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CHAPTER 8 Interest Rate Futures RefinementsIn this chapter, we extend the discussion of interest rates futures. This chapter is organized into the following sections: 1. The T-Bond Futures Contract 2. Sellers Options for T-Bond Futures 3. Interest Rate Futures Market Efficiency 4. Hedging with T-Bond Futures

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T-Bond Futures ContractIn this section, the discussion of T-bond futures is extended by analyzing the cheapest-to-deliver bond. Recall that a number of candidate bonds can be delivered against a T-bond future contract. Recall further that short traders choose when to deliver and which combination of bonds to deliver. Some bonds are cheaper to obtain than others. In this section, we learn techniques to identify the cheapest-todeliver bond, including: 1. Cheapest-to-deliver bond with no intervening coupons. 2. Cheapest-to-deliver bond with intervening coupons. 3. Cheapest-to-deliver and the implied repo rate.

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Cheapest-to-Deliver with No Intervening CouponsAssume today, September 14, 2004, a trader observes that the SEP 04 T-bond futures settlement price is 107-16 and thus decides to deliver immediately. That is, the trader selects today, September 14, as her Position Day. Therefore, she will have to deliver on September 16. The short is considering the following bonds with $100,000 face value each for delivery. The short wishes to determine if delivering one or the other bond will produce a larger profit for her. How much should the short receive? Which bond should the short deliver?aturity C oupon P rice 93-15 127-13 S P04 E C 0.9052 1.2113 D ays D ays ay-N ov ay-S et 184 122 184 122

N ber 15, 2028 5.25 ovem N ber 15, 2021 8.00 ovem

To answer these two questions, we need to determine the invoice amount and then which bond is cheapest-todeliver.Chapter 8 3

Cheapest-to-Deliver with No Intervening CouponsRecall that the total price of a bond depends upon the quoted price plus the accrued interest (AI).

Invoice Amount ! DSP ($100,000)(CF ) AIWhere: DSP = decimal settlement pricethe decimal equivalent of the quoted price

CF = conversion factorthe conversion factor as provided by the CBOT

AI

= accrued interestthe Interest that has accrued since the last coupon payment on the bond

Pi

= cash market price

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Cheapest-to-Deliver with No Intervening CouponsThe accrued interest (AI) is computed as follows: Days Since Last Coupon 1 AI ! Days in Half Years # Coupons per year Coupon Rate FaceValue

The days in half-year can be obtained from Table 8.1.T ab l e 8.1 D ay s i n H al f B ear s YDays in HalfB Year Interest Period Interest Paid on 1st or 15th Regular Year January to July February to August March to September April to October May to November June to December July to January August to February September to March October to April November to May December to June 1 year (any 2 consecutive halfB years) 181 181 184 183 184 183 184 184 181 182 181 182 365 Leap Year 182 182 184 183 184 183 184 184 182 183 182 183 366 Interest Paid on Last Day Regular Year 181 184 183 184 183 184 184 181 182 181 182 181 365 Leap Year 182 184 183 184 183 184 184 182 183 182 183 182 366

Source: Treasury Circular No. 300, 4th Rev.

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Cheapest-to-Deliver with No Intervening CouponsSt : t t

.2 % Bond AI = (122/184) (0. ) (0.0 2 ) ($100,000) = $1,740.4 Invoice Amount= 1.07 0 ($100,000) (0. 0 2) + $1,740.4 Invoice Amount = $ 8.00% Bond AI = (122/184) (0. ) (0.08) ($100,000) = $2,6 2.17 Invoice Amount= 1.07 0 ($100,000) (1.2113) + $2,6 2.17 Invoice Amount = $132,866. 2 ,04 .4

The 8% bond has an invoice amount 34% greater than the .2 % bond.

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Cheapest-to-Deliver with No Intervening CouponsSept : pute t e cheapest-t - eliver bond.

The bond that is most profitable to deliver is the cheapest--to-deliver bond. The shorts profit is the difference between the invoice amount and the cash market price. For a particular bond I, the profiti i

is:

= Invoice Amount - (Pi + AIi)

Recall that the invoice amount is:

Invoice Amount ! DSP ($100,000)(CF ) AISubstituting the formula for the invoice amount into the profit equation gives: i = (DFPi) ($100,000) (CFi) + AIi - (Pi + AIi) And simplifying: i = DFPi ($100,000) (CFi) - PiChapter 8 7

Cheapest-to-Deliver with No Intervening CouponsThe cheapest-to-deliver is: .2 % Bond = 1.07 0 ($100,000) (0. 0 2) - $ 3,468.7 = $3,840.2 8.00% Bond = 1.07 0 ($100,000) (1.2113) - $127,0 3.7 = $3,121.00 Thus, in this case the cheapest-to-deliver bond is the .2 % bond.

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Cheapest-to-Deliver with No Intervening CouponsGeneral rules based on interest rates: 1. When interest rates are below 6%, there is an incentive to deliver short maturity/high coupon bonds. 2. When interest rates exceed 6%, there is an incentive to deliver long maturity/low coupon bonds. General rules based on duration: 1. A trader should deliver low duration bonds when interest rates are below 6%. 2. A trader should deliver high duration bonds when interest rates are above 6%.

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Cheapest-to-Deliver with Intervening CouponsThis section examines, cheapest-to-deliver bonds when a bond pays a coupon between the beginning of the cashand-carry holding period and the futures expiration. To find the cheapest-to-deliver bond before expiration, the cash-and-carry strategy is used. The bond with the greatest profit at delivery from following the cash-and-carry strategy will be the cheapest-to-deliver bond. For this analysis Assume that:1. A trader buys a bond a today and carries it until delivery. 2. Interest rates and futures prices remain constant. 3. Consider the estimated invoice amount plus the estimate of the cash flows associated with carrying the bond to delivery.

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Cheapest-to-Deliver with Intervening CouponsThe estimated invoice amount depends on three factors:1. 2. 3. Today's quoted futures price. The conversion factor for the bond we plan to deliver. The accrued interest on the bond at the expiration date.

Acquiring and carrying a bond to delivery involves three cash flows as well:1. 2. 3. The amount paid today to purchase the bond. The finance cost associated with obtaining money today to buy a bond in the future. The receipt and reinvesting of coupon payment.

Figure 8.1 brings all these factors together.

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Cheapest to Delivery and Bond YieldInsert Figure 8.1 here

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Cheapest-to-Deliver with Intervening CouponsEstimated Invoice Amount = DFP0 $100,000 (CF) + AI2 Estimated Future Value of the Delivered Bond = (P0 + AI0)(1 + C0,2) - COUP1(1 + C1,2) For bond I, the expected profit from delivery is the estimated invoice amount minus the estimated value of what will be delivered: = DFP0 ($100,000) (CF) + AI2 - {(P0 + AI0)(1 + C0,2) COUP1(1 + C1,2)} where: P0 AI0 C0,2 COUP1 C1,2 DFP0 CF AI2 = = = = = = = quoted price of the bond today, t = 0 accrued interest as of today, t = 0 interest factor for t = 0 to expiration at t = 2 coupon to be received before delivery at t = 1 interest factor from t = 1 to t = 2 decimal futures price today, t = 0 conversion factor for a particular bond and the specified futures expiration = accrued interest at t = 2Chapter 8 13

Cheapest-to-Deliver with Intervening CouponsTo illustrate these computation consider the following situation. Suppose that today is Sept 14, 2004, and you want to find the cheapest-to-deliver bond for the DEC 04 futures expiration. The bond has a $100,000 face value and a target delivery date of Dec 31, 2004. The futures contract is the DEC 04. The T-bond contract had a settlement price of 106-23 today. The coupon invested repo rates is 7%. Summary Today = Bond face value = Target delivery date = Futures contract = Coupon invested repo rate = Settlement price Sept 14 = Sept 14 $100,000 Dec 31 DEC 04 T-bond 7% 106-23

You are considering two bonds for delivery. The bonds are as follows:

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Cheapest-to-Deliver with Intervening Couponsaturit ove er 1 ove er 1 C oupon ri e C 8 1 8 -1 1 -1 1 rue Interest a s 1 1 rue interest 1 1

tep 1: esti ate the value of AI.5.25% Bond P0 + AI0 = $ 3,468.7 + $1,740.4 = $ 8% Bond P0 + AI0 = $ 127,0 3.7 + $2,6 2.17 = $12 ,74 . 2 ,20 .24

Step 2: estimate the accrued interest that will accumulate from the next coupon date, Nov 1 , 2004 if the planned delivery date is Dec 31, 2004 (46 days).5.25% Bond AI2 = (46/181) (0. ) (0.0 2 ) ($100,000) = $667.13 8% Bond AI2 = (46/181) (0. ) (0.08) ($100,000) = $1,016. 7

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Cheapest-to-Deliver with Intervening CouponsStep 3: compute the estimated invoice amounts.5.25% Bond 1.067187 ($100,000) (0. 0 6) + $667.13 = $ 7,311.63 8% Bond 1.067187 ($100,000) (1.20 4) + $1,016. 7 = $130,082.23

Step 4: compute financing rates.Period: Sept 15 until Dec 31 (108 days) C0,2 = 0.07 (108/360) = 0.0210 Period: Nov 15 until Dec 31 ( C1,2 = 0.07 (46/360) = 0.008 44 days)

Table 8.2 summarizes these calculations.

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Cheapest-to-Deliver with Intervening CouponsTable 8.2 Data for CheapestBtoBDeliver BondsBond 5.25% 8.00% P0 $93,468.75 $127,093.75 AI0 $1,740.49 $2,652.17 C0,2 .0210 .0210 C1,2 .008944 .008944 DFP0 1.0671875 1.0671875 CF (DEC 04) 0.9056 1.2094 AI2 $667.13 $1,016.57

Step : Compute expected profit for each bond.5.25% Bond = (1.067187 0) ($100,000) (0. 0 6) + $667.13- [($ 3,468.7 + $1,740.4 ) (-1.0210-) - ($2,62 ) (-1.008 44-)] = $2,7 1.48 8% Bond = (1.067187 0) ($100,000) (1.20 4) + $1,016. 7[($127,0 3.7 + $2,6 2.17) (1.0210) - $4,000 (1.008 44)] =$1,647.43

The profit from the .2 % bond is higher, so it i