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Highest Conversion Factor

Thread starter #1
Hi David, isn't the answer should be B because

FV=100,N=20,1/Y=6/2,PMT=3.25, it gives us PV = 103.71
Conversion factor = 103.71/100 = 1.03

Here's the question from Hull 2011 Chapter 5&6.

171.3. The following four bonds can be delivered by the party with the short position in a U.S. Treasury bond futures contract:
Bond A: 15 year maturity and 5.0% semi-annual coupon Bond B: 20 year maturity and 6.5% semi-annual coupon Bond C: 25 year maturity and 6.0% semi-annual coupon Bond D: 30 year maturity and 5.5% semi-annual coupon
Which of these bonds has the HIGHEST conversion factor?
a) Bond A
b) Bond B
c) Bond C
d) Bond D

Answer listed in doc -

171.3. D. Bond D We can approximate the CF by pricing the bonds with a yield of 6.0%; e.g, Price of Bond A = -PV(6%/2, 30 periods, $2.5 s.a. coupon, $100 FV) = $87.74 so the CF is approximately $87.74/100 ~= 0.88.
As the coupons of Bonds A & B are below 6.0%, their CF must be less than zero. The CF of Bond C will be nearly 1.0 Only the CF of Bond D will be above 1.0; approximately 1.07 (as price at 6% ~ $107)

Regards,
atandon
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#2
Hi atandon,

Yes, you are correct, it has been previously identified in the forum but I have not had a chance to revert such corrections to the PDF, see http://www.bionicturtle.com/forum/t...td-in-us-treasury-bond-futures-contract.4564/

(thank you for locating this in the correct forum, much appreciated! I see you have other clarifications/questions ... I am recording 2b and 5b today, but I will address them ASAP. Thanks!)
 

PL

Active Member
#4
Hello again David,

I apologize (in advance) if this point of the forum is not the appropriate point for the following question regarding the CTD bond,

It is mentioned in the notes that

if Bond yields > 6% then Favors delivery of low-coupon, long-maturity bonds
if Bond yields < 6% then Favors delivery of high-coupon, short-maturity bonds
if Upward-sloping yield curve then Favors long time-to-maturity bonds
and if Downward-sloping yield curve then Favors short time-to-maturity bonds

Could you please explain the use of 6%, and the rational of the aforementioned 4 points?

Thank you in advance,
PL
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#5
Hi PL,

Those are just Hull's statements of course. Tuckman actually explains it, see my extraction below from Tuckman Chapter 20.

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.

"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
 

Mark W

Active Member
#7
David,

This is a great explanation - really appreciate your never ending efforts to explain these things in an easy manner. :)

Thanks,

Mark
 
#8
The new Tuckman book is great by the way. I have given it praise elsewhere, but Tuckman certainly complements hull when it comes to fixed income. Indeed, I would be surprised if GRAP does not source material from his newest edition for next years exam.
 
#10
IMO Tuckman is clearer than Fabozzi on most topics, including the - according to number of forum posts on the topics - the dreaded "duration family" and convexity questions.
Fabozzi has been around for a long time, but I get the feeling that he has read his book too many times, and the value of his incremental updates have converged asymptotically towards zero. Contrast that with Tuckman's new edition, which has not been updated (until now) for 10 years. Although a lot of material stays the same in some respect, it's a true new edition.

An analogy would be this: you write an essay in word and you think it's pretty good. You then spend some time over the next few days editing things here and there, and adding a few points while removing others. That's the approach of Fabozzi.

I have found that, rather than doing the aforementioned, a better outcome will result from you deleting your original document, and then writing a new one. It will contain much of the same information, only this time you are able to think outside of the framework you had already established, and you can ignore that one sentence you thought was too good to delete even though it didn't fit in. Basically, you are looking at it with fresh eyes. The result: a better, more refined end-product; one in which you had to organize your thoughts systematically while still adding valuable new information.

That's my silly analogy of Tuckman vs. Fabozzi. If you are upgrading, then yes, I would strongly recommend Tuckman. Indeed at times, Fabozzi goes off on a tangent and never really makes his point. It's like he's confusing himself or forgetting what he was trying to say.

This is only my opinion though, and there are those who swear to Fabozzi. I would ask David as well and look at the reviews on amazon (keeping in mind that Tuckman's book is new and has not been updated until recently so may have fewer reviews).
 

Mark W

Active Member
#12
Splendid description!

I will certainly look at getting Tuckman in the very near future - thanks a bunch for the comparison, not silly at all.

Cheers
 
#13
Atalon wrote:

FV=100,N=20,1/Y=6/2,PMT=3.25, it gives us PV = 103.71
Conversion factor = 103.71/100 = 1.03

Shouldn't N=40 since it's a semi-annual paying bond?
naturally if I plug in N=40 a different answer comes up.
 

isagasta

New Member
Subscriber
#14
Hi David,

From your explanation above on the CTD bond question, why would the short prefer more (less) responsive bonds if yields are above (below) 6%??

Is it because with yields above (below) 6%, bond prices will tend to be below (above) par, and therefore more room to improve if yields fall (increase)?

Many thanks.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#15
Hi @isagasta By "more responsive" I think I simply meant "higher duration" and by "higher duration" I meant to summarize "low-coupon, long maturity" bonds. Below is what Hull says but my annotations [] are inserted:
"A number of factors determine the cheapest-to-deliver bond. When bond yields are in excess of 6%, the conversion factor system tends to favor the delivery of low-coupon long-maturity bonds [i.e., longer duration]. When yields are less than 6%, the system tends to favor the delivery of high-coupon short-maturity bonds [i.e., shorter duration]. Also, when the yield curve is upwardsloping, there is a tendency for bonds with a long time to maturity to be favored, whereas when it is downward-sloping, there is a tendency for bonds with a short time to maturity to be delivered." -- Hull, John C.. Options, Futures, and Other Derivatives (Page 141). Pearson Education. Kindle Edition.
otherwise, the basic idea is articulated above in detail at https://www.bionicturtle.com/forum/threads/highest-conversion-factor.5569/post-15926
 

isagasta

New Member
Subscriber
#16
Thanks David, i understand what you mean by "responsive", that is clear. But, why the short side will prefer to choose longer duration bonds if yields are above 6% (or lower duration bonds if yields are below 6%)? What is the intuition behind this way to proceed by the short side?

Thank you.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#17
Hi @isagasta It's what i wrote above. Consider the example CF implied by 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. The short's motivation is simply to buy the cheapest bond and deliver it (getting credit per the CF). If the actual yield is 9.0%, this bond's price is only -PV (9%/2, 20*2, $100*2%/2, 100) = $35.59. Further, at this higher yield (i.e, actual 9.0% versus CF assumes 6.0%), even cheaper options are available:
  • longer maturity (eg) of 25 years: -PV (9%/2, 25*2, $100*2%/2, 100) = 30.83 costs only 30.8 per 100!
  • lower coupon (eg) of 1.0%: -PV (9%/2, 20*2, $100*1%/2, 100) = $26.39 costs only 26.3 per 100! Thanks,
 
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