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Interest Rate Markets Chapter 5 1 Types of Interest Rates • Treasury rates • LIBOR rates • Repurchase rates 2 QUOTES ARE GIVEN BY CONVENTION, USING A DISCOUNT YIELD, d, WHERE: DISCOUNT = FACE VALUE - MARKET PRICE: d 360 DISCOUNT t FV FV P 360 d t FV BOND EQUIVALENT YIELD(BEY): i 365 FV P t P 1 365 dt id 1 360 360 365d 360 dt 3 EXAMPLE: t = 90 days FV = $1,000,000 d = 11% 360 (DISCOUNT) .11 DISCOUNT $27,500. 90 1,000,000 1,000,000 972,500 360 d .1111% 1,000,000 90 BEY i 365 1,000,000 972,500 .11468 90 972,500 365 (.11)90 i (.11) 1 360 360 1 .11468 (11.468%). 4 The market for Repurchase Agreements An integral part of trading T-bills and T-bill futures is the market for repurchase agreements, which are used in much of the arbitrage trading in T-bills. In a repurchase agreement -- also called an RP or repo -- one party sells a security (in this case, T-bills) to another party at one price and commits to repurchase the security at another price at a future date. The buyer of the T-bills in a repo is said to enter into a reverse repurchase agreement., or reverse repo. The buyer’s transactions are just the opposite of the seller’s. The figure below demonstrates the transactions in a repo. 5 Transactions in a Repurchase Agreement Date 0 - Open the Repo: Party A T- Bill PO Party B Date t - Close the Repo T-Bill Party A Party B Pt= P0(1+r0,t ) 6 Example: T-bill FV = $1M. P0 = $980,000. The repo rate = 6%. The repo time: t = 4 days. P1= P0 [(repo rate)(n/360) + 1] = 980,000[(.06)(4/360) + 1] = 980,653.33 7 A repurchase agreement effectively allows the seller to borrow from the buyer using the security as collateral. The seller receives funds today that must be paid back in the future and relinquishes the security for the duration of the agreement. The interest on the borrowing is the difference between the initial sale price and the subsequent price for repurchasing the security. The borrowing rate in a repurchase agreement is called the repo rate. The buyer of a reverse repurchase agreement receives a lending rate called the reverse repo rate. The repo market is a competitive dealer market with quotations available for both borrowing and lending. As with all borrowing and lending rates, there is a spread 8 between repo and reverse repo rates. The amount one can borrow with a repo is less than the market value of the security by a margin called a haircut. The size of the haircut depends on the maturity and liquidity of the security. For repos on Tbills, the haircut is very small, often only one-eighth of a point. It can be as high as 5% for repurchase agreements on longer-term securities such as Treasury bonds and other government agency issues.Most repos are held only overnight, so those who wish to borrow for longer periods must roll their positions over every day. However, there are some longer-term repurchase agreements, called term repos, that come in standardized maturities of one, two, and three weeks and one, two, three, and six months.Some other 9 customized agreements also are traded. Zero Rates A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T. 10 Example (Table 5.1, page 95) Maturity (years) 0.5 Zero Rate (% cont comp) 5.0 1.0 5.8 1.5 6.4 2.0 6.8 11 Bond Pricing • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate • In our example, the theoretical price of a two-year bond (FV = $100) providing a 6% coupon semiannually is: 3e 0.050.5 103e 3e 0.0581.0 0.0682.0 3e 0.0641.5 98.39 12 Bond Yield • The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond • Suppose that the market price of the bond in our example equals its theoretical price of 98.39 • The bond yield is given by solving 3e y0.5 3e y1.0 3e y1.5 103e y2.0 98.39 to get y=0.0676 or 6.76%. 13 Par Yield • The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. • In our example we solve c 0.050.5 c 0.0581.0 c 0.0641.5 e e e 2 2 2 c 0.0682.0 100 e 100 2 to get c = 6.87 (with s.a. compoundin g) 14 Par Yield continued In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date (100 100d)m c A 15 Sample Data for Determining the Zero Curve (Table 5.2, page 97) Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) 100 0.25 0 97.5 100 0.50 0 94.9 100 1.00 0 90.0 100 1.50 8 96.0 100 2.00 12 101.6 16 The Bootstrapping the Zero Curve • An amount 2.5 can be earned on 97.5 during 3 months. • The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding • This is 10.127% with continuous compounding • Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding 17 The Bootstrap Method continued • To calculate the 1.5 year rate we solve 4e 0.10469( 0.5) 4e 0.10536(1.0) 104e R (1.5) 96 to get R = 0.10681 or 10.681% • Similarly the two-year rate is 10.808% 18 Zero Curve Calculated from the Data (Figure 5.1, page 98) 12 Zero Rate (%) 11 10.681 10.469 10 10.808 10.536 10.127 Maturity (yrs) 9 0 0.5 1 1.5 2 2.5 19 BONDS - THE CASH MARKET DEFINITION A BOND IS A PROMISE TO PAY CERTAIN AMOUNTS ON PRESPECIFIED FUTURE DATES. BOND PARAMETERS P = THE BOND CASH PRICE FV= THE FACE VALUE or THE PAR VALUE OF THE BOND M= THE MATURITY DATE OF THE BOND, or THE END OF THE LAST PERIOD OF THE BOND’S LIFE. t= THE TIME INDEX; Ct= THE CASH FLOW FROM THE BOND AT TIME PERIOD t. USUALLY, THE CASH FLOW IS ASSUMED TO OCCUR AT THE PERIOD’S END. t = 1,2,……, M. NORMALLY: Ct= C for t = 1, …., M – 1 and CM= C + FV. C IS CALLED THE COUPON OF THE BOND. CR = THE COUPON RATE. C= (CR)(FV) = THE $ AMOUNT OF THE COUPON. 20 EXAMPLE: A 30 YEAR TREASURY BOND WITH FACE VALUE OF $1,000, PAYS ANNUAL COUPONS AT AN 8%. THUS: M=30; FV = $1,000; CR = 8%; C = (.08)($1,000) = $80 for t= 1, 2, …29 and C = $1,080 for t = 30. An investor who buys this bond will receive $80 every year for the next 30 years. 21 EXAMPLE: THE SAME T-BOND PAYS COUPONS SEMI ANNUALLY. IN THIS CASE, WE HAVE NEW PARAMETERS: m= THE NUMBER OF PAYMENT PERIODS EVERY YEAR. N = THE NUMBER OF COUPON PAYMENTS. Thus: M=30; m = 2; N = 60; FV = $1,000; CR = 8%; C = (.08/2)($1,000) = $40 for t= 1, 2, …59 and C = $1,040 for t = 60. An investor who buys this bond will receive $40 every six months for the next 30 years. DEFINITION: A PURE DISCOUNT BOND PAYS THE FV AT ITS MATURITY BUT PAYS NO COUPONS (C = 0) IN ANY OF THE INTERIM PERIODS. 22 PRICING BONDS: A BOND WITH MATURITY OF M YEARS WITH COUPONS PAID ANNUALLY IS PRICED BY: M Ct P . P is the NPV of the CFs : t t1 (1 r) C C C 3 1 2 P ...; 1 r1 (1 r1)(1 r2 ) (1 r1)(1 r2 )(1 r3) The usual convention is r rt for all t. M Ct FV , P t1 (1 r)t (1 r)M A closed form solution : M C FV P 1 (1 r) . M r (1 r) 23 r = the bond’s yield to maturity. That is, if the investor buys the bonds at the market price and holds it to its maturity, r is the annual rate of return on this investment. 24 FOR A BOND WITH SEMIANNUAL COUPON PAYMENTS: C 2M FV 2 Ρ r t r 2M (1 ) t1 (1 ) 2 2 A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0: P FV (1 r) M 25 EXAMPLES: M = 30; FV = $1,000; CR = 8% PAID SEMIANNUALLY. => C = (.08) 1,000/2=$40. r = 10% 60 t 1 40 1,040 40 1,000 60 1 1 . 05 $810.70 t 60 60 (1 .05) 1.05 .05 1.05 THE SAME BOND WITH ANNUAL COUPON PAYMENTS WOULD BE PRICED: 80 1,000 30 1 1.1 $811.46 1.1 1.130 THE BOND IS SOLD AT A DISCOUNT BECAUSE THE YTM IS GREATER THAN THE CR r = 5% P 40 1,000 1 1.02560 $1,463.63 60 .025 1.025 THE SAME BOND AS A PURE DISCOUNT BOND T BOND WOULD BE PRICED: 1,000 $57.31 30 11 . IF IT WERE A CONSUL WITH ANNUAL COUPONS: P 80 $800 .1 26 Result: The bond sells at par if P = FV. If CR = r the bond sells at par. If CR > r the bond sells at a premium. If CR < r the bond sells at a discount. 27 DURATION DEFINITION: M D OBSERVE THAT: t 1 tC t t 1 r . P C t 1 r t D t P t 1 M M Wt, t 1 t Where: Ct / 1 r Wt 1 and Wt . P t 1 This result leads to the following M interpretation of Duration: t 28 DURATION IS THE WIEGHTED AVERAGE OF COUPON PAYMENTS’ TIME PERIODS, t, WEIGHTED BY THE PROPORTION THAT THE DISCOUNTED CASH FLOW, PAID AT EACH PERIOD, IS OF THE CURRENT BOND PRICE. 29 Duration in continuous time • Duration of a bond that provides cash flow c i at time t i is ci e ti i 1 B n yti where B is its price and y is its yield (continuously compounded) • This leads to B Dy B 30 DURATION INTERPRETED AS A MEASURE OF THE BOND PRICE SENSITIVITY M Ct The bond price : P . t t 1 1 r Its derivative with respect to r or r 1 : dP dP 1 dr d(1 r) (1 r) tC t (1 r) t . dP (1 r) (1 r) d(1 r) P P(1 r) tC t (1 r) t . Rearrangin g, yields : tC t dP t (1 r) P D. d(1 r) P 1 r 31 The final result states that : dP P D. d(1 r) 1 r THE %Δ OF THE BOND PRICE D THE %Δ OF THE YIELD D THE BOND PRICE ELASTICI TY. The negative sign merely indicates that D changes in opposite direction to the change in the yield, r. Next we present a closed form formula to calculate 32 duration of a bond: COMPUTING DURATION r = 10% Coupon Rate M 0 2 4 6 8 10 12 14 16 5 5 4.76 4.57 4.41 4.28 4.17 4.07 3.99 3.92 10 10 8.73 7.95 7.42 7.04 6.76 6.54 6.36 6.21 15 15 11.61 10.12 9.28 8.74 8.37 8.09 7.88 7.71 20 20 13.33 11.20 10.32 9.75 9.36 9.09 8.89 8.74 25 25 14.03 11.81 11.86 10.32 9.98 9.75 9.58 9.45 30 30 14.03 11.92 11.09 10.65 10.37 10.18 10.04 9.94 35 35 13.64 11.84 11.17 10.82 10.61 10.46 10.36 10.28 40 40 13.13 11.70 11.18 10.92 10.76 10.65 10.57 10.51 50 50 12.19 11.40 11.40 10.99 10.91 10.85 10.81 10.78 100 100 11.02 11.01 11.00 11.00 11.00 11.00 11.00 11.00 33 N = NUMBER OF PAYMENTS m = COUPON PAYMENTS PER YEAR Θ = FRACTION OF YEAR TO THE NEXT COUPON PAYMENT (1 D r) 1 Θ 1 2 ….years r N 1 r N r 2 FV Θ N 1 m m C m r 1 r N 1 r 2 FV m C EXAMPLE: r = 10% = .1 FV = $ 100 C = $6 => CR = 6% M = 30 =1 m=1 P = $62.29 100 1.1 1.1 1 (.1)30 (.1) 30 6 D 11.09 30 2 100 (.1) 1.1 1 (.1) 34 6 30 2 APPLICATIONS OF DURATION: 1. Δr ΔΡ ( D)(P) 1 r EXA MPLE: M=30; CR=6%;D=11.09 and P=$62.29. r = 10% r = 11% P 1109 . 62.29 r 10% .01 11 . 6.28 P $56.01 1 r = .08 P = -(11.09)(62.29) (-.02) 12.56 P $74.85 1.1 1 35 Duration Continued • When the yield y is expressed with compounding m times per year BDy B 1 y m • The expression D 1 y m is referred to as the “modified duration” 36 DURATION OF A BOND PORTFOLIO V = The total bond portfolio value Pi = The value of the i-th bond Ni = The number of bonds of the i-th bond in the portfolio Vi = Pi Ni = The total value of the i-th bond in the portfolio V = ΣPiNi The total portfolio value. We now prove that: DP = ΣwiDi . 37 We already saw that duration may be interprete d as the bond price elasticity : dV dV (1 r) V DP . d(1 r) d(1 r) V (1 r) Thus, taking the derivative of the bond portfolio value, V, with respect to the yield - to - maturity leads to : d N i Pi dV dPi Ni . d(1 r) d(1 r) d(1 r) 1 r Multiplyin g and deviding by , Pi yields : 38 dV dPi (1 r) Pi N i . d(1 r) d(1 r) Pi (1 r) Rearrangin g terms in the summation sign : 1 dPi (1 r) N i Pi [ ] (1 r) d(1 r) Pi The terms in the square bracket are the negative of bond' s i duration, while N i Pi Vi . Thus, we rewrite : dV 1 Vi D i . d(1 r) (1 r) 39 Substituti ng this result into dV 1 r DP , we have : d(1 r) V 1 1 r DP Vi D i 1 r V Vi Di Dp , or : V 40 1 Vi D P Vi Di Di w i Di , V V where : Vi wi ; V w i 1. Thus, in conclusion , the protfolio duration D P w i Di is the weighted average of the durations of the bonds in the portfolio. The weights are the proportions the bond value is of the entire portfolio 41 value. Example: a portfolio of two T-bonds: BOND N YTM COUPON T-BOND FV in $M $100 15yrs 6% 5% T-BOND $200 30yrs 6% 15% BOND PRICE $90.2 $449.1 539.3 T-BOND T-BOND D W 0.1673 0.8327 1.0000 D 10.4673 12.4674 = (.1673)(10.4673) +(.8327)(12.4674) = 12.1392 42 Example: a two bond portfolio: BOND FV N YTM CUPON T-BOND $500M 15yrs 6% 5% T-BOND $200M 30yrs 6% 15% BOND PRICE $451,5M $449,1M $900,6M 1 2 T-BOND1 T-BOND 2 D W 0,5013 0,4987 1,0000 D 10,4673 12,4674 = (0,5013)(10,4673) +(0,4987)(12,4674) = 11,45. 43 APPLICATION OF DURATION 2. IMMUNIZING BANK PORTFOLIO OF ASSETS AND LIABILITIES TIME 0 ASSETS $100,000,000 (LOANS) D=5 r = 10% TIME 1 r => 12% LIABIABILITIES $100,000,000 (DEPOSITS) D=1 r = 10% (.02) ΔVA 5 100,000,000 $9,090,909.09 1.1 (.02) ΔVL 1 100,000,000 $1,818,181.82 1.1 44 BUT IF DA = DL THEY REACT TO RATES CHANGES IN EQUAL AMOUNTS. THE BANK PORTFOLIO IS IMMUNIZED , i.e., IT’S VALUE WILL NOT CHANGE FOR A “small” INTEREST RATE CHANGE, IF THE PORTFOLIO’S DURATION IS ZERO or: DP = DA - DL = 0. 45 APPLICATIONS OF DURATION. 3. EXAMPLE: A 5-YEAR PLANNING PERIOD CASE OF IMMUNIZATION IN THE CASH MARKET BOND A B C FV M $100 $1,000 5 yrs $100 $1,000 10 yrs r 10% 10% D P 4.17 $1,000 6.76 $1,000 4.17WA + 6.76WB =5 WA + WB =1 WA = .677953668. WB = .322046332. VP = $200M implies: Hold $135,590,733.6 in bond A, And $64,409,266.4 in bond B. Next, assume that r rose to 12%. The portfolio in which bonds A and B are held in equal proportions will change 46 to: [1 - 4.17 (.02/1.1)] 100M = $92,418,181.2 [1 - 6.76 (.02/1.1)] 100M TOTAL = $87,709,090.91 = $180,127,272.7 INVEST THIS AMOUNT FOR 5 YEARS AT 12%, CONTINUOUSLY COMPOUNDED YIELDS: $328,213,290. ANNUAL RETURN OF: 5 328,213,290 1 9.9%. 200,000,000 47 The weighted average portfolio changes to: (.02) 135,590,733.61 4.17 125,310,490.7 1.1 (.02) 56,492.782 64,409,266.41 6.76 181,803,272.7 1 . 1 AFTER 5 YEARS AT 12%: $331,267,162. ANNUAL RETURN OF: 5 331,267,162 1 10%. 200,000,000 48 BOND FUTURES We will study: Short-term bonds: U.S Government T-Bills and Euro-dollar time deposit rates. U.S Government T-Bonds. 49 AN APPLICATION OF DURATION 4. THE PRICE SENSITIVITY HEDGE RATIO The Hedge Value: V = S + nF. S = The bond’s spot value. F = The futures price. n = The number of futures used in the hedge. Minimize the position’s value sensitivity to interest rate changes. First, we write the change in the hedge value when r changes: Criterion: 50 dV dS dF n . dr dr dr Using the chain rule : dV dS dyS dF dy F n . dr dyS dr dy F dr 51 Solve for n that sets dV/dr = 0. dV 0, dr dS dy S n dF dy F dy S dr . dy F dr Next, we use the following substitutions for dS dF and . dyS dy F 52 From the definition of DURATION: dS 1 + yS DS = S dyS dS SDS dyS 1 yS dF 1 + yF dF FDF DF = F dyF dy F 1 y F Upon substitution in n: 53 SDS 1 yS n FD F 1 yF dyS dr dy F dr 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: dyS dy F dr dr 54 The price sensitivity hedge ratio. SDS (1 y F ) n FD F (1 yS ) 55 The price sensitivity hedge ratio with continuous rates is: SDS n FD F 56 INTEREST RATE FUTURES The three most traded interest rate futures: TREASURY BILLS (CME) USD1,000,000; pts. Of 100% EURODOLLARS (CME) Eurodollars1,000,000; pts. Of 100% TREASURY BONDS (CBT) USD100,000; pts. 32nds of 100% 57 CONTRACT SPECIFICATIONS FOR: 90-DAY T-BILL; 3-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 1st day of spot month Last day of trading DELIVERY DATE on which 13-week T-bill is issued and a 1-year T-bill has 13 weeks to maturity 58 Arbitrage profits in the short-term futures market involve Activities in the futures market and the spot repo market. The following figures explain the cash and carry and the reverse cash and carry strategies: 59 Transactions in a Cash-and-Carry Arbitrage. Repo Market PO Arbitrageur T-Bill Short Position F0,T T-Bill T-Bill Dealer Futures Market Date 0 Repo Market P0(1+r0T) Arbitrageur T-Bill F 0,T > P0(1+r0,T) Receive F Deliver 0,T T-Bill Futures Market Date T 60 Transactions in a Reverse Cash-and-Carry Arbitrage. Repo Market PO Arbitrageur T-Bill Long Position F0,T P0 T-Bill Dealer Futures Market Date 0 Repo Market P0(1+r0,T) Arbitrageur T-Bill Take Delivery F 0,T < P0(1+r0,T) Date T Pay F 0,T T-Bill Futures Market 61 EXAMPLE: A 91- DAY T-BILL ARBITRAGE An arbitrageur observes that a 91-day T-bill yields 9.20 percent, a 182-day bill yields 9.80 percent, and a futures contract requiring delivery of a 91-day T-bill three months hence is priced so as to yield 10.2 percen t. What action would he take ? TIME LINE <-------- r = 9.20% --------> <--------- 10.20% --------> -|----------------------------------------|---------------------------------------|---> 0 91 days 182 days ------------------------------- r = 9.80% -------------------------------------> (1.098)2 = 1.092(1 + r1,2) r1,2= .1040 or 10.40% The no-arbitrage rate: 10.4% is greater than the market rate: 10.2%. CASH - AND - CARRY 62 THE CASH-AND-CARRY STRATEGY . TIME 0 91 DAYS CASH a) ENTER A REPO AGREEMENT BY BORROWING $954,330.56 FOR 91 DAYS. b) BUY THE 182-DAY T-BILL FOR $954,330.56 AND GIVE IT TO THE OTHER PARTY OF THE REPO AGREEMENT. THUS, EFFECTIVELY, THE 91-DAY T-BILL IS SOLD TO THE OTHER PARTY FOR $954,330.56 RECEIVE THE T-BILL WHICH IS NOW A 91-DAY BILL. FUTURES c) SHORT A T-BILL FUTURES F0,t = $976,011.75 DELIVER THE 91-DAY T-BILL FOR $976,011.75 REPAY THE REPO DEALER $975,561.14 PROFIT $450.61 / CONTRACT. 63 Repo Market PO =$954,330.56 Arbitrageur 182-day T-Bill Short FO,T = $976,011.75 P0= $954,330.56 T-Bill Dealer Futures Market Date 0 Repo Market P0(1+r0,t) = $975,561.14 Arbitrageur 91 day T-Bill Profit = $450,61 Deliver 91-day T-Bill F 0,T = $976,011.75 Futures Market Date T 64 IF THE IMPLIED FUTURES RATE WAS 11.2% THEN: THEORETICAL = 10.4% < 11.2% = ACTUAL REVERSE CASH-AND-CARRY TIME CASH 0 a) ENTER A REVERSE REPO FOR $954,330.56. b) SELL 182-DAY SHORT FUTURES c) LONG A 91-DAY T-BILL FUTURES F0,t = $973,809.04 1,000,000 (1.112).25 91 DAYS CLOSING THE REPO YOU RECEIVE $975,561.13 AND CLOSING THE SHORT POSITION YOU DELIVER THE 21 DAY T-BILLS PROFIT: $975,561.13 - $973,809.04 = $1,752.09 TAKE DELIVERY OF 91-DAY T-BILL FOR $973,809.04 65 PO = $954,330.56 Repo Market Arbitrageur 182-day T-Bill Long FO,T = $973,809.04 PO = $954,330.56 Futures Market T-Bill Dealer Repo Market Date 0 P1 = $975,561.13 Arbitrageur 91 day T-Bill PROFIT = $1,752.09 Date T Take Delivery 91-day T-Bill Futures Market F 0,T = $973,809.04 66 We now present several example of hedging with 90-day T-bills futures. In order to determine whether to hedge LONG or SHORT, one must remember that the bonds price is reciprocal to the interest rate and thus, hedging a falling interest rate means hedging an increase in the bond’s price and vice-versa. 67 A LONG HEDGE WITH T-BILLS FEB. 15 CASH FUTURES DO NOTHING LONG 1 JUN T-Bill FUTURES d = 8.20 => P = 100-8.2(91/360) IMMI = 91.32 => d = 8.68 => P = 97.927222 PER 1M P = $979,272 365 91 100 yS [ ] 1 0.0876 97.927222 F = 1,000,000[1-8.68/100(.25)] F = $978,300 100 yF [ ] 97.83 365 91 1 0.0919 (.25)(979, 272)(1.0919) n 1 (.25)(978, 300)(1.0876) 68 MAY 17 d = 7.69 => P = 100 - 7.69(91/360) SHORT 1 JUNE T-BILL FUTURES P = 98.0561 IMMI = 92.54 BUY PER 1M $980,561 F = 1,000,000[1 - 7.46/100(.25)] = OPPORTUNITY LOST <$1,289.17> $981,350 ACTUAL PRICE $977,511 PROFIT $3,050 69 HEDGE COMMERCIAL PAPER ISSUE: A SHORT HEDGE WITH T-bills FUTURES CASH MARKET FUTURES MARKET APR 6 d = 10.17 180 P = 94.915 = (100 - 10.17 ) 360 365 180 1,000,000 yS [ ] 949,150 DS = 180/365 IMMI = 88.23 => d = 11.77 F = 10,000[100-11.77(90/360)] = $970,575 DF = 90/365 1,000,000 ] 1 .1116 yF [ 970,575 365 91 1 .1283 180 / 365 9,491,500 1.1283 n= = 20 90 / 365 970,575 1.1116 Do Nothing. Short 20 T-bills futures 70 JUL 20 ISSUE $10M COMMERCIAL PAPER MATURING IN 180 DAYS d = 11.34 BUY 20 SEP T-bills FUTURES IMMI = 87.47 F=10,000[10012.53(90/360)] P = 94.33 INCOME: $9,433,000 F = $968,675 PROFIT: 20[970,575-968,675] = $38,000 COST(NO HEDGE) : ( 10M 9,433,000 365 )180 - 1 = 12.51% COST(WITH HEDGE): 20(970,575) + [9,433,000 - 20(968,675)] =$ 9,471,000 10M ) ACTUAL COST: ( 9,471,000 365 180 - 1 = 11.65% 71 A FIRM BORROWS $10M AT A FLOATING RATE LIBOR +1% TIME SPOT FUTURES SEP. 15 RECEIVE $10M LIBOR = 8% PAY 9% I1 = $225,000 SHORT 10 T-BILL FUTURES DEC F = 91.75 MAR F = 91.60 JUN F = 91.45 DEC. 15 LIBOR = 9.15% PAY 10.15% I1 = $225,000 LONG 10 DEC. FUTURES: F = 90.85 GAIN: 10[91.75 - 90.85]100 (25) = $22,500 ACTUAL: $202,500 8.10% R 72 A FIRM BORROWS $10M AT A FLOATING RATE LIBOR +1% TIME MAR. 15 SPOT LIBOR = 9.50 PAY 10.50% I2 = $253,750 ACTUAL: $226,250 JUNE 15 LIBOR = 10.05% PAY 11.05% I3 = $262,500 ACTUAL: $225,000 SEP. 15 FUTURES LONG 10 MAR FUTURES F = 90.50 GAIN: 10[91.60 - 90.50]100 (25) = $27,500 9.05% R LONG 10 JUN. FUTURES: F = 89.95 GAIN: 10[91.45 - 89.95]10,000 (.25) = $37,500 9.00% R I4 = $276,250 PAY $10M ACTUAL: $276,250 11.05% R Unhedged average rate: 10.175% Hedged average rate: 9.30%. 73 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. 74 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] 75 Arbitrage with Eurodollar Futures r0,k = REPO RATE FOR 27 DAYS 9.45% r0,T = ED TD RATE FOR 117 DAYS 9.40% rF = ED TD FUTURES RATE: F0,k = 90.65 rF = 9.35% _|_________________|_____________________|_________TIME 0 k T 1+ r F = (1.094 ) 117 90 (1.0945) 1.1238867 27 = 1.0274595 = 1.09385 90 rF = 9.385% > 9.35% 76 Arbitrage with Eurodollar Futures continued DATE SPOT FUTURES MAY 23 Deposit $1,000,000 Short 1 ED futures. in a 117 ED time deposit L = 9.35% 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 77 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. 78 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. 79 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 determine 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. 80 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. DATE CASH FUTURES Q 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 81 PERIOD: 1 2 RATEa: 10.54% 10.75% INTERESTb: $263,500 FUTURESc: NETd: 3 10.44% 9.88% $268,750 $261,000 $247,000 $23,750 $34,000 $24,250 $6,750 $239,750 $234,750 $236,750 $240,250 EFFECTIVE RATEe: 9.59% 9.39% 9.47% UNHEDGED AVERAGE RATE: 10.40% HEDGED AVERAGE RATE: 9.52% a. b. c. d. e. 4 9.61% LIBOR + 100 BPS ($10M)(RATE)(3/12) (PRICE CHANGE)(25)(100)(10) b-c (NET/10M)(12/3)(100%) 82 A STACK HEDGE WITH EURODOLLAR FUTURES: DATA ON NOVEMBER 11, 2000 DEC 00 VOLUME 46,903 MAR 01 29,236 127,714 JUN 01 5,788 77,777 SEP 01 2,672 30,152 DECISION: OPEN INTEREST 185,609 STACK MAR FUTURES FOR JUN AND SEP AND ROLL OVER AS SOON AS OPEN INTEREST REACHES 100,000. 83 12.19.00 THE STACK HEDGE CASH FUTURES 8.44% S 10 DEC 91.41 S 30 MAR 91.61 9.54% L 10 DEC 90.46 1.12.01 9.47% L 20 MAR 90.47 S10MAR S 20 JUN 90.42 S20JUN 2.22.01 9.95% 3.13.01 9.75% 6.19.01 9.44% L 10 JUN 89.78 S10MAR S 10 SEP 89.82 S10JUN S10SEP L 10 MAR 90.25 S10JUN S10SEP L 10 JUN 90.56 S10SEP 9.18.01 8.88% L 10 SEP 91.12 NONE DATE 11.1.00 F. POSITION S10DEC S30MAR S30MAR 84 PERIOD: 1 RATE(%) a: 10.54 INTERESTb: FUTURES($)c: NET($) d: 2 3 4 10.75 10.44 9.88 263,500 268,750 261,000 247,000 23,750 34,000 239,750 234,750 9.59 9.39 UNHEDGED AVERAGE RATE HEDGED AVERAGE RATE EFFECTIVE RATE (%) e: a. b. c. d. e. 28,500 <3,500> 28.500 16,000 <32,500> 236,000 235,000 9.44 9.40 10.40% 9.46% LIBOR + 100 BPS ($10M)(RATE)(3/12) (PRICE CHANGE)(25)(100)(10) b-c (NET/10M)(12/3)(100%). 85