A.7.6 Advanced Extensions

Basic Prerequisites: 02 · Fundamental Factors and 05 · Failure Modes.

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1. Intuition & Practical Objective

The single-factor pages built the axis; this page builds the model. The objective is the question every quantamental portfolio asks: once I have value, profitability, investment, and quality factors, how do I combine them, and how do I know a portfolio’s return is really “alpha” and not just exposure to known factors? Two instruments answer it:

• The Fama–French factor construction - how SMB, HML, RMW, and CMA are literally built (2×3 double-sorts, portfolio arithmetic) from the same accounting data this folder has used all along.
• The factor regression - the test that decomposes any portfolio’s return into a market loading, a value loading, a profitability loading, etc., plus a residual $\alpha$. If $\alpha \approx 0$, the portfolio is just the known factors in disguise; if $\alpha$ is large and stable, you have something new.

The through-line: a factor model turns “this strategy made money” into “this strategy is long value, short growth, and that’s where the return came from.” It is the machine that both combines factors and audits them - which is why 05 · Failure Modes and this page sit side by side.

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2. Mathematical Ground Truth & Derivations

The Fama–French 2×3 construction (1993; extended 2015). Independent sorts: two size groups (small/big, split at the NYSE median market cap) and three groups on each of B/M, operating profitability, and investment (30th/70th percentile breakpoints). The intersections give six value-weighted portfolios; each factor is the average of small and big versions of a long-short:

$$\text{SMB}=\frac{1}{3}[\text{avg(S/L,S/M,S/H)}-\text{avg(B/L,B/M,B/H)}],$$ $$\text{HML}=\frac{1}{2}[\text{(S/H + B/H)}-\text{(S/L + B/L)}],$$

and analogously $\text{RMW}$ (robust minus weak profitability) and $\text{CMA}$ (conservative minus aggressive investment). Because each factor averages small and big portfolios, SMB/HML/RMW/CMA are roughly size-neutral - that neutrality is part of their definition.

The factor-model regression (Fama–MacBeth; Fama–French 2015). A portfolio’s excess return is regressed on the factor excess returns:

$$R_{it}-R_{ft}={\alpha }_i+{\beta }_i(R_{Mt}-R_{ft})+s_i{\text{SMB}}_t+h_i{\text{HML}}_t+r_i{\text{RMW}}_t+c_i{\text{CMA}}_t+e_{it}.$$

The intercept ${\alpha }_i$ is the abnormal return unexplained by the factors. The five-factor model (Fama–French 2015) is rejected by the strict GRS test - it fails on small stocks that invest a lot despite low profitability - but for applied purposes it gives an acceptable description of average returns. HML averages about 0.38%/month in the 2×3 construction; RMW and CMA add positive premiums on top.

Combining factors - composite scores. The simplest combination that the evidence supports (and that 03 · Value & Profitability motivated) is a rank-sum: rank the universe on each factor and average the ranks:

$$\text{Composite}=\frac{1}{K}\sum _{j=1}^K{\text{rank}}_j(\text{firm}),$$

so a firm cheap and profitable and high-quality scores best. Greenblatt’s magic formula and Piotroski’s F-score are both instances; a factor model tells you how much of that composite’s return is incremental.

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3. Computational Implementation - build the 2×3 factors and run the regression, stdlib only

Two halves. First it constructs SMB and HML from a 20-stock cross-section using the genuine 2×3 algorithm (size median, B/M 30/70 breakpoints). Then it builds a 48-month time series for the six cells, forms the market and HML factors, and runs an OLS regression (via the normal equations) of a value portfolio’s return on [market, HML] - showing the portfolio is fully “explained” by the factors, with ~zero alpha.

The output is the whole point of a factor model. HML is +3.53% (value pays); SMB is ≈0 (this cross-section has no size premium). The regression then decomposes the value portfolio: a market loading of 1.00, a value (HML) loading of 0.50 - it is half value-factor - and an alpha of −0.00%. The strategy’s return is fully attributable to the known factors; it has no hidden edge to claim. That is exactly what a factor model is for: converting “it made money” into “it is long value,” and flagging as alpha only what survives after the known factors are removed.

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4. Failure Modes & First-Principles Breakdowns

1. The 2×3 breakpoints are a modeling choice with real consequences. NYSE-median size and 30/70 B/M breakpoints are arbitrary (Fama–French 2015 test 2×2 and 2×2×2×2 variants); HML, RMW, and CMA averages differ across constructions. Quote the construction, or the factor is not comparable.
2. Factors built without controls are confounded. HML built from a B/M-only 2×3 is not neutral to profitability and investment - the regression slopes then don’t isolate clean exposures (Fama–French’s own caveat). Interpreting the HML slope as “pure value” requires controls.
3. Alpha is a residual, and residuals are the hard part. $\alpha \approx 0$ means “explained,” not “no return”; a large $\alpha$ can be a new factor, luck, or a data artifact. Distinguish them with out-of-sample tests and multiple-testing discipline (05 · Failure Modes).
4. The model is rejected by the strict test. The five-factor model fails the GRS test on small high-investment/low-profitability stocks. It is “acceptable for applied purposes,” not the truth - the q-factor model (Hou, Xue & Zhang) and others compete to digest the same anomalies.
5. Composite ranks hide factor-specific risk. Averaging ranks on value and profitability bundles two exposures; the model regression is what lets you see how much of the composite is each - don’t skip it.

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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)
• Hou, Xue & Zhang: “Digesting Anomalies: An Investment Approach” (RFS, 2015)
• Green, Hand & Zhang: “The Characteristics That Provide Independent Information…” (RFS, 2017)
• Fama & French: “Choosing Factors” (JFE, 2018)
• Gray & Carlisle: Quantitative Value

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Factor-model layer: Fundamental Multi-Factor Models (Barra/FF model construction, the professional layer) · Backtesting Hygiene & Deflated Sharpe (how to audit the alpha)
• Portfolio layer: Portfolio Optimization (turning factor tilts into an optimized portfolio)
• Screening: Fundamental Analysis & Screening · Screening · Advanced Extensions