Executive Summary

This document synthesizes key findings on the application of extreme value statistics within the insurance industry, particularly in reinsurance. The central thesis is that while Extreme Value Theory (EVT) is indispensable for modeling large claims and managing tail risk, standard methods require significant adaptation to address the unique data challenges inherent in actuarial practice.

The modeling of largest claims is paramount for reinsurers, as the aggregate sum of claim payments is dominated by its largest components. Data from sectors like marine liability exhibit extremely heavy tails, where classical risk‑management principles are rendered inapplicable, necessitating specialized techniques.

Key challenges in applying EVT to insurance data include:

• Data Incompleteness: Datasets are frequently affected by right‑censoring (due to policy limits or claim settlement delays), truncation (due to maximum possible losses), and missing entries (Incurred But Not Reported claims).
• Model Constraints: The commercial retention and limit levels in reinsurance contracts do not necessarily align with statistically optimal thresholds for tail modeling.
• Evolving Risk Landscapes: Covariate information and non‑stationarities, especially those driven by climate change, demand more sophisticated regression and multivariate models.

To address these issues, the field has developed specialized adaptations:

• Truncation and Tempering Models: These methods account for situations where loss distributions are capped by physical or contractual limits (truncation) or where tail behavior weakens at extreme levels due to factors like claim management (tempering).
• Censoring‑Adapted Estimators: Techniques like the Kaplan‑Meier estimator and modified Hill‑type estimators have been developed to handle censored data, which is common in long‑tailed business lines like liability insurance.
• Full Distribution Models: "Splicing" or "composite" models combine distributions for the body of claims (e.g., Mixed Erlang) with a heavy‑tailed distribution for the tail (e.g., Generalized Pareto), providing a comprehensive fit for the entire loss range. Alternatives like matrix‑Mittag–Leffler distributions offer a unified approach without an arbitrary splicing point.
• Advanced Modeling: Regression techniques incorporate covariate information to model extremes, while multivariate models are crucial for understanding spatial diversification and dependencies between different lines of business, particularly in the context of natural catastrophes.

Ultimately, a profound understanding and tailored application of extreme value statistics are critical for the sound pricing, risk management, and continued insurability of catastrophic risks in a changing climate.

1. The Imperative of Extreme Value Statistics in Insurance

The robust modeling of large claims is a foundational element of actuarial science. Because the aggregate sum of payments in an insurance portfolio is naturally dominated by the largest claims, a thorough understanding of the tail behavior of claim distributions is critical for accurate pricing and effective risk management. This is especially true for the reinsurance sector, for which the handling of extreme events is a core responsibility.

A key concept is the Pareto‑type model for a loss random variable X, where the probability of exceeding a value x is defined as: P(X > x) = x⁻¹/ξ * ℓ(x) Here, ξ > 0 is the extreme value index, and ℓ(x) is a slowly varying function.

An illustrative case is the marine liability data analyzed by Guevara‑Alarcón et al. [45]. Statistical analysis of these large losses reveals an extreme value index between 0.8 and 1.1. This finding is significant because it implies that the existence of the first moment of the distribution is questionable, and the second moment is clearly infinite. Consequently, classical risk‑theoretical principles do not apply, and standard insurance coverage is not feasible. The tail risk in such cases is typically transferred to reinsurers.

Actuarial methods for measuring and controlling risk must be tailored to specific situations and data constraints. Standard Extreme Value Analysis (EVA) methods often require adaptation to address challenges like data availability, censoring, and truncation, which are common in insurance practice.

2. Reinsurance Structures and Inherent Data Challenges

Reinsurance contracts are the primary mechanism through which insurance companies transfer their exposure to very large claims. The structure of these contracts and the nature of actuarial data create specific challenges for statistical modeling.

2.1 Common Reinsurance Forms

• Proportional Reinsurance: In a quota‑share (QS) treaty, the insurer and reinsurer share risk proportionally (Ri = aXi). While simple to implement, this form does not alter the shape of the tail risk for the insurer.
• Excess‑of‑Loss (XL) Reinsurance: This is the most relevant form for managing extremes. The reinsurer pays for the portion of each claim that exceeds a predefined retention level M, often up to a limit L. The reinsured amount is min{(Xi − M)+, L}. This structure is intrinsically linked to the Peaks‑Over‑Threshold (POT) methodology at the heart of EVA. Variations include Stop‑Loss (applied to aggregate claims) and CAT‑XL (applied to aggregate claims from a single catastrophe event).
• Large Claim Reinsurance: Contracts that cover the r largest claims in a portfolio (e.g., ECOMOR reinsurance) are a natural fit for extreme value theory. However, they have not gained widespread popularity due to the complexity of calculating premiums and modeling the retained risk.

2.2 Data Incompleteness and Modeling Constraints

Actuarial datasets for XL reinsurance are subject to several forms of incomplete information that complicate statistical analysis:

• Censoring: In bounded XL treaties, the reinsurer may not have information on the full claim amount beyond the limit L. Furthermore, claims that are "Reported But Not Settled" (RBNS) at an evaluation date are right‑censored, as the final payout is unknown. This is a significant issue in long‑tailed lines like liability and catastrophe insurance, where settlement can take decades.
• Missing Data: "Incurred But Not Reported" (IBNR) claims represent losses that have occurred but have not yet been reported to the insurer, creating gaps in the data.
• Data Scarcity: Extreme events are rare by definition, leading to sparse data. Analysis often requires merging historical data with information from related risks and expert opinion.
• Truncation: The existence of a maximum possible loss, due to contractual limits or physical constraints (e.g., total building values in a catastrophe zone), results in the need for truncation modeling.
• Tempering: The tail behavior of a distribution may be altered by operational factors. For instance, the highest‑level claims may be subject to more intense inspection and management, leading to weaker tails than would be apparent from intermediate‑level claims. This effect is known as tempering.

2.3 Premiums and Risk Measures

The pure premium for an unlimited XL treaty, Π(M), is the expected reinsured amount E{(Xi − M)+}. This premium is directly related to the mean excess function e(M) = E(Xi − M | Xi > M), a fundamental tool in EVA. Risk measures like Value‑at‑Risk (VaR) and Conditional Tail Expectation (CTE) are also built upon these concepts. A key challenge is that the commercial values of retention M and limit L are not necessarily the same as the statistically optimal thresholds at which tail models provide the best fit. This necessitates the construction of "full models" that fit well across the entire distribution, not just the extreme tail.

3. Adaptations of Classical Tail Analysis for Insurance Data

To address the specific constraints of insurance data, classical EVA techniques have been modified to handle truncation and tempering.

3.1 Truncation Modeling

Standard Pareto‑type models can assign probability to unrealistically large or impossible loss amounts. Truncation addresses this by imposing an upper bound T on the loss distribution, which can arise from policy limits or physical constraints. This effect can sometimes be observed in data when a Pareto QQ‑plot is linear for most points but shows non‑linear deviations for the very largest claims.

Key aspects of truncation modeling include:

• POT Approximation: For a truncated loss X, the probability of an exceedance over a threshold t can be approximated as: P(X/t > y | X > t) ≈ (y⁻¹/ξ − β⁻¹/ξ) / (1 − β⁻¹/ξ), where β is related to the ratio of the truncation point T to the threshold t.
• Estimation: Based on this approximation, estimators have been developed for the extreme value index (ξ̂Tk,n), the odds of the truncated probability mass (D̂T), and the truncation point itself (T̂k,n).
• Example (German Flood Risk): Analysis of aggregate annual flood losses in Germany (1980–2013) shows a Pareto QQ‑plot that becomes non‑linear at the top data points. A truncated Pareto QQ‑plot, which accounts for an estimated endpoint, provides a more linear and thus better fit, suggesting the presence of truncation effects.

3.2 Tempering Models

Tempering is an alternative to truncation for modeling distributions where the power‑law behavior does not extend indefinitely. Instead of an abrupt cutoff, the tail of the distribution decays more quickly than a pure power‑law. This can be modeled by multiplying a Pareto‑type survival function by a decay term.

Key aspects of tempering models include:

• Survival Function: The general form for Weibull tempering is P(X > x) = cx⁻αe⁻(βx)τ.
• POT Approximation: For a tempered distribution, the probability of an exceedance over a threshold t can be approximated as: P(X/t > y | X > t) ≈ x⁻αe⁻λ(xτ−1).
• Estimation: Maximum likelihood and weighted least‑squares methods can be used to estimate the parameters α, λ, and τ from the data. These methods can also be used to estimate return periods and extreme quantiles.
• Example (Norwegian Fire Insurance): The Pareto QQ‑plot for this dataset becomes concave for the largest claims, while a Weibull model appears to fit well in this region. This suggests that a tempering model is appropriate. A tempering QQ‑plot using estimated parameters shows a strong linear fit, supporting the model choice.

4. Modeling with Censored Data

In long‑tailed lines of business, such as liability insurance, claim settlement can be a lengthy process. When a portfolio is evaluated, many claims are not yet settled, and their final costs are unknown. This right‑censoring must be accounted for in any statistical analysis.

Key methods for handling censored data include:

• Censoring‑Adapted Hill Estimator: The standard Hill estimator for the extreme value index can be adapted for censoring by dividing it by the proportion of non‑censored data among the top k observations: H(c)k,n = (Σ log(Z(n‑j+1)/Z(n‑k))) / (Σ δ(n‑j+1))
• Kaplan‑Meier Estimator: This non‑parametric estimator (F̂n) for the distribution function is widely used as a basis for other estimators in the presence of censoring.
• Other Estimators: Various other estimators, including those proposed by Worms (ξ̂Wk) and moment‑based estimators (ξ̂Mk), have been developed to handle censored data. These estimators often exhibit higher bias, particularly when a large proportion of intermediate data is censored.
• Extreme Quantile Estimation: The Weissman estimator for extreme quantiles can be adapted for censoring by using a censoring‑adapted extreme value index estimate and the Kaplan‑Meier estimator for the survival function.
• Example ((Loss, ALAE) Data): Analysis of a general liability claims dataset with 34 censored losses shows that different estimators for the extreme value index behave differently. The estimates indicate a heavy tail (ξ between 0.5 and 1.0), but some estimators suffer from substantial bias.
• Expert Information: For censored observations, expert opinion (e.g., actuarial reserves for open claims) can be incorporated into statistical procedures to improve estimation.

5. Full Models for Claims: The Splicing Approach and Alternatives

While EVA focuses on the tail, actuaries often need models that fit the entire range of a loss distribution for calculating premiums and capital requirements. Splicing (or composite) models address this by combining two or more distributions: one for the body of small‑to‑moderate "attritional" losses and another for the tail of "large" losses.

5.1 The Splicing Methodology

• Structure: A typical splicing model combines a light‑tailed distribution for the body (e.g., Exponential, Lognormal, Weibull) with a heavy‑tailed distribution for the tail (e.g., Pareto, Generalized Pareto).
• Mixed Erlang (ME) Splicing: A semi‑automatic and flexible approach uses a mixture of Erlang distributions for the body. The ME class is dense, tractable for calculations (like XL premiums), and can be fitted efficiently using the Expectation‑Maximization (EM) algorithm, even with censored or truncated data.
• Estimation: The splicing point t can be chosen based on an initial EVA, with the remaining parameters estimated via algorithms like EM. The final estimates for the tail parameters are typically close to the initial EVA estimates.
• Risk Measure Calculation: Formulas for calculating VaR and the pure premium Π(u) can be derived for the two‑component spliced model, distinguishing between cases where the threshold u is below or above the splicing point t.
• Example ((Loss, ALAE) Data): For the censored liability loss data, a mean excess plot suggests a change in behavior around 10⁵. A spliced model using three ME components for the body and a Pareto distribution for the tail (with splicing point at the 50th largest observation) provides a very good fit to the data, as shown by the QQ‑plot.

5.2 Alternatives to Splicing

While effective, splicing requires the somewhat arbitrary choice of a threshold. An alternative approach is the use of matrix‑Mittag–Leffler distributions. This class of distributions can model heavy tails without needing an increased number of parameters and avoids the need for splicing, providing a unified and high‑quality fit across the entire range of the distribution.

6. Advanced Modeling Techniques in Insurance

6.1 Regression Modelling with Covariates

The availability of granular data has increased the potential to use covariate information in modeling extremes. This is particularly relevant for insurance applications where risk factors can be identified.

• Methodology: Regression techniques for censored data can be developed using local smoothing methods. For instance, the Akritas–Van Keilegom extension of the Kaplan‑Meier estimator allows for the estimation of conditional distributions.
• Application: This approach can be used to model the extremes of final claim payments in a long‑tailed portfolio as a function of a covariate like the number of development years, making it possible to detect if longer development times are associated with heavier tails. Estimators for the conditional extreme value index can be derived within this framework.

6.2 Multivariate Modelling

Multivariate extremes are critical for reinsurers, who manage risk through aggregate covers across different business lines and spatial diversification across multiple insurers.

• Multivariate Splicing: The splicing approach can be extended to multiple dimensions. A bivariate model might use a multivariate mixed Erlang (MME) distribution for the body and a bivariate Generalized Pareto Distribution (GPD) for the tail.
• Dependence Estimation: For multivariate models, estimating the dependence structure is key. The Pickands dependence function A(t) can be estimated from censored data using a maximum likelihood approach adapted for censoring.
• Application: In the (Loss, ALAE) data example, this framework can be used to model the joint distribution of losses and associated expenses and to calculate the pure premium for complex reinsurance contracts that depend on both variables.

7. Natural Catastrophe Insurance and Climate Change

The importance of multivariate extreme value statistics is most apparent in the insurance of natural catastrophes, which are characterized by very heavy‑tailed losses and strong correlations across claims and perils.

• Peril‑Specific Modeling: Different perils require distinct modeling approaches. For example, modeling flood risk on a river network may use a max‑stable process calibrated with a river‑flow distance concept rather than a standard Euclidean distance.
• Role of Statistical Models: While bottom‑up physical models (e.g., hydrological models for floods) are valuable, they are often not designed to capture extremes fully. The toolkit of extreme value statistics, including regression on physical covariates, remains an essential ingredient in risk analysis.
• Climate Change as a New Challenge: Increasing evidence of climate change introduces non‑stationarity into frequency and severity data for extreme events. This poses a significant challenge for traditional EVA. A sound statistical modeling of extremes under these changing conditions is crucial for the proper risk management of reinsurance companies and for ensuring that coverage for natural hazards remains available and adequately priced.