Conversion factor and Cheapest-to-deliver bond

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?

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 or first call; 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.

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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:

#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?

I/Y is not 3%; it’s 5.433%/2: the 15-year YTM.

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.

#1 How did you calculate these current yields?

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?

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.

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.

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.

You’re quite welcome.

I have a copy of Hull’s book around here somewhere.

Dear S2000 magician: This is a great example. I am glad I saw that. I have two quick questions for you.

1. You start saying “theoretical one is a 6%, 20 year bonds.” I agree with the 6% but what is your reference to 20 years maturity. I am reading the guide of CME group T-Bond Futures… I can’t find that it is a 20 years bond. In my opinion, it would not matter.

2. After computing the conversion factors, you say you added some randomness… I did not understand why? (I did not check the values you computed, I took what you said at face value:-)) My understanding of the conversion factor is that they cannot handle the perfect equivalent computation due to several factors that too complicated to discuss candidly here.

So, it seems to me that you imply you did not want to use your “perfect” conversion factors” and added some randomness. But there is no perfect conversion factor… this is the essence of the problem that creates the “cheapest to deliver” issue. Would you please let me know if I am missing anything here?

Best regards,

Ktopyan

CME groups guide link for T-Bond Futures: https://www.cmegroup.com/education/files/understanding-treasury-futures.pdf

I learned that a long time ago (coincidentally, about 20 years ago) when I was analyzing mortgage-backed securities. I probably read it in Fabozzi.

It matters only in the sense that you need some objective value to deliver. You’re correct in the sense that the conversion factors would be proportional no matter what the theoretical underlying is.

For my example, I could compute the perfect conversion factors, so that all of the deliverable bonds would have the same market value as the theoretical underlying. Because I wanted an example that clearly had a cheapest-to-deliver bond, so after computing the perfect conversion factors, I changed them slightly so that there was a clear-cut, CTD bond.

In the real world, of course, you cannot get perfect conversion factors; in a made-up example, it’s easy.

Thank you so much for this instant answer! I appreciate it greatly.

With your permission, I would like to ask another question: (I am checking if my thinking style is correct or not) Your made-up example has bonds very similar to reality… different coupons, maturities, durations etc… and a slightly up-sloping yield curve.

So, why is it possible to produce the exact conversion factors for your example… what assumptions enable this exact computation that we do not have in real world of T-Bonds futures?

In other words, at this very moment, I might have 6 real T-bonds qualified to be delivered. And, I compute the CF for all those 6 bonds exactly the way you did… and using the original exact conversion factors (CFs), I can find the CTD bond… (i.e. no further adjustments to CFs required.)

I believe the issue is that the CASH PRICE determined by the supply and demand for the available bonds (which takes into account a number of issues that appear as biases) is the reason for CTD. i.e. we always compute the price using a hypothetical 6% yield, but the price we obtained this way will not match the cash price of the bonds due to known biases. Settlement price is for the hypothetical 6% yield bond, not the ones we will purchase to deliver by paying the market price. So, we use a biased benchmark… with an exact CF computations…

In more detail: We have bonds qualified to be delivered with low and high coupon bonds with shorter and longer maturities… so the duration differences will cause biases against certain securities: say, yield > 6 % implies bias to long-duration [low coupon, long maturity - riskier long-duration bonds will be preferred to be delivered] when we use the exact conversion factors. I think this creates a delivery bias originated from the fixed conversion factor’s ignoring the price sensitivity differences of different bonds to the change of yield… but conversion factors fails to take this into account creating a difference between the theoretical price and the cash price.

So, if I am thinking correctly, in your example, you should use exact CFs but provide market prices of your bonds in such a way that for each deliverable bond, the difference between the market price of the bond and the settlement price (obviously, there is only one settlement price regardless of how many bonds are qualified to be delivered) times corresponding CF of the bond in question needs to be computed and the bond with the lowest difference must be selected as the CTD.

Sorry, I am new to this site and I failed to clarify by using proper quotations that my previous post was a response / question to S2000magician’s last posting.

S2000Magician: How did you calculate the CFs in your post 24 April? What is the formula for this and will we be given the CFs in exercises on the exam?

I simply divided the market price of each bond by the theoretical market price of the theoretical underlying.

You’ll be given conversion factors on the exam.