©david dubofsky and 9-1 thomas w. miller, jr. chapter 9 t-bond and t-note futures futures contracts...

©David Dubofsky and 9-1 Thomas W. Miller, Jr.

Chapter 9T-Bond and T-Note Futures

• Futures contracts on U.S. Treasury securities have been immensely successful.

• But, the outlook for Treasury bond futures contracts is bleak, as the government has not issued any new 30-year bonds since October 2001.

CBOT Treasury Volume, Contract Jan - Aug 2002

Treasury Bond 38.0 million10-year Tnote 59.4 million5-year Tnote 32.2 million2-year Tnote 2.2 million


The T-bond Futures Contract

• Underlying asset is: $100,000 (face value) in deliverable T-bonds.

• Futures prices are reported in the same way as are spot T-bonds, in "points and 32nds of 100%" of face value.

• A T-bond futures price of 112-15 equals 112 and 15/32% of face value, or $112,468.75.

• A change of one tick, say to 112-14, results in a change in value of $31.25.


T-Note Futures Prices

• For T-bonds, a tick is 1/32nd. The resulting quote of 112-15 equals 112 and 15/32.

• But, for 5 and 10-year T-notes, a tick is ½ of a 32nd, or $15.625 per tick. The resulting quote, say, of 98.095 = 98 and 9.50/32.

• For 2-year T-Notes, however, tick sizes are ¼ of a 32nd. So,– A quote of 92.072 = 92 and 7.25/32.– A quote of 109.017 = 109 and 1.75/32. – But, the contract size is $200,000 of deliverable T-Notes, so a tick

= $15.625.

• For CBOT futures prices for T-bonds and T-notes, see: http://www.cbot.com/cbot/www/page/0,1398,12+31,00.html .

http://www.cbot.com/cbot/www/page/0,1398,12+31,00.html


What Determines T-bond and T-noteFutures Prices, Basically?

• In a very simple sense, the futures price is the forward price of a Treasury bond, such that it has a forward yield [fr(t1,t1+30)] consistent with:

• [1+r(0,t1+30)]t1+30 = [1+r(0,t1)]t1[1+fr(t1,t1+30)]30

• t1 = time until delivery, in years.

0 t1 t1+30

What is Deliverable? And When?

• Under the terms of the T-Bond futures contract: – Any Treasury Bond that has fifteen or more years to first call, or at

least 15 years to maturity (whichever comes first), as of the first day of the delivery month, is deliverable.

– Therefore, the seller of the futures contract has the option of choosing which bond to deliver.

– Delivery can take place on any day in the delivery month, and the short chooses.

• Under the terms of the 10-year T-note futures contract: – Any U.S. T-note maturing at least 6 1/2 years, but not more than 10

years, from the 1st day of the delivery month is deliverable.


Which T-bond will the Short naively Choose to Deliver?

• Assuming the short will receive the same dollar amount (invoice amount) upon delivery:– The short will select the T-bond that costs less than

any other deliverable T-bond.– So, to fix this problem, the CBOT adjusts the invoice

amounts to try to make all T-bonds equally deliverable.• If the short delivers a low priced bond, the short will receive

less (low conversion factor)• If the short delivers a high priced bond, the short will receive

more (high conversion factor).


The Invoice Amount and Conversion Factors

• A conversion factor for a given T-bond is its price if it had a $1 face value, and was priced to yield 6%.

• For a file of conversion factors, see: http://www.cbot.com/cbot/www/cont_modular/0,2291,14+479+12,00.html#a.

• So a cheap, low coupon bond will have a small conversion factor.

• Therefore, the short will receive less.


interestaccrued

factorconversion

pricesettlementcontractfutures

sizecontract

amountinvoice

http://www.cbot.com/cbot/www/cont_modular/0,2291,14+479+12,00.html#a

So NOW, Which T-bond Will the Short Choose to Deliver?

• The one that costs the least….

• ….and at the same time gets the short the most money upon delivery (i.e., the highest invoice amount).

• That is, the Max [invoice amount – quoted spot price]

• (Accrued Interest is ignored because it is included in both the invoice amount and the gross cash bond price)

• Max [(CF)(F) – (S)]

• During the delivery month, the amount in brackets will always be negative, for every deliverable bond….. WHY?



Another Method to Identify the Cheapest to Deliver T-bond

• Before the delivery month, find the T-bond with the highest Implied Repo Rate. This is given by:

• Can be used to identify the “most likely to be delivered” T-bond.

• When coupons will be paid between today and the delivery day, include them, and the interest earned on them, in the carry return (see eqn. 9.5b).

T

365

]0AI[S

]0AI[S]TAI[(CF)(F) IRR

A Good Concept Check

• Identify three T-bonds that are deliverable into the nearby T-bond futures contract.

• Which of these three bonds is the cheapest to deliver?

• Be able to show your work, and be able to explain why one of the T-bond is more likely to be delivered than the other two.



Questions

• The implied repo rate for every deliverable T-bond must be less than interest rates available in the market (WHY?)

• Reverse cash and carry arbitrage will (almost never) be possible (WHY?)

The Options Held by the Short

• Quality option: can deliver any eligible bond.

• Timing option: can deliver on any day of the delivery month.

• Wild card option: futures cease trading at 2PM, but the short can announce intent to deliver as late as 8PM.

• End of month option: futures cease trading 8 business days before the end of the delivery month, but the short can deliver on any day of the month.


Theoretical T-bond Futures Price

• Once the C-T-D T-bond has been identified,

F = S + CC - CR (- value of shorts’ options)

T0

T0

0

AI 365

t)r(T 1

2

C

365

rT1)AI(S

CF

1 F

AI 365

t-Tr1

2

C -

365

rT 1AIS(CF)(F)

T AI 365

rT 1 0AIS

CF

1 F

TAI365

rT 1AIS(CF)(F)



Using T-bond and T-note Futures to Hedge Interest Rate Risk

• Buy T-bond or T-note futures to hedge against falling interest rates. Sell them to hedge against rising interest rates. (Remember that when interest rates fall, bond prices rise, and when interest rates rise, bond prices fall.)

• Use T-bond futures to hedge against changes in long-term (15+ years) rates. Use 10-year T-note futures to hedge against changes in 8-10 year rates.

r

Inherent risk exposureprofits

Long T-bond futures

Short T-bond futures

r

profits


Dollar Equivalency

• Estimate the loss in value if the spot YTM adversely changes by one basis point, denoted VS.

• Estimate the YTM of the CTD Treasury security if the spot YTM changes by a basis point; assume the CTD’s YTM will change by ‘b’ basis points. Compute the change in the CTD’s price if its YTM changes by b basis points. Denote this as SCTD.

• Estimate the change in the futures price if the CTD’s price changes by SCTD, denoted F per $100 face value. It can be shown that:

• The profit, VF, is then $1000 F.• Compute the number of futures contracts to trade, N, so that

N VF = VS

CTDΔSCF

T)h(0,1ΔF


Bond Pricing, I

• U.S. Treasury bonds and notes are coupon bonds. Their values are computed using:

• C is the semiannual coupon payment.• F is the face value.• Y is the unannualized, or periodic, 6-month yield.• N is the number of 6-month periods to maturity.• This assumes that the first coupon payment is 6 months hence.

NN21 y1

F

y1

C...

y1

C

y1

CValue


Bond Pricing, II

• To calculate the value of a bond , one must discount each cash flow at the appropriate zero rate.

• For example, let r(0,t) be the annual spot rate for a zero coupon bond maturing t years from today. Then,– Let r(0,0.5) = 2%, r(0,1) = 2.5%, r(0,1.5) = 3%, and r(0,2) = 3.3%.

– The semiannual coupon amount is $25, and face value is $1000.

– Maturity is 2 years hence.

• The bond’s value is:

1033.0981.0165

1000

1.0165

25

1.015

25

1.0125

25

1.01

254432


Yield to Maturity (YTM)

• The YTM is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond.

• Input PV = -1033.098, FV = 1000, PMT = 25 (semiannually), N = 4 (this is four semiannual periods) => CPT I/Y = 1.63838% (per six-month period). (1.63838%) (2) = 3.277% = YTM

• With Excel, use =YIELD("9/15/02","9/15/04",0.05,103.3098,100,2,0)

• With FinCAD, use aaLCB_y


Duration

• Duration is the weighted-average of the time cash flows are received from a bond. Example:

• Verified with Excel: =DURATION("6/25/2001","6/25/2003",0.06,0.068755,2,0)

• We will see that Modified Duration is handy for hedging purposes.

• Modified Duration: D/(1 + YTM/2) = 1.9135 / (1.03438) = 1.85.

Time Cash Flow P Value Weight Time * Wgt

0.50 3.00 2.9003 0.0295 0.01471.00 3.00 2.8039 0.0285 0.02851.50 3.00 2.7107 0.0276 0.04132.00 103.00 89.9747 0.9145 1.8289

Sum: 98.3897 1.0000 1.9135


U.S. Treasury Bond Price Quotes

• U.S. T-bond and T-note prices are in percent and 32nds of face value.

• For example (see fig. 9.1), on 12/01/00, the bid price of the 6 3/8% of Sep 01 T-note was 100 and 4/32% of face value.

• If the face value of the note is $1000, then the bid price is $1001.25.

• The asked price of this note is $1001.875.

• N.B. These prices are based on transactions of $1 million or more. In other words, a trader could buy $1 million face value of these notes for about $1,001,875 from a government securities dealer.

• These prices are quoted flat; i.e., without any accrued interest.

• Cash price = Quoted Price + Accrued Interest.


On December 1, 2000, the 6 3/8% of September 2001 was quoted to yield 6.12%.

• You can verify by using the YIELD function in Excel:=YIELD("12/01/00","9/30/01",0.06375,100.1875,100,2,1)

• To Calculate the cash price (quoted price + AI), you would pay:– The ask price is 100:06, or $1001.875 for a $1000 par bond.– Coupons on this note are paid every March 31 and September 30.– The last coupon was paid on September 30, 2000, which is 62 days

before December 1, 2000. – There are 182 days between the last coupon payment date and the

next one on March 31, 2001. – Interest accrues on an actual/actual basis. Thus the buyer pays the

seller accrued interest equal to:

AI = (62/182)*(63.75/2) = $10.859

The cash price (quoted price plus accrued interest) is: QP + AI = $1001.875 + $10.859= $1012.73.


Some Extra Slides on this Material

• Note: In some chapters, we try to include some extra slides in an effort to allow for a deeper (or different) treatment of the material in the chapter.

• If you have created some slides that you would like to share with the community of educators that use our book, please send them to us!


Hedging With T-Bond Futures: Changing the Duration of a Portfolio

• Hedging decisions are essentially decisions to alter a portfolio’s duration.

• By buying or selling futures, managers can lengthen or shorten the duration of an individual security or portfolio without disrupting the underlying securities (an “overlay”).

• That is, adding (buying) T-bond or T-note futures to a portfolio increases its interest rate sensitivity, while selling futures decreases the interest rate sensitivity of the portfolio.

• A portfolio manager will want to decrease (increase) the duration of the portfolio if the manager expects interest rates to increase (decrease).

• A completely hedged portfolio lowers the duration to the duration of a short-term riskless Treasury bill.


The Key Concept: Basis Point Value (BPV).

• The bond portfolio manager can change the duration of the existing portfolio to the duration of a target portfolio. This “immunizes” the portfolio against a change in interest rates.

• That is, if the portfolio manager knows:– how the current and target portfolios respond to interest rate

changes. – how T-bond (or T-note) futures contracts respond to interest

rate changes.

• Fortunately, if interest rates change by a small amount, say one basis point, the value of the portfolio will change predictably.


BPV, II.

• Using the bond pricing formula, the duration formula, and some algebra, the change in the value of a bond or a portfolio of bonds if interest rates change by one basis point can be written:

• When dy = 0.0001 (1 basis point), then dB is called the basis point value (BPV).

dyBy) (1

Duration dB


BPV, III.

• If y is defined to be one-half of the bond’s annual yield to maturity (YTM), then for a bond, or a portfolio of bonds:

0.0001 Bond of ValueMarket YTM/2)(1

DurationdB BPV


BPV, IV.

• The portfolio manager chooses a target duration so that it will have a particular BPV; i.e., a targeted change in value if interest rates change by one basis point.

• Assuming that the CTD bond and the bond portfolio will both experience a one basis point change in yield, the goal is to choose to buy or sell NF futures contracts so that

BPV (target) = NF BPV (futures) + BPV (existing)

• Thus, the BPV of the existing portfolio, the target portfolio, and the futures contract must be computed.


BPV, V.

• To determine the BPV for either a T-bond or T-note futures contract, the cheapest-to-deliver (CTD) security must first be identified.

• The futures price generally tracks the CTD security. • The BPV of the futures price is generally written as a present

value BPV(futures)/[1+h(0,T)] where BPV(futures) is the BPV of the cheapest-to-deliver instrument divided by the CTD’s conversion factor.

• To solve for the appropriate number of futures contracts needed to change the duration of an existing portfolio to a target duration:

NF = [BPV (target) - BPV (existing) ] / BPV (futures)


Example Using BPV

• Facts:– On December 1, 2000, a fixed-income portfolio manager

expects a steep decline in bond yields over the next six weeks.

– Because of these strong beliefs and “aggressive” management, the manager decides to more than double the duration of his fixed-income portfolio.

– The manager wants to avoid disrupting his carefully constructed bond portfolio to profit from the belief that interest rates will decline only over the next six weeks.

– Therefore, the manager decides to buy T-bond futures to increase the duration.


• Inputs:

– Existing Portfolio Duration: 5.7– Target Duration: 12.0– March T-Bond Futures Price: 102-03– Portfolio Value: $100,000,000 – Portfolio Yield to Maturity: 6.27%

• Solution:1. Find the BPV of the existing portfolio and the target

portfolio:

BPV(existing) = (5.7 /(1 + 0.0627/2)) * $100,000,000 * 0.0001 = $55,267.36

BPV (target) = 12 /((1+0.0627/2)) * $100,000,000 * 0.0001 = $116,352.35


2. Calculate the BPV of the T-Bond futures contract.

– It has a face value of $100,000.– Using well-known techniques, one can determine that the

CTD T-bond on December 1st for March futures is the 8.875% of August 2017.

– On December 1st, the conversion factor of this T-bond is 1.2957, duration is 9.83, and YTM is 5.80%. So,if interest rates change by one basis point, then the value of the CTD T-Bond will change by

BPV of CTD = 9.83 / (1+0.0580/2) * $100,000 * 0.0001 = $95.53.

– If the interest rate on 6-week treasury bills is 5%, then h(0,T) = (0.05)(6)/52 = 0.0058. Thus, we find

BPV of Futures: $95.53(1.0058) / 1.2957 = $74.156


Finally,

3. Determine the number of contracts required to achieve the desired portfolio duration:


($116,352.35 - $55,267.36) / $74.156= 823.736

• The bond portfolio manager should buy 824 March T-Bond futures contracts in order to increase the portfolio’s duration to 12 years.

• Note that the portfolio manager can choose any target duration.


Finally, (Really)

• Suppose the manager chooses to have a target duration of zero. This makes the BPV of the target equal zero. Then,


(0 - $55,267.36) / $74.156= -745.285

• The bond portfolio manager should sell 745 March T-Bond futures contracts in order to decrease the portfolio’s duration to 0 years.


Reading Treasury Bond Futures Prices

• Delivery dates exist every 3 months.• Delivery months are in March, June, September, and

December. • On December 1, 2000, the Dec T-bond settle price is 102-02.

This equals 102 and 2/32% of face value, or $102,062.50. • The December 2000 futures price was down 17 ticks, or 17/32. • This means that on December 14, 2000, the December contract

settled at 102-19. • A price change of one tick (1/32) will result in a daily

resettlement cash flow of $31.25.


Treasury Bond Futures, Delivery

• The Delivery Process is Complicated. But, in sum:• Any Treasury Bond that has fifteen or more years to

first call, or at least 15 years to maturity (whichever comes first), as of the first day of the delivery month.

• The seller of the futures contract, I.e., the short, has the option of choosing which bond to deliver.

• Delivery can take place on any day in the delivery month…the short chooses.

• Cash received by the short = (Quoted futures price × Conversion factor) + Accrued interest.

• Conversion Factor? Wha?


The Necessity for the Conversion Factor

• At its website, the CBOT lists deliverable T-bonds and T-notes, by delivery date.

• For example, as of November 29, 2000, there were 34 T-bonds deliverable into nearby T-bond futures contracts.

• By allowing several possible bonds to be delivered, the CBOT creates a large supply of the deliverable asset.

• This makes it practically impossible for a group of individuals who are long many T-bond futures contracts to “corner the market” by owning so many T-bonds in the cash market that the

shorts cannot fulfill their delivery obligation.


• Recall that the invoice price equals the futures settlement price times a conversion factor plus accrued interest.

• The conversion factor is the price of the delivered bond ($1 par value) to yield 6 percent.

• The purpose of applying a conversion factor is to try to equalize deliverability of all of the bonds.

• If there were no adjustments made, the short would merely choose to deliver the cheapest (lowest priced) bond available.

• In theory, if the term structure of interest rates is flat at a yield of 6% then, by applying conversion factor adjustments, all bonds would be equally deliverable.

• In practice, however, there is a “cheapest to deliver” or CTD T-bond. This is the T-bond used to price futures contracts on T-bonds.

©david dubofsky and 9-1 thomas w. miller, jr. chapter 9 t-bond and t-note futures futures contracts...

Documents