"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
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