We examine the information transmission role of stock recommendation revisions by sell-side security analysts. Revisions are associated with economically insignificant mean price reactions and often piggyback on recent news, events, long-term momentum, and short-run contrarian return predictors, typically downgrading after bad news and upgrading after good news. However, the revisions are usually information-free for investors. The findings go against the long-standing view that recommendations are an important means by which analysts assimilate information into stock prices. They disagree with the view of policymakers that analysts’ stock picks materially impact stock prices.

We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous-time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. The members of this family are indexed by the elapsed time between payoff and valuation dates, and they are necessarily related via a mathematical structure called a semigroup. We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale, and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state. We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk-return trade-off.