FORWARD RATE AGREEMENTS

1
FORWARD RATE AGREEMENTS

• An FRA is an agreement between two parties where
  it is agreed that
• ? if interest rates rise above the agreed rate,
  then the seller pays the buyer the increased cost
  on a nominal sum of money
• ? if interest rates fall below the agreed rate,
  then the buyer pays the seller the increased cost
  from the higher rate

2

• An FRA is a cash-settled interbank forward
  contract on the interest rate
• Notation
• A agreed rate
• S settlement rate (market rate at the beginning
  of FRA period)
• N nominal contract amount
• d days in FRA contract

3

• If S gt A then the seller pays the buyer
• N(S - A)(d/365)/1 S(d/365)
• If S lt A then the buyer pays the seller the
  absolute value of this amount
• FRAs are used to close maturity gaps
• FRAs are constructed from interbank bid and ask
  rates on eurocurrency deposits

4

• E.g. A six against nine FRA is an agreement on
  a 3-month interest rate for a 3-month period
• To obtain the ask price for this FRA
• ? borrow in interbank market for nine months at
  9-month ask rate
• ? lend in interbank for six months at bid rate
• ? sell an FRA to lend in the remaining 3-month
  period at a rate higher than the implied forward
  rate

5

• E.g. Bank buys a three against six FRA for 2
  million for 3-month period, beginning 3 months
  from now and ending 6 months from now
• Interest rate is 7.5 and there are 91 days in
  the FRA period
• Suppose that 3 months from now interest rate is
  9
• Bank will receive cash from seller of FRA

6

• Bank receives
• 2m((.09-.075)(91/360)/(1.09(91/360)
• 7,414.65
• Banks net borrowing cost on 2m at end of FRA
  period is
• 2m(.09)(91/360) - 7,414.65(1.09(91/360)
• 37,916.67

7

• This is the same as the net borrowing cost on 2m
  for 91 days at 7.5
• 2m(.075)(91/360) 37,916.67

8
EUROCURRENCY FUTURES

• Eurocurrency futures are standardized contracts
  that are traded in organized exchanges
• The most common eurocurrency futures contract is
  the 3-month eurodollar future
• A Eurodollar futures contract calls for the
  delivery of a 1 million, three-month, Eurodollar
  time deposit
• Eurodollar futures employ a cash settlement
  procedure

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• A Eurodollar futures contract is a bet concerning
  the direction of movements of a futures price
  relative to the interest paid on eurodollar
  deposits
• The price of a eurodollar futures contract is
  based on the three-month LIBOR and is given by
• 100 - interest of a three-month eurodollar
  deposit (in percentage terms)

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• If the LIBOR goes up, then the futures price goes
  down and a party that goes short on futures makes
  money
• If the LIBOR goes down, then the futures price
  goes up and somebody with a long position on a
  futures contract makes money
• If we expect that interest rates will fall, we go
  long on eurodollar futures, and we go short if we
  expect that interest rates will rise
• E.g. Eurodollar futures is 94 ? interest on
  3-month eurodollar for forward delivery is 4

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HEDGING WITH EURODOLLAR FUTURES

• Note that
• ? ? in interest rate of 1 bp means ? in
  eurodollar deposit of 100
• ? in 3 months, ? is 25
• Two types of hedging
• ? Stack hedge
• To hedge a 6-month 1m deposit we use two
  eurodollar futures contracts
• ? Strip hedge
• To hedge a 6-month 1m deposit use two
  subsequent eurodollar futures contracts

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• Eurodollar futures are also used to hedge other
  interest rate assets and liabilities
• E.g. T-bills, commercial paper, bankers
  acceptances,etc.
• Rates on these assets are not perfectly
  correlated with eurodollar rates
• Define
• Slope ? underlying int. rate/? eurodollar
  futures int. rate

13

• Number of eurodollar futures needed to hedge an
  underlying amount is
• n (F/1,000,000) (D/90) slope
• F face amount
• D duration of underlying asset
• E.g. Hedge 10 m of 270-day commercial paper
  with slope .935 we need ? 25 eurodollar contracts

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EXAMPLE OF USING EURODOLLAR FUTURES TO HEDGE
MATURITY GAP RISK

• Suppose that a bank notices that on June 12
• ? It can borrow a 3-month eurodollar has 10
  annual rate
• ? It can lend the money for 6 months at 10 and
  13/16 percent annually
• ? September eurodollar futures trade at 89.23
  (10.77 annual rate)

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• Bank can make a profit as follows
• ? Borrow 1m for 3 months
• ? Lend 1m for 6 months
• ? Interest rate risk from maturity gap
• ?? Implied forward rate is 11.34 annually
• ? Go short on a September eurodollar futures
  (receive a positive cash flow to offset
  potentially higher cost of borrowing above 10.77)

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OPTIONS ON EURODOLLAR FUTURES

• A eurodollar futures call option gives the buyer
  the right to go long a eurodollar futures
  contract
• A eurodollar futures put option gives the buyer
  the right to go short a eurodollar futures
  contract
• E.g. A buyer of a eurodollar futures call option
  decides to exercise the option

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• ? Buyer will go long a eurodollar futures
• ? Seller will go short a eurodollar futures
• or
• ? Buyer will buy a 3-month eurodollar deposit at
  agreed rate
• ? Seller will sell a 3-month eurodollar deposit
  at agreed rate

18

• Eurodollar futures options premiums are quoted in
  percentage points
• Price 25 ? basis point
• E.g., Price of option is quoted as .23, then
  price is 25 ? 23 575

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INTEREST RATE CAPS

• An interest rate call option is exercised when
  the interest rate is above the exercise price
• Interest caps protect against rises in interest
  rates
• When a call is exercised
• ? buyer pays exercise rate
• ? writer pays market rate to buyer of call
• ? writer covers any additional interest costs
  for the buyer

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• A cap is a series of European interest rate calls
  that expire on the dates that interest payments
  on the loan are due
• The individual options are called caplets
• E.g. A firm borrows 25m on January 2 for one
  year
• Interest payments are made every 3 months

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• Interest is LIBOR at the beginning of each period
• Suppose on January 2, LIBOR is 10 and firm wants
  to cap this rate
• Firm buys an interest rate cap with 10 rate for
  the year
• At each date interest is paid, cap is worth
• 25,000,000 (days/360) Max(0, LIBOR - .10)

22

• Firm behaves as follows
• ? For first quarter, firm does not exercise
  option
• ? Suppose on April 2, LIBOR is at 10.68
• ? Firm exercises option cap will pay
• 25,000,000(91/360)(.1068 - .10) 42,972
• ? Interest payment on July 2 is 674,917 (higher
  than 631,944 with 10 rate)
• ? Cap payment covers the difference