Public-Private Programs for Covering Extreme Events: The Impact of Information Distribution on Risk-Sharing

Transcription

1 Michel-Kerjan, Erwann, and Nathalie de Marcellis-Warin Public-Private Programs for Covering Extreme Events: The Imact of Information istribution on Ris-Sharing, Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue : -49 Public-Private Programs for Covering Extreme Events: The Imact of Information istribution on Ris-Sharing Erwann Michel-Kerjan Nathalie de Marcellis-Warin Abstract Recent extreme events have significantly raised the question of the role of ublic and rivate sectors in roviding adequate financial rotection to victims eveloing ublicrivate insurance rograms could constitute one of the most aealing ways to solve the roblem of financing the consequences of those large-scale catastrohes owever, catastrohic riss resent very secific characteristics which challenge any traditional economic aroaches to analyzing them Further, the government may have better information about the ris than insurers eg, national security Currently, little has been done in the economic literature to better understand how this assumtion imacts on how riss are shared between all staeholders in such artnershis This aer analyzes olicy issues related to ris/information sharing between insurers and a dedicated State-baced governmental reinsurer, who are art of a national artnershi rogram The government develos a mandatory coverage against catastrohic riss and decides the level of remiums levied against the insureds sing a game-theoretical aroach, we show that a government can act to induce rivate insurers in the country to articiate in the artnershi instead of leaving the maret By modulating its remium olicy, the government can also led them to adot two different strategies: behave as a simle financial intermediary between the insured and the ublic reinsurer so that the latter suorts the largest ortion of the riss or conserve the largest art of riss to benefit from maret conditions created by the government seeing to avoid its intervention ex ost to bail out the ublic reinsurer The aer also discusses the imacts of government information-sharing strategies on the game equilibrium Illustrations are rovided for natural hazards and terrorism ris Key Words: Extreme events, ublic-rivate artnershi, ris sharing, information sharing, ublic olicy, natural disaster, terrorism Erwann Michel-Kerjan is with the Wharton School of the niversity of Pennsylvania SA Nathalie de Marcellis-Warin is with the Ecole Polytechnique de Montreal and Cirano Canada The authors are grateful to ominique enriet, Claude enry, Arye illman, oward Kunreuther, Pierre Picard, Bernard Sinclair-esgagné, Thierry Warin and various seminar articiants at for valuable discussions and their inut on earlier versions of this aer as well as anonymous referees of the journal for insightful comments The authors also wish to acnowledge the editorial assistance of annah Chervitz Any remaining errors are authors own Michel-Kerjan wishes to than ocheed Martin, the S eartments of omeland Security and Transortation, the EF-Polytechnique de Paris Chair on Sustainable eveloment and the Wharton Ris Management and ecision Processes Center for scientific and financial suort of this roject e Marcellis-Warin thans the Ecole Polytechnique de Montreal, and the VRQ Ris Project at CIRANO for financial suort of this research The authors can be contacted at erwannm@whartonuennedu Michel-Kerjan, corresonding author and nathaliedemarcellis-warin@olymtlca de Marcellis-Warin, resectively

2 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue I Introduction In most develoed countries, insurance constitutes an imortant illar of economic develoment, as it is one of the rincial mechanisms used by individuals and organizations for managing ris Without the caacity of transferring ris to the insurance and reinsurance industry, it is liely that numerous activities would not have had the develoment that we now today In the field of insurance and ris management, however, extreme events resent a set of very secific characteristics as they can have long-term imacts on the social and economic continuity of the countryies they affect This raises the question of the resonsibility of the rivate and ublic sectors in roviding adequate financial rotection to victims of these catastrohes Over the recent years, levels of insured and non-insured economic damage due to the occurrence of catastrohes have dramatically increased, questioning the insurability of catastrohic riss by the rivate maret alone Facing unrecedented large-scale claims, the rivate sector of insurance may be actually induced to severely restrict the insurance suly or even refuse to cover against secific lines of catastrohic riss in the future if not required by law owever, because these events are caable of inflicting debilitating imacts on a large number of eole and firms, covering them often constitutes a national issue Governments, therefore, have a social and economic resonsibility to rovide citizenry and commercial firms with alternative answers This can be done by develoing urely government rograms or by offering the rivate sector of insurance enough incentives to artner with the government to establish a joint rogram This aer discusses this second otion by roviding a theoretical framewor of ublic-rivate artnershi regarding catastrohic ris coverage The aer roceeds as follows Section discusses some of the secific challenges raised by extreme events: increased level of loss due to catastrohes worldwide which questions the insurability of such events by the rivate sector alone, the resulting need for combining strengths of the ublic and rivate sectors to deal with that issue and, more secifically, the ossibility of seeing national government entities more informed about certain riss than the rivate sector This section rovides readers who might not be familiar with those issues with some bacground evidence Section 3 resents a basic model of ublic-rivate ris sharing in the form of a game with Perfect Bayesian Equilibrium PBE The framewor is resented in this section: the model s hyotheses including the creation of a dedicated State-baced governmental reinsurer, the layers action, the objectives of the government and the insurance industry and finally the taxonomy of the PBE candidates This section also resents the assumtions relative to remiums levied against olicyholders This aer analyzes two different ris sharing olicies by the government in more detail: using the insurers as simle indemnification intermediaries or looing for the financial autonomy of the rogram such that the state-guarantee of the ublic reinsurer is never triggered Section 4 gives a characterization of the ooling or searating equilibria that corresond to those two cases, with a secific focus on the ultimate ris-sharing between the ublic and rivate sectors It is worth noting that if a condition is that government does not reveal its nowledge about the riss eg, sensitive information about terrorist grous, strategies leading to the ooling equilibrium are the only viable ones We discuss the results of the model and ossible extensions in Section 4 as well Section 5 illustrates the theoretical aroach with examles: first, the insurance against natural disasters in France as well as an alication of the model for the S maret; second, some ost 9/ national insurance rograms recently develoed for coverage against terrorism Section 6 concludes the aer Aendixes rovide detailed roofs for the two roositions of the aer

3 Public-Private Programs for Covering Extreme Events II Some Persectives of Extreme Events A New imension of oss In a worldwide context, as far as the insurance against natural disasters is concerned, a dual trend has been observed since the beginning of the 990 s: an increase in the number of so-called imortant events and the occurrence of catastrohic events that were unthinable a decade ago 99 s urricane Andrew inflicted $ billion of insured losses and the 994 Northridge earthquae's costs were $7 billion in direct insured losses alone both in 003 dollars Asia also suffered several extreme events over the ast 5 years In 99 and 999, tyhoons Mireille and Bart inflicted $75 billion and $45 billion of insured losses, resectively In 995, the Great anshin earthquae in Kobe illed nearly 6,500 eole Although insured losses are estimated at $3 billion, that figure needs to be ut into ersective and comared to the nearly $00 billion of total damage Swiss Re, 004 More recently, the year 004 hit a record with 48 billions dollars insured losses due to natural disasters worldwide But 005 is already a new record holder, with 83 billion dollars insured losses due to catastrohe worldwide Swiss Re, 006 urricane Katrina alone, which hit the S landfall on August 9, 005, inflicted insured losses estimated in the range of billion of dollars Beyond the level of insured losses, economic disrution can be even more imortant in countries where the level of insurance is low, a situation which is tyical in most develoing countries, limiting the caacity of ris transfer In 005, more than 97,000 eole lost their lives due to natural catastrohes or man-made disasters According to Swiss Re, 94% of these fatalities occurred in the Asian continent but only 3% of the worldwide insured loss costs The ecember 6, 004, tsunami that illed over 50,000 eole in 0 countries illustrates this tragically as well Terrorism ris is another area we consider in this aer Prior to Setember, 00 terrorism coverage in most Asian and Euroean countries as well as in the S came essentially free of charge by virtue of its being included in most standard commercial olicy acages with no additional remium to sea of associated with these events Most insurers had simly not aid close attention to their otential losses from terrorism and/or transferred the largest ortion of the riss to the reinsurers The 9/ terrorist attacs against New Yor City, Washington C and rural Pennsylvania, along with other attacs worldwide, illustrate how the nature of terrorism has changed over the ast decade This change would be directly caused by the evolution of international terrorism over recent years: from local olitical actions to extremist religious-based grous seeing to inflict mass casualties and maximum disrution and fear, offman 998; Enders and Sandler, 006 Recent attacs in Euroe and Asia demonstrated that no develoed country can be viewed as immune, and that terrorism threat will be with us for a long time to go The 9/ terrorist attacs-the second attac against the World Trade Center after the 993 bombing-illed more than 3,000 citizens and first resonders from over 80 countries, illustrating the transnational character of modern-day terrorism and the otential for hyercorrelation of riss Sandler and Enders, 004; Chal, offman, Reville, and Kasusi, 005 After all losses are accounted for, these attacs will be the second costliest insured According to Swiss Re, an imortant event is defined as an event that inflicts insured damage higher than $70 million Over the decade , on average, 40 imortant events er year occurred worldwide Over the decade , the number increased to an average of 00 imortant events er year In 995, 30 imortant events occurred worldwide and no fewer than 60 in 998 Swiss Re, 000 And the increasing activities in natural-hazard rone areas cannot exlain everything In arallel to such an evolution, the cost of damage also increases Between 970 and 999, 46 events worldwide inflicted insured damage higher than billion dollars, 3 of which occurred between 989 and 999 Swiss Re, 000 In the nited States alone, there have been 7 such events between 000 and 005 3

4 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue loss in the history of world insurance, after hurricane Katrina, with current estimates at $33 billion oherty et al, The reinsurers, most of them Euroean, aid nearly two-third of the claims These reinsurance ayments came in the wae of outlays triggered by a series of catastrohic natural disasters over the ast decade and ortfolio losses due to stoc maret declines aving their caital base severely hit, most reinsurers decided to reduce their terrorist offerings drastically or even to sto covering this ris Considering such a new scale of catastrohic riss, the issue of develoing secific rograms for ris coverage is obviously ey to roviding victims with adequate indemnification, therefore assuring the continuity of economic and social activities of the devastated country Catastrohe Ris Sharing: the Need for Public-Private Partnershis With recent disasters and their resulting effect on the solvency of insurance comanies, the question of liquidity and insurers insolvency has become of great imortance Insurers may be reluctant to cover these riss alone Traditionally, the insurance industry avoids these roblems by transferring such riss to the reinsurance maret Borch, 990 owever, the current reinsurance caacity for coverage against natural catastrohes and terrorism is limited, and rices of catastrohe reinsurance tend to seriously increase in the aftermath of major catastrohes 4 The develoment of securitization roerty-catastrohe-ris financial instruments could also rovide an aealing way to finance these riss In the aftermath of urricane Andrew and the Northridge Earthquae in the early 990s, roerty catastrohe reinsurance was in short suly and the rice of reinsurance more than doubled in the S comared with the late 980s For insurers to rovide their clients with the same amount of coverage they offered rior to these events they had to find caital from other sources They collaborated with the investment baning community to develo new classes of financial instruments Alternative ris transfers, such as otions and catastrohe bonds, emerged to cover these losses by transferring art of the riss to the $9 trillion caital marets 5 owever, whereas the develoment of catastrohe modeling and simulation has been effective owing to the advances in information technology, the use of financial instruments remains under utilized Though insurers and reinsurers had over $43 billion in catastrohe bonds outstanding at the end of 003, an increase of more than 50% over 00 Swiss Re, 004 This maret is still considerably below the exectations of insurers, reinsurers, and investment baners, accounting in 00 for less than 3% of worldwide catastrohe reinsurance coverage Grossi and Kunreuther, 005 Still, we have not seen any definitive argument to show that the rivate sector alone can handle the roblem of insolvency in case of extreme financial consequences of natural disasters and large-scale terrorism 3 Not accounting for the fact that the federal Victim Comensation Fund that will have aid nearly $5 billion to 9/ victims and their family Chal et al, Very few reinsurers in lace rovide rotection against industry-wide losses for catastrohic event greater than $5 billion Several arguments exlain the reason why the rices of catastrohe reinsurance are high: insufficient reinsurance caital, reinsurers maret ower, inefficiency of the cororate form of reinsurance, high transaction costs, moral hazard and adverse selection at the insurer level Froot, 999, 00 5 Among the first wors regarding the develoment of derivatives and catastrohe bonds, see, 'Arcy and France, 99; Niehaus and Mann, 99; for more recent ublications, see Cox, Fairchild and Pedersen, 000, Nell and Richter, 004, and ane, 00 Since the 9/ attacs, the ossibility of transferring some terrorism ris to financial marets has also been studied; see Kunreuther and Michel-Kerjan 005 4

5 Public-Private Programs for Covering Extreme Events These low-robability/high-consequence events are difficult to cover for several additional reasons On the suly side, because insurers are dealing with a lac of historical data on catastrohic events as they are new in their nature or scale, the aversion for ambiguity leads them to set remiums higher than they would if they had a erfect nowledge of the distribution of ris Kunreuther, ogarth, and Meszaros, 993 In some instances, the situation is counterbalanced by a regulation of remium rates that requires rivate insurers to sell insurance coverage at a lower rice than is necessary for business Priest, 996; Jaffee and Russell, 997 owever, the insurers will often refer not to cover these riss or deely limit their involvement On the demand side, it is well nown that with time otential urchasers tend to underestimate the real level of riss and thus consider the insurance remiums as being too exensive, limiting the level of coverage Browne and oyt, 000; Ganderton et al, 000; Kunreuther and Pauly, 004 The charity hazard 6 also induces otential urchasers of insurance to limit their coverage or even to refer not to be covered at all As a result of these demand and suly effects, numerous citizens and firms could be left without coverage owever, under the ublic ressure in the aftermath of a catastrohe, the government would have to financially hel the citizens and firms that were victims of the catastrohe through emergency measures, crisis management, disaster relief to uninsured citizens, etc This raises the question of the Samaritan s dilemma associated with federal disaster assistance 7 The continuing increase in the cost of aid for governments 8 has caused olicymaers to loo closely at ossible national insurance rograms to levy ex ante contributions and increase the general concerns on those issues The creation of secific ublic insurance rograms may be one of the solutions eg the S National Flood Insurance Program But it may be very costly for the government in imlementation as well as day-to-day management levying surcharges, estimating damage of a catastrohe, ayment of claims, etc, thus leading to inefficiencies On the other hand, insurers can easily add and manage an extra line of ris in their ortfolio They can also tae advantage of their commercial networ throughout a country to sell the associated olicies and collect remiums Moreover, ayment of claims is usually much faster when organized by the insurers For that reason, collaboration between the government and the insurers could, at least artially, be art of a global solution for catastrohic ris coverage What would be the strengths of the ublic sector in such a artnershi? Government resents at least two ey strengths First, government constitutes a owerful source of wealth redistribution toward losses already occurred What is crucial here is the government s ability to diversify the riss over the entire oulation and to sread ast losses to future generations of taxayers This is a form of cross-time diversification that the 6 efined by Browne and oyt 000 as the tendency of an individual at ris not to urchase insurance or other ris financing as a result of a reliance on exected charity from a government emergency rogram 7 The Samaritan s dilemma was introduced by Nobel laureate James Buchanan The basic idea is that the government the Good Samaritan wants to hel victims after a major loss While such an attitude is liely to generate ublic aroval after a disaster it has otentially negative effects on otential victims behavior rior to the event Indeed, it creates moral hazard roblems by encouraging ris-taing behavior including not urchasing insurance by those who feel they will be financially rotected by the government action after an event See Buchanan Since the late 70 s the Federal government has sent annually an average of $8 billion current on disaster assistance This is far greater than the average annual loss borne by reinsurers on S catastrohe coverage Froot 00 5

6 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue rivate maret cannot achieve because of the incomleteness of inter-generational rivate marets and legal limitations for insurers to accumulate financial reserves Gollier, 00 9 Government involvement in a ublic-rivate rogram of catastrohic ris coverage could thus be a definite advantage in that it can limit the otential of catastrohe losses for insurers, for instance by being a reinsurer of last resort Second, government can constrain adverse selection henomena by the enforcement of insurance urchase for a secific maret segment exosed to the catastrohic riss we are concerned with here ie, mandatory coverage Even if this governmental control of adverse selection does not reduce the ris itself, it constitutes a strong vector of redistribution 0 For all these reasons, the idea of develoing secial arrangements between the ublic and rivate sectors to deal with coverage against catastrohic riss obviously emerges and needs to be analyzed in further details Since the rivate sector of insurance can simly refuse to cover these riss to the extent that there is no regulation reventing this, a ey question to analyze is how a government can wor with insurers to create a secific rogram based uon mutual interests Information istribution The observation of secific marets for coverage against extreme natural hazards and terrorism ris indicates that the distribution of information between government agencies and the rivate sector of insurance can be quite secific There are actually situations where the former is more informed about the riss than the latter Consider, for examle, international terrorism Whatever information exists on ossible attacs or current threats is et secret by government agencies for national security reasons These services will not exlicitly share such information with the rivate sector of insurance The roblem osed by terrorism aears to be one where there is symmetry of non-information on the ris between olicyholders and insurers This information issue constitutes a notable feature of terrorism as a catastrohic ris Terrorism is not only a new ris with limited available historical data for insurers, it is also a ris where the government aears to be the most informed arty Godard et al, 00 With resect to natural disasters, it can be the case that there has been a long-time lac of interest from insurers regarding the secific riss they had not really considered There could be several reasons for this: because the riss reresent only a small art of their ortfolio, because a detailed ris analysis nationwide would be too exensive, 3 because they were not considered catastrohic until an extreme event occurred, and/or because government rograms cover these riss 9 The ublic sector is also able to achieve high diversification by ooling several sources of riss 0 For a recent survey of adverse selection henomena in insurance, see ionne, oherty and Fombaron 000 In using this exression we consider situations where the two sectors are caable of defining and mutually acceting secific configurations of ris-sharing that would resond to their own exectations Of course, this does not mean all government agencies are uniformly informed; for an insightful discussion about recent information sharing failure among intelligence services in the S, see Garicano and Posner In the nited States, for examle, the 973 Flood isaster Protection Act made the national Flood Insurance Program resonsible for identifying all communities nationwide that contained areas at ris of serious flood hazard As of 998, over 8,700 communities had joined the NFIP, for twothirds of which a detailed flood study has been conducted According to Pasteric 998, through fiscal year 997, the cost of this massive study effort has been about $5 billion 6

7 Public-Private Programs for Covering Extreme Events As such a situation government more informed aears counter-intuitive when viewed in the ersective of the economic literature of insurance, we have been interested in discussing how such an assumtion affects the ris-sharing outut in secific ublic-rivate artnershis 4 In this whole context, this aer resents in the next section a simle model of ublic-rivate artnershi in which the government not only has better information but it can also determine the rice of coverage The central question that this aer addresses under these assumtions is What national olicies can a government develo regarding the artnershi it establishes with the rivate sector of insurance to cover extreme events? In other words, how do olicymaers consider the rogram resulting from the artnershi? o they use insurers as simle intermediaries between the government and the citizens/firms they want covered, or do they really loo for an effective ris-sharing with the insurers? If so, how does the trade-off between information-sharing and ris-sharing between the ublic and rivate sector lay? To our nowledge, this roblem of ublic-rivate artnershi in develoing national coverage against catastrohic riss has not been formalized in such a way in the literature III A Model of Public-Private Partnershi Framewor of the Public-Private Partnershi In order to investigate the ublic-rivate artnershi for insuring catastrohic riss, we first characterize the general framewor We consider a country where there is an extremely limited insurance maret for the class of riss we study here To fill the ga, a artnershi is launched between the government and the rivate insurers and a new insurance rogram is established The government votes for the alication of an extra charge against the insureds on secific basic lines of insurance eg roerty and casualty This extra charge equals a certain ercentage of that basic insurance remium If the insurers decide to articiate in the rogram, they collect these extra charges and receive a articiation commission from the government in return For this line of ris, articiating insurers can be reinsured by an unlimited state-guaranteed governmental ublic reinsurer that is established for that urose, GRe This configuration is deicted in Figure The Game We consider a simle game of incomlete information In order to simlify the model, assumtions are made on both the set of catastrohe riss and the ossible actions We are only interested in the events covered by the catastrohe ris rogram during a given eriod of time, for examle, one to five years denotes the total insured damage due to some catastrohes covered by the ublic-rivate rogram eg, terrorism, earthquaes or floods The game has three layers: Nature, the Public Sector and the Private Insurers Player s Action For the sae of simlicity, Player, Nature, can choose only between two levels of lielihood that an extreme event occurs, and, with < 5 Player, the Public Sector or government hereafter, 6 fixes the level of the extra charge rate denoted, taing into account its nowledge of the otential occurrence of the catastrohic damage is here a arameter Referring to the game theory literature, we call the strategy or action layed by the government the extra charge olicy The government 4 Our discussion in Section 3 articularly, Taxonomy of PBE Candidates refers to recent contributions that rovide a more general theoretical analysis of the imact of such reversed asymmetry of information between an insured art and the insuring one 5 The notion of Nature as a layer is relative to the game theoretical literature 6 ere the Treasury 7

8 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue Figure : The Public-Private Relationshi can choose between a low extra charge olicy, denoted by, and a high one, denoted by 7 Π is the total remiums of the basic contracts eg Proerty and Casualty on which the extra charge is alied Therefore, the total extra charges collected for that catastrohic ris line over the eriod of time considered here are Π ie, the total remiums for the rogram Player 3, the rivate insurance industry or insurers, receives the total extra charges collected for that line, Π The insurance industry's behavior is assumed to be summarized by a single action: the reinsurance transfer rate to the governmental reinsurer GRe 8, denoted by The rivate insurers are assumed to be reinsured only by the ublic reinsurer, GRe They choose a transfer rate to GRe in [ ; ] [0,] It is worth noting here that we imose a restriction on that reinsurance transfer rate The introduction of such sub-limits allows the government to imact on the minimal and maximal ris-sharing with the ublic reinsurer a more secific case than insurers allowed to choose within the whole [0,] interval Insurers can ee the largest art of the remiums and riss by laying a low transfer rate or choose to transfer a major ortion to GRe and then tae the action a high transfer rate The high limit of reinsurance quota-share allows the rogram to share a minimum ercentage of the riss with the insurers they ee a minimum ercentage whatever their decision of transfer The low rate limits the amount of losses the insurers would bear in case of catastrohic damage if they had decided to ee the largest ossible ortion of riss 7 The choice of a continuum would not change significantly the results but would mae the determination of equilibria more difficult to be interreted 8 The GRe s equalization reserves are assumed to be zero in the model The introduction of such a arameter maes the model more realistic but leads to results difficult to interret economically 8

9 Public-Private Programs for Covering Extreme Events From a theoretical ersective, as the interval of 's is determined by government, one could suggest the government determine the otimal for them,, and then simly squeeze the interval of ossible reinsurance by GRe toward Such a strategy, however, resents several imortant limitations We mention two of them here First, it limits the main motivation of this aer which is to study different government olicies/alternatives, for each of which we determine the conditions of existence of equilibrium and resulting ris-sharing between insurers and the ublic reinsurer Should one decide the establishment of a unique ossible transfer, the model would be immediately reduced to two trivial cases: a the insurers articiate and both rice and ris transfer variables are fixed by the government; b they refuse to articiate, and the existence of a ublic reinsurer does not mae any sense anymore Second, an a more ractical note, it is far from certain that any insurance association would welcome its members not having any command control in the artnershi In fact, any rivate reinsurer would have hard time doing business if the only alternative it offered to its otential clients would be a tae-it or leave it treaty based on only one ossible level of transfer The introduction of lower and uer limits on imosed by the government in our model satisfies a tye of government control as well as a certain degree of flexibility for the insurers in their action within this artnershi 9 Finally, we also assume that the government offers the insurers a commission m in order to induce them to articiate in the rogram m is a oll commission, which does not deend on the level of ris et by insurers 0 Players Preferences In the standard theoretic framewor, the two main staeholders are i the ublic sector and ii the rivate insurers who decide according to their own references The criteria for action are the following The government wishes all catastrohe victims to be comensated owever, the establishment of such an insurance rogram based on a national redistribution induces several costs First, there is the total amount of extra charges aid by the insured, Π The ayment by the government here the Treasury of the articiation commission m to rivate insurers also has to be taen into account Moreover, as GRe is a ublicly owned comany, the Treasury sees to balance the ublic reinsurer So as the government accets a ossible ex ost ublic intervention ris financing by offering an unlimited guarantee to GRe, we have to consider the effect of such an intervention by introducing a arameter λ, which reresents the shadow rice of ublic funds The greater λ, the more reluctant the 9 We discussed in the aendix the three ossible cases that are liely to occur deending on the size of the interval limits and where the otimal level of transfer, chosen by the insurers, seats equation systems S-a, S-b and S-c 0 As suggested by one of the referees, and in the same vein than our discussion on not having a unique ossible transfer rate decided by the government, an alternative would be to mae m chosen by government deend on the level of ris assumed by the insurers That endogeneity would certainly be an interesting alternative scenario to discuss but it is not studied here In fact, in revious versions of the aer, we studied that case where both m and were endogenous variables on which government loos for maximization of utility FOC on m that guarantees insurers articiation deending on the level of ris By controlling both m and, government might increase its utility under certain circumstances Maing m endogenous also means, however, that m along with gives a signal to insurers about the riss That results in too many sub-cases to study and thresholds that are not always intuitive, mainly due to the two different sources of signal, and m Our final choice has been for a articiation commission only The fact that it does not deend on the level of ris covered by the insurers also translates a will to mae that comonent of the ublic-rivate artnershi rogram more easily manageable over time iscussions with ractitioners in charge of similar rograms in several countries actually revealed that establishing a rogram that was easily manageable was an imortant asect; a ool commission facilitates that 9

10 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue government is to financing ris ence, in this game, the Treasury has to deal with a ermanent trade-off between setting a low extra charge to mae the rogram more oular for citizens and firms, and maing its exenses balanced limiting its ex ost intervention Adding to its own objective, we can consider that the government internalizes all the objectives of the ublic sector as a whole: the insured citizens and firms, GRe, and its own exenses With such rogram in lace, it is obvious that the Treasury would lie rivate insurers to ee the largest ortion of riss in order to avoid having to ay when the reserves of GRe are not high enough In this secific case, we have to tae into account the shadow cost of ublic funds In all other cases, the shadow cost does not aear Neither the benefits nor losses obtained by the insurers nor the benefits of GRe because they are not transferred to the Treasury are affected by this arameter in our model The shadow cost of ublic funds is only considered when the Treasury has to oerate its guarantee to GRe Taing into account such an argument, the definition of utility of the Treasury can be written as follows: = Π Π Π ; = Π Π λ Π ; m if m otherwise The first equation corresonds to the case in which the total amount of extra charges is high enough to comensate all the victims of the catastrohic events with total damage that occur with a robability or during the studied eriod of time The ex ost governmental intervention is not required because Π > The benefit of the ublic reinsurer, which ositively affects the utility of the ublic sector in this model, is the difference between the total remiums received from the rivate insurers and the total amount of reimbursement aid by GRe The second equation corresonds to the case in which the extra charge olicy is not high enough: the remium received is too low to comensate the ortion of the damage the ublic reinsurer would have to ay GRe is virtually banruted at the end of the eriod The state guarantee is triggered and the government ays the excess of insured losses that GRe is not able to ay, that is Π multilied by the factor λ For Player 3, the rivate insurers, we use a mean-variance utility function assuming they are ris averse for catastrohic riss The ortion of remiums et by the insurers and the articiation commission ositively affect their utility The exected ayments and the variance in exected reimbursable damage affect it negatively / is assumed to be ositive catastrohe ris aversion We finally assume in this model that the commission m, which the government offers the insurers to cover their management costs, is always high enough to incite them to articiate ie, V>0 When the insurers believe the robability of damage is, their utility V is: In that sense, the resent game differs from situation where the government wants to maximize the total social welfare aroaches à la affont-tirole in which the Treasury loos to maximize the total social surlus ere the government and rivate insurers are ind of cometing with each other in the catastrohe ris sharing We than one of the reviewers who aroriately highlighted this imortant asect In the literature, insurers are often considered to be ris-neutral For catastrohic riss, the reinsurance demand shows exlicitly their ris aversion We use a mean-variance utility function because it is easily maniulated and it rovides good intuition But we now that this tye of function has an undesirable roerty: it does not resect the first-order stochastic dominance Borch, 968 owever, with a more correct version of the utility function, calculations are much more comlicated and results are similar 30

11 Public-Private Programs for Covering Extreme Events V;; = - Π m Players Information As discussed in the revious section, in this model the government is assumed to be the informed arty whereas the rivate insurers have only rior beliefs on the distribution of riss: there is a roortion of low ris tyes and a roortion of high ris tyes ence, the game is layed with Player 3 having imerfect information, which is reresented by two information sets in the extensive form of the game Once again, the main goal here with the model is to understand the olicy imlications of such reversed asymmetry of information, rather than the sources of it Sequence of ecisions The game taes lace over three eriods See Figure In eriod one, Player Nature chooses between two robability levels, or In the second eriod, Player the government receives the information on the robability of catastrohes during the eriod covered by the rogram and gets to choose between two olicies nodes G and G in the extensive form of the game, a low extra charge olicy or a high level one: { ; } For choosing one of the two olicies, the government taes account of its nowledge of the ris and the reaction from the insurers it anticiates Finally, in the third eriod, Player 3 the rivate insurance industry, who has no information on ris other than its rior belief, receives either the signal of a high extra charge olicy nodes I and I or the signal of a low extra charge olicy nodes I3 and I4 But insurers only receive indirect information on the nature of the riss they are going to insure when the government chooses the extra charge olicy Observing that olicy, they revise their rior beliefs using Bayes rule The government nows the rior beliefs of insurers Then insurers choose a reinsurance transfer rate [ ; ] The game is layed and both the government and the insurers obtain a certain level of ayment according to their actions and criteria We can now draw the comlete extensive form of the game as in Figure Taxonomy of PBE Candidates Considering the comlete set of ossibilities, when the government announces its, this does not necessarily corresond to its real nowledge of exected damage Insurers who observe the government's decision should udate their beliefs and base their choice on the osterior distribution: Φ\, which deends on the signal received and comatible with Bayes' rule Observing { ; }, the insurers can use Bayes' rule to udate to Φ \ efinition A erfect Bayesian equilibrium with ure strategies is an action rofile ; and osterior beliefs Φ \ such that: 3 i { ; }, = arg max ; ; ii { ; }, = arg max Φ \ V ; ; ; iii Su, Φ \ can be obtained from the a riori distribution discrete in the model by using the Bayes' s rule, whenever feasible 3 For a rigorous discussion of erfect Bayesian equilibrium, see Fudenberg and Tirole 99 3

12 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue Figure : The Extensive Form of Game Government Insurers I ;,V ; Nature G I3 ; ; V ; I ; ; V ; G I4 ; ; V ; i and ii are the erfection conditions: i says that the government taes into account the effect of on insurers' decisions and determines its best resonse for each level { ; } ; ii states that the insurers react otimally to governmental decisions given their osterior beliefs about They choose the transfer rate to GRe maximizing their utility iii corresonds to the alication of Bayes' rule by the insurers It should be noted that if is not art of government otimal action for some tye, observing is a robability-0 event, and Bayes' rule does not in down osterior beliefs Every osterior belief Φ \ is then admissible, and every decision, which is the best resonse for certain beliefs, can thus be ut into lay Consider now taxonomy of otential erfect Bayesian equilibria A ooling equilibrium is an equilibrium in which the government chooses the same action whatever the tye In that case, government does not reveal its nowledge about the riss The insurers can not udate their beliefs when they observe the equilibrium action: Φ \ = = Φ \ and Φ \ = = Φ \ A searating equilibrium is an equilibrium in which the government chooses two different actions deending on the tye Observing the nature of the extra charge layed by the government, the insurers now the governmental tye This case is then not accetable for the government if it wants to ee the information secret, for examle, in the case of terrorism 3

13 Public-Private Programs for Covering Extreme Events Moreover, when the observed level of extra charge is inconsistent with the given equilibrium strategy, it is not ossible to use Bayes s rule We deal with this well-nown roblem by assuming that the rivate insurers view any surrising ie, out-of-equilibrium action by the government as truly intentional as oosed to being the result of some mistae They first rule out the level of exected damage at which a rational government would not deart from the roosed equilibrium Once their beliefs are udated accordingly, the rivate insurers utility-maximizing reaction must deter the level extra charge at any other exected damage level An equilibrium obtained in this manner turns out to be unique modulo the rivate insurers current state of mind Government s Extra Charge Policy As introduced above, we focus on two different extra charge olicies that the government can imlement, about which we mae the following assumtions First, the low extra charge equals the actuarial rate for the low ris Π = The advantage of such a low governmental extra charge olicy is to limit ex ante the ayment of the insured citizens and firms, thus maing the rogram oular But in the event of a catastrohe, the total remiums collected in that case is not sufficient to enforce the balance of the indemnification rocess without any ex ost governmental intervention A difficulty for the government in creating this rogram, therefore, may be to find a balance between the exected damages and the extra charge olicy in order to assure the autonomy of the rogram, that is to limit the ex ost ublic intervention This autonomy ends as soon as the total extra charges levied during the eriod of time studied here are not high enough to balance the ublic reinsurer's results In this case, the state guarantee must be triggered in order to assure victim indemnification equation To be certain of enforcing the autonomy of this rogram, the government can define an extra charge olicy that is sufficiently high so that the state guarantee would never be required, that is Π 4 We call such a olicy the governmental high-ris sharing ayment RS ayment of the rogram Indeed, it is worth noting that by so doing, the government defines a remium that could be much higher than the corresonding actuarial remium for high-exected riss Π > In that case, the insurers may be also inclined to be less reinsured by GRe in order to ee a higher art of the high-ris sharing remiums The RS ayment is a ey art for a clear understanding of this model Traditionally, the insurers choose to insure only good riss and refuse to insure bad riss owever, with the RS ayment, the oosite situation may occur here The insurers' retention rate, which reresents their reaction function to the governmental extra charge olicy, could be non-decreasing with their rior beliefs as to the roortion of high riss IV Government Catastrohe Ris Management: Solutions of the Model In this section, we focus on two olicies the government can choose and we determine rissharing outut in the corresonding erfect Bayesian equilibrium Pooling Equilibrium: sing Insurers as Indemnification Intermediaries First, the government can decide to levy the same amount of extra charges whatever the level of ris As discussed above, government that does not want to reveal information about the ris or threat can decide such a strategy 4 By taing into account of GRe s equalization reserve R, the inequality becomes Π R 0 So the government has to define a high extra charge olicy higher than R, with alha the action decided by the insurers 33

14 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue 34 Moreover, we focus on a situation where government does not want to levy high level of extra-charge ex ante against olicyholders For instance, that s often the case at the beginning of a rogram s oeration when the government sees to obtain a consensus with the citizens and firms who will be required to urchase catastrohe insurance with this rogram To encourage that accetance, the government would choose such a low extra charge olicy 5 The question is whether, according to the model, any equilibrium can be achieved with the government always choosing such a olicy If so, what is the ultimate ris sharing between the ublic and the rivate sector and under which conditions does it remain sustainable? emma The best resonse of the rivate insurance industry to a low extra charge olicy decided by the government whatever the tye is to transfer the largest ortion of the riss to the ublic reinsurer Proof: 6 If the government chooses the low level of extra charge rate, insurers will choose the level of transfer, which maximizes their utility according to this olicy Because the government lays the same action whatever the tye, the rivate insurance industry will choose its otimal level of transfer, denoted, by using their rior beliefs on Considering: m V ; Π Π = The first derivative can be written as follows: [ ] V 0 ; ' Π Π = = = Therefore, the sign of the first derivative taen at = is ositive if and only if the inequality holds: Π Π < As Π =, the inequality becomes: Π < 5 The imlementation of a high extra-charge olicy, whatever the level of ris that government agencies are aware of, is not art of the discussion here Indeed, such systematic high beta olicy would raise imortant equity questions which would mae such olicy very doubtful 6 All roofs are ut in aendix This roof is in the main text to hel the reader understanding the notation

15 Public-Private Programs for Covering Extreme Events As Π <, the fraction is strictly negative The inequality is equivalent to < a, with a > 0 As [0,], the inequality always holds true whatever the beliefs out of the equilibrium and the first differential is always ositive Therefore, rivate insurers choose to transfer the largest ortion of the riss to the ublic reinsurer when they receive no information from the governmental action The lemma is very intuitive The insurers, receiving no information from the governmental signal, thus maximize their utility considering their rior beliefs As the insurers now the government has more information on this secific line of riss, they now that the government could be induced to levy less extra charge than the situation really requires owever, to determine whether equilibrium can be reached, it must be shown that the government has an interest in laying that action sing the lemma, the conditions under which, if any, the government has no interest in deviating from that action need to be determined nder the assumtions of the model, Proosition holds The demonstration is resented in the aendix A Proosition : Government sing Insurers as Intermediaries Consider the following actions and beliefs: the government chooses a low extra charge olicy whatever the robability of extreme riss, the rivate insurers choose the largest ossible transfer rate to the ublic reinsurer and the insurers' beliefs are Φ \ = and Φ \ = There exists r and r such that: this ooling candidate is always an equilibrium iff < ; it can be an equilibrium deending on a secial secification of λ the insurers beliefs out of the equilibrium as soon as min{ r ; r }; finally, it will never be an equilibrium iff min { r ; } < Proof: See Aendix A r The rate bounds are r = λ and r λ = 7 Proosition analyzes a first configuration of ris sharing between the government and the insurers in such a rogram In that situation, the government ass rivate insurers to levy ex ante small extra charges on the basic olicies of their insured The insurers, who have no other information on the nature of the ris than their rior beliefs, levy those extra charges and choose to transfer the largest ossible ortion as authorized through the reinsurance treaty to the ublic reinsurer In that case, such a rogram uses the networ and management of rivate insurers as what we could call a financial indemnification intermediary er contra the ayment of a ool articiation commission m for managing that catastrohic ris line in their ortfolio In that case, government may have to ay ex ost a ortion of the damage corresonding to the state guarantee if triggered 7 If we consider the GRe s equalization reserves noted R, the bounds are quite comlicated For instance, [ λ λ R ] and we obtain r = R R r = R [ λ λ R ] R 35

16 Asia-Pacific Journal of Ris and Insurance 006, Volume, Issue It must be noted, however, that this ooling equilibrium is not sustainable if is higher than either r or r Indeed, in that case, taing into account the insurers' beliefs out of the equilibrium, the government would have an interest in deviating 8 ow do the arameters of the model affect the existence of this ooling equilibrium? The boundaries r or r are such that this equilibrium is only sustainable for low values of λ When λ becomes too high, the government may estimate that the cost of financing the indemnification through the unlimited state guarantee becomes too exensive, counterbalancing the imortance of not imosing a too high remium on the insured and/or not revealing the information about the riss Indeed, as the value of the bound rates r and r decreases with the exogenous arameter λ, above a articular level of λ, the government is incited to deviate from this equilibrium, which falls see the roof of the roosition in the aendix An alternative for the government in that case if it wants to continue laying that action whatever the tye of ris is to reduce the maximum ossible transfer rate to the governmental reinsurer ie, decreasing By so doing, olicymaers in government limit the amount of damage that the Treasury would ay if the state guarantee is triggered The lielihood of catastrohes covered by the rogram lays in the same sense: the higher the lielihood of the level of catastrohic damage occurring during the eriod of oeration of the rogram and, the less liely the government is to lay the same low extra charge olicy whatever the tye of ris Searating Equilibrium: the Autonomy of the Program We now consider a totally different situation As discussed above, the continuous increase of governmental ayments for reeated catastrohes can also encourage government to establish rograms that would limit its intervention ex ost In that sirit, the second olicy we consider now is the government seeing the rogram to be financially balanced for high-ris exosure As roven in the receding subsection, the ooling equilibrium can only be obtained with the government laying the action But, in so doing, the government nows that the insurers will tae an intermediary course That is not the government s objective here In other words, the equilibrium candidates for the rogram s autonomy can only be searating equilibria In a searating equilibrium, the government's choice reveals it s the ris tye Moreover, the members of the government ought never to decide a low extra charge olicy when they now that the robability of damage is high Indeed, rivate insurers would decide to transfer the highest art of ris to GRe And, because the total remiums transferred to the ublic reinsurer would not be sufficient to counterbalance the cost of indemnities should catastrohes occur during the eriod covered by the rogram s oeration, the state guarantee would always be triggered in that case Conversely, when a high extra charge olicy is determined although the robability of damage is low, the insurer will ee the highest art of remiums Whereas the government levies more charges than necessary, the insurers will choose to transfer a minimal ortion of remiums This olicy is not beneficial from the ublic sector s oint of view Thus, defining a high extra charge olicy for high-riss will be necessary to guarantee that autonomy As discussed earlier, by so doing, the government may also create conditions 8 When min { r ; } r, we find that conditions for no deviation deend on the secification of the insurers beliefs out of the equilibrium Taing ν \ = as the insurers' beliefs out of the equilibrium is sufficient to guarantee the sustainability of the equilibrium 36