Finance-Theory · Unit 16 · Video 4 · Interactive Practice

How Correlation Bends the Bullet, from +1 to −1

IKey Formulas

Formula	Case	How to read it
$\text{E}[R_p] = \omega_a\mu_a + \omega_b\mu_b$	Expected return	No $\rho$ in it: changing the correlation never moves a portfolio's expected return
$\text{Var}[R_p] = \omega_a^2\sigma_a^2 + \omega_b^2\sigma_b^2 + 2\omega_a\omega_b\sigma_a\sigma_b\rho_{ab}$	Variance	$\rho$ sits only in the cross term; at $\rho = 0$ that term drops out
$\text{SD}[R_p] = \omega_a\sigma_a + \omega_b\sigma_b$	$\rho = +1$	For $0 \le \omega \le 1$ a weighted average of the two SDs: a straight line, no risk reduction
$\text{SD}[R_p] = \|\omega_a\sigma_a - \omega_b\sigma_b\|$	$\rho = -1$	Zero when $\omega_a\sigma_a = \omega_b\sigma_b$ , at $\omega_b = \dfrac{\sigma_a}{\sigma_a + \sigma_b}$ : two straight segments meeting on the vertical axis

Key Insight: The two stocks' means and standard deviations fix the endpoints, and the correlation decides how far the curve between them bends west. For GM and Motorola the 75/25 mix earns about $1.25\%$ a month whatever $\rho$ is, while its risk runs from about $7.1\%$ at $\rho = +1$ down to $2.24\%$ at $\rho = -1$ .

IICorrelation Bends the Bullet

Lowering $\rho$ leaves every portfolio's expected return in place and moves only its risk, bending the curve west.

Correlation ρ 0.37

IIIThe ρ = −1 Triangle

At $\rho = -1$ the weighted SDs subtract, so one mix of two illustrative stocks carries no risk.

Ibis weight ω 1.00

💡 For the illustrative Ibis–Jacana pair, a riskless $0.97\%$ a month, about $11.6\%$ a year, far above a T-bill's yield, would be an arbitrage, a free lunch. That is why no real pair of assets has a correlation of exactly $-1$ .

IVQuiz Questions

Problem 1 · Risk of an Even Split

Given: two illustrative stocks with monthly figures: Ibis has mean $0.7\%$ and standard deviation $4\%$ , Jacana has mean $1.5\%$ and standard deviation $8\%$ , and their correlation is $\rho = 0.5$ . Find the standard deviation of a portfolio with $50\%$ in each.

✅ Correct! The variance is

4 + 16 + 8 = 28

\%^2

, and

\sqrt{28} \approx 5.29\%

: less than the

6\%

average of the two SDs, because

\rho < 1

❌ Close, but… the cross term carries a factor of

2

2(0.5)(0.5)(4)(8)(0.5) = 8

, not

4

. With

4

you get

\sqrt{24} \approx 4.90\%

❌ Not quite.

0.5 \times 4 + 0.5 \times 8 = 6

is the weighted average of the SDs, which is the portfolio SD only at

\rho = +1

. At

\rho = 0.5

the cross term is smaller, so the risk is lower.

❌ Not quite.

\sqrt{4 + 16} \approx 4.47\%

leaves out the cross term, which is right only at

\rho = 0

. Here

\rho = 0.5

adds

8

to the variance.

Show solution

With $\omega_{\text{Ibis}} = \omega_{\text{Jacana}} = 0.5$ , variances in $\%^2$ :

\text{Var}[R_p] = 0.5^2(4^2) + 0.5^2(8^2) + 2(0.5)(0.5)(4)(8)(0.5) = 4 + 16 + 8 = 28

\text{SD}[R_p] = \sqrt{28} \approx 5.29\% \text{ per month}

The expected return is $0.5(0.7) + 0.5(1.5) = 1.1\%$ , exactly halfway between the stocks, but the risk is below the halfway value of $6\%$ .

Problem 2 · What a Lower Correlation Changes

Given: the same illustrative stocks (Ibis $0.7\%$ and $4\%$ , Jacana $1.5\%$ and $8\%$ per month) and a portfolio with $25\%$ in Ibis and $75\%$ in Jacana. Their correlation falls from $\rho = 0.5$ to $\rho = -0.5$ . Which statement describes the change?

✅ Correct! The portfolio moves straight west: the same

1.30\%

return with about one point less risk.

❌ Not quite. The expected return

\omega_a\mu_a + \omega_b\mu_b

has no

\rho

in it, so it stays at

1.30\%

. Only the risk moves.

❌ Close, but… those values use a cross term of

6\rho

. The factor of

2

makes it

2(0.25)(0.75)(4)(8)\rho = 12\rho

❌ Close, but…

5\% = |0.25 \times 4 - 0.75 \times 8|

is the shortcut that holds only at

\rho = -1

. At

\rho = -0.5

the variance is

1 + 36 - 6 = 31

Show solution

Expected return (no $\rho$ ): $0.25(0.7) + 0.75(1.5) = 0.175 + 1.125 = 1.30\%$ at either correlation.

Variance in $\%^2$ , as a function of $\rho$ :

\text{Var}[R_p] = 0.25^2(16) + 0.75^2(64) + 2(0.25)(0.75)(4)(8)\rho = 1 + 36 + 12\rho

At $\rho = 0.5$ : $37 + 6 = 43$ , so $\text{SD} = \sqrt{43} \approx 6.56\%$ .

At $\rho = -0.5$ : $37 - 6 = 31$ , so $\text{SD} = \sqrt{31} \approx 5.57\%$ .

The lower correlation sheds about one point of risk at an unchanged return.

Problem 3 · Two Mixes with the Same Risk

Given: the illustrative Ibis ( $0.7\%$ , $4\%$ ) and Jacana ( $1.5\%$ , $8\%$ ) with correlation exactly $\rho = -1$ , long positions only ( $0 \le \omega_{\text{Ibis}} \le 1$ ). Two portfolios have a standard deviation of exactly $2\%$ per month.

Which Ibis weights give $\text{SD}[R_p] = 2\%$ ?

Which of the two does a mean-variance investor hold?

✅ Correct! Same

2\%

risk, but

1.10\% > 0.83\%

: the even split dominates, so nobody holds a portfolio on the lower segment of the triangle.

❌ Not quite. At

\rho = -1

\text{SD}[R_p] = |4\omega - 8(1 - \omega)| = |12\omega - 8|

. Set that equal to

2

❌ Close, but… the absolute value gives two cases,

12\omega - 8 = 2

and

12\omega - 8 = -2

, and both weights lie between

0

and

1

❌ Not quite.

\omega = \tfrac{2}{3}

is the zero-risk vertex, where

|12\omega - 8| = 0

, and

\omega = \tfrac{1}{3}

gives

|4 - 8| = 4\%

❌ Not quite. Compare the two portfolios' expected returns: at equal risk, the higher mean wins.

❌ Not quite. Equal risk is only half the comparison. Their expected returns differ,

1.10\%

against about

0.83\%

, so one dominates the other.

❌ Not quite. Both portfolios have the same

2\%

risk, so holding more Ibis makes this one no safer, only lower-returning.

Show solution

Step 1: risk at $\rho = -1$ . The variance is a perfect square, so

\text{SD}[R_p] = |4\omega - 8(1 - \omega)| = |12\omega - 8|

Step 2: solve $|12\omega - 8| = 2$ . Either $12\omega - 8 = 2$ , giving $\omega = \tfrac{10}{12} = \tfrac{5}{6}$ , or $12\omega - 8 = -2$ , giving $\omega = \tfrac{6}{12} = \tfrac{1}{2}$ .

Step 3: expected returns.

\omega = \tfrac{1}{2}:\ 0.5(0.7) + 0.5(1.5) = 1.10\%

\omega = \tfrac{5}{6}:\ \tfrac{5}{6}(0.7) + \tfrac{1}{6}(1.5) = \tfrac{3.5 + 1.5}{6} \approx 0.83\%

Step 4: compare. Equal risk, higher expected return: $\omega_{\text{Ibis}} = \tfrac{1}{2}$ dominates. The $\tfrac{5}{6}$ mix lies on the segment from Ibis to the vertex, and every portfolio there has a twin on the upper segment with the same risk and more return.

Problem 4 · Shorting at Perfect Correlation

Given: suppose instead the illustrative Ibis ( $0.7\%$ , $4\%$ ) and Jacana ( $1.5\%$ , $8\%$ ) had correlation $\rho = +1$ . An investor shorts Jacana and puts the proceeds into Ibis: $\omega_{\text{Ibis}} = 1.25$ , $\omega_{\text{Jacana}} = -0.25$ .

What is the portfolio's standard deviation?

Does this portfolio dominate holding Ibis alone?

✅ Correct! Lower risk (

3\% < 4\%

) and lower return (

0.50\% < 0.7\%

): a trade-off along the straight line, not dominance.

❌ Not quite. Put the signed weights into the full variance formula; the cross term is negative because one weight is negative.

❌ Not quite.

1.25 \times 4 + 0.25 \times 8 = 7

treats the short as a long position. With

\omega_{\text{Jacana}} = -0.25

the weighted SDs subtract:

|5 - 2| = 3

❌ Not quite. At

\rho = +1

long-only mixes cannot get below Ibis's risk, but a short position can: the variance is

25 + 4 - 20 = 9

❌ Not quite.

\sqrt{25 + 4}

drops the cross term

2(1.25)(-0.25)(4)(8)(1) = -20

, which is large at

\rho = +1

❌ Not quite. Dominance needs both lower risk and at least as much expected return. Work out

\text{E}[R_p]

with the signed weights.

❌ Not quite. Lower risk alone is not dominance. The short Jacana position also lowers the expected return, to

0.50\%

❌ Not quite. The expected return changes with the weights:

1.25(0.7) - 0.25(1.5) = 0.50\%

, not

0.7\%

Show solution

Variance in $\%^2$ , with the signed weights:

\text{Var}[R_p] = 1.25^2(16) + (-0.25)^2(64) + 2(1.25)(-0.25)(4)(8)(1) = 25 + 4 - 20 = 9

So $\text{SD}[R_p] = \sqrt{9} = 3\%$ . At $\rho = +1$ the variance is the perfect square $(1.25 \times 4 - 0.25 \times 8)^2 = (5 - 2)^2$ , which gives the same answer.

Expected return: $1.25(0.7) - 0.25(1.5) = 0.875 - 0.375 = 0.50\%$ .

The portfolio has less risk than Ibis ( $3\% < 4\%$ ) but also less expected return ( $0.50\% < 0.7\%$ ). At $\rho = +1$ the only way below Ibis's risk is to short Jacana, and that lowers the return along with the risk, so it is a trade-off, not dominance.

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