Consider two firms with one-year probabilities of default of $p _ { 1 } = 0.10$ and $p _ { 2 } = 0.05$ , respectively. The correlation of default of the two firms is $\rho = 0.5$ . What is the conditional probability of default $\operatorname { Pr } \left[ D _ { 2 } \mid D _ { 1 } \right]$ ?
A) 0.38
B) 0.42
C) 0.46
D) 0.50

Question

Consider two firms with one-year probabilities of default of  $p _ { 1 } = 0.10$ and $p _ { 2 } = 0.05$ , respectively. The correlation of default of the two firms is $\rho = 0.5$ . What is the conditional probability of default $\operatorname { Pr } \left[ D _ { 2 } \mid D _ { 1 } \right]$ ?
A) 0.38
B) 0.42
C) 0.46
D) 0.50

Quizplus · Accepted Answer

The conditional probability of default $\operatorname{Pr}[D_2 \mid D_1]$ can be calculated using the formula: $\operatorname{Pr}[D_2 \mid D_1] = \frac{\operatorname{Pr}[D_1 \cap D_2]}{\operatorname{Pr}[D_1]}$. Given the correlation and individual probabilities, $\operatorname{Pr}[D_1 \cap D_2] = p_1 \cdot p_2 + \rho \cdot \sqrt{p_1 \cdot (1-p_1) \cdot p_2 \cdot (1-p_2)}$. Substituting the values, we find $\operatorname{Pr}[D_2 \mid D_1] \approx 0.38$.

Consider Two Firms with One-Year Probabilities of Default Of p1=0.10p _ { 1 } = 0.10p1​=0.10

Consider Two Firms with One-Year Probabilities of Default Of $p _ { 1 } = 0.10$