Question

A collateralized debt obligation (CDO) is backed by 10 subprime mortgages. Five of them are from California, each of which defaults with probability 50%. Three mortgages are from Florida, each of which defaults with probability 60%. Two mortgages are from Nevada, each defaults with probability 40%. A senior tranch in this CDO defaults only if all of these mortgages default. Find the probability that the senior tranch does not default in the following cases: (a) all independent; (b) all mortgages from the same state default (or not default) simultaneously, but mortgages in different states are independent.

Answer 1

(a)

The probability that senior tranch from California is not default is 1 - 0.50 = 0.50

The probability that senior tranch from Florida is not default is 1 - 0.60 = 0.40

The probability that senior tranch from Nevada is not default is 1 - 0.40 = 0.60

Since all are independent so the  probability that the senior tranch does not default is

Hence, the required probability is 0.00072.

(b)

Now we need take into account that a tranch from California is not default, a tranch from Florida is not default and a tranch from Nevada is not default so the required probability is

0.5 *0.4 *0.6 = 0.1200

Hence, the required probability is 0.1200.