Value premiums, which we define as value portfolio returns in excess of market portfolio returns, are on average much lower in the second half of the July 1963–June 2019 period. But the high volatility of monthly premiums prevents us from rejecting the hypothesis that expected premiums are the same in both halves of the sample. Regressions that forecast value premiums with book-to-market ratios in excess of market (BM–BMM) produce more reliable evidence of second-half declines in expected value premiums, but only if we assume the regression coefficients are constant during the sample period. Received: January 21, 2020; editorial decision: July 21, 2020; Editor: Jeffrey Pontiff.

We use bootstrap simulations to examine the properties of long-horizon U.S. stock market returns. We document the rate at which continuously compounded market returns converge toward normal distributions as we extend the horizon from 1 month to 30 years, and the rate at which dollar payoffs converge toward lognormal. We also verify that, though largely irrelevant at short horizons, uncertainty about the expected market return has a substantial impact on uncertainty about long-horizon payoffs.