Using TTC and PIT Default Probabilities in a One-Factor Merton Model
Summary
This note discusses how point-in-time (PIT) and through-the-cycle (TTC) default probabilities fit into a one-factor Merton-style credit model. It presents the conditional default probability given a systemic economic factor and explains that inserting a TTC probability into the conditional formula produces a PIT probability for a specified factor outcome. A related inverse expression is proposed for obtaining PIT probabilities from TTC inputs and a chosen economic state.
The response highlights two practical challenges: estimating asset correlations is difficult, and IFRS 9 use requires a multi-period model for the systemic factor so that PIT probabilities can be projected through time. It mentions Kalman filtering with Basel correlation estimates as one possible aid, and transition-matrix or Markov-chain approaches as alternatives to the Vasicek framework. The note does not give implementation details or resolve how to select future factor paths, so its equations and suggestions are conceptual guidance rather than a complete IFRS 9 modeling procedure.
Key ideas
- Conditioning on the systemic factor converts a TTC default probability into a state-dependent PIT probability.
- Asset-correlation estimation is identified as a major challenge in applying the model.
- IFRS 9 applications require a multi-period model for the systemic factor.
- Kalman filtering with Basel correlation estimates is suggested as one possible approach.
- Transition matrices or Markov chains are presented as alternatives to the Vasicek model.
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# Use of PIT vs TTC PD in a Merton one-factor model
# 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?
$$ PD_i(z) = P(a_i<d_i \mid Z=z) = \dots = \Phi \left( \frac{\phi^{-1}(PD_i)-\sqrt{\rho_i}z}{\sqrt{1-\rho_i}} \right) $$ 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 by nyk (score 2, accepted)
https://quant.stackexchange.com/a/58798
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.