# Modified duration of treasury futures tracking CTD?

If I know TYU7 contract's CTD is T 2.500 05/15/2024 with modified duration of 6.37. I know futures DV01 is calculated by taking the CTD's DV01 divided by conversion factor as shown here. What is the modified duration of TYU7?

I tried backing out the Mod Dur based on the formula for calculating Mod Dur using DV01 for cash instruments, which is $$Mod Dur = \frac{DV01}{0.01*0.01*Price}$$

Given the future's DV01 and its price, this gave me a Mod Dur for futures which is the same as Mod Dur for cash CTD. Kinda weird, isn't it?

## 1 Answer

Let's make a simplifying assumption that futures perfectly track their CTDs, then

$$D_\text{mod, fut} = \frac{1}{f}\frac{df}{dy} = \frac{1}{F_\text{CTD} / \lambda_\text{CTD}} \cdot \frac{dF_\text{CTD} / dy}{\lambda_\text{CTD}} = \frac{1}{F_\text{CTD}}\frac{dF_\text{CTD}}{dy},$$ where $f$ is the futures price, $F_\text{CTD}$ is the CTD's forward price, and $\lambda_\text{CTD}$ is the CTD's conversion factor. So you're right; under the assumption that futures track its current CTD, then its modified duration is identical to the CTD's (forward) modified duration.

However, I cannot emphasize how dangerous the assumption can be. In the current environment where CTDs have 100% delivery probability into the futures, you're fine. But in other environments, multiple underlyings can be contenders for CTD status. In these cases, futures duration can be very different from the current CTD. The proper thing to do is to build a futures model that account for delivery options and calculate risk metrics such as option-adjusted DV01/duration.