A.7.2 The Fundamental Factors

Basic Prerequisites: 01 · From Zero and Core Financial Ratios (the ratios each factor is built from).

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1. Intuition & Practical Objective

The previous page established the machine: rank the universe on a fundamental characteristic, hold one side, collect the premium. This page fills the machine with its factors. There are four families, and each answers a different question about the firm:

• Value - is it cheap? Book-to-market and earnings yield measure price against book capital and earnings.
• Profitability - is it a good business? ROE, and Novy-Marx’s gross profitability, measure the return the business earns on what it employs.
• Investment - is it over-investing? Asset growth measures how aggressively the firm is expanding; the premium is on the conservative side (low growth wins).
• Quality - is the cheapness real, or is the accounting hiding rot? The Piotroski F-score and Sloan’s accruals separate the financially strong from the distressed.

The objective is to see these not as unrelated tricks but as four independent axes of “good and cheap” - and to learn the single mechanical construction (sort into portfolios, measure the spread) that builds every one of them. Fama–French 2015 is the formal statement: value (HML), profitability (RMW), and investment (CMA) are all priced, alongside size (SMB). The accounting statement is the input; the factor is the priced output.

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2. Mathematical Ground Truth & Derivations

Value. With BE book equity, ME market equity, NI net income:

$$\text{B/M}=\frac{\text{BE}}{\text{ME}},\text{E/P}=\frac{\text{NI}}{\text{ME}}.$$

Profitability. With ROE return on equity, and Novy-Marx’s gross profitability $GP/A$:

$$\text{ROE}=\frac{\text{NI}}{\text{BVE}},\frac{GP}{A}=\frac{S-\text{COGS}}{\text{TA}}.$$

Gross profitability is deliberately pre every cost layer except direct cost - the rawest possible measure of the business’s pricing power. Novy-Marx (2013) shows it predicts returns with roughly the same power as book-to-market, but orthogonal to it - a profitable firm can be expensive on B/M yet still earn more.

Investment. Let $g_A$ be one-year growth in total assets:

$$g_A=\frac{{\text{TA}}_t-{\text{TA}}_{t-1}}{{\text{TA}}_{t-1}}.$$

The premium is negative: Fama–French (2015) show conservative (low $g_A$) firms earn more than aggressive (high $g_A$) firms. The q-theory logic (Hou, Xue & Zhang 2015): a firm invests when its marginal cost of capital is low, so high-investment firms are precisely the ones whose future returns are low.

Quality - the Piotroski F-score. Nine binary signals summed to a 0–9 score (profitability, leverage/liquidity, operating efficiency); detailed on 04 · Quality & F-scores. And the accruals factor (Sloan 1996): because

$$\text{NI}=\text{CFO}+\text{Accruals},$$

high-accrual (low-cash) earnings are less persistent and earn lower subsequent returns - the “quality” dark-side factor.

Portfolio-sort construction (the shared skeleton). Rank $N$ firms on characteristic $c_i$, split into $K$ portfolios by rank, equal-weight:

$${\overline{r}}_k=\frac{1}{N_k}\sum _{i\in k}{r}_i,\text{premium}={\overline{r}}_K-{\overline{r}}_1.$$

Every factor above is exactly this estimator with a different $c_i$ - keep that single shape in mind and the factor catalog stops being a list and becomes one repeated thought.

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3. Computational Implementation - all four factor families, stdlib only

Builds value, profitability, investment, and the accruals signal from a twelve-firm panel, sorts each into terciles, and reports the premium - then checks the correlation between value and gross profitability to show they are near-orthogonal axes (the reason they combine well).

Read the profitability row carefully. Raw gross profitability shows no premium here (even slightly negative) - because the profitable firms in this panel are the expensive growth firms (high GP/A, low B/M). That is the exact confound Novy-Marx resolved: profitability predicts returns once value is conditioned on. The negative rank correlation with B/M ($\rho =-0.50$: profitable firms tend to be more expensive) is exactly why the two factors are complements - they correct each other’s blind spots when combined, which is the logic 03 · Value & Profitability builds on.

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4. Failure Modes & First-Principles Breakdowns

1. Univariate factors are confounded. Sorting on one characteristic while ignoring its correlated cousins misattributes the premium - the raw GP/A sort above “shows” no profitability premium precisely because it is tangled with value/expensiveness. Real factor work is multivariate or double-sorted.
2. Sign matters and is not obvious. Value and profitability have positive premiums; investment has a negative one (low growth wins). Mis-taking “grow fast = good” as a factor reverses the actual evidence - the q-theory sign is the whole point.
3. Every factor is a claim that survives only if its confounds are controlled. Fama–French 2015 built HML, RMW, and CMA to be roughly size-neutral; an HML that loads on size isn’t the value factor. Neutrality in construction is part of the definition, not decoration.
4. The premium is a long-horizon average. Any single period can invert the sign. What distinguishes a factor from noise is the persistence of the spread across decades - which is exactly what 05 · Failure Modes stress-tests.

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5. References

• Fama & French: “The Cross-Section of Expected Stock Returns” (JF, 1992)
• Fama & French: “Common Risk Factors in the Returns on Stocks and Bonds” (JFE, 1993)
• Fama & French: “A Five-Factor Asset Pricing Model” (JFE, 2015)
• Novy-Marx, Robert: “The Other Side of Value: The Gross Profitability Premium” (JFE, 2013)
• Hou, Xue & Zhang: “Digesting Anomalies: An Investment Approach” (RFS, 2015)
• Sloan, Richard: “Do Stock Prices Fully Reflect Information in Accruals and Cash Flows About Future Earnings?” (TAR, 1996)

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Value & Profitability · 04 · Quality & F-scores
• Factor-model layer: Fundamental Multi-Factor Models