NO.PZ2022062755000019
问题如下：
A hedge fund that runs a distressed securities strategy is evaluating the solvency conditions of two potential
investment targets. Currently firm RST is rated BB and firm WYZ is rated B. The hedge fund is interested in
determining the joint default probability of the two firms over the next 2 years using the Gaussian default time
copula under the assumption that a 1-year Gaussian default correlation is 0.36. The fund reports that xBB and xB are abscise values of the bivariate normal distribution presented in the table below where
xBB = N-1 (QBB(tBB)) and xB= N-1(QB(tB)) with tBB and tB being the time-to-default of BB-rated and B-rated companies respectively; and QBB and QBB being the cumulative distribution functions of tBB and tB, respectively; and N denotes the standard normal distribution; and M denotes the joint bivariate cumulative standard normal distribution:
Applying the Gaussian copula, which of the following corresponds to the joint probability that firm RST and
firm WYZ will both default before the end of year 2?
选项：
A.M(xBB = 0.0612) + M(xB = 0.1063) – M(xBB = 0.0612)*M(xB = 0.1063)
B.M(xBB = 0.1133) + M(xB = 0.2969) – M(xBB =0.1133)*M(xB = 0. 0.2969)
M(xBB ≤ 0.1133 ∩ xB ≤ 0.2969)
D.M(xBB ≤ -1.209 ∩ xB ≤ -0.533)
解释：
中文解析：
RST和WYZ会同时在第二年年末违约的联合概率是多少？
如下公式所示：
𝑃{[𝑡BB ≤ 2] ∩ [𝑡B ≤ 2]} = 𝑃{[𝑁-1 (𝑄BB (𝑡BB)) ≤ 𝑁-1 (𝑄BB(2))] ∩ [𝑁-1 (𝑄B (𝑡B )) ≤ 𝑁-1 (𝑄B (2))]} = 𝑃{[𝑋BB ≤ −1.209] ∩ [𝑋B ≤ −0.533]}
D is correct.
The required probability is:
𝑃{[𝑡BB ≤ 2] ∩ [𝑡B ≤ 2]} = 𝑃{[𝑁-1 (𝑄BB (𝑡BB)) ≤ 𝑁-1 (𝑄BB(2))] ∩ [𝑁-1 (𝑄B (𝑡B )) ≤ 𝑁-1 (𝑄B (2))]} = 𝑃{[𝑋BB ≤ −1.209] ∩ [𝑋B ≤ −0.533]}
如题
