Conversion factor and Cheapest-to-deliver bond

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allalongthewatc

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Hello All,
It’s been an hour, and I have been reading this topic again and again. Unfortunately, I couldn’t find a good link on this topic. I found a bunch of thesis on this. Can someone please explain this to me?
Thanks in advance.
 
The underlying on T-Bond futures is a theoretical, 20-years-to-maturity, 6% coupon Treasury bond. As there is likely no such bond in circulation, the short is allowed to deliver any Treasury bond with at least 15 years to maturity; there are many such bonds in circulation. Because they will be trading at a variety of prices – none of which is likely to be the same as the theoretical price of the theoretical bond – each real bond has a conversion factor: if you want to deliver a bond that’s trading at a lower price, you have to deliver a greater par value, and vice-versa.
The conversion factors are recalculated daily, and they try to ensure that the market price of all deliverable bonds (including conversion factors) will be the same as the market price of the theoretical bond, but they can only come close to equality. Because the numbers are only close, there will always be one bond that will be the cheapest of the bunch: that’s the cheapest-to-deliver (CTD) bond, and that’s the one that the short will choose to deliver. Which bond is the CTD will change from day to day, but each day there will be a CTD bond.
 
Thanks S2000. Can you please give me an example? Also, what is conversion rate? Can you please explain this to me?
 
Suppose that the current (par) yield curve is (in part):
  • 15 years, 5.433%
  • 16 years, 5.514%
  • 17 years, 5.584%
  • 18 years, 5.647%
  • 19 years, 5.701%
  • 20 years, 5.749%
There are five bonds that can be delivered against a T-Bond futures contract:
  • 15 years to maturity, 4.0% coupon, conversion factor (CF) = 0.83213
  • 16 years to maturity, 5.5% coupon, CF = 0.96398
  • 17 years to maturity, 6.5% coupon, CF = 1.06608
  • 18 years to maturity, 5.0% coupon, CF = 0.90696
  • 19 years to maturity, 7.0% coupon, CF = 1.12498
So, for example, if you deliver the 15-year bond, instead of delivering $1,000,000 par (of the theoretical, 20-year, 6% coupon bond), you’d have to deliver $1,000,000 ÷ 0.83213 = $1,201,734.81 par of the 15-year bond.
The market prices of the deliverable bonds are:
  • 15 years to maturity, 4.0% coupon, $854.29
  • 16 years to maturity, 5.5% coupon, $998.58
  • 17 years to maturity, 6.5% coupon, $1,099.68
  • 18 years to maturity, 5.0% coupon, $927.52
  • 19 years to maturity, 7.0% coupon, $1,149.53
(For comparison, the price of the theoretical bond would be $1,029.60.)
The cost to the short to purchase the required deliverable amount of the 15-year bond would be $1,201,734.81 × ($854.29/$1,000.00) = $1,026,632.44. The costs for all of the deliverable bonds are:
  • 15 years to maturity, 4.0% coupon, $1,026,632.44
  • 16 years to maturity, 5.5% coupon, $1,035,888.71
  • 17 years to maturity, 6.5% coupon, $1,031,517.43
  • 18 years to maturity, 5.0% coupon, $1,022,675.92
  • 19 years to maturity, 7.0% coupon, $1,021,822.74
(For comparison, the market value of the $1,000,000 par of the theoretical bond is $1,029,598.41.)
Thus, the short will choose to deliver the 19-year, 7.0% coupon bond: it’s the cheapest of the bunch.
 
In futures contract cheapest security can be delivered to the long position . This is particularly applicable to treasury bond furures contract . The conversion factor is used to determine the value of the security being delivered. So the seller can choose the security being delivered.
 
Hello S2000magician,
Thank you so much for your detailed response. I have two follow-up questions in your post:
S2000magician wrote:
The market prices of the deliverable bonds are:
  • 15 years to maturity, 4.0% coupon, $854.29
  • 16 years to maturity, 5.5% coupon, $998.58
  • 17 years to maturity, 6.5% coupon, $1,099.68
  • 18 years to maturity, 5.0% coupon, $927.52
  • 19 years to maturity, 7.0% coupon, $1,149.53
(For comparison, the price of the theoretical bond would be $1,029.60.)
Click to expand...
#1 - How did you calculate above prices? I tried calculating above using TVM for the first (assuming semi-annual) bond : I/Y = 3; N=30; PMT = 20; FV = 1000. {I took the fictitious bond as the reference yield.} Calculate PV? I got 803.99955, but not 854.29. Can you please help me?
Secondly, I couldn’t understand the below part. Can you please explain how you calculated the cost of shorting? Should I memorize this formula?
S2000magician wrote:
The cost to the short to purchase the required deliverable amount of the 15-year bond would be $1,201,734.81 × ($854.29/$1,000.00) = $1,026,632.44.
Click to expand...
I would appreciate your help. Thanks in advance.
 
allalongthewatchtower wrote:Hello S2000magician,
Thank you so much for your detailed response. I have two follow-up questions in your post:
S2000magician wrote:The market prices of the deliverable bonds are:
  • 15 years to maturity, 4.0% coupon, $854.29
  • 16 years to maturity, 5.5% coupon, $998.58
  • 17 years to maturity, 6.5% coupon, $1,099.68
  • 18 years to maturity, 5.0% coupon, $927.52
  • 19 years to maturity, 7.0% coupon, $1,149.53
(For comparison, the price of the theoretical bond would be $1,029.60.)
Click to expand...
#1 - How did you calculate above prices? I tried calculating above using TVM for the first (assuming semi-annual) bond : I/Y = 3; N=30; PMT = 20; FV = 1000. {I took the fictitious bond as the reference yield.} Calculate PV? I got 803.99955, but not 854.29. Can you please help me?
Click to expand...
I/Y is not 3%; it’s 5.433%/2: the 15-year YTM.
allalongthewatchtower wrote:Secondly, I couldn’t understand the below part. Can you please explain how you calculated the cost of shorting? Should I memorize this formula?
S2000magician wrote:The cost to the short to purchase the required deliverable amount of the 15-year bond would be $1,201,734.81 × ($854.29/$1,000.00) = $1,026,632.44.
Click to expand...
I would appreciate your help. Thanks in advance.
Click to expand...
The par amount of the 15-year bond that the short would have to deliver is $1,201,734.81. However, the bonds are trading at a price of 85.429% of par (=$854.29/$1,000.00). So the total price will be 85.429% of $1,201,734.81, or $1,026,632.44.
 
Thank you S2000magician for your detailed reply. Your response is gazillion times better than CFA1 material. I have two questions about the assumption you made above.
#1 How did you calculate these current yields?
S2000magician wrote:
Suppose that the current (par) yield curve is (in part):
  • 15 years, 5.433%
  • 16 years, 5.514%
  • 17 years, 5.584%
  • 18 years, 5.647%
  • 19 years, 5.701%
  • 20 years, 5.749%
Click to expand...
Secondly, what is the logic you used to calculate the conversion factor below?
The reason why I am asking this is that it seems to me that because CF is less than 1 for cases < 6%, short are penalized because they would be spending more to sell the bond. Why is it so? I am curious. Is it that the exchange doesn’t want them to trade these? What’s the benefit to short?
S2000magician wrote:
  • 15 years to maturity, 4.0% coupon, conversion factor (CF) = 0.83213
  • 16 years to maturity, 5.5% coupon, CF = 0.96398
  • 17 years to maturity, 6.5% coupon, CF = 1.06608
  • 18 years to maturity, 5.0% coupon, CF = 0.90696
  • 19 years to maturity, 7.0% coupon, CF = 1.12498
Click to expand...
I would really appreciate your help. Thanks in advance.
 
allalongthewatchtower wrote:Thank you S2000magician for your detailed reply. Your response is gazillion times better than CFA1 material. I have two questions about the assumption you made above.
#1 How did you calculate these current yields?
S2000magician wrote:Suppose that the current (par) yield curve is (in part):
  • 15 years, 5.433%
  • 16 years, 5.514%
  • 17 years, 5.584%
  • 18 years, 5.647%
  • 19 years, 5.701%
  • 20 years, 5.749%
Click to expand...
Click to expand...
First, they’re not current yields; they’re yields to maturity. (Make sure that you use the correct terminology; it will make learning this stuff a ton easier.)
I just put some numbers into Excel and made a normal-looking yield curve that started at 2% for the 1-year maturity and went to 6% at 30 years. Nothing special about the numbers.
allalongthewatchtower wrote:Secondly, what is the logic you used to calculate the conversion factor below?
S2000magician wrote:
  • 15 years to maturity, 4.0% coupon, conversion factor (CF) = 0.83213
  • 16 years to maturity, 5.5% coupon, CF = 0.96398
  • 17 years to maturity, 6.5% coupon, CF = 1.06608
  • 18 years to maturity, 5.0% coupon, CF = 0.90696
  • 19 years to maturity, 7.0% coupon, CF = 1.12498
Click to expand...
Click to expand...
I calculated the prices on the five bonds, then calculated the perfect conversion factors that would make all of the bonds cost the same amount, then added a little randomness to those factors. Again, nothing special.
allalongthewatchtower wrote:The reason why I am asking this is that it seems to me that because CF is less than 1 for cases < 6%, short are penalized because they would be spending more to sell the bond. Why is it so? I am curious. Is it that the exchange doesn’t want them to trade these? What’s the benefit to short?
I would really appreciate your help. Thanks in advance.
Click to expand...
The exchange is trying to get short to deliver bonds with exactly the same market value as the theoretical bonds would have; if the theoretical bonds would be worth $1.029 million, they want short to deliver $1.029 million worth of bonds. The Conversion factor tries to do that exactly, but it only comes close. Short takes advantage of the slight errors.
 
Thank you S2000magician! This CTD thing is now clear to me. I did have to google a few things to understand some of the concepts, which took me a few days. Thank you so much for your help. I also borrowed John Hull’s book (old edition) from the public library.
 
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