Learn about the efficient frontier — Stock-Market notes

WHY does the efficient frontier exist?

WHAT it answers: "Given the assets I have, what is the best possible trade-off between risk and return?" HOW we find it: for every target return, solve for weights that minimize variance.

Building block: two-asset portfolio (derive from scratch)

Let weights $w_A, w_B$ with $w_A + w_B = 1$ . Returns $R_A, R_B$ with expected values $\mu_A, \mu_B$ , standard deviations $\sigma_A, \sigma_B$ , correlation $\rho$ .

Step 1 — Expected return. Expectation is linear: $\mu_p = \mathbb{E}[w_A R_A + w_B R_B] = w_A\mu_A + w_B\mu_B$ Why this step? $\mathbb{E}[\cdot]$ passes through sums and constants, so nothing curves here — return is a straight weighted average.

Step 2 — Variance. Use $\text{Var}(X+Y) = \text{Var}(X)+\text{Var}(Y)+2\,\text{Cov}(X,Y)$ : $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\,\text{Cov}(R_A,R_B)$ Since $\text{Cov} = \rho\,\sigma_A\sigma_B$ : $\boxed{\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\,\sigma_A\sigma_B}$ Why this step? The cross term $2w_Aw_B\rho\sigma_A\sigma_B$ is where the magic lives. If $\rho<1$ , this term is smaller than the value that would make risk a straight line — so risk drops.

The Minimum-Variance Portfolio (MVP)

HOW to find the leftmost tip of the bullet: minimize $\sigma_p^2$ over $w_A$ .

Set $w_B = 1-w_A$ , take derivative, set to zero: $\frac{d\sigma_p^2}{dw_A} = 0 \;\Rightarrow\; \boxed{w_A^* = \frac{\sigma_B^2 - \rho\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho\sigma_A\sigma_B}}$ Why this step? $\sigma_p^2$ is a convex (upward) parabola in $w_A$ , so the single stationary point is the global minimum — the tip of the frontier bullet.

Figure — Learn about the efficient frontier

From bullet to frontier

Worked Example 1 — two uncorrelated assets

$\mu_A = 8\%, \sigma_A = 10\%$ ; $\mu_B = 12\%, \sigma_B = 20\%$ ; $\rho = 0$ .

Try $w_A = 0.5$ :

$\mu_p = 0.5(8) + 0.5(12) = 10\%$ — Why? linear averaging.
$\sigma_p^2 = 0.25(100) + 0.25(400) + 0 = 125 \Rightarrow \sigma_p = 11.18\%$ — Why? cross term vanishes since $\rho=0$ ; note $11.18\% < 15\%$ (the naive average of 10 and 20)! Diversification worked.

Find the MVP: $w_A^* = \frac{400 - 0}{100 + 400 - 0} = 0.8$ Why? Formula with $\rho=0$ . So 80% in the safer asset.

$\mu_p = 0.8(8)+0.2(12) = 8.8\%$
$\sigma_p^2 = 0.64(100)+0.04(400) = 80 \Rightarrow \sigma_p = 8.94\%$ — even lower risk than either single asset combination we tried.

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

Same $\sigma_A=10, \sigma_B=20$ . Zero-risk weight solves $w_A\sigma_A = w_B\sigma_B$ : $w_A(10) = (1-w_A)(20) \Rightarrow w_A = \frac{20}{30} = 0.667$ Why? At $\rho=-1$ , $\sigma_p = |w_A\sigma_A - w_B\sigma_B|$ ; set it to zero. Risk fully cancels — a synthetic risk-free asset from two risky ones.

Common mistake Steel-manning the common errors

Error 1: "Portfolio risk = weighted average of the individual risks." Why it feels right: return is a weighted average, so risk "should" be too. The fix: variance carries a cross/covariance term. Only when $\rho=+1$ does risk become a straight average. Otherwise $\sigma_p < w_A\sigma_A + w_B\sigma_B$ .

Error 2: "The whole curve is the efficient frontier." Why it feels right: the bullet is one smooth curve. The fix: only the upper half above the MVP is efficient. The lower half is dominated.

Error 3: "More diversification always lowers risk toward zero." Why it feels right: adding assets kept helping in examples. The fix: systematic (market) risk cannot be diversified away; $\sigma_p$ flattens toward a floor, not zero (unless $\rho=-1$ ).

Recall Feynman: explain to a 12-year-old

Imagine choosing snacks. Some snacks are cheap but boring, some expensive but tasty. If you always eat them together, one being boring is balanced by the other being tasty — so the "bad days" get smoother. The efficient frontier is a chart of the best snack mixes: for how bumpy you're willing to let it be (risk), it shows the tastiest average outcome (return). Any mix that's bumpy and boring gets thrown away — you'd never pick it.

Flashcards

What is the efficient frontier?

The set of portfolios giving max expected return for each risk level (upper half of the minimum-variance frontier, above the MVP).

Why does the risk–return curve bend leftward?

Because when

\rho<1

the covariance cross-term reduces portfolio variance below the weighted average — diversification cancels volatility.

Formula for two-asset portfolio variance?

\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\sigma_A\sigma_B

Is portfolio expected return a weighted average?

Yes, always:

\mu_p = w_A\mu_A + w_B\mu_B

(expectation is linear).

At what correlation does diversification vanish?

\rho = +1

(risk becomes a straight weighted average).

At what correlation can risk hit exactly zero?

\rho = -1

, with weights

w_A\sigma_A = w_B\sigma_B

What is the Minimum-Variance Portfolio?

The leftmost point of the frontier;

w_A^* = \frac{\sigma_B^2-\rho\sigma_A\sigma_B}{\sigma_A^2+\sigma_B^2-2\rho\sigma_A\sigma_B}

What is the Capital Market Line?

The tangent line from

R_f

to the frontier; its tangency point is the market portfolio; slope = Sharpe ratio.

Why is the lower half of the bullet inefficient?

For the same risk you can move straight up to a higher return, so those portfolios are dominated.

Connections

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, efficient frontier ka core idea simple hai. Jab tum do ya zyada stocks ko mix karte ho, toh return toh simple weighted average nikalta hai — par risk (volatility) weighted average se kam ho jaata hai, agar stocks fully saath-saath nahi chalte (yani correlation $\rho < 1$ ). Kyunki jab ek stock neeche jaata hai, doosra upar jaake usko balance kar deta hai. Isi cancellation ki wajah se risk-return ka graph ek seedhi line na banke, left ki taraf mud jaata hai — ek "bullet" jaisa shape.

Us bullet ke sabse left wale point ko Minimum-Variance Portfolio (MVP) kehte hain — yani sabse kam risk wala mix. MVP ke upar wala poora upper edge hi efficient frontier hai. Neeche wala hissa bekaar (dominated) hai, kyunki utne hi risk pe tum seedha upar jaake zyada return le sakte ho — toh koi bewakoof hi neeche wala portfolio choose karega.

Formula yaad rakho: $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\sigma_A\sigma_B$ . Wo aakhri cross-term hi asli jaadu hai — yahi risk ko kam karta hai. Agar $\rho=+1$ ho, toh koi fayda nahi, risk seedha average ban jaata hai. Agar $\rho=-1$ ho, toh risk bilkul zero tak ja sakta hai (perfect hedge). Real markets mein $\rho$ beech mein hota hai, isliye diversification kaam toh karta hai par risk kabhi poora zero nahi hota — market ka systematic risk bacha rehta hai.

Exam aur investing dono mein yeh important hai kyunki yeh batata hai ki tumhare paas jitne assets hain, unse best possible risk-return trade-off kya hai. Aur agar risk-free asset (jaise FD/T-bill) add karo, toh us frontier pe ek tangent line kheencho — wahi Capital Market Line hai, aur uska slope Sharpe ratio hota hai jise hum maximize karte hain.

Test yourself — Portfolio Theory

Connections