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Why Long-Dated Swaps Concentrate Key-Rate Risk Near Maturity

Article Quant Q&A · Author: black_pde
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Summary

The document explains why a long-dated fixed-to-floating interest rate swap can show most of its index-curve sensitivity at its final tenor, even in a multi-curve setup. It frames a vanilla swap as a combination of a fixed-rate bond and a floating-rate note. Because floating coupons reset periodically in line with prevailing rates, the floating leg’s overall DV01 is described as close to zero. The fixed leg does not have the same offset, leaving much of the swap’s duration and risk associated with principal repayment at maturity.

The answer says key-rate sensitivities at other curve tenors may be small rather than exactly zero, depending on curve construction and swap details. It offers a rough DV01 scaling heuristic for long swaps, but explicitly presents it as approximate and dependent on assumptions such as rates near zero. The exchange points to a separate proof without reproducing it, so the intuition is useful but does not fully derive the multi-curve sensitivity result.

Key ideas

  • A vanilla fixed-to-floating swap can be viewed as a fixed-rate bond combined with a floating-rate note.
  • Periodic floating-rate resets make the floating leg’s overall DV01 close to zero in the stated intuition.
  • The fixed leg leaves much of the swap’s duration and sensitivity associated with its maturity payment.
  • Other key-rate sensitivities may be small rather than exactly zero, depending on curve and trade construction.
  • The DV01 scaling example is only a heuristic under stated assumptions.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# Why would a 15Y swap index=EUR3M and discount=OIS, show only a EUR3M-delta at 15Y


# Why would a 15Y swap index=EUR3M and discount=OIS, show only a EUR3M-delta at 15Y












When computing the index-delta for a swap in a multi-curve framework, only the last cash tenor seem to show sensitivity. Could anyone explain with formulas why it is the case ?

For example a 15Y swap with index curve=EUR3M and discount=OIS, the EUR3M-delta will show zeros for all tenors except the 15Y.

Anyone could help with formula and also an intuition?

## Answer by oronimbus (score 3)

https://quant.stackexchange.com/a/53124

Regardless of single- or multi-curve framework, you can always think of a vanilla, fixed-to-float interest rate swap as a linear combination of a long (short) fixed rate bond and a short (long) floating rate note. The floating rate note has an overall DV01 of close to zero since the coupons adjust periodically alongside the discount factors. Since you don't have this offsetting effect on the fixed rate bond, most of the risk/duration/DV01 is concentrated on the redemption date as the principal is paid back. Depending on how you build your curve and the swap is set up, the key rate risk will not be exactly zero on each tenor, but close to it.

If rates are close to zero the DV01 scales roughly with a factor of 1`000 on 10MM notional: for example, 10y swap ~ 10k DV01, 20y swap ~ 20k DV01 etc. Again this is more of heuristic.

A formula and proof for DV01 can be found here.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.