Consider three bonds X, Y , and Z, each of which pays $1000 if it does not default, and nothing in case of default. The default probabilities are 0.3, 0.2, and 0.1 respectively, and the default correlations are all zero. These bonds are bundled together to form an asset pool, and a prioritized structure of claims (tranches) against this pool is issued. There are three tranches (senior, mezzanine, and junior), each of which promise $1000. If there are insufficient funds to make all promised payments, then all losses are applied first to the junior tranche, then to the mezzanine, and finally to the senior tranche.
(a) Write down the list of all possible outcomes and identify which of these outcomes would cause the mezzanine tranche to default.
(b) What is the default probability of the mezzanine tranche?
(c) What is the expected value of payment for the mezzanine tranche?
(a) 1 means default and 0 means default
| X | Y | Z |
| 1 | 1 | 1 |
| 1 | 1 | 0 |
| 1 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 0 |
if any one bond defaults only junior will default
for mezannine to default atleast 2 bond have to default
these outcomes would cause the mezzanine tranche to default :
(1,0,0), (0,1,0), (0,0,1), (0,0,0)
(b)
default probability of the mezzanine tranche = P((1,0,0), (0,1,0), (0,0,1) or (0,0,0))
= (1-0.3)(0.2)(0.1) + (0.3)(1-0.2)(0.1) + (0.3)(0.2)(1-0.1) + (0.3)(0.2)(0.1)
= 0.098
default probability of the mezzanine tranche = 0.098
(c) expected (payment for mezzanine) = $ 1000*P(not default) + 0 * P(default)
= $ 1000*(1-0.098) + 0*0.098
expected (payment for mezzanine) = $ 902
P.S. (please upvote if you find the answer satisfactory)
Consider three bonds X, Y , and Z, each of which pays $1000 if it does...
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