VIEWS: 11 PAGES: 6 POSTED ON: 8/9/2012
FNCE 4040 Spring 2012 NAME__________________ David M. Gross Ph.D. Midterm 2 with Answers Write your answers in the spaces below. Show your work to receive partial credit. 1. (15 Points) A Xylophone company and a Yogurt company have been offered the following annualized borrowing rates. Company Fixed Float Xylophone 3.00% LIBOR + 100 Yogurt 4.00% LIBOR + 150 a) What must be the desired net borrowing structures for each company in order for there to be a gain from a swap? WHY? X’s comparative advantage is in fixed rates: 4.00 – 3.00 = 1.00 Y’s comparative advantage is in floating rates: 150 - 100 = 50 Gain from a swap if X prefers a floating rate and Y prefers a fixed rate b) Calculate the total potential gain from a swap. Fix diff: 4.00% - 3.00% = 1.00% Or look at total borrowing costs: Float diff: (LIBOR + 150) – (LIBOR + 100) = 50 bps Preferred: 4.00% + LIBOR + 100 = LIBOR + 500 Difference = 100 - 50 bps = 50 bps Alternative: 3.00% + LIBOR + 150 = LIBOR + 450 Preferred terms cost 50 bps more c) Design a swap with a financial institution in which the FI gets 6 bps of the gain and the remainder of the gain is split equally between the two companies. On the diagram below show all the percentage cash flow. Calculate the net “percentage cash flows” for each entity. Gains: (50 bps – 6 bps)/2 = 22 bps X’s net = LIBOR + 100 - 22 = LIBOR + 78 Y’s net 4.00% - 0.22% = 3.78% LIBOR LIBOR LIBOR + 150 X FI Y 3.00% 2.22% 2.28% X = -3.00% - LIBOR + 2.22% = -(LIBOR + 78) Y = -(LIBOR + 150) + LIBOR – 2.28% = -3.78% FI = LIBOR – LIBOR + 2.28% - 2.22% = 0.06% Note the FI ALWAYS pays and receives LIBOR for its part of the swaps. d) What are the bid and ask LIBOR swap rates inherent in this swap? The bid and ask are the amounts at which the FI will buy and sell LIBOR: X sells LIBOR for 2.22% Y pays 2.28% for LIBOR Bid = 2.22% and Ask = 2.28% 1 2. (16 Points) You own a swap contract on which you receive LIBOR and pay 6.00% APR Semi- Annual on a notional amount of $700 million. There are exactly 9 months remaining in the swap. The current risk-free rate (LIBOR) for a payment in 3 months is 3.50% CCAR. The current risk-free rate (LIBOR) for a payment in 9 months is 4.00% CCAR. Three months ago, six-month LIBOR was 4.50% APR Semi-Annual. Use the steps below to calculate value of the swap using the forward method (the method covered in class). a) Calculate the implied forward risk-free rate (LIBOR) in 3 months for 6 months in CCAR terms. Next calculate the implied forward risk-free rate (LIBOR) in 3 months for 6 months in APR Semi- Annual terms. In other words: What rate (in APR S-A terms) can you lock in today that will start at time 4 and go to time 10? Forward Rate in CCAR terms: rF = (r2t2 - r1t1)/(t2 - t1) = [0.04(9/12) – 0.035(3/12)]/[(9/12) – (3/12)] = 4.25% Forward Rate in APR S-A terms: rm = m(er/m -1) = 2(e0.0425/2 – 1) = 0.0430 = 4.30% b) Calculate the present value of the net swap payment in 3 months. In 3 months: Pay 6.00% APR S-A on $700 Get 4.50% APR S-A on $700 $700(0.045 – 0.06)/2 = $700(-0.015) = -$5.25 PV = $5.25e-0.035(3/12) = -$5.20 c) Calculate the present value of the net swap payment in 9 months. In 9 months: Pay 6.00% APR S-A on $700 Get 4.30% APR S-A on $700 $700(0.0430 – 0.06)/2 = -$5.95 PV = -$5.95e-0.04(9/12) = -$5.77 d) Calculate the value of the swap. VSwap = -$5.25 + -$5.77 = -$10.97 e) Has the floating rate increased or decreased since the swap was initiated? Are you paying or receiving the floating rate? Has the value of the swap for your perspective increased or decreased since it was initiated? LIBOR was approximately 6.00% when the swap was initiated. It has decreased. You are receiving the floating rate so the swap’s value has decreased. Recall a swap’s value is zero when it is initiated and now the swap has a negative value. 2 3. (16 Points) Moments ago, an FI initiated the two swaps shown in the diagram below. In the swap with Company A, the FI gets 4.00% on US dollars and pays 6.00% on euros. In the swap with Company B, it pays 6.00% on dollars and gets 8.10% on euros. The notional amounts are $300 million and €240 million. For this question, consider these swaps separately from the rest of the FI’s obligations. 6.00% € 8.10% € A FI B 4.00% $ 6.00% $ a) Calculate the annual NET number of dollars paid or received by the FI each period Dollar rates: 4% - 6% = -2% Rates applied to $300 nominal amount: Dollars Paid = -2% x $300 = -$6m b) Calculate the annual NET number of euros paid or received by the FI each period. Euro rates: 8.10% - 6% = 2.1% Rates applied to €240 nominal amount: Euros Received = 2.1% x €240m = €5.04m c) Calculate FI’s annual profit in dollars at the current exchange rate. Calculate the profit in dollars as a percent of the notional dollar amount of swap. Current Exchange rate = $300/€240 = 1.25 $/€ At each payment time, the FI gets €5.04 and owes $6.00 The dollar value of €5.04m = €5.04m x 1.25 $/€ = $6.30m Profit in dollars: $6.30 - $6.00 = $0.30 Profit as a percent of the nominal swap value = $0.30/$300m = 0.0010 = 0.10% = 10 bps Note that 2.10% - 2.00% = 0.10% d) At what exchange rate is the FI’s profit equal to zero? Does this happen when the dollar strengthens or weakens? Calculate the exchange rate that equates €5.04 and $6.00: $6.00/€5.04 = 1.1905 $/€ So at 1.1905 $/€ the €5.04 can be sold for exactly $6.00 so the FI makes no money. The dollar strengthens. 3 4. (8 Points) A company can borrow at 6.00% fixed rate or LIBOR + 310 bps floating rate. The LIBOR swap rates are 2.70% at 2.80%. a) Assume the firm wants to borrow at a fixed rate. Calculate the net borrowing rate if it borrows at floating rate from the market and then swaps for a fixed rate payment. Should it borrow and swap or just borrow? Why? Borrows at a fixed rate from the market: Cost = -6.00% Borrows at a float rate from the market and “buy LIBOR” from the FI: Cost = - (LIBOR + 310) – 2.80% + LIBOR = -5.90% 5.90% < 6.00% so borrow at a floating rate and swap. b) Now assume the firm wants to borrow at a floating rate. Calculate the net borrowing rate if it borrows fixed from the market and then swaps for a floating rate. Should it borrow and swap or just borrow? Why? Borrow at a floating rate from the market: Cost = -(LIBOR + 310) Borrow at a fixed rate from the market and “sell LIBOR” to the FI: Cost = -6.00% + - LIBOR + 2.70% = -(LIBOR + 330) LIBOR + 310 < LIBOR + 330 so just borrow at a floating rate 5. (12 Points) You are bullish on a stock so you have just bought it. You are considering using an option on the stock to either enhancing the long stock position’s yield or, alternatively, add a floor to the long stock position. a) What option position would you add to the long stock in order to enhance the yield? Short Call b) What is the “trade-off” from adding yield-enhancing option? Accept a cap c) What synthetic option position is created by the long stock and option position described in part (a)? Short put d) What option position would you add to the stock in order to create a floor? Long Put e) What is the “trade-off” from adding an option that creates a floor? Greater out-of-pocket or profitable at a higher price f) What synthetic option position is created by the long stock and option position described in part (d)? Long Call 4 6. (15 Points) The price of a non-dividend stock is $102. The price of a six-month $100 European call on the stock is $8.50. The risk-free interest rate is 5.00% CCAR. a) According to put-call parity, what is the no-arb price of a six-month $100 European put on the stock? T = 6/12, c = $8.50, K = $100, S0 = $102, r = 5% c + Ke–rT = p + S0 p = c + Ke–rT - S0 = $8.50 + $100e–.05(6/12) - $102 = $4.03 Note: The Put-Call parity formula is a no-arbitrage condition. If p is not equal to c + Ke–rt - S0 (either greater than or less than) then the no-arbitrage condition is not met. b) Describe the actions of an arbitrageur (if any) today and at expiration if the market price of the put is $4.80. Please calculate the time-zero arbitrage profit, if any. p > c + Ke–rT - S0 Sell the put, Buy the call, Sell the Stock (Lock in purchase price at $100) t=0 t=T Sell Put (+p) +$4.80 Get Loan Amount = $98.30e.05(6/12) = $100.79 Sell Stock (+S0) +$102.00 If ST > $100, exercise call, pay $100 Buy Call (-c) -$8.50 If ST < $100, put is exercised, pay $100 Lend @ 5% for 8 months +$98.30 T = $100.79 - $100 = $0.79 0 = $79e-.05(6/12) = $0.77 Or Lend the PV of $100 at time 0: $100e-.05(6/12) = $97.53 0 = $98.30 – $97.53 =$0.77 NOTE: Short the Stock and Long a Call is a SYNTHETIC LONG PUT. The arbitrageur is Selling a Put for $4.80 and Buying a Synthetic Put for $4.03. c) Describe the actions of an arbitrageur (if any) today and at expiration if the market price of the put is $3.20. Please calculate the time-zero arbitrage profit, if any. p < c + Ke–rT - S0 Buy the put, Sell the call, buy the Stock (Lock in sale price at $100) t=0 t=T Buy Put (-p) -$3.20 Owe Loan Amount = $96.70e.05(6/12) = $99.15 Buy Stock (-S0) -$102.00 If ST > $100, call is exercised, get $100 Sell Call (+c) +$8.50 If ST < $100, exercise put, get $100 Borrow @ 5% for 8 months -$96.70 T = $100 - $99.15 = $0.85 0 = $0.85e-.05(6/12) = $0.83 Or borrow the PV of $100 at time 0: $100e-.1(8/12) = $97.53 0 = $97.53– $96.70 =$0.83 NOTE: Long the Stock and Short a Call is a SYNTHETIC SHORT PUT. The arbitrageur is Buying a Put for $3.20 and Selling a Synthetic Put for $4.03. 5 7. (18 Points) A trader pays $5 for $50 European put on a non-dividend stock. The trader also sells short a $40 European put option on the same stock for $1. Both options have the same expiration date. a) What is the initial cash flow from this position? CF0 = -$5 + $1 = -$4 b) Construct a table showing the Value and Profit of this position at expiration for all values of ST. ST VT = max{$50 - ST, 0} – max{$40 - ST,0} T = VT + CF0 ST ≥ $50 0–0=0 $0 - $4 = -$4 $50 > ST ≥ $40 ($50 - ST) – 0 = $50 - ST $50 – ST - $4 = $46 - ST $40 > ST ($50 - ST) – ($40 - ST) = $10 $10 – $4 = $6 c) Over what values of ST is the profit from this position positive? $46 – ST > 0 ST < $46 d) Graph the position. Include each component of the position and the combined position. Label the premiums, any maximum or minimum values and anytime a line crosses the “x-axis”. e) What is the name of this position? What is the trader’s expectation about the stock? It is Bear Put Spread. The trader expects the price of the stock to fall (or be below 46). f) Compare this position to buying only the $50 put. What would be the difference in the initial cash flow, profit range and max gain or loss? For the Bear Put Spread: CF0 is -$4 which is less than Long the $50 put (-$5) Profitability starts at ST < $46 which is higher than Long the $50 put (ST < $45) Max Gain is capped at $6 which is lass then $45 = $50 – $5 for long the $50 put 6