Suppose we create an asset backed security $($ABS$)$ with four mortgages. These mortgages either pay off or default and the probability of default is $7\%.$ Defaults are independent across the mortgages. Now suppose that we create four tranches $($senior$1,$ senior$2,$ senior$3,$ and equity$).$ Each mortgage has a value of $$200,000.$ The senior$1$ tranche defaults only if all four mortgages default, and the equity tranche defaults if any mortgage defaults.

a$.$ Calculate the probability of default for each of the four tranches. How does the likelihood of a tranche defaulting compare with the risk of the underlying mortgages? What does this say about the risk of senior tranches?

b$.$ If the price of a tranche is equal to its expected value, price all four tranches.