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# Determination of "Cheapest To Deliver"- Bonds

##### Active Member
In reference to; R19.P1.T3.Hull Study Notes

How do we arrive at the conclusion below...am a little stuck on this...any inputs/insights would be much appreciated ..
• If bond yields are less than 6%, this favors delivery of high-coupon, short-maturity bonds; i.e., bonds with lower durations.

• Similarly, If bond yields are greater than 6%, this favors delivery of low-coupon, long-maturity bonds; i.e., bonds with higher durations.

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#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
As far as i can understand,
The short position as per the agreement with the long should should deliver the Bond with a price of Quoted futures price(CTD)*CF+AI ,Quoted futures price(CTD) is the settlement price which is agreed upon in the contract.The short position should buy the bond to deliver at the quoted Bond price+AI from the market to settle the contract therefore the net cost that the short position has in settling the trade=cash received by him from the contract-cash used by him to buy the bond= (Quoted futures price(CTD)*CF+AI)-(quoted Bond price+AI)=Quoted futures price(CTD)*CF-quoted Bond price is the net cost to the short position therefore the short position should choose the bond with maturity>15 yrs(with any CTD) to deliver such that the cost is minimized.
The CTD bond choosen for delivery depends on the expected yield in the future if they are expected to be low(<6%,downward slope,CTD ) then choose CTD bonds with the lower duration(High coupon and short maturity) as the lower yield would increase the cost to deliver the bond for the short position (higher price to buy),lower duration minimizes this cost as price do not rise much, whereas if yield in the future are expected to be high(>6%,upward slope) then choose CTD bonds with the higher duration(low coupon and high maturity) as the higher yield coupled with the higher duration would decrease the cost to deliver the bond for the short position (lower price to buy)
Thanks

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#### Nicole Seaman

Staff member
Subscriber
In reference to; R19.P1.T3.Hull Study Notes

How do we arrive at the conclusion below...am a little stuck on this...any inputs/insights would be much appreciated ..
• If bond yields are less than 6%, this favors delivery of high-coupon, short-maturity bonds; i.e., bonds with lower durations.

• Similarly, If bond yields are greater than 6%, this favors delivery of low-coupon, long-maturity bonds; i.e., bonds with higher durations.
View attachment 437

Please note that I removed the URL that you posted, which linked to the study notes PDF because non-paid members do not have access to those materials so they should not be posted in the forum since they are paid materials. The screen shot that you attached is okay so I left that.

Thank you,

Nicole

##### Active Member
@Nicole Manley My bad Thanks so very much for pointing that out...completely overlooked that....

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
"The short position purchases the bond at the quoted bond price (which reflects the actual yield curve; i.e., not a flat yield curve at 6.0%) and receives, in exchange for delivery (in addition to accrued interest): settlement price * conversion factor. But the conversion factor (abstracting the details) "standardizes" the basket of choices by assuming a flat 6.0% yield curve. So, in real shorthand terms, the short is buying the actual yield curve and receiving credit for a flat 6.0% curve; when yields are above the notional 6.0% and above low coupon rates, prices will be below par and the short tends to prefer bonds that MORE responsive to the actual yield (i.e., higher duration. "low coupon, long-maturity" --> higher duration) ... when yields are below the notional and below high coupon rates, prices will be above par and the short tends to prefer bonds that are LESS responsive to the actual yield (i.e., lower duration. "high-coupon, short-maturity" --> low duration).

For example, a 20-year 2% semi-annual pay coupon bond will have a CF of somewhere around = -PV (6%/2, 20*2, $100*2%/2, 100) =$53.77 = about 0.5377 (not exactly the actual mechanics are shown in the learning XLS). That assumes a flat "notional" yield of 6.0%; but if the market's yield is actually 9.0%, the market price will be nearer to the theoretical (model) price of -PV (9%/2, 20*2, $100*2%/2, 100) =$35.59. So, the short "buys" this for only ~36 per 100 but receives for it on a valuation of ~53 per 100. Actual yields above 6% are favoring long duration bonds because their price drop is greater. As yield increases above (below) 6.0% (i.e., the notional coupon rate), CTD favors higher (lower) duration bonds."

And here is Tuckman:
"As yield increases above the notional coupon rate the prices of all bonds fall, but the price of the bond with the highest duration, namely the 5s of August 15, 2011, falls relative to the prices of other bonds. But, because conversion factors are fixed, the delivery price of the 5s of August 15, 2011, stays the same relative to that of all other bonds. In other words, as yields increase above the notional coupon rate, the cost of delivering the 5s of August 15, 2011, falls more than that of any other bond. Therefore, while all bonds are equally attractive to deliver at a yield of 6%, as yield increases the 5s of August 15, 2011, become CTD. Graphically, the ratio of the price to conversion factor of the 5s of August 15, 2011, falls below that of all other bonds.
As yield falls below the notional coupon rate, the prices of all bonds increase but the price of the bond with the lowest duration, namely the 4.75s of November 15, 2008, increases the least. At the same time, since the conversion factors are fixed the delivery price of the 4.75s of November 15, 2008, stays the same relative to those of other bonds. Therefore, while all bonds are equally attractive to deliver at a yield of 6%, as yield decreases the 4.75s of November 15, 2008, become CTD.

Figure 20.1 is a stylized example in that it assumes a flat term structure. It is for this reason that the CTD is either the 4.75s of November 15, 2008, or the 5s of August 15, 2011, but never the 6.50s of February 15, 2010, except, of course, at 6% when all bonds are jointly CTD. In reality, of course, the term structure can take on a wide variety of shapes that will affect the determination of the CTD. In general, anything that cheapens a bond relative to other bonds makes that bond more likely to be CTD. If, for example, the curve steepens, then long-duration bonds (e.g., the 5s of August 15, 2011) are more likely to be CTD. On the other hand, if the curve flattens, then short-duration bonds (e.g., the 4.75s of November 15, 2008) are more likely to be CTD. Figure 20.2 depicts a different shift in which the 6.50s of February 15, 2010, cheapen by 4 basis points (i.e., their yield increases by 4 basis points) relative to levels in Figure 20.1. As a result the 6.5s of February 15, 2010, become CTD when the general yield level is between about 5.60% and 6.20%. For lower yields the 4.75s of November 15, 2008, remain CTD, and for higher yields the 5s of August 15, 2011, remain CTD." -- Tuckman pages 431-33

#### federico32

##### New Member
Hi David,

What is exactly the "most recent settlement price"?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @federico32 In Hull's glossary, "Settlement Price: The average of the prices that a contract trades for immediately before the bell signaling the close of trading for a day. It is used in mark-to-market calculations." -- Hull, John C. Options, Futures, and Other Derivatives (9th Edition) (Page 835). Prentice Hall. Kindle Edition. Thanks!

#### federico32

##### New Member
Alright thanks. I don't really get, however, how can an interest rate future be used to hedge. I know that if, for example, an investor wants to lock in an interest rate at which to invest, he must shield himself from declining rates by going long in a specific number of interest futures contracts, but I can't see where the T-bond futures would help. If rates declined where would the gain in the T-bond futures position come from? From an increase in the Settlement Price?

The work you do is great by the way, keep it up!

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
Just to give an idea, If S is the value of the T-bonds, as the rates decline the value of S increases whereas if rates increase the value of S decreases. Since the T-bond futures value=S*exp(rT) ,r is the risk free rate and T is the time to maturity ,thus if S increases(rates decline) the T-bond futures value increases whereas if if S decreases(rates incline) the T-bond futures value decreases.
The T-bond futures value depends directly on the underlying T-bonds values,for a long position in T-bond futures, the larger the value of T-bonds the larger the value of T-bond futures. Since the value of T-bonds is inversely dependent on the interest rates therefore if the rates declined then the value of T-bonds increases which results in increase in the value of the T-bond futures.

thanks

#### federico32

##### New Member
Ok thanks for your help. So basically if F=S*e(rT) increases due to lower interest rates, this means that in the payoff formula [Settlement Price*Conversion factor - Quoted price] the Settlement Price will increase? What I have a hard time understanding is the link between the Futures value and the payoff formula.