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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

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  • $\begingroup$ Can you please use Latex? It is difficult to understand your equation. $\endgroup$
    – Gordon
    Feb 8, 2016 at 17:41

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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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  • $\begingroup$ Can you guide me to any derivation of NPV at origination of the floating leg? I think that will clarify in my mind a lot of the questions I have. $\endgroup$
    – Soham
    Feb 9, 2016 at 4:28
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    $\begingroup$ By NPV I simply mean the fair value of the swap on the trade date. Discount floating leg bearing libor with a Libor/Swap curve will lead to the floating leg having fair value (NPV) of 100 (or 1). The fixed rate on the fixed leg is set so that NPV of the fixed leg is also 100. The net of these two legs equals zero. Which is what the theoretical value of the swap should be at trade date. In practice banks introduce various spreads (profit, funding, credit risk, ...) making the swap a liability at inception to the counterparty, but that's a different topic. $\endgroup$
    – PBD10017
    Feb 9, 2016 at 5:20
  • $\begingroup$ Ah okay. I get it now. So the floating leg will be a hypothetical bond with coupon rate as "LIBOR +spread" discounted at LIBOR ? If that is so, then the RHS will be less than 1. $\endgroup$
    – Soham
    Feb 9, 2016 at 10:57
  • $\begingroup$ @Soham No. If you can get LIBOR+spread but discounted with just LIBOR, it's obviously good for you. The NPV will be more than 1. $\endgroup$
    – SmallChess
    Feb 10, 2016 at 3:44
  • $\begingroup$ I am sorry, yes you are right. I was processing two different things together. I meant RHS will be less than 1, in the context of this question where the coupon is LIBOR-x%. Thanks a lot, it answers all the question in my mind. $\endgroup$
    – Soham
    Feb 10, 2016 at 10:17

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