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Determining the theoretical Future Price

jcjc0602

Member
Hi David,

Long time no see! How are you doing!

I have a question about this topic, which is in Hull's Chapter 6.2, Edition 7, and also a topic in FRM part1. In Hull's book, first, he calculated the dirty price by adding quoted bond price and accrued interest rate. Then, using F0=(S0-I)e^rT. After that, F0 was subtracted by another accrued interest rate, and then divided by conversion factor. Is this the quoted future price that theoretically we should get from the exchange or broker based on the assumption that it was the cheapest to deliver and known delivery date? Could you please explain why the calculation should be like that?

Thanks a lot for your time!

Aleksander Hansen

Well-Known Member
My understanding of your question (and the referenced question and how to look at it)... David can (and prob will correct me if I'm wrong)
1. Price + accrued interest between current time and past coupon just gives you the cash price of the bond
2. Subtract PV of coupon to be paid, compound 1-2 over life of futures contract to get cash price of futures written on this 12% bond
3. Quoted price of this futures = Cash price minus the accrued interest
4. quoted futures price in terms of the "standardized" terms, using the applicable conversion factor
In general, this need not be the cheapest to deliver. By construction, this particular example you are referring to assumes the cheapest-to-deliver bond is known.
Otherwise you would need to calculate and compare which one is cheapest-to-deliver.

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Nice! I agree that the bond used does not need to be the CTD: I don't see how the CTD could be anyways known with certainty in advance, as the quoted bond prices (among bonds available for selection by the short) will change over time.

This exercise is tedious but it's worth understanding even from an exam standpoint because, while the price of a T-bond futures has low testability, the exercise is a compact application of individually testable concepts, especially: 1. quoted (clean) versus cash (dirty) and 2. cost of carry

I very much like ahansen's four steps. I can't improve on them, so I will just add commentary to the same four (4) steps above:

Step (3) reverses Step (1) and both would be un-necessary except that we require quoted prices. Step (1) translates quoted price into cash price; after applying cost of carry to infer F(0) from spot, S(0), Step (3) merely reverses this to get back to to quoted price.

Step (2) is the essential application of the cost of carry that ASSUMES a selected cash bond will be the cheapest-to-deliver, it applies:
F(0) = [S(0) - I]*exp(rT); Hull 5.2;
i.e., If, while owning the commodity I receive income (I) before selling it to you, I am willing to reduce the future price by the PV(I).
So step (2) just gets you a futures price, estimated by COC model, on an assumed cash bond.

So my comments on ahansen's four steps:
1. Assume a CTD, translate quoted price --> cash price
2. Apply cost of carry to infer F(0) based on today's cash price, S(0)
3. Translate back from cash priced F(0) to quote-price F(0); i.e., without coupon accrual
4. As above (CF).

I personally can only cope with this tedious exercise by breaking it into pieces:
• If you get the quoted --> cash --> quoted, you can conceptually drop that out and you will be left with:
• A cost of carry model F(0) = [S(0) - I] to infer future cash price F(0), like any commodity with the exception:
• Unlike most spot prices, S(0) which we can observe with certainty, in this case we are guessing (assuming) an S(0) "as if" it represents the future, unknowable CTD bond. I might even suggest, in this case:
cash_F(0) = [cash_S_est(0) - I]*exp(rT); where cash_S_est(0) = spot price of estimated/assumed CTD bond
• (finally the CF ... a different story)

jcjc0602

Member
Thank ahansen and David!

Your explanation helps a lot! I have a further question:
Step 4, why do we need to standardize it by dividing CF? What is the price after dividing CF? Should that price be the theoretical future price quoted from exchange or bloomberg or other data source? Thanks!

Aleksander Hansen

Well-Known Member
The conversion factor standardization is just done since there's a range of maturities for which you can deliver. We have to adjust the price accordingly.
I wouldn't necessarily use the prescriptive 'should': Bloomberg [depending on the screen/function] will quote you both clean and dirty, and you can calculate both plain or the standardized.
Convention is to quote the clean price.
David can correct me if I'm wrong, but my understanding is that you'll be quoted a standardized price.

afterworkguinness

Active Member
My understanding of your question (and the referenced question and how to look at it)... David can (and prob will correct me if I'm wrong)
1. Price + accrued interest between current time and past coupon just gives you the cash price of the bond
2. Subtract PV of coupon to be paid, compound 1-2 over life of futures contract to get cash price of futures written on this 12% bond
3. Quoted price of this futures = Cash price minus the accrued interest
4. quoted futures price in terms of the "standardized" terms, using the applicable conversion factor
In general, this need not be the cheapest to deliver. By construction, this particular example you are referring to assumes the cheapest-to-deliver bond is known.

Otherwise you would need to calculate and compare which one is cheapest-to-deliver.

Great way of explaining it. You put it better than Hull himself.

Roshan Ramdas

Active Member
Hi David,
I have read the full thread in this forum as I had a similar doubt with respect to the final step in the example outlined by Hull wherein the final quoted price is being divided by the know conversion factor.
I still haven't been able to understand the rational of adjusting the final futures price using the conversion factor,...assuming that the CTD is known well in advance. Please, could you shed some light on the relevance of this last step alone.
Thank you,
Roshan

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Roshan Ramdas Below is Hull's Example 6.2 (page 136) input into our learning XLS. Per @Aleksander Hansen 's four steps above, I think you can look at this in two pieces, per my (1) and (2) blocks on the right panel below
1. The first steps (green below) apply cost of carry model to determine the estimated quoted forward price ($119.71) of the bond we guess will be cheapest; i.e., Aleks's "plain" (non standard) price. If the futures contract were simply on this (single) bond itself, this would be the end of it. We can think of this as the cost of carry's implied forward price of the (presumed) cheapest to deliver bond. 2. But, in exchange for delivering the T-bond, the short who delivers is going to be paid, not the futures settlement price, but rather a multiplier on the futures price: Invoice price = (futures price * conversion factor) + accrued interest (see the contract spec @ http://www.cmegroup.com/trading/int...us-treasury-bond_contract_specifications.html "The invoice price equals the futures settlement price times a conversion factor, plus accrued interest. The conversion factor is the price of the delivered bond ($1 par value) to yield 6 percent.").

Because invoice price = futures price * CF, the final "standardization" step just "unwinds" this with: futures price = estimate invoice price / CF, or as McDonald says: arbitrage should compel the Futures price ~= (price of the cheapest to deliver bond)/(CF of cheapest to deliver)

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Roshan Ramdas

Active Member
Thank you David....this explanation really helps.

Koulyo

New Member
Hi Dave (AKA the Master)

Your CF is 1.6 and not 1.4. I hope I am right????

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Koulyo Yes, agreed, the title text in the snapped exhibit (for some reason) contains an incorrect 1.4 CF, but 1.6 CF is used in the calculations. Thank you!

Koulyo

New Member

Hi good people.

I am reading this paragraph and i cant seem to 100% grasp it and I tend not to take things for granted rather to really digest. So can some one explain the below please?

1) " Valuing an interest rate swap in the terms of bonds involves understanding that the value of the floating rate bond will be equal to the notional amount at any of its periodic settlement dates, when the next payment is set to the market floating rate."

2) "We can value the fixed rate bond using the spot rate curve and then discount the next known (floating) rate payment plus the notional amount at the current discount rate."

quoting Hull.

Tons of thanks

SyroneDavid

New Member
Out of curiosity, is the quoted clean price of 115 and the CF of 1.6 assumed? or is there a way we can calculate this on our own.

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QuantMan2318

Well-Known Member
Subscriber
Out of curiosity, is the quoted clean price of 115 and the CF of 1.6 assumed? or is there a way we can calculate this on our own.

The Quoted price would be given in the problem as that is what is quoted on the Terminal for the Futures or the Bonds. The general convention in the US is to quote the clean prices.

As far as the Conversion factor goes, you can calculate the same provided the yield on the 'standard' Bond is provided as the intention here is to convert the available bonds of different maturities into equivalent terms as the standard Bond (here the Standard Bond being the Bond whose covenants including the Yield are the underlying for the Short Futures).

The blokes at the CBOT incorporated all these rules so that there may not be any difficulty in finding the relevant issues what one may deliver. If we restricted the issue to be on same terms as the underlying, there may be a shortfall in the Bonds that can be delivered causing a supply squeeze and hence spike in prices and lower yields.

(If you are interested) If you want to squeeze your competitors and make money, what better way there is than to have huge longs, buy the underlying and corner the market so that the party with the shorts cannot deliver and hence is forced to liquidate the positions at a huge cost to him/her as Jay Gould and some of the other robber barons did.

GARP generally doesn't ask questions on computation of CFs as it will be provided in the questions. If you are curious, you can check out the video on CF by the great @David Harper CFA FRM himself from his rich and nostalgic walk down memory lane

Hope this helped
Thanks

aangermeyer

Member
My understanding of your question (and the referenced question and how to look at it)... David can (and prob will correct me if I'm wrong)
1. Price + accrued interest between current time and past coupon just gives you the cash price of the bond
2. Subtract PV of coupon to be paid, compound 1-2 over life of futures contract to get cash price of futures written on this 12% bond
3. Quoted price of this futures = Cash price minus the accrued interest
4. quoted futures price in terms of the "standardized" terms, using the applicable conversion factor
In general, this need not be the cheapest to deliver. By construction, this particular example you are referring to assumes the cheapest-to-deliver bond is known.
Otherwise you would need to calculate and compare which one is cheapest-to-deliver.

Hi @David Harper CFA FRM ,

What I dont unterstand is why one has to add the AI of the past and subtract the AI of the next coupon (instead of adding it as well). Could you please help me out?

Thanks,

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @aangermeyer I recently recorded a new YouTube video walking through the theoretical T-bond price exercise (using Hull's exact numbers). To further clarify, my explainer includes the diagram below (my video is further below). Hopefully this helps. You can see, we start with an assumption for the quote (flat) price of the CTD bond (= $115.00). Then we need to add the AI to retrieve the cash (aka, full) spot price of the CTD (=$116.978). The cash price is the "true" price, so we need to apply the COC to it; applying the COC gets us to the full, forward price (=119.711). Then we need to subtract AI to retrieve the associated quote price (=114.859). So, the way, that I look at this is: we start with quote price and "unwind" to the cash price (by subtracting AI) in order to accurate price the forward, then "re-wind" (by adding AI) back to the quote price of the assumed CTD bond. Please view my video (below) and let me know, it's my best explainer yet on the concept. Thanks,

patrickbs

New Member
Hi David,

I'm totally familiar with calculation of theoretical T-bond futures price, however since we deduct the accrued interest (AI) from the cash futures price to reach to the quoted T-bond futures price, why in this case we don't use the same approach (i.e. deduct the AI) to calculate a bond futures or forward price (i.e. in the case of a NON T-bond).

Many thanks.

Staff member
Subscriber