For pricing Treasury bond futures, the curriculum only gives one “version” of the formula, where the future value of coupon payments is subtracted from the futures price. In contrast, the bond forward formula has two versions, one in which the present value of coupon payments is subtracted from the spot price, and the other version where the future value of coupon payments are subtracted like in the futures formula.
Is there a reason why there is only one version of the formula for T bond futures?