Fixed income futures

F0(T) = (S0 - PVCI 0,T) * (1+r)T

= [B0(T+Y)+ AI0 - PVCI 0,T] * (1+r)T

I don’t quite get this. On one hand we are deducting the present value of coupon income during the term of the contract for calculating future/forward price, but on the other the spot here itself is in dirty price (it includes AI)

Why are we calculating clean futures price with a dirty spot price?

What am I missing?

If accrued interest is included in the bond price (i.e., dirty price) then you don’t need to make adjustments to the futures price for those amounts, though you’ll still need to deduct the present value of the coupon payments (carry benefits) earned over the term of the futures contract. If the clean prices are given, you’ll need to add AI as of the date you’re pricing the future and subtract the AI as of the date the future is set to expire.

Does this mean that futures prices of equities are non-continuous on/just prior to ex-dividend day (they will gap down) whereas bond futures will be continuous due to accounting for the accrued interest earned over the holding period in the futures price?

In the same way that to buy a bond in market you have to pay up for accrued interest (dirty price) but you don’t pay an explicit accrued dividend for equities?