CAPM time-series and cross-sectional tests on the 25 Fama-French size and book-to-market portfolios, 1964 to 1993, then out to 2023.
Twenty-five portfolios sorted on size and book-to-market, monthly excess returns regressed on the market excess return, 1964 to 1993. For each portfolio: beta, alpha, both t-statistics, and adjusted R-squared.
The design of the test matters. A single-factor model can fit the time series beautifully and still fail completely at pricing the cross-section, and that separation is exactly what these portfolios are built to expose.
Average monthly excess returns rise from roughly 0.3 to 0.4% for low book-to-market portfolios to around 1% for high book-to-market portfolios. That is the value effect, and it is economically large.
Betas across the same low-to-high movement are flat or slightly declining. So the CAPM predicts no increase in expected return, or a small decrease, precisely where the realised increase is largest. The size direction is weaker and not cleanly monotonic, which makes book-to-market the dominant cross-sectional feature in this sample.
High book-to-market portfolios, particularly at small and mid sizes, show positive alphas with t-statistics above 1.96. The pricing restriction is rejected.
At the same time adjusted R-squared is high nearly everywhere. The market factor explains most of the time-series variation in these portfolios while failing to explain the differences in their average returns. Those are different jobs and the model only does one of them.
The right reading of a significant alpha here is model misspecification, not an arbitrage opportunity. Something systematic and correlated with book-to-market is priced, and a one-factor model has no slot for it.
Repeating the actual-versus-predicted scatter over 1933 to 1963 and 1933 to 2023 shows deviations from the 45-degree line in both. The earlier window is the kinder one: fewer significant alphas, but a value spread still worth about 5.96% a year.
Over the full 1933 to 2023 sample the failure is sharper. Eight of twenty-five portfolios carry significant alphas and the value premium is about 5.26% a year. The conclusion is that the cross-sectional failure is structural rather than an artefact of one sample window.