# swaps fixed rate?

I feel i have a decent grasp on this material but was hoping someone could clarify or provide some additional context to the following:

“At initiation of the swap, the fixed rate is selected so that the PV of the floating-rate payments is equal to the PV of the fixed-rate payments, which means the swap value is zero to both parties”

I understand this is an underlying concept to not just swaps, but also forwards and futures…ha, so my grasp can’t be that good, right !?! I can solve for the fixed rate and value a swap, however I would just want to get someone elses take reagarding this.

Does this mean that the fixed/swap rate could be used to discount both the floating and fixed side and would result in the same value at initiation for both ??? Not seeing in the math how the fixed rate applies to the floating rate…

Thanks for any help!!!

It has to be the same as the fixed rate was derived from the floating rate - hence they would be equivalent. If the NP is the same, the value would be the same.

I beg to differ.

Each floating rate payment happens at a different point in time. So each floating rate payment needs to be discounted at the periodic rate. I think there was someone who had worked through an example in the last 1 week or so and posted out here.

Ah, and that someone was using spot rates to estimate cfs. In a swap your coupon is decided by relevant spot rates. For example, 1 year seminannual swap has 2 settlements. The 2nd floating payment is not based on 360 day libor - it is based on 180 libor 180 days from now (hence forward rates). If it did not work, there would be much moolah to be made…

What has to be?

So how exactly does that mean that the fixed rate is such so that the PV of the floating-rate payments is equal to the PV of the fixed-rate payments. Does it theoretically mean that you could discount either cash flows (fixed or floating) by the fixed rate and they would have the same PV ??

If you discount 1) the cash flows by the fixed rate, or 2) the cash flows by your expected floating rates for the life of the swap You should get the same answer.

If you discount both cash flows by fixed rate, you should get the same value (say x). If you discount both cash flows by floating rate, you should get same value (also x). The current spot rate embeds expected future spot rates (aka forward rates which can be computed via bootstrapping). The cash flows for floating should be based on forward rates and not current spot rates.

Awesome, thanks for the help!!!