If two firms have independent default events, the joint probability of default $\pi_{12}$ is equal to the product of their individual probabilities of default, $\pi_1$ and $\pi_2$. Therefore, $\pi_{12} = \pi_1 \pi_2$. Also, the default correlation $\rho_{12}$ between the two firms is exactly zero.

Let's evaluate each option:

• Options A, B, and D all have the default correlation $\rho_{12}$ as a multiplicative factor. Since $\rho_{12} = 0$, these expressions will all evaluate to zero.
• Option C: $\frac{\pi_{12}}{\pi_1} = \frac{\pi_1 \pi_2}{\pi_1} = \pi_2$. Assuming the firm has a non-zero probability of default ($\pi_2 > 0$), this expression evaluates to a non-zero value.

Thus, the correct answer is C.