The structured products division of your investment bank is planning to create a set of products representing claims on its portfolio of commercial loans. The value of each of the loans has an annual variance σ2 and any two loans have a correlation of ρ (0 < ρ < 1).
The bank plans to offer three types of claims on the value of the loan portfolio at time T. The claims differ in the seniority of their claims to the value of the underlying portfolio. Tranche 1 pays the value of the underlying loan portfolio, up to a maximum of D1, and is paid first;
tranche 2 pays the excess value of the loan portfolio after tranche 1 has been paid, up to a maximum of D2; tranche 3 pays the residual value of the portfolio after the claims of the first two tranches have been satisfied.
(a) Explain why if there are N equally-weighted loans in the loan portfolio, the variance of the return on the overall portfolio is σ2/N +(1−1/N)ρσ2. (Hint: use standard formulas relating to variance. If you can’t do this part of the question, take the result as given in subsequent parts.)
(b) Draw separate payoff diagrams for each of the three tranches as a function of the value of the underlying loan portfolio, V . (The locations of any kinks and plateaus should be clearly indicated on the axes.)
(c) Write down the value of each of the three tranches in terms of V,D1,D2, using the notation C(X) to indicate the price of a T-year European call option on the value of the underlying loan portfolio with exercise price X.
(d) You are the manager of the structured products division. Your bank will hold one of the tranches on its own books, and will sell the remaining tranches to investors. Your task is to decide how many loans, N, to include in the portfolio underlying the structured products; each of the loans will be equally weighted in the portfolio. Suppose your bank holds tranche 1 on its books—how many loans will you choose to have in the underlying portfolio? What if you hold tranche 3? Why?
(e) Now suppose that there is a fixed number of loans in the portfolio, N. You announce that you will hold the “toxic” tranche 3, while selling on tranches 1 and 2 to investors; this announcement is intended to reassure investors. After making this announcement, you realize that you can choose the portfolio of loans in such a way as to alter the correlation between different loans: you can choose either a low correlation ρL or a high correlation ρH.
i. Which correlation will you choose? Why?
ii. Which correlation would the investors choose, if they could, and why?