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Question re: accrued interest on bond futures

Am hoping someone can help me understand why we don’t deduct AI at expiry when calculating bond futures price in the following problem. 

I understand that there’s no AI at time of entering the contract.  But at time of expiry, i.e., 8 months from now, there should be AI of $3,500 * (2/6), since 2 months will have elapsed since the latest coupon payment. 

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We note the following information for the cheapest to deliver US Treasury bond for the contract; the bond has a face value of $100,000, pays a 7% semiannual coupon, and matures in 15 years. The bond is priced at $156,000, has no accrued interest, and a yield of 2.5%. The futures contract expires in 8 months, and the annualized risk-free rate is 1.5%. There are multiple deliverable bonds, and the conversion factor for this bond is 1.098.

Q. Based on the information provided, the price of the bond futures contract is closest to:

A. 141,234

B. 140,298

C. 146,689

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The official answer is B, and I don’t understand why accrued interest isn’t deducted at time T.

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I have come across the same conceptual challenge. The answer that I have come across is in the below post’s last topic of “bond prices and accrued interest.

http://www.tvmcalcs.com/index.php/calculators/apps/baiiplus_bond_valuation

Basically, bond prices are quoted by dealers without the accrued Interest. I had thought that I would have to add the accrued interest in order to calculate the forward price but since the bond is quoted without the accrued interest hence the forward price calculation would be without it. Can someone else confirm my understanding?

I’m also a little confused why the accrued interest in that formula is discounted by the conversion factor. on the CME documents for delivery mechanisms this isn’t how it works and I think it may be incorrect. There is also another question in that same set that mentions the conversion factor being unrelated to accrued interest. 

The lack of consistency here is pretty annoying.

I just built an interest rate future model for work and I’m 90% sure you’re right that there would be 2 months of accrued interest that should be taken into account.