Answers to homework questions are discussed in the Sunday online session.


Answers to these four questions are due at 6pm, Thursday 15 Feb 2018. One point per question.




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]?