Amer. J. Agr. Econ. 83(3) (August 2001): 656–661Copyright 2001 American Agricultural Economics Association

Managing Catastrophic Risk ThroughInsurance and Securitization

Olivier Mahul

In recent years, the magnitude of catastrophicproperty/casualty disaster risks has becomea major topic of discussion. Recent catas-trophic events such as Hurricane Andrewand the Northridge earthquake each cost theinsurance industry in excess of $10 billion.These events illustrate the potential stresseswhich the insurance industry is now fac-ing. In addition, concentration of values incatastrophe-prone coastal areas has causedan increase in the amount of damage.

When loss exposures are (partially) cor-related, the design of an optimal risk shar-ing contract is based on a decomposition ofthe risk into systemic, i.e., non-diversifiable,and idiosyncratic, i.e., diversifiable, parts. Thisdecomposition allows us to apply two com-plementary risk sharing rules: risk mutualiza-tion and risk securitization.

In his classic 1962 article, Borch showedthat Pareto optimal risk sharing in an econ-omy with risk-averse agents is one in whicheach shares the aggregate wealth. This is riskmutualization.When aggregate wealth is risk-less and without transaction costs, agents canfully insure. If aggregate wealth is risky, thenindividuals may insure idiosyncratic risk butretain shares in aggregate wealth accordingto their tolerance towards risk. This mutual-ity principle has largely been ignored in theliterature on optimal insurance design (e.g.,Arrow, Raviv). Doherty and Dionne wereamong the first to examine alternative con-tracts for insuring individual risks when insur-ance companies are unable to eliminate riskby pooling. However, these risks can be diver-sified by pooling them with other economicevents that are not usually the subject ofinsurance. This is achieved through the secu-ritization of risk.

Risk securitization is accomplished by issu-ing specific conditional claims and selling

This paper was presented at the ASSA winter meetings (NewOrleans, LA, January 2001). Papers in these sessions are not sub-jected to the journal’s standard refereeing process.

Olivier Mahul is research fellow, Department of Economics,Institut National de la Recherche Agronomique (INRA),Rennes, France.

them directly to financial investors. TheChicago Board of Trade launched optionson natural catastrophes (cat spreads) in 1995and catastrophe-linked bonds (cat bonds)have been issued since 1997. These innova-tive financial instruments are the responseto the traditional insurance and reinsurancemarket’s inability to deal with highly cor-related risks. For instance, simulations con-ducted by modeling firms suggest that dam-ages caused by a major hurricane in Floridacould be at least $75 billion and those dueto an earthquake in California could exceed$100 billion. With prospective event-losseseasily exceeding $50 billion, the capitaliza-tion of the insurance and reinsurance indus-try is at issue. Estimates of total capitaland surplus of U.S. insurers and internationalreinsurers are about $300 billion and $100billion, respectively.1 Although such catas-trophic losses are large enough to placethe insurance industry under severe stress,they are lower than one standard deviationof the daily value traded (about $130 bil-lion on average) in the U.S. capital mar-kets (Cutler and Zeckhauser). Therefore,the pool of financial capacity provided byasset markets is able to bear the most pes-simistic estimated losses caused by a naturalcatastrophe.

From this risk decomposition, the optimalinsurance policies are contracts in which thepolicyholder insures the idiosyncratic compo-nent of his individual risk, but receives fromhis insurer a dividend based on the aggre-gate experience of the insurance pool. Thisis the logic of participating policies. Thesecontracts are proposed in several types ofinsurance, and especially in life insurance.Theaggregate loss is then (partially) hedged onfinancial markets through appropriate hedg-ing instruments.

The model and assumptions are specifiedin the next section, followed by a character-ization of an optimal variable participating

1 This capital and surplus applies to all risks (property/casualty,liability, etc.), and not just catastrophes.

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insurance policy. The findings are thenapplied to the management of crop-yield riskin agriculture. A conclusion highlights themain results concerning risk sharing in thepresence of aggregate risk and presents fur-ther research.

Assumptions and the Model

The static model is examined in the standardexpected utility framework. The agent withnon-random initial wealth w0 faces a risk ofloss l̃ ∈ [0� w0].

2 The individual loss l̃ can bepartitioned into a systemic component x̃ withEx̃ = 0�E denoting the expectation opera-tor, and an idiosyncratic component ε̃ ≥ 0through the deterministic relationship

l = l(x� ε)(1)

with lx ≥ 0 and lε ≥ 0.3 The distribution ofboth risks as well as the deterministic func-tion are assumed to be known by the poli-cyholder and the insurance company and tobe observable costlessly by the insurer.4 Therisk pool is defined by a large population ofagents with identical loss distributions. Theidiosyncratic random losses of two differentagents of the risk pool are assumed to beindependent. However, the systemic compo-nents are equal for all members of the riskpool.

The individual has the possibility to self-construct a variable participating insurancepolicy via the purchase of two separatecontracts: a non-participating policy and afully participating policy. The fully partic-ipating policy is described by the couple[I(� )� P ()] where I(x� ε) is the amount ofindemnity payments when the realized sys-temic and idiosyncratic risks are x and ε,respectively. The indemnity function must benon-negative:

I(x� ε) ≥ 0 for all x and ε(2)

The premium is variable and it depends onthe realization of the systemic loss. Idiosyn-cratic losses are assumed to be insurable at

2 “Tildes” are used to denote random variables and the samevariables without the tildes to denote realizations of randomvariables.

3 Variables in subscript denotes the partial derivative withrespect to this variable.

4 Problems associated with informational asymmetries, i.e.,moral hazard and adverse selection, not analyzed here.

zero transaction cost and the risk pool isassumed to be sufficiently large so that, fromthe law of large numbers, deviations of theaverage loss from the expected loss can beneglected. The premium is thus equal to theexpected indemnity conditional on the real-ization of the systemic risk:

P(x) = EI(x� ε̃)(3)

The non-participating insurance contract isdescribed by the couple [J(� )�Q]. It dif-fers from the fully participating policy in twoways. First, the indemnity schedule dependson the index that is not perfectly correlatedwith the systemic component of the risk pool.The contract is thus exposed to basis risk. Tomodel this, I assume that x̃ = α+βz̃+ b̃, withα ≥ 0, β > 0, Eb̃ = 0 where the b̃ basis riskis assumed to be independent of the index z̃,the risk pool’s systemic component x̃ and theidiosyncratic risk ε̃. For example, x̃ denotesthe aggregate losses of a regional pool andthe z̃ index represents the national aggre-gate losses. To simplify the notation, I furtherassume that α = 0 and β = 1. Thus x̃ = z̃+ b̃.It is easy to relax these assumptions, but noadditional insights are gained. The indemnityfunction of the non-participating policy mustbe non-negative:

J(z� ε) ≥ 0 for all z and ε(4)

Second, by definition of a non-participatingpolicy, the insurance premium is fixed. Inaddition, the cost of insurance is unfair. Therisk premium can be justified by the pres-ence of undiversifiable risk and, thus, it is thepayment required by the risk-averse share-holders. This can also be the consequence offirm-specific costs of risk bearing, e.g., convextax schedule. For the sake of simplicity, thetariff is assumed to be sustained by a compet-itive insurance market with risk-neutral insur-ers and transaction costs that are proportionalto claims:

Q = (1+ λ)EJ(z̃� ε̃)(5)

where the loading factor is positive, λ > 0.The variable insurance participating policy

thus provides indemnity payments [I(x� ε) +J(z� ε)] and it is sold at a price equal to[P(x) + Q] when the realized values of thesystemic and the idiosyncratic componentsand of the index are x, ε and z, respectively.By purchasing this contract, the policyholder

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658 August 2001 Amer. J. Agr. Econ.

will have final wealth of

w̃ = w0 − l(x̃� ε̃)+ I(x̃� ε̃)(6)

+ J(z̃� ε̃)− P(x̃)−Q

The problem of the policyholder with avon Neumann-Morgenstern utility function u,where u′ > 0 and u

′′< 0, is to determine

the indemnity schedule of the variable partic-ipating contract that maximizes his expectedutility of final wealth, Eu(w̃), subject to con-straints (2) to (5).

Optimal Variable ParticipatingInsurance Contract

The design of an optimal variable participat-ing insurance policy is closely related to thepartition of individual risk into systemic andidiosyncratic components. Additive and mul-tiplicative relationships between both partsare successively examined.

Additive Risk Components

The deterministic function l expressed inequation (1) is rewritten as

l(x� ε) = x + ε(7)

This means that if x = 10 then lossesare 10 higher for everyone than the long-run average. The optimal variable participat-ing insurance contract is designed in twosteps. The optimal fully participating insur-ance contract is first designed, then theoptimal non-participating insurance policy isderived. Since the fully participating con-tract is assumed to be sold at an actuariallyfair price whereas the premium of the non-participating policy is unfair, the policyholderpurchases full insurance on the idiosyncraticrisk

I ∗(x� ε) = x + ε(8)

with P = x + Eε̃. From equations (7) and(8), his final wealth expressed in equation (6)becomes

w̃ = w0 − Eε̃− z̃+ J(z̃� ε̃)−Q − b̃(9)

The variability of his final wealth comes fromthe random index z̃ and the uninsurable

and unhedgeable basis risk b̃. Therefore, theindemnity schedule of the optimal fully par-ticipating insurance contract will be only con-tingent on the index, J ∗(z� ε) ≡ K(z) for all zand ε. From Mahul (1999), the optimal unfairnon-participating policy in the presence of anindependent and additive background risk b̃displays full insurance on the index above adeductible D > 0:

J ∗(z� ε) = K(z) = max[z−D� 0](10)

with Q = (1 + λ)E max[z̃ − D� 0]. The opti-mal hedging strategy with index-based con-tract thus requires a long call position at astrike index D and a hedge ratio equal tounity.

Therefore, the optimal variable participat-ing insurance policy provides full insuranceon the ε̃ idiosyncratic risk and full insuranceabove a deductible on the z̃ index.The policy-holder bears part of the systemic risk throughthe partial coverage on the z̃ index, and theb̃ basis risk. This optimal insurance strategycan be replicated by two contracting patterns.First, a mutual insurance company proposesa contract that provides full insurance onthe idiosyncratic risk, through risk mutualiza-tion, and partial insurance on the systemicrisk. This company partially reinsures the sys-temic component of its portfolio risk throughindex-based contracts. In the second alterna-tive, two separate non-participating contractsare available.The first contract is sold at a fairprice and it provides a coverage against theidiosyncratic risk ε̃. The second one is unfairand it provides a coverage against the ran-dom index. It is straightforward to show thatthe agent fully insures the idiosyncratic com-ponent of his risk and the systemic compo-nent is partially hedged through index-basedcall options provided by the financial mar-kets. Securitization can thus be handled inde-pendently from the insurance contract.

Multiplicative Risk Components

The deterministic relationship between thesystemic and idiosyncratic components is nowexpressed by

l(x� ε) = (1+ x)ε(11)

For instance, if x = 01, then everyone’srealized individual loss is 10% higher thanthe long-run average. Assuming an insurance

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market for the ε̃ idiosyncratic risk with actu-arially fair pricing, the optimal fully partici-pating policy displays full insurance on ε̃:

I ∗(x� ε) = (1+ x)ε(12)

with P = (1 + x)Eε̃. The policyholder’s finalwealth becomes

w̃ = w0 − Eε̃− z̃Eε̃(13)

+ J(z̃� ε̃)−Q − b̃Eε̃

where b̃Eε̃ can be viewed as an indepen-dent background risk. From Mahul (1999),the indemnity schedule of the optimal non-participating policy is

J ∗(z� ε) = K(z) = Eε̃max[z−D� 0](14)

with Q = (1+λ)Eε̃E max[z̃−D� 0]. The opti-mal hedging strategy with the index-basedcontract is to buy call options at the strikeindex D with a hedge ratio equal to Eε̃.

The optimal variable participating policythus provides full insurance on the idiosyn-cratic risk and partial insurance on the sys-temic risk. The main difference with theadditive case is the impact of the expectedidiosyncratic component on the optimalhedge ratio and on the uninsurable basis risk.

This variable participating insurance con-tract can be replicated via insurance andfinancial markets as follows. A mutual com-pany offers the optimal variable participatingpolicy. The idiosyncratic component of eachindividual risk is pooled and the systemicrisk is partially hedged on financial marketsthrough securitization. Contrary to the addi-tive case, the consumer is not able to replicatethe variable participating policy by purchas-ing two separate non-participating contractson idiosyncratic risk via insurance marketsand on systemic risk via financial markets.This comes from the fact that the optimalhedge ratio should be equal to the randomvariable ε̃. Such a stochastic hedge ratio isnot available on real-world financial markets.Consequently, the insurance company playsa central role in the management of individ-ual risks by eliminating the idiosyncratic riskthrough mutualization and thus allowing toselect a hedge ratio based on a determinis-tic value, the expectation of the idiosyncraticloss.

Managing Crop Yield Risk in Agriculture

Historical experience shows that multipleperil crop insurance programs, in which indi-vidual farm yields are used to measure yieldloss, fail to operate on an actuarially soundbasis. There is extensive literature investigat-ing the causes of market failure. They areusually explained by the presence of infor-mation asymmetries such as moral hazardand adverse selection (Chambers, Quiggin).In contrast to this literature, Miranda andGlauber argue that systemic risk may be themost serious obstacle in the development ofa private crop insurance market. They notethat, contrary to automobile or fire riskswhich tend to be independent and to priceor rate risks which tend to be highly corre-lated, crop-yield risk lies between these twoextremes. This risk is a combination of sys-temic component stemming primarily fromthe impact of unfavorable weather events andof an idiosyncratic component depending onindividual characteristics. The systemic partis highly correlated among individual farm-level yields (droughts or extreme tempera-ture affect simultaneously a large number offarms), while the idiosyncratic part is almostindependent among individual yields.

Several studies have been recently devotedto the optimal management of agriculturalsystemic risk. The limitations on the abil-ity of the private insurance and reinsuranceindustry to fund catastrophic losses have ledto a proposal for government involvement.Since the 1980s, the Federal Crop InsuranceCorporation has acted as the primary rein-surer for crop insurers.Although governmentsolutions to the capacity problem may havesome potential, they should be used only ifthe private solutions are not forthcoming. Aprivate market to provide additional capac-ity for financing catastrophic risk in agri-culture is developing. In 1995, the ChicagoBoard of Trade launched an innovative con-tract into the agricultural markets, namely,the quantity-based crop yield futures andoptions contracts. The payoff of such con-tracts is based on the aggregate yield ofa surrounding geographical area (see, e.g.,Li and Vukina for a detailed description ofthese contracts). When a linear relationshipexists between the individual yield and thearea yield, Mahul (1999) shows that the opti-mal area yield insurance policy provides fullinsurance on the area yield risk above adeductible and that the marginal indemnity

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function is equal to the individual beta coef-ficient which measures the sensitivity of farmyield to area yield. Under a multiplicativerelationship, the slope of the optimal indem-nity schedule turns out to be less than thebeta coefficient if the producer is prudent(Mahul 2000). Due to the similarity betweeninsurance contracts and options contracts, theoptimal hedge ratio is thus equal to (lowerthan) the beta coefficient under a linear(multiplicative) relationship between individ-ual yield and area yield. Notice that theseresults can also be applied to other types ofindex-based contracts, such as weather-basedsecurities. Nevertheless, these studies assumethat the idiosyncratic risk is uninsurable andunhedgeable and therefore is treated like abackground risk.

Variable participating contracts turn out tobe useful instruments in efficiently manag-ing both components of the individual yieldrisk. The idiosyncratic risk would be managedthrough the mutuality principle and the finan-cial markets would deal with the systemic riskvia securitization. It is interesting to noticethat the idiosyncratic component could bemanaged not only by mutual insurance com-panies but also by cooperatives. Under a lin-ear relationship between both componentsof risk, it has been shown that the optimalvariable participating contract can be repli-cated by two separate contracts. The firstone provides full insurance on the idiosyn-cratic risk and the second one provides par-tial insurance on the systemic risk. However,a multiplicative relationship seems to be amore natural assumption in agriculture. Inthis context, cooperatives could play a centralrole. They would manage the independentidiosyncratic risks through mutualization anduse area yield insurance contracts to hedgeagainst the systemic risk of the pool.

Nevertheless, Crop Yield Insurance con-tracts are facing limited trading interest sincetheir market entry. This relative failure is usu-ally explained by the large amount of yieldbasis risk they contain and therefore they areconsidered unattractive by most U.S. farm-ers. The development of participating poli-cies should contribute to increasing the cor-relation between the index and the aggregaterisk of the cooperative, i.e., to reducing theyield basis risk, and thus to increasing theefficiency of these instruments in the manage-ment of crop-yield risk.

Conclusion

This article has presented a normative modelto investigate the role of insurance and secu-ritization in the management of catastrophicrisk.The variable participating insurance con-tract, defined as a linear combination of fullyparticipating policy and non-participatingpolicy, has turned out to be an optimal hedg-ing tool. Individuals insure the idiosyncraticcomponent of their loss exposure througha participating contract and then the sys-temic component is hedged through securi-tized products offered by financial markets.In the case of additive risk component, theoptimal variable participating policy can bereplicated with a fully participating insurancepolicy offered at a fair price and a separatehedging contract. For the case of multiplica-tive component, the key role of a mutualinsurer has been stressed. This role could beplayed by cooperatives in agriculture. Theycould offer a variable participating policy toinsured farmers, and pass off the systemicrisk, which is not assumed by the policy-holders, in the capital markets. Innovativehedging instruments, like area yield insurancefutures and options contracts, provide such anopportunity.

An important assumption in the currentmodel is that fully participating policy issold at an actuarially fair price. Neverthe-less, the existence of informational asymme-tries, like moral hazard or adverse selection,or the presence of transaction costs precludethe insurer or the cooperative from offeringfully participating policy at a fair price. Fur-ther research should investigate the design ofan optimal variable participating policy whenboth fully participating and non-participatingcontracts face administrative costs.

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Cutler, D.M., and R.J. Zeckhauser. “Reinsur-ance for Catastrophes and Cataclysms.” Paperpresented at the NBER Conference onProperty/Casualty Insurance, Boston MA,February 1997.

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