Question

a)​What is a margin call? To what level should the trader top up her/his account once receiving a margin call?

b)​An investor shorts 100 shares of a stock when the share price is $50 and closes out the position six months later when the share price is $43. The stock pays a dividend of $3 per share during the six months. Does the investor gain or lose? How much does the investor gain/lose?

c)​The spot price of an asset that pays no dividend is $30 and the risk-free rate for all maturities (with continuous compounding) is 10% per annum. What is the three-year forward price?

Answer 1

a)​What is a margin call?

A Margin Call occurs when the value of the investor’s margin account drops and fails to meet the account's maintenance margin requirement. An investor will need to sell positions or deposit funds or securities to meet the margin call. If the investor fails to cover the margin call within 3 trading days, Firstrade will have to liquidate their positions to meet the margin call.

To what level should the trader top up her/his account once receiving a margin call?

This can be explained through an example-

Suppose you have $20,000 worth of securities bought using $10,000 borrowed and $10,000 in cash. When the margin requirement is 30% and the value of the securities drop by 40% to $12,000, since the amount you borrowed from your broker stays at $10,000, your own equity becomes $2,000 which is lower than the 30% minimum margin requirement.

$2,000 / $12,000 = 16.6% < 30%

A margin call occurs when the percentage of the equity in the account drops below the maintenance margin requirement.

$12,000*30% = $3600 → amount of equity you were required to maintain.
$3600 - $2000 = $1600 → You will have a $1,600 margin call.

When a Margin Call occurs, you may either deposit funds or liquidate part of the positions you purchased on margin to cover the margin call.

By depositing funds you decrease the amount of margin and increase your equity. When you deposit $1,600 of cash into your account, your new account balance consists of $3,600 of cash and $8,400 of margin.
$3,600/($3,600 + $8,400) = 30% → reached margin requirement.

By selling stocks, you decrease the amount of margin, therefore increase the percentage of the equity.

Below is the calculation:
X = the amount of stocks you should sell to cover the call.
[($10,000 - X) + $2,000] * 0.3 = $2,000
($12,000 - X) * 0.3 = $2,000
$3,600 - 0.3X = $2,000
$1,600 = 0.3X
X = $1,600/0.3 = $5,333.3 → reached margin requirement.

In general, if you would like to deposit funds, the amount has to be equal to the margin call amount. If you choose to liquidate your stocks to cover the call, the amount you have to sell should be equal to the margin call amount divided by the minimum maintenance requirement.

$1600/30% = $5333.3
→ To maintain the 30% minimum margin requirement, you will need to either sell $5,333.3 worth of securities or deposit $1,600 worth of cash within 3 trading days, or Firstrade must liquidate your positions.

b)​An investor shorts 100 shares of a stock when the share price is $50 and closes out the position six months later when the share price is $43. The stock pays a dividend of $3 per share during the six months. Does the investor gain or lose? How much does the investor gain/lose?

The investor will gain.

The investor gains $7 per share because he or she sells at $50 and buys at $43. However, the investor has to pay the $3 per share dividend. The net profit is therefore 7−3 or $4 per share. 100 shares are involved. The total gain is therefore $400.

c)​The spot price of an asset that pays no dividend is $30 and the risk-free rate for all maturities (with continuous compounding) is 10% per annum. What is the three-year forward price?

Given information:

• Stock spot price = $43
• Risk free rate of interest Rf = 10% per annum with continuous compounding
• Time to maturity = 3 years

The 3-year forward price is the spot price grossed up for 3 years at the risk-free rate. It is 30e^(0.1×3) =$40.50.