When a natural hazard event like an earthquake affects a region and generates a natural catastrophe (NatCat), the following questions
arise: how often does such an event occur? What is its return period (RP)?
We derive the combined return period (CRP) from a concept of extreme value
statistics and theory – the pseudo-polar coordinates. A CRP is the
(weighted) average of the local RP of local event intensities. Since CRP's
reciprocal is its expected exceedance frequency, the concept is testable. As we show, the CRP is related to the spatial characteristics of the NatCat-generating hazard event and the spatial dependence of corresponding local block maxima (e.g., annual wind speed maximum). For this purpose, we extend a previous construction for max-stable random fields from extreme value theory and consider the recent concept of area function from NatCat research. Based on the CRP, we also develop a new method to estimate the NatCat risk of a region via stochastic scaling of historical fields of local event intensities (represented by records of measuring stations) and averaging the computed event loss for defined CRP or the computed CRP (or its reciprocal) for defined event loss. Our application example is winter storms (extratropical cyclones) over Germany. We analyze wind station data and estimate local hazard, CRP of historical events, and the risk curve of insured event losses. The most destructive storm of our observation period of 20 years is Kyrill in 2002, with CRP of
After a natural hazard event such as a large windstorm or an earthquake has occurred in a defined region (e.g., in a country) and results in a natural catastrophe (NatCat), the following questions arise: how often does such a random event occur? What is the corresponding return period (RP, also called recurrence interval)? Before discussing this issue, we underline that the extension of river flood events or windstorms in time and space depends on the scientific and socioeconomic event definition. This definition may vary by peril and is not our topic even though they influence our research object – the RP of a hazard and NatCat event.
The RP of an event magnitude or index is frequently used as a stochastic measure of a catastrophe. For example, there are different magnitude scales for earthquakes (Bormann and Saul, 2009). However, their RP may not correspond with the local consequences since the hypocenter position also determines local event intensities and effects. For floods, regional or global magnitude scales are not in use (Guse et al., 2020). For hurricanes, the Saffir–Simpson scale (National Hurricane Centre, 2020) is a magnitude measure; however, the random storm track also influences the extent of destruction. Extratropical cyclones hitting Europe, called winter storms, are measured by a storm severity index (SSI; Roberts et al., 2014) or extreme wind index (EWI; Della-Marta et al., 2009). Their different definitions result in quite different RPs for the same events. In the rare scientific publications about risk modeling for the insurance industry, such as by Mitchell-Wallace et al. (2017), better and universal approaches for the RP are not offered. In sum, previous approaches are not satisfactory regarding the stochastic quantification of a hazard or NatCat event. This is our motivation to develop a new approach. Building on results of extreme value theory and statistics, we mathematically derive the concept of combined return period (CRP), which is the average of RPs of local event intensities. As we will show by a combination of existing and new approaches from stochastic and NatCat research, the concept of CRP is strongly related to the spatial association/dependence between the local event intensities, their RPs, and corresponding block maxima, such as annual maxima.
Spatial dependence is not suitably considered in previous research about NatCat. The issue is only a marginal topic in the book by Mitchell-Wallace et al. (2017, Sect. 5.4.2.5) about NatCat modeling for the insurance industry. Jongman et al.'s (2014) model for European flood risk considers such dependence explicitly. However, their assumptions and estimates are not appropriate according to Raschke (2015b). In statistical journals, max-stable dependence models have been applied to natural hazards without a systematic test of the stability assumption. Examples are the snow depth model by Blanchet and Davison (2011) for Switzerland and the river flood model by Asadi et al. (2015) for the Upper Danube River system. Max-stable dependence means that the copula (the dependence structure of a bi- or multivariate distribution) and corresponding value of dependence measures are the same for annual maxima as for 10-year maxima or those of a century (Dey et al., 2016). Also, Raschke et al. (2011) proposed a winter storm risk model for a power transmission grid in Switzerland without validation of the stability assumption. The sophisticated model for spatial dependence between local river floods by Keef et al. (2009) is very flexible. However, it needs a high number of parameters, and the spatial dependence cannot be simply interpolated as is possible with covariance and correlation functions (Schabenberger and Gotway, 2005, Sect. 2.4). Besides, the random occurrence of a hazard event is more like a point event of a Poisson process than the draw/realization of a random variable. For instance, the draw of the annual random variable is certain; the occurrence of a Poisson point event in this year is not certain but random.
In the research of spatial dependence by Bonazzi et al. (2012) and Dawkins and Stephenson (2018), the local extremes of European winter storms are sampled by a pre-defined list of significant events. Such sampling is not foreseen in (multivariate) extreme value statistics; block maxima and (declustered) peaks over thresholds (POT) are the established sampling methods (Coles, 2001, Sects. 3.4 and 4.4; Beirlant et al., 2004, Sect. 9.3 and 9.4). Event-wise spatial sampling is a critical task; the variable time lag between the occurrences at different measuring stations, such as river gauging stations, makes it confusing. The corresponding assignment of Jongman et al. (2014) of one local/regional flood peak to peaks at other sites is not convincing, according to the comments by Raschke (2015b). The sampling of multivariate block maxima is simpler. However, the univariate sampling and analysis are also not trivial. An example is the trend over decades in the time series of a wind station in Potsdam (Germany). Wichura (2009) assumes a changed local roughness condition over the time as the reason; Mudelsee (2020) cites climate change as the reason.
The research of spatial dependence of natural hazards is not an end in itself; the final goal is an answer to the question about the NatCat risk. What is the RP of events with aggregate damage or losses in a region equal to or higher than a defined level? By using CRP, we quantify the risk via stochastic scaling of fields of local intensities of historical events and averaging corresponding risk measures. This new approach significantly extends the methods to calculate a NatCat risk curve. Previous opportunities and approaches for a risk estimate are the conventional statistical models that are fitted to observed or re-analyzed aggregated losses (also called as-if losses) of historical events, as used by Donat et al. (2011) and Pfeifer (2001) for annual sums. The advantages of such simple models are the controlled stochastic assumptions and the small number of parameters; the disadvantages are high uncertainty for widely extrapolated values and limited possibilities to consider further knowledge. The NatCat models in the (re-)insurance industry combine different components/sub-models for hazard, exposure (building stock or insured portfolio), and corresponding vulnerability (Mitchell-Wallace et al., 2017, Sect. 1.8; Raschke, 2018); additionally, they offer better opportunities for knowledge transfer such as the differentiated projection of a market model on a single insurer. However, the corresponding standard error of the risk estimates is frequently not quantified (and cannot be quantified). The numerical burden of such complex models is high. Tens of thousands of NatCat events must be simulated (Mitchell-Wallace eta al., 2017, Sect. 1). Thus, the question arises of what the stochastic criterion for the simulation of a reasonable event set in NatCat modeling is. As far as we know, scientific NatCat models for European winter storms (extratropical cyclones) are based on numerical simulations (Della-Marta et al., 2010; Osinski et al., 2016; Schwierz et al., 2010) and are not intensively validated regarding spatial dependence.
To answer our questions, we start with topics of extreme value statistics in Sect. 2, where we recall the concept of max-stability for single random variables, bivariate dependence structures (copulas), and random fields. We also extend Schlather's (2002) first theorem with a focus on spatial dependence. The more recent approaches to area functions (Raschke, 2013) and survival functions (Jung and Schindler, 2019) of local event intensities within a region are implemented therein. In Sect. 3, we derive the CRP from the concept of pseudo-polar coordinates of extreme value statistics and explain its testability, possibility of scaling, and corresponding risk estimate. Subsequently, in Sect. 4, we apply the new approaches to winter storms (extratropical cyclones) over Germany to demonstrate their potential. This application implies several elements of conventional statistics, which are explained in Sect. 5. Finally, we summarize and discuss our results and provide an outlook in Sect. 6. Some stochastic and statistical details are presented in the Supplement and Supplement data. In the entire paper, we must consider several stochastic relations. Therefore, the same mathematical symbol can have different meanings in different subsections. We also expect that the reader is familiar with statistical and stochastic concepts such as statistical significance, goodness-of-fit tests, random fields, and Poisson (point) processes (Upton and Cook, 2008).
Before introducing CRP and its properties, we discuss and extend the concept
of max-stability in extreme value statistics, with a focus on random processes
and fields. Max-stability has its origin in univariate statistics. The
cumulative distribution functions (CDFs)
It is also well-known that a bivariate CDF
The spatial extension of the bivariate situation and corresponding
distribution is the random field
Extreme value statistics is interested in the max-stable dependence structure (copula) between the margins, the unit Fréchet distributed random variables
Schlather (2002) has demonstrated the flexibility of his construction by
presenting realizations of maximum fields for different variants of
Both
The construction of Theorem 1 is already used to model natural hazards in
the geographical space. Smith (1990) has applied the bivariate normal
distribution as
We now illustrate spatial max-stability and its absence by examples of
Eqs. (9) and (10) with standard normal PDF as random function
Examples of simulated fields of local event intensities and enveloping field of maxima (bold green line) generated with standard normal
PDF as
Spatial dependence in relation to the distance measured by
Kendall's
To illustrate the effect on spatial dependence quantitatively, we have
generated local maxima
Beside our extension of Schlather's theorem, we also consider a more recent
approach from NatCat research to understand the spatial characteristics.
Raschke (2013) described an earthquake event by its area function for the peak ground accelerations. This is a cumulative function and measures the set of points in the geographical space (the area) with an event intensity higher than the argument of the function. The area function is limited here to a region and is normalized as follows (
The area function and corresponding characteristics:
We also use the parameters of a random variable
As per Sect. 2, Schlather's first theorem has parallels to NatCat models, is used already in hazard models, and was extended here to the non-max-stable case regarding the spatial dependence and characteristics. Statistical indication for max-stability is the independence of the spatial dependence measure from the block size (e.g., 1 versus 10 years) and independence between CV and expectation of the area function (Eq. 11). Otherwise, non-max-stability is indicated.
Let the point event
Schlather's theorem is also based on and implies the concept of pseudo-polar
coordinates. According to De Haan (1984) and explained well by Coles (2001,
Sect. 8.3.2), two linked max-stable point processes with expected exceedance frequency (Eq. 15) and point events
According to Coles (2001, Sect. 3.8), the pseudo-radius
More than one RP can be averaged since the averaging of two RPs can be done
in serial (and the pseudo-polar coordinates are also applied to more than
two marginal processes). Serial averaging (averaging the last result with a
further RP) also implies a weighting; the first considered RPs would be
less weighted than the last in the final CRP. The general formulation of
averaging of RP with weight
Before the CRP is applied in stochastic NatCat modeling, it should be tested statistically to validate the appropriateness. A sample of CRPs can be tested by a comparison of its exceedance frequency function (Eq. 15) and their empirical variant. Therein the empirical exceedance frequency of the largest CRP in the sample is the reciprocal of the length of the observation period. The second largest CRP is hence associated with twice the exceedance frequency of the largest CRP and so on. It is the same as for empirical exceedance frequency for earthquake (e.g., the well-known Gutenberg–Richter relation in seismology; Gutenberg and Richter, 1956). However, not all small events are recorded; the sample is thinned and incomplete. This completeness issue is well known for earthquakes and is less important here if only the distribution (Eq. 16) of maximum CRPs is tested. There are several goodness-of-fit tests (Stephens, 1986, Sect. 4.4) for the case of known distribution. The Kolmogorov–Smirnov test is a popular variant.
The CRP also offers the opportunity of stochastic scaling. The CRP
The main goal of a NatCat risk analysis is the estimate of a risk curve
(Mitchell-Wallace et al., 2017, Sect. 1), the bijective functional of event loss in a region, and the corresponding RP, which is called the event loss return period (ELRP)
We introduce an alternative method. Under the assumption of max-stability
between ELRP
Schemes of the scaling approach:
According to the delta method (Coles, 2001, Sect. 2.6.4), statistical estimates and their standard error can be approximately transferred in another parameter estimation and corresponding standard error by the determined transfer function and its derivatives. A condition of this linear error transfer is a relatively small standard error of the original estimates. The delta method could be used to compute the reciprocal of ELRP – the exceedance frequency and corresponding standard error – or the exceedance frequency is computed directly by averaging the reciprocal of scaled CRPs, and the transfer proxy acts implicitly. We also apply the idea of linear transfer proxy when we average the modeled event loss for the historical events being scaled to the same defined CRP. The unknown sample of ELRPs, which represents the ELRP's distribution for a fixed CRP, is implicitly transferred to a sample of event loss. If the proxies perform well, the difference to the risk estimates via CRP averaging should be small.
There is a further chain of thoughts as argument for the different variants of averaging. The scaling implies subsets of intensity fields of all possible intensity fields. The links between the fields of a subset are determined by the scaling of their CRPs. Correspondingly, every subset generates a risk curve with CRP (now also an ELRP) versus event loss. We also assume a certain unknown probability per subset that is applied if all these subsets generate the entire risk curve via an integral like the expectation of a random variable (Eq. 12). The corresponding empirical variant (estimator) is the averaging. However, we can average three values: CRP, exceedance frequency, or event loss.
All mentioned estimators for risk curve via scaling and averaging over
We draw attention to the fact that the explained scaling does not change the
CV of Eq. (23); this implies independence between CRP and CV (Sect. 2.4).
Therefore, the presented scaling only applies to the max-stable case of
local hazard. For the non-max-stable case, the scaling factor
We have selected the peril of winter storms (also called extratropical cyclones or winter windstorms) over Germany to demonstrate the opportunities of the CRP because of good data access and since we are familiar with this peril (Raschke et al., 2011; Raschke, 2015a). Our analysis follows the scheme in Fig. 4a, important results are presented in the subsequent sections, and the technical details are explained in Sect. 5. At first, we provide an overview.
We analyzed 57 winter storms over 20 years from autumn 1999 to spring 2019 (Supplement data, Tables 1 and 2) to validate the CRP approach. Different references (Klawa and Ulbrich, 2003; Gesamtverband Deutscher Versicherer, 2019; Deutsche Rück, 2020) have been considered to select the time window per event. In our definition, the winter storm season is from September to April of the subsequent year. It accepts a certain opportunity of contamination of the sample of block maxima by extremes from convective windstorm events and a certain opportunity of incompleteness from extratropical cyclones outside our season definition. The term winter storm is only based on the high frequency of extratropical cyclones during the winter. The seasonal maximum is also the annual maximum of this peril.
The maxima per half season (bisected by the turn of the year) are analyzed to double the sample size and to increase estimation precision. The appropriateness of this sampling is discussed in Sect. 5.1. We considered
records of wind stations in Germany of the German meteorological service (DWD, 2020; FX_MN003, a daily maximum of wind peaks (m s
Results of the analysis:
The intensity field per event is represented by the maximum wind gust for the corresponding time window of the event at each considered wind station. The local RP per event is computed by a hazard model per wind station. This is an implicit part of the estimated extreme value distribution per station, as explained in Sect. 5.1. The resulting CRPs per event and corresponding statistical tests are presented in the following Sect. 4.2. We have considered two weightings per station, capital, and area. Both are computed per wind station by assigning the grid cells with capital data of the Global Assessment Report (GAR data; UNISDR, 2015) via the smallest distance to a wind station. We also use these capital data to spatially distribute our assumed total insured sum of EUR 15.23 trillion for property exposure (residential building, content, commercial, industrial, agriculture, and business interruption) in Germany in 2018. This is based on Waisman's (2015) assumption for property insurance in Germany and is scaled to exposure year 2018 under consideration of inflation in the building industry (Statistisches Bundsamt, 2020) and increasing building stock according to the German insurance union (GDV; GDV, 2019). It is confirmed by the assumptions of Perils AG (2021); however, their data product is not public. We also used loss data of the GDV (2019) for property insurance when we fitted the vulnerability parameters for the NatCat model. These event loss data of 16 storms during a period of 17 years are already scaled by GDV to exposure year 2018.
The spatial characteristics are analyzed in Sect. 4.3 according to Sect 2.4, focusing on the question of if there is max-stability or not in the spatial dependence and characteristics. Finally, we present the estimated risk curve for the portfolio of the German insurance market in Sect. 4.4 including a comparison with previous estimates. Details of the vulnerability model are documented in Sect. 5.2. The concrete numerical steps, the applied methods to quantify the standard error of estimates, and the consideration of the results from vendor models are explained in Sect. 5.3–5.5, respectively.
As announced, we have computed the CRP according to Eq. (20) with the wind gust peaks listed in the Supplement data, Table 2, and local hazard models
according to Eq. (30). Our local hazard models are discussed in Sect. 5.1 and
parameters are presented in the Supplement data, Table 4. An example for
a complete CRP calculation is also provided therein (Table 8). We have
considered two weightings for the CRP, a simple area weighting and a capital
weighting (Supplement data, Table 3). In Fig. 5a, we compare the estimates which do not differ so much; the approach is robust in the example. The most significant winter storm of the observation period is Kyrill that occurred in 2007. It has CRPs of
In Fig. 5b, the results are validated according to Sect. 3.2. The empirical exceedance frequency matches well with the theoretical one for
The seasonal and annual maxima of CRP must follow a unit Fréchet distribution (
Spatial characteristics of winter storms over Germany:
As discussed in Sect. 2.4, the spatial characteristics is an important aspect from a stochastic perspective. Therefore, we have analyzed the relation between distance and dependence measure. We have applied Kendall's
Furthermore, the differences between the estimates of Kendall's
For completeness, we compare the current estimates of Kendall's
We have also computed the area functions and show examples in Fig. 6d for winter storm Kyrill. The different weightings result in similar area functions. Figure 6e plots the CRP and CV of all events. The regression analysis reveals the statistical dependence between CRP and CV. For the linearized regression function, the
Before we estimated risk curves according to the approach of Sect. 3.4, we
must estimate a vulnerability function (Eq. 31) which determines the local loss ratio
Estimates for insured losses from winter storms in Germany:
In Fig. 7b, the three estimated risk curves according to the three estimators in Eq. (28) are presented for max-stable scaling and differ less from each other, which indicates the robustness of our approach. The empiricism is presented by the historical event losses and their empirical RP (observation period 17 years of GDV loss data) and capital-weighted CRP. In addition, we present the range of two standard errors of the estimates of loss averaging which imply the simplest numerical procedure. Details of uncertainty quantification are explained in Sect. 5.4.
The differences between max-stable and non-max-stable scaling in the risk estimates are demonstrated in Fig. 7c. For smaller RP, no significant difference can be stated in contrast to higher RP. This corresponds with the differences between the CV in relation to the CRP for max-stable and non-max-stable cases in Fig. 6f. These are also higher for higher CRP.
We also compare our results with previous estimates in Fig. 7c. For this purpose, we must scale these to provide comparability as well as possible. The relative risk curve of Donat et al. (2011) is scaled simply by our assumption for the total sum insured (TSI) for the exposure year 2018. The vendor models of Waisman (2015) are scaled by the average of ratios between modeled and observed event losses from storm Kyrill since a scaling via TSI was not possible (uncertain market share and split between residential, commercial, and industry exposure). The result of the standard model of European Union (EU) regulations (European Commission, 2014), also known as Solvency II requirements, is also based on our TSI assumption, split into the CRESTA zones by the GAR data. The CRESTA zones (
The risk estimate of Donat et al. (2011) is based on a combination of frequency estimation and event loss distribution by the generalized Pareto distribution, which is fitted on a sample of modeled event losses for historical storms. The corresponding risk curve differs very much from other estimates and overestimates the risk of winter storms over Germany. The standard model of the EU only estimates the maximum event loss for RP of 200 years; the estimated event loss is very high. The vendor models vary but have a similar course as our risk curves. The non-max-stable scaling is in the lower range of the vendor models, whereas the unrealistic max-stable scaling is more in the middle. The concrete names of the vendors can be found in Waisman's (2015) publication. The reader should be aware that the vendors might have updated their winter storm model for Germany in the meantime.
The major result of Sect. 5 is the successful demonstration that the CRP can be applied to estimate reasonable risk curves under controlled stochastic conditions. In addition, we have discovered the strong influence of the underlying dependence model (max-stable or not) and the corresponding spatial characteristics to loss estimates for higher ELRP.
As mentioned, the maximum wind gusts of half seasons of winter storms
(extratropical cyclones) – block maxima – have been analyzed. Therein, the
generalized extreme value distribution (Beirlant et al., 2004, Sect. 5.1) is
applied:
We have validated the sampling of block maxima per half season. The opportunity of correlation between the first and second half-season maxima
has been tested for significance level
To optimize the intensity measure of the hazard model, we have considered the wind speed with power 1, 1.5, and 2 as the local event intensity in a first fit of the Gumbel distribution by the maximum likelihood method. According to these, power 1.5 offers the best fit of wind gust data to the Gumbel distribution. Such wind measure variants were already suggested by Cook (1986) and Harris (1996).
We do not apply the generalized extreme value distribution in Eq. (29) with extreme value index
To provide reproducible results, we also present a computational example for the CRP in Table 1 with reference to all needed equations and information. The entire calculation for storm Kyrill is presented in the Supplement data, Table 8.
Example for the computation of a CRP at Station #303 (Baruth) for storm Kyrill.
To quantify the loss ratio
Our suggested estimator (Eq. 32) has the advantage that it is less affected by the issue of incomplete data (smaller events with smaller losses are not
listed in the data) than the ratio of sums over all events, and the
corresponding standard error can be quantified (as for the estimation of an
expectation). The current point estimate is
An example of our vulnerability function (with the average of
Vulnerability functions: current estimate with the average of local parameters and previous estimates by Heneka and Ruck (2008) and Munich Re (2002) for residential buildings.
Here, we briefly explain the numerical procedure to calculate a risk curve
via averaging the event loss. For any supporting point of a risk curve during an event loss averaging, the ELRP
The historical events are also scaled for a defined event loss and the
corresponding scaled CRP is averaged. However, the “goal seek” function in
MS Excel is applied to find the correct scaled CRP
The uncertainty of the local hazard models influences the accuracy of the
CRP since the CRP is an average of estimates of local RP. The issue is that
there is a certain correlation between the estimated hazard parameters of
neighboring wind stations. We consider this by application of the jackknife
method (Efron and Stein, 1981). According to these, the root of mean squared error (RMSE, which is the standard error if the estimate is bias free as we assume here) of the original estimated parameter
To consider any correlation in the error propagation of CRP estimate, the
maximum of the same half-season
We use the same approach to consider the error propagation from local hazard
models to the risk estimate for the max-stable case in Sect. 4.4. But the
finally estimated parameter
The computed standard errors in Fig. 7b are in the range of 7.5 % to 8.5 % of estimated event loss per defined ELRP. The shares of uncertainty components on the error variance (squared SE) of our risk estimates depend on the RP. On average for our supporting point, these are 15 % for the limited sample of scaled historical events, 24 % for the uncertainty of local hazard parameters, and 61 % for the vulnerability model's parameter. Unfortunately, we do not know a published error estimation for a vendor model for winter storm risk in Germany. Therefore, we can only compare our estimates with Donat et al.'s (2011). Their confidence range indicates a smaller precision than ours.
We have compared our results with vendor models in Sect. 4.4. These have
estimated the risk curve for the maximum event loss within a year. This is a
random variable, and their RP is the reciprocal of the exceedance probability and can never be smaller than 1. We transform the RP of annual maxima to the RP of event loss according to the relations in Sect. 3.1 and the explanations by Coles (2001, p. 249, with
In the beginning, we asked the questions about the RP of a hazard event in a region, the corresponding NatCat risk, and necessary conditions for a reasonable NatCat modeling. To answer our questions, we have mathematically derived the CRP of a NatCat-generating hazard event from previous concepts of extreme value theory, the pseudo-polar coordinates (Eq. 17). This implies the important fact that the average of the RPs of random point events remains a RP with exceedance frequency (Eqs. 8 and 15). Furthermore, we extended Schlather's first theorem for max-stable random fields to the non-max-stable spatial dependence and characteristics. We have also considered the normalized variant of the area function of all local RPs of the hazard event in a region with parameters CRP and CV. The absence of max-stability in the spatial dependence results in a correlation between CRP and CV, which is a further indicator for non-max-stability beside changes in measures for spatial dependence by changed block size (e.g., annual maxima versus 2-year maxima).
The derived CRP is a universal, simple, plausible, and testable stochastic measure for a hazard and NatCat event. The weighting of local RP in the computation of the CRP can be used to compensate for an inhomogeneous distribution of corresponding measuring stations if the physical-geographical hazard component of a NatCat, the field of local event intensity, is of interest. However, the concentration of human values in the geographical space could also be considered in the weighting to obtain a higher association of the CRP with the ELRP of a risk curve. This link implies the conditional expectation (Eq. 18) under the assumption of a max-stable association between CRP and ELRP and offers a new opportunity to estimate risk curves, the bijective function event loses to ELRP, via a stochastic scaling of historical intensity fields and averaging of corresponding risk parameter. The averaged parameter can be the scaled CRP for a defined event loss, corresponding exceedance frequency, or the event loss for a defined/scaled CRP.
The differences between the three estimators are small in our application example, insured losses from winter storms over Germany. In contrast, the influence of the stochastic assumptions regarding the spatial dependence and characteristics (max-stable or not) is significant in the range of higher ELRPs. This highlights the importance of a realistic consideration of the spatial dependence and characteristics of the hazard in a NatCat model. Besides, our risk curves for Germany have a similar course as those derived by vendors (Fig. 7d). The risk assumption by the EU for Germany with ELRP of 200 years is significantly higher than ours. The estimate by Donat et al. (2011) differs significantly and seems to be implausible for higher ELRP. A reason might be their statistical modeling by the generalized Pareto distribution as already applied for wind losses by Pfeifer (2001). The tapered Pareto distribution (Schoenberg and Patel, 2012), also called tempered Pareto distribution (Albrecher et al., 2021), or a similar approach (Raschke, 2020) provide more appropriate proxies for our risk curve's tail.
According to our results, necessary conditions for appropriate NatCat modeling are the realistic consideration of local hazard and their spatial dependence (max-stable or not?). Correspondingly, the spatial characteristics of NatCat events, described here by relation CRP to CV, must be reproduced. In addition, the CRPs of a simulated set of hazard events in a NatCat model should have an empirical exceedance frequency that follows the theory (Eq. 15). Finally, the standard error of an estimate should be quantified, the sampling should be appropriate, and overfitting (over parametrization and parsimony) should be avoided. This principle applies to all scientific models with a statistical component (e.g., Lindsey, 1996).
The advantage of our approach over vendor models is the simplicity and clarity of the stochastic assumptions. The numerical simulations for models in the insurance industry (Mitchell-Wallace et al., 2017, Sect. 1.8) and science (e.g., Della-Marta et al., 2010) need tens of thousands of simulated storms with unpublished or even unknown (implicit) stochastic assumptions. We have only scaled 16 event fields of historical storms with controlled stochastics and could even quantify the standard error.
Our approach to CRP is based on two assumptions. At first, the local and global events occur as a Poisson process. This is a common assumption or approximation in applied extreme value statistics (Coles, 2001, Sect. 7), and the corresponding Poisson distribution of the number of events can be statistically tested (Stephens, 1986, Sect. 4.17). Moreover, the verified clustering (overdispersion) of winter storms over Germany (Karremann et al., 2014) is statistically not relevant for higher RP (Raschke, 2015a). With increasing RP, the number of winter storms occurring converges to a Poisson distribution. Clustering is also influenced by the event definition, which is not the topic here (keyword declustering; Coles, 2001). We also point out that the assumed Poisson process does not need to be homogenous during a defined unit period (year, hazard season, or half season).
The second prerequisite is robust knowledge about the local RP by a local hazard curve. Unfortunately, there are no appropriate and comprehensive models for the local hazard of every peril and region, for example, hail in Europe; we only know local hazard curves for Switzerland by Stucki and Egli (2007), and these were roughly estimated. There are public hazard maps of flooding areas for defined RP; corresponding local hazard curves are rarer.
Furthermore, existing models for local hazard are partly questionable
according to our discussion about local modeling of wind hazard from winter storms in Sect. 5.1. We have assumed a Gumbel case of the generalized
extreme value distribution for local block maxima with extreme value index
Since the current model for the local hazard of winter storms over Germany results in considerable uncertainty, it should be improved in the future. This could be realized by a kind of regionalization of the hazard as already known in flood research (Merz and Blöschl, 2003; Hailegeorgis and Alfredsen, 2017) or a spatial model (Youngman and Stephenson, 2016). Besides, more wind stations could be considered in the analysis with better consideration of incompleteness in the records. An extension of the observation period is conceivable if homogeneity of records and sampling is ensured. A more sophisticated approach might be used to discriminate the extremes of winter storms from other windstorm perils at the level of wind station records. The POT methods (Coles, 2001, Sect. 4.3; Beirlant et al., 2004, Sect. 5.3) could then be used in the analysis even though the spatial sampling is complicated as stated in the “Introduction”.
Further opportunities for improvements in the winter storm modeling are conceivable. The event field might be reproduced/interpolated in more detail, as done by Jung and Schindler (2019). They have considered the roughness of land cover at a regional scale besides further attributes. However, they did not consider the local roughness of immediate surroundings, as Wichura (2009) discussed for a wind station.
Besides, our approach could be used for further hazards such as earthquake,
hail, or river flood. The reasonable weighing would not be trivial for river
flood. It may be that the local expected annual flood loss would be a reasonable
weighting if the final goal is a risk estimate for a region. The numerical
handling of the case that an event does not occur everywhere in the researched region but local RP
We also see research opportunities for the community of mathematical statistics, especially extreme value statistics. Does Eq. (18) for conditional expected RP also apply to the non-max-stable case? A deeper theoretical understanding of non-max-stable random fields with max-stable margins is of great interest from practitioners' perspectives. Research about the link between normalized area functions (expectation versus CV) and spatial dependence could increase understanding of natural hazard and risk, and our construction for the non-max-stable scaling is just a workaround to illustrate the consequences of dependence characteristics; for risk models in practice, a transparent stochastic construction is needed. Furthermore, estimation methods could be extended and examined, such as the bias in estimates of local RP.
A special code was not generated or used. MS Excel carried out our computations. The wind data were downloaded from the server of the German meteorological service (
The supplement related to this article is available online at:
The contact author has declared that there are no competing interests.
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The author thanks the reviewers for helpful comments.
This paper was edited by Yves Bühler and reviewed by Francesco Serinaldi and one anonymous referee.