We characterize and measure a long-term risk-return trade-off for the valuation of cash flows exposed to fluctuations in macroeconomic growth. This trade-off features risk prices of cash flows that are realized far into the future but continue to be reflected in asset values. We apply this analysis to claims on aggregate cash flows and to cash flows from value and growth portfolios by imputing values to the long-run dynamic responses of cash flows to macroeconomic shocks. We explore the sensitivity of our results to features of the economic valuation model and of the model cash flow dynamics. I.

This paper proposes a dynamic risk-based model capable of jointly explaining the term structure of interest rates, returns on the aggregate market and the risk and return characteristics of value and growth stocks. Both the term structure of interest rates and returns on value and growth stocks convey information about how the representative investor values cash flows of different maturities. We model how the representative investor perceives risks of these cash flows by specifying a parsimonious stochastic discount factor for the economy. Shocks to dividend growth, the real interest rate, and expected inflation are priced, but shocks to the price of risk are not. Given reasonable assumptions for dividends and inflation, we show that the model can simultaneously account for the behavior of aggregate stock returns, an upward-sloping yield curve, the failure of the expectations hypothesis and the poor performance of the capital asset pricing model.

The covariance between US Treasury bond returns and stock returns has moved considerably over time. While it was slightly positive on average in the period 1953— 2005, it was particularly high in the early 1980’s and negative in the early 2000’s. This paper specifies and estimates a model in which the nominal term structure of interest ratesisdrivenbyfive state variables: the real interest rate, risk aversion, temporary and permanent components of expected inflation, and the covariance between nominal variables and the real economy. The last of these state variables enables the model to fit the changing covariance of bond and stock returns. Log nominal bond yields and term premia are quadratic in these state variables, with term premia determined mainly by the product of risk aversion and the nominal-real covariance. The concavity of the yield curve–the level of intermediate-term bond yields, relative to the average of short- and long-term bond yields–is a good proxy for the level of term premia. The nominal-real covariance has declined since the early 1980’s, driving down term

City for useful comments and discussion. We also thank Ken French for making the data available through his website. Jason Higbee provided excellent research assistance. The views expressed in this paper are those of the authors and do not necessarily reflect the official positions of the Federal Reserve Bank of St. Louis or the Federal Reserve System. Is the Value Premium a Proxy for Time-Varying Investment Opportunities: Some Time-Series Evidence We uncover a positive stock market risk-return tradeoff after controlling for the covariance of market returns with the value premium. Fama and French (1996) conjecture that the value premium proxies for investment opportunities; therefore, by ignoring it, early specifications suffer from an omitted variable problem that causes a downward bias in the risk-return tradeoff estimation. We also document a positive relation between the value premium and its conditional variance, and the estimated conditional value premium is strongly countercyclical. The latter evidence supports the view that value is riskier than growth in bad times, when the price of risk is high.

I explore the equilibrium value implications of economic models that incorporate reactions to a stochastic environment. I propose a dynamic value decomposition (DVD) designed to distinguish components of an underlying economic model that influence values over long horizons from components that impact only the short run. To quantify the role of parameter sensitivity and to impute long-term risk prices, I develop an associated perturbation technique. Finally, I use DVD methods to study formally some example economies and to speculate about others. A DVD is enabled by constructing operators indexed by the elapsed time between the date of pricing and the date of the future payoff (i.e. the future realization of a consumption claim). Thus formulated, methods from applied mathematics permit me to characterize valuation behavior as the time between price determination and payoff realization becomes large. An outcome of this analysis is the construction of a multiplicative martingale component of a process that is used to represent valuation in a dynamic economy with stochastic growth. I contrast the differences in the applicability between this multiplicative martingale method and an additive martingale method familiar from time series analysis that is used to identify shocks with long-run economic consequences.

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This paper considers the impact of takeover (or acquisition) likelihood on firm valuation. If firms are more

We present a novel approach to depicting asset pricing dynamics by characterizing shock exposures and prices for alternative investment horizons. We quantify the shock exposures in terms of elasticities that measure the impact of a current shock on future cash-flow growth. The elasticities are designed to accommodate nonlinearities in the stochastic evolution modeled as a Markov process. Stochastic growth in the underlying macroeconomy and stochastic discounting in the representation of asset values are central ingredients in our investigation. We provide elasticity calculations in a series of examples featuring consumption externalities, recursive utility, and jump risk.

Dynamicstochasticequilibriummodelsofthemacroeconomyaredesignedtomatch the macro time series including impulse response functions. Since these models aim to be structural, they also have implications for asset pricing. To assess these implications, we explore asset pricing counterparts to impulse response functions. We use the resulting dynamic value decomposition (DVD) methods to quantify the exposures of macroeconomic cash flows to shocks over alternative investment horizons and the corresponding prices or compensations that investors must receive because of the exposure to such shocks. We build on the continuous-time methods developed in Hansen and Scheinkman (2010), Borovička et al. (2011) and Hansen (2011) by constructing discrete-time shock elasticities that measure the sensitivity of cash flows and their prices to economic shocks including economic shocks featured in the empirical macroeconomics literature. By design, our methods are applicable to economic models that are nonlinear, including models with stochastic volatility. We illustrate our methods by analyzing the asset pricing model of Ai et al. (2010) with tangible and intangible capital.

By reinterpreting the symbols, one-period mean-variance portfolio theory can apply to dynamic intertemporal problems in incomplete markets, with non-marketed income. Investors first hedge non-traded income and preference shocks. Then, their optimal payoffs are split between an indexed perpetuity and a “long-run mean-variance efficient ” payoff, which avoids variation over time as well as variation across states of nature. In equilibrium, the market payoff and the average outside-income hedge payoff span the long-run mean-variance frontier, and long-run expected returns are linear functions of long-run market and outsideincome-hedge betas. State variables for investment opportunities and outside income are conveniently absent in these characterizations.