Options and Speculative Markets Interest Rate Derivatives Professor André Farber Solvay Business School Université Libre de Bruxelles

August 23, 2004 OMS 04 IR Derivatives |2 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 |3 Term deposit as a forward on a zero-coupon 0 T = 0.50T* = 0.75 M = 100  = 0.25 M (1+R S ×  ) 100(1+6%× 0.25) = Profit at time T* = [M(R S – r S )  ] = [100 (6% - r S ) 0.25] Profit at time T = [M(R S – r S )  ] / (1 + r S  )

August 23, 2004 OMS 04 IR Derivatives |4 FRA (Forward rate agreement) OTC contract Buyer committed to pay fixed interest rate R fra Seller committed to pay variable interest rate r s 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 CF fra = M[ (r S – R fra )  ] / (1+r S  ) Long position on FRA equivalent to cash settlement of result on forward loan (opposite of forward deposit) An FRA is an elementary swap

August 23, 2004 OMS 04 IR Derivatives |5 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%) / x (4% - 6%) / 1.04 = = Interest on loan: FV(settlement) TOTAL

August 23, 2004 OMS 04 IR Derivatives |6 Treasury bill futures Underlying asset90-days TB Nominal valueUSD 1 million MaturitiesMarch, June, September, December TB Quotation (n days to maturity) –Discount ratey% –Cash price calculation: S t = y  (n/360 ) –Example : If TB yield 90 days = 3.50% St =  (90/360) = TB futures quotation : Ft = TB yield

August 23, 2004 OMS 04 IR Derivatives |7 Example : Buying a June TB futures contract quoted Being long on this contract means that you buy forward the underlying TBill at an implicit TB yield y t =100% % = 3.17% set today. The delivery price set initially is: K = M (100 - y t  )/100 = 1,000,000 [  (90/360)]/100 = 992,075 If, at maturity, y T = 4% (  F T = 96) The spot price of the underlying asset is: S T = M (100 - y T  )/100 = 1,000,000 [  (90/360)]/100 = 990,000 Profit at maturity: f T =S T - K = - 2,075

August 23, 2004 OMS 04 IR Derivatives |8 TB Futures: Alternative profit calculation As forward yield is y t = F t yield at maturity y T = F T = S T profit f T = S T - K = M (100 - y T  )/100 - M (100 - y t  )/100 profit can be calculated as: f T = M [(F T - F t )/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 f T =  F  TICK  F in bp In our example :  F = = - 83 bp f T = -83  25 = - 2,075

August 23, 2004 OMS 04 IR Derivatives |9 Interest Rate Futures (IRF)

August 23, 2004 OMS 04 IR Derivatives |10 3 Month Euribor (LIFFE) Euro 1,000,000 SettleOpen int. July ,507 Sept ,241 Dec ,471 Mr ,896 Wall Street Journal July 2, 2002 Est vol 259,073; open int 1,645,536

August 23, 2004 OMS 04 IR Derivatives |11 Interest rate futures vs TB Futures 3-month Eurodollar (IMM & LIFFE) 3-month Euribor (LIFFE) Similar to TB futures  Quotation F t = y t with y t = underlying interest rate  TICK = M   (0.01/100)  Profit f T =  F  TICK But: TB futuresPrice 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-r T ) with r T underlying interest rate at maturity

August 23, 2004 OMS 04 IR Derivatives |12 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 F t R = (100 – F t ) / 100 r : underlying 3-month interest rate at maturity r T = (100 – F T ) / 100 Cash settlement at maturity: Similar to short FRA except for discounting

August 23, 2004 OMS 04 IR Derivatives |13 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 futuresF   F 0 Loss LoanLossGain F 0 = => R = 5.95% Nominal value per contract = € 1 mio Tick = €25 (for on bp)

August 23, 2004 OMS 04 IR Derivatives |14 Checking the effectiveness of the hedge rTrT 5%6%7% FTFT  F (bp) CF/contract-2, ,625 X ,7506,250131,250 Interest-625, , ,000 Total CF-743,750 Short 50 IRF, F 0 = 94.05, Tick = €25 (for one bp)

August 23, 2004 OMS 04 IR Derivatives |15 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) = contracts

August 23, 2004 OMS 04 IR Derivatives |16 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 |17 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 , 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.

August 23, 2004 OMS 04 IR Derivatives |18 Time scale

August 23, 2004 OMS 04 IR Derivatives |19 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 : = 109 Forward price: Use general formula with S = cash price If no coupon payment before maturity of forward, cash forward F cash = FV(S cash ) If coupon payment before maturity of forward, cash forward F cash = FV(S cash -I) where I is the PV at time t of the next coupon Quoted forward price F quoted : F quoted = F cash - Accrued interest

August 23, 2004 OMS 04 IR Derivatives |20 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 :  0.5 = 109 PV of next coupon : 8  exp(6%  0.5) = 7,76 Cash forward price : ( ) e (6%  0.75) = Accrued interest : 8  0.25 = 2 Quoted forward price: =

August 23, 2004 OMS 04 IR Derivatives |21 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 F T x 100,000

August 23, 2004 OMS 04 IR Derivatives |22 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 |23 Conversion factor: Calculation Step 1: calculate bond value at time T-f (date of last coupon payment before futures maturity): B T-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(B T-f ) - Accrued interest = B T-f (1+y) f - C  f Example: Euro-Bund Future Mar 2000 Deliverable Bond Coupon Maturity Conversion Factor ISIN Code (%) DE DE DE Source:

August 23, 2004 OMS 04 IR Derivatives |24 Cheapest-to-deliver Bond The party with the short position decides which bond to deliver: Receives: F T  k j + AcInt j =(Quoted futures price)  (Conversion factor) + Accrued int. Cost = cost of bond delivered: s j + AcInt j = Quoted price + Accrued interest To maximize his profit, he will choose the bond j for which: Max (F T  k j - s j ) or Min (s j - F T  k j ) j j Before maturity of futures contract: CTD= Max (F  k j - s j ) or Min (s j - F  k j ) j j

August 23, 2004 OMS 04 IR Derivatives |25 Suppose futures= 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 = 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