Bond Futures

I’m just reading through SS61 on futures pricing, and I have found bond futures pricing confusing. In the reading, it says that the future price of bonds is the current price less the pv of coupons to be paid between now and the futures expiration (time T), using the rf rate as discount rate. Intuition tells me that you could just calculate the price of the bond at time T using the bond’s yield, since you will have the maturity, coupon payments, and face value. Can someone please explain to me if my model has any mistakes? Thanks in advance.

Edit. the future price should be the future value of (PV of Bond - pv of coupons)