# Homework Margin Purchases. Dr. Patrick Toche

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1 Homework Margin Purchases Dr. Patrick Toche

2 A dagger indicates a possibly more challenging question. Maintenance Margin 1. You are bullish on Telecom stock. The current market price is \$50 per share, and you have \$5,000 of your own to invest. You borrow an additional \$5,000 from your broker at an interest rate of 8% per year and invest \$10,000 in the stock. (a) What will be your rate of return if the price of Telecom stock goes up by 10% over the next year? (Ignore any expected dividend payments) You buy 200 shares of Telecom for \$10, 000. These shares increase in value by 10%, that is \$1,000. You pay interest of 0.08 \$5,000 = \$400. The rate of return is: \$1,000 \$400 \$5, 000 = 0.12 = 12% (b) How far does the price of Telecom stock have to fall for you to get a margin call if the maintenance margin is 30%? Assume the price fall happens immediately. The value of the 200 shares is 200P. Equity is 200P \$5,000. You will receive a margin call when: 200P \$5, P = 0.30 P \$ You ve borrowed \$20,000 on margin to buy shares in Disney, which is now selling at \$40 per share. Your account starts at the initial margin requirement of 50%. The maintenance margin is 35%. Two days later, the stock price falls to \$35 per share. (a) Will you receive a margin call? You will not receive a margin call. You borrowed \$20, 000 and with 2

3 another \$20,000 of your own equity you bought 1,000 shares of Disney at \$40 per share. At \$35 per share, the market value of the stock is \$35,000, your equity is \$15,000, and the percentage margin is \$15,000/\$35,000 = 42.9%. Your percentage margin exceeds the required maintenance margin. (b) How low can the price of Disney shares fall before you receive a margin call? You will receive a margin call when: 1,000P \$20,000 1, 000P = 0.35 P \$ Suppose that Intel currently sells at \$40 per share. You buy 500 shares, using \$15,000 of your own money and borrowing the remainder of the purchase price from your broker. The rate on the margin loan is 8%. (a) What is the percentage change in the net worth on your brokerage account if the stock price changes overnight to: (i) \$44; (ii) \$40; (iii) \$36? The total cost of the purchase is 500 \$40 = \$20,000. You borrow \$500 from your broker and invest \$15,000 of your own funds. Your margin account starts out with net worth of \$15,000. We can ignore the interest cost for an overnight loan, 0.08/ %. (i) If the stock is selling at \$44, that is \$4 higher than the initial price, your account s net worth rises by: (500 \$44 \$5,000) (500 \$40 \$5,000) 500 \$44 \$5, (\$44 \$40) = \$15, 000 = \$ + 2,000 \$15, % 3

4 Note that the \$5,000 term cancels out for simplicity, we omit it in subsequent calculations. (ii) If the stock is selling at \$40, that is at the same price as the initial price, your account s net worth is unchanged: 500 (\$40 \$40) \$15, 000 = 0% (iii) If the stock is selling at \$36, that is \$4 lower than the initial price, your account s net worth falls by: 500 (\$36 \$40) \$15, 000 = \$ 2,000 \$15, % (b) Suppose the maintenance margin is 25%. How low can the price fall before you get a margin call? The stock value is 500P. Equity is 500P \$5,000. If the maintenance margin is 25%, you will receive a margin call if the price falls to: 500P \$5, P P \$5, P P \$5, P \$13.33 (c) Suppose you finance the initial purchase with as little as \$10,000 of your own money instead. How low can the price fall before you get a margin call? The stock value is 500P. Equity is 500P \$10,000. You will receive a 4

5 margin call if the price falls to: 500P \$10, P P \$10, P P \$10, P \$26.67 With less equity in the account, you are more vulnerable to a margin call. (d) Suppose again you invest \$15,000 of your own money. What is the rate of return on your margined position after one year if the stock price changes to: (i) \$44; (ii) \$40; (iii) \$36? For an investment horizon of one year, the borrowing cost of 0.08 \$5, 000 = \$400 must be deducted. (i) If the stock is selling at \$44, your account s rate of return is: 500 (\$44 \$40) \$400 \$15, 000 = \$ + 1,600 \$15, % (ii) If the stock is selling at \$40, your account s rate of return is: 500 (\$40 \$40) \$400 \$15, 000 = \$ 400 \$15, % (iii) If the stock is selling at \$36, your account s rate of return is: 500 (\$36 \$40) \$400 \$15, 000 = \$ 2,400 \$15,000 = 16% (e) Continue to assume that one year has passed. How low can the price fall before you get a margin call? 5

6 The stock value is 500P. Equity is 500P \$5,000 \$400. You will receive a margin call if the price falls to: 500P \$5, P P \$5, P P \$5, P \$ You deposit \$100,000 of your own capital in a brokered account and borrow an additional \$100, 000 from your broker. You invest everything in a single stock. The current market price of the stock is \$100 per share. Your broker charges an interest rate of r % per year. Your broker s maintenance margin is 30%. You expect the stock to pay a semi-annual dividend payment of \$5 per share. Different exam scripts had different values of the parameters, namely: r = 10% and r = 20%. (a) What would be your annual rate of return if the stock price rose by 10% over the next year? You buy 2,000 shares for a total investment of \$200,000. These shares increase in value by 10%, so the capital gain is \$20,000. You receive two semiannual dividend payments of \$10 per share, for a total of \$20,000. With a rate of interest of 10%, your total interest cost is 0.10 \$100,000 = \$10,000. Your net income is: \$20,000 + \$20,000 \$10,000 = \$30,000 6

7 Your rate of return is: \$30, 000 = 0.30 = +30% \$100,000 With a rate of interest of 20%, your total interest cost is 0.20 \$100,000 = \$20,000. Your net income is: \$20,000 + \$20,000 \$20,000 = \$20,000 Your rate of return is: \$20, 000 = 0.20 = +20% \$100,000 (b) What would be your annual rate of return if the stock price fell by 10% over the next year? You buy 2,000 shares for a total investment of \$200,000. These shares decrease in value by 10%, so the capital loss is \$20,000. You receive two semiannual dividend payments of \$10 per share, for a total of \$20,000. With a rate of interest of 10%, your total interest cost is 0.10 \$100,000 = \$10,000. Your net income is: \$20,000 \$20,000 \$10,000 = \$10,000 Your rate of return is: \$10, 000 \$100,000 = 0.10 = 10% With a rate of interest of 20%, your total interest cost is 0.20 \$100,000 = 7

8 \$20,000. Your net income is: \$20,000 \$20,000 \$20,000 = \$20,000 Your rate of return is: \$20, 000 \$100,000 = 0.20 = 20% (c) You are bullish on the stock. You believe the probability of a 10% rise is 60% and the probability of a 10% fall is 40%. What is your expected rate of return? Is it positive? With a rate of interest of 10%, your expected rate of return is: 60% (+\$30,000) + 40% ( \$10,000) \$100, 000 \$18,000 \$4,000 = = \$14,000 = 0.14 = +14% \$100, 000 \$100,000 The expected rate of return is positive. With a rate of interest of 20%, your expected rate of return is: 60% (+\$20,000) + 40% ( \$20,000) \$100, 000 \$12,000 \$8,000 = = \$4,000 = 0.04 = +4% \$100, 000 \$100,000 The expected rate of return is positive. (d) The day after you purchase the stock, the price falls to \$70. Will you receive a margin call? The value of your 2,000 shares is now 2,000 \$70 = \$140,000. You have not received any dividend payments yet nor have had to pay interest. Equity 8

9 is therefore: \$140,000 \$100,000 \$160, 000 = \$40,000 \$160,000 = 25% The day after your margin has dropped to 25%, below the maintenance margin of 30%: you will receive a margin call! 5. An investor buys \$11,000 worth of stock at \$100 per share, contributing \$7,000 and borrowing the remaining \$4,000 from a broker. (a) Calculate the initial margin percentage. The percentage margin immediately after the margin loan is arranged: margin = value of equity value of securities = \$7,000 \$11, % (b) The price falls to \$60. Calculate the margin percentage after the fall. The investor is long 110 shares (initial stock purchase \$11,000 at \$100 per share). The percentage margin after the price has fallen: margin = value of equity 110 \$60 \$4,000 = = \$2,600 value of securities 110 \$60 \$6, % (c) The broker s maintenance margin is 30%. Do you get a margin call? At about 39% the percentage margin is still above the broker s maintenance margin of 30%, so you do not get a margin call. 6. An investor buys \$15,000 worth of stock at \$50 per share, contributing \$9,000 and borrowing the remaining \$6,000 from a broker. 9

10 (a) Calculate the initial margin percentage. The percentage margin immediately after the margin loan is arranged: margin = value of equity value of securities = \$9,000 \$15,000 = 60% (b) The price falls to \$40. Calculate the margin percentage after the fall. The investor is long 300 shares (initial stock purchase \$15, 000 at \$50 per share). The percentage margin after the price has fallen to \$40: margin = value of equity 300 \$40 \$6,000 = = \$6,000 value of securities 300 \$40 \$12,000 = 50% (c) The price falls to \$30. The broker s maintenance margin is 35%. Do you get a margin call? The percentage margin after the price has fallen to \$30: margin = value of equity 300 \$30 \$6,000 = = \$3,000 value of securities 300 \$30 \$9, % The percentage margin is less than the maintenance margin, 0.33 < 0.35, so you will get a margin call before the price falls as low as \$ The current market price of Telecom stock is \$50 per share. You have \$5,000 of your own to invest. You borrow an additional \$5,000 from your broker at an interest rate of r% per year. Altogether you invest a total of \$10,000 in the stock. Telecom stock pays no dividend. Your broker s maintenance margin is 40%. Different exam scripts had different values of the parameters. Only one set of parameters is reported below: the case where the interest rate is equal to 10%. 10

11 (i) What would be your annual rate of return if the stock price rose by 10% over the next year? You buy 200 shares of Telecom for \$10, 000. These shares increase in value by 10%, that is by \$1,000. You pay interest of 0.10 \$5,000 = \$500. The rate of return is: \$1,000 \$500 \$5, 000 = 0.10 = +10% (ii) What would be your annual rate of return if the stock price fell by 10% over the next year? You buy 200 shares of Telecom for \$10,000. These shares fall in value by 10%, that is by \$1,000. You pay interest of 0.10 \$5,000 = \$500. The rate of return is: \$1,000 \$500 \$5, 000 = 0.30 = 30% (iii) You are bullish on Telecom stock. You believe the probability of a 10% rise is 50% and the probability of a 10% fall is 25%. What is your expected rate of return? Is it positive? The expected rate of return is negative: 50% (+\$1,000) + 25% ( \$1,000) \$500 \$5, 000 \$500 \$250 \$500 = = \$250 = 0.05 = 5% \$5, 000 \$5,000 (iv) The day after you purchase the stock, the price falls by \$10. Will you receive a 11

12 margin call? The value of your 200 shares is now 200 \$40 = \$8,000. Equity is therefore reduced to \$8,000 \$5,000 = \$3,000. But 3/8 < 40%. You will receive a margin call! \$8,000 \$5,000 \$8, 000 = = 37.5% < 40% 8. The current market price of Telecom stock is \$50 per share. You have \$5,000 of your own to invest. You borrow an additional \$5,000 from your broker at an interest rate of r% per year. Altogether you invest a total of \$10,000 in the stock. Telecom stock pays no dividend. Your broker s maintenance margin is 40%. Different exam scripts had different values of the parameters. Only one set of parameters is reported below: the case where the interest rate is equal to 10%. (i) What would be your annual rate of return if the stock price rose by 10% over the next year? You buy 200 shares of Telecom for \$10, 000. These shares increase in value by 10%, that is by \$1,000. You pay interest of 0.10 \$5,000 = \$500. The rate of return is: \$1,000 \$500 \$5, 000 = 0.10 = +10% (ii) What would be your annual rate of return if the stock price fell by 10% over the next year? You buy 200 shares of Telecom for \$10,000. These shares fall in value 12

13 by 10%, that is by \$1,000. You pay interest of 0.10 \$5,000 = \$500. The rate of return is: \$1,000 \$500 \$5, 000 = 0.30 = 30% (iii) You are bullish on Telecom stock. You believe the probability of a 10% rise is 50% and the probability of a 10% fall is 25%. What is your expected rate of return? Is it positive? The expected rate of return is negative: 50% (+\$1,000) + 25% ( \$1,000) \$500 \$5, 000 \$500 \$250 \$500 = = \$250 = 0.05 = 5% \$5, 000 \$5,000 (iv) The day after you purchase the stock, the price falls to \$10. Will you receive a margin call? The value of your 200 shares is now 200 \$10 = \$2,000. Equity is therefore reduced to \$2,000 \$5,000 = \$3,000. The value of equity is negative, you will certainly receive a margin call! 9. An investor buys \$150,000 of stock at \$50 per share, contributing \$90,000 and borrowing \$60,000 from a broker. The broker s annual interest rate is 10% and maintenance margin 40%. (a) Calculate the initial margin percentage. The percentage margin immediately after the margin loan is arranged: margin = value of equity value of securities = \$90,000 \$150,000 = 60% 13

14 (b) The investor is bullish on the stock and believes there is an 80% probability that the stock price will rise overnight by 10% and a 20% probability that it will fall by 10%. What is the expected rate of return overnight? A 10% rise from \$50 takes the price to \$55. A 10% fall from \$50 takes the price to \$45. The investor is long 3,000 shares: \$150, 000 \$50 = 3,000 where \$150,000 is the initial value of the stock purchase and \$50 is the initial stock price. The expected rate of return is therefore: 0.8 (\$55 \$50) 3, (\$45 \$50) 3,000 \$90, 000 (\$4 \$1) 3,000 = = \$9,000 = 0.1 = 10% \$90, 000 \$90,000 An annual rate corresponding to this overnight return of 0.1 is: ( ) = 26,974,702, ,697,470,226,776%!! where 252 is the typical number of trading days in a year. Clearly, it is unlikely to make sense to extrapolate a return obtained overnight! The calculation is presented as a pure exercise. (c) The stock price falls overnight to \$40. Calculate the account s margin percentage after the fall. The percentage margin after the price falls to \$40: margin = value of equity 3,000 \$40 \$60,000 = = \$60,000 value of securities 3,000 \$40 \$120,000 = 50% 14

15 Since the margin percentage is above the maintenance level, the investor does not get a margin call. (d) One year later, the stock price is still \$40. However, the stock has paid quarterly dividends of \$2 per share. What is the annual rate of return? The investor bought 3, 000 shares for a total value of \$150, 000. The value of these shares has fallen to \$120,000. The investor has received cash dividend payments of 4 \$2 3,000 = \$24,000. The investor must pay interest on the loan of 0.1 \$60,000 = \$6,000. The rate of return is therefore: (\$40 \$ \$2) 3, \$60,000 \$90, 000 \$ 2 3, \$60,000 = = \$ 12,000 \$90, 000 \$90, % The rate of return is negative. Since the period is one year, this is an annual rate of return. The loan cost \$6,000 in interest payments. The stock paid \$24,000 in cash dividends. Thus, the negative return is caused entirely by a capital loss, mitigated somewhat by generous dividend payments. 10. An investor buys 3, 000 shares of stock at \$50 per share, contributing 60% of the total and borrowing the rest from a broker. The investor holds the stock long for one year. The stock has paid quarterly dividends of \$2 per share. The broker s annual interest rate is 10% and maintenance margin 40%. The stock price has risen to \$60. (a) Calculate the account s margin percentage at the year end. The investor has contributed \$90,000 and borrowed \$60,000, since: 3,000 \$50 = \$150,000 = 0.6 \$150,000 + \$60,000 = \$90,000 + \$60,

16 The value of the securities at the end of the year contribute to the equity: 3,000 \$60 = \$180,000. The value of the \$60,000 loan must be deducted from the equity. The investor must pay 10% interest on the loan of \$60,000 for a total of: 0.1 \$60,000 = \$6,000. The investor has received 4 quarterly cash dividend payments of \$2 per share for a total of: 4 \$2 3,000 = \$24,000. Therefore, the percentage margin after the price rises to \$60 is: margin = value of equity (\$180,000 \$60,000) \$6,000 + \$24,000 = value of securities \$180, 000 = \$138,000 = % \$180,000 This is above the maintenance margin of 40% and above the initial margin of 60%, which was expected since the price has risen. (b) Calculate the annual rate of return at the year end. The investor contributed \$90, 000 and the rate of return must therefore be calculated as a ratio of that contribution. The value of the investment increased from \$150, 000 to \$180, 000 (the capital gain ). The investor received cash dividend payments of \$24,000. The investor must pay interest on the loan of \$6,

17 The rate of return is therefore: (\$180,000 \$150,000) \$6,000 + \$24,000 rate of return = \$90, 000 = \$48,000 = % \$90,000 The rate of return is obviously positive. Since the period is one year, this is an annual rate of return. Note that the margin and rate of return expressions have different numerators and denominators. Consider the denominators: in the margin calculation, we have the final value of the securities (\$180,000) while in the return calculation we have the value of the initial equity (\$90,000). Consider the numerators: in the margin calculation we deduct the value of the loan (\$60,000) while in the return calculation we deduct the initial value of the securities (\$150,000). 11. An investor buys 3, 000 shares of stock at \$50 per share, contributing 60% of the total and borrowing the rest from a broker. The investor holds the stock long for one year. The stock has paid quarterly dividends of \$2 per share. The broker s annual interest rate is 10% and maintenance margin 40%. The stock price is back to \$50 at the year end. (a) Calculate the account s margin percentage at the year end. The investor has contributed \$90,000 and borrowed \$60,000, since: 3,000 \$50 = \$150,000 = 0.6 \$150,000 + \$60,000 = \$90,000 + \$60,

18 The value of the securities at the end of the year contribute to the equity: 3,000 \$50 = \$150,000. The value of the \$60,000 loan must be deducted from the equity. The investor must pay 10% interest on the loan of \$60,000 for a total of: 0.1 \$60,000 = \$6,000. The investor has received 4 quarterly cash dividend payments of \$2 per share for a total of: 4 \$2 3,000 = \$24,000. Therefore, the percentage margin after the price rises to \$60 is: margin = value of equity (\$150,000 \$60,000) \$6,000 + \$24,000 = value of securities \$150, 000 = \$108,000 = 0.72 = 72% \$150,000 This is above the maintenance margin of 40% and above the initial margin of 60%. This is because the cash dividend payments exceed the interest cost. (b) Calculate the annual rate of return at the year end. The investor contributed \$90, 000 and the rate of return must therefore be calculated as a ratio of that contribution. The value of the investment has remained unchanged at \$150,000 (there is no capital gain ). The investor received cash dividend payments of \$24,000. The investor must pay interest on the loan of \$6,

19 The rate of return is therefore: (\$150,000 \$150,000) \$6,000 + \$24,000 rate of return = \$90, 000 = \$18,000 = 0.2 = 20% \$90,000 The rate of return is positive. Since the period is one year, this is an annual rate of return. Note that the margin and rate of return expressions have different numerators and denominators. Consider the denominators: in the margin calculation, we have the final value of the securities (\$150,000) while in the return calculation we have the value of the initial equity (\$90,000). Consider the numerators: in the margin calculation we deduct the value of the loan (\$60,000) while in the return calculation we deduct the initial value of the securities (\$150,000). 19

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