A Public-Private Insurance Model for Natural Risk Management: an Application to Seismic and Flood Risks on Residential Buildings in Italy
AA Public-Private Insurance Model for Natural RiskManagement: an Application to Seismic and Flood Riskson Residential Buildings in Italy. ∗ Selene Perazzini a , Giorgio Stefano Gnecco a , and Fabio Pammolli b a IMT School for Advanced Studies Lucca b Politecnico di MilanoJune 11, 2020
Abstract
This paper proposes a public-private insurance scheme for earthquakes and floods in Italy inwhich property-owners, the insurer and the government co-operate in risk financing. Our modeldeparts from the existing literature by describing a public-private insurance intended to relieve thefinancial burden that natural events place on governments, while at the same time assisting indi-viduals and protecting the insurance business. Hence, the business is aiming at maximizing socialwelfare rather than profits.Given the limited amount of data available on natural risks, expected losses per individual havebeen estimated through risk-modeling. In order to evaluate the insurer’s loss profile, spatial corre-lation among insured assets has been evaluated by means of the Hoeffding bound for r-dependentrandom variables. Though earthquakes generate expected losses that are almost six times greaterthan floods, we found that the amount of public funds needed to manage the two perils is almost thesame. We argue that this result is determined by a combination of the risk aversion of individualsand the shape of the loss distribution. Lastly, since earthquakes and floods are uncorrelated, wetested whether jointly managing the two perils can counteract the negative impact of spatial cor-relation. Some benefit from risk diversification emerged, though the probability of the governmenthaving to inject further capital might be considerable.Our findings suggest that, when not supported by the government, private insurance might eitherfinancially over-expose the insurer or set premiums so high that individuals would fail to purchasepolicies.
Natural risks pose a broad range of social, financial and economic issues, with potentially long-lasting effects. Historically, governments have mostly addressed the financial effects of natural eventson an ad-hoc basis, but countries are now increasingly focusing on proactive planning before a disasterstrikes (World Bank, 2014). Among others, OECD, G20 (OECD, 2012), the World Bank and GFDRR(World Bank, 2014) claim that governments should guide citizens towards recovery by implementingboth risk reduction and financial protection. In particular, the World Bank (2014) argues that “financialprotection complements risk reduction by helping a government address residual risk, which is either notfeasible or not cost effective to mitigate. Absent a sustainable risk financing strategy, [...], a country ∗ The idea for this project was inspired by author’s collaboration in Struttura di Missione Casa Italia (2017), andfor this reason we would like to thank Giovanni Azzone and his research team from Politecnico di Milano. We wouldalso like to thank Giuseppe Di Capua (INGV), Andrea Flori (Politecnico di Milano), Ilan Noy (Victoria University ofWellington), Valentina Tortolini (IMT School for Advanced Studies Lucca), and Francesco Turino (Fundamentos delAnálisis Económico, Universidad de Alicante) for their valuable comments and advice. a r X i v : . [ ec on . GN ] J un ith an otherwise robust disaster risk management approach can remain highly exposed to financialshocks, either to the government budget or to groups throughout society” .While guaranteeing social assistance, governments should at the same time encourage private ini-tiatives in prevention and financial protection. As emphasized by the OECD (2015), improving publicawareness reduces the human-induced factors that make a major contribution to the cost of disastersand alleviates losses on public finances. In particular, since private insurance is the main risk financ-ing tool for businesses and households, the OECD (2012) recommends that governments “assess theiravailability, adequacy and efficiency to the population and within the economy, as well as their costsand benefits relative to other types of possible risk reduction measures” .A series of challenges hinder the development of the insurance business in protection from naturaldisasters. First of all, Kousky and Cooke (2012) shows that spatial correlation creates the potential forenormous losses at the aggregate level, and insurers therefore need to access a large amount of capital inorder to offer the cover and meet solvency constraints. As a consequence, they are often forced to driveup premiums, which could become so high that it would not be rational for individuals to purchasethe policy. Large insurers can significantly reduce the probability of insolvency by pooling risks frommore independent regions or by transferring a portion of their portfolio through reinsurance. However,while lowering premiums for regions with a higher risk, this solution might raise those of those with alower risk and, especially in a competitive market, low risk-individuals might fail to purchase, thereforeleaving the company with an extremely risky pool. As shown by Charpentier and Le Maux (2014),the free market does not necessarily provide an efficient level of natural-catastrophe insurance, butgovernment-supported insurance allows losses from disasters to be spread equally among policyholdersthanks to the government’s easier access to credit.Climate change also exacerbates these issues: the Geneva Association (2013) warns that returnperiods and correlation among claims for several high-loss extreme events are “ambiguous rather thansimply uncertain” , and raises concerns about the future sustainability of insurance business on naturalrisks. Social assistance policies may also hinder the development of private markets and increase thefinancial burden of natural disasters on public finances due to charity hazard (World Bank, 2014).Against this background, a number of economies have established various forms of public-private co-operation to support the insurance business, and several countries have decided to enter the market byestablishing a public-private company entirely devoted to insuring citizens’ properties against naturaldisasters at a discounted price (e.g. Spain, France, Australia, Turkey, New Zealand, Taiwan, USA,etc..) (Consorcio de Compensación de Seguros, 2008).This work proposes a public-private insurance scheme for Italy. Italy is highly exposed to naturalrisks, especially earthquakes and floods, but there is currently no well-defined loss allocation mecha-nism at national level. A few people insure their properties (Maccaferri et al., 2012) and expect socialassistance from the government instead. Each natural event is evaluated by public authorities when itoccurs, social assistance depends on the decisions of the parties in charge and is therefore commensu-rate with the financial resources available at the time. In recent years public debate has increasinglyshifted towards natural risk management and planning, although at the moment no initiative has beenundertaken.In defining the public-private insurance scheme for Italy, our work addresses three issues: • Loss estimation and lack of data on past losses .Insurance companies need big loss database for premium rating, but there is currently no sourcethat collects information on natural impacts in Italy at national level. Lack of data on the impactsof natural disasters is a widespread issue and in order to overcome this problem, the world’s biggestinsurance companies have developed sophisticated models for loss estimation based on engineeringand geology studies. In this paper, one of these models has been applied to estimate earthquakelosses, while an alternative approach is proposed for floods. • Public-private insurance model .Once losses have been estimated, we define a public-private insurance model. Our model departsfrom the existing literature by addressing a public-private partnership, which therefore modifiesthe fundamental hypotheses of traditional insurance. Our contribution to the literature can be2ummarized in three aspects. First, the purpose of the business is social assistance, and premiumcollection serves solely to risk management and to guarantee quick compensation to the damagedpopulation. Therefore, rates do not include any profit load and are commensurate to citizens’demand. Second, we introduce the government as a social guarantor that contributes to reservesand provides public funds in case reserves are not sufficient for claim compensation. Finally,our model includes spatial correlation by applying the Hoeffding bound for r-dependent randomvariables. • Multi-hazard management.
As well known in finance, merging portfolios is beneficial only if risks are uncorrelated, as floodsand earthquakes are likely to be. It remains to be seen whether the benefits from risk diversifica-tion counteract the negative impact of spatial correlation. The last part of this work extends thepublic-private insurance scheme to multi-hazard management.The paper presents and discusses results for each of these three aspects. We found that seismicrisk produces the highest expected losses at national level, but floods may generate the highest lossesper square metre. The two perils differ in geographic extent: while the seismic risk involves almostall the nation, floods concern approximately two thirds of the territory. Though the seismic riskgenerates expected losses that are almost six times greater than floods, we found that the amount ofpublic funds needed to manage them is almost the same. Our analysis shows that the public-privateinsurer can benefit from risk differentiation by jointly managing earthquake and flood risks through amulti-hazard policy: the amount of public capital needed is lower than would be necessary if the tworisks were managed separately. Another desirable feature emerges: rates for multi-hazard policies aremore geographically homogeneous, and therefore promote fairness perception among the population.However, it emerged that under no circumstances does the maximum premium that individuals arewilling to pay match the insurer capital constraints. Without the government as a guarantor, it wouldtherefore be impossible for the company to offer policies throughout the territory.The paper is organized as follows: Section 2 describes risk assessment models for flooding and seismichazards and concludes presenting expected losses in Italy; Section 3 defines the insurance model forsingle hazard policies and applies it to the two risks; Section 4 extends the model to the analysis ofmulti-hazard policies; Section 5 concludes.
Expected losses are traditionally estimated from records of past events, but when data are tooscarce or not available, alternative techniques are needed. During the last decades, a new family ofmodels inferring losses from the characteristics of soil and structures has emerged (Grossi et al., 2005).According to this branch of literature, risk can be reconstructed as a combination of four components: • Hazard ( H ) provides a phenomenon description based on physical measurements, usually fre-quency, severity and location. • Exposure ( E ) identifies the object at risk. • Vulnerability ( V ) defines the relationship between hazard and exposure, quantifying the impactof the catastrophic event on the property under analysis. • Loss ( L ) converts physical damages into monetary values.Each component is defined on a series of geophysical, engineering or financial variables and relations,and equally contribute to the overall estimate of risk (Mitchell-Wallace et al., 2017). Through a properdefinition and combination of these components, a risk model should describe the geological or environ-mental features of the peril in analysis and should also capture differences in impacts on the relevantstructural typologies. 3lthough this line of research is growing fast, not many models are currently available and not anyperil has been satisfactorily described. Moreover, these models, while not requiring data on losses, needa large amount of information on soil, weather, and housing. In addition to the difficulty of finding thisdata, models strongly depend on geographical and urban features of the area they have been definedon, and therefore can hardly be adapted to other territories (Hufschmidt and Glade, 2010; Scorzini andFrank, 2015).As far as Italy concerns, current literature offers some analysis that allow to appreciate seismicrisk on the whole territory, while little is still known about floods. We therefore refer to the existingliterature for seismic risk, and develop a new model for flood assessment. Our methodology is similarto Cesari and D’ Aurizio (2019).After a brief presentation of the database, the following two subsections present earthquake andflood risk assessment respectively. Although the two model strongly differ, they both combine the fourrisk components as: Expected monetary damage = L × E × (cid:90) V ( H ) d ( H ) . (1)After a general description of the model, each of these subsection discuss the components separately.The section concludes presenting estimated expected losses. Our analysis considers residential housingonly, furniture not included. Multi-hazard risk assessment is postponed to the Section 4. There is currently no database collecting records on impacts from natural disasters in Italy, butsome information on national riskiness is available, thought data quality is sometimes questionable. Inparticular, our models require data about hazard and exposure. • Hazard
While seismic hazard is well documented, flood data are strongly affected by the lack of a singlebody responsible for physical detection.Seismic movements are in fact regularly monitored by the National Institute of Geophysics andVolcanology (INGV), that freely provides daily-updated databases both on past events and aboutseveral seismic indicators. Records are georeferenced and cover almost all the national territory,indicators are presented for different probability scenarios and associated to an accuracy index.Data for the analysis of earthquake has been drawn from INGV’s maps of seismic riskiness. Onthe other side, flood monitoring is demanded to a number of regional authorities - named “basins’authorities” - that independently choose collection methods and indicators. These differences indata collection often leads to inconsistencies and poor comparability among regions (Molinari etal., 2012). The main database on hydrological risk in Italy is the AVI (“Aree Vulnerate Italiane” -“Italian Vulnerable Areas”) archive managed by National Research Council (Guzzetti and Tonelli,2004). The archive collects historical information on flood events in Italy (mainly from 1900 to2002). However, records are mostly gathered from local journals and, unfortunately, are rarelysuitable to scientific analysis: information are provided in a narrative form, georeferencing is poor,physical phenomena description is not uniform and data quality depends on the original source(Molinari et al., 2014). Despite these limitations, the archive is currently among the best repre-sentation of the flood hazard, and has therefore been used here. Information from the archive havebeen integrated with data from “Italian Flood Risk Maps” (EU Directive 2007/60/CE) indicatingthe perimeter of geographic areas that could be affected by floods according to three probabilityscenarios (Decreto legislativo 23 febbraio 2010 n.4, 2010): extreme events with time to return 500years (P1); events with time to return of 100-200 years (P2); events with time to return between20-50 years (P3). • Exposure
As far as exposure concerns, we refer to the “Mappa dei Rischi dei Comuni Italiani” (“Riski-ness Map of Italian Municipalities” - MRCI). This database has been created during a recent4nstitutional project - “Casa Italia” - to the aim of providing the best representation of majornatural risks in Italy (volcanic, seismic, hydrological, geological). Among several risk indicators,the database presents a fairly rich representation of Italian real estate. Additional informationon regional average house’s squared metres and the average dwelling value are estimates by theRevenue Agency (Agenzia delle Entrate, 2015).
Earthquakes and land movements are among the most studied risks in the literature, but most ofthe analysis focus on vulnerability and explore the relationship that links hazard intensity and damageto buildings. As far as Italy concerns, a few analysis investigate the number of deaths, missing personsand/or injured people (Cascini et al., 2008; Salvati et al., 2010; Marzocchi et al., 2012), while, to ourknowledge, risk assessment on residential risk is presented in Asprone et al. (2013) only. The lattermodel follows the structure specified in eq. (1) and has been tested on the L’Aquila earthquake,therefore we are referring to it for seismic loss estimates. Some slight modification of the model hasbeen introduced in order to update the analysis with latest released data on hazard and to considera wider range of potential loss scenario. Moreover, our real-estate database provides a more detailedrepresentation of residential housing, thus allowing for higher accuracy of the estimates.Damages have been estimated per municipality relating the peak ground acceleration (PGA) andits exceedance probability λ ( P GA ) with the existing residential building stock by means of fragilitycurves. Given a certain set of “limit states” ( LS ) representing subsequent level of damage (usually from“no damage” to “collapse”), a fragility curve describes the probability of reaching a given limit stateas a consequence of the observed PGA, P ( LS | P GA ) . Expected loss can be estimated by comparingfragility curves of each LS . Damages are then monetarily quantified by means of a function RC ( LS ) linking the property’s value to the level of damage.Literature offers many fragility curves’ models, and we rely on Asprone et al. (2013) selection forItaly (Table 2). Each model k applies to a number of specific building structures and is defined on N LS k limit states chosen by the authors to describe the impact of earthquakes on the j -th structure.Since many models may address the same j -th structure, losses are estimated by averaging results fromthe K j models describing j .Municipal residential housing stock is divided into five relevant structural typologies - thus fixing j = 1 , . . . , - and seismic losses per square metre l s are computed for each j and each municipality c .Given the probability P k ( LS + 1 | P GA ) of the structural typology j of suffering a damage level LS given a certain PGA, expected losses are estimated as: l sj,c = 1 K j K j (cid:88) k =1 N LSk (cid:88) LS =1 RC ( LS ) (cid:90) ∞ [ P k ( LS | P GA ) − P j ( LS + 1 | P GA )] d F c ( P GA ) == 1 K j K j (cid:88) k =1 N LSk (cid:88) LS =1 RC ( LS ) · (cid:90) ∞ [ P k ( LS | P GA ) − P k ( LS + 1 | P GA )] | d λ c ( P GA ) d ( P GA ) | d ( P GA ) . (2)where F c ( P GA ) = 1 − λ c ( P GA ) is the cumulative density function of PGA for the c -th municipal-ity. According to Asprone et al. (2013), we assume P k ( N LS k + 1 | P GA ) = 0 . Model (2) combines aprobability distribution with domain [0 , ∞ ) and a damage function increasing with P GA . P GA istraditionally expressed in gravity acceleration units g and Asprone et al. (2013) bounds the integrationvariable P GA to [0 , g ] . Since we wanted to include as many scenarios as possible, we extended thedomain to include even most unlikely events, and therefore the considered domain is [0 , ∞ ) .The five municipal losses estimates have been multiplied by municipal exposure and then aggregatedinto municipal total seismic losses L sc . L sc = (cid:88) j =1 l sj,c · E sj,c . (3)5 igure 1: PGA exceedance probability. lllllllll . . . . . . . PGA exceedance probability
PGA e xc eedan c e p r obab ili t y Power Law
Note : the plot shows the PGA distribution of a random municipality. The nine points are data by INGV, and the redline represent fitting with the power law distribution.
Seismic hazard is represented by PGA and its annual probability of exceedance, which are bothavailable on the INGV website (Gruppo di Lavoro MPS, 2004) for most of Italy .INGV released seismic maps for 9 probabilities of exceedance in 50 years (Meletti and Montaldo,2007). Those P GA measurements are presented for points in a 0.05 degree grid drawn on the Italianmap. Grid points are defined by longitude and latitude, and can be associated to a municipality bymeans of reverse geocoding, that led to the definition of a PGA distribution for over 4600 municipalities.Sometimes more points referred to the same municipality, hence their average value has been considered.In order to capture the widest possible representation of the territory, missing municipalities have thenbeen approximated by averaging the neighbours’ PGA values. However, we failed to represent the wholenational territory since Sardinia and many other small islands cannot be captured by neighborhood(missing municipalities can be seen in Figure 7). Our database is thus composed of 7685 municipalities.The 9 INGV measurements describe the tail of λ ( P GA ) for each grid point (a grid point’s P GA curveexample is plotted in Figure 1). Asprone et al. (2013) assumed uniform seismicity in each municipality,but the known curve’s sections in Figure 1 do not seem to reflect this hypothesis. Moreover, sincethe left-side of the curve is missing, classical fitting methodologies led to unsatisfactory results, oftenoverestimating tails. Therefore, parameters of the distribution have been estimated by regression. Bestfitting results have been obtained by the power law distribution.In order for the hazard curves to reflect the soil category at the building foundation, O.P.C.M.3274 (2003) and D.M. 14/01 (2008) state that PGA values at the bedrock should be multiplied bythe stratigraphic S S and topographic S T amplification factors. These factors have been computed byColombi et al. (2010) for all the Italian municipalities and kindly provided by INGV. As seismic events differently affect buildings, relevant structural typologies have been identified onthe basis of the information available.First, the MRCI database divides municipal housing stock into: masonry, reinforced concrete, and Sardinia, Alicudi, Filicudi, Panarea, Pantelleria, Pelagie Islands, Stromboli, Ustica not included. able 1: Number of buildings per seismic structural typology.
Material Building Code Buildings (u=1000)RC gl Reinforced concrete Gravity Load 2853.96RC sl Reinforced concrete Seismic Load 636.92M Masonry Gravity Load 6975.98A gl Other Structures Gravity Load 1406.21A sl Other Structures Seismic Load 260.88other; Asprone et al. (2013) argue that buildings of type “other” contain both components of reinforcedconcrete and masonry structures, so we assumed this category to be a mixture of these two.These structures may then have been built in compliance with modern anti-seismic requirementsor not. Since the database does not include this information, we refer to the construction year andbuilding laws in force. In fact, from 1974 a series of subsequent laws (Legge n. 64, 2 feb, 1974) ledto the progressive re-classification of risk-prone areas, where more restrictive anti-seismic constructionrequirements entered into force, thus substantially modifying buildings’ structures. The process endedin 2003 when anti-seismic laws (O.P.C.M. 3274, 2003) were extended to the whole Italian territory.Thus, we define reinforced concrete and other structures as seismic loaded if built after these lawsentered into force, or gravity loaded otherwise . According to Asprone et al. (2013), we assumedmasonry as seismic loaded only. Therefore, we refer to 5 structural typologies (see Table 1) : masonry( M ), and gravity or seismic loaded reinforced concrete ( RC.gl and
RC.sl ), gravity or seismic loadedother-type structures (
A.gl and
A.sl ).Since l sj,c is the expected seismic loss of the structure type j in the municipality c per square metres, E sj,c is obtained by multiplying the number of buildings B j,c by the average apartment’s surface ¯ s c (Agenzia delle Entrate, 2015) and the average number of apartments per building ¯ A c (ISTAT, census2015): E sj,c = ¯ s c · B j,c · ¯ A c . (4) Seismic vulnerability is represented by fragility curves, that provide the probability of exceedinga certain damage state, given some hazard parameters. Several curves are offered by the seismicengineering literature, each referring to a specific building structural category. We rely on Asprone etal. (2013) selection of curves, that is reported in Table 2. The selection contains 5 models for masonrystructures, 11 for reinforced concrete ones, and 1 for the other typology. Each model k is defined ona different set of N LS k limit states representing building’s structural damage conditions (the last limitstate always corresponds to collapse) and provides one fragility curve for each limit state. Our fragilitycurves are log-normally shaped and require PGA values as unique input. The loss component is represented by the function RC ( LS ) transforming structural damages intomonetary losses. We assume that the property value equals its reconstruction cost - on average 1500euro per square metre, constant among all the municipalities (Agenzia delle Entrate, 2015) - and define RC ( LS ) as a fraction of the total reconstruction cost RC through a function RC ( LS ) : RC ( LS ) = (cid:18) LSN LS k (cid:19) α RC. (5)where each limit state is represented by a positive integer and N LS k is the number of limit states ofmodel k . According to Asprone et al. (2013), we assume α = 1 . As far as the year of construction concerns, ISTAT does not specify the exact year in which the building has beenbuilt, but a time interval which is approximately ten-years long. We assumed that the number of buildings constructedin any year of the interval is constant. able 2: Fragility curves for seismic risk assessment.
Structure Model ( k ) N LS k gravity load seismic load µ σ µ σ Masonry Rota et al. (2008) 3 -2.03 0.36-1.65 0.27-1.35 0.22Ahmad et al. (2011) 4 -1.13 0.35-1.03 0.35-0.85 0.26-0.77 0.23Erberik (2008) 2 -0.47 0.35-0.33 0.35Lagomarsino and Giovinazzi (2006) 3 -1 0.41-0.75 0.34-0.61 0.37Rota et al. (2010) 3 -0.85 0.24-0.7 0.18-0.58 0.14Reinforced Concrete Kappos et al. (2003) 4 -1.78 1.14 -1.32 0.29-1.12 0.8 -0.95 0.27-0.7 0.63 -0.57 0.27-0.59 0.57 -0.24 0.28Spence (2007) 4 -1.01 0.32 -0.87 0.29-0.55 0.32 -0.46 0.28-0.28 0.31 -0.02 0.29-0.09 0.32 0.15 0.27Crowley et al. (2008) 2 -0.77 0.24 -0.8 0.18-0.62 0.26 -0.61 0.22Ahmad et al. (2011) 3 -1.07 0.22 -1.07 0.22-0.91 0.29 -0.91 0.29-0.59 0.26 -0.44 0.26Borzi et al. (2007) 2 -0.74 0.32 -0.56 0.32-0.46 0.34 -0.37 0.33Borzi et al. (2008) 2 -0.68 0.45 -0.41 0.35-0.41 0.36 -0.31 0.35Kostov et al. (2004) 3 -0.48 0.47 -0.44 0.48-0.34 0.48 -0.28 0.49-0.29 0.48 -0.19 0.49Kwon and Elnashai (2006) 2 -1.08 0.22-0.73 0.22Ozmen et al. (2010) 2 -0.37 0.35 -0.36 0.3-0.17 0.23 -0.12 0.15Kappos et al. (2006) 4 -1.57 0.44 -1.14 0.43-0.92 0.44 -0.57 0.43-0.67 0.44 -0.18 0.43-0.51 0.44 0.1 0.43Tsionis et al. (2011) 2 -0.67 0.27 -0.64 0.28-0.22 0.38 0.18 0.79Other Kostov et al. (2004) 3 -0.62 0.5 -0.52 0.49-0.44 0.49 -0.34 0.49-0.35 0.49 -0.24 0.49
Note : this Table reproduces the selection of seismic fragility curves per building structural typology by Asprone et al.(2013). .3 Flood Hydraulic literature offers very little about flood damage in Italy because the lack of uniform dataat national level hinders research in this field. A few studies concern small geographical areas (usuallycities, sometimes sections of river basins) and focus on the estimation of damages in the immediatefollow-up of an event. Most of the analysis study the relationships between some flood’s physicalmeasurements and expected losses, and the most common output are depth-percent damage curves.Machine learning techniques have been recently applied to the creation of river basins hazard maps(Degiorgis et al., 2012; Gnecco et al., 2015). However, these techniques still require quite accurate dataon past loss. Few example of probabilistic risk assessment have been developed for other countries also,and, similarly to Apel et al. (2006), we decided to extend the deterministic post-event models availablein the literature to probabilistic assessment. In this respect, we estimated expected losses by meansof depth-percent damage curves from the existing literature and additional information on hazard andexposure from our database. In particular, two functions characterize our model: depth damage curves g ( · ) and depth probability, that might be represented by the density f δ ( δ ) , the cumulative distribution F δ ( δ ) and the exceedance probability λ ( δ ) = 1 − F δ ( δ ) .Similarly to seismic fragility curves, depth-damage curves refer to structural typologies. In particu-lar, we consider the buildings’ number of storeys and classify the housing stock into 3 classes ( j ) - 1, 2and 3 or more storeys . A sample of depth-damage curves g j ( δ ) has been selected from the engineeringliterature per each structural typology j . Unlike seismic fragility curves, depth-damage curves do notspecify the probability that a given level of depth might produce a certain damage, and return the mostlikely outcome only. Moreover, the selected curves are “depth-percent damage”, and indicate damagesas percentages of property’s total value.Given the building’s reconstruction cost RC , expected flood loss per square metre l fj,c on a j -typebuilding in the municipality c can be estimated as: l fj,c = RC (cid:90) ∞ (cid:104) g j ( δ ) | d ( λ ( δ )) d δ | (cid:105) d δ = RC (cid:90) ∞ (cid:104) g j ( δ ) | d [1 − F δ ( δ )] d δ | (cid:105) d δ = RC (cid:90) ∞ g j ( δ ) f δ ( δ ) d δ. (6)By construction, there is a value δ j,max after which a g j ( δ ) = 100 . Thus, equation (6) can be split intwo parts as: l fj,c = RC · (cid:34) (cid:90) δ max g j ( δ ) f δ ( δ ) d δ + 100 · (cid:90) ∞ δ max f δ ( δ ) d δ (cid:35) . (7)Bayes’ theorem allow us to express f δ ( δ ) as the product of the probability of δ conditional to theoccurrence of at least a flood event f δ | N F ( δ | N F ≥ and the probability that at least one flood eventoccurs in a year: f δ ( δ ) = P ( N F ≥ f δ | N F ( δ | N F ≥ . (8)When estimating losses, we are considering N F ≥ only, thus substituting eq. (8) into eq. (7) leadsto: l fj,c = RC · P ( N F ≥ · (cid:34) (cid:90) δ max g j ( δ ) f δ | N F ( δ | N F ≥ d δ + 100 · (cid:90) ∞ δ max f δ | N F ( δ | N F ≥ d δ (cid:35) . (9)Since (cid:82) ∞ δ max f δ | N F ( δ | N F ≥ d δ = 1 − F δ | N F ( δ max | N F ≥
1) = λ δ | N F ( δ max | N F ≥ , the model becomes: l fj,c = RC · P ( N F ≥ · (cid:34) (cid:90) δ max g j ( δ ) f δ | N F ( δ | N F ≥ d δ + 100 · λ δ | N F ( δ max | N F ≥ (cid:35) . (10)Loss estimates per square metre per municipality and structural typology are multiplied by municipalexposure and aggregated into municipal flood losses L fc L fc = (cid:88) j =1 l fj,c · E fj,c . (11)9 igure 2: Flood frequency distribution. l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l . . . . . . . Flood frequency
Nr flood per year P r obab ili t y l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Cluster 1Cluster 2
Note : the plot divides observations (points) in two clusters: records from municipalities with < P < . and P ≥ . .Both the clusters have been fitted with a negative binomial, as shown by the black and red lines. Flood hazard has been represented by frequency and depth probabilities. Both the distributionshave been estimated from the AVI database and fitted by means of non-parametric techniques due tothe lack of data. Since AVI gathers information from local press, it is likely that most remote eventshave not been captured. In particular, the number of floods listed after 1900 in the AVI archive is muchhigher than those recorded before and therefore we considered events occurred from that date onwardonly. Unfortunately, only events remain and they are too few to fit distributions at municipal level.Frequency has been described by the probability density function of the number of floods in a year f N F ( N F ) . In order to capture differences between the frequency of occurrence among the municipalities,data have been divided into two clusters - A P (120 obs.) and A P (620 obs.) - on the basis of thehydrological hazard index P from MRCI. Figure 2 shows that frequencies f A P N F approximate negativebinomial behaviour in both the two clusters. Despite the curves appear so close, they strongly differ inmean (the average number of floods per year is . in A P and . in A P ).The probability of flood returns in each cluster is then adapted to fit the municipal and individualrisk: since each flood involves a certain number of municipalities within the cluster A P , the municipalprobability of experiencing at least one flood in a year is estimated by multiplying f A P N F times the averagenumber ¯ c f of municipalities flooded in A P over the number of municipalities N A P c in A P : F cN F (1) = (cid:16) − f A P N F (0) (cid:17) ¯ c f N APc c ∈ A P . (12)Floods usually strike several municipalities at the same time, but not all the properties in a floodedmunicipality will be hit by the flood. Therefore, the individual flood frequency does not coincide withthe municipal one. We approximated the individual frequency probability by means of the P index10 igure 3: Depth probability distribution. . . . . Depth PDF, fitting
N = 475 Bandwidth = 0.2545 D en s i t y datalogNGammaGen GammaGen Beta Note : the dotted line is the empirical distribution f δ | N F ( δ | N F ≥ , and colored lines shows fitting. in MRCI , that indicates the percentage of municipal surface flooded in a 20-50 years probabilisticscenario. We indicate the index as ext P . Assuming homogeneously distributed buildings among themunicipal area, the individual probability of flood returns is: P ( N F ≥
1) = F cN F (1) · ext c,P . (13)In addition to frequency, we estimate the probability of water to reach a certain depth during a flood.Depth information are missing for most of the events in the AVI database and sometimes are replacedby hydrometric heights measuring water depth from the river bed. We excluded hydrometric heightsand assumed that depth levels reported in the database always correspond to the maximum reached inthe area, which is a reasonable hypothesis since records in AVI are largely gathered from local press orcompensation claims.We found no significant difference in depth distributions between differently-exposed areas A P but this may be due to the low amount of available data, and therefore decided to estimate a uniquefunction f δ | N F ( δ | N F ≥ for the entire national territory. Since a flood usually hits more municipalities,a number of depth measurements are often reported for the same event but we represented each eventwith the maximum depth reported in the database. Hence, estimates have been computed on 475observations.The depth empirical distribution estimated from AVI data f δ | N F ( δ | N F ≥ is shown in Figure3, where a graphical comparison between some distributions is presented too. Satisfactory fittingshave been reached with the Generalized Beta (GB), the Generalized Gamma (GG) and the Gammadistributions. Table 3 shows that GG and GB’s led to similar sum of squared errors and sum ofabsolute errors, while errors are much higher for the Gamma. The Chi squared goodness of fit testconfirms the higher performance of GG and GB with respect to the Gamma, even though none of them Indicators P3 are not available for the entire Italian territory, since data are missing for part of Marche and Emilia-Romagna Regions. able 3: Flood depth distribution, goodness of fit.
SSE SAEGamma 0.02194857 0.2493763GG 0.01328367 0.1951612GB 0.01444061 0.2024778
Note : this Table shows the sum of squared errors (SSE) and sum of absolute errors (SAE) obtained when fitting flooddepth distribution with Gamma, Generalized Gamma (GG) and Generalized Beta (GB) distributions. reached a positive outcome. However, the likelihood ratio test shows weak evidence that the GG ismore appropriate, therefore the Gamma has been chosen because of computational advantages.
When evaluating structural vulnerability to floods, the number of storeys of the building is a funda-mental feature to take into account. Therefore, buildings have been classified in three groups accordingto the number of storeys - one, two and three or more - in MRCI. Another element significantly affectingbuildings resistance to floods is the presence/absence of a basement floor; since this information is notavailable, we assumed the two features to be equally distributed.Given the number of buildings per structural typology within the municipality B j,c , the averagenumber of apartments per building ¯ A c (ISTAT, census 2015) and the average apartment’s surface ¯ s c (Agenzia delle Entrate, 2015), exposure has been estimated as: E fj,c = ¯ s c · B j,c · ¯ A c . (14) Table 4:
Number of buildings per number of storeys.
Number of Storeys Buildings (u=1000)1 2083.392 5981.263 or more 4123.05
Flood’s vulnerability is evaluated by depth damage curves defined on the building’s number ofstoreys. The most widely adopted curves in hydraulic literature express damage as a percentage ofbuilding’s total value and therefore called “depth-damage curves”. Conversely to the curves expressingdamages in absolute values, percentages curves are not affected by monetary volatility and are morereliable (Appelbaum, 1985).Many studies have led to the definition of different depth-percent damage curves, that are stronglygeographical-dependent (Scorzini and Frank, 2015): being derived from the analysis of historical data,they are in fact defined on the characteristics of the area under analysis and tend to lose accuracy whenapplied to contexts whose urban and territorial features differ too much from the original site.We have selected depth-percent damage curves from six previous works (Appelbaum, 1985; Arrighiet al., 2013; Debo, 1982; Genovese, 2006; Luino et al., 2009; Oliveri and Santoro, 2000), all either definedor tested on Italian data. The selection is reported in Figure 4. Selected curves per structural typologyhave then been averaged into three new curves in order to guarantee higher reliability of results at thenational level. Curves have been fitted by polynomial regressions, as shown in Figure 5.
Structural damages have been converted into monetary terms by means of the function RC . Similarto the seismic model, we assume that the property value is equal to its reconstruction cost - on average12 igure 4: Depth-percent damage curves for flood risk assessment.
Curve_idro_1p[, 1] D a m age ( % t o t a l v a l ue ) One storey
Depth−percent damage curves
Curve_idro_2p[, 1] D a m age ( % t o t a l v a l ue ) Two storeys
Depth C u r v e_ i d r o_3p [, ] Three or more storeys
Genovese(2006)Appelbaum(1985)Debo(1982)Luino(2008)Olivieri, Santoro(2000)Arrighi(2013)no basementbasementbothDepth D a m age ( % t o t a l v a l ue ) Note : selection of depth-percent damage curves for flood risk assessment. Curves are listed per buildings’ number ofstoreys and can refer to dwellings with and/or without basement.
Earthquake and flood losses have been estimated per municipality and structural typology. Seismicrisk is described in Table 5, where results from Asprone et al. (2013) are also reported for compari-son. We can note that, though the model adopted is the same, huge differences emerge between thetwo analysis. Several reasons contribute to these discrepancies and should be discussed for a betterunderstanding results.First of all, (i) estimates are highly sensitive to the probability distribution of hazard intensities, andwhile λ ( P GA ) has been here fitted from INGV data, Asprone et al. (2013) rely on some distributionalassumption. In addition, (ii) we assumed PGA values ranging in [0 , ∞ [ , while the previous analysis13 igure 5: Depth-percent damage curves.
Depth−percent damage curves
Depth D a m age ( % t o t v a l ue ) Note : black lines represent the average values of the curves selected per number of storeys. Red, green and blue linesshow the functions fitted by polynomial regression. considers [0 , g ] only. (iii) INGV data on PGA fails to represent many smaller municipalities thathave here been approximated by means of neighbours’ values and this assumption may have furthercontributed to the differences in results. (iv) Exposure strongly affect results too and while MRCIcollects the number of dwellings per structural typology at the municipal level, Asprone et al. (2013)had information at the provincial level only. Moreover, MRCI refers to the 2011 population census,while the database used by Asprone et al. (2013) date back to 10 years earlier.Arguments (i)-(iv) determine the different loss scenario, and, in particular, Table 5 shows thatestimated loss per square metre obtained by our model are considerably lower than those of Asprone etal. (2013). The main reason is the adoption of a power law distribution that concentrates the probabilityon weaker events. However, our model highlights the gap in expected losses between more and lessfragile buildings more than the older version.Though our losses per square metre are lower than previous findings, the second column of the Table5 ( max( l sj,c ) ) describe similar patterns. By contrast, expected losses per municipality and structuraltypology L sj,c in the third column do not even show the same pattern. As argued before, exposurestrongly affect results and the detailed information on buildings in MRCI allowed us to better repre-sent real estate assets. In fact, Rome is the biggest municipality in Italy, and therefore its exposureproduces expected losses that are extremely higher than those of other municipalities. By contrast, thehomogeneous distribution of provincial structures among the municipalities in Asprone et al. (2013)very likely underestimates the exposure of major areas.The fundamental role of exposure becomes clear when comparing l sj,c and L sj,c geographically. Fig-ure 6 represents the expected loss per square metre on the most vulnerable buildings - the masonrystructures - in each municipality. The map reflects the hazard component of the risk model and clearlyshows the proximity to risk sources. By contrast, this pattern in risk distribution is not evident inFigure 7 showing annual total expected losses per municipality. In fact, the risky dark area delimitedin Figure 6 largely corresponds to the Appennino mountain chain, where several municipalities are14 able 5: Estimated seismic expected losses.
Structure max( l sj,c ) (euro) max( L sj,c ) (Mln euro) tot L sj (Mln euro) λ ( P GA ) ∼ P L
RC.gl 10.53 Castelbaldo (Padova) 216.79 Roma 2223.61RC.sl 3.83 Castelbaldo (Padova) 3.54 Roma 130.70A.gl 4.03 Castelbaldo (Padova) 7.16 Roma 233.76A.sl 3.22 Castelbaldo (Padova) 0.43 Roma 30.73M 12.69 Castelbaldo (Padova) 109.54 Roma 3615.87tot 6234.661Asprone (2013) RC.gl 17.04 Giarre (Catania) 51.5 Roma 1186.8RC.sl 11.34 Navelli (L’Aquila) 8.0 Reggio di Calabria 489.9A.gl 14.51 Giarre (Catania) 25.1 Roma 667.2A.sl 11.71 Navelli (L’Aquila) 2.4 Napoli 174.0M 29.99 Giarre (Catania) 196.4 Roma 8661.8tot 11179.6
Note : the table lists some descriptive statistics about estimated seismic expected losses per structural typology; in order:maximum expected loss per square metre l fj,c , maximum expected loss at the municipal level L sj,c and the total expectedloss L sj . The upper part describes current results, obtained by fitting PGA with a power law distribution; the lower sidereports results by Asprone et al. (2013) for comparison. Table 6:
Estimated flood expected losses.
Structure max( l fj,c ) (euro) max( L fj ) (Mln euro) tot L fj (Mln euro)1 storey 19.61 Vigarano Mainarda (Ferrara) 7.93 S. Michele al T. (Venezia) 105.752 storeys 15.16 Vigarano Mainarda (Ferrara) 36.53 Ferrara 536.143 storeys 11.56 Vigarano Mainarda (Ferrara) 18.24 Rimini 234.01tot 875.90 Note : the Table shows descriptive statistics of flood expected losses per number of storeys. In order: maximum expectedloss per square metre l fj,c , maximum expected loss at the municipal level L fj,c and the total expected loss L fj . sparsely inhabited. On the other hand, densely populated municipalities on the coast do not showextremely high level of loss per square metre but reach the highest expected losses at the aggregatelevel because of large real estates.In order to appreciate the effect of different hazard and exposure components, one can consider re-inforced concrete gravity loaded structures: though the power law distribution gets to a lower max( l sj,c ) ,the associated estimate of the expected loss L sj,c in Rome is four times greater than that obtained inthe previous paper.Our analysis of seismic risk led to total expected loss equal to . million, which is almost halfthe value obtained by Asprone et al. (2013). The value is seven times greater than the expected lossestimated for flood risk, equal to . million per year, thus indicating that the earthquakes are thenatural hazard of main concern in Italy.As far as flood losses concern, main findings are presented in Table 6. Maximum losses per squaremetre l fj,c are higher than the seismic ones, but Figure 8 shows that a great part of the territory doesnot appear to be affected by hydrological risk and most municipalities are associated to values of l fj,c close to . The map shows that the risk mostly affects northern Italy, and in particular the Emilia-Romagna, Veneto and Lombardia regions. More or less the same risk distribution is obtained at theaggregate level in Figure 9, where the effect of exposure highlights additional areas of interest, such asthe north-west coast, north Sardinia and Rome.By comparing Figures 7 and 9, we can observe that north-east Italy is highly affected by both thetwo hazards, though the effect of floods remains consistently limited with respect to that of earthquakes.To conclude, Table 7 ranks the fifteen largest expected municipal losses per each hazard. One can noticethat three cities in Emilia-Romagna are listed for both: Bologna, Ravenna and Rimini.15 igure 6: Seismic expected loss per square metre (masonry buildings).
Note : the minimum value is l sj,c = 0 . , maximum is . , and average value is . euro per square metre. Figure 7:
Expected seismic annual loss per municipality. igure 8: Flood expected loss per square metre (one-storey buildings)
Note : “Not at risk” identifies municipalities where l fj,c = 0 . Among the other municipalities, the minimum loss is . e − .Maximum value is l fj,c = 19 . . On average, expected loss in risky areas (municipalities “Not at risk” not included) is . euro per square metre. Figure 9:
Expected flood annual loss per municipality. able 7: Municipalities with higher expected loss per natural risk.
Seismic Expected LossMunicipality Province Region L sc (Mln euro)1 Roma Roma Lazio 337.462 Napoli Napoli Campania 114.033 Bologna Bologna Emilia-Romagna 105.264 Verona Verona Veneto 59.585 Firenze Firenze Toscana 58.556 Torino Torino Piemonte 45.177 Reggio di Calabria Reggio di Calabria Calabria 43.238 Modena Modena Emilia-Romagna 39.579 Prato Prato Toscana 33.3610 Terni Terni Umbria 33.0311 Ravenna Ravenna Emilia-Romagna 31.6912 Rimini Rimini Emilia-Romagna 30.3813 Messina Messina Sicilia 29.9214 Pistoia Pistoia Toscana 29.5015 Catania Catania Sicilia 29.14Flooding Expected LossMunicipality Province Region L fc (Mln euro)1 Ferrara Ferrara Emilia-Romagna 56.222 Ravenna Ravenna Emilia-Romagna 52.893 Rimini Rimini Emilia-Romagna 45.034 Pisa Pisa Toscana 37.335 San Michele al Tagliamento Venezia Veneto 34.486 Jesolo Venezia Veneto 27.837 Parma Parma Emilia-Romagna 23.518 Bologna Bologna Emilia-Romagna 21.639 San Donà di Piave Venezia Veneto 21.3110 Cesenatico Forlì-Cesena Emilia-Romagna 17.0211 Piacenza Piacenza Emilia-Romagna 16.0512 Cervia Ravenna Emilia-Romagna 15.4013 Verbania Verbano-Cusio-Ossola Piemonte 14.5814 Forlì Forlì-Cesena Emilia-Romagna 13.3315 Abano Terme Padova Veneto 12.94 When constructing an insurance scheme, two fundamental quantities should be carefully evaluated:the premium per policyholder and the amount of reserves needed for the business given the solvencyconstraint. In the private market, insurers aim at profit and constitutes reserves through premium’scollection. Therefore, rates should be sufficiently high to yield profit and avoid unacceptable levels ofloss, while at the same time meeting an acceptable level of risk. Moreover, in order for the policy tobe purchased, premiums should also meet the demand. Premium rating can hence be represented asa typical decision problem of profit and utility maximizing agents (Mossin, 1968; Ehrlich and Becker,1972). However, when the government takes over the market, it radically changes the managementobjectives of the insurance company, and the traditional model is no longer suitable to capture agent’sbehavior. In this respect, this section presents a public-private insurance model for natural disasterswhere homeowners, insurers and government cooperate in risk financing.Although the traditional private insurance-model has to be suitably modified to describe a public-private partnership, the problem can still be represented by comparing two perspectives: on the onehand, individuals who are willing to spend up to a certain amount on coverage; on the other hand, theinsurer who, supported by the government, offers the policy subject to some solvency constraints. Thenext two Subsections address the problem on the demand and supply perspectives respectively.On the demand side, individuals face a standard decision problem and their utility functions can18e defined as in the private-market literature. We keep the standard assumption of perfect informationand rationality of individuals, though these hypotheses are often criticized as inappropriate to describereal world conditions (Goda et al., 2015). These criticisms are extremely important for the privatemarket, but less relevant to our analysis since governments have the ability to modify the behavior ofindividuals by investing in risk education, promoting public awareness or introducing the obligation topurchase the policy.On the supply side, the goal of insurance business is substantially affected by the partnership withthe government. In the free market, insurer’s goal is profit maximization subject to survival and/orstability constraints that require low ruin probability and low probability of high operational costs(Goda et al., 2015). By contrast, when entering the business, the government forces insurers to setthe lowest premium possible given both the demand and the solvency constraints. Our model departsfrom traditional literature by assuming that the business endorses social welfare and therefore ratesdo not include profit-load. On the other side the government also supports the business by relaxingits financial burden: it partially subsidises reserves through capital injections and contributes to thereimbursement whenever stored funds are not sufficient for claim compensation.Several additional issues arise in specifying the supply side. In particular, insurers’ solvency con-straint refer to the aggregate loss distribution, which is difficult to represent due to lack of information.While expected losses can be reconstructed through risk modeling as in Section 2, particular attentionshould be devoted to the variance as spatial correlations strongly affect insurer’s potential of extremelosses. Quantifying correlation is difficult when records of past events are available, and it is practicallyimpossible when they are not. However, it is reasonable to assume that spatial correlation betweenmunicipalities depends on their proximity, so that it can be identified a sufficiently large threshold r such that two municipalities that are at least r -km far away are independent. We assume r = 50 km,and include spatial correlation by means of the Hoeffding bound for r -dependent variables.In addition to risk quantification, private insurers are also affected by state regulations, marketcompetition (Grossi et al., 2005) and social or political decisions that may result in moral hazard andadverse selection (Kunreuther and Pauly, 2009). Coordinating government and insurers actions canprevent these drawbacks, that should therefore not be included in the public-private model.Finally, agents’ attitudes toward risk should also be carefully evaluated. While in the literatureis widespread agreed that homeowners are risk-averse, some evidence suggests that insurers may alsoexhibit risk-aversion (Gollier, 2013). Actuarial practice also encourages cautious behaviors, emphasizingthe importance of adjusting rates by a risk-load component proportional to aggregate loss variance forextraordinary uncertain events such as natural hazards (Kunreuther, 1996; Larsen and Kuzak, 2005).However, by entering the business as a guarantor, the government release the insurer from its strictcapital constraint and there is no need for the insurer to over-protect the reserves. We thus assumerisk-averse homeowners and a risk-neutral insurer.The two agent’s perspectives are combined in Subsection 3.3 where the insurance scheme is defined.The application to Italian data is discussed in the following Subsection, and results for both the twohazards conclude the Section. Four different policies have been estimated and, thought seismic riskgenerates highest expected losses, the analysis shows that almost the same amount of public funds isnecessary to manage the two risks. This section discusses single hazards policies only, multi-hazardanalysis follows in the next Section. Since the seminal papers by Mossin (1968) and Ehrlich and Becker (1972), several premium settingmodels on insurance purchase decision have been developed. These models describe policies offered bythe private sector and set premiums by comparing the risk-averse individual’s willingness to pay with Common shared information between insurer and insured is questionable in real contexts (Cooper and Hayes, 1987;Kunreuther and Pauly, 1985). In addition to lack of data for risk assessment, individuals also have to face limited cognitivecapacity (Kahneman, 2003; Goda et al., 2015) and imperfect rationality: Kunreuther (1996) asserts that policy adoptionconveys individual risk perception; Palm (1995) observes that appreciation of earthquake policies’ benefits depends onpersonal attitude, socioeconomic and demographic characteristics, proximity to physical hazards, and past experience. i has an m i square metres property. The N i individuals gather in municipalities, thus any i belongs to a generic Italian municipality c . A negative event has an annual probability − π c (0) to hitthe Municipality c and ruin the i -th individual property at time t causing a loss l ai,t per square metre.Consider discrete time period t equal to one year.Individual i may incur in a loss l ai,t with probability − π c (0) , i ∈ c . This loss affects his wealth w i,t ,that we assume equal to the house value for simplicity. However, the individual may buy an insurancecoverage and pay a premium p i,t per square metre to get a reimbursement x i,t per square metre in casethat the event occurs. Let us define x i,t as a function of the actual loss l ai,t : x i,t = (cid:40) , with probability π c (0) ,x (cid:16) l ai,t (cid:17) , with probability − π c (0) , < x (cid:16) l ai,t (cid:17) ≤ l ai,t , (15)with i ∈ c and x (cid:0) l ai,t (cid:1) = if l ai,t ≤ D,l ai,t − D if D < l ai,t < E + D,E if l ai,t ≥ E + D, (16)where D and E are the deductible and the maximum coverage provided per square metre by the insurer.The homeowner’s utility of not being insured is traditionally expressed as the sum of two componentsrepresenting the case of no events occurring during the year and a unique loss scenario: U not insured = π c (0) u ( w i,t ) + (1 − π c (0)) u ( w i,t − l ai,t m i,t ) . (17)Similarly, the utility of purchase is defined as: U insured = π c (0) u ( w i,t − p i,t m i,t ) + (1 − π c (0)) u ( w i,t − p i,t m i,t − l ai,t m i,t + x (cid:0) l ai,t (cid:1) m i,t ) . (18)Therefore, assuming rational behaviour, we can assume that the homeowner will buy an insurancecoverage for its property if and only if its utility of purchasing is greater than or equal to that of notpurchasing the policy: U insured ≥ U not insured.Considering any possible loss level, hence any possible phenomena intensity ζ , we can define theprobability π c ( ζ ) that c will experience a ζ -intensity event in a year and that the homeowner i livingin municipality c will suffer a loss l ai,t ( ζ ) expressed as a function of ζ . In case he is owning a residentialinsurance coverage, its claim value will be then: x i,t = (cid:40) , with probability π c (0) ,x (cid:16) l ai,t ( ζ ) (cid:17) , with probability π c ( ζ ) , < x (cid:16) l ai,t ( ζ ) (cid:17) ≤ l ai,t , with i ∈ c. (19)with x (cid:0) l ai,t ( ζ ) (cid:1) = if l ai,t ( ζ ) ≤ D,l ai,t ( ζ ) − D if D < l ai,t ( ζ ) < E + D,E if l ai,t ( ζ ) ≥ E + D (20)and the insured purchase-convenience condition becomes: π c (0) · u ( w i,t ) + (cid:90) ∞ π c ( ζ ) · u ( w i,t − l ai,t ( ζ ) m i,t ) d ζ ≤ π c (0) · u ( w i,t − p i,t m i,t ) + (cid:90) ∞ π c ( ζ ) · u ( w i,t − p i,t m i,t − l ai,t ( ζ ) m i,t + x (cid:0) l ai,t ( ζ ) (cid:1) m i,t ) d ζ. (21)20ccording to the traditional literature on insurance purchasing decision, we assume the individual tobe risk-averse and we represent its preferences by means of the utility function u ( x ) = log( x + 1) . Weset w i,t equal to the house value and assume for simplicity that it corresponds to the reconstructioncost, equal to RC per square metre. The logarithmic specification allows us to simplify the modelconsidering losses per square metre, so we can rewrite eq. (21) as: π c (0) · log( RC + 1) + (cid:90) ∞ π c ( ζ ) log( RC − l ai,t ( ζ ) + 1) d ζ ≤ π c (0) · log( RC − p i,t + 1) + (cid:90) ∞ π c ( ζ ) log( RC − p i,t − l ai,t ( ζ ) + x (cid:0) l ai,t ( ζ ) (cid:1) + 1) d ζ. (22)We assume that the premium p i,t is fixed at t = 0 and does not vary with respect to time, p i,t = p i ,and neither do inhabited square metres, so m i,t = m i . We can compute the highest premium thathomeowners are willing to pay by restricting condition in eq. (22) to the equality, obtaining: π c (0) · log ( RC + 1)( RC − p i + 1) + (cid:90) π c ( ζ ) log ( RC − l ai,t ( ζ ) + 1)( RC − p i − l ai,t ( ζ ) + x (cid:16) l ai,t ( ζ ) (cid:17) + 1) d ζ = 0 . (23)This equality states that the individual is indifferent to the decision to purchase the policy or not,and allows us to derive the risk-based maximum premium p Hi that the individual is willing to pay perstructural typology and municipality. We now consider the supply side, where the insurer and the government cooperate in risk manage-ment. As previously discussed, the goal of the business is maximizing social well-being, while financiallyprotecting the insurer. Therefore, the government forces insurers to apply the lowest possible premiums,given both the demand and the solvency constraints, and offers its support to the business by partiallysubsidising reserves and committing to pay reimbursements whenever the reserve is not sufficient forclaim compensation.As the demand can be represented through the maximum premium that individuals are willing topay, supply is concerned about the constitution of reserves in order to cope with possible future claims.At the beginning of the activity, say t = 0 , the insurer should create a reserve W , that will be increasedevery year by annual premiums p i collected from the N i individuals. Since the government supportsthe insurers, the reserve is partially subsidises by public capitals. Assume for simplicity that all thepremiums are paid at the beginning of the year, while claims are paid when experienced. Hence, aminimum capital requirement W d should be fixed, so that the government will have to pay W d in t = 0 and to refill the fund at the end of the year t if W t goes below this threshold. So, at the beginning ( b )of the year t = 0 the initial reserve W b is created: W b = W d + N i (cid:88) i =1 p i m i , (24)and at the end ( e ) of the year it will be decreased of the total amount of reimbursement paid duringthe year: W e = W b − N i (cid:88) i =1 x i, m i . (25)Since claims x (cid:16) l ai,t ( ζ ) (cid:17) may incur at any random time t and more events may happen close in time,the minimum capital requirement W d is necessary to guarantee money availability for reimbursementwith a sufficiently high probability. Thus, if W e < W d the government will refill it with an additionalamount W r = W d − W e . 21t any subsequent time t , the fund value at the beginning of the year is: W bt = W t − + N i (cid:88) i =1 p i m i with W t − = max( W et − ; W d ) , (26)while at the end it will be: W et = W bt − N i (cid:88) i =1 x i,t m i . (27)However, the insurer is legally asked to meet some solvency constraint and hence need the government toset W d such that the probability of not being able to promptly pay the claims (“insolvency” probability)below a certain low value (cid:15) .Let us assume that a negative event hits any building within a municipality. We assume that everypolicy can generate at most one claim per year and per individual; since reconstructing or restoring abuilding requires long time, this hypothesis is reasonable. Moreover, assume that actual square metrelosses l ai,t are equal for all the individuals within the same municipality and so does x i,t . Consider the N c municipalities in Italy and indicate the total number of inhabited squared metres in the municipality c as M c , we have: M c = (cid:88) i ∈ c m i , (cid:88) i ∈ c x i,t m i = X c,t M c , hence X c,t = (cid:80) i ∈ c x i,t m i M c , (28)so we can compute the total amount of claims as: Y t = N i (cid:88) i =1 x i,t m i = N c (cid:88) c =1 (cid:88) i ∈ c x i,t m i = N c (cid:88) c =1 X c,t M c . (29)Since our policy covers at most one claim per year and per individual, claim occurrence per year andper municipality can be modelled as a Bernoulli random variable ¯ X c,t ∼ Ber ( q c ) Olivieri and Pitacco(2010) ¯ X c,t = (cid:40) with probability q c , with probability − q c . (30)with q c = π c ( ζ > ζ D ) and ζ D such that l ai,t ( ζ D ) = D .We can rewrite Y t as: Y t = N c (cid:88) c =1 X c,t M c = N c (cid:88) c =1 M c ¯ X c,t x (cid:0) l ac,t,j (cid:1) = N c (cid:88) c =1 ¯ X c,t (cid:88) j M j,c x (cid:0) l ac,t,j (cid:1) = N c (cid:88) c =1 ¯ X c,t a c,t , (31)where j indicates the structural typology and M j,c is the number of squared metres of properties oftype j in municipality c .A main issue related to covering natural disasters is the high level of correlation between individualrisks, which makes the description of the probability distribution of Y t non-trivial. There is no physicalbound for energy propagation and this means that we cannot consider municipalities as perfectlyindependent among each other, especially in the earthquakes’ case. By the way, natural phenomenahit neighbour cities, but far enough municipalities fairly never experience the same event. Therefore,it could be found a certain distance r such that municipalities whose centroids are at least r km farare independent. This assumption is similar to the Hoeffding (1963)’s definition of ( r − -dependence,and allows us to follow his work to model the national claim amount Y t .We sample municipalities in N g groups Y g of independent units, namely we create the groups insuch a way that all the municipalities within a group are at least r km apart from each other. Thenumber n g of municipalities in group g varies.The total amount of claims in Italy can thus be obtained as: Y t = Y t + Y t + Y t + · · · + Y N g t , (32)22ith Y gt = (cid:88) c ∈ g ¯ X c,t a c,t , c = 1 , . . . , n g . (33)Each group claim amount Y gt is the sum of n g independent and bounded random variables.Assuming that the hazard distribution does not vary with respect to time too, expected losses donot depend on t , and neither do E [ Y t ] and E [ Y gt ] . Considering that (cid:82) ζ D π c ( ζ ) x (cid:104) l aj,c ( ζ ) (cid:105) d ζ = 0 , the g -th group expected value: E [ Y gt ] = E [ Y g ] = (cid:88) c ∈ g (cid:88) j M j,c (cid:90) ∞ π c ( ζ ) x (cid:2) l aj,c ( ζ ) (cid:3) d ζ = (cid:88) c ∈ g (cid:88) j M j,c · E (cid:2) x (cid:0) l aj,c (cid:1)(cid:3) . (34)The expected total amount of claims in Italy is: E [ Y t ] = E [ Y ] = N g (cid:88) g =1 E [ Y g ] . (35)Now we can define the insolvency probability at time t , and impose an upper bound (cid:15) on it: P rob (cid:40) W t − + N i (cid:88) i =1 p i m i − Y t < (cid:41) < (cid:15) , (36)or equivalently: P rob (cid:40) Y t > W t − + N i (cid:88) i =1 p i m i (cid:41) < (cid:15) . (37)We consider the worst case scenario W t − = W d : P rob (cid:40) Y t > W d + N i (cid:88) i =1 p i m i (cid:41) < (cid:15) . (38)The minimum capital requirement W d that the government should guarantee is then obtained by ap-plying the Hoeffding (1963) bound to our weighted sum of independent and bounded random variables: P rob (cid:40) Y t > N c φ + E [ Y ] (cid:41) < N g (cid:88) g =1 w g e − h φ E (cid:20) e h ng ( Y gt − E [ Y g ] ) (cid:21) , h > , (39)with w g = n g N c .Set W d + N i (cid:88) i =1 p i m i = N c φ + E [ Y ] , (40)and fix the right hand side of eq. (39) equal to (cid:15) : (cid:15) = N g (cid:88) g =1 w g e − h φ E (cid:20) e h ng ( Y gt − E [ Y g ] ) (cid:21) = e − h φ N g (cid:88) g =1 w g E (cid:20) e h ng Y gt e − h ng E [ Y g ] (cid:21) == e − h φ N g (cid:88) g =1 w g e − h ng E [ Y g ] E (cid:20) e h ng Y gt (cid:21) = e − h φ N g (cid:88) g =1 w g e − h ng E [ Y g ] E (cid:20) e h ng (cid:80) c ∈ g ¯ X c,t a c,t (cid:21) . (41)The last expected value in eq. (41) is the moment generating function of the sum of random variables M Y gt (cid:16) h n g (cid:17) : E (cid:20) e h ng (cid:80) c ∈ g ¯ X c,t a c,t (cid:21) = (cid:89) c ∈ g M Y gt (cid:18) h n g (cid:19) = (cid:89) c ∈ g M ¯ X c,t a c,t (cid:18) h n g (cid:19) , (42)23ence eq. (41) can be rewritten as: (cid:15) = e − h φ N g (cid:88) g =1 w g e − h ng E [ Y g ] (cid:89) c ∈ g M ¯ X c,t a c,t (cid:18) h n g (cid:19) . (43)Solving eq. (43) we obtain φ as: φ = 1 h log (cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17) (cid:15) (44)and estimate W d from eq. (40): W d = N c φ + E [ Y ] − N i (cid:88) i =1 p i m i . (45)Eq. (45) may result in a negative value of W d , but we bind possible solutions to W ∗ d ≥ . (46)In case of W d < , we assume that the government will decide to set it equal to and keep an insolvencyprobability even lower than the desired level: (cid:15) ∗ ≤ (cid:15) .Moreover, it is reasonable to suppose that the government aims to minimize the probability to refillthe fund with additional capital W r = W d − W et , so it will need to set a premium sufficiently high toguarantee a low probability bounded from above by (cid:15) to pay that quantity at any time t : P rob (cid:40) W d − W et > (cid:41) = P rob (cid:40) W d − W t − − N i (cid:88) i =1 p i m i + Y t > (cid:41) < (cid:15) . (47)Once again, consider the worst case scenario W t − = W d : P rob (cid:40) W d − W d − N i (cid:88) i =1 p i m i + Y t > (cid:41) = P rob (cid:40) Y t − N i (cid:88) i =1 p i m i > (cid:41) < (cid:15) , (48)or equivalently: P rob (cid:40) Y t > N i (cid:88) i =1 p i m i (cid:41) < (cid:15) . (49)Note that this condition applies a new constraint on the premiums’ value.Given a sufficiently low probability (cid:15) , we can define the minimum amount of total premiums byapplying again the Hoeffding (1963) inequality: P rob (cid:40) Y t > N c γ + E [ Y ] (cid:41) < e − h γ N g (cid:88) g =1 w g e − h ng E [ Y g ] (cid:89) c ∈ g M ¯ X c,t a c,t (cid:18) h n g (cid:19) , h > . (50)Set (cid:15) = e − h γ N g (cid:88) g =1 w g e − h ng E [ Y g ] (cid:89) c ∈ g M ¯ X c,t a c,t (cid:18) h n g (cid:19) (51)and get γ = 1 h log (cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17) (cid:15) (52)which in turn allows us to estimate the minimum allowable value of the sum of premiums (cid:80) N i i =1 p Gi m i : N i (cid:88) i =1 p Gi m i = N c γ + E [ Y ] . (53)24 .3 Insurance model The maximum value p Hi that each individual is willing to pay in eq. (23) and the minimum amountof total premium necessary to avoid excessive government risk-exposure (cid:80) N i i =1 p Gi m i in eq. (53) are thetwo constraints that the supply faces when defining a national insurance scheme. The two equationspose conditions on rates and they may either identify a range of possible values or fail to find a uniquesolution. However, since we are focused on a publicly supported insurance scheme, it is reasonableto assume that the government will keep the premium as low as possible in order not to financiallyover-stress homeowners, though this may imply a higher probability of found refill at each t , thus agreater risk for public resources. Therefore, given the desired probability (cid:15) of government non-financialover-stress we define the optimal premium level p ∗ i as: p ∗ i = min( c, · p Hi with c = (cid:80) N i i =1 p Gi m i (cid:80) N i i =1 p Hi m i . (54)Premiums as defined in eq. (54) are risk-based on municipality hazard and individual structural typol-ogy, thus guaranteeing social fairness. Moreover, the equation implies that: N i (cid:88) i =1 p ∗ i m i = min( c, N i (cid:88) i =1 p Hi m i = min (cid:18) , c (cid:19) N i (cid:88) i =1 p Gi m i = min (cid:18) , c (cid:19) ( N c γ + E [ Y ]) = N c γ ∗ + E [ Y ] , (55)thus γ ∗ is γ ∗ = min (cid:0) , c (cid:1) ( E [ Y ] + N c γ ) E [ Y ] N c ≤ γ (56)and the insurer is thus able to guarantee an upper bound (cid:15) ∗ on the probability to refill the fund equalto: (cid:15) ∗ = (cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17) e h γ ∗ ≥ (cid:15) . (57)Given the desired upper bound on insolvency probability (cid:15) , the optimal capital minimum requirement W ∗ d is then obtained from condition (45): W ∗ d = max (cid:40) N c φ + E [ Y ] − N i (cid:88) i =1 p ∗ i m i ; 0 (cid:41) = N c φ ∗ + E [ Y ] − N i (cid:88) i =1 p ∗ i m i , (58)with φ ∗ = W ∗ d + (cid:80) N i i =1 p ∗ i m i − E [ Y ] N c ≥ φ. (59)Thus, the optimal value (cid:15) ∗ is: (cid:15) ∗ = (cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17) e h φ ∗ . (60)Since (cid:15) decreases as φ increases, the optimal insolvency probability will be at most equal to the leveldesired by the insurer: (cid:15) ∗ ≤ (cid:15) .Moreover, note that: W ∗ d = N c φ ∗ + E [ Y ] − N i (cid:88) i =1 p ∗ i m i = N c φ ∗ + E [ Y ] − E [ Y ] − N c γ ∗ = N c ( φ ∗ − γ ∗ ) . (61)From eq. (57) and (60), γ ∗ and φ ∗ can be defined as: γ ∗ = 1 h log (cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17) (cid:15) ∗ (62)25nd φ ∗ = 1 h log (cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17) (cid:15) ∗ . (63)Given the condition in eq. (46), eq. (61) implies (cid:18)(cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17)(cid:19) h (cid:18)(cid:80) N g g =1 w g e − h ng E [ Y g ] (cid:81) c ∈ g M ¯ X c,t a c,t (cid:16) h n g (cid:17)(cid:19) h · ( (cid:15) ∗ ) h ( (cid:15) ∗ ) h ≥ . (64)In particular, if a parameter h = h = h is chosen, eq. (61) becomes W ∗ d = N c h log (cid:18) (cid:15) ∗ (cid:15) ∗ (cid:19) . (65)Eq. (65) shows that the amount of public resources needed increases with the ratio (cid:15) ∗ /(cid:15) ∗ , and moreimportantly, eq. (64) collapses to: (cid:15) ∗ ≥ (cid:15) ∗ , (66)indicating that insolvency should never be preferred to the disbursement of public funds, thus enforcingthe Government role of social guarantor. The minimum W ∗ d value corresponds to (cid:15) ∗ = (cid:15) ∗ and is equalto 0.However, (cid:15) ∗ /(cid:15) ∗ affects W ∗ d logarithmically, while the capital requirement is largely determined by N c /h . Therefore, W ∗ d is directly proportional to the number of municipalities, and inversely related tothe parameter h , whose value is determined by the Government’s initial preferences (cid:15) and (cid:15) and theoverall risk distribution. Premium model application to the seismic case requires particular attention due to the hazard com-ponent ζ = P GA . We can estimate π ( ζ ) as π ( P GA ) = | d λ ( P GA ) d ( GP A ) | . The absence of seismic movements ζ = 0 corresponds to the case of no seismic event happening in the year, thus we have l i,t (0) = 0 and x ( l i,t ( ζ )) = 0 . This allows us to include the case of no seismic event in the integral term of condition(23): (cid:90) ∞ π c ( P GA ) log ( RC − l ai,t ( P GA ) + 1)( RC − p i,t − l ai,t ( P GA ) + x (cid:16) l ai,t ( P GA ) (cid:17) + 1) d ( P GA ) = 0 . (67)In Section 2.2.1 we have shown that λ c ( P GA ) approximately behaves as a Power Law distribution andtherefore we have: π c ( P GA ) = | d ( λ ( P GA )) d ( P GA ) | = α c P GA − β c , (68)whose domain does not include values in [0 , P GA min c [ , with P GA min c = e log ( αcβc − ) βc − . (69)This implies that, in this case, the integral in condition (67) cannot be evaluated in [0 , + ∞ [ but in [ P GA min c , + ∞ [ only. However, P GA min c take values ranging from . e − to . , and are smallenough to include the case of no seismic loss.The loss function per structural typology l aj,t ( P GA ) is derived from eq. (2): l aj,t ( P GA ) = 1 K j K j (cid:88) k =1 N LSk (cid:88) LS =1 RC ( LS ) · [ P k ( LS | P GA ) − P k ( LS + 1 | P GA )] , (70)26ith P k ( N LS k + 1 | P GA ) = 0 .Condition (23) for seismic risk becomes: (cid:90) ∞ P GA minc α c P GA − β c log RC − l ai,t ( P GA ) + 1 RC − p i,t − l ai,t ( P GA ) + x (cid:16) l ai,t ( P GA ) (cid:17) + 1 d ( P GA ) = 0 . (71) The premium model application to flood is simpler with respect to the seismic. Here, hazard isrepresented by depth ζ = δ and l i,t ( δ ) is obtained by the depth-percent damage curve g j ( δ ) for thenumber of storeys j . The probabilistic component π c ( ζ ) is given by f ( N F ) defined in equation (8),whose estimation has been discussed in section 2.3.1. We define the individual flooding probabilityfrom equation (13) as: P ( N F ≥
1) = (1 − f A P N F (0)) · ext c,P · ¯ c f N APc . (72)The probability of no flood events in a year π c (0) is then defined as: f N F (0) = (cid:20) − (1 − f A P N F (0)) · ext c,P · ¯ c f N APc (cid:21) := π c (0); (73)while π c ( δ ) corresponds to: π c ( δ ) = (1 − f A P N F (0)) · ext c,P · ¯ c f N APc · f δ | N F ( δ | N F ≥ (74)So condition (23) becomes: (cid:20) − (1 − f A P N F (0)) · ext c,P · ¯ c f N APc (cid:21) · log (cid:18) RC + 1 RC − p i,t + 1 (cid:19) + (1 − f A P N F (0)) · ext c,P · ¯ c f N APc ·· (cid:90) ∞ f δ | N F ( δ | N F ≥ · log (cid:32) RC − RC g j ( δ ) + 1 RC − p i,t − RC · g j ( δ ) + x (cid:2) RC · g j ( δ ) (cid:3) + 1 (cid:33) d δ = 0 . (75)We focus on the integral in the second term, and split it into two components: (cid:90) ∞ f δ | N F ( δ | N F ≥ · log (cid:32) RC − RC · g j ( δ ) + 1 RC − p i,t − RC · g j ( δ ) + x (cid:2) RC · g j ( δ ) (cid:3) + 1 (cid:33) d δ == (cid:90) ∞ f δ | N F ( δ | N F ≥ · log (cid:18) RC − RC · g j ( δ ) + 1 (cid:19) d δ + − (cid:90) ∞ f δ | N F ( δ | N F ≥ · log (cid:18) RC − p i,t − RC · g j ( δ ) + x (cid:20) RC · g j ( δ ) (cid:21) + 1 (cid:19) d δ, (76)then, we consider them separately.Since g j is a positive non-decreasing function that becomes constant at level correspondingto a certain depth δ max , the first integral can be simplified to: (cid:90) ∞ f δ | N F ( δ | N F ≥ · log (cid:18) RC − RC · g j ( δ ) + 1 (cid:19) d δ == (cid:90) δ max f δ | N F ( δ | N F ≥ · log (cid:18) RC − RC · g j ( δ ) + 1 (cid:19) d δ + (cid:90) ∞ δ max f δ | N F ( δ | N F ≥ · log (1) d δ == (cid:90) δ max f δ | N F ( δ | N F ≥ · log (cid:18) RC − RC · g j ( δ ) + 1 (cid:19) d δ. (77)27he second integral involves two piecewise functions: g j ( δ ) and x ( g j ( δ )) . Defining δ D such that g j ( δ D ) · RC = D and δ E such that g j ( δ E ) · RC = E + D and considering δ max , we can split it into 4 components: (cid:90) ∞ f δ | N F ( δ | N F ≥ · log (cid:18) RC − p i,t − RC · g j ( δ ) + x (cid:20) RC · g j ( δ ) (cid:21) + 1 (cid:19) d δ == (cid:90) δ D f δ | N F ( δ | N F ≥ · log (cid:18) RC − p i,t − RC · g j ( δ ) + 1 (cid:19) d δ ++ log ( RC − p i,t − D + 1) · (cid:2) F δ | N F ( δ E | N F ≥ − F δ | N F ( δ D | N F ≥ (cid:3) ++ (cid:90) δ max δ E f δ | N F ( δ | N F ≥ · log (cid:18) RC − p i,t − RC · g j ( δ ) + E + 1 (cid:19) d δ ++ log ( E − p i,t + 1) (cid:2) − F δ | N F ( δ max | N F ≥ (cid:3) . (78)Summing up, insured purchasing indifference condition (23) for the flood case study is: (cid:104) − (1 − f A P N F (0)) · ext c,P · ¯ c f (cid:105) · log (cid:18) RC + 1 RC − p i,t + 1 (cid:19) + (1 − f A P N F (0)) · ext c,P · ¯ c f ·· (cid:40) (cid:90) δ max f δ | N F ( δ | N F ≥ · log (cid:18) RC − RC · g j ( δ ) + 1 (cid:19) d δ + − (cid:90) δ D f δ | N F ( δ | N F ≥ · log (cid:18) RC − p i,t − RC · g j ( δ ) + 1 (cid:19) d δ + − log ( RC − p i,t − D + 1) · (cid:2) F δ | N F ( δ E | N F ≥ − F δ | N F ( δ D | N F ≥ (cid:3) + − (cid:90) δ max δ E f δ | N F ( δ | N F ≥ · log (cid:18) RC − p i,t − RC · g j ( δ ) + E + 1 (cid:19) d δ + − log ( E − p i,t + 1) (cid:2) − F δ | N F ( δ max | N F ≥ (cid:3) (cid:41) = 0 . (79) In order to apply the model, M ¯ X c,t a c,t (cid:16) hn g (cid:17) should be defined and perhaps some distributionalassumption should be introduced. The best distributional form depends on the scope of the cover-age, and the analysis might rather compare multiple significant scenarios represented by alternativedistributional hypotheses.An informative choice is focusing on the expected value of claims, and thus assuming that Y t is aweighted sum of Bernoulli random variables. Assuming that the properties within a municipality areperfectly correlated, Y t equal to: Y t = N c (cid:88) c =1 X c,t M c = N c (cid:88) c =1 M c ¯ X c,t (cid:90) ∞ ζ D π c ( ζ | ζ > ζ D ) x (cid:2) l ac,t ( ζ ) (cid:3) d ζ == N c (cid:88) c =1 ¯ X c,t (cid:88) j M j,c (cid:90) ∞ ζ D π c ( ζ | ζ > ζ D ) x (cid:2) l aj,c,t ( ζ ) (cid:3) d ζ = N c (cid:88) c =1 ¯ X c,t a c . (80)Note that now parameters a c do not depend on time t and are constants. The expected reimbursement28f the g -th group E [ Y gt ] in eq. (34) therefore becomes E [ Y gt ] = (cid:88) c ∈ g q c a c = (cid:88) c ∈ g π ( ζ > ζ D ) (cid:88) j M j,c (cid:90) ∞ ζ D π c ( ζ | ζ > ζ D ) x (cid:2) l aj,c,t ( ζ ) (cid:3) d ζ == (cid:88) c ∈ g π ( ζ > ζ D ) (cid:88) j M j,c (cid:90) ∞ ζ D π c ( ζ ) π c ( ζ > ζ D ) x (cid:2) l aj,c,t ( ζ ) (cid:3) d ζ == (cid:88) c ∈ g (cid:88) j M j,c (cid:90) ∞ ζ D π c ( ζ ) x (cid:2) l aj,c,t ( ζ ) (cid:3) d ζ = (cid:88) c ∈ g (cid:88) j M j,c E (cid:2) x (cid:0) l aj,c,t (cid:1)(cid:3) , (81)and the moment generating function of ¯ X c,t a c can be written through the probability generating functionof a weighted sum of Bernoulli variables: M ¯ X c,t a c (cid:18) hn g (cid:19) = G ¯ X c,t a c (cid:18) e hng (cid:19) = (cid:20) (cid:18) e hng a c − (cid:19) q c (cid:21) . (82)However, since Bernoulli variables are bounded in [0 , , each Y gt is bounded between ≤ Y gt ≤ (cid:80) c ∈ g a c = b g . In the seismic model, a sc = (cid:88) j x sj,c (83)with x sj,c = (cid:90) ∞ P GA minc x (cid:32) K j K j (cid:88) k =1 N LSk (cid:88) LS =1 RC ( LS ) · [ P k ( LS | P GA ) − P k ( LS + 1 | P GA )] ·· | d λ c ( P GA ) d ( P GA ) | (cid:33) d ( P GA ) , (84)while expected claims for flood policies are a fc = (cid:88) j x fj,c (85)with x fj,c = x (cid:18)(cid:90) ∞ RC · P ( N F ≥ · g j ( δ ) f δ | N F ( δ | N F ≥ d δ (cid:19) == P ( N F ≥ · x (cid:18)(cid:90) ∞ RC · g j ( δ ) f δ | N F ( δ | N F ≥ d δ (cid:19) == P ( N F ≥ · (cid:32) (cid:90) δ E δ D RC · g j ( δ ) f δ | N F ( δ | N F ≥ d δ + (cid:90) ∞ δ E E · f δ | N F ( δ | N F ≥ d δ (cid:33) == P ( N F ≥ · (cid:32) (cid:90) δ E δ D RC · g j ( δ ) f δ | N F ( δ | N F ≥ d δ + E · λ δ | N F ( δ E | N F ≥ (cid:33) . (86)According to Hoeffding (1963), the bounds in eq. (39) and (50) simplify for the case of boundedweighted random variables. Consider, for instance, the bound as in eq. (39) P rob (cid:40) Y t > N c φ + E [ Y ] (cid:41) < N g (cid:88) g =1 w g e − h φ E (cid:20) e h ng ( Y gt − E [ Y g ] ) (cid:21) , h > . According to Lemma 1 in Hoeffding (1963), since the final term in the right-hand side of the inequalityis convex, we know that: E (cid:20) e h ng ( Y gt − E [ Y g ] ) (cid:21) ≤ e h ng E [ Y gt ] (cid:20) b g − E [ Y g ] b g + E [ Y g ] b g e h ng b g (cid:21) == e h ng E [ Y g ] (cid:20) E [ Y g ] b g (cid:18) e h ng b g − (cid:19)(cid:21) = e L ( h g ) . (87)29 ( h g ) can be specified as L ( h g ) = − h g p g + ln (cid:16) p g (cid:16) e h g − (cid:17)(cid:17) (88)with h g = h n g b g and p g = E [ Y g ] b g . According to the proof of Theorem 2 in Hoeffding (1963), L ( h g ) ≤ h g = 18 (cid:18) h b g n g (cid:19) , (89)hence the bound can be rewritten as P rob (cid:40) Y t > N c φ + E [ Y ] (cid:41) < N g (cid:88) g =1 w g e − h φ (cid:18) e (cid:16) h bgng (cid:17) (cid:19) = N g (cid:88) g =1 w g e − h φ + (cid:16) h bgng (cid:17) . (90)In order to get the best possible upper bound, we find the minimum of the right-hand side of theinequality as a function of φ , thus obtaining h = 4 φn g b g . (91)Substituting the parameter h as defined in eq.(91) into eq.(90), the Hoeffding’s bound simplifies to P rob (cid:40) Y t > N c φ + E [ Y ] (cid:41) < N g (cid:88) g =1 w g e − φ n gb g . (92)Similarly, the bound in eq.(50) can be rewritten as P rob (cid:40) Y t > N c γ + E [ Y ] (cid:41) < N g (cid:88) g =1 w g e − γ n gb g . (93) We now revise the insurance model by applying the distributional assumptions in section 3.4.3.Once again, parameters φ and γ are obtained by fixing the desired probabilities (cid:15) and (cid:15) equal tothe right-hand side of inequalities (92) and (93) respectively.Premiums p Hi are obtained as in section 3.4.1 and 3.4.2, and p Gi are computed as in eq. (53). Theoptimal premium amount (cid:80) N i i =1 p ∗ i m i is again computed according to eq. (54).While φ ∗ and γ ∗ remain unchanged as in eq. (63) and (62), (cid:15) ∗ = N g (cid:88) g =1 w g e − φ ∗ n gb g (94)and (cid:15) ∗ = N g (cid:88) g =1 w g e − γ ∗ n gb g . (95)Optimal values φ ∗ and γ ∗ here cannot be expressed as explicit functions of (cid:15) ∗ and (cid:15) ∗ respectively, henceeq. (61) W ∗ d = N c ( φ ∗ − γ ∗ ) ≥ cannot be directly related to the two probabilities. However, the equation implies φ ∗ ≥ γ ∗ and since (cid:15) ∗ and (cid:15) ∗ are inversely related to φ ∗ and γ ∗ respectively, the condition is satisfied if and only if (cid:15) ∗ ≥ (cid:15) ∗ . (96)30 able 8: Public-private insurance scheme for earthquake risk management. (cid:15) = 0 . , (cid:15) = 0 . Deductible Maximum coverage (cid:80) N i i =1 p ∗ i c W ∗ d (cid:15) ∗ (cid:15) ∗ (per square metre) (Mln) (Mln)0 1500 10735.784 1.394 7970.726 0.010 0.061(0.000) (0.021) (0.080) (0.000) (0.035)0 1200 9725.082 1.505 8563.810 0.010 0.080(0.000) (0.021) (0.073) (0.000) (0.030)200 1500 8837.312 1.576 8582.130 0.010 0.095(0.000) (0.021) (0.068) (0.000) (0.027)200 1200 8221.215 1.652 8771.088 0.010 0.112(0.000) (0.021) (0.065) (0.000) (0.024) Note : results are based on samplings over the N c = 6404 municipalities for which data were fully available. Policiesare defined on deductible and maximum coverage and listed by row, while columns represent the model’s relevant variables.Reported values are mean and coefficient of variation. Table 9:
Optimal seismic premiums per square metre.
Deductible 0 0 200 200Maximum coverage (per square metre) 1500 1200 1500 1200RC.gl min 0.460 0.460 0.034 0.034mean 6.620 6.582 4.413 4.106max 32.261 32.261 30.471 30.471RC.sl min 0.034 0.034 0.007 0.007mean 2.005 1.676 1.351 1.350max 10.226 10.226 8.922 8.683A.gl min 0.027 0.027 0.008 0.008mean 1.902 1.712 1.536 1.424max 10.200 10.197 9.269 9.124A.sl min 0.012 0.012 0.011 0.011mean 1.745 1.535 1.278 1.205max 10.153 10.153 7.810 7.696M min 0.075 0.062 0.041 0.041mean 4.461 3.910 3.975 3.544max 50.182 40.042 31.387 30.926
Note : the Table shows the minimum, average and maximum premium at the municipal level per each combination ofstructural typology (row) and coverage limits (columns).
Similarly to the special case h = h = h , the model indicates that providing additional publicresources should always be preferred to being insolvent. Coherently, W ∗ d = 0 is obtained if and only if (cid:15) ∗ = (cid:15) ∗ . Of course, the fund is again directly proportional to the number of municipalities N c . Initialpreferences (cid:15) and (cid:15) and claim distribution are now reflected by φ ∗ and γ ∗ instead. The insurance model has been applied to the Italian residential building stock according to theassumption discussed in section 3.4. Results here presented refer to initial preferences (cid:15) = 0 . and (cid:15) = 0 . . The former value is representative of solvency requirement in Solvency II, the latter hasbeen fixed slightly greater than (cid:15) according to the model description. In addition, we assumed r = 50 km, thus adopting a precautionary hypothesis on spatial correlation. This criteria allows for multiplesampling solutions, each resulting in different optimal values of the relevant variables. Therefore, finalresults have been averaged over 100 samplings. 31 able 10: Public-private insurance scheme for flood risk management. (cid:15) = 0 . , (cid:15) = 0 . Deductible Maximum coverage (cid:80) N i i =1 p ∗ i c W ∗ d (cid:15) ∗ (cid:15) ∗ (per square metre) (Mln) (Mln)0 1500 1021.072 5.644 7422.276 0.010 0.408(0.000) (0.035) (0.041) (0.000) (0.029)0 1200 823.444 6.957 7567.733 0.010 0.534(0.000) (0.035) (0.040) (0.000) (0.028)200 1500 1020.885 5.607 7366.177 0.010 0.402(0.000) (0.035) (0.041) (0.000) (0.029)200 1200 823.257 6.898 7496.555 0.010 0.547(0.000) (0.035) (0.040) (0.000) (0.027) Note : results are based on samplings on N c = 7772 municipalities for which data were fully available. Policies aredefined on deductible and maximum coverage and listed by row, while columns represent the model’s relevant variables.Reported values are mean and coefficient of variation. Table 11:
Optimal seismic premiums per square metre.
Deductible 0 0 200 200Maximum coverage (per square metre) 1500 1200 1500 12001 storey min 2e-06 1e-06 2e-06 1e-06mean 1.355 1.088 1.355 1.088max 16.140 12.962 16.139 12.9612 storeys min 1e-16 1e-16 1e-16 1e-16mean 0.198 0.161 0.198 0.161max 2.382 1.933 2.381 1.9333 or more storeys min 1e-16 1e-16 1e-16 1e-16mean 0.182 0.147 0.182 0.147max 2.188 1.771 2.187 1.770
Note : the Table shows the minimum, average and maximum premium at the municipal level per each combination ofstructural typology (row) and coverage limits (columns).
Four policies have been considered, differing on the level of deductible (none or 200) and maximumcoverage (none or 1200 per square metre). Note that deductible equal to 0 corresponds to the absenceof it, while maximum coverage equal to per square metre indicates that no maximum coverageapplies.Results for seismic policies are reported in Table 8, where relevant variables are presented in termsof their mean and coefficient of variation (
CoV ). It can be noticed that the optimal premiums alwayscorresponds to the maximum price that individuals are willing to pay, p Hi , as shown by (i) c ≥ ; (ii) thecoefficient of variation of (cid:80) N i i =1 p ∗ i equal to 0; and (iii) (cid:15) ∗ = (cid:15) . When interpreting these findings, thereare two elements that should be carefully evaluated: individuals’ risk aversion and spatial correlation.On one hand, because of risk aversion, individuals are keen to buy policies at a premium greaterthan their expected loss; the more individuals are risk averse, the higher is the amount of premiumthat the insurer is able to collect and, in turn, the lower is the additional capital needed to satisfythe solvency constraint (cid:15) . On the other hand, spatial correlation between individual risks inflate lossvolatility and bumps the tail of the aggregate loss distribution, thereby increasing the amount of capitalcorresponding to (cid:15) . Parameter c > indicates that individual’s risk aversion is not sufficient to tacklethe risk-enhancing effect of spatial correlation at the aggregate level.As a consequence of c > , the premium p Gi that would satisfy the desired solvency constraint (cid:15) and capital re-investment probability (cid:15) does not meet market demand, and would generate a marketfailure. This result suggests a potential weakness of the free market: since the government has easier32ccess to capital markets than private companies, it is reasonable to assume that a private insurer willrequire a probability of capital re-investment at most equal to the one desired by the government; underthis condition, p Gi would be the minimum pure premium that traditional insurers would be able tocharge to the homeowner, and the policy would not be purchased.Limiting coverage might help the insurer controlling risk’s volatility, thus allowing for lower premi-ums. In particular, being earthquakes low frequency-high intensity perils, the aggregate loss distributionis strongly affected by rare events causing severe damages and therefore we expect maximum coverageto reduce the insurer’s financial exposure more than deductibles. In Table 8 we can clearly notice thatincreasing coverage limits reduces the overall minimum amount of reserves that should be guaranteedat the beginning of each year W min = (cid:80) N i i =1 p ∗ i + W ∗ d , but the minimum capital requirement W ∗ d in-creases and (cid:15) ∗ deviates more and more from the desired level. As confirmed by the greater values of c ,individuals are in fact reluctant to deductibles and maximum coverage due to increasing risk aversion .Coverage limits negatively affect individual willingness to pay, that in turn lower their contribution toreserves and the insurer is left with an enhanced financial pressure. As said, limits-reluctance is heregenerated by risk-aversion, but unfortunately c > even for full-coverage policies, thus suggesting theneed for a government intervention on the private market.Results for floods are collected in Table 10. Once again c > and the need for a government inter-vention in the insurance private market is even more strongly suggested (higher value of c ). However,deductibles are here beneficial to the insurer and, in fact, both W ∗ d and (cid:15) ∗ are lower when D = 200 ap-plies. Though findings are completely different from the seismic case study, this evidence still generatesdue to a combination of risk aversion and loss distribution. Being high frequency-low intensity perils,floods mostly cause small claims on relatively low return periods and the aggregate loss distributiontherefore concentrates on low values. On the other side, increasing risk aversion makes individualsextremely averse to high losses and less concerned about low damages that can afford by them own: inTable 11, when applying the deductible D = 200 , p Hi remains substantially unchanged. Combining thetwo effects, deductibles relieve the insurer commitment while not substantially modifying individual’swillingness to pay.By contrast, introducing an maximum coverage worsens the insurer condition by increasing both (cid:15) ∗ and W ∗ d . This effect is clear when comparing the policy ( D = 0 , E = 1500) with the ( D = 0 , E = 1200) or ( D = 200 , E = 1500) with ( D = 200 , E = 1200) . This limit in fact diminishes the risk of theinsurer by lowering the tail of its aggregate loss distribution, but leaves highest level of individual riskto property-owners. Because of increasing risk aversion, the premium individuals are willing to pay istherefore much lowered, and the amount of public funds needed much increased.The most interesting result is obtained when comparing policies with estimated losses in Section2.4: though earthquakes produce expected losses that are more than seven times greater than thosefrom floods, the minimum capital requirement W ∗ d for the two hazards almost coincide. Once again,the shape of the aggregate loss distribution and individual’s increasing risk aversion jointly determinethis surprising result. As low frequency-high intensity perils, earthquakes sometimes produce enormousdamages that individuals are extremely concerned about. Therefore, owners are willing to pay a pre-mium consistently higher than their expected loss. On the other side, floods happen quite more oftenbut their damages are usually minor and can mostly be afforded by homeowners themselves. Peopleare risk averse, and hence keen to buy a policy and get rid of their flood risk, but the amount they arewilling to pay for the insurance service is lower. In other words, both the two premiums are higher thanthe corresponding expected loss but the difference between the premium that the homeowner pays forthe earthquake policy and its expected seismic loss is greater than that of floods: p H,si − E ( L si ) > p H,fi − E ( L fi ) . (97)The higher is the difference between premiums and expected losses, the lower is the additional capitalneeded by the insurance in order to manage the risk, and hence the lower is the capital requirement Without profit load and expenses. Risk aversion has been here represented by means of the utility function u ( x ) = log( x )+1 , whose relative risk aversioncoefficient is increasing in x . igure 10: Optimal premiums per square metre for earthquake policies on masonry buildings.
Note : the map refers to the full coverage ( D = 0 , E = 1500 per square metre) policy, that has been estimated over N c =6404 . The minimum value reported is . and therefore yellow municipalities should be interpreted as approximately . The maximum premium is . . W ∗ d . This becomes clear when considering the ratio (cid:80) Nii =1 p ∗ i W ∗ d . Ratios for seismic policies span between . and . and indicate that the government and the homeowners almost equally contribute to theconstitution of reserves. On the other side, flood risk is much unfairly distributed between the twoagents with ratios [0 . , . .Evidence suggests that spatial correlation strongly affects the development of the private insurancemarket for both the two perils, but larger values of (cid:15) ∗ indicate that flood risk is even more difficult toinsure. A second level risk transfer (such as a reinsurance contract or a catastrophe linked securities)might help reducing (cid:15) by lowering the aggregate loss tail.To conclude, Figures 10 and 11 show annual optimal premiums per square metre p Hi for the mostvulnerable structural typology per each municipality for the two perils respectively. Since premiumsare risk-based, the two maps reflect the hazard component of risk modeling and hence report a patternsimilar to the maps on loss per square metre in Figure 6 and 8. In the previous section a public-private insurance model has been created for single peril policies.As well known in finance, merging portfolios of different risks is beneficial if risks are uncorrelated, asfloods and earthquakes are likely to be. It remains to be seen whether benefits from risk diversificationcounteract the negative impact of spatial correlation emerged in previous results. This section goesthrough what has been discussed so far and extends the analysis to multi-hazard.The first subsection is devoted to risk assessment, and therefore comes back to the risk-modelingintroduced in Section 2. While assessing single hazard risk is challenging, studying possible consequencesfrom several perils is even more complicated. Kappes et al. (2012) identifies two major issues raising in34 igure 11:
Optimal premiums per square metre for flood policies on one-storey buildings.
Note : the map refers to the full coverage policy ( D = 0 , E = 1500 per square metre), that has been estimated over N c = 7772 . a multi-hazard context: finding a common measure suitable to describing all the hazards considered,and understanding the relationship linking them. Regarding the former issue, since it is impossibleto find a geological or atmospheric indicator describing both flood and earthquakes, the two risk havebeen here assessed separately and compared in monetary terms only. The second point refers to thecorrelation between the two phenomena. Based on some empirical evidence in the literature, we arguethat the two risks are uncorrelated.To our knowledge, Marzocchi et al. (2012) is the only work addressing multi-hazard risk assessmentin Italy by studying seismic, volcanic, hydrogeological, flooding, landslide and industrial risk in themunicipality of Casalnuovo. However, this analysis is restricted to a municipality and has been appliedtherein to human and societal risk only, and does not pursue any insurance decision. A differentframework is therefore needed for our case study.The following subsection extends the insurance model in Section 3 to multi-hazard. The model isshaped by redefining supply and demand. In particular, we show that the maximum premium thatindividuals are willing to pay is equal to the sum of the premiums for the two single hazard policies,while the required amount of public capital is less than or equal to the sum necessary when managingthe risk separately.The last subsection presents results, which clearly show that benefit from risk diversification are notsufficient to override the effect of spatial correlation. However, additional positive externalities emerge:for example, premiums for multi-hazard policies are geographically more homogeneous with respect tothe single hazard’s, thus favoring the perception of fair-treatment among the population. As discussed in Section 2, four elements determine losses from natural events : H , E , V and L . Amathematical model for loss estimation should therefore be able to capture the relevant characteristics35f each component and describe those process that link them. Since every natural phenomenon hasspecific characteristics, studying the effects of multiple hazards furthermore complicates risk assessment.In particular, in our two-peril framework, H should encompass both floods and earthquakes andtherefore a common physical measure able to describe both the two perils should be identified. However,given the different characteristics of the two perils, this is impossible, and we are able to compare thetwo risks in money-value only. As a consequence, we are forced to assess the two risks separately,though this approach cannot capture the potential dependence among them.In our multi-hazard risk assessment, we refer to two hazard-indicators ζ s and ζ f , where apexes s and f indicate seismic and flood risks respectively. Each indicator is associated to a certain probabilityof occurrence F s ( ζ s ) and F f ( ζ f ) . Hence, hazard is described by both ζ h and F h ( ζ h ) . As we havepreviously shown, vulnerability functions are defined over a specific hazard-indicator, and their outputcan be easily converted into monetary terms. Referring to the definition of loss in Sections 2.2.4 and2.3.4, for simplicity we convey L and V in a unique function v h ( ζ h ) , with h = s, f . Expected losses persquare metre generated by a peril can hence be estimated as l h = F h ( ζ h ) v h ( ζ h ) E h , h = s, f. (98)As anticipated, multi-hazard expected loss might be affected by potential interactions of the two perils,and therefore we need some assumption on the degree of dependence between floods and earthquakes.Unfortunately, our database do not offer any information about if and how the two perils interact,but some empirical analysis in the literature (Tarvainen et al., 2006; Cesari and D’ Aurizio, 2019)support the hypothesis of independence between seismic and flood risks. However, various degreesof independence are also possible. Following the work of Brunette et al. (2015), we now discuss twopossible independence scenarios. First, we consider the hazards to be mutually exclusive, thus assumingthat floods and earthquakes cannot happen simultaneously and the structure can get damaged by oneperil only; we will refer to this case as “mutual exclusion scenario”. As an alternative, we consider perilsto be “mutually independent”, allowing them to potentially happen together. In this case, the propertymay be damaged by at least one of the two events. • Mutual exclusion scenario
If the two hazards are mutually exclusive the joint probability of an event F MHdep is obtained bysimply summing the single hazard probabilities F MHexc = F s ( ζ s ) + F f ( ζ f ) , (99)and we can compute expected losses per square metre as: l MHexc = F s ( ζ s ) v s ( ζ s ) + F f ( ζ f ) v f ( ζ f ) . (100)We can notice that in case of mutually exclusion the multi-hazard loss per square metre coincideswith the sum of the single hazards expected losses: l MHexc = l s + l f . (101) • Mutual independence scenario
Avoid now any dependence and allow the hazards to happen simultaneously. The joint occurrenceprobability F MHind becomes: F MHind = F s ( ζ s ) + F f ( ζ f ) − F s ( ζ s ) F f ( ζ f ) . (102)Expected losses now arise from flood, earthquakes or a combination of the two. When the twoevents happen together the damages suffered by the property are defined by a new vulnerability36unction v s + f ( ζ s , ζ f ) , therefore expected losses per square metre are obtained as: l MHind = (cid:104) F s ( ζ s ) − F s ( ζ s ) F f ( ζ f ) (cid:105) v s ( ζ s ) + (cid:104) F f ( ζ f ) − F s ( ζ s ) F f ( ζ f ) (cid:105) v f ( ζ f )++ F s ( ζ s ) F f ( ζ f ) v s + f ( ζ s , ζ f ) == F s ( ζ s ) v s ( ζ s ) + F f ( ζ f ) v f ( ζ f ) + F s ( ζ s ) F f ( ζ f ) (cid:104) v s + f ( ζ s , ζ f ) − v s ( ζ s ) − v f ( ζ f ) (cid:105) == l s + l f + F s ( ζ s ) F f ( ζ f ) (cid:104) v s + f ( ζ s , ζ f ) − v s ( ζ s ) − v f ( ζ f ) (cid:105) . (103)We can notice that l MHind > l
MHexc if v s + f ( ζ s , ζ f ) > v s ( ζ s ) + v f ( ζ f ) and this happens when theinteraction of the two events amplifies the damages they cause on the property. We are unable todefine the function v s + f ( ζ s , ζ f ) or to state whether it is smaller or greater than the sum of the twosingle hazard vulnerability functions. However, the low number of reported events suggests that theassociated probability F s ( ζ s ) F f ( ζ f ) is reasonably close to 0. Moreover, assuming the expected multi-hazard loss l MHind equal to l MHexc is a prudential assumption if v s + f ( ζ s , ζ f ) < v s ( ζ s ) + v f ( ζ f ) because itrequires the insurer to create slightly greater funds, thus effectively getting the probability of insolvencyand fund-refill lower than the required level. For these reasons, we estimate expected losses as: l MHind = l s + l f . (104) In Section 3 we have argued that premiums should meet the demand and that maximum rates thatindividuals are willing to pay pose a constraint to an insurance model. Similarly to the single-hazardpolicy, the demand constraint in a multi-hazard framework is therefore given by the equality: u MH not insured = u MH insured . (105)Given the assumption of independence between floods and earthquakes and the individual utility func-tions defined in Section 3, we can now address the multi-hazard purchase decision problem. We referto seismic events by means of the apex s and flood by f , and for simplicity individual loss l i,t areindicated by l si,t for earthquakes and l fi,t for floods. In addition, single hazard and multi-hazard policiesare specified by means of apexes as SH and M H .Given the probability of multiple events’ probabilities as defined in eq. (99) and losses computedas in eq. (101), the reimbursement function in eq. (15) becomes: x MH = , with probability π sc (0) + π fc (0) ,x s = x (cid:16) l si,t (cid:17) , with probability − π sc (0) , < x (cid:16) l si,t (cid:17) ≤ l si,t ,x f = x (cid:16) l fi,t (cid:17) , with probability − π fc (0) , < x (cid:16) l fi,t (cid:17) ≤ l fi,t , with i ∈ c, (106)with x h = x (cid:16) l hi,t (cid:17) = if l hi,t ≤ D,l hi,t − D if D < l hi,t < E,E − D if l hi,t ≥ E, h = s, f. (107)Hence, individual utilities of being and not being insured in eq. (17)-(18) for multi-hazard policies are: u MH not insured = (cid:104) π sc (0) + π fc (0) (cid:105) u ( RC ) + [1 − π sc (0)] u ( RC − l s ) + (cid:104) − π fc (0) (cid:105) u ( RC − l f ) (108)and u MH insured = (cid:104) π sc (0) + π fc (0) (cid:105) u ( RC − p MH ) + [1 − π sc (0)] u ( RC − p MH − l s + x s )++ (cid:104) − π fc (0) (cid:105) u ( RC − p MH − l f + x f ) . (109)37n Section 3.1 the maximum premium that an homeowner is willing to pay for a single hazard policy isthe quantity p SH solving the equality: u SH not insured = u SH insured . (110)with u SH not insured = π SHc (0) u ( RC ) + (cid:2) − π SHc (0) (cid:3) u ( RC − l SH ) (111)and u SH insured = π SHc (0) u ( RC − p SH ) + (cid:2) − π SHc (0) (cid:3) u ( RC − p SH − l SH + x SH ) , (112)for SH = f, s .Comparing M H and SH utilities, we get: u MH insured = u MH not insured = u s not insured + u f not insured = u s insured + u f insured . (113)This equality states that the homeowner utility of buying both the two single hazard policies is equal tothe utility of buying a multi-hazard one. However, when evaluating one peril per time, policies pricesare fixed by solving a consume decision with two options - to buy or not to buy the policy-, but amulti-hazard framework extends the range of possible choice: the individual may decide to buy a M H policy, both the SH policies, one out of the two SH , or neither of them. We know that if the policyis priced at p S H the individual is indifferent between buying or not the single-hazard policy, and eq.(113) states that the sum of the two utilities equals the utility of buying a M H one. We should theninvestigate the option of buying both the two single hazard policies ( s + f ): u s + f insured ≥ u s + f not insured (114) u s + f insured = (cid:104) π sc (0) + π fc (0) (cid:105) u ( RC − p s − p f ) + [1 − π sc (0)] u ( RC − p s − p f − l s + x s )++ (cid:104) − π fc (0) (cid:105) u ( RC − p s − p f − l f + x f ) (115)while u s + f not insured = (cid:104) π sc (0) + π fc (0) (cid:105) u ( RC )+[1 − π sc (0)] u ( RC − l s + x s )+ (cid:104) − π fc (0) (cid:105) u ( RC − l f + x f ) . (116)Note that the premium paid by the owner in this scenario is p s + f = p s + p f . Assuming consumer’sperfect rationality and neglecting any operational cost that a policy may generate, the individualprefers a multi-hazard policy to two single-hazard ones if p MH < p s + p f because it implies that u MH insured > u s + f insured. Therefore u s + f not insured = u MH not insured = u s not insured + u f not insured . (117)which in turn implies: u s + f insured = u MH insured = u s insured + u f insured . (118)Thus, the maximum premium that an individual is willing to pay for a multi-hazard policy makes himindifferent between any purchase choice and is equal to p MH,H = p s + p f . (119) Main differences in risk-pooling single- or multi- hazard policies are determined by the different loss,reimbursement and premium functions, that are now described by eq. (101), (106) - (107) and (119).We now construct the fund W MH for multi-hazard policies by extending the single-hazard model.The reader can find the extended description of the procedure in Subsection 3.2.38he multi-hazard fund at the beginning W MH,bt and at the end W MH,et of the year t , are now: W MH,bt = W MHt − + N i (cid:88) i =1 p MHi m i = W MHt − + N i (cid:88) i =1 (cid:16) p si + p fi (cid:17) m i with W MHt − = max( W MH,et − ; W MHd ) , (120)and W MH,et = W MH,bt − N i (cid:88) i =1 x MHi,t m i = W MH,bt − N i (cid:88) i =1 (cid:16) x si,t + x fi,t (cid:17) m i . (121)Assume that an earthquakes or a flood hits any building within a municipality and that every policycan generate at most one claim per hazard per year. Square metre expected losses l MHi,t are equal for allthe individuals within the same municipality and so does x MHi,t . Given the number of inhabited squaredmetres M c = (cid:80) i ∈ c m i , the total claims value per municipality is: (cid:88) i ∈ c x MHi,t m i = (cid:88) i ∈ c (cid:16) x si,t + x fi,t (cid:17) m i = (cid:16) X sc,t + X fc,t (cid:17) M c , (122)and the total national amount is Y MHt = N i (cid:88) i =1 x MHi,t m i = N c (cid:88) c =1 (cid:88) i ∈ c x MHi,t m i = N c (cid:88) c =1 (cid:16) X sc,t + X fc,t (cid:17) M c = Y st + Y ft , (123)and therefore is equal to the sum of the national claims for earthquakes Y st and floods Y ft computedby means of eq. (29). We model claim probabilities by means of Bernoulli variables ¯ X sc,t ∼ Ber ( q sc ) and ¯ X fc,t ∼ Ber ( q fc ) with q sc = π sc ( ζ s > ζ D ) and q fc = π fc (cid:0) ζ f > ζ D (cid:1) and apply equation (31): Y MHt = N c (cid:88) c =1 (cid:16) ¯ X sc,t a sc,t + ¯ X fc,t a fc,t (cid:17) . (124)Assuming municipalities that are at least km far each other to be independent, we can recall thesample that have been created for single hazard policies. Considering the two hazard separately, wewill have N g groups of municipalities’ seismic risks and other N g groups for floods. Each group willcontain n g municipalities: Y s,gt = (cid:88) c ∈ g ¯ X sc,t a sc,t c = 1 , . . . , n g (125) Y f,gt = (cid:88) c ∈ g ¯ X fc,t a fc,t c = 1 , . . . , n g (126)such that Y MHt = Y s, t + Y s, t + · · · + Y s,N g t + Y f, t + Y f, t + · · · + Y f,N g t . (127)Defining w g = n g N c , the expected total amount of claims in Italy is: E (cid:2) Y MHt (cid:3) = E (cid:2) Y MH (cid:3) = N g (cid:88) g =1 w g (cid:16) E [ Y s,gt ] + E (cid:104) Y f,gt (cid:105)(cid:17) , (128)with E [ Y s,gt ] and E (cid:104) Y f,gt (cid:105) computed as in (34).Applying the Hoeffding (1963) bound as in eq.(38)-(45), we get to the definition of both insolvency39robability and W d . We fix the insolvency probability (cid:15) : (cid:15) = e − h φ N g (cid:88) g =1 w g e − h ng E ( Y s,g + Y f,g ) E (cid:20) e h ng (cid:16) Y s,gt + Y f,gt (cid:17) (cid:21) == e − h φ N g (cid:88) g =1 w g e − h ng E ( Y s,g + Y f,g ) E (cid:20) e h ng ( Y s,gt ) e h ng (cid:16) Y f,gt (cid:17) (cid:21) == e − h φ N g (cid:88) g =1 w g e − h ng E ( Y s,g + Y f,g ) E (cid:20) e h ng ( ¯ X sc,t a sc,t ) e h ng (cid:16) ¯ X fc,t a fc,t (cid:17) (cid:21) , (129)and since seismic and flood risk are independent: (cid:15) = e − h φ N g (cid:88) g =1 w g e − h ng E ( Y s,g + Y f,g ) E (cid:20) e h ng ( ¯ X sc,t a sc,t ) (cid:21) E (cid:20) e h ng (cid:16) ¯ X fc,t a fc,t (cid:17) (cid:21) == e − h φ N g (cid:88) g =1 w g e − h ng ( E ( Y s,g )+ E ( Y f,g )) (cid:89) c ∈ g M ¯ X sc,t a sc,t (cid:18) h n g (cid:19) (cid:89) c ∈ g M ¯ X fc,t a fc,t (cid:18) h n g (cid:19) . (130)The minimum capital requirement for a multi-hazard public insurance is W MHd = N c φ + E [ Y ] − N i (cid:88) i =1 (cid:16) p si + p fi (cid:17) m i (131)with φ = 1 h log (cid:80) N g g =1 w g e − h ng ( E ( Y s,g )+ E ( Y f,g )) (cid:81) c ∈ g M ¯ X sc,t a sc,t (cid:16) h n g (cid:17) (cid:81) c ∈ g M ¯ X fc,t a fc,t (cid:16) h n g (cid:17) (cid:15) (132)The probability of fund-refill (cid:15) and the minimum amount of premiums (cid:80) N i i =1 p MH,Gi m i that the insurerneeds given a certain W d are obtained by applying the Hoeffding (1963) bound as in (47)-(53). Hence,fixing (cid:15) = e − h γ N g (cid:88) g =1 w g e − h ng ( E ( Y s,g )+ E ( Y f,g )) (cid:89) c ∈ g M ¯ X sc,t a sc,t (cid:18) h n g (cid:19) (cid:89) c ∈ g M ¯ X fc,t a fc,t (cid:18) h n g (cid:19) (133)we get N i (cid:88) i =1 p MH,Gi m i = N c γ + E [ Y ] (134)where γ is computed as: γ = 1 h log (cid:80) N g g =1 w g e − h ng ( E ( Y s,g )+ E ( Y f,g )) (cid:81) c ∈ g M ¯ X sc,t a sc,t (cid:16) h n g (cid:17) (cid:81) c ∈ g M ¯ X fc,t a fc,t (cid:16) h n g (cid:17) (cid:15) (135) The model for the definition of a public-private insurance scheme with multi-hazard policies can bedefined as in Section 3.3, therefore here we briefly extend the model to the multi-hazard scenario, butthe reader can refer to the previous Section for technical details.40he two fundamental conditions are now defined by equations (119) and (134). The optimal pre-mium p ∗ MHi is estimated as: p MH ∗ i = min( c, · p MH,Hi with c = (cid:80) N i i =1 p MH,Gi m i (cid:80) N i i =1 p MH,H m i , (136)from which we obtain N i (cid:88) i =1 p MH ∗ i m i = min (cid:18) c, c (cid:19) N c γ + E [ Y ] = N c γ ∗ + E [ Y ] , (137)and the optimal probability of fund-refill (cid:15) ∗ : (cid:15) ∗ = (cid:80) N g g =1 w g e − h ng ( E ( Y s,g )+ E ( Y f,g )) (cid:81) c ∈ g M ¯ X sc,t a sc,t (cid:16) h n g (cid:17) (cid:81) c ∈ g M ¯ X fc,t a fc,t (cid:16) h n g (cid:17) e h γ ∗ , (138)where γ ∗ is γ ∗ = min (cid:0) , c (cid:1) ( E [ Y ] − N c γ ) − E [ Y ] N c . (139)The optimal W MH ∗ d is estimated as in equation (58): W MH ∗ d = max (cid:40) N c φ + E [ Y ] − N i (cid:88) i =1 p MH ∗ i m i ; 0 (cid:41) = N c φ ∗ + E [ Y ] − N i (cid:88) i =1 p MH ∗ i m i == N c ( φ ∗ − γ ∗ ) , (140)with φ ∗ = W ∗ d + (cid:80) N i i =1 p MH ∗ i m i − E [ Y ] N c , (141)and the optimal value (cid:15) ∗ is: (cid:15) ∗ = (cid:80) N g g =1 w g e − h ng ( E ( Y s,g )+ E ( Y f,g )) (cid:81) c ∈ g M ¯ X sc,t a sc,t (cid:16) h n g (cid:17) (cid:81) c ∈ g M ¯ X fc,t a fc,t (cid:16) h n g (cid:17) e h γ ∗ . (142)As in the single-hazard scenario, some distributional assumptions are needed in order to solve themodel. We keep the assumptions as in Section 3.4.3, and therefore we represent Y t as a weightedsum of Bernoulli random variables. We assume that the properties within a municipality are perfectlycorrelated. Hence, the Hoeffding (1963) bound simplifies and the probabilities (cid:15) ∗ and (cid:15) ∗ become: (cid:15) ∗ = N g (cid:88) g =1 w g e − φ ∗ n gb g (143)and (cid:15) ∗ = N g (cid:88) g =1 w g e − γ ∗ n gb g , (144)where b g = (cid:88) c ∈ g a sc + a fc . (145)41 able 12: Multi-hazard public-private insurance scheme. (cid:15) = 0 . , (cid:15) = 0 . Deductible Maximum coverage (cid:80) N i i =1 p ∗ i c W ∗ d (cid:15) ∗ (cid:15) ∗ (per square metre) (Mln) (Mln)0 1500 MH 11185.123 1.694 13281.008 0.010 0.091(0.000) (0.024) (0.068) (0.000) (0.029)S 10356.859 1.424 8194.243 0.010 0.063(0.000) (0.022) (0.080) (0.000) (0.035)F 828.264 6.346 7359.961 0.010 0.403(0.000) (0.036) (0.044) (0.000) (0.029)0 1200 MH 10046.430 1.852 13983.445 0.010 0.120(0.000) (0.024) (0.063) (0.000) (0.025)S 9378.433 1.537 8760.141 0.010 0.082(0.000) (0.022) (0.074) (0.000) (0.030)F 667.997 7.820 7469.278 0.010 0.427(0.000) (0.036) (0.043) (0.000) (0.028)200 1500 MH 9335.925 1.919 13879.044 0.010 0.132(0.000) (0.024) (0.061) (0.000) (0.024)S 8507.814 1.612 8766.645 0.010 0.097(0.000) (0.022) (0.069) (0.000) (0.027)F 828.110 6.305 7305.976 0.010 0.396(0.000) (0.036) (0.044) (0.000) (0.029)200 1200 MH 8575.042 2.047 14183.694 0.010 0.163(0.000) (0.024) (0.058) (0.000) (0.022)S 7907.199 1.692 8945.297 0.010 0.114(0.000) (0.022) (0.066) (0.000) (0.024)F 667.843 7.756 7401.415 0.010 0.435(0.000) (0.036) (0.043) (0.000) (0.028) Note : the Table shows multi-hazard (MH), seismic (S) and flood (F) insurance for the Italian residential building stock.Results have been estimated over samplings on N c = 6217 municipalities for which data were fully available forboth flood and earthquakes. Policies are defined on deductible and maximum coverage and listed by row, while columnsrepresent the model’s relevant variables. Reported values are mean and coefficient of variation. For multi-hazard analysis, only municipalities where both seismic and flood data are available havebeen considered, thereby restricting the database to N c = 6217 .As in Section 3.5, municipalities have been assumed independent if centroids are at least 50 km farand 100 samplings have been considered for final results. The four policies considered for single hazardpolicies have also been estimated for multi-hazard: (i) a full coverage policy ( D = 0 , E = 1500 ); (ii)one with a maximum coverage equal to 1200 per square metre ( D = 0 , E = 1200 ); (iii) one with adeductible equal to 200 ( D = 200 , E = 1500 ); (iv) a policy with both the maximum coverage and thedeductible ( D = 200 , E = 1200 ). Initial preferences have been again fixed to (cid:15) = 0 . and (cid:15) = 0 . .Results are presented in Table 12 together with the corresponding single hazard policies, thathave been re-estimated on the restricted number of municipalities for the sake of comparability. Asexpected, seismic risk dominates the multi-hazard scenario because of the consistently higher impacton the national territory. In particular, since N i (cid:88) i =1 p MH ∗ i = N i (cid:88) i =1 p s ∗ i + N i (cid:88) i =1 p f ∗ i , (146)we can notice that multi-hazard premiums amount (cid:80) N i i =1 p ∗ i is mostly determined by seismic risk andjust a small portion of it is due to floods. Though premiums for the two single hazard policies are42 able 13: Optimal multi-hazard premiums per square metre.
Deductible0 2001s 2s 3s 1s 2s 3s M a x i m u m c o v e r ag e ( p e r s q u a r e m e tr e ) RC.gl min 0.644 0.592 0.591 0.051 0.051 0.051mean 7.541 6.730 6.719 5.322 4.512 4.500max 32.261 32.261 32.261 30.471 30.471 30.471RC.sl min 0.038 0.034 0.034 0.012 0.008 0.008mean 2.932 2.122 2.110 2.195 1.384 1.373max 16.951 10.679 10.638 16.935 9.277 9.248A.gl min 0.036 0.036 0.036 0.021 0.021 0.021mean 2.849 2.038 2.027 2.478 1.667 1.656max 19.136 10.793 10.730 19.111 9.625 9.595A.sl min 0.021 0.021 0.021 0.015 0.015 0.015mean 2.684 1.873 1.862 2.221 1.410 1.399max 18.860 10.283 10.272 18.614 8.165 8.136M min 0.092 0.092 0.092 0.058 0.058 0.058mean 5.383 4.573 4.561 4.898 4.088 4.077max 50.428 50.218 50.215 31.387 31.387 31.387
RC.gl min 0.644 0.589 0.588 0.038 0.035 0.035mean 7.315 6.666 6.656 4.818 4.169 4.159max 32.261 32.261 32.261 30.471 30.471 30.471RC.sl min 0.037 0.034 0.034 0.011 0.008 0.008mean 2.428 1.779 1.770 2.133 1.484 1.474max 13.964 10.583 10.549 13.910 8.962 8.938A.gl min 0.031 0.028 0.028 0.011 0.008 0.008mean 2.469 1.820 1.811 2.175 1.526 1.517max 15.046 10.648 10.596 14.109 9.339 9.315A.sl min 0.015 0.012 0.012 0.015 0.015 0.015mean 2.286 1.637 1.627 1.958 1.308 1.299max 13.952 10.258 10.250 13.608 7.985 7.960M min 0.068 0.063 0.062 0.045 0.042 0.041mean 4.641 3.992 3.983 4.275 3.626 3.616max 40.891 40.168 40.157 30.926 30.926 30.926
Note : the table is divided in four sub-tables, each representing a specific combination of deductible (by column) andmaximum coverage (by row). Each sub-table presents minimum, average and maximum premium at the municipal levelper each combination of structural typology (row) and number of storeys (column). extremely different, the corresponding minimum capital requirement W ∗ d is similar (see Section 3.5),and for the multi-hazard policy W MH ∗ d ≤ W s ∗ d + W f ∗ d . (147)Therefore, multi-hazard policies need for less public capital than managing the two hazards separately,and this finding is attributable to risk differentiation.However, advantages from multi-hazard are evident with respect to flood risk, but a bit controversialwhen we look at the seismic risk. The multi-hazard parameter c is a bit greater than that of the seismiccase and much smaller than in the flood case but, unfortunately, is always c ≤ . Our analysis suggeststhat benefits from risk differentiation are not sufficient for the natural risks to be entirely managed bythe private market and once again, a government intervention is highly recommended. This evidenceis confirmed by the probability (cid:15) ∗ , that shows a behaviour similar to c and is always greater than thedesired level, and (cid:15) ∗ = (cid:15) .As far as coverage limits concern, the minimum amount of public funds W ∗ d and the minimum43 igure 12: Optimal multi-hazard premium per square metre for one storey masonry buildings.
Note : the map represents a full coverage ( D = 0 , E = 1500 per square metre) policy, that has been estimated on N c =6217. The minimum value represented is . and therefore yellow municipalities should be interpreted as approximately . The maximum premium reported is . . probability (cid:15) ∗ are obtained with the full coverage policy, while applying a deductible D = 200 or amaximum coverage E = 1200 lead to similar results. In any case, the worst solution would be applyingthe two limits together, since both the greatest W ∗ d and the highest (cid:15) ∗ are here obtained.In addition to benefits from risk differentiation, a government may prefer multi-hazard policies foranother interesting feature: risk-based premiums are much more geographically uniform than those ofsingle hazards. In fact, Figure 12 mapping premiums for the most vulnerable buildings (masonry-onestorey) shows a quite homogeneous price at the municipal level, while differences are a bit more empha-sised in the corresponding single hazard (see Figures 10 and 11). From the public sector perspective,a uniform rating system is desirable because it weakens the perception of unequal treatment betweenthe property-owners from different areas and therefore allows easier acceptance by the population. Onthe other hand, different risk-based premiums signal the riskiness of the area to its inhabitants andis therefore important to discourage the construction of most vulnerable housing structures and toencourage preventive behavior. The current rating also preserves this desirable feature since premiumsare defined on structural typologies among which rates substantially vary (see Table 13). Seismic and flood risks in Italy have been analysed. Given the limited amount of data available onnatural risks, an alternative approach based on risk-modeling has been applied to estimate expectedmonetary losses. We found that seismic risk results in the highest expected losses at national level, butfloods may generate the highest losses per square metre. The two perils differ in geographic extent:while the seismic risk is relevant for almost all the national territory, floods affect a limited area.In order to cope with the effects of natural risks, a public-private insurance scheme has been pro-44osed. Our insurance model is intended to alleviate the financial burden that natural events placeon governments, while at the same time assisting individuals and protecting the insurance business.Therefore, in our model, propertyowners, an insurer and the government co-operate in risk financing.Though expected losses generated by floods and earthquakes are considerably different, we found thatthe amount of public funds needed to manage the two perils is almost the same. We argue that thisevidence is generated by a combination of individuals’ increasing risk aversion and hazard loss distri-butions. Our analysis also shows that the amount of public capital necessary for risk financing canbe reduced by jointly managing the two risks. Along with this benefit from risk differentiation, multi-hazard policies allow the insurer to apply rates that are more geographically homogeneous, thereforefavoring the perception of fair treatment among the population.Unfortunately, our results show that neither single- or multi-hazard policies are sustainable by theprivate market alone: due to spatial correlation among insured assets, the maximum premiums thatindividuals are willing to pay do not meet the insurer’s solvency or capital constraints for any policyconsidered. Without the government, a private insurer would be forced to drive up premiums, whichwould not meet the demand and would therefore not be purchased. This evidence suggests the needfor the government to intervene in the insurance market for natural disasters.To conclude, our results show that the probability of the government having to inject further capitalmay be moderate. Though the insurance scheme reduces the government’s financial burden due tonatural perils with respect to the current state, adding some layer of risk transfer might be beneficial.For example, CatBonds or some level of reinsurance may reduce losses suffered by the government andtheir volatility.
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