Understanding Key Rate Durations in a Swap

Question

As I understand, the Cashflows of a Payer-Swap are nothing else than being long a floating-rate bond and short a fixed-rate bond. So, to calculate the Key Rate Durations of the Swap I should be able to calculate the Key Rates of both bonds and aggregate them accordingly.

Now it is understood, that the Price of a floating-rate bond can be calculated by discounting the value of the first coupon, plus the notional at the first repricing date: $(C+N)*e^{rt}$

So the price of the floating leg is only dependent on the Key-Rate of the repricing date, let`s say 6M. Additionally, my floating leg would lose value in an upward shock.

Now for the fixed-rate bond, I have of course some Key-Rate sensitivity on all my coupon payments, and then a relatively large sensitivity on the maturity date, where I discount my notional.

For the total Swap this leaves me with a really large sensitivity on the maturity date, some on the repricing date, and less on the coupon dates of the fixed leg.

Intuitively this seems wrong to me, since the the notional is never really exchanged in a Swap. Hence, it seems counterintuitive to have such a large sensitivity on the maturity date. Additionally, since I discount the notional of my floating leg, the Swap actually loses value in an upward shock on the 6M-Rate. This also feels wrong, since this is where I get payed in a Payer-Swap.

Can anyone offer some intuitive understanding regarding the Key Rate Durations of a Swap? Or is this not how you would calculate the Key Rate Durations of a Swap?

If my thought process is not understandable, I will add a numerical example later.

Answer 1

I'll try to explain this pretty much all in words.

In swaps, a more common implementation is to calculate your sensitivity to the standardized pillars of the market observable swap curve and or the zero coupon curve backed out from it. The coupon flows for a particular swap may not land exactly on these pillars, so interpolation is needed, but it is much more convenient because you have multiple observable market quotes on screen for standard maturities but not for odd maturities.

With that being said, you observations are correct. You're asking for intuition so let me try to explain what is going on in words.

"Additionally, since I discount the notional of my floating leg, the Swap actually loses value in an upward shock on the 6M-Rate. This also feels wrong, since this is where I get payed in a Payer-Swap."

The overall value of the swap will increase, but If you have 6M fixings on the floating leg, and the floating leg has just fixed, then the payment you receive is fixed for 6 months. That means when rates go up, you're receiving less for 6 months than the prevailing market rate on the floating leg.

However, at the same time, the value of all other future expected coupons will go up, because the future implied 6m floating fixings have increased, but you're still paying the same 6m fixed rate. It's for this reason that a 2Y swap with 6m floating leg will have a "modified duration" much closer to 1.5 than 2 (2-0.5 = 1.5)

Let us talk about the outsize sensitivity to the maturity date. It is true that the notional isn't exchanged, but you can model a swap differently, without the notional, but this still gives you the same result.

Assume a 5 year swap with annual payments. Assume the yield curve is 4% flat, such that all zero rates are 4%. This also means that all forward rates are 4% initially. Assume annual compounding for this simple example.

For this you need to model all the forward rates for the floating leg. 0Y1Y, 1Y1Y, 2Y1Y 3Y1Y and 4Y1Y, calculated as 2Y1Y = (1 + r_3)^3 / 1 + r_2)^2.

Then you draw up the expected floating rate coupons, using the relevant forward rate for each coupon period. The fixed coupon is simply the fixed rate, 4%.

For each period, you subtract one coupon form the other and discount to the present. The sum of these is your PNL.

What happens when you bump zero rate on the 3Y pillar from 4.00% to 4.01% in isolation as a proxy for key note duration? The calculated 2Y1Y forward rate increases to 4.03% , whilst the calculated 3Y1Y forward rate decreases by a similar amount to 3.97%. Due to discounting differences, this generates a small net PNL.

However, on the last pillar, ie. the maturity date, If you increase the 5Y pillar, the 4Y1Y forward rate increases, and the 5Y1Y forward rate decreases. However the 5Y1Y is not part of your contractual swap, hence you get the full PNL increase on the 4Y1Y forward rate, without an off-setting decrease on the following coupon, hence the maturity pillar still generates the largest PNL impact.

I thinks this is the best way I can describe it.