Short sales

Allow investors to profit from a decline in the price of an asset.

To open the position:

1. Borrow the stock from your broker;
2. Sell the stock and deposit proceeds and margin in account;

To close the position:

1. Buy the stock;
2. And return it to the lending party;

Any dividends distributed by the stock are paid to the lending party.

Example

Assume you want to sell short 1000 shares of a stock currently traded for $100 per share. There is a 50% margin requirement on the short sales.

1. Deposit $50000 in the margin account;
2. You borrow the 1000 shares from your broker;
3. You sell the stock for $100 per share;
4. Deposit the sale proceeds in your margin account;

Ignore fees and transaction costs. Assume that the stock does not pay any dividend.

The initial margin (\(t=0\)) can be calculated as follows: \[ \begin{align} \text{Margin} &= \frac{\text{Equity in the account}}{\text{Stock market value}} \\[0.3cm] &= \frac{\text{Sales proceeds + Initial margin - Stock market value}}{\text{Stock market value}} \\[0.3cm] &= \frac{100000 + 50000 - 100000}{100000} \\[0.3cm] &= 50\% \end{align} \]

The margin will change if the stock price changes.

You can change the slider below to see how the margin changes as price changes. The red dot gives you the margin for a current stock price of $.

n_shares = 1000
price_t0 = 100

viewof market_price = Inputs.range([50, 150], {step: 1, label: "Market price", value:100})

stock = d3.range(50, 151).map(price => ({
  price: price,
  margin: ((n_shares * price_t0 + 50000 - n_shares * price) / (n_shares * price) * 100)
}))

// Calculate current margin for the selected price
current_margin = [{
  price: market_price,
  margin: ((n_shares * price_t0 + 50000 - n_shares * market_price) / (n_shares * market_price) * 100)
}]

current_margin_value = current_margin[0].margin.toFixed(2)


// Plot the margin data
Plot.plot({
  caption: `Margin`,
  x: {label: "Stock market price"},
  y: {domain: [0, 200], label:"Margin"},
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([market_price], {stroke: "gray", strokeDasharray: "4"}),
    Plot.lineY(stock, {x: "price", y: "margin"}),
    Plot.ruleY([current_margin[0].margin], {stroke: "gray", strokeDasharray: "4"}),
    Plot.text([`$${market_price}`],
      {
        x: market_price, 
        y: 80,
        dy: 0,
        dx: 14,
        fontSize: 11
    }),
    Plot.text([`${current_margin_value}%`],
      {
        x: 145, 
        y: current_margin[0].margin,
        dy: -10,
        dx: 0,
        fontSize: 11
    }),
    Plot.dot(current_margin, {x: "price", y: "margin", r:5, fill: "red"}),
  ]
})

Margin call

If the maintenance margin is 30%, how far can the stock price increase before a margin call?

\[ \begin{align} \text{Margin} &= \frac{\text{Sales proceeds + Initial margin - Stock market value}}{\text{Stock market value}} \\[0.3cm] &= \frac{100000 + 50000 - n \times P}{n \times P} \\[0.3cm] &= \$115.38 \end{align} \]