If two default events are independent, their default correlation ($\rho_{12}$) is zero. Furthermore, the joint probability of default ($\pi_{12}$) equals the product of their individual probabilities of default ($\pi_{12} = \pi_1 \times \pi_2$).

Analyzing the options:

• Option A: Contains $\rho_{12}$ as a multiplier, meaning the expression evaluates to zero.
• Option B: Contains $\rho_{12}$ as a multiplier in the numerator, evaluating to zero.
• Option C: Evaluates to $\frac{\pi_{12}}{\pi_1} = \frac{\pi_1 \pi_2}{\pi_1} = \pi_2$. Since the probability of default for firm 2 ($\pi_2$) is typically a non-zero value, this expression will be non-zero.
• Option D: Contains $\rho_{12}$ as a multiplier, so the expression evaluates to zero.

Therefore, Option C is the correct answer.