Understand Markowitz mean-variance optimization

5.5.3 · Stock-Market › Portfolio Theory

YE EXIST KYU KARTA HAI?

Markowitz (1952) se pehle, log ek ek karke "acche stocks" chunte the. Problem ye thi: do stocks alag alag risky ho sakte hain, phir bhi milkar dono se zyada calm ho sakte hain — kyunki wo ek saath nahi chalte. Assets kaise milkar chalte hain (covariance) isko ignore karne ka matlab hai ki tum risk mein zyada pay kar rahe ho. Markowitz ne pehla math diya jo return aur risk dono ko saath treat karta hai aur dikhata hai ki diversification ek free lunch hai (ek limit tak).

HUM KYA MEASURE KAR RAHE HAIN?

KAISE: portfolio return aur risk scratch se derive karo

Portfolio return sirf pieces ka money-weighted average hai. Agar tum asset $i$ mein fraction $w_{i}$ lagate ho: $R_{p} = \sum_{i} w_{i} R_{i} \Rightarrow μ_{p} = E [R_{p}] = \sum_{i} w_{i} μ_{i} = w^{⊤} μ$ Kyun? Expectation linear hoti hai — mean mein koi covariance nahi aata.

Portfolio variance — pehle do assets ke liye derive karo, phir generalize karo. $σ_{p}^{2} = Var (w_{1} R_{1} + w_{2} R_{2})$ $Var (a X + bY) = a^{2} Var (X) + b^{2} Var (Y) + 2 ab Cov (X, Y)$ use karo: $σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} σ_{12}$ Ye step kyun? Cross term $2 w_{1} w_{2} σ_{12}$ hi poori baat hai: agar $σ_{12} < 0$ ho, toh ye risk ko subtract karta hai. Yahi diversification hai.

General $n$ -asset form (covariance matrix $Σ$ mein stack karke): $σ_{p}^{2} = \sum_{i} \sum_{j} w_{i} w_{j} σ_{ij} = w^{⊤} Σ w$

Optimization problem

KAISE solve karo (Lagrange, first principles)

Multipliers $λ, γ$ ke saath Lagrangian banao: $L = \frac{1}{2} w^{⊤} Σ w - λ (w^{⊤} μ - μ^{*}) - γ (w^{⊤} 1 - 1)$ $\partial L / \partial w = 0$ set karo: $Σ w - λ μ - γ 1 = 0 \Rightarrow w^{*} = Σ^{- 1} (λ μ + γ 1)$ Kyun? $\frac{1}{2} w^{⊤} Σ w$ ka gradient $Σ w$ hota hai; isko constraint gradients ke against set karne se optimum milta hai. $λ, γ$ ke liye do constraints solve karo. Saare aisa $w^{*}$ ka set (har $μ^{*}$ ke liye) $(σ_{p}^{2}, μ_{p})$ space mein ek parabola trace karta hai — yahi minimum-variance frontier hai.

Upar ki aadhi (same risk ke liye zyada return) efficient frontier hai. Iska sabse baaya tip global minimum-variance portfolio hai.

Figure — Understand Markowitz mean-variance optimization

Worked example 1 — do assets, diversification ka jadoo

Assets: $μ_{1} = 8%, σ_{1} = 10%$ ; $μ_{2} = 12%, σ_{2} = 20%$ ; correlation $ρ = 0$ toh $σ_{12} = 0$ . 50/50 try karo: $w_{1} = w_{2} = 0.5$ .

$μ_{p} = 0.5 (8) + 0.5 (12) = 10%$ . Kyun? Means ka linear average.
$σ_{p}^{2} = 0.25 (100) + 0.25 (400) + 0 = 125 \Rightarrow σ_{p} = 11.18%$ .

Kyun remarkable hai? Do risks ka naive average $15%$ hota. Kyunki ye uncorrelated hain, real risk sirf $11.18%$ hai — risk average se neeche aa gaya bilkul free mein.

Worked example 2 — min-variance weight dhundho (2 assets)

$σ_{p}^{2} = w^{2} σ_{1}^{2} + (1 - w)^{2} σ_{2}^{2} + 2 w (1 - w) σ_{12}$ minimize karo. $d / d w = 0$ lo: $w^{*} = \frac{σ _{2}^{2} - σ _{12}}{σ _{1}^{2} + σ _{2}^{2} - 2 σ _{12}}$ Ye step kyun? Differentiate karo, $w$ collect karo, linearly solve karo. Upar ke numbers ke saath ( $σ_{12} = 0$ ): $w^{*} = \frac{400 - 0}{100 + 400 - 0} = 0.8$ Toh safe asset mein 80% se sabse kam risk wala mix milta hai: $σ_{p}^{2} = 0.64 (100) + 0.04 (400) = 80 \Rightarrow σ_{p} = 8.94%$ — kisi bhi asset ke $10%$ ya $20%$ se kam!

Worked example 3 — negative correlation isse aur powerful banata hai

Same assets lekin $ρ = - 1$ (perfectly opposite), toh $σ_{12} = ρ σ_{1} σ_{2} = - 1 (10) (20) = - 200$ . $w^{*} = \frac{400 - ( - 200 )}{100 + 400 - 2 ( - 200 )} = \frac{600}{900} = 0.667$ $σ_{p}^{2} = 0.444 (100) + 0.111 (400) + 2 (0.667) (0.333) (- 200) = 44.4 + 44.4 - 88.8 = 0$ . Zero kyun? Perfect negative correlation se tum risk puri tarah cancel kar sakte ho — do risky assets se ek riskless portfolio.

Common mistakes (steel-manned)

Recall Feynman: ek 12 saal ke bacche ko samjhao

Socho ek ice-cream ki gaadi pack kar rahe ho. Barish ke din chhate bikti hain, dhoop mein ice cream. Akele dono gambling hain — kharab mausam se dhanda chaupar. Lekin dono becho aur tum har roz jeetoge: jab ek girta hai, doosra uthta hai. Markowitz ne sirf "aise cheezein milao jo ek saath fail na hon" ka math likha, taaki tumhari total kamaai smooth rahe. Sabse acche mixes — ek chosen average profit ke liye sabse smooth kamaai — ek special curve par hote hain, aur samajhdaar log sirf usi curve se chunte hain.

Active recall

Portfolio expected return formula

μ_{p} = w^{⊤} μ = \sum_{i} w_{i} μ_{i}

(weights mein linear, koi covariance nahi).

2 assets ke liye portfolio variance

σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} σ_{12}

Diversification risk kyun ghata hai?

Cross term

2 w_{1} w_{2} σ_{12}

risk subtract kar sakta hai jab covariance kam/negative ho.

Markowitz program mein kya minimize hota hai?

Portfolio variance

w^{⊤} Σ w

, target return aur weights ka sum 1 ke subject to.

Min-variance weight (2 assets)

w^{*} = \frac{σ _{2}^{2} - σ _{12}}{σ _{1}^{2} + σ _{2}^{2} - 2 σ _{12}}

Lagrange se optimal weight vector

w^{*} = Σ^{- 1} (λ μ + γ 1)

Efficient frontier vs minimum-variance frontier

Efficient frontier = upar ki aadhi (risk per max return); global min-var portfolio iska sabse baaya tip hai.

Do risky assets kab riskless portfolio banate hain?

Jab

ρ = - 1

(perfect negative correlation) ho; tab

σ_{p}

zero ho sakta hai.

Kya assets ke standard deviations jodte hain?

Nahi — sirf agar

ρ = + 1

ho; generally

σ_{p} \leq \sum w_{i} σ_{i}

(σ, μ) space mein frontier ki shape

Ek hyperbola (variance mein parabola), seedhi line nahi.

Connections

Efficient Frontier
Capital Asset Pricing Model (CAPM)
Sharpe Ratio
Capital Market Line
Covariance and Correlation
Diversification
Lagrange Multipliers
Risk-Free Asset and Tangency Portfolio