Market Efficiency, Asset Returns, and the Size of the Risk Premium in Global Equity Markets

Transcription

1 Market Efficiency, Asset Returns, and the Size of the Risk Premium in Global Equity Markets Ravi Bansal and Christian Lundblad January 2002 Abstract An important economic insight is that observed equity prices must equal the present value of the cash flows associated with the equity claim. An implication of this insight is that present values of cash flows 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 five major equity markets, the U.S., the U.K., France, Germany, and Japan. We present evidence that cash flow growth rates contain a small predictable long-run component; this feature, in conjuction 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 specific versions of the model can, in many cases, capture observed price levels. Our evidence suggests that the ex-ante risk premium on the global market portfolio has dropped considerably we show that this fall in the risk premium is related to a decline in the conditional variance of global real cash flow growth rates. Duke University (Bansal) and Kelley School of Business, Indiana University (Lundblad). We have benefited from conversations with Tim Bollerslev, Ron Gallant, George Tauchen, and seminar participants of the 2000 WFA, London Business School, Columbia University, Duke University, Indiana University, Stockholm School of Economics, the 2001 Triangle Econometrics Conference, and Wharton. Send correspondence to: Ravi Bansal, Fuqua School of Business, Duke University, Durham, NC Phone: (919) , (Bansal): rb7@mail.duke.edu, (Lundblad): clundbla@indiana.edu. An earlier version of this paper was titled, Market Efficiency, Fundamental Values, and Asset Returns in Global Equity Markets. All data employed in this paper are available at

2 1 Introduction An important economic insight is that observed equity prices should equal the present value of the cash flows associated with the ownership of the equity claim. The work of Shiller (1981), LeRoy and Porter (1981), West (1988), and Campbell and Shiller (1987, 1988a,1988b), however, poses a challenge to this insight. These authors document the volatility puzzle quantitatively, equity prices are far too volatile to be justified as present values of fundamental cash flows. This result underscores the key feature of the data that cash flow 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 flow growth rates. This feature poses an additional quantitative challenge to present values, and is labeled the correlation puzzle. Present values of the cash flows are determined by the time-series dynamics of the expected cash flow 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 flow 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 first considering the role of the cash flow dynamics, followed by that of fluctuations in the cost of capital. In the data, real growth rates have near zero auto-correlation, hence it is common to assume that cash flow growth rates are i.i.d.. In addition to this assumption, if cost of capital is constant, then news regarding cash flow 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-flow 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 flow growth rates can be modeled as an integrated process (more precisely, an ARIMA(0,1,1) process). It is important to note that in finite 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 different. Expected growth rates in this specification contain a unit root, and consequently, news regarding growth rates have large effects on dividend yields as they permanently alter future expected growth rates. 1 1 At a firm level, it is well documented that cash flow news leads to significant price reaction (see Easton and Zmijewski (1989)) 1

3 Campbell, Lo, and MacKinlay (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 flow 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 flow growth correlations. However, this is unlikely to justify return cross-correlation, as growth rate correlations across economies are quite small. One factor that may account for high return correlation is fluctuations in global risk premia a source of common fluctuations in asset prices. This view is also consistent with Ammer and Mei (1996), who document that much of the asset return covariation between national stock markets is related to news about future risk premia. Indeed, relying on a simple CAPM-GARCH specification, as in Bollerslev, Engle, and Wooldridge (1988), we show that fluctuating global risk premia in conjunction with the assumed cash flow dynamics can reproduce the observed ex-post return cross-correlations and asset return volatility. Further, we show that the persistent component in cash flows is also needed for duplicating asset return cross-correlations in the absence of this, asset price fluctuations are dominated by common cost of capital fluctuations, 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 flow news and expected returns to measure economic and financial integration, respectively, across markets. However, they do not provide any economic mechanism to link these two measures of integration -in this paper, we do provide such a mechanism and show that if there is little economic integration, then financial 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 flow growth rate dynamics and the specification for fluctuating global risk premia, our valuation model can account for about 70-80% of the volatility 2

4 of asset prices (change in dividend yield or returns) and cross-correlations in asset returns. The more standard VAR methods of modeling cash flow growth rates and expected rates of returns to compute present values (as in Campbell and Shiller (1988a)) leads 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 finite samples, this approach fails to capture the persistent component in cash flow growth rates which leads to large asset return variability, and hence also affects 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-1980 s) 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 specific 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) and 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 flow growth of this benchmark asset. We find that modeling the systematic risk in this manner provides a significant improvement over the GARCH specification. The latent volatility model matches the observed equity prices quite well, and captures an economically significant portion of the volatility (about 80%). Additionally, it justifies almost all of the observed cross-correlation, other than for Japan. In contrast to the GARCH specification, this model suggests that the aggregate risk premium in the global economy has fallen significantly in the last decade to about 2%. This difference has important effects on measured fundamental values. We also show that much of the fall in the latent systematic risk can be attributed to a fall in the conditional world market cash flow variance. In parallel and independent work, Fama and French (2000) back out the risk premia from the US equity index values, and also argue that the market risk premium has fallen. In independent papers, Dumas, Harvey, and Ruiz (2000) and Chue (2000) focus on the cross-correlation 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 different cash flow dynamics, their results and conclusions differ 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 3

5 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 flow model and the evidence supporting it, and lays down the specific fundamental restrictions implied by the model. Section 4 discusses the estimation methodology, and provides the empirical findings and diagnostics. Section 5 provides evidence on the valuation implications of our model, and 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 five developed equity markets: France, Germany, Japan, the United Kingdom, the United States, 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 defined as follows: 1+R i,t+1 = P i,t+1 P i,t (1 + DY i,t+1 ) (1) 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 the previous year divided by the current price level, DY t = ; this is similar to the approach taken in Heaton j=0 D t j P t (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 12 th order autoregressive seasonal; our empirical 2 The Datastream World Market Index return has a correlation of 0.99 with the Morgan-Stanley world index (MSCI) return. 4

6 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 flow (i.e., dividends or earnings) is the log of the gross growth rate of the cash flow under consideration. When converting nominal variables to real, we simply subtract from the relevant variable in country i the seasonally adjusted CPI inflation rate (i.e. log of gross inflation) 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 inflation in the U.S., taken here to be the inflation expectation implied by an ARIMA(1,0,1) model on the inflation series. 3 InTable1,wereportsummarystatistics 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 Figure 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 first 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 Tables 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 first difference 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 different markets. Also, note that ex-post real equity return cross-correlations are of similar magnitude as observed for the first difference 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 difference to our empirical results. We have also constructed the ex-ante real interest rate using different methods such as removing the trailing 12-month inflation rate. The implied real interest rate series is very similar to the one backed out using the ARIMA(1,0,1) series. 5

7 3 Present Value and Asset Prices It is well recognized that the fundamental value of the asset is the present value of the cash flow 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 + κ i,0 + κ 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. κ i,0 and κ 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 = κ 0 + E t [ κ j 1 κ 1[g t+1+j r t+1+j ]] (3) 1 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 flow growth rates and the ex-ante rate of return on equity. To derive fundamental values of assets, we first model cash flow 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 flow 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 flow growth deflated by a discount factor, and use boot-strapping procedures to solve for fundamental present values. Campbell and Shiller (1988a, 1988b) 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. 4 1 κ i,1 = and κ (1+exp(d i p i )) i,0 = log(κ i,1 ) (1 κ 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. j=0 6

8 3.1 Cash Flow Dynamics In this section, we provide the description of the assumed time-series model for the growth rate of cash flows, which we demonstrate can have large effects on implied equity prices, volatility, and asset beta s. Statistically, we posit that cash flow 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 flow levels into exponentially smoothed stochastic trend and autoregressive cyclical components. 5 Let g i,t be the real growth rate of cash flows for equity claim i. We assume that the process for g i,t satisfies g i,t =(1 ω i )θ i + ρ i g i,t 1 + η i,t ω i η i,t 1 (4) The variable that affects present values is the conditional mean of the growth rate. For the ARIMA(1,0,1) process the conditional mean x i,t equals (ρ i ω i ) g (1 ω i L) i,t, wherel is the standard lag operator. Consequently, the ARIMA(1,0,1) process can be more conveniently written as g i,t = θ i + x i,t 1 + η i,t (5) It is assumed that g is stationary, and hence ρ and ω (where country subscript i is suppressed) are less than one in absolute value. Using equation (5), it follows that x i,t itself follows an AR(1) process, x i,t =(ρ i ω i )θ i + ρ i x i,t 1 +(ρ i ω i )η i,t (6) The parameter ρ determines the degree of persistence, and ω is the smoothing parameter that affects the construction of x i,t. It is also worth noting two special cases that the ARIMA(1,0,1) representation accommodates. If ρ = ω, 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 ω =0,g follows a standard AR(1) process. To develop intuition regarding the implications of an ARIMA(1,0,1) process for cash flow 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 ]] = ρ n 1 (ρ ω)η t (7) 5 Alternatively, following Kasa (1992), one could model cash flows as determined by a common trend. This specification is closely related to the world business cycle described by Dumas, Harvey, and Ruiz (2000). To explore this specification, we perform cointegration tests, but we do not find 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). 7

9 Economically, equation (7) implies that rational agents may significantly revise their longrun expected growth rates so long as ρ ω 0. In the extreme case when ρ = ω (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 ρ. If ρ = 1, the revision in expectations is identical across all horizons (as in Barsky and DeLong (1993)). When ρ is less than one, the revision is larger for shorter horizons, and almost zero for very long horizons. For the case where the difference between ρ and ω is small and positive and ρ 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 flow growth rate process in equation (6), consider the implications for the log price dividend ratio, z, in equation (3) (assuming that the cost of capital is constant for now): z t = z κ 1 ρ (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 ρ 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 news is: z t E t 1 [z t ]=(ρ ω)η t n=1 κ n 1ρ n 1 = (ρ ω)η t 1 κ 1 ρ (9) Again, when ρ is close to one and larger than ω, the impact of cash flow news on the innovation to the price dividend ratio can be very large even though the ex-post cash flow process seems very close to an i.i.d. process in a finite sample. In contrast, in the i.i.d. case (ρ = ω), 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 flows 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 finite samples, the persistent component of cash flows which is important for understanding asset price volatility Empirical Evidence Regarding the Cash Flow Dynamics In Table 3, we present evidence for an ARIMA(1,0,1) process for both of the two alternative measures of cash flow growth rates, dividends and earnings, for the different equity 8

10 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 significant. The magnitude of the AR coefficient ranges across countries from (U.K. earnings) to (Japan dividends) and the MA coefficient ranges from (U.K. earnings) to (Japan dividends). In general, the AR coefficient is fairly large, and, in all cases, exceeds the MA coefficient, reflecting the fact that the observed dividend or earnings growth autocorrelations are positive and fairly small (the first order autocorrelation is about 6%, on average). Figure 1 shows that the average autocorrelation function observed in the data is fairly close to that implied by the estimated ARIMA(1,0,1) specification for cash flow growth. Also, note that the R 2 in all cases is fairly small (about 5%). The magnitudes of the estimates of ρ suggest that cash flow news affects long-run expectations of cash flow growth. As stated earlier, when ρ = ω, the ARIMA(1,0,1) process collapses to an i.i.d. process. Hence, we need to test the hypothesis that ρ is statistically different from ω. The test of this hypothesis is non-standard as, under the i.i.d null, the parameters of the ARIMA(1,0,1) specification are separately identified only under the alternative. Fortunately, Andrews and Ploberger (1996) provide a likelihood ratio based test statistic for ρ = ω, where they show that this test statistic (referred to as the LR-test) reduces to two times the difference 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 ω i =0is sharply rejected in all cases. This implies that the AR(1) specification for cash flows 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 6 In our estimation of the full model described below, to maintain parsimony in the number of estimated parameters, we restrict ρ and ω to be same across all equity markets. Note, in almost all cases univariate GMM estimates of the ARIMA(1,0,1) specification for cash flow are similar to the likelihood based estimates provided in Table 3. Across all five countries, pooled GMM estimates for ρ and ω 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%. 9

11 alternative to decompose the level of the cash flow series into trend and cyclical components: the Hodrick-Prescott (HP) filter (see Hodrick and Prescott (1997)). For comparison, the trend components extracted from the HP and ARIMA(1,0,1) filters are plotted in Figure 2 for the U.S. First, for both dividends and earnings, it is evident from the figure that the differences in implied trend components across the two filters 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) coefficient of about 0.98, on average). This evidence suggests that the 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 flow. 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 differ for the U.S. and Japan (see Figure 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 flow, 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 flows. Further, it will be shown below that our results, in terms of the implications for volatility and cross-correlations across markets, are not very different whether we use the measured dividends or earnings. 3.2 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 consideration. 7 A fairly direct 7 There are many alternative models that one could use to model the expected risk premia 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 effects relative to the market (see Ng (2001)). Bekaert and Hodrick (1992) consider a latent factor model, and Bansal, Hsieh, and Viswanathan (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 five equity returns under consideration, seems to be an 10

12 alternative to using the CAPM would be to rely on the general equilibrium dynamic market based model discussed in Campbell (1996); however, this would significantly increase the number of parameters to estimate. Furthermore, given that Campbell (1996) shows that the first order effects in determining risk premia are associated with market risk and the evidence in support of the CAPM 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 risk premia are determined cross-sectionally as follows: E t [r i,t+1 ]=r f,t + β i λσm,t [β2 i σ2 m,t + σ2 ɛ 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 σm,t 2 is the conditional variance of the world market portfolio. The beta of the asset is β i,andσɛ 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 λ. To solve for fundamental values, we assume the following dynamic processes underlying the cash flow 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 = θ m + x m,t + η m,t+1 (11) where x m,t is the expected world cash flow growth rate, as defined in equation (5). Analogously, we also assume that each country s cash flow growth rate is; g i,t+1 = θ i + x i,t + η i,t+1 (12) η i,t+1 = ν i η m,t+1 + v i,t+1 where x i,t is country i s expected cash flow growth rate, as stated in equation (5), and ν i is a cash flow beta describing the relationship between country i s cash flow growth and the world s. Next, we model the time-varying world market return volatility using a adequate model for the risk premia in our exercise. However, we also explore a two-factor world CAPM with exchange rate risk (not reported). While we find 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). 8 Note in equation (10), the component 1 2 [β2 i σ2 m,t + σɛ 2 i ] is due to the assumption of log normality (see Campbell (1996)), or in continuous time is an Ito adjustment for log returns. 11

13 GARCH(1,1)-M model for the world market return (see Bollerslev, Engle, and Wooldridge (1988)). Using the fact that β m =1, R m,t+1 R f,t = λσ 2 m,t + ε m,t+1 (13) σ 2 m,t = ς +(δ + γ)σ 2 m,t 1 + γ(ε 2 m,t σ 2 m,t 1) (14) where ε m,t+1 is distributed with mean zero and variance σm,t. 2 This implies that the timevarying price of risk follows a first order autoregressive process, with the autoregressive parameter value equal to (γ +δ). Further if (γ +δ) is large (but less than one), the systematic time-varying price of risk will be persistent and have significant 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 = φ 0 + φ 1 r f,t + ɛ rf,t+1 (15) Given the preceding state variable dynamics in equations (11), (13), (14), and (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 σ 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 flow growth rate, and matching the coefficients, leads to the following solution for the unknown coefficients (see Appendix): A i,1 = 1 (1 κ i,1 ρ i ),A i,2 = [λβ i 1 2 β2 i ] (κ i,1 (γ+δ) 1), and A i,3 = 1 (κ i,1 φ 1 1) (17) Additionally, the fundamental betas, β i, are also endogenously determined by the underlying parameters (for details see Appendix). Assuming all AR(1) coefficients are positive and less than one, 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) coefficients within the autoregressive structure will yield large values for the coefficients 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 flow growth rate. 12

14 3.2.1 Fundamental Market Beta s and Measures of Integration The fundamental market beta of an asset is endogenously determined. The critical input is ν i, which is the cash flow beta describing the relationship between a given country s cash flow growth and that of the aggregate market. To see this more clearly, suppose that κ m,1 and κ i,1, the approximation parameters, are equal, and ignore the effect of the risk free rate on the fundamental market beta (for the complete expression for the fundamental asset beta see equation 40 in the Appendix). 9 In this case, the fundamental return beta satisfies β i = C mc i ν i σ 2 (η m ) C 2 mσ 2 (η m ) = C iν i C m (18) where C i {(1 + κ i,1 (ρ ω)a i,1 )}, which is the return elasticity with respect to cash flow 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 + κ i,1 (ρ ω)a i,1 )}η i,t+1 + κ i,1 A i,2 ɛ σ,t+1 + κ i,1 A i,3 ɛ rf,t+1 (19) News regarding returns is composed of cash flow news, η i,t, news regarding the market risk premium, ɛ σ,t, and that of the risk-free rate, ɛ rf,t. The key components of the fundamental market beta are the cash flow beta (i.e., ν) and the asset return elasticity with respect to cash flow 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 flow 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 flow news and cost of capital news. They measure economic integration by the cross-correlation among the cash flow news components, and financial 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 financial integration can generate return correlation through correlation among the equity premium components, despite economic segmentation (i.e. near-zero correlation among the cash flow news components). Our fundamental valuation method directly addresses this important issue. If a country s cash flow news has close to zero correlation with aggregate market cash flow news (i.e, if ν i 0), then the implied fundamental market beta s will also be close to zero 9 In practice, these assumptions are actually reasonable. κ i,1 s are very close to one another. Additionally, σ 2 (ɛ rf ) is extremely small, and its inclusion matters little quantitatively. 13

15 (see equation (18)) and ex-post returns across markets will have little correlation. Stated differently, if there is little economic integration among countries, then our fundamental expost returns will imply little financial integration among them as well. Hence, our approach provides a direct economic link between 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 risk is homoskedastic, is given by the expression (β i β j σm,t) 2 (20) β 2 i σm,t 2 + σɛ 2 i β 2 j σm,t 2 + σɛ 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 link between the conditional volatility on the aggregate world cash flow process and the conditional volatility of the world market portfolio. This link also allows us to connect the world market s cash flow volatility to the market risk premium, and interpret movements in the market risk premium in section 6 below. Given the fundamental solution, consider the innovation in the market return, ignoring the risk-free rate contribution for exposition: r m,t+1 E t [r m,t+1 ]={(1 + κ m,1 (ρ m ω m )A m,1 )}η m,t+1 + κ m,1 A m,2 ɛ σ,t+1 (21) If we assume that the shocks to volatility, ɛ σ,t+1, are homoskedastic and uncorrelated with cash flow news, then the conditional variance of the market return can be expressed as follows: σm,t 2 = (κ m,1 A m,2 ) 2 σσ 2 + {1+κ m,1 (ρ m ω m )A m,1 )} 2 ση 2 m,t (22) = (κ m,1 A m,2 ) 2 σσ 2 + {1+κ m,1(ρ m ω m ) )} 2 ση 2 1 κ m,1 ρ m,t m 14

16 Conditional volatility of the world market portfolio is simply a magnified version of the world market cash flow conditional volatility, ση 2 m,t. This provides a fundamental justification for the statistical process assumed in Campbell and Hentschel (1992), linking excess returns to dividend volatility. Global cash flow uncertainty determines the volatility of the market portfolio, and hence, return cross-correlations as well. Therefore, the above equation, in conjuction with equation (19), suggests that ex-post returns in each economy are influenced by news regarding cash flows, η i,t+1, and changing economic uncertainty in global output (i.e., cash flows), ση 2 m,t. 3.3 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 specification. In this section, we present an alternative latent stochastic volatility specification 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 flow 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 first assume that the latent market volatility, as in the case with the GARCH(1,1) specification, follows an AR(1) process σ 2 m,t+1 = τ 0 + τ 1 σ 2 m,t + ε σ,t+1 (23) The world cash flow 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 σm,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 λσm,t 2 is the market price of risk, which is latent and is not directly observable. Exploiting the fundamental solution for the world market portfolio, it follows that the latent volatility can be extracted by inverting the valuation restriction σm,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 15

17 Exploiting equation (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 σm,t 2 + A i,3 r f,t (26) The economic intuition in this model specification is identical to the GARCH case, with one difference. 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 satisfied. The fundamental values for all other assets, as in the case with the GARCH specification, are determined by the market volatility and cash flow dynamics. This relative valuation approach is similar to Bossaerts and Green (1989), where they construct a model for relative asset valuation. Fundamental beta s 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) framework for estimation (Hansen 1982). The precise orthogonality conditions we exploit and the construction of the robust VARHAC (den Haan and Levin (1996)) weighting matrix are detailed in the Appendix. For the unrestricted model we consider, we estimate return β i s, cash flow ν is (beta s), the autoregressive coefficient for the risk free rate, φ 1, and the parameters associated with the GARCH-M process for the market return, ψ =(λ, γ, δ). Additionally, we estimate the parameters associated with the ARIMA cash flow dynamics. To keep the number of estimated parameters manageable, we restrict the cash flow parameters, ρ and ω, 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 flow growth rates. Hence, the parameter vector is (ψ,ν,φ 1,ρ,ω,ρ m,ω m ). In sum, the stochastic processes we estimate are as follows: R i,t+1 R f,t = α i + β i [R m,t+1 R f,t ]+ɛ i,t+1 16

18 R m,t+1 R f,t = λσm,t 2 + ε m,t+1 σm,t 2 = ς +(δ + γ)σ2 m,t 1 + γ(ε2 m,t σ2 m,t 1 ) r f,t+1 = φ 0 + φ 1 r f,t + ɛ rf,t+1 g i,t+1 = θ i + x i,t + η i,t+1 η i,t+1 = ν i η m,t+1 + v i,t+1 z i,t+1 = A i,1 (ρ 1)x i,t + A i,2 (γ + δ 1)σm,t 2 + A i,3 (φ 1 1)r f,t + e i,t+1 (27) where e i,t+1 is the forecast error associated with the first difference of the log price dividend (earnings) ratio implied by the equilibrium solution, and the fundamental beta is a restricted function of the estimated parameters. In the last equation, as the A i s are also restricted by the estimated parameters, these forecast errors represent important present value restrictions the model must confront in the data. 10 There is every reason to believe that equity prices contain valuable information regarding cash flow dynamics. Consequently, in addition to the information provided by the cash flow growth rates, our estimation strategy exploits the information contained in z i,t+1 to identify the parameters of the cash flow process. There are 18 parameters to estimate, with 41 moment conditions. GMM distribution theory (Hansen (1982)) provides an asymptotic jointly normal distribution for the parameter estimates. Note that in the interests of parsimony, as in Campbell and Shiller (1988b), we de-mean the growth rate variables to reduce the number of parameters (intercepts) to be estimated. For the latent stochastic volatility model, we employ the analogous set of restrictions implied by this structure using GMM, given the extracted market volatility from equation (25) and the definitions of the solution coefficients. The precise orthogonality conditions we exploit are also detailed in the Appendix. There are 40 moment conditions and 17 parameters. We document the parameter estimates and present value implications next. 4.2 CAPM-GARCH: Parameter Estimates First, in Table 5 we provide evidence that the CAPM adequately captures the risk premia of the international assets under consideration; a standard intercept test that the α s for each index are jointly equal to zero is not rejected with a χ 2 5 = In Tables 6, we present the unrestricted GMM parameter estimates for the world CAPM-GARCH model. In general, 10 Note in estimation, we impose the fundamental valuation restriction on the first difference of the log price dividend (earnings) ratio (i.e., z i,t+1 ) as this quantity is far less persistent relative to its level. In practice, the qualitative implications of using the first difference relative to the level of z i,t+1 for the estimates, and hence our results, are small. 17

19 the parameter estimates are precise. As the cost of capital parameter estimates do not substantially differ, we focus on the estimates obtained when earnings are employed as the cash flow measure. The AR(1) parameter (standard error) on the real risk free rate is (0.016). The market volatility GARCH parameters are precisely estimated with, γ = (0.005) and δ = (0.006), which together imply that the AR(1) parameter (i.e., γ +δ) on the market volatility is 0.989, suggesting a high degree of persistence in volatility. The risk aversion parameter, λ, is imprecisely estimated at 3.49 (2.27) 11. The unconditional standard deviation of the market price of systematic risk λσm,t 2 is only , which is clearly very small. The parameters governing the cash flow dynamics are also precisely estimated. The autoregressive parameter, ρ, for dividend and earnings growth is (0.010) and (0.009), respectively, and the moving average parameter, ω, is (0.017) and (0.013), respectively. The large autoregressive parameter suggests that the component determining the expected cash flow growth rate is very persistent for both dividends and earnings. Note that these estimates do not differ significantly from those discussed earlier in Section 3.1.1, where we only used information contained in cash flows to estimate these parameters. However, it is important to note that equity prices provide additional information regarding the process governing cash flow growth rates. Consequently, any distinction in the parameter values reflects the additional information that equity prices bear on the estimation through the fundamental present value restrictions. Using the GMM test of overidentifying restrictions, the world CAPM-GARCH model is statistically rejected with a χ 2 23 of for dividends and for earnings. However, our diagnostics reported in Table 9 suggest the model is capable of explaining much of the observed asset volatility and cross-correlation. Additionally, given the over-rejection associated with the GMM test documented in Ferson and Foerster (1994) and Hansen, Heaton, and Yaron (1996), caution is required. Table 7 presents the A i s implied by our estimation. While the solution coefficients on the expected cash flow growth rate and the real risk free rate are significant, the coefficient on the market volatility is not due to the documented imprecision associated with the estimation of λ. In Table 8, we report fundamental return beta s implied by the estimated parameters, Ψ, as determined in equation (18). We perform parameter restriction tests that the estimated fundamental beta s are jointly equivalent to the traditional beta s. In all cases, the hypothesis 11 Lundblad (2000) demonstrates the imprecision with which λ is estimated in a univariate GARCH-M framework. 18

20 of equivalence is rejected with a p-value of In this sense, the fundamental valuation restrictions, primarily for the beta s are rejected. However, while the fundamental market beta s are smaller than the traditional, we are able to explain on average roughly 60% (90%) of the empirically observed market beta s with dividends (earnings). In a related context, Campbell and Mei (1993) decompose the asset return process itself to explain the components of the estimated beta, backing out that part of the beta which can be attributed to cash flow news. However, they do not directly employ cash flow information. To the best of our knowledge, this is the first direct empirical evidence based on observed cash flow information regarding asset beta s. Further, note that mismeasurement of cash flows can significantly affect fundamental beta estimates, while having secondary effects on the constructed fundamental value. As can be seen in the present value solution in equation (3.2), the persistent component in cash flows is a critical input into the fundamental value, and is far less likely to be sensitive to cash flow mismeasurement. 12 The expected cash flow growth rate, the market price of risk, and the risk free rate (the relevant state variables) are quite persistent. Consequently, the fundamental solution for z i is also quite persistent. The first order autocorrelation coefficient for the fundamental (implied) z i is close to 0.99, as is also the case for its counterpart in the data. Given the persistence in the observed (and fundamental) level of z i, we present our diagnostics using the first difference of the fundamental log price dividend (earnings) ratio. However, the message from our diagnostics is the same when focusing on the level of the price dividend (earnings) ratio, its first difference, or ex-post returns. 4.3 Diagnostics Volatility and Correlation In Table 9, we present the implications of our fundamental valuation model for the volatility and cross-correlations of z i,t, focusing first on the model s implications for asset volatility. The range for Std( z i,t ) observed in the data (using dividends) is from (France) to (U.S.) (see Table 1). At the estimated parameter values, the CAPM-GARCH model with dividends explains about 65% of this observed volatility (see Table 9, Panel A). Alternatively, when earnings are used as a measure of cash flow, the range for Std( z i,t ) observed in the data is from (France) to (U.S.), and the model can explain about 12 For issues regarding measurement of aggregate cash flows see Ackert and Smith (1993) and Campbell and Shiller (1998). Additionally, Campbell (1991), Campbell and Mei (1993), and Campbell and Ammer (1993), also focus on deriving implications for beta s, but do not directly employ the cash flow data. 19