We derive lower and upper bounds on the conditional expected excess market return that are related to risk-neutral volatility, skewness, and kurtosis indexes. The bounds can be calculated in real time using a cross section of option prices. The bounds require a no-arbitrage assumption, but they do not depend on distributional assumptions about market returns or past observations. The bounds are highly volatile, positively skewed, and fat-tailed. They imply that the term structure of expected excess holding period returns is decreasing during turbulent times and increasing during normal times and that the expected excess market return is on average 5.2%.

In this paper, we develop lower bounds on the variance of the permanent component and the transitory component, and on the variance of the ratio of the permanent to the transitory components of SDFs. Exactly solved eigenfunction problems are then used to study the empirical attributes of asset pricing models that incorporate long-run risk, external habit persistence, and rare disasters. Specific quantitative implications are developed for the variance of the permanent and the transitory components, the return behavior of the long-term bond, and the comovement between the transitory and the permanent components of SDFs.

This paper investigates whether multivariate crash risk (MCRASH), defined as exposure to extreme realizations of multiple systematic factors, is priced in the cross-section of expected stock returns. We derive an extended linear model with a positive premium for MCRASH, and we empirically confirm that stocks with high MCRASH earn significantly higher future returns than stocks with low MCRASH. The premium is not explained by linear factor exposures, alternative downside risk measures, or stock characteristics. Extending market-based definitions of crash risk to other well-established factors helps to determine the cross-section of expected stock returns without further expanding the factor zoo.