WO2000008567A1 - Instrumentalities for insuring and hedging against risk - Google Patents

Abstract

Where frequency and severity of risks, e.g. of catastrophic risks are unknown, as is the case, e.g., for environmental risks, health risks, nuclear reactor risks, and satellite risks, ordinary insurance contracts on occasion are likely to result in claims which an insurer cannot cover. For efficient allocation of risk bearing, an insurance contract is combined or 'bundled' with a derivative security. Insurance is contingent on the frequency of the insured event as observed, and the derivative security has a payoff which depends on that frequency. Thus, the derivative security is a contract which is contingent on an index established as a standardized measure of a risk, e.g. of environmental conditions such as El Niño versus La Niña, and/or temperature measures such as heating degree days (HDD), cooling degree days (CDD) and/or precipitation measures in a specific geographic region and during a specified time period.

Description

INSTRUMENTALITIES FOR INSURING AND HEDGING AGAINST RISK

Technical Field

The invention is concerned with methods, systems and instruments for insuring and hedging against risk, e.g. weather-related risk, including catastrophic risk and other large-scale risk which may be weather-related or otherwise.

Background of the Invention

Concerns are on the rise with large-scale risks, e.g. weather-related risks in view of increasing volatility of weather, climate changes, and population movement to warm coastal areas with attendant changing property prices. Weather affects an estimated $2 trillion of the $9 trillion U.S. economy. Such risks are difficult to diversify using traditional insurance and reinsurance practices, and even though climate changes remain putative, the financial challenge is manifest. In the last several years the property/casualty insurance industry has experienced record claims of some $43 billion in connection with climate volatility. Illustrative of such instances are the Midwest drought of 1988, the Midwest floods of 1993, and flooding along the California coast in 1995. In 1992, Hurricane Andrew caused insured losses of some $25 billion of which $18 billion were covered by insurance. In the aftermath of Hurricane Andrew, a significant number of reinsurance companies either stopped offering insurance entirely or limited their offering by ceasing to underwrite catastrophe reinsurance. As reinsurance supply dried up, reinsurance rates increased almost threefold.

In view of insurance inadequacies in covering catastrophic loss, new strategies have been proposed and implemented, e.g. the creation of financial instruments for betting on the frequency of catastrophes which may be weather related or otherwise. Instruments of this type were introduced by the Chicago Board of Trade (CBOT) under the designation of Catastrophe Futures in 1993. Among similar instruments are bonds having a return which is linked to hurricane frequency and severity in the current season and in a specified geographic area. Also, such return has been tied to an insurance company's losses from hurricanes. Typically, the frequency and severity of catastrophic risks is unknown, as is the case for environmental health risks, health risks, nuclear reactor risks, and satellite risks.

Bilateral contracts contingent on weather risks have been traded by investment banks and brokers/dealers, and offered on a case-by-case basis to insurance companies, energy/utility companies and others whose revenues depend on weather conditions.

Summary of the Invention

To avoid idiosyncratic bilateral contracts, each having a different price for possibly similar services or products, contracts can be based on a standardized index or benchmark. An index can quantify a risk factor to businesses and/or individuals, e.g. atmospheric temperature deviation from a nominal temperature in a specific area and over a specific time interval as expressed by heating degree days (HDD) and cooling degree days (CDD). In correspondence with the quantified risk condition, such an index may be termed "Temperature Index", "Weather Index", "Climate Index" or "El Nino Index", for example.

For hedging against the risk, standardized derivative securities or financial contracts can be drawn contingent on the index, including futures and options, and such contracts can be traded on an exchange, for example.

Furthermore, where insurance is used, for efficient allocation of risk bearing, an insurance contract can be combined or "bundled" with one or more derivative securities. The insurance contract pays an agreed amount contingent on the occurrence of an event, .and the derivative securities have a payoff which depends on an index that represents the aggregate frequency of such events, for example.

Brief Description of the Drawing and Appendices Fig. 1 is a graphic representation of two probability distributions of losses due to hurricanes in El Nino and La Nina years, respectively, in the El Nino

Southern Oscillator (ENSO) cycle.

Fig. 2 is a schematic of a technique in accordance with a preferred embodiment of the invention, wherein an insurer executes trades in a financial instrument here designated as ENSO Index or El Nino Index. Fig. 3 is a flow chart for scientific and computerized determination of heating/cooling degree days.

Fig. 4A is a conceptual diagram for a simple weather contract contingent on a weather index, Fig. 4B is a conceptual diagram for a call contract on the contract of

Fig. 4A.

Included herewith are the appended papers by authors Graciela Chichilnisky and Geoffrey Heal entitled "Managing Unknown Risk" and "Financial Markets for Unknown Risks", respectively.

Detailed Description

The following description is addressed primarily to so-called catastrophe bundles, presupposing a novel risk index and a novel contract contingent on the risk index. For example, if the index is a measure for temperature, the contract will be contingent on a temperature value. While insurance is unsatisfactory when the frequency of a risk is unknown, and securities are unsatisfactory when the risks are individual, a combination of insurance and securities can achieve efficient allocation of risk bearing. Such a combination, here called a catastrophe bundle, can guard against a financial debacle due to overexposure of an insurer, while providing nearly full coverage of the insured. The catastrophe bundle requires the novel risk index which provides a standardized or benchmark measure of the risk, and the novel contract which is contingent on the value of the index. Preferably, the index depends on scientific variables, e.g. temperature or precipitation.

Preferably, a catastrophe bundle is customized based on descriptions of the risk. A computerized mathematical formula can be used in customization of the catastrophe bundle, taking into account a plurality of risk patterns having different actuarial tables. Derivative securities are created with payoffs depending on which description of the risk is applicable, and insurance contracts are created to establish compensation depending on which description of the risk is applicable.

As an example, Fig. 1 shows hurricane incidence depending on the so- called El Nino Southern Oscillator (ENSO) cycle. There are two extreme states of the cycle, known as El Nino and La Nina. In El Nino years, hurricane incidence in the southeastern part of the United States is below average; in La Nina years it is above average. Fig. 1 shows the probabilities for three outcomes or levels of losses, namely 5, 10 and 15 billion dollars, for El Nino and La Nina years, respectively. As shown, under El Nino conditions the respective probabilities are 0.1, 0.2 and 0.1. Corresponding probabilities are higher under La Nina conditions, namely 0.2, 0.3 and 0.2.

For hedging of an insurance risk in view of the lack of an actumal function of El Nino versus La Nina conditions, for example, an'ΕNSO Index" or "El Nino Index" can be used whose value is low under El Nino conditions and high under La Nina conditions. The El Nino Index is an example of an index for a physical parameter, contingent on which a contract can be drawn to pay an agreed amount.

Other environmental indices can be based on precipitation or temperature measures, e.g. heating/cooling degree days for a specific geographic region such as a state or a city, and for a specific period of time. A heating degree day (HDD) is defined for days with an average temperature of less than 65 degrees

Fahrenheit, as 65 degrees Fahrenheit minus the daily average temperature. A cooling degree day (CDD) is defined for days with an average temperature of more than 65 degrees Fahrenheit, as the average daily temperature minus 65 degrees Fahrenheit. Fig. 3 illustrates computerization for determining heating degree days HDD and cooling degree days CDD for an n-day period, e.g., with n=31, for the month of January of a specified year.

Similar indices can be established based on different parameters, e.g. precipitation or yet other climate conditions in a geographic region and for a certain time period. Contracts contingent on an index can be time-dependent, e.g. with reference to a year, month or any specified time period. For example, contracts can be drawn on cumulative HDDs/CDDs over a time period. Such a contract is an example also of a security which is conditional on the incidence of an insured peril, i.e., on which risk description is applicable. With the probabilities in accordance with Fig. 1, in an El Nino year the expected value of hurricane damage is calculated in billions as

(0.1 x $5) + (0.2 x $10) + (0.1 x 15) = $4 Correspondingly calculated, in a La Nina year the expected value is $7 billion.

The following is under the assumption of a 40% chance of an El Nino year, a 60% chance of a La Nina year and a total value of insured property of $30 billion. In a worst-case scenario, when hurricane damage is at its maximum of $15 billion, half of the insured value is at risk.

In an El Nino year the expected loss is 13.33% of the insured risk; in a La Nina year it is 23.33%. Thus, the insurance rates on line, i.e. the premiums as a percentage of the insured amount conditional on being in El Nino and La Nina years would have to be at least 13.33% and 23.33%, respectively, for the insurer to break even in terms of expected value.

But the expected losses are different, depending on the type of year, El Nino or La Nina. Without knowledge of the type of year, the expected loss due to El Nino is the expected loss in an El Nino year times the probability of such a year, i.e. 0.4 x $4 = $1.6 billion. For La Nina the corresponding calculation is 0.6 x $7 = $4.2 billion. Hence, without knowledge of the type of year, the expected losses in El Nino and La Nina years are $1.6 and $4.2 billion, respectively, for a total of $5.8 billion.

As to the premiums that would have to be charged for coverage in each type of year without actual knowledge of the type of year, in order to break even on average they would have to be the premiums contingent on being in each year — i.e. 13.33% and 23.33%, respectively — multiplied by the probabilities of each type of year. Thus, without knowledge of the type of year, the rates on line would have to be at least 0.4 13.33% = 5.33% or 0.6 23.33% = 13.99%, respectively.

If an insurer were to follow conventional procedures of charging premiums based on the over-all expected loss, without distinguishing between the two climate patterns, premiums would be charged to yield the over-all expected loss of $5.8 billion, implying a rate on line of 5.5/30 = 19.33%. This is unsatisfactory, amounting to overcharging in El Nino years when expected claims are $4 billion and the rate on line need be only 13.33%, and undercharging in La Nina years when expected claims are $7 billion and rate on line is 23.33%. In the former case, the insurer receives premium income in excess of the expected claims by $1.8 billion; in the latter, premium income falls short by $ 1.2 billion. For proper matching of assets to liabilities, income from El Nino years should be shifted to La Nina years.

Such shifting can be effected by trading shares of a suitably structured security which is contingent on a novel, standardized index, here termed "ENSO Index" or "El Nino Index" whose value can be related to the incidence of hurricanes, (see Fig.1 , for example), and in which traders can take long and short positions. Such trading has the intended effect in that, under the probabilities and the dollar amounts which the insurer has available or will need, the excess $1.8 billion will be available with a 40% probability and the shortfall $1.2 billion will be required with a 60% probability, and 0.4 x $1.8 billion = 0.6 x $1.2 billion. The respective prices of ENSO Index contracts delivering $1 in both El

Nino and La Nina years will be proportional to the probabilities of these events, so that they will be in the ratio of 0.4/0.6 or 2/3. But $1.2/$l .8 = 2/3, so that at such prices the sale of surplus income in El Nino years will exactly finance the purchase of income to cover the deficit in La Nina years. Accordingly, with an ENSO Index, a preferred pattern of financial transactions can be summarized as follows:

1. Issuing insurance contracts which provide coverage against damage in either El Nino or La Nina years.

2. Selling $1.8 billion of contracts contingent on the ENSO Index having a value corresponding to an El Nino year, at a price of $0.40 per dollar.

3. Buying $1.2 billion of contracts contingent on the ENSO Index having a value corresponding to a La Nina year, at $0.60 per dollar.

By such a combination of trades in securities and insurance policies, here termed a catastrophe bundle, an insurer has complete coverage for themselves as well as their clients, even without knowledge of the odds of loss.

The technique is illustrated by Fig. 2 where t represents a count which starts at t=0 in a year in which insurance is issued for subsequent years t = 1, 2, ...

While the expository example presented above involves just two states (El Nino, La Nina) and three outcomes (losses of $5, $10, $15 billion), a more general derivation can be used to demonstrate the efficacy of catastrophe bundles as follows.

Assuming k states numbered i = 1, ..., k, and n outcomes, numbered j = 1, ..., n, for outcome j the loss will be denoted by x_j5 the probability of state i will be denoted by p„ and for state i the probability of outcome j will be denoted by p_y. The probabilities are such that

Pi. P.J ^≥ °; Σ{.} P. ⁼ !; ∑o) Po = ^{l for a11} *

For outcome j the total probability is

The mean loss is

The loss variance is

For each of the states, the mean loss is

and the corresponding loss variance is

When an insurer is hedged, with a security, against indeterminacy of the state, the loss variance is

σ, CB = ∑m P. ^σ.²-

Without such hedging, the loss variance is σ² = ∑_fJ) q_J (x_J - μ)²

= ∑{.ι ∑(j( P. p.j ((^χj - μ.) ⁺ (μ. - μ))

= ∑₍.) P. [∑iji p.j (^χj - μ.)^{2 + +} 2 - ∑_{j, p_1J (x_J - μ₁)(μ₁ - μ) + + ∑ij) Pu (μ. - μ)²]

⁼ Σ₍₁} P₁ σ,² +

+ 2 ^• ∑„_} p, (μ, - μ) [∑_ϋ} p,_j X_j - μ,] +

= °CB² + Σ{,_} p, (μ, - μ)²-

Accordingly, the variance with catastrophe bundle (CB) is less than the variance without catastrophe bundle, with the difference being directly related to the magnitudes of the spacing of the μ, from μ. Therefore, for each expected return, the use of a novel index, novel contract contingent on the index, and novel bundle of insurance and the contract leads to advantageously reduced risk for the expected return.

The following is further with respect to indices, e.g. for weather risk, demand/supply risk, political risk, etc. which are commercially significant in themselves, as are contracts contingent thereon even aside from catastrophe bundles. The use of an index can be sold or licensed by Exchanges such as the New York Stock Exchange, London Stock Exchange and Bermuda Stock Exchange, for example, providing an industry-wide systematic benchmark measure of a specific risk. Contracts which are contingent on such an index can be used for risk hedging or management that protects the revenue of individuals or corporate entities when excessive losses or costs are incurred due to unfavorable conditions, e.g. climate patterns such as El Niflo or La Nina, excessively warm or cold periods or excessively dry or wet periods. Figs. 4A and 4B illustrate such hedging, using a simple contract (Fig. 4A) and a call on the simple contract (Fig. 4B)

Contracts based on indices can be bought/sold jointly with or independently from insurance contracts. They can have one or more "triggers", e.g. HDDs,CDDs, precipitation, time of year or season, length of time, El Nino or La Nina seasons, as well as industry and over-all demand levels for commodities of interest, e.g. electricity, heating oil and natural gas.

Index values can be determined from suitable data, e.g. meteorological, oceanographic, demographic, political or commercial data. Such determinations may involve computational procedures, e.g. accumulating, averaging and smoothing where computerization can be used to advantage to cope with data. Computerization can be used also in trading contracts which are contingent on an index, with buy or sell orders issuing when profitable in view of an actual value of the index as compared with a contract value.

Managing Unknown Risks

The future of global reinsurance.

Graαela Chichdnisky and Geoffrey Heal

I t has been said that insurance is the last cf the financial services to accept radical change (Denr.ey [1995-1996]). Yet there has been a fundamental shift in the geographic location and in the organization of the reinsurar.ee industry in the last six yean (Chi hilnisky [1996b]). Global environmental nsks ate partly responsible for this change; increased weather volatility and catastrophic risks are difficult to diversify using traditional insurance practices

To provide a map to the future, we need a realistic appraisal of how wc got where we are This is, the story of how humans nave hedged ris<cj. There are two basic and distinct approaches: statistical and economic. The former is typical of :nc insurance industry, the latter typifies the securities industry. Both are needed to manage to<iay's catastrophic nslcs. Neither alone will do. We show how a combination of both leads tc effi¬

GRΛCIELΛ CHICHHMSKY a cient outcomes, ir.d s the way to the future

UNESCO prαt^'euoi ct luthesutus (Chichϋmsky [1996a, 1 96b. 96d]). uc econor.iici, and director of the The volatility of weather, taken together with Progn^β'_' °n infoimaάoβ and population movement to warm coastal areas and changReso rce at Cotunibi. Un;versr ing property prices, has made catastrophic risks highly in Ne* York C.0027). unpredictable. Many scientists believe that climate change could be the source. A recent report by the

GEOFFREY HEAL & ώ« Paul Intergovernmental Panel on Climate Change (IPCC), Giπeα ptofnsor of puobc policy charged by governments with investigating global and corporat rcipoimbilir at th wanning, says that huma s have a "discernible" influCraduate School of Bus crs at ence on global climate Columbii Umvenir in New Yotk In May 1996, insurance executives confronted

(toccη the energy inuustry over global warming, and took rhcir

TMΪ KWBJMλl r WRTTOUO MΛ.NΛC0M JNT 35 case :o the United Nations Geneva meeting on climate msrket. Investment banks are now betting heavily on change in June 1996 (Doulton [1996]). Their case was the reinsurance market. Thev are the owners of most of heard, and for the fin: rime the United States took a the businesses created since 1992. leading position in supporting the developing countries' calls for hard targets in the reduction of greenhouse gas REVOLUTION IN GLOBAL FINANCE emissions in the industrial countries. Environmental markets that trade countries' rights tc emit have been Together with the geographic shift, there has proposed and loom large on the hcri2on.^: been a substantial shift in :he industry's strategy The insuiance derivatives tlwt have been recommended for

FINANCIAL RISKS several years aie starting to play a role.

Ir. 1992, we recommended the creation of an

Although the data on climate change aic not instrument to bet on the frequencies of catastrophes, conclusive, the financial challenge is already reai. In the which the Chicago Board of Trade (CDOT) introduced las; few years the propercy/ casualty insurance industry under the name Catastrophe Futures in 1993 (see has experienced record claims of about US$43 billion Chichϋnisk and Heal [1993]). In 199"^?, Morgan connected with climate volatility. In the United States Stanley started marketing a similar instrument: a bond alone, there was the 1 88 Midwest drought, the 1993 issue whose returns are linked to hurricane frequency Midwest floods, and 1 95 flooding along the California and seventy in the current U.S season. Recently, coast. Hurricane Andrew in 1992 produced about Merrill Lynch structured a transaction for USAA. the US$13 billion of insured losses and cotal losses greater country's largest direct marketer of home and car insurthin US$25 billion (Chich nisky [1996a]). ance, offering USS500 million ir. bonds on the U.S

Andrew was the most devastating natural catascapital markets that are tied to the company's losses trophe ever recorded. It also led :o a wave of financial from hurricanes (see Waters [1996]). catastrophe; the hurricane affected almost every insurFinancial innovation in reinsurance markets is ance company in the United States. Not knowing how slowly developing, but the underlying pressure is to hedge unpredictable risks adds the risk of financial relentless. Everyone knows that access to more liquid catastrophe on top of that of the natural catastrophe, a capital markets is essential to che reinsurance industry. one-two punch that could lead to a societal disaster. The derivatives market is the key to liquid and flexible The -year after Andrew, thirty-eight r.on-U.S. and eight tradir.g of weather risk? U.S ; reinsurers, with names as familiar as Continental f and Ne England Re, either withdrew from the UNKNOWN RISKS bufiness or ceased underwriting catastrophe reinsurance [Chichilftisky [1996b)). Unknown risks are risks whose frequencies we

Facing an impossible challenge, many reinsurers d not know, and for which we are aware of our ignoleft the market. Worldwide reinsurance capacity rance ^Chichilnisky ;i996d]) You could think of these dropped more than 30% between 1989 and 1993, and as risks for which we have more than one actuarial it appears that ever 20% of that is due to Andrew. This table, each equally likely. There is more than one prior naturally led to changes in the marketplace, Insurance estimate of the frequency of the event (sec Cass, companies could not buy enough catastrophe reinsurChichilnisky, and Wu [1996]). ance, no matter how hard they tried. As supply dried Examples of unknown risks are environmental up, prices of course increased dramatically, the rate on health πsks of new and little known epidemics, or risks line went from 6.2% in 1989 to 21.4% in 1994, induced by scientific uncertainty in predicting the fre¬

Higher prices then attracted new capital. This quency and seventy of catastrophic events such as led to a major geographic shift of the industry nuclear reactor and satellite πslcs. These risks are drivContinuing doubts about the future existence of ing major changes in the insurance a reinsurance Lloyd's of London led to a drop in the U.K. market industry today (sec Chichilnisky and Heal [1998]} share, from about 56% in 1989 to 23% in 1995 Since Take a simple example One reliable source g_^ivcs 1993 Bermuda's reinsurance industry evolved from a 2% annuai chance of the occurrence of a hurricane of practically z ro :o its current position of 25% of the a certain type, and anothci a ' 2% chance- Monte Carlo

36 simulations and other procedures can be used to must be known. Loss of life and car accidents are typical attempt to tease from all models a unique statistical examples. Here the law of large numbers operates. approximation to the true frequency. But what if there There is safety in numbers; with a large enough is no true frequency? population, the number of those likely to be affected is

How conic this be: Easily. There may be two kiiown with considerable accuracy. The sample mean is possible climate patterns, both equally likely. This is highly predictable if the distribution for each person or typical of complex and chaotic systems such as the cligroup is known. This is the standard principle on mate (see Chtchihiisky [1 8 ; which insurance operates.

Many climate experts view climate as a fundaReinsurance is simply a way to augment the mentally non-linear phenomenon in which chaotic patpool of those affected so that the law of large numbers terns emerge easily. Such systems can have two "attrac- operates better. All that is needed is a reliable actuarial tors," or two disr.net overall patterns of behavior, each table describing the incidence per person or group, and significantly likely. Each of these ttractors describes a a large pool of msursds to distribute the πsk (sec weather pattern, a reasonable statistical inference of the ChichJmsky and Heal [1993]). frequencies of a major event. In such a chaotic system, If the numbers are no: large enough, it is stanit is scientifically impossible to predict from the initial dard to spread risk through time. The number of peoconditions which of the two patterns the climate will ple affected by a hurricane over a ten-year period is at take: a pattern with two hurricanes a year, or the other least ten times that affected n one year. This requires with a dozen. Because we cannot predict, we face a risk. that the risks be independent through time, ekminaαne ^"We cdl it a chaotic risk because it emerges from the irreversible risks such :s once-and-for-all shifts arising chaotic nature of the climate system. from global warming

The first statistical reaction is to construct a r.ew Hurricanes such as Andrew (1992) and Opal actuarial table by taking an average; assuming the two (1995), however, defy the law of large numbers. They sates, 2% and 12%, are equally likely, this is 7%. But affect large areas all at once, both in physical and in finantaking an average does not help It only ensures that cial terms, and their frequency and severity seem to be one is wrong 1 0% of the time: 50% of the time we changing. The actuarial table itself has become the r»k. are ovcπnsured (the pattern with two hurricanes per Insurance docs not work. W at are the alternatives? year), ind the other 50% we arc uhceπ sured (the pat¬

Derivatives: The Economic Approach tern ^'with a dozen a year). Both have major financial costf . If each hurricane leads to USS2 billion in losses, An alternaαve is the economic approach. This :.ιf averaging method leads to a USS10 billion shortworks best for correlated πsks, in which the same event fall 50% of the time and US$10 billion oveπnsurance occurs for macy people all at once. A chop in the value the other 50% of the time. Hardly a measured way to of the dollar is an example; the event is :he same for manage risks everyone in the U.S. economy. There is no way to pool

Is there a soluαon to this problem? The good news this risk, although, as we all know, wc can hedge r*. by is that there is. It is possible to hedge such unknown risiss using derivatives (currency futures or options). The juccsssfiilly and efficiently. To do so, however, one needs principle used here is negative correlation. One hedges a careful and customized approach that blends both insurby taking a position that is highly correlated with the ance and securities approaches to hedging risks. risk, except with the opposite sign.

For example, an investor with a dollar-based

TWO WAYS TO HEDGE RISK portfolio who fears a drcp in the value of the dollar can buy a futures contract in yen, or a dollar put. If the dollar drops in value, the investor is covered by the increase

Insurance: The Statistical Approach in the value of the denvauve. Bear funds have been

The stettut a! approach to hedging risks, which constructed on this principle. relics on the law of large numbers, is the traditional The economic procedure s radically different foundation of the insurar.ee industry. from the insurance approach in that it does not require a

For this to work, risks must be reasonably indelarge number of people. Ncr does it icquire knowing the pendent across individuals or groups, and the frequencies frequency of the event or the actuarial table. This f. da-

TUt l 'J NAL OI> POUTFOIIC MA ΛGΪM'NT g mentally different method is the way the securities indusHOW DO CATASTROPHE try operates. Irwte. d of yooiirtg risks, one trades risks BUNDLES WORK?

Securities markets are, however, notoriously complex. For example, the procedure of ending risks Catastrophe bundles work best in the hands of just outlined makes no sense for individual risks, such an experienced reinsurer or broker who can customize as death. How would we describe the death of one sinthe instrument to the client^'s needs. In a way, the reingle person within a large economy as one event on surer is selling a package that consists of insurance, a which all of us can trade? To do so would require an security, and a r sk management/consulting tool. unrealistically tøgh number of securities, indeed 2^X, The broker must first identify with the client the where x is the number of people in the economy. In a set of possible descnpnons of the risk. This crucial part world with five billion people, the number of securiof the process involves new techniques of risk manageties could exceed the number of all known particles in ment. It is best handled on a facc-to-facc and customized the universe (see Chichilnnky and Heal [1998]). basis. A mathematical formula is then brought to bear in

Insurance, instead, deals with such risks expe- c stomizing catastrophe bundles to customer needs This diriously If all individuals are in a similar risk class, formula works very well when there is more than one one insurance contnct would suffice. The contrast is pattern of risk and therefore more than one "possible^"' stark, but it makes a point. In a world of unknown actuarial cable, each table being substantially likely. rislu, neither securities nor insurance methods work After this is achieved, derivative securities whose in isolation. payoffs depend on which description of the πsk is 'correct are introduced. These securities serve to hedge

THE IDEAL HEDGE: uncertainty about actuarial tables. Finally, one strucCATASTROPHE BUNDLES tures insurance contracts that establish a compensation arrangement in a way thac depends on which descrip¬

We see that insurance does not work when the tion of the risk is correct. frequency of a risk is unknown, and secur.ϋcs do not Catastrophe bundles are proprietary, and their use work when the risks are individual. If neither of these in a particularly simple case is illustrated in Exhibit \. two approaches works on its own, what does work?

The ideal hedge is a combination of insurance PRICING AND OPTIMAL PORTFOLIOS and ^'.securities; chis can achieve efficient allocation of risJjrbcaπng. We call this a catastrophe bundle because it Fund managers can look at the flip side of this bundles together two types of instruments. It consists of picture and seek a cotnb arion of insurance and securii3t insurance instrument with a novel derivative secun- ties that offer an optimal portfolio in insurance and ty for betting on the frequency itself (see Chichilnisky investment markets. A part of tins instrument is what and Heal [1993]). Merrill Lynch and .Morgan Stanley have floated recently.

The latter type of secunty has emerged and is Secunαzmg such instruments a, of course, the next step. now traded on the CBOT. As we have mentioned, Through the use of catastrophe bundles, the related securities have recently emerged also in the form reinsurance broker can access a large pool of n i ged of bonds floated by Morgan Stanley and ernll Lynch. funds while offering its clients a customized reinsurance

The combination of both instruments ensures service that manages risks optimally, and at very com- that no financial catastrophe will occur, since the reinpetinve pnces. surer is not exposed to more risks than it can afford. At Pricing, of course, is a crucial issue. What is the same time, this approach can be used to provide needed here is to separate two parts of the risks and to nearly full coverage for the insured at a tninimal cost. push each as far as it will go. The contingent insusancc

We show elsewhere that such instruments lead to part cf the .nstru ent should be applied as far as possian efficient allocation of risk-bearing (see Chichilnisky ble, covering the independent part of the risk for whicn and Heal [1993, 1 9S] and Cass. Chichilnisky, and Wu it is optimally suited. Securities are then used for the [1996]). They require a carefully customized approach purpose for which thev arc best: the correlated part of to hedging risk. This ivei the traditional face-to-face the risk. A mathematical formula used to construct the insurance approach an edge over raw technology. catastrophe bundle separates and prices both parts. gς MANA'JINV UNKNOWN H8K3 SU MIR. ItU EXHIBIT 1

CATASTROPHE BUNDLE EXAIVIPLE

Hurπcane worse than $3 billion ones in 15-year or once in 5:

Insurance covers individual property πsks

SeΛinties cover frequency risk

CONVERGENCE OF INSURANCE A tncablc ENSO ir.dcx s a contract that pays an AND SECURITIES MARKETS agreed amount conungent on the value of a physical index. It is similar in concept to the catastrophe futures

It is r.o secret that the securities industry is maktraded on the CBOT, and is an example of a security ing inroads into the reinsurance business. By itself, conditional on the incidence of the insured peril, that hcwever, it canno: succeed, because the individual parts is, on which risk description is correct. of the risks cannot be handled efficiently by securities There are two extreme states of the ENSO markets: they are too cumbersome for individual risks. cycle, known as El Nino and I-a Nina. In El Nino Insurance, based en the law of large numbers, has an years hurricane incidence in the southeastern U.S. is important place in simplifying financial transactions and below average; in La Nϋia years, it is above. hedging known individual risks. . Exhibit 1 shows possible probability distribu¬

, , 'Catastrophe bundles offer one approach :o com- tions of damage due to hurricanes conditional on El puurig the limits of each instrument, and blending Nino or La Nina years. thpm optimally to achieve the most competitive pricing As an example, assume that, an El Nino year, of^ catastrophe reinsurance portfolio there is a 10% chance of a $5 billion loss, a 20% chance

The future of the industry is in the hands o: of a 510 billion loss, and i 10% chance of a $15 billion those who acliievc the optimum balance, through inteloss. The expected value of the damage is therefore (0 1 grating derivative securities with conungent insurance x S5) + (0.2 x 510)+(0 1 x $15) = $4 billion. In a La contracts, and integrating technology with customized Nina year, the probabilities are 2C%, 30%, and 20%, facϊ-to-face know-how. respectively, giving an expected loss of $7 billion Assume that there is a 40% chance of ar. El Nino year,

HURRICANE RISKS AND ΕL NINO: and a 60% chance of a La N;ήa year.³ The total value of AN EXAMPLE insured property is taken as 530 billion, so that a worst case scenario — when the hurricane damage is at

How exactly would catastrophe bundles work? its maximum of 515 billion — half of this value is at risk. We answer that question with a simple but typical In an El Nino year, the expected loss is 13.33% example, drawn from hurricane insurance. Hurricane of the insured risks, and in a La Nina year, it is 23 33%. incidence is conditioned by the ENSO cycle, so we It follows that the rates on line (i.e., premiums as a perconsider, instead of hurricane bonds of the type that centage of the insured amount) conditional on being in have receπdy been issued, a tradablc ENSO index.² This El Ni o and La Nina years would need to be at least incex would achieve everything one needs from hurri13.33% and 23.33%, respectively, to break even in cane bonds, but in a more general and simple fashion. expected value terms.

-m jσWAi or !«::ι<.τFθt:o MANAC£MC T φ EXHIBIT 2 In the former case, the insur¬

HURRICANE PROBABILITIES AND THE ENSO SYSTEM ers are charging premiums in excess of expected losses by $1.8 billion, hardly a competitive strategy, and in the latter case, premium income falls short of expected cl.ums by $1 2 billion, clearly a dangerous and unsustainaclc position. Neither case is satisfa ory. To match assets to liabilities properly, insurers need to shift income from El Niήo to La Nina years

This is where securities conditional on incidence, on description of the risk, come into the pic¬

As we have already noted, expected losses arc ture. They can be used to transfer different, depending on what type of year we arc in. income between El Nuio and La Nina years so that the Before we know what kind of year will occur, we surplus in the former cover the deficit in the latter. We therefore have an expected loss due to El Niήo equal to need a security whose value depends on the incidence the expected loss m an El Nino year times the probaofhurncar.es, for the purposes cf this example, we take bility of such a year, ., (0.4 x $4) » $1.6 billion. For this to be a tradable ENSO mdex. This would be a La Nina, the equivalent calculation is (0.6 X $7) = $4.2 contract whose value depends on the value of the billion. Hence, ex ante, before we knew winch year we ENSO index and m which traders can take long or are or will be in, the expected losses in El Niήo and La short positions. By trading tius security, the insurer in Nina years are, respectively, SI -6 billion and $4.2 bilour example can m effect trade income in El Niϋo lion, giving a total of $5.8 billion as the annual expectyears for income in La Nina yean. ed loss altogether. The odds work out r.iccly. The insurer wants to

We can now compute the premiums mat would sell $1 8 billion in in El Nine year, its surplus of prehave?^' to be charged for cover in each type of year mium income over expected claims, which occurs with be&re the type of year is known, in order to break a 40% chance. Correspondingly, it needs to buy $1 2 e fen on average. These would have to be the premibillion of income in La Nina years, to cover the shortums contingent on being in each year — seen above to fall between premium income and expected claims. In be 13,33% and 23.33% for El Niήo and La Nina — our example, this happens 60% of the time. multiplied by the probabilities of each cvpe of year. The prices for ENSO index contracts delivering Thus the ex ante rates on hne (before it is known $1 in H Niήo and La Nina yean will be proportional whether we are in an El Niήo or a La Nina year) have to the probabilities of these events, and so w ll be in the to be at least (0.4 x 13.33%) = 5 33% or (0.6 κ ratio of 0.4/O.6 or 2 But $1 2 bill:on/$l .S billion = 23.33?*} = 13 99%, .respectively. 2/3, so that at such prices the sale of surplus income in

If insurers follow the obvious and traditional El Niήo years will exactly finance the purchase of procedure of charging premiums based on the overall uicome to cover the deficit in La Nina years. expected loss and not distinguishing between the two Overall, then, we have a pattern of transactions climate patterns, they will charge premiums that wjll a: follows- bring in their overall ex ante expected loss of $5.8 billion, implying a rate on line of 5.8/30 = 19.33%. This ! Issuing insurance contracts which provide cover is unsatisfactory because n El Niήo years they are overagainst damage in either El Nmo or La Nina years. charging (expected claims are $4 billion; the rate en 2 Selling $1.8 billion of contracts contingent on the line need be only 13.33%); La Nina years, t ey are ENSO index havmg a value corresponding to an El undercharging (expected claims are $7 billion; a rate en Niήo year, at a price of $0 40 per dollar. hne of 23 33% is needed). 3. Buying $^*. 2 bilhcn of contracts contingent on the

90 MΛNΛGINC UNKNOWN RftKS SJMWIR !^•»» ENSO index having a value corresponding to a La ^Tha 8 a sraplitt ancn. There xtt also years that wc nσ- Nina year, at $0.60 per doilar. 'he:, jo-called ncutrai years. The numbers we tsc in this example ire purc.y illustrative.

This specific combination of trades in securities and insurance policies described in these steps is what REFERENCES we refer to as "catastrophe bundles." Through trading

3oulton. L "Debate Warms Up " FMΠOOI Tma. .May 29. 1990, catastrophe bundles, insurers can arrange complete

? 10 cover for themselves and their clients at minimum cost, in spite of not knowing wbat the odds of loss will be. Cass, David, Gπciela Ch chιlrθ y, and Ho-.vlou Wi "individual They achieve this by a specific tailor-made combination Risks and Mutual Insurance " £cenomctr.ιa, Vol. 64. No. 2 i March of insurance contracts and sccuπαes. All these contracts 1996). pp. 332-341 are conditional on the incidence of the insured risk. Chichilnisky, Graciela "Catasirophe Bundles Can Deal with

How different is this approach from the practice Unknown Risks." Btst's Roncw, February 1996a, pp. 1-3. today? The securities issued today securitize insurance or reinsurance πsks, and therefore bring more liquidity "Financial Innovation :n Propetty Catwtrophe emiuiaiiee to the reinsurance market. This is an improvement. But The Convergence of Insurance and Capital .Markets." Risk these securities still leave open the possibility that the Financing Newsle'.tσ, Vol 13, No 2 (June 1996b). pp. 1-7. insurer is either offeriu non-compcuove rates or tak"The Greening of Rrctton Woods " Ftrunaal Times, Jiiuary ing on a dangerous exposure. Today's securities do no: 2, 1996c. tackle the essence of the problem.

The key to catastrophe bundles is to recognize "MaKcts with Endogenous Uncertainty: Theory and that when there arc several possible actuarial tables, all Policy." Theory and Deα^'jion, Vol. 41 (199<Sd), pp. 99-13'. reasonably likely, we have to supplement insurance . "A Radical Shift in Managing Risks: Practical Applications of introducing and trading securities dependent on them. Complexity Theory ^*' C«ι*t»y«n<ιes, American Academy of A specific combination of insurance and securities, and Actuaries, Janair.'-Fcbroary 19*, pp. 28-32. an equally specific pricing policy, are required for an optimal allocation of πsks on compcr.nve terms. Cbichxinisky. Gracich, and Geoffrey M. Heal. "Financial Markets for Unknown Risks." In Ci Chichilnisky, G.M. Heal, and A. Verccli. eds., Sm u ibi ly, Dvtumus tni Urxαtαmtγ Amsterdam:

ENDNOTES Kluwer Academic Publishers. 199H, pp. 277-294.

'Chichilr.isky |1996c] advances a propc&l for a global . "Clobal Environmental Risks." utnal of Ortnomu Market on greenhouse cat -mmi r and an International Sank for Pαspαtvα, Fall 1993, pp. 65-86 Environmental Settlements to handle executions, cleaang. and s*C- demetia ai well as regulate borrowing and lending rates. Denney, Valerie. "Editor's Note " C.'ebtl Rrnsutat t. Dccemter

ΕNSC sunds fee the £1 Nino-Southern Oscillator, the 1995-Febrearv 1*96, p. 14 name given co ie w.at.icr pattern that originates in the equatorial Pacific nd influences rainfall and storui inadcr.ee from Australia to Wa^ets, R. "Investor Get Chance :o Gamble or. Weather: U S southern Africa. An indicator of the state v( the ENSO cycle is a s a Iπsuraπte Group Links Bonds 13 Huiπcans Lusses " Pinαt ttil TTKIC surface temperature (SST) index for the equatorial Pacific. J y 3C. 1996

SUMMW \VΛ THl .UVKNA Of P B.TFCUO MΛNAGΪMtNT O} GRACIELA CHICHILNISKY AND GEOFFREY HEAL

3.5. Financial Markets for Unknown Riste

1. Introduction

New risks seem to be an unavoidable in a period of rapid change. The last few decades have brought us the risks of global wanning, nuclear meltdown, ozone depletion, failure of satellite launcher rockets, collision of supertankers, AIDS and Ebola.¹ A key feature of a new risk, as opposed to an old and familiar one, is that one knows little about it In particular, one knows little about the chances or the costs of its occurrence. This makes it hard to manage these risks: existing paradigms for the rational management of risks require that we associate probabilities to various levels of losses. This poses particular challenges for the insurance industry, which is at the leading edge of risk management. Misestimarion of new risks has lead to several bankruptcies in the insurance and reinsurance businesses.² In this paper we propose a novel framework for providing insurance cover against risks whose parameters are unknown. In fact many of the risks at issue may be not just unknown but also unknowable: it is difficult to imagine repetition of the events leading to global warming or ozone depletion, and, therefore, difficult to devise a relative frequency associated with repeated experiments.

A systematic and rational way of hedging unknown risks is proposed here, one which involves the use of securities markets as well as the more traditional insurance techniques. This model is quite consistent with the current evolution of the insurance and reinsurance industries, which are beginning to explore the securitization of some aspects of insurance contracts via Act of God-bonds, contingent drawing facilities, catastrophe futures and similar innovations. In fact, our model provides a formal framework within which such moves can be evaluated. An earlier version of this . framework was presented in [6]; Chichilnisky [3] gives a more industry-oriented analysis.

This merging of insurance and securities market is not surprising: traditionally economists have recognized two ways of managing risks. One is risk

^* We are graceful to Peter Bernstein, David Cass and Frank Hahn for valuable comments on an earlier version of this paper.

277

C. Chichilnisky et aL (eds.), Sus unabiliiy: Dynamics and Uncertainty, 277-294. 278 G. Chichilnisky and G. Heal pooling, or insurance, invoking the law of large numbers for independent and identically distributed (IID) events to ensure that the insurer's loss rate is proportional to the population loss rate. This will not work if the population loss rate is unknown. The second approach is the use of securities markets, and of negatively correlated events. This does not require knowledge of the population loss rate, and so can be applied to risks which are unknown or not independent. In fact, securities markets alone could provide a mechanism for hedging unknown risks by the appropriate definition of states, but as we shall see below this approach requires an unreasonable proliferation of markets. Using a mix of the two approaches can economize greatly on the number of markets needed and on the complexity of the institutional framework. In the process of showing this, we also show that under certain conditions the market equilibrium is anonymous in the sense that it depends only on the distribution of individuals across possible states, and not on who is in which state.

The reason for using two types of instrument is simple. Agents face two types of uncertainty: uncertainty about the overall incidence of a peril, i.e., how many people overall will be affected by a disease, and then given an overall distribution of the peril, they face uncertainty about whether they will be one of those who are affected. Securities contingent on the distribution of the peril hedge the former type of uncertainty: contingent insurance contracts hedge the latter.

Our analysis implies that insurance companies should issue insurance contracts which depend on the frequency of the peril, which we call a statistical state. The insurance companies should offer individuals an array of insurance contracts, one valid in each possible statistical state. Insurance contracts are, therefore, contingent on statistical states. Within each statistical state, of course, probabilities are known. Therefore, companies are writing insurance only on known risks, something which is actuarially manageable. Individuals then buy the insurance that they want between different statistical states via the markets for securities that are contingent on statistical states. The following is an illustration for purchasing insurance against AIDS, if the actuarial nsks of the disease are unknown. One would buy insurance against AIDS by (1) purchasing a set of AIDS insurance contracts each of which pays off only for a specified incidence of AIDS in the population as a whole, and (2) making bets via statistical securities on the incidence of AIDS in the population. Likewise, one would obtain cover against an effect of climate change by (1) buying insurance policies specific to the risks faced at particular levels of climate change, and (2) making bets on the level of climate change, again using statistical securities. The opportunity to place such bets is currently provided in a limited way by catastrophe futures markets which pay an amount depending on the incidence of hurricane damage.

The present paper draws on recent findings of Chichilnisky and Wu [5] and Cass et al. [4], both of which study resource allocation with individual risks. Catastrophe Futures 279

Both of these papers develop further Malinvaud's [15, 16] original formulation of general equilibrium with individual risks, and Arrow's [1] formulation of the role of securities in the optimal allocation of risk-bearing. Our results are valid for large but finite economies with agents who face unknown risks and who have diverse opinions about these risks: in contrast, Malinvaud's results are asymptotic, valid for a limiting economy with an infinite population, and deal only with a known distribution of risks. Our results use the formulation of incomplete asset markets for individual risks used to study default in [5, section 5. c]. The risks considered here are iinknown and possibly unknowable, and each individual has potentially a different opinion about these risks, while Chichilnisky and Wu [5] and Cass et al. [4] assume that all risk is known.

2. Notation and Definitions

Denote the set of possible states for an individual by 5, indexed by s = 1 , 2. . . . , 5. Let there be H individuals, indexed by h = 1, 2, . . . . H. All households have the same state-dependent endowments: endowments depend solely on the household's individual state s, and this dependence is the same for all households. The probability of any agent being in any state is unknown, and the distribution of states over the population as a whole is also unknown. A complete description of the state of the economy, called a social state, is a list of the states of each agent There are S^H possible social states. A social state is denoted σ : it is an -vector. The set of possible social states is denoted Ω and has S^ff elements. A statistical description of the economy, called a statistical state, is a statement of the fraction of the population in each

J statistical states. Clearly many social states map into a given statistical state. For example, if in one social state you are well and I am sick and in another, I am well and you are sick, then these two social states give rise to the same statistical state. Intuitively, we would not expect the equilibrium prices of the economy to differ in these two social states. One of our results shows that under certain conditions, the characteristics of the equilibrium are in fact dependent only on the statistical state.

How does the distinction between social and statistical states contribute to risk management? Using the traditional approach, we could in principle trade securities contingent on each of the S^H social states. Clearly this would require a large number of markets, a number which grows rapidly with the number of agents. The institutional requirements can be greatly simplified. When the characteristics of the equilibrium depend only on the statistical state, one can trade securities which are contingent on statistical states, i.e., contingent on the distribution of individual states within the population, and still attain efficient allocations. We will trade securities contingent on whether 280 G. Chichilnisky and G. Heal

4 or 8% of the population are in state 5, but not on which people are in this state. Such securities, which we will call statistical securities, plus mutual insurance contracts also contingent on the statistical state, lead (under the appropriate conditions) to an efficient allocation of risks. A mutual insurance contract contingent on a statistical state pays an individual a certain amount in a given individual state if and only if the economy as a whole is in a given statistical state.

Let Z_jte denote the quantity of good j consumed by household h in social state a : Z_hσ is an N-dimensional vector of all goods consumed by h in social state σ, z_hσ = z_itiσ, j = 1 , . . . , N and z_h is an NS^H -dimensional vector of all goods consumed in all social states by h, z^ = z^, σ ζ Ω.³

Let s{h, σ) be. the state of individual h in the social state σ, and r_t(σ) be the proportion of all households for whom s(h, σ) = s. Let r(σ) — τ_\ (σ), . . . , rs{σ) be the distribution of households among individual states within the social state σ, i.e., the proportion of all individuals in state s for each 5. r(σ) is a statistical state. Let R be the set statistical states, i.e., of vectors r(σ) when σ runs over Ω. R is contained in S¹ , the product of 5-dimensional simplices, and has f j elements. n^Λ is household h's probability distribution over the set of social states Ω, and π denotes the probability of state σ. Although we take social states as the primitive concept, we in fact work largely with statistical states. We, therefore, relate preferences, beliefs and endowments to statistical states. This is done in the next section: clearly any distribution over social states implies a distribution over statistical states.

The following anonymity assumption is required: r(σ) - r(<7') → π* = πj, .

This means that two overall distributions σ and σ' which have the same statistical characteristics are equally likely. Then π defines a probability distribution Ilj? on the space of statistical states R. 1 can be interpreted, as remarked above, as /ι's distribution over possible distributions of impacts in the population as a whole. The probability that a statistical state r obtains and that simultaneously, for a gjyen household h a particular state 5 also obtains, πj_r, is⁴ πj_r = π_r ^hr₅ with ∑ Π* Π*. (i) s

Tne probability n that, for a given h, a particular individual state s obtains is, therefore, given by

where r_s is the proportion of people in individual state .s in statistical state r. Note that we denote by II . the conditional probability of household h being Catastrophe Futures 281 in individual sta_te s, conditional on the economy being in statistical state r. Clearly ∑_s πj_|r = 1. Anonymity implies that π},_P = r„ i.e., that the probability of anyone being in individual state s contingent on the' economy being in statistical state r is the relative frequency of state s contingent on statistical state r.

3. The Behavior of Households

Let e be the endowment of household h when the individual state is 5. We assume that household h always has the same endowment in the individual state s, whatever the social state. We also assume that all households have the same endowment if they are in the same individual state: endowments differ, therefore, only because of differences in individual states. This describes the πsks faced by individuals.

Individuals have von Neumann-Morgenstem utilities:

W^h{z_h) = ∑ U^h _σ U^h{z_hσ). σ

This definition indicates that household h has preferences on consumption which may be represented by a "state separated" utility function W*¹ defined from elementary state -dependent utility functions.

We assume like Malinvaud [15] that preferences are separable over statistical states. This means that the utility of household h depends on σ only through the statistical state r(σ). If we assume further ttiat in state σ household h takes into account only its individual consumption, and what overall frequency distribution r(σ) appears, and nothing else, then its consumption plan can be expressed as z = Z_hST its consumption depends only on its individual state s and the statistical state r. Summation with respect to social states σ in the expected utility function can now be made first within each statistical state. Hence we can express individuals' utility functions as:

r,s which expresses the utility of a household in terms of its consumption at individual state s within a statistical state r, summed over statistical states. This expression is important in the following results, because it allows us to represent the utility of consumption across social states σ as a function of statistical states r and individual states s only. The functions U are assumed to be C², strictly increasing, strictly quasiconcave, and the closure of the indifference surfaces {*/ }^{" l}(x) C int(Λ"⁺) for all x 6 R^÷. The probabilities Il are in principle different over households. 282 G. Chichilnisky and G. Heal 4. Efficient Allocations

Letp" be a competitive equilibrium price vector of the Arrow-Debreu economy E with markets contingent on all social states⁵ and let z^* be the associated allocation. We will as usual say that z^* is Pareto efficient if it is impossible to find an alternative feasible allocation which is preferred by at least one agent and to which no agent prefers z'. Letp^* and z_σ ^* be the components ofp^* and z" , respectively, which refer to goods contingent on state σ.

We now define an Arrow-Debreu economy E, where markets exists contingent on an exhaustive description of all states in the economy, i.e. for all social states σ 6 Ω. We, therefore, have NS^H contingent markets. An Arrow-Debreu equilibrium is a price vector p^* = (p_σ) ,p € R^N+, σ € Ω, and an allocation z" consisting of vectors z_h' = (z^_σ) , z_h ^* _σ € R^N+, σ 6 Ω. h = I , . . . , H such that for all Λ, z_h' maximizes

subject to a budget constraint

P tøϊ - e x) = 0 (4) and all markets clear:

∑ [z_h' - e_h) = 0. (5)

Proposition 1 considers the case when households agree on the probability distribution over social states, this common probability being denoted by II. It follows that they agree on the distribution over statistical states. It shows that in this case, the competitive equilibrium prices p' and allocations z^* are the same across all social states σ leading to the same statistical state r.⁶

PROPOSITION 1. When agents have common probabilities, i.e., U^h = IF VΛ, ^', then equilibrium prices depend only on statistical states. Consider an Arrow-Debreu equilibrium of the economy E, p^* = (p^* ), z" = (z*), σ € Ω. For every state σ leading to a given statistical state r, i. e., such that r (σ) « r, equilibrium prices and consumption allocations are the same, i. e., there exists a price vector p^* and an allocation z^* such thatVσ : r(σ) = r, p_σ" = p". and z_σ = z_r ^*, where p_r' 6 R^N+ and z' € R^Nl depend solely on r. Proof. In the Appendix.

DEFINITION. An economy E_. is regular if at all equilibrium prices in E the Jacobian matrix of first partial derivatives of its excess demand function has full rank [I I]. Regularity is a generic property [10, 11].

We now consider the general case, which^' allows fόrTI^Λ ≠ IP if Λ ≠ j. Proposition 1 no longer holds: the reason is that households may not achieve Catastrophe Futures 283 full insurance at an equilibrium. However, Proposition 2 states that if the economy is regular, if all households have the same preferences and if there are two individual states, there is always one equilibrium at which prices are the same at all social states leading to the same statistical state. This confirms the intuition that the characteristics of an equilibrium should not be changed by a permutation of individuals: if I am changed to your state, and you to mine, everyone else remaining constant, then provided you and I have the same preferences, the equilibrium will not change.

PROPOSITION 2. An Arrow-Debreu equilibrium allocation of the economy E i-p' . z") is not fully insured if ϊl^h ≠ U^k for some households h, k with U^h ≠ U^k in (2). In particular, household h has a different equilibrium

5. Equilibrium in Incomplete Markets for Unknown Risks

Consider first the case where there are no assets to hedge against risk, so that the economy has incomplete asset markets. Individuals cannot transfer income to the unfavorable states. Examples are cases when individuals are not able to purchase hurricane insurance, as in some parts of the south eastern United States and in the Caribbean. Market allocations are typically inefficient in this case, since individuals cannot transfer income from one state to another to equalize welfare across states. Which households will be in each individual state is unknown. Each individual has a certain probability distribution over all possible social states σ, II^Λ.Tn each social state σ each individual is constrained in the value of her/his expenditures by her his endowment (which depends on the individual state s (h, σ) in that social state). In this context a general equilibrium of the economy with incomplete markets_. Ej consists of a price vector p" with NS¹* components and H consumption plans z_h" with NS^H components each, such that z_h ⁹ maximizes W^h (z_/J:

W^h (z_k) = ∑ ϊi^h _σU^h (z_hσ) (6) σ subject to

Vσ [t_he ~ e/ur) = 0 for each σ 6 Ω (7) and

H

∑ (zh - e_κ) = 0. (8) h= \ 284" G. Chichilnisky and G. Heal

The above economy E is an extreme version of an economy with incomplete asset markets (see, e.g., [13]) because there are no markets to hedge against risks: there are S^H budget constraints in (7).

6. Efficient Allocations, Mutual Insurance and Securities

In this section we study the possibility of supporting Arrow-Debreu equilibria by combinations of statistical securities and insurance contracts, rather than by using state contingent contracts. As already observed, this leads to a very significant economy in the number of markets needed. In an economy with no asset markets at all, such as £_/, the difficulty in supporting an Arrow-Debreu equilibrium arises because income cannot be transferred between states. On the basis of Propositions 1 and 2, we show that households can use securities defined on statistical states to transfer into each such state an amount of income equal to the expected difference between the value of Arrow-Debreu equilibrium consumption and the value of endowments in that state. The expectation here is over individual states conditional on being in a given statistical state. The difference between the actual consumption-income gap given a particular individual state and its expected value is then covered

(H ' S 1

THEOREM I . Assume that all households in E have the same probability π over the distribution of risks in the population. Then any Arrow-Debreu equilibrium allocation (p', z*) ofE (and, therefore, anyPareto Optimum) can be achieved within the general equilibrium economy with incomplete markets Eι by introducing a total of LA mutual insurance contracts to hedge against individual risk, and A- statistical securities to hedge against social risk. In a regular economy with two individual states and identical preferences, even if agents have different probabilities, there is always an Arrow-Debreu equilibrium (p' . z') in E which is achievable within the incomplete economy Ei with the introduction of LA mutual insurance contracts and A statistical securities.

Proof In the Appendix.

6.1. Market Complexity

We can now formalize a statement made before about the efficiency of the institutional structure proposed in Theorem 1 by comparison with the standard Arrow-Debreu structure of a complete set of state-contingent markets. We use here complexity theory, and in particular the concept of NP-completeness. The key consideration in this approach to studying problem complexity is how fast the number of operations required to solve a problem increases with the size of the problem. Catastrophe Futures 285

DEFINITION. If the number of operations required to solve a problem must increase exponentially for any possible way of solving the problem, then the problem is called "intractable " or more formally, NP-complete. If this number increases polynomially, the problem is tractable. Further definitions .are in [12].

The motivation for this distinction is of course that if the number of operations needed to solve the problem increases exponentially with some measure of the size of the problem, then there will be examples of the problem that no computer can or ever could solve. Hence there is no possibility of ever designing a general efficient algorithm for solving these problems. However, if the number of operations rises only polynomially then it is in principle possible to devise a general and efficient algorithm for the problem.

Theorem 2 investigates the complexity of the resource allocation problem in the Arrow-Debreu framework and compares this with the framework of Theorem 1. We focus on how the problem changes as the economy grows in the sense that the number of households increases, and consider a very simple aspect of the allocation problem, which is as follows. Suppose that the excess demand of the economy Z (p) is known. A particular price vector p^* is proposed as a market clearing price. We wish to check whether.or not it is a market clearing price. This involves computing each of the coordinates of Z (p) and then comparing with zero. This involves a number of operations proportional to the number of components of Z (p); we, therefore, take the rate at which the dimension of Z (p) increases with the number of agents to be a measure of the complexity of the resource allocation problem. In summary: we ask how the difficulty of verifying market clearing increases as the number of households in the economy rises. We show that in the Arrow-Debreu framework this difficulty rises exponentially, whereas in the framework of Theorem 1 it rises only polynomially.

THEOREM 2. Verifying market clearing is an intractable problem in an Arrow-Debreu economy, i.e.. the number of operations required to check if a proposed price is market clearing increases exponentially with the number of households H. However, under the assumptions of Theorem 1, in the economy Ei supplemented by LA mutual insurance contracts and A statistical securities, verifying market clearing is a tractable problem, i.e., the number of operations needed to check for market clearing increases only polynomially with the number of households.

Proof. The number of operations required to check that a price is market clearing is proportional to the number of market clearing conditions. In E we have NS^H markets. Hence the number of operations needed to check if a proposed price is market clearing must rise exponentially with the number of households H. Consider now the case of Er supplemented by LA mutual insurance contracts and A securities. Under the assumptions of Theorem 1, 286 G. Chichilnisky and G. Heal

_•by Propositions 1 and 2, we need only check for market clearing in one social state associated with any statistical state, as if markets clear in one social state leading to a certain statistical state they will clear in all social states leading to the same statistical state. Hence we need to check a number of goods markets equal to N.A, plus markets for mutual insurance contracts and securities. Now

where (H, S) is a polynomial in H of order (5 - 1). Hence A itself is a polynomial in H whose highest order term depends on H^s~^l, completing the proof. D

7. Catastrophe Futures and Bundles

We mentioned in the introduction that securities contingent on statistical states are already traded as "catastrophe futures" on the Chicago Board of Trade, where they were introduced in 1994. Recently, hurricane bonds and earthquake bonds have been introduced, additional examples of statistical securities. (The concept was discussed by Chichilnisky and Heal in 1993 [6].) Catastrophe futures are securities which pay an amount that depends on the value of an index of insurance claims paid during a year. One such index measures the value of hurricane damage claims: others measure claims stemming from different types of natural disasters. The value of hurricane damage claims depends on the overall incidence of hurricane damage in the population, but is not of course affected by whether any particular individual is harmed. If, therefore, depends, in our terminology, on the statistical state, on the distribution of damage in the population, but not on the social state. Catastrophe futures are thus financial instruments whose payoffs are conditional on statistical state of the economy: they are statistical securities. According to our theory, a summary version of which appeared in [6] in 1993, they are a crucial prerequisite to the efficient allocation of unknown risks. And as the incidence and extent of natural disaster claims in the U.S. has increased greatly in recent years, risks such as hurricane risks are in effect unknown risks: insurers are concerned that the incidence of storms may be related to trends in the composition of the atmosphere and incipient greenhouse warming. However, catastrophe futures .are not on their own sufficient for this: they do not complete the market Mutual insurance contracts, as described above, are also needed. These provide insurance conditional on the value of the catastrophe index. The two can be combined into "catastrophe bundles", see [3]. Catastrophe Futures 287 8. Conclusions

We have defined an economy with unknown individual risks, and established that a combination of statistical securities and mutual insurance contracts can be used to obtain an efficient allocation of risk-bearing. Furthermore, we have shown that this institutional structure is efficient in the sense that it requires exponentially fewer markets that the standard approach via state- contingent commodities. In fact the state-contingent problem is "intractable" with individual risks (formally, NP -complete) in the language of computational complexity, whereas our approach gives a formulation that is polynomially complex. This greatly increases the economy's ability to achieve efficient allocations. Another interesting feature of this institutional structure is the interplay of insurance and securities markets involved. Its simplicity leads to successful hedging of unknown risks and predicts some convergence between the insurance and securities industries.

9. Appendix

given t at the economy is in the statistical state r. Now

so that Ez_fo is a feasible consumption vector for each h in the statistical state r. Next we show that by strict concavity, moving for each h and each σ from 288 G. Chichilnisky and G. Heal z_h*_σ (which depends on σ) to Ez_hr (which is the same for all σ € Ω), is a strict Pareto improvement. This is because

W» [z_h'_σ) = ∑ TI_σU> (z_k'_σ) = ∑ n_r ∑ Tl^U (z_h'_σ) .

By strict concavity of preferences,

u^h (E_Zhσ) .

Since Ez_hσ is Pareto superior to z" with z_h'_σι ≠ z^*^,, such a z^* cannot be an equilibrium allocation. Hence z_h ^* _σι = z^ = ^^'for all h = 1 , . . . , H. Note that this implies that in an equilibrium, household h consumes the same allocation z'_r across all individual states s in a given statistical state, i.e. it achieves full insurance. Since p^* supports the equilibrium allocation z^ψ , and z_h'_σ = z^ it follows that p^* ₍ = p^*. when r {σ_\) = r (σ ), because utilities are assumed to be C¹ and, in particular, to have a unique gradient at each point which, by optimality, must be collinear both with p^* ₍ and with p",, , i.e. P_σ, = P ₂ — Pr- This implies that at an equilibrium, household Λ faces the same prices p" at any σ with r (σ) = r. □

^~

Catastrophe Futures 289 _s (./j_{2 ι σ}, ) = ₅ (ft _j , ₂) . Assume that there exists an equilibrium price for E, p* 6 R^NsH , such that its components in states σ\ and σi are different i.e. Pi ≠ Pi, • Define now a new price p^* € iϊ^ " , called a "conjugate" of p", . which differs from p^* only in its coordinates in states σ\ and σi, which are permuted as follows: V σ ≠ σ_\ , σi, = p^* , ^*,, = p^*,, , and _% = p" , . We shall now show that ^* is also an equilibrium price for th^*e economy E. At ^*, household h_\ has the same endowments and faces the same prices in states σ and σ as it did at states σi and σ_\ respectively at price p"; at all other states σ e Ω, h_\ faces the same prices and has the same endowments facing p* and facing p^*. The same is true of household h . Furthermore, h\ and hi have the same utilities and probabilities at σ_\ and σi because r (σ_\) — r (σ ) and probabilities are anonymous. Therefore, the excess demand vectors of h_\ in states σ_\ and σi at prices p^* equal the excess demand vectors of h in σ₂ and σ\ respectively, at prices p", and at all other states σ 6 Ω the excess demand vectors of h\ are the same at prices p^* and p\ Reciprocally: the excess demand vectors of h in σ_\ and σ at prices p" equal the excess demand vectors of h_\ in σi and σ_\ respectively at prices p", and in all other states σ, the excess demand vectors of h are the same as they are with prices p^*. Formally:

*hισ, (P" ) = ^z ₂σ₂ P") . -Sλ, j (T) = * ι.σ, (p^*)

and c € Ω. σ 7^ σi . σ₂."

²Λι<r (P^*) = ^zA,<r ^'( ") > ^Λ, (p^*) = ∑ ,_σ (p^*) .

The excess demand vectors of all other households h ≠ hi , hi are the same for p^* and p^* . Therefore, at p" the aggregate excess demand vector of the economy is zero, so that p is an equilibrium. The same argument shows that permuung the two components pl₍ , p_σ ^* ₂ of a price p" at any two social states σ . σι leading to the same statistical state τ (σ_\) leads from an equilibrium pπce p^* to another equilibrium price T. This is because if two social states σ\ and σ₂ lead to the same statistical state and there are two individual states _\

f ^* d b i f 290 G. Chichilnisky and G. Heal

V_j > l . p = pϊ , then there are two price equilibria, i.e. k = 2; however, since the number of price equilibria must be odd,⁹ there must exist p^*, with ι > 1, .and p* ≠ p\. Consider now the conjugate of p^* ₍ with respect to the first two social 'states σ_\ , σi which correspond to the same statistical state and have different components in p_j" , and denote this conjugate pj₍ . Repeat the procedure until all equilibria are exhausted. In each step of this procedure, two different price equilibria are found. Since the number of equilibria must be odd, it follows that there must exist a j < k for which all conjugates of equal p^* : this is the required equilibrium which assigns the same equilibrium prices p_σ" = ρ , to all σ_\ , σ with r (σ_\) = r (σ;), completing the proof. 0

THEOREM 1. Assume that all households in E have the same probability II over the distribution of risks in the population. Then any Arrow-Debreu equilibrium allocation (p^* , z^*) ofE (and, therefore, any Pareto Optimum) can be achieved within the general equilibrium economy with incomplete markets Ei by introducing a total of LA mutual insurance contracts to hedge against individual risk, and A statistical securities to hedge against social risk. In a regidar economy with two individual states and identical preferences, even if agents have different probabilities, there is always an Arrow-Debreu equilibrium (p" . ∑^*) in E which is achievable within the incomplete economy Ei with the introduction of LA mutual insurance contracts and A statistical securities.

Proof. Consider first the case where all households have the same probabilities, i.e., π^Λ — IF = π. By Proposition 1, an Arrow-Debreu equilibrium of E has the same prices p_σ' = p^* and the same consumption vectors z_h"_σ — z_h'_τ for each h, at each social state σ with r (σ) — r. Define Ω (r) as the set of social states mapping to a given statistical state r, i.e. Ω (r) = {σ 6 Ω : r (σ) = r}. The budget constraint (4) is

P^' {z^{' ~ β}h) = ∑ P ( ,σ ^{~ e}hσ) = ∑ ; ∑ [ ^~ 6/ισ) = 0. ^{σ r} σ€Ω(r)

Individual endowments depend on individual states and not on social states, so that e σ = eh.s(σ) = e^,; furthermore, by Proposition 1 equilibrium prices depend on r and not on σ, so that for each r the equilibrium consumption vector z_hσ can be written as z_hs. The individual budget constraint is, therefore, ΣT _T ∑₅(_r) (^zhs - ^β/w)» where summation over s (r) indicates summation over all individual states s that occur in any social state leading to r, i.e. that are in the set Ω (r). Let #Ω (r) be the number of social states in Ω (r). As π_s|_r = r_s is the proportion of households in state s within the statistical state r, we can finally rewrite the budget constraint (4) of the household h as:

#Ω (r) ∑ P_r ∑ #Ω (Γ) II_s|r {z_hs - e_hs) = 0. (9) Catastrophe Futures 291

Using (2), the household's maximization problem can, therefore, be expressed as: max ∑ U_srU^h {z^_r) subject to (9)

and the equilibrium allocation z_h' by definition solves this problem. Similarly, we may rewrite the market clearing condition (5) as follows:

∑ (^ - e_/,) = ∑ (^ - e .)) = 0. ^V^ Ω. h Λ

Rewriting the market clearing condition (5) in terms of statistical states r, and within each r, individual states 5, we obtain:

^ r₅H (z _r - e;_r) = 0, Vr € H (10) s or equivalently:

∑ ϊl_s{rH (z_h'_r - e _r) = 0, Vr € A s

Using these relations, we now show that any Arrow-Debreu equilibrium allocation z' = (z _r) is within the budget constraints (7) of the economy Ei for each σ € Ω, provided that for each σ £ Ω we add the income derived from a statistical security A_r , r = r (σ), and, given r (σ), the income derived from mutual insurance contracts m _r = mj_(ff)r(ff), s = 1, . . . , S. We introduce A statistical securities and LA mutual insurance contracts in the general equilibrium economy with incomplete markets Ei . The quantity of the security A_τ purchased by household h in statistical state r, when equilibrium prices are ^*, is: βi' = ∑ ϊl_s]rp_r' (z_h'_r - e_h3) . (11)

The quantity α ^* has a very intuitive interpretation. It is the expected amount by which the value of equilibrium consumption exceeds the value of endowments, conditional on being in statistical state r. So on average, the statistical securities purchased deliver enough to balance a household's budget in each statistical state. Differences between the average and each individual state are taken care of by the mutual insurance contracts. Note that (10) implies that the total amount of each security supplied is zero, i.e., _Λ α ^* = 0 for all r, so that this corresponds to the initial endowments of the incomplete economy £/. Furthermore, ∑_r ^a _r ^* — by (9), so that each household h is within her/his budget in Ej.

We now introduce a mutual insurance contract as follows. The transfer made by individual h in statistical state r and individual state 5, when prices are ", is: m^h,; = _P; (z_k'_r - e_hr) - a ' . (12) 292- G. Chichilnisky and G. Heal

Note that, as remarked above, mj_r" is just the difference between the ac^tual income-expenditure gap, given that individual state s is realized, and the expected income-expenditure gap αf " in statistical state r, which is covered by statistical securities. In each statistical state r, the sum over all h and s of all transfers mj equals zero, i.e. the insurance premia match exactly the payments: for any given r,

∑ HIi_slrm^h _s; = ∑ HTl_{s r}p; (z_h'_r - e_hs) - ∑ H J^* ∑ π_f ^',_r

= 0 (13) because ∑, H^,. = 1. Therefore, the {m};} meet the definition of mutual insurance contracts. Finally, note that with N spot markets, A statistical securities {α_r} and / mutual insurance contracts {m_3r} p_r' [z_h'_τ - ej) = rn^_r + cf;, σ 6 Ω with r (σ) = r, s = s (σ) (14) so that (7) is satisfied for each σ € Ω. This establishes that when all households have the same probabilities over social states, all Arrow-Debreu equilibrium allocation z^* of E can be achieved within the incomplete markets economy Ei when A securities and LA mutual insurance contracts are introduced into Ei, and completes the proof of the first part of the proposition dealing with common probabilities.

Consider now the case where the economy E is regular, different households in E have different probabilities over social states but have the same preferences, and S = 2. By Proposition 2, we know that within the set of equilibrium prices there is one p^* in which at all social states σ € Ω (r) for a given r, the equilibrium prices are the same, i.e. p* = p~. In particular, if E has a unique equilibrium (p' , ∑"), it must have this property. It follows from the above arguments that the equilibrium (p", z") must maximize (2) subject to (9). Now define the quantity of the security A_τ purchased by a household in the statistical state r by

and the mutual insurance transfer made by a household in statistical^' state r and individual state s, by m_s ^h; = _P; (z_k'_sr - ei) - α_r ^Λ". (16)

As before, ∑._r αj?" = 0 and for any given r, ∑_hψS ϊή_lrHm^ =

2 ,s ^rsHm¹^ = 0, so that the securities purchased correspond to the ini^tial endowments of the economy Ei and at any statistical state the sum of the premia and the sum of the payments of the mutual insurance contracts match, completing the proof. CD Catastrophe Futures 293

Notes

1. A dealy viral disease.

2. Many were associated with hurricane Andrew which at S 18 billion in losses was the most expensive catastrophe ever recorded. Some of the problems which beset Lloyds of London arose from underestimating environmental risks.

3. All consumption vectors are assumed to be non-negative.

4. See [ 16, p. 387, para. 1].

5. Defined formally below.

6. Related propositions were established by Maiinvaud in an economy where all agents are identical, and risks are known.

7. The condition that all agents have the same preferences is not needed for this result. However, it simplifies that notation and the argument considerably. The general case is treated in the worl ing papers from which this article derives.

8. The condition that all agents have the same preferences is not needed for this result but simplifies the notation and the proof considerably. In the working papers from which this article derives, the general case was covered.

9. This follows from Dierker [11, p. 807] noting that his condition D is implied by our assumption that preferences are strictly increasing (see Dierker's remark following the statement of property D on p. 799).

References

1. Arrow, K.. J. "The Role of Securities in an Optimal Allocation of Risk-Bearing", in Econo eirie. Proceedings of the Cσlloque sur les Fondements et Applications, de la Theorie du Risque en Economeirie, Paris, Centre National de la Recherche Scientifiq e, 1953. pp. 41 -48. English translation in Review of Economic Studies 31, 1964, 91-96.

2. Arrow, . i. and R. C. Lind. "Uncertainty and the Evaluation of Public Investments", American Economic Review 60, 1970, 364—378.

3. Chichilnisky, G. "Catastrophe Bundles Can Deal with Unknown Risks", Bests ' Review, February 1996. 1-3.

4. Cass, D.. G. Chichilnisky and H. M. Wu. "Individual Risks and Mutual Insurance", CARESS Working Paper No. 91-27, Department of Economics, University of Pennsylvania, 1991.

5. Chichilnisky, G. and H, M. Wu. "Individual Risk and Endogenous Uncertainty in Incomplete Asset Markets", Working Paper, Columbia University and Discussion Paper, Stanford Institute for Theoretical Economics, 1991.

6. Chichilnisky, G. and G. M. Heal. "Global Environmental Risks", Journal of Economics Perspectives- 7(4), 1993, 65-86.

7. Chichilnisky G., J. Dutta and G. M. Heal. "Price Uncertainty and Derivative Securities in General Equilibrium", Working Paper, Columbia Business School, 1991.

8. Chichilnisky, C., G. M. Heal, P. Streufert and J. Swinkels. "Believing in Multiple Equilibria", Working Paper, Columbia Business School, 1992.

9. Debreu, G. The Theory of Value, New York, Wiley, 1959;

10. Debreu, G. "Economies with a Finite Set of Equilibria", Econometrica 38, 1 70, 387-

392. ! I. Dierker, E. "Regular Economies", in Handbook of Mathematical Economics, Vol. II,

Chapter 17, K. J. Arrow and M. D. Intrilligator, eds., Amsterdam, North-Holland, 1982, pp. 759-830. 12. Gary, M. R. and D. S. Johnson. Computers and Intractability: A Guide to A/P-

Completeness, New York, W.H. Freeman and Company, 1979. 294 G. Chichilnisky and G. Heal

13. Geanakoplos, J. "An Introduction to General Equilibrium with Incomplete Asset Markets", Journal of Mathematical Economics 19, 1990, 1-38.

14. Heal, G. M. "Risk Management and Global Change", Paper presented at the First Nordic Conference on the Greenhouse Effect, Copenhagen, 1992.

15. Maiinvaud, E. "The Allocauon oflndividual Risk in Large Markets", Journal of Economic Theory A, 1972, 312-328.

16. Maiinvaud, E. "Markets for an Exchange Economy with Individual Risk", Econometrica 3, 1973, 383^09.

Claims

1. A method for facilitating insuring/hedging against a risk condition, comprising establishing/using an index which is a measure for the condition.

2. The method of claim 1, wherein using is under a contract.

3. The method of claim 2, wherein the contract is a license contract.

4. The method of claim 1, wherein using is in a contract.

5. The method of claim 4, wherein the contract comprises an option contract.

6. The method of claim 4, wherein the contract comprises a futures contract.

7. The method of claim 1, wherein the condition is measured scientifically.

8. The method of claim 1, wherein the condition comprises a political condition.

9. The method of claim 1, wherein the condition comprises an environmental condition.

10. The method of claim 9, wherein the environmental condition comprises an atmospheric condition.

11. The method of claim 10, wherein the atmospheric condition comprises a temperature condition.

12. The method of claim 11, wherein the temperature condition comprises heating/cooling degree days (HDD/CDD) in a pre-specified geographic region over a pre-specified time period.

13. The method of claim 9, wherein the environmental condition comprises an oceanographic condition.

14. The method of claim 13, wherein the oceanographic condition comprises an El Nino Southern Oscillator condition.

15. A method in insuring/hedging against a risk condition, comprising buying/selling a contract which is contingent on an index which is a measure for the risk condition.

16. The method of claim 15, wherein the contract comprises an option contract.

17. The method of claim 15, wherein the contract comprises a futures contract.

18. The method of claim 15, wherein the condition is measured scientifically.

19. The method of claim 15, wherein the condition is a political condition.

20. The method of claim 15, wherein the condition comprises an environmental condition.

21. The method of claim 20, wherein the environmental condition comprises an atmospheric condition.

22. The method of claim 21, wherein the atmospheric condition comprises a temperature condition.

23. The method of claim 22, wherein the temperature condition comprises heating/cooling degree days (HDD/CDD) in a pre-specified geographic region over a pre-specified time period.

24. The method of claim 20, wherein the environmental condition comprises an oceanographic condition.

25. The method of claim 24, wherein the oceanographic condition comprises an El Nino Southern Oscillator condition.

26. A method in insuring/hedging against a risk, comprising, in combination: buying/selling at least one insurance contract for a risk condition; and buying/selling at least one security which is contingent on an index which is a measure for the risk condition; wherein the combination yields a payment in an amount which depends on one or more triggers.

27. The method of claim 26, wherein the security is an option contract.

28. The method of claim 26, wherein the security is a futures contract.

29. The method of claim 26, wherein the security is priced as a function of probability of different catastrophic regimes and on incidence of loss in the regimes.

30. The method of claim 26, wherein one of the triggers is based on a correlated risk and another on an uncorrelated event.

31. The method of claim 30, wherein the uncorrelated event comprises a pre- specified scientific pattern.

32. The method of claim 30, wherein the uncorrelated event comprises a pre- specified political pattern. Jo

33. The method of claim 30, wherein the uncorrelated event comprises a pre- specified environmental pattern.

34. The method of claim 33, wherein the environmental pattern comprises an atmospheric pattern.

35. The method of claim 34, wherein the atmospheric pattern comprises a temperature pattern in a pre-specified geographic region and over a pre-specified time period.

36. The method of claim 34, wherein the atmospheric pattern comprises a precipitation pattern in a pre-specified geographic region and over a pre-specified time period.

37. The method of claim 33, wherein the environmental pattern comprises an oceanographic pattern.

38. The method of claim 37, wherein the oceanographic pattern comprises an El Nino Southern Oscillator pattern.