# What Makes a Bond Cheapest to Deliver Against the Futures Contract?

The 'CTD' bond is the one with the lowest price relative to the price the futures seller can charge the buyer for it.
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Several weeks ago, I wrote about how the price of the Treasury bond futures contract traded on the

corresponds to the yield of the 30-year Treasury bond. I

explained that the futures contract tracks the price of the so-called cheapest-to-deliver Treasury bond. I didn't explain what makes a particular bond cheapest to deliver, thinking no one would care.

Silly me.

Richard Battikha

and

Daniel Chisholm

, here's the lowdown, thanks again to Keith Schap of the CBOT.

I'm going to assume in this discussion an understanding of duration. If you don't understand duration, please have a look at a recent

column on the subject. It isn't as difficult as you may think.

There are two issues here. First, why isn't the cheapest-to-deliver bond the one with the lowest quoted price?

Recall that the seller of the futures contract has to deliver to the buyer \$100,000 face value of Treasury bonds that "have a maturity of at least 15 years from the first day of the delivery month."

That sentence is from the CBOT's

specifications for the bond contract. The very next sentence begins to explain why the bond with the lowest quoted price may not be the cheapest to deliver. It says: "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 8%."

Translation: The invoice price is the price the buyer of the futures contract pays for the underlying bonds at delivery. The fact that it includes accrued interest isn't unusual: Whenever one person buys a bond from another, he pays the seller accrued interest. Accrued interest is the portion of a bond's next coupon payment that the seller is entitled to, based on when he sells it. If a bond makes coupon payments every six months, and I buy it from you three months before its next coupon payment, I have to pay you half of that coupon payment.

As for the conversion factor system, its purpose is to put all of the deliverable bonds on more or less equal footing. All else equal, bonds with big coupons have higher prices than bonds with small coupons. The conversion factor system equalizes the prices of the deliverable bonds by pricing them to an 8% yield. A deliverable bond's conversion factor tells you what the price of that bond would be, per \$1 of par value, at a yield of 8%. Bonds with coupons larger than 8% have conversion factors greater than 1, while bonds with coupons smaller than 8% have conversion factors less than 1.

So, is the cheapest-to-deliver bond the one for which the futures seller can get the highest invoice price? Not necessarily, because futures sellers have to buy the bonds they are going to deliver against the contract. The cheapest-to-deliver bond is the bond with the lowest price relative to the invoice price. If it costs more than the invoice price, it is closer to its invoice price than any other deliverable bond. If it costs less than the invoice price, it is further from its invoice price than any other deliverable bond. In other words, the cheapest-to-deliver bond is the bond that results in the smallest loss or greatest profit for the futures seller.

Have a look at this table. It provides all the relevant data on both the bond that is currently cheapest-to-deliver against the futures contract (the Feb. 15, 2015 bond) and the most recently issued 30-year Treasury bond. The full price is what the futures seller would have to pay to acquire the bond, and the futures invoice price is what he'd receive for it. As you can see from the last column, delivering the Feb. 15, 2015 bond would cost the seller 1.32, or \$1,320 per contract, while delivering the active bond would cost 9.50, or \$9,500 per contract.

The second issue is: What characteristic of a bond causes it to result in the largest profit or the smallest loss for the futures seller at delivery? That has to do with interest rates and duration.

Recall that the current front-month futures contract, the September contract, equalizes the deliverable bonds by pricing them to an 8% yield. Why 8%? Because once upon a time, the 30-year Treasury bond yielded 8%. But the Treasury last issued a 30-year bond with an 8% coupon in 1991. Since then, prevailing interest rates have come down. Yields have been rising lately; still, the most recently issued 30-year Treasury bond has a 5.25% coupon.

When interest rates fall, bond prices rise. And the bonds with the longest durations rise most. Conversely, when interest rates rise, bond prices fall, with the longest-duration instruments falling most.

So the bonds that have benefited least from the broad decline in interest rates since 1991 are the short-duration instruments -- bonds with relatively short maturities and relatively large coupons. Bonds like the current cheapest-to-deliver issue.

At any given time, the cheapest-to-deliver issue will be the one with the lowest converted price, where the converted price is the bond's price divided by its CBOT conversion factor.

As a result, when interest rates are below 8%, a short-duration issue will be the cheapest to deliver; its price will have risen least. Conversely, if interest rates were to rise above 8%, causing prices to fall, the cheapest-to-deliver issue would be one of the ones whose price fell furthest -- a long-duration issue.

As you may know, the CBOT has decided that beginning with the March 2000 contract, Treasury futures will equalize the deliverable bonds by pricing them to a 6% yield. All that means is that the same rule applies relative to the 6% yield level: When yields are below 6%, the cheapest-to-deliver bond will have a short duration; when they are above 6%, a long-duration bond will be cheapest to deliver.

Have a look at this table. The first row of numbers gives the particulars on the bond that was cheapest to deliver against the March 2000 contract at yesterday's futures close. The shaded box shows that of the three bonds in the table, it has the lowest converted price. The second row of numbers gives the particulars on the bond that would have been cheapest to deliver at yesterday's close if yields were 25 basis points lower (and prices were higher). As you can see, it's a shorter-duration instrument whose price would have benefited least from the decline in rates. Finally, the third row of numbers tells all about the bond that would have been the cheapest to deliver if yields were 25 basis points higher (and prices were lower) -- a longer-duration instrument whose price would have fallen most.