Understand Markowitz mean-variance optimization — Stock-Market notes

WHY does this exist?

Before Markowitz (1952), people picked "good stocks" one at a time. The problem: two stocks can each be risky, yet combined be calmer than either — because they don't move together. Ignoring how assets co-move (covariance) means you overpay in risk. Markowitz gave the first math that treats return and risk jointly and shows diversification is a free lunch (up to a limit).

WHAT are we measuring?

HOW: derive portfolio return and risk from scratch

Portfolio return is just the money-weighted average of the pieces. If you put fraction $w_i$ in asset $i$ : $R_p = \sum_i w_i R_i \quad\Rightarrow\quad \mu_p = E[R_p] = \sum_i w_i \mu_i = \mathbf{w}^\top \boldsymbol{\mu}$ Why? Expectation is linear — no covariances appear in the mean.

Portfolio variance — derive for two assets, then generalize. $\sigma_p^2 = \text{Var}(w_1 R_1 + w_2 R_2)$ Use $\text{Var}(aX+bY) = a^2\text{Var}(X)+b^2\text{Var}(Y)+2ab\,\text{Cov}(X,Y)$ : $\boxed{\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\,\sigma_{12}}$ Why this step? The cross term $2w_1w_2\sigma_{12}$ is the whole point: if $\sigma_{12}<0$ , it subtracts risk. That's diversification.

General $n$ -asset form (stacking into a covariance matrix $\Sigma$ ): $\sigma_p^2 = \sum_i\sum_j w_i w_j \sigma_{ij} = \mathbf{w}^\top \Sigma\, \mathbf{w}$

The optimization problem

HOW to solve (Lagrange, first principles)

Build the Lagrangian with multipliers $\lambda,\gamma$ : $L=\tfrac12\mathbf w^\top\Sigma\mathbf w-\lambda(\mathbf w^\top\boldsymbol\mu-\mu^*)-\gamma(\mathbf w^\top\mathbf 1-1)$ Set $\partial L/\partial \mathbf w = 0$ : $\Sigma\mathbf w-\lambda\boldsymbol\mu-\gamma\mathbf 1=0 \ \Rightarrow\ \mathbf w^* = \Sigma^{-1}(\lambda\boldsymbol\mu+\gamma\mathbf 1)$ Why? The gradient of $\tfrac12\mathbf w^\top\Sigma\mathbf w$ is $\Sigma\mathbf w$ ; setting it against the constraint gradients gives the optimum. Solve the two constraints for $\lambda,\gamma$ . The set of all such $\mathbf w^*$ (over every $\mu^*$ ) traces a parabola in $(\sigma_p^2,\mu_p)$ space — the minimum-variance frontier.

The upper half (higher return for same risk) is the efficient frontier. Its leftmost tip is the global minimum-variance portfolio.

Figure — Understand Markowitz mean-variance optimization

Worked example 1 — two assets, the diversification magic

Assets: $\mu_1=8\%,\sigma_1=10\%$ ; $\mu_2=12\%,\sigma_2=20\%$ ; correlation $\rho=0$ so $\sigma_{12}=0$ . Try 50/50: $w_1=w_2=0.5$ .

$\mu_p=0.5(8)+0.5(12)=10\%$ . Why? Linear average of means.
$\sigma_p^2 = 0.25(100)+0.25(400)+0 = 125 \Rightarrow \sigma_p=11.18\%$ .

Why remarkable? A naive average of the two risks would be $15\%$ . Because they're uncorrelated, real risk is only $11.18\%$ — risk fell below the average for free.

Worked example 2 — find the min-variance weight (2 assets)

Minimize $\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_{12}$ . Take $d/dw=0$ : $w^*=\frac{\sigma_2^2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}$ Why this step? Differentiate, collect $w$ , solve linearly. With numbers above ( $\sigma_{12}=0$ ): $w^* = \frac{400-0}{100+400-0}=0.8$ So 80% in the safe asset gives the lowest-risk mix: $\sigma_p^2=0.64(100)+0.04(400)=80\Rightarrow\sigma_p=8.94\%$ — lower than either asset's $10\%$ or $20\%$ !

Worked example 3 — negative correlation supercharges it

Same assets but $\rho=-1$ (perfectly opposite), so $\sigma_{12}=\rho\sigma_1\sigma_2 = -1(10)(20)=-200$ . $w^*=\frac{400-(-200)}{100+400-2(-200)}=\frac{600}{900}=0.667$ $\sigma_p^2=0.444(100)+0.111(400)+2(0.667)(0.333)(-200)=44.4+44.4-88.8=0$ . Why zero? With perfect negative correlation you can fully cancel risk — a riskless portfolio from two risky assets.

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine packing an ice-cream cart. Umbrellas sell on rainy days, ice cream on sunny days. Each alone is a gamble — bad weather ruins it. But sell both and you win every day: when one drops, the other rises. Markowitz just wrote the math for "mix things that don't fail at the same time," so your total earnings stay smooth. The best mixes — smoothest earnings for a chosen average profit — sit on a special curved line, and smart people only pick from that line.

Active recall

Portfolio expected return formula

\mu_p=\mathbf{w}^\top\boldsymbol\mu=\sum_i w_i\mu_i

(linear in weights, no covariance).

Portfolio variance for 2 assets

\sigma_p^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\sigma_{12}

Why does diversification reduce risk?

The cross term

2w_1w_2\sigma_{12}

can subtract risk when covariance is low/negative.

What is minimized in the Markowitz program?

Portfolio variance

\mathbf w^\top\Sigma\mathbf w

, subject to target return and weights summing to 1.

Min-variance weight (2 assets)

w^*=\dfrac{\sigma_2^2-\sigma_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_{12}}

Optimal weight vector from Lagrange

\mathbf w^*=\Sigma^{-1}(\lambda\boldsymbol\mu+\gamma\mathbf 1)

Efficient frontier vs minimum-variance frontier

Efficient frontier = upper half (max return per risk); global min-var portfolio is its leftmost tip.

When can two risky assets form a riskless portfolio?

When

\rho=-1

(perfect negative correlation); then

\sigma_p

can equal 0.

Do standard deviations of assets add?

No — only if

\rho=+1

; generally

\sigma_p\le\sum w_i\sigma_i

Shape of frontier in (σ, μ) space

A hyperbola (parabola in variance), not a straight line.

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Markowitz ka core idea simple hai: kisi bhi stock ko akele mat judge karo — dekho wo poore portfolio ke risk ko kya karta hai. Do stock alag-alag risky ho sakte hain, par agar wo ulti direction mein move karte hain (negative covariance), to mix karke total risk kam ho jaata hai. Isko diversification kehte hain, aur yahi "free lunch" hai investing mein.

Formula yaad rakho: portfolio ka return to seedha weighted average hai ( $\mu_p=\sum w_i\mu_i$ ), lekin risk mein ek twist hai. Variance = "squares plus a cross" — har asset ka apna variance, aur ek cross term $2w_1w_2\sigma_{12}$ . Yeh cross term hi magic hai: agar covariance negative hai to yeh risk ko minus kar deta hai. Isliye standard deviations kabhi seedhe add nahi hote.

Optimization ka game yeh hai: ek target return fix karo, aur us return ko dete hue minimum possible variance wala portfolio dhoondo. Sab aise best portfolios ko join karo to ek curve banti hai — efficient frontier. Smart investor sirf isi curve par baithta hai, kyunki uske neeche koi bhi point "wasteful risk" hai (utna hi return kam risk mein mil sakta tha).

Exam aur real life dono mein: agar do assets ka correlation $-1$ ho jaye, to tum poori tarah risk cancel karke riskless portfolio bana sakte ho — theoretically $\sigma_p=0$ . Yahi baat proof karti hai ki covariance kitna powerful hai. Toh mantra: mix cheezein jo saath mein fail na hon.

Test yourself — Portfolio Theory

Connections