How Floating-Rate Spreads Affect a Swap’s Par Fixed Rate

Summary

The document addresses whether a swap’s fixed rate remains independent of a spread deducted from its floating leg, such as LIBOR minus a stated amount. The answer explains that the par fixed rate is set so the swap’s fixed and floating legs have approximately equal present values at inception. The apparent independence comes from an equation that implicitly assumes the floating leg’s value without the spread; once the spread changes that leg’s value, the fair fixed rate must adjust to restore balance.

The practical implication is that the fixed-rate payer cannot treat the floating spread as a separate yield stream while leaving the swap’s agreed fixed rate unchanged and the initial value at zero. The response gives a conceptual valuation explanation and a simplified discounting equation, but no detailed market calibration, schedule conventions, or numerical example. Its conclusion applies to the par rate at inception; subsequent swap values can move as rates, discount factors, and other relevant market inputs change.

Key ideas

• A swap’s par fixed rate is chosen to make the fixed and floating legs’ present values approximately equal at inception.
• A spread deducted from the floating rate changes the floating leg’s value.
• The fixed rate must adjust to account for that spread if the swap is to begin near zero value.
• The simplified explanation does not specify market conventions or later valuation changes.

Tags

# Is the value of fixed swap leg independent of X, where the Floating Rate is say, LIBOR minus X%?


# Is the value of fixed swap leg independent of X, where the Floating Rate is say, LIBOR minus X%?












In my texts of swap valuation, the fixed leg is decided by calculating the following equation, say for a swap agreement where:

Fixed Leg : $s(1)=s(t)$ Floating Leg : 1 year LIBOR - 25bps Term = 2 years

s(t) is calculated as:

$\frac{s(1)}{[1+r(1)]} + \frac{1+ s(1)}{[1+r(2)]^{2}} = 1$

So in effect, s(1) is independent of LIBOR plus/minus X.

Kindly explain, what is the real life implications of it? Does it imply that the payer of the SWAP( one who gives fixed leg) can arrange another cash flow stream with X% yield?

Thank you! Soham

## Answer by PBD10017 (score 1, accepted)

https://quant.stackexchange.com/a/23177

No. $s_1$ is dependent on $X$ in the sense that the value of the swap at inception must equal zero (or close to it). This is what your equation is actually showing. $$ \frac{s_1}{1+r_1} + \frac{s_1}{(1+r_1)^2} = 1$$ The $1$ is also the NPV of the floating leg assuming no spread. Banks calibrate $s_1$ so that the NPVs of the floating and fixed legs are equal or close to equal.

If your swap is receiving $Libor - X$, then $s_1$ will by adjusted to make the initial fair value zero or close to zero so that: $$NPV_{floating \space leg} \approx NPV_{fixed \space leg} $$

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