I am trying to run a montecarlo simulation based on this model in order to assess whether a loan/company will default or not. Basically the model is
$$X_i = \sqrt{1-\rho}Z_i + \sqrt{\rho} Y$$
where $$ Z_i \sim \mathcal{N}(0, 1) $$ represents the credit $i$ and $$ Y \sim \mathcal{N}(0, 1) $$ $Z_i$ are independent of each other and of the common factor $Y$.
The time to maturity for each credit is divided in year and for each each year I have generated $10.000$ variables for $Z$ and $10.000$ variables for $Y$ and then combined them to get the value for $X$. So for each year I have $10.000$ values of X and from this the model says that I have to assess whether a default has occured or not. This is done by comparing the result with a predefined threshold value of (historical) default rate $a$ as follows
$$\Phi(X_i) < a $$ So if $\Phi(X_i) < a $ then we have a default, otherwise not. I am very unsure about this part and here is where I am lost and do not know how to proceed. What I do not understand is $what$ is it that I am supposed to compare with the threshold value?
Have I misunderstood the model? Is the simulation run correctly?
Thanks for the help in advance