Forward price versus forward value: inception and expiration

In short

• The forward price $F_0(T)$ is agreed at t = 0 and never changes. The forward value $V_t(T)$ is what the contract is worth to one side at time t, and it does change.
• At inception the no-arbitrage forward price makes the contract worth zero to both parties: $V_0(T) = 0$.
• With no costs or benefits of holding the underlying, $F_0(T) = S_0(1+r)^T$: the spot price grown at the risk-free rate.
• At expiration the value equals the settlement amount: $V_T(T) = S_T - F_0(T)$ for the long, $F_0(T) - S_T$ for the short.
• One side's mark-to-market (MTM) gain is always the other side's MTM loss. Futures settle MTM daily; forwards usually settle only at maturity.