Answer-first summary for fast verification
Answer: 1.90%
## Explanation The unconditional default probability during year 5 is calculated using the formula: $$\exp\left(-\bar{h}_4 \times 4\right) - \exp\left(-\bar{h}_5 \times 5\right)$$ Where: - $\bar{h}_4$ = average hazard rate for first 4 years = 1.1% = 0.011 - $\bar{h}_5$ = average hazard rate over 5 years (unknown) **Step 1: Calculate 5-year survival rate** - 5-year cumulative default probability = 6.2% = 0.062 - 5-year survival rate = 1 - 0.062 = 0.938 **Step 2: Calculate survival probability to end of year 4** $$\exp(-0.011 \times 4) = \exp(-0.044) = 0.956954$$ **Step 3: Calculate unconditional default probability during year 5** $$0.956954 - 0.93800 = 0.01895 \text{ or } 1.895\%$$ **Why other options are incorrect:** - **A (1.71%)**: Incorrectly calculates survival rate as $\exp(-0.062) = 0.939883$ - **B (1.80%)**: Incorrectly calculates as $0.062 - 4 \times 0.011$ - **D (1.98%)**: This represents the conditional default probability during year 5 The correct answer is **1.90%** (rounded from 1.895%).
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A credit risk analyst at a wholesale bank is estimating annual default probabilities of a 5-year loan that has just been extended to a corporate borrower. The analyst determines from rating agency data that the 5-year cumulative default probability of bonds from this borrower with identical terms and seniority is 6.2%, and uses this information to calculate the 5-year survival rate for the borrower. If the borrower's average hazard rate for the first 4 years of the loan is 1.1%, what is the unconditional default probability of the borrower during year 5 of the loan?
A
1.71%
B
1.80%
C
1.90%
D
1.98%
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