Answer: $ Q(X_{BB} \leq -1.209 \cap X_B \leq -0.533) $

## Explanation In Gaussian copula modeling for credit risk: - The joint probability of default is expressed using the cumulative distribution function of a bivariate normal distribution - The variables $X_{BB}$ and $X_B$ represent the latent variables (asset values) for the two firms - The default thresholds are determined by the inverse of the cumulative default probabilities For option D: - $X_{BB} \leq -1.209$ represents the default threshold for firm RST (BB-rated) - $X_B \leq -0.533$ represents the default threshold for firm WYZ (B-rated) - The joint probability $Q(X_{BB} \leq -1.209 \cap X_B \leq -0.533)$ correctly represents the probability that both firms default within the specified time frame The other options are incorrect: - **A and B**: These use addition formulas that don't represent joint probability in copula modeling - **C**: Uses the raw default probabilities (0.1133 and 0.2969) rather than the transformed default thresholds The Gaussian copula approach transforms individual default probabilities into standard normal thresholds, then calculates the joint probability using the bivariate normal cumulative distribution function.

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