Treasury Futures

[welsh_owen]

Hi all,

I'm currently working through chapter 12 of the notes, specifically looking at the section on Tresury futures. I found the Q&A section at the end of the chapter very useful at explaining how long-bond futures prices are calculated in practice.

I like the idea that the futures price is usually determined by using the market price of the cheapest bond in the relveant market that satisifes the necessary criteria (i.e. over 15 years and non-callable) divided by the conversion factor (assuming my understanding of the circular argument is correct here).

I wondered whether there would ever be a case where the short party would choose not to deliver the cash amount corresponding to the cheapest to deliver bond? I'm assuming here that the actual amount delivered is just reduced/increased to account for the (bond price - futures price * c.f) in each case and this would be stated in the terms of the futures contract?

Assuming the CTD bond is used to value the futures contract presumably the 'cost of delivery' will be close to zero in the majority of cases. I find it quite strange why the short party would ever choose to deliver anything other than the cash amount for the short bond (+/- 'cost of delivery' where appropraite of course). I did consider actual delivery problems (i.e. if the short party couldn't actually purchse the bond to then deliver it) but as these appear to be 'cash settled' I don't believe this should present a problem in deciding on the amount delivered.

Following on from my initial question I wondered whether a more appropriate measure to calcualte conversion factors would be to use a current yield curve each day and then value each bond based on their stream of coupon payments? Presumbably this isn't done due to the work required and different providers using different mathematical procedures to construct yield curves?

 

[Mike Lewry]

Like all futures, the vast majority of contracts will be closed out prior to delivery, so the cheapest-to-deliver bond issue is largely theoretical for determining the futures price, where we use the current CTD bond.

Due to the approximate nature of the conversion factor calculation, there is no reason why the cost of delivery for the CTD bond would be close to zero. If the interest rate used in the calculation is significantly different from current yields, then we could have a significant cost of delivery, either plus or minus.

The conversion factor is a simplified approach, which gives a proxy to the correct value of the bond. The exchange publishes a spreadsheet with all the conversion factors for the possible bonds for each delivery date - as you say, to do this based on actual yield curves would be much more complex.

 

[Adam]

Hi there,
I have a related question on conversion factor. It is in section 6.2 of Hull's book (9th-global). It says that if the short side decides to deliver the bond, then the cash to receive is "the product of the conversion factor and the most recent settlement price for the futures contract".
My questions are

• Settlement price is for daily mark to market. Right?
• If delivery is taken, then it should be the locked-in price (i.e. F_0) at the issue of futures contract that is paid instead of the most recent settlement price. Right?

I guess I am missing something here. Could anyone help clarify, please?

It also says that "When bond yields are in excess of 6%, the conversion factor system tends to favor the delivery of low-coupon long-maturity bonds". Can I interpret this as follows? I don't get the point on "low-coupon".

1. The bigger the conversion factor is, the better (everything else being the same);
2. If the bond yield is higher than 6% (e.g. 8%), then the conversion factor will be higher then the market price (lower discounting rate being used for the calculation of conversion factor); Note this is regardless of coupon or maturity.
3. For two bonds which are only different in terms of maturity, the longer-maturity will have longer duration. Hence, when discount rate changes from 8% to 6%, the longer-maturity bond will see bigger increase in price and conversion factor.
4. For two bonds which are only different in terms of coupon rate, the bigger the coupon rate, the higher the price and the higher the conversion factor. But why the text says it should be low-coupon?

Thank you!
Regards,
Xu

 

[Deepesh]

Hi there,
I have a related question on conversion factor. It is in section 6.2 of Hull's book (9th-global). It says that if the short side decides to deliver the bond, then the cash to receive is "the product of the conversion factor and the most recent settlement price for the futures contract".
My questions are

• Settlement price is for daily mark to market. Right?
• If delivery is taken, then it should be the locked-in price (i.e. F_0) at the issue of futures contract that is paid instead of the most recent settlement price. Right?

I guess I am missing something here. Could anyone help clarify, please?

It also says that "When bond yields are in excess of 6%, the conversion factor system tends to favor the delivery of low-coupon long-maturity bonds". Can I interpret this as follows? I don't get the point on "low-coupon".

1. The bigger the conversion factor is, the better (everything else being the same);
2. If the bond yield is higher than 6% (e.g. 8%), then the conversion factor will be higher then the market price (lower discounting rate being used for the calculation of conversion factor); Note this is regardless of coupon or maturity.
3. For two bonds which are only different in terms of maturity, the longer-maturity will have longer duration. Hence, when discount rate changes from 8% to 6%, the longer-maturity bond will see bigger increase in price and conversion factor.
4. For two bonds which are only different in terms of coupon rate, the bigger the coupon rate, the higher the price and the higher the conversion factor. But why the text says it should be low-coupon?

Thank you!
Regards,
Xu

Hi Xu,

For your first question, it seems to me that all the mark to market gains/losses up until now would have been reflected in the margin account of the person with the short position. Hence the most recent settlement price is used instead of the locked in price.


For your second question, I would interpret as follows:

The important factor in determining the CTD Bond is the percentage change in price between bond price calculated at current implied yields and the exchange specified yield of 6%. For a higher coupon bond, its duration will be lower than a corresponding lower coupon bond since the payment would be weighted more towards the end of the term for the lower coupon bond.
This would imply higher percentage change in price for the low coupon bond (Approximate % change in price is Duration * change in yield).
Hence when yields are in excess of 6% low coupon bonds are favored.

 

[Adam]

Hi Xu,

For your first question, it seems to me that all the mark to market gains/losses up until now would have been reflected in the margin account of the person with the short position. Hence the most recent settlement price is used instead of the locked in price.


For your second question, I would interpret as follows:

The important factor in determining the CTD Bond is the percentage change in price between bond price calculated at current implied yields and the exchange specified yield of 6%. For a higher coupon bond, its duration will be lower than a corresponding lower coupon bond since the payment would be weighted more towards the end of the term for the lower coupon bond.
This would imply higher percentage change in price for the low coupon bond (Approximate % change in price is Duration * change in yield).
Hence when yields are in excess of 6% low coupon bonds are favored.

Thank you for your explanation, Deepesh.

For Q1, I get what you said. Essentially, it sounds like that the delivery is not the same as futures maturing? Then my question becomes: under what situation the future price specified/locked-in at the contract issue date would be used?

For Q2, I agree with you that lower coupon bond will have bigger price change. However, from the formula, the decision is made to minimize (Quoted bond price - Settlement Price * Conversion Factor). Hence, I would think it is the absolute size of conversion factor that is important. Please see a little example as below (sorry for the poor formatting; couldn't figure out how to make it look normal). That is, overall, the higher coupon bond would give bigger conversion rate. Right?

Bond 1 Bond 2
Face Value 100 100
Bond Yield 8% 8%
Term (Y) 3 3
Coupon (semi-annual) 9% 11%

Conversion Factor Discount Rate 6% 6%

Time CF 1 CF 2
-
0.50 4.50 5.50
1.00 4.50 5.50
1.50 4.50 5.50
2.00 4.50 5.50
2.50 4.50 5.50
3.00 104.50 105.50

Price at 8% 103.03 108.29
Price at 6% 108.37 113.80
Price Change 5.19% 5.09%

Conversion Factor 1.08 1.14

 

[Deepesh]

Thank you for your explanation, Deepesh.

For Q1, I get what you said. Essentially, it sounds like that the delivery is not the same as futures maturing? Then my question becomes: under what situation the future price specified/locked-in at the contract issue date would be used?

For Q2, I agree with you that lower coupon bond will have bigger price change. However, from the formula, the decision is made to minimize (Quoted bond price - Settlement Price * Conversion Factor). Hence, I would think it is the absolute size of conversion factor that is important. Please see a little example as below (sorry for the poor formatting; couldn't figure out how to make it look normal). That is, overall, the higher coupon bond would give bigger conversion rate. Right?

Bond 1 Bond 2
Face Value 100 100
Bond Yield 8% 8%
Term (Y) 3 3
Coupon (semi-annual) 9% 11%

Conversion Factor Discount Rate 6% 6%

Time CF 1 CF 2
-
0.50 4.50 5.50
1.00 4.50 5.50
1.50 4.50 5.50
2.00 4.50 5.50
2.50 4.50 5.50
3.00 104.50 105.50

Price at 8% 103.03 108.29
Price at 6% 108.37 113.80
Price Change 5.19% 5.09%

Conversion Factor 1.08 1.14

Hi Xu,

For Q1, my understanding is that the locked-in price is relevant only for the first mark to market (MTM) adjustment. The latest settlement price should be that price for which most recent MTM adjustment has occurred for all parties with outstanding open positions.
Since both, long and the short parties, would have their gains/losses adjusted in the margin account, the latest settlement price is used. Speaking in terms of terminology, the delivery of the bonds underlying the futures happens on maturity of the futures. So delivery of futures is not different from futures maturing. It is a usual practice to close out any open positions before maturity to avoid delivering or taking delivery of the bonds.

For Q2, first point to note is the bond with a high conversion factor (CF) doesn't necessarily result in a CTD bond (since a high CF would result in a high quoted bond price). It is the interaction of these two factors along with the Settlement price of futures that determines the CTD bond.
Doing the calculations in a spreadsheet and varying these factors brought in clarity. In case where yields are more than 6%, high coupon bonds are preferred where settlement price is higher than a particular break-even price. Similarly low coupon bonds are preferred where settlement price is less than a particular break-even price.
In case of a high yield scenario, the settlement price is more likely to be below the break-even price. Hence a low coupon bond is more likely to be a CTD bond.

 

[Adam]

Hi Xu,

For Q1, my understanding is that the locked-in price is relevant only for the first mark to market (MTM) adjustment. The latest settlement price should be that price for which most recent MTM adjustment has occurred for all parties with outstanding open positions.
Since both, long and the short parties, would have their gains/losses adjusted in the margin account, the latest settlement price is used. Speaking in terms of terminology, the delivery of the bonds underlying the futures happens on maturity of the futures. So delivery of futures is not different from futures maturing. It is a usual practice to close out any open positions before maturity to avoid delivering or taking delivery of the bonds.

For Q2, first point to note is the bond with a high conversion factor (CF) doesn't necessarily result in a CTD bond (since a high CF would result in a high quoted bond price). It is the interaction of these two factors along with the Settlement price of futures that determines the CTD bond.
Doing the calculations in a spreadsheet and varying these factors brought in clarity. In case where yields are more than 6%, high coupon bonds are preferred where settlement price is higher than a particular break-even price. Similarly low coupon bonds are preferred where settlement price is less than a particular break-even price.
In case of a high yield scenario, the settlement price is more likely to be below the break-even price. Hence a low coupon bond is more likely to be a CTD bond.

Hi Deepesh,
Again thank you for your explanation.
For Q1, my only confusion now is: by definition a long futures is enable the holder to buy something at X in T months. If we agree that delivery is the same as maturing, then by definition it means that the holder can but at X (which is locked in at issue). In other words, if we consider an equivalent forward contract (no daily settlement), then the maturity price should be X. Right?
For Q2, I miss the point where bond price also differs if coupon is different.

Hi Xu,

For Q1, my understanding is that the locked-in price is relevant only for the first mark to market (MTM) adjustment. The latest settlement price should be that price for which most recent MTM adjustment has occurred for all parties with outstanding open positions.
Since both, long and the short parties, would have their gains/losses adjusted in the margin account, the latest settlement price is used. Speaking in terms of terminology, the delivery of the bonds underlying the futures happens on maturity of the futures. So delivery of futures is not different from futures maturing. It is a usual practice to close out any open positions before maturity to avoid delivering or taking delivery of the bonds.

For Q2, first point to note is the bond with a high conversion factor (CF) doesn't necessarily result in a CTD bond (since a high CF would result in a high quoted bond price). It is the interaction of these two factors along with the Settlement price of futures that determines the CTD bond.
Doing the calculations in a spreadsheet and varying these factors brought in clarity. In case where yields are more than 6%, high coupon bonds are preferred where settlement price is higher than a particular break-even price. Similarly low coupon bonds are preferred where settlement price is less than a particular break-even price.
In case of a high yield scenario, the settlement price is more likely to be below the break-even price. Hence a low coupon bond is more likely to be a CTD bond.

Hi Deepesh,
Thank you again for your explanation.
For Q1, my only confusion now is: by definition a long futures is to enable the holder to buy something at price X in T months. And if you say delivery is the same as maturity, then by definition it means that the holder should be able to buy that something at price X (determined at issue date). Similarly, if you think about an equivalent forward (without daily settlement), then the maturity price should be X (instead of settlement price, which changes every day). Right?
For Q2, I realized that I missed the point that bond price also change if coupon changes. It is not just conversion factor that determines the outcome. Could you explain why "In case of a high yield scenario, the settlement price is more likely to be below the break-even price"?
Regards.

 

[Deepesh]

For futures, at maturity last settlement price (LSP) + MTM gains/losses till date should be equal to the locked-in price. Hence the holder will effectively buy at the locked-in price.

In case of forwards the concept of MTM is not applicable. Hence the locked-in price instead of LSP should be used.


For Q2, the final settlement price should be very close to bond price at maturity of the futures. Bond prices should be low in a high yield scenario.
On the contrary in case of a low yield scenario, bond prices are likely to be high. Hence they are less likely to be below the break even price compared to a high yield scenario.