Modelling the term-structure of default risk under IFRS 9 within a multistate regression framework

TL;DR

This work tackles the challenge of modelling the lifetime probability of default (PD) under IFRS 9 by examining a term-structure of default risk within a multistate framework. It benchmarks three modelling approaches—a baseline time-homogeneous Markov chain, beta regression (BR) for time-varying transition probabilities, and multinomial logistic regression (MLR) with loan-level covariates—to derive dynamic PD estimates across loan lifetimes. Empirical analysis on a large South African mortgage dataset shows that both BR and MLR substantially outperform the MC baseline, with MLR providing the strongest, loan-level predictions and a close match to empirical PD term-structures; BR offers competitive portfolio-level predictions with strong fit. The results support more timely and accurate expected credit loss (ECL) estimation under IFRS 9, while offering diagnostics and a scalable framework to manage model risk and regulatory requirements.

Abstract

The lifetime behaviour of loans is notoriously difficult to model, which can compromise a bank's financial reserves against future losses, if modelled poorly. Therefore, we present a data-driven comparative study amongst three techniques in modelling a series of default risk estimates over the lifetime of each loan, i.e., its term-structure. The behaviour of loans can be described using a nonstationary and time-dependent semi-Markov model, though we model its elements using a multistate regression-based approach. As such, the transition probabilities are explicitly modelled as a function of a rich set of input variables, including macroeconomic and loan-level inputs. Our modelling techniques are deliberately chosen in ascending order of complexity: 1) a Markov chain; 2) beta regression; and 3) multinomial logistic regression. Using residential mortgage data, our results show that each successive model outperforms the previous, likely as a result of greater sophistication. This finding required devising a novel suite of simple model diagnostics, which can itself be reused in assessing sampling representativeness and the performance of other modelling techniques. These contributions surely advance the current practice within banking when conducting multistate modelling. Consequently, we believe that the estimation of loss reserves will be more timeous and accurate under IFRS 9.

Figures (10)

• Figure 1: The state space in which loans may reside at any point of their lifetimes.
• Figure 2: Illustrating the construction process of the outcome variable in BR-modelling for the P$\rightarrow$D transition type, having used the entries within the time-dependent transition matrix $T(t')$.
• Figure 3: Illustrating the estimation of two MLR-models across various loans within the starting state $k\in\{\text{P, D}\}$, as a function of two covariates, loan amount [Principal] and the central bank policy rate [Repo_rate].
• Figure 4: Comparing the 12-month default rates over time across the various datasets. The Mean Absolute Error (MAE) between each sample and the full set $\mathcal{D}$ is overlaid in summarising the line graph discrepancies over time.
• Figure 5: Histograms and empirical densities of the sojourn times per transition type for the following starting states: performing P in (a), and default D in (b).