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## Interest Rate Futures Chapter 6

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**Day Count Conventions in the U.S. (Pages 102-103)**Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (in period) 30/360 Actual/360**S = The bond’s spot value.**F = The futures price. n = The number of futures used in the hedge.**The objective is to take a position in n futures such that a**small change in the interest rate, r, will have no effect on the hedge value. Notationally:**Solve for n that sets dV/dr = 0.**Next, we use the following substitutions for**From the definition of DURATION:**Upon substitution in n:**Usually, the yields sensitivities to the interest rate, r,**are assumed to be the same for the spot yield and for the futures yield . Thus:**Theprice sensitivity hedge ratio,**Example: S = $50,000,000 F=$80,000 DS=8; yS=7.98%; DF=8.3; yF= 7.62%**The price sensitivity hedge ratio with continuous rates is:**In our example:**INTEREST RATE FUTURES**The two most traded interest rate futures: TREASURY BONDS (CBT) USD100,000; pts. 32nds of 100% EURODOLLARS (CME) Eurodollars1,000,000; pts. Of 100%**LONG-TERM INTEREST RATE FUTURES**The U.S. T-BOND FUTURES trades on the CBOT The underlying assets are Treasury Bonds with long-term maturity. It is among the most successful futures contracts of all existing contracts. On any given day, there are 40 to 50 different T-bonds traded in the cash market. Most of these are deliverable against a T-bond futures position (contract specifications below), which makes this market extremely liquid.**SPECIFICATIONS OF U.S. TREASURY BOND FUTURES CONTRACTS**EXCHANGECBOT TICKET SYMBOL US CONTRACT SIZE $100,000 FACE VALUE CONTRACT MONTHS MAR. JUN. SEP. DEC. PRICE QUOTATION POINTS AND 1/32 OF A POINT. PRICES ARE BASED ON 6% COUPON RATE WITH 20 YEARS TO MATURITY TICK SIZE 1/32 OF A POINT, = $31.25 DELIVERABLE GRADES U.S. T-BONDS THAT ARE NOT CALLABLE FOR AT LEAST 15 YEARS AND HAVE A MATURITY OF AT LEAST 15 YEARS FROM THE FIRST BUSINESS DAY OF THE DELIVERY MONTH. LAST TRADING DAY 7TH BUSINESS DAY PRECEDING THE LAST BUSINESS DAY OF THE DELIVERY MONTH. DELIVERY METHOD FEDERAL RESERVE BOOK-ENTRY WIRE-TRANSFER SYSTEM. PRICE LIMITS NONE**Delivery**• The SHORT position prerogatives: • On which day to deliver? • The timing option**Delivery**• Which T-bond to deliver? • The bond option • Many deliverable T-bonds are available • With (call) Maturity > 15years • The short will deliver the most profitable (deliverable) T-bond available. • This bond is called: • The Cheapest –To-Delivery bond.**The Cheapest –To-Delivery bond**SHORT: On date t: Opened a short position for Ft,T. On date T: Buys T-bond for its Cash Price and deliver it for an adjusted futures price. The original futures price, Ft,T was Calculated Based on CR = 6% and time to Maturity 20 yrs, without accrued Interest.**The Cheapest –To-Delivery bond**The delivered T-bond CR is most likely different than 6% and its Maturity may be any number greater than 15. Finally, because of the Daily Marking-to Market process, the original Ft,T does not apply any more. Instead, the last Settlement price applies.**The Cheapest –To-Delivery bond**The SHORT’s payment is calculated as follows: [Last settlement price][Conversion Factor] + Accrued Interest. The SHORT’s cost of delivery is: The Quoted T-bond price + Accrued Interest.**The Cheapest –To-Delivery bond**• The SHORT’s profit is: • [Last settlement price][Conversion Factor] • [The Quoted T-bond price]**The Cheapest –To-Delivery bond**Definition: Of all the deliverable T-bonds in the market The Cheapest to Delivery yeilds the Min{BT(Quoted) – FT(last settle)[CF]} CF = The delivered T-bond’s Conversion Factor.**The Conversion Factor puts the quoted price of the delivered**T-bond on the same footing with the original futures price. It is the NPV of the difference between the two scaled by the value of one contract; $100,000 as follows: CF = NPV(differnece)/(100,000)**CBOT T-BOND CONVERSION FACTORS**YRS = Years to maturity or 1st callability. M = Number of remaining months CR = Coupon of the delivered T-bond. CF = Conversion factor. Round off M to: M*= 0,3,6, OR 9.**Case 1: M* = 0**Case 2:M* = 3**Case 3:M* = 6**Case 4: M* = 9**EXAMPLE:**THE CF ON DEC 1999 FOR DELIVERING THE 11 3/4s WITH MATURITY: NOV 15, 2015. ON 12.1.99 YRS = 15 until 2014. M = 11 14 DAYS are ignored. YRS = 15 M is rounded off to M* = 9 . First compute: CF6 = 1.575012319.**CF6 = 1.575012319.**Next compute:**A SHORT T-BOND HEDGE A bond portfolio manager decides to**sell $10M FV of 11 7/8 T-bonds on March 28. Currently, FEB 26, the bond sells for S=$101/$100FV. DATECASH FUTURES FEB. 25 10M FV T-BONDS SELL 160 JUN T-bond CR = 11 7/8 FUTURES. S = 10,100,000 F=70-16 Ds = 7.83 Df = 7.20 Ys = 11.74% Yf = 14.92%**DATECASH FUTURES**MAR. 28 S = 95.6875/$100FV LONG 160 JUN T-bond $9,568,750 FUTURES Opportunity loss <$531,250> F = 61 - 23 Futures gain: [(70-16)-(61-23)]160 =(8-25)160=($8,781.25)160 =$1,405,000. Total selling price: 9,568,750 + 1,405,000 = $10,973,750**LONG HEDGE WITH T - BOND FUTURES**DATECASHFUTURES MAR. 29 LONG 110 SEP T-BOND Fs. F = 78-21 BY REGRESSION: n*= 110. JUL. 15 S=107 19/32 SHORT110 SEP T-BOND Fs $10,759,375 BUY BONDS. F = 86-6 Gain from futures:110[(86-6) – (78-21)] =110[7-17] =110[$7,531.25] = $828,437.5 THUS, THE EFFECTIVE PURCHASE PRICE OF THE T- BONDS IS: $10,7593,750 - $ 828,437.5 = $9,930,937.5.**HEDGING A CORPORATE BOND ISSUE**FEB. 24. DECISION: ISSUE $50M CORPORATE BONDS AT PAR VALUE ON MAR. 24. EXPECTATIONS: CR = 13.76% M = 20yrs D = 7.22 DATECASHFUTURES 2.24 DS = 7.83 SHORT 674 FUTURES. yS = 13.6% F(JUN) = 68-11 S=$50M. DF=7.83; yF = 13.6%**DATE SPOTFUTURES**3.24 ISSUE BONDS LONG 674 JUN T-BOND Fs CR=13.26% F(JUN) = 55-25 S=$90.74638/$100FV V(BOND ISSUE) = $45,373,190 Gain from futures: 674[(68-11)-(55-25)] =674[12-18] =674[$12,562.5] =$8,467,125. TOTAL VALUE=$53,840,315.**SHORT-TERM INTEREST RATE FUTURES**EURODOLLAR FUTURES The U.S. T-BILLS FUTURES The underlying assets For the Eurodollars futures are 3-months Eurodollars time deposit. It is the most successful futures contracts of all existing contracts. Clearly, the underlying asset for the U.S. Gov. T-bills are T-bills.**CONTRACT SPECIFICATIONS FOR:**90-DAY T-BILL3-Month EURODOLLAR FUTURES SPECIFICATIONS 13-WEEK US T-BILL 3-MONTH EURODOLLAR TIME DEPOSIT SIZE USD1,000,000 Eurodollars1,000,000 CONTRACT GRADE new or dated T-bills CASH SETTLEMENT with 13 weeks to maturity YIELDS DISCOUNT ADD-ON HOURS ( Chicago time) 7:20 AM-2:00PM 7:20 AM - 2:00PM DELIVERY MONTHS MAR-JUN-SEP-DEC MAR-JUN-SEP-DEC TICKER SYMBOL TB EB MIN. FLUCTUATION .01(1 basis pt) .01(1 basis pt) IN PRICE USD25/pt USD25/pt LAST TRADING DAY The day before the 2nd London business day first delivery day before 3rd Wednesday DELIVERY DATE 1st day of spot month Last day of trading on which 13-week T-bill is issued and a 1-year T-bill has 13 weeks to maturity**EURODOLLAR FUTURES**These are futures on the interest earned on Eurodollar three-month time deposits. The rate used is LIBOR - London Inter-Bank Offer Rate. These time deposits are non transferable, thus, there is no delivery! Instead, the contracts are CASH SETTLED.**EURODOLLAR FUTURES PRICE**The IMM (CME) quotes the IMM index. Let the quote be denoted by Q then, the Futures price is given by: F = 1,000,000[1 – (1 – Q/100)(.25)]. On the delivery date – the third Wednesday of the delivery month – the quote for the CASH SETTLEMENT is given by the 90-day LIBOR: Q/100 = 1 – L/100 F = 1,000,000[1 - .25L/100]**EURODOLLAR FUTURES PRICE:**The IMM index = 95.53 for the JUN 2001. F = 1,000,000[1 – (1 – 95.53/100)(.25)] =$988,825 On the delivery date the quote for the CASH SETTLEMENT is given by: The 90-day LIBOR: 8% F = 1,000,000[1 - .25(8)/100]=$980,000**Arbitrage with Eurodollar Futures continued**DATESPOTFUTURES MAY 23 Deposit $1,000,000 Short 1 ED futures. in a 117 days ED time L = 9.35% deposit to earn 9.40% over the 117 days. Jun 19 Borrow $1,000,000 Cash settle (Long) for the current L. at the current L. This is equivalent to borrow the money for 9.35% SEP 17 Receive Repay the loan 1,000,000[1+.094(117/360)] 1,000,000[1+.0935(90/360)] =1,030,550 = 1,023,375 Arbitrage profit = $7,175**How to calculate the profit from a ED futures?**Assume that initially, Qt = 90.54. Thus, Ft = 1,000,000[1 – (1 – 90.54/100)(.25)]. At a later date, k, the index dropped by exactly 100th of a point; that is, Qk = 90.53. Fk = 1,000,000[1 – (1- 90.53/100)(.25)]. It is easily verified that the difference between the two futures prices is exactly: $25. Thus, we have just seen that Every 100th of the quote Q is $25.**Hedging with Eurodollars futures.**Eurodollar futures became the most successful contract in the world. Its enormous success is attributed to its ability to fill in the need for hedging that still remained open even with a successful market for T-bond and T-bill futures. The main attribute of the 90-day Eurodollars futures is that, unlike the T-bills futures, it is risky. This risk makes it a better hedging tool than the risk-free T-bill futures.**The examples below demonstrate how to hedge with ED futures**using a STRIP, or a STACK. In most of the loans involved in these hedging strategies, the interest today determines the payment by the end of the period. Only interest payments are paid during the loan term and the last payment include the interest and the principal payment.**A STRIP HEDGE WITH EURODOLLARS FUTURES**On November 1, 2000, a firm agrees to borrow $10M for 12 months, beginning December 19, 2000 at LIBOR + 100bps. DATECASHFUTURESQ 11.1.00 LIBOR 8.44% Short 10 DEC 91.41 Short 10 MAR 91.61 Short 10 JUN 91.53 Short 10 SEP 91.39 12.19.00 LIBOR 9.54% Long 10 DEC 90.46 3.13.01 LIBOR 9.75% Long 10 MAR 90.25 6.19.01 LIBOR 9.44% Long 10 JUN 90.56 9.18.01 LIBOR 8.88% Long 10 SEP 91.12**PERIOD: 1 23 4**RATEa: 10.54% 10.75% 10.44% 9.88% INTERESTb: $263,500 $268,750 $261,000 $247,000 FUTURESc: $23,750 $34,000 $24,250 $6,750 NETd: $239,750 $234,750 $236,750 $240,250 EFFECTIVE RATEe:9.59% 9.39% 9.47% 9.61% UNHEDGED AVERAGE RATE: 10.40% HEDGED AVERAGE RATE: 9.52% a. LIBOR + 100 BPS b. ($10M)(RATE)(3/12) c. (PRICE CHANGE)(25)(100)(10) d. b - c e. (NET/10M)(12/3)(100%)**A STACK HEDGE WITH EURODOLLAR FUTURES:**DATA ON NOVEMBER 11, 2000 VOLUMEOPEN INTEREST DEC 00 46,903 185,609 MAR 01 29,236 127,714 JUN 01 5,788 77,777 SEP 01 2,672 30,152 DECISION: STACK MAR FUTURES FOR JUN AND SEP AND ROLL OVER AS SOON AS OPEN INTEREST REACHES 100,000.**THE STACK HEDGE**DATE CASHFUTURESF. POSITION 11.1.00 8.44% S 10 DEC 91.41 S10DEC S 30 MAR 91.61 S30MAR 12.19.00 9.54% L 10 DEC 90.46 S30MAR 1.12.01 9.47% L 20 MAR 90.47 S10MAR S 20 JUN 90.42 S20JUN 3.13.01 9.75% L 10 MAR 90.25 S20JUN 3.22.01 9.95% L 10 JUN 89.78 S10JUN S 10 SEP 89.82 S10SEP 6.19.01 9.44% L 10 JUN 90.56 S10SEP 9.18.01 8.88% L 10 SEP 91.12 NONE**PERIOD: 1234**• RATE(%)a: 10.54 10.75 10.44 9.88 • INTERESTb: 263,500 268,750 261,000 247,000 • FUTURES($)c: 23,750 91,000 12,500 <32,500> • NET($) d: 239,750 177,750 248,500 279,500 • EFFECTIVE RATE (%)e:9.59 7.11 9.94 11.18 • UNHEDGED AVERAGE RATE 10.40% • HEDGED AVERAGE RATE 9.455% • a. LIBOR + 100 BPS • b. ($10M)(RATE)(3/12) • c. (PRICE CHANGE)(25)(100)(10) • d. b - c • (NET/10M)(12/3)(100%).