Valuing a forward during its life

In short

• MTM value of a long forward at time t: $V_t(T) = S_t - F_0(T)(1+r)^{-(T-t)}$. The short's value is the negative of this.
• Discount over the remaining time, T − t, not the original life T.
• A rise in the spot price helps the long and hurts the short. An instant jump at t = 0 changes the value one-for-one: $V_0 = S_0^{+} - S_0$.
• A rise in the risk-free rate lowers the present value of the forward price, which raises the long's value and lowers the short's.
• The value is zero whenever $S_t$ equals the present value of $F_0(T)$; the spot and forward prices also imply a risk-free rate.