1.3.3 Factor Construction

Basic Prerequisites: 02 · The FF Factor Model (the regression the constructed factors feed into).

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1. Intuition & Practical Objective

The factors of the previous page are not handed down; they are built by sorting. The practical objective of this page: reproduce the exact Fama–French recipe so you can construct SMB, HML, RMW, and CMA from raw data (or from Ken French’s library), understand why each step exists, and then feed the result into the time-series regression. A factor is a long-short hedge portfolio - a traded spread between the extreme groups of a sort - so its return is the price of that style risk.

The four FF factors and the characteristics they sort on:

Factor	Name	Sort on	Long side	Interpretation
MKT	Market	-	the market	the one-factor benchmark
SMB	Size	market cap (ME)	small	small-cap risk premium
HML	Value	book-to-market (B/M)	high B/M (cheap)	value risk premium
RMW	Profitability	operating profitability (OP)	robust (high OP)	profitability premium
CMA	Investment	asset growth / investment (Inv)	conservative (low Inv)	punishes over-investment

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2. Mathematical Ground Truth & Derivations

2.1 The canonical 2×3 construction (FF 2015, §4; FF 1993)

At each rebalance (in June, using prior-fiscal-year accounting with a lag to avoid look-ahead):

1. Size breakpoint: the NYSE median market cap splits stocks into Small and Big (S/B). Only NYSE stocks set breakpoints - the sample then covers NYSE+AMEX+NASDAQ, so extreme microcaps cannot dominate the splits.
2. Second-sort breakpoints: the NYSE 30th and 70th percentiles of B/M (for HML), operating profitability (for RMW), or investment (for CMA) give three groups (Low/Mid/High).
3. Independent 2×3 sorts → six value-weighted portfolios (value-weighted so the factor isn’t swamped by microcap noise).

Define $R_{xy}$ as the value-weighted return of the portfolio in size class $x\in \{S,B\}$ and signal class $y$. Then:

$$\text{HML}=\frac{R_{SH}+R_{BH}}{2}-\frac{R_{SL}+R_{BL}}{2}\text{(high B/M minus low B/M, averaged over small \& big),}$$ $$\text{RMW}=\frac{R_{SR}+R_{BR}}{2}-\frac{R_{SW}+R_{BW}}{2}\text{(robust minus weak profitability),}$$ $$\text{CMA}=\frac{R_{SC}+R_{BC}}{2}-\frac{R_{SA}+R_{BA}}{2}\text{(conservative minus aggressive investment),}$$

and the size factor is the average of the three size spreads - one from each of the B/M, OP, and Inv sorts (FF 2015 §4):

$$\text{SMB}=\frac{{\text{SMB}}_{B/M}+{\text{SMB}}_{OP}+{\text{SMB}}_{Inv}}{3}.$$

2.2 Why 2×3 and value-weighted? (first-principles design choices)

• Two size × three signal groups balances power against noise: the 30/70 breakpoints keep enough stocks in the extreme HML groups to be diversified, while independent (not sequential) sorts let each signal vary holding the others roughly fixed.
• Value-weighting keeps factor returns economically meaningful - a market-cap-weighted long-short is tradeable at low cost, unlike an equal-weight microcap spread (which Tsay notes would be dominated by illiquid names).
• The 6-month accounting lag (June rebalance on prior-December fiscal data) enforces point-in-time availability and is the FF defense against look-ahead bias.

2.3 Relation to the factor-mimicking portfolio

Every factor return is a linear combination of underlying portfolio returns - a factor-mimicking portfolio. In the BARRA formalism (Tsay §9.3.1), the factor realization is ${\hat{f}}_t={\omega }^{\top }{r}_t$ with portfolio weights $\omega =(X^{\top }{V}^{-1}X{)}^{-1}{X}^{\top }{V}^{-1}$; in the FF formalism the weights are the simple $\pm \frac{1}{2}$ (or $\pm \frac{1}{3}$) patterns above. Both are the same object: a traded portfolio whose return tracks a priced characteristic. See 04 · Cross-Sectional Models.

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3. Computational Implementation - building SMB and HML from a cross-section

We reproduce the 2×3 independent sort on a synthetic 200-stock monthly cross-section: split on median size, split on 30/70 B/M percentiles, form six value-weighted portfolios, and compute SMB and HML. Stdlib only.

This is the whole construction in one script: two independent sort dimensions, six value-weighted cells, and the two long-short differences that are SMB and HML. RMW and CMA are built identically, substituting operating profitability and investment for B/M in the second sort.

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4. Failure Modes & First-Principles Breakdowns

1. Breakpoint universe ≠ investment universe. If you set breakpoints on all stocks instead of NYSE-only, a flood of tiny NASDAQ names drags the “Big” bucket down to illiquid microcaps and the factor becomes untradeable. The NYSE-breakpoint rule exists precisely to prevent this.
2. Sort dimension overlap (B/M vs OP vs Inv are correlated). Because value, profitability, and investment are correlated, HML, RMW, and CMA are not mutually neutral - each carries an unknown mix of the others’ exposures, which makes the separate regression slopes hard to interpret (FF 2015 explicitly warns about this). The 2×2×2×2 four-sort variant controls for all simultaneously.
3. Value-weighting vs equal-weighting flips results. An equal-weighted small-cap long-short is dominated by the most extreme, often illiquid, names; the “factor” then prices noise and microcap liquidity rather than a systematic risk premium.
4. Point-in-time discipline. Using restated accounting or current (rather than lagged) book values leaks information into the factor and inflates its apparent premium. The 6-month lag is not an accident.

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5. References

• Fama & French, “Common Risk Factors in the Returns on Stocks and Bonds” (JFE, 1993)
• Fama & French, “A Five-Factor Asset Pricing Model” (JFE, 2015)
• Tsay, Analysis of Financial Time Series
• Ken French Data Library

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6. Connected Graph Bridges

• Back: 02 · The FF Factor Model · Index Hub
• Forward: 04 · Cross-Sectional Models (Barra-style construction of factor returns from exposures)
• Fundamentals: Quantitative Fundamental Investing · 03 · Value & Profitability
• Sibling: Momentum Factors