Answer: 9.16%

The question involves a Poisson process, which is a statistical distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. In this scenario, the fixed-income trading desk analyst has observed that the number of defaults in the bond portfolio follows a Poisson process with an average of four defaults per year. To find the probability of at most one default occurring next year, we use the Poisson probability mass function (PMF), which is given by: \[ P(K = n) = \frac{e^{-\lambda} \lambda^n}{n!} \] where: - \( P(K = n) \) is the probability of \( n \) events occurring in the given time period. - \( \lambda \) is the average rate of events per time period (in this case, four defaults per year). - \( n \) is the number of events we are interested in (0 or 1 default in this case). - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. However, there is an error in the provided explanation. It incorrectly states that the average number of defaults is 2 per year, but the question states it is 4 per year. The correct calculation should be as follows: For \( n = 0 \) (no default): \[ P(K = 0) = \frac{e^{-4} \cdot 4^0}{0!} = \frac{e^{-4}}{1} = e^{-4} \approx 0.0183 \] For \( n = 1 \) (one default): \[ P(K = 1) = \frac{e^{-4} \cdot 4^1}{1!} = \frac{4e^{-4}}{1} = 4e^{-4} \approx 0.0733 \] Adding these probabilities gives us the probability of at most one default: \[ P(at most one default) = P(K = 0) + P(K = 1) \approx 0.0183 + 0.0733 = 0.0916 \] Converting this to a percentage, we get approximately 9.16%, which matches the correct answer provided in the file content (C). The explanation in the file content, despite the initial error in stating the average number of defaults, correctly applies the Poisson distribution formula to calculate the probability of at most one default occurring next year.

Author: LeetQuiz Editorial Team