This paper proposes a dynamic risk-based 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 cross-section of long-lived 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 long-term bonds, growth firms are high-duration assets while value firms are low-duration 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 time-varying 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, risk-adjusted 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 book-to-market. Moreover, the CAPM alpha on the value-minus-growth 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 book-to-market 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 time-varying 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 exponential-affine in the state variables, which allows for economically interpretable closed-form expressions for prices and risk premia. Motivated by these expressions, Brennan et al. empirically evaluate whether expected returns on a cross-section 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 long-run 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 high-duration 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 book-to-market, earnings-to-price, or cash-flow-to-price, 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 (zero-coupon equity). Because zero-coupon equity has a well-defined 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 zero-coupon equity claims. We then introduce a cross section of long-lived assets, defined by their shares in the aggregate dividend. These assets are themselves portfolios of zero-coupon 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 time-series and cross-sectional implications of our model. We calibrate the model to the time series of aggregate returns, dividends, and the price-dividend ratio. After choosing parameters to match aggregate time-series facts, we examine the implications for zero-coupon equity. We find that the parameters necessary to fit the time series imply risk premia, Sharpe ratios, and alphas for zero-coupon 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 near-term 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 near-term 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 long-horizon 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.