When valuing a swap, why do we multiply the summation of the discount factors with the difference between the swap rates at initiation and swap rate now ?
Basically how does this formula come about ?
A fixed-for-floating interest-rate swap is essentially the exchange of a fixed-rate bond for a floating-rate bond; because we’re assuming that there is no cash payment at the inception of the swap, the market prices of those bonds must be equal. We also assume that the floating rate is the market rate.
Because the floating-rate is reset to the market rate at each coupon date, the price of the floating-rate bond is par; we’ll assume that par is $1.
The price of the fixed-rate bond is sum of the discounted cash flows: each coupon times its discount factor, plus par times the final discount factor. Furthermore, this, too, must be $1:
1 = C(Z1) + C(Z2) + . . . + C(Z_n_) + 1(Z_n_)
Now . . . the algebra:
1 – Z_n_ = C(Z1) + C(Z2) + . . . + C(Z_n_) = C(Z1 + Z2 + . . . + Z_n_) = C(ΣZ_i_)
C = (1 – Z_n_) / ΣZ_i_
Voilà!
I have a full article on pricing plain vanilla interest rate swaps: http://financialexamhelp123.com/pricing-plain-vanilla-interest-rate-swaps/.
(Full disclosure: there’s a fee for reading the articles on my website; however, I explain pricing and valuing all species of swaps so clearly that you’d be silly not to read the articles.)
Not sure I understand OP’s question correctly but if I do, I have a similar problem.
I understood at swap valuation from 2015 curriculum that we value a swap against the floating rate bond as Magician describes.
But now in this rewritten Derivatives chapter there are examples where we calculate a new swap rate based on the new LIBOR rates (called “equilibrium swap rate”) and the MTM will be the PV of the difference between the original swap rate and this equilibrium swap rate.
At this point I am lost.