If an individual with expected utility and a reasonable level of wealth rejects a small actuarially favorable gamble, it implies a very high degree of risk aversion. It also predicts (counterfactually) the rejection of more sizable and very attractive bets. If additional background uncertainty affects wealth, this result also applies to non-expected utilities. The authors describe a set of reasonable conditions under which an individual may reject the small bet but accept the large bet, even in the presence of background uncertainty. The two critical assumptions that the authors use are rank-dependent utility and a discrete distribution for background risk. Plausible calibrations can reconcile large/small bet risk attitudes and the empirical evidence on limited stock market participation in the presence of labor income risk.

This article tests a simple consumption‐based asset pricing model by approximating the true asset pricing kernel using low‐order orthonormal polynomials based on the model's state variables. Approximated kernels based solely on next period's consumption growth are not rejected by overall measures of model fit, but they produce statistically and economically large pricing errors. Approximated kernels based on two quarters of future consumption growth and technology shocks have substantially improved overall fit. In particular, the best of these kernels are capable of eliminating the small firm effect.

This article tests a simple consumption-based asset pricing model by approximating the true asset pricing kernel using low-order orthonormal polynomials based on the model's state variables. Approximated kernels based solely on next period's consumption growth are not rejected by overall measures of model fit but they produce statistically and economically large pricing errors. Approximated kernels based on two quarters of future consumption growth and technology shocks have substantially improved overall fit. In particular, the best of these kernels are capable of eliminating the small firm effect.

This article tests a simple consumption-based asset pricing model by approximating the true asset pricing kernel using low-order orthonormal polynomials based on the model's state variables. Approximated kernels based solely on next period's consumption growth are not rejected by overall measures of model fit but they produce statistically and economically large pricing errors. Approximated kernels based on two quarters of future consumption growth and technology shocks have substantially improved overall fit. In particular, the best of these kernels are capable of eliminating the small firm effect.

If a nonlinear risk premium in a conditional asset pricing model is approximated with a linear function, as is commonly done in empirical research, the fitted model is misspecified. We use a generic reduced-form model economy with moderate risk premium nonlinearity to examine the size of the resulting misspecification-induced pricing errors. Pricing errors from moderate nonlinearity can be large, and a version of a test for nonlinearity based on risk premiums rather than pricing errors has reasonable power properties after properly controlling for the size of the test. We conclude by examining the importance of moderate nonlinearity in the context of the investment-specific technology shock models of Papanikolaou (2011) and Kogan and Papanikolaou (2014).

We examine a wide range of two-date economies populated by heterogeneous agents with the most common forms of nonexpected utility preferences used in finance and macroeconomics. We demonstrate that the risk premium and the risk-free rate in these models are sensitive to ignoring heterogeneity. This follows because of endogenous withdrawal by nonexpected utility agents from the market for the risky asset. This finding is important precisely because these alternative preferences have frequently been proposed as possible resolutions to various asset pricing puzzles, and they have all been examined exclusively in a representative agent framework.

We examine a wide range of two‐date economies populated by heterogeneous agents with the most common forms of nonexpected utility preferences used in finance and macroeconomics. We demonstrate that the risk premium and the risk‐free rate in these models are sensitive to ignoring heterogeneity. This follows because of endogenous withdrawal by nonexpected utility agents from the market for the risky asset. This finding is important precisely because these alternative preferences have frequently been proposed as possible resolutions to various asset pricing puzzles, and they have all been examined exclusively in a representative agent framework.

Do stocks bear a crash risk premium? We examine the empirical performance of the tail index measure from Kelly and Jiang (2014). We find that the tail index explains the cross-section of the discount rate component of returns, but not the cash-flow component. Moreover, in the time series the tail index is uncorrelated with theoretically motivated measures of aggregate uncertainty and systemic risk. In contrast, the tail index Granger causes and is Granger caused by the level of the term structure, and the slope of the term structure Granger causes tail risk. Received June 22, 2016; editorial decision December 23, 2017 by Editor Raman Uppal.

[The dynamics of the unobservable short rate are frequently estimated directly using a proxy. We examine the biases resulting from this practice (the "proxy problem"). Analytic results show that the proxy problem is not economically significant for single-factor affine models. In the two-factor affine model of Longstaff and Schwartz (1992), the proxy problem is only economically significant for pricing discount bonds with maturities of more than five years. We also describe two different numerical procedures for assessing the magnitude of the proxy problem in a general interest rate model. When applied to a nonlinear single-factor model, they suggest that the proxy problem can be economically significant.]