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25 -0 Chapter Twenty Five Derivatives and Hedging Corporate Finance Ross Westerfield Jaffe Risk 25 -0 Chapter Twenty Five Derivatives and Hedging Corporate Finance Ross Westerfield Jaffe Risk 25 Sixth Edition Prepared by Gady Jacoby University of Manitoba and Sebouh Aintablian American University of Beirut Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -1 Chapter Outline 25. 1 Forward Contracts 25. 2 Futures Contracts 25. 3 25 -1 Chapter Outline 25. 1 Forward Contracts 25. 2 Futures Contracts 25. 3 Hedging 25. 4 Interest Rate Futures Contracts 25. 5 Duration Hedging 25. 6 Swap Contracts 25. 7 Actual Use of Derivatives 25. 8 Summary & Conclusions Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -2 25. 1 Forward Contracts • A forward contract specifies that a certain 25 -2 25. 1 Forward Contracts • A forward contract specifies that a certain commodity will be exchanged for another at a specified time in the future at prices specified today. – It’s not an option: both parties are expected to hold up their end of the deal. – If you have ever ordered a textbook that was not in stock, you have entered into a forward contract. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -3 25. 2 Futures Contracts: Preliminaries • A futures contract is like a 25 -3 25. 2 Futures Contracts: Preliminaries • A futures contract is like a forward contract: – It specifies that a certain commodity will be exchanged for another at a specified time in the future at prices specified today. • A futures contract is different from a forward: – Futures are standardized contracts trading on organized exchanges with daily resettlement (“marking to market”) through a clearinghouse. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -4 Futures Contracts: Preliminaries • Standardizing Features: – Contract Size – Delivery Month 25 -4 Futures Contracts: Preliminaries • Standardizing Features: – Contract Size – Delivery Month • Daily resettlement – Minimizes the chance of default • Initial Margin – About 4% of contract value, cash or T-bills held in a street name at your brokerage. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -5 Daily Resettlement: An Example Suppose you want to speculate on a rise 25 -5 Daily Resettlement: An Example Suppose you want to speculate on a rise in the US$/¥ exchange rate (specifically you think that the dollar will appreciate). Currently US$1 = ¥ 140. The 3 -month forward price is US$1=¥ 150. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -6 Daily Resettlement: An Example • Currently US$1 = ¥ 140 and it 25 -6 Daily Resettlement: An Example • Currently US$1 = ¥ 140 and it appears that the dollar is strengthening. • If you enter into a three-month futures contract to sell ¥ at the rate of US$1 = ¥ 150 you will make money if the yen depreciates. The contract size is ¥ 12, 500, 000 • Your initial margin is 4% of the contract value: Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -7 Daily Resettlement: An Example If tomorrow, the futures rate closes at US$1 25 -7 Daily Resettlement: An Example If tomorrow, the futures rate closes at US$1 = ¥ 149, then your position’s value drops. Your original agreement was to sell ¥ 12, 500, 000 and receive US$83, 333. 33: But ¥ 12, 500, 000 is now worth US$83, 892. 62: You have lost US$559. 28 overnight. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -8 Daily Resettlement: An Example • The US$559. 28 comes out of your 25 -8 Daily Resettlement: An Example • The US$559. 28 comes out of your US$3, 333. 33 margin account, leaving US$2, 774. 05 • This is short of the US$3, 355. 70 required for a new position. Your broker will let you slide until you run through your maintenance margin. Then you must post additional funds or your position will be closed out. This is usually done with a reversing trade. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -9 Selected Futures Contracts Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited 25 -9 Selected Futures Contracts Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -10 Futures Markets • The Chicago Mercantile Exchange (CME) is by far the 25 -10 Futures Markets • The Chicago Mercantile Exchange (CME) is by far the largest. • In Canada: – Montreal Futures Exchange (MFE) – Winnipeg Commodity Exchange (WCE) • Others include: – – The Philadelphia Board of Trade (PBOT) The Mid. America Commodities Exchange The Tokyo International Financial Futures Exchange The London International Financial Futures Exchange Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -11 The Winnipeg Commodity Exchange Canola Futures • Expiry cycle: January, March, May, 25 -11 The Winnipeg Commodity Exchange Canola Futures • Expiry cycle: January, March, May, July, September, November. • First delivery day is the first business day of the delivery month. • Last trading day is seven clear business days prior to the end of the delivery month. • Trading hours 9: 30 a. m. to 1: 15 p. m. CT. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -12 The Chicago Mercantile Exchange • Expiry cycle: March, June, September, December. • 25 -12 The Chicago Mercantile Exchange • Expiry cycle: March, June, September, December. • Delivery date third Wednesday of delivery month. • Last trading day is the second business day preceding the delivery day. • CME hours 7: 20 a. m. to 2: 00 p. m. CST. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -13 CME After Hours • Extended-hours trading on GLOBEX runs from 2: 30 25 -13 CME After Hours • Extended-hours trading on GLOBEX runs from 2: 30 p. m. to 4: 00 p. m dinner break and then back at it from 6: 00 p. m. to 6: 00 a. m. CST. • Singapore International Monetary Exchange (SIMEX) offers interchangeable contracts. • There are other markets, but none are close to CME and SIMEX trading volume. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -14 National Post Futures Price Quotes Highest and lowest prices over the lifetime 25 -14 National Post Futures Price Quotes Highest and lowest prices over the lifetime of the contract. Highest price that day Closing price Opening price Mc. Graw-Hill Ryerson Daily Change Lowest price that day Number of open contracts Expiry month © 2003 Mc. Graw–Hill Ryerson Limited

25 -15 Basic Currency Futures Relationships • Open Interest refers to the number of 25 -15 Basic Currency Futures Relationships • Open Interest refers to the number of contracts outstanding for a particular delivery month. • Open interest is a good proxy for demand for a contract. • Some refer to open interest as the depth of the market. The breadth of the market would be how many different contracts (expiry month, currency) are outstanding. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -16 25. 3 Hedging • Two counterparties with offsetting risks can eliminate risk. 25 -16 25. 3 Hedging • Two counterparties with offsetting risks can eliminate risk. – For example, if a wheat farmer and a flour mill operator enter into a forward contract, they can eliminate the risk each other faces regarding the future price of wheat. • Hedgers can also transfer price risk to speculators and speculators absorb price risk from hedgers. • Speculating: Long vs. Short Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -17 Hedging and Speculating Example You speculate that copper will go up in 25 -17 Hedging and Speculating Example You speculate that copper will go up in price, so you go long 10 copper contracts for delivery in three months. A contract is 25, 000 pounds in cents per pound and is at US$0. 70 per pound or US$17, 500 per contract. If futures prices rise by 5 cents, you will gain: Gain = 25, 000 ×. 05 × 10 = US$12, 500 If prices decrease by 5 cents, your loss is: Loss = 25, 000 × -. 05 × 10 = -US$12, 500 Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -18 Hedging: How many contacts? You are a farmer and you will harvest 25 -18 Hedging: How many contacts? You are a farmer and you will harvest 50, 000 bushels of corn in three months. You want to hedge against a price decrease. Corn is quoted in U. S. cents per bushel at 5, 000 bushels per contract. It is currently at 230 cents for a contract three months out and the spot price is US$2. 05. To hedge you will sell 10 corn futures contracts: Now you can quit worrying about the price of corn and get back to worrying about the weather. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -19 25. 4 Interest Rate Futures Contracts • • Pricing of Treasury Bonds 25 -19 25. 4 Interest Rate Futures Contracts • • Pricing of Treasury Bonds Pricing of Forward Contracts Futures Contracts Hedging in Interest Rate Futures Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -20 Pricing of Government of Canada Bonds Consider a Government of Canada bond 25 -20 Pricing of Government of Canada Bonds Consider a Government of Canada bond that pays a semiannual coupon of $C for the next T years: – The yield to maturity is r … 0 1 2 3 2 T Value of the bond under a flat term structure = PV of face value + PV of coupon payments Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -21 Pricing of Government of Canada Bonds If the term structure of interest 25 -21 Pricing of Government of Canada Bonds If the term structure of interest rates is not flat, then we need to discount the payments at different rates depending upon maturity … 0 1 2 3 2 T = PV of face value + PV of coupon payments Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -22 Pricing of Forward Contracts An N-period forward contract on that Government of 25 -22 Pricing of Forward Contracts An N-period forward contract on that Government of Canada Bond … 0 N N+1 N+2 N+3 N+2 T Can be valued as the present value of the forward price. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -23 Futures Contracts • The pricing equation given above will be a good 25 -23 Futures Contracts • The pricing equation given above will be a good approximation. • The only real difference is the daily resettlement. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -24 Hedging in Interest Rate Futures • A mortgage lender who has agreed 25 -24 Hedging in Interest Rate Futures • A mortgage lender who has agreed to loan money in the future at prices set today can hedge by selling those mortgages forward. • It may be difficult to find a counterparty in the forward who wants the precise mix of risk, maturity, and size. • It’s likely to be easier and cheaper to use interest rate futures contracts however. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -25 25. 5 Duration Hedging • As an alternative to hedging with futures 25 -25 25. 5 Duration Hedging • As an alternative to hedging with futures or forwards, one can hedge by matching the interest rate risk of assets with the interest rate risk of liabilities. • Duration is the key to measuring interest rate risk. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -26 25. 5 Duration Hedging • Duration measures the combined effect of maturity, 25 -26 25. 5 Duration Hedging • Duration measures the combined effect of maturity, coupon rate, and YTM on bond’s price sensitivity – Measure of the bond’s effective maturity – Measure of the average life of the security – Weighted average maturity of the bond’s cash flows Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -27 Duration Formula Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited 25 -27 Duration Formula Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -28 Calculating Duration Calculate the duration of a three-year bond that pays a 25 -28 Calculating Duration Calculate the duration of a three-year bond that pays a semi-annual coupon of $40, has a $1, 000 par value when the YTM is 8% semiannually. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -29 Calculating Duration is expressed in units of time; usually years. Mc. Graw-Hill 25 -29 Calculating Duration is expressed in units of time; usually years. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -30 Duration The key to bond portfolio management • Properties: – Longer maturity, 25 -30 Duration The key to bond portfolio management • Properties: – Longer maturity, longer duration – Duration increases at a decreasing rate – Higher coupon, shorter duration – Higher yield, shorter duration • Zero coupon bond: duration = maturity Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -31 25. 6 Swaps Contracts: Definitions • In a swap, two counterparties agree 25 -31 25. 6 Swaps Contracts: Definitions • In a swap, two counterparties agree to a contractual arrangement wherein they agree to exchange cash flows at periodic intervals. • There are two types of interest rate swaps: – Single currency interest rate swap • “Plain vanilla” fixed-for-floating swaps are often just called interest rate swaps. – Cross-currency interest rate swap • This is often called a currency swap; fixed for fixed rate debt service in two (or more) currencies. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -32 The Swap Bank • A swap bank is a generic term to 25 -32 The Swap Bank • A swap bank is a generic term to describe a financial institution that facilitates swaps between counterparties. • The swap bank can serve as either a broker or a dealer. – As a broker, the swap bank matches counterparties but does not assume any of the risks of the swap. – As a dealer, the swap bank stands ready to accept either side of a currency swap, and then later lay off their risk, or match it with a counterparty. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -33 An Example of an Interest Rate Swap • Consider this example of 25 -33 An Example of an Interest Rate Swap • Consider this example of a “plain vanilla” interest rate swap. • Bank A is a AAA-rated international bank located in the U. K. and wishes to raise US$10, 000 to finance floating-rate eurodollar loans. – Bank A is considering issuing five-year fixed-rate eurodollar bonds at 10 -percent. – It would make more sense to for the bank to issue floatingrate notes at LIBOR to finance floating-rate eurodollar loans. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -34 An Example of an Interest Rate Swap • Firm B is a 25 -34 An Example of an Interest Rate Swap • Firm B is a BBB-rated U. S. company. It needs US$10, 000 to finance an investment with a five -year economic life. – Firm B is considering issuing five-year fixed-rate eurodollar bonds at 11. 75 -percent. – Alternatively, firm B can raise the money by issuing fiveyear floating-rate notes at LIBOR + 0. 5 -percent. – Firm B would prefer to borrow at a fixed rate. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -35 An Example of an Interest Rate Swap The borrowing opportunities of the 25 -35 An Example of an Interest Rate Swap The borrowing opportunities of the two firms are: Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -36 An Example of an Interest Rate Swap 10. 375% Bank LIBOR – 25 -36 An Example of an Interest Rate Swap 10. 375% Bank LIBOR – 0. 125% Bank A Mc. Graw-Hill Ryerson The swap bank makes this offer to Bank A: You pay LIBOR – 0. 125% per year on US$10 million for five years and we will pay you 10. 375% on US$10 million for five years © 2003 Mc. Graw–Hill Ryerson Limited

25 -37 An Example of an Interest Rate Swap 0. 5% of US$10, 000 25 -37 An Example of an Interest Rate Swap 0. 5% of US$10, 000 = US$50, 000. That’s quite a cost savings per year for five years. 10. 375% Swap Bank LIBOR – 0. 125% Bank 10% A Mc. Graw-Hill Ryerson Here’s what’s in it for Bank A: They can borrow externally at 10% fixed and have a net borrowing position of -10. 375% + 10% + (LIBOR – 0. 125%) = LIBOR – 0. 5% which is 0. 5% better than they can borrow floating without a swap. © 2003 Mc. Graw–Hill Ryerson Limited

25 -38 An Example of an Interest Rate Swap The swap bank makes this 25 -38 An Example of an Interest Rate Swap The swap bank makes this offer to company B: You pay us 10. 5% per year on US$10 million for five years and we will pay you LIBOR – 0. 25% per year on US$10 million for five years. Mc. Graw-Hill Ryerson Swap Bank 10. 5% LIBOR – 0. 25% Company B © 2003 Mc. Graw–Hill Ryerson Limited

25 -39 An Example of an Interest Rate Swap Here’s what’s in it for 25 -39 An Example of an Interest Rate Swap Here’s what’s in it for B: Swap Bank They can borrow externally at LIBOR + 0. 5 % and have a net 0. 5 % of US$10, 000 = US$50, 000 that’s quite a cost savings per year for five years. 10. 5% LIBOR – 0. 25% borrowing position of 10. 5 + (LIBOR + 0. 5 ) - (LIBOR – 0. 25 ) = 11. 25% which is 0. 5% better than they can borrow floating. Mc. Graw-Hill Ryerson Company B LIBOR + 0. 5% © 2003 Mc. Graw–Hill Ryerson Limited

25 -40 An Example of an Interest Rate Swap The swap bank makes money 25 -40 An Example of an Interest Rate Swap The swap bank makes money too. Swap 10. 375% Bank 10. 5% LIBOR – 0. 125% Bank A 0. 25% of US$10 million = US$25, 000 per year for five years. LIBOR – 0. 24% LIBOR – 0. 125 – [LIBOR – 0. 25]= 0. 125 10. 5 – 10. 375 = 0. 125 Company B 0. 250 Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -41 An Example of an Interest Rate Swap The swap bank makes 0. 25 -41 An Example of an Interest Rate Swap The swap bank makes 0. 25% Swap 10. 375% Bank LIBOR – 0. 125% 10. 5% LIBOR – 0. 25% Bank Company A B A saves 0. 5% B saves 0. 5% Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -42 An Example of a Currency Swap • Suppose a U. S. MNC 25 -42 An Example of a Currency Swap • Suppose a U. S. MNC wants to finance a £ 10, 000 expansion of a British plant. • They could borrow dollars in the U. S. where they are well known and exchange dollars for pounds. – This will give them exchange rate risk: financing a sterling project with dollars. • They could borrow pounds in the international bond market, but pay a premium since they are not as well known abroad. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -43 An Example of a Currency Swap • If they can find a 25 -43 An Example of a Currency Swap • If they can find a British MNC with a mirror-image financing need they may both benefit from a swap. • If the spot exchange rate is S 0($/£) = US$1. 60/£, the U. S. firm needs to find a British firm wanting to finance dollar borrowing in the amount of US$16, 000. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -44 An Example of a Currency Swap Consider two firms A and B: 25 -44 An Example of a Currency Swap Consider two firms A and B: firm A is a U. S. -based multinational and firm B is a U. K. -based multinational. Both firms wish to finance a project in each other’s country of the same size. Their borrowing opportunities are given in the table below. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -45 An Example of a Currency Swap Bank US$8% £ 11% US$9. 4% 25 -45 An Example of a Currency Swap Bank US$8% £ 11% US$9. 4% £ 12% Mc. Graw-Hill Ryerson Firm A US$8% B £ 12% © 2003 Mc. Graw–Hill Ryerson Limited

25 -46 An Example of a Currency Swap A’s net position is to borrow 25 -46 An Example of a Currency Swap A’s net position is to borrow at £ 11% Swap US$8% Bank £ 11% US$9. 4% £ 12% Mc. Graw-Hill Ryerson Firm A US$8% B A saves £. 6% £ 12% © 2003 Mc. Graw–Hill Ryerson Limited

25 -47 An Example of a Currency Swap B’s net position is to borrow 25 -47 An Example of a Currency Swap B’s net position is to borrow at US$9. 4% Swap Bank US$8% £ 11% US$9. 4% £ 12% Firm A US$8% B £ 12% B saves $. 6% Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -48 An Example of a Currency Swap The swap bank makes money too: 25 -48 An Example of a Currency Swap The swap bank makes money too: Swap US$8% £ 11% US$8% Firm A Mc. Graw-Hill Ryerson Bank 1. 4% of US$16 million financed with 1% of £ 10 million per year for five years. US$9. 4% £ 12% Firm £ 12% At S 0($/£) = $1. 60/£, that is a gain of US$124, 000 B per year for 5 years. The swap bank faces exchange rate risk, but maybe they can lay it off (in another swap). © 2003 Mc. Graw–Hill Ryerson Limited

25 -49 Variations of Basic Swaps • Currency Swaps – – fixed for floating 25 -49 Variations of Basic Swaps • Currency Swaps – – fixed for floating for floating amortizing • Interest Rate Swaps – zero-for floating – floating for floating • Exotica – For a swap to be possible, two humans must like the idea. Beyond that, creativity is the only limit. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -50 Risks of Interest Rate and Currency Swaps • Interest Rate Risk – 25 -50 Risks of Interest Rate and Currency Swaps • Interest Rate Risk – Interest rates might move against the swap bank after it has only gotten half of a swap on the books, or if it has an unhedged position. • Basis Risk – If the floating rates of the two counterparties are not pegged to the same index. • Exchange Rate Risk – In the example of a currency swap given earlier, the swap bank would be worse off if the pound appreciated. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -51 Risks of Interest Rate and Currency Swaps • Credit Risk – This 25 -51 Risks of Interest Rate and Currency Swaps • Credit Risk – This is the major risk faced by a swap dealer—the risk that a counter party will default on its end of the swap. • Mismatch Risk – It’s hard to find a counterparty that wants to borrow the right amount of money for the right amount of time. • Sovereign Risk – The risk that a country will impose exchange rate restrictions that will interfere with performance on the swap. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -52 Pricing a Swap • A swap is a derivative security so it 25 -52 Pricing a Swap • A swap is a derivative security so it can be priced in terms of the underlying assets: • How to: – Plain vanilla fixed for floating swap gets valued just like a bond. – Currency swap gets valued just like a nest of currency futures. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -53 25. 7 Actual Use of Derivatives • Because derivatives don’t appear on 25 -53 25. 7 Actual Use of Derivatives • Because derivatives don’t appear on the balance sheet, they present a challenge to financial economists who wish to observe their use. • Survey results appear to support the notion of widespread use of derivatives among: – – large publicly traded firms, Canadian multinational companies, nonregulated companies, gold and silver, paper and forest, pipelines, and agricultural companies. • Foreign currency and interest rate derivatives are the most frequently used. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited

25 -54 25. 8 Summary & Conclusions • This chapter shows a number of 25 -54 25. 8 Summary & Conclusions • This chapter shows a number of hedging strategies. • A short hedge involves an agreement to sell the underlying asset in the future. • A long hedge involves an agreement to buy the underlying asset in the future. • Swaps can also be used to hedge; a swap can be viewed as a portfolio of futures with different maturities. Mc. Graw-Hill Ryerson © 2003 Mc. Graw–Hill Ryerson Limited