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Answers to homework questions are discussed in the Sunday online session.
1. Simulate the number of defaults in the following five-firm portfolio 100,000 times:
Firm |
PDi |
Correlation Matrix ri,j |
||||
1 |
0.5 |
1 |
0.05 |
0.1 |
0.15 |
0.2 |
2 |
0.4 |
0.05 |
1 |
0.25 |
0.30 |
0.35 |
3 |
0.3 |
0.10 |
0.25 |
1 |
0.40 |
0.45 |
4 |
0.2 |
0.15 |
0.30 |
0.40 |
1 |
0.50 |
5 |
0.1 |
0.20 |
0.35 |
0.45 |
0.50 |
1 |
What is the standard deviation of the number of defaults? What is the standard deviation of the number of defaults if all the off-diagonal correlations are equal to zero instead of the values shown?
2. Suppose that the default rate of a portfolio has the triangular distribution: [ ] = 2 − 2 .
Suppose that in this portfolio is a function of : [ ] = 1/2. Derive [ ]. On the same diagram for inputs between 0 and 1, plot [ ], [ ], [ ].
Questions 3 and 4 suppose a portfolio with three firms that have PD’s and PDJ’s as follows:
PD1 |
PD2 |
PD3 |
PDJ1,2 |
PDJ1,3 |
PDJ2,3 |
0.1 |
0.2 |
0.3 |
0.06 |
0.06 |
0.06 |
3. Find the three values correlation, r1,2, r1,3, and r2,3 and find the three values of default correlation, Corr[D1, D2], Corr[D1, D3], and Corr[D2, D3].
4. What is the probability that each of the three firms defaults, Pr[ D1 =1, D2 = 1, D3 = 1]? (Hint: this requires a triple integral of the multi-normal PDF.) Given this probability, what is Pr[ D1 =1, D2 = 1 | D3 = 1], Pr[ D1 =1, D3 = 1 | D2 = 1], and Pr[ D2 =1, D3 = 1 | D1 = 1]?