Chapter 3, Problem 25C

Summary Introduction

(A)

Initial Margin:

Sometimes, securities are bought with initial margin. It is the percentage of purchase price of stock that the investor has to pay with his own cash. 50% of initial margin means, the investor has to pay 50% or more of the stock price up front and remaining amounts can be taken as a loan.

Short sale:

In case of short sale, the investor borrows the shares of a stock and first sells them. Later on, the investor has to buy the shares of the same stock to replace to what was borrowed initially. Short sale happens when the investor predicts that stock price decreases in future. So, they earn money by the difference of buying and selling price of the stock. Here, investor has to give dividends (if any dividends distributed) to the broker with whom he has borrowed the shares of a stock.

Maintenance margin:

It is the minimum equity amount that must be maintained in a margin account. If the stock value falls below the equity value, then the value of stock provides no longer the enough collateral to cover the loan from the broker. In order to protect it, the broker asks to set the maintenance margin.

Margin call:

If the percentage of the margin falls below the maintenance level, the broker will issue a call margin. When a call margin is issued it is required by the investor has to add new cash to margin account.

To compute:

The rate of return for the given stock after one year of short sale, assuming no dividends paid, when

1. The stock price is $44.
2. The stock price is $40.
3. The stock price is $36.

Answer to Problem 25C

1. The rate of return on short position is -13.33%.
2. The rate of return on short position is 0%.
3. The rate of return on short position is 13.33%.

Explanation of Solution

Total stock price is $\$20,000\left(=\$40\times 500\right)$ where per stock price is $40 and 500 shares of stock have to be purchased. Initial margin requirement is $15,000, which should be given to broker to establish the margin account. So, $15,000 is the equity value.

Rate of return should be found on the equity value. In case of short sale, profit is the initial price less ending price less dividends, if any distributed. Here no dividends are paid.

i.

Given:

Initial price= $40

Ending price = $44

Number of shares=500

Equity value=$15,000

Dividends=$0

Calculation:

  $\begin{array}{c}Rateofreturnonshortposition=\frac{\left(Initialprice-Endingprice-Dividends\right)\times No.ofshares}{Equityvalue}\times 100\%\\ =\frac{\left(\$40-\$44-\$0\right)\times 500}{\$15,000}\times 100\%\\ =-13.33\%\end{array}$

ii.

Given:

Initial price= $40

Ending price = $40

Number of shares=500

Equity value=$15,000

Dividends=$0

Calculation:

  $\begin{array}{c}Rateofreturnonshortposition=\frac{\left(Initialprice-Endingprice-Dividends\right)\times No.ofshares}{Equityvalue}\times 100\%\\ =\frac{\left(\$40-\$40-\$0\right)\times 500}{\$15,000}\times 100\%\\ =0\%\end{array}$

iii.

Given:

Initial price= $40

Ending price = $36

Number of shares=500

Equity value=$15,000

Dividends=$0

Calculation:

  $\begin{array}{c}Rateofreturnonshortposition=\frac{\left(Initialprice-Endingprice-Dividends\right)\times No.ofshares}{Equityvalue}\times 100\%\\ =\frac{\left(\$40-\$36-\$0\right)\times 500}{\$15,000}\times 100\%\\ =13.33\%\end{array}$

Summary Introduction

(B)

Initial Margin:

Sometimes, securities are bought with initial margin. It is the percentage of purchase price of stock that the investor has to pay with his own cash. 50% of initial margin means, the investor has to pay 50% or more of the stock price up front and remaining amounts can be taken as a loan.

Short sale:

In case of short sale, the investor borrows the shares of a stock and first sells them. Later on, the investor has to buy the shares of the same stock to replace to what was borrowed initially. Short sale happens when the investor predicts that stock price decreases in future. So, they earn money by the difference of buying and selling price of the stock. Here, investor has to give dividends (if any dividends distributed) to the broker with whom he has borrowed the shares of a stock.

Maintenance margin:

It is the minimum equity amount that must be maintained in a margin account. If the stock value falls below the equity value, then the value of stock provides no longer the enough collateral to cover the loan from the broker. In order to protect it, the broker asks to set the maintenance margin.

Margin call:

If the percentage of the margin falls below the maintenance level, the broker will issue a call margin. When a call margin is issued it is required by the investor has to add new cash to margin account.

To compute:

The price of stock below of which the investor would get a margin call

Answer to Problem 25C

If the stock price falls below $40 per share, the investor would get a margin call.

Explanation of Solution

Here maintenance margin is 25%. Let P be the stock price. The value of stock at P price for 500 shares will be 500P. And the equity in the margin account will be $500P-\$15,000$ . Here the percentage margin is $\left(\frac{500P-\$15,000}{500P}\right)$ . The price at which percentage margin equals to the maintenance margin of 25% is determined by solving the following equation for P.

  $\begin{array}{l}\frac{500P-\$15,000}{500P}=0.25\\ 500P-\left(500\times 0.25\right)P=\$15,000\\ P=\$40\end{array}$

So, if the stock price falls below $40 per share, the investor would get a margin call.

Summary Introduction

(C)

Initial Margin:

Sometimes, securities are bought with initial margin. It is the percentage of purchase price of stock that the investor has to pay with his own cash. 50% of initial margin means, the investor has to pay 50% or more of the stock price up front and remaining amounts can be taken as a loan.

Short sale:

In case of short sale, the investor borrows the shares of a stock and first sells them. Later on, the investor has to buy the shares of the same stock to replace to what was borrowed initially. Short sale happens when the investor predicts that stock price decreases in future. So, they earn money by the difference of buying and selling price of the stock. Here, investor has to give dividends (if any dividends distributed) to the broker with whom he has borrowed the shares of a stock.

Maintenance margin:

It is the minimum equity amount that must be maintained in a margin account. If the stock value falls below the equity value, then the value of stock provides no longer the enough collateral to cover the loan from the broker. In order to protect it, the broker asks to set the maintenance margin.

Margin call:

If the percentage of the margin falls below the maintenance level, the broker will issue a call margin. When a call margin is issued it is required by the investor has to add new cash to margin account.

Redo work of part a.

To compute:

The rate of return for the given stock after one year of short sale, assuming $1 dividends paid, when

1. The stock price is $44.
2. The stock price is $40
3. The stock price is $36

Redo work of part b.

To compute:

The price of stock below of which the investor would get a margin call

Answer to Problem 25C

Part a

1. The rate of return on short position is -16.67%.
2. The rate of return on short position is -3.33%.
3. The rate of return on short position is 10%.

Part b

If the stock price falls below $41.33 per share, the investor would get a margin call.

Explanation of Solution

Part a

Total stock price is $\$20,000\left(=\$40\times 500\right)$ where per stock price is $40 and 500 shares of stock have to be purchased. Initial margin requirement is $15,000, which should be given to broker to establish the margin account. So, $15,000 is the equity value.

Rate of return should be found on the equity value. In case of short sale, profit is the initial price less ending price less dividends, if any distributed.

i.

Given:

Initial price= $40

Ending price = $44

Number of shares=500

Equity value=$15,000

Dividends=$1per share

Calculation:

  $\begin{array}{c}Rateofreturnonshortposition=\frac{\left(Initialprice-Endingprice-Dividends\right)\times No.ofshares}{Equityvalue}\times 100\%\\ =\frac{\left(\$40-\$44-\$1\right)\times 500}{\$15,000}\times 100\%\\ =-16.67\%\end{array}$

ii.

Given:

Initial price= $40

Ending price = $40

Number of shares=500

Equity value=$15,000

Dividends=$1 per share

Calculation:

  $\begin{array}{c}Rateofreturnonshortposition=\frac{\left(Initialprice-Endingprice-Dividends\right)\times No.ofshares}{Equityvalue}\times 100\%\\ =\frac{\left(\$40-\$40-\$1\right)\times 500}{\$15,000}\times 100\%\\ =-3.33\%\end{array}$

iii.

Given:

Initial price= $40

Ending price = $36

Number of shares=500

Equity value=$15,000

Dividends=$1 per share

Calculation:

  $\begin{array}{c}Rateofreturnonshortposition=\frac{\left(Initialprice-Endingprice-Dividends\right)\times No.ofshares}{Equityvalue}\times 100\%\\ =\frac{\left(\$40-\$36-\$1\right)\times 500}{\$15,000}\times 100\%\\ =10\%\end{array}$

Part b

Calculation:

Here maintenance margin is 25% and dividend is $1 per share. Let P be the stock price. The value of stock at P price for 500 shares will be 500P. And the equity in the margin account will be $500P-\$15,000-\$500$ . Here the percentage margin is $\left(\frac{500P-\$15,500}{500P}\right)$ . The price at which percentage margin equals to the maintenance margin of 25% is determined by solving the following equation for P.

  $\begin{array}{l}\frac{500P-\$15,500}{500P}=0.25\\ 500P-\left(500\times 0.25\right)P=\$15,500\\ P=\$41.33\end{array}$

So, if the stock price falls below $41.33 per share, the investor would get a margin call.