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Answer: M(xBB≤ - 1.209 n XB≤-0.533)
The correct answer to the question is option D. The explanation for this is based on the concept of the Gaussian copula, which is used to model the joint probability of two or more events occurring simultaneously. In this case, the events are the default of two firms, RST and WYZ, before the end of year 2. The Gaussian copula uses the standard normal distribution to transform the individual default probabilities into standard normal space. The table provided in the question gives us the cumulative default probabilities for each firm and their corresponding standard normal percentiles (N-1(QBB(t)) and N-1(Q(t))) for different time periods. To find the joint probability of both firms defaulting before the end of year 2, we need to consider the time-to-default for each firm and their respective standard normal percentiles at that time. The table shows that for firm RST, the default probability at the end of year 2 is 6.12%, which corresponds to a standard normal percentile of -1.209. For firm WYZ, the default probability at the end of year 2 is 11.33%, which corresponds to a standard normal percentile of -0.533. The Gaussian copula formula for the joint probability of two events is given by: \[ P([tBB ≤ 2] \cap [tB ≤ 2]) = P([N^{-1}(QBB(tBB)) ≤ N^{-1}(QBB(2))] \cap [N^{-1}(Q(tB)) ≤ N^{-1}(QB(2))]) \] By substituting the standard normal percentiles for the end of year 2, we get: \[ P([XBB ≤ -1.209] \cap [XB ≤ -0.533]) \] This is the probability that both firm RST and firm WYZ will default before the end of year 2, which corresponds to option D. The other options (A, B, and C) do not correctly apply the Gaussian copula formula or do not properly consider the transformation using the standard normal distribution.
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A hedge fund operating under a distressed securities strategy is currently assessing the solvency risk of two potential investment targets. As of now, firm RST holds a credit rating of BB, while firm WYZ is rated B. The hedge fund intends to determine the probability that both firms will simultaneously default within the next 2 years, applying a Gaussian default time copula model. The analysis assumes a 1-year Gaussian default correlation of 0.36. The fund has provided a table containing the abscissa values X and xg from the bivariate normal distribution. Specifically, XBB is defined as N^-1(QBB(tgB)) and Xg is defined as N^-1(Qe(t)), where tgB and tg represent the time-to-default for BB-rated and B-rated companies, respectively; QBB and Q denote the cumulative distribution functions for tBB and tg, respectively; and N denotes the standard normal distribution, while M represents the joint bivariate cumulative standard normal distribution:
| Firm RST | Firm WYZ | Firm RST | Firm WYZ |
|---|---|---|---|
| Cumulative Default Probability | Cumulative Default Probability | Standard Normal Percentiles | Standard Normal Percentiles |
| Year | Qe(t) | QBB(t) | N^-1(QBB(t)) |
| 1 | 5.21% | 5.21% | -1.625 |
| 2 | 6.12% | 11.33% | -1.209 |
| 3 | 5.50% | 16.83% | -0.961 |
| 4 | 4.81% | 21.64% | -0.784 |
| 5 | 4.22% | 25.86% | -0.648 |
Using the Gaussian copula approach, which option correctly represents the joint probability that both firm RST and firm WYZ will default before the end of year 2?
A
M(XBB= 0.0612) + M(xB= 0.1063) - M(×BB= 0.0612)*M(XB= 0.1063)
B
M(xBB= 0.1133) + M(xB= 0.2969) - M(xBB= 0.1133)*M(xB = 0.2969)
C
M(XBB≤ 0.1133 n XB≤ 0.2969)
D
M(xBB≤ - 1.209 n XB≤-0.533)