Lecture 8 Agenda: Treasury bond futures 1. Treasury bond futures ~ Definition: ~ Cheapest-to-Deliver (CTD) Bond: ~ The wild card play: ~ Interest rate futures pricing: ~ 3-month Eurodollar futures: ~ The Libor Zero Curve (Swap Zero Curve):. Duration-based hedging ~ Duration ~Convexity ~Duration-based hedging ratio
1. Treasury bond futures: ~ Definition: T-bond futures are traded in CBOT. The par value is $100,000. The value of the bond is calculated based on a hypothetical 0-year bond at 6% interest rate. The bond can be used in delivery: 1. Government bond maturity > 15 years on the first day of the delivery month.. Government bonds are not callable within 15 years. Conversion factors: T-bond futures allow the futures sellers to choose to deliver any bond qualified to be delivered. When a particular bond is delivered, a parameter known as its conversion factor defines the priced received by the party with the short the futures. Conversion factor = Bond value at 6% interest rate / ar value = Bond value at 6% interest rate / $100 ~ Time to maturity is rounded down to the nearest three months. If after rounding, the bond lasts for exact 6 months, the first coupon is assumed to be paid in six months. If not, the first coupon is assumed to be paid after three months and accrued interest is subtracted. A 10% coupon bond with 0 years and months to maturity: Rounded to exactly 0 years: Conversion factor: $146.3 A 8% coupon bond with 18 years and four months to maturity: Rounded to 18 years and 3 months: An 18 year bond with 6% interest: 15.83 Discount 3-month back: 15.83/( (1+3%) 3/6 )=13.99 Subtract accrued interest for 3 month: $ Conversion factor: 1.199 1
~ Cheapest-to-Deliver (CTD) Bond: ~ rofit for sellers: Cash inflow to sellers: (Bond futures price Conversion factor) + Accrued interest Cash outflow to sellers: Bond spot price + Accrued interest rofit = (Bond futures price Conversion factor) - Bond spot price Given quoted futures price is $93.5 Bond Quoted spot price Conversion Factor 1 99.5 1.038 143.5 1.5188 3 119.75 1.615 ~ Factors affecting CTD: If yield > 6%: deliver the bonds with higher sensitivity to interest rate change. (i.e., lower C, longer maturity) If yield < 6%: deliver the bonds with lower sensitivity to interest rate change. (i.e., higher C, shorter maturity) If yield curve is upward sloping: Deliver bonds with higher sensitivity to interest rate change. If yield curve is downward sloping: Deliver bonds with lower sensitivity to interest rate change.
~ The wild card play: ~ T-bill futures sellers can take advantage of delivery rule: :00 pm 4:00 pm 8:00 pm T-B futures T-B spot Intention to delivery Invoice price on the futures is calculated based on :00 pm ~ The practice gives an option know as the wild care play to T-bond futures sellers. If T-B spot price drops after :00, sellers will issue the intention to delivery. The extra profit S pm S 4pm. rofit pm = (Bond futures price pm Conversion factor) - Bond spot price pm rofit 4pm = (Bond futures price pm Conversion factor) - Bond spot price 4pm 3
~ Interest rate futures pricing: ~ Assume that timing of delivery and choice of the bond are known. F 0 = (S 0 -I)e rt Given: a T-bond with 1% coupon is priced at $10 time to maturity is 70 days conversion factor for the bond is 1.4 yield is constant at 10% F=? C 0 C Maturity C 60days 1 days 148 days 35days $6 $6 $6 Quoted spot price: $10 Cash price: $10 +$6* 60/(1+60) = $11.978 I= V of $6 in 1 days = $6e -10%*1/365 = $5.803 ($11.978 -$5.803) e 10%*70/365 = $15.094 $15.094 includes the accrued interest $6*148/183=$4.85 The quoted futures price for the specific bond should be=15.094-4.85=10.4 Convert the price to the hypothetical T-bond futures price: 10.4/1.4=85.887 4
~ 3-month Eurodollar futures: Annualized future interest rate for 3-month Eurodollar futures = (ar ) / ar : the quoted futures price for 3-month Eurodollar futures The settlement price at maturity = [ar - 90/360(ar-)] Mar June Eurodollar futures=95.53 The settlement price of 95.53: [100-0.5(100-95.53] =$98.885 Annual discount rate: (100-95.53)/100=4.47% Relationship between forward and futures interest rates: If contract size is $ 1 million: The price is $988,85 For short maturities, the Eurodollar futures interest rate can be assumed to be the same as the corresponding forward interest rate, because interest rates may not vary too much. However for longer maturities, the difference between forward and futures rates becomes wider. The relation becomes: Forward rate = Futures rate ½ σ t 1 t σ is the std. of short-tem interest rate in one year. σ = 1.% 8-year Eurodollar futures price = 94 Forward rate? Futures rate: Annual futures rate = 6% (based on actual/360, compounding quarterly) =6.083% (based on actual/365, compounding quarterly) =6.038% (based on actual/365, continuously compounding) Forward rate = 6.038- ½ 1.% 8*8.5 = 5.563% 5
~ The Libor Zero Curve (Swap Zero Curve): The yield curve calculated based on zero coupon bonds. In practice, this curve is used to calculate derivatives prices. F i = (R i+1 T i+1 R i T i ) / (T i+1 -T i ) R t+1 = [ F i (T i+1 -T i ) + R i T i ] / T i+1 F i is the forward rate at the period i estimated based on 3-month Eurodollar futures contract. R 400 = 4.80% 400F 491 = 5.3% (calculated based on a 3-month Eurodollar futures contract) R 491 = (5.3*91 + 4.8*400) / 491 =4.893% T 0 T 400 T 491 R 400 = 4.80% 400F 491 =5.3% R 491 = 4.893% 6
. Duration-based hedging ~ Definition: The weighted average number of years necessary to recover the initial cost of the bond. It measures how soon investors can recover the initial investment. It measures how sensitive of a bond value is respect to the change of interest rate. The longer a bond s duration, the longer the time investors can recover, and the greater its sensitivity to interest rate change, and the higher the interest rate risk. ~ Estimation of duration: D = N t C 1 (1 i) t t t m D p = w i D i i 1 i n = t 1 t C (1 i) t t 1 = 1 (1 i) n t C t t 1 (1 i) t = ( 1 i) D Modified duration (MD): MD can estimate the percentage change in the bond s price in response to a 1% point change in bond yields. 1 / i = D (1 i) = MD 1 = D (1 i) i = MD i 7
~ Convexity: The sensitivity of Duration to Interest rate change. Error Convexity is related to the second differential of respect to k the change of duration. i Convexity = CV =( / i ) / Relationships: -i, -C, +N, +NO Application of convexity: a) Correction: = MD i + 1 CV ( i) b) Improve investment decisions: i 8
~ Duration-based hedge ratio: Assume: Yield curve ~ parallel shift Hedging ratio: If it is not a parallel shift, i for bond portfolio i for futures contracts, because maturities for portfolio and futures may not be the same. : bond portfolio value at maturity of hedging futures contract MD: Modified duration at maturity CV: convexity of the portfolio at maturity F: Interest rate futures contract price MD F : Modified duration of the underlying assets for the futures at maturity CV F : convexity of the underlying assets for the futures at maturity (We have to choose the bonds which are most likely delivered) F F = MD i + 1 CV ( i) = -MD F i + 1 CVF ( i) N F = 1 N = / F = [ MD i + CV ( i) 1 ] / F[ -MD F i + CVF ( i) ] q E(N) = j 1 rob j N j In practice, E(N) N j Eurodollar futures contracts are for exposures to short-term interest rates. T-bond futures contracts are for exposures to long-term interest rates. 9
A portfolio manager plans to use a T-bond futures contract to hedge a bond portfolio over the next three months. The portfolio is worth $100 million and will have a duration of 4.0 years in three months. The futures price is 1, and each futures contract is on $100,000 of bonds. The bond that is expected to cheapest to deliver will have a duration of 9 years at the maturity of the futures contract. a. What position in futures contracts is required? b. What adjustments to the hedge are necessary if after one month the bond that is expected to be cheapest to deliver changes to one with a duration of seven years? c. Suppose that all rates increase over the three months, but long-term rates increase less than short-term and medium-term rates. What is the effect of this on the performance of the hedge? 10