premiums. As a consequence, this implication explains the possibility that the variance premium of a leader economy predicts other countries(cid:146)equity returns which in turn implies that the leader country variance premium plays the key role in predicting equity return correlations across countries. The third contribution of this paper is that it provides new empirical evidence on the two main qualitative implications of my model. That is, I investigate the fundamental linkages between the variance premiums across countries as well as the interplay between the variance premiums and international equity returns. To do so, I (cid:133)rst provide evidence that the variance premiums are highly correlated across countries as suggested by the common loads of volatility of volatility in the variance premiums suggested by my model. Then, I investigate the second main implication of my model which suggests that the leader country variance premium plays the key role in predicting local and foreign equity returns. On the one hand, I confront the evidence on the poor performance of the local variance premiums in predicting local returns for countries other than the US. Thus, I provide new evidence that only the US variance premium predicts equity returns for all countries in the sample except perhaps for Japan. The predictive power of the US variance premium over international equity returns holds for horizons between 1 to 6 months, and reaches its maximum at the quarterly horizon. In addition, I show that the US variance premium outperforms all other countries(cid:146)variance premiums in predicting local and foreign equity returns. On the other hand, I provideevidencethatinternationalequityreturnstendtocomovemoreintenselyfollowingepisodes of increasing US variance premium. The predictive power of the variance premium for both equity returns and cross-country return correlations holds for horizons between 3 and 6 months and is additional to that of traditional (local or US) variables such as the term spread and the dividend yield. Theremainderofthepaperisorganizedasfollows:Section2introducesthemainde(cid:133)nitionsand data used throughout the paper. Section 3 provides international single-country evidence on the regularities related to the variance premium. Section 4 introduces the international consumption based general equilibrium model and analyzes its qualitative implications. Section 5 investigates the empirical evidence in light of the implications of my model. Finally, Section 6 concludes. 2 Data and De(cid:133)nitions In this section, Iintroduce the data used to estimate the monthly variance premiums for the following countries: US, Germany, Japan, UK, Switzerland, The Netherlands, Belgium and France. The variance premium is de(cid:133)ned as the di⁄erence between the risk neutral and the physical expectation of the market return variation between time t and one month forward t+1 for each market. It is estimated, as it has become standard in the related literature, using model-free measures for the expectations of the total return variation. 3

I approximate the risk neutral expectation of the market return variation as (the square of) the model-freeoptionsimpliedvolatility(IV)indexforeachmarket. ThemethodologyfortheIVindex was initially proposed by Carr and Madan (1998) and Britten-Jones and Neuberger (2000). The IV index has shown to provide a much better approximation to the expected risk neutral return variationthanpreviouslyBlack-Scholesbasedmeasures(Bollerslev,GibsonandZhou,(2011)). The IVindicesareconstructedfromaportfolioofEuropeancallswheretheunderlyingisarepresentative market index for each country as in C (t+1; K ) C (t;K) iv = 2 1 j;t Bj(t;t+1) (cid:0) j;t dK; j;t K2 Z0 where C are the prices of calls with strikes from zero to in(cid:133)nity, and B (t;t+1) are the local j;t j prices of zero-coupon bonds with one month ahead maturity. The availability of the IV index for the countries analyzed is limited by the recent development of their option markets. The index was (cid:133)rst reported for the US by the Chicago Board Options Exchange (CBOE), the VIX, in 1993 (with data from 1990). The VIX was adapted to the modelfree methodology in 2003, and was then called the New-VIX. An index for the German market, the VDAX, was released by the German Stock Exchange (Deutsche Beurse and Goldman Sachs) in 1994 (with data from 1992). The Swiss Exchange introduced the index for Switzerland, the VSMI, in 2005. Currently, Eurex estimates and reports both VDAX and VSMI following a uni(cid:133)ed New- VIX methodology. The Center for the Study of Finance and Insurance (CSFI) at Osaka University launched an index for Japan, the VXJ, with data from 1995. Finally, in 2007, Euronext announced IV indices for France (VCAC), Belgium (VBEL), the UK (VFTSE, in partnership with FTSE), and The Netherlands (VAEX) with data from 2000.3 Considering the data restrictions for the European markets, the empirical analysis in this paper is centered on the sample period between 2000 and 2009. In order to construct the variance premiums, an expectation of the total return variation under the physical measure has to be estimated. I estimate a measure based on the (cid:133)rst order autoregressive forecast of the total realized return variation or realized variance from the following equation: rv = (cid:13) +(cid:13) rv +(cid:15) ; j;t+1 o 1 j;t t where the realized variance is calculated summing the squared daily equity returns for each market as in Nt rv = (r )2; j;t j;ti t Xi=1 where r are daily local returns within month t. I rely on daily returns since data at a higher j;ti frequency are not available for all countries in the sample.4 3Both, the UK (FTSE) and France (French March des Options Negociables de Paris) had previously introduced IV indices separately. 4It has been shown in the literature that the use of intradaily returns outperforms lower frequency data in the estimation of the realized variance (Andersen et. al., (2001), Barndor⁄-Nielsen and Shephard, (2002); and Meddahi, 4

In order to make the results comparable to those in the literature, and as a preventive solution to the possible underperformance of this benchmark measure, all results are checked using three alternativeapproximationsoftheexpectedrealizedvariance. Inthe(cid:133)rstmeasure, Iusethemartingale measure where the expected realized variance is approximated as the current realized variance (E (rv ) = rv ). In the second one, I estimate a forecast of the realized variance that includes t t+1 t the local IV index as in the following equation: rv = (cid:13) +(cid:13) rv +(cid:13) iv +(cid:15) : j;t+1 o 1 j;t 2 j;t t Finally, in the third one, I estimate a forecast of the realized variance that includes the range-based variance for each country as in rv = (cid:13) +(cid:13) rv +(cid:13) RangeV +(cid:15) ; j;t+1 o 1 j;t 2 j;t t where RangeV is the range-based variance calculated as j;t 1 Nt RangeV = range2; j;t 4ln2 ti t Xi=1 where range is the daily di⁄erence between the highest and the lowest price of the index.5;6 ti In order to estimate the variance premiums, the monthly data (end of the month) for the IV indices as well as the daily returns for the underlying index returns for all countries are obtained from Datastream. All returns are expressed in local currencies.7 In order to obtain the local excess returnstoinvestigatethereturnpredictability,Iconsiderthe3-monthsT-billratesforeachcountry. Finally, I also control for two variables traditionally used to predict excess returns, the dividend yield and the term spread, calculated as the di⁄erence between the 1 year T-bill and the 3-months T-bill rate for each country. The T-bill rates as well as the dividend yields for all countries are also obtained from Datastream. 3 Variance Premium: Single-Country Evidence In this section, I investigate whether the stylized facts observed for the variance premium in the US also hold internationally. In a (cid:133)rst step, I analyze the positive and time-varying nature of the variancepremium. Then,Iinvestigatetheabilityofthelocalvariancepremiuminpredictingequity returns in each country separately. In order to get an idea of the magnitude and the time-varying nature of the variance premiums, Figure 1 displays the (benchmark) time series for all countries considered. The main statistics of (2002)). 5Martens and van Dijk (2007) provide a description of the range based estimation of volatility. Jacob and Vipul (2008) analyze the extension of the range based measure to forecast the variance. 6Also as a preventive measure to reduce the noise implicit in country-speci(cid:133)c variance premiums, I also check the robustness of my results to considering a proxy for the world VP (with and without the US) similar to that in Bollerslev, et. al. (2011). 7The results are checked for robustness when all returns are expressed in US dollars. 5

these series are summarized in Table 1. This table also displays the IV indices and their underlying equity market indices for each country. The volatility premiums [volp = iv rv ] are also j;t t j;t+1 (cid:0) included in the table in order to visualize the magnitude of the premiums in annual percentages. p The average volatility premium ranges between 1.7% for Belgium to 3.8% for Japabn. In order to getanintuitiveideaofthesemagnitudesintermsofonemonthmaturityat-the-moneyputoptions, the 3.8% volatility premium in Japan translates into a price di⁄erence of 18% in a Black Scholes world. That is, one month at-the-money put options priced at 26.75% implied volatility, which is the average IV index for Japan, are 18% more expensive than the same options priced at 22.87% implied volatility, which is the average realized volatility for this country in this sample. [Insert Figure 1 here.] [Insert Table 1 here.] TheinformationinTable1andFigure1suggeststhatthevariancepremiumsdisplaysigni(cid:133)cant time variation. In particular, the premiums show several episodes of high volatility and notorious spikes around the same periods of time which translate into large Kurtosis for all series. The (cid:133)rst high-variance-premiumsepisodeoccursaroundtheendofthetechnologicalboomin2000. Asecond episode occurs at the end of 2002. This second episode coincides with the high macroeconomic uncertainty reported in the second semester of 2002 in the US ((cid:133)rst semester of 2003 for Germany. An episode also related to the corporate accounting scandals around those years). Finally, the most notorious variance premium spikes occur around the recent subprime crisis. Not surprisingly, the minimum and maximum values for all series, except for Germany, occur in the last quarter of 2008. For Japan, for example, the variance premium reached 3,398.2 (annual percentage squared) in October 2008.8 In order to assess the positive nature of the average variance premiums, Figure 2 summarizes the results for a test on the signi(cid:133)cance of the mean variance premium for all countries. This (cid:133)gure displays the average variance premiums and their respective con(cid:133)dence intervals for the four alternative measures introduced in Section 2. The evidence suggests that the average variance premium is positive and signi(cid:133)cant for all countries analyzed and all alternative measures considered, except perhaps when the martingale measure is used.9 This evidence supports the idea that agents also price market volatility in countries other than the US. These results are new evidence that extends that found for the US by Britten-Jones and Neuberger (2000), Jiang and Tian (2005), Bakshi and Madan (2006), Carr and Wu (2009), Todorov (2010), Bollerslev, Gibson and Zhou (2011), Bekaert and Engstrom (2010), and BTZ, among others.10 This paper is, to the best of my knowledge, the (cid:133)rst to show that these stylized facts also hold in other developed markets.11 8See Bollerslev, Gibson and Zhou (2011), and Corradi, et. al. (2009) for a more detailed analysis of the relation between the variance premium and the business cycle in the US. 9The martingale measure is the benchmark measure in BTZ and Bollerslev, et. al. (2011). 10A group of papers have also provided preliminary evidence of this regularity using Black(cid:150)Scholes-based implied volatility. See, for instance, Bakshi, Cao and Chen (2000), Christo⁄ersen, Heston and Jacobs (2006), and Bollerslev and Zhou (2006). 11Thisiscertainlynotthe(cid:133)rstonein analyzingtheinformationalcontentofoption marketsinternationally. Some 6

[Insert Figure 2 here.] In the rest of this section, I test another US-based stylized fact, namely that the local variance premium predicts local equity returns.12 Given the new evidence presented above on the existence of a volatility premium in all countries analyzed, I investigate the role of the variance premium in predicting returns for all countries in the sample. To do so, Figure 3 reports the estimation results for the following regressions: (r r ) = (cid:13) +(cid:13) vp +(cid:13) dy +(cid:13) ts +(cid:15) ; f j;t;t+h 0;;j;h 1;;j;h j;t 2;;j;h j;t 3;;j;h j;t j;h;t (cid:0) where (r r ) represents future compounded annualized excess returns h-months ahead, dy f j;t;t+h j;t (cid:0) is the local dividend yield, and ts is the local term spread. j;t [Insert Figure 3 here.] The evidence in Figure 3 con(cid:133)rms most of the results previously found in the literature for the US.Thatis,theUSvariancepremiumpredictsreturnsspeciallyforhorizonsbetween3to6months. In fact, the evidence shows that the US variance premium explains up 15% of the total variation in future equity returns at the quarterly frequency. The predictive power, as well as the coe¢ cient of the variance premium in these regressions, follows a hump-shaped pattern and becomes irrelevant for horizons around one year. However, the evidence suggests that the local variance premium plays a modest or insigni(cid:133)cant role in predicting returns in any other country analyzed except perhaps for Belgium. For example, the results show that for Germany, Japan, the UK and the Netherlands, the R2 is modest and hardly ever above 1%. Not surprisingly, for these countries, the variance premium does not predict equity returns for any horizon considered. For Belgium the R2 is as high as 10% for the one-month horizon; and the predictive power of the variance premium follows a linearly decreasing pattern as the horizon increases.13 Finally, for France, although the R2 are also modest, the predictability follows a pattern similar to that found for the US. In the case of France however, both the R2 and the variance premium coe¢ cient are only signi(cid:133)cant at the 2-months horizon.14 In sum, although this is, to the best of my knowledge, the (cid:133)rst paper to present evidence on the role of the variance premium in predicting returns for countries other than the US, the singlepreliminary evidence that volatiliy risk is priced in an international setting can be found in Mo and Wu (2007) and Driessen and Maenhout (2006). Implied volatility in international markets has also been analyzed in Konstantinidi, Skiadopoulos, and Tzagkaraki (2008), Siriopoulos and Fassas (2009), and Jiang, Konstantinidi and Skiadopoulos (2010). 12See for instance BTZ, Zhou (2010) and Drechsler and Yaron (2011). 13It is worth pointing out that the variance premium in Belgium shows the lowest Sharpe ratio (almost half that for the rest of the countries). This could preliminary suggest that the variance premium is particularly volatile in Belgium. This in turn implies a somehow noisier measure in this country, potentially driven by the liquidity of the Belgian option market. 14The empirical evidence on the inability of the local VP to predict local excess returns for all countries in our sampleexceptforBelgiumandtheNetherlandshasalsobeenobtainedinindependentconcurrentworkbyBollerslev, et. al. (2011). 7

countryevidenceispuzzling. My(cid:133)ndingsareontheonehandconsistentwiththeexistenceoftimevarying variance premiums for a large sample of countries. On the other hand, they suggests that the variance premium does not predict returns in countries other than the US. The concurrence of thesetwo(cid:133)ndingscannotbeexplainedbytheexistingdomesticmodelswherethevariancepremium implicitly explains the variation in the local equity premium. This puzzling evidence is nonetheless the motivation for the international general equilibrium model introduced in the following section. The model proposed is able to qualitatively explain the poor evidence for the role of the local variance premium in predicting returns outside the US. This model suggests that the variance premium of a leader country plays a dominant role in predicting returns for all other countries; a key implication for which I provide empirical evidence in the subsequent section. 4 A Two-Country Model for the role of the Variance Premium in International Equity Markets The domestic nature of the existing models in the literature prohibits the analysis of the variance premium in an international setting. These models cannot provide an explanation for the poor role of the local variance premium in predicting returns in countries other than the US as shown in the previoussection. Therefore,Iproposeaninternationalconsumption-basedgeneralequilibrium(GE) model where the variance risk is priced in the global as well as in the local portfolios. My model yields several new qualitative implications for the role of the variance premium in international markets. The most relevant implication of the model is that the variance premium of a leader economy plays a dominant role in predicting equity returns in all portfolios. In addition, the model implies that the leader country variance premium also plays a role in explaining equity and option markets correlations across countries. In this section, I present the basic setup of the model as well as its main implications.15 I do not attempt to estimate nor to test my model but rather to use its qualitative implications to investigate the inability of the variance premium to predict local equity returns in countries other than the US. Therefore, I propose a numerical simulation of the model in order to understand its implications and illustrate the mechanism behind it. These numerical simulations provide the link between the single-country evidence, the implications of the model and the empirical evidence presented in the following section. 4.1 Model Setup and Assumptions Themodelpresentedhereisatwo-countryextensionofthatinBTZ.Itpreservestwokeyingredients in BTZ(cid:146)s model: the use of recursive preferences, and the time-varying nature of macroeconomic uncertainty characterized by the volatility of consumption. However, my model adds to the literature by extending the intuition that (cid:133)nancial markets dislike macroeconomic uncertainty (BY and Bansal,et. al.,(2005))toaninternationalsetting. Therefore,Iincludetheadditionalsourcesofrisk embedded in the consumption process of each country, namely the country-speci(cid:133)c time-varying 15In order to save space, the detailed solution of my model is presented in Appendix A 8

volatility and the volatility of volatility (VoV) of consumption.16 The setup of the model requires several additional assumptions. First, the two countries are assumed to be of a "considerable" size. That is, they both play a role in determining the global consumption growth which is a weighted average of the two countries(cid:146)consumption growth. Second, one of the countries is assumed to be "the leader". The consumption process for the leader country is assumed to be entirely driven by local shocks, while the consumption process for the second country, "the follower", is also a⁄ected by the shocks generated in the leader country. Finally, I assume fully integrated equity markets. That is, there exists a unique representative agent holding a global portfolio with positions in the two equity markets. The assumptions of fully integrated equity markets and potentially integrated economiesseemadequategiventheparticularcharacteristicsofthesampleconsideredinthispaper. Formally, each country consumption process is modeled similar to BTZ. The log of the consumption growth g for the leader country (labeled as 1) follows j;t g = (cid:22) +(cid:27) z ; (1) 1;t+1 1;g 1;t g1;t+1 (cid:27)2 1;t+1 = a (cid:27) +(cid:26) (cid:27) (cid:27)2 1;t +pq 1;t z (cid:27)1;t+1 ; q 1;t+1 = a q +(cid:26) q q 1;t +’ q pq 1;t z q1;t+1 ; whereas the consumption process for the follower country (labeled as 2) follows g = (cid:22) +(cid:30) (cid:22) +(cid:30) (cid:27) z +(cid:27) z ; (2) 2;t+1 2;g g 1;g (cid:27) 1;t g1;t+1 2;t g2;t+1 (cid:27)2 2;t+1 = a (cid:27) +(cid:26) (cid:27) (cid:27)2 2;t +pq 2;t z (cid:27)2;t+1 ; q 2;t+1 = a q +(cid:26) q q 2;t +’ q pq 2;t z q2;t+1 : The global consumption growth is a weighted average of the two countries(cid:146)consumption process as in g = !g +(1 !)g ; w;t 1;t 2;t (cid:0) where ! is the weight of the leader country in the global economy. In order to simplify the model, the parameters in the volatility and VoV processes in Eqs. (1) and (2) are assumed to be the same across countries. I also assume that there are neither within nor cross-country statistical correlations in the shocks. The only correlations assumed in my model are those implied by the parameters (cid:30) (level) and (cid:30) (volatility) in Eq. (2). These two parameters g (cid:27) control the extent to which the follower country is a⁄ected by the shocks generated in the leader country. In particular, (cid:30) implies that the consumption process of the follower country is a⁄ected (cid:27) 16Bekaert, Engstrom and Xing (2009) survey the evidence on time-varying volatility of consumption for the US. Bansal, et. al. (2005) provide empirical evidence of time-varying macroeconomic uncertainty for the US, Germany, Japan, and the UK. BTZ also (cid:133)nd preliminary empirical evidence on the existence of time-varying VoV for the US. 9

not only by the local macroeconomic uncertainty, but also by that generated in the leader economy. More importantly, the fact that both economies are exposed to the same sources of macroeconomic uncertainty yields the systematic component in both countries(cid:146)variance premiums.17 The unique world representative agent is endowed with Epstein Zin Weil preferences (Epstein and Zin, 1989; and Weil, 1989). That is, her life-time utility function is given by the following equation: U t = [(1 (cid:0) (cid:14))C t 1 (cid:0)(cid:18) (cid:13) +(cid:14)(E t [U t 1 +(cid:0)1 (cid:13) ])(cid:18) 1 ]1 (cid:0) (cid:18) (cid:13); (3) where 0 < (cid:14) < 1 is the time discount rate, (cid:13) 0 is the risk aversion parameter, and (cid:18) = 1 (cid:13) for (cid:21) 1 (cid:0)1 (cid:0) 1 is the intertemporal elasticity of substitution (IES).18 These preferences have the property (cid:21) of assigning non-zero market prices to shocks not directly related to aggregate consumption. This property is crucial to investigate other risk factors such as news related to volatility which is the main objective of this paper. 4.2 Model-Implied Variance Premiums Given the solution of the model in Appendix A, it can be shown that the two countries(cid:146)VoV isolate the variance premium in the global and the local portfolios. The expression for the global portfolio(cid:146)s variance premium is given by19 VP = E Q [Var ] EP[Var ]; w;t t rj;t+1 (cid:0) t rj;t+1 where Var is the conditional variation of returns between time t and t+1 for portfolio j for rj;t j = 1;2;w (see appendix A). The variance premium can be approximated as20 VP ((cid:18) 1)(cid:20) (V q +V q ); (4) w;t w;1 w;1 1;t w;2 2;t (cid:25) (cid:0) 17Theparameters(cid:30) ,and(cid:30) canofcoursebesetto0;acasethatIwillalsoanalyzeinthenumericalsimulationof g (cid:27) themodel. Now,although(cid:30) turnsouttohaveaninsignifcante⁄ectontheroleofinternationalvariancepremium,it g is kept to maintain the possibility of a common level component in consumption. Alternative ways of characterizing the systematic component of the variance premiums outside the simpli(cid:133)cations of a two-country model are being explored in my current research agenda. 18To be coherent with the idea of agents that fear an increase in macroeconomic uncertainty, is assumed to be higher than 1. This assumption accomodates some empirical asset pricing regularities, among them: (i) a positive variancepremium;(ii)thefeedbacke⁄ectbetweenPDratiosandconsumptionvolatility;and(iii)alowrisk-freerate (BY and BTZ). See also Mehra and Prescott (1985) for reasonable values of (cid:13). 19It is important to keep in mind that this is actually the drift di⁄erence of the conditional variance between the two measures. In the case of Gaussian shocks, the level di⁄erence (VarQ(r ) VarP(r )) would be zero (see t+1 t+1 (cid:0) Drechsler and Yaron, (2011)). I intentionally omit the use of models that generate a level di⁄erence in the variance premiumtomaintainthesimplicityoftheexpressionsspeciallygiventhatthemainattentionwillbecenteredonthe qualitative implications of my model and not on the calibration of its parameters. 20The risk neutral probability is replaced by its log-linear approximation: E t Q((cid:27)2 r;t+1 ) (cid:25) log[e(cid:0) rt;tE t (emt=1+(cid:27)2 r;t+1)] (cid:0) 2 1 Var t ((cid:27)2 r;t+1 ): Bear in mind that a closed form solution to the risk neutral variance cannot be obtained in this setting. 10

where ((cid:18) 1)(cid:20) V represents the load of q on VP . For the global portfolio, these loads are w;1 j;k k;t j;t (cid:0) characterized by the following expressions: V = (!+(1 !)(cid:30) )2A +(A2 +A2 ’2)(cid:20)2’2A ; w;1 (cid:27) w;1 w;1 w;2 q 1 q w;2 (cid:0) V = (1 !)2A +(A2 +A2 ’2)(cid:20)2’2A ; w;2 w;3 w;3 w;4 q 1 q w;4 (cid:0) where A ; A ; A and A are respectively the loads of the risk factors (cid:27)2 ; q ; (cid:27)2 ; j;1 j;2 j;3 j;4 1;t+1 1;t+1 2;t+1 q on the wealth-consumption ratio of each portfolio. These loads are derived in detail in 2;t+1 Appendix A. For the leader country, the variance premium is given by VP = E Q [Var ] EP[Var ] (5) 1;t t r1;t+1 (cid:0) t r1;t+1 ((cid:18) 1)k (V q +V q ); w;1 1;1 1;t 1;2 2;t (cid:25) (cid:0) V = A +(A2 +A2 ’2)(cid:20)2 ’ 2A ; 1;1 w;1 1;1 1;2 q 1;1 q 2 V = A2 +(A2 ’2)(cid:20)2 ’2A ; 1;2 1;3 1;4 q 1;1 q w;4 while for the follower country VP = E Q [Var ] EP[Var ] (6) 2;t t r2;t+1 (cid:0) t r2;t+1 ((cid:18) 1)(cid:20) (V q +V q ); w;1 2;1 1;t 2;2 2;t (cid:25) (cid:0) V = (cid:30)2A +(A2 +A2 ’2)(cid:20)2 ’2A ; 2;1 (cid:27) w;1 2;1 2;2 q 2;1 q w;2 V = A +(A2 +A2 ’2)(cid:20)2 ’2A : 2;2 w;3 2;3 2;4 q 2;1 q w;4 Eqs. (4) to (6) imply that the VoV of both countries are the unique sources of the variance premiums in all portfolios. Actually, for (cid:18) < 1, the two countries(cid:146)VoV load positively on the variance premiums. That is, V 0 for j;k = 1;2;w (see appendix A). Consequently, the global j;k (cid:20) and local variance premiums are positive if (cid:18) < 1. Note that while the load of foreign VoV in the leader country variance premium is explained by the recursive nature of the utility function given fully integrated equity markets, the leader country VoV load on the follower country variance premium has the following two sources: the recursive nature of the preferences, and the implied sensitivity to the leader country macroeconomic uncertainty (See Eq. (2)). As an immediate consequence of the common components in the variance premium of all portfolios, the variance premium covariance across countries is uniquely characterized by the two countries(cid:146)VoV. The expression for the variance premium covariance derived from Eqs. (5) and (6) can 11

be written as follows: Cov (VP ;VP ) = ((cid:18) 1)2k2 ’2(V V q +V V q ) (7) t 1;t+1 2;t+1 w;1 q 1;1 2;1 1;t 1;2 2;2 2;t (cid:0) where the VoV of both countries loads positively on the variance premium covariance across countries as long as (cid:18) < 1. 4.3 Model-Implied Equity Premiums In order to understand the relation between the variance premiums and the dynamics of returns, in this section, I (cid:133)nd the expressions for the equity premiums. The global equity premium is characterized by the following expression: EP = E (r r ) (8) w;t t w;t+1 f;t (cid:0) 1 = (cid:13)(cid:27)2 (cid:27)2 w;t (cid:0) 2 w;t +(1 (cid:18))k (P q +P q ); w;1 w;1 1;t w;2 2;t (cid:0) where r is the (log) gross return for portfolio j (j = 1;2;w), r is the global risk-free rate, j;t+1 f;t (cid:27)2 = !(cid:27)2 +(1 !)(cid:27)2 is the volatility of the world consumption, and ( 1(cid:27)2 ) is the geometric w;t 1;t (cid:0) 2;t (cid:0)2 rwt adjustment term. The term (1 (cid:18))k P represents the load of q on EP . For the global w;j j;k k;t j;t (cid:0) portfolio these loads are given by P = k (A2 +A2 ’2); w;1 w;1 w;1 w;2 q P = k (A2 +A2 ’2): w;2 w;1 w;3 w;4 q Equation (8) shows the three model-implied components of the global risk premium. The (cid:133)rst component is the classic risk-return trade-o⁄(cid:13)(cid:27)2 . This (cid:133)rst component is also present when the w;t agents are endowed with CRRA preferences. There are two additional components, one for the VoV generated in each country. The VoV components of the equity premium represent the true premium for variance risk since they are driven by the shocks to the volatility and the volatility of volatility of consumption in both countries. In the case of the global portfolio, the VoV of both countries load positively on the equity premium if (cid:18) < 1. That is, (1 (cid:18))k P 0, for j = 1;2 w;1 w;j (cid:0) (cid:21) (see Appendix A). These positive loads are in line with the concept that, at least for the global portfolio, agents are positively compensated for the risk generated by the time-varying nature of the VoV. The expressions for the equity premiums for each country are given by EP = E (r r ) (9) 1;t t 1;t+1 f;t (cid:0) 1 = (cid:13)(! +(1 !)(cid:30) )(cid:27)2 (cid:27)2 (cid:0) (cid:27) 1;t (cid:0) 2 r1;t +(1 (cid:18))k (P q +P q ); w;1 1;1 1;t 1;2 2;t (cid:0) 12

and EP = E (r r ) (10) 2;t t 2;t+1 f;t (cid:0) 1 = (cid:13)(cid:30) (!+(1 !)(cid:30) )(cid:27)2 +(cid:13)(1 !)(cid:27)2 (cid:27)2 (cid:27) (cid:0) (cid:27) 1;t (cid:0) 2;t (cid:0) 2 r2;t +(1 (cid:18))k (P q +P q ); w;1 2;1 1;t 2;2 2;t (cid:0) where P = k (A A +A A ’2); j;1 j;1 w;1 j;1 w;2 j;2 q P = k (A A +A A ’2); for j = 1;2: j;2 j;1 w;3 j;3 w;4 j;4 q As for the global portfolio, the equity premium in each country is characterized by a volatility of consumption component, and two VoV components, one for each country. In particular, the VoV components in Eqs. (9) and (10) represent the true premium for local and foreign variance risk. Comparing the expressions for the Variance premium (Eqs. (5) and (6)) with those for the equity premiums (Eqs. (9) and (10)) yields the basic intuition for the role of local and foreign variance premium in predicting equity returns in any country. The intuition is as follows: the VPs reveal the VoV in both countries which in turn drives (in part) the time variation in the equity premiums. It is important to bear in mind that although the VoV is not a necessary condition to generate a variance risk premium, introducing the VoV isolates the risk premium on volatility and di⁄erentiate it from the consumption risk premium. It seems natural from Eqs. (8) to (10) to expect the VoV to also explain the time variation in the covariance of returns across countries. The expression for the covariance of returns is given by Cov (r ;r ) = (cid:30) (cid:27)2 +CO q +CO q ; (11) t 1;t+1 2;t+1 (cid:27) 1;t 1 1;t 2 2;t where CO is the load of q on the covariance of returns. These loads are given by j j;t CO = (cid:20) (cid:20) (A A +A A ’2); 1 1;1 2;1 1;1 2;1 1;2 2;2 q CO = (cid:20) (cid:20) (A A +A A ’2): 2 1;1 2;1 1;3 2;3 1;4 2;4 q 4.4 Numerical Implications of the Two-Country Model In this section, I present some numerical simulations of my model in order to investigate the mechanism of transmission of VoV shocks across countries. The purpose of these simulations is to analyze the qualitative implications of my model for the variance premiums and for the interactionbetweenthevariancepremiumsandtheequityreturns. Understandingthesequalitative implications provides a natural step between the model and the empirical evidence presented in the next section. 13

The base scenario for the numerical simulations is displayed in Table 2. In this scenario, the parameters in the preference function are calibrated as in BTZ. In order to simplify the interpretation of results, I consider the hypothetical case where the world is composed of only two countries: the US, and Germany. Just for the purpose of illustrating the mechanism behind the model, the US is considered as the leader economy.21 For these two countries, the parameters in Eqs. (1) and (2) are calibrated as follows: (cid:22) is estimated as the average IP growth in each country during the period j;g 1973-2009; (cid:22) is estimated as the IP growth unconditional variance for the same period; and the j;(cid:27) rest of the parameters are taken from BTZ (homogeneous parameters for the two countries). The Campbell and Shiller constants k and k are estimated using data for the Price-Dividend (PD) o 1 ratioforeachcountryaswellasfortheDatastreamworldportfolio. Thelog-linearizationconstants are estimated as k = eE(PD) , where E(PD) is the unconditional mean of the (log) PD ratio, and 1 1+eE(PD) k = k ln(1 k ) (1 k )ln(1 k )(CampbellandCochrane,1999). Bearinmindthatk andk 0 1 1 1 1 o 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) should actually be made dependent on the theoretical wealth-consumption ratio (see Appendix A). However, I use the unconditionally expected PD ratio, to make these two parameters independent from the set of parameters considered in each case.22 [Insert Table 2 here.] 4.4.1 Variance Premium Dynamics According to the (cid:133)rst main implication of my model, both countries(cid:146)VoV load positively on all portfolios(cid:146)variance premiums (see Eqs. (4) to (6)). In order to show this implication, Figure 4 displays the (unconditionally expected) VoV loads on the variance premium for all portfolios. The (cid:133)gure shows the components of the variance premiums for alternative values of the risk aversion ((cid:13)), the weight of the leader country (!), and the correlation of consumption ((cid:30) ). The simulations (cid:27) show that the implied size of the US VoV load dominates that of Germany in all cases considered. The dominance of the US VoV increases with the relative size of the leader economy (!), and with the relative dependence of the follower economy ((cid:30) ). The contribution of the follower economy (cid:27) VoV in the variance premiums, on the other hand, is almost insigni(cid:133)cant no matter the size nor the independence of the consumption process in this economy. [Insert Figure 4 here.] The simulations also suggest that the magnitude of the expected variance premiums monotonously increases with the risk aversion, and decreases with the relative size of the riskiest market.23 The riskiest market is assumed, for coherence, to be that in the follower country. However, for all cases considered, the average variance premium is quantitatively far from that empirically observed for these two countries (see Table 1). The limitation to quantitatively re(cid:135)ect the observed 21In the following section, the identi(cid:133)cation of the leader economy will be fully given by the empirical evidence. 22A full calibration of my model is out of the scope of this paper. My attention is rather centered in its qualitative implications. These implications explain in turn the main empirical (cid:133)ndings of this paper such as the local predictability puzzle and the ability of the US variance premium to predict all other countries equity returns. 23It is easy to show that the same monotonous relation holds for the IES . Results for the relation between and the model implications are available upon request. 14

premium in models with recursive preferences has been previously documented by Drechsler and Yaron (2011) in a single-country setting. In unreported results, I show that the model-implied variance premium correlation across the twocountriesisabove0.98forallsimulations. Thisresultsistobeexpectedgiventhehighcommon component of the leader country VoV in all variance premiums. Actually, for all cases considered, the model implies that the leader country VoV accounts for more than 99% of the total crosscountry variance premium covariance. Surprisingly, the result on the dominant role of the leader country(cid:146)s VoV holds no matter the relative size of the follower economy ((1 !) < 0:5) or its (cid:0) implied correlation with the leader economy. In sum, the numerical simulations show that the VoV generated in the leader economy accounts for most of the systematic component of the variance premiums. Therefore, the VoV generated in the leader economy plays the key role in explaining the variance premium for all portfolios. This in turn implies that the leader country VoV is also the key driver of the expected variance premium correlation across countries. 4.4.2 Return Dynamics According to the second main implication of my model, the two countries(cid:146)VoV that uniquely characterize the variance premiums also drive the time variation in equity premiums. Figure 5 displaysthemodel-impliedcomponentsoftheequitypremiumfortheglobalandthelocalportfolios foralternativesetsofparameters. ThesimulationsshowthattheleadercountryVoVloaddominates that of the follower country in all portfolios(cid:146)equity premiums for all cases considered. They also reveal that in some cases the VoV of the leader country loads negatively on the follower country(cid:146)s equity premium. This case only occurs when economies are poorly correlated as can be seen in Panel J and K. However, the follower country VoV loads negatively on the leader country(cid:146)s equity premiumforallcasesconsidered, exceptofcoursefortheextremecasewherethesizeofthefollower economy is insigni(cid:133)cant (Panels C,F,I, and L). [Insert Figure 5 here.] The possibility of VoV loading negatively on the equity premiums can actually be explained by the mechanism of transmission of shocks to VoV implied by the model. According to this mechanism, a positive shock to VoV in the follower country has a negative impact on the leader country(cid:146)s equity premium. This e⁄ect can be interpreted as a macroeconomic uncertainty induced (cid:135)ight-to-safety from the follower to the leader economy. The possibility of an uncertainty (cid:135)ightto-safety in this direction is actually generated by the fact that the leader country consumption process is, by construction, not sensitive to the shocks generated in the follower country (Eq. (1)). Investing in equities in the leader country is then expected to become a more attractive investment alternative with respect to this foreign source of risk. In contrast, an uncertainty (cid:135)ight-to-safety in the other direction (leader to follower) is not always possible. This is due to the fact that the follower country consumption process is a⁄ected by the shocks in the leader economy (Eq. (2)). Therefore, a (cid:135)ight-to-safety in this direction is only possible if the economies are assumed to be quite independent from each other. For example, in the case of totally independent economies in 15

PanelJ,anyequitymarketisfreefromtheuncertaintyriskgeneratedintheforeigneconomy. Thus, in this extreme case, the VoV of one country will always load negatively on the other country(cid:146)s equity premium. As a consequence of the second main implication of my model, both countries(cid:146)VoV also play a roleinexplainingthecovarianceofequityreturnsacrosscountries. Asexpected, evenifthefollower economy has a large (relative) size, the VoV of the leader country dominates. The dominance of the leader country VoV increases with the relative size of its economy (!), and the degree of dependence across the two economies ((cid:30) ). In line with the simulations in Figure 5, the VoV (cid:27) generated in the follower country may even load negatively on the covariance of returns. Actually, in the case of totally independent ormildly correlated economies, the simulations con(cid:133)rm that even VoV generated in the leader economy might load negatively on the covariance of returns.24 Finally, the simulations in Figure 6 show the relation between the correlation of the economies andthemodel-impliedcorrelationacrossequitymarkets. Thesesimulationsre(cid:135)ectthedocumented disparity between the correlation of equity markets and the correlation of economies. They show that the equity return correlation is in some cases higher than the implied correlation of consumption. In particular, the simulations suggest that for moderately risk averse agents ((cid:13) > 2) and moderately correlated economies, the implied correlation across equity markets is larger than that implied by the correlation of consumption. This result arrives as a direct consequence of the recursive nature of the representative agent(cid:146)s preferences. [Insert Figure 6 here.] Insum, thenumericalsimulationsshowthattheVoVgeneratedintheleadereconomyplaysthe key role in explaining the time variation in the equity premium of all portfolios. As a consequence of this implication, the leader economy VoV also plays a dominant role in explaining the time variation in equity return correlations across countries. The simulations also show some consequences derived from the model setup. In particular, from the assumptions of integrated markets where the representative agent is endowed with recursive preferences and one economy behaves as a follower. For example, the model introduces the possibility of a macroeconomic uncertainty induced (cid:135)ight-to safety, whichinturnintroducesthepossibilitythattheVoVofonecountrycovariesnegativelywith the equity premium of another country. 5 The Variance Premium and International Equity and Option Markets: Empirical Evidence In this section, I present the empirical evidence based on the qualitative implications of the GE model analyzed in Section 4. Using the variance premiums for all countries in the sample, I investigate their role in (i) explaining the time variation in the variance premium for all other countries, (ii) predicting the variance premium correlations across countries, (iii) predicting not only the local equity returns, but also those in other countries, and (iv) predicting the correlation of equity returns across countries. 24These simulations are left unreported in order to save space. 16

5.1 Cross-country Variance Premium Correlations A(cid:133)rstimplicationofmymodelisthatthevariancepremiumsarehighlycorrelatedacrosscountries. The high variance premium correlation is due to the common load of the leader country VoV in all variance premiums (leader, follower, and global portfolio). This in turn implies that the leader country variance premium plays a key role in predicting the variance premium correlations across countries. In order to analyze this implication, I (cid:133)rst provide evidence for the variance premium correlations across countries. Then, I investigate the role of each country(cid:146)s variance premium in predicting the variance premium correlations with any other country. Table 3 displays the variance premium correlations across all countries in the sample. In line with the (cid:133)rst implication of my model, all countries but Japan show correlations above 0.5. In particular, the US and the UK show a high correlation coe¢ cient of 0.73. Among European markets, France and The Netherlands show the highest correlation coe¢ cient in the sample: 0.89. However, Japan(cid:146)s variance premium shows a relatively low, or even negative, correlation with the variancepremiumofanyothermarketexceptsperhapswithSwitzerland.25 TheevidenceforJapan standsinsharpcontrasttotheimplicationsofthemodel. Infact, mymodelcanonlyaccommodate positive variance premium correlations. This is in turn derived from the ability of my model to characterize only positive variance premiums. [Insert Table 3 here.] The results on the high variance premium correlations has been previously documented in the literatureforashortersampleofcountries. Forexample,Bekaert,HoerovaandScheicher(2009)(cid:133)nd evidence of high risk aversion and uncertainty correlation between Germany and the US. Although their measures are not directly the variance premiums, their empirical methodology uncovers the risk aversion and uncertainty time series using the observed IV and realized volatilities for these twocountries. Sugihara(2010)also(cid:133)ndsevidenceofstronglinkagesinvolatilitypremiumsbetween the US, Germany and Japan. He actually (cid:133)nds empirical evidence that the correlation between these three markets is stronger around certain episodes; in particular, after the subprime crisis. However, in this paper, I not only extend the evidence for a larger sample of countries but also provide a fundamental explanation for the dynamics of the variance premium correlation across countries. In particular, my model relates the high variance premium correlation across countries to a systematic component which is mainly driven by the leader country variance premium. A direct consequence of the common component in all variance premiums is that the variance premium correlations are predicted by the leader country(cid:146)s variance premium. In order to test this consequence, Table 4 reports the estimated coe¢ cients (cid:13) for the following regressions: 1;jk (cid:26) (vp ;vp ) = (cid:13) +(cid:13) vp +(cid:15) ; t j;t;t+1 k;t;t+1 0;jk 1;jk k;t jk;t where the correlation coe¢ cient for the period t to t + 1 is calculated using daily data for the variance premiums of the two countries for the month starting immediately after the realization of 25The highly idiosyncratic dynamic of the variance premium in Japan has been previously documented in the literature (see, for instance, Driessen and Maenhout, (2006)). 17

vp .26;27 The evidence suggests that the US variance premium predicts the one-month-ahead k;t 1 (cid:0) variance premium correlation between the US, Germany, and Japan ((cid:133)rst horizontal block of results).28 The results also show that the US variance premium does not necessarily outperform all other countries(cid:146)variance premium. For example, the (cid:133)rst vertical block of results in the table suggests that the variance premiums in Germany, Japan, the UK, Switzerland and The Netherlands can also forecast the variance premium correlation between these countries and the US.29 [Insert Table 4 here.] In sum, the evidence in this section suggests that the variance premium correlations across countries increase following episodes of increasing variance premiums. It also suggests that the model-implied dominant role of a leader country variance premium might restrict the interpretation of the potential ability of other countries in predicting one-month ahead variance premium correlations. 5.2 Cross-Country Equity Return Correlations The second main implication of my model is that the variance premiums covary with the equity premiums(Eqs. (9)and(10)). ThisisduetothefactthattheVoVshocksthatuniquelycharacterize the variance premiums also load on both countries(cid:146)equity premiums. In particular, the model impliesthattheleadercountry(cid:146)sVoVdominatesthatofthefollowercountryinallequitypremiums. As a consequence, the variance premium of a leader country should outperform that of the follower country in predicting local and foreign returns. In this section, I provide evidence for the role of foreign variance premiums in predicting equity returns for all countries in the sample. Table 5 reports the estimation results for the following regressions: (r r ) = (cid:13) +(cid:13) vp +(cid:13) dy +(cid:13) ts +(cid:15) ; f j;t;t+3 0;j;k 1;j;k k;t 1;j;k j;t 1;j;k j;t j;k;t (cid:0) where (r r ) represents future compounded annualized excess returns 3-months ahead, dy f j;t;t+3 j;t (cid:0) isthelocaldividendyield, andts isthelocaltermspread.30 Ontopofthelocalpredictabilityevij;t dencediscussedinSection3,theresultsinTable5suggestthatonlytheUSvariancepremiumplays a signi(cid:133)cant role in predicting equity returns for all other countries in the sample. Nevertheless, for other pairs of countries, the predictive power of the foreign variance premium over international equity returns holds. This is the case for the signi(cid:133)cant predictive power of the Japanese variance 26The following month (t;t+1) is assumed to be the period 22 days after the realization of vp . k;t 27Equation(7)actuallyhasanimplicationonthevariancepremiumcovariance. Toavoidapotentialscaleproblem, andmakeresultseasiertointerpret,Ionlyreportcross-countrycorrelations. Anexpressionforthevariancepremium correlation from Eq. (7) is direct, although not necessarily linear in VoV. 28In unreported results, I actually show that, except for the variance premium measure based on the martingale assumption, the predictive role of the US variance premium over its correlation with Germany and Japan holds for all alternative variance premium speci(cid:133)cations considered. 29Giventhehighcorrelationinvp acrosscountries,itwouldbehardtodisentanglethesimultaneousroleofvp t US;t with any other vp since multiple regressions will be highly a⁄ected by multicolinearity. j;t 30The evidence suggests that the predictive power of the variance premium is stronger at the quarterly horizon. Thisresultisinlinewiththe(cid:133)ndingsinBTZfortheUSandisdiscussedindetailintheinternationalsettingbelow. 18

premiumovertheequityreturnsofBelgiumandFrance. Itisalsothecaseforthe(oftenborderline) predictive power of the variance premium of all countries, except for Switzerland and Japan, over the US equity returns.31 [Insert Table 5 here.] In order to investigate more in depth the predictive power of the US variance premium over international equity returns, Figure 7 reports the estimation results for the following regressions: (r r ) = (cid:13) +(cid:13) vp +(cid:13) dy +(cid:13) ts +(cid:15) ; f j;t;t+h 0;j;h 1;j;h US;t 1;j;h j;t 1;j;h j;t j;h;t (cid:0) where (r r ) represents future compounded annualized excess returns h-months ahead. The f j;t;t+h (cid:0) resultssuggestthathepredictivepoweroftheUSvariancepremiumforallcountriesexceptperhaps forJapanresemblesthehump-shapedpatternfoundbyBTZfortheUS(localreturnpredictability). Thispatternre(cid:135)ectsthefactthatthevariancepremiumshouldbeadominantpredictorforhorizons where the VoV is the main source of variation in equity returns. The extension of this evidence for other countries indicates that the US VoV is the dominant source of variation in all countries(cid:146) equity returns for horizons between 3 and 6 months. The (cid:133)gure also suggests that the predictive power of the US variance premium is complementary to that of local term spreads and dividend yields.32 When compared to Figure 3, the evidence also suggest that the US variance premium outperforms the local variance premiums in predicting equity returns for all countries considered. In unreported results, I show that the ability of the US variance premium to predict one-quarter ahead foreign returns holds if a noise signal is added to the original variance premium. For all countries, except perhaps for the Netherlands and Japan, the standard deviation of the noise signal has to be at least 50% that of the original US variance premium before its predictive power disappears.33 Moreover, the predictive power of the US variance premium holds for all alternative variance premium speci(cid:133)cations considered, except perhaps for the range-based estimation.34;35 [Insert Figure 7 here.] 31In fact, in unreported results, I show that not even a proxy for the world variance premium (with and without theUS)isabletosigni(cid:133)cantlypredictequityreturnsforallothercountriesinthesample. Thisevidenceisincontrast withthatinBollerslev,et.al. (2011). They(cid:133)ndavalueweightedVP,wheretheVPismeasuredusingtheMartingale assumption as in BTZ, to have predictive power over future equity returns for all countries in our sample except for Belgium and the Netherlands for a sample period between 2000 and 2010. 32The hump-shaped predictability pattern, as well as the signi(cid:133)cance of the US variance premium in predicting foreign equity returns is robust to considering the US term spread and dividend yield. Results for these regressions are available upon request. 33For the Netherlands and Japan, adding any noise to the US variance premium almost immediately weakens its predictive power. In contrast, for the UK, the standard deviation of the noise signal has to be at least 70% that of the original variance premium before its predictive power disappears. 34When the range based forecast of realized volatility is used, the US variance premium predicts returns only for the UK, Belgium and France. 35Results for the robustness tests are left unreported in order to save space and center the discussion. The results forthenoisestresstests,thealternativevariancepremiumspeci(cid:133)cations,samples,currencies,aswellasforalternative variance covariance matrix approximations (In particular, Hodrick, (1992)) are available upon request. 19

As a consequence of the systematic component of equity premiums, the leader country variance premium should also be a useful predictor of equity return correlations across countries (Eq. (11)). In order to test this consequence, Figure 8 reports the estimation results for the following regressions: (cid:26) (r ;r ) = (cid:13) +(cid:13) vp +(cid:15) ; t j;t;t+h US;t;t+h 0;jk 1;j;US US;t jk;t where (cid:26) (r ;r ) is the h-months ahead equity return correlation between any country t j;t;t+h US;t;t+h and the US. The results suggest that the US variance premium predicts equity return correlations between the US and any other country in the sample except for Japan and Belgium. As for the equity returns, the ability of the US variance premium to forecast return correlations holds for horizons between 3 and 6 months for most of the countries. Actually, for the equity correlation between the US and Germany, the US variance premium has predictive power for horizons up to 12 months. In unreported results, I also show that the US variance premium outperforms all other countries in the sample in predicting equity return correlations. [Insert Figure 8 here.] In sum, the evidence in this section supports the qualitative implications of my model for the role of the variance premium in predicting international equity returns. In particular, this evidence con(cid:133)rms the predominant role of the US variance premium in predicting foreign equity returns and return correlations across countries. Therefore, the evidence supports the theoretical solution implied by my model to the local return predictability puzzle in Section 3. That is, the local variance premium cannot predict returns in countries other than the US because the role of the variance premium in those countries is dominated by the variance premium in a leader country: the US. 6 Conclusions This paper presents several new (cid:133)ndings related to the variance risk premium for a total of eight countries. First, I provide new evidence that the variance premiums display signi(cid:133)cant time variation and are, on average, positive for all countries analyzed. However, I also provide evidence that except for the US and Belgium, the local variance premiums do not predict local equity returns. This evidence is in sharp contrast to the existing theoretical models where the variance premium explains the time variation in equity returns. Motivated by the puzzling single-country evidence, I propose an international model to understand the role of the variance premium in explaining international equity returns. My model is a two-country general equilibrium model which extends that in Bollerslev, Tauchen and Zhou (2009). It yields relevant qualitative implications that explain the inability of the variance premium in predicting local returns in countries other than the US. In particular, my model implies that the variance premium generated in a leader economy plays a key role in explaining the time variation inequityreturnsinthetwocountries. Therefore, theleadercountryvariancepremiumoutperforms the follower country variance premium in predicting not only equity returns, but also equity return 20

correlations across countries. The dominant role of the leader country variance is a consequence of the common components in the variance premiums of all countries. In particular, a consequence of the dominant load of macroeconomic uncertainty shocks generated in the leader economy in the variance premiums of both countries. Finally, I provide new empirical evidence for the qualitative implications of my model for the eight countries in the sample. I show that the US variance premium has predictive power over the equityreturnsforallcountriesinthesample. ThepredictivepoweroftheUSvariancepremiumover international equity returns is (i) stronger for horizons between 3 and 6 months, (ii) additional to that of traditional local (or US) variables, and (iii) clearly outperforms the local variance premium themselves. Finally, I also show that the US variance premium predicts the correlation of equity returns between the US and all countries in the sample, except for Japan and Belgium. 21

APPENDIX A Detailed Solution of the Two-country Model This appendix explains in detail the solution to the model in Section 4. Each country return process is assumed to be a claim on the local consumption growth, while the global portfolio return is a claim on the weighted global consumption gw = !g1 +(1 !)g2, t t t (cid:0) where ! is the weight of the leader country. Following Campbell and Shiller (1988), the returns are linearized as r = (cid:20) +(cid:20) z z +g ; for j = 1;2;w; (12) j;t+1 j;0 j;1 j;t+1 j;t j;t+1 (cid:0) where z denotes the log of the wealth-consumption ratio of the asset that pays the consumption j;t endowment C . As it is standard in the asset pricing literature, I conjecture a solution for f j;t+i g 1i=1 z as a function of the state variables of both countries as follows: j;t z = A +A (cid:27)2 +A q +A (cid:27)2 +A q : (13) j;t+1 j;0 j;1 1;t+1 j;2 1;t+1 j;3 2;t+1 j;4 2;t+1 Based on this solution, the basic asset pricing equation is imposed in order to determine the components of z . The basic asset pricing equation is the (cid:133)rst order condition from the agent j;t+1 maximization problem given by E [(exp(m +r )] = 1; t t+1 j;t+1 where m is the (log of) intertemporal marginal rate of substitution. For the case of Epsteint+1 Zin-Weil preferences, and given that markets are assumed to be perfectly integrated, the unique marginal rate of substitution is given by (cid:18) m = (cid:18)log(cid:14) g +((cid:18) 1)r t+1 t+1 t+1 (cid:0) (cid:0) = b +b g +b r ; mo mg w;t+1 mr w;t+1 where 0 < (cid:14) < 1 is the time discount rate, (cid:13) 0 is the risk aversion parameter, and (cid:18) = 1 (cid:13) for (cid:21) 1 (cid:0)1 (cid:0) 1 is the intertemporal elasticity of substitution (IES). (cid:21) Solving for the world portfolio yields the following expressions for the components of z : j;t+1 (cid:18)log(cid:14)+(1 (cid:13))(!(cid:22) +(1 !)((cid:22) +(cid:30) (cid:22) )) 1;g 2;g g 1;g A = (cid:0) (cid:0) w;0 (cid:18)(1 (cid:20) ) w;1 (cid:0) (cid:20) +(cid:20) A a +(cid:20) A a +(cid:20) A a +(cid:20) A a w;0 w;1 w;1 (cid:27) w;1 w;2 q w;1 w;3 (cid:27) w;1 w;4 q + ; (1 (cid:20) ) w;1 (cid:0) 22

(1 (cid:13))2(!+(1 !)(cid:30) )2 A = (cid:0) (cid:0) (cid:27) ; 1 2(cid:18)(1 (cid:20) (cid:26) ) w;1 (cid:27) (cid:0) (1 (cid:20) (cid:26) ) (1 (cid:20) (cid:26) )2 (cid:18)2(cid:20)2 ’2A2 A+ w; ; 2(cid:0) = (cid:0) w;1 q (cid:6) q (cid:18) (cid:0) (cid:20)2 w ’ ;1 2 q (cid:0) w;1 q w;1 ; w;1 q (1 (cid:13))2(1 !)2 A = (cid:0) (cid:0) ; w;3 2(cid:18)(1 (cid:20) (cid:26) ) w;1 (cid:27) (cid:0) and (1 (cid:20) (cid:26) ) (1 (cid:20) (cid:26) )2 (cid:18)2(cid:20)2 ’2A2 A+ w; ; 4(cid:0) = (cid:0) w;1 q (cid:6) q (cid:18) (cid:0) (cid:20)2 w ’ ;1 2 q (cid:0) w;1 q w;3 : w;1 q Toavoidtheloadoftime-varyingvolatilities(cid:27) ,and(cid:27) fromgrowingwithoutbounds,Ionlykeep 1;t 2;t A (A ). Thepositiverootdiscardedisexplosivein’ ,i.e.,lim A+ ’ = 0(lim A+ ’ = (cid:0)w;2 (cid:0)w;4 q ’ q! 0 w;2 q 6 ’ q! 0 w;4 q 6 0). A (A ) will be a solution to the model as long as (1 (cid:20) (cid:26) )2 (cid:18)2(cid:20)2 ’2A2 ((1 (cid:0)w;2 (cid:0)w;4 (cid:0) w;1 q (cid:21) w;1 q w;1 (cid:0) (cid:20) (cid:26) )2 (cid:18)2(cid:20)2 ’2A2 ). It is easy to show from these expressions that all state variables load w;1 q (cid:21) w;1 q w;3 negatively on the global wealth-consumption ratio. That is, A , A , A , A 0 as long as w;1 w;2 w;3 w;4 (cid:20) (cid:18) < 1. Solving for the leader country 1 yields the following expressions: (cid:20) +(cid:20) A a +(cid:20) A a +(cid:20) A a2 +(cid:20) A a +(cid:22) 1;0 1;1 1;1 (cid:27) 1;1 1;2 q 1;1 1;3 (cid:27) 1;1 1;4 q 1;g A = 1;0 (1 (cid:20) ) 1;1 (cid:0) (cid:20) +((cid:20) 1)A +(cid:20) A a +(cid:20) A a + w;0 w;1 w;0 w;1 w;1 (cid:27) w;1 w;2 q (cid:0) +(cid:20) A a +(cid:20) A a +!(cid:22) +(1 !)((cid:22) +(cid:30) (cid:22) ) w;1 w;3 (cid:27) w;1 w;4 q 1;g 2;g g 1;g (cid:0) ; (cid:0) (1 (cid:20) ) 1;1 (cid:0) (1 (cid:18))(1 (cid:13))2(!+(1 !)(cid:30) )2+(cid:18)(1 (cid:13)(!+(1 !)(cid:30) ))2 A = (cid:0) (cid:0) (cid:0) (cid:27) (cid:0) (cid:0) (cid:27) ; 1;1 2(cid:18)(1 (cid:20) (cid:26) ) 1;1 (cid:27) (cid:0) (1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2 A+ 1; ; 2(cid:0) = (cid:0) 1;1 q (cid:20)2 (cid:0) ’2 1;1 w;1 w;2 q 1;1 q ((1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2)2 (cid:20)2 ’2(((cid:18) 1)2(cid:20)2A2 ’2+ (cid:0) 1;1 q (cid:0) 1;1 w;1 w;2 q (cid:0) 1;1 q (cid:0) 1 w;2 q +2((cid:20) (cid:26) 1)((cid:18) 1)A +((cid:20) A +((cid:18) 1)(cid:20) A )2) s w;1 q w;2 1;1 1;1 w;1 w;1 (cid:0) (cid:0) (cid:0) ; (cid:6) (cid:20)2 ’2 1;1 q (1 (cid:18))(1 (cid:20) (cid:26) )A + 1(cid:13)2(1 !)2 A = (cid:0) (cid:0) w;1 (cid:27) w;3 2 (cid:0) ; 1;3 (1 (cid:20) (cid:26) ) 1;1 (cid:27) (cid:0) 23

and (1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2 A + 1; ; 4(cid:0) = (cid:0) 1;1 q (cid:20)2 (cid:0) ’2 1;1 w;1 w;4 q 1;1 q ((1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2)2 (cid:20)2 ’2[((cid:18) 1)2(cid:20)2 ’2A2 + (cid:0) 1;1 q (cid:0) 1;1 w;1 w;4 q (cid:0) 1;1 q (cid:0) w;1 q w;4 +2((cid:20) (cid:26) 1)((cid:18) 1)A +((cid:20) A +((cid:18) 1)(cid:20) A )2] s w;1 q w;4 1;1 1;3 w;1 w;3 (cid:0) (cid:0) (cid:0) : (cid:6) (cid:20)2 ’2 1;1 q Finally, for the follower country 2; solving the basic asset pricing equation yields (cid:20) +(cid:20) A a +(cid:20) A a +(cid:20) A a +(cid:20) A a +(cid:22) +(cid:30) (cid:22) 2;0 2;1 2;1 (cid:27) 2;1 2;2 q 2;1 2;3 (cid:27) 2;1 2;4 q 2;g g 1;g A = 2;0 (1 (cid:20) ) 2;1 (cid:0) (cid:20) +(cid:20) A +(cid:20) A a +(cid:20) A a + w;0 w;1 w;0 w;1 w;1 (cid:27) w;1 w;2 q +(cid:20) A a +(cid:20) A a A +!(cid:22) +(1 !)((cid:22) +(cid:30) (cid:22) ) w;1 w;3 (cid:27) w;1 w;4 q w;0 1g 2;g g 1;g (cid:0) (cid:0) ; (cid:0) (1 (cid:20) ) 2;1 (cid:0) (1 (cid:18))(1 (cid:13))2(!+(1 !)(cid:30) )2+(cid:18)((cid:30) (cid:13)(!+(1 !)(cid:30) ))2 A = (cid:0) (cid:0) (cid:0) (cid:27) (cid:27) (cid:0) (cid:0) (cid:27) ; 2;1 2(cid:18)(1 (cid:20) (cid:26) ) 2;1 (cid:27) (cid:0) (1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2 A 2 + ;2 ; (cid:0) = (cid:0) 2;1 q (cid:20)2 (cid:0) ’2 w;1 2;1 w;2 q 2;1 q ((1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2)2 2’2(cid:20)2 (((cid:18) 1)((cid:20) (cid:26) 1)A + (cid:0) 2;1 q (cid:0) w;1 2;1 w2 q (cid:0) q 2;1 (cid:0) w;1 q (cid:0) w;2 +1(((cid:18) 1)(cid:20) A +(cid:20) A )2+ 1’2((cid:18) 1)2(cid:20)2 A2 ) s 2 (cid:0) w;1 w;1 2;1 2;1 2 q (cid:0) w;1 w;2 ; (cid:6) ((cid:20)2)2’2 1 q (1 (cid:18))(1 (cid:20) (cid:26)2)A + 1(1 (cid:13)(1 !))2 A = (cid:0) (cid:0) w;1 (cid:27) w;3 2 (cid:0) (cid:0) ; 2;3 (1 (cid:20) (cid:26)2) 2;1 (cid:27) (cid:0) and (1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2 A + 2; ; 4(cid:0) = (cid:0) 2;1 q (cid:20)2 (cid:0) ’2 w;1 2;1 w;4 q 2;1 q ((1 (cid:20) (cid:26) )+(1 (cid:18))(cid:20) (cid:20) A ’2)2 2’2(cid:20)2 [2((cid:20) (cid:26) 1)((cid:18) 1)A + (cid:0) 2;1 q (cid:0) 1 2;1 w;4 q (cid:0) q 2;1 w;1 q (cid:0) (cid:0) w;4 +[((cid:18) 1)(cid:20) A +(cid:20) A ]2+’2((cid:18) 1)2(cid:20)2 A2 ] s (cid:0) w;1 w;3 2;1 w;3 q (cid:0) w;1 w;4 : (cid:6) (cid:20)2 ’2 2;1 q Again, following the reasoning for the world portfolio, it only makes sense to keep A and A . (cid:0)j;2 (cid:0)j;4 24

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smuimerP )ytilitaloV dna( ecnairaV .scitsitatS yrammuS :1 elbaT ehtrof) vr( E vi = plovsaylhtnomdetaluclac)segatnecreplaunnani(smuimerpytilitalovegarevaehtstroperelbatehT 1+t;j t t;j t;j (cid:0) ni( smuimerp ecnairav eht rof scitsitats yrammus eht stroper osla tI .9002 ot 0002 doirep elpmas eht rof deredisnoc seirtnuoc thgie kramhcneb eht erehw;2) vr( 2vi = pv sa detamitse si yrtnuoc hcae ni muimerp ecnairav ehT .)segatnecrep derauqs launna 1+tj (cid:0)t;j t;j evissergerotuaredro-tsr(cid:133)detamitseehtsi)1(RA .tsacerofevissergerotuaredrotsr(cid:133)stisiecnairavdezilaerdetcepxeehtrofnoitac(cid:133)iceps rof htob ,slevel ecned(cid:133)noc %b01 dna 5 ,1 dradnats eht ta ecnac(cid:133)ingis tneserper *** dna **,* .muimerp ecnairav eht fo tneic ¢eoc naem dradnats a mrofrep I ,muimerp ecnairav dna ytilitalov egareva eht roF .tneic ¢eoc )1(RA eht dna naem eht fo ecnac(cid:133)ingis eht evissergerotua eht fo ecnac(cid:133)ingis eht no ecnedive eht nevig( sgal 21 htiw tseW-yeweN gnisu snoitaived dradnats eht tcerroc dna tset CAH tseW-yeweN eht gnisu detcerroc osla era tneic ¢eoc )1(RA eht fo ecnac(cid:133)ingis eht ssessa ot srorre dradnats ehT .)tneic ¢eoc .)7891 ,tseW dna yeweN( sgal ylhtnom 21 htiw )noitcerroc noitalerrocotua dna yticitsadeksoreteH( muimerP ecnairaV ytilitaloV muimerP )1(RA .wekS .truK .veD .tS .xaM .niM )% .qs( naeM )%( naeM xednI xednI VI yrtnuoC 83:0 54:0 09:11 55:752 26:5511 53:5621 23:421 81:3 005 P&S XIV SU (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) (cid:0) 82:0 01:2 65:01 82:882 07:2161 49:356 52:241 24:2 03 XAD XADV REG (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) 47:0 19:3 46:12 34:584 42:8933 64:342 53:752 78:3 522 IEKKIN IXV PAJ (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) 55:0 80:1 82:8 71:232 85:3201 60:018 86:251 04:3 001 ESTF ESTFV KU (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) 17:0 21:2 07:7 09:842 80:4511 81:482 84:041 27:2 IMS IMSV IWS (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) 34:0 67:0 40:01 27:923 63:0241 97:7531 59:461 42:3 52 XEA XEAV LN (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) 22:0 27:0 62:61 93:062 43:0811 24:1941 48:76 17:1 02 LEB LEBV EB (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) (cid:0) 72:0 63:0 13:81 82:223 54:5081 27:1471 78:511 13:2 001 CAC CACV RF (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) 29

Table 2: Base Scenario for the Numerical Implications of the Two-Country Model The table reports the values for the two-country model parameters considered as the base scenario to test its numerical implications. In this scenario, all parameters in the preference function (Eq. (3)) are taken fromBTZ.Thecountry-speci(cid:133)cparametersinEqs. (1)and(2)areestimatedasfollows: (cid:22) isestimatedas j;g the average IP growth for the sample 1973-2009; (cid:22) is estimated as the IP growth unconditional variance j;(cid:27) for the sample 1973-2009. Finally, the parameters k and k in the log-linearization of returns (Eq. (12)) o 1 are estimated using data for the Price-Dividend (PD) ratio for each country as well as for the Datastream world portfolio. The log-linearization constants are estimated as k = eE(PD) , where E(PD) is the 1 1+eE(PD) unconditional mean of the (log) PD ratio, and k = k ln(1 k ) (1 k )ln(1 k ) (Campbell and 0 1 1 1 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) Cochrane, 1999). Value Param. Global US GER Description (cid:22) 1:6 10 3 8:3 10 4 Mean consumption growth g (cid:0) (cid:0) (cid:2) (cid:2) a 1:2 10 6 6:6 10 6 Long-run consumption volatility (cid:27) (cid:0) (cid:0) (cid:2) (cid:2) (cid:26) 0:98 0:98 Speed of reversion consumption volatility (cid:27) a 2:0 10 7 2:0 10 7 Long-run VoV q (cid:0) (cid:0) (cid:2) (cid:2) (cid:26) 0:80 0:80 Speed of reversion VoV q k 0:12 0:13 0:12 Campbell-Shiller k 0 0 k 0:97 0:97 0:97 Campbell-Shiller k 1 1 2:50 Intertemporal elasticity of substitution log(cid:14) 1:00 Discount factor Table 3: Variance Premium Correlations across countries The table reports the correlation coe¢ cients among the monthly variance premiums for all countries for the sample period 2000 to 2009. US GER JAP UK SWI NL BE FR US 1:00 0:56 0:08 0:74 0:37 0:76 0:62 0:78 (cid:0) GER 1:00 0:24 0:74 0:79 0:82 0:42 0:78 JAP 1:00 0:18 0:61 0:07 0:07 0:14 (cid:0) UK 1:00 0:70 0:85 0:64 0:77 SWI 1:00 0:67 0:48 0:51 NL 1:00 0:72 0:89 BE 1:00 0:57 FR 1:00 30

seirtnuoC ssorca snoitalerroC muimerP ecnairaV gnitciderP :4 elbaT :noisserger gniwollof eht ni (cid:13) stneic ¢eoc detamitse eht stroper elbat ehT k;j;1 ; (cid:15)+ pv (cid:13)+ (cid:13) = ) pv; pv( (cid:26) t;k;j 1 t;k k;j;1 kj;0 1+t;t;k 1+t;t;j t (cid:0) swohsPAJnmulocehtrednunoitamrofnieht,elpmaxeroF .sworehtniseirtnuoceht,kdnasnmulocehtniseirtnuocehtera jerehw yrtnuoc rettal eht yb detsacerof si )k( swor eht ni yrtnuoc rehto yna dna )j( napaJ neewteb noitalerroc muimerp ecnairav eht woh eht fo smuimerp ecnairav eht rof atad yliad gnisu detaluclac si 1+t ot t doirep eht rof tneic ¢eoc noitalerroc ehT .muimerp ecnairav detcerroc era snoisserger lla ni srorre dradnats ehT . pv fo noitazilaer eht retfa yletaidemmi gnitrats htnom eht rof seirtnuoc owt 1 t;k (cid:0) derauqsylhtnomninekaterasmuimerpecnairaveht,terpretniotreisaestneic ¢eocehtekamotredronI .sgal21htiwtseW-yeweNyb .segatnecrep ) pv; pv((cid:26) 1+t;t;k 1+t;t;j 2R .gvA RF EB LN IWS KU PAJ REG SU pv 1 t;k (cid:0) 22:8 04:64 43:03 45:62 26:82 35:54 25:05 SU (cid:3)(cid:3) (cid:3)(cid:3) )14:0( )84:1( )16:1( )24:1( )85:1( )30:2( )50:2( 80:2 41:0 03:2 40:2 61:1 88:1 05:2 75:4 2R 23:01 23:42 22:91 25:6 13:13 61:22 32:83 REG (cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) )98:0( )70:1( )48:1( )26:0( )85:2( )04:1( )25:3( 28:1 35:0 92:1 19:1 71:0 15:3 69:0 04:4 2R 48:73 99:74 99:63 67:55 36:64 09:03 54:73 PAJ (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) )89:2( )80:3( )76:2( )12:4( )15:3( )77:2( )13:2( 31:5 40:4 17:5 42:4 89:7 64:6 59:2 25:4 2R 33:8 32:0 88:42 42:3 08:66 52:72 54:93 KU (cid:3) (cid:3)(cid:3)(cid:3) (cid:3) (cid:3)(cid:3)(cid:3) (cid:0) (cid:0) )75:0 ( )10:0 ( )48:1( )52:0( )49:2( )77:1( )82:3( (cid:0) (cid:0) 90:2 32:0 00:0 51:2 30:0 00:6 19:1 23:4 2R 97:8 32:2 07:81 61:51 41:24 59:61 94:12 IWS (cid:3) (cid:3) (cid:3)(cid:3) (cid:3)(cid:3) )39:0( )70:0( )47:1( )87:1( )90:2( )06:1( )13:2( 00:1 62:0 10:0 22:1 38:0 25:2 20:1 21:1 2R 02:5 24:92 08:51 17:22 66:41 80:12 23:12 LN (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) )85:0( )55:1( )81:1( )79:1( )85:0( )12:2( )31:2( 73:1 71:0 48:1 59:0 04:2 04:0 22:2 36:1 2R 07:52 70:43 23:81 48:31 29:41 80:6 04:6 EB (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )06:0 ( )46:0 ( )05:0 ( )63:0 ( )33:0 ( )31:0 ( )71:0 ( (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 43:0 85:0 90:1 13:0 81:0 51:0 30:0 30:0 2R 84:2 59:5 02:12 27:6 42:91 99:1 95:0 RF (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )21:0 ( )25:0 ( )33:1 ( )84:0 ( )78:0( )81:0 ( )40:0 ( (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 03:0 10:0 81:0 82:1 51:0 94:0 10:0 00:0 2R 31

snruteR ytiuqE lanoitanretnI gnitciderP :5 elbaT :snoisserger gniwollof eht ni (cid:13) tneic ¢eoc detamitse eht stroper elbaT ehT k;j;1 ; (cid:15)+ st (cid:13)+ yd (cid:13)+ pv (cid:13)+ (cid:13) = ) r r( t;k;j t;j k;j;1 t;j k;j;1 t;k k;j;1 k;j;0 3+t;t;j f (cid:0) )lacol( eht st dna sdleiy dnedivid )lacol( eht era yd ,snruter ssecxe )dezilaunna dednuopmoc( shtnom-3 era ) r r( erehw t;j t;j 3+t;t;j f (cid:0) ybdetcerrocerasrorredradnatsehT .etarllib-tshtnom3ehtdnallib-Traey1ehtneewtebecnere⁄idehtsadetaluclacsdaerpsmret ecnairav s(cid:146)yrtnuoc hcae fo rewop evitciderp eht rof 2R egareva eht stroper osla elbat ehT .21 = l sgal fo rebmun a htiw tseW-yeweN .muimerp ) r r( h+t;t;j f (cid:0) 2R .gvA RF EB LN IWS KU PAJ REG SU pv 1 t;k (cid:0) 40:0 40:0 30:0 30:0 30:0 30:0 40:0 50:0 SU (cid:3)(cid:3) (cid:3)(cid:3) (cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) )35:2( )05:2( )69:1( )16:2( )58:3( )76:1( )05:2( )49:4( 31:21 77:21 11:9 48:6 93:11 61:51 39:7 49:01 78:22 2R 00:0 00:0 10:0 00:0 10:0 10:0 00:0 20:0 REG (cid:3) (cid:0) (cid:0) )92:0( )52:0( )59:0 ( )71:0( )56:0( )45:0 ( )21:0 ( )07:1( (cid:0) (cid:0) (cid:0) 71:6 78:5 68:2 22:4 31:6 66:7 03:4 42:5 50:31 2R 20:0 20:0 20:0 10:0 00:0 10:0 30:0 10:0 PAJ (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )57:2 ( )00:2 ( )54:1 ( )10:1 ( )85:0 ( )28:0 ( )17:1 ( )51:1 ( (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 87:7 78:9 87:5 87:5 47:7 75:7 50:5 76:9 08:01 2R 20:0 20:0 00:0 20:0 20:0 00:0 10:0 30:0 KU (cid:3)(cid:3) )19:0( )35:0( )30:0 ( )69:0( )59:0( )90:0( )62:0( )73:2( (cid:0) 27:6 03:7 24:3 75:3 03:7 36:8 30:4 93:5 41:41 10:0 10:0 30:0 10:0 00:0 20:0 40:0 00:0 IWS (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) )99:0 ( )85:0 ( )93:1 ( )55:0 ( )13:0 ( )67:0 ( )54:1 ( )53:0( (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 44:6 34:6 22:3 37:5 53:6 82:7 57:4 29:7 48:9 2R 20:0 20:0 00:0 10:0 20:0 10:0 20:0 30:0 LN (cid:3)(cid:3)(cid:3) )41:1( )12:1( )42:0( )90:1( )94:1( )06:0( )09:0( )68:2( 00:8 49:7 64:5 76:3 97:7 50:01 87:4 46:6 07:71 2R 10:0 30:0 10:0 10:0 10:0 10:0 10:0 10:0 EB (cid:3)(cid:3)(cid:3) (cid:3) )39:0( )60:3( )56:0( )50:1( )05:1( )95:0( )48:0( )56:1( 75:6 24:6 35:6 08:3 16:6 01:8 24:4 86:5 20:11 2R 20:0 20:0 10:0 10:0 20:0 10:0 20:0 30:0 RF (cid:3)(cid:3)(cid:3) )75:1( )72:1( )66:0( )92:1( )46:1( )77:0( )99:0( )48:2( 17:8 68:9 55:5 71:4 70:8 74:01 61:5 72:7 41:91 2R 32

Figure 1: Estimated (model-free) Variance Premiums The (cid:133)gure shows the Variance Premiums vp in annual squared percentages for the eight countries in the t sample(seeTable1)forthesampleperiodbetween2000and2009. Thevariancepremiumineachcountryis estimated as vp = iv2 (rv )2;where the benchmark speci(cid:133)cation for the expected realized variance j;t j;t(cid:0) jt+1 is its (cid:133)rst order autoregressive forecast. The shaded areas represent NBER recession episodes for the US. b 33

snoitac(cid:133)iceps evitanretlA .smuimerP ecnairaV egareva eht fo ecnac(cid:133)ingiS :2 erugiF elpmas eht rof )serauqs dlob ni( snoitac(cid:133)iceps evitanretla 4 rof yrtnuoc hcae rof smuimerp ecnairav egareva eht stroper erug(cid:133) ehT egareva ehtfo ecnac(cid:133)ingis ehtrof slavretni ecned(cid:133)noc %59ehttroperoslaI ,yrtnuoc hcae dnaerusaem hcaeroF .9002ot0002 doirep redro tsr(cid:133) sti sa detamitse si ecnairav dezilaer detcepxe eht erehw ))1(RA( erusaem kramhcneb eht si 1 erusaeM .muimerp ecnairav vr yb deixorp-llew si erusaem lacisyhp eht rednu ytilitalov eht fo noitatcepxe eht taht semussa 2 erusaeM .tsacerof evissergerotua t secidni VI eht sedulcni taht noisserger a morf detamitse si ecnairav dezilaer detcepxe eht ,3 erusaem nI .noitpmussa elagnitram ro taht noisserger a morf detamitse si ecnairav dezilaer detcepxe eht ,4 erusaem ni ,yllaniF . (cid:15)+ vi (cid:13)+ vr (cid:13)+ (cid:13) = vr ni sa t t;j 2 t;j 1 o 1+t;j eht si VegnaR erehw ; (cid:15)+ VegnaR (cid:13)+ vr (cid:13)+ (cid:13) = vr ni sa yrtnuoc hcae rof ecnairav desab-egnar ylhtnom eht sedulcni t;j t t;j 2 t;j 1 o 1+t;j dna tsehgih eht neewteb ecnere⁄id yliad eht si egnar erehw ;2egnar tN 1 = VegnaR sa detaluclac ecnairav desab egnar it it 1=it 2nl4 t;j .xedni eht fo ecirp tsewol eht P 34

snruteR ytiuqE lacoL gnitciderP ni muimerP ecnairaV lacoL eht fo elor ehT :3 erugiF :snoisserger gniwollof eht ni (cid:13) stneic ¢eoc detamitse eht stroper erugiF ehT h;j;1 ; (cid:15)+ st (cid:13)+ yd (cid:13)+ pv (cid:13)+ (cid:13) = ) r r( t;h;j t;j h;j;;3 t;j h;j;;2 t;j h;j;;1 h;j;;0 h+t;t;j f (cid:0) eht era st dna sdleiy dnedivid lacol eht era yd ,snruter ssecxe )dezilaunna dednuopmoc( shtnom-h era ) r r( erehw t;j t;j h+t;t;j f (cid:0) gnitsacerof ylhtnom redisnoc I .etar llib-t shtnom 3 eht dna llib-T raey 1 eht neewteb ecnere⁄id eht sa detaluclac sdaerps mret lacol srorre dradnats detcerroc tseW-yeweN eht rof slavretni ecned(cid:133)noc %59 eht tneserper saera dedahs ehT .shtnom 21 ot pu snoziroh ot redro nI .noisserger hcae rof 2R eht sixa yradnoces eht ni stroper osla erug(cid:133) ehT . 21;h2 xam = l sgal fo rebmun a htiw g f ecnairav eht ylno hcihw ni snoisserger rof detroper era 2R eht ,muimerp ecnairav eht fo rewop evitciderp eht yfitnedi yletarapes : (cid:15)+ pv (cid:13)+ (cid:13) = )fr r( ni sa deredisnoc era smuimerp t;h;j t;j h;j;1 h;j;0 h+t;t;j (cid:0) 35

deunitnoC .snruteR ytiuqE lacoL gnitciderP ni muimerP ecnairaV lacoL eht fo elor ehT :3 erugiF 36

sretemaraP evitanretlA .smuimerP ecnairaV eht no VoV fo sdaol :4 erugiF hcae fo muimerp ecnairav eht no )VoV( ytilitalov fo ytilitalov(cid:146)seirtnuoc owt eht fo sdaol lanoitidnocnu latot eht stroper erug(cid:133) ehT detaluclac era sdaol lanoitidnocnu ehT : (cid:30) dna ,! ;(cid:13) sretemarap fo seulav evitanretla rof )6( ot )4( .sqE yb deilpmi sa oiloftrop (cid:27) :soiloftrop elbissop eerht eht fo eno hcae j dna ;REG;SU = k rof ) q(E V (cid:20))1 (cid:18)( dna ) q(E V (cid:20))1 (cid:18)( sa t;REG k;j 1;w t;SU k;j 1;w (cid:0) (cid:0) = ) q(E yb nevig si VoV egareva eht ,)2 elbaT( oiranecs esab eht ni sretemarap eht neviG .oiloftrop labolg eht dna ,REG ,SU t;SU :6 01 0:1 = qa = ) q(E (cid:0) (cid:2) q (cid:26) (cid:0) 1 t;REG 37

deunitnoC .sretemaraP evitanretlA .smuimerP ecnairaV eht no VoV fo sdaol :4 erugiF 38

sretemaraP evitanretlA .smuimerP ytiuqE no sdaol VoV dna ytilitaloV noitpmusnoC :5 erugiF rof )01( ot )8( .sqE yb deilpmi sa smuimerp ytiuqe s(cid:146)oiloftrop elbissop lla fo stnenopmoc lanoitidnocnu latot eht stroper erug(cid:133) ehT ) 2(cid:27)1 ( mret tnemtsujda laitnenopxe eht ,noitaterpretni eht etatilicaf ot redro nI : (cid:30) dna ,! ;(cid:13) sretemarap fo seulav evitanretla t;1r 2(cid:0) (cid:27) era noitpmusnoc fo ytilitalov eht fo sdaol eht ,2 elbaT ni oiranecs esab eht ni sretemarap eht neviG .erug(cid:133) eht morf dedulcxe si fo sdaol ehT .5 01 0:6 = SU;(cid:27)a = ) (cid:27)(E = ) (cid:27)(E )noitpmusnoc fo ytilitalov egareva eht( taht gnimussa yb detaluclac (cid:0) (cid:2) :6 (cid:27) (cid:26) 0 (cid:0) 1 1 0:1 t; = REG qa = ) t;SU q(E = ) q(E )VoV egareva eht( taht gnimussa yb detaluclac era VoV (cid:0) (cid:2) q (cid:26) (cid:0) 1 t;REG t;SU 39

deunitnoC .sretemaraP evitanretlA .smuimerP ytiuqE no sdaol VoV dna ytilitaloV noitpmusnoC :5 erugiF 40

Figure 6: Cross-Country Return Correlations and Model-implied correlation of consumption The (cid:133)gure shows the model-implied unconditional correlation of consumption ((cid:26)(g ;g )) and the US;t GER;t model-implied equity return correlation ((cid:26)(r ;r )) between Germany and the US for several alter- US;t GER;t native values of parameters (cid:13); !, and (cid:30) . (cid:27) 41

snruteR ytiuqE lanoitanretnI gnitciderP ni muimerP ecnairaV SU eht fo elor ehT :7 erugiF :snoisserger gniwollof eht ni (cid:13) tneic ¢eoc detamitse eht stroper erugiF ehT h;j;1 ; (cid:15)+ st (cid:13)+ yd (cid:13)+ pv (cid:13)+ (cid:13) = ) r r( t;h;j t;j h;j;1 t;j h;j;1 t;SU h;j;1 j;0 h+t;t;j f (cid:0) eht era st dna sdleiy dnedivid lacol eht era yd ,snruter ssecxe )dezilaunna dednuopmoc( shtnom-h era ) r r( erehw t;j t;j h+t;t;j f (cid:0) gnitsacerof ylhtnom redisnoc I .etar llib-t shtnom 3 eht dna llib-T raey 1 eht neewteb ecnere⁄id eht sa detaluclac sdaerps mret lacol srorre dradnats detcerroc tseW-yeweN eht rof slavretni ecned(cid:133)noc %59 eht tneserper saera dedahs ehT .shtnom 21 ot pu snoziroh ot redro nI .noisserger hcae rof 2R eht sixa yradnoces eht ni stroper osla erug(cid:133) ehT . 21;h2 xam = l sgal fo rebmun a htiw g f ecnairav SU eht ylno hcihw ni snoisserger rof detroper era 2R eht ,muimerp ecnairav eht fo rewop evitciderp eht yfitnedi yletarapes : (cid:15)+ pv (cid:13)+ (cid:13) = )fr r( ni sa deredisnoc si muimerp t;h;j SU;j h;j;1 h;j;0 h+t;t;j (cid:0) 42

deunitnoC .snruteR ytiuqE lanoitanretnI gnitciderP ni muimerP ecnairaV SU eht fo elor ehT :7 erugiF 43

seirtnuoc ssorca snoitalerroC nruteR ytiuqE gnitciderP ni muimerP ecnairaV SU eht fo elor ehT :8 erugiF :noisserger gniwollof eht ni (cid:13) stneic ¢eoc detamitse eht swohs elbat ehT SU;j;1 ; (cid:15)+ pv (cid:13)+ (cid:13) = ) r; r( (cid:26) t;kj 1 t;SU SU;j;1 kj;0 h+t;t;SU h+t;t;j t (cid:0) noitalerroc ehT .SU eht dna yrtnuoc yna neewteb noitalerroc nruter ytiuqe daeha shtnom-h eht si ) r; r( (cid:26) erehw h+t;t;SU h+t;t;j t yletaidemmi gnitrats htnom eht rof seirtnuoc owt eht rof snruter ytiuqe yliad gnisu detaluclac si 1+t ot t doirep eht rof tneic ¢eoc dradnats detcerroc tseW-yeweN eht rof slavretni ecned(cid:133)noc %59 eht tneserper saera dedahs ehT . pv fo noitazilaer eht retfa 1 t;k .noisserger hcae rof 2R eht sixa yradnoces eht ni stroper osla erug(cid:133) ehT . 21;h2 xam = (cid:0) l sgal fo rebmun a htiw srorre g f 44

deunitnoC .seirtnuoc ssorca snoitalerroC nruteR ytiuqE gnitciderP ni muimerP ecnairaV SU eht fo elor ehT :8 erugiF 45