# 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?

## 1 Answer

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...

• I disagree with that. A bond and a swap with the same fixed rate have the same duration and convexity. The missing principal payment is a red herring because the swap has a floating leg which offsets this effect.
– dm63
Dec 30, 2020 at 23:20
• You're right. My bad, I hallucinated. Corrected my answer. Thanks Dec 31, 2020 at 0:12
• Ok great thanks
– dm63
Dec 31, 2020 at 2:27
• I think it's important to note further that the swap starts at 0 mtm, while one has to pay to buy the bond: borrow money from your treasury and pay interest on this funding leg. If you pay fixed for funding, then the funding leg has interest risk that mostly offsets the bond's interest rate risk. If you pay floating for the funding, then indeed you can ignore the interest rate risk of the funding leg. Dec 31, 2020 at 16:36
• Taking this into account Dimitri, it seems then as if the above argument is not correct. Say I am long the bond, and short the swap (paying fixed) in a way that I believe is duration neutral. If I take into account the IR risk of the funding leg for the bond, which let’s say is fixed, then that would leave me with additional duration exposure, and convexity as well yes? Jan 2, 2021 at 18:35