Factor
Modeling

What this week covers

W4 introduced the major equity factors — Market, Size (SMB), Value (HML), Momentum (UMD), Quality, Low Vol — and noted that they have multi-year drawdowns. This week is how to actually use them: regress portfolio returns on factor returns to read off exposures and separate alpha from beta.

By Friday you should be able to:

• Write down the Fama-French three-factor model and explain how SMB and HML are constructed (not just defined).
• Run a multi-factor regression on a portfolio's monthly returns and interpret the alpha, betas, and R².
• Construct a custom factor from a stock characteristic and test whether it adds explanatory power over the standard set.

The reading is the original Fama-French (1993) paper and Carhart (1997); Palomar Ch 7 §2 covers the regression interpretation. Cochrane (2001) Ch 12–13 has the standard-error theory if you want to go deeper.

The factor model framework

A factor model decomposes an asset's excess return into a weighted sum of common-factor returns plus an idiosyncratic residual. Both Merton's ICAPM and Ross's APT motivate the same form — factors must carry an economic reason to predict returns.

\[ R_{i,t} - R_{f,t} \;=\; \alpha_i \;+\; \sum_{k=1}^{K} \beta_{i,k}\, F_{k,t} \;+\; \varepsilon_{i,t} \quad (1) \]

A second, cross-sectional equation links expected returns to factor risk premia \( \lambda_k \):

\[ \mathbb{E}[R_i - R_f] \;=\; \lambda_0 \;+\; \sum_{k=1}^{K} \beta_{i,k}\, \lambda_k \quad (2) \]

Under a correctly specified model, \( \lambda_0 = 0 \) — no return without exposure. The \( \beta_{i,k} \) is the asset's loading on factor \( k \); the \( \lambda_k \) is the risk premium the market pays per unit of that exposure.

What plays the role of \( F_{k,t} \) depends on the factor type:

• Factor-mimicking portfolios (Fama-French SMB, HML; Carhart MOM): long-short return spreads. \( F_{k,t} \) is just the time-\( t \) return of the spread. This is the case we focus on.
• Macroeconomic factors (Chen, Roll & Ross 1986): inflation, term spread, industrial production. \( F_{k,t} \) is the surprise — realized value minus the predictable component.

Two estimation routes match these two cases (covered below): time-series regression for traded-portfolio factors, Fama-MacBeth two-pass for macro factors. QT uses time-series exclusively — we're auditing what the live book is exposed to, not pricing new assets.

From CAPM to Fama-French to FF5

Every model below is the same equation above with a different choice of factors.

Model	Factors	What it adds
CAPM (Sharpe-Lintner, 1964)	Mkt − Rf	The baseline. Only systematic (market) risk should be compensated. Empirically inadequate — small and value stocks earn more than CAPM predicts.
Fama-French 3 (1993)	Mkt − Rf, SMB, HML	Adds Size and Value premia. Explains roughly 90%+ of US-equity cross-sectional return variance.
Carhart 4 (1997)	+ UMD (a.k.a. MOM)	Adds Momentum (Up-Minus-Down). Essential when evaluating any trend-following or momentum strategy.
Fama-French 5 (2015)	+ RMW, CMA	Adds Profitability (Robust-Minus-Weak) and Investment (Conservative-Minus-Aggressive). Subsumes much of HML; better empirical fit post-2000.

Written out, the four-factor Carhart model is:

\[ R_{p,t} - R_{f,t} \;=\; \alpha_p \;+\; \beta_{\mathrm{MKT}}(R_{M,t} - R_{f,t}) \;+\; \beta_{\mathrm{SMB}}\,\mathrm{SMB}_t \;+\; \beta_{\mathrm{HML}}\,\mathrm{HML}_t \;+\; \beta_{\mathrm{MOM}}\,\mathrm{MOM}_t \;+\; \varepsilon_{p,t} \]

This is the default QT uses for portfolio decomposition. FF5 is worth running as a robustness check.

How Fama-French actually builds SMB and HML

W4 told you what SMB and HML mean ("Small Minus Big", "High Minus Low book-to-market"). The factor returns Fama and French publish on their data library are not just SMB = small-cap index minus large-cap index. They come from a specific 2×3 sort that isolates each premium from the other.

Every June, the universe (NYSE + AMEX + NASDAQ common stocks) is sorted on two characteristics simultaneously:

1. Size — market cap, split at the NYSE median into Small and Big.
2. Book-to-market — split at the NYSE 30th and 70th percentiles into Low, Medium, High. (Some papers use median or 10/90 splits; 30/70 is the Fama-French convention.)

The intersection gives six value-weighted portfolios: SL, SM, SH, BL, BM, BH. The factor returns are then:

\[ \mathrm{SMB}_t = \tfrac{1}{3}(R_{\mathrm{SL}} + R_{\mathrm{SM}} + R_{\mathrm{SH}})_t \;-\; \tfrac{1}{3}(R_{\mathrm{BL}} + R_{\mathrm{BM}} + R_{\mathrm{BH}})_t \] \[ \mathrm{HML}_t = \tfrac{1}{2}(R_{\mathrm{SH}} + R_{\mathrm{BH}})_t \;-\; \tfrac{1}{2}(R_{\mathrm{SL}} + R_{\mathrm{BL}})_t \]

Two design choices in here are worth internalizing:

• Long-short construction. Each factor is dollar-neutral by construction — a long basket minus a short basket of equal weight. That isolates the spread in returns due to the characteristic, not the market beta.
• 2×3 sort, not 2 separate 1D sorts. SMB averages across the three B/M groups (and HML across the two size groups), so each factor's loading on the other characteristic is mechanically neutralized. This is why Fama-French betas are interpretable as independent exposures.

Building your own factor

The Fama-French recipe generalizes. Any cross-sectional stock characteristic can be turned into a factor by sorting and going long-short.

1. Pick a characteristic. It must be (a) measurable from public data, (b) cross-sectionally varying — every stock has a different value, (c) tied to an economic mechanism you can defend (recall W1: hypothesis before backtest).
2. Rank and sort. At each rebalance, rank the universe on the characteristic. The split is a design choice — median (2 buckets), 30/70 (3 buckets, the Fama-French convention), 10/90, or quintiles/deciles. Finer sorts give cleaner separation between top and bottom; coarser sorts give larger, more diversified buckets.
3. Form the long-short portfolio. Long the top quintile (or decile), short the bottom. Value-weight within each quintile. The factor return is the spread. \[ F_{t}^{\mathrm{custom}} = R_{\mathrm{top}, t} - R_{\mathrm{bottom}, t} \]
4. Test orthogonality. Regress your factor against the existing Carhart 4 factors: \[ F_{t}^{\mathrm{custom}} = a + b_1 \mathrm{MKT}_t + b_2 \mathrm{SMB}_t + b_3 \mathrm{HML}_t + b_4 \mathrm{MOM}_t + u_t \] If the intercept \( a \) is significantly positive and \( R^2 \) is low, your factor carries return that the standard set doesn't capture — it's a real candidate. If \( R^2 \) is 0.85, you've reinvented an existing factor with a new name.
5. Test robustness. Split the sample (pre-2010 / post-2010), look at sub-sectors, run in different countries. A factor that works only in one regime is unlikely to survive live.

Two estimation routes

Equations (1) and (2) suggest two regression strategies. Which one you use depends on whether your factors are traded portfolios or general macroeconomic surprises.

Running the regression on a portfolio

Once factors exist, decomposing a portfolio's returns is just OLS:

import pandas as pd, statsmodels.api as sm

# portfolio_excess = monthly portfolio return - monthly Rf
# factors = DataFrame with columns ['MKT_RF', 'SMB', 'HML', 'MOM']

X = sm.add_constant(factors)
model = sm.OLS(portfolio_excess, X).fit(
    cov_type='HAC', cov_kwds={'maxlags': 6}  # Newey-West, accounts for autocorrelation
)
print(model.summary())

Three notes on the regression itself:

• Use monthly, not daily, data. Daily factor returns are noisier and microstructure effects dominate. The Fama-French data library publishes monthly series specifically because of this.
• Use Newey-West standard errors. W5 covered the autocorrelation problem; the HAC adjustment here is the fix. Without it, the t-stats on the betas are wrong.
• Lookback ≥ 60 months. Five years of monthly data is the Fama-MacBeth standard; 36 months is the bare minimum. With fewer observations the betas have standard errors larger than the betas themselves.

Interpreting α, β, R²

Worked example: decomposing a portfolio

A hypothetical multi-strategy portfolio, 48 months of returns. Regress against Carhart 4-factor + a custom Quality factor (QMJ). OLS with Newey-West errors:

Term	Estimate	t-stat	Interpretation
α (monthly)	0.04%	0.6	Not statistically distinguishable from zero. No skill premium after factors.
βMKT	0.85	14.2	High market exposure — the portfolio moves 85¢ for every $1 of market.
βSMB	0.22	2.4	Significant small-cap tilt.
βHML	-0.18	-1.9	Marginal growth tilt (negative value loading).
βMOM	0.35	3.8	Strong momentum exposure.
βQMJ	0.15	1.6	Borderline quality tilt; not statistically robust.
R²	0.87	—	87% of variance explained by the 5 factors.

Reading the result

• The portfolio's return is almost entirely a function of its factor exposures (\( R^2 = 0.87 \)) — there's no measurable alpha (t-stat on \( \alpha \) is 0.6).
• The dominant exposures are market (0.85) and momentum (0.35). Whoever runs this book is leaning hard on trend.
• A modest small-cap tilt and a hint of growth orientation. Not a deliberate factor bet — likely a side-effect of the universe selection.
• The custom quality factor adds nothing significant — for this portfolio, FF3 + MOM is enough.

Common mistakes