Buying on margin

Buying on margin is a way to buy stocks with borrowed money. You’ll borrow from your broker and the stocks that you buy will be deposited in your margin account as collateral.

There is a maximum amount of money that you can borrow. In many countries this maximum is regulated. For instance in the USA the maximum is 50%.

To reduce the default risk the broker will require a maintenance margin and when the margin drops below the maintenance margin, the investor will receive a margin call (notification that you must deposit additional funds).

Example

n_shares = 100
d = 0.08

Suppose an investor has $6000 and wants to buy shares of a stock trading at $100 a share. They contact the broker to arrange a call loan.

The broker will ask the investor to provide $6,000 as initial margin,and will give a loan of $4000. With the $10000 the investor can buy those shares, that will be deposited in the margin account as a guarantee for the loan.

Ignore fees and transaction costs. Assume that the stock does not pay any dividend and the interest rate on the loan is % per annum.

The initial margin ($t=0$) can be calculated as follows: \[ \begin{align} \text{Margin}_{t=0} &= \frac{\text{Equity in the account}}{\text{Stock market value}} \\[0.3cm] &= \frac{\text{Stock market value} - \text{Borrowed amount}}{\text{Stock market value}} \\[0.3cm] &= \frac{10000 - 4000}{10000} \\[0.3cm] &= 60\% \end{align} \]

The margin will change if the stock price changes and by the passage of time. Even if the stock price remains constant, the margin will change over time as interest accrues.

You can change the sliders below to see how the margin changes as time passes, and price changes. The red dot gives you the margin at time $t$= for a current stock price of $.

viewof time = Inputs.range([0, 1], {step: 0.1, label: "Time (t)", value: 0})
viewof market_price = Inputs.range([50, 150], {step: 1, label: `Market price at time (t)`, value:100})

// Calculate stock data
stock = {
  let data = [];
  for (let price = 50; price <= 150; price++) {
    let margin = (n_shares * price - 4000 * Math.pow(1 + d, time)) / (n_shares * price) * 100;
    data.push({price: price, margin: margin});
  }
  return data;
}

// Calculate current margin
current_margin = {
  let margin = (n_shares * market_price - 4000 * Math.pow(1 + d, time)) / (n_shares * market_price) * 100;
  return [{price: market_price, margin: margin}];
}

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

Plot.plot({
  caption: `Margin at time t=${time}`,
  x: {label: "Stock market price"},
  y: {domain: [10, 85], label:"Margin"},
  marks: [
    Plot.ruleY([10]),
    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"}),
  ]
})

The margin can be calculated as follows: \[ \begin{align} \text{Margin}_{t} &= \frac{n \times P - 4000 \times (1+d)^t}{n \times P} \\[0.3cm] \end{align} \]

Where:

$t$ is
$n$ is the number of shares you have in the margin account:
$P$ is the current market price of the stock:
$d$ is the loan interest rate: %

which results in a margin of %.

Margin call

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

Assume the price drop is instantaneous, i.e. it occurred immediatly after the stock was bought.

\[ \begin{align} \text{Margin} &= \frac{\text{Stock market value} - \text{Borrowed amount}}{\text{Stock market value}} \\[0.3cm] 30\% &= \frac{n \times P - 4000}{n \times P} \\[0.3cm] P &= \$57.14 \end{align} \]