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.

Much of Mike Jensen's research is foundational, including his early publications, which focus on empirical asset pricing. For example, Jensen's alpha, which he developss in Jensen (1968 and 1969) to evaluate mutual fund managers, is the foundation for most measures of investment performance. Similarly, in Fama et al (1969), Jensen and coauthors present the first event study, Thereafter, event studies play a major role in finance, accounting, and legal research. Finally, Black, Jensen, and Scholes (1972) develop a key insight about the importance of interdependence of sampling errors in the precision of asset pricing tests.

Our goal is to develop insights about the maximum squared Sharpe ratio for model factors as a metric for ranking asset pricing models. We consider nested and non-nested models. The nested models are the capital asset pricing model, the three-factor model of Fama and French (1993), the five-factor extension in Fama and French (2015), and a six-factor model that adds a momentum factor. The non-nested models examine three issues about factor choice in the six-factor model: (1) cash profitability versus operating profitability as the variable used to construct profitability factors, (2) long-short spread factors versus excess return factors, and (3) factors that use small or big stocks versus factors that use both.

Average stock returns for North America, Europe, and Asia Pacific increase with the book-to-market ratio (B/M) and profitability and are negatively related to investment. For Japan, the relation between average returns and B/M is strong, but average returns show little relation to profitability or investment. A five-factor model that adds profitability and investment factors to the three-factor model of Fama and French (1993) largely absorbs the patterns in average returns. As in Fama and French, 2015, Fama and French, 2016, the model's prime problem is failure to capture fully the low average returns of small stocks whose returns behave like those of low profitability firms that invest aggressively.

A five-factor model directed at capturing the size, value, profitability, and investment patterns in average stock returns performs better than the three-factor model of Fama and French (FF, 1993). The five-factor model׳s main problem is its failure to capture the low average returns on small stocks whose returns behave like those of firms that invest a lot despite low profitability. The model׳s performance is not sensitive to the way its factors are defined. With the addition of profitability and investment factors, the value factor of the FF three-factor model becomes redundant for describing average returns in the sample we examine.

Variables with strong marginal explanatory power in cross-section asset pricing regressions typically show less power to produce increments to average portfolio returns, for two reasons. (1) Adding an explanatory variable can attenuate the slopes in a regression. (2) Adding a variable with marginal explanatory power always attenuates the values of other explanatory variables in the extremes of a regression’s fitted values. Without a restriction on portfolio weights, the maximum Sharpe ratios in the GRS statistic of Gibbons, Ross, and Shanken (1989) provide little information about an incremental variable’s impact on the portfolio opportunity set.

In the four regions (North America, Europe, Japan, and Asia Pacific) we examine, there are value premiums in average stock returns that, except for Japan, decrease with size. Except for Japan, there is return momentum everywhere, and spreads in average momentum returns also decrease from smaller to bigger stocks. We test whether empirical asset pricing models capture the value and momentum patterns in international average returns and whether asset pricing seems to be integrated across the four regions. Integrated pricing across regions does not get strong support in our tests. For three regions (North America, Europe, and Japan), local models that use local explanatory returns provide passable descriptions of local average returns for portfolios formed on size and value versus growth. Even local models are less successful in tests on portfolios formed on size and momentum.

Standard asset pricing models assume that: (i) there is complete agreement among investors about probability distributions of future payoffs on assets; and (ii) investors choose asset holdings based solely on anticipated payoffs; that is, investment assets are not also consumption goods. Both assumptions are unrealistic. We provide a simple framework for studying how disagreement and tastes for assets as consumption goods can affect asset prices.

Valuation theory says that expected stock returns are related to three variables: the book-to-market equity ratio (Bt/Mt), expected profitability, and expected investment. Given Bt/Mt and expected profitability, higher expected rates of investment imply lower expected returns. But controlling for the other two variables, more profitable firms have higher expected returns, as do firms with higher Bt/Mt. These predictions are confirmed in our tests.