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Answer: 7.89%
## Explanation Using Bayes' theorem, we need to find the conditional probability that the longevity bond defaults given that the market decreases by 20% over 1 year. Let: - **A** = bond default - **B** = 20% decrease in market returns From the probability matrix: - **P[A ∩ B]** = probability of bond default AND market decreases by 20% = 3% - **P[B]** = probability of market decreases by 20% = 35% (no default) + 3% (default) = 38% Using Bayes' theorem: \[ P[A|B] = \frac{P[A \cap B]}{P[B]} = \frac{0.03}{0.38} = 0.0789 \rightarrow 7.89\% \] **Why other options are incorrect:** - **A (3.00%)**: This is the joint probability P[A ∩ B], not the conditional probability P[A|B] - **B (4.00%)**: This is the unconditional probability that the bond defaults (1% + 3% = 4%) - **D (10.53%)**: This incorrectly uses the unconditional probability of default in the numerator: \(\frac{0.04}{0.38} = 0.1053\)
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A portfolio manager is assessing whether the 1-year probability of default of a longevity bond issued by a life insurance company is uncorrelated with returns of the equity market. The portfolio manager creates the following probability matrix based on 1-year probabilities from the preliminary research:
| Market returns | 20% increase | 20% decrease |
|---|---|---|
| Longevity bond | No default | Default |
| 61% | 1% | |
| 35% | 3% |
Given the information in the table, what is the probability that the longevity bond defaults in 1 year given that the market decreases by 20% over 1 year?
A
3.00%
B
4.00%
C
7.89%
D
10.53%