# OMS 2005-2006 04 IR Derivatives

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```					Options and Speculative Markets
2005-2006
Interest Rate Derivatives
Professor André Farber
Université Libre de Bruxelles
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.
August 23, 2004                        OMS 04 IR Derivatives            |2
Term deposit as a forward on a zero-coupon

M (1+RS × )
100(1+6%× 0.25) = 101.50

0                 T = 0.50                      T* = 0.75

 = 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 )
August 23, 2004                          OMS 04 IR Derivatives              |3
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) ] / (1+rS )

• 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 R=6%
• 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      r=8%                        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

August 23, 2004                       OMS 04 IR Derivatives          |7
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 futures:TICK = 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

August 23, 2004                             OMS 04 IR Derivatives       |8
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

August 23, 2004                                OMS 04 IR Derivatives   |9
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
August 23, 2004                         OMS 04 IR Derivatives      |10
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

FT  Ft 3
• Cash settlement at maturity:            M             
100     12
M
100(1  rT )  100(1  R)     3
100                  12
3
 M  (R  r) 
12
Similar to short FRA except for discounting
August 23, 2004                             OMS 04 IR Derivatives           |11
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 <0 Gain               F F>0 Loss
• Loan                  Loss                        Gain

• F0 = 94.05 => R = 5.95%
• Nominal value per contract = € 1 mio
• Tick = €25 (for on bp)

August 23, 2004                      OMS 04 IR Derivatives       |12
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

August 23, 2004                        OMS 04 IR Derivatives          |13
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  (1+r  3/12)
• Effective Rate = [(Total Debt Payment/50,000,000)-1]  (12/3)
• €X in June is equivalent to €X(1+r) 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/(1+5.95% x 3/12) = 98.53 contracts

August 23, 2004                        OMS 04 IR Derivatives           |14
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

August 23, 2004                    OMS 04 IR Derivatives   |15
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
Delivery Months
The three successive months within the cycle March, June, September and December.
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.

August 23, 2004                                                   OMS 04 IR Derivatives                                 |16
Time scale

Current date t
Maturity of
forward T

Next coupon
Last coupon
August 23, 2004                    OMS 04 IR Derivatives                 |17
Quotation

•    Spot price                                 •    Forward price:
Cash price =                                    Use general formula with S = cash price
Quoted price + Accrued interest                 If no coupon payment before maturity of
Example: 8% bond with 10.5 years to maturity         forward, cash forward Fcash = FV(Scash)
( 0.5 years since last coupon)              If coupon payment before maturity of forward,
Quoted price : 105                                   cash forward Fcash = FV(Scash -I)
Accrued interest : 8  0.5 = 4                      where I is the PV at time t of the next
coupon
Cash price : 105 + 4 = 109
Quoted forward price Fquoted :
Fquoted = Fcash - Accrued interest

August 23, 2004                                OMS 04 IR Derivatives                |18
Quotation: Example

•    8% Bond, Quoted price: 105      •     Cash spot price :
•    Time since last coupon:                 105 + 8  0.5 = 109
•          6 months                  •     PV of next coupon :
•    Time to next coupon :                   8  exp(6%  0.5) = 7,76
•          6 months (0.5 year)       •     Cash forward price :
•    Maturity of forward:            •     (109 - 7.76) e(6%  0.75) = 105.90
•    9 months (0.75 year)            •     Accrued interest :
•    Continuous interest rate: 6%    •      8  0.25 = 2
•     Quoted forward price:
•     105.90 - 2 = 103.90

August 23, 2004                     OMS 04 IR Derivatives                  |19
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

August 23, 2004                         OMS 04 IR Derivatives            |20
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

August 23, 2004                           OMS 04 IR Derivatives        |21
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-(1+y)-n] + (1+y)-n
• Step 2: Conversion factor k = bond value at time T :
• k = FV(BT-f) - Accrued interest = BT-f (1+y)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

August 23, 2004                        OMS 04 IR Derivatives              |22
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

August 23, 2004                       OMS 04 IR Derivatives           |23
•    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

August 23, 2004                           OMS 04 IR Derivatives           |24
Duration

•     Duration of a bond that provides cash flow c i at time t i is

 ci en                       yti

D   ti                                      
i 1  B                                   
where B is its price and y is its yield (continuously compounded)

B
  Dy
B

August 23, 2004                                     OMS 04 IR Derivatives       |25
Duration Continued

• When the yield y is expressed with compounding m times per year

• The expression                BDy
B  
1 y m
is referred to as the “modified duration”

D
1 y m

August 23, 2004                            OMS 04 IR Derivatives   |26
Convexity

The convexity of a bond is defined as
n

1 B    2        c t      i i
2
e    yti

C                   i 1
By 2
B
so that
B          1
  Dy  C (y ) 2

B          2

August 23, 2004                          OMS 04 IR Derivatives   |27
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

August 23, 2004                         OMS 04 IR Derivatives           |28

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