Why Matched Treasury Bonds and Payer Swaps Have Similar Rate Sensitivity

Summary

The document offers an intuition for comparing the duration and convexity of a Treasury bond with those of a payer interest rate swap having a similar fixed rate and maturity. It frames the difference between the bond and the swap as a floating-rate bond: when the fixed coupons are matched, the remaining exposure is the principal payment paired with the swap’s floating leg. Because a floater has little sensitivity to rate changes, the difference between the two instruments has close to zero rate sensitivity.

Under this simplified comparison, the bond and payer swap therefore have approximately the same DV01 and convexity. The explanation is a rule of thumb, not a full valuation treatment. It depends on the instruments and cash flows being suitably matched; the document does not discuss adjustments for conventions, discounting, funding, or other market details.

Key ideas

• Matching fixed cash flows makes the bond-minus-payer-swap position resemble a floating-rate bond.
• A floating-rate bond has relatively little sensitivity to changes in rates.
• The matched Treasury bond and payer swap therefore have similar DV01 and convexity in this framing.
• The comparison is a simplified intuition and does not address market or contract-specific adjustments.

Tags

# Gamma/Convexity of a Swap vs a similar bond


# Gamma/Convexity of a Swap vs a similar bond












As a rule of thumb, how would the duration and convexity of a 30y UST bond paying X% compare to the duration and convexity of a matched maturity vanilla interest rate swap, with a similar fixed rate.

Will have a lower duration and higher convexity than the corresponding swap? Intuitively, why would this be?

## Answer by Soumirai (score 1, accepted)

https://quant.stackexchange.com/a/60258

Intuitively, the difference between your UST and a payer swap with same coupons is a floating-rate bond. The coupons cancel out, and you're left with principal payment and the floating leg of the swap. This floater bond has close to no rates sensitivity. Put differently:

$\frac{d}{dr}(Bond - PayerSwap) = 0$

So

$\frac{d}{dr}Bond = \frac{d}{dr}PayerSwap$

Both the bond and the payer swap have the same DV01 and convexity...

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.