8. We consider a reference portfolio of four investment grade names with the following one-year CDS rates: c(1) = 56 c(2) = 80 c(3) = 137 c(3) = 12 The recovery rate is the same for all names at R = 25.

The notional amount invested in every CDO tranche is $100. Consider the questions:

(a) What are the corresponding default probabilities?

(b) How would you use this information in predicting defaults?

(c) Suppose the defaults are uncorrelated, what is distribution of the number of defaults during one year?

(d) How much would the 0–33% tranche lose under these conditions?

(e) Suppose there are three tranches: • 0–33% • 33–66% • 66–100% How much would each tranche pay over a year?

(f) Let iTaxx(t) be the index of CDS spreads at time t, where each name has a weight of .25. How can you calculate the mezzanine delta for a 1% change in the index? (g) Suppose the default correlation goes up to 50%, answer questions (1)–(4) again.