Pricing a futures contract at inception

In short

• A futures contract is a standardised, exchange-traded forward commitment. Like a forward, its value at inception is zero: $V_0(T) = 0$.
• With no costs or benefits of holding the underlying, the futures price is $f_0(T) = S_0(1+r)^T$, identical to the forward price $F_0(T)$.
• For indexes (equity, bond, commodity, credit) and FX, continuous compounding is preferred: $f_0(T) = S_0e^{rT}$.
• With income I and costs C in present value terms: $f_0(T) = [S_0 - PV_0(I) + PV_0(C)](1+r)^T$. Costs raise the futures price; benefits lower it.
• A futures price well below the level implied by cash costs and benefits hints at a convenience yield.