Finance-Theory · Unit 17 · Video 1 · Interactive Practice

Never Hold One Stock: Three-Stock Frontiers and Why Stocks Co-Move

IKey Formulas

Formula	Name	What it says
$\mathsf{E}[R_p] = \sum_{i} \omega_i\,\mathsf{E}[R_i]$ , $\sum_i \omega_i = 1$	Portfolio mean	One set of weights gives one portfolio, one point in the $(\sigma_p, \mathsf{E}[R_p])$ plane
$\sigma_p^2 = \sum_{i}\sum_{j} \omega_i\,\omega_j\,\sigma_{ij}$ , $\sigma_{ij} = \rho_{ij}\,\sigma_i\,\sigma_j$	Portfolio variance	Every cell of the covariance matrix counts; $\sigma_p$ is not the weighted average of the $\sigma_i$ unless every $\rho_{ij} = 1$
$p$ dominates $q$ if $\mathsf{E}[R_p] \ge \mathsf{E}[R_q]$ and $\sigma_p \le \sigma_q$ (one strictly)	Dominance	$p$ lies in the north-west quadrant of $q$
$\omega = (0, \ldots, 0, 1, 0, \ldots, 0)$	A single stock	Still a feasible portfolio; with three or more stocks it sits inside the bullet, so it is dominated

Key Insight: An extra security can never hurt: putting weight zero on it leaves every old portfolio available, so the efficient frontier can only move north-west (or stay put).

IIEvery Mix Is One Point

Each set of weights is one portfolio; on the bullet, drawn as in the video with short sales allowed, anything inside it has a better portfolio straight up and straight left.

IIIOne More Security

Adding Motorola keeps every GM–IBM portfolio available (weight zero), so the upper branch can only move north-west.

Risk level σ (% per month) 6.34

IVWhy Stocks Move Together

A shared business cycle, and panics that sell all equities at once, raise correlation and shrink what diversification removes.

Business-cycle share of each stock's variance 40%

💡 Without the panic, the business-cycle share in this illustrative model is the true correlation between any two stocks; 48 simulated months only estimate it. Diversification still lowers risk until the share reaches 100%, the case $\rho = 1$ .

VQuiz Questions

Problem 1 · Who Dominates Whom?

Given (illustrative): three fictional stocks, annual returns.

Stock	Mean	σ	Correlations
Aster (A)	$6\%$	$14\%$	$\rho_{AB} = 0.3$
Birch (B)	$9\%$	$20\%$	$\rho_{AC} = 0.2$
Cedar (C)	$13\%$	$30\%$	$\rho_{BC} = 0.4$

Taken one at a time, does any of the three stocks dominate another?

The portfolio $Q$ with weights $(0.4,\ 0.3,\ 0.3)$ has $\mathsf{E}[R_Q] = 9\%$ and $\sigma_Q \approx 15.2\%$ . Which single stocks does $Q$ dominate?

✅ Correct! No stock dominates another, yet

Q

matches Birch's mean with about

4.8

points less risk, so 100% Birch is dominated, exactly as 100% IBM was in the video.

❌ Lower risk is only half of dominance. Aster's lower

\sigma

comes with a lower mean (

6\%

against

9\%

), so it is south-west of Birch, not north-west.

❌ Higher mean is only half of dominance. Cedar earns more but carries

30\%

risk, so it is north-east of the others: neither better nor worse.

❌ Not quite. Dominance needs a mean at least as high and a

\sigma

at least as low.

❌ Check Aster.

Q

beats Aster on mean, but

15.2\% > 14\%

, so

Q

is north-east of Aster: no dominance.

❌ That is the intuition the video overturns. Because

\rho < 1

, the mix's

\sigma

falls below the weighted average of the three

\sigma

s; compare

Q

with Birch.

❌ Not quite. Compare

Q

(15.2\%,\ 9\%)

with each stock's

(\sigma, \text{mean})

one by one.

Show solution

Stocks against each other. Sorted by risk, $14\% < 20\% < 30\%$ , the means also rise, $6\% < 9\% < 13\%$ . Each step to the right is also a step up, so every pair is north-east / south-west of each other and no stock dominates another (like GM and IBM in the video).

$Q$ against each stock.

Birch $(20\%, 9\%)$ : same mean, $15.2\% < 20\%$ , so $Q$ dominates Birch.
Aster $(14\%, 6\%)$ : $Q$ has a higher mean but a higher $\sigma$ , so no dominance.
Cedar $(30\%, 13\%)$ : $Q$ has less risk but a lower mean, so no dominance.

$Q$ dominates Birch only.

Problem 2 · The Weighted-Average Trap

Given: the Problem 1 stocks. A portfolio holds $70\%$ Aster and $30\%$ Birch (no Cedar).

What is its standard deviation $\sigma_p$ ?

Does this portfolio dominate 100% Aster?

✅ Correct! Moving

30\%

into a riskier stock lowered risk, because

\rho_{AB} = 0.3

is well below

1

: 100% Aster is dominated even by a two-stock mix.

❌ That is the weighted average of the $\sigma$ s,

0.7(14) + 0.3(20)

. It would be right only if

\rho_{AB} = 1

; with

\rho_{AB} = 0.3

the variance formula gives less.

❌ The covariance term is missing.

\rho_{AB} = 0.3 \ne 0

, so add

2\,\omega_A\,\omega_B\,\sigma_{AB}

with

\sigma_{AB} = 84

❌ Close: the cross term appears twice. The variance table has two off-diagonal cells,

\sigma_{AB}

and

\sigma_{BA}

, hence the factor

2

❌ Not quite. Use

\sigma_p^2 = \omega_A^2\sigma_A^2 + \omega_B^2\sigma_B^2 + 2\,\omega_A\omega_B\,\rho_{AB}\,\sigma_A\sigma_B

, then take the square root.

❌ Check the number. That reasoning treats

\sigma

as a weighted average; the variance formula gives about

12.9\% < 14\%

❌ Wrong comparison. The question is about Aster,

(14\%, 6\%)

; Birch is not the benchmark here.

❌ Negative correlation is not needed. Any

\rho

comfortably below

1

can do it; here

\rho_{AB} = 0.3

is enough.

❌ Not quite. Compare

(\sigma_p, \mathsf{E}[R_p])

with Aster's

(14\%, 6\%)

Show solution

Mean: $\mathsf{E}[R_p] = 0.7(6\%) + 0.3(9\%) = 4.2\% + 2.7\% = 6.9\%$ .

Covariance: $\sigma_{AB} = \rho_{AB}\,\sigma_A\sigma_B = 0.3 \times 14 \times 20 = 84$ (in $\%^2$ ).

Variance:

\sigma_p^2 = 0.7^2(196) + 0.3^2(400) + 2(0.7)(0.3)(84) = 96.04 + 36 + 35.28 = 167.32

\sigma_p = \sqrt{167.32} \approx 12.9\%

Against Aster: $6.9\% > 6\%$ and $12.9\% < 14\%$ , so the mix lies north-west of Aster and dominates it.

Problem 3 · A Three-Stock Mix against Birch

Given: the Problem 1 stocks. A portfolio holds $\omega = (0.3,\ 0.3,\ 0.4)$ in Aster, Birch and Cedar.

What is its standard deviation $\sigma_p$ ?

Where does this portfolio sit relative to 100% Birch, $(20\%, 9\%)$ ?

✅ Correct!

(17.0\%,\ 9.7\%)

is north-west of

(20\%,\ 9\%)

: more return and less risk than putting everything in Birch.

❌ That is the weighted average of the $\sigma$ s,

0.3(14) + 0.3(20) + 0.4(30)

, correct only if every correlation were

1

❌ Close: each cross term appears twice. The three off-diagonal pairs each contribute

2\,\omega_i\omega_j\sigma_{ij}

❌ The covariances are missing. With positive correlations the three cross terms add

92.88

to the variance.

❌ Not quite. Add the three weighted variances and the three doubled weighted covariances, then take the square root.

❌ Check the risk.

17.0\% < 20\%

, so the mix is to the left of Birch, not the right.

❌ Not quite. Its mean is

0.3(6) + 0.3(9) + 0.4(13) = 9.7\%

; compare both coordinates with Birch's.

Show solution

Covariances (in $\%^2$ ): $\sigma_{AB} = 0.3(14)(20) = 84$ , $\sigma_{AC} = 0.2(14)(30) = 84$ , $\sigma_{BC} = 0.4(20)(30) = 240$ .

Mean: $\mathsf{E}[R_p] = 0.3(6) + 0.3(9) + 0.4(13) = 1.8 + 2.7 + 5.2 = 9.7\%$ .

Variance: the diagonal terms, then the doubled cross terms:

0.09(196) + 0.09(400) + 0.16(900) = 17.64 + 36 + 144 = 197.64

2(0.3)(0.3)(84) + 2(0.3)(0.4)(84) + 2(0.3)(0.4)(240) = 15.12 + 20.16 + 57.6 = 92.88

\sigma_p^2 = 197.64 + 92.88 = 290.52, \qquad \sigma_p = \sqrt{290.52} \approx 17.0\%

Against Birch: $9.7\% > 9\%$ and $17.0\% < 20\%$ , so the mix is north-west of Birch and dominates it.

Problem 4 · A Fourth Stock Nobody Would Hold Alone

Given: a fourth fictional stock, Dogwood, joins the menu: mean $5\%$ , $\sigma = 40\%$ , uncorrelated with Aster, Birch and Cedar. On its own it is dominated by Aster. What happens to the efficient frontier?

✅ Correct! Every old portfolio is still available with

\omega_D = 0

. Here the frontier actually improves: a small long position in the uncorrelated Dogwood lowers the minimum-variance

\sigma

from about

12.8\%

to about

12.2\%

❌ Dominated alone does not mean useless in a mix. GM, IBM and Motorola are each dominated too, yet frontier portfolios hold all three; Dogwood's zero correlation makes it a diversifier.

❌ Nothing forces you to hold it. With

\omega_D = 0

every old portfolio is still feasible, so no part of the frontier can get worse.

❌ Not forced to include it. The old minimum-variance portfolio is still available; the new one is at least as good, and here it is better.

❌ Not quite. Ask what happens to the old portfolios when Dogwood's weight is zero.

Show solution

Nothing is lost. Any portfolio of Aster, Birch and Cedar is also a four-stock portfolio with $\omega_D = 0$ , with the same mean and the same $\sigma$ . So at every level of risk the best attainable mean cannot fall, and at every mean the least attainable $\sigma$ cannot rise: the frontier moves north-west or stays put.

Something is usually gained. Being dominated on its own says nothing about a stock's value in a mix. Dogwood is uncorrelated with the other three, so a small weight adds little variance of its own while spreading risk. Solving the frontier numerically (short sales allowed), the minimum-variance $\sigma$ falls from about $12.8\%$ to about $12.2\%$ , with roughly $9\%$ in Dogwood.

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