Ruin probability for the quota share model with~phase-type distributed claims

Paper: arXiv 2303.07705

Authors: Krzysztof Burnecki, Zbigniew Palmowski, Marek Teuerle, Aleksandra Wilkowska

Abstract

In this paper, we generalise the results presented in the literature for the ruin probability for the insurer–reinsurer model under a pro-rata reinsurance contract. We consider claim amounts that are described by a phase-type distribution that includes exponential, mixture of exponential, Erlang, and mixture of Erlang distributions. We derive the ruin probability formulas with the use of change-of-measure technique and present important special cases. We illustrate the usefulness of the introduced model by fitting it to the real-world loss data. With the use of statistical tests and graphical tools, we show that the mixture of Erlangs is well-fitted to the data and is superior to other considered distributions. This justifies the fact that the presented results can be useful in the context of risk assessment of co-operating insurance companies.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents rigorous mathematical derivations of ruin probability using phase-type distributions and change-of-measure techniques in risk theory. It validates the theoretical framework empirically by fitting a mixture of Erlangs to real-world insurance loss data using statistical goodness-of-fit tests.

Research Flowchart

  flowchart TD
    A[Research Goal: Generalise Ruin Probability<br/>for Quota Share Reinsurance Model] --> B[Model Formulation & Data Input:<br/>Phase-Type Claims & Real Losses]
    B --> C[Derive Formulas via<br/>Change-of-Measure Technique]
    B --> D[Fit Phase-Type Distributions<br/>Exponential, Erlang, Mixtures]
    C --> E[Statistical Tests & Graphical Assessment]
    D --> E
    E --> F[Key Outcome: Mixture of Erlangs fits best<br/>& optimizes co-insurer risk assessment]