Learn correlation and covariance

5.5.2 · Stock-Market › Portfolio Theory

WHAT is covariance?

HOW: covariance ko scratch se derive karna

Yeh idea se shuru karo ki "kya deviations ka sign agree karta hai?"

$X$ ka apne centre se deviation: $d_{X} = X - μ_{X}$ .
$Y$ ka deviation: $d_{Y} = Y - μ_{Y}$ .
Inhe multiply karo. Agar dono positive ya dono negative hain, toh $d_{X} d_{Y} > 0$ . Agar sign disagree kare, toh $d_{X} d_{Y} < 0$ . Toh product ka sign literally record karta hai "same direction ya opposite."
Tendency summarise karne ke liye saare outcomes pe average lo: $Cov (X, Y) = E [d_{X} d_{Y}] .$

Sample version ( $n$ historical data points se), jo aap actually compute karte hain: $Cov (X, Y) = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \overset{x}{ˉ}) (y_{i} - \overset{y}{ˉ})$ $n - 1$ kyun? Humne data ko do baar use kiya — ek baar $\overset{x}{ˉ}$ estimate karne ke liye, ek baar deviations ke liye — toh humne ek degree of freedom "use up" kar liya; $n - 1$ se divide karna resulting downward bias ko correct karta hai.

WHAT is correlation?

HOW: correlation $\pm 1$ se bounded kyun hai

Standard deviations se divide karo taaki standardised variables milein $Z_{X} = (X - μ_{X}) / σ_{X}$ (mean 0, variance 1). Tab $ρ_{X Y} = E [Z_{X} Z_{Y}]$ .

Yeh hamesha-nonnegative quantity consider karo: $E [(Z_{X} - Z_{Y})^{2}] \geq 0.$ Expand karo: $E [Z_{X}^{2}] - 2 E [Z_{X} Z_{Y}] + E [Z_{Y}^{2}] = 1 - 2 ρ + 1 \geq 0 \Rightarrow ρ \leq 1.$ Similarly $E [(Z_{X} + Z_{Y})^{2}] \geq 0$ se $ρ \geq - 1$ milta hai. Isliye $- 1 \leq ρ \leq 1$ . (Yeh Cauchy–Schwarz inequality ka disguised form hai.)

Figure — Learn correlation and covariance

WHY this matters: portfolio variance

Worked examples

Common mistakes (Steel-manned)

Recall Ek 12-saal ke bachche ko explain karo (Feynman)

Do doston ko see-saw par imagine karo. Covariance poochtha hai: jab ek upar jaata hai, kya doosra bhi upar jaata hai, ya neeche? Positive number = dono saath upar jaate hain; negative = ek upar, doosra neeche. Lekin covariance mein strange units hote hain, toh yeh kehna mushkil hai ki yeh "bahut zyada" hai ya nahi. Correlation wahi idea hai lekin fairly score kiya gaya $- 1$ se $+ 1$ tak: $+ 1$ = perfect same-direction team, $- 1$ = perfect opposite see-saw, $0$ = koi pattern nahi. Investing mein, aap aise doston chahte ho jo tab zig kare jab baaki zag kare — kyunki jab ek stock drop kare, doosra aapki basket ko wapas utha le.

Flashcards

Covariance kya measure karta hai?

Do variables ke means se deviations ka average product — unke co-movement ki direction aur raw magnitude.

Covariance ki definition formula?

Cov (X, Y) = E [(X - μ_{X}) (Y - μ_{Y})]

Covariance ka computational form?

Cov (X, Y) = E [X Y] - μ_{X} μ_{Y}

Correlation ko covariance se kaise define karte hain?

ρ_{X Y} = Cov (X, Y) / (σ_{X} σ_{Y})

Correlation coefficient ki range?

- 1 \leq ρ \leq + 1

σ_{X} σ_{Y}

se divide kyun karte hain?

Units/scale remove karne ke liye, jo ek dimensionless, comparable strength score deta hai jo

\pm 1

se bounded ho.

Kya

ρ = 0

independence imply karta hai?

Nahi — yeh sirf linear relationship ko rule out karta hai; nonlinear dependence phir bhi exist kar sakti hai.

Two-asset portfolio variance formula?

σ_{P}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ_{A B} σ_{A} σ_{B}

Sabse zyada diversification benefit kaunsa correlation deta hai?

Sabse zyada negative (

ρ = - 1

portfolio variance ko 0 tak le ja sakta hai).

Sample covariance mein

n - 1

kyun use karte hain?

Bessel's correction — mean data se estimate kiya gaya tha, toh ek degree of freedom lost ho jaata hai; downward bias correct karta hai.

∣ ρ ∣ \leq 1

prove karne wali inequality?

Cauchy–Schwarz, standardised variables ke saath

E [(Z_{X} \pm Z_{Y})^{2}] \geq 0

ke zariye.

Connections

Variance and Standard Deviation — building blocks $σ_{X}, σ_{Y}$ .
Diversification — negative correlation risk kyun reduce karta hai.
Portfolio Variance — jahaan covariance directly enter karta hai.
Efficient Frontier — correlation frontier ki curve ko shape karta hai.
Beta and CAPM — beta $= Cov (R_{i}, R_{m}) / Var (R_{m})$ .
Cauchy–Schwarz Inequality — woh math jo $ρ$ ko bound karta hai.

WHY do we even need this?