# Hedging amortising interest rate swap with vanilla swaps

Is it possible to hedge an amortising interest rate swap (linearly decreasing notional) with a series of vanilla interest rate swaps? With the amortising swap originated today at par rate and the vanilla swaps also being at par.

I tried by having a series of swaps equal to the number of fixed leg payments and matching the cashflow on the fixed leg side. This is simple enough, but the end result is series of swaps each with different notional - which means that there is a notional mismatch on the floating leg and while the PVs of both the amortising swap and the series of swaps are the same they are clearly not equivalent.

Equally if we start from the floating side and determine the notionals on the swaps to be in line with the amortisation schedule, we end up with mismatches on the fixed side.

Does it mean that amortising swap cannot be hedged using vanilla swaps? Needing adding bonds to the mix?

I find that a highly surprising result, but cannot identify how to do it.

No, any swap can be hedged by linear combinations of other swaps. You never need bonds to hedge swaps.

Suppose you had a more complex example of uneven payment frequencies on the fixed and floating side of a 2Y IRS:

Rec Fixed side (annual payments at 1% coupon) with notionals 100, then 50.
Pay Float side (semi-ann payments on IBOR) with notionals, 100, 75, 50, 25.


How do you hedge this?

Pay a 2Y Ann-Semi IRS with notional 50 at 1%.
Pay a 1Y Ann-Semi IRS with notional 50 at 1%.

You are left with two periods not quite matched on the float side so:

Rec a 0.5Y0.5Y Semi-Semi IRS with notional 25 at 0%
Rec a 1.5Y0.5Y Semi-Semi IRS with notional 25 at 0%.


The last two swaps are forward starting IRSs. The coupon of 0% negates the impact of the fixed leg. Note if the coupon frequencies matched you would not even need to have these offsetting elements.

Market makers don't do this of course, they aggregate all swaps in the portfolio and put hedges in place that mitigate most risks but there will always be some small residual elements since it is not cost effective to try and hedge every single cashflow.

• The example will only work for perfectly flat curve though. Normally amortising par rate is not the same as par swap rate, which means that if notional matches then the cash flow doesn't or vice versa. Am I missing something? – NotCaring Mar 6 '18 at 22:59
• If you insist that your swap hedges are done with at-market coupons then you cannot perfectly hedge your amortizing swap. If you are allowed to select any coupon (so there would be an upfront payment to compensate for the off market coupon) then you can do it. – dm63 Mar 7 '18 at 10:55
• @NotCaring the amortisation schedule is always known at trade time, my amortised notionals are examples. Whatever other amortisation schedule you proposed I could tweak those given notionals to reflect the same underlying amortised swap. dm63 is right, although if you insist on using at-market coupons you cant hedge really any swap 'perfectly' since you'll always be left with some form of annuity from residual fixed cashflows. – Attack68 Mar 7 '18 at 12:40
• @Attack68 I see that I missed that in my question - the question was about hedging using par coupons when the swap is at par as well. Hence my confusion. Thank you for pointing that out. – NotCaring Mar 7 '18 at 16:13
• You can generally hedge the underlying risks though just not match the cash flows. Eg a trader hedges the full delta risk of a 1mm$cashflow in 1y by paying fixed in 1mm$ versus OIS for 1y and the PV of the cashflow is locked in, i.e. hedged. This scenario does not even contain any (cross) gamma, which is why swaps are considered linear derivatives – Attack68 Mar 8 '18 at 16:44