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Aggregate Volatility Risk and Empirical Factors: An International Study∗ Woongki Lee† James L. Park‡ Korea University ...

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Aggregate Volatility Risk and Empirical Factors: An International Study∗ Woongki Lee†

James L. Park‡

Korea University Business School Korea University

Korea University Business School Korea University

Bumjean Sohn§ Korea University Business School Korea University

This Draft: January 16, 2018

Abstract We examine four regions (North America, Europe, Japan, and Asia Pacific) and find that the aggregate volatility risk is robustly priced across stocks in all regions but Japan. Within the ICAPM framework, we investigate the relationship between the aggregate volatility risk and the conventional empirical factors such as size, value, and momentum. We find evidence that the aggregate volatility risk is closely connected with the momentum profits. Our theoretical framework coupled with return and volatility spillover effect provide interesting explanation for the coexistence of global and local factors.

JEL Classification: G11, G12, G17 Keywords: Aggregate Volatility Risk; Two-Component Volatility Model; ICAPM



We are grateful for supports from the Asian Institute of Corporate Governance (AICG) and the Institute for Business Research and Education (IBRE). † Department of Finance, Korea University Business School, Korea University, 145 Anam-Ro, Seongbuk-Gu, Seoul, 02841, Republic of Korea. email: w [email protected] ‡ Department of Finance, Korea University Business School, Korea University, 145 Anam-Ro, Seongbuk-Gu, Seoul, 02841, Republic of Korea. email: [email protected] § Corresponding Author: Department of Finance, Korea University Business School, Korea University, 145 AnamRo, Seongbuk-Gu, Seoul, 02841, Republic of Korea. email: [email protected]

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Introduction

According to Harvey et al. (2015), the literature in asset pricing has reported more than 240 pricing factors by 2012. Most of these factors are empirically motivated, and we try to understand the nature of the systematic risks these factors represent. Unfortunately, we still don’t have a definitive answer to the issue for the size, value, and momentum factors from which the research seeking new factors took off. Various approaches have been taken to address the issue. One of the popular routes taken is to relate the empirical factors to more economically intuitive business cycle variables. However, as Shanken and Weinstein (2006) shows, most factors based on these macroeconomic or financial variables are not robustly priced across assets in the first place.1 Another way is to follow a theoretical guidance in understanding the nature of the systematic risks. Among many asset pricing theories, the ICAPM of Merton (1973) is one of very popular choice due to its intuitive appeal. However, it might also be true that the popularity is due to its empirical flexibility. ICAPM has somewhat loose restrictions on the identity of state variables, which seems to allow researchers to cite ICAPM as their theoretical background without actually testing the restrictions. Cochrane (2001) recommends such tests to avoid the fishing license critique. Campbell (1993) introduces a discrete time version of the ICAPM and suggests that state variables should predict the market return and the innovations of the state variables are legitimate pricing factors. In Campbell’s (1993) economy in which all asset returns are homoskedastic, the market return fully represents the investment opportunity set. Investors in this economy are concerned about the future changes in the market risk and would like to hedge against such risk. Thus, pricing factors conveying information about the future changes in the market risk would be priced across asset. This has important implications for asset pricing. However, the predictability of the market return is questionable or not robust to say the least. Bossaerts and Hillion (1999) find the presence of in-sample predictability in an international stock market dataset but discover no out-of-sample forecasting power. Paye and Timmermann (2006) provide evidence of instability for the vast majority of countries by detecting structural breaks in 1

Precisely speaking, the innovation of macroeconomic (e.g., industrial production growth and inflation) or financial (e.g., default spread and term spread) variables are examined to find out that industrial production growth factor is the only one to be priced robustly.

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the coefficients of predictors such as the lagged dividend yield, short interest rate, term spread and default premium. In Ang and Bekaert (2007), excess return predictability at long horizons by the dividend yield is not statistically significant, not robust across countries, and not robust across different sample periods. Welch and Goyal (2008) comprehensively reexamine the performance of conventional predictors and find that they perform poorly both in-sample and out-of-sample. Even if the market return is predictable, typical R2 ’s are very low. This implies that the innovation of the predictor contains very noisy information about the future changes in the market risk. Extending Campbell’s (1993) ICAPM by allowing heteroskedastic asset returns, Chen (2003) shows not only the market return but also the market volatility should be considered in representation of the investment opportunity set. Unlike the market return, the market volatility is well known to be persistent and predictable. The innovation of the market volatility predictors would convey more precise information about the future changes in the market volatility, and hence it is expected to be priced more robustly when compared with the market return innovation factor. Ang et al. (2006) show that innovation of the market volatility measured by VIX index is robustly priced with negative risk price across assets in the U.S. Adrian and Rosenberg (2008) propose two-volatility component model and show that innovations of both short- and long-run volatility components have statistically significant and negative prices of risk. Sohn (2009) investigates the relationship between the aggregate volatility risk and conventional empirical factors and finds the aggregate volatility risk is closely linked to the momentum profits. Campbell et al. (2017) present evidence that low-frequency movements in equity volatility are tied to the default spread and priced in the cross-section of stock returns. We will use the term aggregate volatility risk interchangeably with the market volatility risk. Our study in the systematic risk should be distinguished from one on the (market) variance risk in the literature of option pricing. In the option pricing literature, the variance risk premium is commonly defined and estimated as the difference between the risk-neutral expected value of the realized variance (i.e., variance swap rate) and the realized variance, and it has been reported to be negative. The research is primarily about the relationship between the expected returns of options and the uncertainty in the underlying asset’s return variance in the market for index options and

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individual options.2 This paper contains three main contributions. First, our first goal is to see if the aggregate volatility risk is priced in regional capital markets of Asia Pacific, Europe, Japan, and North America. To the best of our knowledge, ours is the first international study of the aggregate volatility risk. As with any empirical results, there is a possibility that the finding of Ang et al. (2006) and Adrian and Rosenberg (2008) is dependent only on a particular sample of a country or a region. If the aggregate volatility risk is recognized in international markets, it is more likely that our theoretical framework is doing a good job in describing or explaining what matters to investors. As a first attempt, we view that each regional markets are segmented and we test the local aggregate volatility risk with the local test assets. Using an updated international daily dataset of Fama and French (2012), we estimate the two-component volatility model of Adrian and Rosenberg (2008). We then time aggregate both components to the monthly components for further analysis. The Campbell’s (1993) VAR factor model is used to generate the monthly innovation factors from the monthly volatility components. We show that the aggregate volatility risk is robustly priced with negative risk price in all regions except for Japan. As for Japan, none of the pricing factors (including the size, value, and momentum) are priced and, as we look into this matter closely later, this is due to the fact that the mean return spread of the test portfolios in Japan is not sufficiently large enough to be explained by any. Our second contribution is that the aggregate volatility risk is closely connected to the momentum profits. Within our ICAPM framework, we can directly test the connection if the state variables of which innovations are the empirical factors are known and observable. However, such state variables are not observable, and we need to test other implications from the ICAPM. As in Petkova (2006), we conduct both time-series and cross-sectional analysis. We investigate if the empirical factors convey information about the future changes in the market volatility by examining contemporaneous time-series regressions of aggregate volatility innovations on the empirical factors. The aggregate volatility innovations share common components with the momentum factor in Asia 2

For reference, see Buraschi and Jackwerth (1999), Buraschi and Jackwerth (2001), Coval and Shumway (2001), Bakshi and Kapadia (2003), and Carr and Wu (2003). Bakshi and Madan (2006), Bollerslev et al. (2009), and Chabi-Yo (2012) provide economic theories to understand the variance risk premium in equilibrium. Carr and Wu (2003) and Gonz´ alez-Urteaga and Rubio (2016) try to find out what explains the cross-sectional variation of the variance risk premia.

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Pacific and North America. For our cross-sectional study, we adopt 100 test portfolios composed of 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. It turns out that momentum sorting generates the largest mean return spread meaning that it is crucial for the performance of a pricing factor to explain the mean return spread by the momentum. The robust pricing of the aggregate volatility risk factors in Asia Pacific, Europe, and North America implies that aggregate volatility risk factors share pricing information with the momentum factor. Unfortunately, the results from the crosssectional regressions with orthogonalized factors do not offer supporting evidence as clear as the one in Petkova (2006). However, this does not exclude the connection between the aggregate volatility risk and the momentum profits. The results simply suggest that the state variable of which the innovation is the momentum factor does convey future market volatility information additional to the current market volatility. In our cross-sectional study, we also follow Griffin et al. (2003) and Liu and Zhang (2008) to find out how much of the traded factor profits can be explained by the factor model of interest. Our model up for the test is a three factor model of excess market return and the aggregate volatility risk factors. The traded factors examined are the size (SMB), value (HML), momentum (UMD), and these are all returns of zero-cost portfolios. Thus, these factors should also be priced by pricing factors. A three factor model of the market excess return and the short- and long-run market volatility factors explains 52%, 19%, 109%, 56% of momentum profits in Asia Pacific, Europe, Japan, and North America, respectively. Finally, our third contribution pertains to literature of international asset pricing.

Our

theoretical framework suggests the innovation of any variable predicting the market volatility (or, short- and long-run components) should be priced across assets in an economy. Within this framework, return and volatility spillover effect across countries has an interesting implication. For example, many report unidirectional return and volatility spillover from the U.S. to the rest of the world, and this implies that the aggregate volatility risk factors in the U.S. should also work in other countries. Since our dataset is regional, we investigate if the aggregate volatility risk factors in North America are priced in other regions, and we find this is true except for Japan where no factor is priced. This empirical finding also shed new light on the discussion of whether financial

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assets are priced globally or locally. Section 2 introduces our theoretical framework. Section 3 describes the dataset and the empirical methodologies adopted. Section 4 presents all the empirical results from estimation of the two-component volatility model to both time-series and cross-sectional analyses. Section 5 concludes.

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Theoretical Motivation

In this section, we discuss the theoretical background for the aggregate volatility risk and its relationship with conventional empirical factors such as the size, value, and momentum. Our main theoretical framework is Campbell’s (1993) discrete version of the ICAPM and its extension. This explains why the aggregate volatility risk would be priced across assets and what systematic risks the empirical factors should represent. Campbell’s (1993) ICAPM suggests the state variables should predict the market return and the innovations of the state variables should be priced. Chen (2003), Sohn (2009), and Campbell et al. (2017) introduce heteroskedasticity in the asset returns and show not only the market return but also the market volatility should be added to the set of sufficient statistics for the investment opportunity set. In this theoretical framework, the market return and the market volatility completely represent the investment opportunity set. The state variables predict either the market return or the market volatility and the innovations of the state variables would be priced across assets because the innovations contain the information about the future changes in the investment opportunity set. In particular, Sohn (2009) derives an equation for the risk premium of an risky asset:

Et [ri,t+1 − rf,t+1 ] +

Vii,t (γ − 1)2 = γVim,t + (γ − 1)Vih,t − Viη,t 2 2(σ − 1)2

(2.1)

where γ, σ, ri,t+1 and rf,t+1 are the coefficient of relative risk aversion, elasticity of intertemporal substitution, log return of asset i and risk-free asset, respectively.

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And, Vii,t =V art (ri,t+1 ),

Vim,t =Covt (ri,t+1 , rm,t+1 ), and, ∞   X Vih,t = Covt ri,t+1 , (Et+1 − Et ) ρj rm,t+1+j

(2.2)

j=1

and ∞    X Viη,t = σ 2 Covt ri,t+1 , Et+1 − Et ρj V art+j [rm,t+j+1 ]

(2.3)

j=1

where ρ is a constant that comes from the log-linearization of the budget constraint. Eq.(2.2) and (2.3) show that the market return and the market volatility are what matters to investors when it comes to the risk involving the future changes in the investment opportunity set. There would be a trading demand in investors’ perspectives to hedge out the unexpected changes in the market return and the market volatility. The equation for the risk premium shown in Eq.(2.1) provides the signs for the price of the future market risk and the future market volatility risk. The sign of the price of risk for the innovation of the market return predictor should be positive because γ −1 > 0 for any representative investor with a reasonable degree of the risk aversion. The sign of the price of risk for the innovation 2

(γ−1) of the market volatility predictor should be negative because − 2(σ−1) 2 < 0. For example, an asset

of which return is expected to positively correlates with the future market return would command a positive risk premium because it is exposed to the future market risk. Consider another asset whose return is expected to positively correlate with the future market volatility. Such asset would provide an hedging opportunity to investors against the deteriorating investment opportunity (i.e. increase in the market volatility) and would command risk discount (or, negative risk premium). Our theoretical framework suggests that any empirical factor should be the innovation of some unknown state variable that predicts either the market return or the market volatility. In other words, the conventional empirical factors such as the size, value, and momentum are supposed to be the innovations of some unknown predictors of the market return or the market volatility. Our theoretical framework shed light on the nature of the systematic risks these empirical factors represent. This is an important issue since we still try to find risk-based explanations for these

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empirical factors. However, the proposed relationship between the investment opportunity set (as represented by the market return and the market volatility) and the empirical factors is not easy to show empirically because the state variables of which the innovations are the empirical factors are unknown. As to this matter, we will discuss further in Section 3. Ang et al. (2006) provide an empirical evidence for our theoretical framework. They show that the innovation of the market volatility is priced across assets in the U.S. As is well known in the market volatility literature, the market volatility is a persistent process and the market volatility itself is a good predictor of the future market volatility. Thus, the innovation of the market volatility contains information about the future changes in the market volatility. Adrian and Rosenberg (2008) also support our theoretical framework. They propose a two-component volatility model and show that the innovations to both the short- and long-run components are robustly priced across assets in the U.S. Furthermore, the empirical results of Ang et al. (2006) and Adrian and Rosenberg (2008) show the price of the aggregate volatility risk is negative. We adopt the two-component volatility model of Adrian and Rosenberg (2008) to examine the aggregate volatility risk in the international capital markets.3

There are two reasons for

adopting a two-component volatility model. First, two-component volatility models perform better in capturing time-variation of the market volatility. Since Engle and Lee (1999), many studies have found that two-component volatility models outperform one-component specifications in explaining equity market volatility. Engle and Rosenberg (2000), Alizadeh et al. (2002), Bollerslev and Zhou (2002), Chacko and Viceira (2003), and Chernov et al. (2003) find that two-component volatility specifications outperform one-factor models for market return volatility. Second, two volatility components allow additional dimension to what matters to investors regarding the market volatility. The (monthly) short-run volatility component is supposed to represent month to month uncertainty in the market while the (monthly) long-run volatility component is supposed to represent more slowly moving business cycle. Schwert (1989) finds the market volatility is closely linked to NBER recession indicator. Engle et al. (2013) propose two-component volatility model (GARCH-MIDAS) and show that their long-run component is driven by inflation and industrial production growth. 3

Even though Ang et al. (2006) provide the empirical evidence for the aggregate volatility risk, the main focus is on the relationship between the mean return of an asset and its idiosyncratic risk. Ang et al. (2009) expand the asset universe to an international setting but did not control for the aggregate volatility risk.

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Hence, the innovation of the short-run volatility component would convey information about the future changes in the market uncertainty while the innovation of the long-run volatility component would contain information about the future changes in economic activity. Our two-component volatility model is specified as follows

e rm,t+1 = θ1 + θ2 st + θ3 lt +



vt εt+1

√ ln vt = st + lt

(2.4) (2.5)

p 2/π) p = θ8 + θ9 lt + θ10 εt+1 + θ11 (|εt+1 | − 2/π)

st+1 = θ4 + θ5 st + θ6 εt+1 + θ7 (|εt+1 | −

(2.6)

lt+1

(2.7)

where the standardized error term ε is assumed to be distributed normally with mean zero and variance one. θ2 and θ3 would verify the risk-return tradeoff relation. θ5 and θ9 would show the persistency of the component processes. θ6 and θ10 would accommodate the asymmetric effect of returns on volatility. θ7 and θ11 would allow the shock of different magnitude to the volatility components. Note that the specification for both the short-run and long-run components are identical. Thus, the process persistency revealed in θ5 and θ9 would distinguish the short-run component from the other.4 We combine the ICAPM in Eq.(2.1)-(2.3) and the two-component volatility model in Eq.(2.4)(2.7) to investigate the aggregate volatility risk and its relation with the empirical pricing factors in the international capital markets. In short, we test if the aggregate volatility risk is priced across assets in various regions of the world. If so, we would like to learn if the empirical factors are priced because they contain the information about the future changes in the market return or the market volatility (the short-run component s or the long-run component l). 4 Adrian and Rosenberg (2008) have no constant term in the short-run component process, but that does not mean the estimated component process without the constant term would necessarily be the short-run component. In their case as well, the identification of the short-run and long-run components would rely on the persistency of the process.

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3

Data and Empirical Methodology

For our study of the aggregate volatility risk, we adopt the updated dataset used in Fama and French (2012) available from Prof. Kenneth French’s Data Library. The dataset has its advantages and disadvantages for our purpose. It covers the breadth of stocks and countries over various regions around the world yet it combines 23 markets into four regions to ensure sufficient number of stocks would be included in each test portfolio for diversification. In addition, the choice of regions takes into account the market integration to keep similar risk characteristics within a region. It includes a variety of small stocks and this also distinguishes the dataset from others used in most of previous studies. For each region, it provides Fama-French three factors, the momentum factor, the risk-free rate, and four different sets of test portfolios; 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. These pricing factors and the test portfolios are provided for both daily and monthly frequency and for four regions: Asia Pacific, Europe, Japan, and North America. Meanwhile, there are some disadvantages as well. The first is the short sample period which starts from July 1990. This choice is to keep the breadth of the stocks in their dataset especially for the small stocks, but it might be problematic for the size factor because the literature reports that the size factor is very weak or even nonexistent from early 1980’s. Another potential disadvantage is the U.S. dollar denomination of all regional returns. Accordingly, the risk-free rates are identical for all regions; one-month U.S. T-bill rate. This is a reasonable approach in the U.S. investors perspective. However, the U.S. dollar denominated returns might distort the risk characteristics of the stocks in a market less integrated with the U.S. by incorporating the currency risk on the top of the systematic risks in the stocks. Perhaps the biggest disadvantage for our research purpose is that the dataset only offers regional data. While this is the inevitable choice to have sufficient number and variety of stocks in each test portfolios, this makes it hard to use additional economic or financial variables along with the dataset. This limits our research since one of the key state variables in our theoretical framework is the market return predictor which is typically a macroeconomic or financial variable is 10

not available and not easy to add to the dataset. As is well known in the literature, the conventional predictors for the market return are dividend yield, default spread, term spread, and the risk-free rate. These are not available in the dataset except for the risk-free rate. Also, it is not clear how to aggregate these variables in the regional level. However, since our focus is on the aggregate volatility risk, we do without the market return predictors.5 For the empirical investigation, we first need to estimate the Adrian and Rosenberg (2008) model as specified in Eq.(2.4)-(2.7). As in the original paper of Adrian and Rosenberg (2008), we follow the insight from Merton (1980) and estimate the model using daily market excess returns in each of four regions; Asia Pacific, Europe, Japan, and North America. The use of more frequent daily data rather than monthly ones would improve the estimation precision. The fitted daily sample period is from July 1990 to December 2016. Once the short-run component st and the longrun component lt are estimated, we time aggregate them to the monthly frequency counterpart, q (m) (m) (m) where st and lt , for all our later analysis. Figure 1 presents the time-series of 12 · vt (m)

vt

is monthly aggregate of the market volatility vt in Eq.(2.5). As the figure shows, we observe

unprecedented level of the market volatility in 2008 due to the global financial crisis. This is considered to be once in a century event by many economists. Thus, we decide to exclude the year 2008 from all our analysis. We investigate if the aggregate volatility risk is priced in each of the four regions of our interests. In addition, we would like to find out the relationship between the aggregate volatility risk and the conventional empirical factors such as the size, value, and momentum. To explore the pricing ability of the aggregate volatility risk, we would eventually need to run cross-sectional regressions. However, before the cross-sectional analysis, we need to do some time-series analysis to (1) generate the innovation factor containing the information about the future changes in the market volatility and (2) examine the pricing information of the aggregate volatility risk factors in terms of the conventional empirical factors. To implement the ICAPM laid out in Eq.(2.1)-(2.3), we adopt the VAR factor model of 5 The only market return predictor for each region readily available is the risk-free rate. We examined if the regional risk-free rate predicts the regional market return in both the AR setting and the VAR setting, and we found no such predictability. This is perhaps due to the fact that the risk-free rates for all regions are identical to one-month U.S. T-bill rate.

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Campbell (1993), Campbell (1996) and Petkova (2006). In the spirit of the VAR factor model of Petkova (2006), we consider the following VAR system for each region: 

zt+1 = Πzt + νt+1

(m)

e where rm,t is the excess market return, and st

e rm,t



   (m)   s   t     (m)   lt   , zt =     SM Bt      HM L  t    U M Dt (m)

and lt

(3.1)

are the monthly short-run and long-run

components of the market volatility for a given region. SM Bt , HM Lt , U M Dt are the regional size, value, and momentum factors. Note that the VAR is first-order and this is not restrictive for further analysis since a higher-order VAR can always be stacked into first-order form in the manner discussed by Campbell and Shiller (1988). This representation is only for theoretical discussion. In actual empirical work, we let the data determine the optimal VAR order. Also, note that all the variables are demeaned for notational convenience. The estimation of the VAR factor model in Eq.(3.1) serves our two-fold purpose. As we e , or s(m) , or l(m) . We want discussed in Section 2, a legitimate state variable should predict rm

to verify the prediction ability of s(m) and l(m) for these target variables. If lagged s(m) and l(m) e , or s(m) , or l(m) , the innovations of s(m) and l(m) would be priced across are good predictors of rm

assets. The Granger causality test through the VAR estimation would allow us to look into the prediction relations among the variables in the VAR system. And, the estimated residuals of the variables predicting the target variables would be our factors of interests. This becomes more clear

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in the following:

(Et+1 − Et )

∞ X

j=1 ∞ X

(Et+1 − Et )

(Et+1 − Et )

= e01 ρΠ(I − ρΠ)−1 νt+1

(3.2)

(m)

= e02 ρΠ(I − ρΠ)−1 νt+1

(3.3)

(m)

= e03 ρΠ(I − ρΠ)−1 νt+1

(3.4)

e ρj rm,t+1+j

ρj st+1+j

j=1 ∞ X

ρj lt+1+j

j=1

where ei is a vector of all zeroes except for the i-th element being one. Another use of the VAR factor model specified in Eq.(3.1) has to do with the inclusion of the empirical factors. As in Campbell (1996) and Petkova (2006), we can triangularize the VAR system to orthogonalize the residuals. Using these orthogonalized factors, we can investigate the incremental explanatory power of the factor loadings of the empirical factors. In other words, the setup allows us to examine if the empirical factors would still be priced when the parts that correlates with the market excess return, short-run component innovation, long-run component innovation are separated out. In our theoretical framework, the empirical factors are the innovations of some unknown state (m)

e , or s variables predicting rm,t t

(m)

, or lt

. However, since the state variable corresponding to each of

the empirical factors is unknown, we cannot directly verify if the empirical factors can be explained or understood in our framework. Hence, we examine the contemporaneous relationship between the empirical factors and the innovations of the short-run and long-run component of the market volatility. On the top of the time-series studies, we examine the cross-section of stock returns in each region. We run cross-sectional regressions in two different sets of specifications. First, for each region, three baseline regional factor models are tested over 100 regional test portfolios. Those are Fama-French three factor model, Carhart’s four factor model, and the Aggregate Volatility Risk (m)

Model (specified as the market excess return, the innovation in st

(m)

, the innovation in lt

). These

results will directly show if the aggregate volatility risk is priced in the regions, and enable us to compare it with the benchmark models in the literature. The second set of cross-sectional

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regressions is done with the orthogonalized factors from the triangularized VAR system.

As

mentioned before, these results will allow us to learn about the incremental pricing information in the empirical factors orthogonal to the market excess return and the aggregate volatility risk factors. Furthermore, we explore two interesting implications of our theoretical framework to learn more about the aggregate volatility risk and its relationship with the empirical factors. First, we follow Griffin, Ji, and Martin (2003) and Liu and Zhang (2008) to find out how much of the profits or the premium of the empirical factors can be explained by the Aggregate Volatility Risk Model. Second, the literature of international asset pricing has long reported the return and volatility spillover effect between countries and regions. Many have showed the interdependence among U.S., Japan, and major European markets. For example, in our regional data, if the market excess return and the volatility in North America predict those in Asia Pacific, North American aggregate volatility factors should be priced in Asia Pacific region. This is actually what is implied from our ICAPM framework and it will support our model. This phenomenon itself will have interesting implications on how the global factors work and so on.

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Empirical Results

In this section, we study stock markets in Asia Pacific, Europe, Japan, and North America. We follow the empirical procedures described in Section 3, and analyze the results in the theoretical framework provided in Section 2.

4.1

Two-Component Volatility Model Estimation

We estimate the two-component volatility model of Adrian and Rosenberg (2008) as specified in Eq.(2.4)-(2.7) with maximum likelihood estimation method. We fit the model with the daily market excess return series from July 1990 to December 2016 (excluding the year 2008) for each region. Table 1 presents the parameter estimates, the corresponding standard errors, and summary statistics. All the key parameters for the volatility component processes are statistically significant.

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For all four regions, the persistency of the long-run component is higher than that of the shortrun component (i.e., θ5 < θ9 ), and θ9 for all regions are close to one. Figure 2 presents the estimated (demeaned) short-run component st process in four regions, and st looks pretty much random without any trend. On the other hand, the long-run component lt looks like the monthly conditional volatility shown in Figure 1 without the huge spike in 2008. Interestingly, the variability of the short-run components in Asia Pacific region and Japan is noticeably smaller than that in Europe and North America in Figure 2. This implies that day-to-day variations of the investment opportunity set in terms of the market volatility are higher in Europe and North America. On the other hand, the standard deviation of the long-run component lt is higher in Asia Pacific region and North America, implying the changes of the economic uncertainty is higher for these regions during our sample period. As in Adrian and Rosenberg (2008), we find that negative returns increase short- and long-run volatility more than positive returns for all regions. Adrian and Rosenberg (2008) argues that the asymmetric impact of market return innovations on market volatility captures the time-varying skewness of market return. The asymmetric effect for the short-run component (θ6 ) is more than twice as large in magnitude as this effect for the long-run component (θ10 ). Thus, we expect short-run component to be closely related to market skewness. The parameter θ2 and θ3 show the risk-return tradeoff relationship in the fashion of Engle et al. (1987) and Bollerslev et al. (1988). None of these parameters are statistically significant at 5% level in Europe and North America. In Asia Pacific region, θ2 is statistically significant with a negative sign. The negative sign does not support the risk-return tradeoff relationship, but this is what many researchers find from empirical work. θ2 and θ3 are statistically significant at 10% and 5% level, respectively, for Japan and they are both positive.

4.2

Time-Series Study of The Pricing Factors

In this section, we construct the pricing factors representing the aggregate volatility risk of each region, and we study their relation with other conventional pricing factors in each region. In the Section 4.2.1, we estimate the VAR system in Eq.(3.1) and show that the innovations (sres and

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lres) of the two volatility components are legitimate pricing factors under our theoretical framework in Section 2. In the Section 4.2.2, we study the pricing information contained in sres and lres.

4.2.1

Predictive Relations

We aggregate the daily short- and long-run components st and lt over each calendar month to (m)

construct the monthly components st

(m)

and lt

. With these monthly volatility component series,

the VAR system in Eq.(3.1) is estimated. The VAR order is determined by the Likelihood Ratio (m)

(m)

tests. The innovations of st

and lt

in the estimated VAR system are srest = e02 νt and

lrest = e03 νt , respectively. (m)

Within our theoretical framework, sres and lres would be legitimate pricing factors if st (m)

lt

(m)

(m)

(m)

and

(m)

e e predict rm,t+1 , or st+1 , or lt+1 because {rm,t+1 , st+1 , lt+1 } constitutes the sufficient statistics

for the investment opportunity set in time t + 1. In order to verify these predictive relations, we conduct the Granger Causality F-tests in the VAR system. Table 2 shows the results. In Table 2, the first row under the regional title shows the variables for the VAR equations, and the first column from the left shows the predictors. The optimal VAR orders determined for each region range from three to five, and the numbers shown in the table are the F-statistics. For concise presentation, the table only shows the VAR equations of our interests. Primarily, we are (m)

interested in the prediction ability of st

(m)

and lt

(m)

. Since lt

is a very persistent process, the

long-run volatility component is naturally a good predictor of its own series, which is strongly supported by the F-statistics in the table (statistically significant at 1% level for all four regions.). As for the short-run volatility component, it predicts the long-run component in Europe and North America at the significance level of 5%. At the significance level of 10%, it predicts its own series in North America and the excess market return in Japan. Based on these empirical evidence, we adopt the sres and lres as legitimate pricing factors, and we will call the following model the Aggregate Volatility Risk Model:

E[ri ] − rf = λm βim + λs βis + λl βil

(4.1)

where ri and rf are the returns on risky asset i and risk-free asset, respectively, and β’s are from 16

the following time-series regression:

e e ri,t = αi + βim rm,t + βis srest + βil lrest + i,t

(4.2)

Note that αi is not a pricing error and need not be zero because sres and lres factors are not in the return form. This is why we cannot just focus on the time-series studies like Fama and French (2012) and Fama and French (2017), but need to look at the cross-sectional studies. Recall the risk premium equation in Eq.(2.1). βim corresponds to Vim , and βis and βil to Viη . Vih part is missing in our specification.6 As we already discussed in Section 3, the data for regional market return predictors (which are normally macroeconomic or financial variables in the region) are not available, and we decided to do without them to focus on the aggregate volatility risk.

4.2.2

Contemporaneous Relations

In this section, we examine the pricing information in sres and lres by running contemporaneous regressions of the aggregate volatility factors on the conventional empirical factors. The direct way of examining the relationship between the market volatility and the empirical factors is to include the state variables of which innovations are the empirical factors in the VAR system and see the prediction relation. However, as we discussed in Section 3, such state variables are unknown, and, hence, we follow Petkova (2006) to investigate the contemporaneous relations. By the VAR estimation in Section 4.2.1, we now have a complete set of pricing factors and review their statistical properties before our investigation of the contemporaneous relations among the factors. Table 3 provides the summary statistics for all pricing factors. Over the sample period from July 1991 to December 2016 (excluding 2008), the regional market risk premiums are all positive and statistically significant except for Japan. They range from 0.751% per month for Europe to 0.969% for Asia Pacific while Japan’s market risk premium stays low at 0.176% without statistical significance. sres and lres are the VAR innovation series and the mean values of these series are supposed to be zero, which is why t-stats are not provided. However, actual mean 6

Precisely speaking, Vim and Viη are covariances, so it takes division by the variance of the respective innovation factor to make them equivalent to the betas. However, the sign will still remain the same because the variance is always positive.

17

(m)

values deviate slightly from zero for the sample period difference. Monthly st

(m)

and lt

series are

constructed for July 1990 - December 2016 (excluding 2008). However, due to the differences in the optimal VAR order across regions, their innovation series sres and lres are estimated for different length. The innovation series shown in Table 3 are taken for July 1991 - December 2016 (excluding 2008) to have the same sample length as those of the other conventional empirical factors for the cross-sectional studies in the later sections. As for the empirical factors, there is no size premium in any region during our sample period. Average SMB returns are statistically not significant across all regions. This finding is not so surprising because many have already reported the disappearance of the size effect. Dichev (1998) and Horowitz et al. (2000) find no evidence of a size effect in the 1981-1995 and 1979-1995 periods, respectively. Hirshleifer (2001) suggests 1984 as the year in which the disappearance of the size effect first materialized. Since our sample period for the Table 3 starts from 1991, our finding is consistent with the literature. The value and momentum premiums are quite robust across the regions except for Japan. In fact, nothing is statistically significant in Japan. The value premium is statistically significant in Asia Pacific and Europe while the momentum premium in Asia Pacific, Europe, and North America. Excluding Japan, the momentum factor has a higher premium than the value factor, and it has the highest kurtosis among all factors. It shows negative skewness across all regions. Considering how the momentum factor is constructed, these statistical properties suggest that the momentumsorting would provide the highest mean return spread and the pricing factors explaining this mean spread would be strongly priced across assets in cross-sectional studies. As a way to test whether the empirical factors proxy for innovations in state variables that predict the short- and long-run volatility components, we examine the contemporaneous regressions of sres and lres on the empirical factors. Since s(m) and l(m) predict the future values of s(m) or l(m) , the innovations of these series (sres and lres) have information about the future changes in s(m) and l(m) . The contemporaneous regression below will reveal how much sres and lres share this information with the empirical factors:

Innovt = c0 + c1 SM Bt + c2 HM Lt + c3 U M Dt + t 18

(4.3)

where Innovt is either srest or lrest . Table 4 shows the results. The most prevailing empirical relation we find in the table is that lres covary negatively with all the empirical factors across all regions without any exception. Most statistically significant coefficients of the empirical factors from the regressions with sres are negative as well. This makes sense because, according to Adrian and Rosenberg (2008), the prices of risk for both sres and lres are negative and those for all the empirical factors are positive. Another interesting observation from Table 4 is that the coefficient to the SMB is statistically significant in more regions than any other, and this is a bit puzzling since the size effect does not exist within our sample as we already discussed in Table 3. Our interpretation is that both sres and lres share information with the size factor but this common variation part with SMB is not the part of sres and lres that would be priced across assets at least in our current sample period. If we could extend Prof. Kenneth French’s dataset to far back to 1960’s, the common variational part might play a role in pricing assets. As for other empirical factors, the coefficient to UMD is statistically significant in Asia Pacific and North America, and that to HML in Europe. Table 3 shows UMD and HML are statistically significant in the corresponding regions. Having the pricing information of UMD in Asia Pacific and North America and that of HML in Europe, the aggregate volatility factors are expected to be priced in these regions. There are a certain limitation in this investigation. We adopt s(m) and l(m) as the only set of the predictors for their own future values, and we examine the contemporaneous relationship between the innovations of these series and the empirical factors. However, there should certainly be other good predictors for s(m) and l(m) . The innovations of these other predictors would have components orthogonal to the innovations of s(m) and l(m) , which should also be considered in investigating the empirical factors.

4.3

Cross-Sectional Regression

We put the regional aggregate volatility risk factors directly to test across regional test assets. The aggregate volatility risk factors (sres and lres) are the innovations to the state variables s(m) and

19

l(m) and they are not the returns of any portfolio. It means that the time-series test in the long tradition from Black et al. (1972) and Gibbons et al. (1989) is not a viable option for us. For a given regional factor model, we estimate the factor loadings with one multiple time-series regressions with the full sample. we then estimate the prices of risk in the cross-sectional regressions as in Fama and MacBeth (1973). We do the second stage regressions without a constant term following Adrian and Rosenberg (2008). Regarding the constant term in cross-sectional regression, Cochrane (2001) comments that it is a trade-off problem between efficiency and robustness, and there is no theory that suggests one should do it with or without the constant. We choose to do it without the constant term, and there are upsides and downsides to this choice. The upside is that the price of market risk become positive and statistically significant. The cross-sectional regressions with the constant term normally leads to negative and statistically insignificant price of the market risk with U.S. data and many sets of international data. Regardless of the statistical significance, positive price of the market risk conforms with our general intuition. The downside is low R2 in the cross-sectional regression and we don’t get to estimate the pricing error. For test assets, we use 100 regional portfolios composed of 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. In theory, each of these portfolios is a tradeable asset yet with minimal idiosyncratic shock from diversification. Each set of 25 portfolios is constructed to have maximum range of exposures to two supposedly systematic risks (or, more precisely, firm characteristics). Thus, taking just one set of portfolios for test assets sometimes leads to counter-intuitive empirical results. For example, the price of risk for HML is estimated to be negative for all regions and statistically significant for Asia Pacific and Europe when it is estimated in Fama-French three factor model over a single set of 25 size and momentum sorted portfolios.7 By having multiple sets of portfolios exposed to supposedly systematic risks in various dimensions, we expect to increase the power of the test and obtain risk price estimates that looks more reasonable. The only empirical concern is that all four sets have common sorting criteria of size and this might cause the test portfolios to be overly exposed to size risk. However, that turns out to be groundless. Consistent with the size premium shown in Table 3, the size factor is not priced across our test portfolios and most of regional test portfolios. 7

See Table IA.1 - IA.4 in the Internet Appendix.

20

We compare the performance of three baseline factor models: Fama-French three factor model (FF3), Carhart’s four factor model (C4), and the Aggregate Volatility Risk model (AVR) as specified in Eq.(4.1)-(4.2). Table 5 presents the results and Figure 3 shows the performance of the AVR model. Above all, the aggregate volatility risk is robustly priced across all the regions except for Japan. sres has negative and statistically significant price of risk for all regions except for Japan, and lres also has negative and statistically significant price of risk for Europe and North America. To fully appreciate this result, several things should be noted. First, the aggregate volatility risk factors are priced across very challenging set of test portfolios. Each set of 100 regional test portfolios is more challenging for the pricing factors than the conventional size and book-to-market sorted test portfolios we typically adopt for studies in the U.S. None of SMB and HML in FF3 model are priced with statistical significance across all regions.8 The price of SMB risk is statistically not significant in any model and in any region. This is consistent with the time-series study in Table 3 that shows the size premium is not significant. Considering that one of the advantage of our dataset updated from Fama and French (2012) is that it covers many small cap companies, no size effect in our sample can be thought to be consistent with the literature reporting the disappearance of the size effect. The price of HML risk is not statistically significant in FF3 model, but becomes statistically significant only with the UMD in Asia Pacific and North America. Second, the aggregate volatility risk factors seem to perform very well due to its ability to explain the momentum-sorted test portfolios. The momentum factor is also priced as robustly as the aggregate volatility risk factors. Since the momentum is one of the portfolio sorting criteria, the good performance of the momentum factor is due to its ability to explain the large mean return spread of the momentum-sorted portfolios. This looks clear from Figure 3 which shows realized average excess return and mean excess return fitted from the AVR model. The vertical distribution of the test portfolios in all regions except Japan clearly shows size and momentum sorted portfolios offer the largest mean return spread among all the test portfolios. Since the size effect is not observed in our sample, the spread must be due to the momentum, and this means that it is very important for the pricing factors to explain this momentum driven mean return spread in order to perform well across these test assets. 8

This is also true even for SMB and HML in FF3 model over regional 25 size and book-to-market sorted portfolios in Europe, Japan, and North America. See Table IA.1 - IA.4 in the Internet Appendix.

21

Third, Japan is a graveyard for all the baseline factor models. The aggregate volatility risk factors are not priced in Japan, and they are not alone. No factor is priced in Japan in crosssectional studies in Table 5, nor is any factor premium statistically significant in time-series studies in Table 3. Figure 3 shows that the range of mean returns or the mean return spread in Japan is very narrow compared to other regions. For Japan, there seems not to be a large enough mean return spread to be explained in the first place, and, hence, no factor is priced. Lastly, the AVR model is missing the market return innovation factors from our ICAPM framework in Eq.(2.1)-(2.3). As we discussed earlier, conventional regional market return predictors (typically, regional macroeconomic or financial variables) are not available in our dataset, and this will limit the performance of the AVR model. Regarding these predictors (e.g., dividend yield, term spread, default spread, and the risk-free rate), economic theories suggest that variables capturing time-varying risk premia are prime candidates for forecasting volatility, and Christiansen et al. (2012) support it empirically. Since state variables predicting the market return are highly likely to predict the market volatility as well, it is not easy to tell if the innovation of such state variable is priced because it conveys the information about the future changes in the market return or the market volatility. Petkova (2006) shows that SMB and HML are proxies of the innovations of these macroeconomic of financial variables. Currently in Table 5, the FF3 model performs better than the AVR model in terms of R2 , but the AVR model with the innovation factors of these variables is expected to do better than the FF3 model while the question of whether the additional innovation factors convey the information about the future market return or volatility would still remain. To examine the incremental explanatory power of the empirical factors, we orthogonalize the innovations in the VAR system in Eq.(3.1) in the spirit of Campbell (1996) and Petkova (2006). In Campbell (1996), the innovation νt is orthogonalized with Cholesky decomposition of the covariance e , SM B , HM L , and matrix of νt . Empirically, in our setup, the return-based factors such as rm,t t t

U M Dt and their corresponding innovations in νt have very high correlations above 0.9. Hence, e , sres , lres , SM B , HM L , U M D } rather than ν itself to construct the we orthogonalize {rm,t t t t t t t

orthogonalized innovation factors. This would allow us to examine the incremental explanatory power of the empirical factors while keeping the spanning space of the set of the orthogonalized factors comparable to FF3 and C4 models.

22

Table 6 presents the cross-sectional regression results with the orthogonalized innovation factors. The results show that uHM L and uU M D stand strong even though these are the component e , sres, lres, and SM B. To our disappointment, u orthogonal to rm sres and ulres lose statistical

significance. If the pricing information HM L and U M D carry is subsumed under that in sres and lres, uHM L and uU M D would have lost their statistical significance. However, this does not imply that HM L and U M D are not the proxies for the innovations of s(m) and l(m) because the state variable of which innovation is HM L or U M D can still predict future s(m) and l(m) on the top of lagged s(m) and l(m) .

4.4

Explaining Traded Factor Profits

As another way to understand the empirical factors within our ICAPM framework, we follow Griffin et al. (2003) and Liu and Zhang (2008). We are particularly interested in these papers because both papers try to explain momentum profits. Our time-series study and cross-sectional study suggest that the aggregate volatility risk is closely related to the momentum profits. Many papers provide risk-based explanations of momentum profits. Ahn et al. (2003) show that their nonparametric risk adjustment can account for roughly half of momentum profits. Pastor and Stambaugh (2003) document that a liquidity risk factor accounts for half of momentum profits. Liu and Zhang (2008) also shows that macroeconomic risk factor (especially, the growth rate of industrial production) explains more than half of momentum profits. All these papers just examine only the U.S. data. Griffin et al. (2003) whose approach is very similar to that of Liu and Zhang (2008) investigate international capital markets to find that an unconditional model based on the Chen et al. (1986) factors does not provide any evidence that macroeconomic risk variables can explain momentum profits. We also look into the international capital markets (regional markets, though), but we do find our AVR model does a pretty good job in explaining the momentum profits around the world. The empirical factors of our interests are returns of portfolios constructed from a zero-cost portfolio. Since they all take the form of returns, the AVR model should be able to price each of these factors. We run the time-series regressions as specified in Eq.(4.2) for each of these regional

23

factors to obtain the regional factor loadings. Then, using the prices of risks of the AVR model estimated in Table 5, we can compute the fitted factor risk premium in the following way:

E[fx ] = λm βxm + λs βxs + λl βxl

(4.4)

where fx is a traded factor x constructed from a zero-cost portfolio. Table 7 presents the empirical results. The ‘Risk Premium’ row shows the fitted risk premium and the ‘Average’ row shows the time average of the factors already presented in Table 3. The AVR model explains 52%, 19%, 109%, 56% of momentum profits in Asia Pacific, Europe, Japan, North Amercia, respectively, though the momentum premium in Japan is not statistically significant. These results are consistent with our earlier time-series study in Section 4.2.2. Table 4 shows that the aggregate volatility risk factors covary with the momentum factor in Asia Pacific and North America in which the AVR model do best in explaining the statistically significant momentum profits. As for the value profits, the AVR model does a good job only in Europe with 79% explanation, which is also consistent with the time-series study in Section 4.2.2. Table 4 also shows the aggregate volatility risk factors covary with the HML only in Europe. The size premiums are not statistically significant in any of the regions, so we do not discuss.

4.5

Implications for International Asset Pricing

From the seminal work of Grubel (1968), the relationship among the returns of various stock markets around the world was analyzed in a series of studies such as Levy and Sarnat (1970), Grubel and Fadner (1971), Ripley (1973), Solnik (1974a), Solnik (1974b), and Hilliard (1979). Empirically and theoretically, these works provided strong evidence supporting the benefits of international diversification. During this time, aside from the stock market comovements, lead-lag relations (return spillover) among the stock markets were also studied. Granger and Morgenstern (1970) used spectral analysis on weekly data for stock indices in eight countries and concluded that there was little or no interrelationship between different stock market exchanges around the world. Agmon (1972) found no significant leads or lags on a monthly basis among the common stocks of Germary, Japan, the U.K., and the U.S.

24

However, the market integration took off and the interdependence in returns across different markets grew over the years. In fact, even early studies such as Longin and Solnik (1995) showed that the correlation across major countries increased from 1960 to 1990. The evolution toward more integrated equity return exposures has been substantiated in later studies (e.g., Baele (2005), Bekaert et al. (2007), Eun and Lee (2010).) Regarding the return spillover between markets, Eun and Shim (1989) used a nine-market VAR system to find that innovations in the U.S. are rapidly transmitted to other markets in a clearly recognizable fashion, whereas no single foreign market can significantly explain the U.S. market movements. As to research in volatility comovement and spillover, it got a little late start because ARCH and GARCH models (Engle (1982) and Bollerslev (1986)) were developed in 1980’s. A series of research such as King and Wadhwani (1990), Hamao et al. (1990), Engle and Susmel (1993), Lin et al. (1994), King et al. (1994), Bae and Karolyi (1994), Karolyi (1995), and Koutmos and Booth (1995) discussed correlation, time-varying covariance, common volatility component, lead-lag relationship, transmission mechanism of stock market volatilities across various stock markets. Contrary to the studies on stock market returns in the international setting, the research in international stock market volatilities recognized and studied active interactions among the stock market volatilities across countries from early on. This is partially because of the predictable nature of the stock market volatility, and partially because the research in volatility comovement and spillover got a little late start to include the sample period when the market integration already took off. The return and volatility spillover across countries have an interesting implication for our ICAPM framework. Recall that, within our theoretical framework, any pricing factor is supposed to be the innovation of a state variable predicting the market return or the market volatility. Thus, for example, the market volatility spillover from the U.S. to U.K. (meaning the U.S. market volatility predicts the U.K. market volatility) should imply the innovation of the U.S. market volatility would be priced across the assets in the U.K. because it carries the information about the future changes in the U.K. volatility as well. As a matter of fact, many papers report unidirectional return and volatility spillover effect from the U.S. to the rest of the world. Rapach et al. (2013) show lagged U.S. returns significantly 25

predict returns in numerous non-U.S. industrialized countries, while lagged non-U.S. returns display limited predictive ability with respect to U.S. returns. Liu and Pan (1997) studied the mean and volatility spillovers from U.S. and Japanese stock markets to four other Asian stock markets and found that the U.S. market is more influential than the Japanese market in transmitting returns and volatilities to the other four Asian markets. Alaganar and Bhar (2002) examined the information flow between dually listed Australian stocks’ trades in Australia and U.S. using a bivariate GARCH model and found unidirectional information flow from U.S. market to Australian market. Savva et al. (2009) examined the spillover among U.S., German, U.K. and French markets using dynamic correlation framework and found that European markets (only U.K. and German) are affected by the U.S. market. We explore this implication from the return and volatility spillover in Table 8 and 9. We examine our regional data to see if they also show the return and volatility spillover effect similar to that reported in the country level data.

The market return series in North America are

almost identical to those in the U.S. since the market capitalization of the U.S. stock market is overwhelmingly larger than that of Canada. However, the market returns in other regions might have some different aspect because European region includes 16 countries and Asia Pacific region four. Table 8 presents the return and volatility spillover effect between North America and other regions. We conduct Granger causality F-tests similar to the ones we did in Table 2 but with the VAR system shown below: h iT (m) (m) (m) (m) e e zt = rm,(N ; s ; l ; r ; s ; l A),t m,(X),t (N A),t (N A),t (X),t (X),t

(4.5)

where the first three elements are the excess market return, monthly short- and long-run volatility components of North America, and the last three are the corresponding terms in some other region X. In Table 8, the first row shows the variables for the VAR equations, and the first column from the left shows the predictors. The optimal VAR order for each region is determined by the likelihood ratio test, and the numbers shown in the table are the F-statistics. The results in Table 8 strongly support the unidirectional spillover from the U.S. to Asia Pacific

26

and Europe. None of the excess market return, short- and long-run volatility components in Asia Pacific predict those in North America. The excess market return and the short-run volatility component in Europe predict North American long-run volatility component at 10% and 5% level, respectively. However, the excess market return and the short-run volatility component in North America strongly predict both the short- and long-run volatility components in Europe at 1% level. In fact, even for Japan, the empirical evidence for the spillover from the U.S. to Japan is more pronounced than vice versa. Another interesting observation in Table 8 is that most of the prediction relationship holds in a way that North American excess market return and the short-run volatility component predict the long-run volatility component of other regions. This seems not to be due to the persistence of the long-run component (even though you may well be worried about the possible spurious results to some extent). If that was the case, we would see statistically significant prediction relationships among the long-run components across regions. The excess market return in North America strongly predicts both volatility components in North America in all VAR specifications (Asia Pacific, Europe, Japan), and this relation seem to reflect the leverage effect within North America. Thus, the excess market return in North America seems to predict the long-run volatility components of other regions via short-run volatility component in North America. The short-run volatility component in the North America is supposed to reflect short-run market uncertainty which is driven by immediate yet noisy response of investors regarding uncertainty in current and future economic activities in North America. The expected uncertainty would materialize in North America and then in other regions with close economic links over time. This might be a probable transmission mechanics for the prediction relations discussed. It is also worthy of noting that the excess market return in North America does not predict that in Asia Pacific or Europe, which seems inconsistent with what is found in Rapach et al. (2013). Although Rapach et al. (2013) investigates prediction relations among the excess market returns across countries rather than regions, we expected to observe the similar pattern because North American excess market return is practically the U.S. excess market return and they reported the excess market return in the U.S. predicted that in almost every country in Europe examined. There may be a couple of reasons for disappearance of such prediction relations within our dataset. First,

27

the regional grouping of the country-level data might still play the role because the prediction relations between each country of a region and the U.S. can be very different. Second, probably more importantly, all the returns in our dataset are in USD while those in Rapach et al. (2013) are in local currency. Due to the USD conversion in returns, the regional excess market returns incorporate foreign exchange rate risk which might offset the predictive relation. Table 9 shows the results from Fama and MacBeth (1973) regressions of North American AVR model on the test assets in other regions. The empirical procedure is the same as in Table 5 except that the same North American AVR factors are used for each set of other regional test assets. Our empirical evidence in Table 8 implies that North American AVR factors should be priced, and that is, surprisingly, what is observed in Table 9 in Asia Pacific and Europe. Again, as we discussed previously, nothing is priced in Japan because the test portfolios in Japan do not generate large enough mean return spreads to be explained in the first place. Our work in this section is based on the theoretical framework discussed in Section 2 and the well-established empirical finding of return and volatility spillover. Our work here has an interesting implications for international asset pricing and shed new light especially on the long debate over the relationship between global and local factors.9 Theoretically, it would be more convenient if the markets are completely segmented or completely integrated. Then, under some regularity conditions, only one set of either local factors or global factors would matter in pricing assets in a stock market. However, in practice, both local and global factors matter in asset pricing, and this has been a challenge for developing models because the two are associated with different views of market integration. Lewis (2011) states that there are three types of models that explain both global and local sources of risk. First, a model with purchasing power parity deviations allows local and global risk factors when exchange rate risk is priced. Second, a model for emerging markets in progress of capital market liberalizations provides coexisting local and global risk factors. Third, a model that accommodates information differences across markets shows equity returns depend on both local and global sources of risk. All these models allow for investors to have access to international markets yet retain exposure to local shocks that cannot be diversified away. 9

Refer to Karolyi and Stulz (2003) for a summary of this debate.

28

Our work suggests another possibility. First, assume practically segmented financial markets. What I mean by practically segmented financial markets is that investors, whether domestic or foreign, for a country’s financial market hold the portfolio optimal only for the country’s financial market (e.g., in terms of CAPM, investors hold a combination of the risk-free asset and the market portfolio for the country.). Within this financial market structure, international investors would hold a global portfolio of many locally optimal portfolios. They choose not to hold the globally optimal portfolio. For example, an investor who invests stocks in U.S., U.K., and Hong Kong would holds a portfolio of S&P500 index fund, FTSE 100 index fund, and Hang Seng index fund, and this actually is what many international investors do for cost efficiency and limited resources for foreign markets. Also, even in the internet era when information and money can be transmitted to anywhere in the world in an instant, there are numerous regulatory barriers to have most domestic investors stay inside the border. The assumption of the practically segmented financial markets makes the assets in each market priced primarily by local risk factors, which is supported strongly in empirical works such as Ferson and Harvey (1993), Griffin (2002), Bekaert et al. (2009), Fama and French (2012), and Fama and French (2017). Second, assume return and volatility spillover effect across countries. Even with the practically segmented financial markets, the return and volatility spillover can occur through channels of various trades of goods rather than securities between countries. For example, a stock market in a country with many large corporations exporting goods to the U.S. will respond positively for any good news on the U.S. economy. The trade channel story would also help explain the varying degrees of financial market integration between countries by the trade relations. Theoretically, Rizova (2010) presents a simple Lucas-tree two-country, two-good, free-trade model to show the interactions between international trade and stock markets. The main implication is that stock markets forecast trade flows and that stock markets react immediately and fully to news about trading partners. Her model assumes open financial market, but the same implications seem possible with stochastic production of traded goods for the segmented financial market. Finally, assume that the investment opportunity set in the financial market is fully described by the market return and the market volatility as in our theoretical model. Then, under such economy, the assets in each local economy would also be priced with global risk factor which

29

is value-weighted average of local risk factors. In such world, the return and volatility spillover would be observed from large economies to smaller economies. Thus, the returns and volatilities of the large economies would predict those of the smaller economies, and the innovations thereof would be legitimate pricing factors for each of the small economies. These large economies also tend to have large stock market, and hence the local risk factors of the large economies would be most heavily weighted in the global risk factors.10 Thus, the global factors, practically local factors of large economy, would be priced in many relatively smaller economies. Our approach also explains why the global risk factors perform worse than the local risk factors even when they are statistically significant. In sum, the global risk factors are priced not because investors around the world try to hedge against it, but because they carry information about the future changes in the local investment opportunity set. The development of the theoretical model encompassing all the features discussed above and the following empirical work are beyond the scope of our paper but clearly suggest an interesting direction for future research.

5

Conclusion

With Campbell’s (1993) ICAPM extended to accommodate heteroskedastic asset returns, we motivate the aggregate volatility risk model and show that the aggregate volatility risks are robustly priced across a large set of test portfolios in each of Asia Pacific, Europe, and North America. In this theoretical framework, investors care about the current market risk and the uncertainty in changes of future investment opportunity set which is completely described by the market return and the market volatility. Thus, the innovation of any variable that predicts either the market return or the market volatility would be priced across assets. Unfortunately, our dataset updated from Fama and French (2012) is regional and hence the conventional market return predictors (which tend to be also the market volatility predictors; e.g., dividend yield, term spread, default spread, and so on.) are not available. However, Petkova (2006) already showed that SMB and HML are the proxies of the innovations of these macroeonomic or financial variables in the U.S. We focus on the aggregate market volatility risk. We show sres and lres are robustly priced across 10

The correlation of the global market return and North American market return in Fama and French (2012) is above 0.9.

30

very challenging regional test portfolios. We investigate if the empirical factors such as SMB, HML, and UMD represent the systematic risk of the aggregate volatility risk. The direct test can be conducted if the state variables of which innovations are SMB, HML, and UMD are observable. Unfortunately, such state variables are unknown and unobservable. Hence, we try various ways to test the implications of our ICAPM on these empirical factors. In our time-series study, we show that the aggregate volatility risk factors (sres and lres) share a common component with UMD in Asia Pacific and North America and with HML in Europe. They are also shown to covary with SMB in all regions except Europe but SMB is not priced in any region. Our regional cross-sectional studies reveal that momentum sorting provides the largest mean return spreads, which implies explaining these momentum-generated mean spreads is crucial for the performance of factors. Across all regions except for Japan, the aggregate volatility risk factors and momentum factor are robustly priced. In addition, the aggregate volatility risk (AVR) model explains 52%, 19%, 109%, 56% of momentum profits in Asia Pacific, Europe, Japan, and North America, respectively. Finally, we propose an alternative theory for explaining the coexistence of global and local factors. Our theoretical framework coupled with return and volatility spillover effect observed empirically provides interesting insights for international asset pricing. We discuss how global factor might work even when no investor holds globally optimal portfolio.

31

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38

39

-0.071 0.105

0.010 0.027

Estimate Std. Err.

0.023 0.018

Estimate Std. Err.

Estimate Std. Err.

0.128 0.373

Estimate Std. Err.

-0.068 0.063

0.392 0.235

-0.034 0.101

-1.239 0.164

-0.010 0.031

0.147 0.067

0.003 0.047

0.082 0.050

Market Excess Return θ1 θ2 θ3

-0.029 0.033

0.050 0.095

-0.010 0.024

0.877 0.016

0.758 0.064

0.868 0.018

0.605 0.035

-0.074 0.004

-0.041 0.006

-0.051 0.004

-0.074 0.006

θ8

-0.001 0.005

0.001 0.001

0.003 0.008

0.979 0.004

0.995 0.001

0.984 0.002

0.011 0.006

0.002 0.002

0.994 0.001

Panel D: North America

0.026 0.011

Panel C: Japan

0.043 0.007

Panel B: Europe

0.011 0.006

-0.017 0.003

-0.016 0.004

-0.009 0.003

-0.020 0.003

0.035 0.003

0.069 0.008

0.034 0.004

0.063 0.004

Long-Run Component θ9 θ10 θ11

Panel A: Asia Pacific

Short-Run Component θ5 θ6 θ7

0.023 0.111

θ4

Mean Std.

Mean Std.

Mean Std.

Mean Std.

-0.243 0.151

0.202 0.069

-0.087 0.123

0.057 0.095

0.057 0.323

0.023 0.229

0.022 0.272

-0.196 0.290

0.943 0.881

1.810 1.063

1.110 0.905

0.952 0.793

Summary Statistics s l v

The table reports the parameter estimates of the two-component volatility model of Adrian and Rosenberg (2008) with four regional datasets from Fama and French (2012). The model is specified in Eq.(2.4)-(2.7), and fitted over the daily sample from July 1990 to December 2016 excluding 2008 by the maximum likelihood estimation method.

Table 1: Parameter Estimates for Adrian and Rosenberg (2008) Model and Summary Statistics

40

(m)

(m)

(m)

Adj. R2 (%)

e rm (m) s l(m) SMB HML UMD

VAR Eq.:

5.93

0.780 1.722 0.954 0.769 3.687∗∗∗ 3.620∗∗∗

e rm

l(m)

5.80

72.68

0.435 1.842 1.219 1.750 2.285∗ 117.453∗∗∗ 0.611 1.534 3.054∗∗ 1.599 2.833∗∗ 1.016

s(m)

Asia Pacific

−1.08

0.642 0.476 0.253 0.672 0.895 2.421∗

e rm

l(m)

7.37

90.73

2.349∗ 0.176 0.636 2.912∗∗ 0.823 473.675∗∗∗ 0.535 1.292 0.291 0.300 1.259 2.025

s(m)

Europe

6.22

2.872∗∗ 2.049∗ 3.047∗∗ 0.424 1.436 1.197

e rm

9.43

5.082∗∗∗ 1.216 1.148 0.731 1.825 1.474

s(m)

Japan

58.97

0.776 1.498 24.382∗∗∗ 2.154∗ 1.487 1.628

l(m)

−0.85

0.554 0.505 0.128 1.278 1.838 1.360

e rm

l(m)

10.36

90.45

6.154∗∗∗ 9.066∗∗∗ 2.004∗ 2.452∗∗ 0.179 382.353∗∗∗ 1.370 0.977 0.149 0.468 0.317 1.487

s(m)

North America

e e , st , lt are the monthly The table reports F-statistics of the Granger causality test in the VAR system of {rm,t ; st ; lt ; SM Bt ; HM Lt ; U M Dt } where rm,t regional market excess return, the monthly short-run (regional) market volatility component aggregated from the daily estimates of st , and the monthly long-run (regional) market volatility component aggregated from the daily estimates of lt . SM Bt , HM Lt , U M Dt are monthly size, value, and momentum factors of the corresponding regions from Prof. Kenneth French’s website. The VAR orders are determined by the likelihood ratio tests. For the brevity of the presentation, (m) (m) e the table shows only a part of each region’s F-statistics for prediction of {rm,t , st , and lt }. The top row shows the identifications of these VAR equations. Significance at the 1% level is denoted by ***, at the 5% level by **, and at the 10% level by *. The adjusted R2 is reported in percentage form.

(m)

Table 2: Granger Causality F-Tests

Table 3: Summary Statistics of Pricing Factors e The table presents the summary statistics for the pricing factors to be studied in this paper. rm is the monthly regional market excess return. sres and lres are the innovations of the monthly short-run component, s(m) , and the monthly long-run component, l(m) , of the regional market volatility. SMB, HML, UMD are the monthly size, value, and momentum factors of the corresponding regions from Prof. Kenneth French’s website. Mean and Std. Dev. are the mean and standard deviation of the returns or innovation series, and t-stat is the ratio of Mean to its standard error. The sample period is from July 1991 to December 2016 excluding 2008.

Factors

Mean

t-stat

Std. Dev.

Skewness

Kurtosis

Panel A: Asia Pacific e rm sres lres SMB HML UMD

0.969 -0.001 0.006 -0.110 0.639 0.849

(2.93) (-0.63) (3.59) (3.23)

5.669 0.826 3.044 2.998 3.056 4.510

-0.052 0.270 0.543 0.259 1.505 -2.977

4.729 3.390 3.565 5.333 13.939 22.074

Panel B: Europe e rm sres lres SMB HML UMD

0.751 -0.003 -0.001 0.046 0.398 0.913

(2.80) (0.35) (2.74) (3.89)

4.601 1.624 1.741 2.248 2.492 4.022

-0.241 0.698 0.598 -0.069 0.351 -1.382

3.722 3.684 3.947 4.035 5.687 10.840

Panel C: Japan e rm sres lres SMB HML UMD

0.176 -0.002 0.020 0.016 0.311 0.067

(0.55) (0.09) (1.81) (0.26)

5.517 0.763 2.771 3.197 2.949 4.465

0.302 0.296 0.629 0.185 -0.130 -0.350

3.459 3.401 3.264 4.619 5.185 5.622

Panel D: North America e rm sres lres SMB HML UMD

0.840 0.004 0.017 0.111 0.292 0.593

(3.58) (0.62) (1.50) (2.09)

4.028 1.891 2.054 3.088 3.326 4.853

41

-0.504 0.058 0.582 0.705 0.526 -0.183

3.964 2.562 4.186 11.809 7.493 11.859

Table 4: Pricing Information of the Aggregate Volatility Risk Factors The table presents contemporaneous time-series regressions of innovations in the monthly short-run component, s(m) , and the monthly long-run component, l(m) , of the regional market volatility on the size, value, and momentum factors. The innovations (sres and lres) are computed from the VAR system estimated in Table 2. The adjusted R2 is reported in percentage form. The sample period is from July 1991 to December 2016 excluding 2008. The numbers in the parenthesis are t-stats adjusted for serial dependence. Dep. Var.

Const.

SMB

HML

UMD

Adj. R2

Panel A: Asia Pacific sres lres

-0.002 (-0.04) 0.111 (0.65)

-0.004 (-0.26) -0.178 (-3.26)

-0.019 (-1.16) -0.078 (-1.40)

0.015 (1.36) -0.088 (-2.32)

0.53 3.94

Panel B: Europe sres lres

0.016 (0.15) 0.069 (0.67)

0.005 (0.12) -0.038 (-0.86)

-0.116 (-2.86) -0.122 (-2.95)

0.030 (1.19) -0.021 (-0.83)

3.56 1.93

Panel C: Japan sres lres

-0.010 (-0.24) 0.029 (0.19)

-0.042 (-3.27) -0.132 (-2.74)

0.025 (1.69) -0.015 (-0.27)

0.010 (1.05) -0.034 (-0.95)

3.45 1.42

Panel D: North America sres lres

-0.023 (-0.21) 0.072 (0.62)

-0.180 (-4.84) -0.111 (-2.79)

42

0.049 (1.42) -0.045 (-1.20)

0.055 (2.40) -0.050 (-2.04)

9.71 3.45

43 28.92

23.24

R2 (%)

0.409 (2.59) [2.23]

0.954 (3.21) [3.15]

0.053 (0.34) [0.28]

HML

0.081 (0.40) [0.42]

UMD

0.221 (1.06) [1.13]

SMB

16.33

-0.943 (-1.32) [-1.12]

0.973 (2.75) [2.85]

lres

0.985 (2.92) [2.89]

AVR

-0.423 (-3.96) [-3.57]

0.922 (2.71) [2.71]

C4

sres

e rm

FF3

Asia Pacific

28.82

0.169 (0.71) [0.73]

0.161 (1.26) [1.28]

0.714 (2.68) [2.64]

FF3

42.98

0.989 (3.66) [3.62]

0.421 (1.76) [1.85]

0.097 (0.76) [0.77]

0.762 (2.87) [2.82]

C4

Europe

21.83

-1.440 (-2.65) [-2.06]

-1.073 (-3.51) [-3.02]

0.713 (2.58) [2.57]

AVR

34.10

0.116 (0.42) [0.46]

0.149 (0.76) [0.79]

0.140 (0.39) [0.38]

FF3

41.19

0.079 (0.25) [0.26]

0.186 (0.73) [0.80]

0.136 (0.69) [0.72]

0.141 (0.39) [0.38]

C4

Japan

23.11

-0.371 (-0.51) [-0.46]

-0.213 (-1.29) [-1.21]

0.174 (0.48) [0.46]

AVR

35.30

0.461 (1.64) [1.77]

0.291 (1.73) [1.78]

0.733 (3.33) [3.30]

FF3

47.15

0.712 (2.38) [2.35]

0.598 (2.17) [2.30]

0.182 (1.09) [1.13]

0.812 (3.74) [3.67]

C4

28.11

-1.150 (-2.52) [-2.27]

-0.807 (-2.59) [-2.36]

0.771 (3.50) [3.47]

AVR

North America

The table reports the risk price estimates and their t-stats of three regional factor models in each region. FF3, C4, AVR are the Fama-French three-factor Model, Carhart’s four-factor Model, and the Aggregate Volatility Risk Model in the corresponding region, respectively. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. Each regional factor model is fitted over 100 regional test portfolios composed of 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The sample is of the monthly frequency and the sample period is from July 1991 to December 2016 excluding 2008. The numbers in the parenthesis are Fama-MacBeth t-stats, and the ones in the brackets are t-stats with Shanken (1992) corrected standard errors. The adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Table 5: Price of Risk Estimation for Baseline Factor Models

Table 6: Price of Risk Estimation for Orthogonalized Factors The table reports the risk price estimates and their t-stats of all-inclusive factor model in each region. The factors are orthogonalized in a similar fashion as in Campbell (1996) and Petkova (2006). The set of factors e {rm,t , sres, lres, SM Bt , HM Lt , U M Dt } in which sres and lres are obtained from the residuals of the VAR system in Table 2 is orthogonalized using the Cholesky decomposition of the covariance matrix of the factors. The resulting e orthogonalized factors are denoted rm , usres , ulres , uSM B , uHM L , uU M D , respectively. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. Each all-inclusive regional (orthogonalized) factor model is fitted over 100 regional test portfolios composed of 25 size and book-tomarket, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The fitted sample is of the monthly frequency and the sample period is from July 1991 to December 2016. The numbers in the parenthesis are Fama-MacBeth t-stats, and the ones in the brackets are t-stats with Shanken (1992) corrected standard errors. The adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Asia Pacific

Europe

Japan

North America

e rm

1.007 (3.00) [2.97]

0.764 (2.87) [2.83]

0.149 (0.41) [0.40]

0.809 (3.68) [3.66]

usres

-0.880 (-1.20) [-1.14]

1.447 (1.05) [0.82]

-1.414 (-1.33) [-1.28]

-1.799 (-2.90) [-2.52]

ulres

1.361 (1.14) [1.06]

-1.513 (-1.90) [-1.54]

1.048 (1.09) [0.98]

1.179 (1.49) [1.29]

uSM B

0.303 (0.63) [0.64]

0.874 (2.34) [1.99]

0.062 (0.17) [0.17]

-0.296 (-1.26) [-1.25]

uHM L

0.729 (2.39) [1.97]

0.668 (1.31) [1.36]

0.316 (0.70) [0.74]

0.773 (2.25) [2.29]

uU M D

1.699 (4.80) [4.24]

1.913 (5.43) [4.64]

0.151 (0.38) [0.40]

1.159 (4.32) [3.67]

R2 (%)

31.55

48.35

44.90

52.40

44

45

SMB

-0.095 -0.319 -0.182

0.214

-0.110 (-0.63)

-1.95

Traded Factor:

βm βsres βlres

Risk Premium

Average (t-stat)

Premium / Avg

-0.15

0.639 (3.59)

-0.096

0.173 0.713 -0.039

HML

Asia Pacific

0.52

0.849 (3.23)

0.444

-0.421 -1.634 -0.173

UMD

4.55

0.046 (0.35)

0.211

-0.268 -0.692 0.236

SMB

0.79

0.398 (2.74)

0.315

0.063 -0.082 -0.127

HML

Europe

0.19

0.913 (3.89)

0.172

-0.406 -0.628 0.147

UMD

9.50

0.016 (0.09)

0.151

0.010 -0.583 -0.067

SMB

0.00

0.311 (1.81)

0.000

-0.078 -0.052 -0.005

HML

Japan

1.09

0.067 (0.26)

0.073

-0.147 -0.397 -0.037

UMD

3.43

0.111 (0.62)

0.382

0.014 -0.448 -0.008

SMB

-0.09

0.292 (1.50)

-0.026

-0.242 -0.060 -0.097

HML

0.56

0.593 (2.09)

0.331

-0.179 0.196 -0.546

UMD

North America

where f is a traded factor x constructed from a zero-cost portfolio. βxm , βxs , βxl are estimated from one multiple time-series regression with the full sample. The sample period is from July 1991 to December 2016 excluding 2008. The regional risk prices (λm , λs , λl ) are taken from Table 5. The averages of the regional traded factors and their t-stats are taken from Table 3.

E[fx ] = λm βxm + λs βxs + λl βxl

The table presents the ratio of the fitted risk premium over the empirical mean estimate of the traded factors for each region. The fitted factor risk premium is based on the Aggregate Volatility Risk Model as in Eq.(4.1)-(4.2) and computed as follows:

Table 7: Fitted Risk Premium vs. Average of the Traded Factors

Table 8: Granger Causality F-Tests between North America and Other Regions The table reports F-statistics of the Granger causality test in the VAR (m) (m) (m) (m) (m) (m) e e e system of {rm,(N ; lt } where rm,(N A),t ; s(N A),t ; l(N A),t ; rm,t ; st A),t , s(N A),t , l(N A),t are the monthly NA market excess return, the monthly short-run NA market volatility component aggregated from the daily estimates of s(N A),t , (m) e and the monthly long-run NA market volatility component aggregated from the daily estimates of l(N A),t . rm,t , st , (m) lt are the regional counterparts. The VAR orders are determined by the likelihood ratio tests. The top row lists the VAR equation variables, or the variable being predicted. Significance at the 1% level is denoted by ***, at the 5% level by **, and at the 10% level by *. The adjusted R2 is reported in percentage form. VAR Eq.:

e rm,(N A)

(m)

(m)

s(N A)

e rm

l(N A)

s(m)

l(m)

Panel A: Asia Pacific e rm,(N A)

0.191

6.130∗∗∗ 10.549∗∗∗ 0.533

0.328

2.342∗

(m) s(N A) (m) l(N A) e rm

0.070

2.229∗

0.416

3.259∗∗

0.111

0.062

369.041

0.254

0.275

0.873

0.428

0.259

0.797

0.817

0.874

1.086

s(m)

0.903

0.969

1.084

1.345

1.517

2.796∗∗

l(m)

0.165

0.867

0.205

0.408

1.588

91.579∗∗∗

1.09

1.93

73.03

Adj. R2 (%)

−3.75

12.00

4.472∗∗∗ 0.116 ∗∗∗

90.34

Panel B: Europe e rm,(N A)

0.393

7.587∗∗∗ 12.214∗∗∗ 1.047

3.413∗∗∗ 8.130∗∗∗

s(N A)

0.098

1.952

4.913∗∗∗ 0.791

3.618∗∗∗ 7.141∗∗∗

l(N A)

0.118

0.165

e rm

0.819

1.269

∗∗∗

215.767

2.152∗

1.037

2.012∗

0.998

1.172

2.173∗

s

0.944

0.911

2.947

0.893

0.664

6.779∗∗∗

l

0.120

1.004

1.055

0.661

0.282

123.075∗∗∗

Adj. R2 (%)

−3.37

12.38

∗∗

0.183

−4.39

90.92

10.70

91.60

Panel C: Japan e rm,(N A)

0.312

7.353∗∗∗ 12.036∗∗∗ 2.493∗∗

s(N A)

0.946

1.874

l(N A)

0.264

0.619

1.893

3.118∗∗

2.590

1.362

2.370∗

424.954∗∗∗ 0.374

0.458

1.698

∗∗∗

3.937

∗∗

∗∗

e rm

0.572

0.256

0.402

1.285

2.576

0.595

s

0.846

0.187

0.399

2.237∗

0.926

2.214∗

l

2.506∗∗

1.417

2.726∗∗

3.002∗∗

0.821

15.856∗∗∗

3.60

5.46

60.47

Adj. R2 (%)

−1.10

11.84

90.40

46

Table 9: Regional Cross-Sectional Regressions with North American Factors The table reports the risk price estimates and their t-stats for the Aggregate Volatility Risk Model in each region. e The same set of North American factors (rm,(N A) , sres(N A) , lres(N A) ) are used for Asia Pacific, Europe, and Japan. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. The North American factor model is fitted over 100 regional test portfolios composed of 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The fitted sample is of the monthly frequency and the sample period is from July 1991 to December 2016. The numbers in the parenthesis are Fama-MacBeth t-stats, and the ones in the brackets are t-stats with Shanken (1992) corrected standard errors. The Adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Region:

Asia Pacific

Europe

Japan

e rm,(N A)

0.716 (1.98) [1.47]

0.388 (1.41) [1.34]

0.154 (0.22) [0.20]

sres(N A)

-0.852 (-2.32) [-1.72]

-1.129 (-2.61) [-1.98]

-0.639 (-0.93) [-0.78]

lres(N A)

-1.999 (-3.14) [-2.25]

-1.775 (-3.51) [-2.70]

-1.230 (-1.25) [-1.07]

Adj. R2 (%)

12.43

19.18

-9.69

47

Figure 1: Monthly Market Volatility (Annualized) The figure plots two measures of the annualized standard deviation of the market excess return at a monthly frequency from July 1990 to December 2016 for four regions: Asia Pacific, Europe, Japan, and North America. The first measure is derived from daily squared returns. It is the square root of the monthly sum of daily squared returns (realized variance). The second measure is the conditional volatility from the Adrian and Rosenberg (2008) model. The shaded bar depicts the year 2008.

48

Figure 2: Short-Run Component of the Market Volatility The figure presents the (demeaned) short-run components st of Adrian and Rosenberg (2008) as estimated in Table 1 for four regions: Asia Pacific, Europe, Japan, and North America.

49

Figure 3: Fitted Risk Premium vs. Average Excess Returns The figure presents the average excess returns for 100 test portfolios against the predicted risk premiums from the Aggregate Volatility Risk Model in Table 5 for four regions: Asia Pacific, Europe, Japan, and North America. The following are the test portfolio identifiers; S for size, B for book-to-market, P for operating profitability, I for investment, and M for momentum. The portfolio numbers go from 1 to 5; from low/small to high/big.

50

Internet Appendix I.A

Regional Cross-Sectional Regressions with Different Set of Test Portfolios

I.A.1

Asia Pacific

I.A.2

Europe

I.A.3

Japan

I.A.4

North America

27.35

26.70

Adj. R2 (%)

0.595 (3.50) (3.21) 3.065 (3.79) (2.84)

0.513 (3.02) (2.88)

HML (FM) (SH)

-0.091 (-0.46) (-0.47)

UMD (FM) (SH)

0.066 (0.32) (0.34)

14.99

-0.325 (-0.41) (-0.40)

lres (FM) (SH)

SMB (FM) (SH)

-0.056 (-0.31) (-0.31)

0.957 (2.74) (2.81)

sres (FM) (SH)

1.030 (3.07) (3.03)

e rm (FM) (SH)

0.887 (2.61) (2.60)

Size-BM

Test Assets:

24.02

-0.409 (-2.29) (-1.65)

0.359 (1.78) (1.84)

0.977 (2.88) (2.86)

24.48

0.258 (0.42) (0.38)

-0.376 (-1.78) (-1.32)

0.340 (1.64) (1.71)

0.988 (2.95) (2.91)

Size-OP

15.13

-1.383 (-1.60) (-1.48)

-0.209 (-1.68) (-1.69)

0.964 (2.78) (2.84)

21.62

0.984 (2.47) (2.40)

0.031 (0.13) (0.14)

0.889 (2.61) (2.61)

24.05

1.512 (2.62) (2.15)

0.991 (2.49) (2.24)

-0.056 (-0.24) (-0.25)

0.985 (2.90) (2.90)

Size-Inv

16.69

-0.734 (-0.87) (-0.84)

-0.238 (-1.52) (-1.56)

0.999 (2.86) (2.94)

31.46

-2.033 (-3.80) (-2.91)

0.501 (2.20) (2.17)

1.100 (3.25) (3.18)

34.27

0.817 (2.69) (2.71)

-0.721 (-2.25) (-1.84)

0.282 (1.29) (1.36)

1.059 (3.09) (3.09)

Size-Mom

21.03

0.348 (0.55) (0.29)

-0.773 (-7.10) (-3.98)

1.130 (3.10) (3.14)

The table reports the risk price estimates and their t-stats of three factor models in Asia Pacific region for each of four sets of regional test portfolios. The three factor models are Fama-French Three Factor Model, Carhart’s Four Factor Model, and the Aggregate Volatility Risk Model. The four sets of test assets are 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. All data are of the monthly frequency and the sample period is from July 1991 to December 2016. Two t-stats are provided for each risk price estimate: Fama-MacBeth t-stats and t-stats with Shanken (1992) corrected standard errors. The Adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Table IA.1: Cross-Sectional Regressions for Asia Pacific Region

42.45

39.39

Adj. R2 (%)

0.354 (1.62) (1.61) 1.440 (2.09) (1.84)

0.403 (1.78) (1.82)

HML (FM) (SH)

0.010 (0.08) (0.08)

UMD (FM) (SH)

0.019 (0.15) (0.15)

28.83

-0.953 (-1.18) (-1.02)

lres (FM) (SH)

SMB (FM) (SH)

-0.579 (-1.69) (-1.58)

0.800 (3.00) (2.93)

sres (FM) (SH)

0.782 (2.97) (2.89)

e rm (FM) (SH)

0.741 (2.78) (2.74)

Size-BM

Test Assets:

30.06

-0.329 (-1.39) (-1.20)

0.225 (1.71) (1.78)

0.760 (2.83) (2.82)

37.64

3.286 (4.56) (3.43)

0.514 (1.69) (1.42)

0.060 (0.46) (0.45)

0.795 (2.99) (2.94)

Size-OP

23.05

-0.152 (-0.39) (-0.38)

-0.448 (-1.43) (-1.50)

0.805 (2.99) (2.96)

35.25

0.677 (2.35) (2.63)

0.030 (0.24) (0.24)

0.751 (2.83) (2.78)

35.73

0.069 (0.10) (0.09)

0.636 (2.09) (2.32)

0.037 (0.29) (0.29)

0.754 (2.84) (2.79)

Size-Inv

30.35

-2.168 (-2.76) (-1.69)

-1.115 (-2.66) (-1.74)

0.815 (3.06) (2.97)

32.73

-1.645 (-3.27) (-2.98)

0.420 (2.98) (2.91)

0.814 (3.09) (3.00)

52.37

0.953 (3.50) (3.49)

0.756 (1.55) (1.56)

0.160 (1.15) (1.16)

0.747 (2.81) (2.76)

Size-Mom

19.89

-2.720 (-4.60) (-2.41)

-1.962 (-5.29) (-2.97)

0.600 (2.03) (1.88)

The table reports the risk price estimates and their t-stats of three factor models in European region for each of four sets of regional test portfolios. The three factor models are Fama-French Three Factor Model, Carhart’s Four Factor Model, and the Aggregate Volatility Risk Model. The four sets of test assets are 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. All data are of the monthly frequency and the sample period is from July 1991 to December 2016. Two t-stats are provided for each risk price estimate: Fama-MacBeth t-stats and t-stats with Shanken (1992) corrected standard errors. The Adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Table IA.2: Cross-Sectional Regressions for European Region

39.98

38.55

Adj. R2 (%)

0.318 (1.43) (1.46) 1.162 (1.15) (1.18)

0.295 (1.30) (1.35)

HML (FM) (SH)

0.087 (0.45) (0.47)

UMD (FM) (SH)

0.083 (0.43) (0.45)

19.99

0.688 (0.94) (0.65)

lres (FM) (SH)

SMB (FM) (SH)

-0.206 (-1.14) (-0.82)

0.214 (0.57) (0.56)

sres (FM) (SH)

0.168 (0.46) (0.45)

e rm (FM) (SH)

0.146 (0.40) (0.39)

Size-BM

Test Assets:

34.95

-0.677 (-1.46) (-1.57)

0.339 (1.45) (1.55)

0.198 (0.54) (0.53)

35.21

0.584 (0.60) (0.61)

-0.588 (-1.26) (-1.30)

0.315 (1.31) (1.43)

0.200 (0.54) (0.54)

Size-OP

27.12

-0.962 (-1.02) (-0.95)

-0.183 (-0.86) (-0.84)

0.165 (0.45) (0.44)

37.12

-0.078 (-0.22) (-0.25)

0.206 (1.01) (1.05)

0.137 (0.38) (0.37)

36.82

0.206 (0.30) (0.31)

-0.051 (-0.15) (-0.16)

0.197 (0.97) (1.01)

0.144 (0.40) (0.39)

Size-Inv

28.82

-0.405 (-0.38) (-0.33)

-0.240 (-1.35) (-1.21)

0.149 (0.41) (0.40)

40.90

-0.119 (-0.22) (-0.22)

0.288 (1.27) (1.31)

0.129 (0.35) (0.35)

50.60

0.030 (0.10) (0.10)

0.030 (0.06) (0.06)

0.251 (1.07) (1.11)

0.118 (0.32) (0.32)

Size-Mom

21.90

-0.643 (-0.96) (-0.88)

-0.231 (-1.27) (-1.21)

0.166 (0.45) (0.42)

The table reports the risk price estimates and their t-stats of three factor models in Japan for each of four sets of regional test portfolios. The three factor models are Fama-French Three Factor Model, Carhart’s Four Factor Model, and the Aggregate Volatility Risk Model. The four sets of test assets are 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. All data are of the monthly frequency and the sample period is from July 1991 to December 2016. Two t-stats are provided for each risk price estimate: Fama-MacBeth t-stats and t-stats with Shanken (1992) corrected standard errors. The Adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Table IA.3: Cross-Sectional Regressions for Japan

46.84

45.61

Adj. R2 (%)

0.376 (1.47) (1.52) 2.620 (4.50) (3.69)

0.339 (1.32) (1.38)

HML (FM) (SH)

0.116 (0.71) (0.71)

UMD (FM) (SH)

0.145 (0.88) (0.89)

23.59

-1.594 (-3.01) (-2.18)

lres (FM) (SH)

SMB (FM) (SH)

-0.264 (-0.84) (-0.63)

0.854 (4.00) (3.80)

sres (FM) (SH)

0.886 (4.13) (3.98)

e rm (FM) (SH)

0.789 (3.68) (3.56)

Size-BM

Test Assets:

39.01

0.920 (2.54) (2.76)

0.115 (0.69) (0.70)

0.756 (3.38) (3.41)

39.92

1.210 (1.87) (1.51)

0.961 (2.61) (2.71)

0.101 (0.60) (0.60)

0.794 (3.63) (3.58)

Size-OP

35.80

1.435 (2.22) (1.32)

-0.832 (-2.74) (-1.71)

0.736 (3.27) (3.24)

44.79

0.771 (2.67) (2.79)

0.099 (0.59) (0.60)

0.817 (3.79) (3.69)

45.91

2.796 (3.71) (2.90)

0.595 (2.08) (2.03)

0.101 (0.60) (0.60)

0.895 (4.14) (4.04)

Size-Inv

29.11

-2.022 (-3.34) (-2.26)

-0.481 (-1.47) (-1.02)

0.860 (3.96) (3.87)

31.65

-0.267 (-0.64) (-0.69)

0.659 (3.19) (3.29)

0.688 (3.09) (3.07)

56.09

0.682 (2.27) (2.26)

0.781 (2.30) (2.33)

0.256 (1.37) (1.48)

0.783 (3.59) (3.52)

Size-Mom

43.35

-1.742 (-3.09) (-2.42)

-0.958 (-2.90) (-2.35)

0.814 (3.75) (3.65)

The table reports the risk price estimates and their t-stats of three factor models in North American region for each of four sets of regional test portfolios. The three factor models are Fama-French Three Factor Model, Carhart’s Four Factor Model, and the Aggregate Volatility Risk Model. The four sets of test assets are 25 size and book-to-market, 25 size and operating profitability, 25 size and investment, and 25 size and momentum sorted portfolios. The risk prices are estimated by Fama and MacBeth (1973) regressions. The factor loadings are estimated with one multiple time-series regression using the full sample. The second stage cross-sectional regressions are conducted without a constant. All data are of the monthly frequency and the sample period is from July 1991 to December 2016. Two t-stats are provided for each risk price estimate: Fama-MacBeth t-stats and t-stats with Shanken (1992) corrected standard errors. The Adjusted R2 follows Jagannathan and Wang (1996) and is reported in percentage form.

Table IA.4: Cross-Sectional Regressions for North American Region