Understand diversification benefits — Hinglish notes

5.5.1 · Stock-Market › Portfolio Theory

Diversification KYA hai?

Key statistical fact: risk additive NAHI hoti. Expected returns linearly add up hoti hain, lekin standard deviations nahi hoti — kyunki jab ek asset zig karta hai, doosra zag kar sakta hai, aur woh movements partially cancel ho jaate hain.

Yeh KYUN kaam karta hai? (First-principles derivation)

Chaliye two-asset portfolio risk formula ko scratch se banate hain. Yeh sab kuch ka dil hai.

Setup. Weights $w_{1}$ aur $w_{2}$ jahan $w_{1} + w_{2} = 1$ . Random returns $R_{1}, R_{2}$ . Portfolio return:

$R_{p} = w_{1} R_{1} + w_{2} R_{2}$

Step 1 — Expected return (linear KYUN hai?) Expectation ek linear operator hai, isliye:

$E [R_{p}] = w_{1} E [R_{1}] + w_{2} E [R_{2}] = w_{1} μ_{1} + w_{2} μ_{2}$ Yeh step kyun? $E [a X + bY] = a E [X] + b E [Y]$ hamesha hota hai — koi assumptions nahi chahiye. Return sirf average ho jaata hai.

Step 2 — Variance (linear KYUN nahi?) Ek sum ka variance ek cross term ke saath expand hota hai:

$Var (R_{p}) = E [(R_{p} - E [R_{p}])^{2}]$

$R_{p} - E [R_{p}] = w_{1} (R_{1} - μ_{1}) + w_{2} (R_{2} - μ_{2})$ substitute karo aur square karo:

σ_{p}^{2} = w_{1}^{2} E [(R_{1} - μ_{1})^{2}] + w_{2}^{2} E [(R_{2} - μ_{2})^{2}] + 2 w_{1} w_{2} E [(R_{1} - μ_{1}) (R_{2} - μ_{2})]

Yeh step kyun? Ek binomial ko square karne se $a^{2} + b^{2} + 2 ab$ milta hai. Pehle do terms variances hain; last term woh jagah hai jahan jaadu chhupa hua hai.

Step 3 — Pieces ko naam do. Define karo $Cov (R_{1}, R_{2}) = σ_{12} = E [(R_{1} - μ_{1}) (R_{2} - μ_{2})]$ aur $ρ_{12} = \frac{σ _{12}}{σ _{1} σ _{2}}$ (correlation, hamesha $[- 1, 1]$ mein).

Correlation KAISE benefit drive karta hai

Cross term $2 w_{1} w_{2} ρ_{12} σ_{1} σ_{2}$ dekho (weights positive hain):

$ρ_{12}$	Cross term	Result
$+ 1$	maximum positive	Koi fayda nahi: $σ_{p} = w_{1} σ_{1} + w_{2} σ_{2}$ (bas ek weighted average)
$0$	zero	Meaningful fayda: $σ_{p} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2}$
$- 1$	maximum negative	Maximum fayda: sahi weights par risk zero ho sakti hai

Proof ki $ρ = + 1$ se koi fayda nahi. $ρ = 1$ set karo: $σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} σ_{1} σ_{2} = (w_{1} σ_{1} + w_{2} σ_{2})^{2}$ Toh $σ_{p} = w_{1} σ_{1} + w_{2} σ_{2}$ — ek perfect square, exactly weighted average. Kuch cancel nahi hota.

Proof ki $ρ = - 1$ se saari risk khatam ho sakti hai. $ρ = - 1$ set karo: $σ_{p}^{2} = (w_{1} σ_{1} - w_{2} σ_{2})^{2}$ Yeh $0$ hota hai jab $w_{1} σ_{1} = w_{2} σ_{2}$ , yaani $w_{1} = \frac{σ _{2}}{σ _{1} + σ _{2}}$ . Ek perfectly hedged, riskless combo.

Figure — Understand diversification benefits

Kai assets: systematic vs unsystematic risk

$N$ assets ke equally-weighted portfolio ke liye ( $w_{i} = 1/ N$ ), har ek ki variance $σ^{2}$ aur average pairwise covariance $\overset{c}{ˉ}$ :

$σ_{p}^{2} = \frac{1}{N} σ^{2} + (1 - \frac{1}{N}) \overset{c}{ˉ}$

Yeh step kyun? $N$ variance terms hain har ek $(1/ N)^{2}$ weighted, jo $σ^{2} / N$ dete hain, aur $N (N - 1)$ covariance terms hain har ek $(1/ N)^{2}$ weighted, jo $\frac{N - 1}{N} \overset{c}{ˉ}$ dete hain.

80/20 takeaway: Diversification ka zyaadatar fayda pehle ~20–30 stocks se hi aa jaata hai; 500th stock add karna muskil se kaam aata hai kyunki $σ^{2} / N$ already bahut chhota hota hai. Energy low-correlation assets choose karne par lagao, sirf count badhane par nahi.

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum ice cream aur umbrellas bechte ho. Sunny days mein ice cream bikti hai; rainy days mein umbrellas bikti hain. Tum almost hamesha kuch na kuch kamate rehte ho — tumhari income steady rehti hai chahe har business akela wild ho. Yahi diversification hai: aisi cheezein chuno jinka bura din ek saath na aaye, aur tumhara total smoother rahega. Lekin tum kabhi woh risk nahi hata sakte jo sabko ek saath affect kare (jaise poore town mein power cut) — woh "market risk" hai jiske saath tum stuck ho.

Active Recall Flashcards

Portfolio expected returns linearly kyun add hoti hain lekin risks kyun nahi?

Expectation ek linear operator hai isliye

E [R_{p}] = \sum w_{i} μ_{i}

; variance mein ek cross (covariance) term hota hai jo linear nahi hai.

Two-asset variance formula mein diversification benefit kahan rehti hai?

Cross term

2 w_{1} w_{2} ρ_{12} σ_{1} σ_{2}

mein — lower correlation use shrink karta hai.

ρ = + 1

par

σ_{p}

kya hota hai?

Plain weighted average

w_{1} σ_{1} + w_{2} σ_{2}

— zero benefit.

ρ = - 1

par, kaunsa weight two-asset portfolio ko riskless banata hai?

w_{1} = σ_{2} / (σ_{1} + σ_{2})

, jisse

σ_{p} = 0

milta hai.

Equal weights ke liye

N \to \infty

par

σ_{p}^{2}

kiske paas jaata hai?

Average pairwise covariance

\overset{c}{ˉ}

(systematic/market risk floor).

Kaunsi risk diversify karke hataayi ja sakti hai aur kaunsi nahi?

Unsystematic (company-specific) ko remove kar sakte ho; systematic (market) ko nahi.

"200 stocks rakho" aksar wasteful kyun hai?

Fayda

\overset{c}{ˉ}

se neeche bounded hai; ~20–30 low-corr names ke baad marginal reduction tiny hai (80/20).

Do stocks 50/50,

σ_{1} = 20%, σ_{2} = 15%, ρ = 0.3

σ_{p}

kya hai?

\approx 14.19%

vs naive average

17.5%

Kya diversification ek free lunch hai? Kyun?

Haan — yeh risk kam karta hai bina linear weighted-average expected return kam kiye.

Connections

Correlation and Covariance — benefit ka engine
Efficient Frontier — inhi risk/return combos ko plot karna
Capital Asset Pricing Model (CAPM) — sirf systematic risk (beta) price karta hai
Systematic vs Unsystematic Risk
Portfolio Variance and Standard Deviation
Modern Portfolio Theory (Markowitz)