Options, Futures, and Other Derivatives, 9th Edition Solution Manual

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CHAPTER 1IntroductionPractice QuestionsProblem 1.8.Suppose you own 5,000 shares that are worth $25 each. How can put options be used toprovide you with insurance against a decline in the value of your holding over the next fourmonths?You should buy 50 put option contracts (each on 100 shares) with a strike price of $25 and anexpiration date in four months. If at the end of four months the stock price proves to be lessthan $25, you can exercise the options and sell the shares for $25 each.Problem 1.9.A stock when it is first issued provides funds for a company. Is the same true of an exchange-traded stock option? Discuss.An exchange-traded stock option provides no funds for the company. It is a security sold byone investor to another. The company is not involved. By contrast, a stock when it is firstissued is sold by the company to investors and does provide funds for the company.Problem 1.10.Explain why a futures contract can be used for either speculation or hedging.If an investor has an exposure to the price of an asset, he or she can hedge with futurescontracts. If the investor will gain when the price decreases and lose when the price increases,a long futures position will hedge the risk. If the investor will lose when the price decreasesand gain when the price increases, a short futures position will hedge the risk. Thus either along or a short futures position can be entered into for hedging purposes.If the investor has no exposure to the price of the underlying asset, entering into a futurescontract is speculation. If the investor takes a long position, he or she gains when the asset’sprice increases and loses when it decreases. If the investor takes a short position, he or sheloses when the asset’s price increases and gains when it decreases.Problem 1.11.A cattle farmer expects to have 120,000 pounds of live cattle to sell in three months. The live-cattle futures contract on the Chicago Mercantile Exchange is for the delivery of 40,000pounds of cattle. How can the farmer use the contract for hedging? From the farmer’sviewpoint, what are the pros and cons of hedging?The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, thegain on the futures contract will offset the loss on the sale of the cattle. If the price of cattlerises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Usingfutures contracts to hedge has the advantage that it can at no cost reduce risk to almost zero.

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Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices.Problem 1.12.It is July 2010. A mining company has just discovered a small deposit of gold. It will take sixmonths to construct the mine. The gold will then be extracted on a more or less continuousbasis for one year. Futures contracts on gold are available on the New York MercantileExchange. There are delivery months every two months from August 2010 to December 2011.Each contract is for the delivery of 100 ounces. Discuss how the mining company might usefutures markets for hedging.The mining company can estimate its production on a month by month basis. It can then shortfutures contracts to lock in the price received for the gold. For example, if a total of 3,000ounces are expected to be produced in September 2010 and October 2010, the price receivedfor this production can be hedged by shorting a total of 30 October 2010 contracts.Problem 1.13.Suppose that a March call option on a stock with a strike price of $50 costs $2.50 and is helduntil March. Under what circumstances will the holder of the option make a gain? Underwhat circumstances will the option be exercised? Draw a diagram showing how the profit ona long position in the option depends on the stock price at the maturity of the option.The holder of the option will gain if the price of the stock is above $52.50 in March. (Thisignores the time value of money.) The option will be exercised if the price of the stock isabove $50.00 in March. The profit as a function of the stock price is shown in Figure S1.1.Figure S1.1Profit from long position in Problem 1.13Problem 1.14.Suppose that a June put option on a stock with a strike price of $60 costs $4 and is held untilJune. Under what circumstances will the holder of the option make a gain? Under whatcircumstances will the option be exercised? Draw a diagram showing how the profit on ashort position in the option depends on the stock price at the maturity of the option.

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The seller of the option will lose if the price of the stock is below $56.00 in June. (Thisignores the time value of money.) The option will be exercised if the price of the stock isbelow $60.00 in June. The profit as a function of the stock price is shown in Figure S1.2.Figure S1.2Profit from short position In Problem 1.1Problem 1.15.It is May and a trader writes a September call option with a strike price of $20. The stockprice is $18, and the option price is $2. Describe the investor’s cash flows if the option isheld until September and the stock price is $25 at this time.The trader has an inflow of $2 in May and an outflow of $5 in September. The $2 is the cashreceived from the sale of the option. The $5 is the result of the option being exercised. Theinvestor has to buy the stock for $25 in September and sell it to the purchaser of the optionfor $20.Problem 1.16.An investor writes a December put option with a strike price of $30. The price of the option is$4. Under what circumstances does the investor make a gain?The investor makes a gain if the price of the stock is above $26 at the time of exercise. (Thisignores the time value of money.)Problem 1.17.The Chicago Board of Trade offers a futures contract on long-term Treasury bonds.Characterize the investors likely to use this contract.Most investors will use the contract because they want to do one of the following:a)Hedge an exposure to long-term interest rates.

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b)Speculate on the future direction of long-term interest rates.c)Arbitrage between the spot and futures markets for Treasury bonds.Problem 1.18.An airline executive has argued: “There is no point in our using oil futures. There is just asmuch chance that the price of oil in the future will be less than the futures price as there isthat it will be greater than this price.” Discuss the executive’s viewpoint.It may well be true that there is just as much chance that the price of oil in the future will beabove the futures price as that it will be below the futures price. This means that the use of afutures contract for speculation would be like betting on whether a coin comes up heads ortails. But it might make sense for the airline to use futures for hedging rather thanspeculation. The futures contract then has the effect of reducing risks. It can be argued that anairline should not expose its shareholders to risks associated with the future price of oil whenthere are contracts available to hedge the risks.Problem 1.19.“Options and futures are zero-sum games.” What do you think is meant by this statement?The statement means that the gain (loss) to the party with the short position is equal to theloss (gain) to the party with the long position. In total, the gain to all parties is zero.Problem 1.20.A trader enters into a short forward contract on 100 million yen. The forward exchange rateis $0.0080 per yen. How much does the trader gain or lose if the exchange rate at the end ofthe contract is (a) $0.0074 per yen; (b) $0.0091 per yen?a)The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0074per yen. The gain is1000 0006 millions of dollars or $60,000.b)The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0091per yen. The loss is1000 0011 millions of dollars or $110,000.Problem 1.21.A trader enters into a short cotton futures contract when the futures price is 50 cents perpound. The contract is for the delivery of 50,000 pounds. How much does the trader gain orlose if the cotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 centsper pound?a)The trader sells for 50 cents per pound something that is worth 48.20 cents per pound.Gain( 0 50000 4820)50 000900$$$=−=.b)The trader sells for 50 cents per pound something that is worth 51.30 cents per pound.Loss( 0 51300 5000)50 000650$$$=−=.Problem 1.22.A company knows that it is due to receive a certain amount of a foreign currency in fourmonths. What type of option contract is appropriate for hedging?

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A long position in a four-month put option can provide insurance against the exchange ratefalling below the strike price. It ensures that the foreign currency can be sold for at least thestrike price.Problem 1.23.A United States company expects to have to pay 1 million Canadian dollars in six months.Explain how the exchange rate risk can be hedged using (a) a forward contract; (b) anoption.The company could enter into a long forward contract to buy 1 million Canadian dollars insix months. This would have the effect of locking in an exchange rate equal to the currentforward exchange rate. Alternatively the company could buy a call option giving it the right(but not the obligation) to purchase 1 million Canadian dollar at a certain exchange rate in sixmonths. This would provide insurance against a strong Canadian dollar in six months whilestill allowing the company to benefit from a weak Canadian dollar at that time.Further QuestionsProblem 1.24(Excel file)Trader A enters into a forward contract to buy gold for $1000 an ounce in one year. TraderB buys a call option to buy gold for $1000 an ounce in one year. The cost of the option is$100 an ounce. What is the difference between the positions of the traders? Show the profitper ounceas a function of the price of gold in one year for the two traders.Trader A makes a profit ofST̶1000 and Trader B makes a profit of max(ST̶1000, 0)–100whereSTis the price of gold in one month. Trader A does better ifSTis above $900 asindicated in Figure S1.3.Figure S1.3:Profit to Trader A and Trader B in Problem 1.24

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Problem 1.25In March, a US investor instructs a broker to sell one July put option contract on a stock. Thestock price is $42 and the strike price is $40. The option price is $3. Explain what theinvestor has agreed to. Under what circumstances will the trade prove to be profitable? Whatare the risks?The investor has agreed to buy 100 shares of the stock for $40 in July (or earlier) if the partyon the other side of the transaction chooses to sell. The trade will prove profitable if theoption is not exercised or if the stock price is above $37 at the time of exercise. The risk tothe investor is that the stock price plunges to a low level. For example, if the stock pricedrops to $1 by July (unlikely but possible), the investor loses $3,600. This is because the putoptions are exercised and $40 is paid for 100 shares when the value per share is $1. Thisleads to a loss of $3,900 which is offset by the premium of $300 received for the options.Problem 1.26A US company knows it will have to pay 3 million euros in three months. The currentexchange rate is 1.4500 dollars per euro. Discuss how forwardand optionscontractscan beused by the company to hedge its exposure.The company could enter into a forward contract obligating it to buy 3 million euros in threemonths for a fixed price (the forward price). The forward price will be close to but notexactly the same as the current spot price of 1.4500. An alternative would be to buy a calloption giving the company the right but not the obligation to buy 3 million euros for a aparticular exchange rate (the strike price) in three months. The use of a forward contract locksin, at no cost, the exchange rate that will apply in three months. The use of a call optionprovides, at a cost, insurance against the exchange rate being higher than the strike price.Problem 1.27(Excel file)A stock price is $29. An investor buys one call option contract on the stock with a strike priceof $30 and sells a call optioncontracton the stock with a strike price of $32.50. The marketprices of the options are $2.75 and $1.50, respectively. The options have the same maturitydate. Describe the investor's position.This is known as a bull spread and will be discussed in Chapter 11. The profit is shown inFigure S1.4.

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Figure S1.4:Profit in Problem 1.27Problem 1.28The price of gold is currently $600 per ounce. Forward contracts are available to buy or sellgold at $800 for delivery in one year. An arbitrageur can borrow money at 10% per annum.What should the arbitrageur do? Assume that the cost of storing gold is zero and that goldprovides no income.The arbitrageur should borrow money to buy a certain number of ounces of gold today andshort forward contracts on the same number of ounces of gold for delivery in one year. Thismeans that gold is purchased for $600 per ounce and sold for $800 per ounce. Assuming thecost of borrowed funds is less than 33% per annum this generates a riskless profit.Problem 1.29.Discuss how foreign currency options can be used for hedging in the situation described inExample 1.1 so that (a) ImportCo is guaranteed that its exchange rate will be less than1.6600, and (b) ExportCo is guaranteed that its exchange rate will be at least 1.6200.ImportCo can buy call options on£10,000,000 with a strike price of 1.6600. This will ensurethat it never pays more than $16,600,000 for the sterling it requires. ExportCo can buy putoptions on£30,000,000 with a strike price of1.6200. This will ensure that the price receivedfor the sterling will be above00,600,48$000,000,3062.1=.Problem 1.30.The current price of a stock is $94, and three-month call options with a strike price of $95currently sell for $4.70. An investor who feels that the price of the stock will increase istrying to decide between buying 100 shares and buying 2,000 call options (20 contracts).Both strategies involve an investment of $9,400. What advice would you give? How high doesthe stock price have to rise for the option strategy to be more profitable?

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The investment in call options entails higher risks but can lead to higher returns. If the stockprice stays at $94, an investor who buys call options loses $9,400 whereas an investor whobuys shares neither gains nor loses anything. If the stock price rises to $120, the investor whobuys call options gains2000(12095)940040 600$−−=An investor who buys shares gains100(12094)2 600$−=The strategies are equally profitable if the stock price rises to a level,S,where100(94)2000(95)9400SS−=−−or100S=The option strategy is therefore more profitable if the stock price rises above $100.Problem 1.31.On July 17, 2009, an investor owns 100 Googleshares. As indicated in Table 1.2,the shareprice is$430.25 and a Decemberput option with a strike price $400 costs $21.15. Theinvestor is comparing two alternatives to limit downside risk. The first involves buying oneDecemberput option contract with a strike price of $400. The second involves instructing abroker to sell the 100 shares as soon as Google’s price reaches $400. Discuss the advantagesand disadvantages of the two strategies.The second alternative involves what is known as a stop or stop-loss order. It costsnothingand ensures that $40,000, or close to $40,000, is realized for the holding in the event thestock price ever falls to $40. The put option costs $2,115and guarantees that the holding canbe sold for $4,000any time up to December. If the stock price falls marginally below $400and then rises the option will not be exercised, but the stop-loss orderwill lead to the holdingbeingliquidated. There are some circumstances where the put option alternative leads to abetter outcome and some circumstances where the stop-loss order leads to a better outcome.If thestock price ends up below $400, the stop-loss order alternative leads to a betteroutcome because the cost of the option is avoided. If the stock price fallsto $380inNovember and then rises to $450 by December, the put option alternative leads to a betteroutcome. The investor is paying $2,115for the chance to benefit from this second type ofoutcome.Problem 1.32.A trader buys a European call option and sells a European put option. The options have thesame underlying asset, strike price and maturity. Describe the trader’s position. Under whatcircumstances does the price of the call equal the price of the put?The trader has a long European call option with strike priceKand a short European putoption with strike priceK. Suppose the price of the underlying asset at the maturity of theoption isTS. IfTSK, the call option is exercised by the investor and the put option expiresworthless. The payoff from the portfolio isTSK−. IfTSK, the call option expiresworthless and the put option is exercised against the investor. The cost to the investor isTKS−. Alternatively we can say that the payoff to the investor isTSK−(a negative

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amount). In all cases, the payoff isTSK−, the same as the payoff from the forward contract.The trader’s position is equivalent to a forward contract with delivery priceK.Suppose thatFis the forward price. IfKF=, the forward contract that is created has zerovalue. Because the forward contract is equivalent to a long call and a short put, this showsthat the price of a call equals the price of a put when the strike price isF.

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CHAPTER 2Mechanics of Futures MarketsPractice QuestionsProblem 2.8.The party with a short position in a futures contract sometimes has options as to the preciseasset that will be delivered, where delivery will take place, when delivery will take place, andso on. Do these options increase or decrease the futures price? Explain your reasoning.These options make the contract less attractive to the party with the long position and moreattractive to the party with the short position. They therefore tend to reduce the futures price.Problem 2.9.What are the most important aspects of the design of a new futures contract?The most important aspects of the design of a new futures contract are the specification of theunderlying asset, the size of the contract, the delivery arrangements, and the delivery months.Problem 2.10.Explain how margins protect investors against the possibility of default.A margin is a sum of money deposited by an investor with his or her broker. It acts as aguarantee that the investor can cover any losses on the futures contract. The balance in themargin account is adjusted daily to reflect gains and losses on the futures contract. If lossesare above a certain level, the investor is required to deposit a further margin. This systemmakes it unlikely that the investor will default. A similar system of margins makes it unlikelythat the investor’s broker will default on the contract it has with the clearinghouse memberand unlikely that the clearinghouse member will default with the clearinghouse.Problem 2.11.A trader buys two July futures contracts on frozen orange juice. Each contract is for thedelivery of 15,000 pounds. The current futures price is 160 cents per pound, the initialmargin is $6,000 per contract, and the maintenance margin is $4,500 per contract. Whatprice change would lead to a margin call? Under what circumstances could $2,000 bewithdrawn from the margin account?There is a margin call if more than $1,500 is lost on one contract. This happens if the futuresprice of frozen orange juice falls by more than 10 cents to below 150 cents per lb. $2,000 canbe withdrawn from the margin account if there is a gain on one contract of $1,000. This willhappen if the futures price rises by 6.67 cents to 166.67 cents per lb.Problem 2.12.Show that, if the futures price of a commodity is greater than the spot price during thedelivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist ifthe futures price is less than the spot price? Explain your answer.If the futures price is greater than the spot price during the delivery period, an arbitrageur

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buys the asset, shorts a futures contract, and makes delivery for an immediate profit. If thefutures price is less than the spot price during the delivery period, there is no similar perfectarbitrage strategy. An arbitrageur can take a long futures position but cannot force immediatedelivery of the asset. The decision on when delivery will be made is made by the party withthe short position. Nevertheless companies interested in acquiring the asset will find itattractive to enter into a long futures contract and wait for delivery to be made.Problem 2.13.Explain the difference between a market-if-touched order and a stop order.A market-if-touched order is executed at the best available price after a trade occurs at aspecified price or at a price more favorable than the specified price. A stop order is executedat the best available price after there is a bid or offer at the specified price or at a price lessfavorable than the specified price.Problem 2.14.Explain what a stop-limit order to sell at 20.30 with a limit of 20.10 means.A stop-limit order to sell at 20.30 with a limit of 20.10 means that as soon as there is a bid at20.30 the contract should be sold providing this can be done at 20.10 or a higher price.Problem 2.15.At the end of one day a clearinghouse member is long 100 contracts, and the settlement priceis $50,000 per contract. The original margin is $2,000 per contract. On the following day themember becomes responsible for clearing an additional 20 long contracts, entered into at aprice of $51,000 per contract. The settlement price at the end of this day is $50,200. Howmuch does the member have to add to its margin account with the exchange clearinghouse?The clearinghouse member is required to provide202 00040 000$$=as initial margin forthe new contracts. There is a gain of (50,200−50,000)100=$20,000 on the existingcontracts. There is also a loss of(51 00050 200)2016 000$−=on the new contracts. Themember must therefore add40 00020 00016 00036 000$−+=to the margin account.Problem 2.16.On July 1, 2010, a Japanese company enters into a forward contract to buy $1 million withyen on January 1, 2011. On September 1, 2010, it enters into a forward contract to sell $1million on January 1, 2011. Describe the profit or loss the company will make in dollars as afunction of the forward exchange rates on July 1, 2010 and September 1, 2010.Suppose1Fand2Fare the forward exchange rates for the contracts entered into July 1, 2010and September 1, 2010, andSis the spot rate on January 1, 2011. (All exchange rates aremeasured as yen per dollar). The payoff from the first contract is1()SF−million yen and thepayoff from the second contract is2()FS−million yen. The total payoff is therefore1221()()()SFFSFF−+−=−million yen.

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Problem 2.17.The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futuresprice for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes.Which is more favorable for an investor wanting to sell Swiss francs?The 1.1000 forward quote is the number of Swiss francs per dollar. The 0.9000 futures quoteis the number of dollars per Swiss franc. When quoted in the same way as the futures pricethe forward priceis1 1 10000 9091 =. The Swiss franc is therefore more valuable in theforward market than in the futures market. The forward market is therefore more attractivefor an investor wanting to sell Swiss francs.Problem 2.18.Suppose you call your broker and issue instructions to sell one July hogs contract. Describewhat happens.Hog futures are traded on the Chicago Mercantile Exchange. (See Table 2.2). The broker willrequest some initial margin. The order will be relayed by telephone to your broker’s tradingdesk on the floor of the exchange (or to the trading desk of another broker).It will be sent by messenger to a commission broker who will execute the trade according toyour instructions. Confirmation of the trade eventually reaches you. If there are adversemovements in the futures price your broker may contact you to request additional margin.Problem 2.19.“Speculation in futures markets is pure gambling. It is not in the public interest to allowspeculators to trade on a futures exchange.” Discuss this viewpoint.Speculators are important market participants because they add liquidity to the market.However, contracts must be useful for hedging as well as speculation. This is becauseregulators generally only approve contracts when they are likely to be of interest to hedgersas well as speculators.Problem 2.20.Identify the three commodities whose futures contracts in Table 2.2 have the highest openinterest.Based on the contract months listed, the answer is crude oil, corn, and sugar (world).Problem 2.21.What do you think would happen if an exchange started trading a contract in which thequality of the underlying asset was incompletely specified?The contract would not be a success. Parties with short positions would hold their contractsuntil delivery and then deliver the cheapest form of the asset. This might well be viewed bythe party with the long position as garbage! Once news of the quality problem became widelyknown no one would be prepared to buy the contract. This shows that futures contractsarefeasible only when there are rigorous standards within an industry for defining the quality ofthe asset. Many futures contracts have in practice failed because of the problem of definingquality.

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Problem 2.22.“When a futures contract is traded on the floor of the exchange, it may be the case that theopen interest increases by one, stays the same, or decreases by one.” Explain this statement.If both sides of the transaction are entering into a new contract, the open interest increases byone. If both sides of the transaction are closing out existing positions, the open interestdecreases by one. If one party is entering into a new contract while the other party is closingout an existing position, the open interest stays the same.Problem 2.23.Suppose that on October 24, 2010, you take a short position in an April 2011 live-cattlefutures contract. You close out your position on January 21, 2011. The futures price (perpound) is 91.20 cents when you enter into the contract, 88.30 cents when you close out yourposition, and 88.80 cents at the end of December 2010. One contract is for the delivery of40,000 pounds of cattle. What is your total profit? How is it taxed if you are (a) a hedger and(b) a speculator?Assume that you have a December 31 year end.The total profit is40 000(0 91200 8830)1 160$−=If you are a hedger this is all taxed in 2011. If you are a speculator40 000(0 91200 8880)960$−=is taxed in 2010 and40 000(0 88800 8830)200$−=is taxed in 2011.Further QuestionsProblem 2.24Trader A enters into futures contracts to buy 1 million euros for 1.4million dollars in threemonths. Trader B enters in a forward contract to do the same thing. The exchange (dollarsper euro) declines sharply during the firsttwomonthsand then increases for the third monthto close at 1.4300. Ignoring daily settlement, what is the total profit of each trader? When theimpact of daily settlement is taken into account, which trader does better?The total profit of each trader in dollars is 0.03×1,000,000 = 30,000. Trader B’s profit isrealized at the end of the three months. Trader A’s profit is realized day-by-day during thethree months. Substantial losses are made during the first two months and profits are madeduring the final month. It is likely that Trader B has done better because Trader A had tofinance its losses during the first two months.Problem 2.25Explain what is meant by open interest. Why does the open interestusually decline during themonthpreceding the delivery month? On a particular day there are 2,000 trades in aparticular futures contract. Of the 2,000 traders on the long side of the market, 1,400 wereclosing out position and 600 were entering into new positions. Of the 2,000 traders on theshort side of the market, 1,200 were closing out position and 800 were entering into newpositions. What is the impact of the day's trading on open interest?

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Open interest is the number of contract outstanding. Many traders close out their positionsjust before the delivery month is reached. This is why the open interest declines during themonth preceding the delivery month. The open interest went down by 600. We can see this intwo ways. First, 1,400 shorts closed out and there were 800 new shorts. Second, 1,200 longsclosed out and there were 600 new longs.Problem 2.26One orange juice future contract is on 15,000poundsof frozen concentrate. Suppose that inSeptember 2009 a company sells a March 2011 orange juice futures contract for 120 centsper pound. In December 2009 the futures price is 140 cents. In December 2010 the futuresprice is 110 cents. In February2011 the futures price is 125 cents. The company has aDecember year end. What is the company's profitor losson the contract? How is it realized?What is the accounting and tax treatment of thetransaction is the company is classified as a)a hedger and b) a speculator?The price goes up during the time the company holds the contract from 120 to 125 cents perpound. Overall the company therefore takes a loss of 15,000×0.05 = $750. If the company isclassified as a hedger this loss is realized in 2011, If it is classified as a speculator it realizes aloss of 15,000×0.20 = $3000 in 2009, a gain of 15,000×0.30 = $4,500 in 2010 and a loss of15,000×0.15 = $2,250 in 2011.Problem 2.27.A company enters into a short futures contract to sell 5,000 bushels of wheat for 250 centsper bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What pricechange would lead to a margin call? Under what circumstances could $1,500 be withdrawnfrom the margin account?There is a margin call if $1000 is lost on the contract. This will happen if the price of wheatfutures rises by 20 cents from 250 cents to 270 cents per bushel. $1500 can be withdrawn ifthe futures price falls by 30 cents to 220 cents per bushel.Problem 2.28.Suppose that there are no storage costs for crude oil and the interest rate for borrowing orlending is 5% per annum. How could you make money on August 4, 2009 by tradingDecember 2009 and June 2010contracts on crude oil? Use Table 2.2.The December 2009settlement price for oil is $75.62 per barrel. The June 2010settlementprice for oil is $79.41per barrel.You could go long one December 2009oilcontract andshort one June 2010 contract. In December2009you take delivery of the oil borrowing$75.62per barrel at 5% to meet cash outflows. The interest accumulated in six months isabout75.62×0.05×0.5 or $1.89. In December the oil is sold for $79.41per barrelwhich ismore than the amount thathas to be repaid on the loan. The strategy therefore leads to aprofit. Note that this profit is independent of theactual price of oil in June 2010 or December2009. It will be slightly affected by the daily settlement procedures.Problem 2.29.What position is equivalent to a long forward contract to buy an asset atKon a certain dateand a put option to sell it forKon that date?

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The equivalent position is a long position in a call with strike priceK.Problem 2.30.(Excel file)The author’s Web page (www.rotman.utoronto.ca/~hull/data) contains daily closing pricesfor the December 2001 crude oil futures contract and the December 2001 gold futurescontract. (Both contracts are traded on NYMEX.) You are required to download the data andanswer the following:a)How high do the maintenance margin levels for oil and gold have to be set so thatthere is a 1% chance that an investor with a balance slightly above the maintenancemargin level on a particular day has a negative balance two days later (i.e. one dayafter a margin call). How high do they have to be for a 0.1% chance. Assume dailyprice changes are normally distributed with mean zero.b)Imagine an investor who starts with a long position in the oil contract at thebeginning of the period covered by the data and keeps the contract for the whole ofthe period of time covered by the data. Margin balances in excess of the initial marginare withdrawn. Use the maintenance margin you calculated in part (a) for a 1% risklevel and assume that the maintenance margin is 75% of the initial margin. Calculatethe number of margin calls and the number of times the investor has a negativemargin balance and therefore an incentive to walk away. Assume that all margin callsare met in your calculations. Repeat the calculations for an investor who starts with ashort position in the gold contract.The data for this problem in the 7thedition is different from that in the 6thedition.a)For gold the standard deviation of daily changes is $15.184 per ounce or $1518.4percontract. For a 1% risk this means that the maintenance margin should be set at3263.224.1518or 4996 when rounded. For a 0.1% risk the maintenancemargin should be set at0902.324.1518or 6636 when rounded.For crude oil the standard deviation of daily changes is $1.5777 per barrel or $1577.7per contract. For a 1% risk,this means that the maintenance margin should be set at3263.227.1577or 5191 when rounded.For a 0.1% chance the maintenancemargin should be set at0902.327.1577or 6895 when rounded. NYMEXmight be interested in these calculations because they indicate the chance of a traderwho is just above the maintenance margin level at the beginning of the period havinga negative marginlevel before funds have to be submitted to the broker.b)For a 1% risk the initial margin is set at 6,921foroncrude oil. (This is themaintenance marginof 5,191divided by 0.75.) As the spreadsheet shows,for a longinvestor in oilthere are157margin calls and 9times (out of 1039days) where theinvestor is tempted to walk away.For a 1% risk theinitial margin is set at6,661forgold.(This is 4,996divided by 0.75.) As the spreadsheet shows, for a short investor ingold there are 81 margin calls and 4 times (out of 459days) when the investor istempted to walk away. When the 0.1% risk level is used there is 1 timewhen the oilinvestor might walk away and 2times when the gold investor might do so.

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CHAPTER 3Hedging Strategies Using FuturesPractice QuestionsProblem 3.8.In the Chicago Board of Trade’s corn futures contract, the following delivery months areavailable: March, May, July, September, and December. State the contract that should beused for hedging when the expiration of the hedge is ina)Juneb)Julyc)JanuaryA good rule of thumb is to choose a futures contract that has a delivery month as close aspossible to, but later than, the month containing the expiration of the hedge. The contractsthat should be used are therefore(a)July(b)September(c)MarchProblem 3.9.Does a perfect hedge always succeed in locking in the current spot price of an asset for afuture transaction? Explain your answer.No. Consider, for example, the use of a forward contract to hedge a known cash inflow in aforeign currency. The forward contract locks in the forward exchange rate—which is ingeneral different from the spot exchange rate.Problem 3.10.Explain why a short hedger’s position improves when the basis strengthens unexpectedly andworsens when the basis weakens unexpectedly.The basis is the amount by which the spot price exceeds the futures price. A short hedger islong the asset and short futures contracts. The value of his or her position therefore improvesas the basis increases. Similarly it worsens as the basis decreases.Problem 3.11.Imagine you are the treasurer of a Japanese company exporting electronic equipment to theUnited States. Discuss how you would design a foreign exchange hedging strategy and thearguments you would use to sell the strategy to your fellow executives.The simple answer to this question is that the treasurer should1.Estimate the company’s future cash flows in Japanese yen and U.S. dollars2.Enter into forward and futures contracts to lock in the exchange rate for theU.S. dollar cash flows.However, this is not the whole story. As the gold jewelry example in Table 3.1 shows, thecompany should examine whether the magnitudes of the foreign cash flows depend on theexchange rate. For example, will the company be able to raise the price of its product in U.S.

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dollars if the yen appreciates? If the company can do so, its foreign exchange exposure maybe quite low. The key estimates required are those showing the overall effect on thecompany’s profitability of changes in the exchange rate at various times in the future. Oncethese estimates have been produced the company can choose between using futures andoptions to hedge its risk. The results of the analysis should be presented carefully to otherexecutives. It should be explained that a hedge does not ensure that profits will be higher. Itmeans that profit will be more certain. When futures/forwards are used both the downsideand upside are eliminated. With options a premium is paid toeliminate only the downside.Problem 3.12.Suppose that in Example 3.4 the company decides to use a hedge ratio of 0.8. How does thedecision affect the way in which the hedge is implemented and the result?If the hedge ratio is 0.8, the company takes a long position in 16 NYM December oil futurescontracts on June 8 when the futures price is $68.00. It closes out its position on November10. The spot price and futures price at this time are $75.00 and $72. The gain on the futuresposition is(7268 00)16 00064 000−=The effective cost of the oil is therefore20 0007564 0001 436 000−= or $71.80 per barrel. (This compares with $71.00 per barrel when the company is fullyhedged.)Problem 3.13.“If the minimum-variance hedge ratio is calculated as 1.0, the hedge must be perfect." Is thisstatement true? Explain your answer.The statement is not true. The minimum variance hedge ratio isSF It is 1.0 when0 5=and2SF=. Since1 0the hedge is clearly not perfect.Problem 3.14.“If there is no basis risk, the minimum variance hedge ratio is always 1.0." Is this statementtrue? Explain your answer.The statement is true. Using the notation in the text, if the hedge ratio is 1.0, the hedger locksin a price of12Fb+. Since both1Fand2bare known this has a variance of zeroand must bethe best hedge.Problem 3.15“For an asset where futures prices are usually less than spot prices, long hedges are likely tobe particularly attractive." Explain this statement.A company that knows it will purchase a commodity in the future is able to lock in a priceclose to the futures price. This is likely to be particularly attractive when the futures price isless than the spot price. An illustration is provided by Example 3.2.

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Problem 3.16.The standard deviation of monthly changes in the spot price of live cattle is (in cents perpound) 1.2. The standard deviation of monthly changes in the futures price of live cattle forthe closest contract is 1.4. The correlation between the futures price changes and the spotprice changes is 0.7. It is now October 15. A beef producer is committed to purchasing200,000 pounds of live cattle on November 15. The producer wants to use the December live-cattle futures contracts to hedge its risk. Each contract is for the delivery of 40,000 pounds ofcattle. What strategy should the beef producer follow?The optimal hedge ratio is1 20 70 61 4=The beef producer requires a long position in2000000 6120 000=lbs of cattle. The beefproducer should therefore take a long position in 3 December contracts closing outtheposition on November 15.Problem 3.17.A corn farmer argues “I do not use futures contracts for hedging. My real risk is not theprice of corn. It is that my whole crop gets wiped out by the weather.”Discuss this viewpoint.Should the farmer estimate his or her expected production of corn and hedge to try to lock ina price for expected production?If weather creates a significant uncertainty about the volume of corn that will be harvested,the farmer should not enter into short forward contracts to hedge the price risk on his or herexpected production. The reason is as follows. Suppose that the weather is bad and thefarmer’s production is lower than expected. Other farmers are likely to have been affectedsimilarly. Corn production overall will be low and as a consequence the price of corn will berelatively high. The farmer’s problems arising from the bad harvest will be made worse bylosses on the short futures position. This problem emphasizes the importance of looking atthe big picture when hedging. The farmer is correct to question whether hedging price riskwhile ignoring other risks is a good strategy.Problem 3.18.On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 pershare. The investor is interested in hedging against movements in the market over the nextmonth and decides to use the September Mini S&P 500 futures contract. The index iscurrently 1,500 and one contract is for delivery of $50 times the index. The beta of the stockis 1.3. What strategy should the investor follow? Under what circumstances will it beprofitable?A short position in50 000301 326501 500 = contracts is required. It will be profitable if the stock outperforms the market in the sense thatits return is greater than that predicted by the capital asset pricingmodel.Problem 3.19.Suppose that in Table 3.5 the company decides to use a hedge ratio of 1.5. How does the

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decision affect the way the hedge is implemented and the result?If the company uses a hedge ratio of 1.5 in Table 3.5 it would at each stage short 150contracts. The gain from the futures contracts would be55.2$70.150.1=per barreland the company would be$0.85 per barrel better off.Problem 3.20.A futures contract is used for hedging. Explain why the daily settlement of the contract cangive rise to cash flow problems.Suppose that you enter into a short futures contract to hedge the sale of a asset in six months.If the price of the asset rises sharply during the six months, the futures price will also rise andyou may get margin calls. The margin calls will lead to cash outflows. Eventually the cashoutflows will be offset by the extra amount you get when you sell the asset, but there is amismatch in the timing of the cash outflows and inflows. Your cash outflows occur earlierthan your cash inflows. A similar situation could arise if you used a long position in a futurescontract to hedge the purchase of an asset and the asset’s price fell sharply. An extremeexample of what we are talking about here is provided by Metallgesellschaft(see BusinessSnapshot 3.2).Problem 3.21.The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expectedreturn on the investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?a)0 050 2(0 120 05)0 064+ −=or 6.4%b)0 050 5(0 120 05)0 085+ −=or 8.5%c)0 051 4(0 120 05)0 148+  −=or 14.8%Further QuestionsProblem 3.22A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6correlation with gasoline futures pricechanges. The company will lose$1 million for each 1cent increase in the price per gallon of the new fuel over the next three months. The newfuel's price change hasa standard deviation that is 50% greater than price changes ingasoline futures prices. If gasoline futures are used to hedge the exposure what should thehedge ratio be? What is the company's exposure measuredin gallons of the new fuel? Whatposition measured in gallons should the company take in gasoline futures? How manygasoline futures contracts should be traded?The hedge ratio should be 0.6 × 1.5 = 0.9. The company has an exposure to the price of 100million gallons of the new fuel. If should therefore take a position of 90 million gallons ingasoline futures. Each futures contract is on 42,000 gallons. The number of contracts requiredis therefore9.2142000,42000,000,90=

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or, rounding to the nearest whole number, 2143.Problem 3.23A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. Duringthe last year the risk-free rate was 5% andequities performed very badly providing a returnof −30%. The portfolio manage produced a return of −10% and claims that inthecircumstances it was good. Discuss this claim.When the expected return on the market is −30% the expected return on a portfoliowith abeta of 0.2 is0.05 + 0.2×(−0.30− 0.05) = −0.02or–2%. The actual return of–10% is worse than the expected return. The portfoliomanagerhas achieved an alpha of–8%!Problem 3.24.It is July 16. Acompany has a portfolio of stocks worth $100 million. The beta of theportfolio is 1.2. The company would like to use the CME December futures contract on theS&P 500 to change the 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 theportfolio from 1.2 to 1.5. What position in futures contracts should it take?a)The company should short(1 20 5)100 000 0001000250−or 280 contracts.b)The company should take a long position in(1 51 2)100 000 0001000250− or 120 contracts.Problem 3.25.(Excel file)The following table gives data on monthly changes in the spot price and the futures price fora certain commodity. Use the data to calculate a minimum variance hedge ratio.Spot Price Change0 50+ 0 61+ 0 22− 0 35− 0 79+ Futures Price Change0 56+ 0 63+ 0 12− 0 44− 0 60+ Spot Price Change0 04+ 0 15+ 0 70+ 0 51− 0 41− Futures Price Change0 06− 0 01+ 0 80+ 0 56− 0 46− Denoteixandiyby thei-th observation on the change in the futures price and the change inthe spot price respectively.

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0 961 30iixy==222 44742 3594iixy==2 352iix y=An estimate ofFis22 44740 960 51169109−=An estimate ofSis22 35941 300 49339109−=An estimate ofis22102 3520 961 300 981(102 44740 96 )(102 35941 30 )− =−− The minimum variance hedge ratio is0 49330 9810 9460 5116SF== Problem 3.26.It is now October 2010. A company anticipates that it will purchase 1 million pounds ofcopperin each of February 2011, August 2011, February 2012, and August 2012. Thecompany has decided to use the futures contracts traded in the COMEX division ofthe CMEGroupto hedge its risk. One contract is for the delivery of 25,000 pounds of copper. Theinitial margin is $2,000 per contract and the maintenance margin is $1,500 per contract. Thecompany’s policy is to hedge 80% of its exposure. Contracts with maturities up to 13 monthsinto the future are considered to have sufficient liquidity to meet the company’s needs. Devisea hedging strategy for the company.Assume the market prices (in cents per pound) today and at future dates are as follows. Whatis the impact of the strategy you propose on the price the company pays for copper? What isthe initial margin requirement in October 2010? Is the company subject to any margin calls?DateOct 2010Feb 2011Aug 2011Feb 2012Aug 2012Spot Price372.00369.00365.00377.00388.00Mar 2011Futures Price372.30369.10Sep 2011Futures Price372.80370.20364.80Mar2012Futures Price370.70364.30376.70Sep 2012Futures Price364.20376.50388.20To hedge the February 2011purchase the company should take a long position in March2011contracts for the delivery of 800,000 pounds of copper. The total number of contractsrequired is800 00025 00032=. Similarly a long position in 32 September 2011contractsis required to hedge the August 2011 purchase. For the February 2012purchase the companycould take a long position in 32 September 2011contracts and roll them into March 2012contracts during August 2011. (As an alternative, the companycould hedge the February2012purchase by takinga long position in 32 March 2011contractsand rolling them into

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March 2012 contracts.) For the August 2012purchase the company could take a long positionin 32 September 2011and roll them into September 2012 contracts during August 2011.The strategy is therefore as followsOct. 2010:Enter into long position in 96 Sept. 2008 contractsEnter into a long position in 32 Mar. 2008 contractsFeb 2011:Close out 32 Mar. 2008 contractsAug 2011:Close out 96 Sept. 2008 contractsEnter into long position in 32 Mar. 2009 contractsEnter into long position in 32 Sept. 2009 contractsFeb 2012:Close out 32 Mar. 2009 contractsAug 2012:Close out 32 Sept. 2009 contractsWith the market prices shown the company pays369 000 8(372 30369 10)371 56+ −=for copper in February, 2011. It pays365 000 8(372 80364 80)371 40+ −=for copper in August 2011. As far as the February 2012purchase is concerned, it loses372 80364 808 00−= on the September 2011futures and gains376 70364 3012 40−=onthe February 2012futures. The net price paid is therefore377 000 88 000 8 12 40373 48+   −  =As far as the August 2012purchase is concerned, it loses372 80364 808 00−= on theSeptember 2011futures and gains388 20364 2024 00−=on the September 2012futures.The net price paid is therefore388 000 88 000 824 00375 20+   −  =The hedging strategy succeeds in keeping the price paid in the range 371.40 to 375.20.In October 2010the initial margin requirement on the 128 contracts is1282 000$or$256,000. There is a margin call when the futures price drops by more than 2 cents. Thishappens to the March 2011 contract between October 2010and February 2011, to theSeptember 2011 contract between October 2010 and February 2011, and to the September2011 contract between February 2011 and August 2011.Problem 3.27.(Excel file)A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager isconcerned about the performance of the market over the next two months and plans to usethree-month futures contracts on the S&P 500 to hedge the risk. The current level of theindex is 1250, one contract is on 250 times the index, the risk-free rate is 6% per annum, andthe 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 marketover the next two months?b)Calculate the effect of your strategy on the fund manager’s returns if the level of themarket in two months is 1,000, 1,100, 1,200, 1,300, and 1,400. Assume that the one-month futures price is 0.25% higher than the index level at this time.a)The number of contracts the fund manager should short is50 000 0000 87138 201259250=Rounding to the nearest whole number, 138 contracts should be shorted.

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b)The following table shows that the impact of the strategy. To illustrate thecalculations in the table consider the first column. If the index in two months is 1,000,the futures price is1000×1.0025.The gain on the short futures position is therefore(12591002 50)2501388 849 250$−=The return on the index is32 12=0.5% in the form of dividend and250 125020%−= −in the form of capital gains. The total return on the index istherefore19 5%−. The risk-free rate is 1% per two months. The return is therefore20 5%−in excess of the risk-free rate. From the capital asset pricing model weexpect the return on the portfolio to be0 8720 517 835%%−= −in excess of therisk-free rate. The portfolio return is therefore16 835%−. The loss on the portfolio is0 1683550 000 000or $8,417,500. When this is combined with the gain on thefutures the total gain is $431,750.Index now12501250125012501250Index Level in Two Months10001100120013001400Return on Index in Two Months-0.20-0.12-0.040.040.12Return on Index incl divs-0.195-0.115-0.0350.0450.125Excess Return on Index-0.205-0.125-0.0450.0350.115Excess Return on Portfolio-0.178-0.109-0.0390.0300.100Return on Portfolio-0.168-0.099-0.0290.0400.110Portfolio Gain-8,417,500 -4,937,500 -1,457,5002,022,5005,502,500Futures Now12591259125912591259Futures in Two Months1002.501102.751203.001303.251403.50Gain on Futures8,849,2505,390,6251,932,000 -1,526,625 -4,985,250Net Gain on Portfolio431,750453,125474,500495,875517,250

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CHAPTER 4Interest RatesPractice QuestionsProblem 4.8.The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year bondthat will pay coupons of $4 every six months currently sells for $94.84. A two-year bond thatwill pay coupons of $5 every six months currently sells for $97.12. Calculate the six-month,one-year, 1.5-year, and two-year zero rates.The 6-month Treasury bill provides a return of6 946 383%=in six months. This is26 38312 766% =per annum with semiannual compounding or2 ln(1 06383)12 38%=per annum with continuous compounding. The 12-month rate is11 8912 360%=withannual compounding orln(1 1236)11 65%=with continuous compounding.For the 112year bond we must have0 1238 0 50 1165 11 54410494 84Reee−  − − ++=whereRis the 112year zero rate. It follows that1 51 53 763 5610494 840 84150 115RReeR− − ++===or 11.5%. For the 2-year bond we must have0 1238 0 50 1165 10 115 1 5255510597 12Reeee−  − −  −+++=whereRis the 2-year zero rate. It follows that20 79770 113ReR−==or 11.3%.Problem 4.9.What rate of interest with continuous compounding is equivalent to 15% per annum withmonthly compounding?The rate of interest isRwhere:120 15112Re=+i.e.,0 1512 ln 112R=+0 1491=The rate of interest is therefore 14.91% per annum.

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Problem 4.10.A deposit account pays 12% per annum with continuous compounding, but interest is actuallypaid quarterly. How much interest will be paid each quarter on a $10,000 deposit?The equivalent rate of interest with quarterly compounding isRwhere40 1214Re=+or0 034(1)0 1218Re=−=The amount of interest paid each quarter is therefore:0 121810 000304 554=or $304.55.Problem 4.11.Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%,4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. Estimatethe cash price of a bond with a face value of 100 that will mature in 30 months and pays acoupon of 4% per annum semiannually.The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price is0 04 0 50 042 1 00 044 1 50 046 20 048 2 5222210298 04eeeee−  −  −  − −  ++++=Problem 4.12.A three-year bond provides a coupon of 8% semiannually and has a cash price of 104. Whatis the bond’s yield?The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yield isthe value ofythat solves0 51 01 52 02 53 044444104104yyyyyyeeeeee− − − − − − +++++=Using theGoal Seektool in Excel0 06407y=or 6.407%.Problem 4.13.Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%,and 7% respectively. What is the two-year par yield?Using the notation in the text,2m=,0 07 20 8694de− ==. Also0 05 0 50 06 1 00 065 1 50 07 2 03 6935Aeeee−  −  −  −  =+++=The formula in the text gives the par yield as(1001000 8694)27 0723 6935−=To verify that this is correct we calculate the value of a bond that pays a coupon of 7.072%per year (that is 3.5365 every six months). The value is0 05 0 50 06 1 00 065 1 50 07 2 03 5363 53653 536103 536100eeee−  −  −  −  +++=

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verifying that 7.072% is the par yield.Problem 4.14.Suppose that zero interest rates with continuous compounding are as follows:Maturity( years)Rate (% per annum)12.023.033.744.254.5Calculate forward interest rates for the second, third, fourth, and fifth years.The forward rates with continuous compounding are as follows: toYear 2:4.0%Year 3:5.1%Year 4:5.7%Year 5:5.7%Problem 4.15.Use the rates in Problem 4.14 to value an FRA where you will pay 5% for the third year on$1 million.The forward rate is 5.1% with continuous compounding or0 051 115 232e%−=with annualcompounding. The 3-year interest rate is 3.7% with continuous compounding. From equation(4.10), the value of the FRA is therefore0 037 3[1 000 000(0 052320 05)1]2 078 85e− −=or $1,964.67.Problem 4.16.A 10-year, 8% coupon bond currently sells for $90. A 10-year, 4% coupon bond currentlysells for $80. What is the 10-year zero rate? (Hint: Consider taking a long position in two ofthe 4% coupon bonds and a short position in one of the 8% coupon bonds.)Taking a long position in two of the 4% coupon bonds and a short position in one of the 8%coupon bonds leads to the following cash flowsYear0 9028070Year10 200100100−= −−=because the coupons cancel out. $100 in 10 years time is equivalent to $70 today. The 10-year rate,R, (continuously compounded) is therefore given by1010070Re=The rate is1100ln0 03571070=or 3.57% per annum.Problem 4.17.Explain carefully why liquidity preference theory is consistent with the observation that the

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term structure of interest rates tends to be upward sloping more often than it is downwardsloping.If long-term rates were simply a reflection of expected future short-term rates, we wouldexpect the term structure to be downward sloping as often as it is upward sloping. (This isbased on the assumption that half of the time investors expect rates to increase and half of thetime investors expect rates to decrease). Liquidity preference theory argues that long termrates are high relative to expected future short-term rates. This means that the term structureshould be upward sloping more often than it is downward sloping.Problem 4.18.“When the zero curve is upward sloping, the zero rate for a particular maturity is greaterthan the par yield for that maturity. When the zero curve is downward sloping the reverse istrue.” Explain why this is so.The par yield is the yield on a coupon-bearing bond. The zero rate is the yield on a zero-coupon bond. When the yield curve is upward sloping, the yield on anN-year coupon-bearing bond is less than the yield on anN-year zero-coupon bond. This is because thecoupons are discounted at a lower rate than theN-year rate and drag the yield down belowthis rate. Similarly, when the yield curve is downward sloping, the yield on anN-yearcoupon bearing bond is higher than the yield on anN-year zero-coupon bond.Problem 4.19.Why are U.S. Treasury rates significantly lower than other rates that are close to risk free?There are three reasons (see Business Snapshot 4.1).1.Treasury bills and Treasury bonds must be purchased by financial institutions to fulfill avariety of regulatory requirements. This increases demand for these Treasury instrumentsdriving the price up and the yield down.2.The amount of capital a bank is required to hold to support an investment in Treasurybills and bonds is substantially smaller than the capital required to support a similarinvestment in other very-low-risk instruments.3.In the United States, Treasury instruments are given a favorable tax treatment comparedwith most other fixed-income investments because they are not taxed at the state level.Problem 4.20.Why does a loan in the repo market involve very little credit risk?A repo is a contract where an investment dealer who owns securities agrees to sell them toanother company now and buy them back later at a slightly higher price. The other companyis providing a loan to the investment dealer. This loan involves very little credit risk. If theborrower does not honor the agreement, the lending company simply keeps the securities. Ifthe lending company does not keep to its side of the agreement, the original owner of thesecurities keeps the cash.Problem 4.21.Explain why an FRA is equivalent to the exchange of a floating rate of interest for a fixedrate of interest?A FRA is an agreement that a certain specified interest rate,KR, will apply to a certain

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principal,L, for a certain specified future time period.Suppose that the rate observed in themarket for the future time period at the beginning of the time periodproves to beMR. If theFRA is an agreement thatKRwill apply when the principal is invested, the holder of theFRA can borrow the principal atMRand then invest it atKR.The net cash flow at the end ofthe period is then aninflowofKR Land an outflow ofMRL. If the FRA is an agreement thatKRwill apply when the principal is borrowed, the holder of the FRA can invest the borrowedprincipal atMR. The net cash flow at the end of the period is then an inflow ofMRLand anoutflow ofKR L. In either case we see that the FRA involves the exchange of a fixed rate ofinterest on the principal ofLfor a floating rate of interest on the principal.Problem 4.22.“An interest rate swap where six-month LIBOR is exchanged for a fixed rate 5% on aprincipal of $100 million is a portfolio of FRAs.” Explain.Each exchange of payments is an FRA where interest at 5% is exchanged for interest atLIBOR on a principal of $100 million. Interest rate swaps are discussed further in Chapter 7.Further QuestionsProblem 4.23(Excel file)A five-year bond provides a coupon of 5%per annumpayable semiannually. Its price is 104.What is the bond's yield? You may find Excel's Solver useful.The answer (with continuous compounding is 4.07%Problem 4.24(Excel file)Suppose that LIBOR rates for maturities of one month, twomonths, three months, fourmonths, five months and six monthsare 2.6%, 2.9%, 3.1%, 3.2%, 3.25%, and 3.3% withcontinuous compounding. What are the forward rates for future one month periods?The forward rates for the second, third, fourth, fifth and sixth months are (see spreadsheet)3.2%, 3.5%, 3.5%, 3.45%, 3.55%, respectively with continuous compounding.Problem 4.25A bank can borrow or lend at LIBOR. The two-month LIBOR rate is 0.28% per annum withcontinuous compounding. Assuming that interest rates cannot be negative, what is thearbitrage opportunity if the three-month LIBOR rate is 0.1% per year with continuouscompounding. How low can the three-month LIBOR rate become without an arbitrageopportunity being created?The forward rate for the third month is 0.001×3 − 0.0028×2 = − 0.0026 or−0.26%. If weassume that the rate for the third month will not be negative we can borrow for three months,lend for two months and lend at the market rate for the third month. The lowest level for thethree-month rate that does not permit this arbitrage is 0.0028×2/3 = 0.001867 or 0.1867%.Problem 4.26A bank can borrow or lend at LIBOR. Suppose that the six-month rate is 5% and the nine-

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month rate is 6%. The rate that can be locked in for the period between six months and ninemonths using an FRA is 7%. What arbitrage opportunities are open to the bank? All rates arecontinuously compounded.The forward rate is08.025.050.005.075.006.0=−or 8%. The FRA rate is 7%. A profit can therefore be made by borrowing for six months at5%, entering into an FRA to borrow for the period between 6 and 9 months for 7% andlending for nine months at 6%.Problem 4.27.An interest rate is quoted as 5% per annum with semiannual compounding. What is theequivalent rate with (a) annual compounding, (b) monthly compounding, and (c) continuouscompounding.a)With annual compounding the rate is21 02510 050625−=or 5.0625%b)With monthly compounding the rate is1 612(1 0251)0 04949−=or 4.949%.c)With continuous compounding the rate is2ln1 0250 04939=or 4.939%.Problem 4.28.The 6-month, 12-month. 18-month,and 24-month zero rates are 4%, 4.5%, 4.75%, and 5%with semiannual compounding.a)What are the rates with continuous compounding?b)What is the forward rate for the six-month period beginning in 18 monthsc)What is the value of an FRA that promises to pay you 6% (compounded semiannually)on a principal of $1 million for the six-month period starting in 18 months?a)With continuous compounding the 6-month rate is2ln1 020 039605=or 3.961%.The 12-month rate is2ln1 02250 044501=or 4.4501%. The 18-month rate is2ln1 023750 046945=or 4.6945%. The 24-month rate is2ln1 0250 049385=or4.9385%.b)The forward rate (expressed with continuous compounding) is from equation (4.5)4 938524 69451 50 5− or 5.6707%. When expressed with semiannual compounding this is0 056707 0 52(1)0 057518e −=or 5.7518%.c)The value of an FRA that promises to pay 6% for the six month period starting in 18months is from equation (4.9)0 049385 21 000 000(0 060 057518)0 51 124e− − = or $1,124.Problem 4.29.What is the two-year par yield when the zero rates are as in Problem 4.28? What is the yieldon a two-year bond that pays a coupon equal to the par yield?The value,Aof an annuity paying off $1 every six months is0 039605 0 50 044501 10 046945 1 50 049385 23 7748eeee−  − −  − +++=

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The present value of $1 received in two years,d, is0 049385 20 90595e− =. From the formulain Section 4.4 the par yield is(1001000 90595)24 9833 7748−=or 4.983%.Problem 4.30.The following table gives the prices of bondsBond Principal ($)Time to Maturity (yrs)Annual Coupon ($)*Bond Price ($)1000.50.0981001.00.0951001.56.21011002.08.0104*Half the stated coupon is paid every six monthsa)Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24months.b)What are the forward rates for the periods: 6 months to 12 months, 12 months to 18months, 18 months to 24 months?c)What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds thatprovide semiannual coupon payments?d)Estimate the price and yield of a two-year bond providing a semiannual coupon of 7%per annum.a)The zero rate for a maturity of six months, expressed with continuous compounding is2ln(1298)4 0405%+=. The zero rate for a maturity of one year, expressed withcontinuous compounding isln(1595)5 1293+=. The 1.5-year rate isRwhere0 040405 0 50 051293 11 53 13 1103 1101Reee−  − − ++=The solution to this equation is0 054429R=. The 2.0-year rate isRwhere0 040405 0 50 051293 10 054429 1 52444104104Reeee−  − −  −+++=The solution to this equation is0 058085R=. These results are shown in the tablebelowMaturity (yrs)Zero Rate (%)Forward Rate (%)Par Yield (s.a.%)Par yield (c.c %)0.54.04054.04054.08164.04051.05.12936.21815.18135.11541.55.44296.07005.49865.42442.05.80856.90545.86205.7778b)The continuously compounded forward rates calculated using equation (4.5) areshown in the third column of the tablec)The par yield, expressed with semiannual compounding, can be calculated from theformula in Section 4.4. It is shown in the fourth column of the table. In the fifthcolumn of the table it is converted to continuous compounding

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