Analysis of an aggregate loss model in a Markov renewal regime

Paper: arXiv 2401.14553

Authors: Pepa Ramírez-Cobo, Emilio Carrizosa, Rosa Elvira Lillo

Abstract

In this article we consider an aggregate loss model with dependent losses. The losses occurrence process is governed by a two-state Markovian arrival process (MAP2), a Markov renewal process process that allows for (1) correlated inter-losses times, (2) non-exponentially distributed inter-losses times and, (3) overdisperse losses counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real operational risk database, the aggregate loss model is estimated by fitting separately the inter-losses times and severities. The MAP2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-losses times distribution leads to higher capital charges.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper uses advanced stochastic modeling (MAP2 and Markov renewal processes) combined with heavy-tailed distributions (double-Pareto Lognormal) to capture dependencies in operational risk. It validates the methodology using a real operational risk database, demonstrating the practical capital charge impacts of accounting for overdispersion compared to standard Poisson assumptions.

Research Flowchart

  flowchart TD
    A["Research Goal: Assess impact of dependent inter-loss times & overdispersion on operational risk capital charges"] --> B["Input Data: Operational Risk Database"]
    
    B --> C["Model Estimation Strategy"]
    
    C --> D["Fit MAP2 to Inter-Loss Times<br/>(via Likelihood Maximization)"]
    C --> E["Fit Double-Pareto Lognormal<br/>to Loss Severities"]
    
    D --> F["Aggregate Loss Model Integration"]
    E --> F
    
    F --> G["Key Finding: Accounting for MAP2 dependence & overdispersion leads to higher capital charges vs. Poisson model"]