State numerical answers to precision of 2 significant digits. For example, if the exact answer is /2, then the answer to 2-digit precision is 1.6.
1. For the following collection of five firms, simulate 10,000 runs to find the standard deviation of the number of defaults. Simulate again to find the standard deviation of the number of defaults when all off-diagonal correlations are set equal to zero instead of the values shown.
Firm
PDi
Correlation Matrix i,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
Question 2. In general, the standard deviation of the number of defaults—the risk, simply put—rises with correlation. Plot the standard deviation of the number of defaults in 1,000 simulation runs as a function of , where every off-diagonal element in the previous matrix is replaced by the value of .
Question 3. Assume the following portfolio. Exposures are stated in USD. Questions can be answered by simulation or calculation; each method provides a check on the other.
Loan
Loan 1
Loan 2
Loan 3
Loan 4
Loan 5
Loan 6
Loan 7
Firm
Firm 1
Firm 2
Firm 3
Firm 4
Firm 5
Firm 4
Firm 5
PD
0.1 0.2 0.3 0.4 0.5 0.4
0.5
ELGD
0.1 0.2 0.3 0.4 0.5 0.6
0.7
Exposure
700
600
500
400
300
200
100
Corrrelation matrix H
Firm 1 Firm 2 Firm 3 Firm 4 Firm 5
Firm 1 1 0.15 0.2 0.25 0.3
Firm 2 0.15 1 0.25 0.3 0.35
Firm 3 0.2 0.25 1 0.35 0.4
Firm 4 0.25 0.3 0.35 1 0.45 Firm 5 0.3 0.35 0.4 0.45 1
What are the values of these four quantities?
• Prob[ D4 = 1 and D5 = 1]? (What is PDJ for these two firms?)
• Prob[ D4 = 1 and D5 = 1| D3 =1 ]? (That is, what is the probability that both Firm 4 and Firm 5 default, given that Firm 3 defaults?)
• What is the portfolio expected loss rate as a fraction of the $2800 exposure?
• What is the correlation between D3 and D4?
Question 4. Simulate for 10,000 runs the default rate of the portfolio of Question 3 and plot a histogram. Separately, simulate the loss rate (total loss / total exposure) assuming that every LGD equals its expected value. Show a scatter plot of each possible loss rate (horizontal) and the number of simulation runs the loss is experienced (vertical).
Summarize all outcomes in a histogram with loss rates separated into bins of width 2%.
