# Why z-spread differs from CDS spread in 1 period example

Suppose we have a 5% (paid-annually) coupon bond with 1-year to maturity. We also have a 1-year CDS with a single payment paid annually (running spread with zero upfront). Assume that the underlying credit can only default at maturity ($$t=1$$). In case of default, the bond recovers $$R = 0.40$$ $$\times$$ (principal + pre-petition interest = \$105). The 1-year risk-free rate is 3%. The 5% risky bond is trading at \$90.

The YTM of the risky-bond is 16.67% $$\implies$$ z-spread = 13.67%. The implied probability of default is $$P(\tau\leq1)=\frac{N * DF(1) - P}{N * DF(1) * (1-R)}$$ where $$N=105, DF(1) = \frac{1}{1+r} = 0.971$$, and $$P = 90$$. We calculate $$P(\tau\leq1)=19.52\%$$. The CDS spread $$S=(1-R)*P(\tau\leq1)=11.7\%$$.

Why does CDS spread not equal the z-spread of the bond? The basis here is ~2%. I think this comes from the bond price being below par. For the bond, loss in default is only $$P-R*(F)$$ not $$1-R$$, but am trying to better quantify and understand this basis. I was reading about adjusted z-spread, but not sure if that applies here.

## 1 Answer

The CDS spread costs you 11.7% in order to ensure that the holder gets the remaining 60% of principal and interest in return. In the end, the payment you are getting in default is 60%-11.7% = 48.3%. The CDS payment you would need to ensure you get the risk-free rate in both scenarios (90*1.03=92.7) is 12.3. Note: 105-12.3 = 92.7. Additionally, this would give a payment of 50.7 in the event of default; note the ratio of 12.3 to 50.7 is the same as the 11.7 to 48.3). Now note, that 12.3/90 = 13.67, similar to the OAS of the bond.

You calculated everything correctly, you just didn't match the magnitude required to offset all risks.

Previous post: I’ll give a shorter, non-computational response now and may expand if I have time later. I’m short, the probability of default and it’s resulting recovery rate does not usually equal the probability of a CDS credit event and it’s resulting recovery rate. Accordingly, the spread from a CDS and bond are inherently different.

Is this a real life situation, or a problem from a textbook where Recovery and Probability are assumed to be the same between CDS events and bond defaults?

• This is from a class. $P$ is the price of the bond $R$ is the recovery, so they are not the same. Commented Nov 2, 2018 at 2:52
• Sorry, by P, I meant probability. If it's for class, real world differences don't appear under simplified assumptions. I'm currently thinking that the primary difference comes from CDS spreads not being a component of yield at all - all payments here happen at time T. Firstly to reconcile, you have to pay the premium whether the company defaults or not, so in order to get the remaining 60% in recovery, you would need to pay additional premium (if you pay 11.7%, you'll only get 60%-11.7% = 48.3% in event of default). Commented Nov 2, 2018 at 20:34