Learn about the efficient frontier — Hinglish notes

5.5.4 · Stock-Market › Portfolio Theory

Efficient frontier KYUN exist karta hai?

KYA yeh answer karta hai: "Mere paas jo assets hain, unke saath risk aur return ke beech best possible trade-off kya hai?" KAISE hum isse dhundhte hain: har target return ke liye, aisi weights solve karo jo variance minimize kare.

Building block: two-asset portfolio (scratch se derive karo)

Weights $w_{A}, w_{B}$ leke chalo jahan $w_{A} + w_{B} = 1$ . Returns $R_{A}, R_{B}$ jinke expected values $μ_{A}, μ_{B}$ , standard deviations $σ_{A}, σ_{B}$ , correlation $ρ$ hain.

Step 1 — Expected return. Expectation linear hota hai: $μ_{p} = E [w_{A} R_{A} + w_{B} R_{B}] = w_{A} μ_{A} + w_{B} μ_{B}$ Yeh step kyun? $E [\cdot]$ sums aur constants se pass ho jaata hai, isliye yahan kuch curve nahi hota — return ek seedha weighted average hai.

Step 2 — Variance. $Var (X + Y) = Var (X) + Var (Y) + 2 Cov (X, Y)$ use karo: $σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} Cov (R_{A}, R_{B})$ Kyunki $Cov = ρ σ_{A} σ_{B}$ : $σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ σ_{A} σ_{B}$ Yeh step kyun? Cross term $2 w_{A} w_{B} ρ σ_{A} σ_{B}$ mein hi saara magic hai. Agar $ρ < 1$ , toh yeh term uss value se chhota hai jo risk ko straight line banata — isliye risk kam ho jaata hai.

Minimum-Variance Portfolio (MVP)

KAISE bullet ki sabse left wali tip dhundhen: $σ_{p}^{2}$ ko $w_{A}$ ke upar minimize karo.

$w_{B} = 1 - w_{A}$ set karo, derivative lo, zero ke barabar karo: $\frac{d σ _{p}^{2}}{d w _{A}} = 0 \Rightarrow w_{A}^{*} = \frac{σ _{B}^{2} - ρ σ _{A} σ _{B}}{σ _{A}^{2} + σ _{B}^{2} - 2 ρ σ _{A} σ _{B}}$ Yeh step kyun? $σ_{p}^{2}$ ek convex (upar ki taraf) parabola hai $w_{A}$ mein, isliye yeh akela stationary point global minimum hai — frontier bullet ki tip.

Figure — Learn about the efficient frontier

Bullet se frontier tak

Worked Example 1 — do uncorrelated assets

$μ_{A} = 8%, σ_{A} = 10%$ ; $μ_{B} = 12%, σ_{B} = 20%$ ; $ρ = 0$ .

$w_{A} = 0.5$ try karo:

$μ_{p} = 0.5 (8) + 0.5 (12) = 10%$ — Kyun? linear averaging.
$σ_{p}^{2} = 0.25 (100) + 0.25 (400) + 0 = 125 \Rightarrow σ_{p} = 11.18%$ — Kyun? cross term zero ho jaata hai kyunki $ρ = 0$ ; note karo $11.18% < 15%$ (10 aur 20 ka naive average)! Diversification kaam aaya.

MVP dhundho: $w_{A}^{*} = \frac{400 - 0}{100 + 400 - 0} = 0.8$ Kyun? $ρ = 0$ ke saath formula. Toh 80% safer asset mein.

$μ_{p} = 0.8 (8) + 0.2 (12) = 8.8%$
$σ_{p}^{2} = 0.64 (100) + 0.04 (400) = 80 \Rightarrow σ_{p} = 8.94%$ — humne jo bhi single asset combination try kiya tha, usse bhi kam risk.

Worked Example 2 — perfect negative correlation ( $ρ = - 1$ )

Same $σ_{A} = 10, σ_{B} = 20$ . Zero-risk weight solve karta hai $w_{A} σ_{A} = w_{B} σ_{B}$ : $w_{A} (10) = (1 - w_{A}) (20) \Rightarrow w_{A} = \frac{20}{30} = 0.667$ Kyun? $ρ = - 1$ par, $σ_{p} = ∣ w_{A} σ_{A} - w_{B} σ_{B} ∣$ ; isse zero set karo. Risk poori tarah cancel — do risky assets se ek synthetic risk-free asset.

Common mistake Common errors ko steel-man karo

Error 1: "Portfolio risk = individual risks ka weighted average." Kyun sahi lagta hai: return hai ek weighted average, toh risk bhi "hona chahiye." Fix: variance mein ek cross/covariance term hota hai. Sirf $ρ = + 1$ par hi risk straight average banta hai. Warna $σ_{p} < w_{A} σ_{A} + w_{B} σ_{B}$ .

Error 2: "Poori curve efficient frontier hai." Kyun sahi lagta hai: bullet ek smooth curve hai. Fix: sirf MVP ke upar wala upper half efficient hai. Neeche wala aadha dominated hai.

Error 3: "Zyada diversification hamesha risk zero tak le jaata hai." Kyun sahi lagta hai: examples mein assets add karte rehne se help hoti rahi. Fix: systematic (market) risk diversify nahi ho sakta; $σ_{p}$ ek floor ki taraf flatten ho jaata hai, zero tak nahi (jab tak $ρ = - 1$ na ho).

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum snacks choose kar rahe ho. Kuch snacks saste par boring hain, kuch mehenge par tasty. Agar tum unhe hamesha saath khaate ho, toh ek ka boring hona doosre ke tasty hone se balance ho jaata hai — toh "bure din" smooth ho jaate hain. Efficient frontier un best snack mixes ka chart hai: tum kitni upar-neeche (risk) ke liye taiyaar ho, yeh dikhata hai ki average mein kitna tasty (return) milega. Koi bhi mix jo upar-neeche bhi ho aur boring bhi — woh throw away kar do — tum kabhi nahi chhunte usse.

Flashcards

Efficient frontier kya hai?

Un portfolios ka set jo har risk level ke liye max expected return deta hai (minimum-variance frontier ka upper half, MVP ke upar).

Risk–return curve kyun left ki taraf bend karti hai?

Kyunki jab

ρ < 1

hota hai toh covariance cross-term portfolio variance ko weighted average se neeche le jaata hai — diversification volatility cancel karta hai.

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}

Kya portfolio expected return ek weighted average hai?

Haan, hamesha:

μ_{p} = w_{A} μ_{A} + w_{B} μ_{B}

(expectation linear hota hai).

Kis correlation par diversification khatam ho jaata hai?

ρ = + 1

(risk ek straight weighted average ban jaata hai).

Kis correlation par risk exactly zero ho sakta hai?

ρ = - 1

, weights

w_{A} σ_{A} = w_{B} σ_{B}

ke saath.

Minimum-Variance Portfolio kya hai?

Frontier ka leftmost point;

w_{A}^{*} = \frac{σ _{B}^{2} - ρ σ _{A} σ _{B}}{σ _{A}^{2} + σ _{B}^{2} - 2 ρ σ _{A} σ _{B}}

Capital Market Line kya hai?

R_{f}

se frontier tak tangent line; iska tangency point market portfolio hai; slope = Sharpe ratio.

Bullet ka neeche wala aadha inefficient kyun hai?

Same risk ke liye tum seedha upar chadh ke zyada return pa sakte ho, isliye woh portfolios dominated hain.

Connections

Modern Portfolio Theory
Diversification
Covariance and Correlation
Capital Market Line
Sharpe Ratio
Capital Asset Pricing Model
Systematic vs Unsystematic Risk