This is the fourth 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.
 Introduction to Interest Rate Swaps
 Fixed legs
 Floating legs
 Swap Curve building Part I
 Swap Curve building Part II
Zero Curve
In the previous articles we described basic swap terminology, created coupon schedules and calculated fixed and floating coupon amounts. We also present valued our cashflows and calculated forward rates from our Zero Curve. A zero curve is a series of discount factors which represent the value today of one dollar received in the future.
In this article we are going to build up the short end of our discount factor curve using LIBOR rates.
Here are the rates we are going to use. They represent USD Libor as of November 10, 2011.
ON

0.1410%

T/N

0.1410%

1W

0.1910%

2W

0.2090%

1M

0.2490%

2M

0.3450%

3M

0.4570%

4M

0.5230%

5M

0.5860%

6M

0.6540%

7M

0.7080%

8M

0.7540%

9M

0.8080%

10M

0.8570%

11M

0.9130%

Our first step will be to calculate the start & end dates for each of our LIBOR. Our TN settles in one day, and the other rates all settle in two days. We also will need to calculate the exact number of days in each period. Keep in mind that November 12th was a Saturday so our TN rate ends on the Monday, November 14th.
Our formula for converting rates (simple interest) to discount factors is
Where R is our LIBOR rates and T is our time calculated by the appropriate daycount convention, which in this case is Actual/360.
So our first discount factor reflecting the overnight rate is:
which equals: 0.999996083348673.
Bootstrapping
For our subsequent rates, they settle in the future. So when we calculate their discount factors, we will need to discount again from their settle date. See the image below to see the time frame each rate represents.
Because we need the previous discount factors to calculate the next discount factor in our curve, the process is known as bootstrapping.
To calculate the discount factor for TN:
Which equals; 0.999988250138061 x 0.999996083348673 = 0.999984333532754
We continue the process for each time period, to build up the short end of our curve.
Note from 1W onwards the DF from Nov 14 is applied as the final multiplier (in the same manner as Nov 11 was used when calculating DF TN). The reason is that the start date of the 1W period (and subsequent periods) is Nov 14 hence we need to use the DF that ends Nov 14 to build from.
We have shown how to convert LIBOR rates into a discount factor curve, while taking into consideration the settle dates of the LIBOR rates.
Next Article: Building the long end of the curve using Par Swap Rates.
For more information see our pricing plans.
