Answer: 7.89%

The correct answer to the question is C, which is 7.89%. This is determined using Bayes' theorem, which is a fundamental concept in probability theory. The theorem allows us to find the probability of an event occurring, given the occurrence of another event. In this scenario, we are interested in the probability that the longevity bond defaults (event A) given that the market returns decrease by 20% (event B). Bayes' theorem is expressed as: \[ P[A|B] = \frac{P[A \cap B]}{P[B]} \] Where: - \( P[A|B] \) is the probability that the bond defaults given a 20% market decrease. - \( P[A \cap B] \) is the probability that the bond defaults and the market decreases by 20%. - \( P[B] \) is the total probability of the market decreasing by 20%. From the provided table, we have: - \( P[A \cap B] = 3\% \) (the probability that the bond defaults and the market decreases by 20%). - \( P[B] = 35\% + 3\% = 38\% \) (the total probability of the market decreasing by 20%). Plugging these values into Bayes' theorem gives us: \[ P[A|B] = \frac{0.03}{0.38} \approx 0.0789 \] Which translates to a 7.89% probability that the bond defaults given a 20% market decrease. This calculation is why option C is the correct answer. The other options (A, B, and D) represent different probabilities that do not align with the conditional probability we are trying to calculate.

Author: LeetQuiz Editorial Team