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?

  • $\begingroup$ 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. $\endgroup$ Commented Oct 12, 2020 at 14:20
  • $\begingroup$ What doea PIT mean $\endgroup$
    – develarist
    Commented Oct 21, 2020 at 15:58
  • $\begingroup$ PIT = Point-in-time $\endgroup$
    – Nasser Bin
    Commented Oct 22, 2020 at 8:08

1 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.

  • $\begingroup$ Thank you very much. It is indeed what I was looking for: the Kalman filter. $\endgroup$
    – Nasser Bin
    Commented Oct 20, 2020 at 13:09
  • $\begingroup$ 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? $\endgroup$
    – Nasser Bin
    Commented Oct 20, 2020 at 13:16
  • $\begingroup$ @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. $\endgroup$
    – nyk
    Commented Oct 21, 2020 at 0:59
  • $\begingroup$ 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. $\endgroup$
    – Nasser Bin
    Commented Oct 21, 2020 at 7:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.