Interest rate swap

[ZebuAlex]

Hoping someone can explain the following in the ActEd notes.
It's says the value of the interest rate swap is
V = B_fl - B_fix
and gives eqns for the values of fixed and floating rate bonds:
B_fix = sum ke^-{r_i t_i} + Le^-{r_n t_n}
B_fl = (L+k*)e^{-r_1 t_1}
This is fine. It then says that the swap rate, k, is set so that the value initially is zero, leading to
k = (1-e^-{r_n t_n})/(sum e^-{r_i t_i})
I'm not seeing how this second step follows from the first. Could anyone enlighten me?
I can see how to derive k if you don't use the formula they gave you, and just rely on interest only payments, but the V formula includes the principle too.

 

[Mike Lewry]

The swap rate will be set at the start of the swap and so we don't need to worry about being in between interest payment dates. So the value of the floating leg (plus notional principal) is just 1, per unit principal.

The value of the fixed leg (plus notional principal) is the value of the coupons at rate k plus the value of the principal.

Setting these two equal and rearranging to make k the subject of the equation gives the expression above for k.

We could deduct the value of the notional principal from both sides of the equation if we wished, but as they just offset each other, it wouldn't change the final expression for k.

 

[ZebuAlex]

Yes, I get how to derive a formula for k...it's just that the notes seem to imply that it 'follows' from the formula they gave for B_fl and B_fix by just setting them equal, but I can't really see how that's the case.

I guess it's not particularly important though. Thanks!