a) Calculation as per lecture with binomial expansion.

# defaults 0 1 2 3 4P 0.6561 0.2916 0.0486 0.0036 0.0001

Prob. of default for different tranches: A: 0.0037 B: 0.0523 C: 0.3439

b) Rating for tranches can be looked up in table, using probability of default from above Tranche A: BBB (investment grade) Tranche B: CCC+ (speculative grade) Tranche C: too risky for rating

The rating agency also estimates that investors expect a yield of 4% for AAA rated certificates, 5%for other investment-grade certificates and 12% for speculative-grade certificates. Using this information, calculate value of tranches and compare to the value of the underlying loans.

The CMO is still profitable, even if the C tranche cannot be placed with investors.

c) Discussion of specific vs. idiosyncratic risk factors in mortgages during boom and bust (In lecture, influence of correlations in default probability only shown for 2 mortgages, not 4). So discussion of impact of correlations would be sufficient. Most important change that needs to be mentioned: Separation of risk into different tranches less strong with high correlations.

Just in case, in case students try to calculate the number of defaults and % of N defaults for correlations of 0.2 and 0.4 (not necessary to answer this question, but would be exceptional to see): Number of defaults in portfolio of 4 % of N defaults

N Boom (corr of 0.2) Crisis (corr of 0.4)

0 65.66% 78.05%

1 29.08% 11.26%

2 4.86% 5.34%

3 0.38% 3.20%

4 0.01% 2.15%