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Answer: A corporate bond that is currently rated A, and maintains the same rating up to the end of year 1, has a 1.8% chance of default in year 2.
## Explanation **Option A is correct** because it accurately calculates the conditional default probability for a bond that maintains rating A: - Probability of staying A-rated in year 1: 90% - Probability of default from A rating: 2% - Year 2 default probability = 90% × 2% = 1.80% **Option B is incorrect** because the calculation is wrong: - Probability of staying B-rated in year 1: 76% - Probability of default from B rating: 9% - Correct calculation = 76% × 9% = 6.84% (not 8.1%) **Option C is incorrect** because it underestimates the 2-year default probability for a B-rated bond: - Actual 2-year default probability = 17.54% - Breakdown: - B → Default (year 1): 9.00% - B → A → Default: 5% × 2% = 0.10% - B → B → Default: 76% × 9% = 6.84% - B → C → Default: 10% × 16% = 1.60% - Total = 9.00% + 0.10% + 6.84% + 1.60% = 17.54% **Option D is incorrect** because it overestimates the 2-year default probability for a C-rated bond: - Actual 2-year default probability is less than 27.0% - Breakdown: - C → Default (year 1): 16.00% - C → A → Default: 1% × 2% = 0.02% - C → B → Default: 14% × 9% = 1.26% - C → C → Default: 69% × 16% = 11.04% - Total = 16.00% + 0.02% + 1.26% + 11.04% = 28.32% This question demonstrates the importance of understanding conditional probabilities and multi-period default calculations using transition matrices.
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A junior risk analyst at an investment bank is using a rating migration matrix to calculate the default probability of a corporate bond issuer. The 1-year credit migration matrix is presented below:
| Current rating | Rating in the next period | |||
|---|---|---|---|---|
| A | B | C | Default | |
| A | 90% | 8% | 0% | 2% |
| B | 5% | 76% | 10% | 9% |
| C | 1% | 14% | 69% | 16% |
Which of the following statements is correct?
A
A corporate bond that is currently rated A, and maintains the same rating up to the end of year 1, has a 1.8% chance of default in year 2.
B
A corporate bond that is currently rated B, and maintains the same rating up to the end of year 1, has an 8.1% chance of default in year 2.
C
A corporate bond that is currently rated B will default over a 2-year period with a probability of 15.9%.
D
A corporate bond that is currently rated C will default over a 2-year period with a probability of 27.0%.