Your bank has been approached by an important client to structure a CDO, using three bonds: A, B and C. You are given the following information:

1. The size of the CDO pool should be $100 million, with 50% invested in bond A, and 25% each in bonds B and C respectively.
2. Securities backed by the pool are to be divided into a senior tranche and an equity tranche.
3. Each bond in the CDO has a probability of default of 7% over the first five-year period. If a bond does not default in the first five years, the probability that it will default in the next five years is 3%. The defaults of any two bonds are independent of each other.
4. The coupon payment for each of the three bonds is 6% per year, compounded annually, but paid at the end of ten years.
5. In the case of default, the loss given default of each bond is 60% at the end of the ten-year period. Assume that this rate is applied to both the principal and the compounded coupon received at the end of ten years.

1.Draw the tree illustrating the outcomes and the associated probabilities at the end of five years. (6 points)

2.Draw the tree illustrating the outcomes and their probabilities at the end of ten years. How does this tree differ from the one in Part 1)? (6 points)

3.How many different outcomes are there in the cumulative probability distribution of the cash flows at the end of ten years from the portfolio underlying the CDO? What is the cash flow corresponding to each outcome? (12 points)

4.Using your answer to part 3), what is the probability of observing at least 1 default, but no more than two defaults, over the ten-year period? (6 points)

5.What outcomes are associated with a probability of loss of less than 3% at the end of the maturity of ten years? If the credit rating agencies require the senior tranche to have a probability of loss less than 1% to be given a AAA rating. What are the maximum and minimum sizes of the senior tranche in terms of cash flows at the end of ten years? (12 points)

6.Calculate the present value of the senior tranche if the AAA bonds with a ten year maturity are yielding 3% today? (6 point)

7.Given part 5) and part 6), what are the present values of equity tranche maximum and minimum cash flows at maturity? (6 points)

8.Suppose, for this part only, that defaults across bonds can be correlated. Consider bond defaults over the ten-year period as a whole. Bond A and bond B satisfy the following relationships: Given that bond A defaults, bond B defaults with probability 0.5. Given that bond A does not default, bond B does not default with probability 0.95. Bond C is uncorrelated with bonds A and B. Redo part 2) to part 7). (22 points)

9.Suppose the issuer of bond C purchased a CDS contract, so that bond C is fully protected against default. Suppose the CDS contract costs 200 basis points per year, payable on a compounded basis at the end of the ten years. What is the maximum size of the AAA tranche now? (24 points)