Finance-Theory · Unit 17 · Video 4 · Interactive Practice

The Stock Nobody Can Find: Negative Correlation and Alpha

IKey Formulas

Formula	Name	What it says
$\mathrm{E}[R] = \dfrac{\mathrm{E}[P_{t+1}]}{P_t} - 1$	Expected return from prices (no dividend)	Video example: $110/100 - 1 = 10\%$ ; once buyers push today's price to $105, $110/105 - 1 \approx 4.8\%$
$R_{\text{short}} = -R_{\text{index}}$	Shorting the index	Correlation $\rho = -1$ with the index, but $\mathrm{E}[R_{\text{short}}] = -\mathrm{E}[R_{\text{index}}] < 0$
$\mathrm{E}[R] = r_f + \dfrac{\mathrm{E}[R_T] - r_f}{\sigma_T}\,\sigma$	Tangency line: the do-it-yourself baseline	Every mix of the tangency portfolio $T$ with T-bills; its slope is $T$ 's Sharpe ratio
$\mathrm{E}[R_M] - \Big(r_f + \dfrac{\mathrm{E}[R_T] - r_f}{\sigma_T}\,\sigma_M\Big)$	Alpha, as the video draws it	How far a manager's portfolio $M$ sits above the line at its own $\sigma_M$ ; positive exactly when $M$ 's Sharpe ratio beats $T$ 's

Key Insight: An opportunity everyone can see does not stay available. Buying a stock with $\rho < 0$ and $\mathrm{E}[R] > 0$ raises $P_t$ and lowers its expected return; and since you can reach the tangency line alone, a manager's fee has to buy distance above that line.

IIThe Four Corners of Correlation and Return

Every popular candidate misses the upper-left corner: negative correlation with the market together with a positive expected return.

Candidate

IIICompeted Away

With the expected future price fixed at $110, every dollar buyers add to today's price lowers the expected return.

Today's price

P_t

$100

IVAlpha: Above the Line You Can Build Alone

A manager adds value only by putting you north-west of the tangency line from $r_f = 4\%$ through $T$ (illustrative, annual).

💡 In the video, Berkshire Hathaway's realized Sharpe ratio of about $0.76$ was higher than that of any other stock or mutual fund with a history of more than 30 years. The practical difficulty is identifying such managers before they become famous.

VQuiz Questions

Problem 1 · Expected Return from Prices

Given: A stock pays no dividend. Its expected price next year is $78 and it costs $65 today. Find its expected return.

✅ Correct!

78/65 - 1 = 1.2 - 1 = 0.20

. If buyers then bid the price up, this number falls.

❌ Not quite.

78/65 = 1.2

is one plus the return; subtract the 1.

❌ Not quite. You divided the $13 gain by next year's price. A return is measured against what you pay today,

P_t

❌ Not quite. $13 is the expected gain in dollars, not a percentage; divide it by today's price.

❌ Not quite. Use

\mathrm{E}[R] = \mathrm{E}[P_{t+1}]/P_t - 1

Show solution

With no dividend, the expected return comes from prices alone:

\mathrm{E}[R] = \frac{\mathrm{E}[P_{t+1}]}{P_t} - 1 = \frac{78}{65} - 1 = 1.2 - 1 = 0.20 = 20\%

Equivalently, the expected gain of $13 divided by today's price of $65 is $0.20$ .

Problem 2 · The Cheap Answer: Short the Index

Given: You hold $1,000,000 in an index fund with (illustrative, annual) expected return $9\%$ and standard deviation $16\%$ . To get some negative correlation you short $250,000 of the same index and hold the proceeds in T-bills at $r_f = 4\%$ . Find the expected return and standard deviation of your whole position.

What is the expected return of the whole position?

What is its standard deviation?

✅ Correct! The short cancels a quarter of your index holding: you now hold

0.75

of the index and

0.25

in T-bills, a point on the T-bill line. No clever stock required.

❌ Not quite. The short is not free: it earns

-9\%

in expectation on the $250,000, so it cancels part of the index's expected return.

❌ Close. You subtracted the short's expected loss but forgot that the $250,000 of proceeds earns

4\%

in T-bills.

❌ Not quite. The T-bill interest does not come on top of the full index return: the short earns

-9\%

on the $250,000 it covers.

❌ Not quite. Net out the long and short index positions first, then mix what is left with T-bills.

❌ Not quite. The short is not a separate risk that leaves the index’s risk unchanged: it cancels a quarter of your index holding, and only the net

0.75

carries risk.

❌ Not quite.

0.25 \times 16\% = 4\%

is the risk of the short leg on its own. What remains is the net index holding of

0.75

❌ Not quite. A short in the same index moves exactly opposite to the long (

\rho = -1

), so its risk cancels rather than adds:

1.25 \times 16\%

treats it as extra exposure.

❌ Check the risk. Only the net index holding carries risk, and T-bills carry none:

\sigma = |w|\,\sigma_{\text{index}}

Show solution

Long $1,000,000 of the index and short $250,000 of it nets to $750,000 of the index; the $250,000 of proceeds sits in T-bills. As fractions of your $1,000,000: $w = 0.75$ in the index, $0.25$ in T-bills.

\mathrm{E}[R] = r_f + w\,(\mathrm{E}[R_{\text{index}}] - r_f) = 4\% + 0.75 \times (9\% - 4\%) = 7.75\%

\sigma = w\,\sigma_{\text{index}} = 0.75 \times 16\% = 12\%

The short position alone does have $\rho = -1$ with the index, but its expected return is $-9\%$ . Held alongside the index it simply means holding less of the market: the same point you reach by putting $75\%$ in the index and $25\%$ in T-bills.

Problem 3 · Which Manager Has Alpha?

Given: $r_f = 4\%$ , and the tangency portfolio $T$ has $\mathrm{E}[R_T] = 10\%$ , $\sigma_T = 15\%$ (illustrative, annual). Fund M offers $\mathrm{E}[R] = 9\%$ with $\sigma = 12\%$ ; Fund N offers $\mathrm{E}[R] = 16\%$ with $\sigma = 30\%$ .

Which fund puts you above the tangency line?

By how many percentage points does that fund sit above the line, at its own $\sigma$ ?

✅ Correct! At

\sigma = 12\%

the line gives

8.8\%

, and Fund M delivers

9\%

. Fund N sits exactly on the line: you could match it alone by borrowing.

❌ Not quite. A high mean is not alpha. At

\sigma = 30\%

the line gives

4\% + 0.4 \times 30\% = 16\%

, exactly Fund N's mean: you can match it by putting

2

T

and borrowing

1

r_f

❌ Not quite. Compare each fund with the line at the fund's own risk, not with

T

's mean. Fund M takes less risk than

T

❌ Not quite.

9\% - 4\%

is the excess return over T-bills; the line itself already rises above

r_f

\sigma

grows.

❌ Close. That is the gap between Sharpe ratios,

5/12 - 0.4

. The question asks for the vertical distance in percentage points of expected return.

❌ Not quite. Compute the line's value at each fund's

\sigma

❌ Not quite. Subtract the line's value at

\sigma = 12\%

from the fund's mean.

Show solution

The line's slope is $T$ 's Sharpe ratio: $(10\% - 4\%)/15\% = 0.4$ , so the line is $\mathrm{E}[R] = 4\% + 0.4\,\sigma$ .

Fund M ( $\sigma = 12\%$ ): the line gives $4\% + 0.4 \times 12\% = 8.8\%$ ; the fund gives $9\%$ , so it is $9 - 8.8 = 0.2$ points above. Its Sharpe ratio $5/12 \approx 0.417$ exceeds $0.4$ .

Fund N ( $\sigma = 30\%$ ): the line gives $4\% + 0.4 \times 30\% = 16\%$ , exactly the fund's mean. Its Sharpe ratio is $12/30 = 0.4$ , the same as $T$ 's: you can build the same point yourself with $w = 30/15 = 2$ in $T$ , borrowing the other $1$ at $r_f$ .

Problem 4 · What the Fee Must Buy

Given: Same $r_f = 4\%$ and $T$ ( $10\%$ , $15\%$ ). Fund S (illustrative) offers $\mathrm{E}[R] = 12\%$ with $\sigma = 18\%$ before fees and charges 2 and 20: $2\%$ a year on the $1,000,000 you invest at the start of the year, plus $20\%$ of the gross profit, with no hurdle rate (the performance fee applies from the first dollar of profit). You invest $1,000,000 and this year the fund earns exactly $12\%$ gross.

What are your total fees for the year?

Is the fund's edge over the tangency line worth that fee?

✅ Correct! The fund does sit above the line, but by far less than it charges; alpha is what the fee has to buy.

❌ Not quite. That is only the 2% management fee; add 20% of the $120,000 profit.

❌ Not quite. That is only the performance fee; add the 2% of assets.

❌ Not quite. The 2% is charged on the $1,000,000 of assets, not on the profit.

❌ Not quite. Fees = 2% of assets + 20% of gross profit.

❌ Not quite. Fund S also takes more risk than

T

; compare it with the line at

\sigma = 18\%

, then compare that gap with the fee.

❌ Not quite. What reaches you is the return after fees. The fund is worth hiring only if it puts you above the line after paying it.

❌ Check the Sharpe ratio.

(12\% - 4\%)/18\% \approx 0.444

, above

T

0.4

: before fees the fund is above the line.

❌ Not quite. Find the fund's distance above the line at

\sigma = 18\%

and compare it with the fees as a percentage of assets.

Show solution

Fees. Management fee $0.02 \times$ $1,000,000 = $20,000. Gross profit $0.12 \times$ $1,000,000 = $120,000, and the performance fee is $0.20 \times$ $120,000 = $24,000. Total: $44,000, which is $4.4\%$ of assets.

Edge over the line. At $\sigma = 18\%$ the tangency line gives

4\% + \frac{10\% - 4\%}{15\%} \times 18\% = 4\% + 0.4 \times 18\% = 11.2\%

so before fees Fund S sits $12 - 11.2 = 0.8$ points above the line (Sharpe ratio $8/18 \approx 0.444 > 0.4$ ). After fees you receive $12\% - 4.4\% = 7.6\%$ this year: the fees are more than five times the fund's edge, so the fee is buying far more than the manager's alpha delivers.

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