An International Dynamic Asset Pricing Model
- Peter Isard, Andrew Rose, and Assaf Razin
- Published Date:
- January 2000
ROBERT J. HODRICK
Graduate School of Business, Columbia University, NY, NY 10027, and the National Bureau of Economic Research
DAVID TAT-CHEE NG
Department of Economics, Columbia University, NY, NY 10027
Department of Economics, Columbia University, NY, NY 10027
We examine the ability of a dynamic asset-pricing model to explain the returns on G7-country stock market indices. We extend Campbell’s (1996) asset-pricing model to investigate international equity returns. We also utilize and evaluate recent evidence on the predictability of stock returns. We find some evidence for the role of hedging demands in explaining stock returns and compare the predictions of the dynamic model to those from the static CAPM. Both models fail in their predictions of average returns on portfolios of high book-to-market stocks across countries.
The starting point of this paper is the observation of Flood, Hodrick, and Kaplan (1986) that stock returns are predictable. Flood, Hodrick, and Kaplan (1986) argued that one could not test for bubbles in stock prices without controlling for the rational movements in stock prices that are caused by fluctuations in expected rates of return. They used annual data and demonstrated that dividend yields, the ratio of the annual dividend to the current stock price, provided statistically significant forecasts of future stock returns, especially at the two-year horizon. Subsequent research by Campbell (1991) and Hodrick (1992) demonstrates that a vector autoregressive approach with monthly data generates significant time-varying forecasts of stock returns at any future horizon and has better small-sample properties than the direct long-horizon approach.
While theories of stock return predictability have been available since Merton (1973), the links between the theories of stock return predictability and the empirical analysis of this issue have been weak.1 The intuition of Merton’s model is that risk-averse consumer-investors want to hedge against adverse changes in their consumption and investment opportunity sets. Thus, the cross-section and time-series patterns of expected asset returns are determined by more than just the covariances of the returns on the assets with the return on the market portfolio, as in the static capital asset pricing model (CAPM). Expected returns also depend on the covariances of the returns on the assets with the state variables that describe the consumption and investment opportunity sets. The generality of this specification provided little basis for a tight econometric test of Merton’s theory.
Predictability of international equity returns is inconsistent with the static CAPM. Some researchers, Harvey (1991), Bekaert and Hodrick (1992), and Ferson and Harvey (1993) for example, have characterized the predictability of country returns and developed tests of conditional CAPM’s in which the conditional expected return on an asset is related to the conditional expected return on the market portfolio with a conditional, time-varying beta. The main problem with this approach is that it lacks a sound theoretical foundation. Once the conditional distributions are allowed to vary, Merton’s (1973) logic indicates that additional hedge portfolios, other than the market, may be priced. This paper develops a formal test of an international, intertemporal, asset-pricing model in which state variables may be priced.
Our analysis builds on the work of Campbell (1993, 1996). The failure of the consumption-based, asset-pricing literature of the 1980’s motivated Campbell (1993) to develop a theoretical model that avoids direct measurements of consumption. Campbell (1996) extended this theoretical model and generated empirically testable specifications directly linking the cross-section of expected stock returns to time variation in expected returns on the market portfolio. Campbell (1996) applied his model to a cross-section of returns on 25 portfolios of U.S. stocks taking the value-weighted U.S. market return as the market portfolio.2 While the theory was marginally rejected, the pricing errors of the model were small on most portfolios. Most of the explanatory power of the model was in the market portfolio, as is predicted by the CAPM.
The purpose of this paper is to determine whether variation in expected returns on the world market portfolio induces significant hedging motives, which are reflected in average returns across countries. We extend Campbell’s (1996) model to an international environment, and we utilize recent evidence of Lamont (1998) to specify the state variables in the forecasting equation.
Our analysis is effectively a constrained arbitrage-pricing model. Korajczyk and Viallet (1989) and Ferson and Harvey (1993) have tested multi-factor, international, arbitrage-pricing models. The problem with these specifications is that one must pre-specify the identity and number of factors with little theoretical guidance. Our framework imposes the restriction that the factors affecting the cross-sectional asset pricing must be those that forecast market returns.
As with many studies of international asset pricing, we assume that national equity markets are integrated, with no barriers to investment, transaction costs or differential taxes. We will also assume a representative agent framework, which raises the issue of aggregation across countries and the failure of purchasing power parity, see Adler and Dumas (1983).
Section II provides a review of Campbell’s (1996) model. Section III describes the data. Section IV discusses forecasting of real and excess nominal stock returns and compares the results of our analysis with Lamont’s (1998) analysis. Section V discusses the econometric aspects of the asset-pricing model, while Section VI provides the estimation and testing of the model. Section VII contains concluding remarks.
II. A Review of Campbell’s (1996) Model
Campbell’s (1996) model uses a representative agent framework, and hence, it shares the weaknesses of the consumption C A PM and Euler equation methods that were developed in the 1980’s to test intertemporal asset-pricing models.3 Nevertheless, there is an advantage to the approach because Campbell uses log-linearization to eliminate observations on consumption from the asset-pricing model. Ultimately, the model relates expected returns on assets to covariances of returns with the market portfolio, as in the traditional static CAPM, and to covariances of returns with changes in the expected future returns on the market portfolio.4
The representative agent’s budget constraint is written in real terms as
where Wt is real wealth, Ct is real consumption, and Rm, t+1 is the gross return on aggregate wealth, which is the return on the market portfolio.
The result is
where lower case letters are natural logarithms of upper case counterparts, ρ ≡ 1—exp (c—ω), and kω is a constant.
When the log consumption-wealth ratio is stationary, equation (2) implies that the innovation in logarithm of consumption can be written as the innovation in the discounted present value of the return on the market minus the innovation in the discounted present value of consumption growth:
Equation (3) indicates that an unanticipated increase in consumption today must be due to an innovation in the return on wealth, either today or an expected increase in the future, or it must coincide with a planned reduction in the growth rate of consumption in the future.
Campbell (1993,1996) next uses a log-linear Euler equation to eliminate expected future consumption growth from the right-hand side of equation (3). The consumer’s preferences are modeled as in Epstein and Zin (1989) and Weil (1989) with separate parameters for the coefficient of relative risk aversion, γ, and for the elasticity of intertemporal substitution, σ. The objective function is defined recursively by
where θ ≡ (l - γ)/[l - (l/σ)]. Campbell (1996) notes that θ can have either sign, θ → 0 as γ → 1, θ → ∞ as σ → 1, and θ → 1 as γ → 1/σ.
Epstein and Zin (1989) solve for the Euler equations associated with maximizing equation (4) subject to the budget constraint given in (1). When asset returns are conditionally homoskedastic and log-normally distributed, the Euler equations may be written in log-linear form as
In equation (6), rf, t+1 is the riskless real interest rate, Vii = var (ri, t+1—Etri, t+1), Vic = cov(ri, t+1, - Etri, t+1, ct+1 - Etct+1), and Vim = cov(ri, t+1, - Etri, t+1, rm, t+1 - Etrm, t+1).
Equation (5) indicates that consumption growth is linearly related to the expected return on the market portfolio, and this fact is used in conjunction with the linearized budget constraint to eliminate consumption from the asset-pricing model. Equation (6) indicates that the continuously compounded risk premium on an asset plus one-half of the asset’s variance is determined by the covariances of the asset’s return with consumption and with the return on the market.
Equation (7) indicates that the innovation in consumption, which enters the asset-pricing equation (6), can be eliminated by focusing on the innovations in the return on the market portfolio and in the discounted expected future returns. Intuitively, consumption responds one-for-one to the current return on the market portfolio because the objective function is homogeneous of degree one. The response of an innovation in consumption to an innovation in expected future returns depends on income and substitution effects. When the elasticity of substitution, σ, is less than one, the income effect dominates causing current consumption to rise. When σ is greater than one, the substitution effect dominates; and current consumption falls.
Equation (9) demonstrates that an asset’s risk premium (adjusted for one-half its own variance) depends on the asset’s covariance with the market portfolio, with weight γ, and on its covariance with the innovation in discounted expected future market returns, with weight (γ - 1).
The coefficient of relative risk aversion determines the compensation that investors demand for covariance risks. When γ = 1, the model’s predictions coincide with those of the static CAPM. When γ > (<)1, investors require higher (lower) expected returns on assets that covary positively with innovations in discounted expected future returns. Campbell (1996) notes that a positive covariance carries a mixed blessing. On the one hand, investors like assets that have good payoffs when expected future returns are high. But, on the other hand, investors dislike the fact that such assets provide poor hedging against deterioration in future investment opportunities. When γ is greater than one, the latter effect dominates the former.
While the derivation of equation (9) follows Campbell (1996) in assuming conditional homoskedasticity, Campbell (1993) discusses several conditions under which such an equation can be derived in a more realistic environment of conditional heteroskedasticity.5 The most straightforward approach is to assume that the elasticity of substitution, σ, equals one. In this case, equation (9) holds in terms of conditional expected returns with all second moments replaced by conditional second moments. Below, we derive and test the unconditional implications of such a specification.6
III. The Data
Table 1 presents some summary statistics for the primary data set that will be used in our study. We use monthly data from 1970:1 to 1998:4. The first variable is the real return on the MSCI world market portfolio (RRET). The units are percent per month, so the mean return of 0.546% corresponds to an annualized value of 6.55%. The next four variables are the additional state variables that are the predictors of real returns. The first is the logarithm of the annualized payout-ratio of world dividends relative to world earnings (LOGDE). The second is the logarithm of the annualized world dividend-price ratio (LOGDP). The third is the relative bill rate (RREL), which is the one-month U.S. interest rate minus its previous twelve-month moving average, in percent per month. The fourth variable is the Treasury bond term spread (TSPD) between the 30-year U.S. interest rate and the one-year U.S. interest rate, in percent per month.
The next seven variables in Table 1 are the dollar rates of return on the MSCI G7 country portfolios in excess of the one-month U.S. Treasury bill rate of return. The countries are the United States (US), Japan (JP), Germany (GE), the United Kingdom (UK), France (FR), Canada (CA), and Italy (IT). These variables are also measured as percent per month. The mean excess rates of return range from Italy’s 1.056% per annum to the U.K.’s 6.732% per annum. The U.S. has the lowest standard deviation, and Italy has the highest.
IV. The Forecasting Model
The first task in developing testable restrictions from equation (9) is to determine that the real return on the world equity market portfolio is forecastable. Since the theoretical model requires multi-period forecasts of the equity return, it is useful to stack the equity return as the first element of a k-dimensional vector of state variables, zt, and use a vector autoregression (VAR) as in the following:
The representation of the VAR as a first-order system is not restrictive, as the variables can always be stacked into a first-order companion form. We report the results of the first-order specification as this is the order chosen by the Schwarz (1978) criterion.
Panel A of Table 2 presents a five-variable, first-order vector autoregression. The first variable is the real return on the MSCI world market portfolio (RRET). The other four variables are the payout ratio, LOGDE; the dividend yield, LOGDP; the relative bill rate, RREL; and the term spread, TSPD. We use RREL and TSPD of the U.S. because we are pricing the excess dollar returns on the G7 equities.
|Coefficients on Regressors||predictability||L-test|
|Panel A: Fundamental VAR|
|Panel B: Forecasting Equations|
Lamont (1998) finds that the payout ratio is an important predictor of the U.S. S&P 500 excess return. Rozeff (1984), Campbell and Shiller (1988), Fama and French (1988), and Flood, Hodrick and Kaplan (1986) emphasize the importance of the dividend yield in forecasting stock returns at short and long horizons. Campbell (1991) and Hodrick (1992) use the relative bill rate. Fama and French (1989) find that the term structure of interest rates predicts stock and bond returns.
For the return forecasting equation, the value of the chi-square statistic with five degrees of freedom is 15.56, which indicates that real equity returns are predictable at standard levels of statistical significance. Nevertheless, the adjusted R2 for the real return equation is only 3.3%. This pattern of strong statistical significance with low percentage predictability is to be expected in monthly data. In efficient markets, most of the observed return will be unexpected, unless economic agents are extremely risk averse.
The most important features of the other forecasting equations are the coefficients on the own lags, which indicate that there is strong positive serial correlation in the log payout ratio, the log dividend-price ratio, and the term premium. There are also important off-diagonal terms indicating significant dynamics between the dividend yield, the payout ratio, the relative bill rate, and the term spread.
Panel A of Table 2 also reports the Cumby-Huizinga (1992) L-tests for residual serial correlation of the first eight lags in each equation. Except for the relative bill rate equation, a first-order VAR appears to be a good approximation to the data-generating process.
Panel B of Table 2 reports unconstrained excess return forecasting equations for the G7 countries. Here again, the evidence for predictability of the excess returns is not overwhelmingly strong. These equations are provided as a comparison to the constrained equations that will enter the asset-pricing model.
Table 3 reports the covariances and correlations of the innovations in the VAR. The correlations are in bold above the diagonal. The innovation variance of the real stock return is much larger than the other innovation variances, and the innovation in the real stock return is highly negatively correlated with the innovation in the log dividend yield. The log dividend yield is positively correlated with the log payout ratio and the relative bill rate.
Our results on the predictability of returns present a striking contrast to those of Lamont (1998). We explore these differences in Table 4, which adopts a quarterly measurement interval for comparison purposes.7 Lamont (1998) forecasts the excess nominal return on the S&P 500 and reports a significantly higher R2 for the sample period 1947:Q1 to 1994:Q4. We use Lamont’s (1998) data and our programs to produce the first line of Table 4, which coincides with the first line of Lamont’s (1998) Table VI. Aside from the higher R2 in the return equation, Lamont (1998) also reports that the dividend-payout ratio positively forecasts returns, which he interprets as a business-cycle effect. The argument is that the payout ratio is countercyclical as earnings fluctuate more than dividends across business cycles. Since expected returns are thought to be countercyclical, being higher in recessions than in booms, the coefficient linking the payout ratio to returns should be positive. Line 2 of Table 4 demonstrates that this effect persists in Lamont’s data for the sample period 1970:Q2 to 1994:Q4, which is the part that overlaps most with our sample. Line 3 of Table 4 demonstrates that the effects are not diminished by the inclusion of the term spread in the regression. Line 4 of Table 4 presents an analogous regression to line 3, but the data are changed. Line 4 uses the excess nominal return on the MSCI-US portfolio, which has a correlation of .997 with the S&P 500 return, and employs the associated MSCI-US dividend and earning series. LOGDE loses significance, and its sign changes from positive to negative. The R2 for the return equation also falls from. 12 to .06. Line 5 of Table 4 demonstrates that extending the sample to 1998:2 causes the dividend yield to lose significance. The last line of Table 4 presents the forecasting regression for the MSCI-World excess nominal return, which is similar to the forecasting equation for the real return in Table 1.
|Lamont (S&P)||0.209||0.066||0.078||0.086||-0.892||0.13||47:Q1 -|
|Lamont (S&P)||0.284||0.037||0.105||0.103||-0.777||0.12||70:Q2 -|
|Lamont (S&P)||0.352||0.034||0.126||0.093||-0.433||3.712||0.12||70:Q2 -|
|MSCI- US||0.275||0.015||0.076||-0.033||-3.669||5.104||0.06||70:Q2 -|
|MSCI- US||-0.238||0.035||0.010||-0.137||-4.032||5.190||0.06||70:Q2 -|
|MSCI - World||-0.058||-0.024||0.022||-0.075||-1.834||6.310||0.03||70:Q2 -|
Innovations in Forecasts of Future Stock Returns
The next step in the analysis is to calculate the innovation in the discounted expected future stock returns, since this is the other factor in the asset-pricing framework. To generate the innovation in the discounted present value of the market equity returns from the VAR, define the k-dimensional indicator vector el, whose first element is one and whose other elements are zero. Because Etzt+j = Ajzt, the innovation in the discounted expected future value of the stock market returns is
The last line of equation (11) defines the vector λh to be a nonlinear function of the VAR coefficients equal to e1‘ρA (I—ρA)-l. The subscript h is used to identify this coefficient with the hedging demands of investors.
The differences in the variances of the innovations and the correlations between innovations make it hard to interpret the estimation results for a VAR factor model unless the factors are orthogonalized and scaled in some way. To deal with this problem, Campbell (1996) employs Sims’s (1980) triangular orthogonalization of the innovations. He also scales the innovations such that they have the same variance as the innovation to the real return. We adopt a similar procedure. Thus, the orthogonalized innovation in the real return is unaffected, and the orthogonalized innovation in the payout ratio is the component of the payout ratio that is orthogonal to the real return. The orthogonalized innovation in the dividend yield is the component of the dividend yield that is orthogonal to the real return and the payout ratio, etc. Since fluctuations in the payout ratio are dominated by fluctuations in earnings, the orthogonalized innovation is essentially an earnings shock that is uncorrelated with returns. The orthogonalized innovation in the dividend yield becomes a dividend shock, and the innovations in the relative bill rate and the term premium become shocks to short-term and long-term interest rates that are orthogonal to returns, earnings and dividends.
Table 5 reports the coefficients of λh for the raw innovations and for the orthogonalized innovations using a ρ = 0.9949. Asymptotic standard errors are reported in parentheses under the coefficients. We find that shocks to the dividend yield, the relative bill rate, and the term premium Jiave positive influences on the innovation in discounted expected future returns, while the innovation in the stock return and the payout ratio have a negative effect. Although none of the coefficients is particularly large relative to its standard error, the χ2(5) statistic that tests whether all five coefficients are zero is 10.940, which has a p-value of .053. Hence, the group of variables as a whole does have some predictive power.
|Panel A: Coefficients of λh|
|Panel B: Covariance of News Variable|
Campbell (1996) notes that the coefficient on the orthogonalized innovation to the real return in the λh vector represents the percent of an innovation in returns that is expected to be reversed over the long run. Our point estimate implies considerably less reversion in international stock returns than Campbell finds in post-war U.S. data. Our coefficient is—0.23, whereas Campbell reports a coefficient of—0.92. We also find somewhat different results on the importance of the term premium. Here, we find a coefficient on the orthogonalized term premium of 0.65 while Campbell reports 0.21. Since the scale of the orthogonalized variables is quite similar because our variance is 16.42 versus Campbell’s 17.04, fluctuations in the term premium are much more important for the international equity markets than for the U.S. domestic market.
Panel B of Table 5 reports the covariances and correlations of the two sources of risk in the asset-pricing model, el’εt+1 and λ’hεt+1. We find that innovations to discounted expected future real returns are slightly more than half as variable as innovations to current returns, whereas Campbell (1996) finds them to be approximately equal, and we find a correlation between the two factors of—.324, whereas Campbell’s is—.915. The decline in the correlation between el’εt+1 and λ’hεt+1 may be due to the decline in the importance of the dividend yield in predicting future stock returns.
V. The Econometric Asset-Pricing Model
The next task is to develop the econometric specification to test the asset-pricing model. We use Hansen’s (1982) generalized method of moments (GMM), which requires that the econometrician define a set of orthogonality conditions specified by the theoretical model.
Three basic sets of orthogonality conditions are used. The first set is derived from the requirements that the errors in the VAR equations are required to be orthogonal to the right-hand-side forecasting variables:
The second set of orthogonality conditions involves the equations that define the innovations in the excess country returns. Here, as in Panel B of Table 1, the excess rate of return for country i at time t + 1 is projected onto the same information set, (1, z’t), that is used to forecast the real return on the world market portfolio:
Thus, the orthogonality conditions for the i-th country excess rate of return are
The third set of orthogonality conditions involves the asset-pricing equations. Since the asset-pricing equations are written in terms of conditional expectations at time t, and the innovations to the first two sets of orthogonality conditions are defined above, the asset-pricing equations (9) can be written as
By substituting for the expected return and using realizations for conditional expectations, we can define the innovation in the i-th asset-pricing equation relative to the time t information set as
In the first set of empirical results, we only impose the orthogonality conditions
where ut+1 is the vector of the ui, t+1’s. We do not impose additional restrictions because all of the orthogonality conditions of the model must be estimated simultaneously.
With seven country returns and five forecasting variables, there are thirty orthogonality conditions in the first set, forty-two in the second set, and seven in the third for a total of seventy-nine. There are also thirty parameters in the first set, forty-two in the second, and one in the third for a total of seventy-three parameters. Thus, the entire system of equations has six more orthogonality conditions than parameters, and the test of the overidentifying restrictions of the model is a χ 2(6).
VI. Estimation of the Model
Table 6 reports the seventy-three parameter estimates of the model associated with the VAR forecasting equations, the G7 excess-return forecasting equations, and the asset-pricing equations, as well as the constrained prices of risks. As might be expected, there is little change in the parameters of the VAR forecasting equations or the excess return forecasting equations relative to their respective unconstrained estimates that are reported in Table 1. The key parameter of the model is the coefficient of relative risk aversion, γ. Its estimated value is 5.06, with a standard error of 2.54. Often, in the consumption-based asset-pricing literature and in Campbell (1996), the estimated coefficients of relative risk aversion are implausibly large, but our value is quite reasonable.
|Coefficients on Regressors|
|Panel A: VAR|
|Panel B: Forecasting equations|
|Panel C: Constrained Prices of Risks and Estimate of γ|
The overidentifying restrictions of our model are not rejected, since the χ 2(6) statistic has a value of 1.23, which corresponds to a p-value of 0.975. Thus, Campbell’s dynamic asset-pricing model explains the cross-section of the G7 country equity returns. When we estimate the model without imposing the constraint that the coefficient on Vih is (γ—1), the estimated value of the coefficient is 3.056 with a standard error of 12.330. The resulting chi-squared test that the coefficient on Vih equals the coefficient on Vim minus one has a value of 0.004. The fact that the coefficient on Vih is insignificantly different from zero suggests that more restricted models may adequately represent the data.
Tests of CAPM Restrictions
There are several conditions that allow the predictions of the dynamic model to collapse to the predictions of the CAPM. This section investigates whether we can reject these conditions or not.
The first way that the dynamic model reduces to the CAPM is if γ = 1, since then expected returns are determined only by the covariance of an asset with the market portfolio. A GMM likelihood ratio test of the hypothesis that γ = 1 involves estimation of the model under this restriction with the same GMM weighting matrix as is used without the restriction. The difference in the values of the criterion functions is a χ 2(1) = 4.512, which coincides with a p-value of 033 indicating that it is unlikely that γ = 1.
The second way that the model reduces to the CAPM is if the covariances of the asset returns with the hedge portfolio are all zero. The test of these restrictions is a χ 2(7) = 54.035 with a p-value less than 0001 indicating very strong evidence that the covariances with the hedge portfolios are important in pricing the cross-section of asset returns.
The third way that the dynamic model can be reduced to the CAPM is if each Vih is proportional to its Vim with the same factor of proportionality. The test of these restrictions is a χ 2(6) = 4.512. Hence, we are unable to reject that the dynamic model collapses to the CAPM for this set of assets. The importance of the hedge portfolios in the pricing of the assets indicates that the hedge portfolios might be important in pricing other assets in which the proportionality does not hold.
Estimated Prices of Risks
The asset-pricing model implies that expected returns on assets adjust to reflect the covariances of asset returns with the return on the market portfolio and with the other underlying factors that forecast the market portfolio. These covariances are the asset’s risk characteristics, and the compensations or prices of the risks are constrained by the model. The price of risk that arises from the traditional covariance of an asset’s return with the market return is γ + (γ—l)λh1, where λhk represents the k-th element of λh. The price of the k-th risk that arises from the covariance of an asset’s return with the innovation in a variable that forecasts the market return is (γ—l)λhk. Thus, this model provides a clear link between the coefficient of relative risk aversion, γ, and the prices of risks that arise from covariances between asset returns and various factors.
The prices of risks are given in Panel C of Table 6 for the orthogonalized innovations. The covariance of a return with the return on the market portfolio indeed commands the largest price of risk, which justifies the important role of this variable in the traditional CAPM. The estimated price the risk caused by covariance of the return with the market return is 4.065 with a standard error of 1.541. The covariance of a return with the term premium also incurs a large price of risk, a value of 2.411 with a standard error of 1.834. The prices of other covariance risks are an order of magnitude smaller than these coefficients and have standard errors larger than the coefficients. The results for the prices of risks for the market return and the term premium are comparable to those estimated by Campbell (1996), but he finds much larger prices of risks for the dividend yield and the relative bill rate. These findings are consistent with Korajczyk and Viallet (1989) and Ferson and Harvey (1993) who also find significant risk prices for factors other than the risk generated by covariance with the return on the international market portfolio.
It is also possible to estimate unconstrained prices of risks. That is, the asset-pricing model is simply postulated to be
where the πk parameters represent the five unconstrained prices of risks associated with covariances between the asset returns and the innovations in the factors that forecast the market return. This specification is essentially a five-factor asset-pricing model as in Ross’s (1976) arbitrage pricing theory (APT). Table 7 presents the estimated parameters of this model, and the unconstrained prices of risk are given at the bottom. Compared to the results of the constrained model, there is a slight reduction in the price of risk associated with the covariance of an asset’s return with the real return on the world market portfolio. There are large changes in the prices of the other covariance risks, which are also very imprecisely estimated. Because this model uses the same orthogonality conditions as the constrained model and estimates four additional parameters, the overidentifying restrictions of this unconstrained model also are not rejected. The χ 2(2) statistic has a value of 0.133, which corresponds to a P-value of 0.936.
|Coefficients on Regressors|
|Panel A: VAR|
|Panel B: Forecasting equations|
|Panel C: Unconstrained Prices of Risks|
Pricing Errors from the Dynamic Model
Table 8 provides the average adjusted excess rates of return for the seven country portfolios for two samples, which have units of percent per month. The first is our base sample of 1970:01 to 1998:04. The second is a sub-sample from 1975:01 to 1995:12, which corresponds to the Fama-French (1998) sample. The average adjusted return corresponds to where the estimated residuals are used. This is simply the average risk premium, which is adjusted for the Jensen’s inequality effect that arises because we are using log returns instead of gross returns. The dynamic pricing model predicts that the average adjusted return contains two parts: a part corresponding to risk induced by covariance with the world market portfolio, γ Vi, m, and a part corresponding to risk induced by covariance with the innovation in discounted expected future returns, (γ—l)Vi, h. The model does very well at pricing the average adjusted returns. The pricing error for the U.S. is only 0.001 percent per month or 0.012 percent per annum. The Canadian pricing error is the largest at—0.045 percent per month for the basic sample or—0.54 percent per annum. The pricing errors are even smaller, in general, for the Fama-French (1998) sub-sample.
|Ave. Adj.||γ Vi, m||(γ - l)Vi, h||Pricing Error|
|Ave. Adj.||γ Vi, m||(γ - l)Vi, h||Pricing Error|
Since the estimates of Vi, h, are uniformly negative, each equity return covaries negatively with the innovation in discounted expected future market returns. As we noted above, when γ > 1, investors view such negative covariation as desirable for hedging purposes. Hence, these terms contribute negatively to the assets’ required rates of return.
CAPM Specification Tests
Our inability to reject the restrictions of the CAPM and the imprecision in the estimates of the unconstrained prices of risks, for the factors other than the market portfolio, suggest that the traditional CAPM may be an adequate representation of the expected returns on the seven country portfolios in the sense of having comparable pricing errors. To investigate this conjecture, we set up the following system of equations:
for i = 1,…, 7. The system is estimated by making each of the error terms orthogonal to a constant. Since there are nine parameters in the system of equations and fifteen orthogonality conditions, the test of the overidentifying restrictions is a χ2(6).
This version of the CAPM is estimated in Panel A of Table 9. For the full sample, the estimated value of γ for is 3.776, with a standard error of 1.478. The χ2(6) statistic that tests the overidentifying restrictions of the model is 1.079, which corresponds to a p-value of 0.982. Hence, the CAPM is not rejected by the data. The average returns are larger in the sub-sample, but inference about the validity of the CAPM is similar.
|Panel A: Parameter Estimates|
|Panel B: Pricing Components and Errors|
|Ave.Adj.||γ V i, m||Pricing||Ave. Adj.||γ V i, m||Pricing|
The pricing errors of the CAPM for the two samples are given in Panel B of Table 9. The largest pricing error is again smaller than 0.05 percent per month.
Predictability of returns is generally inconsistent with the CAPM. Only if γ = 1, or if Vi, h is proportional to Vi, m, does Campbell’s (1996) theory collapse to a conditional CAPM. If the static CAPM is true, the pricing errors should not be predictable with conditioning information. We test this predictability restriction by examining the additional orthogonality conditions that the asset-pricing errors from the CAPM model should be orthogonal to zt. Since there are seven assets, five elements in zt, and no additional parameters to estimate, we can calculate a χ2(35) statistic directly from the value of the GMM objective function for these thirty-five orthogonality conditions. The value of this statistic is 69.553, with a p-value of 0.0005. This indicates considerable evidence against the static CAPM.
Of course, we also did not require the pricing errors of the dynamic model to be orthogonal to the zt information set. When we calculate the analogous χ2(35) statistic for these restrictions on the dynamic model, we find a test statistic of 178.87, with a p-value smaller than 00001. Thus, the dynamic model also fails this test of dynamic asset pricing.
High Book-to-Market Returns
Fama and French (1998) further demonstrate the inadequacy of the static international CAPM by constructing portfolios of stocks for each country based on book-to-market ratios.8 They find that portfolios with high book-to-market (HBM) ratios are particularly troublesome for the CAPM to price. The average returns on high book-to-market firms are significantly higher than those predicted by the CAPM. Our version of these results is presented in Table 10, which excludes Canada because of data availability. Notice that the average adjusted returns on the H BM portfolios are considerably higher than the returns on the value-weighted MSCI market portfolios, except in the case of Italy. For example, the average adjusted return for the German H BM portfolio is 0.213 percent per month higher than the German market portfolio. For the U.K. the incremental value is 0.369. Table 9 indicates that this range is also the magnitude of the pricing errors from the CAPM, as well. The estimation uses the estimated value of γ from the analogous sample for the CAPM estimation with the country market portfolios, and the mean parameters for the HBM return equations are estimated prior to examining the pricing equations.
|Ave. Adj.||γ V i, m||Pricing Error|
Table 11 examines whether the dynamic model can successfully price the HBM country portfolios. The answer here, too, is negative. The pricing errors from the dynamic model are comparable to the pricing errors from the CAPM.
|Ave. Adj.||γ V i, m||(γ - 1) V i, h||Pricing Error|
This paper develops the implications of stock-return predictability for cross-sectional international asset pricing using the framework developed by Campbell (1996). The dynamic model explains the average returns on the dollar-denominated excess returns on the market portfolios of the G7 countries quite well, but the static CAPM does also. The hedging terms of the dynamic model are non-zero, yet we are unable to reject that they are proportional to covariances with the market portfolio, in which case they have no direct role in the crosssectional international asset pricing. Because the static CAPM is inconsistent with return predictability, which exists in the data, our specification tests demonstrate that instrumental variables can predict the asset-pricing errors from the static CAPM. Unfortunately, these same specification tests reveal that the dynamic model fails along the same dimensions. Neither model can price the high book-to-market country portfolios that Fama and French (1998) construct.
A rational explanation of the data requires a bridge between the intertemporal predictability of international equity returns and the cross-section of asset returns. Harvey (1991) and Ferson and Harvey (1993) develop an eclectic, empirical approach. Harvey (1991) postulates a conditional CAPM, and Ferson and Harvey (1993) develop conditional beta-pricing models in which expected returns equal the sum of several betas times their associated prices of risks. They allow the betas to vary over time depending on local information, but they require the prices of risks to vary only with global information. The problem with such approaches is that they are not tightly linked to theory.
Fama and French (1996) argue that the static CAPM is flawed, but they develop multifactor explanations of the data. In domestic U.S. asset pricing, they allow expected returns to depend on sensitivities of returns to three factors. The first is the excess return on a broad market portfolio. The second is the difference between the return on a portfolio of small-capitalization stocks and the return on a portfolio of large-capitalization stocks. The third is the difference between the return on a portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks. For international asset pricing, Fama and French (1998) use two factors: the excess return on the world market portfolio, and the difference in returns between the world high and low book-to-market portfolios. Since the Fama-French (1998) model successfully prices the problematic returns on high book-to-market country portfolios, a risk-based theoretical explanation for the presence of the second risk factor seems plausible. But, Fama and French (1998) note that their tests do not cleanly identify the consumption-investment state variables that would link their analysis to intertemporal asset-pricing theories.
The Campbell (1996) model provides a tight theoretical framework that allows us to test the implications of a truly dynamic asset-pricing model of international equity returns. We demonstrate that this particular dynamic model provides only a marginal improvement over the static CAPM. Our extension of the model to price international assets involves some strong assumptions. Perhaps the strongest is the assumption of a representative international investor. Adler and Dumas (1983) note that failure of purchasing power parity implies that an international representative investor does not exist. They demonstrate that in such a world expected equity returns depend on hedging terms involving foreign exchange rates. Vassalou (1997) provides some tests of unconditional restrictions implied by this approach by developing indexes of exchange rates and inflation rates. She finds support for the pricing of foreign exchange risk in equity returns. Dumas and Solnik (1995) postulate a version of the Adler-Dumas model that allows time-varying world prices of foreign-exchange risk that are postulated to depend on the conditioning information of the econometrician. They, too, find that conditioning information is important and that foreign-exchange risk is priced. De Santis and Gerard (1998) develop tests of the Adler- Dumas model using a conditional approach that estimates parametric conditional means and covariances. Since there is strong evidence against purchasing power parity, modifying the model to allow for different real returns in different countries is an extension that is pursued in Ng (1999).
Another weakness of our approach is its assumption that the value-weighted real stock return correctly measures the return on real wealth. Both Jagannathan and Wang (1996) and Campbell (1996) stress the importance of labor income and human capital in asset pricing. While Jagannathan and Wang (1996) and Campbell (1996) treat labor income as tradeable in a domestic asset-pricing context, such an assumption appears untenable in a multicountry model. Unfortunately, the required extension of the Campbell (1996) model to allow non-traded human wealth in different countries appears non-trivial.
As Flood, Hodrick, and Kaplan (1986) note, asset return predictability is not inconsistent with rational, maximizing behavior. Indeed, modern asset-pricing theories link this return predictability to hedging motives of investors. Different sensitivities of asset returns to the underlying state variables that generate time-varying market returns cause risk premiums on the assets to differ. This paper investigates the links between return predictability and the cross-section of country returns. Hedging demands may play a role in cross-sectional asset pricing, but there are still many puzzles that remain to be investigated.
We thank the participants at the conference, especially Marjorie Flavin, Paul Kaplan, and Louis Scott, for their comments. We also thank Geert Bekaert, John Campbell, Ken Froot, Laurie Hodrick, and Owen Lamont for their comments, as well as the participants in seminars at Columbia University, Georgetown University, Stanford University, the NBER International Finance Group, and the Federal Reserve Bank of New York. We are very grateful that Owen Lamont and Kenneth French each provided us with part of the data.
The model also incorporates human capital as an additional asset.
The Euler equation approach of Hansen and Singleton (1982) provides researchers with econometric tools that allow tests of models in which expected returns on stocks were allowed to fluctuate, but the approach requires correct specification of a representative agent’s utility function and direct observations on the consumption of this agent. Both aspects are problematic.
Campbell (1996) also allows for an additional factor in the asset-pricing model related to human wealth. He argues that the return on the real wealth portfolio is appropriately measured as the value-weighted sum of the return on assets and the return on human wealth. Jagannathan and Wang (1996) also include labor income in their analysis. While there may be an asset-pricing role for labor income, in a global economy the variable would have to be some measure of the world’s labor income. We have made no attempt to measure such a variable.
While it is tempting to substitute σ = 1 in equation (7) and conclude that intertemporal considerations are not important in this case, which would imply that equation (9) is wrong, the correct derivation follows Giovannini and Weil (1989) who note that L’Hopital’s rule must be applied because θ → ∞ as σ → 1.
We thank Owen Lamont for providing the data used in the comparisons to his article in Table 4.
We thank Kenneth French for providing the data on the HBM portfolios used in this part of our paper.
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BY LOUIS SCOTT
In this paper, Hodrick, Ng, and Sengmueller use cross section and time series data to test the implications of a dynamic asset pricing model. The objective of the paper is to develop a model that explains both the predictability of stock returns over time and one of the anomalies in the finance literature, the empirical observation that portfolios of stocks with high book to market ratios have higher average returns. Much of the empirical asset pricing research on stock returns relies on ad hoc empirical models or extensions of the single period Sharpe-Lintner capital asset pricing model (CAPM). Hodrick, Ng, and Sengmueller work with an empirical model that is derived within the framework of an intertemporal CAPM. Merton (1973), in the development of the intertemporal CAPM, showed that expected returns are influenced by desires to hedge unfavorable changes in the investment opportunity set. In this paper, the changes in the investment opportunity set are changes in future expected returns. The intertemporal CAPM is the appropriate framework for modeling and explaining the predictability of returns over time, and this approach is superior to the frequently encountered ad hoc approach of specifying a Sharpe-Lintner CAPM with dynamic changes in expected returns and risk premia.
The model is successful empirically at explaining the predictability of international stock returns, but it is not able to explain the higher average returns on portfolios with high book-to-market ratios. Hodrick, Ng, and Sengmueller use a vector autoregression to identify changes in expected returns; the four variables that are useful for predicting stock returns are the dividends to earnings ratio, the dividend yield, the one month Treasury-bill rate relative to a moving average, and the slope of the term structure. The R2 for the stock return regression is only 3 percent, which is statistically significant, but the predictable variation of stock returns is small relative to its total variation. The slope of the term structure variable appears to be the most important variable for predicting stock returns. In most of the previous work on stock return predictability, dividend yields have been useful as predictors of stock returns, but not in this study in which the sample is extended to 1998.
The empirical result that dividends yields have not predicted subsequent stock returns is new. In numerous studies using stock return data back to the 1920’s, researchers have found that dividend yields predict subsequent stock returns. In regressions of future stock returns on the dividend yield, the coefficient on the dividend yield is typically positive and statistically significant, and the R2 is small, but statistically significant. Dividend series are relatively smooth over time so that most of the variation in dividend yields is due to the stock price in the denominator. When stock prices rise relative to dividends, the dividend yield drops, and subsequent stock returns tend to be lower. Conversely, when stock prices fall relative to dividends, the dividend yield increases, and subsequent stock returns tend to be higher. In Figure 1, I have reproduced the dividend yield for the S&P 500 from 1970 to the beginning of 1999. Dividend yields have fluctuated between 3 percent and 6 percent over most of this period, but since 1992, there has been a steady decrease and the S&P 500 dividend yield is now (January 1999) below 2 percent. Since 1992, stock prices have generally risen and stock returns have been strongly positive, which is a departure from the previous relationship. What happens next? There are, of course, several explanations for the recent behavior of dividend yields. One is that the stock market is overvalued and dividend yields will return to historical norms when there is a major correction. The alternative explanation is that firms have been paying out less of their earnings in the form of cash dividends and have been using the extra cash to buy back shares of stock and invest. Earnings will grow at a faster rate in the future and dividends will eventually catch up so that dividend yields will gradually increase and return to historical averages. This issue will, of course, be settled by future research.
Figure 1.Dividend yield for the S&P 500.
In Figure 2, I have reproduced the price-earnings ratio for the S&P 500 because it is a widely followed indicator of the stock market. This P/E ratio has varied between 10 and 20, but in recent months it has surpassed a level of 30, which is a record high. The challenge is to build a model of rational, optimizing agents that explains the observed variability of P/E ratios. One explanation is variability in the market risk premium, but one must use some extreme variability in this risk premium to explain the variability observed in Figures 1 and 2. One possibility is to work with a dynamic asset pricing model, similar to the model used by Hodrick, Ng, and Sengmueller, and have factors driving earnings, dividends, and the marginal utility of wealth, and solve the model for equilibrium stock prices. Models of this form are difficult to solve, but are necessary in order to understand the behavior of dividend yields and P/E ratios. A framework for analyzing this issue is the Euler equation from which dynamic asset pricing models are derived:
Figure 2.Price-earnings ratio for the S&P 500.
In this model, the stock price is a solution to a first order difference equation, and a solution with a Flood-Garber bubble is
where At is a martingale, Et(At+j) = At. The last term is the Flood-Garber bubble and there are many possible solutions or models for its behavior. One example is a bubble that grows exponentially and every period there is a small probability that it will collapse. The bubble solution can generate the observed variability in dividend yields and P /E ratios. The open issue is whether one can generate this variability with the fundamental part of the solution.
BY PAUL D. KAPLAN
The premise of this paper is that asset return predictability is not inconsistent with rational, maximizing behavior. Campbell’s versions of the intertemporal capital asset pricing model seem to be the most promising approach to producing a model derived from rational, maximizing behavior in which asset returns exhibit predictability. This paper provides an elegant framework for tracing the implications of Campbell’s models for the cross-sectional variation of expected returns.
The merit of this paper is that it tries to tightly tie a multivariate model of time-varying expected returns with asset pricing theory. In contrast, the multivariate models of Fama and French (1996, 1998) are not tied to any specific theory.
The weakness of this paper is that it tries to impose too much structure on the data. The failure to reject the model of average expected returns is likely due to a lack of statistical power. The failure to also reject the static CAPM confirms this view. Furthermore, when all of the instrumental variables are included in the asset pricing restrictions, both the dynamic model and the static CAPM are rejected. This is because including all of the instrumental variables raises the power of the test. The failure of the model to explain the average returns of the high book-to-market portfolios also shows that the data do not conform to the restrictions imposed by the model.
One suggestion that I have for refining the empirical model is to let ρ be a free parameter rather than a predetermined value. It would be interesting to see if the point estimate comes out plausible.
This paper provides an excellent example of what it means to say that predicable asset returns can be the outcome of rational, maximizing behavior. However, most of the money management industry is based on the premise that returns are predicable because of market inefficiencies and irrational behavior. Empirical refutation of that view continues to elude us, even with models such as this one.
Alessandro Prati noted that the sample periods used in estimating the model parameters—from 1970 to 1998 or 1975 to 1995—included changes in capital mobility and exchange rate regimes that were particularly important for the European members of the G-7 countries. Such changes presumably influenced the covariances of returns, certainly the cross-section covariances, so one would expect that the parameters that were estimated would not have been stable over time. Richard Meese wondered why the authors had not drawn more on the work of Michael Adler and Bernard Dumas in considering the types of factors that one might expect to be priced in an international setting.
Marjorie Flavin pointed out that there were several ways to interpret the empirical findings. The authors had assumed that the consumption CAPM model was true, had built on the work of John Campbell in deriving an asset-pricing model that could be estimated without using data on consumption, and had then used that model to test the hypothesis that the coefficient of relative risk aversion (γ) was equal to unity. Testing γ = 1 was equivalent to testing the validity of the traditional static CAPM model, given the assumptions that had been made in deriving the asset-pricing model, but Flavin argued that the authors’ rejection of the hypothesis γ = 1 should not be interpreted as a rejection of static CAPM. An alternative possibility was that the consumption CAPM model should be rejected, and that the static CAPM model with γ > 1 provided a perfectly reasonable view of the world.
Ng thought that the comments by Prati, Meese, and Flavin were well taken and indicated that he and his co-authors were intending to address them in their future work. He would be looking at the Adler-Dumas model in his doctoral dissertation. Hodrick added that the problem with the Adler-Dumas framework was that covariances of asset returns with every exchange rate in the world had to be priced. One of his colleagues, Maria Vassalou, had tried to estimate the Adler-Dumas model by developing indexes of exchange rates, which was an ad hoc modification of the Adler-Dumas model but might be a promising direction for investigation. With regard to the point that Flavin had made, Hodrick noted a subtlety that he hadn’t brought up in his presentation—namely, that the exercise of moving from the conceptual framework to testing the model involved changing the weighting matrix on the rates of return.
Flood wondered how to think about the substitutions that Campbell and Hodrick, Ng, and Sengmueller had made to derive specifications that did not involve consumption terms. These substitutions were motivated by the results of previous studies that had found that consumption correlations don’t work very well. Was there some “terrible error” with the earlier consumption equations, and if so, how should one view where that error had gone?
Flavin responded that Campbell’s rationalization for substituting out the consumption terms was based on discrepancies between measured consumption and the concept of consumption that was envisioned in the theoretical framework. In an idealized world, measured consumption might be a good proxy for the marginal utility of wealth, but Campbell and others had taken the view that it was not a good proxy in reality. Flavin herself felt that the lack of empirical support for consumption CAPM frameworks based on measured consumption might reflect more than measurement problems and called into question the consumption CAPM model. Grossman and Laroque (1990), for example, had made some allowances for the adjustment costs associated with durables consumption, and had found that during periods in which consumers do not adjust the sizes of their houses, the consumption CAPM model fails while the traditional CAPM model holds.
Hodrick recalled a seminar at Carnegie-Mellon in the early 1980s, when Lars Hansen and Kenneth Singleton presented the results of their first attempts to estimate the consumption CAPM model and were confronted with the objection that such attempts were hopeless owing to problems associated with durables consumption, aggregation issues, and so forth. He nevertheless thought that the framework provided a useful parable. The intuition really came from Robert Merton’s model: when there is time variation in rates of return, investors will want to hedge against it. That’s a general principle. But in Merton’s general framework, the coefficients of the model are functions of the state, and the model is too complex to be estimable. It was nice to have a more simplified conceptual framework in which hypotheses could be tested.
The fact that the tests did not find dividend yields to be important was a problem in Hodrick’s view. There had been a large move toward repurchases in the stock market, as opposed to dividends, for reasons on which Hodrick did not want to elaborate. It was also the case that calculations of the ex ante expected rates of returns on the market, conditional on forecasts of future earnings, had been suggesting that equity premiums were very low. This was true not only in the United States, but also around the world; and it had been true for the past five or ten years. There are two schools of thought on this. One is that it reflects irrational exuberance. The other is that a structural change has occurred and we are now in a world in which risks are being shared differently and more internationally. According to the latter view, although we may not yet have a good understanding of the structural change that has taken place, increases in volatility associated with events like the Russian crisis wind up affecting all assets around the world, which is very different than the effects that such events might have had in the 1970s.
Dale Henderson wondered how much credence could be attached to the view that a structural break had been brought about by the widening of access to mutual funds. What was the order of magnitude of the change that had occurred in the amount of wealth that can be easily invested in equities? Flavin thought that the proportion of people who have easy access to equity investments had increased considerably, but because the distribution of wealth is skewed, the proportion of wealth that had easy access to equity investments might not have increased dramatically.