CounterpointGlobalInsights

ProbabilitiesandPayoffs

ThePracticalitiesandPsychologyofExpectedValue

CONSILIENTOBSERVER|February19,2025

IntroductionAUTHORS

FindingsecuritieswithgapsbetweenpriceandvalueistheMichaelJ.Mauboussin

foundationofgeneratingexcessreturnsinactiveinvestmentmichael.mauboussin@

management.1Thedifferencebetweenpriceandvalue,

commonlycalled“variantperception”or“edge,”comesfromDanCallahan,CFA

dan.callahan1@

havingasubstantiatedviewthatdivergesfromwhatthemarket

reflects.2Intheory,thesizeofaninvestmentwithinaportfolio

maximizesthebenefitofedgewhileconsideringrisk.

Thisisallsimpleinprinciplebutdifficultinpractice.Oneofthe

mainchallengesisdiscerningthegapbetweenpriceandvalue.

Priceistherelativelyeasypart.Buyingorsellingsecuritiesincurs

transactioncosts,andthemagnitudeofthosecostsdependson

factorssuchastheliquidityofthesecurity.3Butpriceis

transparentandinvestorscanestimatemarketimpact.

Valueisthehardpart.Thisisbecausevalueisreally“expected

value,”whichrepresentsarangeofpotentialpayoffswith

associatedprobabilities.Investingisaninherentlyprobabilistic

activity.Theconceptofexpectedvalueraiseslotsofissuesthat

wewillexplore.

Oneofthemostchallengingaspectsofunderstandingexpected

valueisthatexcessreturnscanbetheproductofhighprobability

eventswithrelativelylowpayoffs,orlowprobabilityeventswith

relativelyhighpayoffs.4Inotherwords,howoftenyouarerightis

notallthatmatters.Whatisvitalishowmuchmoneyyoumake

whenyouarerightversushowmuchyoulosewhenyouare

wrong.

Wehavecalledthisthe“BabeRutheffect.”5Ruthisconsidered

oneofthegreatestbaseballplayersofalltimeandyetwasthe

careerleaderinstrikeouts,ameasureofoffensivefailure,when

heretired.Atthesametime,hissluggingpercentage,which

assessesbattingproductivity,remainsthehighestinthehistoryof

MajorLeagueBaseball.Whatwasgoodinhisresultsmorethan

offsetwhatwasbad.Wewilllookatthefrequencyandmagnitude

ofpayoffsacrossassetclasses.

.

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Somemarketshaveseenashiftinappetitefromhighprobability,lowpayoffopportunitiestolowprobability,highpayoffones.Inbettingonhorseraces,therehasbeenarisein“exotic”wagers,whichcanincludeseveralhorsesandmultipleraces,versussimplewin,place,andshowbets.6Insportsbetting,parlaybets,alsowagersonmultipleoutcomes,havegrownrelativetosimplepointspreadorover/underbets.7Andtherehasbeenasurgeinthetradingofshort-datedoptionsinequityoptionsmarkets.8

Inthisreport,wediscusssomeoftheissueswiththecalculationofexpectedvalue,whatthepayoffpicturemeansforinvesting,theimplicationsofvolatilitydrag,thepsychologyofdealingwithprobabilitiesandpayoffs,andhowtheseideascanbehelpfulforinvestinginvariousassetclasses.

Wefocusonequitiesprimarilybutthethinkingappliestocreditandderivativesaswell.

WhattoExpectfromExpectedValue

Acalculationofexpectedvaluerequiresaquantificationofpotentialpayoffsandtheprobabilityofeachpayoffoccurring.Thesumoftheprobabilitiesmustbe100percent.Theexpectedvalueisthesumoftheproductofeachpayoffanditsassociatedprobability(seeexhibit1forasimplifiedexampleoftheexpectedvalueofadrug).

Exhibit1:ExpectedValueCalculationofaHypotheticalDrug

Scenario

BreakthroughAboveaverageAverage

BelowaverageDog

Probability

10%

20%

40%

20%

10%

100%

Payoff$2,500,000

$1,200,000

$137,500

$20,000

$10,000

Weightedvalue$250,000$240,000 $55,000$4,000

$1,000

Expectedvalue

$550,000

Source:CounterpointGlobalbasedonDavidKelloggandJohnM.Charnes,“RealOptionsValuationforaBiotechnologyCompany,”FinancialAnalystsJournal,Vol.56,No.3,May/June2000,76-84.

Expectedvaluecalculationsspanfromthesimpletotheverycomplex.AsWarrenBuffett,chairmanandchiefexecutiveofficerofBerkshireHathaway,hassaid,“Taketheprobabilityoflosstimestheamountofpossiblelossfromtheprobabilityofgaintimestheamountofpossiblegain.Thatiswhatwe’retryingtodo.It’simperfect,butthat’swhatit’sallabout.”9

Economiststypicallytranslateexpectedvalueintoexpectedutility,anideathatDanielBernoulli,amathematician,introducedin1738.Utilityisameasureofsatisfactionandvariesfrompersontopersonbasedonindividualpreferences.Mostpeopleexhibitriskaversion,meaningthatthemarginalutilityofwealthdiminishesaswealthincreases.10Expectedvalueisakeyconceptindecision-makingunderuncertainty,buteconomistsrecognizethatindividualsmakechoicesbasedondifferentutilityfunctions,whichleadtoarangeofpreferences.

Mostteachersstarttheirlessonsaboutexpectedvalueusingexampleswithsetprobabilitiesandpayoffs.Forinstance,theexpectedvalueofthetossofafaircointhatpays$2forheadsand$1fortailsis$1.50([0.50×$2]+[0.50×$1]=$1.50).Butinvestingisvastlymorecomplexthanthetossofacoin,therollofadie,ortheturnofaplayingcard.Themindsetcarriesoverbutthemathdoesnot.Overapplyingthesesimplecasestothemorecomplicatedonesiscalledthe“ludicfallacy”—ludusisLatinforgame.11

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FrankKnight,aneconomist,madethispointbydistinguishingbetween“risk”and“uncertainty.”Withrisk,accordingtoKnight,“thedistributionoftheoutcomeinagroupofinstancesisknown.”Thisisnottruewithuncertainty“becausethesituationdealtwithisinahighdegreeunique.”12Riskincludesthenotionofharmwhereasuncertaintyneednotreflectloss.MostofwhatinvestorsdealwithisKnightianuncertainty,althoughitisbesttothinkoftheabilitytosetprobabilitiesandpayoffsalongacontinuumfromtheobvioustotheimpossible.

Duringapressbriefingin2002,DonaldRumsfeld,thenU.S.SecretaryofDefense,answeredaquestionbydistinguishingbetween“knownknowns”(“thingsweknowweknow”),“knownunknowns”(“weknowtherearethingswedonotknow”),and“unknownunknowns”(“theoneswedon’tknowwedon’tknow”).Headdedthattheunknownunknownsisthe“categorythattendstobethedifficultone.”13

RichardZeckhauser,aneconomistandchampionbridgeplayer,writes,“Theessenceofeffectiveinvestmentistoselectassetsthatwillfarewellwhenfuturestatesoftheworldbecomeknown.”14Henotesthattheefficientmarkethypothesispositsthatprobabilitiesandpayoffsareestablishedand,asaresult,smartinvestingisanexerciseinoptimization.

Thekeytofinancialsuccesswhendealingwithunknownsandignorance,agooddescriptionofmostinvesting,istheabilitytoassessprobabilitiesandpayoffs.Decisiontheorybecomesmoreimportantthanoptimization.

Withthesethoughtsasbackground,wewilltakeacloserlookatpayoffsandprobabilities,thedeterminantsofexpectedvalue.

Payoffs.Payoffsreflectthefuturestatesoftheworldandcanrangefromtheverysimpletothehighlycomplex.Acommonmistakeindecision-makingis“overprecision,”aformofoverconfidencethatoccurswhensomeoneistooconfidentintheirviewsandthereforefailstoconsiderasufficientlywiderangeofalternatives.15Herearesomepointstokeepinmindwhenassessingthelikelihoodthatparticularfuturestatesoftheworldwillcometopass:

?Considertheshapeofthedistributionofpayoffs.BenoitMandelbrot,arenownedmathematician,usedtheterms“mild”and“wild”todistinguishbetweentherangesoffuturestates.16Mildstatescangenerallybecapturedwithanormal,bell-shapeddistribution.Thedistributionoftheheightofpeople,forinstance,ismild,withtheratiobetweenthetallestandshortesthumansonrecordbeingfive-to-one(leftpanelofexhibit2).Statisticalconceptssuchasmean(average)andstandarddeviationareusefulinexpressingmildstates.

Wildstatesareoftenpowerlaws,wherefewverylargeoutcomeshaveadisproportionateimpactonthedistribution.17Examplesincludethedistributionofwealthandcitysize(rightpanelofexhibit2).Asacaseinpoint,theratiobetweenthepopulationofthelargestcity(NewYork,NewYork)andtheone-thousandth(DekalbCity,Illinois)intheU.S.is205-to-1.Meanandstandarddeviationarenotusefulinexpressingtheoutcomesofthesesystems.

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Wild:CitySize

Population(Log)

10,000,000

1,000,000

100,000

10,000

1101001,000

Rank(Log)

Exhibit2:MildandWildStates:HumanHeightandCitySize

Frequency(Percent)

11

10

9

8

7

6

5

4

3

2

1

0

Mild:Height

ActualNormal

-4-3-2-101234

StandardDeviation

Source:StatisticsOnlineComputationalResourceHumanWeight/HeightDatasetandU.S.CensusBureau.

Note:Heightof25,000people;citiesintheUnitedStates.

NassimTaleb,anauthor,popularizedtheideaofa“blackswan,”whichhedefinesasaneventthatisanoutlier,isconsequential,andthathumanstrytoexplainafterthefact.18Blackswansarefromthedomainofunknownunknowns.

Manyoutcomesthatinvestorscallblackswansarereallywhathecallsgrayswans,orknownunknowns.Forexample,alargeanddevastatingearthquakewouldbeanoutlierandconsequential.Butgeologistshaveagoodsenseofthedistributionofearthquakemagnitudeseveniftheydonotknowexactlywhenorwhereanearthquakewilloccur.

TheStoics,ancientphilosopherswhobelievedinalifeledwell,advocated“premeditatiomalorum”—thepre-meditationofevils.Seneca,aStoic,wrotethefollowingaboutadverseeventsthatareunanticipated:“Thefactthatitwasunforeseenhasneverfailedtointensifyaperson’sgrief.Thisisareasonforensuringthatnothingevertakesusbysurprise.”19Thepointistoprepareforalleventualities.

?Bemindfulofthe“grandah-whoom.”20Phasetransitions,wheresmallchangesinacauseleadtolargeeffects,arepervasiveincomplexsystemssuchasbusinessesandmarkets.Thinkofcoolingwaterthatstartsatatemperaturejustabovefreezing.Asthetemperaturedropsbelowthepointoffreezing—ah-whoom—theliquidturnsintoasolid.Amodestchangehasalargeimpact.

JayForrester,aprofessoratMassachusettsInstituteofTechnologywhotaughtsystemdynamics,developedthebeergametoillustratehowsmalldecisionscanamplifyintobigeffects.Therearefourteams(manufacturer,distributor,supplier,andretailer)andassumedlagsbetweenwhentheordersarereceivedandwhenthebeerisdelivered.Thegoalistomeetconsumerdemandwhileminimizingbackordersandinventory.

Bullwhipeffects,whererelativelysmalldistortionsindemandcreateinefficiencythroughoutthesupplychain,commonlyemergefromplayingthegame.BullwhipeffectsoccurredinmultiplesupplychainsduringandfollowingtheCOVID-19pandemic.Onewidelydiscussedcasewastoiletpaper:aninitialspikeinconsumerpurchasescausedretailerstoordermuchmoreproduct,whichsignaledhighdemandto

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manufacturerswhothenrampedproduction.Demandthennormalized,whichledtooversupplyatwarehouses,distributioncenters,andstores.That,inturn,ledtolowerretailerorders.

Marketsalsohavephasetransitionswhenthereisadiversitybreakdown.Investorswithdiverseapproachesinteractingwithoneanothergenerallyproduceaccurateassetprices,asthewisdomofcrowdspredicts.21Butfromtimetotime,diversitybreaksdownandthebeliefsofinvestorsalign,resultinginboomsorbusts.

Theessentialinsightisthatthetrendintheassetpriceremainsinplaceevenasdiversitydeclinesandfragilityrises.Itisonlyatacriticalpointthatthereisastrongreversal—thebubblepops—anddiversityisrestored.Hereagain,asmallchangeinthestateofthesystemleadstoalargeeffectonthesystem.

?Controlandreversibility.Becausepayoffsreflectfuturestatesoftheworld,itisimportanttounderstandwhenthepayoffsareexpectedtohappen(timehorizon),whetherthedecision-makercanalterthepayoffs(control),andiftheinvestmentcanbeexitedatanacceptablecost(reversibility).22Reversibilityiscloselytiedtoliquidity,thecostofturningcashintoanassetoranassetintocash.Thatcostislowinliquidmarketsandhighinilliquidmarkets.

Toillustrate,considerthedifferencesbetweenaninvestmentmadebyacompanyandonemadebyanequityinvestor.Investmentsbycompaniessuchasbuildingadatacenteroracquiringanothercompanytendtobelong-termbecausethecostofreversalishigh.Offsettingthatilliquidityissomecontroloverthepotentialpayoffs.Companiescanactifthepotentialpayoffsappeartobefollowinganunsatisfactorypath,includingtweakingaproductoffering,changingthepricing,refiningthemarketingstrategy,orreplacingthemanagersinchargeofthebusiness.

Publicequityinvestorshavemuchmoreliquiditybutcommonlyhavelimitedcontroloverpayoffs.Notethatevenactivistinvestors,whoseektoimprovethepayoffsbypromotingchangeatthecompaniestheyinvestin,oftenneedtohaveasizeablestakeinthecompanytoestablishcredibility.Morecontrolrequireslessreversibility.

?Asymmetricpayoffs.Fundamentalinvestorscommonlyseekopportunitieswherethemagnitudeofpayoffsonthedownsidearesmallerthanthoseontheupside.Inotherwords,therearemorepotentialgainsthanlosses(naturally,theprobabilityofthepayoffsisalsocrucial).Insomecases,certainmeasuresofvaluationmaysuggestalimittodownsidepayoffs.Theseincludecashbalances,tangiblebookvalue,andfreecashflowyield.

Whileourfocusisonequities,theupsidepayoffsaregenerallycappedforbonds.Theupsidepayoffforastraightbondheldtomaturityisthepresentvalueofcouponpaymentsplusthereturnofprincipal.Forthisreason,equityinvestorstendtofocusonupsidepayoffsandbondinvestorsareinclinedtodwellontheavoidanceofloss.ThisiswhyBenjaminGrahamandDavidDodd,authorsofSecurityAnalysis,calledbondselection“primarilyanegativeart.”23

DanielKahnemanandAmosTversky,professorsofpsychology,developed“prospecttheory”inpartasanefforttoexplaintheobservationthatpeoplesuffermorefromlossesthantheyenjoygainsofcomparablesize,leadingto“lossaversion.”24Relativetoareferencepoint,peopletendtoberisk-averseintherealmofgainsandrisk-seekingintherealmoflosses(seeexhibit3).Empiricaldatabackthepointthatprospecttheoryexplainsinvestorbehaviorbetterthanclassicexpectedutilitytheorydoes.25

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Exhibit3:KinkedUtilityFunctionAsDescribedByProspectTheory

Utility

Loss

Gain

Utility

Source:CounterpointGlobalbasedonDanielKahnemanandAmosTversky,“ProspectTheory:AnAnalysisofDecisionunderRisk,”Econometrica,Vol.47,No.2,March1979,263-292.

Note:Assumesutilityofwealthequalsthelogofwealth,U(w)=ln(w),andalossaversioncoefficientof2.0.

Therearecategoriesofinvestmentswherethepayoffshavealowprobabilityofalargepositiveornegativeoutcome.Specifically,investorscanbuyinvestmentswithcharacteristicssimilartoalottery(losealittleandpotentiallymakealot)orsellthosesimilartoinsurance(makealittleandpotentiallylosealot).

Researchshowsthatinvestorscommonlyoverpricestockswithlotterycharacteristicsbecausetheyoverweighttheprobabilityofahighpayoff.26Somefinancialeconomistshaveconcludedthatitisbettertosell,ratherthanbuy,investmentswithlottery-andinsurance-typepayoffs.27

Thisthinkingcanbeexpandedfromindividualopportunitiestoinvestmentstrategies.Sellinginvestmentswithlotteryorinsurancepayoffsmeansmakingalittlemoneymostdaysandlosinglotsofmoneyfromtimetotime(blowup).Buyinginvestmentswithlotterypayoffsmeanslosingalittlemoneymostdaysandmakinglotsofmoneyeverynowandthen(bleed).

NassimTalebbelievesinthebleedstrategyandarguesthatextremeoutcomesareunderpriced.Butheconcedesthatafinancialfirmmayprefertomakemoneysteadilyevenattheriskofablowup.28AnexampleisLong-TermCapitalManagement,ahedgefund,whichhadreturnswellinexcessofthemarketfrom1993toearly1998butthenplummeted.

?Internalversusexternalfactors.Whenconsideringpayoffsandprobabilitiesforthestockofacompany,investorscommonlyandappropriatelyfocusprimarilyonthedriversofvalueforthatfirm.Forexample,ananalystmayconsiderdifferentscenariosformeasuresofcorporateperformancesuchassalesgrowthandoperatingprofitmarginsandestimatethepayoffpershareforeachscenario.

Oneofthemostimportantfindingsinfinanceisthatchangesinstockmarketpricesaregreaterthanwhatisjustifiedbychangesinfundamentals.29Apairofacademicpaperslookedatthelargestmovesinthe

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S&P500,anindexofabout500ofthelargeststocksintheU.S.,from1941to2012andthenexaminedtheexplanationsofferedbythebusinesspressafterthefact.30

Theyfoundthatexternalshocks,generallyrelatedtointernationalrelationsorpoliticaldevelopments,couldexplainsomeofthemoves.Calltheseexogenousrisks.Butmorestrikingly,alargepercentageofthebigmovesdidnotseemtohaveacorrespondingcausaleventbutratherseemedtohavecomefromwithinthesystem.Theauthorsofoneofthestudieswrote,“Onmostofthesizablereturndays,however,theinformationthatthepresscitesasthecauseofthemarketmoveisnotparticularlyimportant.Pressreportsonsubsequentdaysalsofailtorevealanyconvincingaccountsofwhyfutureprofitsordiscountratesmighthavechanged.”31Calltheseinternal,orendogenous,risks.32

Aswesawwithah-whoommoments,largescalechangescanoccurfromwithinthesystemwithoutanobviousexternalcause.Thisempiricalrealityisworthconsideringwhenassessingpayoffs.

Probabilities.Philosophers,statisticians,andmathematicianshavedebatedthemeaningofprobabilityforcenturies.Somehavearguedthatprobabilityisasubjectiveassessmentthatfailstoreflectarealquantity,andhencedoesnotreallyexist.33Thatsaid,itisusefultoconsiderprobabilitiesinevaluatingopportunities.Herearesomepointstokeepinmindwhenassessingthelikelihoodthatparticularfuturestatesoftheworldwillcometopass:

?Methodstosetprobabilities.Therearebroadlythreeapproachestosettingprobabilities:frequentist,propensity,anddegreesofbelief(subjective).34Thesecampsdonotalwaysseeeyetoeye.35

Thefrequentistsetsprobabilitiesbasedonalargesampleofoutcomesforaparticularreferenceclass.Thelikelihoodofasixappearingwiththerollofadieisone-in-sixbasedonahugenumberofobservationsofdierolls.

Thepropensityapproachjudgesprobabilitybasedonthepropertiesoftheobjectunderconsideration.Theprobabilityofrollingasixis16.7percent,reflectingthephysicalnatureofadieasaperfectcube.

Degreesofbeliefmeasuresthesubjectiveprobabilityanindividualassignstoanoutcome.Thisprobabilitycanbequantifiedthroughananalyst’swillingnesstobet.36Ananalystwhobelievesthelikelihoodofrollinga6is16.7percentandisneutraltoriskwouldbeindifferentbetweendoingnothingandbetting$1onrollinga6ifthepayoffwas$6($1=.167×$6).Investmentanalystsdealmostlywithsubjectiveprobabilities.Aninitialdegreeofbeliefiscalleda“prior.”Thepriorthatadiewillshowasixisbasedonaperson’sassessmentbeforeanynewinformationisrevealed.

Onesensibleapproachforinvestorsistousebaseratesasawaytoinformpriorprobabilitiesandthenupdatethoseprobabilitiesasadditionalinformationbecomesavailable.Abaseratereflectstheprobabilitiesandpayoffsforaspecificreferenceclass.Toillustrate,exhibit4showsthedistributionofthree-yearcompoundannualgrowthratesofsalesforU.S.companiesoverthepast63years.

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Exhibit4:Three-YearCompoundAnnualSalesGrowthRatesforU.S.Companies,1962-2024

40

35

Frequency(Percent)

30

25

20

15

10

5

0

<(12)(12)-(8)(8)-(4)(4)-00-44-88-1212-1616-2020-24>24

3-YearCAGR(Percent)

Source:FactSet,Compustat,andCounterpointGlobal.

Note:CompanieslistedontheNewYorkStockExchange,NASDAQ,andNYSEAmericanstockexchanges,withaminimum$1millionofsalesin1962dollars;nominalgrowth;CAGR=compoundannualgrowthrate.

Oneessentialpointisthatprobabilitiesandpayoffsaredynamic.Thatmeansthatnewinformationwilljustifyarevisioninpriorprobabilities.TheformalwaytodothisiswithBayes’Theorem,whichtellsyoutheprobabilitythatapriorbeliefistrueconditionalonsomeeventhappening.Whilethemathisuseful,whatismoreimportantisanopennesstoupdatingyourviews.37

Researchsuggeststhatconfirmationbias,thetendencytodismiss,discount,ordisavownewinformationinfavorofapriorview,canimpedeproperupdating.38Ithelpstothinklikea“fox”—onewhoknowsalittleaboutalot—ratherthana“hedgehog”—onewhoknowsonebigthing.Foxesupdatetheirviewsmorereadilythandohedgehogs,whoprefertofitthefactstotheirworldview.39

Investorsdealmostlywithsubjectiveprobabilities.Theseareusefulifsetcarefullyandrevisedappropriately.Butthereisanadditionallayerofnuance:confidenceinprobability.

?Confidenceinprobability.Probabilityandconfidencearedistinctconceptsthatoftengetcombined,unwittingly,ininvestmentanalysis.Youcanthinkofprobabilityasanestimateofthechancesofapayoffandconfidenceas“thedegreetowhichananalystbelievesthatheorshepossessesasoundbasisforassessinguncertainty.”40Psychologistscalltheprobabilityassignedtoapayoff“firstorderuncertainty.”Areasonablerangeofprobabilitiesforfirstorderuncertaintyiscalled“secondorderuncertainty.”Itreflectsuncertaintyaboutanuncertainpayoff.

JeffreyFriedman,aprofessorofgovernment,andRichardZeckhauserdescribethreedimensionstoconfidence:reliabilityofavailableevidence,rangeofreasonableopinion,andresponsivenesstonewinformation.

Reliabilityofavailableevidenceanswersthequestion,“CanIdefendthisestimatewithasubstantialamountofinformation?”Alargeamountofrelevantknowledgeprovidesasoundbasisforassessingrisk

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anduncertainty.Factandopinionarebothimportantininvesting.Thisdimensionfocusesonfact,whichshouldcarrymoreweightthanopinioninassessingconfidence.

Rangeofreasonableopinionaddressesthequery,“Mightreasonablepeoplegivesubstantiallydifferentanswerstothisquestion?”Thiscomesintoplaywheneverananalystisconsideringprobabilitiesandpayoffsthatrelatetoacomplexadaptivesystem,whereinputsandoutputsarenotlinkedlinearly.

Complexadaptivesystemsdescribeanetworkofadaptiveagentsthatinteractwithoneanothercreatingasystemthatisemergent.Inthesecases,analysisoftheunderlyingagentsdoesnotpredictoutcomes.41Prominentexamplesincludeclimatesystems,stockmarkets,andtheeconomy.Thisiswhytheaccuracyofforecastsinthesedomainstendstobepoor.

Responsivenesstonewinformationreflectsonthequestion,“IsmyviewlikelytochangesubstantiallyifIstudythesubjectfurther?”Theanswerisbasedonhowfirmlyananalystholdsapriorviewandwhetherthebenefitofnewinformationisworththecostandtimetoaccessit.Thisintroducestheconstraintsofresourcesandtime.Responsivenesstonewinformationcompelsthedecision-makertothinkaboutthecostandbenefitofpursuingadditionalinformation.

Confidenceinprobabilitiescanbeimportantwhenmakinginvestmentdecisions.Forinstance,twoopportunitiesmayhavethesamediscounttoexpectedvalue,buttheconfidenceintheprobabilitiesforonemayexceedthatoftheother.Thatinsightmaybeimportantforpositionsizingwithinaportfolioorforriskassessment.

Wordstoprobability.Aperceivedvariantperceptionmotivatesmostinvestmentdecisionsbasedonfundamentals.Thechallengeisthatthearticul