Understand CAPM and beta — Stock-Market notes

WHY does CAPM exist?

WHY: After Markowitz portfolio theory, we know diversification kills company-specific risk. So if the market only pays you for undiversifiable risk, we need a number that measures "how much of the market's swings does this stock inherit?" That number is beta. CAPM turns beta into an expected return.

WHAT: A linear relationship between an asset's expected return and its exposure to the overall market.

HOW: Split total risk into two parts, argue one part is free to remove, price only the other.

Derive beta from first principles

Step 1 — What does "sensitivity to the market" mean? Regress the asset return $r_i$ on the market return $r_m$ : $r_i = \alpha + \beta_i\, r_m + \varepsilon_i$ Why this step? A straight-line fit; the slope $\beta_i$ tells us "if the market moves 1%, the asset moves $\beta_i$ % on average." $\varepsilon_i$ is the leftover (idiosyncratic) noise, uncorrelated with the market.

Step 2 — Slope of a regression line. The least-squares slope is always $\dfrac{\text{Cov}(x,y)}{\text{Var}(x)}$ . Here $x=r_m$ , $y=r_i$ : $\boxed{\beta_i=\frac{\text{Cov}(r_i,r_m)}{\text{Var}(r_m)}}$ Why this step? Covariance measures how they move together; dividing by $\text{Var}(r_m)$ normalises so the market itself has $\beta=1$ .

Step 3 — Sanity check the market's own beta. $\beta_m=\frac{\text{Cov}(r_m,r_m)}{\text{Var}(r_m)}=\frac{\text{Var}(r_m)}{\text{Var}(r_m)}=1$ Why this step? Confirms the yardstick: the market has $\beta=1$ by construction. Risk-free asset has $\beta=0$ (no covariance).

Derive the CAPM equation

Idea: Expected return must be a straight line in beta, passing through two points we already know.

A $\beta=0$ asset (risk-free) must return $r_f$ . → point $(0,\,r_f)$ .
A $\beta=1$ asset (the market) must return $E[r_m]$ . → point $(1,\,E[r_m])$ .

A straight line through these two points has slope $=\dfrac{E[r_m]-r_f}{1-0}=E[r_m]-r_f$ and intercept $r_f$ :

$\boxed{\,E[r_i]=r_f+\beta_i\big(E[r_m]-r_f\big)\,}$

Why this works: CAPM proves (via mean-variance optimisation) that in equilibrium every asset lies on this line — the Security Market Line (SML). If a stock plots above the line it's underpriced (buy!); below the line, overpriced.

Worked examples

Common mistake Steel-man the common errors

"Higher total volatility = higher CAPM return." Why it feels right: more ups and downs feels riskier, so surely it should pay more. The fix: CAPM only prices $\beta$ (systematic part). Volatility from idiosyncratic noise ( $\varepsilon_i$ ) is uncompensated because a diversified investor deletes it. Example 4 proves it.

"Beta is the correlation." Why it feels right: both measure "moving together." The fix: $\beta=\rho_{im}\,\sigma_i/\sigma_m$ . Correlation is scale-free; beta is scaled by relative volatility, so beta can exceed 1 while correlation cannot.

"A negative beta stock is bad." Why it feels right: negative sounds like a loss. The fix: $\beta<0$ (e.g. gold, some hedges) means it rises when the market falls — valuable insurance. CAPM gives it $E[r_i]<r_f$ , and investors happily accept it.

Recall Feynman: explain to a 12-year-old

Imagine the stock market is a big trampoline everyone is bouncing on. Beta is how much your bounce copies the big group bounce. Beta 1 = you bounce exactly with the crowd. Beta 2 = you fly twice as high (and crash twice as low). Beta 0 = you're standing on the floor, ignoring the trampoline. CAPM is a rule that says: you only get extra candy for the bouncing you can't avoid (the crowd bounce). If you personally wobble because you ate too much sugar, that's your problem — no candy for that, because your friends could just look away and it cancels out.

Active recall

What does CAPM stand for?

Capital Asset Pricing Model.

What is the CAPM equation?

E[r_i]=r_f+\beta_i(E[r_m]-r_f)

Define beta as a formula.

\beta_i=\text{Cov}(r_i,r_m)/\text{Var}(r_m)

What is beta of the market portfolio?

Exactly 1.

What is beta of the risk-free asset?

0 (no covariance with the market).

Which risk does CAPM reward — systematic or idiosyncratic?

Only systematic (undiversifiable) risk.

Why isn't idiosyncratic risk rewarded?

Because it can be diversified away for free; the market won't pay for removable risk.

Second formula for beta using correlation?

\beta_i=\rho_{im}\,\sigma_i/\sigma_m

A stock plots ABOVE the Security Market Line — over- or under-priced?

Under-priced (offers more return than its beta warrants) → buy.

What does the slope of the SML represent?

The market risk premium

E[r_m]-r_f

r_f=3\%

E[r_m]=11\%

\beta=0.5

, find

E[r_i]

3\%+0.5(8\%)=7\%

Can beta be negative and what does it mean?

Yes; the asset moves opposite the market (hedge/insurance), with

E[r_i]<r_f

Connections

Markowitz Portfolio Theory — CAPM builds on diversification.
Systematic vs Idiosyncratic Risk — the split CAPM exploits.
Security Market Line — geometric form of CAPM.
Sharpe Ratio — reward per unit of total risk, contrast with beta.
Efficient Frontier and the Capital Market Line — where the market portfolio comes from.
Cost of Equity (WACC) — CAPM is used to estimate it.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, CAPM ka funda simple hai: market tumhe sirf us risk ke liye paisa (extra return) deta hai jo tum diversify karke hata nahi sakte. Agar ek stock apni company ke reasons se upar-neeche hota hai (idiosyncratic risk), toh koi baat nahi — tum 50 aur stocks kharid ke us wobble ko cancel kar sakte ho, isliye market uske liye extra candy nahi deta. Jo bacha, wahi market risk, aur usko naapte hain beta se.

Beta matlab: "market 1% hilta hai toh yeh stock kitna hilta hai?" Beta = Cov(stock, market) / Var(market). Market ka apna beta hamesha 1, aur risk-free asset ka beta 0. Formula yaad rakho: $E[r]=r_f+\beta(E[r_m]-r_f)$ — matlab free floor ( $r_f$ ) plus beta guna bonus (market premium). High beta = zyada return chahiye, low/negative beta = kam ya risk-free se bhi kam return (kyunki woh hedge ka kaam karta hai).

Ek common galti: log sochte hain "zyada volatile stock = zyada return". Nahi bhai! CAPM sirf beta dekhta hai. Agar volatility idiosyncratic hai (beta 0), toh return sirf $r_f$ milega, chahe stock kitna bhi uchhle. Yeh baat interview aur real investing dono mein kaam aati hai.

Kyun important? Kyunki isse hum kisi bhi stock ka fair expected return nikaal sakte hain, aur agar actual return isse zyada mile toh stock "under-priced" hai — buy signal. Yahi CAPM ki asli power hai.

Test yourself — Portfolio Theory

Connections