Exercise HULL 06.11

[GVD]

Hi David,

I am referring to question 06.11 in the exercises, it is one of the hardest questions in the Hull questions.


It is July 30, 2005. The cheapest-to-deliver bond in a September 2005 Treasury bond futures contract is a 13% coupon bond, and delivery is expected to be made on September 30th, 2005. Coupon payments on the bond are made on February 4 and August 4 each year. The term structure is flat, and the rate of interest with semiannual compounding is 12% per annum. The conversion factor for the bond is 1.5. The current quoted bond price is $110. Calculate the quoted futures price for the contract.

I understand how to get to the dirty price at 30 July: it is adding up accrued interest up to 30 July to the quoted (clean price of 110) subtract the present value of payment of coupon in 5 days (Augsut 4) to get to a dirty price of 112.028 at July 30.

I don't understand the last step where the accrued interest 6.5*(57/184) is subtracted from 112.028 to get a clean price. In the first place I don understand how this future accrued interest of (6.5*(57/184)) sneaked into the current dirty price of 112.08?

Thanks

 

[ShaktiRathore]

Hi
I think there is a typo here (lets David confirm) the dirty price of 112.028 is for Sep 30 not July 30 so we subtract the AI from the last settlement date of Aug 4 til today Sep 30 of 6.5*(57/184) to get the clean price on Sep 30.
Dirty price for july 30 would be clean price of 110+ AI of 6.5*(176/181) and not 102..028
Thanks

 

[Praveen_India]

Hi David/Shakti,

I have a similar doubt too. Please refer to attached.

In cell D20 we calculated Future price as below:

Futures Price=(Dirty Price - PV of coupon) * EXP^(Interest rate * 270/365)

Once we arrive at this price, why is AI for 148 days is subtracted from the Future Price?

Because as per the above formula FP is arrived at by continuously compounding (Dirty Price - PV of coupon) at 10% for 270 days. How did Future price accrue the interest (coupon), that we are subtracting it to arrive at quoted price?

Kindly throw some light.

Thanks,
Praveen

 

[ShaktiRathore]

Hi
Dirty price is the present value of all future cash flows including coupons. (Pv at t=0 be x then pv at t=T at y% is (1+y)^T*x)To arrive dirty price 270 days in future we compound dirty price today and pv of coupon at next settlement at 10% the yield which shall give the pv of all the future cash flows after 270 days in future which is nothing but dirty price 270 days in future. Futures price is used to denote clean price so we subtract AI since last coupon pay and subtract it from dirty price to get clean price which is nothing but futures prive.
Thanks

 

[Praveen_India]

Hi Shakti,

"To arrive dirty price 270 days in future we compound dirty price today and (you meant minus in place of 'and') pv of coupon at next settlement at 10% the yield which shall give the pv of all the future cash flows after 270 days in future which is nothing but dirty price 270 days in future."

Kindly can you elaborate bit more on it. Really appreciate your time.

Thanks,
Praveen

 

[ShaktiRathore]

Pv of coupon is the Pv of coupon on next settlement date (t=270-148=122)i.e. Coupon paying date , if p1 is the dirty price today at t=0 then at t =270 pv of remainin future cash flows is (p1-pv of coupon)*exp(yield*270/365) which is nothing but dirty price by definition, now we have AI since last settlement date for 148 days that we subtract from this dirty price at t=270 to get the clean price.
I hope u got it what i am trying to say.
Thanks

 

[Praveen_India]

Hi Shakti,

Ok, so if we do not subtract the AI from (p1-pv of coupon)*exp(yield*270/365), it will be the Dirty price of bond and if we subtract the the AI, it will be Futures price as well as quoted bond price.

And also i understood that (p1-pv of coupon)*exp(yield*270/365) is nothing but the dirty price of bond 270 days from now, i.e PV of all futures cashflows happening after 270 days calculated at yield of 'r' (as per [(S0-I)*EXP (rT)] or 'yield' as you have mentioned in your formula.

I think I am getting it..

Thanks a lot Shakti. Appreciate your time.

Praveen

 

[David Harper CFA FRM]

Please note the updated source Q&A is located here @ https://forum.bionicturtle.com/threads/hull-06-11.3777
and the correct calculation should be reflected in this XLS @ https://www.dropbox.com/s/y8x1mzu5lp44wn3/hull_06.11.xlsx?dl=0
i.e.,

 

[GVD]

Hi,

Revisiting this, I probably understand the question better now:
If one shorts he future contract and settle in cash on 30 th September to cover the future, the short has to:
Buy now: at 110 + accrues interest of 6.5*(177/182)
Get coupon / dividend with pv 6.49
This is a dirty price....subtract the accrued interest at 30th Sptember to get clean price and divide clean price by CF to get quoted price.