Tarheel%20Consultancy%20Services

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Tarheel Consultancy Services

• Manipal, Karnataka

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Corporate Training and Consulting
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Course on Fixed Income Securities

• For
• XIM -Bhubaneshwar

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For

• PGP-II
• 2003-2005 Batch
• Term-V September-December 2004

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Module-I

• Part-VI-A
• Fundamentals of Swaps

6
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.

7
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.

8
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.

9
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.

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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

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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.

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Summary of Cash Flows
Time Days in the 6M interval Fixed Rate Amount payable by Citibank LIBOR Amount payable by HSBC
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
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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
• ------ 3,087,671
• 365

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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.

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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.

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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.

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Illustration
Settlement Date Amount Payable by Citibank Amount Payable by HSBC Net Amount Payable by Citibank
6M 2975343 2851370 123973
12M 3024658 3087671 -63013
18M 2975343 2975343 0
24 M 3024658 2772603 252055
12000000 11686987 313013
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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.

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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.

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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 would
  be made quarterly.

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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

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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.

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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.

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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.

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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 risk.

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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.

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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.

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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.

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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.

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Advantages (Cont)

• For instance if it were to rollover a series of
  short term deposits, it would beeffectively
  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 fixed rate of interest from
  its loan.

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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.

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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.

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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.

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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
  prevailing commercial paper rate.

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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.

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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.

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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 flow streams are
  based on floating rates.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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 default risk is a major concern.

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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.

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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.

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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.

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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.

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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.

52
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 yield
  to arrive at the all-in rate.

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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
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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
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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 we use the terms paying and receiving.

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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.

57
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
  8.85if he is paying the fixed rate.

58
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.

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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.

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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

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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.

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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 were
  launched by market associations.

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Documentation (Cont)

• The two principal multilateral initiatives have
  originated from
• The British Bankers Association (BBA)
• The International Swap Dealers Association (ISDA)

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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.

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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.

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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.

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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.

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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.

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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 Master agreement)

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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.

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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.

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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 counterparties, for
  which they were paid a fee.

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(No Transcript)
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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.

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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.

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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.

77
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.

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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.

79
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.

80
The Role of Brokers (Cont)

• The series of two-way prices broadcast by brokers
  back to the customers is called a Brokers 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
  Name-Passing brokers.

81
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.

82
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.

83
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.

84
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 every
  month has 30 days.

85
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.

86
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.

87
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 receive a payment.
• Else it would make a payment.

88
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.

89
Possible Scenarios
LIBOR Payoff from FRA Interest on Loan Total Effective Interest Annualized Cost with FRA Annualized Cost 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
90
Sample Calculation

• LIBOR 12
• Payoff from FRA
• 20,000,000 x (0.12-0.10) x 90
• ____
• 360
• 100,000

91
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

92
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
• 20,000,000

93
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.

94
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.

95
A Binomial Tree
16.45

• 10.5

14.95
13.06
13.47
11.60
12
10.17
9.76
8.35
8.74
6.57
6.96
5.20
3.46
96
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.

97
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.

98
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

99
A Two-Period in Arrears FRA

• .52x.52x(.1347-k)2X.52x.48x(.1017-k)
  .48x.48x(.0696-k) 0
• ? K 0.1032

100
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)
• ________ ____________ 0?k .1041
• 1.12 1.0874

101
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)