[1]CharpentierArthur [2]JamesMolly [3]AliHani

1]UQAM, Université du Québec à Montréal (UQAM), Montréal (Québec), Canada
2]EURo Institut d’Actuariat (EURIA), Université de Brest, France 3]Willis Re,
Paris, France

Arthur Charpentier<EMAIL_ADDRESS>
# Predicting Drought and Subsidence Risks in France

(July 2021; )

###### Abstract

The economic consequences of drought episodes are increasingly important,
although they are often difficult to apprehend in part because of the
complexity of the underlying mechanisms. In this article, we will study one of
the consequences of drought, namely the risk of subsidence (or more
specifically clay shrinkage induced subsidence), for which insurance has been
mandatory in France for several decades. Using data obtained from several
insurers, representing about a quarter of the household insurance market, over
the past twenty years, we propose some statistical models to predict the
frequency but also the intensity of these droughts, for insurers, showing that
climate change will have probably major economic consequences on this risk.
But even if we use more advanced models than standard regression-type models
(here random forests to capture non linearity and cross effects), it is still
difficult to predict the economic cost of subsidence claims, even if all
geophysical and climatic information is available.

CC BY-NC 4.0

For the insurance industry, climate change is a challenge since risks are
increasing, in terms of frequency and intensity, as discussed in
mccullough2004managing, mills2007synergisms, charpentier2008insurability or
schwarze2011natural. In this article, we will see if it is possible to
predict, for a given year, the costs associated with drought, and more
specifically here, clay shrinkage induced subsidence, in France.

###  0.1 Drought and climate change

In a seminal book published twenty years ago, bradford2000drought started to
address the problem of getting a better understanding of the connections
between drought and climate change, already suggesting that the frequency and
the intensity of such events could increase in the future. And in the more
recent book, Iglesias provided additional evidence about the influence of
climate change on meteorological droughts in Europe. Ionita studied the
temporal evolution of three drought indices over 120 years (the standardized
precipitation index – SPI, the standardized precipitation evapo-transpiration
index – SPEI, and the self-calibrated Palmer drought severity index – scPDSI).
This updated study regarding the trends and changes in drought frequency in
Europe concluded that most of the severe drought events occurred in the last
two decades, corresponding to the time after the publication of Lloyd-Hughes,
for example. Similarly, SPINONI201550 and SPINONI2017113 (studying more
specifically Europe, following their initial, all over the world in Spinoni)
observed that for both for frequency and severity, the evolution towards drier
conditions is more relevant in the last three decades over Central Europe in
spring, the Mediterranean area in summer, and Eastern Europe in autumn (using
also multiple indices, over 60 years).

Regarding economic impacts, Hagenlocher_2019 provided a state of the art of
scientific publications over the past twenty years (presenting the outcomes of
a systematic literature review of people-centered drought vulnerability and
risk conceptualization and assessments). naumann2021increased shows that (in
Europe), drought damages could strongly increase with global warming and cause
a regional imbalance in future drought impacts. They provide some forecasts,
under the assumption of absence of climate action (+$4^{\circ}$C in 2100 and
no adaptation), annual drought losses in the European Union and United Kingdom
combined are projected to rise to more than 65 billion € per year compared
with 9 billion € per year currently, still two times larger when expressed
relative to the size of the economy. Note that this corresponds to the general
feeling of the insurance industry: SR2 suggests that, regarding climate
change, drought trends and socio-economic factors, not only will it persist,
but even accelerate further.

But at the same time, Naumann_2015 pointed out that related direct and
indirect impacts are often difficult to quantify. A key issue is that the lack
of sufficient quantitative impact data makes it complicated to construct a
robust relationship between the severity of drought events and related
damages. Insurance coverage for drought has been intensively studied, when
related to agriculture. Iglesias mentions some drought insurance schemes, with
either indemnity based mechanisms, but also drought index based insurance, in
Section 2.8. vroege2019index provides an overview of index-based insurances in
Europe and North America, in the context of droughts, while bucheli2021optimal
focuses on Germany. Note that tsegai2019drought addresses the importance of
designing insurance products which does not only address drought impacts but
also minimize land degradation. Besides this theoretical work, some countries
provide actual covers for such risks. For example in Spain, it is possible to
insure rain-fed crops against drought, as discussed in ENESA, but in most
country, drought coverage is only concerned with respect to agricultural
(crop) insurance, as such as frost (see also perez2017crop).

###  0.2 From drought to subsidence

In this article, we will use data from several insurance companies in France,
regarding a very specific drought related risk, that is clay shrinkage induced
subsidence (even if we predict only a part of the evaluation of cost
assessment of land subsidence, as pointed out in kok2021framework). If clay
shrinkage induced subsidence is now a well known risk (or at least recognized
as a major risk, see for instance Doornkamp, or brignall2002assessing which
assessed the potential effects of climate change on clay shrinkage-induced
land subsidence), insurance coverage for subsidence is still uncommon. Almost
twenty years ago, as indicated in mccullough2004managing, while perils related
to earth movement were traditionally excluded from most property policies,
several states (in the United States of America) have mandated coverage for
some subsidence related claims (with several limitations). And recently,
herrera2021mapping proved that subsidence permanently reduces aquifer-system
storage capacity, causes earth fissures, damages buildings and civil
infrastructure, and increases flood susceptibility and risk. From an insurer’s
perspective, SwissRe pointed out that as incidents of soil subsidence increase
in frequency and severity with climate change, there is a need for systematic
managing of the risks through a combination of loss prevention and risk
transfer initiatives (such as insurance).

In France, subsidence is a phenomenon covered by all private property
insurance covers, and that enters the scope of the government backed French
natural catastrophe regime provided by the Caisse Centrale de Réassurance
(CCR). It is the second most important peril in terms of costs that the system
covers (the first being floods, see GP for a recent discussion about flood
events in France, in the context of climate change). Subsidence risk is
defined (DefSubsidence) as the displacement of the ground surface due to
shrinkage and swelling of clayey soils. It is due mainly to the presence of
clay in the soil which swells in humid conditions and shrinks in dry ones,
thus creating instabilities in the terrain under constructions causing cracks
to appear on the floor and walls which can jeopardise the solidity of the
building. France, having a temperate climate, has saturated clayey soils,
making subsidence predominant during droughts.

However, the past few years have seen this risk exacerbated by the extreme
heat waves and lack of rainfall in France (see CCRbilan), causing more and
more subsidence claims, with little hope of this tendency stopping, given the
current climate change context. Indeed, since 1989, 38% of the total costs of
claims are concentrated over the period 2015-2019, that is 15% of the total
time that subsidence coverage has been in place (as discussed in MRN35).
Furthermore, ClimSec shows that the frequency and intensity of heat waves and
droughts will inevitably increase in the coming century in continental France
and new areas that so far have been protected from drought will be at risk.
Additionally, AFA_CC predicts that the cost of geotechnical droughts will
nearly triple in 2040. More recently, the French Geological Survey (BRGM,
Bureau de Recherches Géologiques et Minières) published a study,
ClimChangeSubs, describing extreme historical subsidence events as well as
forecasts using various climate change scenarios. It found that the first
third of the century will suffer from unusual droughts in both their intensity
and spatial expansion: one in three summers between 2020 and 2050 and one in
two summers between 2050 and 2080 are to be as extreme as the summer of 2003
in continental France (the worst subsidence event ever registered by the CCR,
see corti2009simulating that focused on the 2003 heatwave in France). Looking
at the most pessimistic scenario, a 2003 type event might occur half of the
time between 2020 and 2050. One should recall that in 2003, the heat wave
caused damage due to the shrinking and swelling of clay, for which
compensation (via the natural disaster insurance scheme, and then via an
exceptional compensation procedure for rejected cases) was estimated at
approximately 1.3 billion € in Frecon, while over the period 1989-2002, the
average annual cost of geotechnical drought for the natural disaster insurance
scheme was more than five times smaller, with 205 million €.

Furthermore, subsidence is a risk with a long declaration period, with on
average 80% of the number of claims declared two years after the event. This
delay is due to the lengthy acceptance process of subsidence natural
catastrophe declarations, upon which the validity of most claims is dependant.
Although most insurers are reinsured against this peril with the CCR, the
retention rate remains high (50%), it is thus necessary for insurers to
develop their own view of this risk in order to estimate their exposure to
this growing hazard. However, the inherent characteristics of subsidence make
it a risk that is complex to model: it has slow kinetics and an absence of
precise temporal definition, making subsidence models sparse on the market.

SR2 mentioned since 2016, annual inflation-adjusted insured losses have
continuously exceeded 600 million €, with an average annual loss close to 850
million , which corresponds to around 50% of the CatNat premiums collected,
and makes subsidence possibly one of the most costly natural risk in France.
Overall, a high to medium clay shrink-swell hazard affects one-fifth of
metropolitan France’s soils and 4 million individual houses, as mentioned in
ANTONI. And SR3 mentioned that this is not just in France, increased
subsidence hazard gains attention in other countries, too (even if this
article focuses solely on France).

###  0.3 Agenda

The purpose of this study is to provide a regression-based model that will
allow to predict annual frequency and severity of subsidence claims to be
made, based on market data and climatic indicators. The model created in this
study bases future predictions on past occurrences, thus a historical
insurance database was necessary to calibrate the models, alongside indicators
of the severity of historical events that could be reproduced into the future.
These indicators were created using climatic and geological data that capture
the specifics of past events and regional information. The creation of this
database and the choice of indicators will be described in section 1. Using
this historical data, various models are implemented, chosen to adapt to the
particularities of the data, in order to improve the precision of the
predictions. There will be three layers to model the costs of subsidence
claims (1) a drought event should be officially recognised (corresponding to a
binary model, specificities of the French insurance scheme will be discussed
in the next section) (2) if there is a drought, the frequency is considered
(corresponding to a counting model, classically a Poisson model) (3) for each
claim the severity is studied (corresponding to a cost model, here some Gamma
model). As we will see, using so-called zero-inflated models, the first two
models can be considered simultaneously. Various tree based models were also
tested in an attempt to obtain more realistic predictions. In section 2, we
will present those models, and we will analyse predictions obtained on the
frequency, and more specifically discuss the geographical component of the
prediction errors. And finally, in section 3, we will present some models to
predict to total costs of subsidence events, in France, and again, study mode
carefully the prediction errors, in 2017 and 2018.

##  1 Subsidence risk in France and our dataset

A yearly claim and exposure dataset, sourced from several different French
insurers, relative to Multi-Peril Housing Insurance (Multi-Risque Habitation)
for individual houses in metropolitan France, for the period 2001 to 2018, was
used here for frequency and severity of past events. This dataset was enriched
with additional information based on geophysical indices usually used to model
droughts.

Our dataset was aggregated at the town level111 Here, we use here the word
“town” to designate a “commune” (in French), or “municipality”. There are
37,613 towns in metropolitan France (as characterized by their INSEE-“commune”
code)., with no information about the particularities of each individual
contract that could influence the claims (such as number of stories, size,
orientation, etc.), creating the need for additional information about the
inherent risks of the town and its climatic and geophysics exposure. One of
the main issues when modelling subsidence is the absence of precise temporal
and geographical definition of a subsidence events. Thus, to combat these
issues, indicators must be created to grasp the geographical and temporal
characteristics of past events.

In Section 1.1, after describing briefly the specificity of the French
insurance scheme, we will explain the variables of interest we will model
afterwards, namely the occurrence of a natural disaster (based on official
data in France), the number of houses and building claiming a loss, and the
amount of the losses (those last two based on data from three important
insurance companies in France, representing about 20% of the French market).
Then, in Section 1.2 and 1.3, we will describe possible explanatory variables
(for the occurrence of a disaster, the percentage of houses claiming a loss
and the severity). And finally, in Section 1.4 we discuss the use of other
variables, mainly geophysical information since we want to predict subsidence,
and not droughts in general.

###  1.1 Specificity of the French Catnat system

The French “Régime d’Indemnisation des Catastrophes Naturelles” (also called
the “CatNat regime”) started in 1982 (see GP for an historical perspective),
even if drought damages were added to the (informal) list of perils in 1989,
as explained in magnan1995catastrophe or more recently Bidan_Cohignac_2017.
The main idea of the mechanism is that any property damage insurance contract,
for individuals as well as for companies, includes mandatory coverage for
natural disasters. The assets concerned are buildings used for residential or
professional purposes and their furniture, equipment, including livestock and
crops, and finally motor vehicles. These assets are insured by multi-risk home
insurance, multi-risk business insurance and motor vehicle insurance. The
contractual guarantees (storm, hail and snow, fire…) are also very often
attached to household and business contracts. Livestock outside the barn and
unharvested crops, on the other hand, are covered differently. It should be
stressed here that the respective scope of insurable and non-insurable risks
is not defined by law, but is established by case law. Indeed, the natural
disaster insurance system is said to be “à péril non dénommé” (or unnamed
peril) in the sense that there is no exhaustive list of all risks that are
covered. The effects of natural disasters are legally defined in France as
“uninsurable direct material damage caused by the abnormal intensity of a
natural agent, when the usual measures to be taken to prevent such damage
could not prevent their occurrence or could not be taken” (article L.125-1 of
the insurance legal code). In practice (and it will be very important in our
study) the state of natural disaster is established by an interministerial
order signed by the Ministries of the Interior and of Economic Affairs. This
order is based on the opinion of an interministerial commission. This
commission analyzes the phenomenon on the basis of scientific reports and thus
establishes jurisprudence regarding the threshold of insurability of natural
risks. More specifically, in the context of our study regarding drought,
requests for recognition of the state of natural disaster are examined for
damage caused by differential land movements due to drought and soil
rehydration. And a list of towns that have been declared a natural disaster
are listed and associated guaranties are then applied, both for individual and
commercial policyholders, by (private) insurance companies.

In comparison to other natural catastrophes, subsidence has certain
particularities. From an insurance point of view, the typical event-based
definitions of a natural catastrophe, that is possible for cyclones or
avalanches, does not apply. Indeed, it has slow kinetics, making it difficult
to determine direct links of causality between the event and the claims.
Damage can be caused long after the dry periods. However, that link of
causality is the very definition of natural catastrophe recognition which
makes the implementation of different criteria to determine the causality link
of the event essential.

Subsidence was first observed, as a major risk, in France after the drought of
1976 which caused important damage to buildings. After a similar event in
1989, subsidence was integrated into the French natural catastrophe regime, in
the sense that policyholders can claim a loss to their insurance companies.
According to MRNsec, between 1989 and 2018, more than 11,300 towns have
requested natural catastrophe recognition for subsidence and over 9,500 were
granted it. The natural catastrophe declarations are published on average 18
months post-event, instead of 50 days for other natural catastrophes and the
duration of an event is on average of 50 days for subsidence and of 5 days for
other perils (like floods, avalanches or landslides, among many others). The
total cost of subsidence losses reached 11 Billion € mid-2018, which is
roughly 16,300€ per claim. The number of towns that have had their request
declined has increased since 2003. Overall, the proportion of acceptance is of
61%, however, just taking years subsequent to 2003, the proportion is of only
50%. As mentioned earlier, this might be explained by the fact that, according
to the law, those events should be caused by “the abnormal intensity of a
natural agent ”. Thus, if a town is claiming losses every years, it ceases to
be “abnormal ”, and claims might be rejected then.

The evolution of the number of natural catastrophes since 1989, which is the
year of the oldest natural catastrophe in the dataset, shows that since the
1990s, the number of orders has been quite variable, however, 4 years seem to
be abnormally hardly-hit: 2003, 2005, 2011 and 2018. Note finally that SwissRe
claims that, since subsidence and drought are related to temperature, climate
change will increase frequency and intensity of drought.

###  1.2 General considerations regarding drought indices

There exists many different options in terms of drought indicators to
characterise their severity, location, duration and timing (see Svoboda for
some exhaustive descriptions). The impact of droughts can vary, depending on
the specificities of each drought, captured differently by each indicator. It
is thus important to select an indicators with its application in mind.
However, the availability of data also plays an important role in the
selection, as it limits which indicator can be constructed.

The criteria to characterise the severity of shrinkage-swelling episodes
evolved in 2018, as the old criteria were outdated and overly technical,
making them difficult to interpret and explain to the public, see
ProcedureCatNat. The new system is based on two factors:

  * •

A geotechnical factor pertaining to the presence of clay at risk of swell-
shrink phenomenon, in place since 1989. This criterion enables the
identification of soils with a predisposition to the phenomenon of shrinkage-
swelling depending on the degree of humidity. The analysis is based on
technical data established by the Bureau des Recherche Géologiques et Minières
(BRMG), see DefSubsidence. Areas of low, medium and high risk are considered
to determine whether the communal territory covered by sensitive soils (medium
and high-risk areas) is greater than 3%. This will be discussed in Section
1.4. However, the intensity of shrinkage-swelling is not only due to the
characteristics of the soil but also to the weather.

  * •

A meteorological criterion defined as a hydro-meteorological variable giving
the level of humidity in superficial soils (1m of depth) at 8-kilometre
precision level. This variable establishes the humidity of the soil for each
season at a communal level called the Soil Water Index (SWI) which varies
between 0 and 1, where 0 is a very dry soil and 1 is a very wet soil. A
humidity indicator is calculated for every month based on the average of the
indicators of the three previous months. This will be discussed in Section
1.2. For example, the indicator for July is fixed using the mean of the
indexes for May, June and July, thus considering the slow kinetics of the
drought phenomenon that can appear over a few months.

To determine whether a drought episode is considered abnormal, the SWI
established for a given month is compared to the indicators for that same
month over the previous 50 years. It is considered “abnormal” if the indicator
presents a return period greater or equal to 25 years, as explained in
ProcedureCatNat. It should be stressed here that this return period is defined
locally, and not nationally. If one of the months of a season meets the above
criteria, in a specific area, then the whole season is eligible to a natural
catastrophe declaration for the whole town. If the natural catastrophe
criteria are met and the Inter-Ministerial commission declares a subsidence
natural catastrophe for the town, if the claims are in direct link with the
event and the goods were insured with a property and casualty insurance policy
then it will be covered by standard insurance policies. The presence of a
threshold set at a 25-year return period indicates that over time, if a
commune is regularly hit by extreme droughts, those events will be less and
less likely to be declared natural catastrophes as they will lose their
exceptional character.

However, over the years, the subsidence criteria have been changed and updated
many times,to consider the new kinds of droughts that arose. The first main
modification was in 2000, when a criterion based on the hydrological
assessment of the soil was added to the criterion assessing the presence of
clay in the soil which was the sole criteria before. However, 2003 was hit by
an extreme and unusual drought limited only to Summer which was not captured
by the criteria in place. Indeed, the uses of the criteria in place at the
time would have led to most of the towns requesting a declaration to be
refused. Thus, a new criterion was created specifically for 2003. In 2004, the
criteria were updated once again, to consider droughts like that of 2003.
However, in 2009 a new indicator was applied based on three seasonal Soil
Water Indexes (SWI) (Winter, Spring and Summer) as well as the presence of
clay in the soils. Finally, in 2018 the SWI criteria were updated to simplify
the thresholds and create four seasonal indicators as presented previously and
the shrinkage and swelling of clay exposure map was also updated.

###  1.3 Drought indices used as covariates

These indicators can be classed into three big families : The first are
meteorological indicators, of which the most common amongst drought-related
studies are the Standardised Precipitation Index (SPI), the Palmer Drought
Severity Index (PDSI) and the Standardised Precipitation and
Evapotranspiration Index (SPEI). These indices are based on precipitation
data, as well as temperature and available water content for the two last
ones. The SPI is the most widely used because it requires little data (only
monthly precipitations needed) and is comparable in all climate regimes (see
Kchouk for a recent review of drought indices). However, those indexes do not
capture drought through soil moisture, whereas other indexes such Agricultural
and Soil Moisture indicators do. Some common indicators from that family are
the Normalised Difference Vegetation Index (NDVI), the Leaf Area Index (LAI)
or the Soil Water Storage (SWS) which require more complex data such as
spectral reflectance, leaf and ground area, soil type, available water content
and more. Finally, the last smaller family of indicators are hydrological
indexes, some examples of which are the Streamflow Drought Index (SDI),
Standardised Runoff Index (SRI) or the Standardised Soil Water Index (SSWI) -
which is used by Météo-France to characterise droughts and applied in the
characterisation of soil dryness and climate change in (soubeyroux2012) -
which require streamflow values, runoff information or soil water data.

Only a small portion of the above indicators could be considered given the
limited data available. Indeed, the data used was the monthly average water
content, the monthly average soil temperature and the monthly average daily
precipitations at a 9km grid resolution globally, from the ERA-5 Land monthly
database (ERA5), between January 1981 and July 2020. Therefore, only the SSWI
and SPI were selected as they only require soil water content and
precipitation respectively and are simple to implement. Note that this SSWI,
recently discussed in hess-2021-209, was inspired by Hao and Farahmand.

The use of precipitation data alone is the greatest strength of SPI, as it
makes it very easy to use and calculate. The ability to calculate over
multiple timescales also allows SPI to have a wide scope. There are many
articles related to SPI available in the scientific literature, which gives
novice users a wealth of resources they can count on for help. Unfortunately,
with precipitation as the only input, SPI is deficient when it comes to taking
into account the temperature component, which is important for the overall
water balance and water use of a region. This drawback can make it more
difficult to compare events with similar SPI values, as highlighted in
Svoboda.

Similarly, the sole use of soil moisture, makes it a simple indicator to use,
but also a deficient one. However, as pointed out in soubeyroux2012, the SPI
and SSWI are complementary : although they do have similarities, they show
some great difference, for example they show that the droughts of 2003 are not
linked to an extreme precipitation deficit so can not be measured solely by
the SPI. In order to add an extra layer of detail to better characterise
droughts, a third indicator was created, based on the soil temperature
available, inspired by the same methodology as the SPI. In the rest of this
article, it will be referred to as the standardised soil temperature index
(SSTI).

To obtain indexes that are comparable all over France, the data must be
transformed. Indeed, in its raw state, a dry period is difficult to
distinguish amongst a simply dry climate: the same magnitude of low
precipitation in areas with very dry climates will have a very different
impact on the soil and on subsidence claims than in wet climates. The SPI
methodology SPImethod allows the creation of normalised indicators (see also
Guttman for an historical discussion). It is calculated using 3-month
cumulative precipitation probabilities by calibrating a gamma distribution to
the data, which is then transformed into a standard normal distribution. Thus,
it allows the quantification of the seasonal deviation of precipitation
compared to the historical mean. A three month step was chosen as it reflects
short- and medium -term drought conditions and provides a seasonal estimation.

In order to obtain indexes with the available data mentioned previously, the
same methodology - which is the SPI computation methodology - was applied to
the monthly average soil wetness, the monthly average soil temperature and the
monthly average daily precipitation separately, thus yielding the three-month
SPI, SSWI and SSTI.

The aim of this study is to predict claims at a yearly time scale, creating
the need for a yearly indicator. The 12-month sliding SSWI, SPI or SSTI were
not chosen as they would not capture the seasonality of droughts. Instead, to
obtain yearly indicators, the extremes are taken over the 4 yearly seasonal
indicators, with the following formulas222Here, we use either the minimum or
the maximum, depending if drought events are related either to low or high
values of seasonal indices.:

$\displaystyle\text{ESSWI}_{z,t}=\min_{s\in\mathcal{S}}\left(\text{SSWI}_{s,z,t}\right),$
(1)
$\displaystyle\text{ESSTI}_{z,t}=\max_{s\in\mathcal{S}}\left(\text{SSTI}_{s,z,t}\right),$
(2)
$\displaystyle\text{ESPI}_{z,t}=\min_{s\in\mathcal{S}}\left(\text{SPI}_{s,z,t}\right)$
(3)

where $\mathcal{S}$ denotes the set of seasons,
$\mathcal{S}=\\{\text{Spring},\text{Summer},\text{Autumn},\text{Winter}\\}$,
$t\in\\{1981,1982,\cdots,2020\\}$ and $z$ denotes the given location. The ESPI
is the Extreme Standardised Precipitation Index, the ESSTI is the Extreme
Standardised Soil Temperature Index and the ESSWI is the Extreme Standardised
Soil Water Index. This methodology - to our knowledge - has not been applied
in the past for the creation of the soil temperature and precipitation indexes
in the context of subsidence claim prediction.

(a) SSWI

(b) SPI

(c) SSTI

Figure 1: Indicators for 2018, with SSWI (Standardised Soil Water Index), SPI
(Standardised Precipitation Index) and SSTI (Standardised Soil Temperature
Index) from left to right.

###  1.4 Other spatial explanatory variables

The concentration and presence of clay in the topsoil plays an important role
in the occurrence of subsidence as it is caused by the shrinkage and swelling
of these particular kinds of soil. Thus, to express the risk of an area, a map
was obtained from the Land Use and Cover Area Statistical Survey (LUCAS)
database published by the ESDAC, which collects harmonised data about the
state of land use and cover over the European Union. This map gives the
concentration of clay in the topsoils (soils at 0-20cm depth) and is available
at a 500m grid resolution over the European Union.

The soil clay concentration was then aggregated by town keeping the highest
concentration of clay in the town, which gives the maps in figure 2(a). Other
aggregation functions were considered (such as the average), but they were
less predictive than the maximum. This map shows that the areas with the
highest clay concentrations appear to be in the North-East of France, in
Charente-Maritime (West), around Toulouse (South-West) and along the
Mediterranean Sea (South-East).

Finally, a binary categorical variable, which takes the values 1 if the town
has historically made a request for a natural catastrophe declaration and 0
otherwise, was sourced from CCRs’ historical data, CCRerretescatnat. The risk
map obtained is visible in figure 2(b), as at 2018.

(a) Topsoil clay concentration

(b) Natural catastrophe occurence

Figure 2: Topsoil clay concentration and historical natural catastrophes
accepted requests. Those two variables will be used for our predictive model.

These variables were then aggregated to the exposure dataset. Thus, the
calibration dataset was composed for each claim year of the “INSEE code” of
the town (from the official classification, that can be seen as the ZIP code
used in the U.S.), the year of the claim, the number of claims, the cost of
claims (in €), the number of policies, the total sums insured (in €), the clay
concentration in the soil, the ESPI, the ESSTI, the ESSWI and Cat (the binary
categorical variable giving the occurrence of a historical natural catastrophe
declaration request, prior to the year of study). All the models calibrated
are based on those five variables mentioned above.

##  2 Model calibration for the frequency

Using simulations at regional scale, corti2009simulating, corti2011drought
pointed out that it is possible to use simulation to get a good representation
of the regions affected by drought-induced soil subsidence, but substantial
differences between simulated and observed damages in some regions. In this
section, we will describe a simple spatio-temporal model for resilience, to
model either the frequency or the intensity of such events (on a yearly
basis). Regression type models will be considered as in blauhut2016estimating.
The main difference is that the interest was to model occurrence, and a
logistic regression was sufficient. Here, since we focus on frequency and
intensity, some Poisson-type regression models will we considered, first, and
then some Gamma models, for the severity, to predict the economic cost of
subsidence. In Section 2.3 we will present regression based models, that we
will extend in Section 2.4 to ensemble models, namely with bagging of
regression trees, as suggested in breiman96bagging to take into account some
possible non-linearity, as well as cross-effects. And in section 2.5, we will
study more carefully the errors. But before, we need to introduce some
criteria to select an appropriate model.

###  2.1 Model selection criteria

Various validation performance measures are used to compare the different
models. They are chosen to optimise the model selection based on the qualities
that are desired from the model which are: a good capacity at predicting the
correct number of claims in the right areas, the ability of not predicting
claims where there are none historically all the while keeping the simplest
possible model. The following performance measures are thus used:

  * •

Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC),
which use maximum likelihood and penalises models with respectively too many
variables and too many variables with respect to the number of observations.
These criteria are only used for the parametric regression models.

  * •

Root Mean Square Error (RMSE) – or simply the sum of the squares of errors,
which penalises greatly extreme errors, i.e. extreme deviations in
predictions. This will be used for costs, and not counts.

###  2.2 Cross-Validation method

In order to assess the predictive power of the calibrated models, cross-
validation was used. The main goal of the models is to predict accurately the
number of claims, per town, on a yearly basis, thus a yearly cross-validation,
that is derived from the basic cross-validation principle, will be used.

The idea of cross validation is related to in-sample against out-of-sample
testing. In a nutshell, the model is fitted on a subset of the data, and
validation is computed on observations that were left out. This approach is
interesting to assess how the results of some statistical analysis will
generalize to an ‘independent’ data set. Note that it is possible to remove
one single observation, and to validate the model on that one – this is the
leave-one-out approach – or to split the original dataset on $k$ subgroups, to
remove one subgroup to fit a model, and predict on that specific subgroup, and
rotate – this is the $k$-fold cross-validation approach. In the case of
spatio-temporal data,

  * •

use some regions, used as subgroups, then use some spatial $k$-fold approach
(as in Pohjankukka) where the model is fitted on $k-1$ regions, and validation
is based using some metric on the error on the region that was left out – it
can be the sum of the squares of the difference, or any metric discussed in
the previous section,

  * •

use cross-validation in time, but because of particular properties of the time
dimension, cross-validation is performed by removing the future from the
analysis (as in BERGMEIR201870): at time $t$, we use observations up to time
$t$ to fit a model, then get a prediction for time $t+1$. In some sense, it is
a classical leave-one-out procedure, except that we cannot use observations
after time $t+1$ to get a prediction at time $t+1$.

###  2.3 Regression-based models

A first attempt at modelling the yearly number of claims was made using
Generalised Linear Models (GLM) with Poisson, Binomial and Negative Binomial
distributions, which are the most adapted to the calibration data, on counts,
as a starting point. These models offer a simple and interpretable approach to
model data by assuming that the response variable
$\boldsymbol{Y}=(Y_{z,t})\in\mathbb{R}^{n\times T}$ is generated by a given
distribution and that its mean is linked to the $q$ explanatory variables
$\boldsymbol{X}=(\boldsymbol{X}_{z,t})\in\mathcal{X}^{n\times T}$ (where
classically $\mathcal{X}=\mathbb{R}^{q}$), through a link function. In this
model, we have $n$ spatial locations (the number of towns), $T$ dates (the
number of years), and $q$ possible explanatory variables.

If we want to model the number of houses claiming a loss in a given location,
the Poisson GLM is defined as
$Y_{z,t}\sim\mathcal{P}(E_{z,t}\cdot\lambda_{z,t})$, for a location $z$ and a
year $t$, where $E_{z,t}$ is the exposure (the number of contracts in the
town) and $\lambda_{z,t}$ is the yearly intensity, per house,

$\lambda_{z,t}=\exp\left[\beta_{0}+\beta_{1}x_{1,z,t}+\cdots+\beta_{k}x_{q,z,t}\right]$
(4)

where $x_{j,z,t}$’s are features used for modeling, such as the ESSWI, the
ESSTI, etc. The prediction, performed at year $t+1\in\\{2001,...,2019\\}$
based on a calibration set of years $\\{2001,...,t\\}$, is

${}_{t}\widehat{N}_{z,t+1}=E_{x,t+1}\cdot{}_{t}\widehat{\lambda}_{x,t+1}$ (5)

and, based on estimator $\widehat{\boldsymbol{\beta}}_{t}$ obtained on the
training dataset with observations of years $\\{2001,...,t\\}$, using maximum-
likelihood techniques, and

${}_{t}\widehat{\lambda}_{z,t+1}=\exp\left[\widehat{\beta}_{0,t}+\widehat{\beta}_{1,t}x_{1,z,t+1}+\cdots+\widehat{\beta}_{k,t}x_{q,z,t+1}\right]$
(6)

where geophysics covariates $x_{j,z,t+1}$ are known. In Table 1 several sets
of parameters estimates $\widehat{\boldsymbol{\beta}}_{t}$ are given (with $t$
varying from 2008 until 2018).

Table 1: Evolution of the parameters in the regression (Poisson regression). Numbers in brackets are the standard deviations of the parameters. Note that two ESSPI parameters are not significant here (95% level). year | $t$ | 2008 | 2010 | 2012 | 2014 | 2016 | 2018
---|---|---|---|---|---|---|---
Intercept | $\widehat{\beta}_{0,t}$ | -13.668 | -13.460 | -13.522 | -13.735 | -13.932 | -14.357
|  | (0.074) | ( 0.071) | ( 0.062) | ( 0.06) | ( 0.059) | ( 0.049)
ESSTI | $\widehat{\beta}_{1,t}$ | 1.522 | 1.420 | 1.511 | 1.494 | 1.539 | 1.661
|  | (0.017) | ( 0.015) | ( 0.013) | ( 0.013) | ( 0.013) | ( 0.012)
ESSWI | $\widehat{\beta}_{2,t}$ | -0.711 | -0.700 | -0.601 | -0.709 | -0.750 | -0.707
|  | (0.011) | ( 0.011) | ( 0.009) | ( 0.009) | ( 0.009) | ( 0.008)
clay | $\widehat{\beta}_{3,t}$ | 0.021 | 0.020 | 0.024 | 0.025 | 0.025 | 0.035
|  | (0.001) | ( 0.001) | ( 0.001) | ( 0.001) | ( 0.001) | ( 0.001)
cat | $\widehat{\beta}_{4,t}$ | 3.924 | 3.950 | 3.957 | 3.957 | 4.003 | 3.902
|  | (0.056) | ( 0.055) | ( 0.049) | ( 0.048) | ( 0.047) | ( 0.038)
ESSPI | $\widehat{\beta}_{5,t}$ | -0.046 | -0.010 | 0.016 | 0.074 | 0.127 | -0.048
|  | (0.013) | ( 0.011) | ( 0.009) | ( 0.009) | ( 0.009) | ( 0.007)

From Table 1, we can see that the model is rather stable over time, which is
an interesting feature from a modeler’s perspective: if we predict more claims
related due to subsidence, it is mainly coming from the underlying factors
than from a change in the impact of each variable. It can be mentioned that
$\widehat{\beta}_{3,t}$ (associate the clay) is significantly increasing (with
a $p$-value of 2%).

This Poisson model is rather classical to model counts, and it is said to be
an equi-dispersed model, in the sense that the variance of $Y$ is equal to the
average value. This model can be extended in two directions, instead of having

  * •

the binomial model, where $Y_{z,t}\sim B(E_{z,t},p_{z,t})$, where $E_{z,t}$ is
the exposure and $p_{z,t}$ is the probability that, for a given year $t$ and
location $z$, a claim is made for a single house, and the prediction for
$p_{z,t+1}$ is

${}_{t}\widehat{p}_{z,t+1}=\frac{\exp\left(\widehat{\beta}_{0,t}+\widehat{\beta}_{1,t}x_{1,x,t+1}+\cdots+\widehat{\beta}_{k,t}x_{q,x,t+1}\right)}{1+\exp\left(\widehat{\beta}_{0,t}+\widehat{\beta}_{1,t}x_{1,x,t+1}+\cdots+\widehat{\beta}_{k,t}x_{q,x,t+1}\right)}$
(7)

In that case, we have an under-dispersed model in the sense that, by
construction, we must have $\text{Var}[Y_{z,t}]<\mathbb{E}[Y_{z,t}]$

  * •

the negative binomial model, where $Y_{z,t}\sim NB(E_{z,t},p_{z,t})$, where
$E_{z,t}$ is the exposure and $p_{z,t}$ is the probability, with standard
notations for the negative binomial probablity function. In that case, we have
an over-dispersed model in the sense that
$\text{Var}[Y_{z,t}]>\mathbb{E}[Y_{z,t}]$

Calibrating the GLMs and averaging the indicators over the years spanning from
2001 to 2018, using the yearly cross-validation method, the results on the
left of Table 2 were obtained. This table shows that the Negative Binomial
model has the lowest AIC and BIC.

In order to better consider the characteristics of the claims data, Zero-
Inflated models were tested using Poisson and Negative-Binomial distributions.
In these models the joint use of logistic and count regression allows the
integration of the over-representation of non-events in the data. More
formally, in Zero-Inflated models, given a location and a time $(z,t)$, we
assume that there is a probably to have $0$ claims. In our context, it could
mean that the town has not been recognized as hit by a drought event. The
occurrence of a drought is modeled with a logistic model, with probability
$p_{z,t}$, here. Then, if there is a drought event, the number of claims is
driven by some specific distribution (Poisson, Binomial, or Negative
Binomial), as introduced by Lambert. If we consider some Poisson regression,
it means that

${\displaystyle\mathbb{P}(Y_{z,t}=0)=p_{z,t}+(1-p_{z,t})e^{-\lambda_{z,t}\cdot
E_{z,t}}}$ (8)

(the second part comes from the fact there there could be a drought, but the
number of counts of claims is null) and

${\displaystyle\mathbb{P}(Y_{z,t}=y)=(1-p_{z,t}){\frac{[\lambda_{z,t}\cdot
E_{z,t}]^{y}e^{-\lambda_{z,t}\cdot E_{z,t}}}{y!}},\text{ where
}y=1,2,3,\cdots}$ (9)

where $\lambda_{z,t}$ and $p_{z,t}$ are related to covariates through
expressions as Equation (4) and (7), respectively. Note that such models can
easily be estimated with standard statistical packages. The results of the
Zero-Inflated models are visible on the right of Table 2. It shows that the
Zero-Inflated Negative-Binomial model is better than the Zero-Inflated Poisson
model.

Table 2: Quality measures for the different GLM distributions. |  |  |  | zero-inflated
---|---|---|---|---
| Binomial | Poisson | NB | Poisson | NB
AIC | 115,051 | 114,189 | 100,491 | 71,154 | 54,375
BIC | 115,113 | 114,252 | 100,564 | 71,259 | 54,510

Figure 3 shows the yearly predictions for the zero-inflated models, alongside
the previously tested GLMs333For purposes of confidentiality, the total number
of claims per year are withheld, but the proportions on the $y$-axis are
valid..

Figure 3: Yearly predictions for the Zero-Inflated models and GLMs

When looking at the yearly total predictions compared to reality, as observed
in figure 3, all the GLM predictions are very similar apart from the negative
binomial model which overpredicts (massively) in 2018, however they all
overestimate claims in 2018 and underestimating the number of claims in 2003,
2011, 2016 and 2017. This graph shows that the predictions closest to the
reality line are for the Negative-Binomial Zero-Inflated model. That model
also has the best metrics when comparing to those of the GLMs. Thus, Zero-
Inflated models appear to provide a better fit than the GLMs.

This section showed that the Zero-Inflated models, in particular the Negative-
Binomial Zero-Inflated model, outperformed the GLMs in terms of number of
claims and model selection criteria. In the next section, we will see of
alternatives to regression-type models can be considered, since the “linear
model ” assumption might be rather strong here.

###  2.4 Tree-based models

Tree-based models are popular models for data analysis and prediction and
offer an alternative to the previous parametric models. Popularised by
CARTtrees, regression trees produce simple and easily interpretable split
rules.

  * •

A Regression Tree is such that

${}_{t}\widehat{Y}_{z,t+1}=\sum_{\ell=1}^{L}\widehat{\omega}_{\ell,t}\boldsymbol{1}(\boldsymbol{x}_{z,t+1}\in\mathcal{L}_{\ell})$
(10)

where $\\{\mathcal{L}_{1},\cdots,\mathcal{L}_{L}\\}$ is a partition of
$\mathcal{X}$, and $\mathcal{L}_{j}$’s are called leaves. In a tree with two
leaves, $\\{\mathcal{L}_{1},\mathcal{L}_{2}\\}$, there is a variable $j$ such
that $\mathcal{L}_{1}$ is the half-space of $\mathcal{X}$ characterized by
$x_{j,t}\leq s$ while $\mathcal{L}_{2}$ is characterized by $x_{j,t}>s$ for
some threshold $s$. For the “classical” regression tree, the split is based on
the squared loss function, in the sense that we select $s$ to maximise the
between-variance, or equivalently, minimise the within-variance. It is
possible to extend this approach by using, instead of the squares of residuals
(corresponding to the squared loss function), the opposite of the log-
likelihood of the data. This can be performed using the rpart R package (see
CARTtrees for further details on regression trees).

If trees are simple to interprete, they are usually rather unstable: when
fitting a tree on subsets of observation, it is common to get different
splitting variables, and therefore different trees. The idea of bagging (as
defined in breiman96bagging) is to use a boostrap procedure to create samples
(resampling the observations with replacement), and then to aggregate
predictions. In the case where the number of covariates is not too large, this
will be also called a Random Forests, from RFbreiman.

Some Tree-Based models are tested in order to attempt to improve the previous
predictions. Three tree-based approaches were used, here:

  * •

A “classical” Random Forest (RF),

${}_{t}\widehat{Y}_{z,t+1}=\frac{1}{m}\sum_{i=1}^{m}{}_{t}\widehat{Y}^{(i)}_{z,t+1}\text{
where
}{}_{t}\widehat{Y}^{(i)}_{z,t+1}=\sum_{\ell=1}^{L_{i}}\widehat{\omega}_{\ell,t}^{(i)}\boldsymbol{1}(\boldsymbol{x}_{z,t+1}\in\mathcal{L}^{(i)}_{\ell})$
(11)

where each tree – corresponding here to different $i$’s – is computed using a
squared loss function, on different boostrap samples (obtained by resampling
$n$ observations, with replacement, out of the initial $n$ ones). This can be
performed using randomForest R package.

  * •

A Poisson Random Forest (RFP) which considers count data, with the aim to
better capture the distribution of the data. Poisson Random Forests are a
modified version of Breiman’s Random Forest allowing the use of count data
with different observation exposures. This is done by modifying the splitting
criterion so that it maximised the decrease of the Poisson deviance and an
offset has been introduced to accommodate for different exposures. This Random
Forest was calibrated using the rfPoisson function available in the
rfCountData package in R.

These Random Forests were all tuned in order to obtain optimal number of
trees, number of variables tested at each split and maximum number of nodes.
However, the tuning was limited given the length of the tuning process, which
may reduce the quality of the models.

The figure 4 shows the total yearly predictions for each tree-based model
alongside the Zero-Inflated Negative Binomial model and the real claims.

Figure 4: Comparison of yearly predictions for the tree-based models, with
ZINB, classical RF, and RFP.

Figure 4 shows that the closest predictions to the real observations (the
plain line) seem to be those of the Zero Inflated model and the Poisson Random
Forest, although all models but the Zero-Inflated model underestimates 2011
and 2016. With our cross-validation approach, we have poor results for early
years (only data from 2002 were used to derive a model for 2003). The standard
Random Forest overpredicts 2018.Thus, the Zero-Inflated model and the Poisson
Random Forest appear to have the best predictions.

Through this section, two different Random Forests were presented; however,
the one with the best results is the Poisson Random Forest which rendered
similar results to the Negative-Binomial Zero-Inflated model. The complex
calibration process and the unclear influence and importance of the variables
on the output of the Poisson Random Forest make it a less attractive choice of
model compared to the Zero-Inflated regression, which has a clear variable
influence and a simple prediction formula. One can thus wonder whether such a
loss in interpretability is worth such a small gain in term of predictions.

###  2.5 Mapping the predictions

In order to improve the predictions, and only predict the actually impacted
areas, a methodology was developed to optimise the removal of all the very low
claim predictions. This methodology was applied to both the Zero-Inflated
model and the Poisson Random Forest. The total predictions changed little, and
the geographical distribution of both models’ predictions are observable for
2018 in figure 5.

(a) Poisson RF

(b) Zero-Inflated

(c) Observed claims

(d) Nat.Cats.

Figure 5: Observed and Predicted number of claims for 2018

In 2018, both models have more or less the same claim distribution, with large
amounts of claims along the Mediterranean, in the North and in Pays-de-la-
Loire. However, the distribution is slightly different when looking at the
centre of France. Indeed, the Random Forest seems to predict more claims in
that area than the Zero-Inflated Negative Binomial model. Comparing these
results with the real claims for 2018, it can be seen that the historical
claims are nearly exclusively concentrated in the centre of France with a few
claims along the Atlantic and Mediterranean coast and in the North-East of
France. Thus, both models seem to over predict the claims around the coasts of
France and - slightly for the Poisson Random Forest and vastly for the Zero-
Inflated Negative Binomial model - under-predict in the centre of France.
However, it can be noted that the over-predicted areas do mostly fall within
areas that have non-recognised natural catastrophe declarations which could
mean that the area was hit but not compensated and thus that the models have
difficulty assessing the difference between areas that will be recognised as
natural catastrophes, or not. The same conclusion can be made when looking at
the predictions for 2017, visible in figure 6.

(a) Poisson Random Forrest (RFP)

(b) Zero-Inflated (ZINB)

(c) Observed claims

(d) Nat.Cats.

Figure 6: Observed and Predicted number of claims for 2017

In these maps, the predicted claims do appear to be in the correct areas in
the South and South-West of France for both models, however, there are
overpredictions in the East, centre and North of France, which also appear to
be areas with non accepted natural catastrophe declarations. Both models seem
to predict the correct areas but also additional zones that often are areas
that have refused natural catastrophe declaration, which means that those
areas were impacted by subsidence but not sufficiently to enter into the scope
of the natural catastrophe regime. Indeed, as the acceptance criteria changed
frequently between 2001 and 2018, the models cannot capture the natural
catastrophe aspect of a claim seeing as one that may have been acceptable in
2017 may no longer be today. Both models predict more or less the same
geographical distribution of claims, posing the question of the usefulness and
practicality of using a complex model such as the Poisson Random Forest
compared to the Zero-Inflated Negative-Binomial model.

Another option that would permit the classification of the predicted claims
into accepted and refuse natural catastrophe categories, would be to use the
Shrinkage and Swelling of clay risk map Georisque:GASPAR, published by
Geological and mining research bureau (BRGM). This map categorises the
exposure of a given point of France between four categories: not at risk, low
risk, average risk and high risk. The acceptance of a natural catastrophe
declaration in France is feasible if more than 3% of the surface of a town is
in a zone with average or high risk. Thus, if the predicted claims map and the
risk map, aggregated by town, were overlapped, the classification of the
predicted claims would be possible between potentially accepted and most
likely refused natural catastrophe declaration.

##  3 Cost predictions

In the previous section, we’ve seen different models to predict the frequency
(the number of claims per town), and here, we consider some Gamma model for
average cost per claim, leading us towards a compound model for the total cost
per town, as introduced in Adelson, and used for example in hydrology in
REVFEIM1984357 or Svensson2017, or for droughts in Khaliq.

###  3.1 Modeling average and total economic costs

For a given location $z$ and year $t$, the total cost (from the insurer’s
perspective) is a compound sum, in the sense that

$Y_{z,t}=\sum_{i=1}^{N_{z,t}}Z_{i,x,t}=\begin{cases}Z_{1,x,t}+\cdots+Z_{N_{z,t},x,t}\text{
if }N_{z,t}>0\\\ 0\text{ if }N_{z,t}=0,\end{cases}$ (12)

with a random sum of random costs. Here $N_{z,t}$ is the frequency, modeled in
the previous section, and $Z_{i,x,t}$’s are individual economic losses per
house. It is possible to consider some Tweedie GLM, as introduced in
Jorgensen1997, corresponding to some compound Poisson model, with Gamma
average cost. Nevertheless this is only a subclass of the general compound
models. Note that Tweedie models are related to some power-parameter, since
they are characterize by a relationship
$\mathbb{E}[Y]=\text{Var}[Y]^{\gamma}$. If $\gamma=1$, $Y$ is proportional to
some Poisson distribution (average costs are non-random) while $\gamma=2$
means that $Y$ is proportional to some Gamma distribution (frequency is non-
random). For inference, we used $\gamma=1.5$ which corresponds the lowest AIC.
For that Tweedie model, SSTI, SSWI, as well as Clay and Cat covariates were
used.

If the model is interesting, it is less flexible than have two separate
models, one for the frequency, and one for the average cost at location $z$,
for year $t$, and to write

$Y_{z,t}=N_{z,t}\cdot\overline{Z}_{x,t}$ (13)

where (as before) $N_{z,t}$ is the frequency, as modeled in the previous
section, and we can use our data, aggregated at the town level, to model the
average cost (per house) $\overline{Z}_{x,t}$. The prediction is then

${}_{t}\widehat{Y}_{z,t+1}={}_{t}\widehat{N}_{z,t+1}\cdot{}_{t}\widehat{Z}_{z,t+1}$
(14)

The results of these methods are rather similar. Figure 7 shows the yearly
predictions for the Tweedie model and the average cost of claims method using
the GLM and both the Zero-Inflated and Poisson Random Forest models,
previously calibrated.

Figure 7: Yearly cost predictions, with a ZINB model + Gamma costs, RFP model
+ Gamma costs, and a Tweedie model (Poisson + Gamma costs).

Figure 7 shows that the Tweedie model makes predictions that are similar to
the previous cost of claims predictions, however this model over-estimates
less the year 2018 and (clearly) over-estimates more the years 2007 and 2012.
On the other hand, the years 2003, 2011 and 2016 are still severely
underestimated by the three predictions, although less so by the Tweedie
model.

###  3.2 Mapping the predictions

As in section 2.5, it is possible to visualize the prediction, and to map
${}_{2016}\widehat{Y}_{z,2017}$ and ${}_{2017}\widehat{Y}_{z,2018}$, on Figure
8. As expected, if we are not able to predict correctly the frequency, the
cost is overestimated. Overall (as we can see on Figure 7), in 2017, we
obtained a good prediction, in France, but we can visualize some spatial
differences. Most of the comments made in section 2.5 remain valid, and
clearly, predicted the economics losses in a specific area is not a simple
task.

(a) Real 2017

(b) Predicted 2017

(c) Real 2018

(d) Predicted 2018

Figure 8: Observed and Predicted average cost of claims in 2017 and 2018

The increase in number and severity of subsidence claims in the past years has
created a need for insurers to better grasp their knowledge of this risk.
However, the implementation of subsidence models is time consuming and
requires detailed data. This study proposed a method of approaching the costs
and frequency of claims due to subsidence based on historical data. This was
applied through two main components: the development of new drought indicators
using Open Data and the use of parametric and tree-based models to model this
risk. Modelling subsidence requires the integration of meteorological and
geological indicators to ascertain the factors predisposing a policy to
subsidence. Without this information, the inherent risk to which a dwelling is
exposed cannot be perceived. For this reason, geological and meteorological
data was obtained from Open Data sets. This data was paired with insurance
exposure and claim data to obtain a complete dataset used to calibrate the
models. However, the data available was only at a communal mesh, making the
results less precise. Indeed, subsidence is a very localised risk whose
modelling would benefit from policy level data.

Overall, the methods enable the predictions of an estimate of the number of
claims, cost of claims and their geographical distributions. Although they
sometimes lack in precision, the models give a good indication of the severity
of a given year. The uncertainty in the predictions may be explained by the
non-homogeneous data on which the models were calibrated. Indeed, the natural
catastrophe declaration criteria evolved many times over the calibration
period, rendering the historical data biased. The predictions would also have
benefited from data at a finer mesh to take into account the individual
particularities of each policy such as the presence of trees close to the
construction, the slope of the terrain, the orientation of the slope, etc.
Finally, more precise temporal definition of the events could have been used
if the claims were available at a monthly scale rather than yearly. In that
case, seasonal indicators could have been created, rather than yearly ones, to
capture more precisely the time scale of an event.

This study has allowed for the creation and implementation of new drought
indicators for a subsidence claim model which performs well considering the
limitations induced by the lack of information and precision of the
calibration data set. The created model provides a better understanding of the
phenomenon and allows other subsidence exposure evaluations to be challenged
which will improve an insurance company’s management of this risk, given the
lack of available subsidence models on the market. This methodology could be
improved in the future by using policy level data with a more accurate
temporal definition of the claims in order to better apply the created
indicators. The framework of this study is also in itself interesting and
could be repurposed for other applications. Indeed, the studied models could
be extended to other claim prediction problematics, which are often confronted
with similar issues of over representation of non-recognized events.

No

No

No

No

No

HA provided insurance data and provided technical support. MJ developed the
model code and performed the simulations. AC prepared the manuscript with
contributions from all co-authors

The authors declare that they have no conflict of interest

###### Acknowledgements.

The authors wish to thank Hélène Gibello, Sonia Guelou, Florence Picard and
Franck Vermet for comments on a previous version of the work. Arthur
Charpentier received financial support from the Natural Sciences and
Engineering Research Council of Canada (NSERC-2019-07077) and the AXA Research
Fund.