Risk-aversion-in-the-small-and-large

RiskAversionintheSmallandintheLargeAuthor(s):JohnW.PrattSource:Econometrica,Vol.32,No.1/2(Jan.-Apr.,1964),pp.122-136Publishedby:TheEconometricSocietyStableURL::02/03/201107:22YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTermsandConditionsofUse,availableat.://=econosoc..EachcopyofanypartofaJSTORtransmissionmustcontainthesamecopyrightnoticethatappearsonthescreenorprintedpageofsuchtransmission.JSTORisanot-for-profitservicethathelpsscholars,researchers,andstudentsdiscover,use,andbuilduponawiderangeofcontentinatrusteddigitalarchive.Weuseinformationtechnologyandtoolstoincreaseproductivityandfacilitatenewformsofscholarship.FormoreinformationaboutJSTOR,[e-mail]ollaboratingwithJSTORtodigitize,preserveandextendaccesstoEconometrica.(January-April,1964)RISKAVERSIONINTHESMALLANDINTHELARGE'BYJOHNW.PRATTThispaperconcernsutilityfunctionsformoney.Ameasureofriskaversioninthesmall,theriskpremiumorinsurancepremiumforanarbitraryrisk,andanaturalconceptofdecreasingriskaversionarediscussedandrelatedtooneanother.Risksarealsoconsideredasaproportionoftotalassets.1.SUMMARYANDINTRODUCTIONLETU(X)BEautilityfunctionformoney.Thefunctionr(x)-u(x)/u'(x)willbeinterpretedinvariouswaysasameasureoflocalriskaversion(riskaversioninthesmall);neitheru(x)northecurvatureofthegraphofuisanappropriatemeasure.Nosimplemeasureofriskaversioninthelargewillbeintroduced.Globalriskswill,however,beconsidered,anditwillbeshownthatonedecisionmakerhasgreaterlocalriskaversionr(x)thananotheratallxifandonlyifheisgloballymorerisk-averseinthesensethat,foreveryrisk,hiscashequivalent(theamountforwhichhewouldexchangetherisk)issmallerthanfortheotherdecisionmaker.Equivalently,hisriskpremium(expectedmonetaryvalueminuscashequivalent)isalwayslarger,andhewouldbewillingtopaymoreforinsuranceinanysituation.Fromthisitwillbeshownthatadecisionmaker'slocalriskaversionr(x)isade-creasingfunctionofxifandonlyif,foreveryrisk,hiscashequivalentislargerthelargerhisassets,andhisriskpremiumandwhathewouldbewillingtopayforinsurancearesmaller.Thiscondition,whichmanydecisionmakerswouldsub-scribeto,involvesthethirdderivativeofu,asr'Oisequivalenttou'u'u2.Itisnotsatisfiedbyquadraticutilitiesinanyregion.Allthismeansthatsomenaturalwaysofthinkingcasuallyaboututilityfunctionsmaybemisleading.Exceptforonefamily,convenientutilityfunctionsforwhichr(x)isdecreasingarenotsoveryeasytofind.Helpinthisregardisgivenbysometheoremsshowingthatcertaincombinationsofutilityfunctions,inparticularlinearcombinationswithpositiveweights,havedecreasingr(x)ifallthefunctionsinthecombinationhavedecreasingr(x).Therelatedfunctionr*(x)=xr(x)willbeinterpretedasalocalmeasureofaver-siontorisksmeasuredasaproportionofassets,andmonotonicityofr*(x)willbeprovedtobeequivalenttomonotonicityofeveryrisk'scashequivalentmeasuredasaproportionofassets,andsimilarlyfortheriskpremiumandinsurance.Theseresultshavebothdescriptiveandnormativeimplications.Utilityfunctionsforwhichr(x)isdecreasingarelogicalcandidatestousewhentryingtodescribethebehaviorofpeoplewho,onefeels,mightgenerallypaylessforinsuranceagainst1ThisresearchwassupportedbytheNationalScienceFoundation(grantNSF-G24035).ReproductioninwholeorinpartispermittedforanypurposeoftheUnitedStatesGovernment.122RISKAVERSION123agivenriskthegreatertheirassets.Andconsiderationoftheyieldandriskinessperinvestmentdollarofinvestors'portfoliosmaysuggest,atleastinsomecontexts,descriptionbyutilityfunctionsforwhichr*(x)isfirstdecreasingandthenincreas-ing.Normatively,itseemslikelythatmanydecisionmakerswouldfeeltheyoughttopaylessforinsuranceagainstagivenriskthegreatertheirassets.Suchadecisionmakerwillwanttochooseautilityfunctionforwhichr(x)isdecreasing,addingthisconditiontotheothershemustalreadyconsider(consistencyandprobablyconcavity)inforgingasatisfactoryutilityfrommoreorlessmalleablepre-liminarypreferences.Hemaywishtoaddafurtherconditiononr*(x).Wedonotassumeorassertthatutilitymaynotchangewithtime.Strictlyspeak-ing,weareconcernedwithutilityataspecifiedtime(whenadecisionmustbemade)formoneyata(possiblylater)specifiedtime.Ofcourse,ourresultspertainalsotobehavioratdifferenttimesifutilitydoesnotchangewithtime.Forinstance,adecisionmakerwhoseutilityfortotalassetsisunchangingandwhoseassetsareincreasingwouldbewillingtopaylessandlessforinsuranceagainstagivenriskastimeprogressesifhisr(x)isadecreasingfunctionofx.Noticethathisactualexpenditureforinsurancemightneverthelessincreaseifhisrisksareincreasingalongwithhisassets.Theriskpremium,cashequivalent,andinsura