We quantify investors' preferences over the dynamics of shocks by deriving frequency-specific risk prices that capture the price of risk of consumption fluctuations at each frequency. The frequency-specific risk prices are derived analytically for leading models. The decomposition helps measure the importance of economic fluctuations at different frequencies. We precisely quantify the meaning of "long-run" in the context of Epstein-Zin preferences -centuries -and measure the exact relevance of business-cycle fluctuations. Finally, we estimate frequency-specific risk prices and show that cycles longer than the business cycle -long-run risks -are significantly priced in the equity market.

We show that the stock market downturns of 2000–2002 and 2007–2009 have very different proximate causes. The early 2000s saw a large increase in the discount rates applied to profits by rational investors, while the late 2000s saw a decrease in rational expectations of future profits. We reach these conclusions by using a VAR model of aggregate stock returns and valuations, estimated both without restrictions and imposing the cross-sectional restrictions of the intertemporal capital asset pricing model (ICAPM). Our findings imply that the 2007–2009 downturn was particularly serious for rational long-term investors, whose losses were not offset by improving stock return forecasts as in the previous recession.

As illustrated in the tale of “the dog that did not bark,” the absence of news and the passage of time often contain information. We test whether markets fully incorporate this information using the empirical context of mergers. During the year after merger announcement, the passage of time is informative about the probability that the merger will ultimately complete. We show that the variation in hazard rates of completion after announcement strongly predicts returns. This pattern is consistent with a behavioral model of underreaction to the passage of time and cannot be explained by changes in risk or frictions.

In affine asset pricing models, the innovation to the pricing kernel is a function of innovations to current and expected future values of an economic state variable, for example consumption growth, aggregate market returns, or short-term interest rates. The impulse response of this priced variable to fundamental shocks has a frequency (Fourier) decomposition, which captures the fluctuations induced in the priced variable at different frequencies. We show that the price of risk for a given shock can be represented as a weighted integral over that spectral decomposition. The weight assigned to each frequency then represents the frequency-specific price of risk, and is entirely determined by the preferences of investors. For example, standard Epstein-Zin preferences imply that the weight of the pricing kernel lies almost entirely at extremely low frequencies, most of it on cycles longer than 230 years; internal habit-formation models imply that the weight is shifted to high frequencies. We estimate the frequency-specific risk prices for the equity market, focusing on economically interesting frequencies. Most of the pricing weight falls on low frequencies -corresponding to cycles longer than 8 years -broadly consistent with Epstein-Zin preferences.

We document a form of excess volatility that is difficult to reconcile with standard models of prices, even after accounting for variation in discount rates. We compare prices of claims on the same cash flow stream but with different maturities. Standard models impose precise internal consistency conditions on the joint behavior of long- and short-maturity claims and these are strongly rejected in the data. In particular, long-maturity prices are significantly more variable than justified by the behavior at short maturities. We reject internal consistency conditions in all term structures that we study, including equity options, currency options, credit default swaps, commodity futures, variance swaps, and inflation swaps.

This paper studies the pricing of volatility risk using the first-order conditions of a long-term equity investor who is content to hold the aggregate equity market instead of overweighting value stocks and other equity portfolios that are attractive to short-term investors. We show that a conservative long-term investor will avoid such overweights to hedge against two types of deterioration in investment opportunities: declining expected stock returns and increasing volatility. We present novel evidence that low-frequency movements in equity volatility, tied to the default spread, are priced in the cross section of stock returns.

This article studies how systemic risk and financial market distress affect the distribution of shocks to real economic activity. We analyze how changes in 19 different measures of systemic risk skew the distribution of subsequent shocks to industrial production and other macroeconomic variables in the US and Europe over several decades. We also propose dimension reduction estimators for constructing systemic risk indexes from the cross section of measures and demonstrate their success in predicting future macroeconomic shocks out of sample.

Standard estimators of risk premia in linear asset pricing models are biased if some priced factors are omitted. We propose a three-pass method to estimate the risk premium of an observable factor, which is valid even when not all factors in the model are specified or observed. The risk premium of the observable factor can be identified regardless of the rotation of the other control factors if together they span the true factor space. Our approach uses principal components of test asset returns to recover the factor space and additional regressions to obtain the risk premium of the observed factor.

We estimate how households trade off immediate costs and uncertain future benefits that occur in the very long run, 100 or more years away. We exploit a unique feature of housing markets in the United Kingdom and Singapore, where residential property ownership takes the form of either leaseholds or freeholds. Leaseholds are temporary, prepaid, and tradable ownership contracts with maturities between 99 and 999 years, while freeholds are perpetual ownership contracts. The price difference between leaseholds and freeholds reflects the present value of perpetual rental income starting at leasehold expiration, and is thus informative about very long-run discount rates. We estimate the price discounts for varying leasehold maturities compared to freeholds and extremely long-run leaseholds via hedonic regressions using proprietary data sets of the universe of transactions in each country. Households discount very long-run cash flows at low rates, assigning high present value to cash flows hundreds of years in the future. For example, 100-year leaseholds are valued at more than 10% less than otherwise identical freeholds, implying discount rates below 2.6% for 100-year claims.

We provide evidence on the relationship between aggregate uncertainty and the macroeconomy. Identifying uncertainty shocks using methods from the news shocks literature, the analysis finds that innovations in realized stock market volatility are robustly followed by contractions, while shocks to forward-looking uncertainty have no significant effect on the economy. Moreover, investors have historically paid large premia to hedge shocks to realized but not implied volatility. A model in which fundamental shocks are skewed left can match those facts. Aggregate volatility matters, but it is the realization of volatility, rather than uncertainty about the future, that has been associated with declines.