Understand CAPM and beta — Hinglish notes

5.5.5 · Stock-Market › Portfolio Theory

CAPM exist kyun karta hai?

WHY: Markowitz portfolio theory ke baad, hum jaante hain ki diversification company-specific risk ko khatam kar deti hai. Toh agar market sirf undiversifiable risk ka payment karta hai, toh humein ek aisa number chahiye jo measure kare ki "is stock mein market ke swings kitne aate hain?" Woh number hai beta. CAPM beta ko ek expected return mein convert karta hai.

WHAT: Ek asset ke expected return aur overall market ke saath uski exposure ke beech ek linear relationship.

HOW: Total risk ko do parts mein baanto, argue karo ki ek part free mein hataya ja sakta hai, aur sirf doosre part ki pricing karo.

Beta ko first principles se derive karo

Step 1 — "Market ke saath sensitivity" ka matlab kya hai? Asset return $r_{i}$ ko market return $r_{m}$ par regress karo: $r_{i} = α + β_{i} r_{m} + ε_{i}$ Yeh step kyun? Ek straight-line fit; slope $β_{i}$ batata hai ki "agar market 1% move kare, toh asset average mein $β_{i}$ % move karta hai." $ε_{i}$ bacha hua (idiosyncratic) noise hai, jo market se uncorrelated hai.

Step 2 — Regression line ka slope. Least-squares slope hamesha $\frac{Cov ( x , y )}{Var ( x )}$ hota hai. Yahan $x = r_{m}$ , $y = r_{i}$ : $β_{i} = \frac{Cov ( r _{i} , r _{m} )}{Var ( r _{m} )}$ Yeh step kyun? Covariance measure karta hai ki woh saath mein kitna move karte hain; $Var (r_{m})$ se divide karne se normalise ho jaata hai taaki market khud ka $β = 1$ ho.

Step 3 — Market ke apne beta ka sanity check. $β_{m} = \frac{Cov ( r _{m} , r _{m} )}{Var ( r _{m} )} = \frac{Var ( r _{m} )}{Var ( r _{m} )} = 1$ Yeh step kyun? Yardstick confirm karta hai: market ka $β = 1$ construction se hota hai. Risk-free asset ka $β = 0$ hai (koi covariance nahi).

CAPM equation derive karo

Idea: Expected return ko beta mein ek straight line hona chahiye, jo do already known points se guzre.

Ek $β = 0$ asset (risk-free) ko $r_{f}$ return karna chahiye. → point $(0, r_{f})$ .
Ek $β = 1$ asset (market) ko $E [r_{m}]$ return karna chahiye. → point $(1, E [r_{m}])$ .

In do points se guzarne wali straight line ka slope $= \frac{E [ r _{m} ] - r _{f}}{1 - 0} = E [r_{m}] - r_{f}$ aur intercept $r_{f}$ hai:

$E [r_{i}] = r_{f} + β_{i} (E [r_{m}] - r_{f})$

Yeh kyun kaam karta hai: CAPM prove karta hai (mean-variance optimisation ke through) ki equilibrium mein har asset is line par hoti hai — Security Market Line (SML). Agar koi stock line ke upar plot ho toh woh underpriced hai (kharido!); line ke neeche ho toh overpriced hai.

Worked examples

Common mistake Common errors ko steel-man karo

"Zyada total volatility = zyada CAPM return." Kyun sahi lagta hai: zyada ups aur downs feel hote hain riskier, toh surely zyada pay karna chahiye. Fix: CAPM sirf $β$ (systematic part) ko price karta hai. Idiosyncratic noise ( $ε_{i}$ ) se aane wali volatility uncompensated hai kyunki ek diversified investor use delete kar deta hai. Example 4 prove karta hai.

"Beta correlation hota hai." Kyun sahi lagta hai: dono "saath move karna" measure karte hain. Fix: $β = ρ_{im} σ_{i} / σ_{m}$ . Correlation scale-free hai; beta relative volatility se scale hota hai, isliye beta 1 se zyada ho sakta hai jabki correlation nahi ho sakta.

"Negative beta wala stock bura hota hai." Kyun sahi lagta hai: negative sunne mein loss jaisa lagta hai. Fix: $β < 0$ (jaise gold, kuch hedges) matlab woh tab badhta hai jab market girta hai — valuable insurance hai. CAPM isko $E [r_{i}] < r_{f}$ deta hai, aur investors khushi se accept karte hain.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho stock market ek bada trampoline hai jis par sab log bounce kar rahe hain. Beta yeh hai ki teri bounce bade group ki bounce ko kitna copy karti hai. Beta 1 = tum exactly crowd ke saath bounce karte ho. Beta 2 = tum do guna zyada oopar jaate ho (aur do guna zyada neeche girte ho). Beta 0 = tum zameen par khade ho, trampoline ko ignore kar rahe ho. CAPM ek rule hai jo kehta hai: tumhe extra candy sirf us bouncing ke liye milti hai jis se tum bach nahi sakte (crowd bounce). Agar tum personally isliye hilte ho kyunki tumne zyada sugar khayi, toh yeh tumhara problem hai — uske liye koi candy nahi, kyunki tumhare doston ne sirf door dekha aur woh cancel ho gaya.

Active recall

CAPM ka full form kya hai?

Capital Asset Pricing Model.

CAPM equation kya hai?

E [r_{i}] = r_{f} + β_{i} (E [r_{m}] - r_{f})

Beta ko formula se define karo.

β_{i} = Cov (r_{i}, r_{m}) / Var (r_{m})

Market portfolio ka beta kya hota hai?

Exactly 1.

Risk-free asset ka beta kya hota hai?

0 (market ke saath koi covariance nahi).

CAPM kaun sa risk reward karta hai — systematic ya idiosyncratic?

Sirf systematic (undiversifiable) risk.

Idiosyncratic risk ko reward kyun nahi milta?

Kyunki ise free mein diversify karke hataya ja sakta hai; market removable risk ke liye pay nahi karega.

Correlation use karke beta ki doosri formula kya hai?

β_{i} = ρ_{im} σ_{i} / σ_{m}

Ek stock Security Market Line ke UPAR plot ho — over- ya under-priced?

Under-priced (apne beta se zyada return offer karta hai) → kharido.

SML ka slope kya represent karta hai?

Market risk premium

E [r_{m}] - r_{f}

Agar

r_{f} = 3%

E [r_{m}] = 11%

β = 0.5

, toh

E [r_{i}]

nikalo.

3% + 0.5 (8%) = 7%

Kya beta negative ho sakta hai aur iska kya matlab hai?

Haan; asset market ke opposite move karta hai (hedge/insurance), aur

E [r_{i}] < r_{f}

hota hai.

Connections

Markowitz Portfolio Theory — CAPM diversification par build karta hai.
Systematic vs Idiosyncratic Risk — woh split jo CAPM use karta hai.
Security Market Line — CAPM ki geometric form.
Sharpe Ratio — total risk per unit reward, beta se contrast.
Efficient Frontier and the Capital Market Line — jahan se market portfolio aata hai.
Cost of Equity (WACC) — CAPM use hota hai ise estimate karne ke liye.