Derivatives Interest Rate Derivatives

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DerivativesInterest Rate Derivatives

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

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Interest Rate Derivatives

• Forward rate agreement (FRA) OTC contract that
  allows the user to "lock in" the current forward
  rate.
• Treasury Bill futures a futures contract on 90
  days Treasury Bills
• Interest Rate Futures (IRF) exchange traded
  futures contract for which the underlying
  interest rate (Dollar LIBOR, Euribor,..) has a
  maturity of 3 months
• Government bonds futures exchange traded futures
  contracts for which the underlying instrument is
  a government bond.
• Interest Rate swaps OTC contract used to convert
  exposure from fixed to floating or vice versa.

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Term deposit as a forward on a zero-coupon
M (1RS ?)
100(16 0.25) 101.50
T 0.50
T 0.75
0
? 0.25
M 100
Profit at time T M(RS rS)? 100 (6 -
rS) 0.25 Profit at time T M(RS rS)? / (1
rS ?)
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FRA (Forward rate agreement)

• OTC contract
• Buyer committed to pay fixed interest rate Rfra
• Seller committed to pay variable interest rate rs
• on notional amount M
• for a given time period (contract period) ?
• at a future date (settlement date or reference
  date) T
• Cash settlement at time T of the difference
  between present values
• CFfra M (rS Rfra) ? / (1rS ?)
• Long position on FRA equivalent to cash
  settlement of result on forward loan (opposite of
  forward deposit)
• An FRA is an elementary swap

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Hedging with a FRA

• Cy X wishes to set today 1/3/20X0
• the borrowing rate on 100 mio
• from 1/9/20X0 (T) to 31/8/20X1 (1 year)
• Buys a 7 x 12 FRA with R6
• Settlement date 1/9/20X0
• Notional amount 100 m
• Interest calculated on 1-year period
• Cash flows for buyer of FRA
• 1) On settlement date r8 r 4
• Settlement 100 x (8 - 6) / 1.08
  100 x (4 - 6) / 1.04
• 1.852 - 1.923
• Interest on loan - 8.00 -4.00
• FV(settlement) 2.00 -2.00
• TOTAL - 6.00 -6.00

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Treasury bill futures

• Underlying asset 90-days TB
• Nominal value USD 1 million
• Maturities March, June, September, December
• TB Quotation (n days to maturity)
• Discount rate y
• Cash price calculation St 100 - y ? (n/360 )
• Example If TB yield 90 days 3.50
• St 100 - 3.50 ? (90/360) 99.125
• TB futures quotation
• Ft 100 - TB yield

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Example Buying a June TB futures contract
quoted 96.83

• Being long on this contract means that you buy
  forward the underlying TBill at an implicit TB
  yield yt 100 - 96.83 3.17 set today.
• The delivery price set initially is
• K M (100 - yt??)/100
• 1,000,000 100 - 3.17 ? (90/360)/100
  992,075
• If, at maturity, yT 4 (?FT 96)
• The spot price of the underlying asset is
• ST M (100 - yT??)/100
• 1,000,000 100 - 4.00 ?(90/360)/100
  990,000
• Profit at maturity fT ST - K - 2,075

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TB Futures Alternative profit calculation

• As forward yield is yt 100 - Ft
• yield at maturity yT 100 - FT 100
  - ST
• profit fT ST - K M (100 - yT??)/100
  - M (100 - yt??)/100
• profit can be calculated as fT M (FT -
  Ft)/100 ?
• Define TICK ? M ?? ?(0.01/100)
• Cash flow for the buyer of a futures for ?F
  1 basis point (0.01)
• For TB futuresTICK 1,000,000 ? (90/360)
  ?(0.01/100) 25
• Profit calculation
• Profit fT ?F ? TICK ?F in bp
• In our example ?F 96.00 - 96.83 - 83 bp
• fT -83 ? 25 -
  2,075

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3 Month Euribor (LIFFE) Euro 1,000,000
Wall Street Journal July 2, 2002
Settle Open int.
July 96.56 43,507
Sept 96.49 422,241
Dec 96.26 338,471
Mr 03 96.09 290,896
Est vol 259,073 open int 1,645,536
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Interest rate futures vs TB Futures

• 3-month Eurodollar (IMM LIFFE)
• 3-month Euribor (LIFFE)
• Similar to TB futures
• ? Quotation Ft 100 - yt
• with yt underlying interest rate
• ? TICK M ?? ?(0.01/100)
• ? Profit fT ?F ? TICK
• But
• TB futures Price converges to the price of a
  90-day TB
• TB delivered if contract
  held to maturity
• IRF Cash settlement based on final contract
  price
• 100(1-rT)
• with rT underlying
  interest rate at maturity

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IRF versus FRA

• Consider someone taking a long position at time t
  on an interest rate future maturing at time T.
• Ignore marking to market.
• Define R implicit interest rate in futures
  quotation Ft
• R (100 Ft) / 100
• r underlying 3-month interest rate at
  maturity
• rT (100 FT) / 100
• Cash settlement at maturity

Similar to short FRA except for discounting
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Hedging with an IRF

• A Belgian company decides to hedge 3-month future
  loan of 50 mio from June to September using the
  Euribor futures contract traded on Liffe.
• The company SHORTS 50 contracts. Why ?
• Interest rate ? Interest rate?
• Short futures F? ?F lt0 Gain F? ?Fgt0 Loss
• Loan Loss Gain
• F0 94.05 gt R 5.95
• Nominal value per contract 1 mio
• Tick 25 (for on bp)

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Checking the effectiveness of the hedge
Short 50 IRF, F0 94.05, Tick 25 (for one bp)
rT 5 6 7
FT 95 94 93
?F (bp) 95 -5 -105
CF/contract -2,375 125 2,625
X 50 -118,750 6,250 131,250
Interest -625,000 -750,000 -875,000
Total CF -743,750 -743,750 -743,750
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A further complication Tailing the hedge

• There is a mismatch between the timing of the
  interest payment (September) and of the cash
  flows on the short futures position (June).
• Net borrowing 50,000 Futures profit
• Total Debt Payment Net borrowing ? (1r ? 3/12)
• Effective Rate (Total Debt Payment/50,000,000)-
  1 ? (12/3)
• X in June is equivalent to X(1r?) in
  September.
• So we should adjust the number of contracts to
  take this into account.
• However, r is not known today (in March).
• As an approximation use the implied yield from
  the futures price.
• Trade 100/(15.95 x 3/12) 98.53 contracts

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GOVERNMENT BOND FUTURES

• Example Euro-Bund Futures
• Underlying asset Notional bond
• Maturity 8.5 10.5 years
• Interest rate 6
• Contract size 100,000
• Maturities March, June, September, December
• Quotation (as for bonds) -
• Clean price (see below)
• Minimum price movement 1 BASIS POINT (0,01 )
• 100,000 x (0,01/100) 10
• Delivery see below

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Example Euro-BUND Futures (FGBL)

• Contract Standard A notional long-term debt
  instrument issued by the German Federal
  Government with a term of 8½ to 10½ years and an
  interest rate of 6 percent. Contract Size EUR
  100,000 Settlement A delivery obligation
  arising out of a short position in a Euro-BUND
  Futures contract may only be satisfied by the
  delivery of specific debt securities - namely,
  German Federal Bonds (Bundesanleihen) with a
  remaining term upon delivery of 8½ to 10½ years.
  The debt securities must have a minimum issue
  amount of DEM 4 billion or, in the case of new
  issues as of 1.1.1999, 2 billion euros.
• Quotation In a percentage of the par value,
  carried out two decimal places.
• Minimum Price Movement 0.01 percent,
  representing a value of EUR 10. Delivery Day
  The 10th calendar day of the respective delivery
  month, if this day is an exchange trading day
  otherwise, the immediately following exchange
  trading day. Delivery Months The three
  successive months within the cycle March, June,
  September and December. Notification Clearing
  Members with open short positions must notify
  Eurex which debt instruments they will deliver,
  with such notification being given by the end of
  the Post-Trading Period on the last trading day
  in the delivery month of the futures contract.

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Time scale
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Quotation

• Spot price
• Cash price
• Quoted price Accrued interest
• Example 8 bond with 10.5 years to maturity (?
  0.5 years since last coupon)
• Quoted price 105
• Accrued interest 8 ? 0.5 4
• Cash price 105 4 109

• Forward price
• Use general formula with S cash price
• If no coupon payment before maturity of forward,
  cash forward Fcash FV(Scash)
• If coupon payment before maturity of forward,
  cash forward Fcash FV(Scash -I)
• where I is the PV at time t of the next
  coupon
• Quoted forward price Fquoted
• Fquoted Fcash - Accrued interest

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Quotation Example

• 8 Bond, Quoted price 105
• Time since last coupon
• 6 months
• Time to next coupon
• 6 months (0.5 year)
• Maturity of forward
• 9 months (0.75 year)
• Continuous interest rate 6

• Cash spot price
• 105 8 ? 0.5 109
• PV of next coupon
• 8 ? exp(6 ? 0.5) 7,76
• Cash forward price
• (109 - 7.76) e(6 ? 0.75) 105.90
• Accrued interest
• 8 ? 0.25 2
• Quoted forward price
• 105.90 - 2 103.90

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Delivery

• Government bond futures based on a notional bond
• In case of delivery, the short can choose the
  bonds to deliver from a list of deliverable bonds
  ("gisement")
• The amount that he will receive is adjusted by a
  conversion factor
• INVOICE PRICE
• Invoice Principal Amount
• Accrued interest of the delivered bond
• INVOICE PRINCIPAL AMOUNT
• Conversion factor x FT x 100,000

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Conversion factor Definition

• price per unit of face value of a bond with
  annual coupon C
• n coupons still to be paid
• Yield 6
• n number of coupons still to be paid at
  maturity of forward T
• f time (years) since last coupon at time T

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Conversion factor Calculation

• Step 1 calculate bond value at time T-f (date of
  last coupon payment before futures maturity)
• BT-f PV of coupon PV of principal
  (C/y)1-(1y)-n (1y)-n
• Step 2 Conversion factor k bond value at time
  T
• k FV(BT-f) - Accrued interest BT-f (1y)f -
  C? f
• Example Euro-Bund Future Mar 2000
• Deliverable Bond Coupon Maturity Conversion
  Factor
• ISIN Code ()
• DE0001135101 3.75 04.01.09 0.849146
• DE0001135119 4.00 04.07.09 0.859902
• DE0001135127 4.50 04.07.09 0.894982
• Source www.eurexchange.com

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Cheapest-to-deliver Bond

• The party with the short position decides which
  bond to deliver
• Receives FT ? kj AcIntj
• (Quoted futures price) ? (Conversion factor)
  Accrued int.
• Cost cost of bond delivered sj AcIntj
• Quoted price Accrued interest
• To maximize his profit, he will choose the bond j
  for which
• Max (FT ? kj - sj) or
  Min (sj - FT ? kj)
• j
  j
• Before maturity of futures contract CTD
• Max (F ? kj - sj) or
  Min (sj - F ? kj)
• j
  j

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• Suppose futures 95.00 at maturity
• Short has to deliver bonds among deliverable
  bonds
• with face value of 2.5 mio BEF
• If he delivers bond 242 above, he will receive
• 2.5 mio BEF x .95 x 1.0237 2.431 mio BEF
• His gain/loss depends on the price of the
  delivered bond at maturity
• As several bonds are deliverable, short chooses
  the cheapest to deliver

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Duration

• Duration of a bond that provides cash flow c i at
  time t i is
• where B is its price and y is its yield
  (continuously compounded)
• This leads to

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Duration Continued

• When the yield y is expressed with compounding m
  times per year
• The expression
• is referred to as the modified duration

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Convexity

• The convexity of a bond is defined as

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Duration Matching

• This involves hedging against interest rate risk
  by matching the durations of assets and
  liabilities
• It provides protection against small parallel
  shifts in the zero curve