Learn Fama-French three - five factor models — Stock-Market notes

WHY does this model exist? (Steel-manning CAPM's failure)

CAPM predicts every asset lies on the Security Market Line: $E[R_i] - R_f = \beta_i\,\big(E[R_m]-R_f\big)$

WHAT went wrong: When you sort stocks by size and by book-to-market ( $B/M$ ) and look at realized average returns, small & value stocks beat what their $\beta$ alone predicts. The residual (alpha) is not noise — it is persistent and cross-sectional.

WHY the CAPM felt right: One factor is elegant and "the market is the only diversifiable-away-nothing risk." Fair. The fix: if a return premium is pervasive across many stocks and time, it is a priced systematic risk, so it deserves its own factor — not to be dumped in alpha.

The Three-Factor Model — built from scratch

We want to explain a portfolio's excess return $R_i - R_f$ . Start with CAPM and add two mimicking portfolios that isolate the size and value effects.

HOW they are constructed (2×3 sort):

Split all stocks by market cap into Small / Big (median breakpoint).
Independently split by $B/M$ into Low (30%) / Medium / High (30%).
This gives 6 value-weighted portfolios: SL, SM, SH, BL, BM, BH.
Then: $\text{SMB}=\tfrac{1}{3}(SL+SM+SH)-\tfrac{1}{3}(BL+BM+BH)$ $\text{HML}=\tfrac{1}{2}(SH+BH)-\tfrac{1}{2}(SL+BL)$

The Five-Factor Model (2015) — why add two more?

Even the 3-factor model left alpha in stocks sorted by profitability and investment. So Fama & French added:

Figure — Learn Fama-French three - five factor models

Worked Examples

Common Mistakes

#flashcards/stock-market

What are the three factors in the Fama-French 3-factor model?

Market excess return (Rm−Rf), SMB (size), HML (value).

What does SMB stand for and capture?

Small Minus Big; the size premium (small-cap stocks earning more than large-cap).

What does HML stand for and which stocks does positive HML favor?

High Minus Low book-to-market; positive HML = value (cheap) stocks over growth.

What two factors are added in the 5-factor model?

RMW (profitability: Robust Minus Weak) and CMA (investment: Conservative Minus Aggressive).

In the equation, what is the difference between a loading (e.g. s_i) and a premium (E[SMB])?

Loading = regression sensitivity (dimensionless slope); premium = the factor's expected return. Expected extra return = loading × premium.

What does alpha represent in a factor model?

The average return NOT explained by the included factors; ideally ≈0. Not automatically "skill".

Why were profitability and investment added?

The dividend-discount/valuation identity implies expected return rises with profitability and falls with investment, for given price & B/M.

How is SMB constructed to be a "pure" size bet?

Average small-cap portfolios across all B/M levels minus average big-cap across all B/M, so the value effect cancels.

CAPM predicts return from what single quantity?

Market beta only: E[Ri]−Rf = βi(E[Rm]−Rf).

A growth stock typically has what sign of HML loading?

Negative h (low book-to-market).

Recall Feynman: explain to a 12-year-old

Imagine sorting toy cars by two things: how tiny they are and how cheap they were. You notice tiny cars and bargain cars tend to roll faster downhill. CAPM only checked "how heavy is the hill" (the market). Fama & French said: also check "is it a tiny car?" (SMB) and "was it a bargain?" (HML). Later they added "does its engine make lots of money?" (RMW) and "does it waste money building stuff?" (CMA). Now, if a car goes fast, you can explain why instead of calling it magic (alpha).

Connections

CAPM and the Security Market Line — the one-factor ancestor this generalizes.
Arbitrage Pricing Theory (APT) — the multi-factor theoretical framework FF is an instance of.
Momentum Factor (Carhart 4-factor) — the famous factor FF originally left out.
Book-to-Market and Value Investing — economic story behind HML.
Behavioral Explanations vs Risk-Based Explanations — is the premium risk or mispricing?
Regression and R-squared — the statistical machinery of factor loadings.
Smart Beta and Factor Investing — how funds package these tilts.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, CAPM bolta tha ki kisi stock ka return sirf ek cheez se explain hota hai — market ke saath uska beta. Lekin real data mein dikha ki chhoti companies (small-cap) aur sasti/value companies (high book-to-market) extra return dete hain, jo akela beta explain nahi kar pata. Isliye Fama aur French ne do naye factor add kiye: SMB (Small Minus Big = size premium) aur HML (High Minus Low = value premium). Ye ek simple regression hai jisme portfolio ke excess return ko in factors pe fit karte ho.

Baad mein 2015 mein unhone do aur factor add kiye — RMW (profitable companies, Robust Minus Weak) aur CMA (kam invest karne wali conservative companies). Iska logic dividend-discount formula se aata hai: same price pe zyada profit wali company ka expected return zyada, aur zyada paisa spend/invest karne wali ka kam. Toh ye factors hawa mein nahi banaye, valuation equation se derive hote hain.

Sabse important baat samajhna: loading (jaise $s_i$ , $h_i$ ) aur premium (jaise $E[SMB]$ ) alag cheezein hain. Loading matlab sensitivity — stock kitna us factor jaisa behave karta hai. Premium matlab us factor ka average return. Expected extra return = loading × premium. Aur alpha ka matlab manager genius nahi hai — alpha sirf woh return hai jo tumhare factors se explain nahi hua. Behtar model lagao toh alpha aksar gayab ho jaata hai.

Exam aur real investing dono ke liye ye kaam ka hai: agar koi fund achha perform kar raha hai, factor regression laga ke dekho — kya woh sirf small-value tilt hai (jo tum khud sasti index se le sakte ho), ya sach mein alpha hai? Yehi "smart beta" investing ki jaan hai.

Test yourself — Factor & Behavioral Finance

Connections