风险的敏感度分析英文文献 Sensitivity Analysis of Risk Toleranc

DecisionAnalysisVol.7,No.3,September2010,pp.313–321issn1545-8490
eissn1545-8504
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0313informs®doi10.1287/deca.1100.0183©2010INFORMSSensitivityAnalysisofRiskToleranceBjørnSandvikDepartmentofEconomics,UniversityofBergen,N-5007Bergen,Norway,[e-mail]-PetersenBodøGraduateSchoolofBusiness,N-8049Bodø,Norway,[e-mail]-aversedecisionmakerwithexponentialutilityfacingadecisionbetweentwoinvestmentsXandY,theremayexistmorethanonecriticalvalueofrisktoleranceforwhichthedecisionisreversedfromoneinvestmenttotheother.OurmainresultestablishesthatifYispreferredtoXbyallrisk-seekingdecisionmakers,thenthereisatmostonesuchcriticalvalue.Weextendthisresulttolinearplusexponentialutilityfunctions.Theresultsreducetheinputrequiredfromadecisionmaker.Keywords:decisionanalysis;riskaversion/tolerance;sensitivityanalysis;stochasticdominanceHistory:ReceivedonDecember19,2008.AcceptedonApril16,2010,after1revision.1.IntroductionAdecisionanalystusuallyneedstoaskaseriesofquestionstoassessadecisionmaker’sriskprefer-ences.Thismaybethecaseeveniftheassessmentisbasedonaparticularutilityfunctionsuchastheexpo-nentialone.Intheexponentialcase,usingsensitivityanalysisofrisktolerancebeforeaskingsuchquestionsmayshortentheprocessconsiderably.Ratherthandeterminingthevalueofthedecisionmaker’srisktol-erance,itmaybesufficienttoaskthesinglequestionwhethertherisktoleranceisbeloworaboveacriti-calvalue,calculatedinadvancebyadecisionanalyst.Thisarticleprovidesresultsthatmayreducetheinputneededfromthedecisionmaker,andmaybeappliedincaseswherethedecisionmakerisunknown.1Itisassumedthroughoutthatthedecisionmakerwishestoobeytheaxiomsofexpectedutility,andsatisfiesthe“strongone-switchcondition”(Bell1988;BellandFishburn2001,Theorem1).Consequently,preferencesareconsistentwiththeutilityfunctionofwealthuw=w−exp−w,≥0,0.Thislinearplusexponentialfunctionhasbeenrecom-mendedasthegenericutilityforwealthinassess-mentsandeconomicanalyses(Bell1995).Inthe1ThisisinlinewithPaté-CornellandDillon(2006,p.227):“Oneofthechallengesistocreateaframeworkandtogenerateresultsthatareappropriateforunknowndecisionmakers.”following,iscalledthereturncoefficient.Pratt’s(1964)measureofriskaversion,rw=−uw/uw=2exp−w/+exp−w,isstrictlydecreasing,andrwif0.If=0,thenwehaveconstantriskaversion,rw=,andbydefinition,istheriskcoefficientand=1/isthecorrespondingrisktolerance.Adecisionproblemiscalledaone-stepproblemifitcanberepresentedbyadecisiontreehavingnootherdecisionnodethantherootofthetree.Fur-thermore,thechoicebetweentwoinvestmentsXandYiscalledabinaryproblem.Iftherearethreeormoreinvestments,theproblemisamultipleone.Oursufficientconditionforuniquenessofacriticalvalueofistransitive.Thus,inamultipleproblemwiththreeinvestmentsX,Y,andZ,ifbothpairsX,YandY,Zsatisfythecondition,thensodoesthepairX,Z.Thismakesextensionfrombinarytomultipleone-stepproblemseasy.Generally,thereisaninvestmentXwithadis-cretedistributionfunctionFdeterminedbynstates1nwithprobabilitiespi=p1pnandann-vectorofstatevaluesxi=x1xninincreasingorder,x1≤···≤xn.Similarly,letYbeanalterna-tiveinvestmentwithdistributionfunctionGthatformstates1mhasprobabilitiesqj=q1qmandvaluesy1≤···≤ym.313SandvikandThorlund-Petersen:SensitivityAnalysisofRiskTolerance314DecisionAnalysis7(3),pp.313–321,©2010INFORMSSupposethatapairXYdeterminesabinary(one-step)problem.Underconstantriskaversion=0,theutilityfunctionisexponential,souxi=−exp−xi,andtheexpectedutilityofXequalsUX=n
i=1−exp−xipi(1)andcorrespondinglyforY.Thepurposeofthisarti-cleistostudyaone-waysensitivityanalysisofrisktolerance(oroftheriskcoefficient)insuchprob-lems,giventheassumptionofconstantriskaversionand,thus,theutilityfunctionin(1).Theorem1pro-videssufficientconditionsfortwocases:(a)UX=UYholdsforexactlyonecriticalvalue=00,or(b)UX≤UYissatisfiedforall0.Therearemultipleinvestmentreversalsiftheequationin(a)holdsforatleasttwodistinctvaluesof.In§2,exam-plesofsuchreversalsarepresented.Section3con-tainsthenecessaryconceptsandthemainresultsforthecase=0.Furthermore,usingatwo-waysensi-tivityanalysisofthecoefficients0,weextendtheresultstolinearplusexponentialutilityfunctionsin§4.In§5,weconcludewithabriefdiscussionoftheimplicationsoftheresultsfordecisionmak-ers.Proofsrelyonthepartialorderingknownas“majorization”(MarshallandOlkin1979).Thisorder-ing,othermathematicaldetails,andproofsarecon-tainedintheappendix.2.Investment-DecisionReversalsLetU=UXY=UX−UYbetheutilitydifferencebetweenXandY,andletUXYbethethderivative,=12.Generally,inabinaryproblemgivenbyXYandpreferencesasin(1),aone-waysensitivityanalysisofrisktolerance(orriskcoefficient)amountstodeterminingthevaluesofforwhichUXY≤0.Ifthisinequalityholdswithstrictinequalityforsomeanddoesnotholdforothervalues,thenbycontinuitythereisatleastoneinvestmentreversalwithrespectto.Considertwospecialcases.First,UXY≤0holdsforall0ifandonlyifYstochasticallydominatesXoftheinfinitedegree(Prat