# Basis Swap Curves and Pricing

## Basis Swap – Part 2

We have previously covered the introduction to Basis Swaps and answered the question of how basis swaps work. In this article we will delve a bit deeper into the world of basis swaps and try and understand exactly why they are so popular. We will also look at the market conventions for these instruments and have a brief look at some of the curve construction intricacies that come up for these instruments. Note that the main area of concern for us in this article is the interest rate basis and not cross currency basis. We will cover the currency **basis swap** in a future article, for now you can read an article on the cross currency swap here if interested. A free spreadsheet that will be made available showing the methodology of how to calculate the *basis swap* spread from 2 vanilla swaps.

### Basis Swap Spreads

The __basis swap__ spread is the difference between 2 money market rates. That is to say for example the spread between the 3-month Libor rate and the 1-month Libor rate compounded to 3-months. This would referred to in the market place as the 3s1s basis and this basis would generally be positive – that is to say the 1-month rate is lower than the 3-moth rate. This is generally the case as most banks would prefer to lend for 1-month rather than 3-months (this comes back to the liquidity preference theory of yield curves). On a very general level the basis will be downward sloping – that is to say the spread decreases as you up in tenors.

### Basis Swaps – Motivations

In the previous article we discussed that banks use basis swaps to hedge against interest rate fluctuations on their books. That is to say a bank may be funded on one basis and has assets in another on their balance sheet. The risk of course is that the basis widens (or narrows). As a result banks use basis swaps to hedge this basis risk.

This is easier to understand with the example of a Bank that has a pool of loans that yield 1-m Libor + 5%, say over a term of 5 years. This is funded at 3-month Libor + 40 b.p. for the entire term (with a debt issue). There is now a mismatch between the 2 basis here. This can be easily be hedged by entering into a 3s1s basis swap for the term. The bank will now receive 3-month-Libor from the Basis swap and Pay 1-month Libor. The diagram below gives a very nice summary of the net flows.

The above principle would equally apply to the situation of a mortgage portfolio that a bank holds – typically getting monthly repayments tied to 1-m Libor while funding is at 3m. Again a basis swap will allow the risk of the basis to be hedged.

In the corporate world a lot of funding is linked to the 6-m Libor rate, and at the same time the corporates have assets tied to the 3-m Libor. So in a very similar way to above they can use a basis swap to hedge against the basis changing and affecting their bottom line.

## Basis Swap Curves

The next important step in understanding Basis swaps is to understand the fact that there may well be several different curves that you need to construct in order to value a Basis Swap correctly. This has become all the more important since the 2008 crisis when Basis really started coming to the fore as a major risk factor.

Before we talk about building the curves lets initially set out the basic framework. In order to value any swap instrument (even simple vanilla) we should have 2 curves. Quite often people tend to use one curve to get the forecast / forward rates and use the same curve to also discount with. This can lead to problems and is quite incorrect. There is some debate about this whole area and a discussion on this can often be quite circular, but let us highlight some basic principles that should be followed :

- 1 Forecasting curve (with the appropriate Libor – so for a 3m-Libor you would have a 3m-Libor curve)

- 1 Discounting curve – This should be generally built from liquid instruments and be a representation of where you would be funded at. There has been a move at some institutions especially for Money Market books to have a discounting curve that is based on OIS rates. However it would make very little sense to discount an FX Forward book with OIS. So a common sense rule should apply.

So we have established that for IRS in general we should have 2 curves. However when pricing a Basis Swap we need 3 curves. We will need the discounting curve which applies to both legs. If we were pricing a 3s1s we would have to do the following.

- Build a 3m zero curve – this should be reasonably straight forward as the market will have this more or less directly. This will now be called the “Base Curve”

- Next we will get the basis swap spreads for the 3s1s (from quotes in the market) for each of the tenors , ending up with what is called a spread curve.

- We will then build our 1-m zero by subtracting the 1-m spread curve from the 3-m curve to then build our 1-m zero curve.

- We now have 2 curves which we can forecast our forward rates off.

The above is a reasonably abbreviated version of what is a very involved and complicated subject more often than not. The exact nature of what happens depends on a case by case basis – market conventions depending on locations, liquidity of instruments, funding levels for the institution etc. The above is more a very quick guide to how you should go about building your set of curves which you will then price your basis swap off.

## Basis Swap Pricing

Once you have the relevant curves the pricing of the basis swap becomes a relatively trivial task. You would forecast your series of different forward rates, work out the flows and then discount the flows back using the discount curve. On inception the Basis swap should be priced at par similar to a vanilla swap (once the spread is added – so for 6s3s spread would be added to the 3-m fixing). Again you will have to be careful about when the payments occur – but typically for a 3s6s basis swap on USD the 3-m leg would be compounded flat (to 6m) and both flows would then happen at 6-months.

In our previous article we presented an example of using 2 vanilla par swaps to determine the **basis swap spread** for a 3s6s example. Now you can download the basis swap excel spreadsheet example by clicking here. Note that we present both a simple example with no compounding and also an example in the excel sheet where the 3-m rate is flat compounded up to 6-m. You will notice this makes a difference of slightly over 2 basis points to the spread that is calculated.

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