Prices and allocations in financial markets Theory

PricesAndAllocationsInFinancialMarkets:TheoryandEvidencePeterBossaertsz,CharlesPlottxandWilliamZame{Thisversion:5November2000Preliminary!zCaliforniaInstituteofTechnologyandCEPRxCaliforniaInstituteofTechnology{UCLA1PricesAndAllocationsInFinancialMarkets:TheoryandEvidencePeterBossaerts,CharlesPlottandWilliamZameAbstract:Thepricesandallocationsintwosetsofassetmarketsexperimentsarestudied.Onesetisthecertaintyequivalentoftheother.Inbothsets,marketsevidentlypriceriskcorrectly(expectedexcessreturnsareproportionaltocovariancewithaggregaterisk)evenwhenallocationsarewrong(individualallocationsdonotre°ecttheportfolioseparationpredictedbytheory).Weexplainthisprice-allocationparadoxbyaddingperturbationtermstostandardgeneralequilibriumtheory.Usingdatafromthecertainty-equivalentexperiments,weshowthattheextenttowhichportfolioseparationobtainsisapoormeasureofthee±ciencyofthe¯nalallocations.Instead,mostoftheutilitylossesstemfromthewrongmixofriskfreeversusriskysecurities.Pricechangesinbothsetsofexperimentsstronglycorrelatewithafullyoptimalaggregateexcessdemand.Wealsodiscoverthatallocationale±ciencyimprovesdramaticallyinthecertaintyequivalentexperiments,yet¯ndmixedevidenceofthisintheuncertaintyexperiments.Becauseutilitylossesallbutdisappearinthecertaintyequivalentexperiments,thelackofallocationalimprovementsintheuncertaintyexperimentscannotbeattributedtopoorincentives.Wedo¯ndstrongevidence,however,thatsubjects'behaviorre°ectunderstandingofmean-variancee±ciency:giventhevolatilityoftheirportfolio,subjectsmanagetoimproveexpectedreturnwhenpotentialimprovementsemerge.Thisexplainswhywe¯ndthatpricechangesaresigni¯cantlyrelatedtoamean-varianceoptimalaggregateexcessdemand,andwhypricingtendstowardsthepredictionsoftheCapitalAssetPricingModel(CAPM).2PricesAndAllocationsInFinancialMarkets:TheoryandEvidencekPeterBossaerts,CharlesPlottandWilliamZame1IntroductionThispaperreportsontheoreticalandexperimentalinvestigationsintopricingandrisksharingincompetitive¯nan-cialmarkets,bothstatic(eventualoutcomes)anddynamic(interplaybetweenexcessdemandandsubsequentpricechanges).Intheprocess,theinvestigationsprovideevidencetorejectwidespreadscepticismthat\thestakesaretoosmall,and,hence,experimentationwith¯nancialmarketsisdoomedtogeneratemeaninglessresults,whileifsuccessesareobtained,theseneedtobeattributedtochance.Themotivationfortheinvestigationswasasetoflarge-scale¯nancialmarketsexperimentswhichproducedstrongevidenceoffastconvergenceofpricestowardsasimplegeneralequilibriumpricingmodel,namely,theCapitalAssetPricingModel(CAPM).Statisticaltestsrejectathighsigni¯cancelevelsthattheconvergenceofpricestowardsCAPMintheseexperimentshappenedbychance.Yet,theevidencesuggestedthatsubjects'holdingsofsecuritieswereatoddswiththeallocationspredictedbytheCAPM,andthatthesituationdidnotimproveovertime.Atthecoreofthisallocationalpredictionisthat,inequilibrium,allagentsshouldholdthesame,diversi¯edportfolioofriskysecurities,namely,themarketportfolio.Thisprice-allocationparadoxispuzzling,becausethepricingpredictionintheCAPMderivesdirectlyfromtheallocationalprediction.Inthetheory,allocationscome¯rst;pricingisnext.Incontrast,intheexperiments,pricesswiftlymovedtotheequilibriumprediction,whereasallocationalpredictionsseemednevertobeveri¯ed.Fromawelfarepointofview,thevalueofcompetitive¯nancialmarketsasamechanismtoshareriskshouldthusbeputintoquestion.We¯rstpresentapotentialtheoreticalexplanationfortheprice-allocationparadox.Weshowthat¯nancialmarketscanreachequilibriuminpricingtermsevenifeveryagentsubmitserroneousdemands.Thedemandperturbations(deviationsbetweensubmitteddemandsandoptimaldemands)satisfycertainrestrictions.Thisgeneralequilibriumtheorywithperturbationsthereforeexplainswhy¯nancialmarketsmaygetpricingrightlongbeforeallocationale±ciencyisreached.Ourtheoryisdiametricallyopposedtothe(partialequilibrium)viewthatforpricestobekFinancialsupportwasprovidedbytheNationalScienceFoundation,theCaliforniaInstituteofTechnology,andtheR.G.JenkinsFamilyFund.3\correct,thereneedstobearationalmarginalinvestorwhofullyoptimizes.Inourtheory,allinvestorsmaybe\irrational.Therearemanyreasonsforthepresenceoftheperturbations.Oneisthatittakesagentstimetolearntooptimizetheirdemands,oreventolearntheirownattitudetowardsriskwithrepecttowhichtooptimizedemands.Anotherplausiblereasonisthatthestakesintheexperimentswheretheprice-allocationparadoxwas¯rstdiscoveredweresimplytoosmall{subjects¯ndtheincentivestofullyoptimizetoosmallandthereforesubmitonlyapproximatelyoptimaldemands.Anequallylikelyreasonisthatdemandperturbationsemergeasanaturalconsequenceoffrictionsinthetradingmechanismoftheexperiments(acomputerizedopenbooksystem).Stillanother,potentiallymorecontroversialreasonisthatagents'attitudestowardsriskconformtothestandardexpectedutilityparadigmonlyonaverage(acrossagents).Toshedsomelightontherelativeimportanceofthesereasons,weranasetofexperimentsthataretheoreticallyequivalentbutwhichdidnotinvolveanyuncertainty.Inthesecertaintyequivalentexperiments,subjectsweregiventheveryquadraticpayo®functionsthatgeneratedthepr