{
  "abstract": "Banks test their probability-of-default (PD) grades by comparing the default rate in each grade with the assigned PD, usually by a binomial or Jeffreys test that treats borrowers as independent. Defaults are not independent: a common economic factor moves all of them. I prove that in the one-factor Gaussian model the variance of a grade's default rate equals the binomial variance plus a term equal to the variance of the conditional PD, which does not shrink with the number of borrowers. The information in one year of data is therefore capped: with an asset correlation of 0.04, the Basel value for credit cards, a grade of any size is worth at most 40 to 85 independent borrowers at the PDs studied here. As a consequence, the binomial test rejects a correct PD in 40% to 89% of years for grades of 1,000 to 10,000 borrowers, and its size tends to 1 as the grade grows. Detecting a PD understated by 25% needs 20 years of data at a 5% PD and 36 years at 1%. I apply the tests to a seven-grade logistic PD model fitted to the UCI credit-card default data (30,000 accounts, hold-out AUC 0.787). Within the single observation period the hold-out default counts agree with the grade PDs (joint chi-square p = 0.116). I explain why that within-period test is valid but says nothing about calibration in future years.",
  "abstract_shown": true,
  "affiliations": [
    "Former Officer, CGGB"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Manoj Kumar Aguru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgfv79z3tfcr7.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23653368",
  "fulltext_url": null,
  "id": "bgfv79z3tfcr7",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23653368",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23653368/files/05_pd_backtest_correlated_defaults.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgfv79z3tfcr7.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:20:56Z",
      "native_id": "23653368",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23653368"
    }
  ],
  "subjects": [],
  "title": "Backtesting Probability-of-Default Grades When Defaults Are Correlated: Exact One-Factor Acceptance Intervals, a Variance Bound on the Information in a Single Year, and an Application to 30,000 Credit-Card Accounts",
  "title_text": "Backtesting Probability-of-Default Grades When Defaults Are Correlated: Exact One-Factor Acceptance Intervals, a Variance Bound on the Information in a Single Year, and an Application to 30,000 Credit-Card Accounts",
  "type": "preprint",
  "updated_at": "2026-11-08T12:20:56Z",
  "url": "https://research.draconitic.com/paper/bgfv79z3tfcr7",
  "venue": "Zenodo",
  "versions": []
}