Understand factor investing fundamentals — Stock-Market notes

WHAT is a factor?

WHY do premiums exist? Two competing stories (both partly true):

Risk-based: value stocks are cheap because they are genuinely riskier (distress, illiquidity), so investors demand higher return. No free lunch — you get paid to hold pain.
Behavioral: investors overreact/underreact (herding, overconfidence), mispricing stocks systematically, and the factor harvests that mistake. This can persist because arbitrage has limits (costs, career risk).

HOW we express returns: from CAPM to multi-factor

Step 1 — The single-factor model (CAPM), derived

We assume a stock's excess return (return above the risk-free rate $r_f$ ) moves linearly with the market's excess return.

$R_i - r_f = \alpha_i + \beta_i (R_m - r_f) + \varepsilon_i$

Why this form?

$R_i - r_f$ : we work in excess returns because only the reward above the guaranteed rate is a compensation for risk.
$\beta_i$ : the sensitivity — how much stock $i$ moves for each 1% the market moves. Derived as a regression slope: $\beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$ Why? Slope of the best-fit line minimizing $E[\varepsilon^2]$ is exactly cov over var — that's the least-squares result.
$\alpha_i$ : the intercept — return unexplained by the market. In an efficient CAPM world, $\alpha_i = 0$ .
$\varepsilon_i$ : idiosyncratic noise, $E[\varepsilon_i]=0$ , diversifiable.

Step 2 — CAPM leaves patterns behind

Empirically, small-cap and cheap ("value") stocks earned more than their $\beta$ predicted → a positive $\alpha$ that wasn't random. That leftover $\alpha$ was actually a hidden factor exposure.

Step 3 — Add factors (Fama–French style)

We augment the equation with factor-mimicking portfolios, each a long-minus-short basket:

$R_i - r_f = \alpha_i + \beta_{m}(R_m - r_f) + \beta_s\,\text{SMB} + \beta_v\,\text{HML} + \beta_{mom}\,\text{WML} + \varepsilon_i$

Figure — Understand factor investing fundamentals

The factor premium in numbers

Worked Examples

Worked example Example 2 — Computing a market beta

Over 3 periods, stock returns $= [2\%, -1\%, 4\%]$ ; market $= [3\%, -2\%, 5\%]$ . $\bar R_i = 1.667\%,\ \bar R_m = 2\%$ . Deviations $R_i$ : $[0.333, -2.667, 2.333]$ ; $R_m$ : $[1, -4, 3]$ . $\operatorname{Cov} = \frac{0.333(1) + (-2.667)(-4) + 2.333(3)}{3} = \frac{0.333 + 10.667 + 7}{3} = 6.0$ $\operatorname{Var}(R_m) = \frac{1 + 16 + 9}{3} = 8.667$ $\beta = 6.0 / 8.667 = \mathbf{0.69}$

Why cov/var? It's the least-squares slope: how much of the market's variance the stock co-moves with.

Common Mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine sorting kids by height in gym class. Tall kids tend to be picked first for basketball — "tallness" is a factor that predicts something. In the stock market, some traits — being cheap, being small, being on a winning streak — tend to predict which stocks earn a bit more over time. Investing by factors is like saying "I'll always draft the tall, fast, and cheap-to-pick players," and on average my team scores more. But some seasons the tall kids get injured — so it doesn't win every game, only over many, many games.

Recall — Flashcards

What is a factor in factor investing?

A broad, persistent, investable characteristic (e.g., value, size, momentum) that systematically explains return differences across securities.

Why does a factor premium exist?

Either risk-based compensation (you're paid to hold genuinely riskier assets) or behavioral (harvesting others' persistent mispricing that limits-to-arbitrage can't erase).

Formula for market beta?

\beta_i = \operatorname{Cov}(R_i,R_m)/\operatorname{Var}(R_m)

— the least-squares regression slope.

Why do we work in excess returns (

R-r_f

Only the reward above the guaranteed risk-free rate is compensation for risk; the risk-free portion carries no risk premium.

What does SMB stand for and measure?

Small Minus Big; the size premium (small caps minus large caps).

What does HML measure?

High Minus Low book-to-market; the value premium (cheap stocks minus expensive/growth stocks).

Why are factor portfolios long–short?

To cancel common market movement and isolate the pure return attributable to the characteristic itself.

Difference between alpha and factor premium?

Factor premium is systematic return from factor loadings (cheaply replicable); alpha is return left over after removing all factor exposures — genuine unique skill.

Expected excess return in a multi-factor model?

E[R_i]-r_f = \sum_k \beta_{i,k}\lambda_k

— sum of factor loadings times factor premiums.

Five criteria of a genuine factor?

Persistent, pervasive, robust, investable, and has an economic rationale (guards against the "factor zoo").

Connections

Capital Asset Pricing Model (CAPM)
Fama-French Three-Factor Model
Momentum Anomaly
Efficient Market Hypothesis
Limits to Arbitrage
Behavioral Biases (Overconfidence, Herding)
Smart Beta ETFs
Portfolio Beta and Diversification

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, factor investing ka core idea simple hai: stock ka return sirf "market upar gaya toh main bhi upar" nahi hota. Return ke peeche kuch repeatable characteristics hote hain — jaise stock ka sasta hona (value), chhota hona (size), ya winning streak pe hona (momentum). In characteristics ko hum factors bolte hain, aur inke saath ek chhota extra return judaa hota hai jise factor premium kehte hain.

Ye premium milta kyun hai? Do reasons: ya toh us factor mein sach mein zyada risk hai (jaise sasti companies distress mein ho sakti hain, isliye market unhe zyada reward deta hai), ya phir baaki investors baar-baar wahi behavioral galti karte hain (herding, overreaction) aur factor us galti ko exploit karta hai. Dono ka mix hota hai. Isliye hi CAPM ke ek market factor se aage badh ke Fama-French ne size aur value factors add kiye — kyunki sirf market beta se sab kuch explain nahi ho raha tha.

Formula yaad rakho: $E[R] - r_f = \sum \beta_k \lambda_k$ . Matlab tumhara expected extra return = har factor pe tumhara loading ( $\beta$ ) times uska premium ( $\lambda$ ), sabko jodo. Beta nikaalte hain regression slope se: cov divided by variance. Aur factor portfolios hamesha long-minus-short hote hain (jaise HML = value minus growth) taaki common market movement cancel ho jaaye aur sirf pure factor ka effect bache.

Ek important warning: factor premium guarantee nahi hai har saal ke liye — value factor 2010s mein ek decade tak underperform kiya. Yahi toh reason hai ki premium exist karta hai — agar aasaan hota toh sab kar lete aur arbitrage khatam kar deta. Aur "alpha" ko factor premium samajhne ki galti mat karo: agar tumhara outperformance kisi factor loading se explain ho raha hai, woh skill nahi, systematic factor return hai jo tum sasta ETF se bhi khareed sakte ho.

Test yourself — Factor & Behavioral Finance

Connections