Investments I HW 2 - Solutions Sailesh Tiwari 26 September, 2008 Problem 1 (BKM Ch3. Ques.3) (a) The Trader buys 300 Internet Dreams shares at $40 per share. Total cost of the shares is 300 $40 = $12000. She had to borrow $4000 from her broker to finance this deal so presumably she had $8000 of her own money in the account and that is what the margin is. Equity in the account V alue of the stock The equity in the account is the amount of her own money that the buyer is putting into ths deal. So, 8000 = 0.667 = 66.7% 12000 (b) If the share price falls to $30 then the value of the stock she holds declines from $12,000 to 300 $30 = $9000. Now what will happen to equity? First note that the amount borrowed from the borrower has accumulated interest for a year. So that $4000 that was originally borrowed is now, with the interest, $4000*(1+0.08) = $4000 * 1.08 = $4320. With the value of the stock at $9000, the stockholders equity has declined to $9000 - $4320 = $4680. So now, 4680 = 0.52 = 52% 9000 1
The margin on her account is still over the maintenance margin requirement of 30% so she will not receive a margin call. (c) Rate of return on his investment? If she sold all her stocks at the end of the year she d get $9000. From the proceeds of this sale, she would have to pay off the amount she borrowed from the broker with the interest. This amount we calculated in (b) to be $4320. So the equity of the trader is $9000 - $4320 = $4680. But the initial investment or the initial equity that the trader started out with was $8000. So, Ending Equity Original Equity Rate of return = Original Equity 4680 8000 = = 0.415 = 41.5% 8000 Problem 2 (BKM Ch. 3. Ques. 7) (a) You have $5000 of your own money. You borrow $5000 from the broker and use the total to buy Telecom stock at $50 per share. How many shares did you buy if you spent all your money? You bought 10000 = 200 shares. Now at the end of the year, the price 50 of each of these shares goes up by 10%. This means the price of the share becomes 50 (1 + 0.1) = $55. The value of your shares becomes 200 55 = $11000. So what is the rate of return? Net gain RR = Initial investment P roceeds from the sale Amount due to the broker Initial equity = Initial equity What is the amount due to the broker? It is the initial borrowed amount plus the 8% interest on it. Or So then, 5000 (1 + 0.08) = $5400 RR = 11000 5400 5000 5000 2 = 600 = 0.12 = 12% 5000
(b) Let P be the number such that if the price of the stock fell up to and below P, you d get a margin call under a margin maintenance requirement of 30%. We know that, Equity in the account V alue of the stock If the price of the stock dropped to P the percentage margin on the account would be, 200 P 5000 = 200 P So you would get a margin call if this percentage margin fell below 0.3. So P can be found by the following equation: 200 P 5000 0.3 = 200 P Re-arrange things a little bit, 60P = 200P 5000 or, 140P = 5000 P = $35.7 Which means the price would have to fall below $35.7 in order for you to get a margin call on this account. Problem 3 (BKM Ch. 3. Ques. 8) (a) You sell short 100 shares at current market price of $50 per share so your account with the broker is credited with $50*100 = $5000. Again, this $5000 is not completely yours yet. At some point in the future, typically at the time you decide to close your short position the broker takes some money from your account to replace the 100 shares that you borrowed to short today. In this question, the value of your short position then is $5000. So if the broker requires a 50% initial margin to have a short position in the first place, then it must be that your account with the broker had at least $2500. So margin required = $2500. 3
(b) When you sell stocks short (just like when you buy put options) you are betting on the price of the stock to go down. So obviously it s not a good thing if the stock price goes up. If your broker has a maintenance margin of 30% on short accounts, this means that if the price of the stock goes up above a certain P, you will receive a margin call. What is P in this case? We have, Equity V alue of Shares Owed = 0.3 2500 + 5000 100P or, = 0.3 100P or, 7500 = 100P + 30P or, P = 7500 130 = 57.69 Problem 4 (BKM Ch. 3. Ques. 17) (a) The initial margin requirement is 50%. This implies that if you borrowed $20,000 to buy Disney shares on margin, you probably had to put in $20,000 of your own equity into the deal. So you bought $40,000 worth of Disney shares. How many shares did that get you? You got 40,000 = 1000. Now, two days later the stock 40 price for Disney fell to $35. What is your percentage margin? Equity in the account V alue of the stock What is the equity in the account? It is (1000*35-20000), that is the value of the stock that you hold less the amount that you owe to your broker. So then, 1000 35 20000 1000 35 = 15000 35000 = 0.4285 = 42.9% 4
Since the percentage margin on your account is still above 35% which is the maintenance margin, you will not receive a margin call. (b) How low can the value of the share fall before you get a margin call? Let s denote with P the price at which the percentage margin on the account exactly equals 35%. Then, 1000 P 20000 0.35 = 1000 P or, 650P = 20000 or, P = 20000 650 = $30.77 So you will not receive a margin call as long as the price of the stock remains above $30.77. Problem 5 (BKM Ch. 3. Ques. 19) Here you have short sales in a bit more realistic framework: you have to deal with dividends and transaction fees (broker commissions)! Jan 1 - You shorted 100 shares of Zenith. March 1 - Zenith pays dividends. April 1 - you cover your short positions or you close your short position. So what happens at each date? Jan 1 - the sale of the shares brings in 100*$14 = $1400 into your account. But the broker charges a commission of $0.5 per transaction, which is $0.50 * 100 = $50. So your account is effectively credited $1400 - $50 = $1350. On March 1, total dividends paid on the shares you shorted is debited from your account. Zenith paid $2 per share in dividends so the total dividend was $200. So you have $200 taken away from your account. What you have left is $1350 - $200 = $1150. Then, on April 1, you close your position. This entails buying the 100 stocks that you shorted at the current price and replacing them in the the owners account. So you spent $9*100 = $900 on this purchase. Additionally, you also had to pay the broker s commissions which was, 5
again, $0.50 per share or a total of $0.5*100 = $50. So on April 1, (900 + 50) = $950 was debited from your account. Which means the value of your account at that date, assuming there was nothing else in there before, is $1150 - $950 = $200. 6