An International Asset Pricing Model with Time-Varying Hedging Risk

An International Asset Pricing Model with

Time-Varying Hedging Risk

JOW-RAN CHANG AND MAO-WEI HUNG* College of Management, National Taiwan University E-mail: hung@handel.mba.ntu.edu.tw

Abstract. This paper employs a two-factor international equilibrium asset pricing model to examine the pricing relationships among the world's ®ve largest equity markets. In addition to the traditional market factor premium, a hedging factor premium is included as the second factor to explain the relationship between risks and returns in the international stock markets. Moreover, a GARCH parameterization is adopted to characterize the general dynamics of the conditional second moments. The results suggest that the additional hedging risk premium is needed to explain rates of return on international equities. Furthermore, the restriction that the coef®cient on the hedge-portfolio covariance is one smaller than the coef®cient on the market-portfolio covariance can not be rejected. This suggests that the intertemporal asset pricing model proposed by Campbell (1993) can be used to explain the returns on the ®ve largest stock market indices.

Key words: international asset pricing, hedging risk, GARCH JEL Classi®cation: C32, F30, G12, G15

I. Introduction

The relationship between risk and return has been the focus of recent ®nance research. Numerous papers have derived various versions of the international asset pricing model. For example, Solnik (1974) extends the static capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965) to an international framework. His empirical ®ndings reveal that national factors are important in the pricing of stock markets. Furthermore, Korajczyk and Viallet (1989) propose that the international CAPM outperforms its domestic counterpart in explaining the price behavior of equity markets.

In a fruitful attempt to extend the conditional version of the static CAPM, Harvey (1991) employs the generalized method of moments (GMM) to examine an international asset pricing model that captures some of the dynamic behavior of the country returns. De Santis and Gerard (1997) test the conditional CAPM for international stock markets, but they apply a parsimonious generalized auto-regressive conditional heteroscedasticity (GARCH) parameterization as the speci®cation for second moments. Their results indicate that a one-factor model cannot fully explain the dynamics of international expected returns and price of market risk is not signi®cant.

On the other hand, recent studies have applied the arbitrage pricing theory (APT) of Ross (1976) to an international setting. For instance, Cho, Eun, and Senbet (1986) employ factor analysis to demonstrate that additional factors other than covariance risk

*Send all correspondence to: Professor Mao-wei Hung, Department of International Business, College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan.

are able to explain the international capital market. Ferson and Harvey (1993) investigate the predictability of national stock market returns and its relation to global economic risk. Their model includes a world market portfolio, exchange rate ¯uctuations, world interest rates, and international default risk. They use multi-factor asset pricing models with time-varying risk premiums to examine the issue of predictability. But, one of the drawbacks of the APT approach is that the number and identity of the factors are determined either ad hoc or statistically from data rather than derived from asset pricing models directly.

In a dynamic economy, it is often believed that if an investor anticipates information shifts, he will adjust his portfolio to hedge these shifts. To capture the dynamic hedging effect, Merton (1973) develops a continuous-time asset pricing model which explicitly takes into account hedging demand. In contrast to the APT framework, there are two factors which are theoretically derived from Merton's model: a market factor and a hedging factor. Stulz (1981) extends the intertemporal model of Merton (1973) to develop an international asset pricing model. However, an empirical investigation is not easy to implement for the continuous-time model. In a recent paper, Campbell (1993) develops a discrete-time counterpart of Merton's model. Motivated by Campbell's results, this paper adopts a conditional two-factor asset pricing model to explain the cross-section pricing relationships among international stock markets. In our setup, assets are priced using their covariance with the market portfolio as well as with the hedging portfolio that accounts for changes in the investment set. Moreover, a GARCH parameterization is employed to characterize the general dynamics of the conditional second moments. With the framework of the our proposed international two-factor asset pricing model, the international CAPM model is misspeci®ed and estimates of the CAPM model are subject to the omitted variable bias.

This paper makes several contributions. First, we present a testable intertemporal capital asset pricing model proposed by Campbell. Hence, we can examine whether Campbell's model explains the intertemporal behavior of a number of international ®nancial markets. Second, we estimate a parameterized speci®cation for second moments and many variables of interest related to the conditional second moments can be recovered.1_{Third, our approach can be used to measure the relative importance of} hedging risk premiums compared to the market premium at each point in time. The results show that the market risk premium is positive. Since hedging strategy reduces the risk, the hedging risk premium should be negative. The larger the risk reduction by the hedging strategy, the bigger the absolute value of the hedging risk premium. Finally, we suggest that the hedging risk should be priced in addition to the market risk in the international asset pricing model. We show that estimates of price of market risk obtained from the De Santis and Gerard (1997) conditional CAPM model may be biased downward due to the omission of the hedging risk which is negatively correlated to the market risk.

The remainder of the paper is organized as follows. Section II derives the intertemporal model of international asset pricing based on Campbell (1993). Section III discusses the empirical methodology of the paper. In Section IV, we describe the data and the main empirical evidence. We report the hedging risk premium and compare its

behavior to that of the market risk premium in Section V. Finally, Section VI concludes the paper.

II. Asset pricing model

This section describes the international asset pricing model that we employ to estimate and test the pricing relationships among the world's ®ve main equity markets. In this paper, we assume that equity markets are internationally integrated. This implies that no matter where assets are traded, assets with the same risk have the same expected excess return. In this setting, there is no barrier to investment and transactions costs and taxes are the same across different countries. Moreover, we further assume there exists a representative agent who uses U.S. dollar as the numeraire. In other word, we hope that purchasing power parity is tenable which makes intertemporal capital asset pricing model holds for international assets.2_{The model we use is a two-factor model based on Campbell (1993).} We ®rst review the theory of non-expected utility proposed by Weil (1989) and Epstein and Zin (1991). Then, we use a log-linear approximation to the budget constraint to derive the international asset pricing model used in the paper.

A. Non-expected utility

We consider an economy in which a single, in®nitely lived representative international agent chooses consumption and portfolio composition to maximize utility and uses U.S. dollar as the numeraire. There is one good and N assets in the economy. The international agent in this economy is assumed to be different to the timing of the resolution of uncertainty over temporal lotteries. The agent's preferences are assumed to be represented recursively by

V_tˆ W…C_t; m‰ ~V_{t ‡ 1}jI_tŠ†; …1†

where W… ? ; ? † is the aggregator function, C_tis the consumption level at time t, and E_tis the mathematical expectation conditional on the information set at time t. As shown by Kreps and Porteus (1978), the agent prefers early resolution of uncertainty over temporal lotteries if W… ? ; ? † is convex in its second argument. Alternatively, if V… ? ; ? † is concave in its second argument, the agent will prefer late resolution of uncertainty over temporal lotteries.

The aggregator function is further parameterized by V_tˆ " …1 ÿ b†C1ÿ1=st ‡ b  E ~V_{t ‡ 1}1ÿg 1ÿ1=s 1ÿg # 1 1ÿ1=s ˆ  …1 ÿ b†C…1ÿg†=yt ‡ b  E t ~ V_{t ‡ 1}1ÿg 1=yy=…1ÿg† : …2†

The parameter b is the agent's subjective time discount factor and g can be interpreted as the Arrow-Pratt coef®cient of relative risk aversion. It can also be shown that s measures the elasticity of intertemporal substitution. For instance, if the agent's coef®cient of relative risk aversion …g† is greater than the reciprocal of his elasticity of intertemporal substitution …1=s†, then he prefers early resolution of uncertainty. Conversely, if the reciprocal of the agent's elasticity of intertemporal substitution is larger than his coef®cient of relative risk aversion, he prefers late resolution of uncertainty. If g is equal to 1=s, the agent's utility becomes an isoelastic, von Neumann-Morgenstern utility and he is indifferent to the timing of the resolution of uncertainty.

Furthermore, y is de®ned as y ˆ …1 ÿ g†=‰1 ÿ …1=s†Š in accordance with Giovannini and Weil (1989). There are three special cases that are worth mentioning. First, y ? 0 when g ? 1. Second, y ? ? when s ? 1. Third, y ˆ 1 when g ˆ 1=s. In this case, equation (2) becomes the von Newmann-Morgenstern expected utility

V_tˆ …1 ÿ b† E_tX? j ˆ 1 bj_C~1ÿg t ‡ j " # 1 1 ÿ g : …3†

B. Log-linear budget constraint

We now turn to the characterization of the budget constraint of the representative investor who can invest his wealth in N assets. The gross rate of return on asset i held throughout period t is given by R_{i;t ‡ 1}. Let

R_{m;t ‡ 1}:XN

i ˆ 1

a_i;tR_{i;t ‡ 1} …4†

denote the rate of return on the market portfolio, and a_i;tbe the fraction of the investor's total wealth held in the i-th asset in period t. There are only N ÿ 1 independent elements in a_i;tsince the constraint

XN

i ˆ 1

a_i;tˆ 1 …5†

holds for all t. The representative agent's dynamic budget constraint can be given by W_{t ‡ 1}ˆ R_{m;t ‡ 1}…W_tÿ C_t†; …6† where W_{t ‡ 1} is the investor's wealth at time t. The budget constraint in equation (6) is nonlinear because of the interaction between subtraction and multiplication. The investor is able to affect future consumption ¯ows by trading in the risky assets. Campbell linearizes the budget constraint by dividing equation (6) by W_t, taking the log, and then using a ®rst-order Taylor expansion around the mean log consumption/wealth ratio, logC

If we de®ne the parameter r ˆ 1 ÿ exp…c_tÿ w_t†, the approximation to the intertemporal budget constraint is Dw_{t ‡ 1}%r_{m;t ‡ 1}‡ k ‡  1 ÿ1 r  …c_tÿ w_t†; …7†

where the log form of the variable is indicated by lowercase letters and k is a constant. Combining equation (7) with the following equality,

Dw_{t ‡ 1}ˆ Dc_{t ‡ 1}‡ …c_tÿ w_t† ÿ …c_{t ‡ 1}ÿ w_{t ‡ 1}†; …8† we obtain a difference equation in the log consumption-wealth ratio, ctÿ wt. Campbell

shows that if the log consumption-wealth ratio is stationary, i.e., limj

? [ ftyrj

…ct ‡ jÿ wt ‡ j† ˆ 0, then the approximation can be written as

c_{t ‡ 1}ÿ E_tc_{t ‡ 1}ˆ …E_{t ‡ 1}ÿ E_t†X? j ˆ 0 rj_r m;t ‡ 1 ‡ jÿ …Et ‡ 1ÿ Et† X? j ˆ 1 rj_Dc t ‡ 1 ‡ j: …9†

Equation (9) can be used to express the fact that an upward surprise in consumption today is determined by an unexpected return on wealth today, or by news that future returns will be higher, or by a downward revision in expected future consumption growth. C. Euler equations

In this setup, Epstein and Zin (1989) derive the following Euler equation for each asset: 1 ˆ E_t  b  C_{t ‡ 1} C_t _ÿ1=s_y 1 R_{m;t ‡ 1} _1ÿy R_{i;t ‡ 1}  : …10†

Assume for the present that asset prices and consumption are jointly lognormal or apply a second order Taylor expansion to the Euler equation. Then, the log version of the Euler equation (10) can be represented as

0 ˆ y log b ÿ_syE_tDc_{t ‡ 1}‡ …y ÿ 1†E_tr_{m;t ‡ 1}‡ E_tr_{i;t ‡ 1} ‡1₂  y s 2 Vcc‡ …y ÿ 1†2Vmm ‡ V_iiÿ2y_s…y ÿ 1†V_cmÿ2y_sV_ci‡ 2…y ÿ 1†V_im  ; …11†

r_{j;t ‡ 1}†Vj ˆ i; m, and V_im denotes Cov_t…r_{i;t ‡ 1}; r_{m;t ‡ 1}†. In equation (11), we do not assume that asset prices and consumption are conditional homoskedastic. That is to say, the conditional second moments in equation (11) are time-varying.

Replacing asset i by the market portfolio and rearranging equation (11), we obtain the relationship between expected consumption growth and the expected return on the market portfolio E_tDc_{t ‡ 1}ˆ s log b ‡1₂  y s  V_cc‡ ysV_mmÿ 2yV_cm  ‡ sE_tr_{m;t ‡ 1} ˆ s log b ‡1 2  y s  Var_t…Dc_{t ‡ 1}ÿ sr_{m;t ‡ 1}† ‡ sE_tr_{m;t ‡ 1}: …12† In the same manner, the second term in equation (12) is not a constant now.

When we subtract equation (11) for the risk-free asset from that for asset i, we obtain E_tr_{i;t ‡ 1}ÿ r_{f ;t ‡ 1}ˆ ÿVii

2 ‡ y

V_ic

s ‡ …1 ÿ y†Vim; …13†

where r_{f ;t ‡ 1}is a log riskless interest rate. Equation (13) expresses the expected excess log return on an asset (adjusted for the Jensen's inequality effect) as a weighted sum of two terms. The ®rst term, with a weight y, is the asset covariance with consumption divided by the intertemporal elasticity of substitution, s. The second term, with a weight 1 ÿ y, is the asset covariance with the return from the market portfolio.

D. Substituting consumption out of the asset pricing model

Now, we combine the log-linear Euler equation with the approximated log-linear budget constraint to get an intertemporal asset pricing model without consumption. Substituting equation (12) into equation (9), we obtain

c_{t ‡ 1}ÿ E_tc_{t ‡ 1}ˆ r_{m;t ‡ 1}ÿ E_tr_{m;t ‡ 1}‡ …1 ÿ s†…E_{t ‡ 1}ÿ E_t†X? j ˆ 1 rj_r m;t ‡ 1 ‡ j ÿ1_2syX? j ˆ 1 rj_‰Var t ‡ 1…Dct ‡ 1 ‡ jÿ srm;t ‡ 1 ‡ j† ÿ Vart…Dct ‡ 1 ‡ jÿ srm;t ‡ 1 ‡ j†Š: …14†

The last term in equation (14) is related to the conditional variance of market return. Campbell (1993) notes several ways in which consumption can be substitute out of the model with changing variances. One of them is set the elasticity of intertemporal substitution s ˆ 1, then y is in®nite, and the conditional variance of (14) must be zero. In

this case, pricing formula can be derived in an environment of conditional heteroskedastically. However, when we test the model empirically in the next section, we assume that the conditional variance of market return follows a GARCH process which is uncorrelated with any asset returns.3_{Hence, based on equation (14), the conditional} covariance of any asset return with consumption can also be rewritten in terms of the covariance with the return on the market and revisions in expectations of future returns on the market which is given by

Covt…ri;t ‡ 1; Dct ‡ 1†:Vicˆ Vim‡ …1 ÿ s†Vih; …15†

where

V_ih ˆ Cov_t…r_{i;t ‡ 1}; …E_{t ‡ 1}ÿ E_t†X?

j ˆ 1

rj_r

m;t ‡ 1 ‡ j†:

Substituting equation (15) into equation (13), we obtain an international asset pricing model that is not related to consumption:

E_tr_{i;t ‡ 1}ÿ r_{f ;t ‡ 1}ˆ ÿV₂ii‡ gV_im‡ …g ÿ 1†V_ih: …16† Equation (16) states that the expected excess log return in an asset, adjusted for a Jensen's inequality effect, is a weighted average of two covariance: the covariance with the return from the market portfolio and the covariance with news about future returns on invested wealth.

III. Empirical methodology A. A vector auto-regressive model

In order to complete the empirical implementation, we ®rst construct a state variable system to estimate the hedging portfolio. We adopt the vector auto-regressive (VAR) approach of Campbell (1991). We assume that the world market index return is the ®rst element of a K-element state variable vector z_t. The other elements of z_tare variables that are known to the market at the end of the period t and are related to the forecasting of future world market returns. In addition, we assume that the vector z_tfollows a ®rst order VAR

z_{t ‡ 1}ˆ Fz_t‡ u_{t ‡ 1}; …17†

where F is a K6K matrix that is known as the companion matrix of the VAR. Then we can use the ®rst order VAR to generate simple multi-period forecasts of future returns as

Etzt ‡ 1 ‡ i ˆ Ft ‡ 1zt: …18†

Besides, we de®ne a K-element constant vector e. The ®rst element of e is one and other elements are all zero. Therefore, we can write rm;t as rm;tˆ e0zt and rm;t ‡ 1ÿ

Etrm;t ‡ 1ˆ e0ut ‡ 1. It follows that the discounted sum of forecast revisions in world

market return now can be represented as …E_{t ‡ 1}ÿ E_t†X? j ˆ 1 rj_r m;t ‡ 1 ‡ jˆ e0 X? j ˆ 1 ri_Fj_u t ‡ 1 ˆ e0_{rF…I ÿ rF†}ÿ1_u t ‡ 1 ˆ y0 hut ‡ 1; …19†

where y0_h is de®ned as e0_{rF…I ÿ rF†}ÿ1 _{which measures the importance of each state}

variable in forecasting future returns on the world market. Then, equation (16) can now be written as

E_tr_{i;t ‡ 1}ÿ r_{f ;i ‡ 1}ˆ ÿV₂ii‡ gV_im‡ …g ÿ 1†XK

ik ˆ 1

y_h;kV_ik: …20†

We de®ne V_ik:Var_t…r_{i;t ‡ 1}; u_{k;t ‡ 1}†, where u_{k;t ‡ 1}is the kth element of u_{t ‡ 1}. Equation (20) implies that the expected log excess return on asset i, adjusted for the effect of Jensen's inequality, is linear in the covariance of the return with the K factor. The set of restrictions on the risk prices of the factor is the most important contribution of the intertemporal optimization problem.

B. Modeling risk premiums

Equation (16) appears to be the natural relation to use in empirical investigation of an intertemporal asset pricing model because it takes into account the investor's use of newly acquired information in creating a portfolio, taking into account his hedging strategy. The model requires equation (16) to hold for every asset including the market portfolio and hedging portfolio. Therefore, we assume that each asset return satis®es the following system of pricing restrictions

E_tr_{1;t ‡ 1}ÿ r_{f ;t ‡ 1}ˆ ÿV11

2 ‡ gV1m‡ …g ÿ 1†V1h …21†

...

Let r_tdenote the N61 time series vector which includes N risky assets, r_iVi ˆ 1; . . . ; N. In matrix notation, equation (21) and equation (17) can be re-expressed in terms of random log excess return as

zt r_tÿ r_{f ;t}? i " # ˆ Fzt ÿ 1 ÿ1 2hd;t‡ ghm;t‡ …g ÿ 1†hh;t " # ‡ ut x_t " # …22† e_tˆ ut xt " # jI_tÿ1*N…0; H_t†; H_tˆ H 11 K6K H12K6N H21 N6K H22N6N " #

where i is an N61 vector of one, H_tis the conditional covariance matrix of asset returns, h_d;tis the diagonal element of H22

N6N which denotes the conditional variance of each risk

asset, h_m;t is the 1st column of H21

N6K which denotes the conditional covariance of each

asset with the market portfolio, and h_h;tˆPK_{k ˆ 1}y_h;kh_k;t, where h_k;tis the kth column of H21

N6Kwhich denotes the conditional covariance of each asset with the hedging portfolio.

Three tests of asset pricing restrictions as special cases of equation (22) are implemented. Firstly, we test validity of the conditional CAPM that implies that g must be signi®cantly different from zero. The null hypothesis of g ˆ 0 is tested against the alternative hypothesis g = 0. Secondly, the test of hedging risk as an important factor in asset pricing model is implemented. This test implies that g is not equal to one in equation (22). Otherwise, risk premiums of assets are determined only by the covariance with the market portfolio. Finally, we test the validity of Campbell's intertemporal model in international equity markets. This is to test the restriction that the coef®cient on the hedging-portfolio covariance in (21) is one smaller than the coef®cient on the market-portfolio covariance.

C. Covariance dynamics

Equation (22) follows directly from the system of intertemporal asset pricing model. To implement the tests of the above hypothesis, the dynamics of the variance-covariance structure in equation (22) must be speci®ed. A multivariate GARCH process based on the work of Ding and Engle (1994) is used to obtain a testable version of the model.

For simplicity, we assume that the innovation vector e_tfollows a popular GARCH(1,1) process. The time-varying conditional covariance matrix therefore can be parameterized as

where G, a, and b denote N6N matrices of parameters. It is dif®cult to estimate the model due to the large number of unknown parameters. In practice, it is necessary to further restrict the speci®cation for H_tto obtain a numerically tractable formulation. One useful special case is to assume that both a and b are restricted to be diagonal matrices. In such a parameterization, the conditional covariance between e_i;t and e_j;t depends only on past values of e_{i;t ÿ 1}? e_{j;t ÿ 1}, and not on the products or squares of other residuals. Therefore, equation (23) can be written in a simple form:

Htˆ G ‡ aa0 et ÿ 1e0t ÿ 1‡ bb0 Ht ÿ 1; …24†

where a, b are N61 vector which includes the diagonal elements of a and b, respectively, and the symbol * denotes the Hadamard (element by element) matrix product. However, the diagonal assumption is still too dif®cult to estimate, unless we can reduce the number of unknown parameter again. To this end, we assume that the et process is covariance

stationary following Ding and Engle (1994). Consider the following system of equations z_t r_t   ˆ E_{t ÿ 1} zt r_t   ‡ e_te_tjI_{t ÿ 1}*N…0; H_t†: …25† If the et process is covariance stationary, its unconditional variance-covariance matrix is

equal to

H₀ˆ G  …ii0_{ÿ aa}0_{ÿ bb}0_†ÿ1_{: …26†}

Then, equation (26) is replaced by H_tˆ H₀ …ii0_{ÿ aa}0_{ÿ bb}0_{† ‡ aa}0_{ e}

t ÿ 1e0t ÿ 1‡ bb0 Ht ÿ 1: …27†

In a covariance stationary and diagonal construction with N assets, the number of unknown parameters in the conditional variance equation is reduced to 2N. The unconditional variance-covariance matrix, H0, is not directly observable. Here we set it

equal to the sample covariance matrix of the return.

We use equations (22) and (27) as the benchmark model. Let c be the unknown parameters in the model. Then, under the assumption of conditional normality, the log-likelihood function can be written as

ln L…c† ˆ ÿTN₂ ln 2p ÿ1₂XT

t ˆ 1

ln jH_t…c†j ÿ1₂XT

t ˆ 1

e_t…c†0H_t…c†ÿ1e_t…c†: …28† This paper estimates the model and computes all tests using the quasi-maximum likelihood estimation (QMLE) approach proposed by Bollerslev and Wooldridge (1992). QMLE estimation provides consistent estimates of the parameters. The standard errors for the estimated coef®cients that are calculated under the normal assumption need not be

correct if the true data generating process is non-normal. Hence, we use the robust Lagrange multiplier (LM) test to test the alternative model.

IV. Data and empirical results

Monthly dollar denominated index returns for U.S., U.K., Germany, Canada, Japan, and the world portfolio are employed to investigate the proposed international asset pricing model in the paper. These returns are reported by Morgan Stanley Capital International (MSCI) and the sample period is from January 1980 through December 1997.

Table 1 reports the summary statistics for the U.S. dollar log index returns in excess of the return on 30-day U.S. T-bills. The summary statistics include means, standard deviations, skewness, kurtosis, Bera-Jarque (1982) statistics, and the sample correlation. The magnitudes of the means, volatilities and correlations are very similar to those previously documented in other studies. The kurtosis values indicate that the unconditional distribution of excess log return has heavier tails than normal distribution for all countries. Furthermore, the Bera-Jarque statistics also show that the hypothesis of normality is rejected in our sample. Hence, in order to obtain robust standard errors, we will use the QMLE method to estimate the asset pricing model in the paper.

Descriptive statistics for the state variables are reported in Table 2. We select a set of state variables that have been widely used in the literature. These instruments include the

Table 1. Summary statistics of excess log returns

Summary statistics

Mean(*100) S.D.(*100) Skewness Kurtosis B-J

World 0.3319 4.2041 ÿ 0.7673* 2.3944** 72.4598** U.S. 0.4346 4.3627 ÿ 1.0075** 5.0423** 264.1372** U.K. 0.3604 5.8577 ÿ 0.4718* 1.9214** 41.0485** Germany 0.3011 6.1143 ÿ 0.4590* 1.3195** 23.1478** Canada ÿ 0.1141 5.5120 ÿ 0.9846** 4.4684** 213.6004** Japan 0.2438 7.0089 0.0332 0.3482* 1.1256* Correlations

World U.S. U.K. Germany Canada Japan

World 1.0000 0.7956 0.7486 0.5861 0.7046 0.7294 U.S. 0.7956 1.0000 0.5821 0.4230 0.7193 0.2570 U.K. 0.7486 0.5821 1.0000 0.5096 0.5999 0.4047 Germany 0.5861 0.4230 0.5096 1.0000 0.3697 0.3257 Canada 0.7046 0.7193 0.5999 0.3697 1.0000 0.2814 Japan 0.7294 0.2570 0.4047 0.3257 0.2814 1.0000

The monthly dollar denominated log index returns for ®ve countries and the world market portfolio are from MSCI. The sample period is from January 1980 through December 1997.

change in the U.S. term premium which is equal to the return on U.S. 30-year Treasury notes in excess of the one-year Treasury notes rate (TERM), the world dividend yield measured by the logarithm of the monthly MSCI world dividend yield (DIV) and the short term U.S. T-bill rate which is the one-month T-bill rate (TB). All of these variables have been found to measure the information that investors use to set prices in the market. The term premium (TERM) follows Campbell (1996) and others. Dividend yield (DIV) is a component of the return of stocks and hence it is a good forecasting variable for capturing predictions of stock returns. Campbell (1996) ®nds that the dividend yield has some predictive power for future stock returns. The short-term bill rate (TB) which has been used by Fama and Schwert (1977), Ferson (1989), and Ferson and Harvey (1991) is capable of predicting monthly returns of bonds and stocks.

We start our investigation by constructing the dynamic behavior of the state variables. Table 3 reports the estimates of the coef®cients in the one-lag VAR. The ®rst row of Table III shows that the monthly forecasting equation for the excess log return of world market

Table 3. VAR summary: dynamics of risk factor Dependent Variables/

Regressors rm;tÿ rf ;t TERM 1 TB DIV R2 w2(4) p-value

rm;tÿ rf ;t ÿ 0.0293 ÿ 0.1257 ÿ 0.1327 0.0603 0.0839 19.34 (0.0674) (0.0570) (0.0359) (0.0174) 0.00 TERM ÿ 0.0829 0.9250 ÿ 0.0138 0.0099 0.8905 1717 (0.0623) (0.0527) (0.0332) (0.0161) 0.00 TB 0.2210 ÿ 0.2406 0.8056 0.0686 0.9164 2313 (0.1208) (0.1022) (0.0644) (0.0313) 0.00 DIV ÿ 0.0272 0.1465 0.1470 0.9209 0.9836 12715 (0.0674) (0.0570) (0.0359) (0.0174) 0.00 Note: Standard errors are in parentheses.

We adopt the Vector Auto-Regressive(VAR) approach of Campbell (1991). We assume that the real world market index return is the ®rst element of the state variable vector zt. The other elements of ztare variables

that are known to the market at the end of the period t and are related to forecasting future world market returns. In addition, we assume that the vector ztfollows a ®rst order VAR

zt ‡ 1ˆ Fzt‡ ut ‡ 1:

Table 2. Summary statistics of state variables

Mean Std. Dev. Maximum Minimum

TERM 0.1100 0.1126 0.3608 ÿ 0.2908

TB 0.5602 0.2495 1.4050 0.2250

DIV ÿ 6.0103 0.3134 ÿ 5.3496 ÿ 6.5785

Summary statistics for state variables (in percentages per month) for the sample period from 1980:1 to 1997:12. The state variables include the U.S. term premium (TERM) which is equal to the yield on 30-year U.S. T-notes in excess of the yield of the 1-year U.S. T-notes, the 30-day U.S. T-bill returns (TB), and the dividend yield (DIV) which is the logarithm of the monthly MSCI world dividend yield.

portfolio, rm;tÿ rf ;t. Since there is little serial correlation in the monthly world market log

return, the coef®cient for the lagged rm;tÿ rf ;t is small and insigni®cant. However, the

coef®cient for the short-term bill rate (TB) and world dividend yield (DIV), are signi®cantly negative and positive, respectively. The coef®cient for the term premium (TERM) is negatively signi®cant. The remaining rows of Table 3 report the dynamics of the state variables. It can be seen that both TERM and DIV behave like a persistent AR(1) process with coef®cients of 0.9250 and 0.9209, respectively.

After constructing the VAR system, we can use the results in Table 3 to calculate the long-run forecasts of world market returns. Revisions in these forecasts are linear

combinations of shocks to the state variables, which are de®ned by the vector y_h in

equation (19). Table 4 shows the elements of vector y_h. Shocks to world market return and TB have negative effects on long-run stock return forecasts. But TERM and DIV have positive effect. The hypothesis of all four coef®cients are zero can be rejected by the w2₍₄₎

statistic which is equal 19.34.

As discussed in Section II, the intertemporal model applied to international ®nancial markets implies that the conditional expected log return on any asset is linearly related to the covariance of asset returns with the world market returns as well as with news about the discounted value of all future market returns which is called the hedging portfolio return. If the world market risk and news about the discounted value of all future market risk are the only two relevant factors, the price of covariance risk g should be signi®cantly positive and signi®cantly less than one. It should also be noted that the value of g should not be equal to zero or one.

In order to avoid the measurement error, equations (22) and (27) are the econometric system employed to simultaneously estimate the parameters of VAR and asset pricing model. The results are reported in Table 5. The estimate of g is 8.4130 and is signi®cantly different from zero. This implies that the conditional expected international asset return varies with world market volatility. This evidence supports the conditional version of international CAPM. Next, consider the estimates of parameters in the multivariate

Table 4. The coef®cient of yhthat generate the hedging portfolio

Shocks to

Vector rm;tÿ rf ;t TERM TB DIV

yh ÿ 0.1226 0.2775 ÿ 0.2820 0.4385

(0.0818) (0.3877) (0.2413) (0.3163)

w2_{4† 19.34}

p-value 0.00

This table reports the estimates of coef®cients of the vector de®ning the innovations in the discounted present value of the news about future world market returns in the equation (19). The state variables include the U.S. term premium (TERM) which is equal to the yield on 30-year U.S. T-notes in excess of the yield of the 1-year U.S. T-notes, the 30-day U.S. T-bill returns (TB), and the dividend yield (DIV) which is the logarithm of the monthly MSCI world dividend yield. The w2_{4† statistics examines whether the four}

GARCH process. All the elements in the vectors a and b are statistically signi®cant at any conventional level. Besides, the estimates satisfy the stationary conditions, a_ia_j‡ b_ib_j51Vi; j, for all the variance and covariance processes.

There are several conditions under which the model collapses to the conditional CAPM. The ®rst case is that g is equal to one. Since we have the unrestricted estimates, a Wald test would be a convenient way to proceed. The value of the criterion function is a w2_{1† ˆ 20:21, which corresponds to a p-value of 0.00. This implies that we can reject the}

Table 5. Quasi maximum likelihood estimates (QMLE) of the intertemporal model of international asset pricing 8.4130 (2.0607) g a b rm;tÿ rf ;t 0.2394 0.9677 (0.0136) (0.0038) TERM 0.3069 0.9512 (0.0055) (0.0017) TB 0.3076 0.9511 (0.0045) (0.0014) DIV 0.3105 0.9504 (0.0025) (0.0007) U.S. 0.2446 0.9642 (0.0173) (0.0053) U.K. 0.2189 0.9767 (0.0220) (0.0074) Germany 0.2421 0.9664 (0.0350) (0.0128) Canada 0.1938 0.9766 (0.0345) (0.0147) Japan 0.2175 0.9741 (0.0297) (0.0100)

This table reports the estimates of the multivariate GARCH(1,1) model, using monthly excess log returns to ®ve equity markets, world market portfolio, hedging portfolio, and the state variables. Data are obtained from the MSCI. Each mean equation relates the country index excess log return to its world market covariance and to its hedging portfolio covariance and is written as

zt rtÿ rf ;t? i " # ˆ Fzt ÿ 1 ÿ1 2hd;t‡ ghm;t‡ …g ÿ 1†hh;t " # ‡ ut xt " # etˆ ut xt " # jIt ÿ 1*N…0; Ht†;

where g denotes the price of the world market covariance or coef®cient of relative risk aversion. The conditional covariance matrix is formulated as

Htˆ H0 …ii0ÿ aa0ÿ bb0† ‡ aat et ÿ 1ett ÿ 1‡ bbt Ht ÿ 1

hypothesis that hedging demand is not an important factor in pricing international stock returns. Moreover, we test the pricing restriction of the intertemporal pricing model of Campbell in an international framework that the coef®cient on the hedging-portfolio covariance in equation (22) is one smaller than the coef®cient on the market-portfolio covariance. The robust LM test in Table 6 shows that we can not reject this hypothesis using this data. Secondly, the conditional CAPM holds if the covariance between the assets and the hedging portfolio are all zero. Figure 1, using the numbers in the Table 5, shows that the absolute value of estimates of Vihare larger than the critical value line which imply

that V_ihare almost signi®cantly different from zero. Finally, the conditional CAPM could arise when the covariance between assets and the hedging portfolio are all proportional to the covariance with the market portfolio. Figure 2 reports the relationship between V_imand V_ih. The graph shows that V_ih is strongly negatively correlated with V_im. Hence, international stock returns are determined by their covariance with world market return. This may be due to the fact that the world market return provides an important information about future investment opportunities rather than the information about its own risk pro®le. This result is the similar to the cross-section data in U.S. of Campbell (1996).

Table 6. Speci®cation tests of the intertemporal model of international asset pricing

Null Hypothesis w2 _{df p-value}

The coef®cient on hedging portfolio covariance is one smaller than the coef®cient on market portfolio covariance?

0.005 1 0.9427

Since the restriction of coef®cient on the hedging portfolio does not arise in some versions of theory, we estimate the intertemporal model without restrictions on hedging portfolio and the results are reported in the Table 7. The estimate of g₁ is signi®cant different from zero. This result is similar to that of intertemporal pricing model. However, the unrestricted parameter of hedging portfolio is different from the restricted model which is positive but insigni®cant. The parameters of second moment are statistically signi®cant at any conventional level.

In the international APT model, the risk prices for world market return, TERM, TB, and DIV are reported on the Table 8. The risk prices for world market return, and DIV are positive and statistically signi®cant. The risk price of TERM is insigni®cantly negative. However, The parameters of GARCH process are statistically signi®cant at any conventional level.

V. Hedging risk and market risk

Since our approach provides estimates of the conditional second moments, we can also directly calculate the premiums associated with the hedging risk and the market risk. Table 9 reports summary statistics of the risk premiums for the ®ve national equity markets. The traditional CAPM uses only the market risk premium to price assets, whereas the intertemporal model also compensates for the hedging risk. Table 9 shows some striking facts about the risk characteristics of international equity returns. First, the pattern of the market risk premium is similar to that of the hedging risk premium. Second, there is a

strong negative cross-sectional correlation between the market risk premium and the hedging risk premium. Assets with a large positive market risk premium tend to have a large negative hedging risk premium. The ®nding that there is a negative relation between the market risk premium and the hedging risk premium can also be seen in the estimated

Table 7. Quasi maximum likelihood estimates (QMLE) of the intertemporal model without restriction on hedging portfolio g1 8.5997 (2.3016) g2 27.3156 (20.5798) a b rm;tÿ rf ;t 0.2453 0.9661 (0.0127) (0.0037) TERM 0.3032 0.9524 (0.0055) (0.0017) TB 0.3044 0.9522 (0.0046) (0.0014) DIV 0.3076 0.9513 (0.0028) (0.0008) U.S. 0.2459 0.9634 (0.0172) (0.0054) U.K. 0.2183 0.9759 (0.0206) (0.0067) Germany 0.2450 0.9667 (0.0329) (0.0114) Canada 0.1914 0.9775 (0.0339) (0.0143) Japan 0.2292 0.9710 (0.0262) (0.0088)

This table reports the estimates of the multivariate GARCH(1,1) Model, using monthly excess log returns to ®ve equity markets, world market portfolio, hedging portfolio, and the state variables. Data are obtained from the MSCl. Each mean equation relates the country index excess log return to its world market covariance and to its hedging portfolio covariance and is written as

zt rtÿ rf ;t? i " # ˆ Fzt ÿ 1 ÿ1 2hd;t‡ g1hm;t‡ g2hh;t " # ‡ ut xt " # etˆ ut xt " # jIt ÿ 1*N…0; Ht†;

where g1denotes the price of the world market covariance or coef®cient of relative risk aversion and g2denotes

the price of the hedging risk. The conditional covariance matrix is formulated as Htˆ H0 …ii0ÿ aa0ÿ bb0† ‡ aa0 et ÿ 1e0t ÿ 1‡ bb0 Ht ÿ 1

mean-reversion of the market portfolio of Campbell (1996). In addition, this result may explain why the estimates of the conditional CAPM relation obtained from De Santis and Gerard (1997) may be biased downward due to the omission of hedging risk from the conditional market risk.

Before discussing how the dynamics of each premium vary across world stock markets, we ®rst focus on the behavior of the U.S. market. As identi®ed by Wall Street Journal,

Table 8. Quasi maximum likelihood estimates (QMLE) of the international apt model

rm;tÿ rf ;t TERM TB DIV Price of Risk 4.1427 ÿ 2.3961 ÿ 0.5694 6.2920 (0.9927) (1.8859) (0.9332) (2.1457) a b rm;tÿ rf ;t 0.2440 0.9664 (0.0128) (0.0037) TERM 0.3068 0.9513 (0.0058) (0.0018) TB 0.3080 0.9510 (0.0048) (0.0015) DIV 0.3112 0.9502 (0.0031) (0.0009) U.S. 0.2432 0.9638 (0.0176) (0.0055) U.K. 0.2182 0.9759 (0.0211) (0.0070) Germany 0.2471 0.9661 (0.0327) (0.0115) Canada 0.1915 0.9772 (0.0344) (0.0149) Japan 0.2287 0.9714 (0.0248) (0.0083)

This table reports the estimates of the multivariate GARCH(1,1) model, using monthly excess log returns to ®ve equity markets, world market portfolio, and hedging portfolio. Data are obtained from the MSCI. Each mean equation relates the country index excess log return to its world market covariance and to its hedging portfolio covariance and is written as

zt rtÿ rf ;t? i " # ˆ Fzt ÿ 1 ÿ1 2hd;t‡Pkkˆ1pkhk;t " # ‡ ut xt " # etˆ ut xt " # jIt ÿ 1*N…0; Ht†;

where pkdenotes the price of the state variable which includes world market risk. The conditional covariance

matrix is formulated as

Htˆ H0 …ii0ÿ aa0ÿ bb0† ‡ aa0 et ÿ 1e0t ÿ 1‡ bb0 Ht ÿ 1

there were some periods of decline in the U.S. market. The two most short severe declines in the sample were 1987:9±1987:11 and 1990:6±1990:10. There were some long mild declines from 1981:1±1982:7 and from 1983:7±1984:7. Some of the evidence seems to validate the claim that the hedging risk is negatively correlated to the market risk. In particular, we ®nd that two of the largest correlations are associated with the severe market declines in our sample, 1987:9±1987:11 and 1990:6±1990:10. In general, the plots indicate that increases in negative correlation of the hedging risk and the market risk are obvious

Table 9. Summary of risk premiums

Total Premium Market Premium Hedging Premium

Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.

U.S. 3.5039 1.9562 14.168 6.7052 ÿ 10.664 5.8526

U.K. 6.8713 2.8779 18.809 7.5345 ÿ 11.938 6.4010

Germany 4.8570 3.4594 14.564 7.2802 ÿ 9.7075 6.4592

Canada 4.5458 2.2700 15.809 7.0565 ÿ 11.263 5.7641

Japan 10.149 2.7099 21.980 8.9050 ÿ 11.831 6.9778

This table reports the annualized means and standard deviations for the risk premiums. The total premium (TP) is the sum of the market premium (MP) and the hedging premium (HP). The de®nition of risk premium is as follows.

TP ˆ g cov…ri;t; rm;t† ‡ …g ÿ 1† cov…ri;t; rh;t†

MP ˆ g cov…ri;t; rm;t†

HP ˆ …g ÿ 1† cov…ri;t; rh;t†

for short severe declines in the U.S. market, but long mild declines are often not very signi®cant. This can be seen from the following two mild and long declines over the periods: 1981:1±1982:7 and 1983:7±1984:7. This may be due to the fact that an international investor does not know exactly when the decline will end. Hence, when he

Figure 4. Risk premium: Canada stock.

expects the end of the decline, the absolute value of the hedging risk premium will decrease.

As shown in ®gure 4, the patterns of the risk premiums for Canada is similar to those for the U.S. That is because the economic connection between Canada and the U.S. are very close. Moreover, the two European markets, U.K. and Germany, display similar dynamics

Figure 6. Risk premium: German stock.

of risk premiums to those in the U.S. However, there is some additional small volatility in these two European markets. This small volatility may be due to business cycles in the European markets themselves. The Japanese market also exhibits similarities to the U.S. market. But, the size and dynamics of the premiums are somewhat different. The market risk premium and the hedging risk premium in 1990:6±1990:10 are larger than those in 1987:9±1987:11. This may result from the fact that the volatility of the Japan market is in¯uenced by the dynamics of both world and Asian economies.

VI. Conclusion

This paper investigates the time varying pricing restrictions of an intertemporal asset pricing model motivated by Campbell (1993) for ®ve national markets. Campbell's model is a two-factor model in which assets are priced using their covariance not only with the market portfolio, but also with the hedging portfolio that accounts for changes in the investment set.

The evidence supports the hypothesis that the hedging risk premium is important in an international setting. In other words, an additional hedging risk premium is needed to explain rates of return on international equities. Furthermore, the pricing restrictions of Campbell's model can not be rejected using the international stock return data, that is the coef®cient of hedge-portfolio covariance is one smaller than the coef®cient of market-portfolio covariance.

In addition, we estimate a parameterized speci®cation for the second moments and we can clearly note the negative relation between the hedging risk and the market risk in each time period. Moreover, we suggest that the coef®cient of the market risk in the conditional CAPM model may be biased downward due to the omission of the hedging risk which is negatively correlated to the market risk.

Notes

1. After we had ®nished the paper, we learned that Hodrick, Ng and Sengmueller (1998) also apply Campbell's model to international stock market. They use the GMM approach to examine the predictability of stock returns.

2. Alder and Dumas (1983) and Stulz (1984) review the literature on international asset pricing and discuss the restrictions of asset pricing models in an international setting.

3. Nieuwland (1991) also extends Campbell (1993) to discuss the case in which the variance of market returns follows a GARCH process.

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