Source: Fama, E. F. & French, K. R. (2015). A five-factor asset pricing model. Journal of Financial Economics 116(1), 1–22. · DOI: 10.1016/j.jfineco.2014.10.010
TL;DR
Adds two factors to the Fama-French three-factor model: RMW (robust minus weak operating profitability) and CMA (conservative minus aggressive investment), both motivated by the dividend-discount/valuation identity (Eq. 3) that links expected return to B/M, expected profitability, and expected investment. In the July 1963–December 2013 U.S. sample, the resulting five-factor model (Rm−Rf, SMB, HML, RMW, CMA) describes average returns better than three factors — and, strikingly, HML becomes redundant: its average return is absorbed by its exposures to the other four factors, especially RMW and CMA.
What anomaly it documents
Predictors (the five factors): market (Rm−Rf), size (SMB), value (HML, book-to-market), profitability (RMW, operating profitability), investment (CMA, asset/book-equity growth).
Direction: holding B/M fixed, firms with higher operating profitability and firms that invest less (conservative) earn higher average returns; high B/M, small, and low-Rm−Rf exposures carry the usual signs.
Shape: linear factor exposures; the regression is
Rit − RFt = ai + bi(RMt − RFt) + si·SMBt + hi·HMLt + ri·RMWt + ci·CMAt + eit (Eq. 5).
A correctly specified model implies the intercept ai = 0 for all assets.
SMB, HML, RMW, CMA are built from independent 2×3 sorts. Each year stocks are split into 2 size groups, crossed with 3 groups (top 30% / middle 40% / bottom 30%) on the second variable: B/M for HML, operating profitability for RMW, investment for CMA. Value-weight portfolios are formed at the intersections.
RMW = average return on robust-profitability portfolios minus weak-profitability portfolios. CMA = average return on conservative (low investment) minus aggressive (high investment) portfolios — constructed exactly like HML but with the second sort on OP or investment.
The paper also tests 2×2 and 2×2×2×2 (joint, all four variables controlled) factor definitions; the model's performance is not sensitive to which factor definitions are used.
Evidence and replication
Tested on portfolios formed on Size, B/M, OP, and Investment, using the GRS statistic (Gibbons, Ross, Shanken) to test whether all intercepts are jointly zero.
The five-factor model improves most on profitability- and investment-sorted portfolios relative to the three-factor model.
HML redundancy: in a regression of HML on the other four factors, its large average return is absorbed (especially by RMW and CMA), so adding HML adds little once RMW and CMA are present in this sample.
Main failure: the model cannot capture the low average returns of small stocks whose returns behave like those of firms that invest a lot despite low profitability (strong negative RMW and CMA slopes) — no specification fully prices this corner.
Why it might work
Derived from the Miller-Modigliani valuation identity (Eq. 3): for fixed price, higher expected earnings imply higher expected return and higher expected investment implies lower expected return; B/M is a noisy proxy because price also reflects earnings/investment forecasts. RMW and CMA are the "natural choices" implied by this decomposition, whether pricing is rational or not.
Limitations and risks
HML redundancy is sample-specific and debated; dropping HML loses information about value in other periods/markets.
Does not include momentum; the small/aggressive/low-profitability microcap failure persists across factor definitions.
A competing investment-based specification is the Hou-Xue-Zhang q-factor model.
Key references
Fama, E. F. & French, K. R. (2015) — A Five-Factor Asset Pricing Model — Journal of Financial Economics 116(1), 1–22.
Novy-Marx, R. (2013) — The Other Side of Value: The Gross Profitability Premium — Journal of Financial Economics.
Hou, K., Xue, C. & Zhang, L. (2015) — Digesting Anomalies: An Investment Approach — Review of Financial Studies.
Gibbons, M., Ross, S. & Shanken, J. (1989) — A Test of the Efficiency of a Given Portfolio — Econometrica.
Provenance: verified/generated from the paper's full text.
Reference replication on ConvexPi
An open, verified replication of this strategy is maintained at convexpi/replications. It recomputes the strategy from underlying building blocks and scores it out of sample (the McLean & Pontiff test):
