1.3.1 Fundamental Multi-Factor Models from Zero

Basic Prerequisites: Linear Algebra & Matrices (a projection is a regression; orthogonality).

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1. Intuition & Practical Objective

This page builds the why of factor models with no prior factor/asset-pricing knowledge needed. The objective is one idea: a stock’s return can be split into a low-dimensional, systematic part (explained by a handful of common factors that reward bearing their risk) and an idiosyncratic part (the firm’s own news, which is not compensated). Everything in this folder - SMB, HML, Barra, the covariance model - is a formalization of that split.

Start with the dumbest question: why does the cross-section of stock returns differ at all? The CAPM answer - one factor, the market, captured by beta - was the state of the art until Fama & French 1992 showed it is incomplete. Their result was sharp and inconvenient: across 1963–1990, beta had essentially no power to explain average returns (a shot “straight at the heart of the SLB model”), while two variables - size and book-to-market - did. Three “aha”s:

1. One factor is not enough. The CAPM’s beta alone fails (FF1992 Table III: the average slope on $\beta$ alone is 0.15%/month, only 0.46 standard errors from 0). But size ($\ln$ ME, slope $-$0.15%/month, $t=-2.58$) and book-to-market are reliably priced. So the priced-risk space is richer than “the market.”
2. A factor is built by sorting, not by forecasting. You do not predict which stock wins; you sort the whole cross-section on a fundamental characteristic (size, B/M) and hold the extreme side as a long-short hedge portfolio. The realized return of “long cheap minus short expensive” is the HML factor - a factor is a traded portfolio, not a fitted number.
3. Factor exposure (beta) is a regression, not a belief. Once the factor portfolios exist, each stock’s loading is estimated by time-series OLS of its return on the factors. A stock with high HML-loading is a value stock; the model says it earns the value premium as compensation for bearing value risk.

The one-sentence essence. “A factor model decomposes return into $\text{systematic risk}\times \text{loading}+\text{alpha}+\text{noise}$, where the systematic factors are traded long-short portfolios built by sorting on real characteristics, so you can attribute performance to risk (beta) instead of skill (alpha).“

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2. Mathematical Ground Truth & Derivations

The split, in one equation (the general factor model, Tsay eq. 9.1):

$$r_{it}={\alpha }_i+{\beta }_{i1}{f}_{1t}+\cdots +{\beta }_{im}{f}_{mt}+{\epsilon }_{it},$$

or in matrix form over all assets, $r_t=\alpha +B{f}_t+{\epsilon }_t$ (Tsay eq. 9.2). Its variance decomposition (Tsay eq. 9.3–9.4) is the single most useful identity in the folder:

$$\text{Cov}(r_t)=B\text{Cov}(f_t)B^{\top }+D,D=\text{diag}\{{\sigma }_1^2,\dots ,{\sigma }_k^2\},$$

i.e. asset covariance = (exposures × factor covariance × exposures${}^{\top }$) + idiosyncratic diagonal. The beta’s job is to absorb as much covariance as possible so that the leftover $D$ is nearly diagonal (idiosyncratic noise is roughly uncorrelated across stocks).

The three factor families (Tsay §9 intro, citing Connor 1995):

• Macroeconomic - observable economic factors (GDP surprises, CPI), factor realizations measured directly, loadings by OLS/multivariate regression.
• Fundamental - factors are attributes of the asset: the BARRA approach treats standardized fundamentals (B/M, earnings yield) as the exposures $\beta$ and estimates the factor returns $f_t$ cross-sectionally; the Fama–French approach builds traded hedge portfolios from the attributes (SMB, HML). (This is the family of this folder.)
• Statistical - latent factors estimated from the covariance of returns by PCA / factor analysis (Tsay §9.4–9.5).

Two-pass estimation intuition (the Fama–MacBeth shape). First pass: in the time-series dimension, regress each asset’s returns on the factor returns to get loadings ${\beta }_i$. Second pass: in each cross-section, regress the realized returns on the loadings to recover the factor risk premiums $\lambda$. Both passes are just OLS projections - the factor model lives in linear algebra, not in exotic statistics.

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3. Computational Implementation - sorting is the factor

The single most convincing way to see the theory is to sort a cross-section on a fundamental characteristic and measure the spread between the extreme portfolios - the raw material of SMB/HML before any regression. Stdlib only.

The +2.63% value spread is the HML factor in miniature: sort on B/M, hold the expensive-to-cheap side, and the realized return difference is the factor return. The full construction - two sort dimensions, value-weighting, NYSE-only breakpoints - is on 03 · Factor Construction.

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4. Failure Modes & First-Principles Breakdowns

1. “More characteristics = better model” is wrong. A characteristic is only a factor if it survives the sort, the regression, and out-of-sample decay tests. Most published return-predictive characteristics are redundancy around a handful of real factors (Green–Hand–Zhang) - the seed of the factor zoo.
2. Sorting is relative, not absolute. A factor premium is a cross-sectional ordering, not a level. A single firm’s B/M is nearly meaningless; its rank against the current cross-section is the signal. Sorting on a stale or look-ahead-contaminated characteristic silently corrupts the factor.
3. Beta is not causation. A stock loading on value earns the value premium on average, but an individual stock can destroy value for years while carrying high HML-loading. Factor models price risk on average, not individual outcomes.

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5. References

• Fama & French, “The Cross-Section of Expected Stock Returns” (JF, 1992)
• Tsay, Analysis of Financial Time Series
• Connor, Gregory: “The Three Types of Factor Models” (FAJ, 1995)

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6. Connected Graph Bridges

• Base: Linear Algebra & Matrices (projection = regression) · Econometrics & Time Series
• Fundamentals: Quantitative Fundamental Investing (where the characteristics come from) · 02 · Fundamental Factors
• Continue: 02 · The FF Factor Model · Index Hub