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

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.

💡 Research Summary

This paper introduces a novel dependent risk model for insurance portfolios in which the frequency vector of claim counts is modeled as a tree‑structured Markov random field (MRF) with Poisson marginals. The authors build on the binomial‑thinning operator to construct a recursive representation of the count vector N = (N_v) over a tree T = (V, E). Each node v carries its own mean parameter λ_v, allowing heterogeneous marginal intensities, while a dependence parameter α_{pa(v),v} governs the correlation between a parent node and its child. The construction ensures that N_v ∼ Poisson(λ_v) for all v, and that the joint distribution is invariant to the choice of root, preserving the undirected nature of the model.

Key theoretical contributions include: (1) Theorem 2.2, which proves the root‑free joint distribution and establishes admissible parameter constraints α_{pa(v),v} < min(λ_v/λ_{pa(v)}, λ_{pa(v)}/λ_v); (2) Proposition 2.6, which provides explicit closed‑form expressions for the joint probability mass function (pmf) and probability generating function (pgf) that factorize along the tree. These expressions enable exact likelihood evaluation in linear time with respect to the number of nodes, a stark contrast to common‑shock models whose parameter space grows exponentially with dimension.

The paper tackles two actuarial tasks. First, it derives the exact distribution of the aggregate loss S = Σ_v X_v, where each individual loss X_v is a compound Poisson sum of severities B_{v,j}. By exploiting the pgf of N, the authors develop efficient algorithms for computing tail risk measures such as TVaR without resorting to approximations. Second, they address risk allocation using (i) Euler’s principle to obtain TVaR contributions (pre‑allocation) and (ii) conditional‑mean risk‑sharing (post‑allocation). Both allocations are expressed in terms of the tree‑structured counts, allowing for closed‑form or fast numerical solutions.

A further contribution is the asymptotic analysis on infinite trees (Bethe lattices). The authors establish weak convergence results for the normalized aggregate loss as the tree grows without bound, providing insight into the behavior of large‑scale portfolios and linking the model to classical results in statistical physics.

Parameter estimation is addressed in Section 5. Because the pmf is explicit, maximum‑likelihood estimation is feasible; the authors propose an EM‑type algorithm that treats the latent “innovation” counts L_v and the thinned parent contributions as hidden variables. The admissible parameter space Λ is explicitly characterized, ensuring that the thinning operation never exceeds the feasible number of inherited events.

The methodology is illustrated on a real‑world dataset of extreme rainfall events in Quebec, Canada. Extreme daily rainfall counts above a high threshold are modeled as Poisson, justified by extreme‑value theory (Coles, 2001). Each weather station is a node; spatial adjacency defines the tree edges. Calibration yields λ_v reflecting local storm frequency and α_{pa(v),v} capturing spatial propagation of extreme events. Compared with a traditional multivariate Poisson model based on common shocks, the tree‑structured MRF achieves a higher log‑likelihood and more accurate TVaR predictions. Risk allocation results clearly differentiate high‑risk mountainous stations from low‑risk plains, offering actionable guidance for reinsurance treaty design and regional capital allocation.

In summary, the paper makes a substantial contribution to actuarial science by marrying graphical models with Poisson count data. It delivers a flexible yet computationally tractable framework for modeling dependence in claim frequencies, provides exact tools for aggregate risk assessment and allocation, extends the theory to infinite networks, and validates the approach on meaningful climate‑risk data. This work opens avenues for applying tree‑based MRFs to other domains where count data exhibit hierarchical or spatial dependence, such as epidemiology, cyber‑risk, and operational loss modeling.