Understand correlation between instruments — Hinglish notes

4.2.10 · Stock-Market › What to Trade

What Is Correlation?

YE range kyun? Correlation, covariance ko har asset ki volatility se normalize karta hai, ek dimensionless measure banata hai. Math ise -1 aur +1 ke beech force karta hai.

Derivation from First Principles

Chaliye correlation formula ko step by step banate hain.

Step 1: Returns define karo Do instruments A aur B ke liye n periods mein:

Returns: $R_{A} = {r_{A, 1}, r_{A, 2}, .., r_{A, n}}$
Returns: $R_{B} = {r_{B, 1}, r_{B, 2}, ..., r_{B, n}}$

Step 2: Dekho ki ve saath kaise vary karte hain Covariance joint movement ko capture karta hai: $Cov (A, B) = \frac{1}{n - 1} \sum_{i = 1}^{n} (r_{A, i} - \overset{r}{ˉ}_{A}) (r_{B, i} - \overset{r}{ˉ}_{B})$

n-1 kyun? Sample variance ke liye Bessel's correction (unbiased estimator). Hum sample data use kar rahe hain, poori population nahi.

ISKA matlab kya hai? Har term $(r_{A, i} - \overset{r}{ˉ}_{A}) (r_{B, i} - \overset{r}{ˉ}_{B})$ ye hota hai:

Positive jab dono returns apne means se upar ya neeche hain (saath chalte hain)
Negative jab ek upar aur ek neeche ho (opposite chalte hain)
Zero jab mean par ho

Step 3: Volatility se normalize karo Covariance assets ke scales par depend karta hai. ₹5000 wale stock mein ₹50 wale se zyada absolute moves hote hain. Hume ek scale-free measure chahiye.

Returns ka standard deviation: $σ_{A} = \frac{1}{n - 1} \sum_{i = 1}^{n} (r_{A, i} - \overset{r}{ˉ}_{A})^{2}$

Normalize kaise karein? Covariance ko dono standard deviations se divide karo:

YE kaam kyun karta hai? $σ_{A} σ_{B}$ se divide karne par covariance [-1, +1] mein scale ho jaata hai. Ise "covariance per unit of volatility" maano. Cauchy–Schwarz inequality guarantee karta hai ki $∣ Cov (A, B) ∣ \leq σ_{A} σ_{B}$ , isliye $∣ ρ ∣ \leq 1$ hamesha.

Expanded form: $ρ_{A, B} = \frac{\sum _{i = 1}^{n} ( r _{A, i} - r ˉ _{A} ) ( r _{B, i} - r ˉ _{B} )}{\sum _{i = 1}^{n} ( r _{A, i} - r ˉ _{A} ) ^{2} \cdot \sum _{i = 1}^{n} ( r _{B, i} - r ˉ _{B} ) ^{2}}$

Figure — Understand correlation between instruments

Real-World Examples

Day	Infosys Return	TCS Return
1	+2.1%	+1.8%
2	-1.5%	-1.2%
3	+0.8%	+1.1%
...	...	...

Step 1: Mean returns calculate karo

$\overset{r}{ˉ}_{Infosys} = 0.4%$
$\overset{r}{ˉ}_{TCS} = 0.5%$

Step 2: Deviations aur products calculate karo

Day 1: $(2.1 - 0.4) (1.8 - 0.5) = 1.7 \times 1.3 = 2.21$
Day 2: $(- 1.5 - 0.4) (- 1.2 - 0.5) = - 1.9 \times - 1.7 = 3.23$

YE multiplications kyun? Positive products ka matlab hai ki ve ek hi direction mein gaye (dono upar ya dono neeche).

Step 3: Sum karo aur n-1 se divide karo

$Cov = \frac{\sum ( \dots )}{19} = 2.35$ (numerator sum ≈ 44.7 over the 20 days)

Step 4: Standard deviations calculate karo

$σ_{Infosys} = 1.8%$
$σ_{TCS} = 1.5%$

Step 5: Correlation compute karo $ρ = \frac{Cov}{σ _{A} \times σ _{B}} = \frac{2.35}{1.8 \times 1.5} = \frac{2.35}{2.7} = 0.87$

YE step kyun? Hume $ρ \leq 1$ rakhna hai. Kyunki maximum possible covariance $σ_{A} σ_{B} = 2.7$ hai, covariance 2.7 se zyada nahi ho sakta. 0.87 ka correlation $Cov = 0.87 \times 2.7 = 2.35$ correspond karta hai — consistent aur valid hai. (Ek purane draft mein galti se Cov = 3.09 use kiya gaya tha, jo ek impossible ρ = 1.14 > 1 deta; ye red flag hai ki aapka covariance galat hai.)

Interpretation: Strong positive correlation (0.87). Jab Infosys badhta hai, TCS bhi zyada tar badhta hai. Ve same sector mein hain, similar macro factors face karte hain.

-0.3 ka matlab kya hai?

Negative: Ve opposite directions mein chalte hain
Weak: Relationship strong nahi hai; gold tab bhi badh sakta hai jab Nifty badhe

Negative kyun? Gold ek "safe haven" asset hai. Jab stock markets crash hote hain (darr), investors gold ki taraf bhagte hain. Jab markets boom karte hain (greed), ve equities ke liye gold bechte hain.

Trading implication: Dono rakhne se diversification milti hai. Agar Nifty 10% girta hai, gold 3% badh sakta hai, aapke portfolio ko cushion deta hai.

Negative kyun? Oil airlines ki sabse badi cost hai. Jab crude spike karta hai, airline profits ghatte hain aur stocks girate hain.

ISKO trade kaise karein?

Hedge: Long airline stocks hain? Insurance ke liye crude oil futures kharido
Pairs trade: Agar correlation break ho (oil upar, airlines upar), mean reversion par bet lagao

Step-by-step pairs trade:

Divergence identify karo: Oil 20% upar, IndiGo unchanged (historical ρ = -0.6 ke basis par IndiGo 12% neeche expected)
IndiGo short karo (catch-up decline ki umeed se)
Oil futures long karo (protection agar oil badhta rahe)
Jab correlation normalize ho tab exit karo

Common Mistakes

Kyun sahi lagta hai: Negative correlation ka matlab hai jab ek upar jaata hai, doosra neeche jaata hai. Ye cause aur effect jaisa lagta hai.

Fix: Correlation association measure karta hai, causation nahi. Dono kisi teesre factor (risk sentiment, inflation expectations) se driven ho sakte hain. Correlation coincidental bhi ho sakta hai.

Steel-man: Aap observe karte ho ki har baar gold badhta hai, Nifty kaafi dino mein girta hai. Ye temporal pattern causation suggest karta hai, lekin correlation akela prove nahi karta. Aapko ek mechanism chahiye: "Risk-off sentiment → stocks becho, gold kharido."

Kyun sahi lagta hai: Aap historical data par correlation calculate karte ho, aur wo high hai. Extrapolate karna reasonable lagta hai.

Fix: Correlation dynamic hai. Ye inke saath badalta hai:

Market regimes: Crashes ke dauran, saari correlations +1 ki taraf spike karti hain (sab saath girate hain)
Company events: Infosys scandal → TCS decouple ho jaata hai
Macro shifts: Nayi regulations ek sector ko alag tarah affect karti hain

ISKO handle kaise karein: Rolling correlation use karo (jaise 60-day window) changes track karne ke liye. Har mahine recalculate karo.

Kyun sahi lagta hai: Pearson correlation linear relationships measure karta hai. Low ρ ka matlab hai koi straight-line pattern nahi.

Fix: Assets ka strong non-linear relationship ho sakta hai even jab ρ near zero ho. Example: VIX (volatility index) aur Nifty. Jab Nifty sharply girta hai, VIX spike karta hai (non-linear, convex). Zero linear correlation ka matlab statistical independence nahi hota — sirf joint normality ke under hi aise hota hai. Do variables bilkul dependent ho sakte hain lekin phir bhi ρ = 0 ho sakta hai.

Kya karein: Rank correlation (Spearman's ρ) ya visual scatter plots se supplement karo.

Kyun sahi lagta hai: Diversification risk reduce karta hai. Zyada uncorrelated assets = kam risk.

Fix: Do issues hain:

Diminishing returns: 15-20 stocks se aage, additional diversification minimal benefit deta hai
Systemic risk rehta hai: Market crash mein, correlations +1 ki taraf converge karti hain. Aapka "diversified" portfolio saath crash karta hai.

Steel-man: Aap sahi ho ki low correlation normal times mein madad karta hai. Lekin aap tail risk underestimate kar rahe ho. Behtar hai: Asset classes mix karo (stocks, bonds, commodities) sirf stocks nahi.

How to Use Correlation in Trading

1. Portfolio Construction

Goal: Diye gaye risk ke liye return maximize karo
Method: ρ < 0.5 wale assets combine karo. Agar A aur B ka ρ = 0 hai, portfolio variance = $w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2}$ (cross-term vanish ho jaata hai). Agar ρ = 1, variance = $(w_{A} σ_{A} + w_{B} σ_{B})^{2}$ (kaafi zyada).

Do-asset portfolio volatility ka formula: $σ_{P} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ_{A, B} σ_{A} σ_{B}$

Cross-term kyun? $2 w_{A} w_{B} ρ_{A, B} σ_{A} σ_{B}$ joint movement capture karta hai. Jab ρ = 0, ye vanish ho jaata hai (linear risks reinforce nahi karte).

2. Risk Management

Concentrated risk: Agar aapke saare stocks ka ρ > 0.8 hai, aap sector risk ke liye exposed ho
Hedge: Negatively correlated assets add karo (gold, inverse ETFs)

3. Pairs Trading

Highly correlated pairs dhundho (ρ > 0.9)
Jab correlation temporarily break ho, mean reversion par bet lagao
Example: HDFC Bank aur ICICI Bank zyada tar saath chalte hain. Agar HDFC earnings miss par 5% girta hai lekin ICICI unchanged hai, ICICI short karo (catch-up decline ki umeed mein)

4. Regime Detection

Portfolio mein average correlation monitor karo
Rising correlations: Market stress (risk-off), exposure reduce karo
Falling correlations: Normal market, risk add karna safe hai

Calculation Tools

Excel mein quick correlation:

=CORREL(A2:A21, B2:B21)

Rolling correlation: 60-day windows use karo, 1 din shift karo, time series plot karo.

Python (algorithmic traders ke liye):

import pandas as pd
df['rolling_corr'] = df['stock_a'].rolling(60).corr(df['stock_b'])

Ya: "Correlation -1 aur 1 ke beech hoti hai, jaise mera mood 'ugh' aur 'yay' ke beech swing karta hai."

Recall Feynman Technique: Ek 12-Saal Ke Bacche Ko Samjhao

Socho tum aur tumhara best friend har roz school jaate ho. Correlation ye notice karna jaisa hai: "Kya hum ek saath thokar khaate hain?"

Agar har baar jab tum thokar khaate ho, tumhara dost bhi khaata hai, toh ye perfect positive correlation (+1) hai. Tum sync mein chal rahe ho.

Agar jab bhi tum thokar khaate ho, tumhara dost stable rehta hai (aur ulta bhi), toh ye perfect negative correlation (-1) hai. Tum ekdum opposite ho.

Agar tumhara thokar khaana tumhare dost ke thokar khaane se straight-line way mein bilkul related nahi hai, toh ye zero correlation (0) hai. Lekin dhyan raho — shayad tum dono sirf same weird crack ke paas thokar khaate ho, bas ek aisi twisted pattern mein jise simple "same time" rule pakad nahi sakta. Toh zero correlation ka matlab hamesha bilkul unrelated nahi hota!

Stocks mein, correlation batata hai: "Agar Stock A girta hai, kya Stock B bhi girega?" Agar ve correlated hain, toh ve un doston ki tarah hain jo saath thokar khaate hain. Agar dono ho tumhare paas, toh double trouble. Lekin agar ve negatively correlated hain, toh ek girta hai jab doosra uthta hai — tum safe ho!

Connections

Diversification-and-portfolio-theory – Correlation diversification ki neenv hai; Markowitz portfolio theory correlation matrices use karta hai
Beta-and-systematic-risk – Beta ek stock aur market ke beech correlation measure karta hai
Hedging-strategies – Negative correlation hedging enable karta hai (jaise long stocks, short index futures)
Pairs-trading – Historically correlated instruments mein temporary correlation breakdowns ko exploit karta hai
Volatility-and-standard-deviation – Correlation volatilities ke saath milke portfolio risk compute karta hai
Covariance-matrix – Multi-asset correlation ke liye covariance matrices chahiye hoti hain
Asset-allocation – Strategic allocation cross-asset correlations par depend karti hai (stocks-bonds-gold)

#flashcards/stock-market

Do instruments ke beech +1 ka correlation coefficient kya matlab rakhta hai?

Ve perfect linear relationship ke saath bilkul ek hi direction mein chalte hain.

-1 ka correlation coefficient kya matlab rakhta hai?

Instruments bilkul opposite directions mein chalte hain (perfect negative correlation).

A aur B ke beech correlation coefficient ka formula kya hai?

ρ = Cov(A,B) / (σ_A × σ_B), jahan Cov covariance hai aur σ standard deviation hai.

Correlation paane ke liye hum covariance ko standard deviations se kyun divide karte hain?

Measure ko normalize karne aur scale-free banane ke liye, use Cauchy–Schwarz inequality ke zariye [-1, +1] range mein constrain karne ke liye.

Kya ρ = 0 ka matlab hai ki do instruments statistically independent hain?

Nahi. Iska sirf matlab hai koi LINEAR relationship nahi. Unka strong non-linear relationship ho sakta hai. Independence se zero correlation milti hai, lekin ulta nahi (sirf joint normality ke under).

Agar Tech Stock A aur Tech Stock B ka correlation 0.85 hai, toh ye ek trader ko kya batata hai?

Ve strongly saath chalte hain; dono rakhne se tech sector ke andar little diversification milti hai.

Ek valid correlation kabhi +1 se zyada ya -1 se kam kyun nahi ho sakta?

Kyunki |Cov(A,B)| ≤ σ_A × σ_B (Cauchy–Schwarz), isliye ratio [-1, +1] se bounded hai. Is range se bahar computed ρ calculation error signal karta hai.

"Correlation stable hai" assumption dangerous kyun hai?

Correlation market regimes, company events, aur macro shifts ke saath badalta hai; crashes ke dauran correlations aksar +1 ki taraf spike karti hain.

Gold aur Nifty ke beech negative correlation portfolio construction mein kaise madad karta hai?

Jab Nifty girta hai, gold aksar uthta hai, portfolio losses ko cushion karta hai aur overall volatility reduce karta hai.

Crude oil aur airline stocks ke beech correlation negative kyun ho sakta hai?

Oil airlines ki major cost hai; jab crude badhta hai, airline profits girti hain aur stock prices girti hain.

Pairs trading mein position mein kab enter karte hain?

Jab historically correlated instruments temporarily diverge karein (correlation break ho), mean reversion par bet lagate hue.

Aapke portfolio mein rising average correlation kya signal karta hai?

Market stress ya risk-off environment; exposure reduce karne ya hedges add karne ka time.