# 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?

• An anecdote: In 2007, 2008 and 2009 the corporate default rate went from 0.37%, to 1.80% and 4.18% (the peak). Yet in each of these 3 years the interest rate on 10 Yr treasuries was lower than the previous year as the economy sank into recession. – Alex C Jun 23 '18 at 3:15
• Yes it does, but it depends how off market it is. Clearly if you are paying a massive premium stream relative to the credit risk you want rates to rise and vice versa. This is however very second order relative to the credit 01 due to the loss leg's contribution. If I get a moment will write a proper answer with a shift f9 formula for ya ;) – Mehness Jun 27 '18 at 17:17
• in extremis - if the credit has a hazard rate of zero, you just have the annuity's dv01 – Mehness Jun 27 '18 at 17:19

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.