Finance-Theory · Unit 15 · Video 1 · Interactive Practice

Portfolio Weights, Short Sales and Buying on Margin

IKey Formulas

Formula	Name	What it tells you
$\omega_i = \dfrac{N_i P_i}{N_1 P_1 + N_2 P_2 + \cdots + N_n P_n}$	Portfolio weight	The dollar holding of security $i$ (shares $N_i$ times price $P_i$ ) divided by the total of all holdings; $N_i$ can be zero or negative
$\omega_1 + \omega_2 + \cdots + \omega_n = 1$	Weights add up to one	The denominator is the sum of all the numerators
$\omega_k < 0 \;\Rightarrow\; \displaystyle\sum_{\omega_j > 0} \omega_j > 1$	Negative weight forces leverage	The remaining weights add up to $1 - \omega_k > 1$ , and any other negative weight pushes the positive ones higher still
$\omega_{\text{bond}} = -\dfrac{\text{loan}}{\text{equity}}$	Margin loan	On margin every weight is measured against your equity, and the borrowed money is a short position in a riskless bond

Key Insight: A negative weight is a loan — of shares (a short sale) or of money (a margin loan, i.e. a short riskless bond) — and whenever one appears, the positive weights add up to more than 100%: the portfolio is leveraged.

IIVisualization 1 — From Share Counts to Weights

Each weight is a dollar holding divided by the total, so selling one stock short makes the other weights grow.

IIIVisualization 2 — A Short Sale Is a Loan Paid in Shares

Selling borrowed Nutmeg shares to buy more Myrtle is borrowing to invest, so the Myrtle weight passes 100%.

Step 0

💡 Your debt is 400 Nutmeg shares, not a fixed $12,000: if Nutmeg rose from $30 to $36, you would owe $14,400.

IVVisualization 3 — The Same Stocks on Margin

Holding the same $240,000 of stock with less equity rescales every weight and adds a negative riskless-bond weight.

Equity (% of holdings) 100%

VQuiz Questions

Problem 1 · Weight of One Holding

Given: A $240,000 account holds three fictional stocks: 600 shares of Larch at $45, 1,500 shares of Myrtle at $72, and 1,500 shares of Nutmeg at $70. Find the portfolio weight of Myrtle.

✅ Correct! The Myrtle holding is worth $108,000, which is 45% of the $240,000 total.

❌ That is Myrtle's share of the share count. 1,500 of the 3,600 shares is about 41.7%, but a weight is a share of dollars: each holding is

N_i P_i

, not

N_i

❌ That compares prices only. $72 is about 38.5% of the three prices added together, but the weight needs the dollar holding, shares times price.

❌ Check the denominator. You divided Myrtle by the other two holdings only. The denominator is the sum of ALL the holdings, Myrtle included.

Show solution

Dollar holdings $N_i P_i$ : Larch 600 × $45 = $27,000; Myrtle 1,500 × $72 = $108,000; Nutmeg 1,500 × $70 = $105,000.

Total: $27,000 + $108,000 + $105,000 = $240,000, the whole account.

Myrtle's weight: $108,000 / $240,000 = 0.45 = 45%. (The other two are 11.25% and 43.75%, and $11.25\% + 45\% + 43.75\% = 100\%$ .)

Problem 2 · Weights With a Short Sale

Given: You have $240,000 of your own money. You sell short 400 shares of the fictional stock Osier at $75 and put your $240,000 plus the proceeds into the fictional stock Larch at $60 a share. Find the two portfolio weights.

✅ Correct! The weights add up to

112.5\% - 12.5\% = 100\%

, and the positive weight exceeds 100%: the short sale has leveraged the Larch position.

❌ These do not add up to one. Larch holds more than your own money: $270,000 of stock measured against $240,000 of net investment.

❌ The short position is not an asset. Its share count is −400, so its dollar holding is −$30,000; the denominator is the NET total, $270,000 − $30,000 = $240,000.

❌ Close, but check the sign. A short sale is a negative share count, so its weight is negative; with

+12.5\%

the weights would sum to 125%.

Show solution

Short sale: $N_{\text{Osier}} = -400$ , so the Osier holding is −400 × $75 = −$30,000. The sale brings in $30,000 of cash.

Larch: $240,000 + $30,000 = $270,000, which buys 4,500 shares at $60.

Total (the denominator): $270,000 − $30,000 = $240,000, your own money.

Weights: $270,000 / $240,000 = 112.5% for Larch and −$30,000 / $240,000 = −12.5% for Osier. They sum to 100%, and the positive weight is above 100% because the Osier shares are effectively a loan.

Problem 3 · Margin Account Weights

Given: An account holds 1,000 shares of Larch at $36, 1,200 shares of Myrtle at $85 and 2,000 shares of Nutmeg at $31 (all fictional). The investor put up $80,000 of their own money and bought the rest on margin, with a loan from the broker. Find the weight of Myrtle and the weight of the riskless bond.

Weight of Myrtle

Weight of the riskless bond

✅ Correct! The stock weights add up to 250% and the bond weight is −150%, so the total is 100% of the $80,000 of equity.

❌ That is the cash-account weight. $102,000 is 51% of the $200,000 of stock, but on margin the denominator is the investor's equity, not the value of the stock.

❌ That divides by the loan. The equity is the investor's own $80,000; the other $120,000 is borrowed.

❌ That is the weight of ALL the stock. $200,000 / $80,000 = 250% is the three stock weights together; Myrtle is only part of it.

❌ Check the Myrtle weight. Myrtle's dollar holding divided by the equity.

❌ That is the loan as a share of the stock. Weights are measured against equity: divide the loan by the $80,000 of equity.

❌ Check the bond weight. A loan is a short position in the riskless bond, so its weight is negative, and it is measured against the equity.

Show solution

Holdings: Larch 1,000 × $36 = $36,000; Myrtle 1,200 × $85 = $102,000; Nutmeg 2,000 × $31 = $62,000; total $200,000 of stock.

Equity: $80,000, which is 40% of the $200,000 of stock. Loan: $200,000 − $80,000 = $120,000, a short position of −120,000 riskless bonds at $1.

Myrtle: $102,000 / $80,000 = 127.5%. Bond: −$120,000 / $80,000 = −150%.

Check: Larch 45%, Myrtle 127.5%, Nutmeg 77.5% add up to 250% of stock, and $250\% - 150\% = 100\%$ .

Problem 4 · Two Short Positions

Given: A portfolio of four fictional stocks has weights $\omega_{\text{Larch}} = 80\%$ , $\omega_{\text{Myrtle}} = -30\%$ and $\omega_{\text{Osier}} = -10\%$ ; the fourth stock is Nutmeg. Find the Nutmeg weight and the sum of the positive weights.

Weight of Nutmeg

Sum of the positive weights

✅ Correct! The two short positions together are −40%, so the positive weights must add up to

1 - (-0.40) = 140\%

❌ One short position is missing. All four weights must add up to 100%, Osier's −10% included.

❌ The short weights were treated as positive. Subtracting a negative weight adds:

100\% - 80\% + 30\% + 10\%

❌ Check the Nutmeg weight. Solve

80\% - 30\% + \omega_{\text{Nutmeg}} - 10\% = 100\%

❌ It is ALL the weights that add up to 100%. With negative weights present, the positive ones must add up to more than that.

❌ That adds the short weights in as well. 180% is the sum of the absolute values of all four weights; count only the positive ones.

❌ Check the sum. Add only the weights that are above zero.

Show solution

The weights add up to one:

0.80 + (-0.30) + \omega_{\text{Nutmeg}} + (-0.10) = 1 \;\Rightarrow\; \omega_{\text{Nutmeg}} = 1 - 0.40 = 0.60 = 60\%

Positive weights: Larch and Nutmeg, $80\% + 60\% = 140\%$ . Equivalently, the positive weights equal one minus the negative ones: $1 - (-0.30 - 0.10) = 1.40$ .

Each short position pushes the positive weights higher, so with two of them the long side is 140% of the money invested: a leveraged portfolio.

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