Interpreting Swap-Leg DV01 and Amortization Cash Flows

# in bloomberg, how do they calc dv01 per leg of a IRS


# in bloomberg, how do they calc dv01 per leg of a IRS












note: i am asking about dv01 per leg. i see the fixed leg has virtually all the dv01, floating has hardly any, which tells me they are treating each leg as a bond in some way. it looks like they calc it assuming notional exchange at start and end of each "couponlet" , but, if there is an amortising swap , and notionals are different on each leg , then what are they doing with principal payments? i guess that they must ignore them

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/60126

Perhaps instead of starting out looking at the dv01 (the P&L impact on the entire instrument of a parallal shift of the entire curve - a high-level summary, really), you can drill down and look at the sensitivities of each cash flow to each curve insrument. This way you can see where the interest rate risk is coming from, and how it adds up to the dv01s. If you can't do it in conveniently with a given vendor tool, then you need a better tool (and may want to build one). It's a bit like looking at the net brightness and color of a photograph and trying to guess the details that contributed.

If the interest rate swap has no amortizations, then: if you assume that the legs include a principal exchange at maturity that offset each other, then indeed the fixed leg looks like a fixed-coupon bond, and has almost all the interest rate risk of the swap, while the floating leg looks like a FRN, and has little interest rate risk. If you can further drill down, then you notice that the fixed leg has the largest risk amount coming from its principal payment at maturity, and is most sensitive to the sap rate at its maturity. The floating leg has exactly the same offsetting risk amount from its principal payment at maturity, but here it is offset by the risks from the unset floating coupons. Indeed, the largest interest rate sensitivity of the floating leg (to short-term rate) that is not offset comes from the first coupon, if it is already set. Also if the (non-vanilla) floating leg is paying index+spread, then the spread contributes interest rate risk not offset by anything in the floating leg, but offset by the fixed leg. You can't see this just looking at the dv01s.

If you do not assume principal exchange, then nothing changes at the level of the swap, but the interest rate risk of the legs changes. Without the principal, the fixed leg now has less interest rate risk remaining, coming only from its coupons. The floating leg now has more interest rate risk remaining, also coming only from its coupons, and no longer being offset by the principal. If your tool does not let you see all this, get (or make) a better tool.

If the interest rate swap is amortizing: suppose for concreteness that we start with 10 million notional, amortize 6 million in 3 years, and mature the remaining 4 million in 5 years. Economically, this is exactly the same as a portfolio of two non-amortizing swaps: 6 million maturing in 3 years and 4 million maturing in 5 years. The counterparties exchange coupons on 6+4=10 million notional in the first 3 years and on 4 million notional in the last 2 years. As in the previous paragraphs, if you assume principal exchange, then the fixed leg, looking just like an amortizing fixed-coupon bond, will have most of its interest rate sensitivity at the times of the amortizaions to the swap rates at the tenors of the amortizations, as above. If you don't assume principal exchanges, then the interest rate risk just comes from the coupons. As there is less notional left toward maturity, the interest rate risk from the coupons will be less than it would have been on the original notional. You need to work through a few numerical examples to get a feel for this.

Assuming further principal exchange at the end of each coupon period, as long as they offset each other, would not change the economics of the swap, but I don't see how they help.