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Hedging Treasury Curve Positions with DV01 and Modified Duration

Article Quant Q&A · Author: A1122
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Summary

The discussion distinguishes a position’s net interest-rate sensitivity from its exposure to changes in the shape of the Treasury curve. It asks how to size one curve trade against another after each trade has been hedged using Treasury futures, and whether modified duration can measure the resulting hedge. The answer explains that DV01 directly measures dollar price sensitivity to a small yield move, while modified duration expresses sensitivity relative to price.

To match two legs, equate their dollar sensitivities after accounting for each leg’s notional or par amount. When using modified duration, multiply it by market value to get the corresponding dollar sensitivity. This gives a sizing method for a small parallel yield change, but the response does not develop a hedge for nonparallel curve moves or demonstrate that a DV01-neutral position has no curve risk. Its formulas therefore address first-order dollar sensitivity, not every risk in a curve spread.

Key ideas

  • DV01 measures the dollar price change associated with a small yield move.
  • Modified duration is a price-relative measure of sensitivity; multiplying it by market value gives dollar sensitivity.
  • Size hedge legs by matching their dollar sensitivities after accounting for notional and price.
  • A DV01 hedge addresses the yield sensitivity represented in the calculation but does not by itself describe curve-shape risk.

Tags

Fixed incomeFuturesRisk management
Full text
# Curve to curve hedging for treasury


# 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?

## Answer by Helin (score 1)

https://quant.stackexchange.com/a/34532

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.

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.