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On a risk model with tree-structured Poisson Markov random field frequency, with application to rainfall events

Hélène Cossette, Benjamin Côté, Alexandre Dubeau, Etienne Marceau

TL;DR

This paper develops a tree-structured risk model where risk frequencies follow a Poisson marginal distribution arranged on a tree via a binomial thinning mechanism, yielding a flexible yet scalable multivariate Poisson MRF. It derives exact joint distributions, explicit portfolio loss transforms, and a link to common shocks, enabling precise aggregate risk analysis and exact risk allocations under TVaR and covariance-based rules. The estimation workflow combines correlation-based MST tree recovery with maximum likelihood parameter fitting, and the methodology is validated on extreme rainfall data across weather stations, illustrating scalability to high dimensions and exact tail-risk computations via FFT. The work demonstrates interpretability and computational efficiency for high-dimensional risk pooling in environmental applications, while outlining future extensions to severities, GLMs, and broader distribution families.

Abstract

In many insurance contexts, dependence between risks of a portfolio may arise from their frequencies. We investigate a dependent risk model in which we assume the vector of count variables to be a tree-structured Markov random field with Poisson marginals. The tree structure translates into a wide variety of dependence schemes. We study the global risk of the portfolio and the risk allocation to all its constituents. We provide asymptotic results for portfolios defined on infinitely growing trees. To illustrate its flexibility and computational scalability to higher dimensions, we calibrate the risk model on real-world extreme rainfall data and perform a risk analysis.

On a risk model with tree-structured Poisson Markov random field frequency, with application to rainfall events

TL;DR

This paper develops a tree-structured risk model where risk frequencies follow a Poisson marginal distribution arranged on a tree via a binomial thinning mechanism, yielding a flexible yet scalable multivariate Poisson MRF. It derives exact joint distributions, explicit portfolio loss transforms, and a link to common shocks, enabling precise aggregate risk analysis and exact risk allocations under TVaR and covariance-based rules. The estimation workflow combines correlation-based MST tree recovery with maximum likelihood parameter fitting, and the methodology is validated on extreme rainfall data across weather stations, illustrating scalability to high dimensions and exact tail-risk computations via FFT. The work demonstrates interpretability and computational efficiency for high-dimensional risk pooling in environmental applications, while outlining future extensions to severities, GLMs, and broader distribution families.

Abstract

In many insurance contexts, dependence between risks of a portfolio may arise from their frequencies. We investigate a dependent risk model in which we assume the vector of count variables to be a tree-structured Markov random field with Poisson marginals. The tree structure translates into a wide variety of dependence schemes. We study the global risk of the portfolio and the risk allocation to all its constituents. We provide asymptotic results for portfolios defined on infinitely growing trees. To illustrate its flexibility and computational scalability to higher dimensions, we calibrate the risk model on real-world extreme rainfall data and perform a risk analysis.

Paper Structure

This paper contains 39 sections, 10 theorems, 47 equations, 18 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Tree-structured MRFs with Poisson marginal distributions
  3. Main characteristics of tree-structured MRFs with Poisson marginal distributions
  4. A subfamily of the multivariate Poisson distribution based on common shocks
  5. Aggregate analysis of the portfolio
  6. Risk sharing
  7. Computation of risk allocations
  8. Asymptotic results on linear risk sharing
  9. Estimation procedure
  10. Tree structure estimation
  11. MPMRF maximum likelihood estimators
  12. Data illustration
  13. Datasets 2 and 3 construction
  14. Frequency models comparison
  15. MPMRF risk model on the 10-station dataset
  16. ...and 24 more sections

Key Result

Theorem 2.2

Consider a tree $\mathcal{T} = (\mathcal{V}, \mathcal{E})$, and let $\mathcal{T}_r$ be its rooted version, for some $r \in \mathcal{V}$. Given a vector of mean parameters $\boldsymbol{\lambda}=(\lambda_v,\,v\in\mathcal{V})$ where $\lambda_v>0$ for every $v\in\mathcal{V}$ and a vector of dependence p Then, $\boldsymbol{N}$ is a MRF with a unique joint distribution whichever the chosen root of $\mat

Figures (18)

  • Figure 1: Filial relations in a rooted tree.
  • Figure 1: Parameters $\gamma_{\mathcal{W}}$ for each set $\mathcal{W}$ of vertices in Figure \ref{['fig:five-nodes']}.
  • Figure 1: Three undirected tree structures for the sensitivity analysis.
  • Figure 2: Tree $\mathcal{T}$ of Example \ref{['ex:MPCS']} and $\boldsymbol{N}$ components' common shock representations.
  • Figure 2: Vertex numbers, meteorological stations, and climate ID suffixes, and the datasets in which they are used.
  • ...and 13 more figures

Theorems & Definitions (19)

  • Definition 2.1: MRF
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Example 2.8
  • Remark 3.1
  • Definition 4.1: OGFEA
  • ...and 9 more