Suppose we create an asset backed security (ABS) with five mortgages. These mortgages either pay off or default, and the probability of a default is .1. Defaults are independent across the mortgages. Now suppose that we create five tranches (senior1, senior2, senior3, mezzanine, and equity). The senior1 tranche defaults only if all five mortgages default, and the equity tranche defaults if any mortgage defaults.
(a) Calculate the probability of default for each of the five tranches. How does the likelihood of a tranche defaulting compare with the risk of the underlying mortgages? [Note that you need to calculate the probability that, say, any two (or three, or four, etc.) of five mortgages default. This requires use of the binomial distribution, and you could use Excel or a similar program to aid your computation.]. What does this say about the risk of senior tranches?
(b) Suppose we now form a new security made up of mezzanine tranches. That is, we combine five securities with the same probability of default you calculated for the mezzanine tranche in part a. Call this a CDO. Again tranche this new security into five parts with the same pattern of seniority. Calculate the probability of default of the various tranches of the CDO.
(c) Suppose that the probability of default of the underlying mortgages is really .15. How does this change the probability of the default of the tranches of the CDO? How much riskier (say, in percentage terms) does the mezzanine tranche of the CDO get given this 50% increase in the default probability?
(d) What if there were 100 mortgages, 10 to a tranche, and the probability of default of the underlying loans is .05. Consider tranche10, which defaults if 10 or more mortgages default. What is the default probability of that tranche? What of the CDO made up of tranche10 securities? What happens if the underlying probability of default rises to .06?