Alexandre Mornet
email: alexandre.mornet@allianz.fr

Thomas Opitz

Michel Luzi

Stéphane Loisel
email: stephane.loisel@univ-lyon1.fr

Wind Storm Risk Management -Sensitivity of Return Period Calculations and Spread on the Territory

Keywords: Storm Index, Wind Speed, Insurance, Extreme Value Theory, Extreme Dependence, Return period, Solvency II

Models and forecasts of damage from wind storms are a major issue for insurance companies. In this article, we focus on the calculation sensitivity of return periods for extreme events.

Numerous elements come into play, such as data quality (location of insured buildings, weather report homogeneity), missing updates (history of insurance portfolios, change of ground roughness, climate change), the evolution of the model after an unprecedented event such as Lothar in Europe and temporal aggregation (events defined through blocks of 2 or 3 days or blocks of one week). Another important aspect concerns storm trajectories, which could change due to global warming or sweep larger areas. We here partition the French territory into 6 storm zones depending on extreme wind correlation to test several scenarios. We use a storm indexdefined in [20]to show the difficulties met to obtain reliable results on extreme events.

Introduction

The study of storms is both a matter for natural phenomena linked to climatic and geographical factors, and on the other hand, for its consequences in terms of catastrophes linked to economical, historical, and human factors. The management of risk during extreme events is something which has an influence on the whole of society [START_REF] Hochrainer-Stigler | Risk management against extremes in a changing environment[END_REF], far beyond insurance.

First, to understand our approach, it is necessary to consider the problematic of the management and evaluation of storms. The economic and structural information available in an insurance firm show the volatility and intensity of damages. Yet they can only account for historical data that are limited and difficult to update [START_REF] Katz | Extreme events in a changing climate: Variability is more important than averages[END_REF]. This is why we have decided to complete our data with meteorological reports. We have taken into account different variables that may be linked to storms and finally to focus on wind speeds ( [START_REF] Donat | High resolution refinement of a storm loss model and estimation of return periods of loss-intensive storms over Germany[END_REF], [START_REF] Klawa | A model for the estimation of storm losses and the identification of severe winter storms in Germany[END_REF] and [START_REF] Pinto | Changing European storm loss potentials under modified climate conditions according to ensemble simulations of the ECHAM5/MPI-OM1 GCM[END_REF]).

However the evaluation of the impact of a storm is not as simple and direct as we could wish.

Several difficulties we have come across will be discussed here in order to account for the variability of the results we have obtained. If it is possible to get data on the disasters and portfolios on relatively precise levels (city or postal address), we generally only have statements on the national or departmental levels, as soon as we use market data or old-time history. Furthermore, if we refer to meteorological data, we have only a partial number of stations, roughly one or two in each department which makes it difficult to work with great precision. We can also note a great variability of recorded speeds according to the geographical situation of the station [START_REF] Sacré | A new statistical approach to extreme wind speeds in France[END_REF]. The stations at high altitude or on the coast often show speeds that are greater than in the rest on the department and that must be balanced according to their distance from main towns. Then, for the same station, the homogeneity of the recordings on a long period is not necessarily confirmed. The study of the breaks [START_REF] Thomson | Estimating the slope and standard error of a long-term linear trend fitted to adjusted annual temperatures[END_REF] has allowed us to notice discrepancies, perhaps due to a change in equipment, but that have to be connected with other phenomena such as different roughness of soils or climatic changes which can also disturb the data. Considering those difficulties, one possible solution is to regroup the French departments according to the extreme speeds of winds. Paradoxically, one might be tempted to try and work in a more subtle way, at least to avoid comparing wind speeds several dozen miles from risks.

Then here, we choose the opposite solution. Our division is justified by the fact that there are departments with very few important damages (because there are few contracts). We must add to this our limited number of significant events and the heterogeneity of distances between the risk and the station. A station approach by department would not be clear enough .We have chosen a distribution into six zones of risks, which allows us to act on an intermediate scale between the department (dimension 95) the whole of France. For this, we use the k-medoids clustering algorithm [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF]. The work on these zones allows us to test various methods of agglomeration of departments (extreme dependence [START_REF] Coles | Dependence measures for extreme value analyses[END_REF], maximal speeds, and geographical distance). It allows us to consider various scenarios of the extent of storms as in [START_REF] Brian | New perspectives on the northern hemisphere winter storm tracks[END_REF] and [START_REF] Mlr Liberato | Explosive development of winter storm xynthia over the subtropical north atlantic ocean[END_REF]. These amplitudes might change with climatic change (arrival of storms of the Medicane(1 ) type for example) or sweep over various zones successively.

Such projections are a coherent means to foresee the small number of major storms in our base of observation.

On a more technical level, the building of a storm index implies the choice of a time-pace. Here we must separate the grouping of days in order to stick to the notion of the event retained as a definition by insurers and of clusters that correspond to a period favorable for the starting point of successive events. We have realized that, contrary to the evaluation of changes, which makes necessary the addition of catastrophes during the whole event [START_REF] Mornet | Index for predicting insurance claims from wind storms with an application in france[END_REF], the maximal daily evaluation of our index is the best information about the relative intensity of storms. Of course, these decision will have an impact on the projection of storms. We have compared the results of statistic approaches according to the use of daily or weekly data. In our models, we have also tried to show the sensitiveness of the calculations of the return period to the inclusion or not of the most important events like Lothar and/or Martin in 1999.

For an insurer, the management of the damages due to the wind depends very much on the return period that is attributed to the most extreme storms. Such events, of course, are very few and the story of claims proves inadequate to give sound answers to questions related to the pricing, to volatility of results, therefore on the need of capital stock. The communication on the uncertainty that goes with the prevision of extreme events is important and is often neglected. Yet, there are technical means to determine that uncertainty, but a sociological reluctance to the aleas limits its diffusion. It was the case during Juno storm in 2015 that was largely overestimated as Winkler [START_REF] Robert | The importance of communicating uncertainties in forecasts: Overestimating the risks from winter storm juno[END_REF] explains. The purpose of this article is to bring a new light on the modelling and calculation of these extreme events by basing on the storm index as defined in our previous articles [START_REF] Mornet | Index for predicting insurance claims from wind storms with an application in france[END_REF].

Here we wish to insist on the great sensitiveness of results to the hypotheses we have returned, while proposing complete concrete results leaning on meteorological data we have retained, at the same time proposing concrete results leaning on the available over the period of abaut40 years, and less disturbed by the actualization problems than insurance data. This sensitiveness will be illustrated both by the calculation of return periods and by the estimation of needs in capital stock.

2 About wind speed

Definition

The reference value of the speed and direction at a given measurement point and moment are the immediate speed and direction averages at long enough intervals (usually 10 minutes) which precede the reported measurement time. Weather reports refer to medium wind, while possibly mentioning the observed measurement or the expected intensity of gusts (whose values can exceed the medium wind measurements by more than half). It is also possible to measure the vertical component of the wind with a three-dimensional anemometer. Thus, we can study the subsiding and rising air movements. However, our data sources are not that exhaustive, and we settle for learning as much as possible from available data.

Problem of breaks in data series

One issue calling for particular attention is the reliability of meteorological data. Indeed, when we focus on a long period of time, measurement devices and data restitution methods can change and bias our results. The presence of a changepoint can lead to a significant loss of precision [START_REF] Thomson | Estimating the slope and standard error of a long-term linear trend fitted to adjusted annual temperatures[END_REF]. It is consequently important to detect ruptures in the structure of data series and to correct them.

Through observation of annual means/medians/variances, for each station, we try to spot a change in the series.

Break detection in our data

We detect potential breaks in the central tendency of wind speed data using a standard software implementation (changepoint package of the R software library [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF]). Relevant methods are described in Killick et al. [START_REF] Killick | Optimal detection of changepoints with a linear computational cost[END_REF]. Formally, for a data sequence (y 1 , ..., y n ), it is said that a break τ ∈ {1, ..., n -1} exists if the statistical properties of subsets {y 1 , ..., y τ } and {y τ +1 , ..., y n } are different according to certain criteria. Multiple break detection necessitates minimizing the target function

m+1 i=1 [C (y(τ i-1 + 1) : τ i )] + βf (m) (1)
with C a cost function for each segment and βf (m) a penalty to avoid over-segmentation. Here, we use the hierarchical and bisection-based PELT (Pruned Exact Linear Time) algorithm. PELT first does a rupture test on the whole period. If a break is detected, data is divided in two subsets at the break position. Then, detection is conducted on both subsets and iterated until no more breaks are found. To avoid seasonality problems, we here focus on the annual averages of each station. Besides, since our index is built from the highest speeds, detection is here made on the threshold exceedance of the empirical quantile q at 80%. Until 1983, the average exceedance is 39.87 m/s and starting from 1984, it is 35.07 m/s. We standardize the series (at the bottom of Figure 1) by adjusting the exceedances of the first sequence with respect to the second one, more recent, whose reports we assume to be more reliable. We underline that this approach is robust yet relatively simple, so we have to remain cautious when interpreting results which involve data from before 1980. Old data remains less reliable and noisier even after preprocessing.

YEAR

Several explanations can be provided for the origin of these breaks. Changing the topology around the station following, for example, the construction of a building nearby, or changing the type of anemometer used are sources of disturbance for the measurement. Sacré et al [START_REF] Sacré | A new statistical approach to extreme wind speeds in France[END_REF] explain that the Papillon anemometers used in the 1970s caused problems, especially for high wind speeds. Figure 2 represents the transmitter of the device in question [START_REF] Papillon | Les appareils de mesure de la vitesse du vent. anémomètres de pression et anémomêtres de vitesse. description des anémomètres actuellement utilisés par l'office national météorologique: Anémomètre électromagnétique papillon; anémomètre magnétique à main richard[END_REF]. Mechanical anemometers are sensitive to frost and humidity, which leads us to underestimate wintry wind speeds in the mountains. Current state-of-the-art devices use transmitters-receivers from directional ultrasounds, but measurements can still be prone to technical problems. The Nice airport station for example presents recurring errors despite its wind profiler, which turns out to be unable to detect land breeze and mountain breeze from the nearby Var valley. This distribution of stations has to be put into relation with population density and the number of policies in the Allianz portfolio for each department (Figure 4). This relation must be incorporated when seeking a connection between meteorological information and insurance claims. The department scale is not necessarily the best option for building a storm index. In the next section, we will see that it is pertinent to work on larger and more homogeneous zones by grouping departments according to their exposition to high wind speeds.

Classifying departments into homogenous storm risk zones

We use daily wind speed records over 50 years. Our goal is to aggregate the data on larger territories than a department to facilitate the search of a connection between wind speeds and costs. Indeed, working on France as a whole would not take the heterogeneities of the territory into account, whereas an approach by department causes a problem in terms of availability, reliability and heterogeneity of data. Our intermediate approach allows looking at the results for each of these larger zones, which would have been difficult for 95 departments. The French territory shows great variety of relief and climate; therefore, it is appropriate to delineate zones likely to be simultaneously affected. We can further recover information on typical storm trajectories and on regions relatively unscathed by wind damages. Furthermore, statistical uncertainty in department-based results would have been rather strong, whereas grouping together "similar" departements enables us to reduce them. For insurance, this approach also allows us to detect the regions where the risk is the highest or the lowest compared with the national average. For a fixed value of k = 2, 3, ..., the k-medoids algorithm of [START_REF] Kaufman | Finding groups in data: an introduction to cluster analysis[END_REF] returns k zones by minimizing the distances between a central department and associated departments. Medoids are the central representatives of a zone, i. e. the central department in our case. The choice of distance is crucial; we here build a distance measure based on a dependence index λ of bivariate extremes, measuring the dependence between distribution tails of two data series. This index represents the probability of simultaneous extreme realizations and therefore its values are between 0 (asymptotic dependence) and 1 (total dependence). For two random variables, X and Y having as distribution functions F X and F Y , the index is defined by

λ = lim v↑1 P(Y > F -1 Y (v)|X > F -1 X (v)) , (2)
provided that this limit exists. We calculated this index from daily wind speeds by fixing as a threshold the quantiles at 80, 90 and 99%. This corresponds to retaining for each department successively the 3700, 370 and 185 highest speeds. For a spatial representation of dependencies, we transform this index into a distance compatible with the k-medoids algorithm. We propose

d 1 (x, y) = 1 -λ. (3)
Naveau et al [START_REF] Cooley | Variograms for spatial max-stable random fields[END_REF] uses a similar distance defined for annual maxima in order to assemble precipitation measuring sites. This method makes it possible to get quite homogenous groups geographically.

Nevertheless, some departments can be very far from their medoid, which prevents us from having zones clearly delimited. For this reason, we undertook to slightly change the distance used by adding to it a component dependent on the geometrical Euclidian distance (d geo ) between each point considered based on a projection of longitude-latitude coordinates into the plane. Moreover, this component also allows taking into account potential differences in the management of local administrations with respect to storms. Calculated intersite distances have been rescaled in order to make them comparable with d(x, y), and we propose a new distance

d 2 (x, y) = p × d 1 (x, y) + (1 -p) × d geo (x, y), (4)
where p taken between 0 and 1 allows regulating the influence of geometric distance on our extremesbased distance. The inclusion of a geographic distance criterion makes it possible to counterbalance issues of inaccurate anemometer, at least partly. Based on preliminary analyses, we use distance The first divisions appear in Figure 5. The groups 1, 2 and 3, which respectively count 15, 22 and 22 departments, are the least solid. Looking at their silhouettes in detail, we note negative values corresponding to departments not closely attached to their medoid. However, groups 4, 5 and 6 present entirely positive silhouettes with the best result for the group 4 (0.17). On the map, this strength is expressed through the dimension of points located at the center of departments.

d
Figure 6 reports the geographic evolution of the 6 zones when the probability threshold increases from 0.80 to 0.98, shifting focus to extreme and unusually high wind speeds. According to the silhouette, the average strength slightly decreases (from 0.09 to 0.07) and group 5 is the only one with an entirely positive histogram. Group 2, in northern France, is largest one, regrouping 26 departments. Zone 6 henceforth spreads from Brittany to the center. In the south, the division is less homogenous with departments isolated from their medoid.

Finally, Figure 7 was obtained by fixing the quantile to exceed at 99%. is less homogenous than before, we can still notice that obtained zones correspond quite well to the French large river basins. These river catchments are fundamental in the study of natural risks [START_REF] Thouret | Vulnérabilité aux risques naturels en milieu urbain: effets, facteurs et réponses sociales[END_REF].

The vulnerability to a storm depends on structural, geographical and seasonal factors at once. Our work on risk zones aims at better exposing the spatial structure of these vulnerabilities. first map (on the left) presents the most homogeneous zones on the whole and expresses a territorial sensitivity to small-scale national storms. When we focus on the most extreme exceedances, the division becomes less accurate geographically but it is more likely to show possible exceptional storm trajectories. Besides dependence of extremes and geographical distance, we have further added a third criterion: the resemblance of entire series of wind speed by department. We then built d tot , which is the Euclidian norm of the difference of wind series at two stations. Our goal is to enhance the strength of the groups and thus delineate at best, by combining those three approaches, homogenous zones where wind circulation is significantly similar. The values of these different distances have been scaled in order to make them comparable. We arrive at the following definition of our distance:

d 3 (x, y) = p 1 × d(x, y) + p 2 × d geo (x, y) + (1 -p 1 -p 2 ) × d tot (x, y), (5)
with p 1 = p 2 = 0.33 chosen in a preliminary analysis. This new formula is used as before with the three quantiles serving as a threshold in the distance of extremes. We consistently notice an improvement in the strength of zones. For q 80 the average strength of all groups reaches 0.16. Group 1, which counts 21 departments, is the least solid (0.05). By observing its silhouette in detail, we find negative values, which correspond to departments not closely bound to their medoid. In contrast, other groups present mostly positive silhouettes with the best result for group 2 (0.28) in the region north of Paris. For q 98 the division is substantially the same. The average strength decreases by 3 points (from 0.16 to 0.13), and groups 2 and 5 stay the most homogenous. Group 1, in Southern

France, remains the most important by gathering 19 departments, but also the least solid. Zone 3 gains one department towards the North. Zone 4 settles more distinctly on Eastern France from the North to the South and zone 6 broadens with two additional departments. Finally, for q 99 , the average strength of the silhouette only decreases by one point to 0.12. Groups thus formed are quite different from the previous approaches (and their numbering also changed). 23 departments along the Eastern frontier of France are gathered, from Alsace to Corsica. The zone around the Paris now counts the most departments [START_REF] Sacré | A new statistical approach to extreme wind speeds in France[END_REF]. The most homogenous zone is the fourth in the South-West with a silhouette at 0.21; it has the same contour as zone 5 in the previous divisions. The new zone 5 ends up centered on Brittany and becomes smaller in terms of surface area. The three maps appear in Figure 9.

Those different approaches lead us to the concept of materiality of perils, expressed through potential drifts caused by storm damages on a geographical and temporal scale. We now investigate how much the morphology of zones depends on few influential events, the most extreme in particular. Therefore, we take the simulation of the k-medoids algorithm but without wind speeds recorded between December 24th and 28th 1999, which are associated to the storms Lothar and Martin. We for the quantile at 98% and 0.11 for the quantile at 99%). The group attribution and hence the geographic shape of zones might change as a consequence. For the first two thresholds (q 80 and q 98 ), changes are negligible; only for the highest threshold (q 99 ), we observe noticeable differences. The most obvious concerns zone 6 around Brittany which clearly broadens at the expense of zones 2 and 3. Zone 4 in Eastern France still extends from North to South but does not spread to the center.

Finally, in the South, zones 1 and 5 remain stable, which can seem normal given that they have been the least seriously affected by the storms Lothar and Martin.

Hazard management is here complicated by the fact that elements outside the scope of variables available to insurers influence the number of claims. A natural phenomenon conveys a field of action (space), intensity (impact) and recurrence (frequency). It is possible to use spatial statistics ([9], [START_REF] Bonazzi | The spatial structure of european wind storms as characterized by bivariate extreme-value copulas[END_REF])

to establish damage simulations adapted to each of the wind zones previously defined. For the rest of the study, the division with the quantile at 99% and all the wind speeds (Figure 11) will be used.

Distribution of the portfolio on the zones

To understand the relative weight of each zone in terms of the insurance portfolio, we consider three quantities: the number of policies, the sum of charges and the frequency of claims. Particular interest lies in the average annual sum of charges for each zone. To estimate this average and its variability according to the presence or not of major storms as in 1999, we use the bootstrap method by creating 2500 different samples, each representing the daily charges over the whole estimation period. We must note that in our case with strongly heavy-tailed distributions this non-parametric method has some flaws. A small number of events has a very strong influence on the results here, which impedes formal statistical consistency [START_REF] Hall | Asymptotic properties of the bootstrap for heavy-tailed distributions[END_REF]. However, the method remains interesting since it is based on a minimum number of model hypotheses. We use it for a first empirical study before moving to models with better theoretical support based on the theory of extreme values. again present the bootstrap distribution for zone 6. This zone is smaller than zone 2. Indeed, it only counts 13 departments versus 25 in the latter. The shape of the distribution is slightly different. We notice several peaks, as if composed of a mixture of distributions. This zone actually illustrates the shortcomings of the bootstrap with heavy-tailed data. The periodicity in the bootstrap distribution may be due to a single huge event, and peaks appear when the event is drawn 0, 1, 2, . . . times. For instance, it is known from elementary bootstrap theory that a particular event does not occur in a bootstrap sample with a probability of about exp(-1) = .37 when n (the number of observations) is high. Moreover, we can see the importance of geographical scale besides the temporal scale for the stability of results.

This bootstrap approach is based on daily data. It does not convey the notion of event and it overlooks dependencies between days. For comparison, we perform a bootstrap based on weekly blocks of aggregated charges to better capture events spreading over several days. In the last part of Table I, we present the quantiles at 2.5% and 97.5% of charges in average annual and in millions of euros. We can see a higher variability in the distribution of the average due to a grouping of clustered events for several days. The estimations for the low quantiles (2.5%) are lower than the ones in the daily approach. For instance, in zone 2, we have a decrease of more than 30% from 9.2 to 6.3 millions.

For an upper quantile, the estimations based on weekly data are higher than the daily ones. Still in zone 2, we move from 55.7 to 66.2 million euros, i.e. an increase of almost 19%. On Figure 13, we observe again the distribution of annual charges for zones 2 and 6 but this time from a weekly aggregation. The problems with the bootstrap approach appear amplified. The claim rate varies according to the observation period. We can finally conclude from this approach that we need more refined methods, such as a parametric model (GPD), taking clusters into account.

Spatial representation of main storms

In a first approach, we give a spatial representation of the distribution of claims during major events in terms of costs and compare them with available climatic variables. The different variables we have checked are temperatures, wind speeds, precipitation and North Atlantic Oscillation (NAO).

A preliminary analysis showed that only wind speeds are geographically correlated with the path of storms. Moreover, a wind index can provide better information because it enables to detect regions where unusually high wind force occurred. Damages are linked to the strength of the storm and the condition of the buildings, whose architecture or construction is more or less resistant to bad weather conditions depending on the region. The wind index formula we use is the one defined in [START_REF] Mornet | Index for predicting insurance claims from wind storms with an application in france[END_REF]. Based on a series of daily maximum wind speed measurements w d (s) for a sequence of days d ∈ D at station s, it is defined as follows:

I d w (s) = [w d (s) -w q (s)] + α , (6)
where w q is the q%-quantile of w d (s) for d ∈ D.The use of the exceedances of a quantile in the construction of a risk index allows us to focus on the most extreme values. This method is quite common [START_REF] Cameron | Summarizing risk using risk measures and risk indices[END_REF]. We then present for each department the observed event in terms of the sum of costs, maximum wind speeds and wind index over a period of 3 days. Nevertheless, a high index does not always coincide with big damages for Allianz as we can see in the departments situated in the center of France, but this is due to a lower density of urbanisation and risks in this zone. In conclusion, the variable "wind speed" can explain a large part of the damages, but not enough to fully describe such a complex phenomenon as a storm. This complexity necessitates to complement the estimates with a discussion about uncertainties, specifically for extreme events [START_REF] Robert | The importance of communicating uncertainties in forecasts: Overestimating the risks from winter storm juno[END_REF]. The wind originates from the difference of atmospheric pressure, and to understand wind damage it is important to distinguish two types illustrated in Figure 16. Synoptic wind is generated by continental or global phenomena; it is sustained, occurs on a large scale and can be violent. Local wind is due to temperature disparities; it is thermal wind. Indeed, the behavior and evolution of the wind flow will be different according to its type. Synoptic wind is generally slowed down by an obstacle (natural or artificial), whereas thermal wind is strengthened by the relief. Wind direction is also important; combined with speed, they form velocity. Knowing these parameters and the duration of wind gusts could improve the understanding of storms.

Modeling

Definition of a storm index

Building a model from extreme event data that are rare by definition leads to preliminary questions about the choice of variables and their use, since the same raw data can yield very different results.

Several approaches can be considered to describe the impact of a storm in space and time. We will focus on three criteria and on their influence on the model. The first one is the length of the observation history. The second concerns temporal aggregation in terms of either daily or weekly charges and the resulting change of perception of storm frequency. The third criterion is related to the existence and knowledge of the event considered as the most extreme in the observation period.

We here test the sensitivity of the fitted distribution and of the return period in particular with respect to the presence or not of cyclones like Lothar and Martin in our data.

In this section, we propose an improved and more flexible version of the storm index, defined in a previous article [START_REF] Mornet | Index for predicting insurance claims from wind storms with an application in france[END_REF], and with its extrapolation beyond the values observed. The index can be studied over a longer period than damage data, it presents less nonstationarities and can exploit a finer spatial resolution. In each of the six risk zones A(k) previously delimited, we build a specific storm index combining wind indexes and exposure for each department in terms of portfolio risk. For k ∈ {1, ..., 6} we define I S (k) as

I S (k) = s∈A(k) R(s) × max d∈E I d w (s) N d , (7)
where in each station I d w (s) is the wind index of day d, and the number of risks R(s) is weighted by the number of contracts. We also take into account the size of the damaged area (geographic aggregation), the duration of the storm event E2 and the number of active stations for day d, N d .

We stress that we choose the maximum and not the sum of daily indexes for the temporal clusters associated to each storm. As opposed to insurance costs where are added over the duration of the event, wind peaks better reflect the relative intensity of storms events. We then obtain a global storm index on the whole country by adding up the six storm indexesm weighted both by the portfolio and by a parameter vector B = {β 1 , ..., β 6 } with β 1 = 1 fixed. This additional parameter allows us to over-or underweight each zone according to the impact of a hypothetical storm passing through the territory in terms of property damages. The formula becomes

I S = 6 k=1 I S (k) × β k , (8)
We have chosen the parameters of this index such that it reflects at best the distribution of the most extreme damages in terms of rank. Our ranking for the most important storms since the 1970s is based on [START_REF] Luzi | Travaux sur les historiques tempêtes[END_REF]. Based on this ranking, we propose a final modification to the index values such that they become approximately proportional to aggregated claim amounts. Therefore, to make the index value distribution more heavy-tailed, we use the exponential function and a last parameter γ, estimated with a Least Squares method. The link between our storm index and the cost C for the insurer now is

C = exp (γ × I S ) . (9)

Statistical modeling of the storm index

Extreme values theory [START_REF] Embrechts | Modelling extremal events for insurance and finance[END_REF] has spawned two complementary approaches: the block maximum and the threshold exceedance. Here, we use threshold exceedance modeling through the Generalized Pareto Distribution (GPD). With this model and for an appropriate threshold u, the distribution function of Z = (X -u), knowing that X > u, is defined by:

H(z) = 1 -exp - z σ , ξ = 0, or H(z) = 1 -1 + ξz σ -1 ξ , ξ = 0,
according to the value of the shape parameter ξ controlling the tail of the distribution. We have used the extRemes package [START_REF] Gilleland | A software review for extreme value analysis[END_REF] of the software R to estimate the different distributions of extremes.

We have consider two different observation periods: the first one over twenty years (1993-2013), the second one over forty years , with data for the first period being the most homogenous.

Insurance records do not reach back to such long periods. We work with the storm index I S and fit the parametric GP distribution to its tail. To begin, the threshold u must be fixed, here done by considering two standard visual methods. The first one plots estimates of shape and scale parameters according to different thresholds and seeks a threshold level above which estimates for different thresholds remain within the confidence interval of all estimates above the threshold. The second one is the mean excess plot confidence intervals.

Figure 17 shows the resulting diagnostic plots for threshold selection. Both graphs on the left represent variations of σ and ξ with their confidence intervals at 95% according to the considered threshold values in the range between u = 5 and u = 40, where the index has been calibrated such that its observed maximum (Lothar storm in December 1999) is 100. By observing the confidence intervals increase in size for larger u, we here choose u = 20 as the upper bound above which the variability of estimates becomes too important. The graph on the right depicts the mean excess above We further propose an analysis of sensitivity with respect to Lothar and Martin, the two most important events of our observation period. We observe how return periods change if we remove Lothar and/or Martin from the data. Our overall aim is to understand the robustness of the model (or lack thereof) when quantifying events with return periods of over 70 years or even 200 years based on a data record of less than 50 years.

u = 10 and period = 1970-2013

We first estimate the parameters of the Generalized Pareto distribution with a threshold u = 10 and over the period 1970-2013. The threshold is exceeded on 62 occasions, that is to say around 1.4 times per year over the observation period. The parameters are estimated according to the maximum-likelihood method (MLE). Figure 18 provides a QQ plot of data quantiles against fitted model quantiles, a comparison of quantiles from model-simulated data against the data, a nonparametric density estimate of the data along with the fitted model density and a return level plot. The parameter ξ is positive (0.20 in Table II), which corresponds to a heavy tail and a distribution function belonging to the Fréchet maximum domain of attraction. Both graphs on the upper row of 18 are QQ plots. On the left one, we note a slight gap in the strong quantile values but a good global alignment. On the right one, the alignment of the dots is excellent to compare the model probabilities and empirical probabilities, except for a single outlier point. The density function of the model is confronted with a kernel estimate on the bottom left of the graph. We notice a good correspondence away from the lower bound 0 where the density estimate is not reliable. All return level estimates stay within the 95% confidence interval. This is done on the log-scale for the abscissa so that the type of GP distribution can be seen from the shape, where heavy-tailed distributions are concave, exponential-tailed distributions are straight lines, and distributions with finite upper bound are convex). In our case, the estimate seems to indicate the upper-bounded Weibull maximum domain of attraction. The red dot in Figure 19 indicates that storm as strong as Lothar occurs approximately every 93 years. However, when interpreting results beyond the observation period of forty years, uncertainty is high and the lower bound of the confidence interval decreases strongly, calling for cautious interpretation.

u = 10 and period = 1993-2013

Still with the same threshold, we now want to compare previous results with those obtained on a period twice as short, spanning 20 years between 1993 and 2013. During this period, the threshold is exceeded only 20 times, one average once a year. The estimation of the parameters appears on Table III, and we see that standard errors have become higher. The scale parameter σ descreased from 10.78 to 9.87. The parameter ξ increases to 0.50, therefore we still deal with a heavy-tailed distribution. The return period of an event such as Lothar strongly deminishes, as we can notice on Figure 20. We now obtain that a level 100 storm on our index could occur every 31 years. We note here that the length of the observation period greatly influences the calculation of the return period. With this limited data record, we tend to overestimate the probability that such an event happens. We thus focus for the rest of this study on the longest available data record from 1970 to 2013.

u = 20 and period = 1970-2013

The following study is based on the 1970 to 2013 period, but with a higher threshold u = 20. This threshold is exceeded by our storm index on 26 occasions, on average 0.6 times per year. The estimates of the model parameters are reported on Table IV.

The estimated scale parameter σ increased from 10.78 to 12.96. The shape parameter ξ is estimated as 0.21, a value very close to the one obtained with the threshold u = 10 over the same period.

However, the standard error increases. In terms of return levels (Figure 21), we get a storm as strong terms of insurance claims in France. The estimations of the model parameters are given in Table V. The estimated scale parameter σ increases from 10.79 to 11.02. The parameter ξ is still positive (0.10) but decreases compared with the Lothar model (0.20). This decline can be explained quite naturally: the distribution tail becomes less heavy because the strongest value has been removed.

The standard error also increases. Regarding the return periods of the model (Figure 22), a storm as important as Lothar occurs once about every 280 years. Without Lothar, we multiply the return period of such an event by 3, a drastic change. This example substantiates the fragility of this type of modeling, whose forecasting radically changes when a new record value occurs. The scale parameter σ slightly decreases from 11.02 to 10.61. The parameter ξ is positive (0.16) and barely decreases compared with the complete data model (0.20). In terms of return levels (Figure 23), a storm as important as Lothar occurs once every 160 years. Without including Martin, the return period is 1.7 times higher for Lothar. In this numerical experiment, the model is again strongly determined by the omission of a major storm. The scale change is not as important as with the omission of Lothar but it remains considerable.

Summary and complement

The GPD model heavily depends on the most extreme value for the estimation of the parameter ξ, which controls the tail distribution. The estimation of ξ decreases in the absence of Lothar from the data record, and the heavy-tailed model comes close to a light-tailed model (ξ = 0) if we take the standard error into account. However, the estimates of the scale parameter σ are stable. Therefore, return periods of extreme events strongly increase. The observation period also plays a crucial role;

we noticed that a too short duration can entail a strong underestimation of return periods. For astorm as strong as Lothar, this period varies between a minimum of 31 years and a maximum of 280 years in our numerical experiments. The removal of an important intermediary event like Martin also impacts the results of the model through an increase of return periods. Table VII shows a summary of the return period variations for an event similar to Lothar according to the considered methods and scenarios. To complete this analysis, we have added the estimations for a loss event which would exceed Lothar by a factor of 1.5, L(1.5). To determine the value of the index I S associated to this level of loss, we reproduced the Formula 9.

L(1.5) = log(1.5) γ + 100 [START_REF] Hall | Asymptotic properties of the bootstrap for heavy-tailed distributions[END_REF] Table VII compares the return period variations according to the magnitude of the storm. For the first two estimations (between 1970 and 2013, or all data), there is an almost linear increase of the event frequency according to the induced damages. This increase reaches a maximum of 490 years in the scenarios with the omission of a major storm. The strongest variation is observed on the period Moreover, the perspective recommended by Solvency II to insurance companies imposes to work with quantiles at the 99.5% level, corresponding to vents with a return period of 200 years.

Capital requirements

Storms have a major impact on portfolios for property damage. In 1999, Lothar and Martin represented an aggregated claims equivalent to an average year. In a single year, insurance companies therefore had to support the equivalent of around two years of losses, which is not without risks for their own capital. Within the European project Solvency II, supervisory authorities seek to determine rules to limit the risk of financial ruin for insurance companies. The first pillar of this regulation defines the norms for calculating regulatory own capital. The SCR (Solvency Capital Requirement) corresponds to the need in capital required for the insurer to keep his commitments over the span of a year. A confidence interval at 99.5% was retained for annual costs. Here, hedging mechanisms (reinsurance) allow to smooth extreme situations by establishing a form of mutualization in space and time. They are technical and a corresponding market exists. However, the needs in own capital are generally measured after reinsurance.

In view of the complexity of these systems which are specific to each company, we focus on the sensitivity of raw results within Solvency II. We study the target capital needed to absorb the shock caused by a risk resulting from extreme and irregular events. The European Supervision Authority of Insurance and Pension defined a standard formula of the SCR. For an annual variable expense C,

the formula is SCR = VaR 99.5 (C) -E(C),
with E(C) the average annual expense calculated on the observation period and VaR 99.5 (C) the value at risk at 99.5% of the distribution of C.

Calculation of the average annual expense

To understand how much exceptional expenses as in 1999 influence the average annual expense, we propose to include it in our calculations according to the different return periods obtained in the previous section. According to the historical records [START_REF] Douvillé | Tempêtes, grêle et neige : Résultats 2010[END_REF] of the FFSA, 1999 alone represents 30% of the total costs over the period 1982-2012. In the following example, we will consider that the updated total expense of a company over a 30-year period amounts to 5 billion euros. The annual expense of 1999 would have reached 1.5 billion euros. The average annual charge due to 1999 then equals 116.7 millions. Table VIII shows that the sum of costs updated for Lothar represents more than two thirds of the total of the period costs. Depending on the chosen return period, the average annual expense varies between 165.1 million (31 years) and 122.1 million euros (280 years). The contribution of year 1999, which includes the two main events of the period, is evaluated at 48.4 million or 5.4 million, depending on two extreme scenarios.

These differences show us that it is difficult to obtain reliable results. Using tools for smoothing and extrapolation may create a false illusion of mastering the problem of rare events. Models may be more or less appropriately chosen, but essentially there always is a lack of precision when evaluating the return period of rare events with high impact.

Measurement of the dispersion of claims

Can we measure the dispersion of claims?

To tackle the problem of dispersion, we need to bear in mind the context of insurance procedures.

Companies generally work on annually aggregated net charges after reinsurance. Calculating net results poses the problem of quantifying the reinsurance regime which depends on technical, strategic and business conditions that are variable over time. Therefore, in our theoretical study we prefer to remain at the level of gross reinsurance. However, the elements at hand should allow the simulation of reinsurance, which necessitates the knowledge of costs per events. Ideally, we should work on this type of measurement to correctly address the issue of dispersion measurement.

Which observations are available to perform measurements?

Regarding the most important insurance events, we can access historical records covering around 40 years [START_REF] Luzi | Travaux sur les historiques tempêtes[END_REF]. This benchmark of major events for the four last decades is based on the basis of the whole insurance. The update over a relatively long period on a storm insurance scale is difficult. Among other aspects, we need to consider the evolution of construction indexes (FFB or RI), of the relative weight of individual and business segments in the portfolio, of the coverage proportion of the storm guarantee which is only mandatory since July 1990, or of the progress of real estate in France. Figure 24 reports the updated costs of the main storms since 1970 according to market data. The year 1999, exceptional and unprecedented in France, substantiates the high variability of the storm guarantee. We can also work on a company basis over shorter periods with a fully detailed claim record which provides more complete information about major events. Since 1998, Allianz lists all the individual wind-related claims with a departmental and daily precision. It represents over 520.000 claims, mostly for individuals (310.000). On a premium and claim basis, Allianz's market-share for the storm guarantee in mainland France is about 10% over the whole period. Moreover, costs carried by this company are highly correlated (0.99) to the market costs. Nevertheless, the update difficulties remain present.

Which assessment can we make?

As shown in section 5.1, data records are too short to derive average expenses, let alone to reconstruct the complete distribution of the annual costs. In general, three groups of claims can be distinguished.

A group including the largest number of events (low segment), another one including around a dozen of events per year (intermediary segment), and a last group concerning only the most extreme storms from the observation period (high segment). Concerning the terminology of such events, we talk about frequent, intermediary or extreme/rare events. In practice, these three groups are not modeled through a unique distribution, but according to a mixture of different and adapted distributions.

Moreover, uncertainty is not identical the three segments of the overall distribution. Available data are usually allow us to satisfactorily reconstruct the low and intermediary parts of the cost distribu-tion, which is not the case for the most extreme costs.

Current solutions and a new approach

Various solutions are already used by the market actors. They can be distinguished in terms of the database used for their construction and of the calculation method. The distribution function is sometimes estimated empirically from the insurer's records. In other cases, it is based on models coupled with Monte-Carlo techniques to simulate a great number of years. We distinguish three types of approaches: classic models (annual frequency and intensity function, by level of costs); market models (using event catalogues, damage functions, location information about the portfolio)

and our model built from daily costs and storm index statements.

Do these solutions provide uncontestable answers?

Theoretical, technical or material difficulties will always compromise the robustness of results. One should be aware of these difficulties and associate them to the models to better understand their qualities and shortcomings and to be able to interpret their results cautiously.

In classic models, functions are defined from a limited number of observations. The fragility increases with the level of costs. The definition of extreme distributions is derived from a very limited number of observations. We sometimes have to settle for a single annual observation, or even no observation at all. By extrapolation of this poorly known distribution of extremes we can obtain the full distribution.

Market models try to be abstract away from observed data by using external data like catalogues of artificial events and damage functions. Validating a model in this context is difficult since it cannot be compared to observations. For instance, it is difficult to validate a catalogue of 18.000 events, generated for France, based on only 600 observations. How can we validate damage functions when no observation is recorded on the selected criteria? No claim file in France has recorded wind speed and damage rate. These models did not inform us on the data related to damage functions.

The work carried out in 2012 and 2013 by [START_REF] Luzi | Travaux sur les historiques tempêtes[END_REF] showed us how much the results delivered by one of these models were disconnected from reality (in particular for the average on non-extreme costs and on geographical restitutions).

Our choice to work with daily data allows us to preserve a maximum of information in a context of historical data records available only over a relatively short time span. This approach then requires a transition from daily to an annual cost distributions. Since we refute the independence of consecutive daily events, we must find a way to estimate dependence over several days and integrate it into our model.

Our model

We have chosen to model updated daily costs according to the GPD law. We first consider the costs without covariates and then with the storm index in covariates. In Table IX, we present an estimate of settings for the distribution law, first in the entire France, then according to six storm zones previously defined. In all these models, the shape parameter ξ is higher than 1 for the costs, which corresponds to a distribution of very heavy-tailed extremes. The mathematical expectation of such a distribution is infinite, which shows how strong the influence of extremes is on all the categories of claims. Note that the difference between the ξ-estimates of the costs and of the storm index is explained through the exponential function in formula 9: taking the exponential reflects the explosion of costs which takes place for the most extreme storms. By estimating the parameters according to geographical storm zones, the shape parameter varies between 1.03 for zone 1 and 1.26 for zone 4. Regarding estimated scale parameters, variations are approximately proportional to the exposition to risks according to the portfolio. These models can be improved by integrating the storm index as a covariate into the shape parameter ξ or the scale parameter σ. To decide on how to integrate the covariate, we use whose shape parameter is linearly related to the storm index. On Table X, we report estimates for the model with the storm index integrated as covariate into the shape parameter via ξ(I S ) = ξ 0 +ξ 1 I S .

The covariate leads to a better model according to model selection criteria AIC and BIC, which both improve. We now have fitted an appropriate GPD model to high daily threshold exceedances, yet we aim to build a model for the overall annual costs. Sinc the shape index of the tail distribution is very high (ξ around 1), we can conclude that the lower and intermediary parts of the distribution have little influence on the annual distribution since the few strongest daily values will dominate the sum of daily values. To properly transform the daily distribution into an annual one while respecting temporal dependence among successive days, we need the distribution of the number of exceedances over a year. Under the assumption of independent days, we would have a binomial distribution with parameters n = 365 and p ∈ (0, 1)) the probability of exceedance. Since n is high and p is low, we can approximate it through the law of rare events, the Poisson distribution with intensity parameter λ = n × p. It has a variance/expectation ratio λ/λ = 1, known as its dispersion index.

Owing to dependence between days, we get clusters of extreme events, which leads to overdispersion with a dispersion index > 1. This stronger variability in the number of exceedances is translating into a stronger variability of annual costs. We here choose the negative binomial distribution as a classical distribution well adapted to overdispersion. It can be seen as an extension to the Poisson distribution with an additional parameter to take into account the degree of overdispersion.

As the main sources of clustering in threshold exceedances of daily aggregated claim amounts due to wind storms, we can consider (in decreasing order of temporal extent) seasonal behavior, atmospheric perturbations stretching over several days or weeks, and extreme storms stretching over at least two days. Of course, these three sources are closely related.

Seasonal behavior in wind speeds and the resulting insurance claims lead to a natural cluster behavior on a relatively large temporal scale, see Figure 26. Most extreme events, here given as exceedances over a fixed high threshold, occur in summer (June to August) and winter (December to February). We present estimates of the extremal index, based on the Ferro-Segers estimator ( [START_REF] Christopher | Inference for clusters of extreme values[END_REF]), on the left of Figure 27. The extremal index is a measure of serial dependence of extremes in stationary time series. We should note that estimation here certainly is biased towards too strong dependence insurance coverage limits impose a finite upper bound to this distribution. Still, in our context yearly aggregated claim amounts will be strongly dominated by the few most extreme daily events having occurred during the year, and often already the single most extreme daily event may be dominating, i.e., it will be of comparable size than the sum of all other events. where the January block is sampled from the 15 available January blocks, and so on. The advantage of this modeling approach is its robustness since we do not have to explicitly model clusters of extreme daily events. For instance, removing the influence of seasonal behavior and taking into account atmospheric perturbations ranging over several weeks would be rather complex.

Figure 29 shows the distribution of the yearly aggregated claim amounts (transformed to the log 10 -scale), based on 100.000 simulations according to our model. We investigated four threshold choices, corresponding to 3, 6, 9, 12 exceedances per year on average. Values at Risk for probabilities 0.95 and 0.995 are indicated. The choice of threshold has some impact on quantiles (between 10 8.78 and 10 9.01 for p = 0.95, and between 10 9.11 and 10 9.60 for p = 0.995), with a systematic shift towards higher quantile estimates when choosing higher thresholds. We further observe that the impact of the resampling scheme below the threshold is negligible for these high quantiles, since the aggregates of resampled values below the threshold rarely exceeded 10 8 over one year. When using only 3 exceedances on average, we can discern a second mode in the distribution that disappears for the higher numbers of exceedances. This second mode is due to the resampling scheme below the threshold, which yields a less smooth distribution when the threshold for exceedances is set very high. Our results confirm that the uncertainty in the estimation of quantities required by capital reserve rules like those stated in Solvency II makes application of these rules difficult for certain types of risks.

Moreover, global warming and its potential impact in the coming years increases uncertainty. Some studies such as those of the AFA3 or of the CCR4 announce a strong increase in costs for all climatic hazards over the next 25 to 35 years. However, the contribution of global warming to these additional costs is not the same according to the studies (between 20 and 30%), and it seems to concern only the damage related to floods and droughts. No clear tendency is therefore identified for storms.

It is also difficult to associate an increase in wind speeds of X% to an increase of the damage of Y %, as we can see in some models of the market. In the simulations based on our storm index, the consequences of a variation of wind speeds are different depending on the magnitude of the event, the size of the affected area and its location.

Concerning temporal clusters, we state that certain atmospheric conditions can foster the development of storms or prevent them. Historical insurance data are available over a few decades and already implicitly represent these conditions. They contain cluster periods like the spring of 1990 or the month of December of 1999, and more quiet periods. A more explicit model could allow us to better take into account clustering behavior, but the main question remains the validation of involved parameters.

Conclusion

Our storm index was designed to estimate and compare the scale of damages caused by wind phenomena during the most extreme events observed over several decades. The interest of this approach lies in the fact that we do not face the same difficulties and needs when measuring variables like wind or damage and their extremes. We underline that for small and medium-sized events insurers do not have to use complex models instead of raw data since their data record of 15 to 20 years is sufficient to answer questions related to the distribution of claims. Regarding extreme events, insurers can transfer the risks to reinsurers, who determine the exposition and express it through reinsurance premiums. Nevertheless, some issues remain since usually some high-impact events in the portfolio are not entirely reinsured. Moreover, reinsurance might not prevent insurance companies from analyzing their exposition on extreme levels, for instance with a view towards justifying their insurance premiums. In any case, it is helpful to exploit additional information like meteorogical data for major events.

In this work, we have first discussed the nature of wind speeds and problems related with their measurement. Six geographical risk zones have been delimited in France with the k-medoids algorithm taking into account extreme dependence, absolute wind speeds and Euclidian distance between the centers of departments, leading to zones that are relatively homogeneous in terms of wind conditions. Due to the heterogeneity of distances between risks and stations and the limited number of significant events and risks, statistical methods lack robustness when working at the finer departmental scale. This approach has allowed us to build a flexible storm index that reflects well the insurance data. We have compared daily and weekly clustering approaches with the bootstrap method. We finally checked different scenarios and methods to model the storm risk.

The Generalized Pareto Distribution (GPD) for threshold exceedances is the main ingredient of our model. Estimated parameters inform us about the nature of the distribution tail, and crucial information is gained from the corresponding return periods associated to major events. They can be used to calculate insurers' average annual costs. It still remains difficult to determine return periods for very extreme events. To meet Solvency II requirements, our experiments show that uncertainty can be almost prohibitively high to extract reliable results. Modeling hypotheses and parameter choices have a great influence on the index value and on the extrapolation of distributions beyond the observed range of values. We analyzed this sensitivity through our results, with a focus on return periods for the most extreme storm events.

Incorporating external data such as wind speeds to establish risk zones remains a good solution for the management of storm risk. However, the volatility of insurance results still depends on the choice of frequency and intensity of the most extreme events, for which also the history of meteorogical records is too short to determine them with satisfactory certainty. We still need to seek for new ways and methods to significantly improve data records concerning extreme events. Even the best models cannot remedy incomplete and relatively short data records. Uncertainty remains high, which is also due to often unverifiable model hypotheses which might create a false illusion of accuracy.

Figure 1 :

 1 Figure 1: Wind speed exceedances in Nîmes -before (above) and after (below) readjusting

Figure 2 :

 2 Figure 2: Papillon anemometer transmitter

Figure 3 :

 3 Figure 3: Maps of active stations -from 1963 (on the left) and from 1998 (on the right)

Figure 4 :

 4 Figure 4: Maps of population (on the left) and number of Allianz' policies (on the right)

Figure 5 :

 5 Figure 5: Partition of France and silhouette according to 80% wind speed quantile

Figure 6 :Figure 7 :

 67 Figure 6: Partition of France and silhouette according to 98% wind speed quantile

Figure 8 :Figure 8

 88 Figure 8: Distribution of the 6 wind zones with d 2 -according to 0.8, 0.98 and 0.99 thresholds (from left to right)

Figure 9 :

 9 Figure 9: Distribution of the 6 wind zones with d 3 -according to 0.8, 0.98 and 0.99 thresholds (from left to right)

Figure 10 :

 10 Figure 10: Distribution of the 6 wind zones without Lothar -according to 0.8, 0.98 and 0.99 thresholds (from left to right)

Figure 11 :

 11 Figure 11: Final division for the 6 wind zones

Figure 12 :

 12 Figure 12: Bootstrap estimates of claims -in zone 2 (up) and zone 6 (down)

Figure 13 :

 13 Figure 13: Bootstrap estimates of claims with week aggregation -in zone 2 (up) and zone 6 (down)

Figure 14 :

 14 Figure 14: Storm Lothar on 12/1999 in France -claims, wind speed and wind index (from left to right)

Figure 15 :

 15 Figure 15: Storm Martin on 12/1999 in France -claims, wind speed and wind index (from left to right)

Figure 16 :

 16 Figure 16: Two types of wind behavior (synoptic on the left, thermal on the right)

Figure 17 :

 17 Figure 17: Threshold selection diagnostic plots for the GP fit to the storm index

Figure 18 :

 18 Figure 18: Results of the GPD analysis for the storm index.

Figure 19 :

 19 Figure 19: Return period for the storm index.

Figure 22 :

 22 Figure 22: Return period for the storm index sans Lothar.

Figure 23 :

 23 Figure 23: Return period for the storm index without Martin.

Figure 24 :

 24 Figure 24: Updated storm costs in France since 1970

Figure 25 Figure 25 :

 2525 Figure[START_REF] Thomson | Estimating the slope and standard error of a long-term linear trend fitted to adjusted annual temperatures[END_REF] which confronts the logarithm of costs exceeds with the storm index for the twenty-five most important exceeds. The distribution of points is close to linear, in particular for the most extreme

Figure 26 :

 26 Figure 26: Number of threshold exceedances in claim numbers by month

Figure 27 :

 27 Figure 27: Estimates of the extremal index (left) and of the dispersion index (right)

Figure 28 :

 28 Figure 28: Hill plot of daily claims

Figure 29 :

 29 Figure 29: Simulated yearly claim distribution on a log 10 -scale with VaR at 0.95 (blue) and 0.995 (red)

Figure 30 :

 30 Figure 30: Simulated yearly claim distribution with VaR at 0.95 (blue) and 0.995 (red)

  [START_REF] Coles | Dependence measures for extreme value analyses[END_REF] with the k-medoids algorithm, set p = 0.7 and divide mainland France into k = 6 zones. The

	less marked depending on the dimension of its central point and the line which binds it to the central
	department (medoid). We determine the strength of each group from the silhouette which gathers
	information in the form of a histogram whose bar size is proportional to the affiliation strength to the
	group. The silhouette s(dep) ∈ [-1, 1] of a department dep is the difference between a(dep), defined
	as the average of distances with the other departments of its area, and b(dep), defined as the minimum
	of this average distance calculated by reference to all the other areas. This difference is standardized
	by the maximum of a(dep) and b(dep) such that s(dep) = (a(dep) -b(dep)/max(a(dep), b(dep)).
	The algorithm does not directly optimize the silhouette (which would be too heavy in calculation),
	and negative values may exist. The silhouette is a means to check if the algorithm globally succeeds
	in delineating zones; ideally, there are no negative values. In front of each histogram, the number of
	departments which constitute a group is reported.
	n = 93				6 clusters Cj
					j : nj | avei∈Cj si
					1 : 15 | 0.03
					2 : 22 | 0.06
					3 : 22 | 0.09
					4 : 11 | 0.17
					5 : 14 | 0.10
						6 : 9 | 0.13
	0.0	0.2	0.4	0.6	0.8	1.0
			Silhouette width s i
	k-medoids algorithm results for the three selected thresholds (0.8, 0.98 and 0.99) are presented in the
	Figures 5, 6 and 7. On the maps, we can see that the affiliation of a department to a group is more or

  The average strength of the silhouette further decreases to 0.05. Groups thus formed are quite different from the previous ones. The departments along the Eastern frontier of France are reunited with Corsica. The only

	n = 93				6 clusters Cj
					j : nj | avei∈Cj si
						1 : 12 | 0.05
						2 : 26 | 0.05
						3 : 11 | 0.07
						4 : 16 | 0.07
						5 : 13 | 0.10
						6 : 15 | 0.07
	0.0	0.2	0.4	0.6	0.8	1.0
			Silhouette width s i

Table I :

 I Distribution of annual portfolio and claim numbers (in 10 6 e) by zone TableIprovides an overview of the number of policies and claims in each of the six zones. The portfolio distribution is nearly proportional to the number of departments constituting the groups. Distributions show high variability. Estimated costs are generally proportional to the size of the group, except for zone 6 whose quantile at 97.5% is of similar order as zone 2 and 3 (around 55 million euros). Large interquantile ranges appear also for the number of claims, for instance between 2800 and 13000 claims per year in zone 2. As opposed to what we had noted on costs, zone 6 (with claims ranging between 800 and 7100) no longer appears as overexposed when compared to other zones. Claim frequency is better aligned with exposition in terms of portfolio risk. In summary, we see that the observation period will always have a strong influence on the results of a model including extreme events.

				Histogram of t
									80
			0.03
									60
		Density	0.02					t*	40
			0.01
									20
			0.00
			0	20	40	60	80		-3	-2	-1	0	1	2	3
					t*					Quantiles of Standard Normal
				Histogram of t
			0.10
			0.08
		Density	0.04 0.06
			0.02
			0.00
			0	20	40	60	80
					t*
	Zones	1		2		3		4		5		6
	Nb. Contracts 155475 265222 248957 147899 52033 130413
	Ratio	16%		27%		25%		15%		5%		13%
	Nb. Dep.	17		25		24		10		6		13
	Average claim number
	Daily sum
	q 2.5	1539	2806		2689	899		517	821
	q 97.5	6498	12868	11480	7580		3353	7117
	Average cost (million euros)
	Daily sum
	q 2.5	4.8		9.2		8.3		2.4		1.8	2.6
	q 97.5	36.1	55.7		52.7	33.9		20.1	48.4
	Weekly sum
	q 2.5	3.6		6.3		6.7		1.9		1.4	2.2
	q 97.5	42.3	66.2		57.4	35.3		21.2	51.4

Zone 2 accounts for the most departments and the biggest part of the portfolio. The distribution of insurance policies between zones is homogenous. The following lines contain quantiles at 2.5% and 97.5% of the costs and of the number of claims for each zone calculated from the bootstrap sample.

Table II

 II

	: Maximum Likelihood Estimates : I S with u=10 and period=1970-2013
	MLE	Std. Err.
	Scale (σ): 10.78 2.13
	Shape (ξ): 0.20	0.15
	Negative log-likelihood : 222.08

Table III :

 III Maximum Likelihood Estimates : I S with u=10 and period=1993-2013

	MLE	Std. Err.
	Scale (σ): 9.87 4.18
	Shape (ξ): 0.50 0.38
	Negative log-likelihood : 75.76

Table V :

 V Maximum likelihood estimates : I S without Lothar

	MLE	Std. Err.
	Scale (σ): 11.02 2.15
	Shape (ξ): 0.10	0.15
	Negative log-likelihood : 213.54

Table VI .

 VI

	Table VI: Maximum likelihood estimates : I S without Martin
	MLE	Std. Err.
	Scale (σ): 10.61 2.04
	Shape (ξ): 0.16	0.14
	Negative log-likelihood : 214.60

Table VII :

 VII Calculations of return period according to different scenarios

	Period	1970-2013			1993-2013
	Threshold : u	10	20	10wL 10wM 10
	Lothar Return Period 93	88	280	160	31
	L(1.5) Return Period 136 128 490	252	84
	1993-2013 with a multiplication by 2.7, where we move from 31 to 84 years. Such long time intervals
	may be beyond imagination, but they have direct consequences on the quote and reinsurance costs.

Table VIII :

 VIII Evaluation of the average annual expense (in millions of euros)

	1999 Return Period	31	88	93	160	280
	Claims : 1999	48.4	17.0	16.1	9.4	5.4
	Average annual claims 165.1 133.7 132.8 126.1 122.1

Table IX

 IX

		: Maximum Likelihood Estimates : claims (global and areas)
	Zone	Global Negative log-likelihood		1228.63
	Scale e+06 (σ): 1.51	AIC				2461.26
	Shape (ξ):	1.12	BIC				2465.92
	Zone	1	2	3	4	5	6
	Scale e+05 (σ): 4.28	5.88 3.26 2.10	0.96 3.29
	Shape (ξ):	1.03	1.17 1.12 1.26	1.14 1.09

Table X :

 X Maximum likelihood estimates : claims with storm index as covariate in the shape parameter

	Zone	Global
	Scale (σ):	1.51e+06
	Shape (ξ 0 ):	1.50e-03
	Shape (ξ 1 ):	9.25e-02
	Global Negative log-likelihood 1210.78
	Global AIC	2427.55
	BIC	2434.55

Medicane is a contraction of Mediterranean and hurricane. This climatic phenomenon has existed for the past two decades. It is due to the conjunction of cold air at altitude and an unusual warming of the Mediterranean sea.

For each event E ⊂ D we select the daily maximum, for example, during the storm Klaus: E = {01/23/09, 01/24/09, 01/25/09}.

AFA (Association Française de l'Assurance : French Insurance Association)

CCR (Caisse Centrale de Réassurance : Reinsurance Central Fund)

Acknowledgements

For this study, the authors have received a research grant which allowed us to work on the records of 192 meteorological stations of Météo France over a period covering 50 years. This work was supported by Allianz France. The authors thank the research chair Actuariat Durable sponsored by Milliman for additional support, as well as Sidney Resnick and Gennady Samorodnitsky for relevant comments.

as Lothar every 88 years. This prediction is slightly lower than the one from the first model, but remains of the same order of magnitude. We replicate the base model (u = 10) over the longest period of time ), but we now emulate a situation where the storm Lothar did not occur. In other words, the most substantial damage since the 1970s would does not exceed the level of storm Martin, which is two times lower than Lothar in


Maria-Francesca Costabile
email: [costabile@di.uniba.it

D Fogli
email: [fogli@ing.unibs.it

Catherine Letondal
email: letondal@pasteur.fr

P Mussio
email: mussio]@ing.unibs.it

A Piccinno°t

M F Costabile°

A Piccinno°
email: piccinno]@di.uniba.it

Domain-Expert Users and their Needs of Software Development

come   L'archive ouverte pluridisciplinaire

Introduction

The ever increasing spread of computer environments in the information society is determining a continuous growth of the end-users population as broad as possible. Such end-users have different needs and backgrounds, and operate in different contexts. The following definition of end-users is given in [START_REF] Cypher | Watch What I Do: Programming by Demonstration[END_REF]: "A user of an application program. Typically, the term means that the person is not a computer programmer. A person who uses a computer as part of daily life or daily work, but is not interested in computers per se." It is evident that several categories of end-users can be defined, for instance depending on whether the computer system is used for work, for personal use, for pleasure, for overcoming possible disabilities, etc. Brancheau and Brown analyse the status of what they call end-user computing and define it as "… the adoption and use of information technology by people outside the information system department, to develop software applications in support of organizational tasks" [START_REF] Brancheau | The Management of End-User Computing: Status and Directions[END_REF]. In this survey, they primarily analysed the needs of users that are experts in a specific discipline, but not in computer science. In our experience, we have often worked with end-users that are experts in their field, that need to use computer systems for performing their work tasks, but that are not and do not want to become computer scientists. This has motivated the definition of a particular class of end-users, that we call domain-expert users (d-expert in the following): they are experts in a specific domain, not necessarily experts in computer science, who use computer environments to perform their daily tasks. They have also the responsibility for induced errors and mistakes. In this paper, we focus on such users and analyse the activities they usually perform or are willing to perform with computers. This analysis shows that d-experts have real needs of performing some programming activities that result in the creation or modification of software artefacts.

In the rest of this paper, Section 2 describes some features of d-experts. Section 3 reports about activities d-experts have the need to perform. Section 4 describes real situations in which programming needs occur and Section 5 concludes the paper.

Characterizing domain-expert users

In scientific and technological domains, d-experts communicate with each other through documents, expressed in some notations, which represent abstract or concrete concepts, prescriptions, and results of activities. Recognizing users as d-experts means recognizing the importance of their notations and dialects as reasoning and communication tools. As designers, we are challenged to create virtual environments, in which users interact using a formal representation of their traditional languages and with virtual tools that recall the real ones with which users are familiar. This is a hard challenge, because d-expert communities develop in time from the experience different notations, which reflect the different technical methods, languages, goals, tasks, ways of thinking, and documentation styles.

Often, dialects arise in a community, because the notation is applied in different practical situations and environments. For example, mechanical drawings are organized according to standard rules, which are different in Europe and in USA. Explicative annotations are written in different national languages. Often the whole document (drawing and text) is organized according to guidelines developed in each single company. The correct and complete understanding of a technical drawing depends on the recognition of the original standard as well as on the understanding of the national (and also company developed) dialects. Similar cases are quite common: d-experts of a same community constitute different sub-communities depending not only on user skill, culture, knowledge, but also on specific abilities (physical/cognitive), tasks, and context. Recognizing the diversity of users calls for the ability to represent a meaning of a concept with different materialization, e.g. text, images or sound, and to associate to a same materialization a different meaning according, e.g., to the context of interaction.

An important phenomenon, often observed in Human-Computer Interaction (HCI), is that "using the system changes the users, and as they change they will use the system in new ways" [START_REF] Nielsen | Usability Engineering[END_REF]. In turn, the designer must evolve the system to adapt it to its new usages; we called this phenomenon co-evolution of users and systems [START_REF] Arondi | Supporting co-evolution of users and systems by the recognition of Interaction Patterns[END_REF]. In [START_REF] Bourguin | Beyond the Interface: Co-evolution inside Interactive Systems -A Proposal Founded on Activity Theory[END_REF] it is observed that these new uses of the system determine the evolution of the user culture and of her/his models and procedures of task evolution, while the requests from users force the evolution of the whole technology supporting interaction.

Co-evolution stems from two main sources: a) user creativity: the users may devise novel ways to exploit the system in order to satisfy some needs not considered in the specification and design phase; and b) user acquired habits: users may follow some interaction strategy to which they are (or become) accustomed; this strategy must be facilitated with respect to the initial design.

Activities of domain-expert users

When working with a software application, d-experts feel the need to perform various activities that may even lead to the creation or modification of software artefacts, in order to get a better support to their specific tasks, thus being considered activities of End-User Development (EUD) in accordance with the following definition: "EUD is a set of activities or techniques that allow people, who are not professional developers, at any stage to create or modify a software artefacts for their own or shared use" (EUD-Net, 2002). The need of EUD is a consequence of user diversity and user evolution discussed in the previous section. Within EUD, we may include various tailoring activities described in the literature, and reported in the following.

Tailoring activities

Tailoring activities are defined in different ways in the literature; they include adaptation, customization, end-user modification, extension, personalization, etc. These definitions partly overlap with respect to the phenomena they refer to, while often the same concepts are used to refer to different phenomena.

In [START_REF] Wulf | Let's see your Search-Tool!" -Collaborative use of Tailored Artifacts in Groupware[END_REF], tailorability is defined as the possibility of changing aspects of an application's functionality, during the use of an application, in a persistent way, by means of tailored artefacts; the changes may be performed by users that are local experts. Tailorability is very much related to adaptability. In [START_REF] Trigg | Adaptibility and Tailorability in NoteCards[END_REF], a system is adaptable if an end-user "produces new system behaviour without help from programmers or designers". There are four levels for being adaptable: 1) flexible -objects and behaviours can be interpreted and used differently; 2) parameterizable -alternative behaviours can be chosen by the user; 3) integrable -the system can be integrated with other components, internal or external, 4) tailorable -users are allowed to change the system itself by building accelerators, specializing behaviour, or adding new functionality. Thus, tailoring involves the creation of new functionalities by end-users.

In [START_REF] Mackay | Triggers and Barriers to Customizing Software[END_REF] and in [START_REF] Nardi | A small matter of programming: perspectives on end user computing[END_REF] empirical studies are reported on activities performed by end-users, and generally defined as tailoring activities. Mackay analyses how users of a UNIX software environment try to customise the system, intending as customisation the possibility of modifying software to make persistent changes. She finds that many users do not customise their applications as much as they could. This also depends on the fact that it takes too much time and deviates from other activities. Nardi conducted empirical studies on users of spreadsheets and CAD software. She found out that those users actually perform activities of end user programming, thus generating new software artefacts; these users are even able to master the formal languages embedded in these applications when they have a real motivation for doing so.

Mørch specifies three main categories of tailoring: customisation, integration, and extension [START_REF] Mørch | Three Levels of End-User Tailoring: Customization, Integration, and Extension[END_REF]. Customisation usually consists of a set of preferences configurable by the user through a preference form, in which a user can set parameters for the various configuration options the application supports. Integration goes beyond customization by allowing users to add new functionality to an application, without accessing the underlying implementation code. Instead, users tailor an application by linking together predefined components within or across the application. Extension refers to the case in which the application doesn't provide, by itself or by its components, any functionality that accomplishes a specific user need, thus adding a new functionality generates a radical change in the software. In the above categorization, there are instances that cut across categorical boundaries.

Two classes of domain-expert activities

The brief overview given in the previous section shows that different meanings are associated to tailorability and adaptability. To avoid ambiguity, we propose two classes of d-expert activities:

Class 1. It includes activities that allow users, by setting some parameters, to choose among alternative behaviours (or presentations or interaction mechanisms) already available in the application; such activities are usually called parameterisation or customization or personalization.

Class 2. It includes all activities that imply some programming in any programming paradigm, thus creating or modifying a software artefact. Since we want to be as close as possible to the human, we will usually consider novel programming paradigms, such as programming by demonstration, programming with examples, visual programming, macro generation.

In the following, we provide examples of activities of both classes from experiences of participatory design workshops in two domains, biology and earth science.

Activities belonging to Class 1 are: -Parameterization. It is intended as specification of unanticipated constraints in data analysis. In this situation, observed very often, the d-expert wishes to guide a computer program by indicating how to handle several parts of the data in a different way; the difference can just lay in associating specific computation parameters to specific parts of the data, or in using different models of computations available in the program. In biology, this is related to protocol design.

-Annotation. D-experts often write comments next to data and result files in order to remember what they did, how they obtained their results, and how they could reproduce them.

The following activities belong to Class 2:

-Modelling from the data. The system supporting the d-expert derives some (formal) models from observing data, e.g. in [START_REF] Blackwell | See What You Need: Toward a visual Perl for end users[END_REF] a kind of regular expression is inferred from selected parts of aligned sequences, or in [START_REF] Arondi | Supporting co-evolution of users and systems by the recognition of Interaction Patterns[END_REF] patterns of interactions are derived.

-Programming by demonstration. D-experts show examples of properties occurrences in the data and the system infers from them a (visual) function.

-Formula languages. This is available in spreadsheets and could be extended to other environments, such as Biok (Biology Interactive Object Kit) that is a programmable application for biologists [START_REF] Letondal | Programmation et interaction[END_REF]. The purpose of Biok is twofold: to analyze biological data such as DNA, protein sequences or multiple alignments, and to support tailorability and extensions by the end-user through an integrated programming environment.

-Indirect interaction with application objects. As opposed to direct manipulation, a command language can be provided to script objects.

-Incremental programming. It is close to traditional programming, but limited to changing a small part of a program, such as a method in a class. It is easier than programming from scratch.

-Extended Annotation. A new functionality is associated with the annotated data. This functionality can be defined by any technique previously described.

Examples of EUD applications

In this section we describe some situations that show the real need of environments with EUD capabilities, as emerged in our work with biologists and earth scientists. Experience acquired at the Pasteur Institute during several years indicates that in the field of biology software for academic research there are two types of software development: 1) large scale projects, developed in important bioinformatics centres, such as the European Bioinformatics Institute; 2) local development, by biologists who know some programming language, in order to deal with daily tasks, such as managing data, analysing results, or testing scientific ideas. We are here interested in the second type of development, since it can be considered end-user development. Moreover, it is worth mentioning that many biologists do not know anything about programming, and yet feel the need of modifying the application they use to better fit their needs. Below is a list of real programming situation examples, drawn from interviews with biologists, news forum, or technical support at the Pasteur Institute. These situations occurred when working with molecular sequences, i.e., either DNA or protein sequences: scripting, i.e. search for a sequence pattern, then retrieve all the corresponding secondary structures in a database; parsing, i.e. search for the best match in a database similarity search report but relative to each subsection; formatting, i.e. renumber one's sequence positions from -3000 to +500 instead of 0 to 3500; variation, i.e. search for patterns in a sequence, except repeated ones; finer control on the computation, i.e. control in what order multiple sequences are compared and aligned (sequences are called aligned when, after being compared, putative corresponding bases or amino-acid letters are put together); simple operations, i.e. search in a DNA sequence for some characters.

Considering the domain of earth science, we worked with scientists and technicians who analyse satellite images and produce documents such as thematic maps and reports, which include photographs, graphs, etc., and textual or numeric data related to the environmental phenomena of interest. Two sub-communities of d-experts are: 1) photo-interpreters who classify, interpret, and annotate remote sensed data of glaciers; 2) service oriented clerks, who organize the interpreted images into documents to be delivered to different communities of clients. Photo-interpreters and clerks share environmental data archives, some models for their interpretation, some notations for their presentation, but also have to achieve different tasks, documented through different subnotations and tools. Therefore, their notations can be considered two dialects of the Earth Scientist & Technologist general notation.

Conclusions

In this paper, we have focused on a specific category of end-users, called domain-expert users, and have analysed the activities they usually perform with computers as well as the activities they would like to perform in order to get a better support from computer systems to their daily work.

The two examples in the previous section make the needs of d-experts emerge. In the case of biologists unpredictable needs may arise at any time, which require some kind of programming, even if such programming is rather simple most of the time. However, existing software does not usually provide any programming capability. Thus, the biologists have often to program everything from scratch, which is usually very difficult for them. In the case of earth science, photo-interpreters need software tools for interactive image processing and extended annotation. They interactively identify and classify parts of glacier images and associate to them an extended annotation, constituted by a textual part and a program created by selecting some computations on the basis of the observed data. In this way, photo-interpreters create new software artefacts, which are managed through user-defined widgets. Clerks will use these widgets to interact with the programs made available by the photo-interpreters to produce the required documents. Thus, it is a challenge for us, as designers of computer systems more accessible to their end-users, to develop programming paradigms and software environments that are adequate to the needs of end-users.

The support of EUD-Net Thematic Network, sponsored by EC, projectIST-2001-37470, is acknowledged.