# 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

• Can you please use Latex? It is difficult to understand your equation. Feb 8, 2016 at 17:41

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: $$NPV_{floating \space leg} \approx NPV_{fixed \space leg}$$