ABSTRACT
 Top of page
 ABSTRACT
 I. Evidence on the Value Premium
 II. The Model
 III. Implications for Equity Returns
 IV. Model Intuition
 V. Conclusion
 Appendix
 REFERENCES
We propose a dynamic riskbased model that captures the value premium. Firms are modeled as longlived assets distinguished by the timing of cash flows. The stochastic discount factor is specified so that shocks to aggregate dividends are priced, but shocks to the discount rate are not. The model implies that growth firms covary more with the discount rate than do value firms, which covary more with cash flows. When calibrated to explain aggregate stock market behavior, the model accounts for the observed value premium, the high Sharpe ratios on value firms, and the poor performance of the CAPM.
This paper proposes a dynamic riskbased model that captures both the high expected returns on value stocks relative to growth stocks, and the failure of the capital asset pricing model to explain these expected returns. The value premium, first noted by Graham and Dodd (1934), is the finding that assets with a high ratio of price to fundamentals (growth stocks) have low expected returns relative to assets with a low ratio of price to fundamentals (value stocks). This finding by itself is not necessarily surprising, as it is possible that the premium on value stocks represents compensation for bearing systematic risk. However, Fama and French (1992) and others show that the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965) cannot account for the value premium: While the CAPM predicts that expected returns should rise with the beta on the market portfolio, value stocks have higher expected returns yet do not have higher betas than growth stocks.
To model the difference between value and growth stocks, we introduce a crosssection of longlived firms distinguished by the timing of their cash flows. Firms with cash flows weighted more to the future endogenously have high price ratios, while firms with cash flows weighted more to the present have low price ratios. Analogous to longterm bonds, growth firms are highduration assets while value firms are lowduration assets. We model how investors perceive the risks of these cash flows by specifying a stochastic discount factor for the economy, or equivalently, an intertemporal marginal rate of substitution for the representative agent. Two properties of the stochastic discount factor account for the model's ability to fit the data. First, the price of risk varies, implying that at some times investors require a greater return per unit of risk than at others. Second, variation in the price of risk is not perfectly linked to variation in aggregate fundamentals. We show that the correlation between aggregate dividend growth and the price of risk crucially determines the ability of the model to fit the cross section.
We require that our model match not only the cross section of assets based on price ratios, but also aggregate dividend and stock market behavior. First, we assume that log dividend growth is normally distributed with a timevarying mean and calibrate the dividend process to fit conditional and unconditional moments of the aggregate dividend process in the data. Firms are distinguished by their cash flows, which we specify as stationary shares of the aggregate dividend. This modeling strategy, also employed by Menzly, Santos, and Veronesi (2004), ensures that the economy is stationary, and that firms add up to the market. Second, we choose stochastic discount factor parameters to fit the time series of aggregate stock market returns. These choices imply that expected excess returns on equity are time varying in the model, that there is excess volatility, and that excess returns are predictable. We find that the model can match unconditional moments of the aggregate stock market and produce dividend and return predictability close to that found in the data.
To test whether our model can capture the value premium, we sort firms into portfolios in simulated data. We find that risk premia, riskadjusted returns, and Sharpe ratios increase in the value decile. The value premium (the expected return on a strategy that is long the extreme value portfolio and short the extreme growth portfolio) is 5.1% in the model compared with 4.9% in the data when portfolios are formed by sorting on booktomarket. Moreover, the CAPM alpha on the valueminusgrowth strategy is 6.0% in the model, compared with 5.6% in the data. These results do not arise because value stocks are more risky according to traditional measures: Rather, standard deviations and market betas increase slightly in the value decile and then decrease, implying that the extreme value portfolio has a lower standard deviation and beta than the extreme growth portfolio. Our model therefore matches both the magnitude of the value premium and the outperformance of value portfolios relative to the CAPM that obtain in the data.
In its focus on explaining the value premium through cash flow fundamentals, our model is part of a growing literature that emphasizes the cash flow dynamics of the firm and how these relate to discount rates. In particular, in a model in which firms have assets in place as well as real growth options, Berk, Green, and Naik (1999) show that acquiring an asset with low systematic risk leads to a decrease in the firm's booktomarket ratio and lower future returns. More recently, Gomes, Kogan, and Zhang (2003) explicitly link risk premia to characteristics of firm cash flows in general equilibrium and Zhang (2005) shows how asymmetric adjustment costs and a timevarying price of risk interact to produce value stocks that suffer increased risk during downturns. These models endogenously derive patterns in the cross section of returns from cash flows, but they do not account for the classic finding of Fama and French (1992) that value stocks outperform, and growth stocks underperform, relative to the CAPM.
Our model for the stochastic discount factor builds on the work of Brennan, Wang, and Xia (2004) and Brennan and Xia (2006) and is closely related to essentially affine term structure models (Dai and Singleton (2003), Duffee (2002)). As Brennan et al. show, their model for the stochastic discount factor implies that claims to single dividend payments are exponentialaffine in the state variables, which allows for economically interpretable closedform expressions for prices and risk premia. Motivated by these expressions, Brennan et al. empirically evaluate whether expected returns on a crosssection of assets can be explained by betas with respect to discount rates. Here we make use of similar analytical methods to address a different goal, namely, endogenously generating a value premium based on the firm's underlying cash flows.
Our paper also builds on work that uses the concept of duration to better understand the cross section of stock returns. Using the decomposition of returns into cash flow and discount rate components proposed by Campbell and Mei (1993), Cornell (1999) shows that growth companies may have high betas because of the duration of their cash flows, even if the risk of these cash flows is mainly idiosyncratic. Berk, Green, and Naik (2004) value a firm with large research and development expenses and show how discount rate and cash flow risk interact to produce risk premia that change over the course of a project. Their model endogenously generates a long duration for growth stocks. Leibowitz and Kogelman (1993) show that accounting for the sensitivity of the value of longrun cash flows to discount rates can reconcile various measures of equity duration. Dechow, Sloan, and Soliman (2004) measure cash flow duration of value and growth portfolios; they find that empirically, growth stocks have higher duration than value stocks and that this contributes to their higher betas. Santos and Veronesi (2004) develop a model that links time variation in betas to time variation in expected returns through the channel of duration, and show that this link is present in industry portfolios. Campbell and Vuolteenaho (2004) decompose the market return into news about cash flows and news about discount rates. They show that growth stocks have higher betas with respect to discount rate news than do value stocks, consistent with the view that growth stocks are highduration assets. These papers all show that discount rate risk is an important component of total volatility, and, further, that growth stocks seem particularly subject to such discount rate risk. Our model shows how these contributions can be parsimoniously tied together with those discussed in the paragraphs above.
Finally, this paper relates to the large and growing body of empirical research that explores the correlations of returns on value and growth stocks with sources of systematic risk. This literature explores conditional versions of traditional models (Jagannathan and Wang (1996), Lettau and Ludvigson (2001a), Petkova and Zhang (2005), Santos and Veronesi (2006)) and identifies new sources of risk that covaries more with value stocks than with growth stocks (Lustig and Van Nieuwerburgh (2005), Piazzesi, Schneider, and Tuzel (2005), Yogo (2006)). Another strand of literature relates observed returns of value and growth stocks to aggregate market cash flows or macroeconomic factors (Campbell, Polk, and Vuolteenaho (2003), Liew and Vassalou (2000), Parker and Julliard (2005), Vassalou (2003)). The results in these papers raise the question of what it is, fundamentally, about the cash flows of value and growth stocks that produces the observed patterns in returns. Other work examines dividends on value and growth portfolios directly (Bansal, Dittmar, and Lundblad (2005), Cohen, Polk, and Vuolteenaho (2003), and Hansen, Heaton, and Li (2004)) and finds evidence that the cash flows of value stocks covary more with aggregate cash flows. The results in these papers raise the question of why the observed covariation leads to the value premium. By explicitly linking firms' cash flow properties and risk premia, this paper takes a step toward answering this question.
The paper is organized as follows. Section I updates evidence that portfolios formed by sorting on prices scaled by fundamentals produce spreads in expected returns. We show that when value is defined by booktomarket, earningstoprice, or cashflowtoprice, the expected return, Sharpe ratio, and alpha tend to increase in the value decile. The differences in expected returns and alphas between value and growth portfolios are statistically and economically large.
Section II presents our model for aggregate dividends and the stochastic discount factor. As a first step toward solving for prices of the aggregate market and firms, we solve for prices of claims to the aggregate dividend n periods in the future (zerocoupon equity). Because zerocoupon equity has a welldefined maturity, it provides a convenient window through which to view the role of duration in our model. The aggregate market is the sum of all the zerocoupon equity claims. We then introduce a cross section of longlived assets, defined by their shares in the aggregate dividend. These assets are themselves portfolios of zerocoupon equity, and together their cash flows and market values sum up to the cash flows and market values of the aggregate market.
Section III discusses the timeseries and crosssectional implications of our model. We calibrate the model to the time series of aggregate returns, dividends, and the pricedividend ratio. After choosing parameters to match aggregate timeseries facts, we examine the implications for zerocoupon equity. We find that the parameters necessary to fit the time series imply risk premia, Sharpe ratios, and alphas for zerocoupon equity that are increasing in maturity. In contrast, CAPM betas and volatilities are nonmonotonic, and thus do not explain the increase in risk premia. This suggests that our model has the potential to explain the value premium. We then choose parameters of the share process to approximate the distribution of dividend, earnings, and cash flow growth found in the data, and produce realistic distributions of price ratios. When we sort the resulting assets into portfolios, our model can explain the observed value premium.
Section IV discusses the intuition for our results. We show that the covariation of asset returns with the shocks depends on the duration of the asset. Consistent with the results of Campbell and Vuolteenaho (2004), growth stocks have greater betas with respect to discount rates than do value stocks. This is the duration effect: Because cash flows on growth stocks are further in the future, their prices are more sensitive to changes in discount rates. Growth stocks also have greater betas with respect to changes in expected dividend growth. Value stocks, on the other hand, have greater betas with respect to shocks to nearterm dividends. The price investors put on bearing the risk in each of these shocks determines the rates of return on value and growth stocks. While shocks to nearterm dividends are viewed as risky by investors, shocks to expected future dividends are hedges under our calibration. Moreover, though discount rates vary over time, shocks to discount rates are independent of shocks to dividends and are therefore not priced directly. Thus, even though longhorizon equity is riskier according to standard deviation and market beta, it is not seen as risky by investors because it loads on risks that investors do not mind bearing.
I. Evidence on the Value Premium
 Top of page
 ABSTRACT
 I. Evidence on the Value Premium
 II. The Model
 III. Implications for Equity Returns
 IV. Model Intuition
 V. Conclusion
 Appendix
 REFERENCES
Much of the previous literature shows that portfolios of stocks with high ratios of prices to fundamentals have low future returns compared to stocks with low ratios of prices to fundamentals.^{1} In this section, we update this evidence by running statistical tests on portfolios formed on ratios of market to book value, price to earnings, price to dividends, and price to cash flow. We show that in all cases, the sorting produces differences in expected returns that cannot be attributed to market beta. Moreover, the alpha relative to the CAPM tends to increase in the measure of value. In our model, firms are distinguished by their cash flows, thus earnings, dividends, and cash flows are equivalent. For this reason, it is of interest to investigate whether the value effect is apparent in portfolios formed according to different measures of value.
Table I reports summary statistics for portfolios of firms sorted into deciles on each of the three characteristics described above and on booktomarket. Data, available from the website of Ken French, are monthly, from 1952 to 2002. We compute excess returns by subtracting monthly returns on the 1month Treasury Bill from the portfolio return. The first panel reports the mean excess return, the second the standard error on the mean, the third the standard deviation of the return, and the fourth the Sharpe ratio. Means and standard deviations are in annual percentage terms (multiplied by 1,200 in the case of means and in the case of standard deviations). Each panel reports results for the earningstoprice ratio, the cashflowtoprice ratio, the dividend yield, and the booktomarket ratio.
Table I. Summary Statistics for Growth and Value Portfolios Portfolios are formed by sorting firms into deciles on the dividend yield (D/P), the earnings yield (E/P), the ratio of cash flow to prices (C/P), and the booktomarket ratio (B/M). Moments are in annualized percentages (multiplied by 1,200 in the case of means and in the case of standard deviations). The data are monthly and span the 1952 to 2002 period. Portfolio  G 1  Growth to Value  V 10  V–G 10–1 

2  3  4  5  6  7  8  9 

Panel A: Mean Excess Return (% per year) 
E/P  4.71  5.02  6.97  7.04  7.00  9.18  9.94  11.18  11.68  12.95  8.25 
C/P  5.05  6.07  6.49  6.73  8.48  7.72  8.85  9.18  11.47  11.81  6.77 
D/P  7.35  6.41  7.28  7.41  6.49  7.60  7.73  9.49  8.84  7.45  0.10 
B/M  5.67  6.55  6.98  6.51  8.00  8.33  8.27  10.08  9.98  10.55  4.88 
Panel B: Standard Error of Mean 
E/P  0.78  0.64  0.62  0.59  0.62  0.61  0.60  0.61  0.65  0.73  0.62 
C/P  0.76  0.64  0.61  0.63  0.62  0.60  0.60  0.60  0.61  0.69  0.59 
D/P  0.78  0.69  0.66  0.64  0.62  0.60  0.59  0.58  0.56  0.56  0.69 
B/M  0.71  0.64  0.64  0.62  0.59  0.59  0.59  0.61  0.63  0.74  0.61 
Panel C: Standard Deviation of Excess Return (% per year) 
E/P  19.35  15.93  15.49  14.78  15.43  15.04  14.87  15.29  16.11  18.11  15.40 
C/P  18.99  15.95  15.24  15.75  15.43  14.95  14.96  14.98  15.14  17.24  14.57 
D/P  19.36  17.11  16.31  15.85  15.43  15.00  14.58  14.37  13.93  13.83  17.08 
B/M  17.77  15.89  15.82  15.42  14.65  14.73  14.74  15.11  15.71  18.46  15.15 
Panel D: Sharpe Ratio 
E/P  0.24  0.32  0.45  0.48  0.45  0.61  0.67  0.73  0.73  0.72  0.54 
C/P  0.27  0.38  0.43  0.43  0.55  0.52  0.59  0.61  0.76  0.69  0.46 
D/P  0.38  0.37  0.45  0.47  0.42  0.51  0.53  0.66  0.63  0.54  0.01 
B/M  0.32  0.41  0.44  0.42  0.55  0.57  0.56  0.67  0.64  0.57  0.32 
Panel A of Table I shows that for all measures except the dividend yield, the mean excess return is higher for the upper deciles (value) than for the lower deciles (growth). Panel B shows that the average return on the portfolio that is long the extreme value portfolio and short the extreme growth portfolio is highly statistically significant, again except when portfolios are formed by sorting on the dividend yield. Panel C shows that the standard deviation of the excess return tends to decrease in the decile number, and thus move in the opposite direction of the mean return. Finally, Panel D shows that the Sharpe ratio increases in the decile number. For example, when portfolios are formed by sorting on the earningstoprice ratio, the bottom decile (growth) has a Sharpe ratio of 0.24. The Sharpe ratio increases as the earningstoprice ratio increases and the top decile (value) has a Sharpe ratio of 0.72. Thus value stocks not only deliver high returns, they deliver high returns per unit of standard deviation.
The results in Table I suggest that portfolios formed by sorting on earningstoprice, cashflowtoprice, the dividend yield, and booktomarket may be closely related. This is confirmed in Table II, which shows the correlation of the bottom and top deciles. For the bottom decile (growth), the correlations are 0.93 or above; for the top decile (value), the correlations are 0.74 or above. In both cases, deciles formed by sorting on the dividend yield are less highly correlated with the deciles formed by sorting on the other three variables than the deciles formed by sorting on the other three variables are with each other. This is consistent with the results in Table I, which shows that portfolios formed by sorting on the dividend yield behave somewhat differently from portfolios formed by sorting on the other variables.
Table II. Correlation of Returns on Extreme Value and Growth Portfolios Portfolios are formed by sorting firms into deciles on the dividend yield (D/P), the earnings yield (E/P), the ratio of cash flow to prices (C/P), and the booktomarket ratio (B/M). The data are monthly and span the 1952 to 2002 period.  E/P  C/P  D/P  B/M 

Panel A: Top Decile (Value) 
E/P  1.00  0.94  0.76  0.85 
C/P  0.94  1.00  0.74  0.85 
D/P  0.76  0.74  1.00  0.75 
B/M  0.85  0.85  0.75  1.00 
Panel B: Bottom Decile (Growth) 
E/P  1.00  0.98  0.93  0.96 
C/P  0.98  1.00  0.93  0.97 
D/P  0.93  0.93  1.00  0.94 
B/M  0.96  0.97  0.94  1.00 
Following the same format as Table I, Table III shows alphas, standard errors on alphas, betas, standard errors on betas, and R^{2} statistics when portfolios are formed by sorting on each measure of value. Alpha is the intercept from an ordinary least squares (OLS) regression of portfolio excess returns on excess returns of the valueweighted CRSP index, multiplied by 1,200. Beta is the slope from this regression. The alpha for the portfolio that is long the extreme value portfolio and short the extreme growth portfolio is statistically significant for all four sorting variables. Panel A of this table confirms the classic result that value stocks have high alphas relative to the CAPM. Moreover, the story is consistent across all sorting variables, including the dividend yield: Alphas are negative for growth stocks, positive for value stocks, and increasing in the decile number. As Panel C shows, betas tend to decline in the decile number, except for the extreme value portfolio. Thus, value stocks have positive alphas relative to the CAPM, and relatively low betas.
Table III. Performance of Growth and Value Portfolios Relative to the CAPM Intercepts and slope coefficients are calculated from OLS timeseries regressions of excess portfolio returns on the excess return on the valueweighted CRSP index. Portfolios are formed by sorting firms into deciles on the dividend yield (D/P), the earnings yield (E/P), the ratio of cash flow to prices (C/P), and the booktomarket ratio (B/M). Intercepts are in annualized percentages (multiplied by 1,200). The data are monthly and span the 1952 to 2002 period. Portfolio  G 1  Growth to Value  V 10  V–G 10–1 

2  3  4  5  6  7  8  9 

Panel A: α_{i} (% per year) 
E/P  −3.09  −1.62  0.69  0.95  0.74  3.25  4.08  5.33  5.60  6.22  9.31 
C/P  −2.70  −0.54  0.19  0.24  2.33  1.79  3.01  3.46  5.75  5.34  8.04 
D/P  −0.58  −0.73  0.62  0.98  0.44  1.77  2.03  4.11  3.96  3.44  4.01 
B/M  −1.66  −0.17  0.33  0.22  2.12  2.37  2.59  4.30  4.05  3.97  5.63 
Panel B: Standard Error of α_{i} 
E/P  1.12  0.74  0.86  0.75  0.86  0.95  0.95  1.07  1.18  1.38  2.14 
C/P  1.03  0.78  0.76  0.80  0.94  0.93  0.98  1.06  1.11  1.28  2.01 
D/P  1.03  0.80  0.88  0.88  1.00  1.00  0.96  1.07  1.19  1.47  2.05 
B/M  0.90  0.65  0.69  0.84  0.86  0.83  1.01  1.07  1.15  1.53  2.12 
Panel C: β_{i} 
E/P  1.18  1.01  0.95  0.92  0.95  0.90  0.89  0.89  0.92  1.02  −0.16 
C/P  1.17  1.00  0.95  0.98  0.93  0.90  0.89  0.87  0.87  0.98  −0.19 
D/P  1.20  1.08  1.01  0.97  0.92  0.88  0.86  0.82  0.74  0.61  −0.59 
B/M  1.11  1.02  1.01  0.95  0.89  0.90  0.86  0.87  0.90  1.00  −0.11 
Panel D: Standard Error of β_{i} 
E/P  0.02  0.01  0.02  0.01  0.02  0.02  0.02  0.02  0.02  0.03  0.04 
C/P  0.02  0.01  0.01  0.02  0.02  0.02  0.02  0.02  0.02  0.02  0.04 
D/P  0.02  0.02  0.02  0.02  0.02  0.02  0.02  0.02  0.02  0.03  0.04 
B/M  0.02  0.01  0.01  0.02  0.02  0.02  0.02  0.02  0.02  0.03  0.04 
Panel E: R^{2} 
E/P  0.83  0.89  0.84  0.87  0.84  0.80  0.80  0.75  0.73  0.71  0.02 
C/P  0.85  0.88  0.87  0.87  0.81  0.80  0.78  0.75  0.73  0.72  0.04 
D/P  0.86  0.89  0.85  0.84  0.79  0.77  0.78  0.72  0.63  0.43  0.27 
B/M  0.87  0.91  0.90  0.85  0.83  0.84  0.76  0.75  0.73  0.65  0.01 
To summarize, this section shows that, in the data, value stocks have higher expected excess returns and higher Sharpe ratios than do growth stocks. Value stocks have large positive alphas while growth stocks have negative alphas. Moreover, value stocks do not have higher standard deviations or higher betas than do growth stocks. Thus, any explanation of the value premium must take into account the fact that value stocks do not appear to be riskier than growth stocks according to traditional measures of risk. These empirical results hold not only when value is defined by the booktomarket ratio, but also when value is defined by the earningstoprice or cashflowtoprice ratios.
III. Implications for Equity Returns
 Top of page
 ABSTRACT
 I. Evidence on the Value Premium
 II. The Model
 III. Implications for Equity Returns
 IV. Model Intuition
 V. Conclusion
 Appendix
 REFERENCES
To study implications for the aggregate market and the cross section, we simulate 50,000 quarters from the model. Given simulated data on shocks ε_{t+1} and state variables x_{t+1} and z_{t+1}, we compute ratios of prices to aggregate dividends for zerocoupon equity from (9) and the price–dividend ratio for the aggregate market from (17).
We calibrate the model to the annual data set of Campbell (1999), which begins in 1890, updating Campbell's data (which end in 1995) through the end of 2002. To ensure that our simulated values are comparable to the annual values in the data, we aggregate up to an annual frequency. Annual flow variables (returns, dividend growth) are constructed by compounding their quarterly counterparts. Price–dividend ratios for the market and for firms (described below) are constructed analogously to annual price–dividend ratios in the Campbell data set: We divide the price by the current dividend plus the previous three quarters of dividends on the asset.
Section A describes the calibration of our model to the aggregate time series. Section B gives the model's implications for the behavior of the aggregate market and dividend growth and discusses the fit to the data. Section C gives the implications for prices and returns on zerocoupon equity. While zerocoupon equity has no analogue in the data, it allows us to illustrate the properties of the model in a stark way. Section D discusses the calibration of the share process that determines the prices of longlived assets (“firms”), and describes implications of the model for portfolios formed by sorting on scaled price ratios.
A. Calibration
Following Menzly et al. (2004), we calibrate the model to provide a reasonable fit to aggregate data. We then ask whether the model can match moments of the cross section. In order to accurately capture the characteristics of our persistent processes, we use the centurylong annual data set of Campbell (1999), which we update through 2002. The riskfree rate is the return on 6month commercial paper purchased in January and rolled over in July. Stock returns, prices, and dividends are for the S&P 500 index. All variables are adjusted for inflation. The Data Appendix of Campbell (1999) contains more details on data construction.
We set r^{f} equal to 1.93%, the sample mean of the riskfree rate. Similarly, we set g equal to 2.28%, which is the average dividend growth in the sample. Calibrating the process z_{t}, which determines expected dividend growth, is less straightforward as, strictly speaking, this process is unobservable to the econometrician. However, Lettau and Ludvigson (2005) show that if consumption growth follows a random walk and if the consumption–dividend ratio is stationary, the consumption–dividend ratio captures the predictable component of dividend growth. The consumption–dividend ratio can therefore be identified with z_{t} up to an additive and multiplicative constant.^{6} In our annual sample, the consumption–dividend ratio has a persistence of 0.91 and a conditional correlation with dividend growth of –0.83; these are, respectively, our values for φ_{z} and the correlation between z_{t} and Δd_{t}. We set  σ_{d}  to match the unconditional standard deviation of annual dividend growth in the data.^{7} Our empirical results imply a standard deviation of z_{t} that is small relative to the standard deviation of dividend growth. Despite the fact that dividend growth is predictable at long horizons by the consumption–dividend ratio, the consumption–dividend ratio has very little predictive power for dividend growth at short horizons. Moreover, the autocorrelation of dividend growth is relatively low (−0.09). We show that  σ_{z}  = 0.0016 (0.0032 per annum) produces similar results in simulated data.
The remaining parameters are , and  σ_{x} . Because the variance of expected dividend growth is small, the autocorrelation of the price–dividend ratio is primarily determined by the autocorrelation of x. We therefore set , as 0.87 is the autocorrelation of the price–dividend ratio in annual data. We set  σ_{x}  to 0.12, or 0.24 per annum, to match the volatility of the log price dividend ratio. We choose so that the maximal Sharpe ratio, when x_{t} is at its longrun mean, is 0.70. This produces Sharpe ratios for the cross section that are close to those in the data. Setting the maximum Sharpe ratio equal to 0.70 implies . As we discuss in the subsequent section, this produces an average Sharpe ratio for the market that is 0.41, which is somewhat higher than the data equivalent of 0.33. However, expected stock returns are measured with noise, and 0.41 is still below the Sharpe ratio of postwar data.
To determine the vectors σ_{d}, σ_{z}, σ_{x}, we assume without loss of generality that the 3 × 3 matrix [σ′_{d}, σ′_{z}, σ′_{x}]′ is lower triangular. Thus ε_{1,t+1}=ε_{d,t+1}, so that the first element of σ_{d} equals  σ_{d}  and the second and third elements equal zero. The vector σ_{z} has nonzero first and second elements determined by  σ_{z}  and σ_{d}σ′_{z}, and zero third element. We focus on the case in which x_{t+1} is independent of Δd_{t+1} and z_{t+1}, so the first and second elements of σ_{x} equal zero, and the third equals  σ_{x} . Table IV summarizes these parameter choices.
Table IV. Parameters of the Model Model parameters are calibrated to aggregate data starting in 1890 and ending in 2002. The model is simulated at a quarterly frequency. The unconditional mean of dividend growth g, the riskfree rate r_{f}, the persistence variables φ_{x} and φ_{z}, and the conditional standard deviations  σ_{d} ,  σ_{z} , and  σ_{x} , are in annual terms (i.e., 4g, φ^{4}_{x}, 2 σ_{d} ). Parameters g, r_{f}, and  σ_{d}  are set to match their data counterparts. Parameters φ_{z} and the correlation between shocks to z and shocks to Δd are set to match their data counterparts, assuming that the conditional mean of dividend growth is determined by the log consumption–dividend ratio in the data. The parameter  σ_{z}  is set to match the autocorrelation and predictability of dividend growth in the data,  σ_{x}  is set to match the volatility of the price–dividend ratio, and φ_{x} is set to match the persistence of the price–dividend ratio. Variable  Value 

g  2.28% 
r_{f}  1.93% 
 0.625 
φ_{z}  0.91 
φ_{x}  0.87 
 σ_{d}   0.145 
 σ_{z}   0.0032 
 σ_{x}   0.24 
Correlation of Δd and z shocks  −0.83 
Correlation of Δd and x shocks  0 
Correlation of z and x shocks  0 
Implied Volatility Parameters 
σ_{d}  [0.0724, 0, 0] 
σ_{z}  [−0.0013, 0.0009, 0] 
σ_{x}  [0, 0, 0.12] 
Given our parameter choices, it is possible to infer the process for x_{t} based on the observed price–dividend ratio and consumption–dividend ratio. The consumptiondividend ratio can be used to construct an empirical proxy for z_{t}.^{8} For each timeseries observation on the price–dividend ratio and z_{t}, we find a corresponding x_{t} by numerically solving (17). Figure 1 plots the resulting series for x_{t}, along with several macroeconomic time series that recent theory suggests should be related to aggregate risk aversion. These macroeconomic time series are: my, the deviation from the cointegration relationship between human wealth and outstanding home mortgages constructed by Lustig and Van Nieuwerburgh (2005); α, the share of nonhousing consumption in total consumption constructed by Piazzesi et al. (2005); and cay, the consumption–wealth ratio of Lettau and Ludvigson (2001b). All series are demeaned and standardized. Figure 1 shows that all three series are positively correlated with x_{t}. Longrun fluctuations in x_{t} appear to be related to longrun fluctuations in both my and α, while cay (which is constructed using data on prices as well as macroeconomic quantities) also picks up shortrun fluctuations in x_{t}.
Table V shows results of contemporaneous regressions of the implied x_{t} on the variables described above. This table confirms that x_{t} is positively and significantly related to all three macroeconomicbased risk aversion measures.
Table V. Results from Contemporaneous OLS Regressions of x on Macroeconomic Variables The variable my is the deviation from the cointegration relationship between human wealth and outstanding home mortgages as in Lustig and Van Nieuwerburgh (2005), cay is the consumption–wealth ratio of Lettau and Ludvingson (2001), and α is the share of nonhousing consumption in total consumption as in Piazzesi, Schneider, and Tuzel (2005). The annual data span the period 1947 to 2002.  β  tStatistics  R^{2} 

my  2.80  6.13  0.54 
cay  21.32  3.44  0.28 
α  29.30  6.19  0.30 
B. Implications for the Aggregate Market and Dividend Growth
Table VI presents statistics from simulated data, and the corresponding statistics computed from actual data. The volatility of the price–dividend ratio is fit exactly and the autocorrelation of the pricedividend ratio is very close (0.87 in the data versus 0.88 in the model). This is not a surprise because  σ_{x}  and φ_{x} are set so that the model fits these parameters. The model produces a mean price–dividend ratio equal to 20.1, compared to 25.6 in the data. Matching this statistic is a common difficulty for models of this type: Campbell and Cochrane (1999), for example, find an average pricedividend ratio of 18.2. As they explain, this statistic is poorly measured due to the persistence of the price–dividend ratio. The model fits the volatility of equity returns (19.2% in the model vs. 19.4% in the data), though it produces an equity premium that is slightly higher than in the data (7.9% in the model vs. 6.3% in the data). As with the mean of the price–dividend ratio, the average equity premium is measured with noise. In the long annual data set, the annual autocorrelation of excess returns is slightly positive (0.03). In our model, the autocorrelation is slightly negative (−0.02). The autocorrelation of dividend growth is small and negative (−0.04), just as in the data (−0.09).
Table VI. Simulated Moments for the Aggregate Market and Dividend Growth The model is simulated for 50,000 quarters. Returns, dividends, and price ratios are aggregated to an annual frequency. The data are annual and span the period 1890 to 2002.  Data  Model 

E(P/D)  25.55  20.96 
σ(p−d)  0.38  0.38 
AC of p−d  0.87  0.88 
E[R^{m}−R^{f}]  6.33%  7.87% 
σ(R^{m}−R^{f})  19.41%  19.19% 
AC of R^{m}−R^{f}  0.03  −0.04 
Sharpe ratio of market  0.33  0.41 
AC of Δd  −0.09  −0.04 
σ(Δd_{t})  14.48%  14.43% 
Table VII reports the results of longhorizon regressions of continuously compounded excess returns on the log price–dividend ratio in the model and in the data. In our sample, as elsewhere (e.g., Campbell and Shiller (1988), Cochrane (1992), Fama and French (1989), Keim and Stambaugh (1986)), high pricedividend ratios predict low returns. The coefficients rise with the horizon. The R^{2}s start small, at 0.05 at an annual horizon, and rise to 0.31 at a horizon of 10 years. The tstatistics, computed using autocorrelation and heteroskedasticityadjusted standard errors, are significant at the 5% level. The simulated data exhibit the same pattern. The R^{2}s start at 0.06 and rise to 0.28. We conclude that the model generates a reasonable amount of return predictability.^{9}
Table VII. Long Horizon Regressions—Excess Returns Excess returns are regressed on the lagged price–dividend ratio in annual data from 1890 to 2002 and in data simulated from the model. Specifically, we run the regression in the data and in the model. For each data regression, the table reports OLS estimates of the regressors, Newey–West (1987) corrected tstatistics (in parentheses), and adjustedR^{2} statistics in square brackets. Significant data coefficients using the standard ttest at the 5% level are highlighted in boldface.  Horizon in Years 

1  2  4  6  8  10 

Panel A: Full Data 
β_{1}  −0.12  −0.23  −0.37  −0.60  −0.86  −1.09 
tstat  (−2.39)  (−2.44)  (−2.01)  (−2.24)  (−2.97)  (−3.54) 
R^{2}  [0.05]  [0.08]  [0.10]  [0.16]  [0.25]  [0.31] 
Panel B: Data Up to 1994 
β_{1}  −0.21  −0.39  −0.61  −0.89  −1.16  −1.34 
tstat  (−3.45)  (−4.04)  (−3.17)  (−4.08)  (−5.81)  (−6.22) 
R^{2}  [0.07]  [0.13]  [0.19]  [0.30]  [0.41]  [0.44] 
Panel C: Model 
β_{1}  −0.11  −0.21  −0.36  −0.49  −0.58  −0.65 
R^{2}  [0.06]  [0.11]  [0.18]  [0.23]  [0.26]  [0.28] 
Table VIII reports the results of longhorizon regressions of dividend growth on the price–dividend ratio. As Campbell and Shiller (1988) show, dividend growth is not predicted by the price–dividend ratio, contrary to what might be expected from a dividend discount model. This result also holds in our data: The coefficients from a regression of dividend growth on the price–dividend ratio are always insignificant and are accompanied by small R^{2} statistics. In contrast, the consumption–dividend ratio predicts dividend growth in actual data. The coefficients are significant, and the adjustedR^{2} statistics start at 3% for an annual horizon and rise to 25% for a horizon of 10 years.
Table VIII. Long Horizon Regressions—Dividend Growth Aggregate dividend growth is regressed on lagged values of the price–dividend ratio and the consumption–dividend ratio in annual data from 1890 to 2002 and in data simulated from the model. For each data regression, the table reports OLS estimates of the regressors, Newey–West (1987) corrected tstatistics (in parentheses), and adjustedR^{2} statistics in square brackets. Significant data coefficients using the standard ttest at the 5% level are highlighted in boldface.  Horizon in Years 

1  2  4  6  8  10 

Panel A: Data 

β_{1}  0.02  −0.01  −0.04  −0.12  −0.23  −0.31 
tstat  (0.56)  (−0.23)  (−0.34)  (−0.85)  (−1.26)  (−1.61) 
R^{2}  [−0.01]  [−0.01]  [−0.01]  [0.00]  [0.02]  [0.05] 

β_{1}  0.10  0.18  0.34  0.56  0.65  0.68 
tstat  (2.30)  (2.52)  (3.05)  (3.42)  (3.56)  (3.78) 
R^{2}  [0.03]  [0.06]  [0.13]  [0.24]  [0.26]  [0.25] 
Panel B: Model 

β_{1}  0.05  0.09  0.17  0.24  0.29  0.33 
R^{2}  [0.02]  [0.03]  [0.06]  [0.08]  [0.09]  [0.09] 

β_{1}  3.73  7.09  13.19  18.13  22.23  25.81 
R^{2}  [0.04]  [0.07]  [0.13]  [0.18]  [0.21]  [0.24] 
Our model replicates both of these findings. Despite the fact that the mean of dividends is time varying, dividends are only slightly predictable by the price–dividend ratio. A regression of simulated dividend growth on the simulated price–dividend ratio produces R^{2}s that range from 2% to 9% at a horizon of 10 years. By contrast, dividends are predictable by z_{t}. Here, the R^{2}s range from 4% to 24%, close to the values in the data. We conclude our model captures the pattern of dividend predictability found in the data.
C. Prices and Returns on ZeroCoupon Equity
Figure 2 plots the solutions for A(n), B_{z}(n), and B_{x}(n) as a function of n for the parameter values given above. A(n) is decreasing in n, as is necessary for convergence of the market price–dividend ratio. This is also sensible economically: The further the payoff is in the future, the lower the value of the security when the state variables are at their longrun means. What generates the decrease is the positive average price of risk and the riskfree rate r^{f}, counteracted by average dividend growth g and the Jensen's inequality term.
Given that we describe the behavior of B_{z}(n) in Section A, here we focus on B_{x}(n). For all values of n, B_{x}(n) is negative, indicating that an increase in the price of risk x_{t} leads to a decrease in valuations. Also, B_{x}(n) is nonmonotonic, starting at zero, decreasing to below −1, then increasing, eventually converging to a value near −0.5. It is not surprising that B_{x}(n) initially decreases in maturity. This is the duration effect: The longer is the maturity, the more sensitive the price is to changes in the discount rate. More curious is the fact that B_{x}(n) rises after a maturity of 10 years. This is because the duration effect is countered by the increase in B_{z}(n). Because expected dividend growth and dividend growth are negatively correlated, shocks to expected dividend growth act as a hedge. Moreover, as the plot of B_{z} shows, expected dividend growth becomes more important the longer the maturity of the equity. Hence, equity that pays in the far future is less sensitive to changes in x_{t} than equity that pays in the medium term, though both are more sensitive than shorthorizon equity.
Figure 3 plots the ratios of price to aggregate dividends for zerocoupon equity as a function of maturity n. The top panel sets z_{t} to be two unconditional standard deviations (2 σ_{z} /(1 −φ^{2}_{z})^{1/2}) below its unconditional mean, the middle panel to the unconditional mean of zero, and the bottom panel to two standard deviations above the mean. Each panel plots the price–dividend ratio for x_{t} at its unconditional mean and two unconditional standard deviations (2 σ_{x} /(1 −φ^{2}_{x})^{1/2}) around the mean. Prices are increasing in z_{t} for all values of x_{t} and n, and decreasing in x_{t} for all values of z_{t} and n. That is, higher expected dividend growth and lower risk premia imply higher prices.
For most values of z_{t} and x_{t}, prices decline with maturity. Generally, the further in the future the asset pays the aggregate dividend, the less it is worth today. Exceptions occur when x_{t} is two standard deviations below the mean. In this case, the premium for holding risky securities is negative in the short term, so shorthorizon payoffs are discounted by more than longhorizon payoffs. Because x_{t} reverts back to , this effect is transitory and only holds at the short end of the equity “yield curve.” The greater is z_{t}, the longer the effect persists because z_{t} raises the price of longrun equity relative to shortrun equity.
Figure 4 presents statistics for annual returns on zerocoupon equity. The top panel shows that the risk premium ER_{i,t+1}−R^{f} declines with maturity. The effect is economically large: The risk premium is 18% for equity that pays a dividend 2 years from now and 4% for equity that pays a dividend 40 years from now.
The second panel of Figure 4 shows that the return volatility initially increases with maturity, and then decreases at maturities greater than 10 years. The third panel of Figure 4 shows that the unconditional Sharpe ratio decreases monotonically in maturity. These results suggest that the model has the potential to explain the patterns described in Table I. Firms that have more weight in lowmaturity equity will have higher expected returns, higher Sharpe ratios, and possibly lower variance than firms that have more weight in equity of greater maturity.
Figure 5 shows the results of regressing simulated zerocoupon equity returns on simulated market returns. The top panel shows the regression alpha, the middle panel the beta, and the last panel the R^{2}. As in Figure 4, returns are annual. The first panel shows that the alpha relative to the CAPM is decreasing in maturity over most of the range, increasing only slightly for longduration equity. For the shortestduration equity the alpha is as high as 11%. The alpha falls below zero for equity maturing in 5 or more years, but remains above −5%. Thus, the model produces relatively large positive alphas and relatively small negative alphas, just as in the data.
The second panel of Figure 5 shows the regression beta. The beta first increases, and then, beginning with a maturity of about 10 years, decreases slowly as a function of maturity. The betas for zerocoupon equity lie in a relatively narrow range; the lowest beta (for very long horizon equity) is about 0.7, and the highest beta (for equity of about 10 years) is 1.5. The beta for the shortesthorizon equity is about 0.9. This plot shows that at least for shorthorizon equity, high alphas are not necessarily accompanied by high betas. These results suggest that the model has the potential to explain the patterns described in Table III.
While the simplicity of zerocoupon equity makes it a convenient way to illustrate the properties of the model, it does not have a direct interpretation in terms of value and growth. The price–dividend ratio is not well defined because zerocoupon equity only pays dividends during a single quarter. For this reason, we turn to a model of firms, that is, longlived assets that have nonzero cash flows in every period.
D. Implications for the Cross Section of Returns
This section shows the implications of the model for portfolios formed by sorting on price ratios. Following Menzly et al. (2004), we exogenously specify a share process for cash flows on longlived assets. For each year of simulated data, we sort these assets into deciles based on the ratio of price to dividends (or equivalently, earnings or cash flows) and form portfolios of the assets within each decile. This follows the procedure used in empirical studies of the cross section (e.g., Fama and French (1992)). We then perform statistical analysis on the portfolio returns.
D.1. Specifying the Share Process
In order to assess the quantitative implications of the model, we specify longlived assets with welldefined ratios of prices to dividends that together sum up to the market portfolio. Moreover, we require that the crosssectional distribution of dividends, returns, and price ratios be stationary. In order to accomplish this, we follow Lynch (2003) and Menzly et al. (2004) in specifying the share each security has in the aggregate dividend process D_{t+1}. The continuoustime framework of Menzly et al. allows the authors to specify the share process as stochastic, yet still keep shares between zero and one. This is more difficult in discrete time; for this reason we adopt the simplifying assumption that the share process is deterministic.
Consider N sequences of dividend shares s_{it}, for i= 1, … , N. For convenience, we refer to each of these N sequences as a firm, though they are best thought of as portfolios of firms in the same stage of the life cycle. As our ultimate goal is to aggregate these firms into portfolios based on price–dividend ratios, this simplification does not affect our results. Firm i pays s_{it} of the aggregate dividend at time t, s_{i,t+1} of the aggregate dividend at time t+ 1, etc. Shares are such that s_{it}≥ 0 and for all t (so that the firms add up to the market). Because firm i pays a dividend sequence s_{i,t+1}D_{t+1}, s_{i,t+2}D_{t+2}, … , noarbitrage implies that the exdividend price of firm i equals
where P_{nt} is the price of zerocoupon equity maturing at time t+n.
We specify a simple model for shares. Let be the lowest share of a firm in the economy, and assume without loss of generality that firm 1 starts at , namely . We assume that the share grows at a constant rate g_{s} until reaching and then shrinks at the rate g_{s} until reaching again. At this point the cycle repeats. All firms are ex ante identical, but are “out of phase” with one another. Firm 1 starts out at , firm 2 at , Firm N/2 at , and Firm N at . The variable is such that the shares sum to one for all t.^{10} We set the number of firms to 200, implying a 200quarter, or equivalently, 50year life cycle for a firm. While this model for firms is simple and somewhat mechanical, it accomplishes our objective of creating dispersion in the timing of cash flows across firms in a straightforward way.^{11}
The parameter that determines the growth in the share process, g_{s}, is set to 5%, implying an annual growth rate of 20%. We choose this value so that the crosssectional distribution of dividend growth rates in the model matches that in the sample. Because data on earnings and cash flows are not available prior to 1952, we construct the crosssection for data from 1952 to 2002.^{12} The top panel of Figure 6 plots the implied crosssection of average growth rates of dividends for firms in the model, as well as the crosssection of average growth rates in earnings, dividends, and cash flows in the sample. Because the firms in our model have no debt, the dividends in our model may be better analogues to earnings and cash flows in the data, rather than dividends themselves. The bottom panel of Figure 6 shows the distribution of firm price–dividend ratios in the model, and price ratios in the data. While the overall fit is reasonable, the model produces more high pricedividend ratio firms than there are in the data. These firms have high price–dividend ratios because they have extremely low dividends. It is possible to construct models that fit the dividend growth and price ratio distributions more closely by assuming growth is linearly decreasing or imposing a greater lower bound on the dividend share. As Lettau and Wachter (2005) show, the asset pricing implications of these alternative models are very similar to the present constant growth model.
D.2. Portfolio Returns
At the start of each year in the simulation, we sort firms into deciles by their price–dividend ratio. We then form equalweighted portfolios of the firms in each decile. As firms move through their life cycle, they slowly shift (on average) from the growth category to the value category, and then revert back eventually to the growth category. This process is not deterministic because shocks have differential impacts on price–dividend ratios of firms at different stages of the life cycle.
Having sorted the firms into deciles at the beginning of each “year,” we compute statistical tests on returns over the year. The first panel of Table IX shows the expected excess return, the standard deviation, and the Sharpe ratio for each portfolio. These simulation results should be compared to the numbers in Table I, which show corresponding results for the data. The expected excess return on the extreme growth portfolio is 5.0% per annum, while for the extreme value portfolio it is 10.1% per annum.^{13} A similar spread occurs in the data: The lowest booktomarket stocks have a premium of 5.7%, while the highest have a premium of 10.6%. The model generates volatilities between 19% and 17%; the volatilities for booktomarketsorted portfolios vary between 18% and 15% in the data. Moreover, the model predicts that value portfolios have lower volatilities than growth portfolios despite their higher returns, as is the case in the data. The model predicts that the Sharpe ratio rises from 0.26 for the extreme growth portfolio to 0.58 for the extreme value portfolio. In the data, the lowest booktomarket portfolio has a Sharpe ratio of 0.32 while the highest booktomarket portfolio has a Sharpe ratio of 0.57. To summarize, the model implies that value stocks have high expected returns, low volatility, and high Sharpe ratios, just as in the data, and further, the magnitude of the difference between value and growth is comparable to that in the data.
Table IX. Performance of Growth and Value Portfolios in the Model In each simulation year, firms are sorted into deciles on the price–dividend ratio. Returns are calculated over the subsequent year. Intercepts and slope coefficients are from OLS timeseries regressions of excess portfolio returns on the excess market return, and on the excess market return together with the return on a portfolio short the extreme growth decile and long the extreme value decile (HML). Portfolio  G 1  Growth to Value  V 10  V–G 10–1 

2  3  4  5  6  7  8  9 

Panel A: Summary Statistics 
E R^{i}−R^{f}  5.00  5.18  5.47  5.90  6.46  7.15  7.89  8.58  9.16  10.08  5.09 
σ(R^{i}−R^{f})  19.27  19.48  19.64  19.67  19.51  19.08  18.38  17.56  16.99  17.30  8.27 
Sharpe Ratio  0.26  0.27  0.28  0.30  0.33  0.37  0.43  0.49  0.54  0.58  0.62 
Panel B: R^{i}_{t}−R^{f}_{t}=α_{i}+β_{i}(R^{m}_{t}−R^{f}_{t}) +ε_{it} 
α_{i}  −2.60  −2.52  −2.31  −1.93  −1.33  −0.50  0.52  1.59  2.48  3.38  5.98 
β_{i}  1.00  1.01  1.02  1.03  1.02  1.00  0.97  0.92  0.88  0.88  −0.12 
R^{2}_{i}  0.97  0.97  0.97  0.98  0.99  1.00  1.00  0.98  0.96  0.93  0.07 
Panel C: 
α_{i}  0.05  0.04  0.02  0.01  0.01  0.01  0.03  0.05  0.06  0.05  0.00 
β_{i}  0.95  0.96  0.98  0.99  1.00  0.99  0.98  0.95  0.93  0.95  0.00 
γ_{i}  −0.44  −0.43  −0.39  −0.32  −0.22  −0.09  0.08  0.26  0.40  0.56  1.00 
R^{2}_{i}  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
The second panel of Table IX shows alphas and betas relative to the CAPM. Annual excess portfolio returns are regressed on excess returns on the aggregate market. Alpha, beta, and the R^{2} are reported for each decile. As this panel shows, the model can replicate the classic result of Fama and French (1992): Value portfolios have positive alphas relative to the CAPM, while growth portfolios have negative alphas. Moreover, value portfolios tend to have lower betas than growth portfolios. Our model predicts alphas that rise from −2.6 for the extreme growth portfolio to 3.4 for the extreme value portfolio. In the data, the lowest booktomarket portfolio has an alpha of −1.7, while the highest booktomarket portfolio has an alpha of 4.0. Thus, the model generates alphas of the correct magnitude, as well as a sizable spread between value and growth. Moreover, alphas in the model are asymmetric: Growth alphas are smaller in absolute value than are value alphas, as in the data.
The third panel of Table IX shows results of regressing portfolio returns on the market return and on a highminuslow factor (HML) equal to the return on a portfolio short the extreme growth decile and long the extreme value decile. The purpose of this test is to see whether the model analogue to the highminuslow FamaFrench factor describes the cross section of returns in the model, as it does in the data. When we add HML to the regression, the alphas are indeed two orders of magnitude smaller than the alphas relative to the CAPM.
D.3. Relation to Conditional Factor Models
The previous discussion shows that the model replicates the high expected returns, low volatility, high Sharpe ratios, and high alphas of value stocks relative to growth stocks. The model also generates testable predictions. Because only the innovation to dividends is priced, expected returns on stocks should be determined by their conditional correlation with the aggregate dividend process. According to the model, a conditional CAPM does not hold because innovations to market returns are not perfectly conditionally correlated with innovations to dividends. Moreover, a conditional dividend CAPM should provide a better fit to the cross section than a conditional CAPM.
To evaluate these predictions, we compare pricing errors for an unconditional CAPM, an unconditional dividend CAPM, a conditional CAPM, and a conditional dividend CAPM in simulated and actual data. For the simulated data, the assets are the 10 portfolios formed on dividend–price ratios described above; for the actual data the assets are the 10 valueweighted booktomarketsorted portfolios. Theoretically, the conditioning variables should be x_{t} and z_{t}. However, because innovations in the price–dividend ratio are driven by innovations to these variables, the price–dividend ratio works well as a conditioning variable in data simulated from the model. To estimate each factor model, we solve min _{δ}[g(δ)′g(δ)], where g(δ) =E[δ′f_{t}R_{t}− 1] and R is the vector of returns. For the CAPM, f_{t}=[1, R^{m}_{t}]′; for the dividend CAPM, f_{t}=[1, Δd_{t}]′; for the conditional CAPM, f_{t}=[1, R^{m}_{t}, (d_{t−1}−p_{t−1}) R^{m}_{t}, d_{t−1}−p_{t−1}]′; and for the conditional dividend CAPM, f_{t}=[1, Δd_{t}, (d_{t−1}−p_{t−1}) Δd_{t}, d_{t−1}−p_{t−1}]′, where R^{m} is the market return, Δd_{t} is log dividend growth and d_{t}−p_{t} is the log dividend–price ratio on the market. In the data, the valueweighted CRSP portfolio is used to proxy for the market.
Table X reports the annualized square root of the squared average pricing errors for each factor model. The first column reports the results from the data, the second column reports results for which the dividend growth process and the price–dividend ratio are adjusted for repurchases as in Boudoukh et al. (2007), and the last column reports data simulated from the model. In both the data and the model, the unconditional CAPM fares the worst, with the unconditional dividend CAPM performing better. Both conditional factor models perform better than either unconditional model in the data, a finding that the model replicates. Moreover, in both the model and the data, the conditional dividend CAPM implies the lowest pricing errors of all the factor models.
Table X. Minimized Pricing Errors in the Data and in the Model A factor model is estimated by minimizing g(δ)′g(δ), where g(δ) =E[δ′f_{t}R_{t}− 1] and f_{t} is the vector of factors at time t. In the data, the return vector R consists of the 10 valueweighted booktomarket sorted portfolios. In the model, R consists of the 10 portfolios formed by sorting firms into deciles on the pricedividend ratio. For the CAPM, f_{t}=[1, R^{m}_{t}]′; for the dividend CAPM, f_{t}=[1, Δd_{t}]′; for the conditional CAPM, f_{t}=[1, R^{m}_{t}, (d_{t−1}−p_{t−1}) R^{m}_{t}, d_{t−1}−p_{t−1}]′; and for the conditional dividend CAPM, f_{t}=[1, Δd_{t}, (d_{t−1}−p_{t−1}) Δd_{t}, d_{t−1}−p_{t−1}]′, where R^{m} is the market return, Δd_{t} is log dividend growth, and d_{t}−p_{t} is the log dividendprice ratio. In the column “DataRepurchases,” dividends are adjusted for share repurchases. The table reports the annualized square root of the squared average pricing errors. The monthly data span the period 1952–1 to 2003–12.  Data  Avg. Pricing Error DataRepurchases  Model 

CAPM  1.634%  1.634%  0.571% 
Dividend CAPM  1.399%  1.076%  0.266% 
Cond. CAPM  0.930%  0.687%  0.033% 
Cond. Dividend CAPM  0.609%  0.492%  0.014% 
IV. Model Intuition
 Top of page
 ABSTRACT
 I. Evidence on the Value Premium
 II. The Model
 III. Implications for Equity Returns
 IV. Model Intuition
 V. Conclusion
 Appendix
 REFERENCES
What explains the model's ability to capture the value premium? As we suggest in Section II, the value premium arises from the differential correlations of returns on value and growth portfolios with underlying shocks.
Figure 7 plots betas from unconditional regressions of portfolio returns on the three shocks, and the R^{2} from the unconditional regressions. The coefficient on the dividend shock, β_{d}, is positive and greater for value portfolios than for growth portfolios. The coefficient on the shock to expected dividends, β_{z}, is also positive but smaller for value portfolios than for growth portfolios. While a shock to expected dividend growth raises the valuation of all portfolios, (as in the present value models of Campbell and Shiller (1988) and Vuolteenaho (2002)), it especially affects the valuations of growth stocks, which pay dividends in the distant future. Finally, all portfolios are negatively correlated with shocks to the Sharpe ratio variable x_{t}, as indicated by a negative β_{x}. A positive shock to x_{t} raises expected returns, and thus lowers prices and realized returns. Because of the duration effect, β_{x} is greater in magnitude for growth portfolios. The R^{2} coefficients follow the same pattern as the magnitude of the βs.^{14}
The patterns in Figure 7 can be traced back to the properties of zerocoupon equity. Growth firms place more weight on highduration zerocoupon claims than do value firms, and thus they inherit the sensitivity of these highduration claims to shocks to x_{t} and z_{t}. Interestingly, β_{x} does not inherit the nonmonotonicity of B_{x} in Figure 2. This is because, all else equal, equity that pays further in the future is worth less (Figure 3). Mediumhorizon equity may therefore have a greater weight than longhorizon equity, even for growth firms.
The loadings of portfolios on various shocks present an intriguing link with the empirical results of Campbell and Vuolteenaho (2004). Using the vector autoregression (VAR) methodology of Campbell (1991), Campbell and Vuolteenaho decompose unexpected market returns into changes in expectations of future discount rates and changes in expectations of future dividend growth rates. Changes in expected discount rates are computed using the VAR; changes in expected growth rates comprise the residual variation in market returns. Relative to value firms, growth firms have high betas with respect to news about discount rates, but low betas with respect to news about dividends.
While not precisely analogous, shocks to x_{t} are similar in spirit to news about discount rates in the Campbell and Vuolteenaho (2004) framework. It is therefore encouraging that our model produces betas with respect to shocks to x_{t} that are greater in magnitude for growth firms than for value firms. The analogue to dividend growth news is less clear in our model. Campbell and Vuolteenaho compute this as a residual, but Figure 7 shows that the residual variance is not accounted for by shocks to current or expected future dividends. Rather, we find that more of the residual is accounted for by Δd_{t+1} than by z_{t+1}. Thus, it is also encouraging that value portfolios load more on shocks to Δd_{t+1} than growth portfolios.
Figure 7 shows that value and growth portfolios have different loadings on the underlying shocks in the economy. How this translates into risk premia depends on the price of risk of these shocks. Equation (16) provides an illustration of how conditional risk premia on zerocoupon equity vary based on loadings on different shocks. As we discuss in Section III, we estimate that shocks to expected dividend growth z_{t} are negatively correlated with shocks to realized dividend growth. This empirical result implies that expected dividend growth has a negative risk price: Because it is negatively correlated with shocks to realized dividend growth, it serves as a hedge and reduces risk premia.
We assume that shocks to x_{t} are uncorrelated with shocks to realized dividends, and thus carry a zero risk price. This assumption represents a departure from the models of Campbell and Cochrane (1999) and Menzly et al. (2004), in which shocks to the price of risk are perfectly negatively correlated with shocks to aggregate dividends. What role does this assumption play in our analysis?
To answer this question, consider the conditional risk premium for equity that matures next period:
Equity that matures two periods from now has a risk premium of
where represents the conditional correlation between Δd_{t+1} and x_{t+1}, and represents the conditional correlation between Δd_{t+1} and z_{t+1}. The risk premium on equity that matures next period is equal to the quantity of risk (the standard deviation of dividends) multiplied by the price of risk, x_{t}. For equity maturing two periods from now, there is also the risk due to changes in x_{t} and changes in z_{t}. The latter effect is small because σ_{z} is a small fraction of σ_{d}. Whether longhorizon equity has a lower risk premium than shorthorizon equity depends in large part on the sign of the correlation of dividend growth with x_{t}. In particular, ρ_{dx} < 0 leads to relatively high premia for longhorizon equity, while ρ_{dx} > 0 leads to relatively low premia for longhorizon equity.
We make this statement precise by solving the model under three different values for ρ_{dx}. Figure 8 plots risk premia on zerocoupon equity when ρ_{dx}=− 0.5, ρ_{dx}= 0 (our base case), and ρ_{dx}= 0.5. For ρ_{dx}= 0, Panel B shows that risk premia decrease in maturity as long as x_{t} > 0 (as it is most of the time). The reason for this decrease is the negative correlation between Δd_{t+1} and z_{t+1}. In contrast, for ρ_{dx}=− 0.5, Panel A shows that risk premia generally increase with maturity. Longhorizon equity (i.e., growth stocks) have greater risk premia than do shorthorizon equity. This occurs even though ρ_{dz} is negative; even a modest correlation of −0.5 between dividends and the price of risk overrides the effect of ρ_{dz}. The case of ρ_{dx} < 0 is of special interest because it corresponds to the correlation between the price of risk and aggregate dividends in external habit models. In the models of Campbell and Cochrane (1999) and Menzly et al. (2004), shocks to the aggregate dividend (which is identified with consumption) increase surplus consumption, and therefore lower the amount of return investors demand for taking on risk. Indeed, in a term structure context, Wachter (2006) shows that the model of Campbell and Cochrane (1999) implies that longhorizon assets exhibit greater risk premia than do short horizon assets for exactly this reason. Longhorizon assets load more negatively on the shock to discount rates; if discount rates are negatively correlated with consumption (or dividends), then longhorizon assets will command greater risk premia.
An alternative is to set the correlation between d_{t+1} and x_{t+1} to be positive, as illustrated in Panel C. Under this assumption, risk premia fall more dramatically in maturity than when d_{t+1} and x_{t+1} are uncorrelated and the premium for shorthorizon equity is greater.
These results at first suggest that a model that seeks to explain the value premium should set ρ_{dx} > 0, rather than ρ_{dx}= 0 as we assume. However, the sign of ρ_{dx} has timeseries as well as crosssectional implications. We are able to calibrate our model to match the time series of aggregate stock returns, as well as the cross section of value and growth portfolios, because our model produces reasonable risk premia in the aggregate. For ρ_{dx} > 0, this may not be the case. Figure 8 shows that the greater is ρ_{dx}, the lower are risk premia in the economy, for all but the shortestmaturity equity. As an asset that pays cash flows in the future, equity must load negatively on x_{t}. If investors view x_{t}risk as a hedge (ρ_{dx} > 0), this makes equity less risky. On the other hand, if x_{t} moves in the same direction as dividends (ρ_{dx} < 0), equity becomes more risky. Explaining the level of the equity premium is therefore easiest when ρ_{dx} < 0 and hardest when ρ_{dx} > 0. The assumption that ρ_{dx} < 0 is part of what enables Campbell and Cochrane (1999) and Menzly et al. (2004) to explain both the high variance and the high premium that stocks command, with comparatively little variance in fundamentals. Faced with this tension between the time series and the cross section, we choose to set the correlation between dividend growth and x_{t} to zero.
In summary, this section shows that setting ρ_{dx} to zero, in combination with the duration effect and the correlation between current and future dividend growth, makes longhorizon equity less risky than shorthorizon equity. It creates a large premium on value stocks, while at the same time limiting their covariance with the market. We hope that future work will reveal microeconomic foundations that determine this important parameter.
V. Conclusion
 Top of page
 ABSTRACT
 I. Evidence on the Value Premium
 II. The Model
 III. Implications for Equity Returns
 IV. Model Intuition
 V. Conclusion
 Appendix
 REFERENCES
This paper proposes a parsimonious model of the stochastic discount factor that accounts for both the aggregate timeseries behavior of the stock market and the relative risk and return of value and growth stocks. At the root of the model is a dividend process calibrated to match the aggregate dividend process in the data, and a stochastic discount factor with a single factor, x_{t}, proxying for investors' timevarying preference for risk. Timevarying preferences for risk allow the model to capture the excess volatility and return predictability that obtain in the data. Our specification for x_{t} allows for interpretable closedforms solutions for asset prices and risk premia.
A key difference between our model and external habit models, which also feature timevarying preferences for risk, is that x_{t} does not arise from fluctuations in aggregate dividends. This may seem like a small detail but it is key to the model's ability to explain how value stocks can have both higher returns and lower betas than growth stocks. In our model, growth and value stocks differ based on the timing of their cash flows. Growth stocks have more of their cash flows in the future. They are highduration assets, and thus their returns covary more with the price of risk x_{t}. We show that for growth stocks to have relatively low returns, it must be the case that investors do not fear shocks to x_{t}. This only occurs if the conditional correlation of the price of risk with dividend growth is zero or positive. We assume that the correlation is zero. In contrast, external habit models assume a perfect negative correlation so that shocks to the price of risk are feared as much as, if not more than, shocks to cash flows.
Our proposed resolution of the value puzzle is risk based. Value stocks, as shorthorizon equity, vary more with fluctuations in cash flows, the fluctuations that investors fear the most. Growth stocks, as longhorizon equity, vary more with fluctuations in discount rates, which are independent of cash flows and which investors do not fear. As we show, such a resolution accounts for the timeseries behavior of the aggregate market, the relative returns of value and growth stocks, and the failure of the capital asset pricing model to explain these returns.