Part 4 of my notes on empirical asset pricing
The idea in one sentence
If you think some characteristic predicts returns, rank stocks on it, group them into portfolios, and check whether the top group beats the bottom group.
That’s it. Almost every anomaly (momentum, value, profitability, liquidity) was first documented with a sort. The details are where the craft lies.
Why sort instead of just running a regression?
You could regress each stock’s return on its characteristic. Sorts have two advantages:
- Diversification kills noise. One stock’s monthly return is mostly idiosyncratic. A portfolio of hundreds averages that away, so the signal shows up.
- No functional form assumption. A regression assumes a straight-line relationship. A sort lets you see the shape: is the effect monotonic, or concentrated in one extreme?
The output is also something you could actually trade: a long-short (zero-cost) portfolio that buys the top group and sells the bottom one.
The basic recipe
- Pick a formation date and measure the characteristic using only information available at that date.
- Rank stocks and split them into groups (deciles, quintiles, terciles).
- Hold each portfolio for some period and record returns.
- Repeat every month or year, building a time series of portfolio returns.
- Test the spread (top minus bottom): is its average return significantly different from zero? Is it explained by known factors? (That’s next post.)
Example 1: Momentum (Jegadeesh & Titman, 1993)
Jegadeesh & Titman made the univariate sort famous with their J/K strategies:
- J = formation period: rank stocks by their past J-month return (J = 3, 6, 9 or 12).
- K = holding period: buy the top decile (winners), sell the bottom decile (losers), hold for K months (K = 3, 6, 9 or 12).
The clever part is overlapping portfolios. Each month you start a new winner-minus-loser portfolio, but you hold it for K months, so at any time you hold K cohorts, each with 1/K of the capital. This produces a clean monthly return series and doesn’t depend on one arbitrary start date.
They also ran versions that skip a week between formation and holding. Why? The last few days of past returns are contaminated by bid-ask bounce and short-term reversal. Skipping them isolates the genuine momentum signal. (Later work usually skips a month: the “12-2” momentum used by Bessembinder & Zhang is exactly that.)
The result: winners beat losers by roughly 1% per month, and the profits partially reverse in the following years. Losers didn’t even have lower betas, so ordinary market risk doesn’t explain it.
They were also careful about data snooping. They note that an earlier claimed winner-buying rule (Levy, 1967) was found after testing dozens of rules and failed outside its sample (Jensen & Bennington, 1970). That’s a good reminder: if you try enough sorts, one will “work.”
Example 2: Double sorts and NYSE breakpoints (Fama & French, 2015)
Characteristics are correlated. Small stocks tend to be value stocks. Value stocks tend to invest less. A single sort on B/M might really be picking up size. So you sort on two or more things at once.
Fama & French build their 5×5 Size × B/M portfolios with independent sorts: stocks go into size quintiles and, separately, into B/M quintiles. The 25 portfolios are the intersections.
The key detail is that breakpoints come from NYSE stocks only, even though the portfolios contain NYSE, AMEX and NASDAQ stocks. Why? There are thousands of tiny NASDAQ stocks. If you used all-stock breakpoints, most “size quintiles” would be slices of microcaps, and the big-stock economy would be squeezed into one bucket. NYSE breakpoints keep the groups economically meaningful.
The grid shows something a univariate sort would hide. The value effect is much stronger among small stocks:
- Microcaps: average excess return rises from about 0.26% per month (growth) to 1.15% (value).
- Megacaps: from about 0.46% to 0.62%.
Example 3: Triple sorts and when cells go empty
To isolate value, profitability and investment together, you’d want to sort on all four characteristics. But even a 3×3×3×3 sort gives 81 portfolios, most poorly diversified.
Fama & French compromise with 2×4×4 sorts: size (2 groups) × two other characteristics (4 groups each). Even then, correlations bite. Big stocks with both high B/M and high profitability are rare, and that portfolio is often empty before 1974. Their fix is to use separate breakpoints for small and big stocks, which spreads stocks more evenly across cells.
Independent vs. dependent sorts
| Independent sort | Dependent (conditional) sort | |
|---|---|---|
| How | Rank on A and B separately; intersect | Rank on A first; within each A-group, rank on B |
| Pro | Breakpoints for B mean the same thing everywhere | Every cell has a similar number of stocks |
| Con | Correlated characteristics leave some cells nearly empty | “High B” means something different in each A-group |
| Example | Fama & French Size × B/M | Daniel & Titman (1997) sort on size, then B/M, then factor loading (covered in a later post) |
Novy-Marx (2013) uses sorts in a similar spirit: he sorts on gross profitability and shows that controlling for it makes value strategies considerably stronger, especially among large, liquid stocks. The two characteristics work better together than apart.
Details that quietly matter
| Choice | Why it matters |
|---|---|
| Timing / look-ahead | Fama & French sort at the end of June of year t using accounting data for the fiscal year ending in t−1, so the numbers were public by then. Using data before it was released inflates results. |
| Equal vs. value weighting | Equal weighting (Jegadeesh & Titman) gives microcaps a big voice. Value weighting (Fama & French) reflects where the money is. Many anomalies are much weaker value-weighted. |
| Number of groups | More groups give bigger spreads but noisier portfolios. Fama & French note their 2×3 factors use the extremes (dropping the middle 40%), producing larger average spreads than 2×2 splits. |
| Skipping recent returns | Avoids bid-ask bounce and short-term reversal contaminating a signal. |
| Rebalancing frequency | Monthly for fast signals like momentum; annually for accounting characteristics that update once a year. |
My checklist when I see a sort
- Was the characteristic known at formation? No look-ahead.
- Whose breakpoints? NYSE or all stocks?
- Equal- or value-weighted? Does the result survive both?
- Is it driven by microcaps? Look at the size-sorted version.
- Independent or dependent sort, and are any cells nearly empty?
- How many sorts were tried before this one?
Next post: turning sorts into factors (SMB, HML, RMW, CMA), time-series regressions, and what the GRS test actually tells you.
References
Daniel, K., & Titman, S. (1997). Evidence on the characteristics of cross sectional variation in stock returns. Journal of Finance, 52(1), 1–33.
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
Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance, 48(1), 65–91.
Jensen, M. C., & Bennington, G. A. (1970). Random walks and technical theories: Some additional evidence. Journal of Finance, 25(2), 469–482.
Novy-Marx, R. (2013). The other side of value: The gross profitability premium. Journal of Financial Economics, 108(1), 1–28. https://doi.org/10.1016/j.jfineco.2013.01.003