Interest Rate Swap Tutorial, Part 3 of 5, Floating Legs

Interest Rate Swap Example

For our example swap we will be using the following inputs:

  • Notional: $1,000,000 USD
  • Coupon Frequency: Semi-Annual
  • Fixed Coupon Amount: 1.24%
  • Floating Coupon Index: 6 month USD LIBOR
  • Business Day Convention: Modified Following
  • Fixed Coupon Daycount: 30/360
  • Floating Coupon Daycount: Actual/360
  • Effective Date: Nov 14, 2011
  • Termination Date: Nov 14, 2016
  • We will be valuing our swap as of November 10, 2011.
In the previous article we generated our schedule of coupon dates and calculated our fixed coupon amounts.

Calculating Forward Rates

To calculate the amount for each floating coupon we do the following calculation:

Floating Coupon = Forward Rate x Time x Swap Notional Amount

Where:

Forward Rate = The floating rate determined from our zero curve (swap curve)
Time = Year portion that is calculated by the floating coupons daycount method.
Swap Notional = The notional amount set in the swap confirmation.

In the next couple articles we will go through the process of building our zero curve that will be used for the swap pricing. In the meantime we will use the following curve to calculate our forward rates and discount our cashflows.

swap zero curve

The numbers at each date reflect the time value of money principle and reflect what $1 in the future is worth today for each given date.

Let’s look at our first coupon period from Nov 14, 2011 to May 14, 2012. To calculate the forward rate which is expressed as a simple interest rate we use the following formula:
simple interest formula
where:

forward rate discount factor

Solving for R
forward rate formula

In our example we divide the discount factor for May 14, 2012 by the discount factor for Nov 14, 2011 to calculate DF.

0.9966889 / 0.9999843 = 0.9967046

T is calculated using Actual/360. The number of days in our coupon period is 182. 182/360 = 0.505556

R = (1 – 0.9967046) / (0.9967046 x 0.505556) = 0.654%

Our first coupon amount therefore is:

Floating Coupon = Forward Rate x Time x Swap Notional Amount

$ 3,306.33 = 0.654% x 0.505556 x $1,000,000

Below is a table with our forward rate calculations & floating coupon amounts for the rest of our coupons.

swap forward rates

The final step to calculate a fair value for our complete swap is to present value each floating coupon amount and fixed coupon amount using the discount factor for the coupon date.

Present Value of Net Coupon is
(Floating Coupon Amount – Fixed Coupon Amount) x Discount Factor

interest rate swap

Our net fair value of this swap is $ 0.00 as of November 10, 2011.

So far in this tutorial we have gone through basic swap terminology, fixed leg coupon calculations, calculating forward rates for floating leg coupon calculations and discounted our cashflows to value a swap.

Thanks to our sister company Resolution for providing us with this series of posts.

Next Article: Present value of money & bootstrapping a swap curve

Interest Rate Swap Tutorial, Part 5 of 5, building your swap curve

Swap Curve

In the final article in this series, we will continue to build out our discount factor curve using longer datedpar swap ratesPar 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.

zero curve

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.

par swap rates

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:

1 year par swap rate resized 600

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).

swap bootstrapping

Using our example data:

discount factor par swap

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.

zero curve construction

Thanks to our sister company Resolution for providing us with this series of posts.