TREASURY BOND FUTURES

Treasury bond futures are the most popular long-term interest rate futures.

These futures trade on the Chicago Board of Trade (CBOT) and expire in the months of March, June, September, and December in addition to extra months scheduled by the CBOT based on the demand for these contracts.

The last trading day for these contracts is the business day prior to the last seven days of the expiration month. Delivery can take place any time during the delivery month and is initiated by the short side. The first delivery day is the first business day of the delivery month. As with most other futures con- tracts, delivery seldom takes place. The uncertainty about the delivery date poses a risk to the futures buyer that cannot be hedged away.

The underlying asset in a Treasury bond (T-bond) futures contract is any $100,000 face value government bond with more than 15 years to ma- turity on the first day of delivery month and which is noncallable for 15 years from this day. The quoted price of the T-bond futures contract is based on the assumption that the underlying bond has a 6 percent coupon rate, but the CBOT also permits delivery of bonds with coupon rates differ- ent than 6 percent. In fact, a wide range of coupons and maturities qualify for delivery. To put all eligible bonds on a more or less equal footing, the CBOT has developed comprehensive tables to compute an adjustment fac- tor, called conversion factor,that converts the quoted futures price to an in- voice price applicable for delivery. The invoice price for the deliverable bond (6.25) D

D s s

D s

f f f

( ) /

( ) ( / )

( ) ( 1 90 365

2 90 365

3 9

2 2

=

= + −

= + 00 365/ )³−s³

Treasury Bond Futures 159

(not including accrued interest) is the bond’s conversion factor times the fu- tures price.

where FP=Quoted futures price CF=Conversion factor

Similarly to Treasury bond prices, Treasury bond futures prices are quoted in dollars and 32nd of a dollar on a $100 face value. Thus, a quoted futures price of 98.04 represents 98+4/32, or $98,125 on a $100,000 face-value contract.

T-bond futures price quotes do not include accrued interest. Therefore, the delivery cash price is always higher than the invoice price by the amount of the accrued interest on the deliverable bond.

where AIis the accrued interest.

Example 6.4 Suppose on November 12, 2003, the quoted price of the 10 percent coupon bond maturing on August 5, 2019, is 97.08 (or $97,250 on a $100,000 face value). Since government bonds pay coupons semiannually, a coupon of $5,000 would be paid on February 5 and August 5 of each year.

The number of days between August 5, 2003, and November 12, 2003 (not including August 5, 2003, and including November 12, 2003), is 99, whereas the number of days between August 5, 2003, and February 5, 2004 (not including August 5, 2003, and including February 5, 2004), is 181 days.

Therefore, with the actual/actual day-count convention used for Treasury bonds, the accrued interest from August 5, 2003, to November 12, 2003, is

If this bond is the deliverable bond underlying the futures contract, then the cash price in equation 6.27 will be greater than the invoice price by an amount equal to the accrued interest of $2,734.81.

AI=$ ,5 000× 99 =$ , . 181 2 734 81

(6.27) CP Invoice price Accrued interest

FP C

= +

= × FF AI+

(6.26) Invoice price=FP CF×

Conversion Factor

A bond’s conversion factor is the price at which the bond would yield 6 per- cent to maturity or to the first call date (if callable), on the first delivery date of the T-bond futures expiration month. The bond maturity is rounded down to the nearest zero, three, six, or nine months. If the maturity of the bond is rounded down to zero months, then the conversion factor is:

wherecis the coupon rate andnis the number of years to maturity. If the maturity of the bond is rounded down to three months, the conversion fac- tor is:

If the maturity of the bond is rounded down to six months, the conversion factor is:

And, finally, if the maturity of the bond is rounded down to nine months, the conversion factor is:

CF₀and CF₆have the same format, and so do CF₃and CF₉. Therefore, we can combine equations 6.28 and 6.30 into a single equation:

where mis the number of semiannual periods to maturity.

(6.32) CF c

m m

0 2

1 0 03

1 0 03 1 03

1

=  − 1 03

 

 +

. . ( . ) ( . )

(6.31) CF

CF c c

9 6

1 2

2

1 03 4

= +

− ( . )

(6.30) CF c

n n

6 2 2 1 2

1 0 03

1 0 03 1 03

1

=  − 1 03

 

 +

. . ( . ) + ( . ) ⁺⁺¹

(6.29) CF

CF c c

3 0

1 2

2

1 03 4

= +

− ( . )

(6.28) CF c

n n

0 2 2 2

1 0 03

1 0 03 1 03

1

=  − 1 03

 

 +

. . ( . ) ( . )

Treasury Bond Futures 161

Thus, instead of rounding the bond’s maturity down to zero, three, six, or nine months, we can just round it down to zero or three months. If the re- sult is zero, use equation 6.32, otherwise, use equation 6.29. The conversion factor always increases with the coupon rate, holding the maturity constant.

If the coupon rate is more than 6 percent, the conversion factor increases with maturity, but if the coupon rate is less than 6 percent, the conversion factor decreases with maturity. The conversion factor equals one when the coupon rate equals 6 percent, regardless of the maturity.

Example 6.5 Consider a futures contract expiring in the month of December with the underlying deliverable bond given in Example 6.4. Assume that the bond is delivered on the first day of the expiration month, December 1, 2003.

On this day, the bond has 15 years 8 months and 5 days to maturity. Rounding the bond’s maturity on the delivery day down to the nearest zero or three months, the maturity is 15 years and 6 months. We treat this as 31 six-month periods. Applying equation 6.32 for m=31:

Suppose the quoted futures price on this bond is 96.04 (or $96,125 on a $100,000 face value contract). The time elapsed since the previous coupon payment date, August 5, 2003, to the expiration date of the futures contract, December 1, 2003, equals 118 days. Hence, the accrued interest on the bond equals:

Using equation 6.27, the delivery cash price of the T-bond futures contract is:

CP=$96,125 × 1.4+$3,259.67=$137,835.49 Cheapest-to-Deliver Bond

The party with the short position in the T-bond futures contract can deliver any government bond with more than 15 years to maturity and which is noncallable for 15 years from the delivery date. At any given day of the de- livery month, there are about 30 bonds that the short side can deliver.

Which bond should the seller choose to deliver? The answer to this question can be understood as follows.

On the delivery date, the seller receives:

Quoted futures price ×Conversion factor+Accrued interest

$ ,5 000 118 $ , . 181 3 259 67

× =

CF₀ 0 10 _{31 3}

2 1 0 03

1 0 03 1 03

1

=  − 1 03

 

 + .

. . ( . ) ( . )¹¹ =1 4.

TABLE 6.5 Deliverable Bonds for the T-Bond Futures Contract

Quoted Bond Price Conversion Factor

Bond Coupon (%) P CF

1 7.50 98−23 1.0138

2 6.00 97−16 1.0000

3 8.00 99−12 1.0204

4 6.25 97−30 1.0049

TABLE 6.6 Cost of Delivery

Quoted Quoted Conversion Cost of

Bond Price Futures Price Factor Delivery

Bond (P) (FP) (CF) (P−(FP×CF))

1 98.72 97.28 1.0138 0.10

2 97.50 97.28 1.0000 0.22

3 99.38 97.28 1.0204 0.11

4 97.94 97.28 1.0049 0.18

The cost of purchasing a bond to deliver is:

Quoted bond price+Accrued interest

Hence, the seller will choose the cheapest-to-deliverbond, which is the bond for which the cost of delivery is lowest, where the cost of delivery is defined as:

Example 6.6 illustrates the selection of the cheapest-to-deliver bond.

Example 6.6 Assume that the T-bond quoted futures price on the delivery day is 97.09 and that the party with the short position in the contract can choose to deliver from the bonds given in Table 6.5.

The cost of delivering each of these bonds is given in Table 6.6. It gives the quoted bond price and quoted futures price in decimal form. Using equation 6.33, the cheapest-to-deliver bond is bond 1.

(6.33) Cost of delivery=Quoted bond price Quoted f− uutures price Conversion factor

P FP CF

×

= − ×

Treasury Bond Futures 163

Options Embedded in T-Bond Futures

A variety of options are embedded in T-bond futures. We have already dis- cussed the cheapest-to-deliver option. The seller of the futures contract can also choose when to deliver the bond on the designated days in the delivery month. Another option known as the wild card playmakes the T-bond futures price lower than it would be without this option. This option arises from the fact that the T-bond futures market closes at 2:00 P.M. Chicago time, while the bonds continue trading until 4:00 P.M. Moreover, the short side of the futures contract does not have to notify the clearing house about her intention to de- liver until 8:00 P.M. Thus, if the bond prices declines between 2 and 4 P.M., the seller can notify the clearing house about her intention to deliver using the 2:00 P.M. futures price, and make the delivery by buying the cheapest-to- deliver bond at a lower price after 2 P.M. Otherwise, the party with the short position keeps the position open and applies the same strategy the next day.

Treasury Bond Futures Pricing

The uncertainty regarding the many delivery options makes the T-bond fu- tures contract difficult to price. In the following analysis, we assume that both the deliverable bond and delivery date are known, and the wild card play option is not significant in the pricing of T-bond futures. Under these assumptions, the Treasury bond futures price can be approximated by its forward price, and is given as:

where P is the current price of the deliverable T-bond, Iis the present value of the coupons during the life of the futures contract, sis the expiration date of the futures contract, and y(t) is the zero-coupon yield for the term t.

In the following analysis, we assume that the delivery date is the first day of the expiration month of the futures contract, and use the follow- ing notations:

T=Current maturity of the cheapest-to-deliver bond C=Coupon payment of cheapest-to-deliver bond

F=Face value of cheapest-to-deliver bond CF=Conversion factor of cheapest-to-deliver bond

τ =Length of time between expiration date of the futures contract and first cash flow payment after the futures’

expiration date

n=number of coupon payments between current time and futures’

expiration date

(6.34) FP=(P I e− ) ^{s y s×( )}

FIGURE 6.1 Timeline for T-Bond Futures C

0 s +τ– 0.5n s +τ– 0.5(n –1) s +τ– 0.5(n –2) s +τ– 0.5 s s +τ

Current time

Futures’

expiration

T

C C . . . . . C C C+F . . . . .

In addition, assume that there are two coupon payments per year, so that by definition 0≤ τ ≤0.5.

The timeline is shown in Figure 6.1. The price of a futures contract de- pends only on cash flows received after the T-bond is delivered. The deliv- ery cash price of the futures contract is given as:⁴

Using equation 6.27, the quoted price of the futures contract is given as:

Substituting equation 6.35 and the definition of accrued interest in equation 6.36, we get,

Example 6.7 Reconsider the cheapest-to-deliver bond in Example 6.5 with the delivery date of December 1, 2003. Assume that the bond’s quoted price is not given, but that the term structure of zero-coupon yields has the polynomial form defined in Chapter 3, as follows:

where for expositional simplicity the height, slope, and curvature are as- sumed to be the only parameters of the term structure of interest rates. Fur- ther, assume that these parameters are defined as:

y t( )=A0+A t1 +A t2 2

(6.37) FP CF

Ce e

s y s Fe

s t y s t

s

= 1 _{(+ + × )×}^×_{(+ + × )}+

0 5 0 5

( )

. .

τ τ

××

= ×

( − −)



∑









− × −

y s T y T t

T s

e

C CF

( ) ( )

.

0

2 ^τ 0 5 τ

0 0 5.









(6.36) FP=CF¹

(

CP AI−

)

(6.35)

CP Ce

e

s y s Fe

s t y s t

s y

= _{(+ + ×τ 0 5.)×}^×₍^{( )}_{+ + ×τ 0 5.)}+ _{T y T}^×(ss

t T s

e

)

×( )

=

( − −)



∑











0

2 τ

Treasury Bond Futures 165

FIGURE 6.2 Timeline for T-Bond Futures C

0 s s +τ s +τ+ 0.5 s +τ+ 2×0.5

Current time

Futures’

expiration

T

C C . . . . . C+F

A₀=0.02 A₁=0.005 A₂= −0.0001

For the futures contract written on this T-bond, the variables are T=15 years, 8 months, and 23 days; C=$5 on a $100 face value (or $5,000 on a $100,000 face value); F=$100; CF=1.4 (see Example 6.4); s=18 days; τ

=66 days; and n=0 (no coupon payments between current time and futures’

expiration date). Though the asset underlying the Treasury bond futures con- tract is a $100,000 face-value government bond, we do all calculations as- suming a $100 face value. All final prices can be multiplied by 1,000 later.

The timeline of the cash flows associated with this futures contract is shown in Figure 6.2.

s=18 days =0.05 years τ =66 days=0.18 years s+ τ =84 days=0.23 years

T=15 years, 8 months, and 23 days=15.73 years T−(s+ τ) =15 years and 6 months=15.5 years

Using equation 6.37, the quoted futures price is given as:

The yields for different maturities are computed using the following equation and are displayed in Table 6.7.

(6.39) y t( )≈A0+A t1 +A t2 = . + . t− . t

2 0 02 0 005 0 00012

(6.38) FP e

e

y

t y t

= ^{0 05× 0 05} _{(0 23+ ×0 5)× 0 23+ ×0} 1 4

. ( . ) 5

. . .

. ₀ ^{..5 15 73. ( . )15 73} .

31 100 5

1 4 0

( ) ^×

=

 +

 

 − ×

∑

_e ^y

t

.. . . 5 0 18

0 5

−

TABLE 6.7 Zero-Coupon Yields

Maturity Yield Maturity Yield

(0.23 +t×0.5) y(0.23 +t×0.5) (0.23 +t×0.5) y(0.23 +t×0.5)

t =0 0.0211 t =16 0.0544

t =1 0.0236 t =17 0.0560

t =2 0.0260 t =18 0.0576

t =3 0.0284 t =19 0.0592

t =4 0.0307 t =20 0.0607

t =5 0.0329 t =21 0.0621

t =6 0.0351 t =22 0.0635

t =7 0.0373 t =23 0.0649

t =8 0.0394 t =24 0.0662

t =9 0.0414 t =25 0.0674

t =10 0.0434 t =26 0.0686

t =11 0.0454 t =27 0.0698

t =12 0.0473 t =28 0.0709

t =13 0.0491 t =29 0.0720

t =14 0.0509 t =30 0.0730

t =15 0.0527 t =31 0.0739

Substituting the zero-coupon yields from Table 6.7 into equation 6.38, the quoted price of the futures contract based on a $100 face value is equal to $95.10. The cash price is derived from the quoted futures price using equation 6.36 and is equal to $136.33.

Duration Vector of T-Bond Futures

Assuming that the changes in the zero-coupon yields and the instantaneous forward rates are given by equations 6.8 and 6.9, respectively, the percent- age change in the T-bond futures quoted price is given as follows:⁵

(6.40)

∆FP ∆ ∆ ∆

FP = −D( )1 A₀−D( )2 A₁−D( )3 A₂− . . . −D M( )∆∆A_M−1