Approximating the Fair Fixed Rate of an Amortizing Swap
Summary
The document asks how to determine the fair fixed rate of an amortizing swap, whose notional declines on a schedule. The response describes an approximation for swaps without embedded call features: use market par swap rates to build an interpolated curve, value the floating leg as a series of overlapping short instruments, and sum their present values. Then find the fixed-leg rate that gives an equivalent value under the fixed payment schedule.
Under this method, the amortizing swap rate can be viewed as a weighted average of interpolated swap rates, with weights tied to the floating-leg present values. The result depends strongly on the yield curve’s shape between quoted market swap maturities, so interpolation choices and available market data matter. The discussion is qualitative and provides no formula or numerical example. It also cautions that many amortizing swaps include call features, which introduce optionality and make this approximation unsuitable without further modeling.
Key ideas
- An amortizing swap reduces its notional according to a schedule while exchanging fixed and floating payments.
- Without call features, its fair rate can be approximated using an interpolated curve built from market par swap rates.
- The floating leg can be valued as a series of overlapping instruments, then matched by the fixed leg.
- The resulting rate is a weighted average of interpolated swap rates, with weights related to floating-leg present values.
- Curve shape between quoted maturities and embedded optionality limit the approximation.
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Full text
# Fair swap rate of an amortizing swap # 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. - Are there good references for a precise formula? - Are there simple formulas, maybe approximations to derive a fair swap rate for the amortizing swap from the plain vanilla swap? - Are there calculators online or on Bloomberg for such agreements? Thank you! ## Answer by Phil H (score 2, accepted) https://quant.stackexchange.com/a/7407 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. ## Answer by Matt Wolf (score 2) https://quant.stackexchange.com/a/7380 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.
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