How to price an interest rate swap?

A received fixed interest swap receives a fixed interest rate and pays a floating rate.

This is equivalent to investing in a fixed rate bond and issuing a floating rate bond.

A pay fixed interest swap pays a fixed interest rate and receives a floating rate. 

This is equivalent to investing in a floating rate bond and issuing a fixed rate bond.

Let’s consider the value of a received fixed swap:

Vswap=VFIX-VFLT

Since both principals are set to 1 they cancel out each other. What remains are the difference between the coupon payments of both legs. 

The received fixed pay floating interest rate swap is then priced by solving for the par swap rate, the fixed rate such that the price of the fixed rate bond equals the price of the floating rate bond.

We then solve for the coupon (C) of the fixed rate bond in the following equation:

C/(1+r) + C/(1+r)^2+ …+ C/(1+r)^n+1/(1+r)^n= 1

[EQUATION 1]

The right side is the value of the floating rate bond which equals to par (1) only because the floating rate resets at each payment date. The coupon of the floating bond corresponds to the forward rate at the future payment date which it is set at the beginning of the period and paid at the end (advanced set, settled in arrears).

As a consequence the value of the floating bond at each payment date is par value (1).

From [EQUATION 1], we calculate the discount factor for  each period using spot rates so that  DF1= 1/(1+r) , DF2= 1/(1+r)^2 and DFn= 1/(1+r)^n. 

We have now :

C*DF1+ C*DF2+… C*DFn+DFn=1

C*(DF1+DF2+…+DFn)=1-DFn

And finally we get the fixed rate (C) that sets the value of the difference between the fixed and floating leg equals to zero: 

C=(1-DFn)/(DF1+DF2+…+DFn)