# Par Rate on an ongoing swap vs a trade startig at spot

Hope you can help me with the following question:

There are two swaps:
- LIBOR 3M vs. fixed 1Y swap, started in the past, has maturity in the future at time X,
- LIBOR 3M vs. fixed 1Y swap, starts at spot, has maturity in the future at time X,
We assume the cashflows of the two trades are paying (and resetting) on the same days. Let's assume the fixed rate is zero for both the swaps.

Question - should the Par Rates on the two swaps be the same and why?

By definition, the par rate is the fixed rate of a swap such that the swap would have an NPV of zero. More specifically, the par coupon rate is the $c_\text{fixed}$ you solve for from the following equation:
$$\sum_{i=1}^n c_\text{fixed} \Delta_i d(t_i) = \sum_{j=1}^m l_j \delta_j d(t_j),$$ where $n$ is the number of fixed payments, $m$ is the number of floating payments, $d(t)$ is the discount factor for time $t$, $\Delta_i$ and $\delta_i$ are year fractions for the fixed and floating legs, respectively, and $l_j$'s are the LIBOR forward rates.