How CDS Par Spreads Relate to Contract Value and Recovery

Summary

The document asks how the market value of a credit default swap with a fixed coupon relates to its par spread and recovery rate. It starts from the intuition that value varies with the difference between the contract coupon and the par spread, while recognizing that this simple picture may fail as recovery assumptions affect the protection payment.

It gives the general risk-neutral valuation relationship: the par spread balances the expected discounted default payment, scaled by loss given default, against the expected discounted premium payments conditional on survival. Thus recovery enters through the loss fraction, one minus recovery, while default probabilities, discount factors, and payment timing also shape the spread. The text is a question rather than a worked explanation; it does not provide a closed-form simplification or specify assumptions under which a straight-line price-versus-spread relation holds.

Key ideas

• A CDS par spread is the coupon that balances the present values of its protection and premium legs.
• The default payment depends on loss given default, represented by one minus the recovery rate.
• Expected default timing, discounting, and premium payment dates also influence the par spread.
• The document does not establish when CDS price varies linearly with par spread.

Tags

# Graph of price of CDS against par spread


# Graph of price of CDS against par spread












I'm new to credit and I'm trying to wrap my head around the following idea. I understand that the par spread $s$ is the value of the fixed coupon payment at which the fixed and floating legs are equal in present value. Therefore, for a given CDS with coupon $c$, I should expect a price of $c-s$. Therefore, if I plot the price against par spread, I should see a linear graph with $y$-intercept $c$ and $x$-intercept $s$; however, I know there behaviour eventually changes due to the recovery $R$. How do I relate $(1-R)$ with $s$? I know $s$ and $r$ are related via $s(t,T)=\frac{V^{\textrm{floating}}(t,T)}{V^{\textrm{fixed}}(t,T)}=\frac{(1-R)\mathbb{E}^\mathbb{Q}\left[P(t,\tau)\mathbb{1}_{\tau\leq T_n}\Big|\mathcal{F}_t\right]}{\mathbb{E}^\mathbb{Q}\left[\sum_{i=1}^n(T_i-T_{i-1})P(t,T_i)\mathbb{1}_{\tau>T_i}\Big|\mathcal{F}_t\right]}$ but I'm pretty sure it should be simpler than this and I'm overthinking. Any help is appreciated, thanks!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.