Part 7 of my notes on empirical asset pricing
The question
Small stocks and value stocks earn higher average returns. Everyone agrees on that. The fight is over why:
- The risk story (Fama & French, 1993): value stocks move together because they share exposure to some undiversifiable risk, perhaps distress. Investors demand a premium for bearing it. What earns the premium is the covariance: the loading on HML.
- The characteristic story (Daniel & Titman, 1997): the premium attaches to the characteristic itself (a high B/M ratio), whatever the reason (mispricing, investor preferences). How the stock co-moves with other value stocks is beside the point.
This matters for everything in posts #5 and #6. A factor model assumes loadings drive expected returns. If characteristics do, a factor model can fit well and still be telling the wrong story.
Why the usual tests can’t tell them apart
Here’s the trap. Book-to-market and HML loading are highly correlated. Value stocks tend to load on HML, because HML is built from value stocks. So when Fama & French show that HML loadings explain the returns of B/M-sorted portfolios, both stories predict that exact result.
Daniel & Titman make a sharp point: tests that use characteristic-sorted portfolios as factors may fail to reject a factor model even when it’s wrong. The factor inherits the characteristic’s premium, so it looks like it “prices” everything.
To separate the stories, you need stocks where the characteristic and the loading disagree: value stocks that don’t move like value stocks, and growth stocks that do.
The design: characteristic-balanced sorts
This is a clever extension of the triple sort from post #4.
Step 1. Sort on characteristics. Form a 3×3 grid on size and B/M (breakpoints at the 33rd and 67th percentiles). Within each cell, stocks have similar characteristics.
Step 2. Within each cell, sort on the loading. Split each of the 9 cells into 5 sub-portfolios by each stock’s pre-formation HML loading. That gives 45 portfolios. Within a cell, B/M and size are roughly constant, but the HML loading varies.
Step 3. Compare. If loadings drive returns, the high-loading sub-portfolio should earn more than the low-loading one in the same cell. If characteristics drive returns, they should earn about the same.
Two details show real care:
- Loadings are estimated from months −42 to −7, deliberately skipping the 6 months before formation. The stocks’ B/M ratios are measured with prices from 6 months earlier, and returns in that gap jump in a way that would add noise to the loading estimates.
- The factor portfolios use constant weights. Instead of the usual annually rebalanced HML, they fix the weights at the formation date and apply them backward. The idea is to measure how a stock co-moves with today’s value stocks, which should better predict its future covariance.
They also check that the sort worked: post-formation loadings line up monotonically with the pre-formation sort. So the method really does create variation in loadings within each characteristic group.
The results
Loadings don’t predict returns. Across the five loading sub-portfolios, average returns barely differ: the gap between lowest and highest loading is 0.07% per month.
So the factor model misprices them. Run the three-factor regression from post #5 on those 45 portfolios:
- Low-loading portfolios: average alpha about +0.02% per month.
- High-loading portfolios: average alpha about −0.24% per month.
This is exactly the characteristic model’s prediction. The factor model expects high-loading stocks to earn more. They don’t, so they show up with negative alphas.
The cleanest test. In each cell, go long the two low-loading sub-portfolios and short the two high-loading ones. Both legs have about the same size and B/M, so this is a characteristic-balanced zero-cost portfolio. Combine the nine cells:
| Factor model predicts | Characteristic model predicts | Actual | |
|---|---|---|---|
| Average return | Negative (it’s short high loadings) | About zero | −0.12%/month (t = −0.60) |
| Three-factor alpha | About zero | Positive | +0.35%/month (t = 2.30) |
The data match the characteristic model on both rows. Daniel & Titman run the same test on SMB and market-beta loadings with the same result. After controlling for size and B/M, even market beta has no explanatory power.
And what about “distress”?
They also attack the idea of a distress factor directly. Value stocks do co-move strongly. But Daniel & Titman show the co-movement was just as strong before the firms became distressed. Their interpretation: stocks that share exposures, such as the same industry or region, tend to fall on hard times together. That makes them look like a “distress factor” after the fact, when really they just have similar businesses.
Seasonality, a detail worth knowing
Their Table I splits returns into January and non-January months. With value-weighted portfolios and NYSE breakpoints, the size effect is almost entirely a January phenomenon. It’s a reminder that an average premium can hide very uneven timing, and that a shared seasonal pattern can create the look of a common factor.
How this connects to the other readings
- Fama & French (2015) notice the same tension from the other side: their regression slopes don’t always line up with characteristics. Value stocks load positively on RMW (robust profitability) even though they tend to be less profitable. They caution that multivariate slopes and univariate characteristics measure different things.
- Bessembinder & Zhang (2013) take a firmly characteristic-based approach: they control for event-firm characteristic differences directly, not loadings. That’s what dissolves the long-run anomalies in post #3.
- Pástor & Stambaugh (2003) is a covariance story: they argue that the sensitivity to aggregate liquidity, not the level of a stock’s liquidity, is priced. A Daniel–Titman-style test is exactly how you’d challenge that claim.
The debate isn’t settled. Fama & French and coauthors later argued, with a longer sample, that loadings do carry explanatory power. The design is the lasting contribution: to test risk against characteristics, you have to break the correlation between them.
My checklist
- Does the paper claim a premium is “risk”? Then ask whether characteristics and loadings were separated.
- Were the test assets sorted on the same characteristics as the factors? If so, a good fit proves less than it seems.
- Are loadings estimated out of sample, before the formation date?
- Does sorting on loadings actually produce different post-formation loadings?
- Is the premium spread evenly over the year, or concentrated in January?
Next post: matching, difference-in-differences and synthetic control. How Brown & Huang and Acemoglu et al. build credible comparison groups when you can’t run an experiment.
References
Bessembinder, H., & Zhang, F. (2013). Firm characteristics and long-run stock returns after corporate events. Journal of Financial Economics, 109(1), 83–102. https://doi.org/10.1016/j.jfineco.2013.02.009
Daniel, K., & Titman, S. (1997). Evidence on the characteristics of cross sectional variation in stock returns. Journal of Finance, 52(1), 1–33.
Davis, J. L., Fama, E. F., & French, K. R. (2000). Characteristics, covariances, and average returns: 1929 to 1997. Journal of Finance, 55(1), 389–406.
Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33, 3–56.
Fama, E. F., & French, K. R. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116(1), 1–22. https://doi.org/10.1016/j.jfineco.2014.10.010
Pástor, Ľ., & Stambaugh, R. F. (2003). Liquidity risk and expected stock returns. Journal of Political Economy, 111(3), 642–685.