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Answer: 23.5%
## Explanation To calculate the probability that a loan currently rated B will default over a two-year period, we need to consider all possible paths that could lead to default (rating D) within two years. **Step 1: Identify possible paths to default in 2 years** From the transition matrix: - B → D directly in year 1: 0.10 - B → C in year 1, then C → D in year 2: (0.15 × 0.40) = 0.06 - B → B in year 1, then B → D in year 2: (0.75 × 0.10) = 0.075 **Step 2: Calculate total probability** Total probability = Direct default in year 1 + Default via C in year 2 + Default via B in year 2 = 0.10 + 0.06 + 0.075 = 0.235 = 23.5% **Step 3: Verify calculation** The calculation considers all possible transitions: - Year 1: B → D (0.10) - Year 1: B → C (0.15), then Year 2: C → D (0.40) = 0.15 × 0.40 = 0.06 - Year 1: B → B (0.75), then Year 2: B → D (0.10) = 0.75 × 0.10 = 0.075 Total = 0.10 + 0.06 + 0.075 = 0.235 = 23.5% Therefore, the correct answer is **23.5%**.
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As a result of the credit crunch, a small retail bank wants to better predict and model the likelihood that its larger commercial loans might default. It is developing an internal ratings-based approach to assess its commercial customers. Given this one-year transition matrix, what is the probability that a loan currently rated at B will default over a two-year period?
| Rating at Beginning of Period | Rating at End of Period | |||
|---|---|---|---|---|
| A | B | C | D | |
| A | 0.90 | 0.10 | 0.00 | 0.00 |
| B | 0.00 | 0.75 | 0.15 | 0.10 |
| C | 0.00 | 0.05 | 0.55 | 0.40 |
A
17.50%
B
20.0%
C
21.1%
D
23.5%
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