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Explanation:
Expected Loss (EL) = EAD × LGD × PD Given:
To find the 3-year cumulative default probability (PD), we find the transition probabilities by multiplying the matrix by itself. Initial state for B-rated borrower (Year 0) = [0, 1, 0] (Probabilities of A, B, and D respectively) Year 1 state = [0.1, 0.7, 0.2] Year 2 state = [0.1×0.80 + 0.7×0.10, 0.1×0.20 + 0.7×0.70, 0.1×0.0 + 0.7×0.20 + 0.2×1.0] = [0.15, 0.51, 0.34] Year 3 state = [0.15×0.80 + 0.51×0.10, 0.15×0.20 + 0.51×0.70, 0.51×0.20 + 0.34×1.0] = [0.171, 0.387, 0.442]
The cumulative default probability at Year 3 (PD) is 44.2%. EL = 15,000,000 × 60.0% × 44.2% = `$3`,978,000. Therefore, the correct answer is C.
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921.1. The following simplified credit rating transition matrix (aka, migration matrix) displays one-year conditional probabilities for only two credits (A and B). For example, the A-rated credit has an 80.0% probability of remaining A-rated at the end of the year and a 20.0% probability of being downgraded to B-rated but is not expected to default within one year. From year to year, migrations are independent; i.e., the matrix satisfies the Markov property.
Transition Matrix (T)
| A | B | D | |
|---|---|---|---|
| A | 80.0% | 20.0% | 0.0% |
| B | 10.0% | 70.0% | 20.0% |
| D | 0.0% | 0.0% | 100.0% |
A bank has extended a three-year `15.0` million loan to a B-rated corporate borrower. The bank assumes the exposure at default (EAD) is the principal amount of \`15.0` million and estimates a 40.0% recovery rate. If the relevant default probability is the three-year cumulative default probability, then what is the expected loss (EL)?
A
$1,800,000
B
$2,250,000
C
$3,978,000
D
$4,392,000