Finance-Theory · Unit 15 · Video 2 · Interactive Practice

130/30 Funds, a Leveraged Home and Zero-Investment Portfolios

IKey Formulas

Formula	Name	What it says
$\omega_{\text{long}} + \omega_{\text{short}} = 1$ , e.g. $1.30 + (-0.30) = 1$	Capped long-short fund (130/30, 120/20, 180/80)	The weights still sum to one, so a floor on the short weight is also a ceiling on the long weight
$\omega_{\text{home}} = \dfrac{\text{home value}}{\text{equity}}, \quad \omega_{\text{mortgage}} = -\dfrac{\text{loan}}{\text{equity}}$	Leveraged home	The denominator is your net investment (the down payment); the loan is a negative position, and leverage ratio = home value / equity
$R_{\text{equity}} = \omega_{\text{home}} R_{\text{home}} + \omega_{\text{mortgage}} \times 0\%$	Realized return on your equity (interest ignored)	Every percentage move in the price reaches your equity multiplied by the leverage ratio, for gains and losses alike
$N_A P_A + N_B P_B = 0 \;\Rightarrow\; \omega_A = \dfrac{N_A P_A}{0}$ is undefined	Zero net investment	With no net money in, there are no percentage weights: describe the position in dollar amounts

Key Insight: One formula covers all three structures. In each, the negative position (a short sale or a loan) pays for the long exposure above 100%; when the negative position grows to match the long one, the weights' denominator hits zero and dollars must replace percentages.

IICapping the Shorts Caps the Longs

Long weight plus short weight is always 100%, so any cap on the short side also caps the long side.

Short cap 10% Client money

💡 A long-only manager who expects a stock to do badly can only leave it out; a capped short side lets the manager gain when it falls while staying 100% net long. The 130/30 version was very popular before 2008 and beat the S&P in its early years, contracted sharply after 2008, and drew renewed interest in the mid-2020s.

IIIA Mortgage Is Leverage

The loan is a negative weight, so each percentage move in an illustrative home's price is multiplied on your equity.

Home Price move −12%

IVWhen the Net Investment Is Zero

As the short position grows to match the long one, the weights' denominator shrinks toward zero and the weights explode.

Shares of B shorted 240

💡 Zero net investment is not zero risk: if A rises while B falls both legs gain, and if A falls while B rises both lose. The dollar amounts on each side say how much is at stake.

VQuiz Questions

Problem 1 · A 150/50 Fund in Dollars

Given: A manager runs a 150/50 fund with $250 million of client money. Find the fund's long and short positions in dollars.

✅ Correct! $375 million − $125 million = $250 million, exactly the client's money:

1.50 + (-0.50) = 1

❌ Not quite. The long side is 150% of the client's money, not 100%: the cash from the short sales buys the extra 50%. As written, the net would be only $125 million.

❌ Not quite. That splits the money in the ratio 150 : 50. The weights are measured against the net investment, so the long side is 1.5 times $250 million, not three quarters of it.

❌ Not quite. Those are the percentages read as millions of dollars. Multiply each weight by the fund's $250 million.

❌ Not quite. Multiply each weight by the net investment.

Show solution

A 150/50 fund has $\omega_{\text{long}} = 1.50$ and $\omega_{\text{short}} = -0.50$ , and $1.50 + (-0.50) = 1$ .

Long: 1.50 × $250 million = $375 million. Short: 0.50 × $250 million = $125 million of stock borrowed and sold.

Check: $375 million − $125 million = $250 million, the money the clients put in. The $125 million raised by the short sales pays for the long exposure above 100%.

Problem 2 · Ten Percent Down

Given: You buy a $360,000 home with 10% down and borrow the rest. Ignore mortgage interest and costs. The home's price then falls 6%. Find the change in your equity as a percentage of what you put in.

✅ Correct! With 10% down the leverage is 10 : 1, so a 6% price fall wipes out 60% of your equity.

❌ Not quite. −6% is the home's return. Your equity is only $36,000, and it absorbs the whole $21,600 loss.

❌ Close, but… 900% is the size of the mortgage weight. The home's weight is $360,000 / $36,000 = 1000%, and the loan's return is zero.

❌ Not quite. Leverage multiplies the price move; it does not divide it.

❌ Not quite. Compare the dollar loss with your down payment.

Show solution

Down payment (equity): 10% × $360,000 = $36,000. Loan: $324,000.

Weights: $\omega_{\text{home}}$ = $360,000 / $36,000 = 1000% and $\omega_{\text{mortgage}}$ = −$324,000 / $36,000 = −900%, which sum to 100%. The leverage ratio is 10 : 1.

In dollars: the home loses 6% × $360,000 = $21,600, and the loan is unchanged, so equity falls from $36,000 to $14,400, a change of −$21,600 / $36,000 = −60%.

By weights: $R_{\text{equity}} = 1000\% \times (-6\%) + (-900\%) \times 0\% = -60\%$

Problem 3 · Does the Short Side Cushion a Fall?

Given: A 150/50 fund has $40 million of client money. Over a month, every stock it holds long and every stock it has shorted falls 12%. Find the fund's gain or loss.

✅ Correct! The fund is 150% − 50% = 100% net long, so a market-wide fall of 12% costs it 12%, exactly like a long-only fund.

❌ Not quite. That counts only the long side. The shorted stocks fell too, and a short position gains when its stock falls.

❌ Not quite. The short side does not lose when its stocks fall: you buy back for less than you sold, so it gains $2.4 million.

❌ Not quite. $60 million long against $20 million short is not balanced. The fund stays 100% net long, so the short side does not cushion a market fall.

❌ Not quite. Work out each side's dollar result, then add.

Show solution

Positions: long 1.50 × $40 million = $60 million; short 0.50 × $40 million = $20 million.

Long side: 12% × $60 million = $7.2 million lost. Short side: the borrowed stocks are now worth 12% × $20 million = $2.4 million less, so buying them back costs $2.4 million less than the sale raised: a gain of $2.4 million.

Net: −$7.2 million + $2.4 million = −$4.8 million, and −$4.8 million / $40 million = −12%.

By weights: $1.50 \times (-12\%) + (-0.50) \times (-12\%) = -18\% + 6\% = -12\%$

Because the weights sum to 100%, a move that hits every stock equally moves the fund by the same percentage. So the 150/50 structure beats holding the same longs at 100% only when the shorted stocks do worse than the longs.

Problem 4 · Zero Net Investment

Given: You buy 200 shares of stock A at $60 and short-sell 600 shares of stock B at $20, using the short-sale cash to pay for A (ignore margin requirements).

What is the portfolio weight of A?

A then rises 3% and B falls 6%. What is your dollar result?

✅ Correct! The weights do not exist, but the dollar amounts do: long $12,000 of A and short $12,000 of B, and both legs gained.

❌ Not quite. That divides by the gross exposure ($12,000 + $12,000). The weight formula divides by the sum of the signed positions, which here is $0.

❌ Not quite. 100% would need A to be all of the net investment, but the net investment is $12,000 + (−$12,000) = $0.

❌ Not quite. Weights are built from dollar values

N_i P_i

, not share counts, and here those dollar values sum to zero.

❌ Not quite. That is only the long leg. B fell, and a short position gains when its stock falls.

❌ Close, but… check the sign of the short leg: B falling 6% means you buy it back $720 cheaper, a gain.

❌ Not quite. Zero net investment is not zero risk: each side carries $12,000 of exposure.

❌ Not quite. Compute the denominator

N_A P_A + N_B P_B

first.

❌ Not quite. Work out each leg's dollar result, then add.

Show solution

Dollar positions: A is 200 × $60 = $12,000; B is −600 × $20 = −$12,000.

Denominator: $12,000 + (−$12,000) = $0, so $\omega_A$ = $12,000 / $0 is undefined. The portfolio is described in dollars instead: long $12,000 of A, short $12,000 of B.

Long leg: 3% × $12,000 = +$360. Short leg: B's shares are now worth 6% × $12,000 = $720 less, so closing the short gains $720.

Total: $360 + $720 = $1,080. Had A fallen and B risen instead, both legs would have lost.

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