# 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.

You don't need to square your book so that there is no net payment anywhere. You do need to hedge the interest rate risk.

Trying to match the fixed rate payments and ending up with a remainder on the floating side is not a good result; it replaces one dependence on future interest rates with a more complicated dependence.

The biggest risk is on the float leg payments; the fixed leg payments are known, so they are not at a great risk of fluctuating in value.

# Ideally, match the notional structure of the swap

You can hedge the floating rate risk by matching the notionals on a series of swaps; suppose your swap notional has 2 steps from A->B and then B->C. Then we can use 3 swaps with notionals C (the lowest, running to maturity), B-C (covering the delta to the second step) and A-B (covering the delta to the first step).

There is a choice, then, on the fixed side of those swaps. Either we do them at the same fixed coupon as the original swap, and accept that we will pay or receive upfront for the nonzero present value, or we trade those at par for zero present value and accept that there will be a nonzero coupon balance at each cashflow date. In theory these are equivalent as you would expect to borrow/lend any upfront PV, but in practise you might have a view on which you prefer.

# But you may need to compromise for liquidity

Hedging with liquid instruments (10y is much more liquid than 8y, for example) is generally more efficient as spreads are tighter. In tandem with that, the market usually hedges less liquid maturities using more liquid ones anyway (8y hedged with 5y and 10y) so it may prove operationally better to instead calculate your amortising swap's sensitivity to shifts in the key liquid swaps (e.g. 1, 2, 5, 10, 20, 30y), and hedge those sensitivities.

Such hedges (and even our ideal hedge) still need monitoring for their adequacy, but usually that is managed by looking at the whole book rather than individual swaps like this. It would, however, feature in the PV calculations for collateralisation.

# Beware dates

Replacing a swap with one maturity with a set of swaps with different maturities can sometimes come undone with small differences in date schedules, particularly around ends of months, so care should be taken not to accidentally end up needing to cover the gap between one swap paying and another receiving.

# Secondary interest rate risk

The secondary (much smaller) effect of interest rates is to change the discounting of the cashflows; a fixed leg still has a small value dependence on interest rates, so you may want to also cover that in some way. This kind of risk, however, is more easily managed by calculating an overall sensitivity and hedging that by adjusting the swap hedge slightly, rather than by trying to eliminate it entirely.

# Beware principal repayments doing this cross currency

Unlike IRS, cross currency swaps usually trade with an exchange of principal. If your cross currency swap amortises, then there will be principal payments at some points in the schedule, which will need to match those in the hedging swaps. These payments are a much bigger deal than coupon cashflows, and forward exchanges are often not at par.

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? Mar 6, 2018 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, 2018 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, 2018 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. Mar 7, 2018 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, 2018 at 16:44