The joint default probability between 1 and 2, $\pi_{12}$ is given by:
$\pi_{12} = \rho_{12} \sqrt{\pi_1 (1 - \pi_1)} \times \sqrt{\pi_2 (1 - \pi_2)} + \pi_1 \pi_2$
$= 0.05 \sqrt{0.025(1 - 0.025)} \times \sqrt{0.035(1 - 0.035)} + 0.025 \times 0.035$
$= 0.05 \times 0.15612 \times 0.18378 + 0.000875$
$= 0.00231$

Things to Remember

• Joint default probability is the probability that two or more firms will default simultaneously.
• Correlation coefficient $\rho$ measures the strength and direction of a linear relationship between two variables.
• When calculating joint default probability, it is important to consider the individual default probabilities of the firms as well as the correlation between them.
• Default correlation is a key concept in credit risk modeling and is used to assess the likelihood of multiple entities defaulting at the same time.