(Pukthuanthong, Roll, and Subrahmanyam, Review of Financial Studies, 2019)
Pukthuanthong · Roll · Subrahmanyam
Have a factor? Find out what it actually is.
Risk factor. Mispricing. Priced but not systematic. Or neither.
Why it matters
A factor can earn a premium and still be the wrong factor to own.
per year
Full protocol, value-weighted long-short, gross
per year
After 50 bp one-way trading costs
One-way break-even trading cost
1991–2022 out-of-sample implementation. See Part 3 for construction, turnover, costs, and caveats. These are historical research results, not prospective investable returns.
But the surprising result is not just what the protocol selects. It is what it tells you not to trade.
High-covariance-content factors earned 2.5% gross per year and broke even at 52 bp, versus 9.5% and 114 bp for the full selection rule.
See the evidence ↓Full research record
Evidence, method, and caveats
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Try the Protocol
Bring your factors. See how the protocol classifies them.
Upload monthly factor returns to test covariance content, pricing, and reward-to-risk—and see which factors classify as risk, mispricing, priced but not systematic, or neither.
Published-page sample · 1965–2025 results · not an upload or prospective investment result
What did the protocol find?
Operating accruals / assets
Size
| Factor | Covariance | Premium | Reward/risk | Classification |
|---|---|---|---|---|
| Free cash flow / price | Fails Stage 1 | Pass | Above bound | Mispricing |
| Operating accruals / assets | Fails Stage 1 | Pass | Above bound | Mispricing |
| Market | Pass | Flat | -- | Unpriced factor |
| Size | Pass | Flat | -- | Unpriced factor |
Why was free cash flow / price classified this way?
MISPRICING — priced, not required to explain covariance, and above the protocol's reward-to-risk bound. Stage 1 mean |t| 1.41; monthly premium 0.2922%; pricing t-statistic 2.38; factor-plus-market Sharpe 0.8457.
Why were operating accruals classified this way?
MISPRICING — priced, not required to explain covariance, and above the protocol's reward-to-risk bound. Stage 1 mean |t| 1.04; monthly premium 0.5096% in the traded direction; pricing t-statistic −5.25 before orientation; factor-plus-market Sharpe 0.8258.
Why were market and size classified this way?
UNPRICED FACTORS — both explain common return variation but do not earn a significant premium in this sample. Market Stage 1 mean |t| is 9.42 and pricing t-statistic 0.33; size is 3.73 and −0.39.
Stage 1 / 2 / 3 definitions
Stage 1 asks whether the candidate contributes to the covariance matrix. Stage 2 asks whether it is priced. Stage 3 asks whether its reward-to-risk exceeds the protocol's risk-compensation bound. The full report above contains the estimators, test assets, corrections, and caveats.
Compare Factors
This view uses only statistics already shown in the research outputs.
| Factor | Classification | Covariance content | Premium / month | t-statistic | Sharpe | Stage 1 / 2 / 3 |
|---|---|---|---|---|---|---|
| Free cash flow / price | Mispricing | Fails · 1.41 | 0.2922% | 2.38 | 0.8457 | Fail / Pass / Above |
| Operating accruals / assets | Mispricing | Fails · 1.04 | 0.5096% | −5.25 | 0.8258 | Fail / Pass / Above |
| Market | Unpriced factor | Pass · 9.42 | 0.0210% | 0.33 | 0.4404 | Pass / Fail / -- |
| Size | Unpriced factor | Pass · 3.73 | 0.0255% | −0.39 | 0.3411 | Pass / Fail / -- |
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