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Module-I Part-VI-A Fundamentals of Swaps 2/8/2012 Prof. Rushen Chahal Introduction What is a swap? It is basically an exchange of two payment streams that are different from each other. Why do parties enter into a swap? To acquire one stream of payments and to dispose off another stream. 2/8/2012 Prof. Rushen Chahal Introduction (Cont…) What is an interest rate swap? It is a contract where two counterparties commit themselves to exchange, over an agreed time period, two streams of payments, each calculated using a different type of interest rate, but with the same notional principal. 2/8/2012 Prof. Rushen Chahal Introduction (Cont…) Are there other types of swaps? Yes In the case of a currency swap the two streams of payment are denominated in different currencies. In an equity swap one stream is calculated based on an equity price. In a commodity swap one stream is calculated based on a commodity price. 2/8/2012 Prof. Rushen Chahal Illustration Citibank and HSBC agree to exchange over a period of two years, two streams of cash flows at six monthly intervals. Citibank will calculate its payments based on a fixed interest rate of 6% per annum. HSBC will calculate its payments based on the 6M LIBOR that is prevailing at the start of the six monthly period for which the payment is being computed. 2/8/2012 Prof. Rushen Chahal Terms Counterparties: Citibank and HSBC Maturity: 2 Years Interest Rate (1): Fixed 6% per annum Citibank pays HSBC Frequency of payment: Semi-annual Interest Rate (2): 6M LIBOR HSBC pays Citibank Frequency of payment: Semi-annual Notional Principal: 100 MM USD 2/8/2012 Prof. Rushen Chahal Illustration (Cont…) The interest rate is normally fixed at the start of the period to which it applies. But the payment calculated using this rate is made at the end of the period. This is what is meant by `determined in advance and paid in arrears.’ We can also have a system of `determined in arrears and paid in arrears.’ 2/8/2012 Prof. Rushen Chahal Summary of Cash Flows Time Days in Fixed Rate Amount LIBOR Amount the 6M payable payable interval by by HSBC Citibank 0 - 6% - 5.75 - 6M 181 6% 2975343 6.125% 2851370 12M 184 6% 3024658 6% 3087671 18M 181 6% 2975343 5.5% 2975343 24M 184 - 3024658 - 2772603 TOTAL - - 12000000 - 11686987 2/8/2012 Prof. Rushen Chahal Sample Calculations Cash Outflow for Citibank after 6 months: 0.06 x 100,000,000 x 181 ------ = 2,975,343 365 Cash Outflow for HSBC after 12 months: 0.06125 x 100,000,000 x 184 2/8/2012 Prof. Rushen Chahal Notional Principal In the case of an interest rate swap only the interest is exchanged. There is no exchange of principal. The principal is specified purely for the computation of interest. Hence it is termed as a `notional principal’. 2/8/2012 Prof. Rushen Chahal Off-Balance-Sheet Since the principal is not exchanged the swap does not impact the balance sheets of the counterparties. Hence interest rate swaps are referred to as off-balance-sheet transactions. 2/8/2012 Prof. Rushen Chahal Netting In our illustration the counterparties were required to make payments to each other on the same date. Hence the payments are usually netted and only a single amount representing the difference is exchanged. 2/8/2012 Prof. Rushen Chahal Illustration Settlement Amount Amount Net Amount Date Payable by Payable by Payable by Citibank HSBC Citibank 6M 2975343 2851370 123973 12M 3024658 3087671 -63013 18M 2975343 2975343 0 24 M 3024658 2772603 252055 12000000 11686987 313013 2/8/2012 Prof. Rushen Chahal Netting Netting of payments reduces the delivery risk. What is delivery risk? It is a default risk that can arise when an exchange of payments does not occur simultaneously. Thus a delay exposes the counterparty making the earlier payment to the risk that the other party may not honour its commitment. 2/8/2012 Prof. Rushen Chahal Netting (Cont…) In order to facilitate netting, the frequency and timing of fixed rate payments will usually match the frequency and timing of the floating rate counter payments. 2/8/2012 Prof. Rushen Chahal Frequency of Payments The frequency of the floating rate payments is usually set by the tenor of the benchmark rate that is used in the swap. Thus if 6M LIBOR is used as the benchmark then the payments will be made semi-annually, whereas if 3M LIBOR were to be used, the payments quarterly. would be madeProf. Rushen Chahal 2/8/2012 Terms A swap agreement ought to contain the following details The names of the counterparties The maturity date of the swap The fixed interest rate The benchmark for the floating rate The notional principal amount And the frequency of payments 2/8/2012 Prof. Rushen Chahal Frequency of Payments In our illustration, both fixed as well as floating rate payments were made on a semi- annual basis. Such a swap is called a semi/semi. Longer term swaps can even be annual/semi. Short-term swaps can often be quarterly/quarterly. In some cases fixed payments are made annually and floating payments are received quarterly. These are called annual/threes swaps. 2/8/2012 Prof. Rushen Chahal Purpose of a Swap Let us consider the swap between Citibank and HSBC. What did it achieve? In the case of Citibank the fixed interest rate payments were known from the outset. But in the case of HSBC since LIBOR id variable, the cash outflows were subject to uncertainty, except for the first six months. 2/8/2012 Prof. Rushen Chahal Purpose (Cont…) Citibank is paying fixed and receiving floating. Hence it is subject to the risk that the LIBOR will fall during the life of the Swap. HSBC is paying floating and receiving fixed. It is therefore exposed to the risk that the LIBOR will rise during the life of the swap. 2/8/2012 Prof. Rushen Chahal Purpose (Cont…) Thus an interest rate swap exposes both the counterparties to interest rate risk. Such swaps may therefore be used for speculation or for profiting from an expected interest rate change by deliberately taking risk. Or they may be used for hedging against another source of interest rate 2/8/2012 Prof. Rushen Chahal Speculation Assume that Citibank is anticipating rates to rise whereas HSBC is expecting that rates will fall. Thus a swap which requires Citibank to pay fixed and HSBC to pay floating, can be used as a speculative mechanism by both the parties. 2/8/2012 Prof. Rushen Chahal Hedging Assume that Citibank has already borrowed on a floating rate basis. It can use the swap with HSBC to hedge interest rate risk. If rates rise it will have to pay more interest on its original borrowings, but will receive a net cash inflow from the swap. 2/8/2012 Prof. Rushen Chahal Hedging (Cont…) Assume that HSBC has already made a loan on floating rate basis. If so it can use the swap to hedge. If interest rates were to fall it would receive less on its original investment but will receive a cash inflow from the swap. 2/8/2012 Prof. Rushen Chahal Advantages Before swaps became available interest rate risk had to be managed using assets and liabilities in the form of cash instruments. For instance assume that a bank anticipates a fall in interest rates. It could make a medium term fixed rate loan and fund it by taking a series of consecutive short term deposits. 2/8/2012 Prof. Rushen Chahal Advantages (Cont…) For instance if it were to rollover a series of short term deposits, it would be effectively borrowing at a floating rate and lending at a fixed rate. If rates were to fall as expected it would pay a lower rate of interest on its deposits would continue to receive a from fixed rate of interest Chahal its loan. 2/8/2012 Prof. Rushen Advantages (Cont…) An interest rate swap where the bank receives fixed and pays floating can be used to achieve the same result. The swap would yield the same profit would there would be no transfer f principal and consequently no impact on the balance sheet. 2/8/2012 Prof. Rushen Chahal Advantages (Cont…) Since a swap is an off-balance-sheet transaction as opposed to the alternative entailing the use of assets and liabilities, it offers several advantages. There is less credit risk. Only interest payments are at risk whereas in the case of assets and liabilities the full principal is at risk. 2/8/2012 Prof. Rushen Chahal Advantages (Cont…) Swaps are subject to lower capital adequacy requirements because they involve less credit risk. Swaps involve lower transaction costs because less money is being transferred and funded. They offer greater flexibility. 2/8/2012 Prof. Rushen Chahal Types of Interest Rate Swaps Coupon swaps What we have just seen is a coupon swap. It entails the exchange of a payment based on a fixed rate in return for a payment based on a floating rate. Basis swaps In these swaps both streams of payment are calculated using a floating rate index. For instance one stream could be based on the LIBOR whereas the other could be based on the 2/8/2012 prevailing commercial paper rate. Prof. Rushen Chahal Types (Cont…) Asset swaps If one of the payment streams is funded with interest received from an asset, the swap and the asset as a whole are called an asset swap. There is no change in the swap mechanism per se. Strictly speaking we could also have liability swaps. But this term is rarely used. Thus swaps used in conjunction with a liability are merely referred to as interest rate swaps. 2/8/2012 Prof. Rushen Chahal Types (Cont…) Term swap A swap with an original maturity of more than two years is called a term swap. Money market swap A swap with an original maturity of up to one year is called a money market swap. Currency swap It is a swap where each stream on interest is denominated in a different currency. These swaps also involve an exchange of principal. 2/8/2012 Prof. Rushen Chahal Terminology The counterparties to a swap are called payers or receivers. In the case of a coupon swap, the party paying on a fixed rate basis is said to be the `payer in the swap’ and the other counterparty is the `receiver in the swap’. In the case of a basis swap we cannot use this convention since both the cash Prof. based flow streams are Rushen Chahal on floating 2/8/2012 Terminology (Cont…) Thus it is a god practice in the case of basis swaps to describe each counterparty in terms of both the rate it pays as well as the rate it receives. In the inter-bank swap market the terms buyer and seller are used in the case of coupon swaps. Buyers are payers and sellers are receivers. 2/8/2012 Prof. Rushen Chahal Terminology (Cont…) In most coupon swaps the 6M LIBOR is the standard index for the floating rate. Thus these swaps can be defined purely in terms of the fixed rate of interest. For example in the case of the Citibank-HSBC swap, the price of the swap would have been quoted as 6% per annum, which is nothing but the fixed rate. The price of a coupon swap is also called the swap rate. 2/8/2012 Prof. Rushen Chahal Terminology (Cont…) In most markets swap rates are quoted as full percentage figures. Example in our case the rate was 6%. This is called an all-in price. However in certain inter-bank swap markets, particularly the US dollar market, the convention of quoting the price on an all-in basis has been replaced by the convention of quoting the differential between the all-in rate and an accepted benchmark rate. 2/8/2012 Prof. Rushen Chahal Terminology (Cont…) The benchmark rate is usually the rate on the government bond with a remaining period to maturity closest to that of the swap. The difference between the all-in price and the benchmark rate is called the swap spread.. For instance assume that the all-in price is 5.5% for a 5 year swap and that 5 year T- notes are yielding 5.3% per annum. The swap price will be quoted as 20 basis points. 2/8/2012 Prof. Rushen Chahal Terminology (Cont…) The trade date or the fixing date is the date on which the terms are agreed upon. The following terms have to be agreed upon The maturity The swap rate The floating rate index The payment frequency The notional principal On this date the counterparties contractually commit themselves to the transaction. 2/8/2012 Prof. Rushen Chahal Terminology (Cont…) The value date is the date on which the interest payments start to accrue. For swaps involving only the domestic currency the value date is usually the same as the trade date. For foreign currency swaps the value date is usually two days after the trade date. 2/8/2012 Prof. Rushen Chahal Terminology (Cont…) The date on which the floating rate is re-fixed for the next period is called The re-fixing or re-pricing or reset date. The date on which the interest is paid for the preceding period is called the effective date. The effective dates are calculated from the value date. For domestic currency swaps the effective dates are the same as the re-fixing dates. For currency swaps the effective date is two business days after the re-fixing date. 2/8/2012 Prof. Rushen Chahal Swaps versus Other Derivatives Swaps are traded on a bilateral basis in decentralized markets. Thus swaps are OTC instruments. In contrast futures contracts and listed options are exchange traded instruments. On an exchange the clearinghouse becomes the buyer for every seller and the seller for every buyer. 2/8/2012 Prof. Rushen Chahal Swaps vs. Others (Cont…) Both the parties have to provide daily collateral called margins. The role of the clearinghouse and the margining mechanism minimizes the risk of default. In OTC markets there is no clearinghouse, and margining is not compulsory. So 2/8/2012 default risk is a major concern. Prof. Rushen Chahal Swaps vs. Others (Cont…) Futures contracts and listed options are standardized instruments. Standardization reduces transactions costs and provides greater liquidity. OTC contracts are however customized. Activity in exchange traded products is limited to certain instruments. However OTC products like swaps are available for any currency and for any tenor provided a counterparty can be found. 2/8/2012 Prof. Rushen Chahal Swaps vs. Others (Cont…) Futures and listed options are usually available only for short to medium terms. Swaps on the other hand can extend as far as 20 years into the future. 2/8/2012 Prof. Rushen Chahal Trading Swaps The swap market is an OTC market. Trading is conducted primarily by telephone. Indicative prices are disseminated over screen services by agencies like Reuters. 2/8/2012 Prof. Rushen Chahal Negotiations Key financial details are agreed verbally between dealers. Key details are then confirmed y an exchange of telexes or faxes within 24 hours. Full contract documentation is agreed, signed, and exchanged subsequently. 2/8/2012 Prof. Rushen Chahal Negotiations (Cont…) Because of the delay is documentation, swaps are sometimes said to be dealt on an as of basis. However a contract is assumed to be struck based on the initial verbal agreement between dealers without waiting for the exchange of confirmations or documentation. 2/8/2012 Prof. Rushen Chahal Negotiations (Cont…) In the case of coupon swaps which are quoted in terms of a spread over a benchmark yield, the dealers will agree on the spread first. They will then break of negotiations to check whether they have credit lines to each other. If there are no credit problems they will resume negotiations and agree on the benchmark yield. The spread will be added to the benchmark 2/8/2012 Prof. Rushen Chahal Illustration of All-in Prices New Zealand Dollar Swaps Maturity Semi-annual Rate 1 year 8.00-7.85 2 years 8.25-8.05 3 years 8.50-8.30 4 years 8.85-8.65 5 years 9.05-8.85 7 years 9.25-9.05 2/8/2012 Prof. Rushen Chahal Illustration of Swap Spreads US Dollars Spread Annual Interest A/360 2 years 21/25 5.70-5.75 3 years 40/45 6.23-6.28 5 years 46/51 7.01-7.05 7 years 46/51 7.46-7.51 10 years 47/52 7.93-7.97 2/8/2012 Prof. Rushen Chahal Two-way Prices As can be seen, two swap rates are quoted for each maturity. Such prices are quoted between professional dealers and consist of a buying and selling price for the instrument. However the terms buying and selling can be ambiguous in the case of swaps. So Prof. Rushen paying and 2/8/2012 we use the terms Chahal Two-way Prices (Cont…) When you have two prices, which is being paid and which is received? The logic is that the dealer hopes to make a profit if he undertakes a fixed- floating swap with one party and a floating-fixed swap with the other. Thus he would like to pay the lower fixed rate and receive the higher fixed rate. 2/8/2012 Prof. Rushen Chahal Two-way Prices (Cont…) For instance the all-in prices for the 5 year NZ Dollar swap is 9.05-8.85. Thus the dealer will demand 9.05% if he is receiving the fixed rate and will part with 8.85%if he is paying the fixed rate. 2/8/2012 Prof. Rushen Chahal Two-way Prices (Cont…) What about quotations in terms of spreads? For instance a 5 year USD swap is quoted as 46/51. This means that when the dealer is paying fixed he will give 46 basis points over the yield on the most liquid 5 year T-note. If he is receiving fixed he will demand 51 basis points over the 5 year T-note yield. The equivalent all-in rates are 7.01% and 7.05%. 2/8/2012 Prof. Rushen Chahal Swap Documentation What is a contract? It is evidence of an agreement between the counterparties to a transaction. It should provide a detailed definition of a transaction in respect of: Financial terms and conditions That is the rights that the parties enjoy or the obligations that they have accepted. 2/8/2012 Prof. Rushen Chahal Documentation (Cont…) The legal framework should be spelt out. What are the rights of enforcement according to law if there is a default by a counterparty. In this context the definition of default must be clearly spelt out The methods of computing damages should be clearly stated 2/8/2012 Prof. Rushen Chahal Documentation (Cont…) In the early days, swap documentation was extremely complex because the instrument was new and there was a need to provide adequate financial and legal definitions. There was a lack of legal precedent and little in the way of `custom and usage’. Contracts therefore contained extensive legal opinion. 2/8/2012 Prof. Rushen Chahal Documentation (Cont…) Contracts were long winded and often took months to finalize. An attempt has been made subsequently to standardize the documentation. Initial efforts were on a bilateral basis between active market players. Subsequently multilateral initiatives by market were launched Prof. Rushen Chahal associations. 2/8/2012 Documentation (Cont…) The two principal multilateral initiatives have originated from: The British Bankers’ Association (BBA) The International Swap Dealers’ Association (ISDA) 2/8/2012 Prof. Rushen Chahal BBA Documentation In 1985 the BBA promulgated its BBAIRS Terms or the Recommended Terms and Conditions for London Interbank Interest Rate Swaps They were intended to apply to money market swaps traded interbank in London. 2/8/2012 Prof. Rushen Chahal BBA Documentation (Cont…) They provided the following: Financial terms and conditions Sample confirmations Rights of enforcement in the event of default In addition to the documentation, the BBAIRS Terms also set out conventions for conducting negotiations. These terms have now been largely superseded by the more comprehensive documentation drafted by ISDA. But the mechanism for fixing LIBOR which was devised as a part of BBAIRS Terms continues to play a central role in the settlement of swaps. 2/8/2012 Prof. Rushen Chahal BBAIRS Interest Settlement Rate The BBA arranged for Telerate to calculate and publish on a daily basis a list of BBAIRS Interest Settlement Rates for each monthly maturity between one and twelve months for 9 currencies. 2/8/2012 Prof. Rushen Chahal ISDA Documentation In 1985 ISDA published a Code of Standard Wording, Assumptions and Provisions for Swaps known as the ISDA Swaps Code. This was a menu from which counterparties could draw when drafting a contract for US Dollar swaps. 2/8/2012 Prof. Rushen Chahal ISDA Documentation (Cont…) The Code dealt mainly with financial terms and conditions such as calculation of interest and termination payments. It was subsequently revised and expanded to address rights of enforcement and credit provisions. 2/8/2012 Prof. Rushen Chahal ISDA Documentation (Cont…) In 1987 ISDA published two master contracts. For USD interest rate swaps – The Interest Rate Swap Agreement (Rate Swap Master Agreement) For interest rate and currency swaps in or between a variety of currencies – the Interest Rate and Currency Exchange Agreement (Rate and Currency Swap 2/8/2012 Master agreement) Prof. Rushen Chahal ISDA Documentation (Cont…) Once an ISDA Master Contract is in place between two counterparties, the details of new swaps are simply added as appendices. Thus there will always be a single contract in place between two counterparties regardless of the number of swaps transacted. 2/8/2012 Prof. Rushen Chahal ISDA Documentation (Cont…) A master agreement is designed to net the profits and losses being made on all the swaps outstanding between the same two counterparties. 2/8/2012 Prof. Rushen Chahal The Primary Market: The Role of Banks In the early days of the swap market, the intermediaries were investment banks with fairly limited resources. They tried to avoid exposure to default risk by assuming the role of an agent rather than a principal in swap transactions. Hence they merely helped arrange such transactions between the for which counterparties,Prof. Rushen Chahal they were 2/8/2012 2/8/2012 Prof. Rushen Chahal The Role of Banks (Cont…) As the market developed it became necessary for swap intermediaries to assume the role of principals. There were two reasons for this. End users desired anonymity Secondly they were reluctant to deal with non-bank counterparties because of the default risk. 2/8/2012 Prof. Rushen Chahal The Role of Banks (Cont…) Intermediaries initially began to maintain matched books. That is, they would arrange a swap only if there was a more or less equal and opposite swap that was immediately available as a hedge. Such a matching swap is known as a reversal. 2/8/2012 Prof. Rushen Chahal The Role of Banks (Cont…) While running a matched book, the intermediary is exposed to default risk from both sides. Consequently they would charge a risk- related dealing spread in the form of a difference between the fixed interest rate paid to one user and that received from the other user. 2/8/2012 Prof. Rushen Chahal The Role of Banks (Cont…) Due to competition, arrangement fees have become rare unless the swap structure is unusual and complex. Swaps have now become an active tool for asset-liability management. Intermediaries have now become market makers, that is, they provide continuous two-way quotes. 2/8/2012 Prof. Rushen Chahal The Role of Banks (Cont…) Such market makers stand ready to accept temporary exposures to a position, until they are able to find a matching swap. 2/8/2012 Prof. Rushen Chahal The Role of Brokers Dealers often trade in swaps through brokers. These brokers act as agents in locating a counterparty. But they do not actually participate in the transaction. In practice brokers continuously take prices from customers, and then select and broadcast the cheapest selling price and the highest buying price for each maturity. 2/8/2012 Prof. Rushen Chahal The Role of Brokers (Cont…) The series of two-way prices broadcast by brokers back to the customers is called a Broker’s Run. If a customer were to accept one of the prices the broker will pass on the identity of the customer who originated the price. For this reason swap brokers are known as Prof. brokers. 2/8/2012 Name-Passing Rushen Chahal The Role of Brokers (Cont…) Brokers are paid a flat fee or brokerage commission which is related to the size of the deal (the notional principal) and maturity. Typical brokerage fees are flat basis points per annum from each counterparty. Brokerage is paid upfront in the form of the present value of the basis points earned over the life of the swap. 2/8/2012 Prof. Rushen Chahal Forward Rate Agreements An FRA is noting but a forward contract on an interest rate. In the case of derivatives on debt instruments the payoff is determined by the price of the instrument and is consequently indirectly determined by the underlying interest rate. In contrast the payoff from a FRA is directly determined by the interest rate. 2/8/2012 Prof. Rushen Chahal FRA (Cont…) The agreement would have to specify a notional principal and the terms on which the payment is to be made. Typically, the payoff is based on the LIBOR. The payoff is based on the difference between the prevailing value of LIBOR and the contract rate which was agreed upon at the outset. 2/8/2012 Prof. Rushen Chahal FRA (Cont…) The method of computing interest is not standard. In some cases the year is assumed to have 360 days. In other cases it is assumed to have 365 days. The number of days for which the interest is computed is sometimes taken to be the actual number of days. In other cases it is computed assuming that 30 days. every month has Prof. Rushen Chahal 2/8/2012 Illustration A company wants to lock in a borrowing rate for a loan that it will take after 30 days. The loan will be for a period of 90 days. The applicable rate will be the LIBOR prevailing after 30 days plus 1%. The year is assumed to have 360 days. 2/8/2012 Prof. Rushen Chahal Illustration (Cont…) The company would like to protect itself against rising rates. Hence it would like a positive payoff from the FRA if rates were to rise. Hence it would need a long position in the FRA. Assume that the company agrees on a fixed rate of 10% for the FRA. 2/8/2012 Prof. Rushen Chahal Illustration (Cont…) The payoff from the FRA would be: Notional Principal x (LIBOR – 0.10) x 90 ____ 360 So if the LIBOR after 30 days were to exceed 10% the company would 2/8/2012 Prof. Rushen Chahal Illustration (Cont…) Assume that the FRA is structured so that the payment will be made 120 days from today so as to coincide with the payment on the loan. The notional principal is 20 MM USD. 2/8/2012 Prof. Rushen Chahal Possible Scenarios LIBOR Payoff from Interest on Total Annualized Annualized FRA Loan Effective Cost with Cost Interest FRA without FRA 6.00% -200,000 350,000 550,000 11.63 7.29 8% -100,000 450,000 550,000 11.63 9.44 10% 0 550,000 550,000 11.63 11.63 12% 100,000 650,000 550,000 11.63 13.85 14% 200,000 750,000 550,000 11.63 16.10 2/8/2012 Prof. Rushen Chahal Sample Calculation LIBOR = 12% Payoff from FRA: 20,000,000 x (0.12-0.10) x 90 ____ 360 = 100,000 2/8/2012 Prof. Rushen Chahal Sample Calculation (Cont…) Interest due on loan: 20,000,000 x 0.13 x 90 _____ 360 = 650,000 Effective interest paid = 650,000 – 100,000 = 550,000 2/8/2012 Prof. Rushen Chahal Sample Calculation (Cont…) Annualized Interest: 20,000,000 + 550,000 ( ___________________)365/90 – 1 = .1163 20,000,000 Annualized interest without the FRA: 20,000,000 + 650,000 ( __________________)365/90 – 1 = .1385 2/8/2012 20,000,000 Prof. Rushen Chahal Illustration (Cont…) Regardless of the LIBOR after 90 days the cost of the loan with the FRA is 11.63%. Without the FRA the cost of the loan will vary directly with the LIBOR. Thus the loan plus the FRA is essentially a risk free transaction. 2/8/2012 Prof. Rushen Chahal Valuing a FRA To value a FRA we need to specify as to how the interest rate is expected to evolve over time. We will specify a binomial model for the evolution of the interest rate through time. That is given a particular rate, the next period the rate could either go up by a pre-specified factor, or down by a pre- specified factor. Rushen Chahal 2/8/2012 Prof. A Binomial Tree 16.45 14.95 13.47 13.06 11.60 12 10.17 9.76 8.35 10.5 8.74 6.96 6.57 5.20 3.46 2/8/2012 Prof. Rushen Chahal A Binomial Tree (Cont…) At every stage the probability of an up move is 0.52 and that of a down move is 0.48. 2/8/2012 Prof. Rushen Chahal Pricing a FRA An interest rate FRA that pays off at the expiration of the contract is said to payoff in arrears. In the earlier illustration we assumed that the FRA expired after 30 days but that the payoff was received after 120 days so as to coincide with the payment of interest on the loan. These are called delayed settlement FRAs. 2/8/2012 Prof. Rushen Chahal A One Period In Arrears FRA The fixed interest rate should be such that the FRA has a zero value today since neither party has to pay to get into a FRA. So the if we set the expected payoff from the FRA equal to zero: .52(.12-k) + .48(.0874-k) = 0 k = .1044 2/8/2012 Prof. Rushen Chahal A Two-Period in Arrears FRA .52x.52x(.1347-k)+2X.52x.48x(.1017-k) + .48x.48x(.0696-k) = 0 K = 0.1032 2/8/2012 Prof. Rushen Chahal A Delayed Settlement One- Period FRA In this case the payoff from the FRA is one period after its time of expiration. Thus all cash flows have to be discounted for one period at the applicable rate. So: .52(.12-k) .48x(.0874-k) ________ + ____________ = 0k = .1041 1.12 1.0874 2/8/2012 Prof. Rushen Chahal A Delayed Settlement Two- Period FRA .52x.52x(.1347-k) 2x.52x.48(.1017-k) _______________ + _______________ 1.1347 1.1017 + .48x.48x(.0696-K) _______________ = 0 k =.1027 (1.0696) 2/8/2012 Prof. Rushen Chahal

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