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arxiv: 2502.14479 · v5 · submitted 2025-02-20 · 💱 q-fin.RM · q-fin.ST· stat.AP

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

Arno Botha , Tanja Verster , Roland Breedt This is my paper

Pith reviewed 2026-05-23 02:45 UTC · model grok-4.3

classification 💱 q-fin.RM q-fin.STstat.AP
keywords default riskterm-structureIFRS 9multistate modellingMarkov chainbeta regressionmultinomial logistic regressionresidential mortgages
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The pith

Multistate models of increasing complexity successively outperform simpler ones in estimating the term-structure of default risk for loans.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper compares three techniques for modelling the lifetime default risk of loans under IFRS 9 using a multistate regression framework. The techniques are applied in ascending order of complexity: a Markov chain, beta regression, and multinomial logistic regression. Using residential mortgage data, each more advanced model performs better than the prior one when estimating time-dependent transition probabilities from a rich set of inputs. A new suite of model diagnostics was developed to support the comparisons and assess sampling. The results point toward more accurate loss reserve calculations for banks.

Core claim

The paper establishes that modelling the transition probabilities of a nonstationary semi-Markov model for loan default behaviour as explicit functions of macroeconomic and loan-level variables produces successively better term-structure estimates as complexity rises from Markov chain to beta regression to multinomial logistic regression, demonstrated on residential mortgage data and aided by new diagnostic tools.

What carries the argument

Multistate regression-based approach that models transition probabilities explicitly as functions of input variables, applied in ascending order of complexity from Markov chain to beta regression to multinomial logistic regression.

If this is right

  • Each successive model outperforms the previous due to greater sophistication.
  • Novel model diagnostics can be reused to assess sampling representativeness and other modelling techniques.
  • Estimation of loss reserves will be more timeous and accurate under IFRS 9.
  • The nonstationary semi-Markov behaviour of loans can be captured through regression on loan-level and macroeconomic inputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework may extend to other loan portfolios if the diagnostics confirm representativeness across different data sets.
  • Banks could embed the multinomial logistic version directly into IFRS 9 reporting systems to reduce provisioning volatility.
  • The diagnostics themselves could serve as a general check for whether any multistate model respects the original loan cohort distribution.

Load-bearing premise

The assumption that outperformance across the three models can be attributed to increasing sophistication rather than differences in how the same data set was prepared or split for each technique.

What would settle it

Re-running all three models on identical data splits and preparations and finding that the performance gains with complexity disappear or reverse.

Figures

Figures reproduced from arXiv: 2502.14479 by Arno Botha, Roland Breedt, Tanja Verster.

Figure 1
Figure 1. Figure 1: The state space in which loans may reside at any point of their lifetimes. Our state space S is deliberately similar to that of Smith and Lawrence (1995), thereby promoting comparability and simplicity. By transforming the number of payments in arrears into states, the 7-state model from Grimshaw and Alexander (2011) is rather excessive in practice and comes at greater computational cost and sparser data, … view at source ↗
Figure 2
Figure 2. Figure 2: Illustrating the construction process of the outcome variable in BR-modelling for the P→D transition type, having used the entries within the time-dependent transition matrix 𝑇 (𝑡 ′ ). In Eq. 3, 𝑔1 (·) and 𝑔2 (·) are link functions, 𝜷 (𝑘𝑙) =  𝛽 (𝑘𝑙) 1 , . . . , 𝛽(𝑘𝑙) 𝑝1 T and 𝜽 (𝑘𝑙) =  𝜃 (𝑘𝑙) 1 , . . . , 𝜃 (𝑘𝑙) 𝑝2 T are vectors of estimable regression coefficients, 𝜂 (𝑘𝑙) 1𝑡 ′ and 𝜂 (𝑘𝑙) 2𝑡 ′ are linea… view at source ↗
Figure 3
Figure 3. Figure 3: Illustrating the estimation of two MLR-models across various loans within the starting state 𝑘 ∈ {P, D}, as a function of two covariates, loan amount [Principal] and the central bank policy rate [Repo_rate]. target state, which may not be true in reality for some states. For example, the policy rate [Repo_rate] may not be statistically relevant in predicting a transition from the performing state P to the … view at source ↗
Figure 4
Figure 4. 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 D is overlaid in summarising the line graph discrepancies over time. 4.2. Estimating the transition matrix 𝑇 of the Markov chain Having used the subsample D𝑆, we obtain an estimate 𝑇ˆ of the time-homogeneous transition matrix, which serves as our baseline model for … view at source ↗
Figure 5
Figure 5. 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). 4.3. Calibrating six BR-models and two MLR-models towards producing 𝑇-estimates In fitting either a BR– or MLR-model, we follow a thematic variable selection process using repeated regressions across themed subsets of input variables. This interactive pro… view at source ↗
Figure 6
Figure 6. Figure 6: Time graphs of actual vs expected 1-month transition rates for P→D across various techniques, having used D𝑉 respective to each technique. The MAE-based AD-statistic from Eq. 9 is calculated between each actual and expected rate pair in summarising the discrepancies over time. We present time graphs in Figs. 6–7 of the actual vs expected rates for a specific transition type 𝑘 → 𝑙. While these graphs includ… view at source ↗
Figure 7
Figure 7. Figure 7: Time graphs of actual vs expected 1-month transition rates for D→P across various techniques, having used D𝑉 respective to each technique. Graph design follows that of [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The implied term-structure of actual vs expected transition rates of type P→D over calendar time 𝑡 ′ , shown across various techniques. The MAE summarises the discrepancies between each pair of actual and expected rates over time. however interested only in those cumulative transition rates of type P→D, which signify the PD term-structure, as shown in [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cook’s distance plot for the BR-model of transition type P→D. Encircled points indicate highly influential observations. ordinary residuals), which are calculated as 𝑟 (P) i = 𝑦𝑖 − 𝑦ˆ𝑖 √︁ 𝑠ˆ 2 (𝑦𝑖) = 𝑦𝑖 − 𝑦ˆ𝑖 √︁ 𝜇ˆ𝑖(1 − 𝜇ˆ𝑖)/(1 + 𝜙ˆ 𝑖 , where 𝑠ˆ 2 (𝑦𝑖) is the estimated variance across all 𝑦𝑖 , as described in Subsec. A.1. Ferrari and Cribari-Neto (2004) admitted that the distribution of 𝑟 (P) 𝑖 , 𝑖 = 1, . … view at source ↗
Figure 10
Figure 10. Figure 10: Histograms and empirical densities of Pearson residuals 𝑟 (P) i in gauging the fit of the BR-models. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents a comparative study of three multistate techniques for modeling the term-structure of default risk under IFRS 9, ordered by ascending complexity (Markov chain, beta regression, multinomial logistic regression). Using residential mortgage data, it claims that each successive model outperforms its predecessor due to greater sophistication and introduces a novel suite of simple model diagnostics for assessing sampling representativeness and performance.

Significance. If the outperformance is demonstrated via consistent out-of-sample protocols and identical data handling, the work could improve the accuracy and timeliness of lifetime expected credit loss estimates required under IFRS 9, while the proposed diagnostics would offer reusable tools for multistate modeling in banking practice.

major comments (2)
  1. [Abstract] Abstract: The claim that 'each successive model outperforms the previous' supplies no quantitative metrics, error bars, tables, or description of the validation protocol (train/test splits, cross-validation procedure, or hold-out rules), preventing any assessment of whether gains are attributable to model complexity.
  2. [Abstract] Abstract: Attribution of results to 'greater sophistication' requires that the three techniques were applied under identical conditions (same feature sets, sampling procedures, and performance measures); no such information is provided, so the central causal claim cannot be evaluated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and constructive comments on the abstract. We address each point below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that 'each successive model outperforms the previous' supplies no quantitative metrics, error bars, tables, or description of the validation protocol (train/test splits, cross-validation procedure, or hold-out rules), preventing any assessment of whether gains are attributable to model complexity.

    Authors: We agree that the abstract, due to length constraints, omits quantitative metrics and validation details. The full manuscript provides these in the results and methodology sections, including out-of-sample performance tables, error metrics, and the train/test/hold-out protocol applied consistently across models. We will revise the abstract to incorporate key quantitative summary results and a concise description of the validation protocol. revision: yes

  2. Referee: [Abstract] Abstract: Attribution of results to 'greater sophistication' requires that the three techniques were applied under identical conditions (same feature sets, sampling procedures, and performance measures); no such information is provided, so the central causal claim cannot be evaluated.

    Authors: The models were applied under identical conditions, using the same feature sets, sampling procedures, and performance measures, as specified in the methodology section. The abstract's brevity did not explicitly restate this. We will revise the abstract to include a statement confirming that all comparisons used identical data handling and evaluation protocols. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper is an empirical comparative study of three regression techniques (Markov chain, beta regression, multinomial logistic regression) applied to residential mortgage data for term-structure default modeling under IFRS 9. The abstract reports observed outperformance across models but contains no equations, first-principles derivations, or predictions that reduce to fitted inputs by construction. No self-citations, uniqueness theorems, ansatzes, or renamings of known results appear in the provided text. The central claim is an empirical finding on data rather than a closed mathematical loop, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms or invented entities can be extracted. Regression coefficients in beta and logistic models are implicitly fitted to data and would count as free parameters if full text were reviewed.

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Reference graph

Works this paper leans on

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