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?


1 Answer 1


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.

Assuming the two swaps have identical future cash flows and are priced on the same curve(s), the required fixed rate (aka par coupon rate) that would produce zero NPV for both swaps must be identical, regardless of their inception dates.

However, a swap struck in the past is unlikely to have been assigned a fixed rate exactly equaling today's par coupon rate. Indeed, you specified that the fixed rate for both swaps is zero, so unless the par rate for this set of cash flows happens to be zero on the pricing date, the two swaps won't be par swaps.

  • $\begingroup$ Helin, thank you a lot for your answer. I also think that the fact that the par rate is the same or not for these two swaps depends on whether we have the same future cash flows on these two swaps. But, i want to confirm with you, please, that my expectation is correct: $\endgroup$
    – Kriska
    Apr 24, 2018 at 10:44
  • $\begingroup$ > swap_1: pays quarterly, started on 1st March 2018 and ends on 1 Sept. 2018 (so two floating cashflows in the future: 1st C-F on 1June, 2nd C-F on 1Sept.). And let's say 1 semi-annual fixed cashflow, which is zero, so we do not care about it. > swap_2: pays quarterly, ends on 1Sept and has the same payment structure: first cashflow is a stub, pays on 1June, second cash flow is full, pays 1Sept. Fixed leg pays on 1Sept, with rate = 0; let's say today is 24th of April. $\endgroup$
    – Kriska
    Apr 24, 2018 at 10:44
  • $\begingroup$ So it should be correct to say that the par rate of the two swaps are not the same (considering the par rate as the fixed rate that would make the NPV = 0 on each of the two swaps). Am i right? Thank you. $\endgroup$
    – Kriska
    Apr 24, 2018 at 10:46

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