Bm Fi6051 Wk12 Lecture

Derivative Instruments FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2009

Week 12 – US Treasury Futures

Treasury Futures   

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 future represents an agreement to buy/sell some underlying asset in the future for a specified price. Both can be for physical settlement or cash settlement. 

Treasury Futures 

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



This means that Futures are traded where the deliverable is a US T-Note with an (exchange specified) maturity range



For this lecture, we’ll look at T-Note Futures contracts with a 10-year bond

Treasury Futures 

The Futures contract typically has less than 6months to maturity



For a 10-Year Futures note, at expiration of the futures contract, the deliverable maturity “must be no less than 6 years 6 months and no greater than 10 years from the first day of the contract expiration month” (source: CBOT)



The long futures contract holder must pay an invoice price equaling “the futures settlement price times a conversion factor plus accrued interest” (source: CBOT)

Treasury Futures

Treasury Futures 

Let’s look more closely at the quoted futures price:



The futures price at 10:00AM Nov 22nd , 2007 = 113’15 = 113 + (15/32) = 113.4688 This is a % expression of the future face value



Treasury Futures 







The face value of a Futures contract is $100,000. This means a 1% change results in a $1,000 change in the futures contract At the futures expiration, rather than deliver bonds, the vast majority of such contracts are rolled I.e., the offsetting trades in the expiring contract are combined with corresponding new positions in the next contract month Less than one percent of all financial futures traded at the Chicago Board of Trade result in the actual delivery of financial instruments.

Treasury Futures 

On delivery, the short contract holder can deliver any number of bonds specified by the exchange in the futures contract



Because the range of deliverable bonds is large, each having different maturities and coupons, a conversion factor is used to standardize the futures price

Treasury Futures 

At expiration, the long futures contract holder must pay

(Quoted Futures Price x Conversion Factor) + Accrued Interest 

For each $100 face value of bond delivered



E.g. Assume the   



The long contract holder must pay 



quoted futures price is 95’16 The conversion factor is 1.25 The accrued interest on the bond is 2.25 95.50x1.25+2.25 = $121,625

for each $100,000 of face value delivered

Treasury Futures 

How do we calculate the conversion factor?



For starters, CBOT provide a comprehensive conversion factor list for each of the deliverable bonds. This information is widely available and we will calculate it ourselves

Treasury Futures

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

Treasury Futures

Treasury Futures 

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



In our case, we can calculate the bond value at 20

2.125 100 + ∑ i 20 1 . 03 1 . 03 i =1 

≈ $87.21 Dividing by the face value give us the conversion factor = 0.8721

Treasury Futures   

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 be the one where Quoted Price –(Quoted Futures Price x Conversion Factor)



is a minimum

Treasury Futures 







A number of factors decide on which bond is the cheapest to deliver, E.g. when bond yields are less than 6%, high coupon, 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 data distributors (reuters, bloomberg, etc)

Treasury Futures 

Determining the quoted futures price



From lecture 2.2, we know that the futures price on an asset with a known income stream is given by

F0 = ( S 0 − I )e rT

 

Where T is the time to contract maturity† And r is the risk free rate for duration T

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

†

Treasury Futures  

Let’s use an example, Assume that the cheapest to deliver is         

4¼% 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 = 101’31+ (see next Slide) Today’s Date = 22/11/2007

Treasury Futures

Treasury Futures  





First, calculate the bond cash (dirty) price Cash Price = Clean Price + Accrued Interest = 101.9844 + (7/182.5)*(4¼/2) = 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 futures contract

Treasury Futures 

But, before the future contract expires, there are no bond coupons, so I = 0



Now, let’s assume that the risk free rate between today (22/11/07) and contract expiry (19/12/07) is 3.15%

F0 = ( S 0 − I )e rT F0 = (102.0659 − 0)e F0 = 102.3040 

This is the dirty Futures Price

0.0315( 27 / 365 )

Treasury Futures 

On delivery of the bonds (15/12/05), the receiver will owe the accrued coupons on the bond, the clean futures price must be calculated

(

F0 = 102.3040 − ( 4 1 4 / 2 ) . 34 

) 182.5

Where the last part of the equation details the accrued interest on the bond at 19/12/2007

F0 = 101.9092

Treasury Futures 

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

F0 = 101.9092 / 0.8721 F0 = 116.855  

The actual futures price was 113’15 Notwithstanding some potential errors such as daycount counventions and the risk free rate of interest, it is clear that the underlying bond used is not the Cheapest To Deliver

Treasury Futures 

We should therefore continue to price the Futures 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.

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2009, 7th Ed. 

Chapter 6

Derivative Instruments FI6051 Finbarr Murphy Dept. Accounting & Finance University of Limerick Autumn 2008

Week 12 – Duration & Convexity

Duration 

We’ve seen how to construct a yield curve from zero and coupon bearing bonds



We need to understand how this curve moves over time before we can mathematically model its behavior



Similarly, if we can better describe interest rate curve movements both mathematically and fundamentally, we are in a better position to make value judgments on future behavior, and make profits from those judgments.

Duration 

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

Source:RiskGlossary.com

Duration 

Duration† can be defined as the change in the value of a fixed income security that will result from a 1% change in interest rates.



Duration is a weighted average of the maturity of all the income streams of a coupon bearing bond



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

†

Also know as Macaulay Duration

Duration 

A 3-year duration means the bond will decrease by 3% if interest rates (across the curve) increase by 1%



A Bond price is give by: n

B = ∑ ci e

− yti

i =1



Where y is the continuously compounded yield

Duration 

The Duration of the bond is defined as − yti t c e ∑i =1 i i n

D= 

B

Which can be re-written as

 ci e − yti  D = ∑ ti   B  i =1  n



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

Duration 

Remember:

n

B = ∑ ci e

− yti

i =1



So

n ∂B − yti = −∑ ci ti e ∂y i =1



But

D × B = ∑i =1 ti ci e



Therefore

n

∂B = −D × B ∂y

− yti

Duration 

Rewritten



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

∂B − D∂y = B

Duration 

Consider the following bond:  2.5 years to maturity  6.5% coupon paid quarterly  4.5% yield (continuously compounded)

 ci e − yti  D = ∑ ti   B  i =1  n

Duration 

Example Time (years) 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

Cash PV Cash Flow Flow 1.625 1.6068 1.625 1.5888 1.625 1.5711 1.625 1.5535 1.625 1.5361 1.625 1.5189 1.625 1.5019 1.625 1.4851 1.625 1.4685 101.625 90.8118 104.6427

Weight 0.0154 0.0152 0.0150 0.0148 0.0147 0.0145 0.0144 0.0142 0.0140 0.8678 1.0000

Time x Weight 0.0038 0.0076 0.0113 0.0148 0.0183 0.0218 0.0251 0.0284 0.0316 2.1696 2.3323

Duration 

Modified Duration



Market conventions usually express y as semiannual compounded yield rather than continuously compounded yield

D D = 1+ y m *

 

Where D* is the modified duration And m is the compounding frequency per year

Duration 

Portfolio Duration



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



So, you have an estimate for the change in the bond portfolio value given a change in yields for all bonds in the portfolio



We are making the assumption that yields on all bonds in the portfolio change by the same amount

Duration 

Duration applies to small changes in yield δy



The usefulness of duration declines for larger changes in yield

Convexity 

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



In other words, we want a measure of the bond (or portfolio) convexity.

Convexity 

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 

Convexity is given by the following formula

1∂ B C= = 2 B ∂y 2



2 − yti c t ∑i =1 i i e n

B

Taking convexity into consideration,

∂B 2 1 = − D∂y + 2 C (∂y ) B

Portfolio Management 

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



By construction a portfolio of bonds, we can “engineer” our portfolio to have particular performance characteristics under certain characteristics



We can therefore reduce the impact of parallel shifts in the yield curve

Non-parallel shifts 

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

Source:RiskGlossary.com

Further reading 

Hull, J.C, “Options, Futures & Other Derivatives”, 2009, 7th Ed. 

Chapter 4