Suppose you are structuring a CDO consisting of three bonds. Each of the bonds has a probability of default of 8% over the next seven years and a recovery rate of 0.35. The coupon payment on each of the three bonds is 5% per year, compounded annually, but paid at the end of six years. (Assume that the recovery rates apply to the amounts received at the end of six years for the principal and the compounded coupon.) Suppose also that the underlying assets of the CDO consist of equal amount invested in the three bonds, i.e., for a notional amount of $390 million, with equal investment of $130 million in each bond. The CDO is split into two tranches, a senior tranche and an equity tranche. The interest rate during the 6-year period is constant at 6% per annum.
1. Assuming that the default events of the three bonds are uncorrelated, draw the tree illustrating the various outcomes and the associated probabilities.
2. Draw the cumulative probability distribution of the cash flows at the end of six years from the portfolio underlying the CDO.
3. Suppose the credit rating agency requires the senior tranche to have a probability of loss of less than 2% at the end of the maturity of six years to be given an AA rating. What is the maximum and minimum size of the senior tranche in terms of cash flows at the end of six years? If AA bonds with a six-year maturity are yielding 4% today, what is the present value of the senior tranche?
4. Compute the maximum and minimum payoffs at maturity to the equity tranche, as well as its value today, if there is no arbitrage.
5.Suppose that the three bonds are correlated in the following manner. Bond 1 and Bond 2 are correlated such that, if Bond 1 defaults, the probability of Bond 2 defaulting is 0.6. However, if Bond 1 does not default, the probability of Bond 2 not defaulting is also 0.9. Bond 3 is uncorrelated with Bonds 1 and 2. Redo your calculations for questions 1 to 4 above.
6. Go back to the zero correlation case. Suppose a CDS contract is available on one of the issuer of one of the three bonds, Bond 1. Suppose the price of the CDS is 300 basis points per year for six years, payable on a compounded basis at the end of the six years. What is the maximum size of the AAA tranche now, if this one-third of the total portfolio is fully protected against default and the probability of loss is tightened to 1%? (You can assume that there is no counter-party risk in the CDS contract.)
7. Discuss, without calculations, how your answer might change if the credit rating agency used the expected loss percentage rather than the probability of loss as the criterion for the AAA rating.