# If 10s20s steepener have equal DV01 weighting on each swap then why does convexity play a role in MtM

Receiver Swap 10yrs

Notional: 1,000,000

DV01: +1,300

Tenor: 10yrs

Rate: 4%

Payer Swap 20yrs

Notional: 500,000

DV01: -1,300

Tenor: 20yrs

Rate: 5%

Looking at this fictitious example, I want to understand why this position will generate negative mark-to-market whenever the yield curve is shifted (parallel) up or down? The DV01s of both swaps are of equal magnitude (i.e. notionals have been adjusted in such a way to render both swaps having same DV01 magnitude, correct?), so therfore a parallel move in any direction of the yields will mean that the MtM of the swaps offset each other? Surely, convexity does not play a role because the DV01s are of equal magnitude?

I am confused, and any help is greatly appreciated.

• Unless I'm missing something, it shouldn't (if the DV01s exactly offset, as you say). However, if the swap curve parallel shifts up or down, then the impact of that on the DV01s of the two legs will not be equal and opposite, so then you will start to create a directional bias after the fact as the swap curve moves. But this (I think) should be after the fact. The initial move itself should not (I think) affect the M2M. Sep 8, 2019 at 22:07
• DV01 is only a good proxy for PnL for small yield movements. It's the first order effect of yield changes (first derivative). Actual PnL is not linear in DV01. Sep 9, 2019 at 1:55

## 1 Answer

Even though your two swaps have offsetting DV01s, in general they do not have the same convexity.

Swap (or bond) convexity is analogous to option gamma - it's the change in the delta (in this case, DV01) of the derivative when the underlying value changes, which for your swaps is the par rate.

For a truly parallel shift in rates, say 20bps for example, the DV01 of each swap will change due to its own convexity, and they will not change equally - thus, your DV01s will cease to be neutral and the position will have a delta lean.