Coping with Catastrophic Risk: The Role of (Non)-Participating Contracts

Transcription

1 Copng wth Catastrophc Rsk: The Role of Non)-Partcpatng Contracts Olver Mahul INRA Department of Economcs, Rennes France Paper presented at the 9 th Semnar of the European Group of Rsk and Insurance Economsts, Nottngham, September 6-8, Abstract Ths paper examnes how the two fundamental prncples n rsk allocaton,.e., the mutualty prncple and the transfer prncple, can be mplemented n order to manage effcently catastrophc rsk. From a decomposton of the nsurable loss nto dosyncratc and systemc components, we show that the systemc rsk s frst fltered through a partcpatng polcy wth a varable premum based on the realzed systemc loss, and then t s transferred through an nsurance contract provdng a coverage on the varable premum. Ths model s reconsdered to examne the fnancng of catastrophe rsk wth alternatve rsk transfer solutons. Group captves, offerng partcpatng polces, are shown to be market enhancng. Keywords : hedgng, nsurance, mutualty prncple, partcpatng polcy, rsk transfer, systemc rsk.

2 . Introducton The year was a terrble year for catastrophc events. Wth $34 bllon of nsured losses n property nsurance, t s comparable wth losses n 99 hurrcane Andrew) and 999 wndstorms n Europe). Whle the 99 and 999 record losses were due to natural catastrophes, t was the man-made losses that weghted heavest on the nsurers book n. Property and busness nterrupton losses arsng from the terrorst attack of September on New York and Washngton are estmated at $9 bllon Swss Re ). Moreover, the forth most costly nsurance losses n were caused by another man-made catastrophe; the exploson n fertlzer factory n Toulouse France) where nsured losses are estmated at $.4 bllon. Even f nsurance losses from the September terrorst attack are stll dffcult to quantfy, the resultng total loss s certan to be much larger than that ncurred by Hurrcane Andrew. The potental losses from natural catastrophes and, snce the September event, man-made catastrophes lead fnancal researchers to address the fnancng of catastrophc losses and effcent rsk sharng rules between re)nsurance and captal markets. The economc role of nsurance and rensurance companes as prvate mechansms of rsk allocaton s closely related to the concept of nsurablty. The problem arses because catastrophc rsks are not wdely dversfable n an nsurance context. In other words, the rsk poolng mechansm s not hghly effcent n rsk reducton. Two types of solutons have been proposed and put nto practce n order to fulfll the nsurance capacty gap. The frst alternatve s mandatory publc provson of nsurance. It reles on the fnancal ablty of the government to spread losses across many ctzens. It was mposed n France where the Parlament voted n July 98 a law nsttutng a mandatory compensaton system for natural dsasters. The second alternatve s the development of fnancal nstruments to transfer catastrophc rsks to fnancal nvestors through captal markets. It reles on the huge fnancal capacty of captal markets. Ths paper focuses on ths second alternatve and on ts optmal combnaton wth the rsk poolng mechansm. Several studes n the nsurance lterature have examned the problem of desgnng optmal rsk-sharng contract between a rsk-averse nsured and a less rsk-averse nsurer n the presence of a sngle source of rsk see, e.g., Arrow 965, Ravv 979) or when the nsured also bears an unnsurable background rsk see, e.g., Goller 996, Mahul ). When the two sources of uncertanty are nsurable, Ravv 979) shows the optmalty of the nsurance polcy wth a sngle deductble on the aggregate loss. Goller and Schlesnger 995) reexamne ths problem when nsurance markets offer separate contracts for separate addtve) loss exposures and derve the second-best nsurance contract desgn n ths ncomplete market. These rsk-sharng arrangements are based on the transfer prncple; rsk s transferred to an external rsk bearer e.g., an nsurance company or a fnancal nvestor) who has a comparatve advantage to bear t. As notced by Doherty and Donne 993), the extensve lterature on optmal nsurance contract desgn usually gnores a second fundamental prncple n the allocaton of rsk; the mutualty prncple. Ths prncple, developed by Borch 96) and Wlson 968), In addton, lablty and lfe nsurance losses related to the September event are estmated between $6.5 and 39 bllon.

3 states that the Pareto optmal rsk-sharng rule n an economy wth rsk-averse agents and no transacton costs s such that the fnal wealth of any agent depends only on the aggregate wealth. A notable excepton s the paper wrtten by Doherty and Donne 993). In a fourstate expected utlty framework, they provde explanatons about the prolferaton of partcpatng polces provded by mutual nsurers and new nsurance contract desgns from a decomposton of the nsured losses nto systemc.e., undversfable) and dosyncratc.e., dversfable) components. They show that a decomposed rsk transfer that effcently uses ths decomposton to apply the two fundamental prncples weakly domnates a smple transfer n whch the nsured loss s not decomposed. Doherty and Schlesnger ) reconsder ths problem n a framework where consumers have preferences for second-degree stochastc domnance, ncludng the expected utlty theory as a partcular case. They derve a varable partcpaton nsurance polcy defned as a convex mxture of a non-partcpatng contract wth a fxed premum) and a fully partcpatng contract wth a varable premum) n whch the level of partcpaton s endogenous. However, these two studes are based on two central assumptons. Frst, consurance contracts are only at the polcyholder s dsposal. Second, the fully partcpatng polcy s sold at a far prce; the dosyncratc component of the ndvdual rsk can be managed wthn ths polcy at no transacton costs. Such an assumpton seems to be dffcult to ustfy as regard real-world nsurance markets. The prevalence of transacton costs e.g., agency costs, admnstratve costs) n the nsurance ndustry s a well-establshed fact. For many nsurance lnes lke automoble nsurance, transacton costs amount up to 3% of the premum. They would thus prevent nsurance companes from offerng actuarally far nsurance contracts. Ths paper frst examnes, from a decomposton of the ndvdual loss, the optmal rsk-sharng rule between a rsk-averse frm and a rsk bearer n the multple state expected utlty framework. The orgnalty of our approach stems from the fact that the two above mentoned assumptons are relaxed; rsk-sharng contracts are not restrcted to consurance and the fully partcpatng polcy s not necessarly sold at an actuarally far prce. Ths allows us to show how the cost effectveness of nsurance affects the optmal rsk-sharng rule. The catastrophc rsk s shown to be fltered through a partcpatng polcy wth a varable premum based on the realzed systemc loss, and then t s transferred through an nsurance contract whch dsplays full coverage above a deductble on the varable premum. An optmal varable partcpatng nsurance polcy s desgned from optmal fully partcpatng and non-partcpatng contracts. The second purpose of ths paper s to nvestgate the fnancng of catastrophc rsk wth two alternatve transfer solutons; group captves and ndex-based dervatves. Group captves act as purveyors of fully partcpatng polcy and ndex-based contracts provde coverage on the catastrophc component of the loss. The fully partcpatng polcy s shown to be market enhancng. Ths s llustrated from the analyss of the domnance relatonshp usng the effcent fronter of the combnaton of two separate Other explanatons have been provded on the role of partcpatng polces offered by mutual nsurers and ther coexstence wth other knds of organzatons. They could help to redress agency problems Mayers and Smth 986), adverse selecton Smth and Stutzer 99) or moral hazard Smth and Stutzer 995). Our paper does not deal wth such problems of nformatonal asymmetres. 3

4 non-partcpatng contracts and that of the varable partcpatng polcy n the meanvarance framework. The remander of ths paper s organzed as follows. The model s descrbed n secton. Optmal fully partcpatng and non-partcpatng nsurance contracts are desgned n secton 3. Secton 4 examnes the fnancng of catastrophc rsk wth alternatve rsk transfer solutons and shows how group captves can ncrease effcency on fnancal markets. A concluson hghlghts the man results and uses our model to nvestgate optmal rsk management of hybrd covers recently offered by nsurance and rensurance companes.. The Model The theoretcal analyss s developed nto a well-accepted and unfed framework; the expected utlty theory. A rsk-averse frm s endowed wth non-random ntal wealth w whch s subect to a rsk of loss y, wth y w for all y. 3,4 Ths loss can be parttoned nto a systemc component x and an dosyncratc component ε through the determnstc relatonshp ) l ε ) y =, wth x, ε, l x and l ε. 5 The two sources of rsk affectng the ndvdual loss are assumed to be stochastcally ndependent. The decomposton ) s determned accordng to the rsk pool whch the frm s part of. 6 For example, n a pool where members are located n a gven earthquake-prone area, defne the y rsk as the ndvdual rsk exposure to an earthquake. The x rsk would be the common uncertanty component whch affects all the members of the pool,.e., the ntensty of the earthquake. The ε rsk would be the mpact of local parameters on the ndvdual losses whch can be consdered as ndependent among the members of the rsk pool. The second source of rsk s partally) dversfable wthn the rsk pool, whle the frst one s undversfable, at least at the rsk pool s level. However, t may be partly) swapped wth groups of ndvduals who are exposed to dfferent and ndependent) catastrophes or transferred to fnancal nvestors wth a worldwde dversfed busness portfolo. The rsk-averse frm can purchase two types of nsurance contracts; a nonpartcpatng polcy and a fully partcpatng polcy. The non-partcpatng polcy s n the lne of standard nsurance contracts see, e.g., Arrow 965, Ravv 979). It s Q J ε descrbed by couple [ J.,.), ], where Q s the fxed premum and the functon ) 3 The frm can also be assumed to be rsk neutral but t s motvated to hedge by market mperfectons, drect and ndrect costs of fnancal dstress, and convex tax schedule. It thus behaves as f t was rsk averse see Doherty ) for a detaled analyss on why rsk s costly to frms). 4 Tldes are used to denote random varables, and the same varables wthout the tlde denote realzatons of the random varables. 5 A subscrpt symbol denotes a partal dervatve wth respect to that symbol. 6 The sze of ths pool s assumed to be fnte but suffcently large so that, from the law of large numbers, devatons of average dosyncratc loss from expected dosyncratc loss are nsgnfcant. 4

5 s the amount of ndemnty payments made by the nsurer when the realzed ndces are x,ε. The ndemnty schedule s assumed to be nonnegatve: ) ) J for all,ε ) x, and the premum s assumed to be a non-decreasng functon of the expected ndemnty: Q = d EJ x, ε ), 3) ) wth d ) = and d s) for all s, and E denotes the expectaton operator wth respect to the correspondng random varables. The frm can also purchase a fully partcpatng nsurance polcy. It s descrbed by couple [ I.,.), P.) ], where the ndemnty schedule I ε ) s assumed to be nonnegatve: 4) I ε ) for all,ε ) x. The nsurance premum s varable and t depends on the realzed systemc component. We assume that t s based on the ex post average ndemnty pad by the nsurer: P x = c EI ε ), 5) ) ) wth c ) = and c s) for all s. The premum s thus subect to ex post adustments. For nstance, the nsured frm may frst pay a premum based on the expected value of the systemc rsk, P E x ). It s then adusted dependng on the realzaton of the systemc component x ; P < P E x ) would ental the payment of a dvdend to the nsured frm whle P > P E x ) would lead to a premum surcharge pad by the frm. 7 The problem of the rsk-averse frm s to determne the ndemnty schedule and the premum of both fully partcpatng and non-partcpatng nsurance contracts that maxmze ts expected utlty of fnal wealth: max Eu w ), l ε ) + I ε ) P x + J x ε ) Q) I, J, P, Q 6) subect to condtons ) to 5), where u.) s a twce-dfferentable von Neumann-Morgenstern utlty functon, wth u >, u <. The orgnalty of ths paper wth respect to prevous work Doherty and Donne 993, Doherty and Schlesnger ) s twofold. Frst, we allow the decson-maker to determne the desgn of optmal non-partcpatng and fully partcpatng nsurance contracts, wthout restrctng nsurance desgn to consurance. Second, we allow the nsurer to sell the fully partcpatng polcy at an unfar prce. 8 7 Insured frms are assumed to always pay n full ex post addtonal premums n the event of a large catastrophe. Hence, there s no default rsk from the polcyholders. 8 Ths paper s not a drect extenson of Doherty and Schlesnger ) because the current problem s nvestgated n the expected utlty framework whereas ther analyss s developed under a weaker assumpton about preferences under uncertanty, the averson to mean-preservng spreads. 5

6 3. Optmal Insurance Contract Desgns The followng proposton shows how the desgn of optmal fully partcpatng and nonpartcpatng contracts s based on the cost effectveness of nsurance. Proposton. The optmal ndemnty schedules of fully partcpatng and nonpartcpatng nsurance contracts I and J, solutons to problem 6), take the followng form: ) If c s) < d s) for all s, then there exst D and D such that ε ) = max l D,) and J ) = max P D,) I ε. ) If c s) > d s) for all s, then there exsts D such that J l ) D,) 3 ε ) = max ε 3 and I. The proof s gven n the Appendx. Suppose frst that the cost of nsurance s hgher under the non-partcpatng polcy than under the fully partcpatng contract, c s) < d s) for all s. Such an assumpton s ustfed f both contracts face dentcal admnstratve costs but the nsurance premum of the non-partcpatng polcy s also charged by a rsk premum. 9 It can be motvated by the presence of the systemc component whch cannot be dversfed by the shareholders of the nsurance company. As a consequence, they wll ask for a rsk premum whch wll ncrease the cost of captal of the company. Ths cost wll be passed on to the polcyholders through a larger premum rate. The optmal nsurance strategy as descrbed n Proposton ) s as follows. The frm purchases the fully partcpatng contract n order to partally) cover the dosyncratc component of ts ndvdual rsk exposure. Such a contract provdes the frm wth full nsurance aganst the ε rsk above a deductble, whle t bears all the systemc rsk through the varable nsurance premum. In a second step, the nonpartcpatng contract s used to provde partal) coverage only on the x rsk through full nsurance on the varable nsurance premum of the fully partcpatng polcy above a straght deductble. The second part of Proposton s qute ntutve. Suppose the cost of nsurance under the fully partcpatng polcy s hgher than that under the non-partcpatng contract. Ths could be the case f, because of economes of scale, the admnstratve costs plus the rsk premum of the latter are lower than the admnstratve costs of the former. The frm wll only purchase the non-partcpatng polcy dsplayng full nsurance aganst ts ndvdual loss above a straght deductble. Ths corresponds to the standard result n the nsurance lterature when the non-partcpatng polcy s only avalable Arrow 965, Ravv 979). The optmal level of deductblty s characterzed n the followng proposton. 9 Gven assumpton 3) on the nsurance premum, ths rsk premum depends only on the expected ndemnty payoff. 6

7 Proposton. Suppose that c s) < d s) for all s. ) Under the fully partcpatng polcy, the optmal deductble D equals zero f the premum s actuarally far [ c s) = for all s ] whereas t s postve f the premum s unfar [ c s) > for all s ]. ) Under the non-partcpatng polcy, the optmal deductble D equals zero f the premum s actuarally far [ d s) = for all s ] whereas t s postve f the premum s unfar [ d s) > for all s ]. The proof s gven n the Appendx. The optmal deductble satsfes standard results under both nsurance polces,.e., t equals zero under far nsurance and t s postve when nsurance s sold at an unfar prce. Under unfar premums, t should be notced that the optmal deductble of a contract depends not only on the cost of nsurance of ths contract but also on that of the other one. If c s) > d s) for all s then, from Ravv 979), the optmal deductble s zero f d.) and t s postve otherwse. Propostons and characterze the optmal nsurance strategy wth fully partcpatng and non-partcpatng contracts. Ths optmal strategy s llustrated under a multplcatve relatonshp between the two components of the random ndvdual loss, l ε ) = xε wth x >. The ndvdual loss s thus proportonal to the value of the systemc component. In the case of nsurance for natural dsasters, x =. 5 would mean that all the ndvdual losses ncrease by 5 percent because of the occurrence of a catastrophe. In the case of agrcultural revenue nsurance, the ε component could be yeld shortfalls caused by a local weather event n quantty) and the x component could be the crop prce at harvest assumed to be ndependent wth the crop yeld rsk). Whle the former can be dversfed f we exclude natural dsasters), the latter s not dversfable. Such an example looks lke nsurance at replacement cost where a random nflaton rate mpacts all clams n the same proporton. The optmal partcpatng polcy s I ε ) = max xε D,) and the loss borne by the nsured frm after the payment of the ndemnty net of the premum s 7) xε max xε D,) + P = x[ ε max ε D ) ] D + P + P = xε + P f ε D f ε D For suffcently large dosyncratc losses, ε D x, the loss borne by the frm s only based on the systemc loss through the varable premum. The systemc rsk s thus fltered through the fully partcpatng polcy. Suppose the polcy s sold at far prce, as n prevous works Mahul, Doherty and Schlesnger ). From Proposton, the optmal polcy becomes I = xε and the varable premum s P = xeε. As a consequence, the optmal ndemnty schedule net of the premum s I ε ) P = x ε Eε ) and the loss after net ndemnfcaton s xe ε. The frm s thus fully nsured aganst the dosyncratc component of ts loss, whle t s fully exposed to the systemc rsk. x x. 7

8 The optmal non-partcpatng polcy s J Eε max x D Eε, ) =. The contract provdes full nsurance on the systemc loss above a postve deductble. Of course, f ths contract s sold at a far prce, the frm s fully nsured aganst ts loss. Ths corresponds to the standard result n the lterature of nsurance economcs. The fully partcpatng and non-partcpatng nsurance polces can be combned to construct the so-called varable partcpatng nsurance contract. In our framework, ts ndemnty schedule s V ε ) = I ε ) + J ε ) and ts premum s T = Q + P. We thus allow the frm to choose the form of ts partcpaton through the choce of the ndemnty schedules of fully partcpatng and non-partcpatng contracts. From the above analyss, the optmal varable partcpatng polcy satsfes 8) V ε ) = max l D,) + max[ P D,], wth T = P + d E max[ P D,]) where P Ec max l ) D, ) = ε. It should be notced that ths polcy dffers from the varable partcpaton contract proposed by Doherty and Schlesnger ). Beyond the fact that they restrct the nsurance contract desgn to consurance, they mpose the ndemnty of the nonpartcpatng polcy to depend on both realzed dosyncratc and systemc losses. Such a constrant turns out to be sub-optmal from Proposton ). However, Doherty and Schlesnger ) varable partcpaton ndemnty schedule net of ts premum and our net varable partcpatng ndemnty schedule become dentcal when the fully partcpatng contract s sold at a far prce. In ther four-state model where the fully partcpatng polcy s sold at a far prce, Doherty and Donne 993) show the separablty of decsons on dosyncratc rsk and on systemc rsk; the purchase of full nsurance of dosyncratc rsk s ndependent of the cost of nsurance aganst systemc rsk and, therefore, the purchase of partal nsurance of systemc rsk. Such a result s reconsdered n our multple state model. From Proposton ), the form of the optmal fully partcpatng contract turns out to be ndependent of the non-partcpatng contract, even f the former s sold at an unfar prce. The separablty result thus holds about the form of the optmal fully partcpatng polcy; t dsplays full nsurance aganst the ndvdual loss above a deductble. The optmal deductble D wll depend on the non-partcpatng contract, except when the varable premum P s actuarally far because, from Proposton ), the optmal level of deductblty s zero,.e., D =. Ths partcular case corresponds to the result of Doherty and Donne 993). On the contrary, the form of the optmal non-partcpatng contract and ts optmal deductble clearly depend on the fully partcpatng contract through the varable premum, as shown n Proposton ). 4. The Fnancng of Catastrophe Rsk wth ART Solutons 4.. Alternatve Rsk Transfer Solutons Companes have been usng non-tradtonal forms of rsk management for many years, usually called alternatve rsk transfer ART) solutons. These nnovatons am at ncreasng the effcency of the rsk transfer, broadenng the spectrum of nsurable rsks 8

9 and usng the captal markets for addtonal fnancal capacty. Among the mportant forms of ART solutons, we focus on two of them; the captves and the nsurance dervatves. They are based on the mutualty prncple and the transfer prncple, respectvely. ART solutons frst descrbed varous forms of self-nsurance, ncludng captves or rsk-retenton groups. The captve market s of partcular nterest because t rests on the applcaton of the mutualty prncple. A captve s an nsurance or rensurance vehcle. It belongs to a company or a group of companes but t s not actve as an nsurance frm. It s called sngle parent captve and collectve captve, respectvely. Its purpose s to nsure rsk exposures from ts parent company. The role of a captve s twofold. Frst, t provdes a vehcle of self-nsurance for hgh frequency/low severty rsks. These mutual-lke companes allow the frms to montor manageral performance at lower cost than tradtonal nsurance companes Doherty and Donne 993). Second, t s used as a fnancng nstrument for specfc low frequency/hgh severty rsks for whch coverage s lmted or unavalable on the rensurance market. The development of group captves, especally n the U.S., s usually vewed as the drect consequence of the lablty crses n some economc sectors, such as ols or chemcals see, for example, Doherty, Klendorfer and Kunreuther 99). The coverage solutons offered by the tradtonal nsurance and rensurance markets to deal wth emergent collectve rsks were consdered as too lmted or too expensve. Therefore, the lack of capacty and the cost of nsurance forced companes facng the same type of rsks to create alternatve rsk management solutons through the development of captves. Accordng to the 999 ssue of Captve Insurance Company Reports, there were about 4, captves operatng worldwde at year-end 998, generatng a premum volume of approxmately 6% of the global commercal nsurance market see Swss Re 999) for a detaled descrpton of the mportance of captves). More recently, the terrorst attacks n New-York and Washngton have nduced nsurance companes and arlne companes to create group captves n order to manage terrorst rsks. Lkewse, the exploson n AZF fertlzer factory n Toulouse France) n September st has nduced frms exposed to potental ndustral catastrophes to create a specfc pool to cope wth such catastrophc rsks. Followng Hurrcane Andrew and the Northrdge earthquake n the 99s, property catastrophe rensurance was n short supply and, as a consequence, premum rates ncreased sharply. In vew of these capacty lmts and related lmts of nsurablty, the dea arose of makng avalable addtonal capacty for catastrophe rsks outsde the nsurance market. Some nsurers thus began developng a new class of fnancal nstruments that transfer nsurance rsk to captal markets. The frst attempt to use fnancal market nstruments for managng nsurance rsks was made by the Chcago Board of Trade CBOT) whch has been tradng futures on catastrophe loss ndces and related optons snce December 99. The PCS optons contracts traded on the CBOT are based on varous catastrophe loss ndces calculated and publshed on a daly bass usng estmates of nsured loss. Recent years have wtnessed a growng nterest n weather-based dervatves as nstruments for sharng rsk due to weather phenomena. Other types of captve structures are assocaton captves, rsk retenton groups and rent a captve arrangements. 9

10 Snce 997, market partcpants n the electrcty and natural gas sectors have used temperature-based dervatves to offset ther exposure to extreme temperature. Over-thecounter dervatves are based on temperature ndex such as cumulatve heatng degree days HDD) or coolng degree days CDD) for a gven locaton over a specfed perod of tme. On September 999, the Chcago Mercantle Exchange began tradng standardzed monthly cumulatve HDD and CDD futures and optons contracts for several U.S. ctes. Such contracts are beng developed n Europe, and especally n France. The European board of trade Euronext plans to launch weather dervatves based on one natonal and fve regonal ndces n 3. They wll complete the sx weather-based futures launched on the Brtsh board of trade Lffe n December. 3 The effcent ndex for a company n ts hedgng strategy s the result of a tradeoff between bass rsk because these ndces are not perfectly correlated wth ts loss), moral hazard, transacton cost and counterparty rsk. 4.. Optmal hedgng strategy We examne how the two ART mechansms captves and ndex-based dervatves) can be combned to manage effcently the frm s loss exposure. The rsk-averse frm belongs to a collectve captve provdng fully partcpatng contract [ I ε ), P ]. It can also buy nsurance dervatves based on the systemc component x. 4 The ndemnty payoff and the premum of ths fnancal contract are denoted K.) and Q, respectvely. In addton, we assume that fnancal contracts are sold at a prce that s proportonal to ther expected ndemnty payoff: Q = + δ ) EK, wth δ. The problem consdered n ths secton s slghtly dfferent from the prevous one because the non-partcpatng polcy offers coverage only on the x systemc rsk. Ths thus creates a source of market ncompleteness. Followng the same lne of reasonng as n Proposton, one can easly show that the optmal hedgng strategy s to nsure the dosyncratc component through the fully partcpatng polcy offered by the collectve captve and the systemc component through the nsurance dervatve contract offered by the fnancal markets. Formally, under multplcatve dosyncratc and systemc rsks, the optmal fully partcpatng polcy satsfes I ) = max xε D, P = c E max xε D,, 9) ε ) wth [ )] where D = f c.) and D > otherwse, and the optmal ndex-based contract s ) K = max P D,) wth Q = + δ ) E max P,), = = > D where D f δ and D otherwse. The ndemnty schedule and the premum of the optmal varable partcpatng polcy are V ε ) = I + K and Atlanta, Chcago, Cncnnat, Dallas, Des Mones, Las Vegas, New York, Phladelpha, Portland and Tucson. See for more detals. Regonal ndces wll be based on the daly average temperature n fve French ctes: Aacco, Bordeau Lyon, Pars-Orly and Strasbourg. See for more detals. 3 See for more detals. 4 We assume away problems assocated wth bass rsk or counterparty rsk.

11 T = P + Q, respectvely. For all x such that K >, the frst dervatve f the ndex-based contract s ε, ) K = P = c B) df D x where F.) s the cumulatve dstrbuton functon of the ε rsk and B = E max xε D,). Its second dervatve s D ε 3. x x ) K = P = c B) df + c B) f D D From equaton ), the optmal fnancal contract s a non-decreasng functon of the ndex. From equaton ), t s convex f the cost functon of the fully partcpatng polcy s lnear or conve c. Its curvature s ambguous f the cost functon s concave. Ths cost functon thus plays a central role n the curvature of the optmal nsurance dervatve contract. Suppose frst that the premum of the fully partcpatng polcy s unfar and nonconcave,.e., c.) > and c.). The optmal ndex-based contract s an ncreasng and convex functon of the x component. Because real-world fnancal markets offer pecewse lnear hedgng nstruments, lke optons, they preclude from replcatng the frst-best hedgng strategy. Ths thus creates a new source of ncompleteness. It s noteworthy that even f ths contract s sold at a far prce, mplyng D =, optons contracts turn out to be useful nstruments n order to replcate as close as possble the convex curvature of the frst-best soluton K, as long as the premum of the fully partcpatng contract s unfar. Consder now the case where the fully partcpatng polcy s sold at a far prce. Ths entals that D = and, from equaton 9), I = xε wth P = Eε )x. The optmal ndex-based contract thus satsfes =,. 3) K Eε ) max x D Eε ) The frst best soluton s replcated by purchasng E ε call optons at a strke prce D Eε. If ths ndex-based contract s sold at a far prce, δ =, then K Q = Eε ) x Ex ); the optmal strategy s to sell a quantty E ε of unbased futures contracts Partcpatng polcy as a market enhancng nstrument Real-world markets are typcally ncomplete n that they offer separate contracts for separate loss exposures. Let [ S ε ), R ] and [ S, R] denote separate hedgng contracts wth fxed premum aganst the ε rsk and the x rsk, respectvely. We assume that the premums are proportonal to the expected ndemnty and that the cost of nsurance/hedgng s dentcal for nsurance polces dealng wth the dosyncratc rsk and for hedgng contracts copng wth the systemc rsk; P = + δ ), EI x ε ),

12 R ) = +δ ES ε ), R ) = +δ ES ε ) and Q = + δ ) EK. It s noteworthy that the contract [ S ε ), R ] s a specal case of the fully partcpatng polcy [ I ε ), P ] n whch the ndemnty does not depend on the realzed systemc loss. Ths mples that any combnaton of separate non-partcpatng contracts [ S ε ), R ] and [ S, R] s weakly domnated by the varable partcpatng polcy [ V ε ), T ]. Ths weak domnance s llustrated n two specal cases. Suppose frst that the dosyncratc and the systemc components of the loss exposure are addtve; l ε ) = x + ε. There exst D and D such that ε ) = max x + ε D,) and K max P D,) I = wth P = + δ ), EI x ε ). From Goller and Schlesnger 995), there exst d and d such that S ε ) = max ε d,) and S = max x d,). If nsurance contracts coverng the ε rsk are sold at a far prce,.e., δ =, ths mples that D = d. Ths gves = ε ) ). 4) I ) P = x + ε ) x + Eε = ε Eε = S R In addton, we have K = max x + Eε D,) = S wth d ε = D E. The optmal varable partcpatng polcy can thus be replcated wth separate contracts, as shown by Doherty and Schlesnger ). However, the varable partcpatng polcy strctly domnates any combnaton of separate contracts when nsurance aganst the ε rsk s costly,.e., δ. > Consder now the case n whch the dosyncratc and systemc components of the loss exposure are multplcatve; l ε ) = xε. Such restrctons on hedgng contracts abound n the real world. For example, frms facng both prce and producton uncertanty have usually the possblty to cover ther quantty rsk wth an nsurance contract and ther prce rsk wth a fnancal product. The optmal form of separate nonpartcpatng contracts for multplcatve rsks s frst examned. Wth separate contracts aganst postve random losses z and z, the problem of the frm s 5) max Eu S, S, R, R subect to R = + ) δ ES z), S z) for all z, w z z + S z ) R + S z ) R ) =, =, wth δ, =,. It can be shown see the Append that there exsts d, =,, such that the optmal hedgng contract satsfes, for =,, = 6) S z) > for all z d for all z > d and the margnal coverage satsfes for all z > d,

13 7) π ) π ) ) cov u S, z = E z + z wth, Eu where E and cov are the expectaton and the covarance operators, respectvely, wth respect to z, =, and π = w zz + S z) R + S z) R. The margnal coverage of the z loss s thus hgher or lower than the expected loss E z dependng on whether the covarance term s postve or negatve. In addton, the optmal level of deductblty s zero, d =, f and only f the contract s sold at a far prce, δ =. It s of nterest to notce that, contrary to the prevous combnaton of hedgng contracts wth a fully partcpatng polcy, the separaton result does not hold any longer; the form of the nsurance contract aganst the z ε rsk depends on the ndemnty schedule aganst the z x rsk as long as the frm s utlty functon s not quadratc,.e., u. The optmal hedgng strateges wth separate non-partcpatng contracts and wth the varable partcpatng polcy are examned when the dosyncratc and systemc components nteract n a multplcatve manner n the loss exposure, nsurance polces are sold at an unfar prce, δ > and δ >, and the frm s preferences are represented by a quadratc utlty functon. 5 The covarance term n 7) s zero and the optmal hedgng contracts become S z ) = Ez max z d, wth, 8) ) for =,. The optmal hedgng strategy thus entals buyng E z call optons at a strke prce d and E z call optons at a strke prce d. The proft of the nsured frm s NP 9) π = w xε + Ex max ε d,) + Eε max x d,) R R, where z ε and z x. When the fully partcpatng polcy s avalable, the proft of the nsured frm s VP ) π = w xε + max xε D,) P + max P D,) Q. The effcent fronters of these hedgng strateges are depcted n Fgure. The ntal wealth poston of the frm s shown as A. If the x systemc rsk can only be hedged or nsured, the effcent fronter s the lne A-B. Ths lne shows how the frm s wealth can be covered wth a sngle contract aganst the x systemc rsk. A reducton n the varance of wealth s assocated wth a reducton n the expected wealth because the premum s hgher than the actuaral value of the contract, δ >. Ths optmal strategy entals buyng E ε call optons; H = Eε max x d3,) wth d > 3. If the deductble S s set at zero, d, the proft of the frm satsfes π = w x ε + xe ε + δ ) E ε E x 3 = and thus ts expectaton and ts varance are, respectvely, d3= 5 The man actors n ths market are frms, not ndvduals, and lnear mean varance preferences are well accepted n the corporate fnance lterature. 3

14 S d3 = ) Eπ = w +δ ) EεEx S and var π ) = var x ε Eε )) > where var.) s the varance operator. Ths s the coordnates of pont B. There s stll bass rsk remanng because the ε rsk s unhedgeable/unnsurable. If separate nonpartcpatng contracts are avalable for the two ndependent sources of rsk, the effcent fronter s the lne A-C. Ths lne shows the potental choces avalable to the purchaser of the hedge wth separate contracts. Because the hedgng strategy wth only one hedgng contract aganst the x rsk s a specal case of the hedgng strategy wth two separate non-partcpatng polces, the latter domnates the former. When the deductbles are set at zero, d = d, the proft of the frm expressed n 9) becomes = d3 = NP ) π = w xε + Ex ε + Eε x + δ + δ ) ExE ε, wth d = d = NP ε d =, d = 3) Eπ = w + δ + δ ) E Ex NP and var π ) = var x E ε Eε ) > d =, d = Ths s the coordnates of pont C. Incomplete markets caused by separate addtve contracts to manage multplcatve random losses prevent the frm from havng a nonrandom fnal wealth. There s bass rsk remanng due to ths ncompleteness. From equatons ) and 3), we have NP < S Eπ Eπ f, and only f, δ. In d=, d = d = addton the bass rsk s lower when two separate contracts are avalable because NP var < var S π π. When the varable partcpatng polcy s at the frm s ) ) d =, d = d = dsposal, the effcent fronter s the lne A-D. It shows the potental choces avalable to the purchaser of the hedge wth the varable partcpatng polcy. Wth full coverage under each polcy, D = D =, the fnal wealth of the frm expressed n equaton ) becomes VP 4) π = w + δ ) + δ ) ExE ε, wth D =, D = VP D =, D = 5) Eπ = w + δ ) Eε + δ ) Ex VP and var ) = π. D =, D = Ths s the coordnates of pont D. The fnal wealth of the frm becomes non-random,.e., the frm s fully covered aganst both components of the rsky loss. Observe that, from equatons 3) and 5), VP < NP Eπ Eπ f δ and δ, and D =, D = d=, d = D =, D = d=, d = VP = NP Eπ Eπ otherwse. The hedgng strategy wth the varable partcpatng polcy clearly domnates the hedgng strategy wth separate nonpartcpatng contracts. Therefore, group captves dealng wth dosyncratc rsk through fully partcpatng polcy) ncrease market effcency n the management of systemc rsk. Ths market enhancement s twofold. Frst, reducng rsk n terms of varance) s less costly n term of expected wealth) wth the varable partcpatng >, > >. 4

15 polcy than wth separate non-partcpatng contracts. Second, the former strategy offers the nfntely rsk-averse) frm the opportunty to be fully covered aganst random loss, contrary to the latter strategy whch contans unhedgeable bass rsk. [INSERT FIGURE HERE] The superorty of the varable partcpatng contract over the combnaton of separate contracts as llustrated n Fgure s closely related to the costs of hedgng whch has been assumed to be dentcal for the non-partcpatng contract and the fully partcpatng polcy on the ε rsk, and for the hedgng contracts on the x rsk. Suppose now that the prce of hedgng the systemc component s far, δ =, whle the hedgng cost of the fully partcpatng polcy, δ, s hgher than that of the non-partcpatng contract on the dosyncratc rsk, δ. The effcent fronters of hedgng strateges are depcted n Fgure. Because the systemc rsk can be hedged at a far prce, there s no loss n expected wealth when ths contract s purchased and, therefore, the effcent fronter of the sngle contract s the horzontal lne A-B. Poston B shows full coverage on the systemc rsk. The remanng rsk s due to the dosyncratc loss. Ths poston can also be reached wth separate non-partcpatng contracts and wth the varable partcpatng polcy. Lnes B-C and B-D show addtonal coverage on the dosyncratc rsk wth the non-partcpatng contract and the fully partcpatng polcy, respectvely. There s bass rsk remanng n poston C because the separate non-partcpatng contract do not allow the frm to be fully hedged aganst ts loss exposure, whle poston D shows full coverage of the loss exposure. However, lne B-C s above lne B- D because hedgng the dosyncratc rsk s less costly under the non-partcpatng contract than under the fully partcpatng polcy, δ > δ. Therefore, the varable partcpatng contract does not domnate the hedgng strategy wth separate contracts. The choce between these two strateges rests on rsk preferences: hghly rsk-averse frms would prefer the varable partcpatng polcy see ndfference curve IC h ), whle moderately rsk-averse frms would prefer separate contracts see ndfference curve IC m ). The choce s thus based on a tradeoff between bass rsk and transacton costs. [INSERT FIGURE HERE] 5. Concluson Ths paper presents a normatve model to nvestgate the role of fully partcpatng and non-partcpatng nsurance polces n the management of catastrophc rsk. The desgn of these two optmal contracts s derved from a decomposton of the ndvdual loss nto dosyncratc and systemc components. When the cost of nsurance of the fully partcpatng contract s lower than that of the non-partcpatng polcy, the optmal rsksharng arrangement s shown to be as follows. Under the fully partcpatng polcy, the ndemnty schedule dsplays full nsurance above a deductble on the ndvdual loss and the premum depends on the realzed systemc loss. The form of ths contract s ndependent of the non-partcpatng contract. Under the non-partcpatng polcy, the ndemnty schedule dsplays full nsurance above a deductble on the varable premum of the fully partcpatng polcy. The payoff thus depends only on the systemc component of the ndvdual loss. Hence, the systemc rsk s frst fltered through the 5

16 partcpatng polcy and then t s transferred through an nsurance contract provdng partal a coverage on the varable premum. These two separate optmal contracts are combned to desgn an optmal varable partcpatng nsurance polcy n whch the degree of partcpaton s endogenous. Ths rsk-sharng problem s reconsdered to nvestgate the fnancng of catastrophe rsk wth two ART solutons; group captves and ndex-based contracts. We show that the frst-best hedgng contract aganst the systemc rsk cannot be replcated wth exstng pecewse-lnear nstruments e.g., optons) when the premum of the fully partcpatng polcy s unfar. Ths thus creates a source of market ncompleteness. Fully partcpatng nsurance contracts provded by group captves are shown to be market enhancng. Ths s llustrated from the analyss of the domnance relatonshp usng the effcent fronter of the optmal varable partcpatng polcy and the effcent fronter of the hedgng strategy based on two separate non-partcpatng contracts n the meanvarance framework. Our model s also useful to nvestgate the rsk management of hybrd covers. Insurance companes are tradtonally exposed to two types of rsk: nsurance rsk from polcyholders clams and fnancal rsks on assets. Hybrd covers are nnovatve hedgng nstruments whch provde coverage on ont nsurance and fnancal market rsks, e.g., stop loss nsurance cover where the retenton vares n proporton to an equty ndex. Whle the nsurance rsk can be covered va tradtonal technques, the fnancal rsk must be transferred va fnancal hedgng tools. Optmal rsk management of hybrd covers s thus based on an approprate decomposton of the hybrd rsk nto a pure fnance rsk and an ndependent hybrd resdual rsk n order to apply the two fundamental prncples n rsk allocaton. The nsurance rsk can be vewed as an unhedgeable bass rsk from the fnancal nvestor s vewpont. Our model shows how such an hybrd rsk can be effcently covered. The nsurance rsk,.e., the dosyncratc component, s frst partally) covered by a pool of nsurers and the fnancal rsk,.e., the systemc component, s partally) hedged on fnancal markets. 6

17 References Arrow, K.J. 965). Aspects of the Theory of Rsk Bearng. Helsnk: Yro Jahnsson Lectures. Borch, K. 96). Equlbrum n a Rensurance Market. Econometrca 3: Doherty, N. ). Innovaton n Corporate Rsk Management: the Case of Catastrophe Rsk, n G. Donne, ed., Handbook of Insurance Kluwer Academc Publshers), Doherty, N., and G. Donne 993). Insurance wth Undversfable Rsk: Contract Structure and Organzatonal Form of Insurance Frms, Journal of Rsk and Uncertanty 6: Doherty, N., and H. Schlesnger ). Insurance Contracts and Securtzaton, Journal of Rsk and Insurance 69): Doherty, N., P. Klendorfer and H Kunreuther 99). Insurance Perspectves on an Integrated Hazardous Waste Management Strategy, Geneva Papers on Rsk and Insurance 7: Goller, C. 996). Optmum Insurance of Approxmate Losses, Journal of Rsk and Insurance, 633): Goller, C., and H. Schlesnger 995). Second-Best Insurance Contract Desgn n an Incomplete Market, Scandnavan Journal of Economcs 97): Mahul, O. ). Optmum Crop Insurance under Jont Yeld and Prce Rsk, Journal of Rsk Insurance 67): 9-. Mahul, O. ). Managng Catastrophc Rsk Through Insurance and Securtzaton, Amercan Journal of Agrcultural Economcs, 83Août ): Mayers, D., and C. Smth 986). Ownershp Structure and Control: The Mutualzaton of Stock Lfe Insurance Companes, Journal of Fnancal Economcs 6, Ravv, A. 979). The Desgn of an Optmal Insurance Polcy, Amercan Economc Revew 69: Smth, B., and M. Stutzer 99). Adverse Selecton, Aggregate Uncertanty, and the Role for Mutual Insurance Contracts, Journal of Busness 634), Smth, B., and M. Stutzer 995). A Theory of Mutual Formaton and Moral Hazard wth Evdence from the Hstory of the Insurance Industry, Revew of Fnancal Studes 8), Swss Re 999). Alternatve Rsk Transfer for Corporatons: a Passng Fashon or Rsk Management for the st Century?, Sgma. Swss Re ). Natural Catastrophes and Man-Made Dsasters n. Sgma. Wlson, R. 968). The Theory of Syndcates, Econometrca 36, -3. 7

18 Appendx Proof of Proposton The desgn of the optmal non-partcpatng polcy s frst derved when the premum Q s gven. Problem 6) can be solved by usng Kuhn-Tucker condtons for J ε ) for all x, ε ) because ts frst dervatves appear nether n the obectve functon nor n the constrants. The frst-order condton s A) w l + I ε ) P + J Q) + λ µ d EJ ) ) u ε = for all x, ε ), where µ and λ are the Lagrangan multplers assocated respectvely to constrants 3) and ), wth A) λ = f J > otherwse. Condton A) can be rewrtten as A3) u w l + I P + J Q) = µ d EJ ) ) ε for all x, ε ) such that J ε ) >. Ths mples that the margnal utlty of the frm s constant n every state of the world where the ndemnty J s pad: A4) I ) + J = l ε ) + P ε for all, : J > x. Consder now the desgn of the optmal fully partcpatng polcy. The frst-order condton s A5) w l + I P + J Q) + λ µ c EI ) ) u ε = for all x, ε ),where µ.) and λ are the Lagrangan multplers assocated respectvely to constrants 5) and 4), wth A6) λ Condton A5) becomes = f I > otherwse. A7) u w l + I P + J Q) = µ c EI ) ) ε for all x, ε ) such that I ε ) >. Consequently, for every level of systemc rsk x, the margnal utlty of the frm must be constant wth respect to the dosyncratc varable: A8) I ) + J = l ε ) + P ε for all ε : > I. Frst, suppose that the cost of coverage s hgher under the non-partcpatng polcy than under the fully partcpatng contract: c s) < d s) for all s. The frm wll choose to be nsured aganst the ε rsk through the less expensve contract,.e., the fully 8

19 partcpatng polcy and, consequently, J ε =. Then t wll select a coverage aganst the x rsk through the non-partcpatng contract. Ths leads to part ) of Proposton. Second, suppose that c s) > d s) for all s. It s less expensve to purchase the nonpartcpatng polcy than the fully partcpatng contract. Full nsurance aganst the ndvdual loss above a straght deductble s thus optmal. Ths leads to part ) of Proposton. Proof of Proposton Consder the optmal level of deductblty under the non-partcpatng nsurance polcy. The optmzaton of the obectve functon n 6) wth respect to Q yelds the frst-order condton µ = E u π ) A9) ) ε where π = w l + I P + J Q. The frst-order condton A) becomes λ x, = u π + Eu π ε ) d EJ ). A) ) ) ) ε Takng the expectaton wth respect to x, ε ) yelds A) λ ε ) = Eu π ε ) )[ d EJ ε )) ] E. λ, x ε ) = If d s) = for all s, then E. From the defnton of λ n A), ths mples that λ x, = for all x, ε ) almost surely and, consequently, from Proposton, D =. If d s) > for all s, then E λ, x ε ) >. From the defnton of λ n A), ths mples that λ x, > for some x, ε ) wth a postve probablty and, consequently, from Proposton, D. Ths leads to part ) of Proposton. > Consder now the optmal level of deductblty under the partcpatng nsurance polcy. The above procedure can be shown to lead to an ambguous result because of the presence of the varable premum. An alternatve way s to consder the followng maxmzaton problem: Max Eu w l x, ε ) + max l x, ε ) D, P + J ε ) Q. A) ) ) D It s easy to show that the frst-order condton of A) evaluated at D = equal zero f c s) = for all s, and t s postve f c s) > for all s. Ths leads to part ) of Proposton. Notce that ths result s derved assumng the second-order condton holds. 9

20 Optmal separate non-partcpatng nsurance contracts When the premum R s fxed, usng Kuhn-Tucker condtons for S z ) for all z yelds the frst-order condton of the maxmzaton problem 5): A) E u w zz + S z ) R + S z ) R ) + λ z) + δ ) µ = for all z, where µ and λ are the Lagrangan multplers assocated respectvely to premum constrant and the non-negatvty constrant, wth A3) = f S z) > λ z) otherwse. For all z such that S z ) =, we have A4) V z ) E u w z z R + S z ) R ) + ) µ = δ. V decreases wth z because u < and z. Ths mples that there exsts a deductble d such that the optmal nsurance contract satsfes equaton 6). For all z such that S z ) >, dfferentatng A) wth respect to z gves A5) S ' z u π ) u π ) E z ) =, E where π = w z z + S z R + S ) z ) R. The margnal ndemnty functon s postve. Decomposng the numerator of A5) leads to equaton 7). Optmzng problem 5) wth respect to R yelds A6) = E E u w z z + S z ) R + S z ) R ) µ. Introducton A6) n A3) and takng the expectaton wth respect to z gves A7) E z ) = δ E E u w z z + S z ) R + S z ) R ) λ. Ths mples that the optmal deductble d s zero f, and only f, the loadng factor δ s zero.

21 E w A D C B var w ) Fgure. Effcent fronters under a sngle non-partcpatng contract on the systemc rsk lne A-B), separate non-partcpatng contracts lne A-C), and varable partcpatng contract lne A-D); δ >, δ. >

22 E w C IC m B A IC h D var w ) Fgure. Effcent fronters under a sngle contract on the systemc rsk lne A-B), separate non-partcpatng contracts lne A-C), varable partcpatng contract lne A-D); δ > δ, δ =.