Paper: arXiv 2508.15651

Authors: Barbara Dömötör, Ferenc Illés

Abstract

Banks are required to use long-term default probabilities (PDs) of their portfolios when calculating credit risk capital under internal ratings-based (IRB) models. However, the calibration models and historical data typically reflect prevailing market conditions. According to Basel recommendations, averaging annual PDs over a full economic cycle should yield the long-term PD. In practice, the available data are often temporally incomplete - even for high-risk portfolios. In this paper, we present a method for the simultaneous calibration of long-term PDs across all sub-portfolios, based on the single risk factor model embedded in the Basel framework. The method is suitable even for smaller, budget-constrained institutions, as it relies exclusively on the bank’s own default data. A complete dataset is not required - not even for any individual sub-portfolio - as the only prerequisite is the presence of overlapping data before and after the missing values, a mild condition that is typically met in practical situations.

Complexity vs Empirical Score

  • Math Complexity: 6.0/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a clear and practical dynamic extension of the single-factor Vasicek model to address incomplete default data in IRB banks. It offers a mathematically sound calibration method for simultaneous estimation of TTC and PiT PDs without needing external macroeconomic variables.

Research Flowchart

  flowchart TD
    A[Goal: Estimate Long-Term Through-the-Cycle PDs with Incomplete Data] --> B[Inputs: Bank's Internal Default Data & Overlapping Time Series]
    B --> C[Framework: Basel Single Risk Factor Model]
    C --> D[Method: Simultaneous Calibration Across All Sub-Portfolios]
    D --> E[Processing: Handle Missing Historical Values Without External Data]
    E --> F[Outcome: Long-Term TTC-PD Estimates for IRB Credit Risk Capital]

Cited References (18)

  1. A Simple and Consistent Credit Risk Model for Basel II/III, IFRS 9 and Stress Testing when Loan Data History is Short (2024) — link
  2. Through-the-cycle to Point-in-time Probabilities of Default Conversion: Inconsistencies in the Vasicek Approach (2023) — link
  3. An Approximate Multi-Period Vasicek Credit Risk Model (2017) — link
  4. The Interaction of the IFRS 9 Expected Loss Approach with Supervisory Rules and Implications for Financial Stability (2016) — link
  5. A methodology for point-in-time–through-the-cycle probability of default decomposition in risk classification systems (2012)
  6. Cyclical adjustment of point-in-time PD (2010) — link
  7. Credit scoring with macroeconomic variables using survival analysis (2009) — link
  8. Designing and Implementing a Basel II Compliant PIT-TTC Ratings Framework (2008)
and 10 more
  1. The empirical relationship between average asset correlation, firm probability of default, and asset size (2002) — link
  2. An Anatomy of Rating Through the Cycle (2001) — link
  3. Basel Committee on Banking Supervision (2023)
  4. Stressing of migration matrixes for International Financial Reporting Standard 9 and Internal Capital Adequacy Assessment Process calculations (2022) — link
  5. The European Banking Authority (2018) — link
  6. Problems in the application of credit risk models1 (2018)
  7. THE DISTRIBUTION OF LOAN PORTFOLIO VALUE (2002)
  8. A one-parameter representation of credit risk and transition matrices
  9. Basel Committee on Banking Supervision an Explanatory Note on the Basel Ii Irb Risk Weight Functions
  10. Calibrate the

0 of 18 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.