Let P be the price of a stock. The broker has an initial margin requirement of m

Question

Let P be the price of a stock. The broker has an initial margin requirement of m0, where 0 < m0 < 1 for shorting the stock. At this requirement, the investor is able to sell short Q units of this stock and does so.

A. What is the investors initial equity E as a function of m0, P, and Q?

B. Now, the price of the stock changes to P, > 0. What is the investor’s equity now? Write your answer in terms of , m0, P and Q.

C. Find the new margin m0 in terms of and m0.

D. Discuss how the sign (positive, negative) of the new margin depends on and m0.

E. Given a maintanence margin, c, what is the smallest , call it such that, given an initial margin m0, whenever price is greater than or equal to P broker will issue a margin call on the short position.

Explanation / Answer

Answer (a)

Price of the stock = P

Initial margin requirement = m0 (where 0<m0<1)

No of units sold Short = Q

Equity E is an amount to be put up by the customer as deposit to enter into the short sale transaction.

Initial Margin can be calculated using the formula

IM = (Initial Market Value of Securities – Amount of Loan) / Initial market value of Securities

      = Customer Equity / Initial Market Value of Securities

Or   Customer Equity = IM * Initial Market value of Securities

Substituting the values from above

Customer Equity E = m0 * (P * Q)

Answer (b)

If price changes by P (where > 0)

Investors Equity changes to E1 = E + m0 * (P * Q)

E1 = m0 * (P*Q) + m0 * (1+ ) * (P*Q)

E1 = (P*Q) [ m0 + m0 * (1+ )]

E1 = m0 * (P*Q) [1+(1+)]

Answer (C)

If price changes by P (where > 0)

Then the Actual Margin can be calculated as

AM = (Initial Market Value of Stock + Initial Margin – Amount of Loan at current rate) / Current Market Value of Stock

AM = (P* Q) + m0* (P*Q) – (P* Q) / P * Q

AM = [(P*Q) * (1+m0) – (1+)* (P*Q)] / P* Q   ==> AM = (P*Q) *[(1+m0) – (1+ )]/(1+ ) * (P*Q)

AM = [(1+m0) – (1+ )]/(1+ )

Answer (D)

If the stock price increases to P where can be either positive or negative ie., the share price increases or decreases. Then the Actual Margin can be calculated using the formula

AM = [(1+m0) – (1+ )]/(1+ )

Any increase in m0 or either positively or negatively will affect the Actual margin in the same direction.

Answer ( E )

If c is the maintenance margin, * is smallest change in price to issue a margin call

Price at which margin call issued can be calculated using the formula

Price at which margin call issued = [Initial Price (1- Initial Margin)]/ (1-maintenance Margin)

Substituting the values

(*) P = [P *(1-m0)]/(1-c) ==> (*) P = P *{(1-m0)/(1-c)} ==> * = (1-m0)/(1-c)

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