Prevention and insurance of natural disasters
Arnaud Goussebaile

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Arnaud Goussebaile. Prevention and insurance of natural disasters. Economics and Finance. Université Paris Saclay (COmUE), 2016. English. �NNT : 2016SACLX014�. �tel-01476867�

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THÈSEpour l’obtention du grade de
DOCTEUR DE L’UNIVERSITÉ PARIS SACLAY
préparée à
L’ÉCOLE POLYTECHNIQUE
École doctorale no 578: Sciences de l’Homme et de la Société
Spécialité: Sciences Économiques
Par
ARNAUD GOUSSEBAÏLEPREVENTION AND INSURANCE OF NATURAL DISASTERS
Prévention et assurance des catastrophes naturelles

Présentée et soutenue publiquement
le 23 Mai 2016 à Palaiseau

Composition du Jury:
Jean-Marc Bourgeon

Professeur, École Polytechnique

Arthur Charpentier

Professeur, Université du Québec à Montréal Rapporteur

Georges Dionne

Professeur, HEC Montréal

Rapporteur

Erwann Michel-Kerjan Professeur, Wharton School

Examinateur

Pierre Picard

Président du Jury

Professeur, École Polytechnique

Directeur de thèseRemerciements
Je voudrais tout d’abord remercier Jean-Marc Bourgeon d’avoir accepté de diriger
ma thèse. Tout au long de mon doctorat il m’a laissé un grand espace de liberté
dans mon travail de recherche tout en étant toujours présent pour me conseiller.
Son exigence académique a été très formatrice pour moi et m’a permis d’acquérir
un solide bagage intellectuel en sciences économiques, très complémentaire de ma
formation initiale en sciences physiques.
Je veux aussi remercier Arthur Charpentier, Georges Dionne, Erwann MichelKerjan et Pierre Picard pour m’avoir fait l’honneur d’accepter de participer à monjury de thèse. Leurs nombreux travaux de recherche de portée internationale sur laprévention et l’assurance des catastrophes naturelles m’ont guidé dans mon travailde thèse. Je leur suis très reconnaissant du temps qu’ils m’ont consacré; leurs
retours sur mes travaux m’ont permis de les faire progresser et m’ont ouvert desperspectives stimulantes sur les suites à mener.
Je tiens aussi à remercier en particulier Alexis Louaas pour les échanges constructifs que nous avons eu pendant nos doctorats respectifs.

L’ouverture etl’enthousiasme d’Alexis nous ont permis de collaborer fructueusement sur nos centres d’intérêt communs et d’aboutir à des résultats de recherche développés dansla deuxième partie de cette thèse.
J’adresse également tous mes remerciements aux membres du Départementd’Économie de l’École Polytechnique avec qui j’ai eu des échanges enrichissantstant sur le plan académique que sur le plan humain. Je pense notamment à
Pierre Boyer, Yukio Koriyama, Jean-Baptiste Michau et Francisco Ruiz-Aliseda.
iRemerciements
Je pense aussi à l’ensemble des doctorants avec qui j’ai partagé de nombreux
bons moments, avec une mention spéciale pour Esther Delbourg mais aussi JeanneCommault, Gwenael Roudaut et Robert Somogyi. Je pense enfin aux membres del’administration du département, en particulier Eliane Nitiga-Madelaine et Lyza
Racon.
J’ai eu aussi la chance au cours de mon doctorat de rencontrer des chercheursd’autres universités et d’avoir des échanges fructueux avec eux. Je les remercie
pour ces échanges et les retours constructifs qu’ils m’ont fait sur mes travaux. Je
pense notamment à Scott Barrett de Columbia University, David Frame de Rutgers
University, Denis Gromb de HEC Paris, Robert Mendelsohn de Yale University,
Christian Traeger de Berkeley University, Nicolas Treich de l’École d’Économie deToulouse et Bertrand Villeneuve de l’Université Paris-Dauphine.
Le Département d’Économie de l’École Polytechnique m’a accueilli pour effectuer cette thèse. Elle a été financée par l’École Normale Supérieure ainsi que parl’École Polytechnique, notamment via le Labex Ecodec, la Chaire Développement
Durable et l’École Doctorale. Le Laboratoire d’Économie Publique de l’INRA a
aussi contribué à cette thèse en finançant des projets et en m’accueillant dans ses
locaux. Le Department of Agricultural and Resource Economics de l’Universitéde Berkeley et la School of International and Public Affairs de l’Université deColumbia m’ont invité chacun un semestre par l’intermédiaire respectivement deBrian Wright et de John Mutter. Je suis reconnaissant à toutes ces Écoles etUniversités de la confiance et de l’aide matérielle et/ou financière qu’elles m’ont
accordé.
Je remercie enfin mes proches pour toute leur affection, avec une pensée particulière pour mes parents Jean et Patricia, mon frère Thibaut et ma soeur Larapour toutes leurs attentions depuis tant d’années.

ii

RésuméCes trente dernières années, l’ensemble des pertes économiques engendrées parles catastrophes naturelles a crû à un rythme plus soutenu que la richesse mondiale. Cette évolution s’explique par l’accroissement de la population dans lesrégions exposées aux catastrophes naturelles, ainsi que par la faiblesse des actions de prévention mises en place dans ces régions. Le changement climatiquelaisse présager une accélération de cette évolution avec des risques accrûs de tempêtes, d’inondations et de sécheresses entre autres. Par ailleurs, les catastrophes
naturelles sont faiblement assurées puisque seulement un tiers environ des pertesmondiales sont aujourd’hui couvertes. Bien que la part assurée ait augmenté aucours des dernières décennies, le niveau moyen de couverture reste encore trop bas
pour limiter la variabilité de la richesse des populations concernées. C’est pourquoi
la réduction des pertes engendrées par les catastrophes naturelles et l’accroissement
des couvertures d’assurance sont des enjeux majeurs.
Les thèmes de cette thèse sont la prévention et l’assurance des catastrophes
naturelles dont les faibles niveaux actuels sont dus aux nombreuses imperfections
de marché mais aussi aux déficiences des politiques publiques, comme l’expliquele chapitre introductif de ce travail. En s’appuyant sur la modélisation des comportements individuels, des marchés et des politiques publiques, cette thèse a pourobjectif d’étudier quels sont les actions de prévention et les mécanismes d’assurancequi permettraient de diminuer efficacement les pertes et aussi la variabilité de larichesse pour les agents averses au risque.
Le chapitre 1 porte sur les choix de prévention dans le contexte du développeiii

Résumément des villes. Il résulte du modèle d’économie urbaine développé dans ce chapitre
que les zones risquées sont plus développées près du centre-ville que loin du centreville et que l’investissement dans la résilience des bâtiments mène à des villes plus
compactes. Ce modèle met aussi en évidence que subventionner les assurances
entraîne une exposition excessive aux risques en augmentant la densité dans leszones les plus risquées et en abaissant les efforts de prévention. Cette analyse
illustre les effets pervers des subventions en ce domaine et le rôle que doivent jouer
les politiques publiques urbaines telles que les restrictions de densité ou les codes
de construction pour restreindre ces effets pervers.
Les chapitres suivants abordent la problématique des mécanismes d’assurance
lorsque les risques individuels ne sont pas indépendants. En effet, une des caractéristiques majeures des catastrophes naturelles est qu’elles affectent simultanément de nombreux agents économiques.
Le chapitre 2 s’attache à développer un modèle d’équilibre général où les agentséconomiques sont averses au risque et sont exposés à des risques individuels potentiellement dépendants. Il y est établi que, sans imperfection de marché, uneallocation Pareto-optimale des risques est atteinte en présence d’un marché compétitif de compagnies d’assurance et d’un nombre restreint d’actifs financiers. Cerésultat est valide sous réserve que la responsabilité des agents économiques dansleurs engagements soit illimitée. En pratique, leur responsabilité est limitée et lespolitiques publiques requièrent que des réserves financières soient constituées pourlimiter les défauts de paiement dans les états catastrophiques. C’est pourquoi leschapitres 3 et 4 abordent la question du coût des réserves financières et de leurimpact sur la demande de couverture et sur la forme optimale des contrats qui en
découle.
Le chapitre 3 étudie l’impact du coût des réserves financières sur le prix del’assurance et le taux de couverture demandé par des agents averses au risque. Si
le prix de l’assurance est composé d’une prime correspondant au risque individuel
et d’une surprime correspondant au coût des réserves financières (qui permettent àl’assureur de faire face au risque agrégé des assurés), il apparaît qu’à risque collectifiv

Résumédonné, le poids de la surprime est d’autant plus fort dans le prix de l’assurance
que la probabilité individuelle d’être sinistré est faible. Dans ces conditions, il estétabli qu’à risque collectif donné, le taux de couverture demandé par les agentsaverses au risque décroît quand la probabilité individuelle décroît.
Le chapitre 4 analyse la forme optimale des contrats d’assurance pour une
communauté d’agents qui sont averses au risque et qui sont exposés à des risquesindividuels corrélés. S’il n’est pas coûteux pour la communauté de constituer lesréserves nécessaires, le contrat optimal pour un risque individuel donné consisteen une couverture totale, quelques soient les pertes collectives, à laquelle s’ajoute
un dividende qui permet de redistribuer le cas échéant les réserves non utilisées.
Dans le cas contraire, le contrat optimal pour un risque individuel donné consisteen une couverture seulement partielle quand les pertes collectives sont élevées.

v

RésuméviContents
Introduction

1

1 Risk prevention in cities prone to natural hazards

9

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2

Risk-prone city model . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3

Risk-prone city development . . . . . . . . . . . . . . . . . . . . . . 19

1.4

The impact of insurance subsidization . . . . . . . . . . . . . . . . . 24

1.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 The role of insurance companies in a risky economy

31

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2

The model of the risky economy . . . . . . . . . . . . . . . . . . . . 36

2.3

Pareto optimality and Arrow-Debreu economy . . . . . . . . . . . . 38

2.4

Decentralized equilibrium with insurance companies . . . . . . . . . 42

2.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Insurability of low-probability catastrophic risks

49

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2

A model with aggregate loss uncertainty . . . . . . . . . . . . . . . 52

3.3

Insurability without aggregate uncertainty . . . . . . . . . . . . . . 54

3.4

Insurability with aggregate uncertainty . . . . . . . . . . . . . . . . 62

3.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
vii

Contents
4 Pooling natural disaster risks in a community

73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Caribbean countries and natural disasters insurance . . . . . . . . . 77
4.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Optimal insurance and reinsurance . . . . . . . . . . . . . . . . . . 85
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Conclusion

103

Bibliography

105

viii

Introduction
People and assets are exposed to natural risks which have weather, geological,
biological or spatial origin. For instance, weather1 processes can lead to storms,
floods, droughts and extreme temperatures; geological mechanisms can generate
volcanic eruptions and earthquakes with potential tsunamis; while biological risks
involve epidemics and spatial risks include asteroids. These natural phenomena
are called natural disasters when they have significant human or economic consequences within affected regions.2 Depending on the socio-economic context, natural disasters may affect population from a few people to hundred millions of people
and generate a variety of economic damage, ranging from thousands to over hundred billions of dollars. The worst natural disasters in terms of affected people are
weather-related in densely inhabited regions of Asia, in particular large droughts
in India and large floods in China. The worst natural disasters in terms of economic losses have both geological and weather origins and are more widespread in
the world. The costliest event is the 2011 earthquake which generated the tsunami
hitting the Fukushima nuclear power plant in Japan (210 billion dollars of losses).
The second costliest is hurricane Katrina which stroke the region of New Orleansin United States in 2005 (125 billion dollars of losses). Figure 1 illustrates the 15
worst natural disasters in terms of economic losses3 , in the world up to now.
1

"Weather" is used here in a wide sense which includes meteorological, hydrological and cli-

matological phenonema.
2
Information and data on natural disasters in the world can be found on the EM-DAT websiteof the International Disaster Database (http://www.emdat.be/).
3
The economic losses shown on the different graphs include only direct losses, such as thedestruction of assets. Indirect losses, such as the loss of production due to the destruction of
productive assets, are not included.

1

Introduction

Figure 1: Worst natural disasters in terms of economic losses (Source: EM-DAT
database).

World economic losses due to natural disasters have increased exponentiallyin the last three decades and a major part of these losses are weather-related, as
shown by figure 2. For weather-related natural disasters, the real annual growthrate of losses has been of 4.1% in average since 1980, which is 1.1% above the world
GDP growth rate (Aon Benfield, 2014). The fast increase of losses can be explainedby increasing population in risky areas and low prevention measures in those same
regions. In particular, nearly half of the world population lives nowadays within
150km of an ocean coastline and can be subject to floods, as it happened in 2005
with hurricane Katrina in New Orleans or more recently in 2012 with hurricane
Sandy in New York. Beyond relocation in safer areas, prevention actions canconsist for example for flooding risks in elevating or waterproofing structures orin building dams or levees. With rising sea levels, more severe rainfall patterns
and higher temperature due to climate change, the trend of losses due to weatherrelated natural disasters is even expected to worsen in the coming years (IPCC,
2014). In this perspective, reducing natural disaster losses through prevention
actions has become a main challenge for our societies.
2

Introduction

Figure 2: World total losses due to natural disasters (weather-related or not)(Source: EM-DAT database)Moreover, losses due to natural disasters are weakly insured in the world, as
illustrated by figure 3 for weather-related natural disasters. Even though insured
losses have increased faster than total losses with an average annual rate of 7.7%
since 1980, insured losses still represent only a third of total losses (Aon Benfield,2014). In a highly developed country such as the USA, only 45% of households
currently have insurance for weather-related risks. The low penetration of insurance can generate undesirable wealth fluctuation for affected population. In thiscontext, increasing insurance coverage for natural disaster risks is another main
challenge.
The present thesis addresses prevention and insurance issues relative to natural
disaster risks. Prevention and insurance are valuable when their costs of implementation are lower than their benefits obtained through the decrease of losses
and wealth variability for risk-averse agents. In an ideal decentralized world, theoptimal level of prevention would be set by agents (such as households and firms)
and the optimal level of insurance would be purchased by them through efficient3

Introduction

Figure 3: World total and insured losses due to weather-related natural disasters(Source: AON Benfield (2014)).
insurance markets, without public intervention.4 However, there are market imperfections for natural disaster risks and public intervention has a role to play inthis perspective (Gollier, 2005; Kunreuther & Michel-Kerjan, 2009). For instance,some preventive actions can be done at an individual level, such as elevating houses
or making them waterproof, but others have to be done at a collective level, suchas building dams or levees. The public good nature of these prevention actionsor the externalities generated by them justify public intervention. There are also
information issues related to natural disaster risks. Agents can be poorly informed
about risks and public intervention can play a role to acquire and transfer information to them in order to improve behaviors both in terms of prevention andinsurance. Agents can also lack information on the capacity of insurers to pay
claims and public intervention can play a role to constraint insurers to secure
capital and pay claims in catastrophic states. Beyond efficiency issues, public intervention is concerned with redistribution issues. The lack of insurance coverage is
4

The optimal level of prevention corresponds to the marginal cost-benefit tradeoff for a preven-tion measure and the optimal level of insurance corresponds to the full elimination of individual
risks and the share of collective risks.

4Introduction
often partly compensated by insurance subsidization and public relief. In the end,
low current levels of prevention measures and insurance coverage can be explainedby numerous market imperfections and poorly-designed public policies. Throughmodeling of individual behaviors, markets and public policies, this thesis aims at
characterizing prevention actions and insurance mechanisms that could mitigate
efficiently losses and wealth variability for risk averse agents. Because risk prevention choices and risk sharing mechanisms interact, it is essential to analyze them
together (Dionne, 2013). Indeed, insurance prices affect the preventive behaviorof risk exposed agents and prevention measures affect the level of required capitalfor insurance coverage.
The first part of my thesis, corresponding to chapter 1, deals with the impact ofrisk sharing policies on preventive behaviors. The underestimation of risks and the
expectancy of public relief lead risk exposed agents to underinsure and underinvest
in prevention (Kunreuther, 1984, 1996; Raschky & Weck-Hannemann, 2007). In
this perspective, many governments have implemented public policies to increaserisk mutualisation and decrease distress after natural disasters. For instance, public reliefs have been implemented in Canada and Germany, while public insurance
systems with subsidization have been implemented in the USA (National Flood Insurance Program) and France (CatNat program) (Grislain-Letrémy & Peinturier,
2010; Michel-Kerjan, 2010). Even though these policies can partly improve risk
sharing, they lead to even less prevention measures because of the lack of incentives(Courbage et al., 2013). This has been observed for example in the USA with the
NFIP (Bagstad et al., 2007; Browne & Hoyt, 2000). To increase insurance purchaseand prevention measures, more constraining policies can also be implemented. For
instance, the purchase of insurance can be made mandatory to have access tocredit and prevention measures can be enforced through urban policies such as
zoning restrictions and building codes. However, in the USA, insurance contracts
required for access to credit are withdrawn after a few years and constraining urban policies are still weakly implemented (Michel-Kerjan et al., 2012; Kunreuther& Michel-Kerjan, 2013), which confirms that insurance and prevention policies
still have to be improved.
5

Introduction
Chapter 1 of my thesis, entitled Risk prevention in cities prone to natural hazards, analyzes how insurance policies affect preventive choices in terms of location
and resilience in the context of urban development in risk exposed regions. Cities
located in regions prone to natural hazards such as flooding are not uniformly exposed to risks because of sub-city local characteristics. Spatial heterogeneity thus
raises the issue of how these cities have spread and should continue to develop.
Chapter 1 investigates these questions by featuring an urban model in which eachlocation is characterized by a transport cost to the city center and a risk exposure. At market equilibrium, riskier areas are developed nearer to the city center
than further away and investment in building resilience leads to more compact
cities. At a given distance to the city center, riskier areas have lower land prices
and get lower household density and higher building resilience. Actuarially fairinsurance generates optimal density and resilience, while an increase of insurance
subsidization leads to an increase of density in the riskiest areas and a generaldecrease of resilience. This analysis highlights the limits of insurance subsidization in the development of cities and addresses the role of public policy in termsof urban development, such as density restrictions or building codes, to limit risk
over-exposure.
The second part of my thesis, which includes chapters 2, 3 and 4, deals withrisk correlation issues and the ability of insurers to pay claims. Because natural disasters often have wide spatial impacts with tremendous losses, insurers arethemselves exposed to risks. To limit the default of insurers in paying claims incatastrophic states, insurers are required to secure high levels of costly financial
reserve. For natural disaster risks, the main issue for the supply of insurance
thus rests on reinsurance and capital market imperfections (Froot, 2001; Jaffee& Russell, 1997). Because insurers have to secure costly capital to face uncertain catastrophic losses, insurance prices can be high and subject to fluctuations
through time (Bernard, 2013; Weiss, 2007; Winter, 1994). Following years withhigh losses, private insurers increase their premiums to lower their liabilities andrebuild their reserves because purchasing high level of external capital throughreinsurance or capital markets would be costlier. The premium increase was par6

Introduction
ticularly strong at the beginning of the nineties because of hurricane Andrew andNorthridge earthquake respectively hitting Florida in 1992 and California in 1994
(Kousky, 2010). In this context, market-based solutions have emerged throughinsurance-linked securities such as Cat-bonds which give a better access to capital to insurers and reinsurers (Cummins, 2006; Cummins & Barrieu, 2013). Also,
public intervention can play a role to improve weak private insurance supply (Charpentier & Le Maux, 2014). For instance, after hurricane Andrew and Northridge
earthquake, Florida and California implemented public reinsurance through theFlorida Hurricane Catastrophe Fund and the California Earthquake Authority.
Chapter 2 of my thesis, entitled The role of insurance companies in a risky economy, analyzes the role of insurance companies when individual risks are potentiallycorrelated. In a classic Arrow-Debreu economy, complete financial markets consist
in one financial asset per state of nature, which allows a Pareto optimal allocation.
Yet this setting requires a prohibitive number of financial assets. This limit finds
its solution in the emergence of insurance companies. Chapter 2 analyzes theirrole in a model with multiple commodities and agents having different preferencesand distributions of endowments, with potential risk dependence across agents. In
this chapter and in the following chapters, it is assumed that individuals are riskaverse, rational and well-informed about risks. Pareto optimality is reached withstock insurance companies in competition plus one financial asset for every stateof nature corresponding to the same aggregate endowments. In this case, agentscan fully cover their endowment risks thanks to fair multi-risk contracts supplied
by insurance companies. For a given endowment risk, the higher the correlation
with the aggregate risk, the higher the premium. Also, agents can choose theirshare of the aggregate risk through the insurance stock shares and the financial
assets. The latter allow agents to hedge commodity price risks due to the aggregate risk. In this setting without market imperfections, individual risks are fully
covered even though there might be risk correlation. However, because of liability
issues, insurers are required in practice to secure capital. In this case, if capital iscostly to secure, correlated individual risks are more difficult to insure, which isanalyzed in the following chapters.
7

Introduction
Chapter 3 of my thesis, entitled Insurability of low-probability catastrophic risks
and co-authored with Alexis Louaas, analyzes how the probability of a risk affects
the purchase of insurance by risk-exposed individuals. With standard insurance
costs and competitive pricing, agents are more inclined to insure for low-probability
risks than for high-probability risks. Yet, these observations are at odds with the
low insurance take-up rates for low-probability catastrophic risks. The explanationis that the risks for which underinsurance is most prevalent display substantial
aggregate uncertainty. This uncertainty generates an additional fixed cost for
insurers due to required financial reserves, which increases the insurance loading
factor when the loss probability decreases and eventually discourages people frompurchasing coverage. The analysis thus explains why some low-probability risks
such as damages from lightnings are efficiently handled by the insurance sector
whereas others, such as earthquakes or floods, are not.
Chapter 4 of my thesis, entitled Pooling natural disaster risks in a communityand co-authored with Alexis Louaas, examines the optimal design of insurance
contracts when individual risks are correlated across risk-averse agents in a community. The community is equipped with a public insurer which supplies insurance contracts to its members and has access to costly reinsurance outside thecommunity. Without transaction costs inside the community, risk-averse agents
fully insure against their individual risk and share collective risk by getting some
dividend in normal states. With premiums raised ex-ante and generating an opportunity cost, they only partially insure against their individual risk, getting alower indemnity in catastrophic states than in normal states, and potentially getsome dividend in normal states. We illustrate the emergence of the latter contracts
for the community of the Caribbean countries exposed to natural disaster risks.

8Chapter 1
Risk prevention in cities prone tonatural hazards
Abstract: Cities located in regions prone to natural hazards such as flooding are
not uniformly exposed to risks because of sub-city local characteristics (e.g. topography). Spatial heterogeneity thus raises the issue of how these cities have spreadand should continue to develop. The current paper investigates these questions
by using an urban model in which each location is characterized by a transport
cost to the city center and a risk exposure. Riskier areas are developed nearer tothe city center than further away. Investment in building resilience leads to morecompact cities. At a given distance to the city center, riskier areas have lower land
prices and get lower household density and higher building resilience. Actuariallyfair insurance generates optimal density and resilience. An increase of insurance
subsidization leads to an increase of density in the riskiest areas and a generaldecrease of resilience. In this case density restrictions and building codes have to
be enforced to limit risk over-exposure.
Keywords: natural disaster risks, city development, insurance, prevention, urban density, building resilience.
JEL classification: Q54, O18, G22, R52, H23.
9

Chapter 1. Risk prevention in cities prone to natural hazards

1.1

Introduction

In October 2012, hurricane Sandy hit the East Coast of the USA, killing 54 people
and generating more than 50 billion dollars of losses.1 The damage was tremendous
in Greater New York: 17% of the city was flooded and 150,000 homes were damaged (The Economist, 2012, 2013). Insurance indemnities were paid to affectedhouseholds that were insured, and relief had to be organized for those that were
not covered. People whose houses were destroyed wondered if they should abandon
or rebuild them, and if so, how high they should elevate their new homes. Governments wondered if they should authorize development in risky areas like Oakwood
Beach on Staten Island, and if so, according to which building codes. Sandy is
only one example of extreme meteorological events that have caused large flooding damages in the world in the recent years. Among those, Xynthia superstorm
affected the European coast in February 2010, hurricane Katrina struck the New
Orleans region in the USA in August 2005 and Maharashtra heavy rains flooded
the area of Mumbai in India in July 2005. Each time, these events and theirdevastating losses have raised the same questions about the necessity of better
managing urban development in areas prone to natural hazards.
Most risk-prone regions were initially urbanized because of the many advantages they offered to communities. In particular, many cities are located nearseas and/or rivers, as they can provide natural resources and transport facilities.
Nowadays, many industries and services rely on these specificities, and agglomeration forces continue to drive urbanization at these locations (Fujita & Thisse,
2002). However, these locations are often double-edged because of exposure toflooding in the case of extreme meteorological events. Natural hazards coupledwith expanding urbanization have already increased losses in the last few decades,
and these are expected to escalate with the rising sea level and more severe rainfall
patterns due to climate change (IPCC, 2014). At a sub-urban scale in risk-prone
cities, locations are differentiated not only by their distance to valuable amenities
such as the city center but also by exposure to risk due to local characteristics
110

http://www.emdat.be/Chapter 1. Risk prevention in cities prone to natural hazards(e.g. topography for flooding risks). The sub-city spatial heterogeneity raises theessential question of how these cities have spread until now and how they should
continue to develop in the future.
The paper investigates these issues by using an urban model in which eachlocation is characterized by a transport cost to the city center (or to other valuable amenities) and a risk of natural hazard (such as flooding). It focuses on theimpacts of risk spatial variation and insurance subsidization on city development.2
My results are the following. Riskier areas are developed nearer to the city center
than further away. Investment in building resilience leads to more compact cities.
At a given distance to the city center, riskier areas have lower land prices and getlower household density and higher building resilience. Actuarially fair insurance
promotes the optimal development of the city in terms of risk prevention, with
optimal household density and optimal building resilience. I analyze how an increase of insurance subsidization affects the city development. If the subsidy isfinanced by households in the city, it leads to an increase/decrease of density inthe riskiest/safest areas. If the subsidy is financed by households in the country,
it leads to a general increase of density in the city because it attracts households
from other cities. Moreover, in any case, an increase of insurance subsidization
leads to a general decrease of building resilience in the city. These results showthat density and zoning restrictions as well as building codes have to be enforcedin the city to limit risk over-exposure when insurance is subsidized.3
Academics in insurance economics have shown much interest in natural disasters, in particular because of the numerous imperfections in natural disasterinsurance markets (Kunreuther, 1984; Kunreuther & Michel-Kerjan, 2009). On
2

In the present framework, households deliberately purchase full insurance because they arerisk-averse and insurance is supplied at or below actuarially fair prices. The model does notconsider charity hazard or risk perception bias. Note however that the expectancy of assistanceor the under-estimation of risk should have effects similar to insurance subsidization on the city
development in terms of risk over-exposure.
3
Density restriction at one location consists in limiting urban density while zoning restriction
at one location consists in completely forbidding urban development. Building codes consist in
imposing a minimal level of building resilience.

11

Chapter 1. Risk prevention in cities prone to natural hazards
the supply side (Charpentier & Le Maux, 2014; Jaffee & Russell, 1997), diversification issues lead private insurers to supply contracts at prices largely above
actuarially fair rates. On the demand side (Botzen et al., 2015; Kunreuther et al.,2007; Raschky & Weck-Hannemann, 2007), households under-insure even if insurance is fair, in particular because they under-estimate the risk or they expect
free assistance (charity hazard). In this context, policy makers have implemented
natural disaster public policies such as the National Flood Insurance Program(NFIP) in the USA and various programs in Europe like the CatNat in France(Bouwer et al., 2007; Kunreuther & Michel-Kerjan, 2009). To deal with diversification issues, public insurance/relief can complement the weak private insurance
supply (e.g. in the USA) or public reinsurance can help private insurance to supply contracts at lower prices (e.g. in France). However, these policies cannot solve
the weak insurance demand issues without subsidizing insurance or/and making it
mandatory. For instance, the NFIP in the USA subsidizes contracts in risky areas
thanks to taxpayers, and insurance is requested for access to loans. Meanwhile,
the CatNat Program in France subsidizes contracts in risky areas with the othercontracts and insurance is mandatory to avoid adverse selection. If the advantage
of subsidization is to improve insurance demand and risk sharing (Browne & Hoyt,
2000; Grace et al., 2004), the disadvantage is to lead to risk over-exposure because
it does not provide the right incentives for individual risk prevention (Bagstad
et al., 2007; Courbage et al., 2013; Picard, 2008).
Academics in urban economics have focused on natural disaster issues in thecontext of city development. As modeled first by Alonso (1964), households spreadout in the space surrounding the city center to commute there for consumption orwork, and those settled further away incurring higher transport costs are compensated by lower land rent, which explains the increasing housing lot sizes and thedecreasing density with distance to the city center. Polinsky & Shavell (1976) and
Scawthorn et al. (1982) add in their model the existence of a negative amenity
such as exposure to natural hazard. These models show that, at a given distancefrom the city center, the land price decreases when the loss exposure increases.
Many empirical studies have confirmed this effect for natural disaster risks, as
12Chapter 1. Risk prevention in cities prone to natural hazards
summarized in the meta-analysis by Daniel et al. (2009). Because households do
not want to incur too much transport cost or natural disaster cost, Frame (1998)
demonstrates that riskier areas are developed nearer to the city center than further
away, and some risky areas inside the city outer boundary may stay undeveloped.
The tradeoff between transport cost and natural disaster cost has been observed
empirically for instance by Smith (1993) and Atreya & Czajkowski (2014). Frame(1998) also points out that insurance subsidization decreases the land price difference between risky areas and safe areas, as confirmed empirically by Shilling
et al. (1989). Furthermore, Frame (2001) shows theoretically that risk aversion
can lead households to under-develop risky areas. However, many empirical studies, such as Browne & Hoyt (2000), Harrison et al. (2001) and Michel-Kerjan et al.
(2012), suggest that households are more inclined to risk over-exposure because
of insurance subsidization, risk under-estimation or charity hazard, than to risk
under-exposure because of risk aversion.4 In this case, urban regulation should be
enforced to limit over-exposure, in particular in terms of zoning/density restrictions
and building codes (Bagstad et al., 2007; Kunreuther, 1996; Kunreuther & MichelKerjan, 2013). In an urban theoretical model with risk exposure but no transport
costs, Grislain-Letrémy & Villeneuve (2014) show that zoning restrictions can be
Pareto improving in the case of full insurance subsidization. In empirical analysis, Czajkowski & Simmons (2014) and McKenzie & Levendis (2010) respectively
observe that investments in building resilience reduce natural disaster losses andincrease housing values.
The present paper aims to further analyze the role of natural hazard exposure
and insurance subsidization in the development of risk-prone cities with transport
costs. Relative to the previously cited theoretical papers on urban economics, thepresent paper adds building resilience modeling and analyzes how densities and4

Browne & Hoyt (2000) observe that households do not usually buy insurance at fair prices,Harrison et al. (2001) notice that the housing rent difference between risky and safe areas is below
the expected loss difference and Michel-Kerjan et al. (2012) point out that insured households
let their insurance contract lapse after a few years even when those are below fair prices. All
this would not be possible if risk aversion was the dominating factor.

13

Chapter 1. Risk prevention in cities prone to natural hazards
resiliences are affected by natural hazard exposure and insurance subsidization.
This analysis is essential from the perspective of implementing efficient urban
regulation, in terms of zoning restrictions, density restrictions and building codes,
for cities with transport costs and natural disaster risks. The rest of the paper isorganized as follows. Section 2 sets up the model. Section 3 provides an analysis
of city development. Section 4 provides an analysis of the impact of a change ininsurance subsidization. Section 5 concludes.

1.2

Risk-prone city model

I consider a static model of a city with commuting transport costs and natural
hazard exposure, in the spirit of Frame (1998, 2001), Polinsky & Shavell (1976)
and Scawthorn et al. (1982). The city is inhabited by N identical households.5
The sub-city scale grid is modeled by a two-dimensional continuous space with thecoordinate system x = (x1 , x2 ). Because of spatial heterogeneity due to transport
costs and natural hazards, all variables potentially depend on location x. Moreover,each variable has a unique value at each location x because I consider identicalhouseholds and identical housing developers. The city has a pre-established center
located at x = (0, 0), also called the central business district where work andconsumption activities are concentrated.
Households compete to spread out in the space around the city center andcommute between their housing location and the city center. They choose theirhousing location x and the quantity of goods purchased in the city center, aggregated in a composite good denoted z(x). Besides composite good consumption,
households value their housing good consumption, characterized by lot size, measured in land area unit and denoted s(x).6 The utility function of each household,denoted v(.), depends on z(x) and s(x) and is classically supposed to be twice con5

I consider identical households in order to analyze the average development of the city.

Inequality or asymmetric information issues are not the purpose the analysis.
6
The lot size for one household is the land area for this household. For example, for a building
occupying 400m2 of land and inhabited by 10 households, the lot size of one household is 40m2 .

14Chapter 1. Risk prevention in cities prone to natural hazards
tinuously differentiable, strictly increasing in each argument (with ∂z v(0, s) = ∞and ∂s v(z, 0) = ∞) and globally concave. The composite good supplied in the city
center is considered as the numéraire (i.e. price equal to 1 for one unit of good)
and the housing good supplied by housing developers at location x has a housing
unit price denoted ph (x) (i.e. price for one land area unit with housing). The
composite good expenses and the housing rent for one household located at x arethus respectively z(x) and ph (x)s(x).
Besides composite good expenses and housing rent, households incur commuting transport costs and expenses related to natural hazards. One household settlingat location x incurs the given transport cost t(x) because of commuting between
its housing location and the city center (or potentially other valuable amenities).
For example, a city located next to an estuary is depicted in figure 1.1.7 On theland, the darkness of the square units characterizes the commuting transport costt(x) for each household located at x. Darker areas represent locations further fromthe city center with higher transport costs. In stylized models, transport costs areoften considered to be proportional to the distance to the city center. However,real transport costs are more complex than this stylized form, in particular becauseof transport system complexity. Moreover, other potential amenities (e.g. the positive amenities of being near the water-front) should be taken into account in thetransport costs. Note also that transport costs should include different costs, inparticular the direct transport cost but also the time opportunity cost.
One household settling at location x is also exposed to natural hazards (such as
flooding), with the given probability of impact π(x). The level of the loss in caseof impact, denoted l(.), depends on the housing lot size s(x) and on the building
resilience, denoted b(x). The loss function l(.) is assumed to be twice continuously
differentiable. It is decreasing with b at a decreasing rate because the most efficientresilience investments are made first. Besides, if it is reasonably assumed that morehouseholds on a land unit leads to more total losses on this land unit (for a given
7

In figures 1.1 and 1.2, the space is represented by a discrete grid even if the model is contin-

uous.

15

Chapter 1. Risk prevention in cities prone to natural hazards∂lbuilding resilience level), the loss function is such that l(s,b)≥ ∂s(s, b) for anyss and b.8 Note that losses should include direct and indirect losses. The city
depicted in figure 1.1 is also represented in figure 1.2 for natural hazard exposure.
On the land, the darkness of the square units characterizes the probability π(x)of being affected by a natural hazard for each household located at x. The higher
the risk, the darker the location. For flooding risks, locations at lower altitude
are usually more subject to flooding and should be darker. The probability of
being affected by a natural hazard can correspond for example to the probability
that the water level reaches a threshold level that induces significant losses forhouseholds. Besides, I consider that insurance is supplied to households at or
below fair prices because I do not consider any insurance transaction cost andI consider potential insurance subsidy. As households are risk-averse (i.e. theirutility function is concave), they deliberately purchase full insurance coverage andbear a certain cost related to natural disaster risks, which is the insurance premium.
With insurance subsidy corresponding to a fraction λ ∈ [0, 1] of expected losses,
the premium paid by a household located at x is (1 − λ)π(x)l(s(x), b(x)). The
higher λ, the higher the subsidy. The insurance subsidy can be financed either
by the city through a lump-sum tax on household wealth or by another party
outside the city. In the former case, the tax borne by each household in the cityis τ = Nλ

∫∫

(

)

π(x)l s(x), b(x) n(x)dx1 dx2 (in which n(x) is the household density

at location x). In the latter case, the tax borne by each household in the city isτ = 0.9 For one household located at x, transport cost, insurance premium andtax are thus respectively t(x), (1 − λ)π(x)l(s(x), b(x)) and τ .
8With 1s households on a developed land unit, each one having a lot size s, the total loss on

the land unit is 1s l(s, b). If more households on a land unit leads to more total losses on this landunit, the loss function is such that l(ss11,b) ≤ l(ss22,b) for any s11 ≤ s12 and b. In this case, for any()∂ls and ds ≥ 0, l(s+ds,b)≤ l(s,b)s+ds
s , which leads to s l(s, b) + ∂s (s, b)ds ≤ (s + ds)l(s, b) and thenl(s,b)∂lwith a first order development.
∂s (s, b) ≤s
9The latter case is representative of an insurance subsidized by the country which is large

relative to the city. Besides, note that a natural disaster like flooding usually strikes many
locations of a city at the same time and thus has an aggregate risk component at the city level.
However, an insurance system organized at the country level enables to better diversify risk.

16

Chapter 1. Risk prevention in cities prone to natural hazards
x2

sea and rivercity centerx1
increasingtransport cost

Figure 1.1: Commuting transport costs in the city.
x2

sea and rivercity centerx1
increasing
natural hazard risk

Figure 1.2: Natural hazard risks in the city.
As households are identical in terms of preferences and wealth, denoted y, theyreach the same utility level v at equilibrium. Otherwise, households with lower
utility levels at location x would have settled at location x′ where other households
reach a higher utility level, which would have decreased at x and increased at x′
housing unit price until the equilibrium with spatially uniform utility level had
been reached. Following Alonso (1964) and Fujita & Thisse (2002), competition
17

Chapter 1. Risk prevention in cities prone to natural hazards
between households over where to settle leads housing prices to be the solutions ofbid price problems: at each location x, the housing unit price ph (x) corresponds tothe highest price that can be afforded by households. The wealth minus expenses
except housing rent, divided by the lot size, is the maximal amount that can be
paid by one household for one land unit with housing. As households are free tochoose their composite good consumption and reach the utility level v, the housing
unit bid price problem at location x can then be expressed as follows:(

ph (x) = maxz(x))

y − τ − z(x) − t(x) − (1 − λ)π(x)l s(x), b(x)
(

)

s(x)

(1.1)

s.t. v z(x), s(x) = v.
Housing goods are supplied by identical housing developers in competition.
Housing developers observe the housing unit price (1.1) resulting from the competition between households. They compete to acquire land from absentee landowners at the land unit price denoted pl (x) at each location x (i.e. price for one
land area unit without housing). They choose the housing lot size s(x) and thebuilding resilience b(x) for urban development at each location x. Besides thecost of land, they incur the cost of housing lot development, denoted c(s(x), b(x))
for lot size s(x) and resilience b(x) for one household. The cost function c(.) is
assumed to be twice continuously differentiable. It is increasing with b at an increasing rate because the less costly resilience investments are made first. Besides,
if it is reasonably assumed that more households on a land unit leads to moretotal housing development costs on this land unit (for a given building resilience∂clevel), the cost function is such that c(s,b)≥ ∂s(s, b) for any s and b.10 At eachslocation, housing developers are constrained by the availability of one land area
unit. Similarly to the housing unit price, the land unit price pl (x) is determinedat each location x by the highest price that can be afforded by housing developers
because of competition. The housing rent per household minus the development
cost, multiplied by the household density denoted n(x), is the maximal amountthat can be paid by one housing developer for one land area unit. As housing
10

18

The proof is similar to the one in the footnote 8 for the natural disaster loss function.

Chapter 1. Risk prevention in cities prone to natural hazards
developers are free to choose housing lot sizes and building resiliences, and theyobserve the housing unit price (1.1) and face land constraints, the land unit bid
price problem at location x can then be expressed as follows:(pl (x) = maxs(x),b(x)(ph (x)s(x) − c s(x), b(x)

))

n(x)
(1.2)

s.t. (1.1) and n(x)s(x) ≤ 1.
The boundaries of the city correspond to the locations where the land unit pricepl (x) is equal to the land opportunity rent denoted pa (e.g agricultural rent).
Finally, the city is characterized by its number of households:∫∫

N=n(x)dx1 dx2 .

(1.3)

With a given number of households (i.e. N given), (1.3) indirectly determines
the welfare level v in the city. This characterizes in particular a "closed city" interms of population. With a given welfare level (i.e. v given), (1.3) determines
the number of households N in the city. This characterizes in particular an "opencity" in which the welfare level depends on the welfare level outside the city.

1.3

Risk-prone city development

Outside the boundaries of the city, housing development is not profitable (n(x) =0 and pl (x) = pa ). On the boundaries, land may be partly developed (0 ≤s(x)n(x) ≤ 1) because housing development is equally profitable to agriculture(pl (x) = pa ). Inside the boundaries, land is fully developed because housing development is more profitable than agriculture and thus the household density is:n(x) =

1
.
s(x)

(1.4)

As explained in the previous section, households settled in the city reach thesame welfare level v at equilibrium. As the utility function v(.) is strictly increasing
in z, z̃(s, v) can be defined such that v(z̃(s, v), s) = v and z̃(.) is decreasing withs at a decreasing rate because v(.) is concave (proof in appendix 1.6.1). Thus, the19Chapter 1. Risk prevention in cities prone to natural hazards
composite good consumption z(x) purchased by one household settled at location
x can be expressed as a function of the housing lot size s(x) and the uniform
welfare level v:()

z(x) = z̃ s(x), v .

(1.5)

With (1.5), the housing unit bid price problem (1.1) boils down to the housing
unit price:(ph (x) =)()

y − τ − z̃ s(x), v − t(x) − (1 − λ)π(x)l s(x), b(x)
s(x)

.

(1.6)

With the housing unit price (1.6) and the household density (1.4), the land
unit bid price problem (1.2) boils down to:pl (x) = max

(
)
(
)
(
)
y − τ − z̃ s(x), v − t(x) − (1 − λ)π(x)l s(x), b(x) − c s(x), b(x)s(x)s(x),b(x)

.

(1.7)

The housing lot size s(x) and the building resilience b(x) chosen by housing developers at location x inside the boundaries of the city are the solutions of the firstorder conditions of (1.7) (proof in appendix 1.6.1):) ∂c ())∂s v (∂l (s(x), z̃(s(x), v) = (1 − λ)π(x)s(x), b(x) +s(x), b(x) + pl (x),
∂z v
∂s
∂s

−(1 − λ)π(x)

)
)
∂c (
∂l (
s(x), b(x) =s(x), b(x) .
∂b
∂b

(1.8)

(1.9)

(1.8) states that the housing lot size s(x) for one household at location x is chosensuch that it equalizes the marginal rate of substitution to the marginal housing
unit cost (over the composite good price, i.e. the numéraire). The marginal rate of
substitution characterizes the marginal benefit of increasing the housing lot size forthe household, which decreases from +∞ to 0 when s(x) increases from 0 to +∞.
The marginal housing unit cost is composed of the marginal insurance premium
borne by the household, the marginal housing development cost and the land
unit price. (1.9) relates that the building resilience b(x) for housing at location
x is chosen such that it equalizes the marginal benefit of decreasing insurance
premium for the household to the marginal cost of increasing building resilience
20Chapter 1. Risk prevention in cities prone to natural hazards
for the housing developer. Note that if −(1 − λ)π(x)∂b l(s(x), 0) ≤ ∂b c(s(x), 0),b(x) is binding in 0. With s(x) and b(x) being determined by (1.8) and (1.9), (1.7)
then indirectly gives the land unit price:(pl (x) =

)

(

)

(

y − τ − z̃ s(x), v − t(x) − (1 − λ)π(x)l s(x), b(x) − c s(x), b(x)
s(x)

)

.

(1.10)

Proposition 1 With null cross derivation for l(.) and c(.) relative to their twoarguments, the housing lot size s(x) and the building resilience b(x) vary in spaceas follows:11

( (

ds
1 dt
A1 (x) → =+ (1 − λ)s(x) d→dxx)

l s(x), b(x)
s(x)

)

) dπ
∂l (
−
s(x), b(x)
→,
∂s
dx

) dπ
∂l (
db
A2 (x) → = −(1 − λ)s(x), b(x) → ,∂bdx
dx
in which A1 (x) and A2 (x) are positive.

(1.11)(1.12)

Proposition 1 is proved in appendix 1.6.1. (1.11) tells that, at a given riskof natural hazard, the housing lot size s(x) increases while translating further
away from the city center. Thus, the household density n(x) (i.e. the inverseof the housing lot size s(x)) decreases while translating further away from thecity center, as first explained by Alonso (1964). With the reasonable assumptionl(s,b)∂l≥ ∂s(s, b) for any s and b, the coefficient in front of dπ→ in (1.11) is positivesdx

and (1.11) says that, at a given distance to the city center, the housing lot size s(x)increases while translating towards riskier areas if insurance is not fully subsidized(λ < 1). Thus, the household density n(x) decreases while translating towards
riskier areas in this case. As l(.) is decreasing with b, (1.12) points out that thebuilding resilience increases while translating towards riskier areas if insurance is
not fully subsidized (λ < 1).
Proposition 2 The housing unit price ph (x) and the land unit price pl (x) varyin space as follows:
dpl

dx

dx

→ =

11 →

c(s(x),b(x))∂c∂cds∂s (s(x), b(x)) −s(x)∂b (s(x), b(x)) db
→ +
→ +
→,

dph

s(x)

dx

s(x)

dx

(1.13)→

d x corresponds to any small move in space: d x = (dx1 , dx2 ).

21

Chapter 1. Risk prevention in cities prone to natural hazards

dpl

→ =−

dx

1 dt
(1 − λ)l(s(x), b(x)) dπ→ −→.
s(x) d xs(x)dx(1.14)

In proposition 2, (1.14) is obtained by spatial derivation of (1.7) with theenvelop theorem, while (1.13) is obtained by spatial derivation of the combinationof (1.6) and (1.10). (1.14) relates firstly that, at a given risk of natural hazard, theland unit price pl (x) decreases while translating further away from the city center,as first explained by Alonso (1964). (1.14) tells secondly that, at a given distanceto the city center, the land unit price pl (x) decreases while translating towards
riskier areas if insurance is not fully subsidized (λ < 1). Moreover, the higher the
insurance subsidization (λ), the lower the land price difference between risky areasand safe areas. These observations confirm the results of Frame (1998), Polinsky& Shavell (1976) and Scawthorn et al. (1982) in a context including building
resilience. (1.13) indicates that the housing unit price ph (x) is modified throughthree channels while moving in the city: the land unit price pl (x), the housing
lot size s(x) and the building resilience b(x). At a given risk of natural hazard,
the housing unit price ph (x) decreases while translating further away from the city
center because firstly the land unit price decreases, secondly the effect through→the housing lot size is negative (because ds/d x ≥ 0 and with the reasonable∂cassumption c(s,b)≥ ∂s(s, b)) and thirdly the effect through the building resiliences

is null. At a given distance to the city center, translating towards riskier areas
leads to the decrease of housing unit price ph (x) through the decrease of land unit∂cprice pl (x) and the increase of housing lot size s(x) (as far as c(s,b)≥ ∂s(s, b)), whileson the other hand it leads to the increase of housing unit price ph (x) through theincrease of building resilience b(x). This differentiates slightly the spatial variationof land unit price pl (x) and housing unit price ph (x), contrary to the previously
cited papers which do not consider building resilience. Moreover, it is coherent
with the empirical observation by McKenzie & Levendis (2010) that investments
in building resilience increase housing prices.
Proposition 3 If the probability of natural hazard is denoted π ∗ (t) on the city
22

Chapter 1. Risk prevention in cities prone to natural hazardsππ ∗ (t) (with affordable building resilience)π ∗ (t) (with unaffordable building resilience)
Developed.
tFigure 1.3: City boundaries and developed areas, as a function of transport cost t
and probability π of natural hazard.
boundaries, π ∗ (t) is such that:
dπ ∗
1
=−
.
dt
(1 − λ)l(s(x), b(x))(1.15)

Proposition 3 is directly deduced from (1.14) because the land unit price pl (x)is constant and equal to the opportunity rent pa on the city boundaries. (1.15)
expresses that riskier locations are developed near the city center because of lower
transport cost, which confirms the result of Frame (1998) in a context including
building resilience. A location x at a distance t from the city center is developed if
π(x) ≤ π ∗ (t). The outer boundary of the city corresponds to the developed area thefurthest away from the city center. The inner boundaries of the city correspondto the riskiest developed area for each distance to the city center. Figure 1.3
illustrates on a graph, with transport and risk as coordinates, the city boundaries
and the developed areas. The slope of π ∗ relative to t is steeper when building
resilience is implemented. Thus, more households are located near the city center(i.e. the city is more compact) when building resilience is more affordable.
23

Chapter 1. Risk prevention in cities prone to natural hazards

1.4

The impact of insurance subsidization

With actuarially fair insurance (λ = 0), the allocation of resources is Pareto optimal (proof in appendix 1.6.2). Actuarially fair insurance policy leads to the Pareto
optimal allocation of resources because it gives the right incentives to households
and housing developers in terms of density development and building resilience. In
practice, actuarially fair insurance is hardly ever implemented, and policy makers
usually implement insurance subsidy.
Proposition 4 With null cross derivation for l(.) and c(.) relative to their twoarguments, the increase of insurance subsidization has the following impact onurban development at each location x of the city:
(

ds(x)
∂l
l(s(x), b(x))s(x)A1 (x)= π(x)s(x)(s(x), b(x)) −
dλ
∂s
s(x)

A2 (x)db(x)∂l= π(x) (s(x), b(x)),
dλ
∂b

)

+ α(x)

dv
dτ
+
,
dλ dλ

(1.16)(1.17)

2

∂ z̃
∂ z̃
(s(x), v) − s(x) ∂v∂s
(s(x), v) and A1 (x) and A2 (x) are positive.
in which α(x) = ∂v

Proposition 4 is proved in appendix 1.6.2. (1.16) characterizes how the increaseof insurance subsidy (λ) affects the housing lot size s(x) and thus the household
density n(x) at each location in the city. The direct impact corresponds to the firstterm on the right-hand side of (1.16), which is negative with the reasonable as∂lsumption l(s,b)≥ ∂s(s, b) for any s and b. A given increase of λ gives, through thissdirect effect, a density increase which is proportional to the probability π(x). The
indirect impact through the levels of welfare v and tax τ corresponds to the second
and third terms. If the number N of households is fixed (which characterizes a"closed city" with v endogenously determined), the density cannot increase everywhere in the city12 and the increase of insurance subsidy reallocates households
from safer areas to riskier areas because of the direct effect. If the welfare level v
12that ds(x)dλ cannot be negative at all locations.
for all the location in the city.

24∫∫

1 ds(x)s(x)2 dλ dx1 dx2 , which meansdvdτ
Thus, α(x) dλ+ dλin (1.16) cannot be negative

With N fixed, the population constraint (1.3) gives 0 =Chapter 1. Risk prevention in cities prone to natural hazards
is fixed (which characterizes an "open city" with N endogenously determined), v is
not affected by an increase of λ while the impact on τ depends on who bears thecost of insurance subsidization. If the households in the city bear this cost throughthe lump-sum tax τ = Nλ

∫∫

(

)

π(x)l s(x), b(x) n(x)dx1 dx2 , an increase of τ due toan increase of λ makes the city less attractive, which explains why it increases lot
sizes and decreases densities. Thus, in this case, the increase of insurance subsidy leads to a density increase in strongly risky areas and a density decrease in
weakly risky areas. If the households in the city do not bear the cost of insurance
subsidization (i.e. τ = 0), an increase of λ does not have this negative effect on
the city attractiveness. Thus, in this case, the increase of insurance subsidy leads
to a general density increase in the city. These results explain in which direction
density policies should be enforced in risk-prone cities when insurance subsidy isimplemented. Besides, (1.17) points out how the increase of insurance subsidy(λ) modifies the building resilience b(x) at each location in the city. The impact
is negative and proportional to the local probability π(x). A given increase of λleads to a higher building resilience decrease in risky areas than in safe areas. As
a consequence, whether with a closed city or an open city and whoever subsidizes
insurance, the increase of insurance subsidy leads to a general decrease in building
resilience in the city. Note that this decrease is null if the building resilience is already binding in zero (which is the case at a risk-free location). These results showthat resilience policies should be enforced when insurance subsidy is implemented.
Proposition 5 The increase of insurance subsidization has the following impact
on housing and land prices at each location x in the city:dph (x)dpl (x)∂s (s(x), b(x)) −
=
+
dλ
dλ
s(x)
∂c

c(s(x),b(x))
ds(x)
s(x)

dλ

∂c

+ ∂b

(s(x), b(x)) db(x)
,
s(x)
dλ

(
)
dpl (x)1dvdτ=π(x)l(s(x), b(x)) − β(x)
−
,
dλ
s(x)
dλ dλ
(
)

(1.18)(1.19)∂ z̃
in which β(x) = ∂vs(x), v is positive.

In proposition 5, (1.19) is obtained at a given location by derivation of (1.7)
relative to λ with the envelop theorem, while (1.18) is obtained at a given location
25

Chapter 1. Risk prevention in cities prone to natural hazards
by derivation of the combination of (1.6) and (1.10) relative to λ. (1.19) relates
how the increase of insurance subsidy (λ) affects the land price pl (x) at eachlocation in the city. The direct impact corresponds to the first term on the righthand side which is positive and proportional to the local probability π(x). For
a given increase of λ, the riskier the location, the higher the land price increase
through this direct effect. The indirect impact through the levels of welfare v andtax τ corresponds to the second and third terms. If the welfare level v is fixed(which characterizes an "open city" with N endogenously determined), v is notaffected by an increase of λ while the impact on τ depends on who bears the costof insurance subsidization. If the households in the city bear this cost throughthe lump-sum tax τ = Nλ

∫∫

(

)

π(x)l s(x), b(x) n(x)dx1 dx2 , an increase of τ due toan increase of λ decreases their wealth and thus land prices. Thus, in this case,the increase of insurance subsidy leads to a land price increase in strongly risky
areas and a land price decrease in weakly risky areas. If the households in thecity do not bear the cost of insurance subsidization (i.e. τ = 0), an increase of λdoes not have this negative effect on land prices. Thus, in this case, the increaseof insurance subsidy leads to a general land price increase in the city because theincrease of attractiveness of the city is not lowered by a tax on households in thecity. Figure 1.4 illustrates the impact of a subsidy increase on the city boundaries
for an "open city" in the case where the subsidy is borne by households in the cityand in the case where the subsidy is not borne by households in the city. Besides,(1.18) states that the housing price ph (x) is modified through three channels while
increasing insurance subsidy (λ): the land unit price pl (x), the housing lot size s(x)and the building resilience b(x). The direction of the impacts through the land
unit price and the housing lot size depends on the location, similarly to these twovariables. The impact through the building resilience decreases the housing unit
price because the increase of insurance subsidy decreases the building resilienceand thus its cost.

26

Chapter 1. Risk prevention in cities prone to natural hazardsππ ∗ (t)π ∗ (t) (with subsidy increase from outside the city)π ∗ (t) (with subsidy increase from the city)
Developed.
tFigure 1.4: City boundaries and developed areas, as a function of transport cost t
and probability π of natural hazard.

1.5

Conclusion

The paper has analyzed urban development choices in a city prone to natural
disasters. It complements previous studies, in particular by including building
resilience choices. Riskier areas are developed nearer to the city center than further away. Investment in building resilience leads to more compact cities. At
a given distance to the city center, riskier areas have lower land prices and getlower household density and higher building resilience if insurance is not fully subsidized. Actuarially fair insurance leads households to optimally settle in spacein terms of density and resilience. An increase of insurance subsidization leads
to an increase of density in the riskiest areas of the city, in particular displacing
inner boundaries towards riskier areas near the city center. Moreover an increaseof insurance subsidization leads to a general decrease of building resilience in thecity. To avoid excessive exposure to risk in the case of insurance subsidization,
policy makers have to complement their policies by enforcing density and zoning
restrictions as well as building codes. In this perspective, the present paper tells
that, in the case of insurance subsidization, density and zoning restrictions haveto be enforced at least in the riskiest areas of the city, in particular near the city
center where land is attractive because of low transport costs. It also tells that,27

Chapter 1. Risk prevention in cities prone to natural hazards
in the case of insurance subsidization, building codes should be generally enforcedin the city for Pareto improvement.

1.6

Appendix

1.6.1

Risk-prone city development

Characteristics of z̃(.)
The derivation of v(z̃(s, v), s) = v relative to s gives:∂v ∂ z̃ ∂v+= 0,∂z ∂s ∂s(1.20)

∂ z̃
∂s v
=−
.
∂s
∂z v

(1.21)

which can be rewritten:

Because v(.) is increasing with z and s, z̃(.) is decreasing with s. Besides, thederivation of (1.20) relative to s gives:(∂ 2 v ∂ z̃∂z 2 ∂s)2∂v ∂ 2 z̃
∂ 2 v ∂ z̃ ∂ 2 v
+
+2
+
= 0,
∂z ∂s2∂z∂s ∂s ∂s2(1.22)

which can be rewritten with (1.21):(∂v ∂ 2 z̃
∂ 2 v ∂s v=−
∂z ∂s2∂z 2 ∂z v)2+2∂ 2 v ∂s v ∂ 2 v−.
∂z∂s ∂z v ∂s2(1.23)The term on the right-hand side of (1.23) is positive because v(.) is concave and2

the determinant of the Hessian matrix of v(.) is positive. Thus, ∂∂s2z̃ is positive andz̃(.) is decreasing with s at a decreasing rate.
Derivation of (1.8) and (1.9)
With (1.21) and the expression (1.7) of pl (x), the first order conditions of (1.7)
relative to s(x) and b(x) are respectively:
∂s v
∂z v

(

)

(

s(x)

28

)

(

)

∂l
∂c
s(x), z̃(s(x), v) − (1 − λ)π(x) ∂ss(x), b(x) − ∂ss(x), b(x)−pl (x)= 0, (1.24)s(x)Chapter 1. Risk prevention in cities prone to natural hazards−(1 − λ)π(x)) ∂c ()∂l (s(x), b(x) −s(x), b(x) = 0,∂b∂b(1.25)

which respectively give (1.8) and (1.9).
Proof of proposition 1
As the first order conditions of (1.7) correspond to a maximum, the second orderconditions of (1.7) are negative at the solutions s(x) and b(x). Thus, the following
expressions which are called A1 (x) and A2 (x) are positive:
A1 (x) =)) ∂2c ()
∂ 2 z̃ (
∂2l (
s(x),
v
+
(1
−
λ)π(x)
s(x),
b(x)
+
s(x),
b(x)
≥ 0,
∂s2
∂s2
∂s2

A2 (x) = (1 − λ)π(x)) ∂2c (
)
∂2l (
s(x),
b(x)
+
s(x),
b(x)
≥ 0.
∂b2
∂b2

(1.26)(1.27)

With null cross derivation for l(.) and c(.) relative to their two arguments, thespatial derivation of (1.8) and (1.9) respectively gives:
( (

ds
1 dt
A1 (x) → =+ (1 − λ)s(x) d→dxx)

l s(x), b(x)
s(x)

)

) dπ
∂l (
−
s(x), b(x)
→,
∂s
dx

) dπ
db
∂l (
A2 (x) → = −(1 − λ)s(x), b(x) → ,∂bdx
dx
which gives proposition 1.

1.6.2(1.28)(1.29)The impact of insurance subsidization

Optimal allocation
The first welfare theorem predicts the Pareto optimality with λ = 0 because efficient insurance markets would lead to actuarially fair insurance. For the formal
proof, the optimal allocation with uniform welfare level v is obtained by minimizing
the total expenditure of the city with N households:
∫∫ (

min

(

)

(

)

(

))

z̃ s(x), v + t(x) + π(x)l s(x), b(x) + c s(x), b(x) n(x)dx1 dx2s(.),b(.),n(.)

s.t. n(x)s(x) ≤ 1, ∀x∫∫N=(1.30)

n(x)dx1 dx2 .

29

Chapter 1. Risk prevention in cities prone to natural hazards
The first order conditions give similar equations to the decentralized economy with
λ = 0. Thus, the actuarially fair insurance policy (i.e. with λ = 0) implements
the Pareto optimal allocation of resources.
Proof of proposition 4
The proof of proposition 4 is similar to the proof of proposition 1. The positive
A1 (x) and A2 (x) are defined by (1.26) and (1.27). Contrary to proposition 1, thederivation of (1.8) and (1.9) relative to λ at a given location x do not have termswith derivatives of t(x) and π(x) but have terms with derivatives of v and τ . With
null cross derivation for l(.) and c(.) relative to their two arguments, the derivationof (1.8) and (1.9) relative to λ at a given location x respectively gives:
(

ds(x)
∂l
l(s(x), b(x))s(x)A1 (x)= π(x)s(x)(s(x), b(x)) −
dλ
∂s
s(x)

A2 (x))+ α(x)

dv
dτ
+
,
dλ dλ

∂l
db(x)
= π(x) (s(x), b(x)),
dλ
∂b
2

∂ z̃
∂ z̃
in which α(x) = ∂v
(s(x), v) − s(x) ∂v∂s
(s(x), v). This gives proposition 4.

30(1.31)(1.32)

Chapter 2
The role of insurance companies
in a risky economy
Abstract: I consider an exchange economy with multiple commodities and agents
having different preferences and distributions of endowments. In an Arrow-Debreu
setting, complete financial markets consist in one financial asset per state of nature and allow to reach a Pareto optimal allocation. Yet, this setting requires aprohibitive number of financial assets. With a competitive insurance market, it issufficient to have only one financial asset for all the states of nature corresponding to the same aggregate endowments. In this case, agents can fully cover theirendowment risks thanks to fair multi-risk contracts supplied by stock insurancecompanies. For a given endowment risk, the higher the correlation with the aggregate risk, the higher the premium. Besides, financial assets allow agents tohedge commodity price risks due to the aggregate risk and to choose a share of
the aggregate risk.

Keywords: individual risk, aggregate risk, complete markets, general equilibrium, securities, insurance.

JEL classification: D53, D86, G10.
31

Chapter 2. The role of insurance companies in a risky economy

2.1

Introduction

The combination of risk exposure and risk aversion creates the demand for risk
sharing and the emergence of risk sharing mechanisms in a decentralized economy.
In this context, complete financial markets can lead to a Pareto optimal allocationof risks, as firstly shown by Arrow (1953) and Debreu (1959). In a classic ArrowDebreu economy, complete financial markets consist in having one financial asset(also called security) per state of nature, in which a state of nature is characterized
by a full specification of the individual endowments obtained by all the agents inthe economy. Yet, in the real world, this setting requires a tremendous amount ofassets and necessitates to make public the realized individual state of each agent toknow the realized state of nature. These limits find their solution in the emergenceof insurance companies. The present paper shows that markets are complete forrisk sharing with only one financial asset for all the states of nature corresponding
to the same aggregate endowments, when there are in addition competitive insurance companies supplying standard insurance contracts and owned through stock
markets.
I consider a static exchange economy with multiple commodities and multiple heterogeneous agents facing heterogeneous risks, in the sense that agents havedifferent preferences and different distributions of endowments. I define an "ArrowDebreu" state of nature as a full specification of the individual endowments in thisstate obtained by all the agents in the economy. With a financial asset associated to each Arrow-Debreu state and a spot market for each commodity, theeconomy reaches a Pareto optimal allocation. The collection of all the "ArrowDebreu" states displaying the same aggregate endowments in the economy definesa "fundamental" state of nature. I consider that the agents have only access to afinancial asset for each fundamental state and to a spot market for each commodity. Besides, there are M insurance companies in competition. They supply fair
multi-risk insurance contracts to agents, with an indemnity compensating for theindividual commodity losses in exchange for a premium corresponding to the expected indemnity. Insurance contracts are standard in the sense that they do not32Chapter 2. The role of insurance companies in a risky economy
include participation on the insurer profits.1 The insurance companies are owned
by agents in the economy through stock markets. With frictionless markets, I showthat this economy reaches a Pareto optimal allocation.
In the general case, agents are exposed to commodity price risks in addition toindividual endowment risks because individual endowment risks generate aggregate endowment risks. Insurance contracts with insurance companies enable agentsto eliminate their individual endowment risks and security markets enable agentsto eliminate their exposure to price risks. Besides, agents purchase a share of
aggregate risks through insurance stock markets and security markets. Thanks toinsurance companies, the number of financial assets can be tremendously reduced
relative to the classic Arrow-Debreu economy. For illustration, we can consider
a simple setting with 1 commodity, 3 identical agents, 10 individual states, withindividual endowment from 1 to 10 commodity units and probability 1/10 for eachstate. In this case, the number of Arrow-Debreu states is 103 and the number offundamental states is 28 (total endowment reaching potentially each integer between 3 and 30). While Pareto optimality is reached in the Arrow-Debreu economy
with 1000 financial assets, it is also reached in an economy with 28 financial assets
and 2 insurance companies which compete to supply to each of the 3 agents an insurance contract for individual endowment risks. Besides, while the Arrow-Debreu
economy requires to make public the realized individual endowments of each agent
to determine the Arrow-Debreu state and allow transactions, the reduced numberof financial assets requires to make public only the realized aggregate endowmentsin the economy (which are indirectly known trough the spot prices because one
vector of spot prices correspond to one vector of aggregate endowments).
Insurance companies serve as intermediaries that pool individual endowment
risks. Because of competition, they sell fair contracts to be able to catch policyholders on one side and shareholders on the other side. A fair premium for an
insured risk is equal to the expected indemnity, in which the indemnity in each1Standard contracts are from stock companies by opposition to mutual contracts from mutualcompanies, in which there is a participation on the insurer profits.

33

Chapter 2. The role of insurance companies in a risky economy
state of nature depends on the commodity losses and on the commodity prices.
For a given endowment risk, the higher the correlation with the aggregate risk (i.e.
with the commodity price risk), the higher the expected indemnity and thus thepremium. That is why a fair contract includes an aggregate risk loading factor
in the premium.2 The loading factor is potentially different from one agent toanother for the same individual risk and can be positive or negative. As insurance
contracts are fairly priced relative to individual risks and their correlation withthe aggregate risk, agents purchase full coverage for their individual risks. Shareholders of an insurance company are exposed to the risk resulting from the pool ofindividual risks insured by the company. The higher the correlation between theserisks and the aggregate risk, the higher the raised premiums, which compensates
shareholders for bearing a less diversifiable risk.
The economics literature on risk has already addressed the role played by insurance companies for risk sharing. In an exchange economy with one commodity andagents having different preferences and distributions of endowments, Kihlstrom &Pauly (1971) and Ellickson & Penalva-Zuasti (1997) show that the economy canreach a Pareto optimal allocation with one contract per agent or per risk and without any financial assets, but they cannot explain who supplies these contracts. To
go beyond this supply issue, Marshall (1974a,b) explains that agents have to signmutual contracts which enable to eliminate individual risks and share aggregaterisks. Malinvaud (1973) and Cass et al. (1996) consider an exchange economy
with one commodity and groups of agents having identical preferences and facing
independently and identically distributed risks. They show that Pareto optimality is reached with mutual insurance contracts within each group of agents and alimited number of financial assets (i.e. all the Arrow-Debreu states corresponding to rearrangements between identical agents necessitates only one financial asset). Besides, Doherty & Dionne (1993) and Doherty & Schlesinger (2002) explainthat mutual insurance contracts can be replaced by "homemade mutualization",2

Contrary to an insurance contract based on a financial loss, an insurance contract basedon commodity losses naturally includes an aggregate risk loading factor through the productbetween commodity losses and commodity prices for the value of the expected indemnity.

34

Chapter 2. The role of insurance companies in a risky economy
in which agents purchase standard contracts from stock insurance companies toeliminate individual risks and invest in insurance companies through stock markets to share aggregate risks. Penalva-Zuasti (2001, 2008), as well as the present
paper, relaxes the condition of mutual contracts thanks to insurance companies
shared on the stock markets. Penalva-Zuasti (2008) considers an exchange economy with one commodity and agents having different preferences and different
initial wealth but facing independently and identically distributed risks. He showsthat Pareto optimality is reached with standard insurance contracts supplied bystock insurance companies and a limited number of financial assets (i.e. all theArrow-Debreu states corresponding to the same aggregate endowment necessitates
only one financial asset).3 Relative to Penalva-Zuasti (2008), the present paper
relaxes the hypothesis of independently and identically distributed risks.4
The main contribution of the present paper is to extend the analysis of therole played by insurance companies to an economy with multiple commodities andagents having different preferences and different risks with potential risk dependence across agents (i.e. the present model is not restricted to independently andidentically distributed risks). The analysis highlights in particular that individual
3

Penalva-Zuasti (2008) considers actually a dynamic economy with uncertainty resolving

nicely through time and continuous trading on financial markets. In this context, it is evensufficient to have only two financial assets, a risk-free one and a risky one relative to the aggregate risk. Note that the dynamic approach does not change the role of insurance because
insurance contracts are not traded continuously, that is why I remain in a static model in thepresent paper.
4
Another part of the economics literature has focused on the role of financial intermediaries
when market frictions are considered. Gorton & Winton (2003) reviews the literature on therole of financial intermediaries when there are transaction costs or asymmetric information between agents. This topic has been particularly developed for bank-like intermediaries which play
an intermediary role between borrowers and lenders (Freixas & Rochet (1997)). With transaction costs due to asymmetric information, contracts through financial intermediaries can Pareto
dominate direct contracts if financial intermediaries are able to diversify. Indeed, even thoughfinancial intermediaries increase the number of transactions, diversification decreases the uncertainty and thus the asymmetric information issue for the less informed agents. As a consequence,with market imperfections, risk correlation represents a limit for financial intermediaries.

35

Chapter 2. The role of insurance companies in a risky economy
risks do not have to be identical or independent to be insured by insurance companies (in a context without market frictions, similarly to the quoted papers). Even
insurance companies weakly diversified are able to sell contracts at fair prices,because the shareholders diversify their risk thanks to other markets. Besides,
because risks are not independently and identically distributed, the insurance premium for an individual risk is specific to the individual risk and includes an aggregate risk loading factor characterizing the correlation between the individual risk
and the aggregate risk in the economy. For a given risk, the higher the risk correlation with the aggregate risk, the higher the premium. The paper is organized
as follows. Section 2 sets up the model. Section 3 reminds the classic results ofBorch, Arrow and Debreu. Section 4 analyzes the decentralized equilibrium withinsurance companies. The last section concludes.

2.2

The model of the risky economy

I consider a static pure exchange economy with N agents (i = 1, .., N ) and C
commodities (c = 1, .., C). Agents have uncertain endowment, are risk-averseand have objective probabilities over the risks. I denote ei 5 the vector with C
components characterizing the highest possible endowment of each commodity foragent i. Agent i can face Si individual states (si = 1, .., Si ). The endowment loss instate si relative to ei is denoted li (si ). The endowment ei is thus a random variablewith ei (si ) = ei − li (si ) in state si . The preferences satisfy the von NeumannMorgenstern axioms with vi (.) : RC
+ → R the corresponding utility function whichis strictly increasing in each argument, globally concave and twice continuously
differentiable.
Definition 1 An "Arrow-Debreu" state is a full specification of the individual endowments obtained by all the agents in the economy.
The number of Arrow-Debreu states in the economy depends on the number of agents, the number of individual states for each agent and on the risk
5

The vectors are denoted in bold letters in the paper. In mathematic formula, the product of

two vectors correspond to the scalar product.

36

Chapter 2. The role of insurance companies in a risky economy
dependences between agents. In the absence of full dependence between individual risks, the number of Arrow-Debreu states in the economy is: Z =∏i Si (i.e.

Z = S N with identical Si ), because each combination of individual states has apositive probability to occur. I denote by z = 1, .., Z the Arrow-Debreu states andπ(z) ≥ 0 the probability of obtaining the Arrow-Debreu state z (with π(z) > 0 and∑z π(z) = 1).

With objective probabilities over the risks, all the agents consider

this probability π(z) for the Arrow-Debreu state z to occur. For any agent i, I denote by si (.) : [1, Z] → [1, Si ] the function which gives the individual state of agent
i associated to each Arrow-Debreu state. I denote by xi (z) the commodity consumption plan of agent i in the Arrow-Debreu state z. Thus, xi = (xi (1), .., xi (Z))
is the commodity consumption plan of agent i in all the Arrow-Debreu states,
x(z) = (x1 (z), .., xN (z)) is the commodity consumption plan of all the agents in
the Arrow-Debreu state z and x = ((x1 (1), .., x1 (Z)), .., (xN (1), .., xN (Z))) is thecommodity consumption plan of all the agents in all the Arrow-Debreu states.

Definition 2 A "fundamental" state is a full specification of the aggregate endowments in the economy.

All the Arrow-Debreu states, which have the same aggregate endowments for
the C commodities, correspond to the same fundamental state. I denote by t =1, .., T the fundamental states in the economy (with T the number of fundamental
states) and E(t) the total endowment vector in the fundamental state t. The setof Arrow-Debreu states included in the fundamental state t is denoted Ft (with∑t #Ft = Z and ∀z ∈ Ft ,∑i ei (si (z)) = E(t)).

I denote by t(.) : [1, Z] → [1, T ]the function which gives the fundamental state associated to each Arrow-Debreu
state. The probability of obtaining the fundamental state t is denoted π̃(t) (wehave π̃(t) =∑z∈Ft π(z) and∑t π̃(t) = 1). The probability of obtaining the Arrow-Debreu state z conditioned on being in the fundamental state t is denoted π(z|t)(we have π(z) = π(z|t)π̃(t) and∑z∈Ft π(z|t) = 1).

37

Chapter 2. The role of insurance companies in a risky economy

2.3

Pareto optimality and Arrow-Debreu economy

This section aims at reminding the classic results of Borch, Arrow and Debreu.
Firstly, in the spirit of Borch, the characteristics of the Pareto optimal allocations
are analyzed. Secondly, in the spirit of Arrow and Debreu, it is shown that thedecentralized economy with a complete set of financial assets allows to reach a
Pareto optimal allocation. This section is useful to understand in the following
section the functioning of a decentralized economy with insurance companies anda reduced number of financial assets.

2.3.1

Borch Mutuality Principle

The Pareto optimal allocations in an economy are by definition the ones thatmaximize a weighted sum of the welfare of all the agents under the constraintsof the economy. In the present economy, the only feasibility constraints that anallocation must satisfy are the aggregate endowment constraints for all the ArrowDebreu states:∑

xi (z) ≤ E(t(z)), ∀ z.

(2.1)

i

Definition 3 A feasible allocation x is Pareto optimal if there exists a positive µifor each agent i such that the allocation is the solution of the following concave
programming problem with Z inequality constraints:
max
x
s.t.

∑
i

∑

µi

∑

π(z)vi (xi (z))

zxi (z) ≤ E(t(z)), ∀ z.

(2.2)

i

Proposition 6 A feasible allocation x is Pareto optimal if and only if there exist
a positive µi for each agent i and a positive vector λ(z) of dimension C for each
Arrow-Debreu state z such that:µi ∇vi (xi (z)) = λ(z), ∀ i, z,
38

(2.3)

Chapter 2. The role of insurance companies in a risky economy∑xi (z) = E(t(z)), ∀ z,

(2.4)

i

in which ∇vi (xi (z)) is the vector of the C derivatives of vi (xi (z)) relative to eachcommodity.
Proposition 6 is the classic equivalence for the characterization of Pareto optimal allocations, obtained with the first order conditions of (2.2) (with π(z)λ(z)the positive Lagrangian multiplier of constraint z) and the binding constraints of(2.2) (because vi is strictly increasing in each argument). With this notation, λ(z)and π(z)λ(z) can also be called the vector of shadow spot prices and the vectorof shadow contingent prices. (2.3) tells that, in a Pareto optimal allocation, themarginal rate of substitution between any two Arrow-Debreu states for one commodity is equal for each agent to the ratio of Lagrangian multipliers of the two
Arrow-Debreu states for this commodity, which means that it is identical acrossagents.
Besides, one can note that the optimal allocation obtained from problem (2.2)
consists actually in separated optimal allocation problems for the different ArrowDebreu states. It is thus separable in Z programming problems which are forz = 1, .., Z:
max
x(z)

s.t.

∑

µi vi (xi (z))

i∑xi (z) = E(t(z)).

(2.5)

i

Proposition 7 (Borch (1960, 1962))6 In a Pareto optimal allocation, the consumption plans x(z) and the shadow spot prices λ(z) are identical across ArrowDebreu states z corresponding to the same fundamental state t.
6

Borch (1960, 1962) demonstrates in a model with one commodity that individual risks are

eliminated and the aggregate risk is shared in a Pareto optimal allocation. Arrow (1996) confirms
in the case of many commodities that "the allocation of consumption in any given state (i.e. in
any Arrow-Debreu state) depends only on the total in that state (i.e. on the fundamental state)".
Gollier (2004) has some more insight on this mutuality principle in Chapter 21 of his book.

39

Chapter 2. The role of insurance companies in a risky economy
Proposition 7 is due to the fact that the programming problem (2.5) is identical for all the Arrow-Debreu states z corresponding to the same fundamental
state t because they have the same aggregate endowments E(t(z)). That is why
the consumption plans x(z) are identical for all the Arrow-Debreu states z corresponding to the same fundamental state t and can be denoted x̃(t). Moreover,thanks to (2.3), the vectors λ(z) are identical for all the Arrow-Debreu states z
corresponding to the same fundamental state t and can be denoted λ̃(t). Proposition 7 states that the individual risks are fully eliminated in each fundamental
state (i.e. one agent gets the same quantity of commodity in two Arrow-Debreu
states corresponding to the same fundamental state). This result firstly shown by(Borch (1960, 1962)) is usually called the Borch mutuality principle. Besides, (2.3)
tells that the increase of a component of λ̃(t) leads to the decrease of consumptionof the corresponding commodity by all the agents because vi (.) is concave. Thismeans that, if two fundamental states have identical λ̃(t) except for one commodity, the fundamental state that has a higher shadow spot price for this commodity
necessarily has a lower aggregate endowment of this commodity. Thus, aggregate
endowment and shadow spot price for each commodity are inversely related andaggregate endowment uncertainty is translated into shadow spot price uncertainty.

2.3.2

Decentralized equilibrium à la Arrow-Debreu

I now consider a decentralized economy equipped with a contingent market for eachcommodity and each Arrow-Debreu state (i.e. a financial asset for each commodity
in each Arrow-Debreu state). A contingent market for one commodity and one
Arrow-Debreu state corresponds to the market for this commodity contingent on
this Arrow-Debreu state before the Arrow-Debreu state has been revealed. Markets are considered frictionless. In this economy, all the deals are made before theArrow-Debreu state has been revealed. The price vector for commodities contingent on the Arrow-Debreu state z is denoted π(z)p(z). At equilibrium, each agent
i maximizes her utility level under her (binding) budget constraint and the market
40

Chapter 2. The role of insurance companies in a risky economy
conditions clear:
max
xi

s.t.

∑

π(z)vi (xi (z))z∑π(z)p(z)xi (z) =

z

∑

(2.6)
π(z)p(z)ei (si (z)),

z∑xi (z) = E(t(z)), ∀ z.

(2.7)

i

Proposition 8 (Arrow (1953, 1964) and Debreu (1959))7 In the decentralized
economy with a contingent market for each commodity and each Arrow-Debreu
state (C ∗ Z contingent markets), the allocation is Pareto optimal at equilibrium.
Proposition 8 is obtained with proposition 6. By denoting ν1i the positive
Lagrangian multiplier of the constraint in problem (2.6), the consumption plan xi
is such that the first order conditions of (2.6) are verified:νi ∇vi (xi (z)) = p(z), ∀ z.

(2.8)

Thus, at equilibrium, there exist a positive µi for each i (i.e. νi ) and a positive
vector λ(z) for each z (i.e. p(z)) such that (2.3) and (2.4) are verified, which gives
proposition 8. Proposition 8 is an application of the first welfare theorem, firstly
shown by Arrow and Debreu. They have also shown that Pareto optimality isreached with a spot market for each commodity and a security market for each
Arrow-Debreu state (i.e. a financial asset for each Arrow-Debreu state) (C + Z
markets), instead C ∗ Z contingent markets.8 Thanks to proposition 7, proposition(8) tells that the consumption plans x(z) and the spot prices p(z) are identicalacross Arrow-Debreu states corresponding to the same fundamental state. Eitherwith C ∗ Z contingent markets or with C spot markets and Z security markets,
7

Arrow (1953, 1964) and Debreu (1959) have firstly shown this result. Gollier (2004) has

some more insight on the decentralized equilibrium with these types of markets in Chapter 22of his book.
8
A spot market for one commodity corresponds to the commodity market once the ArrowDebreu state has been revealed and in this case the spot market prices correspond to p(z). A
security associated to one Arrow-Debreu state enables to get 1 money unit if this Arrow-Debreu
state is revealed in exchange for π(z) money unit ex-ante.

41

Chapter 2. The role of insurance companies in a risky economy
the Arrow-Debreu economy requires at least one financial asset for each state ofnature, characterized by a full specification of the individual endowments obtainedby all the agents in the economy, which is an unrealistic setting. The first limit isthat it supposes an incommensurable number of financial assets (C ∗ Z or C + Z)
in the real world to hedge all the individual risks. Indeed, in an economy with
N = 7∗109 people, each one facing S = 10 individual states, the required number of
9financial assets is prohibitive with Z = 107∗10 . The second limit is that it requires
to make public the realized individual endowments of each agent in the economy.
Indeed, exchanges made on these markets aim at securing transfers ex-post which
are contingent on the realized Arrow-Debreu state and thus necessitate to makepublic the realized individual state of each agent in the economy. Besides, as
explained by Arrow (1996), agents buy risk sharing contracts to insure against theuncertainty of their individual endowments and not against the uncertainty of otheragent endowments. This corresponds to more usual insurance contracts between
two agents, in which one exposed agent buys coverage against her individual risk
from another agent playing the role of an insurance company, as introduced in thefollowing section.

2.4

Decentralized equilibrium with insurance companies

2.4.1

The setting

I now consider a decentralized economy equipped with a spot market for eachcommodity and a security market for each fundamental state (i.e. a financial asset
for each fundamental state). Moreover, there are M insurance companies in competition, which supply standard insurance contracts and are shared through stock
markets. All markets are frictionless. Spot markets correspond to the commodity
markets once the Arrow-Debreu state has been revealed. In the Arrow-Debreu
state z, the price vector for the C commodities is denoted p(z). With these spot
markets, agents cannot transfer wealth from one Arrow-Debreu state to another42

Chapter 2. The role of insurance companies in a risky economy
because the deals are made after the Arrow-Debreu state has been revealed. Sells
and purchases of securities, as well as purchases of insurance contracts and insurer shares, enable these transfers. I consider a security market for each of the
T fundamental states. The security associated to the fundamental state t gives
1 money unit in all the Arrow-Debreu states z corresponding to t, in exchangefor∑z∈Ft π(z) = π̃(t) money unit before the state of nature z is revealed.

The

quantity of securities "purchased" by agent i is denoted ai = (ai (1), .., ai (T )).9
I consider M insurance companies in competition. They supply fair multi-riskstandard insurance contracts. The insurance contract supplied to agent i consists in compensating her in any state si for her loss li (si ) (i.e. an indemnity
τi (p(z), si (z)) = p(z)li (si (z)) in any state z ∈ Z), in exchange for a premium αipaid before the state of nature z has been revealed. The quantity of insurance purchased by agent i is denoted ni .10 This is an insurance contract in the sense thatthe contract for one agent does not depend on the individual states of other agents(i.e. on the Arrow-Debreu state) but on her individual state si and on aggregate
data (i.e. commodity price vector p(z)). Thanks to competition and no transaction costs, the insurance contract is fair in the sense that the premium denoted αi
is equal to the expected indemnity (αi =

∑

′
′
′
z ′ π(z )p(z )li (si (z ))).

The insurance

contract is multi-risk in the sense that it covers for the different individual risks towhich the agent is exposed.11 The insurance contract is standard and not mutual in
the sense that the premium is paid ex-ante and there is no participation on the insurer profit in the contract. Each insurance company k ∈ M invests the premiums
raised∑j∈Nk αj nj in financial assets, in which Nk is the group of agents purchasing

a contract from the insurance company k. The quantity of securities purchased byinsurer k is denoted bk = (bk (1), .., bk (T )) and thus∑t π̃(t)bk (t) =∑j∈Nk αj nj . In-

surance companies are owned through stock markets. The profit of insurer k ∈ M
9ai (t) can be either positive or negative. A positive ai (t) corresponds to a security purchase.

A negative ai (t) corresponds to a security sell.
10
ni can take any positive value. For instance, ni = 0 corresponds to zero coverage, ni = 1
corresponds to full coverage and a value strictly between 0 and 1 corresponds to partial coverage.
11
Different contracts for different risks could also be offered to each agent, but one multi-risk
contract per agent is sufficient.

43

Chapter 2. The role of insurance companies in a risky economy
in state z for the shareholders is rk (z) = bk (t(z)) −∑j∈Nk τj (p(z), sj (z))nj .
share of the insurance company k purchased by agent i is denoted mki .12

2.4.2

The

The equilibrium

At equilibrium, each agent i maximizes her utility level under her (binding) budget
constraints (before and after the state of nature is revealed), plus the clearing
market conditions (for spot markets, security markets and stock markets):∑max

xi ,ni ,mki ,ai

π(z)vi (xi (z))

zs.t. p(z)xi (z) = p(z)ei (si (z)) + τi (p(z), si (z))ni +αi n i +∑∑rk (z)mki + ai (t(z)), ∀ zkπ̃(t)ai (t) = 0

t

(2.9)
∑

xi (z) = E(t(z)), ∀ z ∈ Z(2.10)i∑ai (t) +i∑bk (t) = 0, ∀ t ∈ T(2.11)k∑mki = 1, ∀ k ∈ M(2.12)

i

Proposition 9 In the decentralized economy with C spot markets for commodities,
T financial assets associated to fundamental states and stock insurance companies
in competition, the allocation is Pareto optimal at equilibrium.
Proof We show the equivalence in the consumption plan x between the ArrowDebreu economy as given by (2.6) and (2.7) and the economy with insurancecompanies:
12

I do not model the managers of insurance companies because, with multiple insurance com-panies in competition and frictionless markets, they cannot extract any rent. Besides, note thatmki cannot be negative.

44

Chapter 2. The role of insurance companies in a risky economy(i) Suppose first that x and p are the solutions of (2.6) and (2.7). Thus, (2.10)
is verified. By choosing ni = 1, mki = N1 , bk (t) =1 ∑p(z)(xi (z) − ei ) − N
xi (z) and∑∑j∈Nk αj nj and ai (t(z)) =k rk (z) (which is possible for ai (t(z)) because p(z),k rk (z) are identical across z corresponding to the same t in thiscase), the first constraint in (2.9) is verified and (2.12) is verified. Then, thefirst constraint in (2.9) with the constraint in (2.6) implies that the second
constraint in (2.9) is verified. Finally, the first constraint in (2.9) with (2.10)
and (2.12) implies that (2.11) is verified.
(ii) Suppose now that x and p are the solutions of (2.9), (2.10), (2.11) and(2.12). Thus, (2.7) is verified and the constraint in (2.6) is verified thanks tothe constraints in (2.9).
The equivalence between the two economies finally tells us that the allocation in
the economy with insurance companies is Pareto optimal at equilibrium, just as
in the Arrow-Debreu economy.

2.4.3

The role of insurance companies and financial assets

Proposition 9 tells us that the decentralized economy with insurance companies incompetition allows to reach a Pareto optimal allocation. The proof of proposition
9 shows that the consumption plan xi chosen by each agent i is the same as theone that is chosen in the Arrow-Debreu economy. It also tells that one way toreach this consumption plan for each agent i is to purchase full insurance coverage (ni = 1), an ownership share of insurance companies (mki = N1 ) and someamount of securities (ai (t) = p(z)(xi (z) − ei ) − N1∑

k rk (z) which may be positive(i.e. purchase) or negative (i.e. sell)), if the insurers invest safely the premiums
raised (bk (t) =∑j∈Nk αj nj

independent from t). The purchase of full insurance

coverage (ni = 1) enables agent i to get in the Arrow-Debreu state z the wealthp(z)ei −∑′′′z ′ π(z )p(z )li (si (z )) instead of the endowment value p(z)(ei −li (si (z))).

Contrary to the endowment value, the wealth with full insurance coverage depends
only on the fundamental state (because p(z) is identical across z corresponding to
the same fundamental state t). Thus, insurance allows each agent to fully elim45

Chapter 2. The role of insurance companies in a risky economy
inate individual risks within each fundamental state. Besides, individual endowment risks generate aggregate endowment risk characterized by the multiplicity
of fundamental states. The aggregate risk materializes for agents through commodity price risk, which affects the agent wealth with full insurance coverage(p(z)ei −∑′′′z ′ π(z )p(z )li (si (z ))).

For instance, the wealth of an agent, who isendowed with only one type of commodity, is very dependent on the price of thiscommodity. This agent may want to protect against price risk to be able to consume other commodities when the commodity price and thus her wealth are low.
By buying or selling some securities (p(z)(xi (z) − ei ) in ai (t)), agent i is protectedagainst the price risk in order to reach the targeted consumption plan xi . Thus,
securities allow each agent to hedge the price risk.13 In her targeted consumption
plan, each agent chooses a fraction of the aggregate risk thanks to shares in stock
insurance companies and sells/purchases of securities. The share of the aggregate
risk targeted by agent i depend on her risk aversion relative to others: the higher
the risk aversion relative to others, the lower the purchased share of the aggregate
risk.

2.4.4

Insurance prices

Insurance companies have to sell fair contracts to catch policyholders on one sideand shareholders on the other side because of competition. On the one hand, competition between insurance companies lead them to decrease their prices at least
until fair prices to catch some policyholders. On the other hand, they cannot decrease them below fair prices, otherwise they would not have enough shareholders
for the insurance stock market to be at equilibrium. As insurance for individual
risks is fairly priced, agents have incentives to purchase full coverage for their individual endowment losses (ni = 1). For agent i, a full coverage insurance contract
gives an indemnity corresponding to the individual loss in any state z ∈ Z, in exchange for a premium equal to the expected indemnity before the state of nature
has been revealed. The insurance premium αi =
13

′
′
′
z ′ π(z )p(z )li (si (z )) for agent

The price risk could also be hedged with other security derivative financial products such as

options.

46∑

Chapter 2. The role of insurance companies in a risky economy
i can also be written:
αi = (1 + γi )p∑π(z ′ )li (si (z ′ )),(2.13)z′in which p =∑′′z ′ π(z )p(z ) is the average commodity price vector and γi is anaggregate risk loading factor such that:
γi =

p

∑
1
π(z ′ )(p(z ′ ) − p)li (si (z ′ )).
′ )l (s (z ′ ))
π(z
′
′
i
i
z
z

∑

(2.14)(2.13) gives an expression for the insurance premium, in which the loading factor γi
expressed in (2.14) captures the correlation between the individual risk (expressedin the uncertainty of li (si (z ′ ))) and the aggregate risk (expressed in the uncertainty of p(z ′ ) − p). For a given loading factor, the higher the individual risk, thehigher the premium. For a given individual risk, the higher the correlation withthe aggregate risk, the higher the correlation with the commodity price risk andthe higher the premium.14 Contrary to the cases with independently and identically distributed risks, the premium and the loading factor are not uniform acrossagents. Bad fundamental states with high aggregate losses are characterized byhigher prices than good fundamental states with low aggregate losses. The loading
factor is positive (respectively negative) if the agent individual risk is positively(respectively negatively) correlated with the price risk. With some aggregate risk,
the factor is positive for the agents facing higher losses in bad fundamental states
than in good fundamental states, whereas it is negative for the agents facing the
opposite. Without any aggregate risk, the factor is null for any agent becausep(z ′ ) = p for any z ′ . Shareholders of an insurance company are exposed to therisk resulting from the pool of individual risks insured by the company. The higher
the correlation between these risks and the aggregate risk, the higher the raised
premiums. Shareholders raise higher premiums in this case at equilibrium because
they cannot diversify a higher part of the company risk. Lastly, as the aggregate
14

If agents were able to reduce their risk through prevention expenses, they would have in-centives through insurance premiums to reduce both their individual risks and their correlation
with the aggregate risk. In other words, insurance premiums would give individual incentives toreduce both the aggregate endowment losses and the aggregate endowment uncertainty.

47

Chapter 2. The role of insurance companies in a risky economy
risk results from the aggregation of individual risks, there are globally more individual endowment risks positively correlated with the aggregate risk, which meansthat the average loading factor in the economy is positive.

2.5

Conclusion

I extend the analysis of the role played by insurance companies to an economy
with multiple commodities and agents having heterogeneous preferences and facing
heterogeneous risks. With competitive insurance companies that supply fair multirisk insurance contracts and are owned through stock markets, Pareto optimality
is reached with a reduced number of financial assets relative to the Arrow-Debreu
setting. Instead of one financial asset per Arrow-Debreu state (characterized by afull specification of the individual endowments of all the agents in the economy), it
is sufficient to have only one financial asset per fundamental state (characterized by
the aggregate endowments in the economy). The higher the risk faced by one agent
or the higher the correlation between her risk and the aggregate risk, the higher
her premium. Correlation between individual risks does not prevent individual
risks to be fully insured, even though it can generate aggregate risk which is notinsurable. Even insurance companies weakly diversified are able to sell contracts
at fair prices, because the shareholders diversify their risk with other insurance
stocks and securities. However, this result is valid only without market failures.
In particular, agents have to be liable for insurance company shares they purchaseand securities they sell, which means that they have to bear insurance company
deficits if some occur and to pay amounts of securities they owe in any state ofnature. In the case of asymmetric information on the capacity of agents to honor
these transactions, public policies require agents to build financial reserve in order
to limit default. Yet financial reserves are costly to build for instance because of
the opportunity cost. In this case, risk correlation becomes an issue for insurancecompanies because it leads to more variability on total claims and thus requires
more financial reserves, as it appears for natural disaster risks.

48

Chapter 3
Insurability of low-probability
catastrophic risksThis chapter is co-authored with Alexis Louaas.

Abstract: We analyze how the risk probability affects the insurance purchaseof rational and well-informed risk-averse individuals. With standard insurance
costs and competitive pricing, we show that agents are more inclined to insure
for low-probability risks than for high-probability risks. Yet, these observations
are at odds with the low insurance take-up rates for low-probability catastrophic
risks (e.g. earthquakes and floods). Our explanation is that the risks for which
underinsurance is most prevalent display substantial aggregate uncertainty. This
uncertainty generates an additional fixed cost for insurers, which increases theinsurance loading factor when the loss probability decreases and eventually discourages people from purchasing coverage.

Keywords: low-probability risks, catastrophic risks, insurance, reserve.

JEL classification: D86, G22, G28, Q54.
49

Chapter 3. Insurability of low-probability catastrophic risks

3.1

Introduction

Why are catastrophic events so difficult to insure? Natural and man-made catastrophes such as earthquakes, floods, nuclear accidents and terrorists attacks havesevere consequences at the individual level, affect large numbers of people at thesame time, but have small individual probabilities of occurrence. The extreme
severity of these risks advocates for a wide-spread use of prevention and insurance. Yet, they are typically excluded from US homeowner policies.1 Private
insurance markets are often non-existant and even when specific contracts benefiting from public subsidies are available, relatively few people purchase them.2 It
is also puzzling to notice that some low-probability risks such as damages fromlightnings are efficiently handled by the insurance sector and covered under the US
standard homeowner policies whereas others, such as earthquakes or floods, arenot. Our model explains this puzzle by showing that the effect of aggregate uncertainty, a well-known threat to insurability, is amplified when the individual loss
probability is low. Low-probability risks characterized by aggregate uncertainty,
such as earthquakes and floods, are therefore predicted to feature lower take-up
rates than low-probability risks without aggregate uncertainty.
We begin by examining the case without aggregate uncertainty. In this framework, we show that low-probability risks are actually easier to insure than highprobability risks. It is known since Mossin (1968) that expected utility maximizing
agents optimally purchase partial coverage when insurance is sold above actuarially fair price. In this case, a decrease in the loss probability has two effects on
the demand for coverage. On the one hand, the cost of providing insurance diminishes, which translates into lower premiums for policyholders. On the other hand,
the likelihood of receiving the indemnity declines as well. Our contribution inthis framework is to show that the cost-reduction effect dominates the likelihoodreduction effect under risk aversion. As the loss probability declines, the ratio of
1
2

http://www.iii.org/article/which-disasters-are-covered-by-homeowners-insurance
Kousky & Kunreuther (2014) document low take-up rates for catastrophe insurance while

Cole et al. (2014) try to explain low rates in the micro-insurance industry.

50

Chapter 3. Insurability of low-probability catastrophic risks
willingness to pay to cost of coverage always increases. In a population heterogeneous in wealth or preferences, take-up rates are therefore predicted to increase as
the loss probability decreases.3 In addition, we show that given the opportunity to
do so, people optimally choose higher levels of coverage for low-probability risks if
their index of absolute risk aversion does not decrease too fast with wealth. These
results are in line with Laury et al. (2009), who experiment in the laboratory theeffect of a change in the loss probability. A previous experiment by McClelland
et al. (1993) also found a decreasing mean ratio of willingness to pay to expectedindemnity, indicating that people tend to be willing to pay higher loadings for
low-probability risks than for high-probability risks.
These observations, however, are at odds with the low insurance take-up rates
for low-probability catastrophic risks. Our explanation is that the risks for which
underinsurance is most prevalent display substantial aggregate uncertainty. Natural and man-made catastrophes feature geographically correlated individual losses
which translates into aggregate loss uncertainty, even within very large pools ofpolicyholders. In order to remain solvent, the insurance provider must either raiseprohibitively high levels of premiums or more realistically, it must have access tocapital to fill the gap between the premiums raised and the amount of claims due
in case of catastrophe. The cost of allocating this capital to a specific line of business4 depends on the size of the potential worst case aggregate loss. This fixed costcauses the loading to increase when the loss probability decreases and eventually
discourages people from purchasing coverage. In a world where agents are heterogeneous in wealth or preferences, a decrease in probability may therefore induce
lower take-up rates. This prediction is much more in line with the observations of
the take-up rates in various settings.
Several literatures have attempted to explain the low insurance take-up rates
for low-probability catastrophic risks. Kunreuther & Slovic (1978) and Kunreuther
et al. (2001) rely on departures from the expected utility paradigm, arguing that
3With the exception of the pathological case of Giffen behaviors which, to our knowledge havenever been observed in an insurance market.
4
See Zanjani (2002) and Froot (2001).

51

Chapter 3. Insurability of low-probability catastrophic risks
low-probabilities are more difficult to process than high probabilities. In the expected utility framework, Kunreuther et al. (2001) show that search costs may
generate a probability threshold below which coverage is not purchased while
Coate (1995) shows how government relief can crowd-out the demand for insurance. Raschky et al. (2013) and Kousky et al. (2013) find empirical support for thishypothesis in various countries. In the context of micro-insurance markets, Coleet al. (2014) suggests that learning is an important determinant of the demand for
insurance. Overall, much less attention has been devoted to the supply side of themarket. Kunreuther et al. (1995) explain that ambiguity may provide a rationale
for high disaster insurance premiums. In the context of micro-insurance markets,
Mobarak & Rosenzweig (2013) argue that basis risk can explain low take-up rates.
Finally, Jaffee & Russell (1997) and Kousky & Cooke (2012) suggest that the low
take-up rates are due to the high premiums required by insurers that have to secure
costly capital for catastrophic events.
Our contribution to the literature is twofold. First, we provide a simple, yetgeneral framework, that provides an alternative explanation to the observed low
take-up rates for disaster risks.5 Second, we explain this apparently puzzlingobservation that some low-probability events are well insured while others are not.
It is in fact the combination of aggregate loss uncertainty with low-probability
that makes earthquakes, floods and terrorism risks difficult to insure.

3.2

A model with aggregate loss uncertainty

This section lays out the framework for the analysis of the insurability of lowprobability events. Our model permits the analysis of correlated risks and encompasses the standard independent losses model as a particular case.
We consider a population of unit mass, in which each individual may or may
5

Jaffee & Russell (1997) discuss the problem of capital allocation cost but provide no for-mal model while Kousky & Cooke (2012) have a simulated model, calibrated to analyze flood
insurance coverage in Broward County, Florida.

52

Chapter 3. Insurability of low-probability catastrophic risks
not be affected by a loss of size L. Similarly to Charpentier & Le Maux (2014), theshare of the population affected by a loss is a random variable q̃. It is distributedwith cumulative distribution function F (.) defined and increasing on [0, 1], withexpected value p and variance σ 2 . From the insurer’s standpoint, there is therefore
a continuum of states of the world, that we call aggregate states, characterized bya fraction q of the population being affected by a loss. Notice that almost alllines in the insurance business feature some kind of aggregate loss uncertainty.
Finding the distribution of the number of claims and therefore of the fraction of
the policyholders who will file a claim, constitutes an important task of actuaries.
Conditionally on the occurrence of a particular aggregate state of the world,
losses are independently distributed across people. Calling x̃i the random lossof agent i, the probability that an individual faces a loss is therefore p.6 The
individual loss is therefore a Bernouilli random variable with parameter p equal tothe expected value of q̃. Individual losses are in general not independent randomvariables. We show in Appendix (3.6.1) that the coefficient of correlation between
2σ
two individual risks can be written δ = p(1−p)≤ 1. Given a fixed expected valuep, increasing the variance of the distribution F always increases the correlation
coefficient. For a given σ 2 , a decrease in p results in an increase in δ if p < 1/2.
This model of loss exposure nests the usual independent losses model as a specific case in which the fraction q is known. In this case, the probability cumulative
distribution function F (.) is such that F (x) = 0 for x ≤ q and F (x) = 1 for x ≥ q.
Individual losses are usual Bernoulli random variables with parameter p = q andsince σ 2 = 0, the coefficient of correlation δ is null.
Finally, we consider that agents are risk-averse with a twice continuously dif′′(x)
ferentiable and concave utility function u(x) and initial wealth w. A(x) = − uu′ (x)denotes the Arrow-Pratt index of absolute risk aversion.

6

The probability that an individual faces a loss is∫10

P(x̃i = L|q̃ = q)dF (q) =∫10

qdF (q) = p.

53

Chapter 3. Insurability of low-probability catastrophic risks

3.3

Insurability without aggregate uncertainty

In this section, we show that in the baseline insurance framework with independent
losses, low-probability events are in fact more likely to be insured than higher
probability events. The intuition behind this result is that individual’s willingness
to pay is concave in the probability, while the cheapest feasible contract is linear.
Therefore, if a risk is insurable somewhere over the parameter space, there is a
probability threshold below which it is insurable and above which it is not.
The insurance provider can be private or public. It is represented by a riskneutral agents that sells an amount of coverage τ ∈ [0, L] at a premium α. Following Raviv (1979) we assume that, in addition to the payment of the indemnity
τ , the insurance provider faces a cost c(τ ) with:c(0) = 0,c′ (0) = b > 0,c′ (τ ) > 0 and c′′ (τ ) ≥ 0,

which represents the various expenses associated with the payment of an indemnity
τ to all affected agents. It can be interpreted as an administrative cost, as acost of expertise, or more broadly as a dead-weight loss, resulting either from anasymmetry of information between the insurance company and the policy holder,
or by imperfect competition. The actuarial and insurance literatures often makethe simplifying assumption that the marginal cost c′ (τ ) is equal to a constant λcalled the loading factor. In this case, the dead-weight cost is simply a fractionof the indemnity. We call loading the ratio α/pτ premium to expected indemnity.
If c′ (τ ) = λ, the loading is 1 + λ and we have c′′ (τ ) = 0, which is indeed aparticular case of our model. A necessary condition for the insurance to provide
such a contract is that its expected profit is positive. This motivates the following
definition.
Definition 4 A contract is called feasible if and only if the insurance provider canrealize at least a zero expected profit.
In the absence of aggregate loss uncertainty, a contract is therefore feasible if andonly if:α ≥ pτ + pc(τ ).
54

Chapter 3. Insurability of low-probability catastrophic risks

3.3.1

Strong insurability

The first notion of insurability that we develop is the following.
Definition 5 A risk is strongly insurable at a level τ if and only if individuals arewilling to purchase the proposed level of coverage τ of some feasible contract.
In order to know whether a risk is strongly insurable or not, it is sufficient toverify that individuals are willing to purchase the zero expected profit contract,
for which the insurer breaks even. If they reject the zero expected profit feasible
contract, agents will also reject all the other more expansive contracts, and if they
accept the zero expected profit feasible contract, then the risk is insurable. In theremaining of the paper, we use the word contract to mean zero expected profit
contract.
Independently of the supply side constraints, the highest price C(p, τ ) that anindividual would pay for a level of coverage τ is given by:pu(w − L + τ − C) + (1 − p)u(w − C) = pu(w − L) + (1 − p)u(w).

(3.1)

First notice that C(0, τ ) = 0 and C(1, τ ) = τ . The willingness to pay for thecoverage of a a zero probability event is zero and the willingness to pay for thecoverage of a sure event is just the coverage itself. Total differentiation of (3.1)
gives:
Cp′ (p, τ ) =u(w) − u(w − L) − [u(w − C) − u(w − L + τ − C)]≥ 0 ∀τ ≤ L,
pu′ (w − L + τ − C) + (1 − p)u′ (w − C)u′ (w − L + τ − C) − u′ (w − C)pu′ (w − L + τ − C) + (1 − p)u′ (w − C)pu′′ (w − L + τ − C) + (1 − p)u′′ (w − C)′
+ (Cp )2 ′≤ 0 ∀τ ≤ L,
pu (w − L + τ − C) + (1 − p)u′ (w − C)′′Cpp(p, τ ) = − 2Cp′′′
represent the first and second order partial derivatives with
where Cp′ and Cpp

respect to p. The agent’s willingness to pay C is therefore increasing and concavein the probability of loss p.
55

Chapter 3. Insurability of low-probability catastrophic risksp(c(τ ) + τ )C(p, τ )C(p, τ )
pτ

pτ
•

.
0

1

p

(a) Actuarially fair coverage

.
0

p∗

1

p

(b) Non-actuarially fair coverage

Figure 3.1: Coverage without aggregate loss uncertainty
Figure 3.1a represents the agent’s willingness to pay and the cost of coverage asa function of the loss probability p for the case c(τ ) = 0. When coverage is sold at
an actuarially fair price, the agent is always willing to purchase it. The surplus she
derives from the transaction however, may vary with the loss probability. Figure
3.1b represents the case where coverage is available at a cost higher than theactuarially fair price. For values of p lower than a threshold p∗ , willingness to pay
is above the price of coverage. An insurance market should therefore emerge inthis case. For values of p higher than p∗ , willingness to pay is below the price ofthe zero expected profit contract, resulting in the absence of any market.
Proposition 10 In the absence of aggregate loss uncertainty, a risk is strongly
insurable at level τ if and only if the individual probability of loss p is below a
threshold p∗ , where p∗ is such that:
C(p∗ , τ ) = p∗ τ + p∗ c(τ ).
For a given level of coverage, low-probability events are therefore more likely
to be covered than high-probability events. Laury et al. (2009) investigate inan experiment how insurance purchase decisions evolve with p. They observe
56

Chapter 3. Insurability of low-probability catastrophic risks
that the fraction of their sample that purchases full-coverage decreases with theprobability of loss p, given a constant expected loss and loading. Proposition
10 confirms that this should indeed be the case but delivers an even stronger
prediction: the fraction of a population that purchases coverage should increase as
p decreases even when the loss L is fixed. To see this, assume that the populationis heterogeneous in wealth and preferences. An agent i endowed with wealth wi
and utility function ui purchases a feasible contract providing coverage τ if andonly if Ci (p, τ ) ≥ pτ + pc(τ ). Each individual therefore has a probability thresholdp∗i above which she stops purchasing insurance. The distribution of wealth andpreferences generates a distribution G over the thresholds p∗i . The fraction of the
population that purchases insurance is P(p∗i ≥ p) = 1 − G(p∗i ), which is decreasing
in p.
An insurance contract providing a level of coverage τ is therefore more likely tobe purchased when the probability of loss p is smaller. In this sense, low-probability
events are more insurable than higher-probability events. However, the exogeneityof the level of coverage τ may appear as a limit to this analysis. If people havecontrol over the level of coverage, the decision is not a take-it-or-leave-it problem
anymore. With this idea in mind, we propose the following notion of insurability.

3.3.2

Weak insurability

The second notion of insurability that we develop is the following.
Definition 6 A risk is weakly insurable if and only if individuals are willing topurchase a positive amount of coverage of some feasible contract.
This notion of insurability is weaker in the sense that a strongly insurable risk isnecessarily weakly insurable. If a person agrees to purchase a level of coverage τrather than no insurance when the loss probability is p, she would never optimallychoose τ = 0. However, she may select a level of coverage lower than τ to reducethe cost of insurance.
57

Chapter 3. Insurability of low-probability catastrophic risks
With endogenous coverage, the agent solves:
max pu(w − L + τ − α) + (1 − p)u(w − α)τs.t. α = pτ + pc(τ ), τ ≥ 0.

(3.2)

The first order condition of this problem is:[1 − (1 + c′ (τ ))p)]u′ (w1 ) = (1 − p)(1 + c′ (τ ))u′ (w2 ),(3.3)

in which w1 = w − L + τ − pτ − pc(τ ) and w2 = w − pτ − pc(τ ) are the levels
of wealth in the loss and no-loss states. It is easy to check that c′ (τ ) > 0 implies
that the agent chooses partial coverage, so that τ < L at any interior solution.
Individuals purchase a positive amount of coverage if and only if:(1 − p)(1 + b)u′ (w) < [1 − (1 + b)p]u′ (w − L).

(3.4)

Re-arranging the terms of (3.4) yields the following Proposition.
Proposition 11 In the absence of aggregate loss uncertainty, a risk is weakly
insurable if and only if its probability p is such that:p<1u′ (w)b−.
1 + b 1 + b u′ (w − L) − u′ (w)(3.5)

For a given p, an event is uninsurable when the marginal cost of coverageb = c′ (0) is too high or the size of the loss L is not sufficiently large to generate
a significant difference between marginal utility in the loss state u′ (w − L) andmarginal utility in the no-loss state u′ (w). Proposition 11 stresses once more that,in the absence of aggregate loss uncertainty, low-probability events are easier toinsure than high-probability events. In addition, proposition 12 and its corollary
give conditions under which a strictly positive optimal coverage increases when
the probability p decreases. When the loss probability diminishes, two effects
interact. On the one hand, the risk of experiencing the loss L diminishes, lowering
the incentive to pay the dead-weight cost. On the other hand, the dead-weight
marginal cost pc′ (τ ) diminishes, making insurance more attractive at the margin.
The total effect on optimal coverage cannot be signed for any utility function, but
the following proposition and corollary enables to identify some interesting andrealistic cases.
58

Chapter 3. Insurability of low-probability catastrophic risks
Proposition 12 In the absence of aggregate loss uncertainty, the optimal coverageτ of a weakly insurable risk is strictly decreasing in p if and only if:
A(w1 ) − A(w2 ) <c′ (τ ).
[τ + c(τ )](1 − p)[1 − (1 + c′ (τ ))p]

The proof of proposition 12 is given in Appendix 3.6.2. Note that the righthand side of the inequality in proposition 12 is positive when 1 − (1 + c′ (τ ))p > 0,
which is necessary to have an interior solution. Since w1 < w2 , the condition
in proposition 12 is trivially satisfied for any increasing or constant absolute risk
aversion functions (IARA or CARA). For the class of decreasing absolute risk
aversion (DARA), the condition puts an upper bound on the variation of risk
aversion between the loss and the no-loss state. The empirical literature most often
fails to reject DARA as a realistic hypothesis such as in Guiso & Paiella (2008) andLevy (1994). We would therefore like to know whether classical utility functions,satisfying the DARA property, also feature an optimal coverage decreasing in theprobability p. This is the purpose of Corollary 12.1 that deals with the case ofHarmonic Absolute Risk Aversion (HARA) functions. We define a HARA utility
function as in Gollier (2004):(

u(x) = ζ η +x )1−γ,γ

whose domain is such that η + γx > 0 and the condition ζ 1−γ> 0 guaranteesγthat the function is indeed increasing and concave. The coefficient of absolute risk
aversion is:(A(x) = η +

x )−1
.
γ

(3.6)

Except for the limit case γ → +∞, the HARA functions satisfy the DARA property when γ > 0, which makes them appealing with respect to the literature
discussed previously.
Corollary 12.1 If a risk-averse agent has preferences represented by a HARA
utility function with γ ≥ 1, then the optimal coverage τ of a weakly insurable risk
is strictly decreasing in p in the absence of aggregate loss uncertainty.
59

Chapter 3. Insurability of low-probability catastrophic risks
The proof of corollary 12.1 is given in Appendix 3.6.3. The HARA class nests
two of the most widely used classes of utility functions. The Constant Absolute
Risk Aversion (CARA) is obtained when γ → +∞. Solving the differential equation (3.6) yields the specification of this function:1
u(x) = −η exp (− x).
(3.7)
η
The second class of functions within the HARA class is the set of Constant Relative
Risk Aversion (CRRA) which is obtained for η = 0. Solving (3.6) in this case yields:
x1−γ
.
(3.8)
1−γ
If people’s preferences can be represented by a CARA or by a CRRA utility funcu(x) =tion with γ ≥ 1, then their optimal coverage is always a decreasing function of p.
For the CRRA case, Szpiro (1986) and Barsky et al. (1997) find values of relativerisk aversion respectively between 1.2 and 1.8 for the first and 4.17 for the second.
According to Gollier (2004) (p.69), the “range of acceptable values of relative risk
aversion [is] [1, 4]”. A complete survey of the literature on risk preferences elicitation would reveal some estimates below one, such as Chetty (2006), but overall our(sufficient but not necessary) condition γ ≥ 1 seems a very plausible assumption.
Besides, Louaas & Picard (2014) have shown the following proposition.
Proposition 13 In the absence of aggregate loss uncertainty and if u′ (w − L) >(1 + b)u′ (w), the optimal coverage converges, as p goes to 0, toward a positive limit
τ such that:u′ (w − L + τ ) = (1 + c′ (τ ))u′ (w).
It may sound surprising that the agent is willing to pay a positive loading factor
for a risk whose probability tends to zero, but the pricing rule α = pτ + pc(τ )implies that the premium tends to zero with the probability. The agent therefore
receives an indemnity with an infinitesimal probability whose price also becomes
infinitesimal.
Finally, figure 3.2 gives an illustration of the previous results in the (w1 , w2 )space. Point A represents the optimal lottery at the highest probability p for which
60

Chapter 3. Insurability of low-probability catastrophic risks
the insured chooses a positive amount of coverage. The thick curve represents
the locus of optimal lotteries as p diminishes and Point B is the limit optimallottery when p → 0. In agreement with proposition 12, the optimal lottery getscloser from the 45 degree line as p decreases, until it (almost) reaches point B
when p → 0. Our theoretical results indicate that under reasonable assumption
on individual’s preferences, low-probability events should be more insured thanhigher probability events. Gollier (1997) shows that the deductible chosen bya risk averse agent is increasing in p when the agent’s utility function features
IARA and p is sufficiently small, which is equivalent7 to purchase more insurance
when the probability decreases. Our results are more general since they apply
everywhere on the domain of p where the agent purchases a positive amount ofcoverage. In addition, we are able to accommodate some of the more realistic
DARA utility functions.
Note that low-probability and high-stake risks are usually characterized by alarge potential loss L. In order to compare insurance choice for these risks tochoices concerning more traditional risks, it may be useful to fix the expected loss,
as in Laury et al. (2009), so that the loss increases as its probability diminishes. In
this case and independently of any assumption on individual’s preferences, agentspurchase more coverage as the probability of loss decreases.
Proposition 14 In the absence of aggregate loss uncertainty, and if the expected
loss pL remains constant, the optimal coverage is a decreasing function of theprobability p for any risk averse individual.
The proof of proposition 14 is given in Appendix 3.6.4. The requirement that
expected loss be fixed makes it easier for the condition of proposition 12 to hold.
Indeed as p diminishes, the loss becomes larger, providing agents with additional
incentives to purchase insurance. This in turn reduces the gap between wealth in
the non loss state w1 and wealth in the loss state w2 .
7

In a model where agents only face two states: loss or no loss, deductible and partial coverage

are strictly equivalent.

61

Chapter 3. Insurability of low-probability catastrophic risks−3

Figure 3.2: w = 10000, L = 5000, u(x) = x 3This section has shown that in the absence of aggregate loss uncertainty, lowprobability risks are easier to insure than high-probability risks. This may explain
why typical homeowners insurance cover perils as unlikely as lightening strikes.
In contrast, low-probability correlated risks such as earthquakes, flooding, windstorms, nuclear hazards and acts of wars are typically excluded from homeownerpolicies. The next section provides an explanation as to why low-probability correlated risks are also difficult to insure.

3.4

Insurability with aggregate uncertainty

This section considers the case of dependent risks. We relax the assumption thatthe fraction q of the population affected by the loss is known. Instead, q̃ is a62

Chapter 3. Insurability of low-probability catastrophic risks
random variable with cumulative distribution function F (q), expected value p andvariance σ 2 . Because the insurance company raises the premiums ex-ante, the total
amount of premium raised is unlikely to match the amount required to pay-off all
claims. When it raises a premium α, it must therefore have access to an amountof capital per policy k such that:k = q̄[τ + c(τ )] − α,(3.9)

where q̄ is the highest fraction of affected policyholders with a non-zero measure.
This capital can be financed with an internal reserve of liquidities, with someform of reinsurance contract8 or by borrowing.9 In any case, the availability ofthis capital comes at a cost per monetary unit called r. When k is a reserve, rrepresents the foregone investment opportunities associated with having to keep
the reserve. If the insurance company is private, r can be measured as the internalrate of return of the insurance company. If the insurance provider is public, r canbe seen as the cost of public funds. When k is a reinsurance or an alternative risk
transfer contract, r is simply the rate of return paid to the seller of the contract.
The zero expected profit feasible contract is therefore such that:α − (τ + c(τ ))∫ 1
0

qdF (q) − kr = 0.

Replacing k by its expression (3.9) yields the price of the contract:τ + c(τ )(p + rq̄).
1+r
In the absence of aggregate loss uncertainty, q̄ = p and we find a lower premiumα=α = pτ + pc(τ ) which corresponds to the premium derived in the previous section.
Aggregate loss uncertainty therefore generates an additional cost determined by
the cost of capital allocation and by the tail of the distribution.10
8

Insurance providers traditionally rely on reinsurance companies to transfer the layers of risk

they do not wish to retain. The past two decades have also seen the emergence of Insurance
Linked Security (ILS) markets, enabling insurance and reinsurance providers to transfer layersof risk to the financial markets.
9This is particularly true for public insurance providers, who have greater access to capitalthrough fiscal policy.
10
We assume here that the insurance company is not allowed to default. In practice, insurancecompanies are subject to regulatory solvency constraints that impose a probability of default

63Chapter 3. Insurability of low-probability catastrophic risks

3.4.1

Strong insurability

Figure 3.3b displays the highest price that an agent is willing to pay C(p, τ ) andthe price of the zero expected profit feasible contract in the presence of aggregate
loss uncertainty. The two curves intersect at p and p, hence defining three zones on
the [0, 1] interval. Coverage is purchased on the domain [p, p]. Below p, willingness
to pay falls below the cost of insurance, which explains the failure to insure lowprobability correlated risks.
Proposition 15 If a risk is strongly insurable at some probability level, then in
the presence of aggregate loss uncertainty, there exist a probability threshold p,
below which the risk becomes uninsurable (in the strong sense), with p such that:p = min{p|C(p, τ ) =τ + c(τ )(p + rq̄)}.
1+rThis result underlies the fact that in the presence of aggregate loss uncertainty,
low-probability risks are indeed more difficult to insure because of the cost of thereserve.

3.4.2

Weak insurability

As in the section without aggregate uncertainty, we now proceed to endogenize thelevel of coverage τ . People are now able to lower the coverage as the probability
decreases, hence mitigating the increase in the loading α/pτ . The agent’s programis:max pu(w + τ − α − L) + (1 − p)u(w − α)τ)s.t. α = τ +c(τ(p + rq̄), τ ≥ 0,
1+rlower than an exogenously defined level η. On the one hand, it would require the insurance
provider to have access to a lower level of capital k = F −1 (1 − η)[τ + c(τ )] − α, which would−1) ∫ F (1−η)
translate in a lower premium α = τ +c(τqdF (q) + rF −1 (1 − η)). On the other hand,
1+r ( 0policyholders would have weaker coverage because of the default risk and default could generate
additional costs. Thus, the optimal default risk depends on the tradeoff between benefit andcosts but is not the focus of the present paper.

64

Chapter 3. Insurability of low-probability catastrophic risks
τ +c(τ )(p + rq̄)
1+rp(c(τ ) + τ )C(p, τ )p(c(τ ) + τ )C(p, τ )

•

•

•

.
0

p∗

1

p

.
0 p

p

p∗

1

p

(a) Non-actuarially fair coverage with(b) Non-actuarially fair coverage withno aggregate loss uncertainty

aggregate loss uncertainty

Figure 3.3: Coverage without and with aggregate loss uncertainty
and the associated first order condition is:p[1 −1 + c′ (τ )1 + c′ (τ )(p + rq̄)]u′ (w1 ) = (1 − p)
(p + rq̄)u′ (w2 ).
1+r
1+r

The problem has an interior solution if and only if:11 Q(p) = −p2 [u′ (w − L) −1+b ′u′ (w)]+p[ 1+ru (w −L)+rq̄(u′ (w)−u′ (w −L))−u′ (w)]−rq̄u′ (w) > 0. Remark thatthe left-hand side of this last inequality is quadratic in p with a negative intercept.
Figure 3.4 represents an example of this quadratic form. There may or may notexist an interval of p where the inequality is satisfied (where the agent purchases a
positive amount of coverage), depending on the relative values of the parameters,
but when such an interval exists, it is delimited downward by a lower bound p∗∗ .
Proposition 16 If a risk is weakly insurable at some probability level, then in
the presence of aggregate loss uncertainty, there exists a probability threshold p∗∗ ,
11

1+b
1+b
From the first order condition, we obtain p[1− 1+r(p+rq̄)]u′ (w−L)−(1−p) 1+r(p+rq̄)u′ (w) >0.

65

Chapter 3. Insurability of low-probability catastrophic risks

0.

p

1
Q(p)

Figure 3.4: Q(p)below which the risk becomes uninsurable (in the weak sense), with p∗∗ such that:p∗∗ = min{p|Q(p) = 0}.
This results emphasizes that even when the agent is free to choose the level ofcoverage τ , there is a probability threshold p∗∗ below which she will stop purchasing
insurance. This also implies the following corollary.
Corollary 16.1 In the presence of aggregate loss uncertainty, the optimal coverage tends to zero as the probability p goes to 0.
This contrasts with proposition 13 of the previous section. In the presence of
aggregate loss uncertainty, the premium α has a component that does not varywith p. As p becomes very small the cost-reduction effect of a decrease in p vanishes
but the likelihood-reduction effect remains and eventually dominates completely,
leading the agent to stop purchasing insurance.
Finally, figure 3.5 shows a calibrated example in the simple case where q̃ takes
value q = 0.5 with probability π and 0 with probability 1 − π. The individual
probability of loss is p = πq. The locus of optimal lotteries as the probability πvaries is represented by the set of dark dots. When π = 1, p = 0.5 is too large forthe agent to purchase any coverage. The optimal lottery is therefore (w, w − L),
which is located at the extreme south-east part of the graph. As π diminishes,
the cost-reduction effect dominates, driving the optimal lottery closer from the 45
66

Chapter 3. Insurability of low-probability catastrophic risks
degree line. At a certain point however, the cost-reduction effect becomes weaker
and the likelihood-reduction effect dominates, leading coverage down to zero in
the limit. The optimal lottery returns to its original (w, w − L) position.

−3

Figure 3.5: w = 10000, L = 5000, u(x) = x 3 , r = 0.05, λ = 0.3, q = 0.5
The actuarial and insurance literatures often use the loading α/pτ as a measureof the cost of insurance. Apart from the case of Giffen behaviors, the higher theloading, the lower the demand for insurance coverage.12)
Proposition 17 In the presence of aggregate loss uncertainty, the loading ττ +c(τ
(1+
(1+r)
rq̄
) is a decreasing function of p and diverges to +∞ as p goes to 0.
p
12

Some DARA utility functions generate Giffen behaviors as shown by Briys et al. (1989).

However, to our knowledge, empirical analysis on the purchase of natural disaster insurance
have not observed such behaviors. For instance, Browne & Hoyt (2000) and Grace et al. (2004)
observe that when insurance price increases, the demand for insurance decreases.

67

Chapter 3. Insurability of low-probability catastrophic risks
In the absence of aggregate loss uncertainty, or if the cost of capital allocation
is null, the loading 1 + c(τ )/τ does not depend on the probability p. In this case,the simplifying assumption c′ (τ ) = λ gives an accurate and useful approximationof the price of insurance. However, it becomes less accurate when, holding otherfactors constant, the loss probability p becomes small. Empirical studies on the
demand for insurance should therefore take this effect into account. Although
we have little doubt that the traditional explanations of low take-up rates havea significant importance in the decision to purchase insurance, their effects would
be overestimated should one omit to control for variations in the loading factor
induced by changes in the loss probability.

3.5

Conclusion

We have shown that the low take-up rates for low-probability event coverage cannot be explained by rational behaviors in the absence of aggregate loss uncertainty.
In the presence of aggregate loss uncertainty however, the cost of capital allocation may represent a significant obstacle for the insurability of low-probability
events. The fact that uncorrelated low-probability events such as lightning strikes
do not posit any insurability issue while correlated low-probability events such as
hurricanes and nuclear accidents do, supports our theory. It is also interesting
to remark that acts of terrorism were included in standard US homeowner policies until early 2002. One month after the 9/11 attacks, the Insurance Services
Office filed a request to exclude acts of terrorism from standard US homeownerpolicies (Kunreuther & Michel-Kerjan (2005)). In a very similar way, the supply of coverage against losses due to earthquakes dramatically dropped after the
1994 Northridge earthquake that devastated California and generated more than
$24 billions in insured losses. These facts suggest that supply side explanations
account for a great share of the insurability problem for low-probability events.
The explanation we propose is that the occurrence of large catastrophes changes
the way actors of the insurance industry perceive the risks they insure. The 9/11
attacks and the Northridge earthquake made insurers realize that the worst case68

Chapter 3. Insurability of low-probability catastrophic risks
scenario was even worst than what they had imagined so far. The impact on theinsurability of low-probability events is particularly salient because even small premium increase may generate important changes on the implied loading factor thatpeople have to pay for coverage. Public-Private Partnership insurance structures,such as the Californian Earthquake Authority, were often created in the aftermathof large catastrophes to face the unavailability of insurance policies at reasonable
prices. Their goal is precisely to reduce the cost of capital allocation by aggregating the risk at the highest possible level and by smoothing the shocks across
time.

3.6

Appendix

3.6.1

Correlation

The coefficient of correlation δ between two individual losses writes for individuals
i and j:Cov(x̃i , x̃j ).
(Var(x̃i )Var(x̃j ))0.5
Since we have an infinite number of agents:13
δ=x̃i x̃j =


 L2
 0

with probability
with probability∫1

2
0 q dF (q)∫1 2
1 − 0 q dF (q),the covariance between two individual losses can be written as:
Cov(x̃i , x̃j ) = E(x̃i x̃j ) − E(x̃i )E(x̃j )
= L

2

∫ 1
0

Remark that:
L

2

∫ 1

2

q dF (q) = L
02

q dF (q) − (L
2

∫ 1(
0
2

∫ 1

qdF (q))2 .

0)(q − p) − p + 2pq dF (q)2

2= L2 [σ − p2 + 2p2 ]= L2 [σ 2 + p2 ].
13

An infinite number of agents enables us to find a simple expression for the coefficient ofcorrelation since it implies that all the aggregate uncertainty comes from the dependence between
the individual losses. But none of our results relies on this assumption.

69

Chapter 3. Insurability of low-probability catastrophic risks
Thus, we have:Cov(x̃i , x̃j ) = L2 [σ 2 + p2 ] − p2 L2= L2 σ 2 .
Since x̃i and x̃j have the same variance equal to p(1 − p)L2 , the coefficient ofcorrelation between two individual losses writes:δ=σ2.
p(1 − p)

To show that δ ≤ 1, remark that:
σ

2

∫ 1

=
0

∫ 1

=
0

(q − p)2 dF (q)q 2 dF (q) − p2

≤ p − p2 .

3.6.2

Proposition 12

We consider the case of an interior solution. From the first order condition (3.3),
we obtain:dτ(1 + c′ (τ ))[u′ (w1 ) − u′ (w2 )]=dp(1 − p)p(1 + c′ (τ ))2 u′′ (w2 ) + [1 − (1 + c′ (τ ))p]2 u′′ (w1 ) − c′′ (τ )[pu′ (w1 ) + (1 − p)u′ (w2 )](+)[τ + c(τ )] [1 − (1 + c′ (τ ))p]u′′ (w1 ) − (1 − p)(1 + c′ (τ ))u′′ (w2 )(1 − p)p(1 + c′ (τ ))2 u′′ (w2 ) + [1 − (1 + c′ (τ ))p]2 u′′ (w1 ) − c′′ (τ )[pu′ (w1 ) + (1 − p)u′ (w2 )]Since u(.) is concave and c(.) convex, we have:
dτ
<0
dp

⇔
τ +c(τ )τ +c(τ )
′
′′
′
′
′′
u′ (w1 ) + 1+c′ (τ ) (1 − (1 + c (τ ))p)u (w1 ) > u (w2 ) + 1+c′ (τ ) (1 − p)(1 + c (τ ))u (w2 )⇔(3.10)

τ +c(τ )τ +c(τ )′′′u′ (w1 )[1 − 1+c′ (τ ) (1 − (1 + c (τ ))p)A(w1 )] > u (w2 )[1 − 1+c′ (τ ) (1 − p)(1 + c (τ ))A(w2 )].

70.

Chapter 3. Insurability of low-probability catastrophic risks
Using the first order condition (3.3), this last inequality holds if and only if:1−

τ + c(τ )1 − (1 + c′ (τ ))p
τ + c(τ )′(1−(1+c(τ))p)A(w)>[1−(1−p)(1+c′ (τ ))A(w2 )],1′′′
1 + c (τ )(1 − p)(1 + c (τ ))1 + c (τ )which can be written as:c′ (τ )
τ + c(τ )>[A(w1 ) − A(w2 )][1 − (1 + c′ (τ ))p].
′(1 − p)(1 + c (τ ))1 + c′ (τ )3.6.3

Corollary 12.1

Since u(.) is a concave function and τ < L we know that u′ (w1 ) > u′ (w2 ). A
sufficient condition for inequality (3.10) to be satisfied is:
1 − [1 − (1 + c′ (τ ))p]

τ + c(τ )τ + c(τ )A(w1 ) > 1 − (1 − p)(1 + c′ (τ ))A(w2 ).
′
1 + c (τ )1 + c′ (τ )This can also be written as:(1 − p)(1 + c′ (τ ))A(w2 ) > [1 − (1 + c′ (τ ))p]A(w1 ).
Using the expression (3.6) of the coefficient of relative risk aversion for a HARA
function, we obtain:

η + wγ2(1 − p)(1 + c′ (τ ))>.
1 − (1 + c′ (τ ))p
η + wγ1(3.11)At an interior solution, the first order condition (3.3) yields:(1 − p)(1 + c′ (τ )) ( η + γ2 )γ
=.
1 − (1 + c′ (τ ))p
η + wγ1w

So for any γ > 1, this implies that (3.11) is verified and optimal coverage is indeed
decreasing in p.

3.6.4

Proposition 14

Assume that L is now a function of p such that a change in p is compensated bya change in L that maintains pL constant, i.e.:L′ (p) = −

L(p)
.
p

(3.12)
71

Chapter 3. Insurability of low-probability catastrophic risks
Let τ ∗ , w1∗ and w2∗ be the optimal values of coverage, loss state wealth and no-loss
state wealth. The first order condition now writes:[1−(1+c′ (τ ∗ ))p]u′ (w+τ ∗ −pτ ∗ −pc(τ ∗ )−L(p)) = (1−p)(1+c′ (τ ∗ ))u′ (w−pτ ∗ −pc(τ ∗ )).
(3.13)
Proceeding as in the proof of proposition 12, we find the necessary and sufficient∗condition for dτdp < 0:c′ (τ ∗ )> [τ ∗ + c(τ ∗ )][A(w1∗ ) − A(w2∗ )] + A(w1∗ )L′ (p).
(1 − p)[1 − (1 + c′ (τ ∗ ))p](3.14)

Using equation (3.12) gives:[τ ∗ + c(τ ∗ )][A(w1∗ ) − A(w2∗ )] + A(w1∗ )L′ (p) = [τ ∗ + c(τ ∗ )][A(w1∗ ) − A(w2∗ )] − A(w1∗ )

L
p

)
L
[τ ∗ + c(τ ∗ )]p
A(w1∗ )[τ ∗ + c(τ ∗ )]p − A(w2∗ )[τ ∗ + c(τ ∗ )]p − A(w1∗ )L= [τ ∗ + c(τ ∗ )][τ ∗ + c(τ ∗ )]p
A(w1∗ ){[τ ∗ + c(τ ∗ )]p − L} − A(w2∗ )= [τ ∗ + c(τ ∗ )](3.15).
[τ ∗ + c(τ ∗ )]p(= [τ ∗ + c(τ ∗ )] A(w1∗ ) − A(w2∗ ) − A(w1∗ )At any interior solution, the left-hand side of (3.13) must be positive. Therefore,
it must be the case that p(1 + c′ (τ ∗ )) < 1. By convexity of c(τ ), we have:p(1 + c′ (τ )) < p(1 + c′ (τ ∗ )) < 1 ∀τ < τ ∗ .
Hence:

∫ τ∗

′

∫ τ∗

p(1 + c (τ ))dτ <
0

dτ .
0

Or equivalently, using c(0) = 0, we have:p(τ ∗ + c(τ ∗ )) < τ ∗ < L.
Therefore:()

A(w1∗ ) [τ ∗ + c(τ ∗ )]p − L < 0,which implies with (3.15):[τ ∗ + c(τ ∗ )][A(w1∗ ) − A(w2∗ )] + A(w1∗ )L′ (p) < 0.
The necessary and sufficient condition (3.14) is therefore always satisfied.
72

Chapter 4
Pooling natural disaster risks in acommunity
This chapter is co-authored with Alexis Louaas.
Abstract: We analyze the design of contracts when individual risks are correlated across risk-averse agents in a community. The community is equipped witha public insurer which supplies insurance contracts to its members and has accessto costly reinsurance outside the community. Without transaction costs inside thecommunity, risk-averse agents fully insure against their individual risk and sharecollective risk by getting some dividend in normal states. With premiums raised
ex-ante and generating an opportunity cost, they only partially insure against theirindividual risk, getting a lower indemnity in catastrophic states than in normal
states, and potentially get some dividend in normal states. We illustrate the emergence of the latter contracts for the community of the Caribbean countries exposed
to natural disaster risks.
Keywords: individual risk, collective risk, insurance contracts, mutual insurance.
JEL classification: D86, G22, G28, Q54.
73

Chapter 4. Pooling natural disaster risks in a community

4.1

Introduction

The Caribbean countries are located in a region of the world widely exposed tolarge natural disaster risks, such as earthquakes, hurricanes and flooding events.
Even though aggregate damages are usually lower than 1 billion dollars per year
in this region, the hurricane season in 2004 affected many countries with morethan 6 billion dollars of aggregate losses and a large earthquake in January 2010
caused in Haiti more than 8 billion dollars of damages.1 In this context, the nonfor-profit Caribbean Catastrophe Risk Insurance Facility (CCRIF) is designed tosupply insurance contracts to the Caribbean countries (CCRIF SPC (2014)). To
deal with the high collective risks due to the spatial correlation of losses, theCCRIF purchases reinsurance outside the community. As reinsurance companies
or other investors on financial markets supply reinsurance contracts2 above fair
prices (Jaffee & Russell (1997), Cummins (2006) and Froot (2001)), the CCRIF
only partially reinsures the collective risks and supplies to its members insurance
contracts which are mutual in the sense that they depend on collective losses. The
indemnities for given individual losses are lower when collective losses are highthan when collective losses are low. Moreover, dividends are given to the insureds
when collective losses are low.3 The CCRIF, created in 2007, is one example
of such facilities that have emerged in different regions of the world exposed tonatural disasters in the last twenty years. The Florida Hurricane Catastrophe Fund
(FHCF) and the California Earthquake Authority (CEA) respectively created in
1993 and 1996 are other examples (Kousky (2010) and Kunreuther & MichelKerjan (2009)).
The present paper analyzes the optimal design of insurance contracts by a
1

Information on natural disaster losses in the Caribbean countries can be found on the EM-

DAT International Disaster Database (http://www.emdat.be/).
2
Investors on financial markets supply insurance-linked securities such as cat-bonds, which
are similar for insurers to standard reinsurance contracts supplied by reinsurance companies.
However, insurance-linked securities have emerged in the nineties because financial markets havelarger financial capacities to supply contracts for very large risks.
3
Dividends are given through premium discounts after a year with low collective losses.

74

Chapter 4. Pooling natural disaster risks in a community
pooling insurance facility when the collective risks are not negligible and reinsurance is above fair prices. We consider a community of identical risk-averse agents(representing for instance the Caribbean countries). Each agent faces two individual states: she can either suffer a loss or not. At the collective level, there
are two states of nature, the normal one and the catastrophic one, respectively
characterized by low and high fraction of the agents affected.4 We consider a nonfor-profit pooling insurance facility for the community (representing for instance
the CCRIF for the Caribbean countries). The insurance facility supplies mutual
insurance contracts to the agents in the community. For one contract, it charges
a premium and pays an indemnity to the insured if affected. The indemnity levelin the normal state may differ from the indemnity level in the catastrophic state.
The insurance contract may also include a dividend if the normal state occurs.
Besides, the insurance facility has access to reinsurance outside the community.
We analyze the characteristics of the optimal insurance and reinsurance contracts
for the community, when reinsurance is above fair prices.
Without any transaction costs inside the community, the optimal insurance
contract consists in full coverage for individual losses in both the normal state andthe catastrophic state. Moreover, it includes a strictly positive dividend in thenormal state because reinsurance is above fair prices. The higher the cost of reinsurance, the higher the premium and the dividend because the insurer substitutes
reinsurance by a higher reserve from the agents to pay the high total indemnitiesof the catastrophic state. However, requiring high amount of premiums ex-ante5
can generate an opportunity cost for the agents in the community. Indeed, thiscapital cannot be used for other purpose (i.e consumption or investment) which
thus may require the agents to borrow more costly external capital. In this case, it
is Pareto improving to implement a contract with a lower indemnity for individual4

We consider only two individual states to keep the model tractable. At the collective level,

we consider two and only two states of nature respectively to model collective risks and to keep
the model tractable.
5
Premiums are required ex-ante to pay reinsurance premiums and to secure a reserve which
avoids participation default. Moreover, raising the premiums ex-ante enables to transfer indemnities quickly to affected agents.

75

Chapter 4. Pooling natural disaster risks in a community
loss in the catastrophic state than in the normal state. Moreover, the optimal
contract still has full coverage and dividend in the normal state if and only if themarginal opportunity cost is low enough relative to the marginal reinsurance cost.
The economics literature has already addressed the question of optimal insurance contract when there are collective risks in a community. Doherty &Schlesinger (1990), Hau (1999), Cummins & Mahul (2004) and Mahul & Wright(2004, 2007) consider the case where the indemnity level in normal states increases
with the premium level but the indemnity level in catastrophic states is null whatever the premium level. In this case, the optimal contract consists in partial
coverage for individual losses in normal states in order to preserve their welfare
level in catastrophic states. However, these papers do not address the issue of theinsurer financial capacity which would explain why indemnities cannot be paidin catastrophic states.6 Charpentier & Le Maux (2014) focuses on the issue by
considering an insurer with an exogenously given amount of reserve besides premiums. In this case, the indemnity level increases with the premium level in normal
states and in catastrophic states because raised premiums increase the financial
capacity. However, the optimal insurance contract consists in full coverage forindividual losses in normal states but not in catastrophic states because of insurer
limited reserve. Relative to Charpentier & Le Maux (2014), we relax the assumption of partially exogenous financial capacity by introducing reinsurance outside
the community. Moreover, we allow a better participation of the insureds in thereserve by introducing dividends in the contracts, which gives more flexibility inthe design of contracts for risk sharing. Indeed, as explained by Borch (1962)
and (Marshall, 1974b), in a community where agents are exposed to individual
risks with collective components, it is Pareto optimal to eliminate individual risks
and to share collective risks (mutuality principle). Malinvaud (1973) and Casset al. (1996) show that a mutual contract with dividend supplied by the insurance company enables to reach the mutuality principle. Penalva-Zuasti (2001)
6

This contingency can be seen as contractual or as a "default risk" with right perception by

insureds and no cost of default. In the quoted theoretical papers, "default risk" is used in thissense.

76

Chapter 4. Pooling natural disaster risks in a community
and Penalva-Zuasti (2008) show that it is also reached with agents purchasing
a standard contract from the insurance company and investing in the insurancecompany through stock market. When reinsurance outside a community is costly,
Doherty & Dionne (1993) and Doherty & Schlesinger (2002) show that the optimal
contract consists in the full elimination of individual risks in each state of nature,plus partial coverage of the collective risks. Relative to Doherty & Dionne (1993)
and Doherty & Schlesinger (2002), we analyze how the cost of reinsurance andthe correlation between individual risks affect the optimal contract. Moreover, we
introduce and analyze the impact of the opportunity cost of capital potentiallygenerated by raising premiums ex-ante.
The first contribution of the present paper is to develop a simple and tractablemodel to analyze the optimal design of insurance contracts by a pooling insurance
facility to manage individual and collective risks. The second contribution is to
study the impact of reinsurance costs and risk correlation on the optimal insurance
contract further than previous works. The third contribution is to consider theopportunity cost of capital potentially generated by raising premiums ex-ante. The
paper is organized as follows. Section 2 presents the example of the Caribbean
countries and their insurance facility. Section 3 sets up the model of a community
with individual and collective risks and the insurance and reinsurance contracts.
Section 4 provides an analysis of the optimal insurance and reinsurance contracts.
Section 5 concludes.

4.2

Caribbean countries and natural disasters insurance

The Caribbean countries are located in a region of the world exposed to importantnatural disaster risks. Figure 4.17 exhibits natural disaster losses in this region in
the last fifty years. Collective losses are widely variable from one year to anotherbecause natural disasters in the region can have large spatial impacts. Year 2010
7

EMDAT International Disaster Database (http://www.emdat.be/)77

Chapter 4. Pooling natural disaster risks in a community
corresponds to the highest losses with a large earthquake affecting Haiti in January with more than 8 billion dollars of damages. Year 2004 corresponds to thesecond highest losses with a particularly dramatic hurricane season affecting many
countries such as the Bahamas, the Cayman Islands, Grenada and Jamaica.

Figure 4.1: Natural disaster losses in the Caribbean countries.
In this context, the Caribbean Catastrophe Risk Insurance Facility (CCRIF)
is a non-for-profit multi-country insurance pool. Created in 2007, it currently
offers disaster-relief insurance policies covering public losses of sixteen Caribbean
countries, protecting them against earthquake, hurricane and excess rainfall losses.
Its effectiveness during the five first years of existence has conducted the program to
be extended to Central American countries, starting from 2016. The facility aims
at pooling the risks faced by its members and reduce the cost the members would
individually face if they directly insured on the reinsurance market. The annual
reports of the CCRIF are publicly available8 and provide useful information about
the catastrophe insurance contracts proposed to the sixteen members. The CCRIF
reports a stable number of 29 or 30 sold policies each year since its inception.9 The
8978

http://www.ccrif.org/content/publications/reports/annual
The CCRIF can sell more than one policy per country per year because insurance policies

Chapter 4. Pooling natural disaster risks in a community
collective risk faced by the CCRIF has remained rather stable as well. Figure 4.2a
displays the cumulative density function of the aggregation of the risks covered by
the CCRIF and reported in its annual reports since year 2007-2008. The darker
lines represent the cumulative distribution of this aggregate risk faced by the poolin the earlier periods of its existence.10 Using the cumulative distribution functions
with the information about the structure of the reinsurance scheme bought by the
CCRIF, we can compute an estimated loading factor paid by the organization as :RαλR = E(L)−1, where αR is the premium paid by the CCRIF to reinsurers and E(L)is the expected loss reinsured. Figure 4.2b displays its evolution through the years
and shows that the CCRIF faces a significant loading factor on reinsurance, whichclearly explains why it only partially reinsures the collective risk. The figure showsthat reinsurers increased their prices a lot in 2010, following the large earthquake
affecting Haiti.

(a) Aggregate risks faced by the CCRIF(b) Reinsurance loading factor

Figure 4.2: Aggregate risks resulting from risk pooling by the CCRIF and estimated reinsurance loading factor faced by the CCRIF.
for the different types of natural disasters are separated.
10
Insured losses in figure 4.2a are much lower than total losses due to natural disasters in figure4.1. This is due to the fact that the CCRIF covers only public losses which represent only asmall fraction of the total losses incurred in a country when a natural disaster occurs. It is also
due to the fact that the Caribbean countries purchase from the CCRIF only partial insurance
for natural disaster risks.

79

Chapter 4. Pooling natural disaster risks in a community
As the CCRIF only partially reinsures the collective risk, it supplies to its
members insurance contracts which are mutual in the sense that they dependon collective losses. In addition to the regular insurance premiums, the facility
requires its members to pay an up-front participation fee. Audited financial statements report that "it is Managements intent that participation fee deposits areavailable to fund losses in the event that funds from retained earnings, reinsurers
and the Donor Trust are insufficient. If deposits are used to fund losses, it is also
Managements intent that any subsequent earnings generated by the Group will be
used to reinstate the deposits to their original carrying value". Figure 4.3 showsthat the total amount of premiums was effectively much higher the first year than
the following years. It has not been necessary to raise high premiums the following years because no extremely large claims had to be paid during these years. In
terms of claims to be paid, the worth year is 2010, during which the CRIFF had totransfer a bit less than 8 million dollars to Haiti for the large earthquake affecting
the country. The yearly insurance contract is similar to a contract with a highpremium requested at the beginning of the year in exchange for an indemnity if theinsured is affected during the year and a dividend at the end of the year if collective
losses are not too catastrophic. In the present case, the dividend is given througha premium discount at the beginning of the following year. Besides, the CCRIF
acknowledges the possibility of lowered indemnities in catastrophic states: "TheCCRIF can currently survive a series of loss events with a less than 1 in 10,000
chance of occurring in any given year. Due to planned premium reductions, thesafety level drops somewhat through the course of our 10-year forward modeling.
However, the lowest projected survivability for the CCRIF in the 10-year modeledperiod is about 1 in 3000 chance of claims exceeding capacity in any one year." Inother words, the CCRIF acknowledges to supply contracts such that the indemnity
for one individual loss level is lower in highly catastrophic states than in the otherstates of nature.

80

Chapter 4. Pooling natural disaster risks in a community

Figure 4.3: Premiums raised and claims paid by the CCRIF.

4.3

The model

4.3.1

The community of agents

We consider a community of N agents identical in terms of preferences, initial
wealth and exposure to risk.11 The preferences of the representative agent satisfy
the von Neumann-Morgenstern axioms, with u(.) the corresponding utility function
which is strictly increasing, globally concave and twice continuously differentiable.
The representative agent has an initial wealth w and is exposed to a potential loss
l. The individual risks can generate a significant collective risk either because N is
not large enough or because individual risks are correlated. To model the collective
risks, we consider two states of nature, one catastrophic and one normal. Ex-ante,
the representative agent knows that with a probability p (such that 0 < p < 1), a
catastrophe occurs and the fraction of agents enduring a loss of size l is qc . In the
11

Heterogeneity of individuals raises questions related to asymmetric information that are outof the scope of our analysis.

81

Chapter 4. Pooling natural disaster risks in a community
normal state, the fraction of agents enduring the same loss l is qn < qc .12 In thistemplate, the individual probability of enduring a loss l is qc in the catastrophic
state and qn in the normal state, and the unconditional individual probability ofenduring a loss l is: q = (1−p)qn +pqc . The individual random wealth without risk
sharing scheme is characterized in figure 4.4. Besides, the collective random wealthof the N agents is characterized in figure 4.5. With N large, the coefficient δ of(q − qn )2
correlation between individual risks is well approximated by: δ = p(1−p)
q(1−q) c(proof in appendix 4.6.1). The higher the difference between the fraction qc ofaffected agents in the catastrophic state and the fraction qn of affected agents inthe normal state, the higher the risk correlation between agents.13 Finally, qn andqc can be expressed as functions of the individual probability q of being affected,
the correlation δ between individual risks and the probability p of catastrophe:q(1−q) 0.5q(1−q) 0.5
qn = q − p( p(1−p)δ) and qc = q + (1 − p)( p(1−p)δ) .

1 − qn
1−p

qn

.
p

1 − qcw

not affected
normal statew−laffectedw

not affectedcatastrophic state

qc

w−l

affected

Figure 4.4: individual random wealth of the representative agent
In this template, average individual loss depends on the state of nature, its valueis qn l in the normal state and qc l in the catastrophic state. Thus, the expected valueof the average individual loss is ql and its variance is q(1 − q)δl2 (proof in appendix
4.6.1). The higher the individual probability q of being affected, the higher the
expected average loss. The more correlated the individual risks, the more volatile12

As pointed out by Malinvaud (1973) and Cass et al. (1996), considering two different in-dividual loss levels in the normal and catastrophic states could be considered as two different
risks.
13
The fully correlated case (δ = 1) is characterized by qn = 0, qc = 1 and 0 < p < 1, in which
everyone endures a loss or no one. The no-correlated case (δ = 0) would correspond to qn = qc ,p = 1 or p = 0, in which there is only one collective state.

82

Chapter 4. Pooling natural disaster risks in a community
N w − qn N l

1−p

.
p

N w − qc N lFigure 4.5: collective random wealth of the N agents
the average loss.14 Figure 4.6 illustrates for two different sets of parameters the
cumulative distribution functions for the individual loss (thick bars) and for the
average individual loss (thin bars). The spread between qn and qc is smaller in 4.6athan in 4.6b, while in both cases p = 0.2 and q = 0.3. The individual probabilityof being affected q is similar for the two sets of parameters, whereas the correlation
across individual risks δ is smaller in 4.6a than in 4.6b, which makes a difference
for risk sharing mechanism as detailed in the paper.
probability

probability

1

1

1−p
1−q

1−p
1−q

0.
0

qn l

qc l

l

0.
0

loss

qn l

qc l

l

(a) p = 0.20, qn = 0.25, qc = 0.50,(b) p = 0.20, qn = 0.20, qc = 0.70,

q = 0.3, δ = 0.048

q = 0.3, δ = 0.190

loss

Figure 4.6: Cumulative distribution functions for individual loss (thick bars) andaverage individual loss (thin bars) for two different sets of parameters in 4.6a and
4.6b.

14

Cummins (2006) and Cummins & Trainar (2009) have more insights on the relation between

the risk correlation and the average loss volatility.

83Chapter 4. Pooling natural disaster risks in a community

4.3.2

Insurance and reinsurance contracts

We consider that the community is equipped with a pooling insurance facility, also
called the insurer. The insurer faces two states of nature, the normal one and thecatastrophic one, in which it respectively has a fraction qn and a fraction qc of
its insureds that have to be indemnified. A standard-type insurance contract isa couple (α, τ ). In this case, α is the premium paid by the agent and τ ≥ 0 is
the indemnity received by the agent if the latter endures a loss l. A mutual-typeinsurance contract is a quadruple (α, τ, ϵ, π). In this case, α is the premium paidby the agent, τ ≥ 0 is the indemnity received by the agent in the normal state if
the latter endures a loss l, τ − ϵ ≥ 0 is the indemnity received by the agent in thecatastrophic state if the latter endures a loss l and π ≥ 0 is the dividend received by
the agent in the normal state. This contract is called mutual-type contract because
each agent shares a fraction of the collective risk of the community. Indeed, ϵ andπ make the insurance contract directly depend on the collective losses, contrary
to the standard contract. The standard contract is a specific case of the mutual
contract with ϵ = 0 and π = 0.15 Besides, we consider that the contract cangenerate an opportunity cost for the insured. When premiums are raised ex-ante
while indemnities and dividends are given ex-post, the secured capital cannot be
used for other purpose (i.e. consumption or investment). Thus, the agents may
have to raise more costly external capital instead of using this capital, which
generates an opportunity cost for the agents. The higher the required premium
α, the higher should be the marginal opportunity cost because the costlier shouldbe the marginal external capital. We denote λl (α) the opportunity cost function
which is increasing and convex relative to the premium α. The agent wealth profile
with a mutual-type contract is represented in figure 4.7.
The insurer has to manage the collective risks generated by the aggregation of
15

The mutual contract defined here is in the spirit of the contracts supplied by the CCRIF to

the Caribbean countries. The premium α corresponds to the regular premium plus the up-front
participation fee in the contracts supplied by the CCRIF. The dividend π corresponds to thepremium discount of the following year if losses are not too catastrophic. The indemnity gap ϵbetween normal state and catastrophic state is also acknowledged by the CCRIF.

84

Chapter 4. Pooling natural disaster risks in a community
1 − qn1−pqn.
1 − qcpqc

w1 = w − α − λl (α) + π
w2 = w − α − λl (α) − l + τ + π
w3 = w − α − λl (α)
w4 = w − α − λl (α) − l + τ − ϵFigure 4.7: agent wealth profile with an insurance contract

the insured individual risks. It can purchase reinsurance outside the community,
to be able to pay the higher total claims of the catastrophic state. Purchasing areinsurance contract, with an indemnity τ R ≥ 0 in the catastrophic state occuringwith a probability p, costs (1 + λR )pτ R , in which λR ≥ 0 is the reinsurance loading
factor.16 For reinsurance to be relevant, we need to have (1+λR )p < 1.17 With theinsurance contracts supplied to the agents and the reinsurance contract purchased
outside the community, the insurer wealth profile is detailed in figure 4.8.
1−p

N α − N qn τ − (1 + λR )pτ R − N π.

p

N α − N qc (τ − ϵ) − (1 + λR )pτ R + τ R

Figure 4.8: insurer profit profile

4.4

Optimal insurance and reinsurance

The optimal insurance and reinsurance contracts for the community consist in
maximizing the expected utility of the representative agent under the budget constraints.
16
17

λ corresponds to frictional costs with reinsurers or investors, as detailed in Froot (2001).
If (1+λR )p ≥ 1, purchasing reinsurance would have no sense for the CCRIF because it would

lose money in both the normal state and the catastrophic state with the reinsurance contract.

85

Chapter 4. Pooling natural disaster risks in a community

4.4.1

Budget constraints

The mutual insurance facility cannot pay claims unless it has secured the funds
either through raised premiums or purchased reinsurance. With the budget constraints in both the normal state and the catastrophic state (budget expressions infigure 4.8), the optimal insurance and reinsurance contracts are thus the solutionof the following maximization problem:max

α,τ,ϵ,π,τ R

E(u(w̃))s.t. N α − N qn τ − (1 + λR )pτ R − N π ≥ 0(4.1)

N α − N qc (τ − ϵ) − (1 + λR )pτ R + τ R ≥ 0
τ ≥ 0, τ − ϵ ≥ 0, π ≥ 0, τ R ≥ 0.
Because utility is increasing with wealth, the budget constraints are binding in thetwo states of nature, the catastrophic one and the normal one. The subtractionof the two binding budget constraints gives the purchased reinsurance indemnity
τ R:
τ R = N qc (τ − ϵ) − N qn τ − N π ≥ 0.

(4.2)

The insurance facility has to purchase a reinsurance indemnity in order to cover
the difference between the amount due in the catastrophic state (N qc (τ − ϵ)) andthe amount due in the normal state (N qn τ + N π). With (4.2), the binding budget
constraints give the required premium:
(

)

(

)

(

)

p(qc − qn ) Rpα= 1+
λ qτ + 1 −λR (1 − p)π − 1 + λR pqc ϵ,
q
1−p

(4.3)

which simplifies, if reinsurance is binding (τ R = 0), to:α = qc (τ − ϵ).

(4.4)

To be able to pay indemnities and dividends, the insurance facility requires thepremium (4.3) (if τ R > 0) or (4.4) (if τ R = 0) for the contract (α, τ, ϵ, π). If it is
not valuable to purchase reinsurance, the insurance facility has to raise premiums(4.4) in order to be able to pay all the indemnities in the catastrophic state. If it
is valuable to purchase reinsurance, the insurance facility has to pass on the cost86

Chapter 4. Pooling natural disaster risks in a communityof reinsurance to insureds, which explains the loading factor p(qcq−qn ) λR in front of
τ in (4.3). As shown by (4.2), allowing a dividend in the normal state (π > 0) ora lower indemnity in the catastrophic state (ϵ > 0) enables to lower the purchaseof reinsurance. With a dividend in the normal state (π > 0), the premium isaffected in two opposite directions. The first channel is straightforward: a higher
dividend implies a higher premium. The second channel is due to the fact that theinsurer has to purchase less reinsurance thanks to the reserve from the insureds
and appears through λR in the coefficient in front of π in (4.3). Note that thefactor in front of π in (4.3) is globally positive because (1 + λR )p < 1. With alower indemnity in the catastrophic state (ϵ > 0), the premium is reduced throughtwo channels. The first channel is straightforward: a lower indemnity implies alower premium. The second channel is due to the fact that the insurer has topurchase less reinsurance and appears through λR in the coefficient in front of ϵin (4.3). If the agents in the community can have direct access to reinsurancewith the same loading factor λR , it is valuable to insure through the insurance
facility because: 1 + p(qcq−qn ) λR < 1 + λR , thanks to partial diversification doneby the insurance facility. This is true with standard insurance contracts and thus
also true with mutual contracts. The higher the cost of reinsurance, the morevaluable the facility. The lower the correlation between participants, the more
efficient the pooling and thus the more valuable the facility. This could explain
why the Caribbean countries would like to extend their facility to South American
Countries in 2016. However, we have assumed that there are no management costs
for the insurance facility. If there are, the pooling insurance facility is valuable
if the cost of implementing the facility generates a loading factor λi such that:
1 + λi + p(qcq−qn ) λR < 1 + λR . In the case of the Caribbean countries, extending
the insurance facility to South American Countries will be valuable if it does not
add too much management costs.

87

Chapter 4. Pooling natural disaster risks in a community

4.4.2

Insurance and reinsurance contracts

With the binding budget constraints, the maximization problem (4.1) for the optimal contracts boils down to:18
max
E(u(w̃))
τ,ϵ,π
(

)

(

)

(

)

p(qc − qn ) R
ps.t. α = 1 +λ qτ + 1 −λR (1 − p)π − 1 + λR pqc ϵq1−pτ ≥ 0, π ≥ 0, (qc − qn )τ − π − qc ϵ ≥ 0.
(4.5)
Note that, with standard insurance contracts (π = 0 and ϵ = 0), the maximization
problem (4.5) corresponds to the standard Mossin problem, in which the optimal
coverage level is obtained by the marginal tradeoff between the aversion to risk
and the cost due to reinsurance ( p(qcq−qn ) λR ).
Without opportunity cost (λl (α) = 0)
We first consider the case in which raising premiums ex-ante does not generate
an opportunity cost for the insured (λl (α) = 0). The first order conditions of(4.5) are derived in appendix 4.6.2. If it is valuable to purchase reinsurance (i.e.
τR= (qc − qn )τ − π − qc ϵ ≥ 0 is not binding), the optimal contract has indemnitiesNτ and τ − ϵ and dividend π such that:18u′ (w2 )u′ (w − α − l + τ + π)=
= 1,
u′ (w1 )u′ (w − α + π)

(4.6)

u′ (w4 )u′ (w − α − l + τ − ϵ)== 1,u′ (w3 )u′ (w − α)(4.7)

u′ (w3 )u′ (w − α)
1 + λR.
==p
λRu′ (w1 )u′ (w − α + π)
1 − 1−p(4.8)

The last inequality constraint in (4.5) corresponds to τ R ≥ 0. The inequality constraint

τ − ϵ ≥ 0 is not written because it is necessarily verified with the other inequality constraints.
Indeed, we have at least as much money for indemnities in the catastrophic state as the amountof money for indemnities and dividends in the normal state.

88Chapter 4. Pooling natural disaster risks in a community
R

If it is not valuable to purchase reinsurance (i.e. τN = (qc − qn )τ − π − qc ϵ ≥ 0 is
binding), the optimal contract has indemnities τ and τ − ϵ plus dividend π suchthat:u′ (w2 )u′ (w − α − l + τ + π)=
= 1,
u′ (w1 )u′ (w − α + π)

(4.9)

u′ (w − α − l + τ − ϵ)u′ (w4 )== 1,
u′ (w3 )u′ (w − α)(4.10)

π = (qc − qn )τ − qc ϵ.

(4.11)

Whether reinsurance is purchased or not, the optimal insurance contract is suchthat: w1 = w2 and w3 = w4 (thanks to (4.6) and (4.7) or (4.9) and (4.10)), whichmeans that τ = l and ϵ = 0. Besides, when λR = 0, (4.8) tells that π = 0, (4.3)∗

gives α = ql and (4.2) gives τ R = N (qc − qn )l. When 0 < λR < λR , (4.8) tellsthat π > 0, (4.3) gives α = (q + p(qc − qn )λR )l + (1 − p − pλR )π and (4.2) gives∗τ R = N (qc − qn )l − N π > 0. λR is determined with (4.8) and the additionalR

constraint τN = (qc − qn )l − π = 0, which tells that π = (qc − qn )l and α = qc l.
∗

When λR ≤ λR , (4.11) tells that π = (qc −qn )l, (4.4) gives α = qc l and reinsurance
is not purchased τ R = 0.
Proposition 18 The optimal insurance and reinsurance contracts are such that:(i) when λR = 0: τ = l, ϵ = 0, π = 0, α = ql, τ R = N (qc − qn )l;∗(ii) when 0 < λR < λR : τ = l, ϵ = 0, π > 0, α = (q + p(qc − qn )λR )l + (1 − p −pλR )π, τ R = N (qc − qn )l − N π > 0;∗(iii) when λR ≤ λR : τ = l, ϵ = 0, π = (qc − qn )l, α = qc l, τ R = 0;
∗

′

R∗

(w−qc l)in which λR is such that uu′ (w−q
= 1−1+λ
.
p
λR ∗
n l)
1−p

Proposition 18 states that the optimal insurance contract has full coverage fora given individual loss in both normal and catastrophic states (τ = l and ϵ = 0)
whatever the cost of reinsurance (λR ). The optimal contract eliminates individual
risks, which is in line with Borch mutuality principle. Besides, proposition 18 states
89

Chapter 4. Pooling natural disaster risks in a communitythat the optimal contract has dividend (π > 0) in the normal state if and only if
reinsurance is supplied above fair prices (λR > 0). If reinsurance is fair (λR = 0),
the insurance facility fully reinsures the collective risk (τ R = N (qc − qn )l) and the
optimal insurance contract is standard, i.e. without any dividend in the normal
state (π = 0). If reinsurance is not fair (λR > 0), a mutual contract (i.e. with
π > 0) is better than a standard contract because it enables the risk-averse agent
to bear a part of the collective risk contrary to the standard contract, which isvaluable because reinsurance is costly. If reinsurance is excessively above fair prices∗(λR ≤ λR ), the insurance facility does not purchase reinsurance (τ R = 0) and the
optimal insurance contract is with dividend in the normal state corresponding
to the indemnity difference between the catastrophic state and the normal state∗(π = (qc − qn )l). If reinsurance is reasonably above fair prices (0 < λR < λR ), theinsurance facility partially reinsures the collective risk (τ R > 0) and the optimal
insurance contract is with dividend in the normal state (π > 0).
∗

Proposition 19 With 0 < λR < λR (and a CARA utility function19 ), we have
R

dπ
dα
dτ
for the optimal insurance and reinsurance contracts: dλ
R > 0, dλR > 0, dλR < 0.

Proposition 19 is proved in appendix 4.6.2. It states that the higher the reinsurance cost (i.e. λR ), the lower the reinsurance purchase and the higher the premiumand the dividend in the normal state. Indeed, to be able to cover individual losses
in the catastrophic state when reinsurance purchase is decreased, the insurance
facility has to increase the reserve financed by the insureds through higher premiums. Moreover, it has higher dividends to give to the insureds if the catastrophic
19

The coefficient of absolute risk aversion of a utility function u(.) is by definition A(.) =′′(.)− uu′ (.). We consider here a utility function with a constant absolute risk aversion A (also called

CARA utility function). If the utility function is not CARA, there is an additional wealth effect.
However, as long as this effect is of secondary order, it does not change the results. Note that
if this effect was not of secondary order, it would have been observed that insurance can be a
Giffen good (i.e. a higher premium leading to a higher purchase of insurance). To our knowledge,
empirical analysis on the purchase of natural disaster insurance have not observed such behaviors.
For instance, Browne & Hoyt (2000) and Grace et al. (2004) observe that when insurance priceincreases, the demand for insurance decreases.

90

Chapter 4. Pooling natural disaster risks in a community∗

state does not occur. In the extreme case where λR reaches λR , reinsurance is
not purchased (τ R = 0) and the premium and the dividend respectively reach thehighest levels α = qc l and π = (qc − qn )l.
Proposition 20 We have for the optimal insurance and reinsurance contracts:
R(i) when λR = 0: dπ= 0, dα= 0, dτdδ > 0;dδdδ∗(ii) when 0 < λR < λR (with a CARA utility function):dπdδ= 0, dα
> 0,
dδ

dτ R
> 0;
dδ
∗

R

(iii) when λR ≤ λR : dπ
> 0, dα> 0, dτdδ = 0.
dδdδProposition 20 is obtained thanks to proposition 18, recalling that qn = q −q(1−q) 0.5q(1−q) 0.5
δ) ((i) and (iii) are obvious and (ii) isp( p(1−p) δ) and qc = q + (1 − p)( p(1−p)
proved in appendix 4.6.2). Firstly, it states that the insurance contract is affectedby a change of correlation δ if and only if reinsurance is not fair (λR > 0). If reinsurance is fair, only the average probability q and the loss l affects the insurance
contract because the collective risk is fully reinsured without any cost. If reinsurance is not fair, the higher the correlation δ, the larger the collective risk and the∗more expensive its coverage. If reinsurance is not too costly (0 < λR < λR ), an
increase of δ is managed by an increase of reinsurance purchase to be able to cover
the higher total indemnities in the catastrophic state and the insurance facility has
to translate the cost of reinsurance to insureds through higher premiums. If rein∗surance is too costly (λR ≤ λR ), an increase of δ is managed by an increase of thereserve through higher premiums and the insurance facility has higher dividends
to distribute if the catastrophe does not occur. In both cases, higher correlation δ
leads to higher premiums.
To sum up, the premium α increases from ql to qc l when λR increases from 0
to high values and it also increases when risk correlation δ increases. Thus, withcostly reinsurance and significant risk correlation, the required premiums can reachhigh levels for insureds if the individual loss l is significant. In this case, whichis relevant for natural disaster risks, raising such levels of premiums ex-ante can91Chapter 4. Pooling natural disaster risks in a community
generate an opportunity cost for insureds, which are considered in the following
section.
With opportunity cost (λl (α) ≥ 0)
We now consider the case in which raising premiums ex-ante generate an opportunity cost for the insureds (λl (α) ≥ 0). As explained in section 4.3.2, we assume
that the opportunity cost function λl (.) is increasing and convex relative to the′premium α. We analyze how the marginal opportunity cost λl (α) affects the op∗

timal insurance and reinsurance contracts. We consider λR ≤ λR , which meansthat purchasing some reinsurance is valuable. The first order conditions of (4.5)
are derived in appendix 4.6.3. If it is valuable to have dividend in the normal stateπ ≥ 0, the optimal contract has indemnities τ and τ − ϵ and dividend π such that:u′ (w2 )u′ (w − α − λl (α) − l + τ + π)=
= 1,
u′ (w1 )u′ (w − α − λl (α) + π)

(4.12)′u′ (w4 )u′ (w − α − λl (α) − l + τ − ϵ)(1 + λl (α))(1 + λR )==, (4.13)l′u′ (w3 )u′ (w − α − λl (α))(1 + λl ′ (α))(1 + λR ) − λ (α)
p(1−qc )l′′λl (α)(1 + λ (α))(1 + λ ) − p(1−qc )u′ (w3 )u′ (w − α − λl (α))
=
=
.
p
′
′
l
u (w1 )u (w − α − λ (α) + π)
(1 + λl ′ (α))(1 − 1−p
λR )R(4.14)

If it is not valuable to have dividend in the normal state (i.e. π ≥ 0 is binding),
the optimal contract has indemnities τ and τ − ϵ and dividend π such that:π = 0,(4.15)′p(1 + λl (α))(1 − 1−pλR )u′ (w2 )u′ (w − α − λl (α) − l + τ )==,l′u′ (w1 )u′ (w − α − λl (α))(1 + λl ′ (α))(1 − p(qc −qn ) λR ) − λ (α)
1−q

1−q

(4.16)′u′ (w4 )u′ (w − α − λl (α) − l + τ − ϵ)(1 + λl (α))(1 + λR )==.
l′u′ (w3 )u′ (w − α − λl (α))(1 + λl ′ (α))(1 − p(qc −qn ) λR ) − λ (α)
1−q

1−q

(4.17)
92

Chapter 4. Pooling natural disaster risks in a community
Proposition 21 The optimal insurance and reinsurance contracts are such that:′∗(i) when 0 < λl (α) < λl : τ = l, ϵ > 0, π > 0, α = (q + p(qc − qn )λR )l + (1 −p − pλR )π − (1 + λR )pqc ϵ, τ R = N (qc − qn )l − N π − N qc ϵ > 0;′∗(ii) when λl (α) = λl : τ = l, ϵ > 0, π = 0, α = (q +p(qc −qn )λR )l −(1+λR )pqc ϵ,τ R = N (qc − qn )l − N qc ϵ > 0;′∗(iii) when λl (α) > λl : τ < l, ϵ > 0, π = 0, α = (q +p(qc −qn )λR )τ −(1+λR )pqc ϵ,τ R = N (qc − qn )τ − N qc ϵ > 0;∗p(1−qc )Rin which λl = 1−p−p(1−q
Rλ .
c )λ

Proposition 21 is derived from the first order conditions of (4.5) written above
plus (4.2), (4.3) and (4.4).20 Firstly, it states that the optimal insurance contract
has lower coverage for a given individual loss in the catastrophic state than in thenormal state (ϵ > 0) when increasing the premium α generates an opportunity′

cost (λl (α) > 0). In this case, it is not valuable to cover fully individual losses
in the catastrophic state, which means that the optimal contract does not fully
eliminate individual risks and does not fulfill the Borch mutuality principle. Thisis a second-best insurance contract when insurance premiums have to be raised
ex-ante and generate an opportunity cost. Besides, relative to the case without an
opportunity cost, proposition 21 states that the optimal contract may not always
have dividend in the normal state. If the marginal opportunity cost is too high′p(1−qc )R
relative to the reinsurance cost (λl (α) ≥ 1−p−p(1−qR λ ), it is not valuable toc )λhave dividend in the normal state, which means that it is more valuable to spend
all the premiums to reinsure rather than to keep some reserves which would begiven back through dividends in the normal state. In this case, it is even valuable
to lower the indemnity in the normal state relative to full coverage (τ < l) to
increase reinsurance and the indemnity in the catastrophic state. We consider in′the following a constant marginal opportunity cost λl (α) = λl .
20 l ∗λ′p(1−qc )R
is obtained with (4.14) equal to 1. Besides, λl (α) > 1−p−p(1−qtells that (4.16)
Rλc )λ

is strictly greater than 1 and τ < l.

93

Chapter 4. Pooling natural disaster risks in a community∗

Proposition 22 With 0 < λl < λl (and a CARA utility function), we havedϵdπdα
for the optimal insurance and reinsurance contracts: dλl > 0, dλl < 0, dλl < 0,
dτ R
> 0.
dλl

Proposition 22 is proved in appendix 4.6.3. It states that an increase of themarginal opportunity cost (λl ) leads to a decrease of the premium to limit theopportunity cost for the insured. Thus, it leads to a decrease of the indemnity inthe catastrophic state and a decrease of the dividend in the normal state. However,to limit the indemnity decrease in the catastrophic state, reinsurance purchase isincreased in this case.
∗

Proposition 23 With 0 < λl < λl (and a CARA utility function), we have fordϵthe optimal insurance and reinsurance contracts: dδ> 0, dπ< 0, dαambiguousdδdδRand dτdδ > 0.
Proposition 23 is proved in appendix 4.6.3. It states that an increase of thecorrelation δ leads to an increase of reinsurance purchase because the collective
risk increases with δ. On the one hand, the premium has to increase because
reinsurance is costly. On the other hand, increasing the premium generates anadditional opportunity cost. That is why the indemnity in the catastrophic state
and the dividend in the normal state are lowered and finally the variation of thepremium is ambiguous.

4.5

Conclusion

In the present paper, we have built a simple model to analyze the type of insurance contracts that emerge when risks are correlated across risk-averse agents in
a community. For the sake of realism, we have considered that the community
simultaneously chooses the type of contract sold to its members and the level ofreinsurance it purchases, given that reinsurance is available at a cost higher thanfair price. In this scheme, the insurer of the community supplies mutual contracts
which are contingent on the state of nature. Without transaction costs in thecommunity, risk-averse agents fully insure against their individual risk and share
94

Chapter 4. Pooling natural disaster risks in a community
collective risk by getting some dividend in normal states of nature. Our model
highlights the tradeoff between reinsurance and mutual types contracts. If reinsurance is costly, the promise of dividends in normal states enables the community
to raise high premiums that are used as reserves to better indemnify in catastrophic states. With premiums raised ex-ante and generating an opportunity cost,risk-averse agents only partially insure against their individual risk, getting a lower
indemnity in catastrophic states than in normal states, and get some dividend innormal states if the marginal cost of the reserve is low relative to the marginal
cost of reinsurance. This analysis helps to understand the limits that risk correlation, costly reinsurance and costly reserve represent for risk sharing and how thecontracts in a community can be improved through higher flexibility. Indeed, contracts with contingent indemnity and dividend enable to share better individual
risks and collective risks. We have illustrated these mechanisms with the example of the Caribbean Catastrophe Risk Insurance Facility (CCRIF) that combines
reinsurance and mutual contracts with indemnity and dividend contingent on thecollective state.

4.6

Appendix

4.6.1

Risk correlation

With the loss represented by the random variable x̃i for individual i and theprobability q = (1 − p)qn + pqc of having a loss l, we have:

x̃i =


 −l

with probability q 0with probability 1 − q

In the normal state, the probability that individual i is affected is qn . Besides, inthe normal state, if individual i is affected, agent j is affected with a probability
qn N −1, which is well approximated by qn
Nwhen N is large. In the catastrophic

state, this is similar with qc instead of qn . Thus, when N is large, we have with a95

Chapter 4. Pooling natural disaster risks in a community
good approximation:x̃i x̃j = l2

with probability (1 − p)qn2 + pqc2

 0with probability 1 − (1 − p)qn2 − pqc2

The correlation between individual risks is:δ=

COV (x̃i , x̃j ).
(V AR(x̃i )V AR(x̃j ))0.5

We have:
COV(x̃i , x̃j ) = E(x̃i x̃j ) − E(x̃i )E(x̃j )= l2 ((1 − p)qn2 + pqc2 ) − (−lq)2= l2 ((1 − p)qn2 + pqc2 − q 2 ),VAR(x̃i ) = E((x̃i )2 ) − E(x̃i )2= l2 q − (−lq)2= l2 q(1 − q).
Then, when N is large, the coefficient of correlation is with a good approximation:(1 − p)qn2 + pqc2 − q 2
q(1 − q)p(1 − p)=(qc − qn )2 .
q(1 − q)δ =With the average individual loss represented by the random variable X̃, we
have:
X̃ =


 q l

with probability p q lwith probability 1 − pcnThis can also be written as X̃ = q̃l where:
q̃ =
96


 q
 q

c

with probability pnwith probability 1 − p

Chapter 4. Pooling natural disaster risks in a community
Hence, the variance of the average individual loss is: Var(X̃) = Var(q̃)l2 , with:q̃ 2 =

 q2

with probability p q2

with probability 1 − pcnVar(q̃) = E(q̃ 2 ) − (E(q̃))2= (1 − p)qn2 + pqc2 − q 2 .
The variance of the average individual loss is then:Var(X̃) = δq(1 − q)l2 .

4.6.2

Without opportunity cost (λl (α) = 0)

Derivation of the FOC of (4.5)
If the inequality constraints are not strictly binding in (4.5), the firstorder conditions of (4.5) relative to τ , ϵ and π are respectively:
−(1 +

−(1−

p(qc − qn ) R
λ )qE(u′ (w̃)) + (1 − p)qn u′ (w2 ) + pqc u′ (w4 ) = 0,q

(4.18)(1 + λR )pqc E(u′ (w̃)) − pqc u′ (w4 ) = 0,(4.19)p
λR )(1−p)E(u′ (w̃))+(1−p)(1−qn )u′ (w1 )+(1−p)qn u′ (w2 ) = 0. (4.20)
1−pFirstly, (4.19) gives:u′ (w4 ) = (1 + λR )E(u′ (w̃)).

(4.21)

Secondly, with q = (1 − p)qn + pqc , the combination of (4.18) and (4.19) gives:u′ (w2 ) = (1 −p
λR )E(u′ (w̃)).
1−p(4.22)

Thirdly, (4.20) gives with the latter equation:u′ (w1 ) = (1 −p
λR )E(u′ (w̃)).
1−p(4.23)
97

Chapter 4. Pooling natural disaster risks in a community
Fourthly, with (4.21), (4.22), (4.23) and the definition of E(u′ (w̃)), we get:u′ (w3 ) = (1 + λR )E(u′ (w̃)).

(4.24)

If the inequality constraints are not strictly binding in (4.5) except(qc − qn )τ − π − qc ϵ ≥ 0, we have then π = (qc − qn )τ − qc ϵ and (4.5) boils
down to:max E(u(w̃))τ,ϵ

s.t. α = qc (τ − ϵ)(4.25)

π = (qc − qn )τ − qc ϵ.
The first order conditions of (4.25) relative to τ and ϵ are respectively:−qc E(u′ (w̃))+(qc −qn )(1−p)((1−qn )u′ (w1 )+qn u′ (w2 ))+(1−p)qn u′ (w2 )+pqc u′ (w4 ) = 0,(4.26)
qc E(u′ (w̃)) − qc (1 − p)((1 − qn )u′ (w1 ) + qn u′ (w2 )) − pqc u′ (w4 ) = 0.

(4.27)Firstly, the sum of (4.26) and (4.27) gives:u′ (w2 ) = u′ (w1 ).

(4.28)

Secondly, (4.27) gives with the latter equation:u′ (w4 ) = u′ (w3 ).

(4.29)

Comparative statics′′(.)
We consider a CARA utility function u(.), i.e. with A = − uu′ (.)> 0 constant.
∗

With 0 < λR < λR , (4.8) gives:
(1 −

p
p
λR )u′′ (w3 )dw3 −u′ (w3 )dλR = (1 + λR )u′′ (w1 )dw1 + u′ (w1 )dλR .
1−p1−p(4.30)

With (4.8), (4.30) can be rewritten:−A(dw3 − dw1 ) = (
98

p
1
+
)dλR ,
R
1 − p − pλ
1 + λR(4.31)

Chapter 4. Pooling natural disaster risks in a community
which finally gives with π = w1 − w3 :
1
dπ
=
.
R
dλ
A(1 − p − pλR )(1 + λR )(4.32)dπ> 0 because (1 + λR )p < 1. Besides, α = (q + p(qc − qn )λR )l + (1 − p − pλR )πdλRand τ R = N (qc − qn )l − N π > 0 respectively give with (4.32):

1
dα
=
p((q
−
q
)l
−
π)
+
,
c
n
dλR
A(1 + λR )(4.33)

dτ R
N
=−
.
R
dλ
A(1 − p − pλR )(1 + λR )(4.34)Rdαdτ RR> 0 because τN = (qc − qn )l − π > 0. dλ
R < 0 because (1 + λ )p < 1.
dλR∗

With 0 < λR < λR , (4.8) gives similarly:
dπ
= 0.
dδ

(4.35)

Besides, α = (q + p(qc − qn )λR )l + (1 − p − pλR )π and τ R = N (qc − qn )l − N π > 0
respectively give with (4.35):dαd(qc − qn ) R
=p
λ l,
dδ
dδ

(4.36)dτ R
d(qc − qn )
=N
l.
dδ
dδ

(4.37)> 0 and dτdδ > 0.
Because d(qcdδ−qn ) > 0, dαdδRWith opportunity cost (λl (α) ≥ 0)

4.6.3

Derivation of the FOC of (4.5)
If the inequality constraints are not strictly binding in (4.5), the firstorder conditions of (4.5) relative to τ , ϵ and π are respectively:′−(1+λl (α))(1+p(qc − qn ) R
λ )qE(u′ (w̃))+(1−p)qn u′ (w2 )+pqc u′ (w4 ) = 0, (4.38)
q99

Chapter 4. Pooling natural disaster risks in a community′(1 + λl (α))(1 + λR )pqc E(u′ (w̃)) − pqc u′ (w4 ) = 0,′−(1+λl (α))(1−(4.39)p
λR )(1−p)E(u′ (w̃))+(1−p)(1−qn )u′ (w1 )+(1−p)qn u′ (w2 ) = 0.
1−p(4.40)Firstly, (4.39) gives:′u′ (w4 ) = (1 + λl (α))(1 + λR )E(u′ (w̃)).

(4.41)

Secondly, with q = (1 − p)qn + pqc , the combination of (4.38) and (4.39) gives:′u′ (w2 ) = (1 + λl (α))(1 −p
λR )E(u′ (w̃)).
1−p(4.42)

Thirdly, (4.40) gives with the latter equation:′u′ (w1 ) = (1 + λl (α))(1 −p
λR )E(u′ (w̃)).
1−p(4.43)

Fourthly, with (4.41), (4.42), (4.43) and the definition of E(u′ (w̃)), we get:(′)λl (α)u′ (w3 ) = (1 + λ (α))(1 + λR ) −E(u′ (w̃)).
p(1 − qc )l′(4.44)

If the inequality constraints are not strictly binding in (4.5) except
π ≥ 0, we have π = 0 (which states u′ (w1 ) = u′ (w3 )) and the first order conditionsof (4.5) relative to τ and ϵ are respectively:′−(1+λl (α))(1+p(qc − qn ) R
λ )qE(u′ (w̃))+(1−p)qn u′ (w2 )+pqc u′ (w4 ) = 0, (4.45)
q′(1 + λl (α))(1 + λR )pqc E(u′ (w̃)) − pqc u′ (w4 ) = 0.

(4.46)Firstly, (4.46) gives:′u′ (w4 ) = (1 + λl (α))(1 + λR )E(u′ (w̃)).

(4.47)

Secondly, with q = (1 − p)qn + pqc , the combination of (4.45) and (4.46) gives:′u′ (w2 ) = (1 + λl (α))(1 −100p
λR )E(u′ (w̃)).
1−p(4.48)

Chapter 4. Pooling natural disaster risks in a community
Thirdly, with (4.47), (4.48) and the definition of E(u′ (w̃)), we get:(′)

p(qc − qn ) R
λl (α)u′ (w1 ) = u′ (w3 ) = (1 + λ (α))(1 −λ )−E(u′ (w̃)).
1−q
1−q
l′

(4.49)

Comparative statics′′(.)
We consider a CARA utility function function u(.), i.e. with A = − uu′ (.)> 0constant.
∗

With 0 < λl < λl , (4.13) gives:
1λl(1+λl )2
′′
(1−
)u
(w
)dw
−
u′ (w4 )dλl = u′′ (w3 )dw3 .
4
4
R
l
R
p(1 − qc )(1 + λ )(1 + λ )p(1 − qc )(1 + λ )(4.50)
With (4.13), (4.50) can be rewritten:−A(dw4 − dw3 ) =1dλl ,p(1 − qc )(1 + λR )(1 + λl )2 − λl (1 + λl )(4.51)

which finally gives with ϵ = w3 − w4 :
dϵ
1
=
.
l
R
dλ
A(p(1 − qc )(1 + λ )(1 + λl )2 − λl (1 + λl ))(4.52)

Similarly, (4.14) gives:
dπ
1
=−
.
l
R
dλ
A(p(1 − qc )(1 + λ )(1 + λl )2 − λl (1 + λl ))(4.53)

Besides, α = (q + p(qc − qn )λR )l + (1 − p − pλR )π − (1 + λR )pqc ϵ and τ R =N (qc − qn )l − N π − N qc ϵ > 0 respectively give with (4.52) and (4.53):
dα
dϵ
R
=
−(1
−
p(1
−
q
)(1
+
λ
))
,
c
dλl
dλl

(4.54)dϵdτ R= N (1 − qc ) l .
ldλdλ(4.55)p(1−qc )R
Note that λl < 1−p−p(1−qand (1 + λR )p < 1 give: p(1 − qc )(1 + λR )(1 + λl ) −Rλc )λλl > 0, which tells the sign of the four latter equations.
101

Chapter 4. Pooling natural disaster risks in a community∗

With 0 < λl < λl , (4.13) and (4.14) give similarly:
dϵ
λl
=
,
dqc
A(p(1 − qc )2 (1 + λR )(1 + λl ) − λl (1 − qc ))(4.56)

dπ
λl
=−
.
dqc
A(p(1 − qc )2 (1 + λR )(1 + λl ) − λl (1 − qc ))(4.57)p(1−qc )Rand (1 + λR )p < 1 give: p(1 − qc )(1 + λR )(1 + λl ) −Note that λl < 1−p−p(1−q
Rλ
c )λ
dϵ
dπ
dπ
λl > 0. Thus, dq> 0 and dq< 0. Because dqdϵn = 0 and dq= 0, we thus have:ccndϵ> 0 and dπ< 0. Besides, α = (q + p(qc − qn )λR )l + (1 − p − pλR )π − (1 + λR )pqc ϵdδdδand τ R = N (qc − qn )l − N π − N qc ϵ > 0 respectively give with (4.56) and (4.57):

)

(

dα
d(qc − qn ) Rdϵdqc+ (1 + λR )pϵ=p
λ l − (1 − p(1 − qc )(1 + λR ))
,
dδ
dδ
dqc
dδ
(

)

dτ R
dϵ dqcdqn= N (l − ϵ) + N (1 − qc )
− Nl
.
dδ
dqc dδ
dδ
R

Thus, dαis ambiguous and dτdδ > 0.
dδ

102(4.58)(4.59)

Conclusion
This thesis addresses prevention and insurance issues relative to natural disasterrisks. It tackles sub-efficient current levels of prevention measures and insurance
coverage caused by market imperfections and poorly-designed public policies.
The first part of the thesis deals with the limits of public policies such as
insurance subsidy or public relief. Even though they can improve risk sharing,these public policies do not give the right incentives in terms of risk exposure.
Chapter 1 highlights that insurance subsidy leads to excessive population density
and insufficient resilience investment in risky areas. To avoid risk over-exposure,
this chapter recommends that public policies should focus on making agents awareof the risk and liable for their choice of risk exposure. Furthermore, if public
policies have to manage inequality issues, they should not favor public aids basedon risk exposure.
The second part of the thesis focuses on risk dependence and costs of financial
reserves. Because insurers have to secure costly capital to face uncertain catastrophic claims, insurance prices can be high for natural disaster risks. Chapter 3
points out that the cost of securing capital limits the insurability of natural disasterrisks, while chapter 4 shows that the design of insurance contracts can be improved
with indemnities and dividends contingent on the collective states. To further improve the insurability of natural disaster risks, we advocate that public policies
should promote international diversification with the development of competitivereinsurance markets and insurance-linked securities such as Cat-bonds.
Lastly, both the time and ambiguity dimensions of natural disaster risks should103

Conclusion
be further investigated. The time dimension is central in particular because natural disaster preventive actions consist usually in long-term investments and natural
disaster insurance markets can be subject to fluctuations through time. The ambiguity dimension relates to the lack of information about risks which has proven to
be a major obstacle for prevention and insurance of natural disaster risks. These
two dimensions are all the more critical in the perspective of climate change which
affects the patterns of natural hazards and our knowledge of them.

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Title: Prevention and insurance of natural disasters
Keywords: natural disasters, risks, insurance, prevention
This thesis deals with prevention and insurance of natural disasters. Low current levels of prevention measures andinsurance coverage are explained by numerous market failures and poorly-designed public policies. Modeling individual
behaviors, markets and public policies, this thesis aims at characterizing prevention actions and insurance mechanismsthat could mitigate efficiently losses and wealth variability for risk averse agents. Chapter 1 investigates preventive
behaviors in the context of city development. It shows that risky areas are more developed nearer to the city center thanfurther away and that investment in building resilience leads to more compact cities. This chapter also highlights theperverse effects of insurance subsidies leading to risk over-exposure and the role that can be played by urban policies
such as density restrictions and building codes. The following chapters focus on insurance mechanisms when individual
risks are not independent, a main feature of natural disaster risks. Chapter 2 shows that, without market failures,
Pareto optimal allocation of risks is reached thanks to stock insurance companies in competition and a reduced numberof financial assets. In practice, agents have limited liability and public policies require agents to secure financial reserves
to limit payment defaults in catastrophic states. That is why chapters 3 and 4 investigate the issue of the cost offinancial reserves. Chapter 3 analyzes how the cost of financial reserves affects the insurance demand of agents exposed
to correlated risks. If the financial reserves for the collective risk leads to an additional premium in the price of insurance
for one agent, it appears that for a given collective risk, the purchased coverage rate decreases when the individual
probability of being affected decreases. Chapter 4 examines the optimal design of insurance contracts when individual
risks are correlated in a community. If it is not costly for the community to build reserves, the optimal contract for agiven individual risk consists in full coverage, whatever the collective losses, plus a dividend if necessary to redistributethe remaining part of the reserves. Otherwise, the optimal contract for a given individual risk consists in partial coveragewhen collective losses are high.

Titre: Prévention et assurance des catastrophes naturelles
Mots-clés: catastrophes naturelles, risques, assurance, prévention
Cette thèse porte sur la prévention et l’assurance des catastrophes naturelles dont les faibles niveaux actuels sontdus aux nombreuses imperfections de marché mais aussi aux déficiences des politiques publiques. En s’appuyant sur lamodélisation des comportements individuels, des marchés et des politiques publiques, cette thèse a pour objectif d’étudier
quels sont les actions de prévention et les mécanismes d’assurance qui permettraient de diminuer efficacement les pertes
et aussi la variabilité de la richesse pour les agents averses au risque. Le chapitre 1 porte sur les choix de prévention
dans le contexte du développement des villes. Il montre que les zones risquées sont plus développées près du centreville que loin du centre-ville et que l’investissement dans la résilience des bâtiments mène à des villes plus compactes.
Ce chapitre met aussi en évidence les effets pervers des subventions à l’assurance qui entrainent une surexposition au
risque et le rôle que doivent jouer les politiques publiques urbaines telles que les restrictions de densité ou les codes deconstruction. Les chapitres suivants abordent la problématique des mécanismes d’assurance lorsque les risques individuels
ne sont pas indépendants, ce qui est une des caractéristiques majeures des catastrophes naturelles. Dans le chapitre 2,
il est établi que, sans imperfection de marché, une allocation Pareto-optimale des risques est atteinte en présence d’unmarché compétitif de compagnies d’assurance et d’un nombre restreint d’actifs financiers. En pratique, la responsabilité
des agents économiques est limitée et les politiques publiques requièrent que des réserves financières soient constituées
pour limiter les défauts de paiement dans les états catastrophiques. C’est pourquoi les chapitres 3 et 4 abordent laquestion du coût des réserves financières. Le chapitre 3 étudie l’impact du coût des réserves financières sur le taux decouverture demandé par les agents exposés à des risques individuels corrélés. Si les réserves financières pour le risquecollectif génèrent une surprime dans le prix de l’assurance individuelle, il apparaît qu’à risque collectif donné, le tauxde couverture demandé décroît quand la probabilité individuelle d’être sinistré décroît. Le chapitre 4 analyse la forme
optimale des contrats d’assurance pour une communauté d’agents qui sont exposés à des risques individuels corrélés. S’iln’est pas coûteux pour la communauté de constituer les réserves nécessaires, le contrat optimal pour un risque individuel
donné consiste en une couverture totale, quelques soient les pertes collectives, à laquelle s’ajoute un dividende qui permetde redistribuer le cas échéant les réserves non utilisées. Dans le cas contraire, le contrat optimal pour un risque individuel
donné consiste en une couverture seulement partielle quand les pertes collectives sont élevées.