The easiest way to view a plain vanilla interest rate swap is as an exchange of a fixed-rate bond for a floating rate bond. The cash flows from a fixed-rate bond are known. The cash flows from a floating rate bond are not known. The swap is priced at a rate which makes the PVs of these two cash flows equal.

Recall the term structure from the bond pricing basics example is:

0________90________180_______270______360

<--4.00%--

< ----------4.25%--------

< ------------------4.375%-----------------

< --------------------------4.50%--------------------------

Step 1) Price a floating rate bond. Think of a 90-day bond as paying a floating rate of 4%. This is similar to the LIBOR rate or a T-bill rate. What is its price? If we know it will pay 1% floating rate coupon in 90 days, then its price is 1.01/[1+(0.04*90/360] or 1.01 * 0.990099 which is $1 or par value. So at any payment date, the value of the floating rate bond will be equal to par.

Step 2) What fixed-rate would make the PV of a fixed-rate bond equal to the par value of the floating rate bond?

Work the bond pricing equation backwards, use an unknown (R) as the

coupon which will make the fixed bond price at par ($1).

R * [0.990099 + 0.979192 + 0.968230 + 0.956938] + 1*

[0.956938] = 1.0, giving R = 0.011074.

Coupon is paid every 90 days. So, multiplying 1.1074% by 360/90 gives 4.43% as the annual fixed rate that creates a fixed-rate par bond.

The price of a floating rate bond paying 1% (variable) is also par. The 4.43% is the rate at which the two bonds price equally at initial, and is the initial swap rate. Note that pricing is from the term structure (and changes daily). Typically, the dealer charges a commission, so she may let the swap rate be 4.5%.

4) Suppose the swap rate was to pay 5% fixed and receive LIBOR (instead of the 4.43% we calculated above).

a)Enter the swap to receive 5% fixed and pay LIBOR.

b)Issues bonds at 4.43% coupon and invest at LIBOR

c)Realize an arbitrage profit of 0.57%

Therefore, R of 4.43% would have to be the initial fixed rate. (Arbitrage argument).

Can you explain each method and especially for STEP 2 I don't understand how that is the answer I keep getting another can you explain the formula please for each?