Curve to curve hedging for treasury

Question

Please correct my conceptual understanding if needed, but I'm trying to calculate the mod duration of treasury curve pieces when the curves are DV01 hedged.

For example:

DV01 of 10 Year Note is 896.1705 and DV01 of ZB futures is 209.0188.

If I +3 10 Year Note and -13 ZB futures, I am very close to being DV01 hedged.

However, I'm sure the mod duration of this position is not close to 0, as I am taking on curve risk. In what way can I calculate mod duration for the example above?

Thank you.

EDIT:

I guess I'm not expressing myself clearly... I've also changed the title to reflect this. say that I have 2 curve pieces:

First curve: +1 5 Year Note -6 ZN futures

Second curve: +1 10 Year Note -4 ZB futures

ASSUME that they are perfectly DV01 hedged.

How do I find how many of the first curve to hedge the second curve? Obviously 10 Year ZB curve have greater ranges, so what's a fundamentally sound way to hedge this using 5 Year ZN curve?

Answer 1

You can use DV01 or mod duration – they yield identical results.

Your objective is to ensure that the two legs of your trade cancel each other out (in $ terms) when the yield curve shifts by a small amount:

$$ \frac{dP_1}{dy_1}\times \text{Notional/Par Amount}_1 = \frac{dP_2}{dy_2}\times \text{Notional/Part Amount}_2. $$

Of course, $dP/dy$ is just DV01, so if you have determined the notional amount on one leg, it's simple algebra to compute the notional requirement on the other leg.

Alternatively, you can use mod duration:

$$ \frac{1}{P_1}\frac{dP_1}{dy_1}\times \text{Notional/Par Amount}_1 \times P_1 = \frac{1}{P_2}\frac{dP_2}{dy_2}\times \text{Notional/Part Amount}_2 \times P_2. $$

Here, $\frac{1}{P}\frac{dP}{dy}$ is the mod duration. Notice that instead of multiplying by notional, it's now notional times $P$ (i.e., market value) on both sides. But again, if you hold notional amount on one leg the same, you can calculate notional on the other leg – just need to take price into account.