Why is Accrued interest at T taken when contract is expiring before the coupon date ?
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That’s the concept of Accrued Interest. Whenever, we are standing not on the coupon date, the seller of the bond is about to charge Accrued Interest, since, he will not be getting the coupon if he sells the bond. So, the amount he’ll charge will be the Spot Price as well as the Accrued Interest. Same logic applies for AI(T). On the maturity of the futures contract, once again, if the maturity is not a coupon date, seller will deliver the bond, but then again, he’ll not get the coupon, hence, he’ll charge accrued interest once again. That’ll be AI(T). So, the seller will charge F*CF + AI(T).