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Answer: 0.07
## Explanation **Step 1: Calculate the variance of one loan** - Loss given default (LGD) = 1 - Recovery Rate = 1 - 0.55 = 0.45 - Loss amount per default = L × LGD = 125,000 × 0.45 = USD 56,250 - Variance of one loan = p(1-p) × (Loss amount)² = 0.06 × 0.94 × (56,250)² - Variance of one loan = 0.0564 × 3,164,062,500 = 178,453,125 **Step 2: Calculate the portfolio variance** For a portfolio with n loans and pairwise correlation ρ: - Portfolio variance = n × σ² + n(n-1) × ρ × σ² - Portfolio variance = 7 × 178,453,125 + 7 × 6 × 0.35 × 178,453,125 - Portfolio variance = 1,249,171,875 + 2,623,260,938 = 3,872,432,813 **Step 3: Calculate portfolio standard deviation** - σₚ = √3,872,432,813 = USD 62,228.87 **Step 4: Calculate the ratio** - Portfolio size = n × L = 7 × 125,000 = USD 875,000 - Ratio = σₚ / Portfolio size = 62,228.87 / 875,000 = 0.0711 ≈ 0.07 **Therefore, the closest ratio is 0.07 (Option A).** This calculation shows how correlation significantly increases portfolio risk, as the standard deviation of losses relative to portfolio size is 7.11%, which is much higher than what would be expected with uncorrelated defaults.
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A credit risk analyst is running scenarios on the impact of correlation on the standard deviation of percentage loss for middle market loan portfolios. The analyst is using a binomial distribution to analyze a portfolio of 7 loans. The principal amount of each loan is USD 125,000. The analyst assumes that each loan has a recovery rate of 55%, a default probability of 6%, and a default correlation of 0.35 with all other loans in the portfolio. Which of the following is closest to the ratio of the portfolio standard deviation of losses to the size of the portfolio position?
A
0.07
B
0.11
C
0.19
D
0.23
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