Prior studies find that a strategy that buys high-beta stocks and sells low-beta stocks has a significantly negative unconditional capital asset pricing model (CAPM) alpha, such that it appears to pay to "bet against beta." We show, however, that the conditional beta for the high-minus-low beta portfolio covaries negatively with the equity premium and positively with market volatility. As a result, the unconditional alpha is a downward-biased estimate of the true alpha. We model the conditional market risk for beta-sorted portfolios using instrumental variables methods and find that the conditional CAPM resolves the beta anomaly.THE SHARPE-LINTNER CAPITAL asset pricing model (CAPM) implies that exposure to market risk, as measured by beta, should be compensated by the market risk premium. Based on the performance of portfolios formed on lagged firm-level beta, however, a number of empirical studies find that the riskreward relation is too flat. For example, Friend and Blume (1970) and Black, Jensen, and Scholes (1972) demonstrate that portfolios of high-beta stocks earn lower returns than implied by the CAPM and therefore have negative alphas, whereas portfolios of low-beta stocks earn positive alphas. French (1992, 2006) extend these results by showing that the beta-return relation becomes even flatter after controlling for size and book-to-market characteristics. Finally, Frazzini and Pedersen (2014) confirm the underperformance of highbeta stocks over a long sample period extending from 1926 to 2012 and develop a "betting-against-beta" strategy, which has drawn substantial interest from academics and practitioners alike. 1 * Scott Cederburg is with the Eller College of Management, University of Arizona. Michael O'Doherty is with the Trulaske College of Business, University of Missouri. We are grateful to Phil Davies and Rick Sias for their detailed suggestions on the paper. We also thank 737 738The Journal of Finance R Figure 1. Cross-sectional distribution of firm betas, July 1927 to December 2012. The figure displays statistics for the cross-sectional distribution of firm betas. The dashed line is the median and the solid lines show the 5 th and 95 th percentiles of firm betas. Firm betas are estimated at the beginning of each month using daily returns over the previous 12 months.

This paper proposes an intertemporal asset pricing model within a long-run risk economy featuring a formal cross section of firms characterized by mean-reverting expected dividend growth. We find considerable empirical support for the cross-sectional implications of the model, as cash flow-and return-based measures of long-run risk exposure are both positively related to returns and offer a partial explanation of the size, value, and momentum anomalies. Interestingly, the model implies a negative relation between exposures to systematic and firm-specific risks in the cross section. Higher cash-flow duration firms exhibit higher exposure to economic growth shocks while they are less sensitive to firm-specific news. Such firms command higher risk premiums but exhibit lower analyst forecast dispersion, idiosyncratic volatility, and distress risk. We find theoretical and empirical support of a long-run risk explanation of these anomalies.

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We study whether stocks are riskier or safer in the long run from the perspective of Bayesian investors who employ the long-run risk, habit formation, or prospect theory models to form prior beliefs about return dynamics. Economic theory delivers important guidance for long-run investment opportunities. Specifically, incorporating prior information from the habit formation or prospect theory models reinforces beliefs in mean reversion and inferences that stocks are safer over longer horizons. Conversely, investors with long-run risk priors perceive weaker mean reversion and riskier equities. Model-based information is particularly important for inferences about uncertainty in the dividend growth component of returns. (JEL C11, G11, G12) * We thank Stijn Van Nieuwerburgh (the editor), two anonymous referees, 1 the models for long-run return dynamics, their success in matching these aspects of the data lend them credibility in this setting. The underlying assumptions in the long-run risk, habit formation, and prospect theory models about preferences and economic dynamics as well as the resulting return dynamics are quite different, which leads to interesting implications for long-horizon variance. In addition, our analysis incorporates a nonnegativity restriction on the equity premium as proposed by Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014). This economic constraint has been shown to improve out-of-sample forecasting performance and sharpen estimates of model parameters. Incorporating a nonnegative equity premium is novel in the context of long-horizon return variance.We study the risk of investing in stocks by examining the predictive return variance over various investment horizons. Predictive variance is typically computed (e.g., Avramov 2002) based on the vector autoregression (VAR) framework of Campbell and Shiller (1988), Kandel and Stambaugh (1996), Campbell and Viceira (1999), and Barberis (2000), among others. Our Bayesian investors consider long-horizon risk in the context of a VAR with stock returns, dividend growth, and model-implied state variables from the long-run risk, habit formation, or prospect theory models. Specifically, the investors form prior beliefs about the VAR parameters based on asset pricing model implications. They also allow for the possibility of model misspecification, such that prior uncertainty about the VAR parameters remains. Finally, the investors combine information from the model-based prior and historical data to make inferences about long-horizon predictive variance.Incorporating model-based prior information can potentially improve inferences about risk over long horizons. As noted earlier, the perceived long-horizon return variance crucially depends on the return dynamics implied by the VAR. It has long been recognized that incorporating prior information into VARs improves the quality of prediction. In the macroeconomics literature, Litterman (1986) and Todd (1984) demonstrate improved predictions by Bayesian VARs with the "Minnesota" ...

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