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.

• This is from a class. $P$ is the price of the bond $R$ is the recovery, so they are not the same. – cpage Nov 2 '18 at 2:52