# Curve to curve hedging for treasury

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?

## 1 Answer

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.

• Hi, thank you for your response.. I have a better picture now. I've added more details about my questions to make it clear that I'm asking about curve to curve hedging.. – A1122 Jun 3 '17 at 0:48
• TBH, now I don't understand your question at all. If they're DV01 hedged, isn't it done? Can you clarify what your goal is? – Helin Jun 3 '17 at 1:18
• if I'm long one 10 Year Zb curve, how many 5 Year Zn curve do I need to short to "curve hedge" 10 Year Zb? – A1122 Jun 3 '17 at 1:26
• It's just the ratio of the two DV01s. If 10-year DV01 is 81.2 & 5-year DV01 is 51.2, then for each 10-year contract, you need 81.2/51.2 = 1.59 of 5-year contracts. – Helin Jun 3 '17 at 1:33
• I'm not trading 5s10s, which would use the hedge ratio you provided. I don't think I'm explaining it correctly. Please excuse me and let me try again. Say that I am long 1 curve which is +1 10 year note - 4 ZB futures. Ok, this is the first curve. I don't like the curve risk associated with this, and I would like to hedge (short) another curve, say 5 year ZN. What ratio of 5 year ZN to 10 year ZB should I use? So I'm essentially creating a 4-legged spread, hedging 10 year ZB with 5 year ZN. – A1122 Jun 3 '17 at 1:35