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Natural disasters are events triggered by natural phenomena such as floods, storms, and earthquakes.
However, what truly turns these events into disasters or catastrophes is their devastating impact on human societies. A hurricane occurring in the middle of the ocean, for example, may go largely unnoticed—regardless of its intensity—if it does not affect human lives or physical assets.
Natural catastrophes pose major challenges for both insurers and governments. Insurers may face substantial losses that can lead to insolvency and even threaten the stability of the national financial system. Governments, in turn, are expected to provide financial assistance to affected households and take responsibility for rebuilding public infrastructure.
Why are NatCat problematic for statistics?
Why are NatCat problematic for statistics?
The key challenge with natural catastrophes is their rarity—both in time and geographic occurrence. This poses a fundamental conflict with the cornerstone of statistics: the law of large numbers.
The statistical problem posed by NatCat events is unique.
Consider rare diseases: although they are uncommon, there may still be around 1,000 known cases worldwide at a given time. While this presents social and economic challenges, it doesn’t necessarily pose a statistical one. A dataset of 1,000 observations is often sufficient for meaningful analysis, and in some cases, the entire affected population can be studied, providing reliable insights.
In contrast, NatCat events in a specific region may have occurred only four or five times in recorded history. With such sparse data, most conventional statistical methods become unusable. In these cases, Extreme Value Theory (EVT) becomes essential—a probabilistic framework specifically designed to analyze and model the behavior of rare and extreme events.
“There is always going to be an element of doubt as one is extrapolating into areas one doesn't know about. But what EVT is doing is making the best use of whatever data you have about extreme phenomena”
Prof. Richard Smith, University of North Carolina
Why treat extreme values separately?
Why treat extreme values separately?
Natural catastrophes are distinct meteorological events that differ fundamentally from everyday weather phenomena. For example, a heavy storm producing 165 mm of precipitation is not simply a more intense version of a rainy day with 5 mm of rainfall.
These events arise from entirely different physical mechanisms. Although we may observe the same variable—precipitation—the underlying data-generating processes are not the same. This is why extreme values should be analyzed separately from the rest of the data: they belong to a different statistical regime and require specialized methods to be properly understood.
This is where EVT comes into the picture.
What is "extreme" in EVT?
What is "extreme" in EVT?
Extreme Value Theory (EVT) has long been a fundamental tool in hydrology—well before it began gaining attention in finance and insurance. As its name suggests, EVT focuses on modeling the behavior of extreme values.
But what exactly qualifies as an "extreme" event or value? Indeed, this notion needs definition.
In EVT, the definition of an extreme value can vary. It might refer to the maximum observation over a certain time period, the value of an observation that exceeds a specified threshold, or even an exceedance relative to another variable. Each of these approaches captures a different aspect of extremity, tailored to the type of risk or data under consideration.
EXTREME VALUE ANALYSIS (EVA) ON CLAIMS DATA
EXTREME VALUE ANALYSIS (EVA) ON CLAIMS DATA
Many insurers and reinsurers hold data on claims resulting from NatCat events. This data is a valuable source of information about each insurer’s specific exposure to such risks. EVA enables the estimation of key quantities—such as return periods, return levels, and exceedance probabilities—using relatively short historical records. These estimates are essential for NatCat risk pricing and management.
What are the strengths of EVA compared to the traditional reliance on third-party NatCat models?
What are the strengths of EVA compared to the traditional reliance on third-party NatCat models?
Lower modeling risk
EVA has sound theoretical basis and many of its tools avoid distributional assumptions, thereby reducing the risk that these do not hold in practice.
Traditional NatCat models have a complex structure, accumulating at every module, modelling risk: the possibility that some of these assumptions may not hold in reality.
Moreover, the thousands of events generated in NatCat models remain simulations, not actual occurrences. No model, regardless of sophistication, can truly replicate the complexity and unpredictability of real extreme events, simply because these models remain calibrated on limited real-world data.
Utilization of the insurer’s own claims data
An insurer’s own claims data is a valuable source of information that is often overlooked in the context of NatCat risk due to its limited historical depth. Typically, claims data related to natural events consist of frequent small losses and occasional large spikes, which mark the significant events of interest for EVA. Any dataset longer than 10 years should not be disregarded.
NatCat models may sometimes bypass actual claims data, relying instead on asset exposure and damage functions. For example, the expected flood damage to a house near a river basin might be estimated based on its inundation index, building codes, and architectural style. These methods often involve averaging, interpolation, or extrapolation—processes that can introduce bias and obscure the real risk profile.
More conservative (prudent) estimates
Extreme Value Analysis (EVA) applied to claims data is likely to yield more conservative loss estimates. This stems from the theoretical assumption that losses are unbounded, whereas in practice, an insurer’s losses are capped by the maximum value of the insured assets.
In contrast, NatCat models typically use damage functions that flatten at higher hazard intensities—meaning that as the intensity of a natural event increases, the incremental damage to assets decreases.
Extrapolating a 200-year event from just 10 years of data using EVA must be done with caution. However, such estimates are almost always higher than those produced by NatCat models. Importantly, EVA provides a robust theoretical framework that allows for the quantification of uncertainty in these estimates.
Conceptual and computational simplicity
EVT is a specialized and sophisticated field, yet many of its practical applications involve only a limited number of parameters—even in high-dimensional settings.
In the case of NatCat claims data, analyses typically involve a few dozen variables. Depending on the application, tasks such as fitting extreme value distributions, computing confidence intervals, estimating quantiles, or calculating exceedance probabilities can be efficiently handled using open-source statistical tools, without requiring significant computational resources.
Moreover, many EVT problems have analytical solutions, which further simplifies and accelerates the analysis process.
Transparency and explainability
We live in a world where the explainability of statistical methods is becoming a competitive advantage over complex systems that may offer high performance but lack transparency.
Although NatCat models follow a clear and logical structure, each of their components—such as hazard modeling, vulnerability, and damage functions—encapsulates data, assumptions, and procedures that are often inaccessible or non-transparent to end users. For example, damage functions may rely on average values across building codes or on datasets not calibrated to the specific exposure profile of a given insurer.
In contrast, applying EVA to claims data involves no hidden information. EVA is grounded in well-established probabilistic theory, allowing its results to be not only explained but rigorously demonstrated. Moreover, it enables the quantification of uncertainty in a reliable and interpretable way, providing an additional layer of insight into risk assessment.
CASE STUDY: MODELING NATCAT LOSSES IN BELGIUM
CASE STUDY: MODELING NATCAT LOSSES IN BELGIUM
July 2025
July 2025
In this case study, we apply several statistical models to NatCat losses reported by Belgian insurers, with the aim of estimating the return level of the July 2021 flood. This flood was the most expensive event of the past decade, with losses four times higher than the previous record.
We analyze a dataset comprising 29 observations, each representing the total insured loss per NatCat event over the last ten years, aggregated across all Belgian insurers. The dataset is published by Assuralia, the association of Belgian insurers, and is publicly available on their website.
All recorded losses exceed EUR 10 million, indicating the use of a reporting threshold, possibly discretionary. To model this data, we compare three statistical distributions:
Three candidate distributions
Three candidate distributions
Log-normal distribution
The log-normal distribution is widely used in modeling financial quantities such as claims and returns. It features moderately heavy tails and is a natural first choice for modeling loss data. It is defined by two parameters, which can be easily estimated from the sample.
Pareto distribution (with positive shape parameter)
The Pareto distribution has heavy tails, which implies a higher probability of extreme losses compared to the log-normal. This makes it particularly suitable for modeling rare but severe NatCat events.
Generalized Pareto distribution (GPD)
A core tool from EVA, the GPD generalizes the Pareto distribution and is especially appropriate when modeling exceedances over a threshold—as is the case here, given the EUR 10 million limit.
The choice between the Pareto and the GPD is subtle but important—both theoretically and practically. While the two distributions are mathematically related through a linear transformation, they differ in their parameterization.
Parameter estimation for the Pareto distribution is straightforward. In contrast, estimating the parameters of the GPD typically requires numerical methods.
A key practical consideration when using the GPD is the selection of the threshold above which data are modeled. This choice significantly influences the results. While there is no universally optimal threshold, several diagnostic tools can guide the selection of an appropriate level.
Return periods and comparison with the Belgian Royal Meteorological Institute's estimates
Return periods and comparison with the Belgian Royal Meteorological Institute's estimates
Consider the total insured loss of €2 311.1 million caused by the July 2021 floods in Belgium. Historical data shows an average of three natural catastrophe events per year exceeding €10 million in insured damage.
Based on this, the estimated return period for such a high-impact event varies significantly depending on the model used:
Log-Normal distribution: ~400 years
Pareto distribution: ~10 years
GPD: up to 50 years (depending on the chosen threshold, our calculations suggest the return period may be even lower)
In the report Closing the Climate Insurance Protection Gap in Belgium (page 4), the Belgian Royal Meteorological Institute (RMI) notes that, besides the fact that the floods in Wallonia in the summer of 2021 were “an eye-opener”, a similar rainfall event has an estimated return period of 10 to 20 years.
While the RMI’s estimates are based on rainfall intensity, and our figures refer to insured losses, it is noteworthy that the return periods derived from the Pareto and GPD models align well with the RMI’s assessment.
This consistency reinforces the value of using extreme value theory for estimating NatCat risk.
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