Understand systematic vs unsystematic risk — Hinglish notes

5.5.7 · Stock-Market › Portfolio Theory

HUM KYA SPLIT KAR RAHE HAIN?

SPLIT KAISE DERIVE KAREIN (scratch se)

Model $R_{i} = α_{i} + β_{i} R_{m} + ε_{i}$ se shuru karo.

Step 1 — Dono sides ka variance lo. $σ_{i}^{2} = Var (α_{i} + β_{i} R_{m} + ε_{i})$ Yeh step kyun? $α_{i}$ ek constant hai, toh yeh variance mein kuch add nahi karta ( $Var (c + X) = Var (X)$ ).

Step 2 — $ε_{i}$ aur $R_{m}$ ki independence use karo. Kyunki $Cov (ε_{i}, R_{m}) = 0$ , sum ka variance = variances ka sum: $σ_{i}^{2} = β_{i}^{2} Var (R_{m}) + Var (ε_{i})$ Kyun? $Var (a X + Y) = a^{2} Var (X) + Var (Y)$ tabhi hota hai jab cross-covariance zero ho.

DIVERSIFICATION UNSYSTEMATIC RISK KO KAISE KHATAM KARTA HAI

$N$ stocks ka equally weighted portfolio banao, weight $w_{i} = 1/ N$ . Portfolio return: $R_{p} = \frac{1}{N} \sum_{i} R_{i} = \frac{1}{N} \sum_{i} (α_{i} + β_{i} R_{m} + ε_{i})$ .

Portfolio ka firm-specific piece hai $\overset{ε}{ˉ} = \frac{1}{N} \sum_{i} ε_{i}$ .

Step 1 — Average noise ka variance. Assume karo ki har $ε_{i}$ ka variance $\overset{σ}{ˉ}_{ε}^{2}$ hai aur woh uncorrelated hain: $Var (\overset{ε}{ˉ}) = \frac{1}{N ^{2}} \sum_{i = 1}^{N} σ_{ε_{i}}^{2} = \frac{1}{N ^{2}} \cdot N \overset{σ}{ˉ}_{ε}^{2} = \frac{σ ˉ _{ε}^{2}}{N}$ Yeh step kyun? Independent terms → variances add hote hain; $1/ N^{2}$ ka front factor weight ko square karne se aata hai.

Step 2 — Limit lo. $lim_{N \to \infty} \frac{σ ˉ _{ε}^{2}}{N} = 0$

Figure — Understand systematic vs unsystematic risk

Worked Examples

Common Mistakes

Active Recall

Total risk ke do components kya hain?

Systematic (market) risk + unsystematic (firm-specific) risk.

Kaunsa component diversify kiya ja sakta hai?

Unsystematic (idiosyncratic) risk — yeh firms ke across independent hai aur zero pe average ho jaata hai.

Risk-decomposition formula do.

σ_{i}^{2} = β_{i}^{2} σ_{m}^{2} + σ_{ε_{i}}^{2}

Large portfolio mein unsystematic risk kyun vanish ho jaata hai?

Independent shocks average out ho jaate hain; average noise ka variance

= \overset{σ}{ˉ}_{ε}^{2} / N \to 0

jaise

N \to \infty

Kaunsa risk risk premium earn karta hai aur kyun?

Sirf systematic risk — tum isse avoid nahi kar sakte, isliye market tumhe isse bear karne ka payment karta hai. Diversifiable risk kuch nahi earn karta.

Stock ke systematic risk sensitivity ko kya measure karta hai?

Uska beta,

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

Kya diversification SAARA risk remove karta hai?

Nahi — systematic floor

β_{p}^{2} σ_{m}^{2}

hamesha rehta hai.

3rd stock add karne se 30th stock add karna kam helpful kyun hai?

Risk reduction

1/ N

scale karta hai, toh benefits diminish hote hain; ~20–30 stocks mein zyaadatar capture ho jaata hai.

Recall Feynman: 12-saal ke bacche ko explain karo

Socho ek class hai jahan har bacche ka grade do cheezoon pe depend karta hai: (1) kya poore school ka acha ya bura saal raha (ek badi toofaan sabke exams cancel kar deti hai) aur (2) har bacche ki apni luck (tum ek test ke din beemar pad gaye). Agar tum poori class ke grades average karo, toh personal bad-luck days cancel ho jaate hain — kuch bacche beemar the, kuch lucky the. Lekin agar toofaan poore school mein aayi, toh average karna help nahi karta — sab ko nuksaan hua. Apni luck = unsystematic risk (group mein chali jaati hai). Toofaan = systematic risk (rehta hai). Aur tumhe sirf toofaan survive karne ka "reward" milta hai, kyunki personal bad luck se bachna free tha (sirf group join karo).

Connections

Beta and CAPM — beta hi systematic risk ka measure hai; CAPM ise price karta hai.
Diversification and Correlation — low correlation hi allow karta hai ki specific shocks cancel ho jayein.
Portfolio Variance and Covariance — general two-asset variance formula jiska yeh specialization hai.
Efficient Frontier — unsystematic risk ko diversify karke banaya jaata hai.
Security Market Line — return vs beta (systematic risk) plot karta hai, total risk nahi.
Sharpe Ratio — total risk ke per unit return reward karta hai; CAPM ke beta se contrast karo.