# Fair swap rate of an amortizing swap

Recently I came across the problem of amortizing swaps. This is an agreement, where fixed payments and floating payments (e.g. 3-months LIBOR + spread) are exchanged based on a notional that is reduced (=amortized) according to a fixed schedule.

1. Are there good references for a precise formula?

2. Are there simple formulas, maybe approximations to derive a fair swap rate for the amortizing swap from the plain vanilla swap?

3. Are there calculators online or on Bloomberg for such agreements?

Thank you!

• For a precise formula I found this note.
– Ric
Mar 6 '13 at 9:58

If there are no call features that Freddy describes, we might be able to approximate an amortizing swap from vanilla par swap rates.

A 3m Libor + spread swap should price at roughly the par swap + the spread. An amortising swap is equivalent to a series of overlapping swaps of slightly different lengths. Given all the market data for par swaps, then, we could approximate the value of the float leg using a curve generated with the swaps, from which we price the overlapping mini-coupon floating rate notes. The sum of their PVs should give the overall PV of the float leg, for which we can find the equivalent fix leg given the fix leg payment schedule. Since the notional for both legs is reducing in step, this should result in a weighted average of the interpolated swap rates, with the weighting defined by the float leg PVs.

The success of that method will strongly depend on the shape of your yield curve between the market swap points.

Again, this assumes no optionality.

• Thanks for your thoughts. I will think about it. There are no optionalities involved.
– Ric
Mar 1 '13 at 12:16

The following is an excellent Hagan paper (I just love his writing style and approach to explain). It covers amortizing swaps as well. A bit hard to find paper but here is a link:

http://www.docin.com/p-414687649.html

Keep in mind most amortizing swaps have embedded call-features and if I remember correctly the origin dates to the desire to hedge floating rate mortgage books where the mortgage originator or investor pays floating and receives fixed on an amortized principal.