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# Page 88 -- Calculating Theoretical futures price for a TBond Futures contract

#### PortoMarco79

##### New Member
Subscriber
Hi

Hi @David Harper CFA FRM

Was working on this one last night.

PV of the coupon, $5.803. Could not get this. Got$5.73x and tried again and calculating $5.85x What am I doing wrong here? I am assuming I am doing something wrong here with the day counts. thanks #### bpdulog ##### Active Member On page 88, we have the following info: -Semi-annual payments -12% coupon -10% interest rate, flat term structure -Next coupon in 122 days -Last coupon paid 60 days ago -Total coupon period = 182 days=122+60 All we are doing is discounting the next$6 coupon to the time since the last coupon that has elapsed, here is the formula for Excel:

=6*EXP((122/182)*-0.05)=5.8022338

#### PortoMarco79

##### New Member
Subscriber
That is so strange. I kept on inputting those figures and getting different numbers. I will try again later.
Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thanks @bpdulog, totally correct!

Here is the XLS, in case it helps with this tedious calculation: http://trtl.bz/0401-hull-6-2 ##### Active Member
Hi @David Harper CFA FRM @bpdulog
I have a couple of questions on this problem statement:-
1) Since the Last "Quoted Futures Price" = $82.04 is obtained by dividing the Quoted Price/ CF = 114.859/1.4, shouldn't the last term be really called the Settlement Price instead of "Quoted Futures Price" ? 2) To calculate the F= (S-I) e^rt, we are effectively adding the 60 (accrued interest) which is the first 60Day Coupon payment to the Spot price of$115 and subtracting the PV 2nd Coupon payment for 122 Days of $5.803. My question is why are we adding the 1st Coupon Payment amount(for 60days) and subtracting the 2nd payment amount for 122days. Isn't out Spot price$115 and should we not be subtracting both $1.978(60day AI) and as well as$ 5.803..? instead of 115 + 1.978- 5.803....?

#### ShaktiRathore

##### Well-Known Member
Subscriber
regarding your second(2nd) question i would clarify ,at present i am 60 days away from last coupon paid and 122 days before the next coupon is paid,so that the accrued interest earned till present since the last coupon paid is=(60/182)*(12/2)=1.978,and therefore the dirty price today is=current Quoted price+accrued interest=115+1.978=116.978.We now calculate the future price as of today on this Bond with futures maturity=time from today till delivery =270 days=T,interest rate=r=10%=0.10,we subtract the present value of the coupon to be paid 122 days from today from the current dirty price since the payment is done from the asset Bond during the maturity only and this shall decrease the value of futures.value of next coupon after 122 days=6, its present value as of today=6*exp(-0.10*122/365)=5.803,Now use the formula with S=Dirty price=116.978,I=PV of next coupon paid during the maturity period=5.803,r=0.10,T=270/365)F= (S-I) e^rT to calculate the value of the futures as of today=F= (S-I) e^rT=> F= (116.978-5.803) e^(0.10*270/365)=119.711.
thanks

##### Active Member
@ShaktiRathore Thanks so much for putting this together. I am further having trouble with the fact that we are dividing the Quoted Price / CF and still calling the outcome Quoted Price....how can Quoted Price be = Quoted Price /CF ...? isn't Quoted Price/CF= Settlement price...? So isn't the 82.04 = the Settlement Price...?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber

Your first question is a good one, I always get confused here. I just realized (remembered) that we've discussed this example, in some detail, four years ago here at https://www.bionicturtle.com/forum/threads/determining-the-theoretical-future-price.5558
(I am linking because with Aleks helped me understand this further than I had previously....)

Per the diagram, I break this pricing exercise down into two steps:
1. Applying the cost of carry, where the underlying commodity is the (assumed) CTD bond; in the example, cost of carry tells us that the theoretical futures price of the CTD bond is $119.711 (dirty) or$114.859 (clean equivalent) (side note: should I be saying subtract AI?).
2. Translate a theoretical futures price of CTD bond to the theoretical price of the future contract. So, to me, the final \$71.79 is a theoretical price of the futures contract based on a COC calculation that determines the associated theoretical price of the underlying (commodity) CTD bond.
I don't think it's a settlement price. The analogy is, say, corn commodity. We observe futures prices for, say, Dec 2016 corn. The cost of carry allows us to calculate a theoretical price of the dec 2016 futures contract, which can be compared to an observed futures contract price. I guess we could think of this as something like "the theoretical estimate of the contract settlement price," but the meaning of "settlement" is very much different than Hull's deliberate use of "theoretical." (theoretical implies a value based on a model such that non-fundamental factors are specifically omitted, while "settlement" denotes traded price, to me, at a minimum). I certainly get stuck on this, too but I hope my musing is helpful!

@David Harper CFA FRM Thanks so very much for pointing me to the other thread. Seeing the piece you discussed and arrived to Futures price ~= (price of the cheapest to deliver bond)/(CF of cheapest to deliver) helped iron out the fuzziness in my head on the last "standardization" step...as to why we were dividing by the CF instead of multiplying by the CF...and more so the confusion around the fact if indeed if we were dividing by the CF, what the resulting outcome was. Thanks so much for shedding light on this. There was indeed so much to learn on Fixed Income instruments...!! phew...  