Answer: Mean = 0.85, standard deviation = 0.84

The correct answer to the question about the mean and standard deviation of the number of bonds defaulting over the next year is option B. This is determined by using the given information and applying the appropriate statistical formulas. In the context of the question, 'n' represents the number of bonds in the portfolio, and 'p' represents the individual default probability. The formulas to calculate the mean and standard deviation for a binomial distribution, which is suitable for this scenario, are as follows: 1. **Mean (E(K)):** This is calculated as the product of the number of bonds (n) and the individual default probability (p). In the given scenario, \( n = 5 \) and \( p = 0.17 \), so the mean is calculated as: \[ \text{Mean} = E(K) = n \times p = 5 \times 0.17 = 0.85. \] 2. **Variance:** The variance for a binomial distribution is calculated as the product of the number of bonds, the individual default probability, and one minus the default probability: \[ \text{Variance} = \text{Variance}(K) = n \times p \times (1-p) = 5 \times 0.17 \times (1 - 0.17) = 0.7055. \] 3. **Standard Deviation:** The standard deviation is the square root of the variance: \[ \text{Standard deviation} = \sqrt{\text{Variance}} = \sqrt{0.7055} \approx 0.8399. \] The explanation provided in the file content aligns with the calculations above, confirming that option B, with a mean of 0.85 and a standard deviation of approximately 0.84, is the correct answer. This question tests the understanding of calculating expected values and variances for binomial distributions, which are key concepts in quantitative analysis as referenced in the Global Association of Risk Professionals' "Quantitative Analysis" textbook, specifically Chapter 3 on Common Univariate Random Variables.

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