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

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?

$$PD_i(z) = P(a_i given $$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?

• As the 'cycle' component is meant to represent a economy-wide credit(risk) cycle, I would second your idea of adding a (cyclic) systemic factor to your model. Oct 12 '20 at 14:20
• What doea PIT mean Oct 21 '20 at 15:58
• PIT = Point-in-time Oct 22 '20 at 8:08

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.

• Thank you very much. It is indeed what I was looking for: the Kalman filter. Oct 20 '20 at 13:09
• For the asset correlation, this paper speaks of a range 12% - 24% in Basel framework for corporate loans. Do you know if it holds for Sovereign? Oct 20 '20 at 13:16
• @NasserBin I don't know whether it is applicable to sovereign. Sovereign is tough because data is very limited. You can dig around Moody's Analytics to see if you can find anything helpful.
– nyk
Oct 21 '20 at 0:59
• I really don't understand how to apply the Kalman filter for $z$. Should I consider the GDP hts and extract this hidden state $z$ from it? I will open it as a separate question as it actually is another story. Oct 21 '20 at 7:15