Introduction

Windstorms are one of the most impactful natural catastrophes in Europe, as their large scale, strong winds, and heavy precipitation cause, on average, dozens of fatalities and billions of euros in economic and insured losses each year1,2. European windstorms are strong, extratropical cyclones that originate either through cyclogenesis in the extratropical region3 or from the extratropical transitioning of tropical cyclones4. As the most powerful windstorms in Europe frequently occur during the winter months, they are often referred to as European winter storms.

Insights into the potential losses from such influential natural catastrophes are important for disaster risk prevention and preparedness policies, effective and efficient (re)insurance arrangements, and climate (change) impact studies. The state-of-the-art method for estimating these losses is the natural catastrophe model5, which consists of three modules: (1) the hazard module, which specifies the type, frequency, and intensity of the natural hazard; (2) the exposure module, which identifies the type and location of exposed assets; and (3) the vulnerability module, which consists of functions that describe the relationship between the natural hazard’s intensity and its associated damage to a variety of exposed assets6. Loss estimations are highly sensitive towards the vulnerability functions because damage to an exposed asset is commonly expressed as a ratio of the total reconstruction value of the asset and the functions apply to all exposed assets. This sensitivity becomes apparent when one considers a seemingly small and easily conceivable adjustment to relatively low outputs of a vulnerability function. For instance, recalibrating a previously anticipated damage ratio associated with a certain storm intensity from 0.5% to 1% would double the loss estimate of a natural catastrophe model. As such, the construction of accurate vulnerability functions is of fundamental importance to the accurate estimation of natural catastrophe losses7. Throughout this study, we refer to the total cost of restoring a building, prior to any insurance adjustments, as ground‐up damage.

Vulnerability functions can be derived analytically by simulating the response of the built environment to natural hazards of various intensities8, heuristically by using expert judgement9, empirically by calibrating statistical models on post-disaster damage data10, or in a hybrid form11. Although vulnerability functions are most often constructed empirically in the academic literature12, few empirical European winter storm vulnerability functions for residential buildings are available, primarily due to a lack of damage data. Most studies that did construct these functions based on empirical data solely had access to highly aggregated damage data13,14. The resulting shortcoming is that those vulnerability functions are less accurate, as essential variation in both the hazard and exposure characteristics are not captured at higher levels of aggregation. The only European winter storm vulnerability function that is calibrated on residential building damage observations at the residence-level stems from Schwierz et al.15, hereafter referred to as the Sw vulnerability function. Theirs is a simplified version of an insurance company’s proprietary vulnerability function, which solely describes the expected or mean damage ratio for the average residential building. Moreover, as Schwierz et al.15 do not provide information on the modelling methodology or on the estimation outcomes, it is challenging to evaluate the reliability and the uncertainties of the vulnerability function’s outcomes.

In this study, we aim to fill this gap by estimating European winter storm vulnerability functions for residential buildings. Based on a large sample of insurance claims (204,118 damage observations at the residence-level over the period 2008–2021) from winter storms in the Netherlands, and on an ultra-high-resolution dataset of their meteorological drivers, conditional vulnerability functions (hereafter referred to as vulnerability functions) are estimated with truncated beta regressions16. Conditional vulnerability functions describe the relationship between a natural hazard’s intensity and damage to exposed assets, conditional upon the asset being damaged12. Being exposed to a certain natural hazard intensity does not guarantee that the building will be damaged. Consequently, to estimate losses to multiple assets, one needs to augment the conditional vulnerability functions with the chances of an asset being damaged at different intensity levels.

There are four main advantages of using beta regressions to construct empirical vulnerability functions. First, they assume the dependent variable to be distributed on the unit interval (0, 1), which makes them more suitable to model damage ratios as compared to the (partially) unbounded distributions that are most often used in the empirical vulnerability function literature11,17. We considered using one-inflated beta regressions18 to accommodate the inclusion of complete loss observations, as these types of regressions assume the dependent variable to be distributed on the half-open unit interval (0, 1]. However, the number of complete loss observations in our sample was deemed insufficient to model separately. Second, aligning the theoretically dependent variable distribution with the empirical distribution also enables the end user to simulate a loss distribution by repeatedly drawing (simulating) random potential damage ratios from the conditional beta distribution of the vulnerability function. In addition to an expected loss estimate, such loss distributions describe the range of plausible losses, which is often referred to as the risk. Considering a loss distribution is important when losses are estimated for small sample sizes and the variance of the damage ratio estimates is large (i.e., when the damage ratio estimate is relatively uncertain). Under these circumstances, the sum of the expected individual losses that stem from the expected damage ratio estimates may deviate substantially from the materialised loss. Third, the beta regression is relatively flexible, as it accommodates the heteroskedasticity and distributional asymmetries that are often found in regressions on fractional data16. Fourth, this parametric approach enables us to account for the unobserved losses below the imposed insurance deductibles, which is a common characteristic of insurance claims data, through truncation adjustments in the model estimation process.

In addition to these methodological innovations, we are first to incorporate multiple damage-driving mechanisms of European winter storms in the vulnerability function. Whereas conventional European winter storm vulnerability functions solely account for damaging wind speeds8,9,10,11,13,14,15—which, for example, can tear off roofing material — the function we produce in this study also represents the damaging effects of precipitation, for instance, from water intrusion. In particular, our vulnerability function uses the daily maximum 3-second wind gust in a 10-minute period at 10 m height (W) and the daily 24-hour cumulative precipitation level (P24) as proxies for the damage-driving mechanisms of winter storms. Thus, next to describing the damaging effects from either wind or precipitation, our function allows for damage estimations from their combined presence. The vulnerability function can, therefore be considered as a compound vulnerability function for European winter storms19.

In general, our new vulnerability function can serve as input for private, academic, and open-source natural catastrophe models to more accurately estimate residential buildings damage from European winter storms. Due to climate change, precipitation levels in winter storms, and hence the frequency of precipitation extremes, are projected to increase across Europe —especially in the northern regions—20,21,22,23,24. This trend increases the role of precipitation in the damage process. Therefore, the compound vulnerability function is particularly suited for climate change impact analyses, as it also captures the damaging effects of precipitation. Our comparison of the expected damage ratio estimates from the compound vulnerability function with those from a vulnerability function that accounts solely for wind speeds shows that the latter underestimates the damage to residential buildings for high-precipitation events. For an average residential building, the expected damage ratio will be underestimated by 6% [21%] {57%} for winter storms that produce wind gusts of 20 m/s and 50 mm [75 mm] {100 mm} of precipitation in 24 hours. This 24-hour cumulative precipitation level only occurs once every 50 years [500 years] {~1000 years} at a specified location in the Netherlands. These estimates are determined by the Generalised Extreme Value and Generalised Logistic distributions that Stichting toegepast onderzoek waterbeheer (STOWA) fitted to the Dutch precipitation statistics25. Additionally, including the reconstruction value (Rv) as parsimonious regressor for relevant building characteristics demonstrates that such characteristics should be considered in vulnerability functions. The reason is that they influence the damage ratio estimate: for a given set of damaging meteorological conditions, residential buildings with higher reconstruction values have less damage relative to their reconstruction value compared to those with lower reconstruction values. Furthermore, in this study, we find a large variance of the damage ratio estimates, which implies that winter storm loss estimates for samples of limited size should be informed by a loss distribution, as previously discussed. The large variance of the damage ratio estimates may stem from the unexplained variability in vulnerability characteristics and the randomness inherent to the damage process. Finally, comparisons with a European winter storm vulnerability function for residential buildings from the literature highlight that there is considerable uncertainty around the wind gust speed threshold at which damage starts to occur15. Having this threshold wrong will lead to either under- or overestimations of damage to residential buildings from European winter storms.

The remainder of this paper is structured as follows. Section 2 presents the regression results and compares the vulnerability functions with the literature. Section 3 provides guidance on their use and concludes. Section 4 introduces the data and methods.

Results

Vulnerability function specifications

Table 1 presents the beta regression results of the vulnerability function specification with the lowest BIC (highest probability of being the true model), which is called the compound vulnerability function. Next to \(\sqrt{{Rv}}\), which represents relevant building characteristics, the compound vulnerability function includes \({W}^{3}\) and \({P24}^{3}\) as proxies for the damage-driving mechanism of winter storms. For comparative purposes, Table 1 also reports the results of the vulnerability function specification with the lowest BIC that does not contain a regressor to account for precipitation damage from winter storms. This more conventional vulnerability function specification is called the univariate vulnerability function. For the univariate vulnerability function, it turns out that the variables that were selected coincided in their transformations with those from the compound vulnerability function. By comparing the regression results of both vulnerability functions, it becomes apparent that there is a role for both wind gusts and the 24-hour cumulative precipitation in explaining the damage-driving mechanisms of winter storms. This finding follows from the positive association that the parameters of both regressors have towards the expected damage ratio in the compound vulnerability function specification and from its BIC being significantly lower than that of the univariate vulnerability function, according to the guidelines provided by Neath and Cavanaugh26. Lastly, the regression parameters of both models differ from zero at any conventional critical value.

Table 1 Regression results
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The reconstruction value

In both the compound and univariate vulnerability functions, the reconstruction value is negatively related to the expected damage ratio. This indicates that, for a given set of damaging meteorological conditions, residential buildings with higher reconstruction values have a smaller fraction of damage as compared to those with lower reconstruction values, albeit having a higher ground-up damage. For instance, the compound vulnerability function estimates a damage ratio of 0.65% and a ground-up damage of €1950 for a residential building with a reconstruction value of €300,000, wind gusts of 20 m/s, and a 24-hour cumulative precipitation level of 50 mm. The residential building with a reconstruction value of €370,000, in turn, has a lower damage ratio of 0.59% but a higher ground-up damage of €2183. In other words, although the ground-up damage from winter storms increases for residential buildings with higher reconstruction values, it increases less than proportionally to the increase in the reconstruction value. A proportional change would imply that a residential building with, for example, twice the reconstruction value of another, would also have a ground-up damage that is two times larger, all else equal. Moreover, the selected transformation of the reconstruction value (i.e., the square root) indicates that this effect diminishes for higher reconstruction values.

Finding a statistically significant parameter for \(\sqrt{{Rv}}\,\) also implies that vulnerability functions that do not account for differences in reconstruction values (or more generally, other relevant building characteristics) will reflect an “averaged out” effect of the omitted regressor. Such vulnerability functions produce less accurate damage ratios for residential buildings at either end of the reconstruction values range: underestimating damage to buildings with low reconstruction values and vice versa.

Wind gusts and precipitation

In both the compound and univariate vulnerability functions, the wind gust regressor is positively related to the expected damage ratio. Hence, higher wind gusts speeds inflict more damage to residential buildings. Observing a cubic transformation of the wind gust regressor is consistent with findings in the literature. For example, Klawa and Ulbrich14 show that wind speeds and economic losses are also linked through a cubic relationship. Interestingly, the cube of a wind speed is proportional to wind power (i.e., the wind’s kinetic energy flux), which suggests that the power of wind is a better determinant of the damage ratio than wind speed itself. The regression results show that the 24-hour cumulative precipitation level was also selected to describe the damage-driving mechanisms of winter storms in our best performing model specification, i.e., the compound vulnerability function. This precipitation regressor is positively related to the expected damage ratio, implying that higher 24-hour cumulative precipitation levels cause more damage to residential buildings. Given that both the 24-hour and one-hour cumulative precipitation levels (P1) were tested as potential precipitation regressor, the analysis reveals that 24-hour cumulative precipitation level proves most suitable. A potential explanation for the underperformance of the one-hour cumulative precipitation metric is that its measurement boundaries (i.e., between each full hour) were set somewhat arbitrarily. Consequently, the measurement period may have been misaligned to properly capture the damaging effect of short but intense precipitation events. In that case, the cumulative precipitation measurement could have “stopped too early”, “started too late” or both. Whilst a cubic transformation for wind speed is common in the literature, we are not aware of an established transformation for precipitation. As such, we explain our finding of a cubic relationship between the damage ratio and the 24-hour cumulative precipitation level as a purely empirical best fit.

The effects of wind gusts and precipitation on damage

Table 2 provides the average marginal effects per regressor for both vulnerability functions. The average marginal effects are calculated as the average of all unit-level marginal effects, where the latter equals a numerically derived partial derivative of the regression equation with respect to a regressor in the model.

Table 2 Average marginal effects
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The average marginal effect of the reconstruction value equals -9.04e−06 in both models. For the compound vulnerability function, the average marginal effect of the wind gusts equals 5.05e−08, for the univariate vulnerability function this is 5.02e−08. The average marginal effect of the 24-hour cumulative precipitation in the compound vulnerability function specification equals 2.68e−09. For the compound vulnerability function, multiplying these average marginal effects with the \({({\rm{maximum}})}^{3}\,\{{\frac{{\rm{maximum}}}{2}\}}^{3}\,\) observed 24-hour precipitation level \({(108.40{\rm{mm}})}^{3}\,\{{54.20{\rm{mm}}\}}^{3}\) and wind gust \({(37.32{\rm{m}}/{\rm{s}})}^{3}\,\{{18.66{\rm{m}}/{\rm{s}}\}}^{3}\), thereby adhering to the selected regressor transformations, shows that the former contributes (0.341% points) {0.043% points} to the damage ratio and the latter contributes (0.262% points) {0.033% points}. Hence, in the context of their observed regressor values, the contribution of the 24-hour cumulative precipitation to the damage ratio is approximately 30% larger than that of the wind gusts. Consequently, not accounting for the damaging effects of precipitation from winter storms will lead to underestimations of the expected damage ratio.

Conventional vulnerability functions underestimate winter storm damage

Conventional winter storm vulnerability functions only consider wind gusts as a proxy for the damage-driving mechanisms of winter storms, thereby departing from this study’s presumed true model: the compound vulnerability function. More importantly, the regression results from the univariate vulnerability function show that the parameters from a conventional model cannot (erroneously) compensate for an omitted precipitation regressor. These underestimations will be relatively pronounced for winter storms that produce low wind gust speeds and high precipitation levels, as the damage estimations can solely account for the (minor) impact of wind. For instance, considering a residential building with an average reconstruction value of €370,000, wind gusts of 20 m/s, and a 24-hour cumulative precipitation level of 50 mm [75 mm] {100 mm}, the compound vulnerability function estimates an expected damage ratio of 0.59% [0.68%] {0.88%} and the univariate vulnerability function estimates 0.56%. For the latter function, this amounts to an underestimation of 5% [21%] {57%}. The underestimation in the expected damage ratio can reach up to 7% [28%] {80%} when the parameter uncertainty of the 24-hour cumulative precipitation is considered at the 99% confidence interval level. The parameter uncertainty of the 24-hour cumulative precipitation is isolated as the difference in the expected damage ratio estimate between the compound vulnerability function and the univariate vulnerability function at their 99% confidence interval levels (in excess over the damage ratio estimates from the mean parameters), evaluated at the stated meteorological conditions.

Figure 1 displays the compound vulnerability function’s output evaluated at the sample’s mean observed reconstruction value (~€370,000) and for differing wind gust speeds and precipitation levels. Although more than half of the damage ratio observations are below 0.5%, it turns out that the mean damage ratio predictions are larger for every combination of meteorological conditions. This results from the subset of higher damage ratio observations, which sufficiently elevate the overall mean above 0.5% for every set of meteorological conditions.

Fig. 1: Compound vulnerability function output.
figure 1

This figure shows the compound vulnerability function’s output for differing wind gust speeds and 24-hour cumulative precipitation levels. In this figure, RP refers to the return period and CI to the confidence interval. The return periods are derived from STOWA25.

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The goodness-of-fit measures

Table 1 reports the RMSE and two pseudo \({R}^{2}\) indices as goodness-of-fit measures. The similarity of the RMSE and pseudo \({R}^{2}\) values between the compound vulnerability function and the univariate vulnerability function are attributed to two factors. First, for both vulnerability functions, the prediction errors are relatively large compared to the magnitude of the expected value predictions. As a result, differences in predictive accuracy between the two models are less discernible when evaluated using the RMSE metric. This is because RMSE reflects the absolute magnitude of prediction errors, and substantial errors can “mask” small gains in accuracy—making one model’s outperformance appear marginal even when it is meaningful in relative terms. Second, a relatively limited number of claims in the estimation sample were induced by high precipitation levels (>50 mm of precipitation in 24 hours) as compared to those that were mainly driven by wind gusts. Consequently, the univariate vulnerability function’s prediction errors on damage driven by high precipitation levels have relatively less weight in the RMSE calculations. The same explanations apply for the obtained pseudo \({R}^{2}\) values, as these indices aim to approach the \({R}^{2}\) metric and this metric, in turn, depends on the (large) prediction errors.

As such, the similarities in goodness-of-fit measures between the two models should not be considered evidence for equal prediction accuracy for winter storms with high precipitation levels. Graphic comparisons between the mean observed damage ratios for such winter storms and the corresponding mean damage ratio predictions from either the compound vulnerability function or the univariate vulnerability function clearly show that the latter is, on average, less accurate. The reason is that the univariate vulnerability function tends to underestimate these damage ratios more severely and more often (see Supplementary Figs. 1 and 2). In quantitative terms, comparing the mean prediction error of the compound vulnerability function with that of the univariate vulnerability function on recorded claims with 24-hour cumulative precipitation levels of 50 mm [75 mm] and above demonstrates that the univariate vulnerability function has a 249% [265%] higher mean prediction error.

The consequence of large prediction errors

The relatively high RMSE and low pseudo \({R}^{2}\) values are partially driven by the underestimation of several large damage ratios. For instance, omitting 1167 observations with damage ratios larger than the 99.5th quantile reduces (increases) the RMSE (the pseudo \({R}^{2}\) values) for the compound vulnerability function by almost 50%. Foremost, they could be indicative of the sizeable variation in vulnerability characteristics that is left unexplained and the randomness inherent to the damage process. Accordingly, these factors also manifest themselves in a large variance of the damage ratio estimates.

The sizeable degree of uncertainty that encompasses the damage ratio estimates also implies that winter storm losses for small samples cannot be accurately predicted by summing all expected individual losses (stemming from the expected damage ratio estimates). This follows from the law of large numbers, which prescribes that the chance of a considerable discrepancy between the expected damage ratio and the average of the materialised damage ratios increases for smaller sample sizes. Instead of relying solely on the sum of expected losses (a point estimate), it is advised to consider a probabilistic range of loss estimates, also known as a loss distribution, for smaller sample sizes. A loss distribution informs the end user on both the expected loss estimate and the range of plausible losses, which is also known as risk. Having knowledge on the latter makes it possible to (probabilistically) account for potential deviations from this expected loss estimate by also considering alternative loss materialisations. The loss distribution can be generated by repeatedly drawing (simulating) random potential damage ratios from a conditional beta distribution and assigning them to the damaged residential buildings. Accordingly, the beta distribution is conditioned on the reconstruction value and meteorological conditions of interest.

Lastly, including a precipitation regressor in the vulnerability functions becomes even more relevant when winter storm losses are estimated for smaller samples. As demonstrated in the following below, omitting the precipitation regressor in winter storm vulnerability functions limits the chances of randomly drawing the relatively large damage ratios that are observed at higher precipitation levels. This, in turn, will restrict the loss distributions in accurately describing the range of plausible winter storm losses. For instance, evaluating the compound vulnerability function’s output at the 99th quantile, conditional upon the average reconstruction value of €370,000, wind gusts of 20 m/s, and a 24-hour cumulative precipitation level of 50 mm [75 mm] {100 mm} shows that 1% of these damage ratios are predicted to be larger than 2.44% [2.63%] {3.02%}. For the univariate vulnerability function, these damage ratio values correspond to the 0.9% [0.62%] {0.28%} quantiles rather than the 1% quantile. Therefore, the chance of randomly drawing damage ratios larger than 2.44% [2.63%] {3.02%} from the univariate vulnerability function is 11% [61%] {257%} smaller than drawing them from the compound vulnerability function, which could cause the simulated loss distribution to underestimate the risk.

Diagnostics

Supplementary Figs. 3–7 present various (residual) diagnostic plots for the compound vulnerability function to further examine its fit. Supplementary Figs. 3 and 4 show the Pearson residuals against the fitted values (i.e., the expected damage ratio estimates) and the indices of the observations, respectively. No significant patterns are observed in these figures, which implies that the model adequately captures the non-linear relationship between the dependent variable and its regressors. Supplementary Fig. 5 depicts a normal quantile-quantile plot, where the residuals are quantile residuals, as proposed by Dunn and Smyth27. This plot highlights that the compound vulnerability function has difficulties in describing both tails of the empirical damage ratio distribution. In particular, it underestimates the right tail of the empirical damage ratio distribution (i.e., there are more large damage ratios than the fitted beta distribution anticipates), whereas it overestimates the left tail of the empirical damage ratio distribution (i.e., there are less small damage ratios than the fitted beta distribution anticipates). In addition, the densities of the quantile residuals and standard normal distribution in Supplementary Fig. 6 also reveal discrepancies between the central regions of the empirical damage ratio distribution and the fitted beta distribution. These observations are confirmed by the descriptive statistics of both the Pearson and quantile residuals in Supplementary Tables 18 and 19, as they show that these residuals are right-skewed and leptokurtic. Nonetheless, both mean residuals are close to zero, thus indicating that the compound vulnerability function can safely be applied to large samples because, under these circumstances, the inaccuracies of the fitted beta distribution have been averaged out.

If, however, the sample is small and simulation methods must be performed, then it is important to point out that the loss distribution will resemble the characteristics of the theoretical damage ratio distribution (i.e., the fitted beta distribution). Consequently, this loss distribution will differ from one that is simulated according to the empirical damage ratio distribution in the same manner as their underlying damage ratio distributions differ. The magnitude of these differences can decrease to an extent for simulations from larger samples. Lastly, Supplementary Fig. 7 displays the leverage values of all observations against their indices to examine their influence on the regression parameters. Although there are many high leverage observations (i.e., having a substantially higher leverage than the mean leverage), they appear to be uninfluential, as the regression parameters are at least robust to removing the top of 0.5% observations (~1000 observations), ranked according to their leverage.

Comparison with the literature

Due to a lack of (granular) damage data, the literature offers few empirical European winter storm vulnerability functions for residential buildings. Figure 2 compares the compound vulnerability function with the only available European winter storm vulnerability function of which we are aware: the Sw vulnerability function. This function was first used in Schwierz et al.15 to estimate winter storm losses on a European scale in current and future climates. The Sw vulnerability function is a simplified version of a winter storm vulnerability function that was originally developed by an insurance company. This European winter storm vulnerability function is available on the open-source weather and climate risk modelling platform CLIMADA28.

Before the outputs are compared, it is important to describe how the compound vulnerability function’s specification distinguishes itself from that of the Sw vulnerability function. In contrast to the Sw vulnerability function, it includes a regressor for the reconstruction value. Our results indicate that this is required to adequately differentiate between the vulnerabilities of different types of residential buildings. Moreover, the compound vulnerability function also includes a regressor to reflect the damaging effects of precipitation. We show that this improves the accuracy of damage ratio estimations for winter storms with heavy precipitation levels. Apart from these differences in vulnerability function specification, the models also differ in terms of the provided auxiliary information. As opposed to the Sw vulnerability function, the compound vulnerability function’s parameters and their uncertainties, and the 99% confidence intervals of the expected damage ratio estimates are reported. This auxiliary information allows for model reliability assessments and ensures that simulations can be performed for smaller samples. Accordingly, as the Sw vulnerability function does not provide parameter uncertainty ranges (for the expected damage ratio estimates), or a distribution specified around the expectation to represent variation, it is advisable to restrict its usage to large samples only.

Fig. 2: A comparison of vulnerability functions in the literature.
figure 2

This figure presents the compound vulnerability function’s output and that from the vulnerability function of Schwierz et al.15. In this figure, RP refers to the return period and CI to the confidence interval. The return periods are derived from STOWA25. The compound vulnerability function performs out-of-sample damage ratio predictions after wind gust speeds of 38 m/s. The out-of-sample boundary is marked by the dotted vertical line.

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Turning to the comparison of the expected damage ratio outputs from the compound vulnerability function and from the Sw vulnerability function, we identify notable differences at virtually all wind gust speeds between both vulnerability functions. To compare their outputs, the compound vulnerability function is evaluated at the sample’s mean reconstruction value (~€370,000) and mean 24-hour cumulative precipitation level (~12 mm), as these predictions most closely resemble the predictions from the Sw vulnerability function. Because the Sw vulnerability function omits these two regressors, its predictions effectively reflect an “averaged out” effect of the reconstruction value and precipitation. Which explains why evaluating the compound vulnerability function at their mean levels most closely aligns with Sw vulnerability function outputs.

The predominant difference between the vulnerability functions revolves around the assumed wind gust speed at which damage starts to occur. Similar to numerous studies (see for example, Donat et al.29 and Pinto et al.30), Schwierz et al.15 set this boundary at the 98th quantile of the local daily maximum wind gust distribution. In their study and ours (coincidentally), this boundary equals wind gusts of approximately 20 m/s. However, the lower wind gust speeds found in our study to be associated with the claims indicate that damage to residential buildings may start earlier. Moreover, the large variance of the damage ratio estimates suggests that there is likely not a universal damage threshold for all types of residential buildings. It is more plausible that these boundaries depend on building vulnerability characteristics and the randomness inherent to the damage process. Nonetheless, our best estimate for such a damage threshold is for wind gusts between 9 m/s and 10 m/s, as this is the threshold from which large numbers of claims are observed. Consequently, this inference implies that winter storm losses will be underestimated if higher boundary speeds are assumed because damage arising from lower wind gust speeds will be overlooked. Note, however, that these boundary values are suitable only for use cases where the incorporated meteorological observations are similar to those in this study. That is, empirically validated and reanalyses-based wind gusts reported per grid at a horizontal resolution of 2.5 km. These boundary values may not apply to, for example, vulnerability functions that are calibrated on meteorological observations with different characteristics (such as those with differing horizontal resolutions).

Additional noteworthy differences are identified between the wind gust speeds of the damage threshold of Schwierz et al.15, (20 m/s) and the maximum observed wind gust speed in this study’s sample (~38 m/s): the compound vulnerability function predicts structurally higher expected damage ratios than the Sw vulnerability function. This pattern reverses itself, however, for the compound vulnerability function’s out-of-sample predictions. In this region, the expected damage ratio predictions from the Sw vulnerability function mostly coincide with the compound vulnerability function’s predictions when the latter is evaluated at 24-hour cumulative precipitation levels larger than zero.

Conclusively, if the compound vulnerability function’s calibration represents the true European winter storm vulnerability function for residential buildings, then the reference function in Schwierz et al.15 underestimates the damage ratio below wind gust speeds of ~42 m/s. This underestimation becomes more pronounced for winter storms with higher precipitation levels. Moreover, the Sw vulnerability function would overestimate the damage ratio above these wind gust speeds for winter storms with relatively low precipitation levels, but accurately estimate the damage ratio for those winter storms with extreme precipitation levels.

Discussion

The limited availability of vulnerability functions in the literature inspired us to deliver vulnerability functions for residential buildings that can be implemented in private, academic, and open-source natural catastrophe models for European winter storms. Well-known use cases of the loss estimates of natural catastrophe models are to support the financial risk management of (re)insurers or to inform their premium pricing process. However, as natural catastrophe risks increase due to economic growth and climate change31, such loss estimates become more relevant for the climate-related risk management and reporting of other (financial) organisations as well32. The compound vulnerability function is especially appropriate for estimating future European winter storm losses, as it explicitly accounts for the damaging effects of precipitation, whose intensity and frequency is expected to be altered by climate change20,21,22,23,24. The vulnerability functions can, therefore, also be a suitable guide to cost-benefit analyses of climate risk reduction measures, as the functions offer insight into their maximum potential benefits i.e., the (expected) damage before the adaptation measures.

Natural catastrophe modellers and other interested parties are encouraged to apply the compound vulnerability function from our study if the availability of 24-hour cumulative precipitation-level data allows for it. When this is not the case, the users may resort to the univariate vulnerability function whilst keeping its outlined limitations in mind. The end users of these vulnerability functions should be aware that their predictions are predominantly appropriate for residential buildings of a similar build quality as those found in the Netherlands and within the range of the meteorological conditions in this study which, furthermore, are of the same horizontal resolution and produced by similar methods. Also note that the required reconstruction value is denoted in euros in 2024 price levels. To facilitate the usage of these vulnerability functions, section S6 in the Supplementary Information provides Supplementary Tables 3-17 with damage ratio outputs from the compound vulnerability function for an average residential building (i.e., with a reconstruction value of €370,000) at differing wind gust speeds, 24-hour cumulative precipitation levels, quantiles of the conditional beta distribution, and confidence intervals for the expected damage ratios.

To estimate losses for multiple buildings, the end user also requires information on the chance of being damaged given certain meteorological conditions alongside these vulnerability functions. Empirical models that estimate the chance of a building being damaged can also be constructed from insurance data and provide opportunities for further research. Especially finding that precipitation levels are a significant contributor to residential building damage signals that such research endeavours could be worthwhile, as it suggests that the chance of being damaged is also influenced by this factor.

Methods

Data

The vulnerability functions are developed by using insurance data from a Dutch financial conglomerate. The insurance dataset provides weather-related residential building claims and policy holder information (e.g., building location) in the Netherlands at the residence-level. The residential building types included in this dataset are: terraced houses, corner houses, semi-detached houses, detached houses, and apartments. The insurance data is matched with meteorological data from the Royal Netherlands Meteorological Institute (KNMI), which is the Dutch national weather service. The matched dataset ranges from 2008 up to and including 2021. Table 3 provides the data’s descriptive statistics.

Damage data

Only weather-related residential building claims from October up to and including March of each year are retained, as this period represents the European winter storm season. Section S1 in the Supplementary Information elaborates on the additional filters used to isolate winter-storm-related claims. The claim amounts in the insurance dataset represent the insured’s compensated damage net of any deductibles imposed by the insurance policies. To reflect the total damage to a building from a winter storm (i.e., the ground-up damage), the average deductible over this period of €200 is added to claims from policyholders who were subjected to a deductible.

The dependent variable of interest is the damage ratio, which represents the ground-up damage to a residential building as a ratio of its reconstruction value. The reconstruction value approximates the cost of rebuilding a residential building according to the latest building codes, in case the building is completely damaged. Its calculation is based on a methodology provided by the Dutch Association of Insurers and mainly depends on the building type (e.g., an apartment or terraced house), the building year, the building volume, and structural characteristics such as the foundation or roof33. Section S2 in the Supplementary Information elaborates on the calculation of the reconstruction value. This normalisation allows for a more accurate comparison between the dependent variable’s observations because it partially corrects for the reconstruction value’s effect on the ground-up damage: buildings with higher reconstruction values (originating from characteristics such as building size and materials) have more potential for damage simply because there is more value that can be damaged.

We observe that damage does not rise proportionally with the reconstruction value. Hence, the effect of the reconstruction value on the ground-up damage differs for differing reconstruction values. Therefore, to more comprehensively normalise the damage ratios, we include the reconstruction value (Rv) as a regressor in the models. To foster the usage of our vulnerability functions, we restricted the number of relevant building characteristic regressors to the reconstruction value alone. In contrast to a reconstruction value (which is required to transform an estimated damage ratio into a loss estimate), more detailed building characteristics data may not be freely or consistently available, especially to catastrophe modellers outside the insurance industry, such as those in academia or government agencies. Therefore, the reconstruction value regressor can be considered a readily available and parsimonious proxy for relevant building characteristics. Hence, only including a reconstruction value allows for a wide applicability of our vulnerability functions. In a similar spirit, we chose not to include location explicitly as a regressor to ensure that the vulnerability functions are easily applicable to regions outside our sample (both within and beyond the Netherlands). Incorporating a location regressor would have explicitly tied the model to specific in-sample locations, thereby making it difficult to generalise elsewhere by requiring an often-challenging comparison between new and in-sample areas. Instead, omitting a location regressor allows the model to “average out” location-specific variation. Accordingly, we also assumed that the resolution of the meteorological data is high enough to accurately control for the spatial variation in meteorological conditions, i.e., spatial variation in the meteorological conditions which could otherwise have been captured by a location regressor.

Both the ground-up damage and the reconstruction value are denoted in 2024 price levels. Section S3 in the Supplementary Information contains the index that is used to adjust the price levels in which the damage data was originally denoted (Supplementary Table 1), along with a description of the methodology and data (Supplementary Table 2) employed to construct the index.

Meteorological data

The daily maximum 3-second wind gust in a 10-minute period at 10 m height is used as one of the proxies for the damage-driving mechanisms of winter storms34,35,36. The wind gusts are based on regional climate reanalysis and are provided in hourly time series per grid at a horizontal resolution of 2.5 km. The reanalysis is based on the numerical weather prediction model HARMONIE-AROME, which is nested in the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis ERA5. Academic literature suggests that these wind gusts are a suitable proxy for the damage-driving mechanisms of winter storms compared to other winter storm characteristics, such as the maximum hourly average wind speed13. The other proxy for the damage-driving mechanisms of winter storms is a precipitation intensity metric, which aims to capture damage resulting from precipitation. There is less consensus on which precipitation metric best reflects the damage-driving process37. Hence, we considered two different precipitation accumulation intervals: the daily maximum one-hour cumulative precipitation and the daily 24-hour cumulative precipitation38. The daily 24-hour cumulative precipitation is defined as the cumulative precipitation measured between 00:00 and 23:59 UTC, whereas the daily maximum one-hour cumulative precipitation is defined as the largest cumulative precipitation measured between each full hour of a specified day. The precipitation measurements are radar based, adjusted by rain gauge observations, and provided in hourly time series per grid at a horizontal resolution of 1 km. Section S4 in the Supplementary Information provides further details regarding the measurement methodologies of the meteorological data.

Data preparations

The damage data are spatially matched to the meteorological data by retaining the values from the meteorological grid cell with the centroid nearest to the centroid of the neighbourhood associated with each claim. In the Netherlands, each neighbourhood corresponds to a unique four-digit and one-letter postal code area. Regarding temporal matching, it is noted that the claim dates in the insurance dataset occasionally do not coincide with the winter storm dates. For instance, we observe that a small portion of the claims from a large storm are reported one day after the storm. To correct for these discrepancies, the highest daily wind gust speed and highest daily one-hour/24-hour cumulative precipitation level are selected over the four days leading up to the claim date, the claim date itself, and the day after.

Because the above-average claim frequencies sharply decline in the days after large storms, it is reasonable to assume that this procedure corrects most of the targeted discrepancies. Nevertheless, the procedure cannot account for discrepancies between the reporting and the storm dates that are larger than the specified buffer. To address the potentially remaining matching errors, we removed 940 observations that had damage ratios larger than the 97.5th quantile (damage ratios > 2.1%) at relatively modest meteorological intensities. The 97.5th quantile is therefore measured over the subset that had relatively modest meteorological intensities. Here, modest meteorological intensities are defined as jointly observing maximum wind gusts below 15 m/s and with 24-hour cumulative precipitation levels below 15 mm. This procedure also treats outliers resulting from inaccurate meteorological values. These may have originated from, for instance, the inability of the meteorological observations to capture very local and damaging meteorological conditions, such as whirlwinds. After removing the outliers, the sample consists of 203,401 claims and their associated damaging meteorological conditions. The regression parameters proved to be robust for a reasonable range of damage ratio and meteorological thresholds. The sample includes three complete loss observations (i.e., having a damage ratio of one). As the beta regression cannot capture a probability mass at one, their damage ratios were set to 0.99.

Table 3 Descriptive statistics of the dependent variable, its components, and regressors
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Amongst the winter storm events that are covered in this sample are six events that the KNMI classified as major storms, i.e., events where average hourly wind speeds of at least 24.5 m/s (10 on the Beaufort scale) are recorded. The sample also contains 8048 observations with 24-hour cumulative precipitation levels between 25 mm and 35 mm, which reflect events with return periods between one and five years, respectively25. There are 1609 observations between 35 mm and 50 mm, with the latter occurring only once every 50 years. At the extreme, there are 81 observations with 24-hour cumulative precipitation levels between 75 mm (~500-year return period) and 100 mm (~1000-year return period) and 14 observations that exceed 100 mm. Figure 3 presents a heatmap of the observed mean damage ratios in percentages, per binned range of meteorological observations. This figure indicates that both high wind gust speeds and high 24-hour cumulative precipitation levels are associated with higher mean damage ratios. This finding suggests that damage ratios can be modelled along these two dimensions using statistical methods such as the beta regression.

Fig. 3: Heatmap of observed damage ratios per binned range of meteorological conditions.
figure 3

This heatmap portrays the observed mean damage ratios in percentages, per binned range of meteorological conditions. The top number in each bin represents the mean damage ratio, whereas the bottom number within parentheses refers to the number of claims captured in each binned range of meteorological conditions. Each bins is colour coded based on the value of its mean damage ratio according to the legend on the right.

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Figure 4 is a two-dimensional histogram of the observed combinations of meteorological conditions. The figure indicates that the highest wind gust speeds do not cooccur with the highest 24-hour cumulative precipitation levels. Moreover, we observe that the winter storm claims are clustered around severe events. For instance, the top 10 events (ranked according to the total number of claims observed on a given day) comprise approximately 20% of all claims in the sample. This clustering, in part, explains the clustering observed in Fig. 4, such as at the combination of wind gusts between 28.5 m/s–29.5 m/s and a 24-hour cumulative precipitation level between 10 mm–15 mm.

Fig. 4: Two-dimensional histogram of wind gust and 24-hour cumulative precipitation combinations.
figure 4

This two-dimensional histogram portrays the observed combinations of meteorological conditions. The bins are colour coded based on the frequency of the observed combination of meteorological conditions.

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Methods

Our vulnerability function development is inspired by Rossetto et al.'s39 data-driven procedure for estimating vulnerability functions. These authors recommend to trial multiple model specifications and to select the vulnerability function according to a fitting criterion. Following their approach, we estimated approximately 400 candidate model specifications with alternating combinations of the introduced regressors (Rv, W, P1, and P24), interactions amongst the meteorological variables (W*P1/P24), and their potential non-linearities. Taking the wind gust regressor as an example, the tested regressor transformations are: \(\sqrt{W},{W}^{2},{W}^{3},{W}^{4},{or}{W}^{5}\). The candidate models are estimated using beta regressions, as introduced by Ferrari and Cribari-Neto16. Beta regressions assume that the dependent variable, \(Y,\) is beta distributed on the open unit interval (0, 1). Amongst other favourable characteristics, the open unit interval restriction of the dependent variable makes these regressions suitable for modelling continuous fractional data, such as damage ratios.

In our analysis, we define \(Y\) as the damage ratio. When \({Y} \sim \,{\mathcal{B}}{{(}}\mu ,\,\phi )\), where \(\mu\) \((0 \,<\, \mu \,<\, 1)\) represents the distributional mean and \(\phi\) \((\phi \,>\, 0)\) functions as a precision parameter, then \({\rm{{\rm E}}}\left(Y\right)=\,\mu\) and \({Var}\left(Y\right)=\,\mu (1-\mu )/(\phi +1)\). The parameter \(\phi\) can be interpreted as a precision parameter since a larger \(\phi\) will result in a smaller variance of \(Y\), provided that \(\mu\) stays fixed. Moreover, as the variance of \(Y\) is a function of \(\mu\), a beta regression model can, by construction, account for heteroskedasticity.

For \({Y}_{i} \sim \,{\mathcal{B}}\left({\mu }_{i},\,\phi \right),{i}=1,\ldots ,n\), the beta regression model can be defined as

$${g}_{1}\left({\mu }_{i}\right)={f}_{1}({x}_{i},\beta )=\,{\eta }_{1i}$$
(1)

where \(\beta ={({\beta }_{1},\ldots {\beta }_{k})}^{T}\) is a \({k}\times 1\) vector of unknown regression parameters \((k < n)\), \({x}_{i}=({x}_{i1,},\ldots ,{x}_{{ik}})\) is the vector of \(k\) regressors, and \({\eta }_{1}={({\eta }_{11},\ldots ,{\eta }_{1n})}^{T}\) is a predictor vector for the mean parameter. Lastly, \({f}_{1}(\cdot ,\,\cdot )\) is a linear or nonlinear twice continuously differentiable function in the second argument, and \({g}_{1}\left(\cdot \right)\) is a strictly monotonic and twice differentiable link function that maps \(\left(\mathrm{0,1}\right){{\longmapsto }}{\mathbb{R}}\). In this study, the logit link function, \(g\left(\chi \right)=\log (\chi /(1-\chi )),\) is used for \({\mu }_{i}\) and \(\phi\) throughout. The candidate vulnerability functions were constructed by modelling the predictor for \({\mu }_{i}\) (i.e., the expected damage ratio) as a linear function of its parameters and regressors Rv, W, P1, and P24, considering their multiple combinations, possible interactions, and non-linearities.

For additional information on the maximum likelihood estimation of the beta regression parameters, their inference, diagnostic measures, and model selection tools, the reader is referred to Ferrari and Cribari-Neto16. Moreover, section S5 in the Supplementary Information provides further detail on the alternative parameterisation of the beta distribution that these authors proposed and on an extension of the beta regression model that considers a variable precision parameter \({\phi }_{i}\), as introduced by Smithson and Verkuilen40 and Simas et al.41.

As damage to residential buildings that fell below the insurance policy’s deductible is not present in the insurance claims dataset, the dependent variable is truncated from the left. Without explicitly accounting for the part of the dependent variable’s distribution that is below the deductible, one may overestimate the damage. All beta regressions are therefore estimated with truncated beta density functions that are conditioned upon the damage being larger than the deductible.

All parameters were estimated by maximum likelihood through a Newton-Raphson algorithm42, using the GAMLSS package version 5.4–22 in R43. In accordance with the estimation output from the GAMLSS package, the precision parameter is reported as \(\sigma\), which is a reparameterisation of \(\phi\) that was proposed by Rigby et al.44. Here, \(\sigma =\,\frac{1}{{(\phi +1)}^{1/2}}\) or \(\phi =\,\frac{1}{{\sigma }^{2}}-1\), with \(0 \,<\, \,\sigma \,<\, 1\) and \({Var}\left(Y\right)={\sigma }^{2}\mu \left(1-\mu \right)\). Hence, a lower \(\sigma\) results in a smaller variance of \(Y\).

The Bayesian information criterion (BIC) is used to select the model with the highest probability of being the true model out of all candidate models45. Given the size of our sample, the BIC favours parsimonious models and, therefore, promotes a fairer comparison between models with different numbers of regressors. The BIC can therefore be used to select the combination and transformation of regressors that underpin the best vulnerability function specification. Goodness-of-fit is assessed by two pseudo \({R}^{2}\) indices: \({R}_{p}^{2}\), which is defined as the square of the Pearson correlation coefficient between the dependent variable and the fitted values for the dependent variable, and \({R}_{{CS}}^{2}\), which represents the pseudo \({R}^{2}\), as proposed in Cox and Snell46. Additionally, the root mean squared error (RMSE) is calculated by taking the square root of the average squared prediction errors of the vulnerability functions. Lastly, 99% confidence intervals for the expected dependent variable were constructed as discussed in Appendix B in Ferrari and Cribari-Neto16.