Interest-Rate Sensitivity and Credit Risk in CDS Positions

# Do CDS have interest rate exposure?


# Do CDS have interest rate exposure?












For hedging purposes, do CDS have interest rate exposure?

I've thought of CDS as a pretty direct proxy for credit risk, but on the other hand say if interest rates rise it would be harder for corporations to finance their debt and thus defaults would pickup.

To put it in concrete terms, what would you estimate the duration of the on-the-run CDX HY contract? Why?

## Answer by Mehness (score 5)

https://quant.stackexchange.com/a/40533

So as per my comments, the answer is yes. It's all about which leg is dominant, the loss payment, or the premium, if any. If you're short risk (bought protection) vs paying premium, then if the premium leg exceeds the loss leg (you're out of the money/OTM), then you want rates to rise to reduce the premium liability. If you're in the money (paying less than the protection's worth), then a drop in rates is good for you as your expected net loss leg 'asset' (vs the smaller premium leg liability) is less heavily discounted .

To see where this comes from let's just use the simplest possible CDS model. Flat, deterministic hazard rate and interest rate, in a reduced form model. It's amazing how much useful information (credit / rates deltas and gammas / cross-gammas etc) can be demonstrated in this simple setup, and it's totally shift f9 :). The MTM of the CDS (buyer's perspective), ignoring accrued and other diversions is given by$$PV_{CDS}=\lambda(1-R) \int_o^t\lambda e^{-(\lambda+r)t} \mathrm d t-p \cdot \int_o^t e^{-(\lambda+r)t} \mathrm d t$$

Where $p$ is the contract premium, $\lambda$ is the hazard rate, $R$ the recovery and $r$ the interest rate.The first term is the loss leg, the second the premium leg. This gives the MTM to be:$$PV_{CDS}=\left\{\lambda(1-R)-p \right\}\cdot \left \{{\frac{1- e^{-(\lambda+r)t}}{\lambda+r}} \right\} $$ The rightmost parentheses contain the credit risky duration of the CDS. The rates sensitivity is entirely driven by this term. Note that the market observed par spread, $s$, say, is simply given by $s=\lambda (1-R)$-if $p$ is struck at this market par spread, the MTM is zero.

I would suggest you try this formula to see what kind of sensitivities you observe for say 1b blips to $r$ and $\lambda$ (or equivalently spread $s=\lambda (1-R)$) with different market spreads vs the contractual premium $p$. You'll observe that the CDS has zero rates dv01 if on market ($p:=\lambda (1-R)$), +ve dv01 if OTM, and -ve if ITM. This is essentially as mentioned just because you have a derivative asset or liability in the different scenarios. Importantly, the rates dv01 is an order of magnitude smaller than the credit CR01, since the sensitivity to rates, as mentioned, is purely driven by the effect on the credit risky duration term, whereas the $\lambda$ sensitivity appears as a first order driver of the loss leg, which effect is much larger.

Lastly, you should see that the buyer of protection is short credit gamma - more of the risk is priced in, so marginal widenings are less important - and this effect is material and often lost on people new to trading the stuff!

Hope that helps, and would be happy to tie out on CDX HY if you'd like. Cheers.