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Answer: 21.34%
The correct answer is C, which represents a risk-neutral probability of 21.34% that the bond rated BBB defaults within the next 3 years. This is calculated using the continuously compounded 3-year spread for the bond, which is 8% per year (0.10 - 0.02). Given the expected recovery rate of 0% in the event of default, the hazard rate is equal to the spread, which is also 8% per year. The formula to calculate the risk-neutral probability of default over time \( t \) is \( 1 - \exp(-\text{hazard rate} \times t) \). Substituting the values, we get \( 1 - \exp(-0.08 \times 3) = 21.34\% \). The other options are incorrect for the following reasons: - Option A (6.55%) is the marginal probability of default in the third year for the 3-year BBB-rated bond, not the cumulative probability over 3 years. - Option B (14.55%) incorrectly uses the credit spreads of the 1-year, 2-year, and 3-year bonds rated BBB and does not properly scale the hazard rates by the factor of time. - Option D (25.92%) incorrectly assumes the hazard rate for the 3-year bond rated BBB is equal to its annual yield of 10%, rather than using the spread adjusted for the risk-free rate.
Author: LeetQuiz Editorial Team
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A risk manager has assigned a junior analyst the task of determining the implied default probability for a corporate bond that holds a BBB rating. For this purpose, the continuously compounded annual yields for various fixed-income securities are given as follows:
Assume that the expected recovery rate for the 3-year BBB-rated bond in the event of a default is 0%. Based on this information, select the option that best approximates the risk-neutral probability of the BBB-rated bond defaulting within the next 3 years.
A
6.55%
B
14.55%
C
21.34%
D
25.92%
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