# 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!

• What do you mean by a "price"? For example, looking at theice.com/cds/MarkitSingleNames.shtml , we see for the same name different prices depending on the running spread - 25 bps, 100 bps, etc, There is no unique "price". Also, "par spread" is somewhat dated. Oct 11, 2021 at 0:22