Hedging With Futre

7/28/2019 Hedging With Futre

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David Dubofsky and 9-1

Thomas W. Miller, Jr.

Chapter 9T-Bond and T-Note Futures

Futures contracts on U.S. Treasury securities havebeen immensely successful.

But, the outlook for Treasury bond futures contractsis bleak, as the government has not issued any new30-year bonds since October 2001.

BOT Treasury Volume,

Contract Jan - Aug 2002Treasury Bond 38.0 million

10-year Tnote 59.4 million




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The T-bond Futures Contract

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

Futures prices are reported in the same way as arespot T-bonds, in "points and 32nds of 100%" of facevalue.

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 achange in value of $31.25.


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T-Note Futures Prices

For T-bonds, a tick is 1/32nd

. The resulting quote of 112-15equals 112 and 15/32.

But, for 5 and 10-year T-notes, a tick is of a 32nd, or$15.625 per tick. The resulting quote, say, of 98.095 = 98 and9.50/32.

For 2-year T-Notes, however, tick sizes are 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 ofdeliverable 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.htmlhttp://www.cbot.com/cbot/www/page/0,1398,12+31,00.html


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

In a very simp le sense, the futures price is the forwardprice 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


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What is Deliverable? And When?

Under the terms of the T-Bond futures contract:

AnyTreasury Bond that has fifteen or more years to first call, or atleast 15 years to maturity (whichever comes first), as of the first dayof the delivery month, is deliverable.

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

Delivery can take place on anyday in the delivery month, and theshort chooses.

Under the terms of the 10-year T-note futures contract:

AnyU.S. T-note maturing at least 6 1/2 years, but not more than 10years, from the 1st day of the delivery month is deliverable.




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Which T-bond will the ShortnaivelyChoose to Deliver?

Assumingthe short will receive the same dollaramount (invoice amount) upon delivery:

The short will select the T-bond that costs less thanany other deliverable T-bond.

So, to fix this problem, the CBOT adjusts the invoiceamounts to tryto 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).

David Dubofsky and 9-6Thomas W. Miller, Jr.


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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.



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


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

The one that costs the least.

.and at the same time gets the short the most money upondelivery (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 theinvoice amount and the gross cash bond price)

Max [(CF)(F) (S)]

During the delivery month, the amount in brackets will always benegative, for every deliverable bond.. WHY?




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Another Method to Identify theCheapest to Deliver T-bond

Before the delivery month, find the T-bond with the highestImplied 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 carryreturn (see eqn. 9.5b).

T365]0AI[S]

0AI[S]

TAI[(CF)(F)

IRR


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A Good Concept Check

Identify three T-bonds that are deliverable into thenearby T-bond futures contract.

Which of these three bonds is the cheapest todeliver?

Be able to show your work, and be able to explain

why one of the T-bond is more likely to be deliveredthan the other two.




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Questions

The implied repo rate for every deliverable T-bondmust be less than interest rates available in the market

(WHY?)

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


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The Options Held by the Short

Quality option: can deliver any eligible bond.

Timing option: can deliver on any day of the deliverymonth.

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

End of month option: futures cease trading 8 businessdays before the end of the delivery month, but the shortcan deliver on any day of the month.




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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

AI365

t)r(T1

2

C

365

rT1)AI(S

CF

1F

AI

365

t-Tr1

2

C-

365

rT1AIS(CF)(F)

TAI365

rT10AISCF

1F

TAI

365

rT1AIS(CF)(F)




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Using T-bond and T-note Futures toHedge Interest Rate Risk

Buy T-bond or T-note futures to hedge against falling interestrates. Sell them to hedge against rising interest rates.(Rememberthat when interest rates fall, bond prices rise, andwhen 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 againstchanges in 8-10 year rates.

r

Inherent riskexposureprofits

LongT-bond futures

Short T-bond futures

r

profits


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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 CTDs YTM will change by bbasis points. Compute the change in the CTDs price if its YTM changesby b basis points. Denote this as SCTD.

Estimate the change in the futures price if the CTDs price changes bySCTD, 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

CTDSCF

T)h(0,1F


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Bond Pricing, I

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

C is the semiannual coupon payment.

F is the face value.

Y is the unannual ized, or p er iod ic, 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


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Bond Pricing, II

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

For example, let r(0,t) be the annual spot rate for a zero couponbond 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 bonds value is:

1033.0981.01651000

1.016525

1.01525

1.012525

1.0125

4432


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Yield to Maturity (YTM)

The YTM is the discount rate that makes the present value of thecash 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


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Duration

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

Verified with Excel:

=DURATION("6/25/2001","6/25/2003",0.06,0.068755,2,0) We will see that Modif ied Durat ionis handy for hedging

purposes.

Modif ied Durat ion: D/(1 + YTM/2) = 1.9135 / (1.03438) = 1.85.

Time Cash Flow P Value Weight Tim e * Wgt

0.50 3.00 2.9003 0.0295 0.0147

1.00 3.00 2.8039 0.0285 0.0285

1.50 3.00 2.7107 0.0276 0.04132.00 103.00 89.9747 0.9145 1.8289

Sum: 98.3897 1.0000 1.9135


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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% ofSep 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. Inother words, a trader could buy $1 million face value of these notes forabout $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.


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On December 1, 2000, the 6 3/8% ofSeptember 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 daysbefore December 1, 2000.

There are 182 days between the last coupon payment date and thenext one on March 31, 2001.

Interest accrues on an actual/actual basis. Thus the buyer pays theseller 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.


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Some Extra Slides on this Material

Note: In some chapters, we try to include some extraslides 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 toshare with the community of educators that use ourbook, please send them to us!


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

Hedging decisions are essentially decisions to alter a portfoliosdurat ion.

By buying orselling futures, managers can lengthen orshortenthe duration of an individual security or portfolio withoutdisrupting the underlying securities (an overlay).

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

A portfolio manager will want to decrease (increase) theduration 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.


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The Key Concept:Basis Point Value (BPV).

The bond portfolio manager can change the durationof the existing portfolio to the duration of a targetportfolio. This immunizes the portfolio against achange 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 interestrate changes.

Fortunately, if interest rates change by a smallamount, say one basis point, the value of the portfoliowill change predictably.


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BPV, II.

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

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

dyBy)(1

DurationdB


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BPV, III.

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

0.0001BondofValueMarketYTM/2)(1

DurationdBBPV


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BPV, IV.

The portfolio manager chooses a target duration so that it willhave a particular BPV; i.e., a targeted change in value if interestrates change by one basis point.

Assuming that the CTD bond and the bond portfolio will bothexperience a one basis point change in yield, the goal is tochoose 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, andthe futures contract must be computed.


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BPV, V.

To determine the BPV for either a T-bond or T-note futurescontract, the cheapest-to-deliver (CTD) security must first beidentified.

The futures price generally tracks the CTD security.

The BPV of the futures price is generally written as a presentvalue BPV(futures)/[1+h(0,T)] where BPV(futures) is the BPV ofthe cheapest-to-deliver instrument divided by the CTDsconversion factor.

To solve for the appropriate number of futures contracts needed

to change the duration of an existing portfolio to a targetduration:

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


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Example Using BPV

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

expects a steep decline in bond yields over the next sixweeks.

Because of these strong beliefs and aggressive

management, the manager decides to more than double theduration of his fixed-income portfolio.

The manager wants to avoid disrupting his carefullyconstructed bond portfolio to profit from the belief thatinterest rates will decline only over the next six weeks.

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


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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 = $55,267.36BPV (target) = 12 /((1+0.0627/2)) * $100,000,000 * 0.0001 = $116,352.35


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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 the8.875% 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 interestrates change by one basis point, then the value of the CTDT-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


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Finally,

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

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

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

The bond portfolio manager should buy 824 March T-Bondfutures contracts in order to increasethe portfolios duration to12 years.

Note that the portfolio manager can choose any target duration.


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Finally, (Really)

Suppose the manager chooses to have a targetduration of zero. This makes the BPV of the targetequal zero. Then,

NF = [BPV (target) - BPV (existing) ] / BPV (futures)(0 - $55,267.36) / $74.156= -745.285

The bond portfolio manager should sell 745 March T-Bond futures contracts in order to decrease theportfolios duration to 0 years.


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Reading Treasury Bond Futures Prices

Delivery dates exist every 3 months.

Delivery months are in March, June, September, andDecember.

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 contractsettled at 102-19.

A price change of one tick (1/32) will result in a dailyresettlement cash flow of $31.25.


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Treasury Bond Futures, Delivery

The Delivery Process is Complicated. But, in sum:

AnyTreasury Bond that has fifteen or more years tofirst call, or at least 15 years to maturity (whichevercomes 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 anyday in the delivery

monththe short chooses.

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

Conversion Factor? Wha?


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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-bondsdeliverable into nearby T-bond futures contracts.

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

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

shorts cannot fulfill their delivery obligation.


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Thomas W Miller Jr

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

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

The purpose of applying a conversion factor is to try to equalizedeliverability 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 bondswould 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.

Hedging With Futre

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