1. We consider a reference portfolio of three investment grade names with the following one-year CDS rates: c(1) = 116 c(2) = 193 c(3) = 140 The recovery rate is the same for all names at R = 40. The notional amount invested in every CDO tranche is $1.50. Consider the questions:

(a) What are the corresponding default probabilities?

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

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

(d) How much would a 0–66% tranche lose under these conditions?

(e) Suppose there are two tranches: 0–50% and 50–100%. How much would each tranche pay over a year if you sell protection?

(f) Suppose all CDS rates are now equal and that we have c(1) = c(2) = c(3) = 100. Also, all defaults are correlated with a correlation of one. What is the loss distribution? What is the spread of the 0–50% tranche?