Market eciency, asset returns, and the size of the riskpremium in global equity markets

Transcription

1 Journal of Econometrics 109 (2002) Market eciency, asset returns, and the size of the riskpremium in global equity markets Ravi Bansal a;, Christian Lundblad b a Fuqua School of Business, Duke University, Durham, NC 27708, USA b Kelley School of Business, Indiana University, Bloomington, IN 47405, USA Received 18 January 2002; accepted 24 January 2002 Abstract An important economic insight is that observed equity prices must equal the present value of the cash ows associated with the equity claim. An implication of this insight is that present values of cash ows must also quantitatively justify the observed volatility and cross-correlations of asset returns. In this paper, we show that parametric economic models for present values can indeed account for the observed high ex post return volatility and cross-correlation observed across ve major equity markets the U.S., the U.K., France, Germany, and Japan. We present evidence that cash ow growth rates contain a small predictable long-run component; this feature, in conjunction with time-varying systematic risk, can justify key empirical characteristics of observed equity prices. Our model also has direct implications for the level of equity prices and specic versions of the model can, in many cases, capture observed price levels. Our evidence suggests that the ex ante riskpremium on the global market portfolio has dropped considerably we show that this fall in the riskpremium is related to a decline in the conditional variance of global real cash ow growth rates. c 2002 Elsevier Science B.V. All rights reserved. JEL classication: F3; G0; C1; C5 Keywords: Asset volatility; Correlation; Cash ows; Riskpremia; Fundamental values 1. Introduction An important economic insight is that observed equity prices should equal the present value of the cash ows associated with the ownership of the equity claim. The work An earlier version of this paper was titled, Market eciency, fundamental values, and asset returns in global equity markets. All data employed in this paper are available at Corresponding author. Tel.: ; fax: addresses: ravi.bansal@duke.edu (R. Bansal), clundbla@indiana.edu (C. Lundblad) /02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 196 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) of Shiller (1981), LeRoy and Porter (1981), West (1988), and Campbell and Shiller (1987, 1988a, b), however, poses a challenge to this insight. These authors document the volatility puzzle quantitatively, equity prices are far too volatile to be justied as present values of fundamental cash ows. This result underscores the key feature of the data that cash ow volatility is quite small relative to equity price volatility. In addition to implications for volatility, present values also restrict cross-correlations of asset returns. In the data, the average cross-correlation in ex post returns is about six times larger than that for the cash ow growth rates. This feature poses an additional quantitative challenge to present values, and is labeled the correlation puzzle. Present values of the cash ows are determined by the time-series dynamics of the expected cash ow growth rates and the cost of capital (i.e., ex ante rate of return). In this paper, we show that a parsimonious time-series model for cash ow growth rates and the cost of capital goes a long way in explaining the observed equity market volatility and return cross-correlations. The main insights that this paper provides can best be understood by rst considering the role of the cash ow dynamics, followed by that of uctuations in the cost of capital. In the data, real growth rates have near zero autocorrelation, hence, it is common to assume that cash ow growth rates are i.i.d. In addition to this assumption, if cost of capital is constant, then news regarding cash ow growth rates is entirely transitory and does not alter future expected growth rates. Consequently, dividend yields are constant and ex post continuous return volatility equals the growth rate volatility. However, as cash ow growth rate volatility is smaller than return volatility, this leads to the volatility puzzle discussed above. In sharp contrast, Barsky and DeLong (1993), argue that cash ow growth rates can be modeled as an integrated process (more precisely, an ARIMA(0,1,1) process). It is important to note that in nite samples, the Barsky and DeLong process for growth rates cannot easily be distinguished from an i.i.d. process (see Shephard and Harvey, 1980), but the economic implications for asset prices are dramatically dierent. Expected growth rates in this specication contain a unit root, and consequently, news regarding growth rates have large eects on dividend yields as they permanently alter future expected growth rates. 1 Campbell et al. (1997) argue that Barsky and DeLong (1993) do not provide any direct empirical support for their growth rate dynamics further, it is not clear if an integrated growth rate process is economically plausible. In this paper, unlike Barsky and DeLong (1993), we provide empirical evidence that growth rates are well modeled as a stationary (i.e., no unit root) ARIMA(1,0,1) process. As cash ow growth rates contain a small predictable (and persistent) component, growth rate news leads to volatile changes in dividend yields and ex post returns. This structure helps address the volatility puzzle and the correlation puzzle discussed above. With constant cost of capital for each economy, the ex post return cross-correlations across economies will be solely determined by the cash ow growth correlations. However, this is unlikely to justify return cross-correlation, as growth rate correlations 1 At a rm level, it is well documented that cash ow news leads to signicant price reaction (see Easton and Zmijewski, 1989).

3 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) across economies are quite small. One factor that may account for high return correlation is uctuations in global riskpremia a source of common uctuations in asset prices. This view is also consistent with Ammer and Mei (1996), who document that much of the asset return covariation between national stockmarkets is related to news about future riskpremia. Indeed, relying on a simple CAPM-GARCH specication, as in Bollerslev et al. (1988), we show that uctuating global riskpremia in conjunction with the assumed cash ow dynamics can reproduce the observed ex post return cross-correlations and asset return volatility. Further, we show that the persistent component in cash ows is also needed for duplicating asset return cross-correlations in the absence of this, asset price uctuations are dominated by common cost of capital uctuations, and hence asset returns are, counter-factually, almost perfectly correlated. The asset valuation model that we develop also provides insights regarding two additional issues. First, authors, such as Ammer and Mei (1996) use cross-correlations in cash ow news and expected returns to measure economic and nancial integration, respectively, across markets. However, they do not provide any economic mechanism to linkthese two measures of integration in this paper, we do provide such a mechanism and show that if there is little economic integration, then nancial integration will be small as well. Second, Longin and Solnik(1995) show that an important feature of global equity market data is that periods of increased market uncertainty are also associated with a rise in the conditional correlation of returns our model, which incorporates time-varying volatility, reproduces this feature of the data as well. Relying on the assumed cash ow growth rate dynamics and the specication for uctuating global riskpremia, our valuation model can account for about 70 80% of the volatility of asset prices (change in dividend yield or returns) and cross-correlations in asset returns. The more standard vector autoregression (VAR) methods of modeling cash ow growth rates and expected rates of returns to compute present values (as in Campbell and Shiller, 1988a) lead to asset values which have very low variability (about 40% of that in the data) and very high (with many in excess of 0.9) cross-correlation in asset returns. In nite samples, this approach fails to capture the persistent component in cash ow growth rates which leads to large asset return variability, and hence also aects asset return cross-correlations. Despite the ability of the model to explain these particularly challenging features of the observed data, the level of fundamental values implied by the model in particular time periods, especially for Japan (in the mid-1980s) and for the U.S. (in ), are far from the observed equity prices. For other countries, such as France and U.K., the model matches the observed equity prices quite well. Partly motivated by the failure to match the observed equity prices in specic time periods for Japan and the U.S., we develop and estimate a model in which the time-varying world market volatility process is assumed to be latent (see Taylor, 1986; Hansen and Hodrick, 1983). Using the valuation restrictions, we show that this latent volatility can be recovered from the observed world equity market prices and the expected cash ow growth of this benchmarkasset. We nd that modeling the systematic riskin this manner provides a signicant improvement over the GARCH specication.

4 198 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) The latent volatility model matches the observed equity prices quite well, and captures an economically signicant portion of the volatility (about 80%). Additionally, it justies almost all of the observed cross-correlation, other than for Japan. In contrast to the GARCH specication, this model suggests that the aggregate riskpremium in the global economy has fallen signicantly in the last decade to about 2%. This dierence has important eects on measured fundamental values. We also show that much of the fall in the latent systematic riskcan be attributed to a fall in the conditional world market cash ow variance. In parallel and independent work, Fama and French (2000) backout the riskpremia from the U.S. equity index values, and also argue that the market risk premium has fallen. In independent papers, Dumas et al. (2000) and Chue (2000) focus on the crosscorrelation among equity returns. However, they do not focus on the joint implications for return volatilities, cross-covariances, the cross-section of equity premia, and the level of equity prices. As they assume dierent cash ow dynamics, their results and conclusions dier from those in this paper (and that in Barsky and DeLong, 1993). For example, unlike the results in this paper, they can only account for a small fraction of the observed return volatility. The paper is organized as follows. Section 2 discusses the data used in the paper. Section 3 provides the general present value relations used in the paper, discusses our cash ow model and the evidence supporting it, and lays down the specic fundamental restrictions implied by the model. Section 4 discusses the estimation methodology, and provides the empirical ndings and diagnostics. Section 5 provides evidence on the valuation implications of our model, and Section 6 presents evidence on the size of the equity premium. Finally, Section 7 provides concluding comments. 2. Data description We collect monthly data, taken from Datastream, on market prices for ve developed equity markets: France, Germany, Japan, the U.K., the U.S., and the World Market Index. 2 For each of these market indices, we also collect data on the dividend yield, earnings yield, and total returns denominated in local currencies. From these, we construct dividend and earnings growth rates; note that as in Fama and French (1995), the measure of earnings is net of depreciation. To measure various quantities in real terms we also collect from International Financial Statistics (IFS) a seasonally adjusted CPI index for each country. The sample period for all the data we collect is from January 1973 to December The total return is dened as follows: 1+R i;t+1 = P i;t+1 P i;t (1 + DY i;t+1 ): (1) 2 The Datastream World Market Index return has a correlation of 0.99 with the Morgan Stanley world index (MSCI) return.

5 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Here P i;t is the equity price and DY i;t represents the dividend yield, the asset s current dividend payment divided by its current price. It is well known there are strong seasonals in the raw dividend series, thus we follow the convention in the literature by measuring the dividend yield as the average of dividends paid on the index over 11 j=0 D t j)=p t ; this is the previous year divided by the current price level, DY t =( 1 12 similar to the approach taken in Heaton (1993), Bollerslev and Hodrick (1995) and Hodrick(1992). For reasons of seasonality, the only data reported in Datastream for the dividend yield and the earnings yield are constructed using the lagged 12-month moving average. Further, we remove additional seasonality from both dividend and earnings growth rates as a 12th order autoregressive seasonal; our empirical results are not sensitive to this additional step. Note that the dividend (earnings) series is constructed using the observed equity market capitalization and the dividend yield (earnings yield) series. These valuation ratios form the focus of our computations for determining present values. The continuous growth rate for the cash ow (i.e., dividends or earnings) is the log of the gross growth rate of the cash ow under consideration. When converting nominal variables to real, we simply subtract from the relevant variable in country i the seasonally adjusted CPI ination rate (i.e., log of gross ination) in country i. Throughout we determine the present value implications for the various markets in real terms. We also construct a real interest rate series by using the one-month Eurodollar rate, and subtracting from it a measure of expected ination in the U.S., taken here to be the ination expectation implied by an ARIMA(1,0,1) model on the ination series. 3 In Table 1, we report summary statistics for the total returns, log price dividend ratios, log price earnings ratios, dividend growth rates, and earnings growth rates. An important feature of the data is the volatility of dividend (earnings) growth rates are on average only about 5% (10%) of the volatility of the log price dividend (earnings) ratio. In Fig. 1, we present the average (across countries) empirical autocorrelation functions for dividend and earnings growth rates; this average is representative of the autocorrelation function for individual countries. Importantly, we observe the average rst-order autocorrelations for either the dividend or earnings growth rates are roughly 0.06, suggesting a very low level of persistence in the observed growth rates themselves. In Table 2, we report the cross-correlations of returns, dividend and earnings growth rates among the various markets in our menu. It is evident from the tables that the correlation across the various equity markets of the rst dierence of the log dividend yield (or the log earnings yield) is on average about six times the average correlation in either observed dividend or earnings growth rates across the dierent markets. Also, note that ex post real equity return cross-correlations are of similar magnitude as observed for the rst dierence of the valuation ratios (the earnings yield and the dividend yield). All data employed in this paper are available at 3 Alternative methods for constructing the ex ante real rate make very little dierence to our empirical results. We have also constructed the ex ante real interest rate using dierent methods such as removing the trailing 12-month ination rate. The implied real interest rate series is very similar to the one backed out using the ARIMA(1,0,1) series.

6 Table 1 Summary statistics ln(p t=d t) ln(p t=d t) ln(p t=e t) ln(p t=e t) r i;t ln(d t+1 =D t) ln(e t+1 =E t) Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std FR (0.293) (0.074) (0.004) (0.003) (0.171) (0.134) (0.004) (0.003) (0.004) (0.003) (0.002) (0.003) (0.003) (0.005) BD (0.165) (0.056) (0.003) (0.003) (0.133) (0.179) (0.004) (0.003) (0.003) (0.003) (0.001) (0.002) (0.003) (0.005) JP (0.301) (0.041) (0.003) (0.003) (0.387) (0.080) (0.003) (0.003) (0.003) (0.003) (0.001) (0.002) (0.002) (0.003) U.K (0.242) (0.118) (0.003) (0.006) (0.090) (0.137) (0.004) (0.006) (0.004) (0.005) (0.001) (0.002) (0.002) (0.003) U.S (0.422) (0.118) (0.003) (0.003) (0.119) (0.204) (0.003) (0.003) (0.003) (0.003) (0.001) (0.000) (0.001) (0.001) WD (0.159) (0.058) (0.002) (0.003) (0.111) (0.063) (0.002) (0.003) (0.003) (0.002) (0.001) (0.001) (0.001) (0.001) All data are reported at the monthly frequency. ln(p t=d t) and ln(p t=e t) are the log price dividend and log price earnings ratios, respectively, where for the latter we scale the log price earnings ratio by the average payout ratio (D=E). indicates the rst dierence of the variable under consideration. r i;t is the continuous real rate of return, and ln(d t+1 =D t) and ln(e t+1 =E t) are the continuous real dividend and earnings growth rates, respectively. The world return, r m;t is the Datastream World Equity Index; its correlation with the commonly employed MSCI World index is All standard errors presented in parentheses are obtained by using a GMM-VARHAC procedure. 200 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002)

7 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Autocorrelation Function (Dividend Growth) 0.20 Autocorrelation Function (Earnings Growth) Average observed Theoretical ARMA: ρ=0.973, ω= Average observed Theoretical ARMA: ρ=0.950, ω= Fig. 1. Cash ow autocorrelation. 3. Present value and asset prices It is well recognized that the fundamental value of the asset is the present value of the cash ow associated with the asset. The present value is determined by the expected growth rate dynamics and the ex ante rate of return on the asset this arithmetic is captured by the approach pursued in Campbell and Shiller (1988a). They show that the log of the ex post total return, that is r i;t+1, can be approximated as r i;t+1 = g i;t+1 + i;0 + i;1 z i;t+1 z i;t ; (2) where g i;t is the continuous growth rate of dividends and z i;t is the log price dividend ratio. i;0 and i;1 are constants related to the Taylor-series approximation. 4 Based on the above approximation they derive the result that z t p t d t = 0 +E t j 1 1 [g t+1+j r t+1+j ] ; (3) 1 j=0 where p t and d t are log equity price and log dividends. The above equation shows that the key determinant of the asset valuation, z i;t, is the dynamics of the expected cash ow growth rates and the ex ante rate of return on equity. 4 i;1 =1=(1+exp(d i p i )) and i;0 = log( i;1 ) (1 i;1 )(d i p i ), where (d i p i ) is the steady-state (or mean) logged dividend yield. In practice, we use the approximation parameter values implied by the average dividend (earnings) yield observed in each market.

8 202 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Table 2 Observed correlations FR BD JP U.K. U.S. WD Correlations: r i;t FR (0.053) (0.054) (0.058) (0.057) (0.057) BD (0.072) (0.066) (0.075) (0.070) JP (0.055) (0.063) (0.053) U.K (0.049) (0.099) U.S (0.049) WD Correlations: ln(p t=d t) FR (0.054) (0.058) (0.059) (0.056) (0.062) BD (0.070) (0.062) (0.073) (0.071) JP (0.056) (0.061) (0.056) U.K (0.050) (0.050) U.S (0.050) WD Correlations: ln(p t=e t) FR (0.050) (0.060) (0.053) (0.059) (0.061) BD (0.065) (0.058) (0.061) (0.065) JP (0.060) (0.066) (0.059) U.K (0.056) (0.051) U.S (0.048) WD Correlations: ln(d t+1 =D t) FR (0.065) (0.058) (0.057) (0.049) (0.062) BD (0.160) (0.052) (0.051) (0.038) JP (0.050) (0.057) (0.048) U.K (0.051) (0.040) U.S (0.045) WD 1.000

9 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Table 2 (continued). FR BD JP U.K. U.S. WD Correlations: ln(e t+1 =E t) FR (0.062) (0.060) (0.036) (0.056) (0.060) BD (0.061) (0.056) (0.072) (0.070) JP (0.038) (0.039) (0.047) U.K (0.047) (0.051) U.S (0.089) WD This table presents cross-correlations of monthly returns, the rst dierence of the two valuation ratios, and the continuous growth rate of cash ows (dividends and earnings). Returns and growth rates are continuous and real. VARHAC adjusted standard errors are provided in parentheses. To derive fundamental values of assets, we rst model cash ow growth rates and then proceed to model the cost of capital. Barsky and DeLong (1993) posit an ARIMA(0,1,1) process for dividend growth rates, that is, growth rates contain a unit root. In Timmermann (1993, 1996), learning about the cash ow growth rate process leads to time variation in expected growth rates. Donaldson and Kamstra (1996) provide a model for the univariate dynamics of the cash ow growth deated by a discount factor, and use boot-strapping procedures to solve for fundamental present values. Campbell and Shiller (1988a, b) posit VAR dynamics for growth rates and ex post rates of returns and then test for certain internal consistency restrictions that follow from (3); we discuss this in greater detail below. Further, all these studies focus on a single equity claim, and consequently do not address the issues associated with the cross-correlation puzzle, which is an important focus of this paper Cash ow dynamics In this section, we provide the description of the assumed time-series model for the growth rate of cash ows, which we demonstrate can have large eects on implied equity prices, volatility, and asset betas. Statistically, we posit that cash ow growth rates are described by an ARIMA(1,0,1) process, an assumption for which we will provide empirical support. We also show that this process for growth rates is equivalent to a decomposition of cash ow levels into exponentially smoothed stochastic trend and autoregressive cyclical components. 5 5 Alternatively, following Kasa (1992), one could model cash ows as determined by a common trend. This specication is closely related to the world business cycle described by Dumas et al. (2000). To explore this specication, we perform cointegration tests, but we do not nd evidence for a single common trend (world business cycle) in either the dividends or earnings series. For dividends, these cointegration results are broadly similar to Kasa (1992).

10 204 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Let g i;t be the real growth rate of cash ows for equity claim i. We assume that the process for g i;t satises g i;t =(1! i ) i + i g i;t 1 + i;t! i i;t 1 : (4) The variable that aects present values is the conditional mean of the growth rate. For the ARIMA(1,0,1) process the conditional mean x i;t equals (( i! i )=(1! i L))g i;t, where L is the standard lag operator. Consequently, the ARIMA(1,0,1) process can be more conveniently written as g i;t = i + x i;t 1 + i;t : (5) It is assumed that g is stationary, and hence and! (where country subscript i is suppressed) are 1 in absolute value. Using Eq. (5), it follows that x i;t itself follows an AR(1) process, x i;t =( i! i ) i + i x i;t 1 +( i! i ) i;t : (6) The parameter determines the degree of persistence, and! is the smoothing parameter that aects the construction of x i;t. It is also worth noting two special cases that the ARIMA(1,0,1) representation accommodates. If =!, the conditional mean of g is a constant, and in fact g can be viewed as an i.i.d. process. On the other hand, if! =0, g follows a standard AR(1) process. To develop intuition regarding the implications of an ARIMA(1,0,1) process for cash ow expectations, and hence equity prices, consider an agent s revision in expected growth rates (for horizon n 1) in response to growth rate news at date t: E t [g t+n ] E t 1 [E t [g t+n ]] = n 1 (!) t : (7) Economically, Eq. (7) implies that rational agents may signicantly revise their long-run expected growth rates so long as! 0. In the extreme case when =! (i.i.d. case), there is no revision in the expected growth rate at all. Also, the permanence of the expectation revision is determined by. If = 1, the revision in expectations is identical across all horizons (as in Barsky and DeLong, 1993). When 1, the revision is larger for shorter horizons, and almost zero for very long horizons. For the case where the dierence between and! is small and positive and is large, growth rate news leads to small, but near permanent, revisions in agent s expectations of future growth rates. To understand the pricing implications of the expected cash ow growth rate process in Eq. (6), consider the implications for the log price dividend ratio, z, in Eq. (3) (assuming that the cost of capital is constant for now): z t =z (x t x); (8) where z and x refer to unconditional means. The volatility of the price dividend ratio is clearly increasing in the persistence of the expected growth rate; when approaches one (as in Barsky and DeLong, 1993), the price dividend ratio becomes extremely volatile. Further, the reaction of the price dividend ratio to growth rate

11 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Table 3 ARIMA(1,0,1) estimation results for dividend and earnings growth Parameter estimates g i;t =(1! i ) i + i g i;t 1 + i;t! i i;t 1 Dividends Earnings i! i R 2 LR-test i! i R 2 LR-test FR : :25 (0.061) (0.080) (0.119) (0.141) BD : :07? (0.053) (0.082) (0.070) (0.087) JP :94?? :16 (0.028) (0.047) (0.036) (0.060) U.K : :82 (0.038) (0.063) (0.074) (0.105) U.S : :50 (0.043) (0.067) (0.066) (0.093) The ARIMA(1,0,1) specication is estimated by maximum likelihood, using the normal distribution function. Standard errors are provided in parentheses. LR-test refers to the Andrews and Ploberger (1996) likelihood ratio test of the null hypothesis that the and! are equal. They show that the test statistic is two times the dierence between the unconstrained (ARIMA) and constrained (i.i.d.) log likelihood values. A above indicates a rejection of the null at the 0.01 level,?? indicates a rejection at the 0.05 level, and? indicates a rejection at the 0.10 level. The critical values for these tests are obtained from Andrews and Ploberger (1996). For a sample size of 250 observations, the 0.01 level critical value is 9.23, 0.05 critical value is 6.13, and 0.10 critical value is 4.74; for 500 observations: 0.01 level critical value is 9.46, 0.05 critical value is 6.09, and 0.10 critical value is Intercepts are not reported. news is z t E t 1 [z t ]=(!) t n=1 n 1 n 1 = (!) t 1 1 : (9) Again, when is close to one and larger than!, the impact of cash ow news on the innovation to the price dividend ratio can be very large even though the ex post cash ow process seems very close to an i.i.d. process in a nite sample. In contrast, in the i.i.d. case ( =!), growth rate news has no impact on the dividend yield, a feature which seems counter-intuitive and empirically implausible (see Easton and Zmijewski, 1989). Collectively, this suggests that the explanation of the volatility puzzle for asset prices is intimately related to large price elasticity with respect to cash ows for parameter estimates presented below, this quantity is well in excess of one. In Section 4.3.3, we show that the standard VAR approach (as used in Campbell and Shiller, 1988a), will fail to detect, in nite samples, the persistent component of cash ows which is important for understanding asset price volatility Empirical evidence regarding the cash ow dynamics In Table 3, we present evidence for an ARIMA(1,0,1) process for both of the two alternative measures of cash ow growth rates, dividends and earnings, for the dierent

12 206 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) equity markets under consideration. The ARIMA model is estimated for each country using maximum likelihood, assuming the normal distribution function. Our results show that both the AR and MA parameters are highly signicant. The magnitude of the AR coecient ranges across countries from (U.K. earnings) to (Japan dividends) and the MA coecient ranges from (U.K. earnings) to (Japan dividends). In general, the AR coecient is fairly large, and, in all cases, exceeds the MA coecient, reecting the fact that the observed dividend or earnings growth autocorrelations are positive and fairly small (the rst-order autocorrelation is about 6%, on average). Fig. 1 shows that the average autocorrelation function observed in the data is fairly close to that implied by the estimated ARIMA(1,0,1) specication for cash ow growth. Also, note that the R 2 in all cases is fairly small (about 5%). The magnitudes of the estimates of suggest that cash ow news aects long-run expectations of cash ow growth. As stated earlier, when =!, the ARIMA(1,0,1) process collapses to an i.i.d. process. Hence, we need to test the hypothesis that is statistically dierent from!. The test of this hypothesis is non-standard as, under the i.i.d. null, the parameters of the ARIMA(1,0,1) specication are separately identied only under the alternative. Fortunately, Andrews and Ploberger (1996) provide a likelihood ratio based test statistic for =!, where they show that this test statistic (referred to as the LR-test) reduces to two times the dierence between the unconstrained (ARIMA) and constrained (i.i.d.) log likelihood values. Additionally, they also provide the distribution for this test statistic and the associated critical values; for convenience, these are also reported in Table 3. In Table 3, we report the Andrews and Ploberger (1996) LR-test to evaluate the hypothesis that dividend (or earnings) growth rates are i.i.d. (equal roots). In almost all cases, the rejection of the null hypothesis is particularly strong, and hence appears to constitute rather sharp evidence against the null of equal roots. Second, the hypothesis that! i = 0 is sharply rejected in all cases. This implies that the AR(1) specication for cash ows is not supported in the data. 6 Persistence in expected growth rates is intimately related to shocks to the trend growth rate on the economy. One way to see this relationship is to rely directly on an extensively used alternative to decompose the level of the cash ow series into trend and cyclical components: the Hodrick Prescott (HP) lter (see Hodrick and Prescott, 1997). For comparison, the trend components extracted from the HP and ARIMA(1,0,1) lters are plotted in Fig. 2 for the U.S. First, for both dividends and earnings, it is evident from the gure that the dierences in implied trend components across the two lters are small. Further, Table 4 shows that the relative variances of the trend-component growth rates are small in size, but extremely persistent (with an AR(1) coecient of about 0.98, on average). This evidence suggests that the 6 In our estimation of the full model described below, to maintain parsimony in the number of estimated parameters, we restrict and! to be same across all equity markets. Note, in almost all cases univariate GMM estimates of the ARIMA(1,0,1) specication for cash ow are similar to the likelihood-based estimates provided in Table 3. Across all ve countries, pooled GMM estimates for and! for dividend growth are (S.E ) and (S.E ), respectively, and are (S.E ) and (S.E ), respectively, for earnings growth. The R 2 is about 3%.

13 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) United States HP Dividend HP Earnings ARMA Dividends, ω = 0.932, ARMA Earnings, ω = Fig. 2. Cash ow trends: Hodrickand Prescott (1997) and ARIMA. persistent component captured by the ARIMA(1,0,1) process is closely linked to a stochastic trend in the overall economy. A common issue that needs to be addressed in the context of equity valuation is the appropriate choice for the cash ow. It is well recognized that neither the measured dividends series nor the earnings series is perfect for valuation. In our data, the stochastic trends in dividends and earnings are comparable for France, Germany, and the U.K. However, there are important periods over which trends in dividends and earnings dier for the U.S. and Japan (see Fig. 2, for the U.S. example). For instance, the recent rise in U.S. equity prices is somewhat better mirrored in the earnings trend. The main results of our paper are driven by the persistence of shocks to the growth rate of the trend component of the cash ow, which should be far less susceptible to mismeasurement and the choice between dividends and earnings. When using earnings, our economic assumption is that the trend for earnings is identical to that of the true dividends. For these reasons, we employ both as measures of cash ows. Further, it will be shown below that our results, in terms of the implications for volatility and cross-correlations across markets, are not very dierent whether we use the measured dividends or earnings Economic models and fundamental values We employ the world capital asset pricing model (CAPM), where we assume the market portfolio is the world market equity index. We will show that this model reasonably describes excess returns for the menu of global equity indices under

14 208 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Table 4 Trend growth rates: autocorrelations and relative volatilities Dividends Earnings HP ARIMA HP ARIMA FR rel-vol BD rel-vol JP rel-vol U.K rel-vol U.S rel-vol is the rst-order autocorrelation of the trend growth rate. Note that rel-vol is equal to 2 (g tr)= 2 (g), the volatility of the trend growth rate relative to the volatility of the cash ow growth rate itself. The trend growth rates are constructed as the logged rst dierence of the respective trend level. HP refers to the two-sided Hodrickand Prescott (1997) Filter, for which we choose the HP smoothing parameter to be , which is the recommended value for the monthly frequency. The weights used in the construction of the ARIMA growth rates are! =0:929 for dividends and! =0:890 for earnings (the corresponding pooled MA parameter estimates presented in footnote 6). consideration. 7 A fairly direct alternative to using the CAPM would be to rely on the general equilibrium dynamic market based model discussed in Campbell (1996); however, this would signicantly increase the number of parameters to estimate. Furthermore, given that Campbell (1996) shows that the rst-order eects in determining riskpremia are associated with market riskand the evidence in support of the CAPM 7 There are many alternative models that one could use to model the expected riskpremia for international equity returns. For example, Adler and Dumas (1983), Dumas and Solnik(1995), and DeSantis and Gerard (1998) argue that exchange rate risks may contribute to the risk premium for equity returns; however, foreign exchange risk may have second-order eects relative to the market (see Ng, 2001). Bekaert and Hodrick (1992) consider a latent factor model, and Bansal et al. (1993) consider a non-linear APT model for jointly explaining equity and bond returns. As discussed later in the paper, when parsimony is highly valued, the static CAPM, at least for the ve equity returns under consideration, seems to be an adequate model for the riskpremia in our exercise. However, we also explore a two-factor world CAPM with exchange rate risk (not reported). While we nd evidence in favor of a time-varying price of foreign exchange risk, the general implications for asset volatility and correlations are almost identical to those presented for the one-factor model (available upon request).

15 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) that we provide below, the CAPM seems to be a reasonable model for capturing the cost of capital given that parsimony is highly valued. Assuming log normality of returns, conditional riskpremia are determined crosssectionally as follows: E t [r i;t+1 ]=r f; t + i 2 m;t 1 2 [2 i 2 m;t + 2 i ]; (10) where non-systematic return volatility is constant. 8 By assumption, the world investor is marginal to all the developed equity markets we consider, and the price of risk is the (scaled) variance of the world market return. Note that r f; t is the real risk-free rate and m;t 2 is the conditional variance of the world market portfolio. The beta of the asset is i, and 2 i is the conditional variance of the non-systematic part of the asset return. The market price of risk is governed by the parameter. To solve for fundamental values, we assume the following dynamic processes underlying the cash ow growth rates, market volatility, and the risk-free rate. First, as described above, we assume the that the growth rate is an ARMA(1,1), from which it follows that g m;t+1 = m + x m;t + m;t+1 ; (11) where x m;t is the expected world cash ow growth rate, as dened in Eq. (5). Analogously, we also assume that each country s cash ow growth rate is g i;t+1 = i + x i;t + i;t+1 ; i;t+1 = i m;t+1 + v i;t+1 ; (12) where x i;t is country i s expected cash ow growth rate, as stated in Eq. (5), and i is a cash ow beta describing the relationship between country i s cash ow growth and that of the world. Next, we model the time-varying world market return volatility using a GARCH(1,1)-M model for the world market return (see Bollerslev et al., 1988). Using the fact that m =1; R m;t+1 R f; t = 2 m;t + m;t+1 ; (13) 2 m;t = & +( + ) 2 m;t 1 + ( 2 m;t 2 m;t 1); (14) where m;t+1 is distributed with mean zero and variance m;t. 2 This implies that the time-varying price of riskfollows a rst-order autoregressive process, with the autoregressive parameter value equal to ( + ): Further if ( + ) is large (but 1), the systematic time-varying price of riskwill be persistent and have signicant impact on present values and on asset cross-correlation. Finally, we assume the real risk-free rate evolves according to an AR(1) process as follows: r f;t+1 = r f; t + rf ;t+1: (15) 8 Note in Eq. (10), the component 1 2 [2 i 2 m;t +2 i ] is due to the assumption of log normality (see Campbell, 1996), or in continuous time is an Ito adjustment for log returns.

16 210 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) Given the preceding state variable dynamics in Eqs. (11) and (13) (15), the present value implications of the model can be evaluated. We conjecture a solution for z i;t, the log price dividend (earnings) ratio, as follows: z i;t = A i;0 + A i;1 x i;t + A i;2 2 m;t + A i;3 r f; t ; (16) where A i;0 ;:::;A i;3 are yet to be determined. Exploiting the processes for the state variables, including the ARIMA process for the cash ow growth rate, and matching the coecients, leads to the following solution for the unknown coecients (see Appendix A): A i;1 = 1 (1 i;1 i ) ; A i;2 = [ i i ] ( i;1 ( + ) 1) and A i;3 = 1 ( i;1 1 1) : (17) Additionally, the fundamental betas, i, are also endogenously determined by the underlying parameters (for details see Appendix A). Assuming all AR(1) coecients are positive and 1, it follows that A i;2 and A i;3 are negative, whereas A i;1 is positive. A rise in the cost of capital lowers the present value, and a rise in expected growth rates raises it. Further, as noted earlier, persistent changes in these variables have a much greater impact; this is now more clearly understood, as values close to one for the AR(1) coecients within the autoregressive structure will yield large values for the coecients of the solution. We have an analytical expression for the fundamental values, z i;t, in terms of the real risk-free rate, the market price of risk, and the expected cash ow growth rate Fundamental market betas and measures of integration The fundamental market beta of an asset is endogenously determined. The critical input is i, which is the cash ow beta describing the relationship between a given country s cash ow growth and that of the aggregate market. To see this more clearly, suppose that m;1 and i;1, the approximation parameters, are equal, and ignore the eect of the risk-free rate on the fundamental market beta (for the complete expression for the fundamental asset beta see Eq. (A.13) in Appendix A). 9 In this case, the fundamental return beta satises i = C mc i i 2 ( m ) C 2 m 2 ( m ) = C i i C m ; (18) where C i {(1 + i;1 (!)A i;1 )}, which is the return elasticity with respect to cash ow news. This can be seen by examining the one step ahead innovation in the return: r i;t+1 E t [r i;t+1 ] = {(1 + i;1 (!)A i;1 )} i;t+1 + i;1 A i;2 ;t+1 + i;1 A i;3 rf ;t+1: (19) News regarding returns is composed of cash ow news, i;t, news regarding the market riskpremium, ;t, and that of the risk-free rate, rf ;t. The key components of the 9 In practice, these assumptions are actually reasonable. i;1 s are very close to one another. Additionally, 2 ( rf ) is extremely small, and its inclusion matters little quantitatively.

17 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) fundamental market beta are the cash ow beta (i.e., ) and the asset return elasticity with respect to cash ow news C i. Our approach to deriving the restrictions for the fundamental market beta is identical to Campbell and Mei (1993) however, unlike their paper we will directly use observed cash ow information to empirically restrict the fundamental market beta of the asset. The fundamental asset values and returns reported in the paper are equity prices and returns derived using the fundamental solution for z i;t, and the complete solution for the fundamental beta. Ammer and Mei (1996) decompose the asset return news into components related to future cash ow news and cost of capital news. They measure economic integration by the cross-correlation among the cash ow news components, and nancial integration as the cross-correlation in news regarding cost of capital components. They do not provide any mechanism to connect these two measures of integration, and point out that nancial integration can generate return correlation through correlation among the equity premium components, despite economic segmentation (i.e., near-zero correlation among the cash ow news components). Our fundamental valuation method directly addresses this important issue. If a country s cash ow news has close to zero correlation with aggregate market cash ow news (i.e., if i 0), then the implied fundamental market betas will also be close to zero (see Eq. (18)) and ex post returns across markets will have little correlation. Stated dierently, if there is little economic integration among countries, then our fundamental ex post returns will imply little nancial integration among them as well. Hence, our approach provides a direct economic linkbetween these measures of integration Conditional second moments Time-varying market risk in our model implies that conditional cross-correlation across equity returns can also be time varying so that the conditional cross-correlation between two positive beta assets will typically rise as the market volatility increases. The conditional cross-correlation, under the assumption that non-systematic riskis homoskedastic, is given by the expression ( i j m;t) 2 : (20) i 22 m;t + 2 i j 22 m;t + 2 j As mentioned earlier, Longin and Solnik(1995), amongst others, have documented that during periods of high uncertainty, conditional correlations across markets are high. The above expression allows us to quantitatively measure the degree to which our fundamental valuation model can duplicate this important empirical feature as well. Note that if the fundamental beta of one of the assets is zero, then the conditional correlation of asset returns is zero as well. Finally, to further develop the intuition regarding the fundamental sources of risk underlying our model, we provide a tight linkbetween the conditional volatility on the aggregate world cash ow process and the conditional volatility of the world market portfolio. This linkalso allows us to connect the world market s cash ow volatility to the market risk premium, and interpret movements in the market risk premium in Section 6. Given the fundamental solution, consider the innovation in the market return,

18 212 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) ignoring the risk-free rate contribution for exposition: r m;t+1 E t [r m;t+1 ]={(1 + m;1 ( m! m )A m;1 )} m;t+1 + m;1 A m;2 ;t+1 : (21) If we assume that the shocks to volatility, ;t+1, are homoskedastic and uncorrelated with cash ow news, then the conditional variance of the market return can be expressed as follows: m;t 2 =( m;1 A m;2 ) {(1 + m;1 ( m! m )A m;1 )} 2 2 m;t {( =( m;1 A m;2 ) )} 2 m;1( m! m ) 2 1 m;1 m m;t: (22) Conditional volatility of the world market portfolio is simply a magnied version of the world market cash ow conditional volatility, 2 m; t. This provides a fundamental justication for the statistical process assumed in Campbell and Hentschel (1992), linking excess returns to dividend volatility. Global cash ow uncertainty determines the volatility of the market portfolio, and hence, return cross-correlations as well. Therefore, the above equation, in conjuction with Eq. (19), suggests that ex post returns in each economy are inuenced by news regarding cash ows, i;t+1, and changing economic uncertainty in global output (i.e., cash ows), 2 m; t Latent stochastic volatility An important input into the model, particularly in determining the equity price level, is the market price of risk, for which the GARCH model considered above is one particular specication. In this section, we present an alternative latent stochastic volatility specication for the market price of risk (see Taylor, 1986). In this model, we continue to assume that the conditional market CAPM model determines the ex ante cost of capital, but the time-varying volatility of the world market portfolio is latent. Exploiting the valuation restrictions for the world market portfolio, we can extract this latent volatility as a linear function of the observed world price dividend ratio and expected cash ow growth rates. The time-varying cost of capital for each asset under consideration is now determined by its market beta and the extracted time-varying market volatility. We rst assume that the latent market volatility, as in the case with the GARCH(1,1) specication, follows an AR(1) process m;t+1 2 = m;t 2 + ;t+1 : (23) The world cash ow dynamics are identical to what we have assumed thus far, an ARIMA(1,0,1) process. Given this, the solution for the price dividend ratio for the world market portfolio is z m;t = A m;0 + A m;1 x m;t + A m;2 m;t 2 + A m;3 r f; t ; (24) where z m;t is the log price dividend ratio for the world market, x m;t is the expected growth rate of earnings for the world, and 2 m;t is the market price of risk, which is latent and is not directly observable. Exploiting the fundamental solution for the world

19 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) market portfolio, it follows that the latent volatility can be extracted by inverting the valuation restriction m;t 2 = 1 (z m;t A m;0 A m;1 x m;t A m;3 r f; t ): (25) A m;2 Exploiting Eq. (25) for measuring the latent volatility, we can rewrite the fundamental solution for all other assets as follows: z i;t = A i;0 + A i;1 x i;t + A i;2 2 m;t + A i;3 r f; t : (26) The economic intuition in this model specication is identical to the GARCH case, with one dierence. The procedure discussed above for extracting latent volatility ensures that the fundamental value for the world market portfolio equals its observed price in the data. In the latent stochastic volatility model we are exhausting the world market portfolio s present value restrictions to extract the latent volatility process, hence the present value implication for the world market portfolio will be exactly satised. The fundamental values for all other assets, as in the case with the GARCH specication, are determined by the market volatility and cash ow dynamics. This relative valuation approach is similar to Bossaerts and Green (1989), where they construct a model for relative asset valuation. Fundamental betas are endogenously determined in exactly the same manner as before. 4. Econometric method and empirical results 4.1. GMM estimation The varied set of economic and statistical restrictions presented above naturally maps into the generalized method of moments (GMM) frameworkfor estimation (Hansen, 1982). The precise orthogonality conditions we exploit and the construction of the robust VARHAC (denhaan and Levin, 1996) weighting matrix are detailed in Appendix A. For the unrestricted model we consider, we estimate return i s, cash ow is (betas), the autoregressive coecient for the risk-free rate, 1, and the parameters associated with the GARCH-M process for the market return, =(; ; ). Additionally, we estimate the parameters associated with the ARIMA cash ow dynamics. To keep the number of estimated parameters manageable, we restrict the cash ow parameters, and!, to be identical across markets. This restriction is not rejected in the data for either dividends or earnings based upon the GMM test of overidentifying restrictions associated with the pooled results presented in Table 3. We also estimate the ARIMA parameters for the world cash ow growth rates. Hence, the parameter vector is ( ; ; 1 ;;!; m ;! m ). In sum, the stochastic processes we estimate are as follows: R i;t+1 R f; t = i + i [R m;t+1 R f; t ]+ i;t+1 ; R m;t+1 R f; t = 2 m;t + m;t+1 ;