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# Theoretical futures price of a bond vs forward price

#### Shantanu Mantri

##### New Member
Hi, David

We use the dirty price when calculating the Theoretical futures price of a bond and later in the final step subtract the accrued interest to get the quoted price.

3 questions :

1.Why doesnt this dirty price usage and acc int subtraction come when we calculate forward prices for bonds? We simply use the clean quoted price subtract the pvc and solve.

2.Can we just use the clean price to calculate the future bond price and then not subtract the accrued interest in the later step?

3.The CTD which they mention is problems and give a conversion factor to calculate the futures settlement price..... How do they already know the CTD ? Is that just an assumption to make up the problem?

#### ShaktiRathore

##### Well-Known Member
Subscriber
hi.
1. we use dirty price to get the bond value in intermittent period between the coupon dates so that dirty price is quoted price+ accrued interest. But the forward values are calculated based on quoted prices because we have already accounted for the coupon interest earned in future in our calculation as PVC already there is no dirty price that plays role here but only quoted price. No need for dirty price here.
2. yes bond future price= (clean price-pvc)*(1+Rf)^T
3. Yes CTD is used to select cheapest to deliver bond among a set of bonds. this is already given as an assumption CF (conversion factor) ,CTD bond price=Bond price/CF. This cf is sued to deliver CTD bond among a set of bonds .

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Shantanu,
1. I think we do use the cash/full/dirty price, don't we? If it's not always exactly specified, I think "spot price of bond" implies the cash/full price. Hull's method for valuation of the theoretical futures price of a T-bond is basically an application of his cost of carry, F = S*exp(c), and the first step is to determine the spot price as quoted price + AI. (although in many pricing examples/methods, you will see the AI omitted implicitly but that is often, like in the CTD, where it can be symmetrically omitted from both sides; i.e., i.e., CTD = MAX[settle price * CF - quote price] is really CTD = MAX[(settle price * CF + AI)- (quote price + AI)] b/c in this application AIs are identical, but the logic is based on CASH received and CASH paid). You can test it in our T-bond futures pricing XLS and see for yourself
2. I don't *think* so: how will you treat, not the accrued interest already, but rather the upcoming (future) coupon payment? We can't ignore, yet to apply the cost of carry, we seem to need to be using cash spot and deducting the PV of the future (cash) coupon. See below a quote from Tuckman that may be relevant; i.e., the quote price varies by day count, but the cash/full price is the "true" price according to the cash flows. It is difficult to manipulate in quotes because, essentially, they derive from the full/cash price
3. Some questions are lazy and do not carefully impute a CF. The true CF computation is shown in our learning XLS, but as the CF is based on a 6% yield, we can simply approximate the CF by pricing the bond with a 6% yield. Good questions should always do this to produce sensible outcomes; e.g., bonds the pay a coupon less than 6% should have CFs less than 1.0.
From Tuckman, which i like for reminding why manipulation of quote price can lead to mistakes:
"The present value of a bond’s cash flows should be equated or compared with its full price, that is, with the amount a purchaser actually pays to purchase those cash flows. Conceptually, denoting the flat price by p, accrued interest by , the present value of the cash flows by , and the full price, as before, by P,

P = p + AI = PV (1.5)

Equation (1.5) reveals an important point about accrued interest: the particular market convention used in calculating accrued interest does not really matter. Say, for example, that everyone recognizes that the convention in place is too generous to the seller because, instead of being made to wait for a share of the interest until the next coupon date, the seller receives that share at settlement. In that case, by equation (1.5), the flat price would adjust downward to mitigate this advantage. Put another way, the only quantity that matters is the invoice price, which determines the amount of money that changes hands." -- Tuckman Chapter 1

#### Shantanu Mantri

##### New Member
Will read through what you have said and come back to you
David and Shakti just want to thank you guys for the quick responses you'll give.
You'll are doing a wonderfull job! Thanks

#### Shantanu Mantri

##### New Member

1. The spot prices of forwards confracts are full/dirty/cash price.(I misunderstood them to be clean prices)!
2. To give the prices of future bonds in clean prices is a convention, so one needs to add the accrued interest, get the dirty price...use this dirty price with COC model to
calculate the "cash/dirty/ futures price f (0) and then subtract the accrued interest to get the price in the usual convention of quoted/clean price!

Hopefully I have drawn the correct conclusions! Please confirm

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thanks Shantanu, it's actually a tricky topic I think.

1) I do agree with you: I think "spot" price of bond implies the full (dirty, cash) price NOT a flat/quoted price. When the cost of carry is applied in F(0) = [S(0) - I]*exp(rT), where (I) is the PV(coupon), I do NOT think the deduction of (I) implies a flat/clean price. That's not the same as quote = full - AI, just because both formulas deduct a coupon. The coupon is cash on the coupon date, and at the time the coupon pays is the unique moment when the full price = quote price, so it only looks like a quoted price. Instead, the reason spot price is full price is simply that the cost of carry is cash-flow based model linking an FV to a PV: the full price is the price paid "on the spot market" in a transaction based on (discounted) cash flows. Related, I think the presumption is generally the full price; e.g., when we price a bond with the calculator, i myself have found it easy to forget, that's a cash (or full) price, not a quote price. There is a good discussion where i had to fix my XLS, it did not reconcile for a long time until I fully groked that the spot pricing was full: http://www.bionicturtle.com/forum/threads/l1-t3-170-clean-versus-dirty-bond-prices.4561/

1a) caveat: we are generalizing when we refer to spot price. None of this changes that a good question will distinguish between quoted and full price; and that we might accuse a question which simply says "price" of ambiguity or imprecision. All we are doing, IMO, is confirming that Hull's methodology makes sense. For example, what is the "settlement price?" In Hull, it's the quoted price, but i've seen it used for cash/full (and i think i could understand if somebody assumed settlement --> cash)

2) Yes, exactly, I think that summarizes the method! I would only add the final step, a further "complication:" your last step ("to get the price in the usual convention of quoted/clean price!") produces the quoted/clean price of the assumed (a guess, we don't know in advance) cheapest-to-deliver bond, but as the short will receive [settlement price * conversion factor], this quoted price is divided by the CF; i.e., since the seller proceeds = settle * CF, quoted price is estimate of settlement = proceeds/CF.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Shantanu - The lower boards are for paid members: we locate our Q&A database (and related follow-on discussions, errata, links to spreadsheets) in the lower protected boards. It's our key and costly IP so it's *not* free to the public. All paid members should have seamless access, thanks,