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This is the fifth in a series of articles that will go from the basics about interest rate swaps, to how to value them and how to build a zero curve.

1. Introduction to Interest Rate Swaps
2. Fixed legs
3. Floating legs
4. Swap Curve building Part I
5. Swap Curve building Part II

Swap Curve

In the final article in this series, we will continue to build out our discount factor curve using longer dated par swap rates. Par Swap rates are quoted rates that reflect the fixed coupon for a swap that would have a zero value at inception. 

Let look at our zero curve that we have built so far using LIBOR rates.

We are now going to build out this curve out to 30 years using par swap rates. These rates are as of Nov 10, 2011, and reflect USD par swap rates for semi-annual LIBOR swaps. The daycount convention is 30/360 ISDA.

Also keep in mind that these rates reflect the settlement conventions, so the one year rate is for an effective date of Nov 14, 2011 and termination of Nov 14, 2012. If we were to price a one year swap from the curve we have built so far, we can derive the 6mo discount factor, but we are currently missing the 1year factor. Since we know the swap should be worth par if we receive the principal at maturity, then the formula for a one year swap is:

Notice that the T's would be adjusted for holidays & weekends and are calculated using the appropriate discount factor. We can rearrange our formula to solve for df(1year).

Using our example data:

We calculate the missing discount factor as: 0.99422634. But, this for a swap which settles on November 14th, and we are building our curve as of November 10th. So we need to multiple this by the discount factor for November 14th to present value the swap to November 10th. So the discount factor we use in our curve for Nov 14, 2012 is 0.9942107.

We continue by calculating discount factors for all the cashflow dates for our par swap rates. The next step is to calculate the discount factor for May 14, 2013. Our first step is to calculate a par swap rate for this date as it is not an input into our curve. We linear interpolate a rate between our 1 year and 2 year rates. 

1.5 year par swap rate = 1 year + (2 year - 1 year)/365 x days 

= .58% + (.60%-.58%)/365 x 181 = 0.589918%

We now can solve for the missing discount factor, continuing our bootstrapping through the curve.

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