Chapter 3: Extra and Questions

1. You put up $10,000 to take a position in a with a price of $20.

(a) If the initial margin rate on the position is 80%, how many shares can you purchase? What dollar amount are you borrowing from the brokerage firm?

Number of Shares = (Dollars invested/Margin Rate)/P0 = ($10,000/0.80)/$20 = 625 shares Amount = $10,000/0.80 – $10,000 = $2,500

Assets Liabilities & Equity Stock = $20 x 625 = $12,5000 Loan = $2,500 Equity = $12,500 - $2,500 = $10,000 [Sanity check: Equity = amount you put up]

(b) You will receive a margin call if the margin rate drops below 30%. Below what price will you receive a margin call? (Assume immediate price change, so you can ignore the and interest on the loan)

Margin Rate = Equity/Stock Value = (Stock Value – Loan)/Stock Value = (Shares x P – Loan)/(Shares x P)

Solve for P and plug in values: P = Loan/[Shares x (1 – Margin Rate)] = $2,500/[625 x (1 – 0.30)] = $5.71

Check Margin rate at P = 5.71: Margin Rate = (Shares x P – Loan)/(Shares x P) = (625 x $5.71 - $2,500)/(625 x $5.71) = $1068.75/$3,568.75 = 30%

(c) Calculate the return on the stock at the margin call price. Calculate the return on your equity at the margin call price. Calculate the ratio of the return on the stock to the return on your equity. How does this ratio compare to the initial margin rate?

Stock Return = $5.71/$20 – 1 = -71.45% Equity Return = $1068.75/$10,000 – 1 = -89.31% Stock Return/Equity Return = (-71.45%)/(-89.28%) = 80% [Sanity check: the ratio equals the initial margin rate]

(d) Assume the stock pays a single annual dividend of $1.00 one year from today. If, at the end of the year, the stock’s price is $15, calculate the total return on the stock over the year.

Stock Return = (P1 + Div)/P0 – 1 = ($15 + 1.00)/$20 – 1 = -20.00% Note that this is equal to the capital gain return plus the : Cap Gain Return = $15/$20 – 1 = -25% Dividend Yield = $1.00/$20 = 5% Overall Stock Return = -25% + 5% = -20%

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(e) If your broker charges 8% on margin , calculate your return on equity (ROE) over the year.

First calculate the new Equity: Assets Liabilities & Equity Stock = $15 x 625 = $9,375 Loan = $2,500(1.08) = $2,700 Div = $1.00 x 625 = $625 Equity = $10,000 - $2,700 = $7,300 Total Assets = $10,000

E quity Return = $7,300/$10,000 – 1 = -2 7%

2. You invest $25,000 and take a SHORT position in a stock with a price of $100.

(a) If the initial margin rate is 50%, how many shares did you short?

Number of Shares = (Dollars invested/Margin Rate)/P0 = ($25,000/0.50)/$100 = 500 shares

Assets Liabilities & Equity Cash from Stock Sale = $100 x 500 = $50,000 Value of Owed = 500 x $100 = $50,000 Margin Cash = $25,000 Equity = $75,000 - $50,000 = $25,000 Total Assets = $75,000

Sanity check: Margin Rate = Equity/Value of Stocks Owed = Equity/(Shares x P) = $25,000/(500 x $100) = 0.50

(b) Assume you will receive a margin call if the margin rate drops below 30%. ABOVE what price will you receive a margin call? (Assume immediate price changes, so you can ignore dividend yield and interest expense)

Margin Rate = Equity/Value of Stocks Owed = (Total Assets – Shares x P)/(Shares x P)

Solve for P and plug in values: P = Total Assets/[Shares x (1 + MR)] = 75,000/[500*(1 + 0.30)] = $115.38

Check Margin rate at P = $115.38: Margin Rate = Equity/Value of Stocks Owed = (Total Assets – Shares x P)/(Shares x P) Margin Rate = ($75,000 – 500 x $115.38)/(500 x $115.38) = $17,310/$57,690 = 30%

(c) Calculate the return on the stock at the margin call price. Calculate you return on equity (ROE) at the margin call price. Calculate the ratio on the return on the stock to the return on your equity. How does this ratio compare to the margin rate?

Stock Return = $115.38/$100 – 1 = +15.38% Equity Return = $17,310/$25,000 – 1 = -30.76%

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Stock Return/Equity Return = 15.38%/-30.76% = -50% Note: the ratio equals the margin rate, with the opposite sign.

(d) Assume the stock pays a single annual dividend of $2.50 one year from today. If, at the end of the year, the stock’s price is $90, calculate the total return on the stock over the year.

Stock Return = (P1 + Div)/P0 – 1 = ($90 + $2.5)/$100 – 1 = $93/$100 – 1 = -7.50%

(e) The broker charges 8% on borrowed shares (calculated on the initial value of the position) and you earn 2% on the cash in the account. Calculate the return on your equity over the year.

First calculate the new equity:

Assets Liabilities & Equity Cash from Stock Sale = $50,000*(1.02) = $51,000 Value of Stocks Owed = 500 x $90 = $45,000 Margin Cash = $25,000*(1.02) = $25,500 Interest on Borrowed Stocks = $50,000*(0.08) = $4,000 Total Assets = $76,500 Dividend Owed = 500 x $2.5 = $1,250 Equity = $76,500 - $45,000 - $4,000 - $1,250 = $26,250

(Note: interest and are listed as “payable” liabilities. Once those are paid, the amount of cash decreases so the equity does not change.)

Equity Return = $26,250/$25,000 – 1 = 5.00% The price dropped 10% and the was levered 2 to 1, but the cost of the short (including paying dividends out) decreased the profit to 5.00%.

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