CAS Exam 9 Spring 2014, Question 15
Study guide: Exam 9
The following credit ratings transition matrix applies to bonds initially rated A or better:
| Initial Rating | Aaa (year-end) | Aa (year-end) | A (year-end) | Baa or worse (year-end) |
|---|---|---|---|---|
| Aaa | 0.95 | 0.04 | 0.01 | 0.00 |
| Aa | 0.05 | 0.90 | 0.04 | 0.01 |
| A | 0.00 | 0.05 | 0.90 | 0.05 |
The ratings transitions for two bonds can be correlated using a Gaussian copula with correlation 0.3.
A company's investment portfolio consists of two bonds:
- One bond initially rated Aa
- One bond initially rated A
The company's actuary samples 1,000,000 pairs of random variables (Z_1, Z_2) from a bivariate normal distribution with correlation coefficient 0.3. The table below summarizes the number of times the sampled pairs were within the indicated ranges:
Range of Z_1 Values |
Z_2 < -2.326 |
Z_2 -2.326 to -1.645 |
Z_2 -1.645 to 0.000 |
Z_2 0.000 to 1.645 |
Z_2 1.645 to 2.326 |
Z_2 > 2.326 |
|---|---|---|---|---|---|---|
| < -2.326 | 556 | 1,261 | 6,091 | 1,967 | 48 | 6 |
| -2.326 to -1.645 | 1,333 | 3,895 | 23,702 | 10,489 | 400 | 50 |
| -1.645 to 0.000 | 6,075 | 23,797 | 231,668 | 175,999 | 10,539 | 1,971 |
| 0.000 to 1.645 | 2,039 | 10,746 | 175,772 | 231,333 | 23,968 | 6,160 |
| 1.645 to 2.326 | 68 | 458 | 10,967 | 23,371 | 3,667 | 1,275 |
| > 2.326 | 7 | 68 | 2,059 | 6,259 | 1,366 | 570 |
Produce a range of estimates for the probability that both bonds in the portfolio will be downgraded to Baa or worse by year-end.
Part a (2 points)
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