Understand correlation risk across positions

4.7.7 · Stock-Market › Risk & Money Management

Correlation risk KYUN exist karta hai?

WHAT chahiye humein: paisa spread karo taaki jab ek cheez neeche jaye, dusri cushion kare.

KYUN fail hota hai: stocks common factors se drive hote hain — interest rates, oil prices, overall market ("beta"), sector news. Agar do positions ek hi driver share karti hain, toh woh ek saath crash karte hain. Diversification tabhi risk reduce karti hai jab positions not perfectly correlated hon.

HOW measure karte hain: correlation coefficient $ρ$ se, jo $- 1$ se $+ 1$ tak range karta hai.

Portfolio risk ko scratch se derive karna

Hum formula seedha dump nahi karte — hum ise build karte hain.

Step 1 — Portfolio return. Do positions weights $w_{A}, w_{B}$ ke saath ( $w_{A} + w_{B} = 1$ ): $R_{p} = w_{A} R_{A} + w_{B} R_{B}$ Yeh step kyun? Portfolio return simply pieces ka weighted average hota hai — money-weighted.

Step 2 — Variance risk ka measure hai. Risk = $R_{p}$ ka variance: $σ_{p}^{2} = Var (w_{A} R_{A} + w_{B} R_{B})$

Step 3 — Variance ke algebra se expand karo. Kisi bhi $X, Y$ ke liye: $Var (a X + bY) = a^{2} Var (X) + b^{2} Var (Y) + 2 ab Cov (X, Y)$ Kyun? Deviation $(a (X - μ_{X}) + b (Y - μ_{Y}))$ ko square karo aur expectation lo; cross term survive karta hai.

Toh: $σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} Cov (R_{A}, R_{B})$

Step 4 — Covariance ki jagah correlation daalo. Kyunki $Cov = ρ σ_{A} σ_{B}$ :

$ρ$ term kya karta hai (the punchline)

Equal weights $w_{A} = w_{B} = \frac{1}{2}$ aur equal risk $σ_{A} = σ_{B} = σ$ lo: $σ_{p}^{2} = \frac{1}{4} σ^{2} + \frac{1}{4} σ^{2} + 2 \cdot \frac{1}{4} ρ σ^{2} = \frac{σ ^{2}}{2} (1 + ρ)$ $σ_{p} = σ \frac{1 + ρ}{2}$

$ρ$	$σ_{p}$	Matlab
$+ 1$	$σ$	No diversification — ek hi stock jaisa
$0$	$σ / 2 \approx 0.71 σ$	Real risk reduction
$- 1$	$0$	Risk fully cancel (perfect hedge)

Figure — Understand correlation risk across positions

Worked examples

Common mistakes (steel-manned)

The 80/20 takeaway

Recall Feynman: 12-saal-ke-bachche ko explain karo

Socho tum ek soccer game pe bet lagate ho. Safe rehne ke liye tum... usi team par usi game mein bhi bet lagate ho. Woh do bets nahi hain — woh ek badi bet hai! Asli safety hai alag-alag games par bet lagana jo ek doosre ko affect na kare. Stock market mein, stocks jo saath upar-neeche jaate hain woh same team ki tarah hain. Unke das hone se bhi ek hi badi bet hai. Sach mein safe rehne ke liye, aisi cheezein rakho jo alag-alag reasons se move karti hain — tab jab ek ka bura din ho, doosre ka acha din ho sakta hai.

Flashcards

Correlation risk ka matlab kya hai?

Woh risk ki positions jo aapko lagta hai diversified hain woh actually saath move karti hain, isliye combined losses expected se bahut zyada hoti hain.

Two-asset portfolio variance ka formula?

σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ σ_{A} σ_{B}

Is formula mein POORA diversification benefit kahan rehta hai?

Correlation cross term

2 w_{A} w_{B} ρ σ_{A} σ_{B}

mein; low/negative ρ portfolio risk ghataata hai.

Equal weight aur equal σ ke liye, σ_p kya hai?

σ_{p} = σ (1 + ρ) /2

ρ = +1 par, ek single stock ke comparison mein portfolio risk kya hai?

Same — bilkul koi diversification benefit nahi.

Equal weights aur σ ke saath ρ = −1 par σ_p kya hai?

Zero — positions perfectly hedge ho jaati hain.

N equal-ρ positions ke liye N→∞ par diversification floor kya hai?

σ_{p} \to σ ρ

; correlation kitna risk remove kar sakte ho uski cap lagaata hai.

"Mere paas 10 stocks hain toh main safe hoon" kyun galat hai?

Diversification correlation par depend karta hai, tickers ki sankhya par nahi; 10 highly correlated names ≈ ek bet.

Market crashes mein correlations ka kya hota hai?

Woh

+ 1

ki taraf spike karte hain, isliye jab sabse zyada zarurat ho tab diversification weak ho jaati hai.

Kya negative correlation free profit deta hai?

Nahi — yeh risk reduce karta hai lekin return bhi cancel karta hai; yeh ek hedge hai, edge nahi.

Connections

Diversification — low correlation ki practical application.
Position Sizing — combined correlated block ko size karo, har naam ko alag nahi.
Beta and Market Risk — correlation drive karne wala common factor.
Value at Risk (VaR) — directly covariance matrix use karta hai.
Hedging Strategies — deliberately negative ρ use karna.
Standard Deviation and Variance — in formulas ke building blocks.
Portfolio Theory (Markowitz) — efficient frontier exactly isi math se banti hai.