bm fi6051 wk12 lecture

8/14/2019 Bm Fi6051 Wk12 Lecture

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Derivative InstrumentsFI6051

Finbarr MurphyDept. Accounting & FinanceUniversity of LimerickAutumn 2009

Week 12 US Treasury Futures


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US Treasury Futures are listed on CBOT Euro (Bund) Futures are quoted on Liffe

This lecture will look primarily at US T-Futures

Remember:

A future is an exchange-traded derivative. A futurerepresents an agreement to buy/sell someunderlying asset in the future for a specifiedprice. Both can be for physical settlement orcash settlement.

Treasury Futures


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Futures on US Treasury notes are traded withunderlying maturities of 2, 5 and 10 years.

This means that Futures are traded where the

deliverable is a US T-Note with an (exchangespecified) maturity range

For this lecture, well look at T-Note Futures

contracts with a 10-year bond

Treasury Futures


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The Futures contract typically has less than 6-months to maturity

For a 10-Year Futures note, at expiration of the

futures contract, the deliverable maturity mustbe no less than 6 years 6 months and no greaterthan 10 years from the first day of the contractexpiration month (source: CBOT)

The long futures contract holder must pay aninvoice price equaling the futures settlementprice times a conversion factorplus accrued

interest (source: CBOT)

Treasury Futures


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Treasury Futures


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Lets look more closely at the quoted futuresprice:

Treasury Futures

The futures price at 10:00AM Nov 22nd , 2007

= 11315= 113 + (15/32)

= 113.4688

This is a % expression of the future face value


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On delivery, the short contract holder can deliverany number of bonds specified by the exchangein the futures contract

Because the range of deliverable bonds is large,each having different maturities and coupons, aconversion factoris used to standardize thefutures price

Treasury Futures


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At expiration, the long futures contract holdermust pay

(Quoted Futures Price x Conversion Factor) + Accrued Interest

For each $100 face value of bond delivered

E.g. Assume the quoted futures price is 9516

The conversion factor is 1.25

The accrued interest on the bond is 2.25

The long contract holder must pay 95.50x1.25+2.25 = $121,625

for each $100,000 of face value delivered

Treasury Futures


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How do we calculate the conversion factor?

For starters, CBOT provide a comprehensiveconversion factor list for each of the deliverable

bonds. This information is widely available andwe will calculate it ourselves

Treasury Futures


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Treasury Futures

"@" indicates the most recently auctioned U.S. Treasury security eligible for deliveryThis is also the 10year benchmark Note at time of writing


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Treasury Futures


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The conversion factor is the face value of all ofdeliverable bonds on the first day of the deliverymonth assuming a 6% semi-annual coupon

In our case, we can calculate the bond value at

$87.21

Dividing by the face value give us the conversionfactor

= 0.8721

Treasury Futures

20

20

1 03.1

100

03.1

125.2+

=ii


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The bonds that can be delivered cost: Quoted Price + Accrued Interest

The short futures contract holder must pay

(Quoted Futures Price x Conversion Factor) + Accrued

Interest

Therefore, the cheapest to deliver bond will bethe one where

Quoted Price (Quoted Futures Price x Conversion Factor)

is a minimum

Treasury Futures


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A number of factors decide on which bond is thecheapest to deliver,

E.g. when bond yields are less than 6%, highcoupon, short maturity bonds are more likely to

be cheapest And visa versa

At any one time, the cheapest-to-deliver bond

(for specific contracts) is details by datadistributors (reuters, bloomberg, etc)

Treasury Futures


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Determining the quoted futures price

From lecture 2.2, we know that the futures priceon an asset with a known income stream is

given by

Where Tis the time to contract maturity

And ris the risk free rate for duration T

Treasury Futures

rTeISF )( 00 =

Because the contract specifies a delivery month, calculations are less concise


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Lets use an example, Assume that the cheapest to deliver is

4% T-Note

Maturity = 15/11/2017

Semi-annual coupon Last Coupon Date = 15/11/2007

Next Coupon Date = 15/05/2008

Futures contract expiry = 19/12/2007

Conversion Factor = 0.8721 Clean Bond Price = 10131+ (see next Slide)

Todays Date = 22/11/2007

Treasury Futures


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Treasury Futures


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First, calculate the bond cash (dirty) price Cash Price = Clean Price + Accrued Interest

= 101.9844 + (7/182.5)*(4/2)

= 101.9844 + 0.0815

= 102.0659

Next, calculate the current value of the future

cash flows, I This involves calculating the present value of the

bond coupons during the life of the futurescontract

Treasury Futures


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But, before the future contract expires, there areno bond coupons, so I = 0

Now, lets assume that the risk free rate

between today (22/11/07) and contract expiry(19/12/07) is 3.15%

This is the dirtyFutures Price

Treasury Futures

rTeISF )( 00 =

)/(.)( 36527031500 0102.0659 eF =30401020 .=F


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The final step is to standardize the futures priceby dividing by the conversion factor

The actual futures price was 11315

Notwithstanding some potential errors such asdaycount counventions and the risk free rate ofinterest, it is clear that the underlying bond usedis not the Cheapest To Deliver

Treasury Futures

8721090921010 ./.=F

8551160 .=F


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We should therefore continue to price theFutures for the 13 other deliverable bonds until

Quoted Price (Quoted Futures Price x Conversion Factor)

is a minimum

See Hull, Page 136, example 6.2 for another

pricing example.

Treasury Futures


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Hull, J.C, Options, Futures & Other Derivatives,2009, 7th Ed. Chapter 6

Further reading


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Weve seen how to construct a yield curve fromzero and coupon bearing bonds

We need to understand how this curve moves

over time before we can mathematically modelits behavior

Similarly, if we can better describe interest rate

curve movements both mathematically andfundamentally, we are in a better position tomake value judgments on future behavior, andmake profits from those judgments.

Duration


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The most typical movement is a parallel shift.I.e. all points on the curve move up or down bythe same amount

Duration

Source: RiskGlossary.com


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Duration can be defined as the change in thevalue of a fixed income security that will resultfrom a 1% change in interest rates.

Duration is a weighted average of the maturityof all the income streams of a coupon bearingbond

So a 5-year zero coupon bond will have aduration of 5 years

Duration

Also know as Macaulay Duration


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The Duration of the bond is defined as

Which can be re-written as

The square bracket term is the ratio of PV of thecash flows to the bond price

Duration

B

ectD

n

i

yt

iii =

= 1

=

=

n

i

yt

ii

B

ectD

i

1


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Remember:

So

But

Therefore

Duration

=

=n

i

yt

iiecB

1

=

= n

i

yt

iiietc

y

B

1

==

n

i

ytii

iectBD1

BDy

B=


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Rewritten

Think about this! The duration gives us a goodindication how the bond will behave when asmall parallel shift in the yield curve occurs

Duration

B

ByD

=


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Example

Duration

Time

(years)

Cash

Flow

PV Cash

FlowWeight

Time x

Weight

0.25 1.625 1.6068 0.0154 0.0038

0.5 1.625 1.5888 0.0152 0.0076

0.75 1.625 1.5711 0.0150 0.01131 1.625 1.5535 0.0148 0.0148

1.25 1.625 1.5361 0.0147 0.0183

1.5 1.625 1.5189 0.0145 0.0218

1.75 1.625 1.5019 0.0144 0.0251

2 1.625 1.4851 0.0142 0.0284

2.25 1.625 1.4685 0.0140 0.0316

2.5 101.625 90.8118 0.8678 2.1696

104.6427 1.0000 2.3323


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Modified Duration

Market conventions usually express y as semi-annual compounded yield rather than

continuously compounded yield

Where D* is the modified duration

And m is the compounding frequency per year

Duration

my

DD

+=1

*


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Portfolio Duration

The duration of a bond portfolio, is defined asthe weighted average of the individual bonds in

the portfolio

So, you have an estimate for the change in thebond portfolio value given a change in yields for

all bonds in the portfolio

We are making the assumption that yields on allbonds in the portfolio change by the same

amount

Duration


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Duration applies to small changes in yield y

The usefulness of duration declines for largerchanges in yield

Duration


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We need a calculation to tell us how the bondwill perform with a larger change in yields

In other words, we want a measure of the bond

(or portfolio) convexity.

Convexity


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What factors influence duration & convexity?

As yields decrease, duration increases

Convexity is greatest when

Longer maturities

lower coupons

Zero-coupon bonds have the highest convexity

Convexity


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Convexity is given by the following formula

Taking convexity into consideration,

Convexity

B

etc

y

B

B

C

iytn

i ii

==

= 1

2

2

21

221 )( yCyD

BB +=


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Duration and convexity are useful indicators ohow bond prices change with changing yields

By construction a portfolio of bonds, we can

engineer our portfolio to have particularperformance characteristics under certaincharacteristics

We can therefore reduce the impact of parallelshifts in the yield curve

Portfolio Management


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Duration and Convexity are not useful when itcomes to non-parallel shifts

Non-parallel shifts

Source: RiskGlossary.com


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