MANAGERIAL AND DECISION ECONOMICS Manage. Decis. Econ. 27: 477–495 (2006) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mde.1283 Information Aggregation in a Catastrophe Futures Market Jason Shachat a, * and Anthony Westerling b a National University of Singapore, Singapore b University of California, San Diego, USA We experimentally examine a reinsurance market in which participants have differing information regarding the probability distribution over losses. The key question is whether the market equilibrium reflects traders maximizing value with respect to their different priors, or whether the equilibrium is one based on a common belief incorporating all participants’ information. When assuming subjects are expected value maximizers, we reject both full information aggregation and no information aggregation equilibria. We discover, as in past individual choice insurance experiments, that buyers under-assess the probabilities of large loss states, or alternatively, subjects assign larger utility values to losses than to comparable gains. After accounting for these decision theoretic concerns, the non-aggregation of information hypothesis explains the data better than full information aggregation. Copyright # 2006 John Wiley & Sons, Ltd. INTRODUCTION It is commonly thought that insurance markets facilitate the efficient sharing of risk, but whether they facilitate the efficient sharing of information is an open question. A defining feature of an insurance market is its underlying uncertainty. It is reasonable to assume that market participants possess differing information regarding the objec- tive probabilities governing states of nature. When these agents participate in a market there are two natural conjectures regarding the nature of the arising competitive equilibrium. First, agents maximize their objectives (holding their priors constant) and the resulting market prices and allocations reflect efficiency with respect to these initial beliefs. Second, market prices and alloca- tions arise that reflect a competitive outcome of agents maximizing their objectives conditional upon a common belief formed by the pooling of the agents’ differing information. In the first conjecture, the invisible hand only optimally coordinates activity treating the initial beliefs as exogenous parameters, while in the second con- jecture the invisible hand does substantially more. The process of market feedback aggregates dis- parate information and generates individually optimal outcomes with respect to the most informed sets of beliefs possible. Such a feature is highly desirable within an insurance market. The study of whether markets efficiently aggre- gate information is well suited for an experimental approach. A laboratory experiment allows for the control of preferences, endowments, and informa- tion structures that are essential in identifying when a market achieves a non-information aggre- gation (NA) equilibrium or a full information aggregation (FA) equilibrium. Several past experi- mental studies have addressed this question in the context of basic asset markets with mixed results. Copyright # 2006 John Wiley & Sons, Ltd. *Correspondence to: Department of Economics, National University of Singapore, Block AS2, # 06-05, 1 Arts Link, Singapore 117570, Singapore. E-mail: [email protected]
Transcript
MANAGERIAL AND DECISION ECONOMICS
Manage. Decis. Econ. 27: 477–495 (2006)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mde.1283
Information Aggregation in a CatastropheFutures Market
Jason Shachata,* and Anthony Westerlingb
aNational University of Singapore, SingaporebUniversity of California, San Diego, USA
We experimentally examine a reinsurance market in which participants have differinginformation regarding the probability distribution over losses. The key question is whether the
market equilibrium reflects traders maximizing value with respect to their different priors, or
whether the equilibrium is one based on a common belief incorporating all participants’information. When assuming subjects are expected value maximizers, we reject both full
information aggregation and no information aggregation equilibria. We discover, as in past
individual choice insurance experiments, that buyers under-assess the probabilities of large
loss states, or alternatively, subjects assign larger utility values to losses than to comparablegains. After accounting for these decision theoretic concerns, the non-aggregation of
information hypothesis explains the data better than full information aggregation. Copyright
# 2006 John Wiley & Sons, Ltd.
INTRODUCTION
It is commonly thought that insurance marketsfacilitate the efficient sharing of risk, but whetherthey facilitate the efficient sharing of informationis an open question. A defining feature of aninsurance market is its underlying uncertainty. It isreasonable to assume that market participantspossess differing information regarding the objec-tive probabilities governing states of nature. Whenthese agents participate in a market there are twonatural conjectures regarding the nature of thearising competitive equilibrium. First, agentsmaximize their objectives (holding their priorsconstant) and the resulting market prices andallocations reflect efficiency with respect to theseinitial beliefs. Second, market prices and alloca-tions arise that reflect a competitive outcome of
agents maximizing their objectives conditionalupon a common belief formed by the pooling ofthe agents’ differing information. In the firstconjecture, the invisible hand only optimallycoordinates activity treating the initial beliefs asexogenous parameters, while in the second con-jecture the invisible hand does substantially more.The process of market feedback aggregates dis-parate information and generates individuallyoptimal outcomes with respect to the mostinformed sets of beliefs possible. Such a featureis highly desirable within an insurance market.
The study of whether markets efficiently aggre-gate information is well suited for an experimentalapproach. A laboratory experiment allows for thecontrol of preferences, endowments, and informa-tion structures that are essential in identifyingwhen a market achieves a non-information aggre-gation (NA) equilibrium or a full informationaggregation (FA) equilibrium. Several past experi-mental studies have addressed this question in thecontext of basic asset markets with mixed results.
Copyright # 2006 John Wiley & Sons, Ltd.
*Correspondence to: Department of Economics, NationalUniversity of Singapore, Block AS2, # 06-05, 1 Arts Link,Singapore 117570, Singapore. E-mail: [email protected]
Plott and Sunder (1988) find aggregation canoccur when market participants have a completeset of Arrow-Debreu securities to trade, or whenthere are homogeneous preferences. In Forsytheand Lundholm (1990) information aggregationoccurs only when traders have experience withmarket institutions and common knowledge ofeach others’ dividends. Plott et al. (2003) find somesuccess for information aggregation in parimutuelmarkets for situations for which Bayes’ Law is notneeded.
Unfortunately, these experiments’ designs andresults do not lend sufficient insight into howeffectively information aggregates in an insurancemarket because of the strikingly different informa-tion structure. In this study we consider a propertyreinsurance market. It is natural to suppose a riskand information structure like that in Figure 1.Purchasers of reinsurance have considerable ex-perience with the high-frequency, low-value claimsprocesses represented by the left side of the figure.Sellers of reinsurance, on the other hand, with along history of business in multiple regions andlines of reinsurance, have better information aboutthe large less likely catastrophe risks representedby the right tail of the probability density in Figure1.1
The presence of low-probability, large-lossstates also is not captured in previous experimentalmarket studies, but is an integral part of aninsurance market. However, there is an extensivebody of survey and experimental work addressinghow individuals make insurance decisions whenfaced with low-probability, high-value risks. Slovicet al. (1977) and Kunreuther et al. (1978) findevidence of either persistent probability biases orconvex utility over losses in insurance experiments.McClelland et al. (1993) find, when agentspurchase insurance from the experimenter in aVickrey auction, evidence of a bimodal response tovery low probability risks, with some participants
disregarding very small risks and others highlysensitive to small risks. None of these experimentsare conducted in a bilateral-market context (i.e.subjects only perform the task of buying insur-ance). Also these experiments do not consider thesituation of differential information.
An empirical example motivates us to drawdistinct elements from the two literatures: a recentinnovation in the US market for catastrophereinsurance. After three recent low probabilitylarge loss events, Hurricane Hugo ($4.2 billion ininsured claims), Hurricane Andrew (claims over$16 billion), and the Northridge Earthquake(claims over $12.5 billion), many insurers tried towithdraw from the catastrophe insurance marketfor earthquake risk in California and wind risk inFlorida.2 However, regulatory measures kept firmsfrom fleeing these markets. At the same time,available reinsurance coverage grew increasinglyscarce,3 as the reinsurance market did not face thesame regulations. These changes created anopportunity for new and innovative entrants tothe reinsurance industry.4 The Chicago Board ofTrade (CBOT) was one of the first non-traditionalentrants, inaugurating trading in CatastropheFutures and Options in December 1992. CBOTofficials were particularly enthusiastic about thepotential success of catastrophe insurance futures.Numerous members of the academic communityshared this enthusiasm. There were many antici-pated benefits of catastrophe insurance futures andone of the strongest was the reduction ofinformation asymmetries.5 Despite the initialoptimism, trading in the CBOT’s catastrophefutures never amounted to much,6 and they areno longer traded today. We hope our experimentsheds some light into this lack of success, and giveinsights into whether any market of this structureleads to information aggregation.
The results of our experiments do not offermuch hope in this regard. First, when we assumeindividuals are expected value maximizers, themarket price and quantity data do not supporteither an NA equilibrium or an FA equilibrium.However, there is strong evidence that prices andquantities rely more heavily upon the realizationof the buyer’s prior information regarding high-probability, low-loss events than the seller’s priorinformation regarding low-probability, high-lossevents. This leads us to investigate the impact thatsubjective probability biases and risk aversions,found in individual choice insurance experiments,
could be having in our markets. We find thatbuyers tend to underestimate the probability ofdisasters while sellers on average assess theseprobabilities correctly. This finding is also con-sistent with an agent model where the correctprobabilities are used by both buyers and sellersbut subjects’ preferences are those given inProspect theory (Kahneman and Tversky, 1979)in which losses loom larger than gains. Oncecontrolling for these preferences, we find that anNA equilibrium typically explains the data morerobustly than does an FA equilibrium.
In the next section we present an example of acatastrophe futures market, which is also the basisof our experiment, and then we present theimplications of the Full Aggregation and Non-aggregation equilibrium concepts. Then we presentour experimental design. After which we presentthe results of our experimental markets. Weconclude with some comments on the implicationsof our work for those who are looking to novelsecurities for insurance solutions.
A SIMPLE MARKET FOR CATASTROPHE
FUTURES AND EQUILIBRIUM
HYPOTHESIS
We now describe the demand and supply condi-tions of an elementary market for a catastropheindex future that we use in our experiments.Primary insurers, who purchase catastrophe fu-tures to help reinsure the risks inherent in theirportfolio of property insurance policies, determinethe demand conditions. Reinsurers, who sell futurecontracts, determine the supply conditions. Thecatastrophe future pays a dividend that is propor-tional to an index of all claims made on theproperty insurance policies sold by primaryinsurers.
Consider a primary insurer who sells propertyinsurance policies that generate a total fixedpremium income of $4.60. There are four differentstates of claim levels which we denote {NL, NH,DL, DH}}N and D are for normal and disasterstates and L and H are for low and high losses.The set of insurance policies has a correspondingset of four possible levels of liabilities, {$2, $4, $10,$20}. In the absence of any other purchases orsales of securities, the primary insurer has a set offour possible net income, {$2.60, $0.60, �$5.40,�$15.40}.
We now introduce a security that trades afterthe primary insurer collects premiums but beforethe level of liabilities is determined. When theamount of liability is determined, the dividends onthe introduced security are paid. Now let usassume there are a total of six such primaryinsurers and, for simplicity, further assume thattheir liabilities are perfectly correlated. An index ofthese insurers’ liabilities has four possible values{$12, $24, $60, $120}. Define a future contract onthis index such that seller of the contract pays thepurchaser a dividend one-twelfth of the realizedvalue of liability index, or the future contract hasfour potential dividend levels {$1, $2, $5, $10}.Notice if a primary insurer purchases two suchfuture contracts he is fully insured and will have anet income of $4.60 less the price paid for the twocontracts regardless of the state. When theexpected net income position is the sole considera-tion, the maximum amount a risk neutral primaryinsurer is willing to pay for a unit of the security isthe expected dividend.
The reality of the property insurance marketdictates that value of assets providing reinsuranceto primary insurers rely upon more than just theexpected dividend. For example, the propertyinsurance market is highly regulated, and regula-tory bodies closely monitor and restrict riskposition of insurers’ portfolios of policies andsecurities.7 To capture the impact of regulatorymandates and incentives to hold conservativefinancial positions we specify that a primaryinsurer derives additional value from the purchaseof future contracts that is independent of therealized dividend. Specifically we denote themarginal amount of this additional valuation forthe first four contracts purchased}as we willrestrict the maximum number of contracts pur-chased to four}is ($0.54, $0.30, �$0.34, �$0.58).Notice that this schedule provides a positivereward for the purchase of contracts that lead toa more fully insured portfolio, and a negativereward for contracts that lead an over-insured andmore risky portfolio. The magnitude of therewards is increasing in the distance one’s portfoliois from the fully insured position.
A primary insurer’s state dependent demandfunctions for each of the four possible liabilityoutcomes is simply the sum of the reward schedulethat is independent of the state and the dividendreceived in the state. This family of state depen-dent demand functions is presented in Figure 2.
Notice that family of demand functions differ bytheir y-axis intercepts. This is due to the fact thatvertical location of the demand curve is deter-mined by the state dividend. Consequently, aprimary insurer’s expected demand curve is definedby the expectation of the intercept value, or inother words the expected dividend. Furthermore, achange in the expected value of the dividend leadsto a vertical shift of the demand curve. Finally, themarket expected demand curve is found by ahorizontal summation of the individual expecteddemand curves.
The sellers in this catastrophe futures market arelarge reinsurers who do not hold any retailproperty insurance policies. In our experimentswe will have six such sellers. The revenue receivedfrom the sale of future contracts is the sole sourceof value for a reinsurer in this market. There aretwo sources of cost for selling contracts. First, thedividend that reinsurer must pay on each contractsold is the state dependent marginal cost for acontract. Second, reinsurers are also subject toregulatory mandates and incentives on theirportfolios like primary insurers. For example, alocal regulator can penalize a reinsurer for notproviding a certain amount of coverage in amarket. We summarize the costs resulting fromthe effects as the state independent marginal costschedule, (�$0.54, �$0.30, $0.34, $0.58). Thenegative values correspond to avoiding the reg-ulatory cost of not providing enough liquidity tothe market, and the positive costs are associatedwith excess volatility in the portfolio.
A reinsurer’s state dependent supply functionsfor each of the four possible liability outcomes issimply the sum of the marginal cost schedule that
is independent of the state and the dividend paid inthe state. The state dependent supply curves arepresent in Figure 3, and like the demand case, onlydiffer by their y-axis intercepts as determined bythe state dividend. Thus the vertical placement of aprimary insurer’s expected supply curve is definedby the expected dividend value and any change inthe expected value of the dividend leads to avertical shift of the expected supply curve. Finally,the market expected supply curve is found by ahorizontal summation of the expected individualsupply curves.
Clearly, the equilibrium prices and quantity ofcontracts will depend upon the probabilities thatbuyers and sellers place on the four possible lossstates. As we described in the introduction, thereare strong reasons to believe that buyers havebetter information regarding high probabilitysmall loss states of the world while sellers havebetter information regarding low probability largeloss states of the world. We now present a simpleway to operationalize this notion. Recall we havefour possible states of the world, {NL, NH, DL,DH} corresponding to the primary insurer’spossible liabilities {$2, $4, $10, $20}. Now Let{0.45, 0.45, 0.05, 0.05} be the prior probabilitiesover these possible losses. Before the market forfuture contracts, buyers receive information thatallows them rule out the high (H) or low (L) lossconditional upon a normal state (N) occurring.Likewise, sellers receive information that allowsthem rule out the high (H) or low (L) lossconditional upon a Disaster state (N) occurring.This process generates in four distinct priorinformation regimes, which we denote LL, LH,HL, HH. The first letter in a pair refers to the
0
2
4
6
8
10
12
0 1 2 3 4 5Contracts
Price
State DH
State DL
State NH
State NL
Figure 2. A primary insurer’s state dependent demand
functions for future contracts.
0 1 2 3 4 5Contracts
0
2
4
6
8
10
12
Price
State DH
State DL
State NH
State NL
Figure 3. A reinsurer’s state dependent supply functions
remaining Normal state and the second letterrefers to the remaining Disaster state. Table 1 givesthe priors the buyers and sellers, respectively, holdat the start of the futures market. Of course ourquestion of interest is whether the competitiveforces of the market will leads to aggregation ofthis disparate information. The final column ofTable 1 presents the prior distribution that resultswhen the buyers’ and sellers’ information isaggregated.
Using the information in Table 1, we can fullyspecify the market demand and supply curves aredepending upon the disparate priors and aggregatepriors. The hypotheses of interest are full informa-tion aggregation (FA) versus non-informationaggregation (NA). The basis of the FA hypothesisis the ability of a market to generate an informa-tion aggregation equilibrium, i.e. the marketgenerates a competitive outcome that reflects thepooling of all diverse information regarding thetrue state of nature. The competitive equilibriumprices and allocations that arise under FAhypothesis are those generated by expected de-mand and supply curves which use the aggregateprior to calculate the expected dividend. The NAhypothesis is generated by the conjecture that themarket generates a competitive outcome reflectingthe agents’ prior beliefs regarding the true state ofnature. The competitive equilibrium prices andallocations that arise under NA hypothesis arethose generated by expected demand and supplycurves which use the respective priors to calculatethe expected dividend.
The impact of these two competing models isgenerated through differing expected dividendvalues. Under the FA conjecture, a competitiveoutcome reflects a common expected dividendvalue based on the pooling of buyers’ and sellers’private information. The expected value is calcu-lated as
E½dðsÞ� ¼ 0:9 ðremaining N-state’s dividendÞ
þ 0:1 ðremaining D-state’s dividendÞ: ð1Þ
On the other hand, if the NA conjecture holdstrue, the market outcome will reflect the followingdistinct expected dividends for the buyer and seller;
E½dðsÞ�buyer ¼ 0:9 ðremaining N-state’s dividendÞ
þ 0:1 ðaverage D-state’s dividendÞ
ð2Þand
E½dðsÞ�seller ¼ 0:9 ðaverage N-state’s dividendÞ
þ 0:1 ðremaining D-state’s dividendÞ:
ð3ÞBuyers’ and Sellers’ expectations of the dividend
values determine the vertical location of supplyand demand curves. Hence, the implications of thecomparative statics of FA versus NA are obtainedfrom the inspection of the competitive equilibriumfor their respective supply and demand curves.Table 2, and Figures 4 and 5 summarize theequilibria for the two models in the four priorinformation regimes.
Figure 4 shows the market supply and demandcurves under the FA premise for the four priorinformation regimes. First, notice that for all fourprior information regimes the equilibrium marketquantity is twelve units. In other words, buyersfully reinsuring their endowed portfolio risk.Turning our attention to price, the FA outcomegenerates distinct equilibrium price tunnels. Themidpoints of these price tunnels represent actuarialfair premiums for reinsurance.
In any prior information regime, the NA modelwill distinctly differ from the FA model in eitherthe equilibrium price or quantity. In the LH andHL regimes, the NA and FA models only differstrongly in equilibrium quantities. The NA modelpredicts that in the LH regime only 6 units aretraded, resulting in an under-provision of reinsur-ance; in the regime HL 18 units are traded, andthere is an over-provision of reinsurance. One canalso observe that under prior information regimesLL and HH the equilibrium prices are distinctunder the FA and NA hypotheses, but full
reinsurance is achieved in both scenarios. How-ever, in these two regimes the NA hypothesis doesnot generate actuarial fair reinsurance premiums:In HH, the midpoint of the price tunnel is belowthe actuarial fair rate and in LL the midpoint isabove the actuarial fair rate.
EXPERIMENTAL DESIGN
In our experiments, twelve participants are ran-domly partitioned into groups of six Buyers andsix Sellers. An experiment consists of a series oftrading periods. In each period, Buyers and Sellers
Table 2. Model Predictions for Equilibrium Prices and Quantities
Disaster stateLow HighLL LH
Normal state Low FA model: 12 units, $1.30–$1.50 12 units, $1.80–$2.00NA model: 12 units, $1.75 6 units, $1.81–$2.19
HL HH
High FA model: 12 units, $2.20–$2.40 12 units, $2.70–$2.90NA model: 18 units, $2.19–$2.21 12 units, $2.25–$2.65
Figure 4. Full-aggregation equilibrium induced supply and demand.
have the opportunity to trade an asset in an oraldouble auction. Before the auction starts, Buyersand Sellers are privately given information rele-vant to the distribution of the dividend. After theauction, the experimenter conducts a probability
experiment that determines the actual dividend forthe trading period.
Consider the time line in Figure 6. Before thestart of each trading period, the experimenter flipsa coin. If the result is heads, the NL state is
Figure 5. No-aggregation equilibrium induced supply and demand.
eliminated. If the result is tails, the NH state iseliminated. Buyers are privately informed of theremaining Normal state (NL or NH) with the use ofa code sheet. Likewise, a second coin toss is usedto eliminate one of the Disaster states. Sellers areprivately informed of the remaining Disaster state(DL or DH).
Next a seven-minute open outcry double auc-tion commences. Buyers may offer bids or acceptasks, and sellers may make asks or accept bids inan oral double auction format. A valid bid or askmust improve upon any standing bid or ask. Oncea bid or ask is accepted, bidding starts over; buyersare then free to open bidding at any non-negativeprice, and sellers are free to make an initial ask atany price between $0 and $20. Bids, asks, andtrades are displayed on an overhead projector asthey are made. After the seven-minute tradingperiod has expired, the final state of nature isresolved by drawing one ball from the bingo cagein view of the participants. If the ball is numbered‘1’ through ‘9’, the result is the remaining Normalstate. If a ‘10’ is drawn, the result is the remainingDisaster state. The ball is returned to the bingocage prior to the next trading period. Buyers thenreceive the random values of the units theypurchased and the random transfers (the netpremium income or premium less the realizedliability), and sellers pay the random costs of theunits they sold.
Figure 7 below is a typical Buyer’s DecisionSheet. In row number 1 Buyer 1 carries overcumulative earnings from the previous period($0.00 since this is the first period). On the leftside of the Buyer’s Decision sheet are four columnslabeled X1, X2, Y1, and Y2, corresponding to thestate-space (NL, NH, DL, DH). In this period thestatement ‘not White’ would inform Buyers thatX1 had been eliminated, and ‘not Blue’ that X2had been eliminated. There are no codes listed forthe ‘Y’ states (DL, DH), since the buyers are notprivy to this information. The values in rownumber 2 are net premiums which apply in eachstate. Similarly, in rows three, six, nine and twelvethe values for each of the four units that Buyernumber 1 may purchase are listed for each of thefour states. For each unit he purchases, Buyer 1enters the purchase price in the appropriate spaceon the far right column. After trading is finishedthe final state is drawn and then completes thedecision sheet. Figure 8 presents a typical Seller’sdecision sheet.
Market participants are inexperienced prior totheir arrival for the experiment. The subjects firstprivately read written sets of buyer or sellerinstructions on the Double Auction procedures.Next subjects privately read instructions on howthe two coins tosses and the draw from the Bingocage determined the state. Then these commoninstructions were read out loud by the experimen-ter as well and the experimenter conducted thenatural probability experiment twice withouttrading. Finally subjects participated in one tothree practice periods that include trading in thesecurity.8
Buyers and Sellers begin the experiment withzero cash endowments. They are permitted to runnegative cash balances without being expelledfrom the experiment, but receive no compensationother than a non-salient show-up fee of five dollarsif their cumulative earnings are negative at the endof the experiment. Subjects were informed of thislimited liability. Unfortunately, this came into playin one of our sessions there was two disasters andin which three sellers ended with negative balances.The number of periods over nine is randomlydetermined, and participants are not informedahead of time which period will be the final period.
RESULTS AND ANALYSIS
We focus our analysis of the experimental datainto two activities. First, we compare how well thedata conforms to our interior predictions for priceand quantity for the two competing models. Pricesand quantities for units traded each period, withfew exceptions; do not match the equilibriumpredictions for either the full-aggregation or theno-aggregation model. Prices typically are lowerthan either model’s predictions and market pricesdo not depend on the sellers’ prior information.The volume of reinsurance contracts also does notreflect either model’s predictions. We do observethat the impact of buyers’ prior information ismore influential on quantity than is the sellers’prior information.
Since prices are generally lower than eitherhypothesis predicts, and buyers’ prior informationhas a greater than expected impact on both priceand quantity, we consider alternative explana-tions. We turn to the experimental and surveyresearch on disaster insurance for possible expla-nations. Given the subjective probability biases
that underestimate the probability of disasterstates found in these literatures, we explore thepossibility that the buyers’ and sellers’ possess thisbias in our experiment. From the experimentalmarket data, we calculate implicit subjectiveprobability beliefs of a disaster for both buyersand sellers under the FA and NA hypotheses. Theresult of this exercise suggests there is a strongbias: the buyers’ implicit beliefs are typically belowthe sellers’ implicit beliefs (which are on averagestatistically indistinguishable from 10%). Once weaccount for this bias, there is evidence that the NA
assumption is more appropriate. We also point outthat there is an alternative to our subjectiveprobability bias conclusion: individuals use theobjective probabilities but differ in the way theyevaluate risky choice. In this scenario we concludethat the implications of prospect theory hold:sellers’ losses loom larger than buyers’ gains.
Data Preliminaries
We start by presenting the data from the fivecatastrophe futures markets in Figures 9–13. We
show the transaction prices for each experiment inchronological order, separated by trading period.For each period, the shaded areas represent thequantities and the range of prices we would expectto observe if markets are in the FA modelequilibrium. The NA model equilibrium pricesand quantities are the clear areas; overlappingregions are cross-hatched. The x-axis gives theperiod, prior information regime, and the tripleFA predicted quantity/NA predicted quantity/observed quantity.
For example, in the first period of Experiment 1in Figure 2.10, the information set is LL. The no-aggregation model prediction of 12 units traded at
$1.75 is represented as a horizontal line 12 unitswide. The full-aggregation model prediction}12units traded between $1.30 and $1.50}is repre-sented by the shaded area. The line representingactual trades shows the first unit traded at $2.25.Subsequent prices fell rapidly to the full-aggrega-tion price range, and the total quantity traded was9 units.
Price and Quantity Data Analysis
A visual inspection of Figures 10–14 quicklyreveals that the observed prices tend to lieoutside the ranges predicted by either model. To
assess the impact prior information has on priceswe obtain the ordinary least squares estimate ofthe coefficients in the following dummy variableequation:
Price ¼ a1LLþ a2LHþ a3HLþ a4HH
The results of this regression, along with NAand FA price predictions, are given in Table 3.
First notice that mean price for each priorinformation regime falls below the predicted rangeexcept in the case of the FA prediction in the LLregime. The second striking result is that priceseems to solely depend upon the buyer’s priorinformation. Specifically, the mean prices in LLand LH are close and the mean prices in HLand HH are close. We conduct an F-test to confirm
this observation. The F-statistic for the hypothesisthat a1 ¼ a2 and a3 ¼ a4 is 2.63 with a p-value of0.073.
These results regarding price are quite surprisinggiven the results of similar treatments in Plott andSunder (1988). In three of their experimentalsessions, subjects are given homogeneous prefer-ences over dividends, thus giving an ordinalranking of states. Strong convergence to the FA
predicted prices occurred by the end of each of thethree sessions.9 The lack of price convergence inour experiment must result from one or somecombination of the following: correlation of priorinformation with buyer and seller roles, pooledinformation does not reveal the true state ofnature, the low probability of large loss states, andhow individuals form assessments in the presenceof this uncertainty.
Before completely dismissing the applicability ofeither model, consider the effects a probability biasmight have on the hypothesized prices. The pricedata imply that market participants may tend tounder-weight the probability of a disaster stateoccurring. Note that under the FA model wewould expect the difference in price between lowand high normal-state information sets to average90¢, and under the NA model, 45¢. As theprobability of a disaster state goes to zero, thesepredictions approach $1.00 and 50¢, respectively.The difference we observe}about 53¢}is suppor-tive of the NA hypothesis. We more vigorouslypursue this idea below.
One of the attractive features of our inducedsupply approach is the ability to discriminatebetween models through the inspection of quan-tities. In the five futures markets, observedquantities tend to diverge from those predictedby either model. The lack of convergence inquantity is readily seen in the Figures 9–13. Wenow ask whether either model can explain theaverage market quantities. Recalling the quantitypredictions of the two models summarized inTable 2, note that under the FA model we expect12 units to be traded in each period. Also note thatunder the NA model the quantity prediction differsin two prior information regimes: in LH thequantity is six and in HL the quantity is eighteen.The FA and NA models both give testable
implications in the following expression:
Qt ¼ aþ nHxt þ dxHt;
where Qt is the market quantity in period t, Hxt isdummy variable for the prior information regimesin which buyers are informed that the low Normalstate is eliminated (i.e. regimes HL and HH), andxHt is a dummy variable for the prior informationregimes in which the seller has been informed thatthe low Disaster state is eliminated (i.e. LH andHH). Under the FA model, a ¼ 12 and n ¼ d ¼ 0and under the NA model a ¼ 12 and n ¼ �d ¼ 6.The OLS estimates of these coefficients arepresented in Table 4. The F-statistic for thisregression (24.301) rejects the hypothesis that themean quantity is independent of the prior infor-mation regime. This is a rejection of the FAcoupled with symmetric subjective probabilitybeliefs of a Disaster state. On the other hand, theestimated model coefficients do not follow thepredictions of the NA model either. The estimatedvalue of a (9.0) is not the predicted 12 units, and at-test indicates a 0.00 probability that a ¼ 12.While the estimated values of n and d aresignificantly different from zero, and have thecorrect sign for the NA model, they are not equalto 6 and �6, respectively. The probability that n,given an estimated value of 4.7, is equal to 6 is0.059 and the probability that d, given anestimated value of �1.5, is equal to �6 is 0.000,
assigning a probability of a disaster state as lessthan 10%. Is this consistent with the data onquantities? If buyers and sellers tend to under-weight the probability of a disaster, we would stillexpect under the FA model a quantity of 12 unitstraded in each period. Under the NA model, wewould expect, as observed, a value for |d| less than6; as the probability of a disaster state goes to zero,d goes to 0 as well. As the perceived probability ofa disaster declines, however, the observed value ofn should increase under the NA model, convergingto 7 as the probability of a disaster state goes tozero, contrary to our result. How then do weaccount for these results? Some possible explana-tions for our results are that the experimentalsubjects’ perceived probability of a disaster statechanges over time, that buyers’ and sellers’ beliefsmay differ, or both.
Subjective Probability Biases
We assess whether subjective probability biasescombined with either the FA or NA model canrationalize our market data. We start by assumingthat the market prices and quantities we observeeach period reflect a competitive equilibrium. Thisassumption relies upon the oral double auction’ssubstantial history of robustly generating compe-titive outcomes in induced supply and demandexperiments. Next we know that the schedules ofprivate marginal valuations and costs give us theslopes of the demand and supply curves. What isnot known is the vertical location of these curvesas these are defined by the experimental subjects’subjective probability beliefs of a disaster state. Wefurther assume that all buyers have the same beliefand that all sellers have the same belief. The size ofa vertical shift given a belief depends upon whetherthere is information aggregation or not. Weproceed by calculating implicit beliefs under boththe FA and NA hypotheses. To summarize, wehave two parameters (the subjective size of thesupply and demand curves’ positive vertical shifts)whose values we can use to calibrate the observedmarket price and quantity.
The answer to the following question is notobvious; are there role-specific probability biaseswhich can explain our results under these twomodels? To address this question, we perform anumerical exercise in which we deduce the implicitprobability biases for buyers and for sellers usingthe FA and NA hypotheses. The are four main
conclusions: the NA model most plausibly ex-plains results in most periods, buyers’ averageimplied beliefs of disaster under the NA hypothesisare below the actual 10% probability, sellers’average probability beliefs of disaster under theNA hypothesis do not differ significantly from10% on average, and correspondingly sellers’implied probabilities are higher than buyers’.
Let pb denote the buyers’ perceived probabilityof a Disaster state and ps denote the sellers’perceived probability of a Disaster state. Substi-tuting into Equations (1)–(3), we get
EðdÞseller ¼ ð1� psÞ ðaverage of the N-state’s dividendsÞ
þps ðremaining D-state’s dividendÞ
for the expected values of the common dividendunder the NA hypothesis. Combining theseequations with the private value and cost incre-ments, we solve for market equilibrium prices andquantities for both models for all the combinationsof probability beliefs (pb,ps) over pb ¼ 0:01; 0:02;. . . ; 1 and ps ¼ 0:01; 0:02; . . . ; 1. From these resultswe identify the range of probability beliefs ofsellers and buyers in our experiments that couldsupport the observed quantities and median pricesfor each period.
The median and range of probability beliefs forbuyers and sellers supporting the observed quan-tities and median prices for each period’s tradesare shown in chronological order in Figures 14 and15, separated by experiment. The dashed verticallines mark occurrences of disaster states. Havingadded two degrees of freedom to our models, thechoice between hypotheses becomes a matter ofjudgement and interpretation, rather than a testof predictions. Nevertheless, there are two featuresof these implied probability beliefs that tend to
support the conclusion that the NA model hasmore explanatory power:
* The implied probability beliefs calculated forthe full-aggregation model are much sparserthan those calculated for the NA model. This isbecause no combination of buyers’ and sellers’probability beliefs support the observed pricesand quantities in 25 out of 54 periods for theFA model, while the same is true in just 14 outof 54 periods for the NA model.
* Buyers’ and sellers’ implied probabilities varymore, and more erratically, over time, and varymore from buyer to seller, under the full-aggregation model than is the case under the
no-aggregation model. This is likely an artificeof the data being forced to fit the model, ratherthan a true representation of the evolution ofparticipants’ probability beliefs. By contrast,the beliefs implied by the NA model tend tomove together. Buyers’ and sellers’ impliedbeliefs tend to move in the same directionunder the NA model, and period-on-periodchanges in beliefs tend to be much less extreme.
Clearly there is variation from period to periodin both the buyers’ and sellers’ subjective beliefs.Table 5 gives some brief statistical analysis of thesets of beliefs under the NA hypothesis. For eachstatistic we conduct a hypothesis test that the mean
Figure 15. Buyer’s and seller’s implied probability beliefs under full-aggregation model.
is equal to 10% versus the alternative that themean is less than 10%. For the sellers’ beliefs wefail to reject the null at all typical levels ofsignificance, however for the buyer we do rejectthe hypothesis. We also conduct a t-test fordifference in means for the two sets of beliefs.Here we reject the null hypothesis that the meansare equal in favor of the alternative that the sellers’mean is larger than the buyers’ mean. (The t-statistic is 2.469, has 78 degrees of freedom and ap-value of 0.008.) The strong negative biaspossessed by buyers corresponds to similar resultsfound in individual choice experiments, for exam-ple Slovic et al. (1977) and Kunreuther et al.(1978), in which subjects purchase insurance fromthe experimenter against small-probability, large-loss events.
Our experiment is the first in which somesubjects sell insurance against small-probability,large losses. It also appears that changing the pointof reference and the framing of the reinsurancetask has eliminated this bias for sellers. However,there is another interesting perspective from whichwe can view these results. Instead of assuming thatindividuals are expected value maximizers whohave probability biases, we could have assumedthat they did not have subjective probability biasesbut that their preferences differ from risk neutral-ity. Under this interpretation we would concludethat the sellers give a greater assessment to thepotential large losses of selling insurance contractthan buyers give to the assessment of the largegains. This interpretation is consistent with theimplications of the Kahneman and Tversky’s(1979) prospect theory of decision making underuncertainty, where relative losses typically loomlarger than relative gains.
CONCLUSION
In this paper we examine an insurance market’sability to generate equilibria which reflect theunion of market participants’ diverse information
regarding the probabilities that govern states ofnature. The correlation of prior information withmarket roles and the structure of uncertainty inthese markets lead us to develop significantchanges to the standard experimental design,introduced by Plott and Sunder (1988), used totest information aggregation. We found that theeconomic environment of a reinsurance marketfailed to generate the equilibrium predictionsunder either the FA model or the NA model. Thisis in contrast to Plott and Sunder’s finding ofinformation aggregation in simpler environments.In evaluating the hypotheses we found strongevidence that the value of the buyer’s priorinformation had more impact on economic out-comes than did the seller’s prior information. Thissuggested alternative explanations.
The uncertainty that characterizes insurancemarkets requires individuals to assess the valueof small-probability, large-loss (gain) states. Aplethora of past studies show that traditionalexpected utility theory’s robustness falters in thesesituations, and that subjective probability biases ornon-expected utility preferences can characterizebehavior. In our setting one cannot distinguishbetween a subjective probability bias and a utilityphenomenon. After we calculate the implicitsubjective probability beliefs in our experimentwe conclude that buyers possess a strong sub-jective probability bias and sellers do not. Thecorresponding utility explanation is that sellers’potential losses from reinsurance contracts loomlarger than buyers’ gains from reinsurance. Final-ly, after we control for these decision theoreticaspects, we see that the NA hypothesis has moreexplanatory power than the FA hypothesis.
These results do not provide optimism thatinsurance markets, such as the catastrophe futuresindex introduced by the CBOT in 1992, can lead tooutcomes in which information is aggregated andrisk is efficiently shared. Given the strong desir-ability of the information aggregation property ininsurance market, it is worthwhile to explorewhether other financial instruments (e.g. PCSoption spreads and Act of God Bonds) and otherinstitutions (such as the long standing bilateralcontractual relationships that governed the re-insurance market prior to 1990) fare better thanthe market we study here.
Our results also suggest future directions in thestudy of information aggregation in general.Specifically, can we explain why the challenging
decision making under uncertainty environment ofcatastrophe insurance impedes the informationaggregation process? If we cannot answer thisquestion, can we at least establish the boundary ofthis breakdown empirically? Furthermore, inprevious experiments in which information aggre-gation occurs, the pooled information reveals thetrue state. In our experiments pooled informationdoes not reveal the true state of nature, and it is ofinterest to assess the impact this has. Clearly, inmost cases of interest, pooled information doesnot reveal the true state. Finally, we believe theintroduction of the induced supply and demandapproach to the study of markets with uncertaintyis an innovation which may permit the perfor-mance of a wider class of experiments. Therobustness of this approach needs to be morethoroughly tested.
NOTES
1. The property insurance market here is assumed tohave little in the way of moral hazard. We believe thiswould muddle the central issue of informationaggregation. Moreover, we feel secure in assumingthat the market participant’s actions do not exertsignificant influence over the probabilities of cata-strophic events such as hurricanes, earthquakes, andfloods.
2. See Nutter (1994), Marlett and Eastman (1997),Lecomte (1996), and Roth (1996).
3. O’Hare (1994) and Kunreuther (1997).4. See Doherty (1997) for a good review of conditions in
the insurance industry at the time.5. D’Arcy and France (1992), Niehaus and Mann
(1992), Harrington et al. (1995), Doherty (1996,1997) discuss benefits of trading in catastrophefutures and insurance derivatives in general. Coxand Schwebach (1992), Cummins and Geman (1995)and Doherty (1997) also address the role ofcatastrophe futures markets in resolving informationasymmetries.
6. Harrington and Niehaus (1999).7. See Lecomte (1996), Nutter (1994), and Roth for
examples of such institutional detail.8. The unfortunate differences in the number of practice
periods resulted from several experiments startinglate due to tardy subjects.
9. For more details of the results from these threesessions see Plott and Sunder (1988, pp. 1100–1102).
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