Use of PIT vs TTC PD in a Merton one-factor model

Question

Under one-factor Merton framework, like Basel, you use unconditional PDs as input of the portfolio model and this "unconditional" means it is a TTC-PD. Given a i-th borrower, the default threshold is just the inverse cumulative standard normal distribution of the unconditional $PD_i$ and you will compare the threshold given by such $PD_i$ with every outcome of the systemic and idiosyncratic component.

Let's imagine to shift the framework: we are now under IFRS9 and for accounting it is requested to use PIT. How should I deal with PIT-PD with a one-factor model? Is it just matter of having different inputs as PDs and define threshold from this new inputs? Or should I consider a particular outcome of the risk-factor that should represent the state of the economy and then evaluate the PIT-PD as a PD conditioned to the risk-factor outcome?

$$ PD_i(z) = P(a_i<d_i \mid Z=z) = \dots = \Phi \left( \frac{\phi^{-1}(PD_i)-\sqrt{\rho_i}z}{\sqrt{1-\rho_i}} \right) $$ given $z$ the the economic state factor outcome.

My idea is to obtain PIT-PD by inverting the Vasicek formula and consider different outcome of the economy (ie. $z$): $$ PD^{PIT}_i(z) = \Phi \left( \phi^{-1}(PD^{TTC}_i) \sqrt{1-\rho_i} + \sqrt{\rho_i}z\right) $$

Does this make sense?

Answer 1

The first equation is already a PIT PD if $\displaystyle PD_{i}$ is substituted by TTC PD. The challenges of using this model are:

(1) $\displaystyle \rho _{i}$, the asset correlation, is very difficult to estimate.

(2) A multi-period model is required for z so that you can use the PIT PDs in IFRS9.

Using Kalman filter and Basel estimates of asset correlations could help you to address the 2 challenges. For details, please refer to the paper by Chatterjee.

Or you can use a transition matrix / Markov Chain approach that helps you step aside from the Vasicek Model. For details, please refer to paper by Varnek and Hampel.