Interpreting Through-the-Cycle and Point-in-Time Default Probabilities in the Vasicek Model
Summary
The document explores the meaning of through-the-cycle (TTC) and point-in-time (PIT) probabilities of default in a two-factor Gaussian Vasicek model. A latent credit variable combines a systemic factor and an idiosyncratic factor; default occurs below a threshold. The unconditional default probability is the threshold’s normal cumulative probability, while conditioning on the systemic factor gives a PIT probability through the Vasicek formula.
The central question is conceptual: if the unconditional probability is generated by both systemic and idiosyncratic risk, why is it treated as TTC when TTC is often understood as insensitive to macroeconomic fluctuations? The text notes that IFRS 9 practice may use IRB transition matrices to obtain TTC estimates before converting them to PIT values, and questions the assumption equating the threshold with the inverse normal of the TTC PD. It accepts the model’s mathematics but offers no resolution; its conclusions are limited to the stated Gaussian model and terminology.
Key ideas
- The Vasicek model represents default with a latent variable combining systemic and idiosyncratic Gaussian factors.
- The unconditional default probability is derived from the default threshold, while conditioning on the systemic factor yields a PIT probability.
- The document questions how an unconditional probability involving systemic risk should be interpreted as TTC.
- The discussion raises a conceptual concern about mapping an IRB TTC estimate to the model’s default threshold.
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# Interpretation of the TTC and PIT probability of default in the Vasicek model
# Interpretation of the TTC and PIT probability of default in the Vasicek model
The well known 2-factor Gaussian model assumes that the default behaviour of a client is ruled by a latent variable defined as:
\begin{align} y=\sqrt{\rho}Z+\sqrt{1-\rho}\xi \end{align} where $Z$ and $\xi$ are independent standard Gaussian r.v. representing the systemic and idiosyncratic risk sources respectively.
The default event is defined as $$D:=\{y\leq K\}$$ for some $K\in\mathbb R$.
The unconditional probability of default, i.e.
$$\mathbb{P}(D)=\Phi(K)=:p,$$
is sometimes referred to as the "TTC" (Through-the-cycle) PD. This is what somehow confuses me, in the IRB world the term "TTC" is used to refer to models which are unsensitive to macro economical (systemic) fluctuations, and where the only source of variability comes from idiosyncratic factors.
In this case $\mathbb{P}(D)$ is defined by taking into consideration the latent variable which depends on both risk sources.
In IFRS9, we sometimes use the IRB transition matrices to derive the TTC PD, and this is used to compute the PIT (point-in-time) PD using the Vasicek's formula, i.e.
$$PD^{PPI}(Z)=\mathbb{P}(D|Z)=\Phi\left(\frac{\Phi^{-1}(PD^{TTC})-\sqrt{\rho}Z}{\sqrt{1-\rho}}\right),$$ where we are "assuming" that $K=\Phi^{-1}(PD^{TTC})$. This assumption I find problematic, $K$ is defined as $K=\Phi^{-1}(p)$ but I find it hard to say that $p$ (defined above) is a TTC probability of default. I understand that under the rather simplistic assumptions of the Gaussian model everything works fine, and I don't have issues with the math, but from a conceptual point of view I struggle connecting the dots.
Could someone clarify if my interpretation is correct, or point out what am I missing?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.