# How to value a long term interest rate swap if the floating leg is USD-LIBOR

To value an IRS, you require a spot/zero curve. If I am correct this zero curve will be the USD-LIBOR curve. However, if you have e.g. a 10-year swap that you are trying to value 2 months into the contract, you need the spot rates, beyond 12 months (LIBOR only goes up until 12 months), in order to value the cashflows.

My question is, how would you 'practically' value a swap if you only have the USD LIBOR yield curve up to 12 months?, how is the curve extended?

I read that, for the medium term, Eurodollar Futures can be used, and for the long term swap rates can be used. I am not sure how you would construct the curve using these different rates. Specifically, because a swap curve/rate includes the spread, i.e. Swap curve = Libor + Spread, but I am only looking for the LIBOR part of that curve.

summary of question: I need to price a USD IRS. If I am correct I require a zero curve which is essentially the LIBOR curve I am not sure how to construct the USD libor curve beyond 12 months even if swaps rates can be used to construct the long end of the curve, I'm not sure how this would be done as you would require a USD LIBOR-based Swap curve (cannot use swaps based on different reference/floating rates, since we require the LIBOR curve) and you need to somehow extract the spread.

• You don't usually use a mixture of labor rates. You construct a forward curve for the required tenor (e.g. 3m libor) and all else will be based on futures (or FRAs) as well as quoted swap rates. The futures will be on the same tenor, as are the swap rates. Do you need to completely start at zero? No access to any interest rate curves, swap pricer etc? Aug 11 '21 at 11:51
• Please also note that the vanilla IRS curve does not entail a spread; rather the quoted par rate (the coupon on the fixed leg of the IRS) is exchanged versus LIBOR flat, and is chosen in a way to ensure a zero PV at initiation. Aug 11 '21 at 11:57
• @AKdemy, By pricing an IRS, you require the discount factors, which are calculated using the zero rates, in this case, the LIBOR curve that starts at 0 and goes up until the required maturity. The forward curve (e.g for the 3 month LIBOR), is also constructed using the zero rates, i.e. LIBOR curve./ So How would a bank price an IRS if libor/zero curve only goes up to 12 months. You cannot calculate the forward rate or discount rate for e.g. the 3rd year cashflows ect. Aug 11 '21 at 12:07
• @KevinT, Do you mean that ALL vanilla IRS will be priced under LIBOR Flat, and no additional spreads will be added by the counterparty? (LIBOR + 2%) Aug 11 '21 at 12:12
• @Michelle: No, I was referring only the swaps that brokers quote in the market. These are traded often, in large sizes, quotes are very liquid, and maturities range up to 60y. You would (1) use the quoted par rates to construct your curve, and then use (2) this curve to value/price any IRS you want (with our without spread, this is then a tailored OTC instrument). Ad (1): the process of getting zero rates from the quoted par swap rates is usually called bootstrapping. You can find a lot of literature on it and also some very good posts here on QSE. Aug 11 '21 at 12:15

A vanilla IRS on USD libor is fixed float with the float variable being 3m Libor. As @KevinT wrote, there is no spread in market quotes. The fixed leg coupon is quoted in a way that the IRS is "fair" at par rate - which means zero cost at initiation.

As of now (before libor cessation), there exists a liquid swap market for 3m libor swaps which will be the main building block in the long end of the curve. Since your swap (assuming you have a standard Libor swap) will have float 3m tenor, the curve must be built in a way to represent accurately the term structure of this specific tenor. This means you do NOT include any other libor tenors.

Generally, the framework will be as follows.

• short end: the 3m Libor spot rate
• mid curve: Futures (or FRAs but for USD mainly futures)
• long end: swaps quotes

Difficulties in building these curves:

• mid curve: you will need to compute convexity adjustment if you use futures
• long end: the market quotes will be dual curve stripped - hence RFR discounted ; that requires you to strip the curves as such as well

On top of this, it is a bit of an art (rather than science) what futures to include, and when to start using swap rates, and what tenors to include, exclude in curve stripping to ensure a reliable and smooth curve (interpolation will also be crucial, especially for non standard tenors and mark to market).

If you want to value these after Libor cessation as well, you will need to think of the fallback rates, or potential zombie Libor rates etc. You can have a look here and here for some details around this.

Overall, getting this right is not trivial if you are not very familiar with curve construction and market conventions (and have access to all the data that is needed). Frequently, if you have access to reliable swap quotes, you will use some of the major vendors like Bloomberg. Fortunately in this case, you will also not need to worry about curve construction (unless you are very sophisticated) because Bloomberg (or any other similar provider) will simply offer this fully automated for you. ICVS 23 will be the 3m Libor curve (and you can refine the settings to your liking if needed), and SWPM -FXFL USD will be the standard template, where you can also add a spread to your floating leg (since it seems you require that).

SWPM would automatically load the appropriate curves (forward and discount), allow you to modify them if needed, and gives you the option of CSA or not. On top of this, you can modify it in (almost) any way that swaps trade in the market.

Lastly, I think the real question (for you) will be what you really need? It is clear that you need to price this. However, what data and tools do you have access to. If none, it may be worth to start asking (yourself) what the best available solutions are (in terms of your budget, usability, reliability). Of course you can do it the hard way and build it all by yourself. In my humble opinion, there are good reasons why Adam Smith already wrote about division of labour. There is no need to reinvent the wheel.

• AKdemy, I did not know that market quoted swaps were based on LIBOR flat (i.e. no spread). Do you know where I can find supporting documents or references to this statement @KevinT. Aug 24 '21 at 11:52
• ISDA is the common definition. I don't think they state explicitly that there is no spread. It's just clear there is no spread in the way it's quoted (only one rate, which is the fixed leg coupon). Bloomberg also doesn't state that there is no spread. I think it's s bit tricky to mention something that is not there as something else may think there is a credit spread or whatever else. Aug 24 '21 at 20:20
• Bloomberg doesn't even specify if dual curve stripping is using fed funds or SOFR and just writes OIS discounting for libor swaps in the description. I think BBG just implicitly assumes everyone knows it's SOFR. That's why I asked if you really need to build this all from scrap. If you get the data, you most likely have a vendor. They will do it all for you. Unless you are very good (know swap markets and the products you price inside out), it is (almost) impossible to best the standard vendor setting. Of course there are lots of details they don't offer but that's is not the discussion here. Aug 24 '21 at 20:28