NBER WORKING PAPER SERIES CATASTROPHE BONDS, REINSURANCE, AND THE OPTIMAL COLLATERALIZATION OF RISK-TRANSFER. Darius Lakdawalla George Zanjani

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1 NBER WORKING PAPER SERIES CATASTROPHE BONDS, REINSURANCE, AND THE OPTIMAL COLLATERALIZATION OF RISK-TRANSFER Darus Ladawalla George Zanjan Worng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA December 2006 The vews expressed n ths paper are those of the authors, and do not necessarly reflect the vews of the Federal Reserve Ban of New Yor, the Federal Reserve System, the RAND Corporaton, or the RAND Center for Terrorsm Rs Management and Polcy. We than Robert Crag, Ken Garbade, Robert M. Hall, Thomas Holzheu, Stewart C. Myers, and semnar partcpants at the 2006 NBER Insurance Project Worshop and the 2006 Rs Theory Semnar for helpful comments and dscussons. The vews expressed heren are those of the authors and do not necessarly reflect the vews of the Natonal Bureau of Economc Research by Darus Ladawalla and George Zanjan. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 Catastrophe Bonds, Rensurance, and the Optmal Collateralzaton of Rs-Transfer Darus Ladawalla and George Zanjan NBER Worng Paper No December 2006 JEL No. G11,G22 ABSTRACT Catastrophe bonds feature full collateralzaton of the underlyng rs transfer, and thus abandon the nsurance prncple of economzng on collateral through dversfcaton. We examne the theoretcal foundatons beneath ths paradox, fndng that fully collateralzed nstruments have mportant uses n a rs transfer maret when nsurers cannot contract completely over the dvson of assets n the event of nsolvency, and, more generally, cannot wrte contracts wth a full menu of state-contngent payments. In ths envronment, nsureds have dfferent levels of exposure to an nsurer's default. When contractng constrants lmt the nsurer's ablty to smooth out such dfferences, catastrophe bonds can be used to delver coverage to those most exposed to default. We demonstrate how catastrophe bonds can mprove welfare n ths way by mtgatng dfferences n default exposure, whch arse wth: 1 contractual ncompleteness, and 2 heterogenety among nsureds, whch undermnes the effcency of the mechancal pro rata dvson of assets that taes place n the event of nsurer nsolvency. Darus Ladawalla The RAND Corporaton 1776 Man Street Santa Monca, CA and NBER darus@rand.org George Zanjan Captal Marets Functon Federal Reserve Ban of New Yor 33 Lberty St. New Yor, NY george.zanjan@ny.frb.org

3 1 Introducton Recent dsaster experence has produced a flurry of economc nqury nto catastrophe nsurance marets. Especally puzzlng s the apparent ncompleteness of catastrophe rs transfer: The prce of rs transfer seems hgh, rs s not spread evenly among nsurers n the manner suggested by Borch s [1] groundbreang theoretcal result, and, n star contrast to Arrow s well-nown characterzaton of optmal nsurance contracts, rensurance consumers do not purchase coverage for hgh layers of rs. Froot [11] documents these puzzles and fngers varous maret mperfectons as possble explanatons. Some vew the catastrophe bond as a promsng vehcle for overcomng mperfectons n the rensurance maret. In prncple, the catastrophe bond opens a drect channel for catastrophe rs to flow to the captal marets, sdesteppng frctons present n the rensurance maret and connectng those who need protecton wth well-funded nvestors eager to provde t. On the other hand, others are septcal that catastrophe securtzaton wll be a panacea. Bouraux and Scott [2] argue that securtzaton terms are unlely to be attractve to buyers of terrorsm coverage and note that the record of rs-lned captal maret nstruments has not been encouragng. Indeed, catastrophe bond ssuance to date has been relatvely lmted, even n the aftermath of events that were expected to stmulate ssuance. It s too early to wrte an eptaph for the catastrophe bond, but the experence to date rases questons about ts theoretcal foundatons and ts lely future role. On closer nspecton, the catastrophe bond seems both paradoxcal and prmtve. Its current form features full collateralzaton and lns prncpal forfeture only to specfc rss, thereby retreatng from the fundamental, tme-tested concept of dversfcaton that allows nsurers to protect nsured value far n excess of the actual assets held as collateral. In a world where frctonal costs e.g., due to taxes, regulatons, or moral hazard mae captal costly to hold, dversfcaton lowers the cost of nsurance. Vewed n ths lght, a fully collateralzed nstrument seems an unlely compettor to tradtonal rensurance products. 1 Ths paper examnes ths ssue by developng a theory of rs collateralzaton. Specfcally, we study the effcent dvson of rs-bearng assets between rensurance company assets and catastrophe bond prncpal both of whch can be used to collateralze promses to ndemnfy consumers. In a narrow sense, the analyss supports the septcal ntuton outlned above. When rensurance companes can wrte any type of contract wth ther nsureds and frctonal costs are dentcal for catastrophe bonds and rensurer assets, catastrophe bonds are at best redundant, and at worst welfare-reducng. If the nsurer has complete freedom to vary ndemnty payments to consumers n every state of the world, t can engneer any possble menu of pay-outs through ts own contracts: Catastrophe bonds add nothng n the absence of contractng constrants. However, rensurers and nsurers do face contractng constrants n practce. Typcal contracts promse an ndemnty payment to a polcyholder who has suffered a covered loss but do not specfy rules ex ante for who gets what n the event of nsurer falure. Instead, the dvson of assets under banruptcy s determned by nsurance recevershp laws, and nsurers ether cannot or do not attempt to contractually specfy how assets wll be dvded n the event of banruptcy. As a result, assets of faled companes are dstrbuted accordng 1 Nehaus [21] observes ths paradox on Page

4 to nflexble and potentally neffcent mechancal rules. Constrants on company asset dstrbuton under default open up a role for catastrophe bonds and other hghly collateralzed vehcles, such as rensurance sde-cars, even f there are no dfferences n the frctonal costs. When nsureds are homogeneous and rs exposures are bnary loss or no loss, optmal nsurance contracts are smlar, and smple rules e.g., a pro rata rule that pays all clamants at the same rate on the dollar n the event of nsurer banruptcy can perform well. Heterogenety, however, exposes the shortcomngs of nsurance contracts n the presence of pro rata rules, whch may msallocate assets n the banruptcy state. 2 Pro rata allocatons can be suboptmal when some nsureds are more concerned about the banruptcy state than others, and ths wll generally be the case. Rensurance buyers hold polces of dfferng qualty even when purchasng these polces from the same rensurer. Some are more exposed to default than others, and those that have greater exposure to banruptcy rs may desre greater collateralzaton of ther potental clams than can be provded under pro rata rules. Ths need opens up a role for catastrophe bonds and other related nstruments n the rs transfer maret, whch can vary the degree of collateralzaton of coverage for specfc nsureds. Thus, catastrophe bonds can mprove welfare when rensurers face constrants on the dstrbuton of assets n banruptcy, and when they must nsure a heterogeneous group of rss. Catastrophe bonds can smooth out allocatons made ragged by the rs of banruptcy. Put dfferently, rensurance captal may wealy domnate the catastrophe bond n terms of rasng average polcy qualty, but such captal can be rendered a blunt nstrument by banruptcy laws: Catastrophe bonds can mprove welfare for those nsureds most exposed to banruptcy rs. The paper s lad out as follows. Secton 2 provdes some bacground and context on catastrophe bonds. Secton 3 then develops a concrete two-consumer example to llustrate the ntuton behnd our results. Secton 4 develops our results formally n the context of a socal plannng problem wth N consumers. Secton 5 dscusses other strateges for protectng consumers aganst default and nterprets them n the context of the model. In partcular, t consders how collateralzaton clauses n rensurance polces nfluence the prorty of clamants under banruptcy, and the extent to whch such clauses substtute for fully collateralzed nstruments such as catastrophe bonds. Secton 6 concludes. 2 Bacground and Motvaton Whle structures vary, a typcal catastrophe bond transacton nvolves a specal purpose vehcle SPV. The SPV sells securtes catastrophe bonds to nvestors, and the proceeds from the sale are deposted n trust and nvested. The SPV then provdes rensurance to a cedng nsurer or rensurer, who pays a premum n exchange. The premum, as well as ncome earned on the trust nvestments, funds nterest payments to nvestors. If a contractually-defned trgger event occurs, part or all of the bond prncpal s forfeted to the 2 Mahul and Wrght [17] also note the neffcency of pro rata rules n the context of a model wth dentcal consumers but generalzed loss dstrbutons. 2

5 cedng company; f no event occurs, the prncpal s returned to nvestors. 3 The catastrophe bond maret s stll evolvng and, n partcular, movng toward hgher layers of rs. Whle the frst catastrophe bonds lned forfeture of prncpal to the ssuer s actual losses an ndemnty trgger, trggers lnng forfeture of prncpal to ndustry losses, model output, or to specfc parameters of the dsaster e.g., the strength of an earthquae centered n a certan geographc regon have grown n popularty. Some deals feature multple event trggers requrng two or more major dsasters wthn a short tme perod to trgger prncpal forfeture see Woo [22]. How mportant s the catastrophe bond maret? Catastrophe bond ssuance n 2005 amounted to a record $2 bllon, wth outstandng prncpal n the neghborhood of $5 bllon. 4 Wth ongong growth n catastrophe bond ssuance, as well as the emergence of rensurance sde-cars whch wll be dscussed later, 2006 s set to be another record year for alternatve rs transfer n the catastrophe maret. On the other hand, catastrophe bonds currently provde only a tny fracton of rensurance capacty. Despte beng around for more than a decade, outstandng catastrophe bond prncpal amounted to less than 2% of global rensurance captal at year-end Relatve to the gddy expectatons of the 1990 s, the volume of catastrophe securtzaton has been dsappontng. Nevertheless, whle ssuance so far has fallen short of optmstc projectons, we argue that catastrophe bonds do serve a well-defned economc role n the rs transfer maret a role dervng from the full collateralzaton underlyng the nstrument. Full collateralzaton allows catastrophe bonds to be useful n cases where tradtonal rs transfer e.g., through rensurance polces s subject to sgnfcant rs of counterparty default. Ths role could expand f frctonal costs assocated wth catastrophe bonds e.g., ssuance costs, secondary maret llqudty fall sgnfcantly n relaton to the frctonal costs assocated wth rensurance equty. However, unless frctonal costs become neglgble, full collateralzaton s lely to lmt the role of catastrophe bonds n crcumstances where dversfcaton opportuntes mae partally collateralzed nstruments attractve. To understand the role of customer-specfc collateral, we must move beyond exstng theory on nsurance prcng. The theory of the nsurance frm has made a great deal of progress n understandng the jont determnaton of multple lne prcng, captal allocaton, and the frm s overall default rs see, e.g., Cummns et al. [6], Myers and Read [20], and Zanjan [23]. However, most models consder the default rs of the frm as a whole. Less progress has been made n studyng dfferences across a frm s polcyholders n ther exposure to default. Dfferences across polcyholders, though, are central to the value of catastrophe bonds. If the object of nterest s a sngle default-related fnancal target for the company as a whole such as the expected polcyholder defct per dollar of labltes a dollar held n the form of a catastrophe bond cannot possbly be preferable to one held as company equty. 3 For more nformaton on the structure of nsurance-lned securtes, see, for example, Canabarro, E., Fnemeer, M., Anderson, R.R., and Bendmerad, F. 1998, Analyzng Insurance-Lned Securtes, Goldman Sachs & Co. Quanttatve Research, October, Source: The Catastrophe Bond Maret at Year-End 2005: Rpple Effects from Record Storms, MMC Securtes. Fgures nclude only publcly dsclosed transactons. 5 Standard and Poor s Global Rensurance Hghlghts, 2006 edton talled over $300 bllon n total adjusted shareholder funds for the ndustry at year-end

6 Snce the dollar held as equty wll be avalable n all states of the world, t wll be avalable to pay for all of the losses that wll be covered by a catastrophe bond and some losses that are not covered by the catastrophe bond. To understand how catastrophe bonds can be used n the absence of compellng frctonal cost dfferences, we must move beyond thnng of a sngle default-related fnancal target for the nsurance company. Instead, we must thn about the company s polces as havng varyng levels of qualty, correspondng to varyng levels of exposure to default, and how catastrophe bonds and equty have dstrbutonal consequences for recoveres by dfferent polcyholder groups n states of default. 3 A Smple Example In the context of a smple two-consumer example, we llustrate how the role for catastrophe bonds depends on the presence of: 1 nonzero banruptcy rs for the nsurer; 2 contractng constrants that prevent the nsurer from optmally allocatng clams payments n the banruptcy state; and 3 heterogenety across consumers, such that one consumer faces greater exposure to nsurer banruptcy rs. Consder the case of two consumers, named A and B. Consumer A faces a 10% chance of losng $100, whle Consumer B faces a 1% chance of losng $100. An nsurer ssues smple contracts to ndemnfy the consumer, fully or partally, n the event of a loss. In the banruptcy state where clams exceed nsurer assets, clams payments are allocated accordng to a mechancal rule by dvdng assets on a pro-rata bass, accordng to the clams made by the nsureds. 6 Suppose we have $150 n assets. How should we allocate them? Consder frst the case where we use all $150 to fund an nsurance company, whch ssues a $100 lmt nsurance polcy to A and a $100 lmt polcy to B. Expected clams n ths example equal: 10% $ % $100 = $11 1 The nsurer s able to pay all clams n full except when both consumers suffer a loss; n that event, the nsurer pays out all $150 of ts assets but declares banruptcy. Therefore, expected clams payments equal: 10% 99%$ % 90%$ % 1%$150 = $ Overall, the nsurer pays $ 10.95, or better than 99 cents, on the dollar. However, the two 11 consumers are unevenly exposed to default. Consumer B ends up beng much more exposed to banruptcy rs on a per dollar bass, because she faces a hgher relatve lelhood of sufferng a loss n the state of the world where the other consumer also suffers a loss. Specfcally, Consumer A expects to lodge $10 worth of clams and to receve payments of: 10% 99% $ % 1% $75 = $ The exact form of the mechancal rule s less relevant than the presence of contractng constrants n the banruptcy state. 4

7 On the other hand, Consumer B expects to lodge $1.00 worth of clams, but receve 1% 90% $ % 10% $75 = $ Thus, Consumer A receves cents on the dollar, whle Consumer B receves only Consumer A s better nsured than Consumer B, and the socal planner, dependng on the objectve functon, mght want to consder redstrbutng coverage from Consumer A to Consumer B. One way of accomplshng ths s to redeploy some of our assets n the form of a catastrophe bond ted to Consumer B. Suppose we now spend $100 fundng the nsurance company, whch sells a $100 lmt nsurance polcy to Consumer A and a $50 lmt polcy to Consumer B. We then use the remanng $50 on a catastrophe bond payable to Consumer B n the event of a loss. Consumer A stll expects to lodge $10 worth of clams, but now receves payments of: 10% 99% $ % 1% = $ On the other hand, Consumer B now expects to lodge $0.50 worth of clams wth the nsurance company, but now also s enttled to receve $50 of catastrophe bond prncpal n the event of a loss: 1% 90% $50 + $50 + 1% 10% 50 $50 + $50 = $ In other words, the recovery dfferental narrows. Consumer A now receves cents of relef per dollar of loss, a slghtly worse rate than before. Consumer B now receves a bt more 98.3 cents once the catastrophe bond prncpal s consdered. In ths example, usng the catastrophe bond nstead of a full rensurance soluton effectvely transfers coverage from one consumer to the other. The transfer occurs only when the rensurer defaults: We have suffcent assets to fully ndemnfy both consumers except when both experence a loss, and the catastrophe bond allows us to affect the dstrbuton of ndemnfcaton n that unfortunate state of the world. Of course, the queston of whether or not ths redstrbuton s desrable depends on partculars such as preferences but the general pont s that the allocaton of assets to consumers n the banruptcy state may be suboptmal n a pure rensurance soluton, and the catastrophe bond s one way of securng the nterests of one consumer over the other. The presence of contractng constrants, the rs of banruptcy, and the presence of consumer heterogenety all play ey roles n drvng ths result. If an nsurer s able to wrte complex contracts that vary ndemnfcaton across all states of the world, t can replcate the payout structure of a catastrophe bond wthout usng the bond tself. For nstance, n the example above, we could replcate the payoffs nvolved under the second approach usng the catastrophe bond smply by captalzng the nsurer wth $150 and ssung polces offerng full $100 ndemnfcaton except n the case where both consumers had losses, n whch case Consumer A would receve $66.67 and Consumer B would receve $ Note that f we allowed polcy lmts to exceed nsurer assets, ths would allow nsurers to nfluence the dvson of resources n the banruptcy state. However, ths s a blunt nstrument for resource allocaton that cannot generally replcate the payouts of catastrophe bonds. For example, wth more than two consumers, nsurer banruptcy s not perfectly correlated wth the loss experence of any one consumer, because there are many possble loss confguratons that trgger banruptcy. Nevertheless, n the general theory developed n Secton 4, we place no constrants on the choce of polcy lmts. 5

8 Contractng constrants that prevent the nsurer from specfyng complcated prorty rules under banruptcy are necessary to preclude ths possblty. Heterogenety also plays an mportant role by renderng mechancal banruptcy rules neffcent. If Consumers A and B were dentcal, an equal pro rata dvson of resources n the banruptcy state would be optmal, and nether consumer would be any more exposed to default rs. Ths example shows how catastrophe bonds can be used to mprove socal welfare by redstrbutng coverage among consumers n unfortunate states of the world, but t falls short of llustratng other aspects of the general trade-off between catastrophe bonds and rensurance. Earler, we emphaszed the costlness of fully collateralzed catastrophe bonds, relatve to less than fully collateralzed nsurance. Yet n ths example, there s no dsadvantage to sequesterng captal n the form of a catastrophe bond snce we mae full use of the collateral assets. In more general versons of the problem, one of the mportant drawbacs assocated wth catastrophe bonds s that the assets dedcated to catastrophe bond prncpal for one consumer wll not be avalable to pay losses experenced by others. In the results that follow, we show that the three condtons dentfed above are necessary for catastrophe bond ssuance to be useful, but not suffcent any benefts assocated wth catastrophe bond ssuance may fal to outwegh the neffcences assocated wth full collateralzaton. 4 Theory The basc approach borrows from Borch s analyss of optmal rs sharng among many consumers: Instead of modellng ndvdual behavor, we study the socal plannng problem and ts solutons. Consder a world wth N consumers. Consumer s endowed wth ntal wealth W and faces the rs of experencng a loss of fxed sze denoted by L. To characterze the possble states of the world, we defne a row vector x of length N, wth the elements all tang a value of zero or one: x = 1 means that consumer experenced a loss, whle x = 0 means that she dd not. Let Ω denote the set of all such vectors of length N wth the elements tang values of one or zero. Each element of Ω corresponds to a complete descrpton of one possble state of the world. The entre set Ω contans all possble such states. The followng set defntons are useful: Ω = {x : x = 1}, the set of all states n whch agent suffers a loss, and Γx = { : x = 1}, the set of all agents that suffer a loss n state x. Thus, usng ths notaton, we may descrbe the probablty of loss faced by consumer as: p = x Ω Prx There are two rs transfer technologes avalable to nsure aganst losses. Frst, we can set up an nsurance company and ssue nsurance polces to consumers, collateralzed by the 6

9 assets of the company. Second, we can ssue a rs-lned securty on behalf of a consumer.e., a catastrophe bond that pays off n the event that the consumer experences a loss. The nsurance company s formed wth assets of A. Throughout our dscusson, we thn of assets as all the resources the nsurer can use to pay clams. Therefore, t ncludes both captal pad n by nvestors and premums pad n by consumers. For our purposes, the ey ssue s whether or not the dollar s avalable for clams payment not how t would be treated by accountng conventons. In the event of a loss or losses, the consumers can draw on the assets to pay clams. When assets exceed clams, what remans after clams are pad reverts to nvestors. On the other hand, f clams exceed assets, the company defaults, and clamants are assumed to be pad accordng to a pro rata rule everyone receves the same rate of recovery per dollar of clam. Each consumer can separately ssue a catastrophe bond 8 to nvestors. The prncpal of the bond s forfeted to the consumer n the event of a loss, but not otherwse. Let B be the bond ssuance of consumer. We smplfy matters by assumng ndemnty trggers where prncpal forfeture s lned drectly to ssuer loss experence. Hence, we avod the complextes of optmal trgger desgn see Doherty and Mahul [7] and the problem of bass rs. We do not drectly model these and other costs assocated wth asymmetrc nformaton, 9 but such costs can be thought of as beng embedded n the frctonal costs assocated wth catastrophe bond prncpal descrbed below. What do nsurance polces and catastrophe bonds cost? The cost of rs transfer can be decomposed nto 1 far ex ante compensaton for clams expected to be pad under the rs transfer agreement and 2 frctonal costs assocated wth establshng and mantanng the rs transfer scheme. We start smply by assumng that the cost of rs transfer amounts to the expected value of clams plus a frctonal cost proportonal to the amount of collateral used n the rs transfer scheme.e., the amount of assets used n the nsurance company, or the amount of catastrophe bond prncpal used. We then generalze the results to the case where far compensaton for expected clams reflects the states of the world where the clams occur: That s, we prce rs transfer accordng to how clams are expected to relate to the expected dstrbuton of returns on other assets avalable n the broader captal marets. We start smply because t turns out that our results are drven entrely by frctonal costs. In the absence of frctonal costs, all consumers wll be fully nsured, and t s rrelevant how rs transfer technologes are combned n provdng ths full nsurance: Frctonal costs provde a motvaton to economze on captal n the process of collateralzng rstransfer. Accordngly, we start wth a model where nsurance rss are zero beta, but where frctonal captal costs lead to lmted rs transfer. We then show that these results hold even when nsurance rss correlate n some way wth captal maret returns and are prced accordngly, but that non-zero frctonal costs are ey to our fndngs. 8 We refer to catastrophe bonds because of ther famlarty, but the analyss that follows obvously apples to other hghly collateralzed nstruments used n rs transfer such as collateralzed rensurance polces and sde-cars. These and other rs transfer alternatves wll be dscussed n Secton 5. 9 See Brandts and Laux [3] for a theoretcal justfcaton for the catastrophe bond maret based on asymmetrc nformaton between nsurers and rensurers. 7

10 4.1 Optmal Collateralzaton wth Frctonal Costs Frctonal costs are magned here as dervng from agency costs, taxes, lqudty costs, or other frctons. Each dollar of assets held n the nsurance company results n per unt frctonal costs of δ A. Each dollar of catastrophe bond prncpal rased has the frctonal cost δ B. Consumers must pay for all frctonal costs and far compensaton for recoveres expected from the rs transfer. Denote the porton of ths total rs transfer cost allocated to consumer as c. Insurance polces are represented as smple promses of ndemnfcaton: The nsurer promses to pay I n the event that consumer experences a loss. We place no constrants on the promsed ndemnty: It may be less than, equal to, or greater than the prospectve loss. However, contractng constrants come nto play n the sense that we do not allow the nsurer to offer a schedule of promsed ndemnfcaton, wth the amount contngent on the loss experences of other nsureds. If the nsurer s able to pay, t pays n full; f not, t defaults, and all clams are pad at the same rate on the dollar. The example of Secton 3 suggests that relaxng ths contractng constrant wll obvate roles for catastrophe bonds or other fully collateralzed nstruments. In the Appendx Secton A, we verfy that ths s n fact the case: When frctonal costs are dentcal δ A δ B, allowng the socal planner to arbtrarly vary the ndemnfcaton promsed n each nsurance polcy elmnates any potental role for catastrophe bonds. We can now defne utlty for consumer accordng to the usual Von Neumann-Morgenstern assumptons as: EU = x Ω PrxU W L + f x I + B c + x/ Ω PrxU W c, 7 where f x represents the proporton of the ndemnty payment actually pad n state x. The socal plannng problem can now be wrtten as: max A,{B },{c },{I },{f x} V = EU 8 subject to: [µ] : c δ A A + δ B B + Prx f x I + x Ω [λ x ] : f x Γx Γx Γx B 9 I A, x 10 [φ x ] : f x 1, x 11 and subject to non-negatvty constrants on catastrophe bond prncpal and polcy lmts. Constrant 9 ensures that consumers total payments for rs-transfer nstruments c cover the frctonal costs of captal and expected losses. Constrant 10 ensures that the 8

11 nsurer always has enough assets on hand to cover actual as opposed to promsed labltes. Fnally, constrant 11 precludes the nsurer from ever payng out more than the polcy lmt. Any dfference n frctonal costs e.g., δ A δ B wll obvously create a potental advantage for one of the technologes, but we wll start by consderng the case where δ A δ B δ. Thus, we start by studyng how the nature of preferences and rs affect the optmal mx of the two rs transfer technologes. The optmalty condtons are derved n the Appendx Secton B.1, as s the followng margnal condton for catastrophe bond ssuance where we use the notaton U x to denote the utlty of consumer n state x: where R = x Ω Prx [1 f x ] U x W w x j = j Γx I j j Γx I j w x j. U x j W x/ Ω λ x 0 12 R s the margnal return on catastrophe bonds the ncrease n utlty assocated wth the frst-dollar of bond ssuance. R 0 f and only f catastrophe bonds cannot mprove on a rensurance-only equlbrum. Specfcally, f R s negatve, ths means that catastrophe bond ssuance was not useful optmal for consumer or, n other words, that B = 0. Study of 12 reveals several mportant characterstcs of optmal catastrophe bond use n the socal plannng problem. Frst, a catastrophe bond s potental to enhance the welfare of the ssung consumer s ntmately lned to the presence of default rs. If consumer does not face any rs of default.e., f x = 1 for all x Ω, catastrophe bond ssuance wll not be useful for that consumer. 10 Second, assumng consumer s confronted wth default, catastrophe bond ssuance on behalf of that consumer has the potental to be useful only f her margnal valuaton of consumpton n the states where the company defaults on her clam exceeds the average valuaton of other consumers who lose n those same states: x Ω Prx [1 f x ] U x W j Γx w x j U x j W > 0. Intutvely, t wll not mae sense to dedcate collateral to consumer f that collateral s actually worth more to her companons n states of default. If her companons value that collateral more hghly on average than she does, t wll be more effcent to add that 10 Ths s equvalent to sayng that R < 0, except n solutons where the nsurance company never defaults on any contract. If the nsurer never defaults, R = 0, mplyng that cat bonds could fgure n a soluton. However, as shown n the Appendx Secton B.2, any such soluton would not be unque: Wthout default, any soluton wth catastrophe bond prncpal can be matched by a soluton wthout catastrophe bond prncpal. 9

12 collateral to the nsurance company or to ssue catastrophe bonds on behalf of the hgh valuaton consumers. Fnally, the value of catastrophe bond ssuance for consumer also depends on the extent of dversfcaton possbltes, as captured n: x/ Ω λ x. If that term s postve, t means those dversfcaton possbltes stll exst. In other words, the company s defaultng n states of the world where consumer does not experence a loss, and, thus, there are other consumers who would enjoy benefts from ncreasng the captalzaton of the nsurance company. Whle ths does not preclude the soluton from featurng catastrophe bond ssuance on behalf of consumer, t maes t more dffcult for the catastrophe bond to be the preferable nstrument for addressng the the rs transfer needs of the consumer n queston: Any benefts obtaned by sequesterng collateral on behalf of consumer and thus sheldng the assets from consumers who place lower valuatons on addtonal coverage n those states where consumer s exposed to default must be weghed aganst the cost of preventng consumers exposed to default n other states of the world from accessng that collateral. The mportance of dfferences across consumers s hghlghted by the case where consumers are homogenous, or ex ante dentcal. The Appendx explores ths case n detal, showng that catastrophe bonds wll not be useful n ths case. The result shown n the Appendx - Secton B.3 can be understood by notng that, assumng that symmetrc solutons apply under homogenety, the margnal utltes of consumers who lose wll be equvalent n each state. So Equaton 12 reduces to: R = x/ Ω λ x 0 In other words, cat bond ssuance wll be strctly suboptmal unless dversfcaton possbltes have been completely exhausted. More precsely, cat bond ssuance for consumer wll be suboptmal f there s a sngle state of the world where addtonal nsurance captal wll beneft someone other than consumer. In the case of homogenety, ths can only happen f all consumers enjoy full ndemnfcaton except n the state where everyone experences a loss. If the socal planner fnds t desrable to ndemnfy consumers to ths extent, she wll be ndfferent between cat bonds and nsurance polces as a means of provdng addtonal coverage n the N-loss state. 11 Thus, the N-consumer case under homogenety exposes the extreme dsadvantage of cat bonds wth respect to dversfcaton. Even when catastrophe bonds are cheaper than 11 An nterestng feature of the model s that contracts under homogenety wll always promse but sometmes fal to delver full ndemnfcaton f the loss amount s less than company assets. Ths contrasts wth the nsurance demand model of Doherty and Schlesnger [8], n whch the consumer buys less than full coverage n the presence of default rs. There are several reasons for ths dfference, ncludng the presence of frctonal costs n our model Doherty and Schlesnger s result referenced actuarally far premums, and also a dfferent defnton of default: Doherty and Schlesnger s default features zero payment, whle our model offers a partal pro-rata payment to nsureds. The pro rata allocaton amounts to a mutualty prncple that encourages full coverage even n the presence of default rs. 10

13 nsurance company assets.e., f δ B < δ A, they could stll be strctly suboptmal f the welfare-maxmzng soluton nvolved tolerance of default beyond the absolute worst case scenaro of N losses. More generally, the return to catastrophe bonds gven n 12 can be rewrtten for the case of dfferent frctonal costs as: R = Prx [1 f x ] U x W U wj x j x λ x + µδ A δ B 0 13 W x Ω x/ Ω j Γx If bond prncpal enjoys frctonal cost advantages relatve to nsurer assets δ B < δ A, addtonal opportuntes for catastrophe bond ssuance may arse. Note, however, that such frctonal cost advantages to the extent they exst are by no means the only consderaton n assessng the potental for the catastrophe bond maret. The extent of heterogenety across consumers n terms of preferences and n terms of rs exposures as well as the presence or absence of dversfcaton opportuntes, stll fgure n the calculus of catastrophe bond ssuance. Catastrophe bonds are often advanced as a method for sdesteppng the frctons n the rensurance maret. But, of course, a dfferent set of frctonal costs exst n the catastrophe bond maret. The ey polcy queston concerns whether supply-sde ntatves to promote catastrophe rs transfer are best focused on the frctonal costs n the rensurance maret or those n the catastrophe bond maret. At frst glance, t seems that the opportuntes for welfare gans are much greater when reducng frctonal costs n the rensurance maret. The value of reducng the frctonal costs of nsurer assets by one unt s the dervatve of the Lagrangan of the welfare functon wth respect to δ A, or, V A µa, whle an analogous reducton n the frctonal costs of catastrophe bond prncpal yelds V B µ Thus, the margnal beneft of reducng frctonal costs n each maret s drectly proportonal to the assets deployed n the respectve maret: Snce far more collateral s held n the form of rensurer assets than n the form of catastrophe bond prncpal, the margnal mpact of frctonal cost reductons n the rensurance maret should be far greater than smlar reductons n the catastrophe bond maret. On the other hand, the cost sde of the polcy equaton.e., what resources must be sacrfced to reduce frctonal costs n each maret s less clear. Indeed, the frctonal cost reducton technologes could dffer substantally across the marets. Snce the catastrophe bond maret s young, there may be low-hangng frut : Investments n nvestor educaton or prmary and secondary bond maret nfrastructure could offer much larger frctonal cost reductons n the catastrophe bond maret than could be possble n the more mature rensurance maret. However, frctonal cost reductons wll not necessarly translate perfectly nto correspondng movements n maret performance. The drawbac of dversfcaton neffcences dscussed above may place mportant lmts on the extent to whch catastrophe rs wll be transferred through catastrophe bonds and smlar vehcles. B. 11

14 4.2 Optmal Collateralzaton wth Generalzed Captal Costs To ths pont, we have abstracted from modellng any rs premum that mght be demanded by nvestors. However, the amount of compensaton due to captal supplers theoretcally depends on how the nsurance rss borne correlate wth returns on other captal assets. Lttle evdence exsts connectng the cost of captal n the nsurance ndustry to the rs characterstcs of the underlyng polcyholder labltes Cummns and Harrngton [5]; Cox and Rudd [4]; ths s true of catastrophe nsurance n partcular, where both academcs e.g., Hoyt and McCullough [14] and practtoners e.g., Ltzenberger, Beaglehole, and Reynolds [16] have found catastrophc rs to be uncorrelated wth stoc maret returns. Nevertheless, a connecton exsts at least n theory, so we explore ths possblty more thoroughly n ths subsecton where the cost of rs transfer depends not only on frctonal costs assocated wth collateral but also on the relatonshp between the rs transferred and the rss n the broader captal marets. In prncple, frctonal costs could be zero, and the cost of rs transfer could thus consst entrely of the equlbrum compensaton for rs demanded by nvestors. However, as noted earler, the cost of rs transfer must nclude a pure frctonal cost that s entrely lost to the captal supplers for the optmal collateralzaton structure to be determnate. If there s no gan to economzng on collateral, t does not matter how rs transfer s collateralzed: All consumers wll be fully nsured, and t does not matter how nsurance polces and catastrophe bonds are combned n provdng that full nsurance. However, f frctonal costs are present, the earler results stll carry through even after the ntroducton of securty marets and equlbrum rs prcng. To add securty marets and equlbrum to our model, we start wth a state-prcng approach bult on the assumpton of no arbtrage, as descrbed n the frst chapter of Duffe [9]. For nsurance marets to be relevant and not redundant, t must be the case that fnancal marets are ncomplete. Specfcally, f t s possble to replcate the payoffs from an nsurance polcy usng exstng securtes n frctonless marets and f those securtes are prced farly, there wll be no need for nsurance polces. Therefore, we wll wor under the assumpton that marets are ncomplete or, specfcally, that nsurance polcy payoffs cannot be replcated usng other fnancal nstruments. Ths s smlar n flavor to the assumpton underlyng Mayers and Smth [18] tradeable portfolo securtes are assumed to have well-defned prces that flow from an equlbrum asset prcng model, but nsurance rss are denoted as nontradeable and thus must be dealt wth separately although, as noted by Mayers and Smth, the decsons are not ndependent. A ey queston not addressed by Mayers and Smth, but crucal for our purposes, s how nsurance polces wll be prced when they are non-tradeable and cannot be replcated wth other fnancal nstruments. The obvous approach s to prce nsurance polces as f they were traded fnancal securtes. That s, ther prces are determned by ther contngent payoffs, weghted by approprate state prces just as wth any other securty. The only complcaton s that, by assumpton, the contngences relevant for nsurance polcy payoffs do not map nto the state space that governs the securty marets so the state prces needed to prce polces wll not follow from the absence of arbtrage among the fnancal nstruments traded n the securty marets. Wth ths n mnd, we extend state prces derved from the assumpton of no-arbtrage n the securty marets to apply to subsets of events wthn 12

15 states. Whle ths extenson may not be a techncal mplcaton of arbtrage prcng theory, t produces a necessary foundaton for nsurance prcng that s logcally consstent wth securty maret prcng. We use the N person model and start by defnng the relevant probablty spaces. Recall the set of vectors that defne consumer loss experence, denoted by Ω, whose members are row vectors x of length N, wth the vector elements all tang a value of zero or one: x = 1 means that consumer experenced a loss, whle x = 0 means that she dd not. We now ntroduce a set of M securtes, each securty wth dstnct payoffs n S states of the world. Let Ψ be the set of those states wth the assocated σ-algebra F Ψ, and let there be state prces, consstent wth the absence of arbtrage, denoted by π s for each s Ψ. Let D be an M S matrx, wth D j descrbng the payoff of the -th securty n the j-th state. We assume that spand R S. Ths condton s typcally nown as a complete marets condton that any arbtrary menu of state-contngent consumpton can be purchased at tme zero. In our case, however, t would be msleadng to characterze marets as complete, snce Ψ does not provde a complete descrpton of the states of the world. Instead, we characterze the full probablty space as Θ, F Θ, µ Θ, wth Θ { = [x1 x2...xn s] x Ω, s Ψ} The state varable Θ s a row vector of length N + 1 that provdes a complete descrpton of one possble state of the world. The frst N elements of descrbe whch consumers experenced losses and whch ones dd not, whle the last element descrbes the state of the securtes marets. The entre set Θ contans all possble states of the world. The followng set defntons are useful: Θ = { : = 1}, the set of all states n whch agent suffers a loss, and Γ = { : = 1}, the set of all agents that lose n state. In addton, for every s and every agent, defne: Υ s0 { : / Θ, N + 1 = s}, the set of all states where agent does not suffer a loss and the securty maret sub-state s s, and Υ s1 { : Θ, N + 1 = s}, the set of all states where agent does suffer a loss and the securty maret sub-state s s. Fnally, note that for any the entre sub-state space defned by s can be wrtten as: Υ s Υ s1 Υ s0 13

16 We now extend the state prces to defne prces for events that are not measurable wth respect to F Ψ. Defne extended state prces as follows. For each s Ψ π = π s, Θ N + 1 = s Recall that we are extendng the state-prces to nclude the prce of clams assocated wth the hazards beng nsured. Ths approach mplctly assumes that there s no varaton n sub-state prces wthn the states prced by the securty maret equlbrum. Ths s not a mathematcal trusm mpled by the absence of arbtrage n securty marets: The absence of arbtrage does not pn down the state prces for events that are not measurable wth respect to F Ψ. However, as dscussed above, ths assumpton provdes a reasonable bass for nsurance prcng that s logcally consstent wth the securty maret equlbrum. We can now defne utlty for Consumer as EU = s Ψ µ Θ U W s L + f I + B + s Ψ Υ s0 µ Θ U W s, where f represents the proporton of the ndemnty payment actually pad n state, and W s s state-contngent securty-maret wealth for consumer. The socal plannng problem can now be wrtten as: max V = A,{B },{c },{I },{f },{W s } EU 14 subject to [µ] : c δ A A + δ B B + µ Θ π f I + B 15 Θ Γ [λ ] : f I A, 16 Γ [φ ] : f 1, 17 [ϕ ] : π s W s W c, 18 s Ψ and the prevous non-negatvty constrants. New to the problem s the last constrant, whch governs portfolo nvestment. The present value of each consumer s contngent consumpton s constraned to no more than tme zero wealth, net of rs transfer costs. Before proceedng further, t s worth notng that the generalzed captal allocaton structure does not alter our basc results for the return on catastrophe bonds see Appendx Secton C.1. In ths settng, the return to catastrophe bonds s gven by: R = Θ µ Θ 1 f U W Γ w U W / Θ λ + µδ A δ B

17 Ths s analogous to equaton 13. The margnal condtons descrbng optmal ssuance of catastrophe bonds do not drectly reference frctonal costs except when δ B δ A, yet frctonal costs are requred for them to be useful. Wthout frctonal costs, both terms of the margnal condton wll be zero n all cases smply because there s no nsurance company default n the absence of frctonal costs, and every consumer enjoys full nsurance. To llustrate ths, we elmnate frctonal costs but contnue to weght clams payments and catastrophe bond default accordng to the approprate state prces: c µ Θ π f I + µ Θ π B. Θ Γ Θ Intutvely, nsurance premums must pay for contngent clams on the assets, but, n the absence of frctonal costs, extra captal always earns ts own eep, because the nsurer can nvest t at maret rates of return. As a result, the only real cost of holdng captal as collateral s the extent to whch you are exposng the owner to addtonal clams rs but ths addtonal rs s compensated n the captal maret. Secton C.2 of the Appendx characterzes the soluton n ths settng wthout frctonal costs n a seres of lemmas that yeld two mportant lessons. Frst, full nsurance wll always be optmal: Every consumer wll be fully covered. Second, the optmal dvson of collateral between rensurance company assets and catastrophe bond prncpal s ndetermnate. In other words, wth full nsurance t ends up beng rrelevant whether the nsurance s provded through rensurance polces or catastrophe bonds: Any combnaton of the two nstruments s optmal, so long as full nsurance s provded. As s often the case, consumers fully nsure when the nsurance s farly prced. Moreover, there s no ncentve to economze on collateral n ths settng collateral s free n the sense that there are no frctonal costs. The only costs wth holdng assets n the rensurer or the SPV are the far value of expected clams payments to polcyholders or cat bond ssuers. Snce there are no penaltes assocated wth over-collateralzaton, nether rs transfer nstrument holds a natural advantage over the other. 5 Other Rs Transfer Optons To ths pont, we have lmted our attenton to catastrophe bonds and tradtonal rensurance polces. Wth ths focus, we rs overloong hedgng strateges based on other rs transfer optons that could potentally yeld welfare mprovements. In ths secton, we consder how other rs transfer strateges ft nto our framewor. 5.1 Collateralzed Rensurance and Sde-cars Rensurance can be collateralzed n at least two senses. The frst, more common sense, s a contract clause requrng the rensurer to collateralze clams oblgatons at the tme they are ncurred but before they are due to be pad. The second s a full or partal collateralzaton of the polcy lmts at the ncepton of the contract. Both are nterestng for our paper 15

18 because they allow the rensurer and ts customers to nfluence how assets are dvded up n the event of banruptcy. We dscuss each n turn. Rensurance contract clauses regardng the collateralzaton of labltes arse most often n the context of transactons between offshore rensurers and U.S. cedents.e., buyers of rensurance. Regulatons regardng statutory credt for rensurance typcally stpulate that an nsurer may not tae credt for antcpated recoveres from unlcensed rensurers unless those antcpated recoveres are fully secured. Acceptable forms of securty nclude funds held n trust and clean, rrevocable, and evergreen letters of credt ssued by fnancal nsttutons deemed acceptable by the regulator n queston. To the extent that some cedents have these contract clauses and others do not, the clauses may be nterpreted as a means of affectng the dstrbuton of assets n banruptcy. Secured clamants effectvely step ahead of unsecured clamants n the lqudaton process. However, t should be noted that the ablty to step ahead s by no means absolute and depends on ex post actons by the nsurer. For example, a transfer of assets to a trust for the beneft of a cedent or to collateralze a letter of credt ssued by a thrd-party for the beneft of a cedent can be challenged as a vodable preference f banruptcy follows soon thereafter. 12 Hence, n practce, a cedent cannot count on securty beng posted when a rensurer s n or near nsolvency, and, even f the rensurer s wllng to post securty the transfer wll be subject to challenge. Ths ssue also surfaces when modellng collateralzaton n a oneperod settng: When clams are submtted smultaneously and banrupt the rensurer, what compels the rensurer or ts recever to honor lablty collateralzaton clauses that have no force durng lqudaton? It s also possble to provde partal or full collateralzaton e.g., a letter of credt of the polcy lmts at contract ncepton. Ths approach s useful f the underwrter does not have a fnancal strength ratng, as s the case wth a number of hedge funds that have entered the property catastrophe rensurance maret n recent years. For an underwrter ssung only fully collateralzed polces, ths form of collateralzed rensurance s effectvely equvalent to a catastrophe bond wth an ndemnty trgger. Another varaton on ths theme s the recent nnovaton of the rensurance sde-car, where nvestors captalze a specal purpose company to provde rensurance to a cedng company, wth the captal beng held n a collateral trust account for the beneft of the cedent see Murray [19] for addtonal nformaton on sde-cars. For underwrters ssung polces that dffer n the degree of collateralzaton, the stuaton s more complcated. Snce collateral posted wll presumably be released n the event that the underlyng polcy s not trggered, t wll subsequently become avalable to pay clamants whose polcy lmts were not fully secured at ncepton. In our framewor, ths approach to collateralzng rs transfer offers the potental for welfare mprovement relatve to catastrophe bonds because of ths ncrease n the avalablty of assets to pay clams. In prncple, varyng the degree of collateralzaton across polces could be used to affect the allocaton of assets durng banruptcy, but t s mportant to note theoretcal lmts. Asde from the two-person case, varyng the degree of collateralzaton assocated wth polces wll generally gve the nsurer only lmted control over the allocaton of assets n 12 For more detals, especally wth respect to letters of credt, see Hall [13] and the NAIC s Recevers Handboo for Insurance Company Insolvences. 16

19 banruptcy. In the multple consumer case, there are multple banruptcy states, wth dfferent consumers affected n dfferent banruptcy states: Varyng ex ante collateral levels does not allow arbtrary prortzaton of clams wthn banruptcy. Practcal lmts also apply. The effectve securty of partal collateralzaton wll not be transparent to the polcyholder n practce: To form an expectaton of relatve prorty n banruptcy, one must now all the detals about the collateralzaton of other polces. For example, f all outstandng polces are collateralzed to the same degree, collateralzaton has no effect, and recoveres would not change f the collateralzaton were termnated. Fnally, n a world where clams are beng submtted n contnuous tme, the rensurer wll not be able to commt all of ts assets to ex ante collateralzaton, snce t would have no funds avalable beyond the collateral supportng any gven polcy to pay a clam on that polcy. 5.2 Thrd-Party Default Insurance In the model, catastrophe bonds protect consumers from the consequences of nsurer default. A possble alternatve approach would be to buy default nsurance from a thrd party, and these possbltes arse f the nsurer has ssued negotable securtes. For example, credt default swaps CDS referencng outstandng transferable bonds or loans of the nsurer 13 are a possble alternatve hedgng devce: Issung polces along wth default protecton could concevably offer welfare mprovements over the strateges consdered n ths paper. There are two aspects of ths strategy that mert comment. Frst, to delver a welfare mprovement over catastrophe bonds, any protecton offered by the CDS would have to be mplctly collateralzed at less than 100% or somehow have lower frctonal costs wth a smlar degree of collateralzaton. On the one hand, less than full collateralzaton mples that there s some rs of counterparty default, so purchasng default nsurance wth a CDS s not a perfect substtute for the catastrophe bond. On the other hand, f the seller of protecton could concevably realze addtonal economes n collateralzaton by tang advantage of dversfcaton opportuntes beyond the nsurance maret. In ths lght, the CDS approach can be understood as an ntermedate devce, located on a contnuum between the fully collateralzed and undversfed catastrophe bond, and the mperfectly collateralzed and dversfed rensurance contract. Second, the CDS hedgng approach wll generally nvolve bass rs when there are more than two polcyholders. Wth two polcyholders, the CDS offers a perfect hedge to a consumer desrng protecton: The contract s trggered only n states where the company defaults and the consumer experences a loss. Wth multple consumers, however, the company may default n states where the consumer does not experence a loss, thereby trggerng a default nsurance payment n a scenaro where the consumer does not need addtonal ndemnfcaton. The consumer becomes overnsured wth respect to default rs. Moreover, the extent of a consumer s recovery on a polcy wll generally vary across states of default n the multple consumer model, and ths varaton may not generally be hedged wth a CDS or wth a short poston n the nsurer s debt, f ths were possble unless the holder of the underlyng debt securty s n the same class as the holder of an 13 Alternatvely, n the absence of a CDS maret, hedgng strateges usng equtes or equty dervatves could substtute. However, equty-based strateges wll generally have more bass rs as defned below. 17