Sample Answers (Chapters 1,2,3,5) PDF

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These are limited problem answers to a few beginning chapters ...

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Chapter 1: Questions and Answers Problem 1.29. On May 8, 2013, as indicated in Table 1.2, the spot offer price of Google stock is $871.37 and the offer price of a call option with a strike price of $880 and a maturity date of September is $41.60. A trader is considering two alternatives: buy 100 shares of the stock and buy 100 September call options. For each alternative, what is (a) the upfront cost, (b) the total gain if the stock price in September is $950, and (c) the total loss if the stock price in September is $800. Assume that the option is not exercised before September and if stock is purchased it is sold in September. a) The upfront cost for the stock alternative is $87,137. The upfront cost for the option alternative is $4,160. b) The gain from the stock alternative is $95,000−$87,137=$7,863. The total gain from the option alternative is ($950-$880)×100−$4,160=$2,840. c) The loss from the stock alternative is $87,137−$80,000=$7,137. The loss from the option alternative is $4,160. Problem 1.30. What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed mining company stock is $50 (USD) on the New York Stock Exchange and $52 (CAD) on the Toronto Stock Exchange. Assume that the exchange rate is such that 1 USD equals 1.01 CAD. Explain what is likely to happen to prices as traders take advantage of this opportunity. Arbitrage involves carrying out two or more different trades to lock in a profit. In this case, traders can buy shares on the NYSE and sell them on the TSX to lock in a USD profit of 52/1.01−50=1.485 per share. As they do this the NYSE price will rise and the TSX price will fall so that the arbitrage opportunity disappears Problem 1.33. A US company knows it will have to pay 3 million euros in three months. The current exchange rate is 1.3500 dollars per euro. Discuss how forward and options contracts can be used by the company to hedge its exposure. The company could enter into a forward contract obligating it to buy 3 million euros in three months for a fixed price (the forward price). The forward price will be close to but not exactly the same as the current spot price of 1.3500. An alternative would be to buy a call option giving the company the right but not the obligation to buy 3 million euros for a particular exchange rate (the strike price) in three months. The use of a forward contract locks in, at no cost, the exchange rate that will apply in three months. The use of a call option provides, at a cost, insurance against the exchange rate being higher than the strike price. Problem 1.35. The price of gold is currently $1,400 per ounce. The forward price for delivery in one year is $1,500. An arbitrageur can borrow money at 4% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides no income.

The arbitrageur should borrow money to buy a certain number of ounces of gold today and short forward contracts on the same number of ounces of gold for delivery in one year. This means that gold is purchased for $1,400 per ounce and sold for $1,500 per ounce. Interest on the borrowed funds will be 0.04×$1400 or $56 per ounce. A profit of $44 per ounce will therefore be made. Problem 1.37. On May 8, 2013, an investor owns 100 Google shares. As indicated in Table 1.3, the share price is about $871 and a December put option with a strike price $820 costs $37.50. The investor is comparing two alternatives to limit downside risk. The first involves buying one December put option contract with a strike price of $820. The second involves instructing a broker to sell the 100 shares as soon as Google’s price reaches $820. Discuss the advantages and disadvantages of the two strategies. The second alternative involves what is known as a stop or stop-loss order. It costs nothing and ensures that $82,000, or close to $82,000, is realized for the holding in the event the stock price ever falls to $820. The put option costs $3,750 and guarantees that the holding can be sold for $8,200 any time up to December. If the stock price falls marginally below $820 and then rises the option will not be exercised, but the stop-loss order will lead to the holding being liquidated. There are some circumstances where the put option alternative leads to a better outcome and some circumstances where the stop-loss order leads to a better outcome. If the stock price ends up below $820, the stop-loss order alternative leads to a better outcome because the cost of the option is avoided. If the stock price falls to $800 in November and then rises to $850 by December, the put option alternative leads to a better outcome. The investor is paying $3,750 for the chance to benefit from this second type of outcome. Problem 1.38. A bond issued by Standard Oil some time ago worked as follows. The holder received no interest. At the bond’s maturity the company promised to pay $1,000 plus an additional amount based on the price of oil at that time. The additional amount was equal to the product of 170 and the excess (if any) of the price of a barrel of oil at maturity over $25. The maximum additional amount paid was $2,550 (which corresponds to a price of $40 per barrel). Show that the bond is a combination of a regular bond, a long position in call options on oil with a strike price of $25, and a short position in call options on oil with a strike price of $40. Suppose ST is the price of oil at the bond’s maturity. In addition to $1000 the Standard Oil bond pays:  0 ST  $25 $40  ST  $25  170(ST  25)  2 550 ST  $40 This is the payoff from 170 call options on oil with a strike price of 25 less the payoff from 170 call options on oil with a strike price of 40. The bond is therefore equivalent to a regular bond plus a long position in 170 call options on oil with a strike price of $25 plus a short position in

170 call options on oil with a strike price of $40. The investor has what is termed a bull spread on oil. This is discussed in Chapter 12. Problem 1.39. Suppose that in the situation of Table 1.1 a corporate treasurer said: “I will have £1 million to sell in six months. If the exchange rate is less than 1.52, I want you to give me 1.52. If it is greater than 1.58 I will accept 1.58. If the exchange rate is between 1.52 and 1.58, I will sell the sterling for the exchange rate.” How could you use options to satisfy the treasurer? You sell the treasurer a put option on GBP with a strike price of 1.52 and buy from the treasurer a call option on GBP with a strike price of 1.58. Both options are on one million pounds and have a maturity of six months. This is known as a range forward contract and is discussed in Chapter 17.

Chapter 2: Questions and Answers Problem 2.30. A company enters into a short futures contract to sell 5,000 bushels of wheat for 750 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Under what circumstances could $1,500 be withdrawn from the margin account? There is a margin call if $1000 is lost on the contract. This will happen if the price of wheat futures rises by 20 cents from 750 cents to 770 cents per bushel. $1500 can be withdrawn if the futures price falls by 30 cents to 720 cents per bushel. Problem 2.31. Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 5% per annum. How could you make money if the June and December futures contracts for a particular year trade at $80 and $86? You could go long one June oil contract and short one December contract. In June you take delivery of the oil borrowing $80 per barrel at 5% to meet cash outflows. The interest accumulated in six months is about 80×0.05×1/2 or $2. In December the oil is sold for $86 per barrel which is more than the $82 that has to be repaid on the loan. The strategy therefore leads to a profit. Note that this profit is independent of the actual price of oil in June and December. It will be slightly affected by the daily settlement procedures. Problem 2.32. What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date? (Draw a picture) The long forward contract provides a payoff of ST − K where ST is the asset price on the date and K is the delivery price. The put option provides a payoff of max (K−ST, 0). If ST > K the sum of the two payoffs is ST – K. If ST < K the sum of the two payoffs is 0. The combined payoff is therefore max (ST – K, 0). This is the payoff from a call option. The equivalent position is therefore a call option.

Chapter 3: Questions and Answers Problem 3.25. Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20 per ounce. What is the effect of the basis on the hedger’s financial position if (a) the trader is hedging the purchase of silver and (b) the trader is hedging the sale of silver? The excess of the spot over the futures at the time the hedge is closed out is $0.20 per ounce. If the trader is hedging the purchase of silver, the price paid is the futures price plus the basis. The trader therefore loses 60×5,000×$0.20=$60,000. If the trader is hedging the sales of silver, the price received is the futures price plus the basis. The trader therefore gains $60,000. Problem 3.26. A trader owns 55,000 units of a particular asset and decides to hedge the value of her position with futures contracts on another related asset. Each futures contract is on 5,000 units. The spot price of the asset that is owned is $28 and the standard deviation of the change in this price over the life of the hedge is estimated to be $0.43. The futures price of the related asset is $27 and the standard deviation of the change in this over the life of the hedge is $0.40. The coefficient of correlation between the spot price change and futures price change is 0.95. (a) What is the minimum variance hedge ratio? (b) Should the hedger take a long or short futures position? (c) What is the optimal number of futures contracts? (a) The minimum variance hedge ratio is 0.95×0.43/0.40=1.02125. (b) The hedger should take a short position. (c) The optimal number of contracts with no tailing is 1.02125×55,000/5,000=11.23 (or 11 when rounded to the nearest whole number) Problem 3.30. It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change beta of the portfolio to 0.5 during the period July 16 to November 16. The index is currently 1,000, and each contract is on $250 times the index. a) What position should the company take? b) Suppose that the company changes its mind and decides to increase the beta of the portfolio from 1.2 to 1.5. What position in futures contracts should it take?

a) The company should short (12  05)  100 000 000 1000  250

or 280 contracts. b) The company should take a long position in (15  12)  100 000 000 1000  250

or 120 contracts. Problem 3.31. A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the S&P 500 to hedge the risk. The current level of the index is 1250, one contract is on 250 times the index, the risk-free rate is 6% per annum, and the dividend yield on the index is 3% per annum. The current 3 month futures price is 1259. a) What position should the fund manager take to eliminate all exposure to the market over the next two months? b) Calculate the effect of your strategy on the fund manager’s returns if the level of the market in two months is 1,000, 1,100, 1,200, 1,300, and 1,400. Assume that the onemonth futures price is 0.25% higher than the index level at this time.

a) The number of contracts the fund manager should short is 50 000 000 087   13820 1259  250 Rounding to the nearest whole number, 138 contracts should be shorted.

b) The following table shows that the impact of the strategy. To illustrate the calculations in the table consider the first column. If the index in two months is 1,000, the futures price is 1000×1.0025. The gain on the short futures position is therefore (1259 1002 50) 250 138  $8849 250 The return on the index is 3 2  12 =0.5% in the form of dividend and 250  1250  20% in the form of capital gains. The total return on the index is

therefore 195% . The risk-free rate is 1% per two months. The return is therefore 205% in excess of the risk-free rate. From the capital asset pricing model we expect the return on the portfolio to be 0 87  205%  17835% in excess of the risk-free rate. The portfolio return is therefore 16835% . The loss on the portfolio is 0 16835 50 000 000 or $8,417,500. When this is combined with the gain on the futures the total gain is $431,750. Index now 1250 1250 1250 1250 1250 Index Level in Two Months 1000 1100 1200 1300 1400 Return on Index in Two Months -0.20 -0.12 -0.04 0.04 0.12 Return on Index incl divs -0.195 -0.115 -0.035 0.045 0.125 Excess Return on Index -0.205 -0.125 -0.045 0.035 0.115 Excess Return on Portfolio -0.178 -0.109 -0.039 0.030 0.100 Return on Portfolio -0.168 -0.099 -0.029 0.040 0.110 Portfolio Gain -8,417,500 -4,937,500 -1,457,500 2,022,500 5,502,500 Futures Now Futures in Two Months Gain on Futures Net Gain on Portfolio

1259 1259 1259 1259 1259 1002.50 1102.75 1203.00 1303.25 1403.50 8,849,250 5,390,625 1,932,000 -1,526,625 -4,985,250 431,750

453,125

474,500

495,875

517,250

Problem 3.32. It is now October 2014. A company anticipates that it will purchase 1 million pounds of copper in each of February 2015, August 2015, February 2016, and August 2016. The company has decided to use the futures contracts traded in the COMEX division of the CME Group to hedge its risk. One contract is for the delivery of 25,000 pounds of copper. The initial margin is $2,000 per contract and the maintenance margin is $1,500 per contract. The company’s policy is to hedge 80% of its exposure. Contracts with maturities up to 13 months into the future are considered to have sufficient liquidity to meet the company’s needs. Devise a hedging strategy for the company. (Do not make the “tailing” adjustment described in Section 3.4.) Assume the market prices (in cents per pound) today and at future dates are as follows. What is the impact of the strategy you propose on the price the company pays for copper? What is the initial margin requirement in October 2014? Is the company subject to any margin calls? Date Spot Price Mar 2015 Futures Price Sep 2015 Futures Price Mar 2016 Futures Price Sep 2016 Futures Price

Oct 2014 372.00 372.30 372.80

Feb 2015 369.00 369.10 370.20 370.70

Aug 2015 365.00

Feb 2016 377.00

Aug 2016 388.00

364.80 364.30 364.20

376.70 376.50

388.20

To hedge the February 2015 purchase the company should take a long position in March 2015 contracts for the delivery of 800,000 pounds of copper. The total number of contracts required is 800 000  25 000  32. Similarly a long position in 32 September 2015 contracts is required to hedge the August 2015 purchase. For the February 2016 purchase the company could take a long position in 32 September 2015 contracts and roll them into March 2016 contracts during August 2015. (As an alternative, the company could hedge the February 2016 purchase by taking a long position in 32 March 2015 contracts and rolling them into March 2016 contracts.) For the August 2016 purchase the company could take a long position in 32 September 2015 and roll them into September 2016 contracts during August 2015. The strategy is therefore as follows Oct. 2014:

Enter into long position in 96 Sept. 2015 contracts Enter into a long position in 32 Mar. 2015 contracts

Feb 2015:

Close out 32 Mar. 2015 contracts

Aug 2015:

Close out 96 Sept. 2015 contracts Enter into long position in 32 Mar. 2016 contracts Enter into long position in 32 Sept. 2016 contracts

Feb 2016:

Close out 32 Mar. 2016 contracts

Aug 2016:

Close out 32 Sept. 2016 contracts

Chapter 5: Questions and Answers Problem 5.26. In early 2012, the spot exchange rate between the Swiss Franc and U.S. dollar was 1.0404 ($ per franc). Interest rates in the U.S. and Switzerland were 0.25% and 0% per annum,respectively, with continuous compounding. The three-month forward exchange rate was1.0300 ($ per franc). What arbitrage strategy was possible? How does your answer change if the exchange rate is 1.0500 ($ per franc). The theoretical forward exchange rate is 1.0404e(0.0025−0)×0.25 = 1.041. If the actual forward exchange rate is 1.03, an arbitrageur can a) borrow X Swiss francs, b) convert the Swiss francs to 1.0404X dollars and invest the dollars for three months at 0.25% and c) buy X Swiss francs at 1.03 in the three-month forward market. In three months, the arbitrageur has 1.0404Xe0.0025×0.25 = 1.041X dollars. A total of 1.3X dollars are used to buy the Swiss francs under the terms of the forward contract and a gain of 0.011X is made. If the actual forward exchange rate is 1.05, an arbitrageur can a) borrow X dollars, b) convert the dollars to X/1.0404 Swiss francs and invest the Swiss francs for three months at zero interest rate, and c) enter into a forward contract to sell X/1.0404 Swiss francs in three months. In three months the arbitrageur has X/1.0404 Swiss francs. The forward contract converts these to (1.05X)/1.0404=1.0092X dollars. A total of Xe0.0025×0.25 =1.0006X is needed to repay the dollar loan. A profit of 0.0086X dollars is therefore made. Problem 5.27. An index is 1,200. The three-month risk-free rate is 3% per annum and the dividend yield over the next three months is 1.2% per annum. The six-month risk-free rate is 3.5% per annum and the dividend yield over the next six months is 1% per annum. Estimate the futures price of the index for three-month and six-month contracts. All interest rates and dividend yields are continuously compounded. The futures price for the three month contract is 1200e(0.03-0.012)×0.25 =1205.41. The futures price for the six month contract is 1200e(0.035-0.01)×0.5 =1215.09. Problem 5.30. A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a six-month forward contract on the stock. a) What are the forward price and the initial value of the forward contract? b) Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract? a) The present value, I , of the income from the security is given by: I  1 e008 212  1 e0 08 5 12  1 9540 From equation (5.2) the forward price, F0 , is given by:

F0  (50  19540)e0 0805  50 01 or $50.01. The initial value of the forward contract is (by design) zero. The fact that the forward price is very close to the spot price should come as no surprise. When the compounding frequency is ignored the dividend yield on the stock equals the risk-free rate of interest.

b) In three months: I  e008 212  0 9868 The delivery price, K , is 50.01. From equation (5.6) the value of the short forward contract, f , is given by

f  (48  09868  50 01e 0 083 12 )  2 01 and the forward price is (48 0 9868)e0 08 3 12  47 96

Problem 5.33. A trader owns a commodity that provides no income and has no storage costs as part of a longterm investment portfolio. The trader can buy the commodity for $1250 per ounce and sell gold for $1249 per ounce. The trader can borrow funds at 6% per year and i...