Quoted Futures Price

Could someone explain what Quoted futures price is all about? When we use the formula FV of (Bond Price + Accrued Interest) - AI - FVCA and then divide it by conversion factor. Why don’t we compare this value with the Quoted futures price of a comparable. And why is AI added to the Quoted futures price to get the adjusted price when we are subtracting AI from our Quoted price?

In the bond futures market, the part who has to deliver the futures at contract expiration can deliver any equivalent bond, that’s why the conversion factor, to make them “comparable”. The AI is added at initiation cause when you want to buy a bond you have to pay the “full price” (bond price + AI), then at expiration of the contract you substract AI (the accrued interest during the time of the futures contract)