IFRS 9 PD Modeling Framework via Empirical Migration Matrix and Credit Cycle Index Approach

This repository implements a migration matrix-based Probability of Default (PD) model aligned with IFRS 9 Expected Credit Loss (ECL) requirements. The model estimates an Empirical Migration Matrix (EMM) from historical loan-level data and applies a Credit Cycle Index (CCI) that derived from the Vasicek (1987) single factor framework to condition the Through-the-Cycle (TTC) matrix into a Point-in-Time (PiT) matrix under multiple macroeconomic scenarios. The resulting PD term structures are suitable for Stage 1, Stage 2, and lifetime ECL calculation.

Overview

his project implements an EMM CCI PD Model designed to support IFRS 9 ECL calculation. The model estimates a TTC migration matrix from observed loan grade transitions, extracts a time series of Credit Cycle Index (Z-Score, Z-Index) that summarises the systematic credit environment at each period, and links it to macroeconomic variables to generate the forward-looking PiT migration matrices and cumulative PD Term structures.

The implementation emphasises:

• Transparent, auditable migration matrix construction suitable for model governance
• Closed-form and optimisation-based CCI Estimation
• Vectorised numerical computation for efficiency and scalability
• Flexible scenario conditioning across baseline, adverse, and severe paths

The resulting PD Term structures can be directly used in Stage 1 and Stage 2 ECL Calculation. The project is intended to serve as a practical reference implementation for credit risk practitioners, model developers, and validators. All calculations are made explicit, facilitating validation, backtesting, and model explainability.

Project Structure

pd_emm_cci_model/
├── model/                                        #Trainned model and parameters (pkl.)
│   ├── rho.pkl
│   ├── fwl_model.pkl
│   └── lifetime_pd_term_structure.pkl  
├── notebooks/
│   ├── 01_data_preparation.ipynb
│   ├── 02_credit_cycle_index.ipynb
│   ├── 03_fwl_model.ipynb
│   └── 04_markov_liftetime.ipynb
├── src/
│   ├── data_prep.py
│   ├── migration_matrix_cci.py
│   ├── regression_model.py
│   ├── lifetime_model.py
│   ├── stats_testing.py
│   └── plot_function.py
├── data/          
│   ├── processed/
|   |   ├── train_data.parquet                    #Not tracked by git
|   |   ├── migration_count.parquet
|   |   ├── average_matrix.parquet
|   |   ├── monthly_cci.parquet
|   |   ├── mev_transformed.parquet
|   |   └── mev_sign_transformed.parquet
│   └── raw/
|   |   ├── usedcar_transaction_score.parquet     #Not tracked by git
|   └── └── mev_data.csv
├── requirements.txt
└── README.md

Project Details

0. Model Segmentation

1. Unbias Model

1.1 Empirical Migration Matrix (TTC)

Empirical Migration Matrix: A Through-the-Cycle (TTC) Migration matrix is estimated from historical loan level data by tracking grade transitions over a defined observation window. For each period, the number of transitions from grade $i$ to grade $j$ is counted and normalised by the total population starting in grade $i$. The long-run average across all periods forms the TTC matrix $\mathbf{P}_{\text{TTC}}$, with the final column representing the observed default rate per grade.

1.2 Credit Cycle Index

CCI Model: The Credit Cycle Index is estimated by fitting a time-varying Z-Score to each period's observed migration matrix. Based on the Vasicek (1987) single-factor model:

$$ P_t(ij) = \Phi\left(\frac{x_{i+1}^{,j} - \sqrt{\rho}, z_t}{\sqrt{1 - \rho}}\right) - \Phi\left(\frac{x_i^{,j} - \sqrt{\rho}, z_t}{\sqrt{1 - \rho}}\right) $$

$$ \min_{z_t} \sum_j \sum_i n_{t,G} \frac{\left[P_t(i,j) - \Delta\left(x_{i+1}^j, x_i^j, z_t\right)\right]^2}{\Delta\left(x_{i+1}^j, x_i^j, z_t\right)\left[1 - \Delta\left(x_{i+1}^j, x_i^j, z_t\right)\right]} $$

The transition probabilities are expressed as threshold crossings of a standard normal distribution (Belkin, Suchower & Forest, 1998). For each period $t$, the CCI ($Z_t$) and asset correlation ($\rho$) are jointly estimated by minimising the weighted squared error between observed and model implied migration probabilities, subject to the constraint $\text{std}({Z_t}) = 1$. A simplified closed-form variant is also provided for cases where only $\rho$ optimisation is required.

2. Forward-looking Model

The forward-looking model processes are finding the relationship between Credit Cycle Index (CCI, Z-Index) with macroeconomics varialbes (MEV). The processes are similar to others ODR Model but changed the dependence variabale from ODR to CCI. In this repository is not covered the forward-looking model but it can refer to this repository for the forward-looking model consideration.

2.1 Model Back-testing

The model back-testing of actual CCI and predicted CCI from the regression model have been displayed in the following section. The visualisation of model back-testing in the following:

Note: It is a ramdom selection model. No expert input in this model.

3. Lifetime Model

Conditional PIT Matrix: For each forecast horizon $h$ and scenario, the TTC matrix is shifted using the forecast $\hat{Z}_{t+h}$ to produce a period-specific PiT migration matrix by the formula below:

$$ P_t(i,j) = \Phi\left(\frac{x_{i+1}^j - \sqrt{\rho}, \hat{z_t}}{\sqrt{1 - \rho}}\right) - \Phi\left(\frac{x_i^j - \sqrt{\rho}, \hat{z_t}}{\sqrt{1 - \rho}}\right) $$

The cumulative multi-periods of PD Term structures are derived by chaining:

$$\mathbf{P}(h) = \prod_{t=1}^{h} \mathbf{P}_t^{\text{PiT}}$$

The last column of $\mathbf{P}(h)$ gives the cumulative PD per grade at horizon $h$, directly applicable to IFRS 9 Stage 1 (12-month) and Stage 2 (lifetime) ECL Calculation.

License

MIT · Built for learning purposes