1. © K.Cuthbertson, D. Nitzsche1 Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche LECTURE Interest Rate Futures

2. © K.Cuthbertson, D. Nitzsche2 Topics u Cash Market for T-Bills u T-Bill Futures Contract u 3m Sterling Futures Contract u Hedging u Arbitrage: Pricing a T-Bill Futures Contract

3. © K.Cuthbertson, D. Nitzsche3 Cash Market for T-Bills

4. © K.Cuthbertson, D. Nitzsche4 Spot Market: T- Bills,Market Price and Yield u Face or par value is FV= $100 u n = days to maturity / days in year u ‘Simple’ yield ( y ~ proportion, p.a.): P = 100 / [ 1 + y (n) ] u Compound yield ( y ~ proportion, p.a.): P = 100 / ( 1 + y ) n u Continuously compounded yield, y P = 100 exp(- y. n )

5. © K.Cuthbertson, D. Nitzsche5 Price from discount Rate: T- Bills u The ‘dollar (or sterling) discount’ is : u D = FV u The price is: u P = FV - D u Also: u P = FV u Price and discount rate are inversely related

6. © K.Cuthbertson, D. Nitzsche6 T-Bill Futures Contract

7. © K.Cuthbertson, D. Nitzsche7 What is a T-Bill Futures Contract ? At expiry, (T), which may be in say 9m time the (long) futures delivers a T-Bill which matures at T+90, with face value M=$100. This allows you to ‘lock in’ at t=0, the forward rate, f 12

8. © K.Cuthbertson, D. Nitzsche8 Figure 5.2 : Interest rate (T-Bill) futures contract t0t0 t*t* T=t 1 t2t2 r1r1 r2r2 t 12, f 12 Futures protection period = t 12 Exposure period, t 0 to t 1 T= t 1 = Maturity of futures contract

9. © K.Cuthbertson, D. Nitzsche9 Buy one Sept, T-Bill futures at F 0 = 98 (no cash is exchanged T = t 1 0 T+90days = t 2 Exposure period (2m) Protection period=t 12 Receive a 90-day T- Bill and pay F 0 T-Bill matures at M =100 (M/F 0 ) 365/90 = ( 1 + annual interest earned over t 12 ) (simple)annual interest earned is approx (2/98) x 4 = 8.16 % T-Bill Futures Contracts

10. © K.Cuthbertson, D. Nitzsche10 What is the known yield “locked in” at t=0, which applies between T (t 1 ) and T+90 = t 2 ) F 0 = M / ( 1 + annual interest earned over t 12 ) 90/365 F 0 = M / ( 1 + f 12 ) t 12 Since f 12 (compound) is observable at t=0, then this is how we price the futures contract (riskless arbitrage is “hidden” in above) Also F 0 = M / ( 1 + f 12 t 12 ) - f 12 is ‘simple interest/yield = M exp(- f 12 t 12 ) - f 12 is ‘contin compound’ Note: For all interest rate contracts, if f falls then F rises Futures Price and ‘futures’ Yield

11. © K.Cuthbertson, D. Nitzsche11 3m Sterling Futures Contract

12. © K.Cuthbertson, D. Nitzsche12 STERLING 3-MONTH CONTRACT (LIFFE) Contract Size Delivery Price Quotation Tick Size (Value) Settlement Initial Margin z = £500,000 Mar/June/Sept/Dec F = (100 - futures rate)  F = 0.01 (= “1-tick”) (£12.5) Cash £500 Can ‘lock in’ interest rate on 3-mth deposits Tick value = £500,000 (0.01 / 100 ) (1/4) = £12.5 If F changes by 0.01 ie.1-tick (eg from 95.00 to 95.01) then value of one contract changes by £12.5 F = 100 - f where f in the “futures /forward rate”(applicable from T to T+3m )

13. © K.Cuthbertson, D. Nitzsche13 Calculation of Tick Value £500,000 ( 0.01 / 100 ) (1/4) = £12.5 z ( ( F 1 - F 0 ) / 100 ) (1/4) The 1/4 appears because a change of 1% pa in f is equal to a change of 1/4 of 1% over 3-mnths (the life of the deposit underlying this futures contract)

14. © K.Cuthbertson, D. Nitzsche14 Hedging

15. © K.Cuthbertson, D. Nitzsche15 Simple Hedge :Short Sterling, Naïve Hedge Ratio Will receive £1m in 2m time and then wishes to place funds on deposit for 3m. Fears a fall in interest rates 15th April(today) r 0 = 10%f 0 = 10.5%F 0 = 89.5 15th June(Hedge ends) r 1 = 8%f 1 = 8.5%F 1 = 91.5 N f = TVS 0 /FVF 0 = £1m/ (0.5m x 0.895) = 2.33 (=2) Lose 2% in cash market and gain 2% on futures

16. © K.Cuthbertson, D. Nitzsche16 Simple Hedge :Short Sterling Loss of interest in cash market = 0.02 x (1/4) x £1m = £5000 Profit on futures contract = 2 x 200 ticks x £12.5 = $5000 Perfect hedge Note: Strictly the cash market loss is based on r 0 = 10% could not have been achieved. (Futures contract used matures in say, December)

17. © K.Cuthbertson, D. Nitzsche17 Risks in a Hedge: Short Sterling Example: 1st Jan and will receive £1.2m on 1st Sept On 1st Sept wish to put proceeds into Commercial Bill for 6-months Underlying in futures is a 3-month deposit Futures matures at end of March, June, Sept, Dec Potential Problems u Cash” amount is not exact multiple of contract size u Margin calls may be required u Nearby contracts matures before Sept and would have to be ‘rolled over’, otherwise use Sept contract u ’Underlying’ = Commercial bill, is not the same as the underlying in the futures (ie. Eurosterling deposit) - Cross Hedge

18. © K.Cuthbertson, D. Nitzsche18 Fig 5.3 : Hedge using US T-Bill Futures Purchase T-Bill future with Sept. delivery date $1m cash receipts Maturity date Sept. T-Bill futures contract 3 month exposure period Desired investment/protection period = 6-months May Aug. Sept. Feb.Dec. Maturity of Underlying in Futures contract

19. © K.Cuthbertson, D. Nitzsche19 Duration based Hedge Ratio F = futures price z = “size” of one futures contract FVF 0 = face value of one futures contract = z F 0 N f = number futures contracts held y s = spot yield, y F = futures yield (usually = f 12 in text book) Using the min var hedge ratio but “replacing”  (  S,  F) and  2 (  F) terms with “duration” and yields we get:

20. © K.Cuthbertson, D. Nitzsche20 Cross Hedge: US T-bill Futures May (Today) Funds of $1m accrue in August to be invested for 6m months in bank or commercial bills Use Sept ‘3m T-bill’ Futures contract Assume parallel shift in the yield curve Q f = 89.2 (per $100 nominal) hence: F 0 = 100 - (1/4)(100 - Q f ) = 97.30 FVF 0 = $1m (F 0 /100) = $973,000 N f = (TVS 0 / FVF 0 ) (D s /D f ) = ($1m / 973,000) ( 0.5/ 0.25) = 2.05 (=2)

21. © K.Cuthbertson, D. Nitzsche21 Cross Hedge: US T-bill Futures

22. © K.Cuthbertson, D. Nitzsche22 Cross Hedge: US T-bill Futures Gain on the futures position = $1m(0.9775 – 0.973)2 = $5,500 or (using tick value of $25 and  Q = 0.01 is ‘1 tick’) = 110 ticks x $25 x 2 contracts = $5,500 The gain on the futures position of $5,500 when invested over 6-months at y 1 = 9.6% is $5,764 hence (using simple interest):

23. © K.Cuthbertson, D. Nitzsche23 Cross Hedge: US T-bill Futures Hedged Return Effective (“simple”) interest = y 1 + = 0.096 + 0.0115 = 0.1075 (10.75%) The 10.75% hedged return is substantially above the unhedged rate (y 2 ) of 9.6% and is reasonably close to the implied (simple) yield on the September futures contract of 11.1% (= (100/97.3 – 1) 4).

24. © K.Cuthbertson, D. Nitzsche24 Arbitrage: Pricing a T-Bill Futures

25. © K.Cuthbertson, D. Nitzsche25 Figure 5.5 : Pricing the futures contract Receive $100 face value of 2-year T-bill Buy 2-year T-bill for $S with face value $100 A. 01 2 r2r2 r2r2 Receive $100 face value of T-bill underlying the F.C. Buy 1-year T-bill for F/(1 + r 1 ) B. 012 r1r1 f 12 Maturity of T-bill receive $F Portfolio A : 2-year T-bill Portfolio B : 1-year T-bill plus interest rate futures contract Go long a T-bill futures (at zero cost today) Pay $F for F.C. on 1-year T-bill

26. © K.Cuthbertson, D. Nitzsche26 Figure 5.5 : Pricing the futures contract The 1-year T-Bill with maturity value F must cost (at t=0) : [5.30]Price 1-year T-Bill = F / (1+ r 1 ) The two portfolios payoff is the same at t=2 and hence must cost the same today: [5.32]F / (1+ r 1 ) = S Price at t=0 of a 2-year T-Bill is : [5.33]S = 100 / (1+r 2 ) 2 = 100 / (1+r 1 ) (1 + f 12 ) Substituting equation [5.33] into equation [5.32] : [5.34] F = 100 / (1+ f 12 )

27. © K.Cuthbertson, D. Nitzsche27 END OF SLIDES