Consider two firms with hazard rates $\lambda _ { 1 } = 0.10$ and $\lambda _ { 2 } = 0.05$ . The correlation of their default times is zero. What is the approximate probability that both firms default within five years?
A) 5%
B) 9%
C) 13%
D) 17%

Question

Quizplus · Accepted Answer

The probability that a firm defaults within five years is given by $1 - e^{-\lambda t}$, where $\lambda$ is the hazard rate and $t$ is the time period. For the first firm, the probability is $1 - e^{-0.10 \times 5} = 1 - e^{-0.5} \approx 0.393$, and for the second firm, it is $1 - e^{-0.05 \times 5} = 1 - e^{-0.25} \approx 0.221$. Since the default times are uncorrelated, the joint probability of both firms defaulting within five years is the product of their individual probabilities, which is $0.393 \times 0.221 \approx 0.087$, or approximately 9%.

Consider Two Firms with Hazard Rates λ1=0.10\lambda _ { 1 } = 0.10λ1​=0.10

Consider Two Firms with Hazard Rates $\lambda _ { 1 } = 0.10$