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Why Identically Dated Swap Cash Flows Share the Same Par Rate

Article Quant Q&A · Author: Kriska
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Summary

The document compares the par rates of two LIBOR 3M versus fixed 1Y swaps: one began in the past, and one begins at spot, with both having the same future payment dates and maturity. It asks whether their par rates should match when priced with a common set of curves.

The answer defines the par coupon as the fixed rate that equates the present values of the fixed and floating legs. If the future cash flows and valuation curves are identical, both swaps require the same fixed coupon for zero net present value, regardless of when they started. The distinction is between that current par coupon and the fixed rate already set on an existing trade: the old contract's coupon need not equal today's par rate, so it may have nonzero value. This conclusion assumes matching future cash flows and consistent curve-based pricing.

Key ideas

  • A par swap coupon is the fixed rate that makes the present values of the two legs equal.
  • Matching future cash flows valued on the same curves imply the same current par rate.
  • A swap's inception date does not alter that result when its remaining cash flows match.
  • An existing swap's contractual coupon may differ from today's par coupon, giving the trade nonzero value.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# Par Rate on an ongoing swap vs a trade startig at spot


# 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?

## Answer by Helin (score 1, accepted)

https://quant.stackexchange.com/a/39417

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.