《金融学》（第二版） 课件 （英文） -Chapter11、12 Hedging,Insuring,and Diversifying；Portfolio Opportunities and Choice

Chapter11:

Hedging,Insuring, andDiversifyingObjectiveExplainmarketmechanismsforimplementinghedgesandinsurance1Chapter11Contents11.1UsingForward&FuturesContractstoHedgeRisks11.2HedgingForeign-ExchangeRiskwithSwapContracts11.3HedgingShortfall-RiskbyMatchingAssetstoLiabilities11.4MinimizingtheCostofHedging11.5InsuringversusHedging11.6BasicFeaturesofInsuranceContracts11.7FinancialGuarantees11.8Caps&FloorsonInterestRates11.9OptionsasInsurance11.10TheDiversificationPrinciple11.11InsuringaDiversifiedPortfolio211.1UsingForwardandFuturesContractstoHedgeRisksForwardContractanagreementbetweentwopartiestoexchangesomethingataspecifiedpriceandtimeThisisanobligationonbothpartiesDistinguishthisfromaright,butnottheobligation,ofapartytoexchangesomethingTonullifythecontract,youtrytonegotiatesecondcontractfora+/-cashconsideration3DefinitionsofTermsForwardPricePrice(agreedtotoday)ofanitemtobepurchased,andpaidfor,atagivenfuturedate

SpotPricePrice(agreedtotoday)ofanitemtobepurchased(andpaidfor)immediatelyFaceValue‘Quantityofdeliverable’times‘forwardprice’4DefinitionsofTermsLongPositionTheagreementtobuytheitem(fromthepersontakingtheshortposition)ShortPositionTheagreementtoselltheitem(tothepersontakingthelongposition)5UsingForwardandFuturesContractstoHedgeRisksTraditionally,nopaymentismadeonaforwardcontractuntilthesettlementdateIfthepartiestoaforwardcontractdonottrusttheother,thenaddclausestoprovideasuretiestoastakeholderperiodicallyrendercontractvaluelessbymakingcashsettlementequaltoitscurrentmarketvalue6UsingForwardandFuturesContractstoHedgeRisks:FuturesContractsFuturescontractsforcommoditiesandfinancialproductsincludessuchclausestoprotectagainstunknowncreditrisks,andweleavethedetailsofthistoChapter14Forclarity,thefollowingexampletreatsfuturesasiftheywerepureforwardcontracts7TheFarmerandtheBaker(Example)Jamelaisafarmerwithawheatcropofabout100,000bushels,1-monthfromharvestMohammedisabakerwhowillneedtorestockhisinventoryofwheatforthecomingyear8TheFarmerandtheBakerJamelaandMohammedwishtoreducepriceuncertaintybecause:Jamelahasamortgagetopayonherfarm,andisconcernedaboutwheatpricesfallinginthenextmonthMohammedwishestocloseanagreementwithasupermarkettosupplybreadatafixedpriceforthecomingyear9TheFarmerandtheBakerJamelaandMohammedagreetoaforwardcontractJamelaagreestodeliver100,000bushelsofwheatat$2.00abushelinonemonth,andMohammedagreestopaythe$200,000ondeliveryAssumingthecropdoesn'tfail,bothpartieshavehedgedtheirobligations10TheFarmerandtheBakerAssumethatJamelaandMohammedlivemilesapart,anddon’tknoweachotherJamelawritesaforwardcontractwithMs.Distributorat$1.99/bushelMohammedwritesaforwardcontractswithMr.Supplierat$2.01/bushelMs.Distributor,Mr.Supplier,andDr.Anotherhedgewithforwardcontractsat$2.00/bushel11TheFarmerandtheBakerMoveforwardamonthWheatpricesarenot$2.00abushel,but$2.20,duetowetconditionsinothergeographicregions12TheFarmerandtheBakerJamela’scropis110,000bushels,anditalsoexceedsthecontractedqualityShedeliversthecontracted100,000bfortheagreed$2.00/bushel,andreceives$200,000Shesellshersurplus10,000bat$2.20+$0.10(aqualitypremium)toalocalbaker,andreceives$23,000Thetotal=$223,00013TheFarmerandtheBaker:

AlternativeStrategyJamelabuys100,000Bushelsofdeliverablequalitywheat@$2.2/bfor-$220,000,deliversittoMs.Distributor,andreceivestheagreed$200,000.Lossof$20,000Shesellsher110,000bat$2.20+$0.10qualitypremiumtoalocalbakerandreceives$253,000Thetotal=$233,000($10,000more)14TheFarmerandtheBaker:

AlternativeStrategyTheproducttobedeliveredwastomeetorexceedacertainqualityJamelawouldhavebeenfoolishtodeliverherwheatwhenalowerqualitywheatwasavailablefordeliveryatalowerprice15TheFarmerandtheBaker:

AlternativeStrategyMohammedbakesapremiumbreadHetoocoulddeviseastrategytoreduceriskusingtheforwardcontract,butalsoreceivepremiumqualitywheat16TheFarmerandtheBaker:

NextDevelopmentTheforwardcontractsalwaysspecifytheminimumdeliverablequality,andoftenaformulafordeliveringotherqualitiesAmarketinforwardcontractsmaybedevisedthatencouragescashsettlement,anddiscouragesphysicaldelivery17TheFarmerandtheBaker:

NextDevelopmentJamelasellsforward200,000bfor1-monthdelivery@2.00/bushelThisshorthasnomonetaryvalueatthistimeAmonthlater,thespotpriceofdeliverablewheatis$2.20/b,andtheforwardisabouttoexpire,andsoisalsotradingat$2.20/bThecontractnowhasamonetaryvalue.Thelongpositionisworth$(2.20-2.00)/bushelJamelasettlesherpositionbypaying$20,00018TheFarmerandtheBaker:

NextDevelopmentJamelahasmadealossof$20,000onhertradeinthehypotheticalwheatforwardsmarketSherecoversthislosswhenshesellsherwheatThisisatruehedge.Shehaslosttheopportunitytoparticipateinariseinthepriceofwheatinreturnfordown-sideprotection19TheFarmerandtheBaker:

NextDevelopmentThebaker’shedgeintheforwardmarketresultedinasettlementof$20,000Whenhetakesphysicaldelivery,heexactlyoffsetshigherspotpriceswiththis$20,000HetradedtheopportunityoflowerwheatpricesforaknownpriceBothgained!Mohammed’sgain(atJamelaexpense)is20/20hindsight,andshouldbeirrelevanttobothofthem20TheFarmerandtheBaker:

NextDevelopmentOmarisrich,andwantstogetricherHepurchasesforward100,000bofwheat@$2.00/bushel,fordeliveryin1-monthAtmaturity,deliverablewheatcosts$2.20/bandhemakesacashsettlement,gaining$20,000Omarisaspeculatorprofitingfromhispurportedunderstandingofthemarket21TheFarmerandtheBaker:

NextDevelopmentRemawantstogetrichtooShesellsforward100,000bofwheat@$2.00/bushelfordeliveryin1-monthAtmaturity,deliverablewheatcosts$2.20/bbutsheisunablepaythe$20,000sheowesRema’sdefaultmustbemadegoodbyoneormoreofthemarket’sparticipants22TheFarmerandtheBaker:

ForwardstoFuturesTomitigatedefault-riskexposureRequireamodestsuretydepositbasedondailyvolatilityMarkcontracttomarketdaily(renderingthemtemporallyvalueless)Thesmallprofit/lossispayableimmediatelyRemainingproblem:Largedailypricemovements23TheFarmerandtheBaker:

ConclusionThefarmerandthebakerhavebotheliminatedspecificrisksthroughperfectlyanti-correlatedassetsSpeculatorsareexposedtoconsiderablerisk,hopingtoenjoyastatisticalprofitTheyprovideliquidityandexpertisethatpushthemarketfurthertowardsefficiency24Farmer’sTotalCashFlowsfromHedging

withFutures25RiskTransfer:ThreePointsWhetherthetransactionreducesorincreasesriskdependsupontheparticularcontextinwhichitisundertakenBothpartiestoarisk-reducingtransactioncanbenefit.Inretrospect,itmayseemasifoneofthepartieshasgainedattheexpenseoftheotherEvenwithnochangeintotaloutputnortotalrisk,redistributingthewaytheriskisbornecanimprovethewelfareoftheindividualsinvolved2611.2HedgingForeign-ExchangeRiskwithSwapContractsSwapContractanagreementbetweentwopartiestoexchangeaseriesofcashflows,atspecificintervals,overaspecifiedperiodoftimetheswappaymentsarebasedonanagreedprincipalamount(thenotionalamount)thereisnoimmediatepaymentofmoneytoeitherpartyascompensationforenteringthecontract27HedgingForeign-ExchangeRiskwithSwapContractsAswapmaycallfortheexchangeofanything,butmostswapsarefortheexchangeofcommoditiescurrenciessecurities’returns28CurrencySwapExampleYouhaveanagreementwithaGermansoftwaredistributorforthemtomarkettheGermanlanguageversionofyourfinancialderivativepricingprogramforapaymentofDM100,000/yearfor10yearsTohedgeforeignexchangerisk,immunizeyourfutureDMto$UStransactionsusingacurrencyswapagreement29CurrencySwapExampleThisswaparrangementisequivalenttoaseriesofforwardforeignexchangecontractsThenotionalamountintheswapcontractcorrespondstothefacevalueoftheimpliedforwardcontracts30CurrencySwapExampleYouarestillatriskaftertheswapDefault:ThereisaprobabilitythattheGermancompanywilldefaultonitsagreement,eitherbygoingbankrupt,orbyexercisingaperformanceclauseinthecontractDefaultdrivenExchangeRisk:Shoulddefaultoccur,youreacquireexchangeriskthroughtheresidualswapagreement31CurrencySwapExampleSupposethespotexchangerateis$0.50/DMYouandthecounterpartyagreethattheforwardexchangeratesshoulddeclinefromthecurrentspotby2%peryear(rounded)for5years,andthenremainstatic32CurrencySwapExampleTheforwardratesarethenagreedtobe:0.49,0.48,0.47,0.46,0.45,0.45,0.45,0.45,0.45,0.45Assumetheactualspotratesare:0.50,0.51,0.53,0.51,0.49,0.48,0.48,0.47,0.45,0.43Theflowsfrom/tothecounterpartyarecomputedas(Forward-Spot)*notionalamount33CurrencySwapExample(0.49-0.50)*100,000DM=$1,000(Year1)(0.48-0.51)*100,000DM=$3,000(Year2)(0.47-0.53)*100,000DM=$6,000(Year3)(0.46-0.51)*100,000DM=$5,000(Year4)(0.45-0.49)*100,000DM=$4,000(Year5)(0.45-0.48)*100,000DM=$3,000(Year6)(0.45-0.48)*100,000DM=$3,000(Year7)(0.45-0.47)*100,000DM=$2,000(Year8)(0.45-0.45)*100,000DM=$0,000(Year9)(0.45-0.43)*100,000DM=$2,000(Year10)34CurrencySwapExampleInthefirstyear,theagreedforwardrateislowerthantheactualspot,resultinginaflowfromyourUSA-basedcompanytothecounter-partyof$1,000ThesaleofDMyields100,000DM*0.50$/DM=$50,000Afterpaymenttocounterparty=$49,00035CurrencySwapExampleWhateverthespotrateinyear1,theagreementabsorbsthevariance,andyoureceiveanet$49,000(guaranteed)Assumingnodefault,theguaranteednetcashflows($’000)areYear1=49,year2=48,year3=47,year4=46,year5throughyear10=4536CurrencySwapExampleIfyoupreferadollarannuity,negotiate:afixedforwardratewiththecounterpartyAssumeasingleforwardrate(ratherthanaschedulebasedonmarketexpectationsformedfromthe$&DMyieldcurves).Withthepassageoftime,theexpectedmarketvalueoftheswapwilldivergemorefromzero.Thisgenerateshighercreditriskforthecounterparty,whichtranslatesintohighercostsforyouanewDMpaymentschedulethat,whencoupledwiththeswap,generatesaconstant$paymentschedule3711.3HedgingShortfall-RiskbyMatchingAssetstoLiabilitiesAssumeacreditunionborrowsusing1-yearCDs,andlendsusing30-yearmortgagesIfinterestratesrisethemarketvalueofthemortgageswillfallmortgagecashflowswon’tfullypaytheCDsResult?Insolvencyandruin!38MarketValueofMortgagesBookValueofMortgages39InterpretationFiveyearslater,interestrateswillhaverisenorfallenThemorerateshaverisenabovethemortgages’couponrate,thelargertheunrealizedcapitalloss,butliabilitiesremainessentiallyconstantThemorerateshavefallenbelowthemortgages’couponrate,thehighertheunrealizedcapitalgain,butthemorelikelyborrowerswillrefinance.(Graphassumesnorefinancing)40CDInterestPaymentsMortgageInterestPayments41InterpretationAsinterestratesrise,sotoodoestheratedemandedbythelendersThemortgageborrowerscontinuetoprovidethesamecashflowTheresultisareductioninthecashflowsthatservicetheCDs42RemedialActiontoPreventfurtherDamageMatchexposureofassetsandliabilitiesSellmortgages&investinshort-termlendingParticipateinGNMA,FNMA,…programsGetlenderstoinvestinlonger-termnotesLendusingadjustableratemortgagesIssuelonger-termbondsHedgeusinginterest-rateforwards,futures,options,orswaps4311.4MinimizingtheCostofHedgingTherearesometimesseveralwaystohedgeatransactionChoosetheonethatminimizesthecostofachievingthedesiredlevelofriskreductionafterconsideringtransactioncosts,andtaxes4411.5InsuringversusHedgingHedging:Acontract“topurchase100,000perfumebottles,sixmonthsfromnow@$0.25/bottle,paymentonreceipt”isaforwardcontract(obligationtopurchase)Insuring:Acontract“topurchaseupto100,000perfumebottles,sixmonthsfromnow@$0.25/bottle,paymentonreceipt”isanotaforwardcontract(rightbutnoobligationtopurchase)45InsuringversusHedgingRecallHedgingissymmetric,yousacrificetheupsiderisktoprotectyouagainstthedownsideriskInsuringisasymmetric,youmaintaintheupsiderisk,butdisposeofthedownsiderisk464711.6BasicFeaturesofInsuranceContractsExclusionsCapsDeductiblesCo-payments4811.7FinancialGuaranteesFinancialguaranteesareinsuranceagainstcreditrisk--therisktoyouthatthecounterpartywilldefaultAloanguaranteeisacontractthatobligestheguarantortomakethepromisedpaymentonaloaniftheborrowerfailstodoso4911.8CapsandFloorsonInterestRatesSomefinancialinstruments,suchasARMs,offeraninterestratethatvarieswithaspecifiedprimerate,theT-billrate,LIBOR,etceteraAclausemayprovideforannualfloors,annualcaps,globalfloors,orglobalcapsoninterestratechanges5011.9OptionsasInsuranceAcall(put)optionistheright,butnottheobligation,topurchase(sell)agivenassetaccordingtoascheduleofpricesandtimesEuropeanoptionshaveasinglestrikeorexerciseprice,andasingleexercisedateAmericanoptionshaveasinglestrikeorexerciseprice,andmaybeexercisedatanytimebeforetheirexpirationormaturitydate51OptionsasInsurance:PutIllustrationYouhave100sharesofXYZstockcurrentlytradingat$100/share,andaplanningtimehorizonof3-monthsYouwishtopayasmallpremiumtoinsurethecurrentpricein3-monthsYouwishtobenefitfromanystockpricerises52OptionsasInsurance:PutIllustrationStrategy:Retainyourholdingof100sharesinXYZcurrentlyvaluedat$10,000Purchaseoneroundlotof100XYZEuropeanputoptionswithastrikepriceof$100forapremiumof$729.51Attheendof3-months,yourholding,asafunctionofXYZnewshareprice,is:5354PutOptiononaBondThevalueofabonddependsupontherisk-freerateforbondsofthatmaturitythevalueofthebond’scollateralPurchasingaputoptiononthebondgivesdownsideprotection,whilepreservingupsidepotential5511.10TheDiversificationPrincipleDiversification:splittinganinvestmentamongmanyriskyassetsinsteadofconcentratingitallinonlyoneDiversificationPrinciple:bydiversifyingacrossriskyassetspeoplecansometimesachieveareductionintheiroverallriskexposurewithnoreductionintheirreturn56DiversificationofUncorrectedRisksAssumethatyouareofferedanumberofinvestmentopportunitiesinvariousbiotechnologyfirmsTheoutcomeofanyoftheinvestmentshasnoeffectonanyoftheothers(independent)Youbelievethateachfirmhasa50%chanceofquadruplingyourinvestment,anda50%chanceoftotalloss57Invest$100,000inanyoneofthefirms:Ifthefirmfails(p=0.50)theexpectedcontributiontoyourpay-outis0.50*$0=$0Ifthefirmissuccessful(p=0.50)theexpectedcontributiontoyourpay-outis0.5*4*$100,000=$200,000ExpectedPay-off=$0+$200,000=$200,00058Invest$50,000inanyofthefirms,&$50,000inanotherIfbothfirmfails(p=0.25)theexpectedcontributiontoyourpay-outis0.25*$0=$0Ifonefirmissuccessful(p=0.50)theexpectedcontributiontoyourpay-outis0.5*4*$50,000=$100,000Ifbothfirmaresuccessful(p=0.25)theexpectedcontributiontoyourpay-outis0.25*4*2*$50,000=$100,000ExpectedPay-off=$0+$100,000+100,000=$200,00059ConclusionInvestinginoneorintwofirmshasthesameexpectedreturnBut...60But...WehavenotanalyzedriskWewillnowcomputethestandarddeviationsofbothstrategies61Standarddeviation,1firmTheStandardDeviationis$200,00062Standarddeviation,2firmsTheStandardDeviationisabout$141,000(c.f.$200,000)63Standarddeviation,equalinvestmentin“n”firmsGeneralizingtheargument,itiseasytoprovethatthestandarddeviationinthiscaseisjust$200,000/SqrareRoot(n)Conclusion:Giventhefactsofthisexample,theriskmaybemadeasclosetozeroaswewishiftherearesufficientsecurities!Inreality,however…nismustbefinite,andpharmaceuticalprojectshaveanon-zerocorrelations64CorrelatedHomogeneousSecuritiesPharmaceuticalprojectsdohavepositivecorrelation(Why?)Loosentheassumptionsmadeaboutthecorrelation,andsetittoρ,andusethegeneralizationof65CorrelatedHomogeneousSecuritiesWeobtaintherelationshipσport=σsec*QSRT(ρ+1/n)Inthecaseofn->Infinity,thereremainstheterm σport=σsec*QSRT(ρ)Thisriskisnotdiversifiable 66676869DiversifiableSecurityRiskNondiversifiableSecurityRisk70AllriskisdiversifiableAllriskisdiversifiable71NegativeCorrelationNotethatasthecorrelationrangesfromonetozerothepercentageofundiversifiableriskfallsthenumberofsecuritiesnecessarytoapproachthislevelincreases…andjustforfun,let’slookatanegativecorrelation7273NondiversifiableRiskThegraphsillustrateanimportantpointForhomogeneoussecurities(atleast),thereisanasymptoticvaluefortheleastriskaportfoliomaycontainForcorrelationsthatarestrictlypositive,thereappearstobealevelofriskthatcan’tbediversifiedaway74NondiversifiableRiskForafundmanager,thecostofholdingherassetsineither(1)awelldiversifiedportfolio,or(2)asinglestock,differbyonly(quitelow)transactioncostsInthisworldofhomogeneoussecurities,shemayreduceriskbydiversifyingsomeoftheriskawayat(almost)nocost75ExcessStandardDeviationbyTime

andNumberofStocks76LanguageThefollowinggroupsofwordaresimilesDiversifiablerisk,individualrisk,security-specificrisk,irrelevantriskNondiversifiablerisk,marketrisk,relevantrisk7711.11Diversification&theCostofInsuranceWhenyoupurchaseinsurance,thepremium,p,maybedividedintotwoportionsa,theactuaryvalueofthegood-faithriskb,thesales,administration,profits,andfraudTheratioa/pnotalwaysashighasonewouldlike78Self-InsuranceAccordingly,acceptingsomeoftheriskyourselfmaybeadvisableinsomesituationsWhenthecorrelationbetweenrisksisnothighwhenthenumberofrisksisrelativelyhighwhentherisksareofthesamemagnitude79Chapter12:

PortfolioOpportunities andChoiceObjectiveTounderstandthetheoryofpersonalportfolioselectionintheoryandinpractice80Chapter12Contents12.1Theprocessofpersonalportfolioselection12.2Thetrade-offbetweenexpectedreturnandrisk12.3Efficientdiversificationwithmanyriskyassets81ObjectivesTounderstandtheprocessofpersonalportfolioselectionintheoryandpractice82IntroductionHowshouldyouinvestyourwealthoptimally?PortfolioselectionYourwealthportfoliocontainsStock,bonds,sharesofunincorporatedbusinesses,houses,pensionbenefits,insurancepolicies,andallliabilities83PortfolioSelectionStrategyTherearegeneralprinciplestoguideyou,buttheimplementationwilldependsuchfactorsasyour(andyourspouse’s)age,existingwealth,existingandtargetlevelofeducation,health,futureearningspotential,consumptionpreferences,riskpreferences,lifegoals,yourchildren’seducationalneeds,obligationstoolderfamilymembers,andahostofotherfactors8412.1TheProcessofPersonalPortfolioSelectionPortfolioselectionthestudyofhowpeopleshouldinvesttheirwealthprocessoftradingoffrisk&expectedreturntofindthebestportfolioofassets&liabilitiesNarrowerdfn:consideronlysecuritiesWiderdfn:housepurchase,insurance,debtBroaddfn:humancapital,education85TheLifeCycleTheriskexposureyoushouldacceptdependsuponyourageConsidertwoinvestments(rho=0.2)Security1hasavolatilityof20%andanexpectedreturnof12%Security2hasavolatilityof8%andanexpectedreturnof5%86PriceTrajectoriesThefollowinggraphshowthethepriceofthetwosecuritiesgeneratedbyabivariatenormaldistributionforreturnsThemoreriskysecuritymaybethoughtofasashareofcommonstockorastockmutualfundThelessriskysecuritymaybethoughtofasabondorabondmutualfund8788InterpretationoftheGraphThegraphisplottedonalogscaleinsothatyoucanseetheimportantfeaturesThemagentabondtrajectoryisclearlylessriskythanthenavy-bluestocktrajectoryTheexpectedpricesofthebondandthestockarestraightlinesonalogscale89InterpretationoftheGraphRecallthelogscale:thevolatilityincreaseswiththelengthoftheinvestmentYoubegintoformtheconjecturethatthechancesofthestockpricebeinglessthanthepricebondishigherinearlieryears90GeneratingMoreTrajectoriesThiswasjustoneofaninfinitenumberoftrajectoriesgeneratedbythesame2means,2volatilities,andthecorrelationIhavenotcheatedyou,thiswasindeedthefirsttrajectorygeneratedbythestatisticsthefollowingtrajectoriesarenotreorderednoreditedInstructor:Onslowercomputerstheremaybeadelay919293…andLotsMore!94FromConjecturetoHypothesisYouareprobablyreadytomakethehypothesisthattheprobabilityofthehigh-risk,high-returnsecuritywillout-performthelow-risk,low-returnincreaseswithtime95But:Ipromisedtobeperfectlyfrankandhonest(pfah)withyouabouttheorderingofthesimulatedtrajectoriesThenexttrajectorytrulywasthenexttrajectoryinthesequence,honest!9697ExplanationThebondandthestockendupataboutthesameprice,whentheexpectedpricesaremorethanamagnitudeapartThereiseitheraverygoodexplanationforthis,orthereisaveryhighprobabilitythatIhavebeenmuchlessthanperfectlyfrankandhonestwithyou98AnotherViewoftheModelAlittlemathematics,andweareabletogeneratethefollowingpricedistributionsforthestockandthebondfor2,5,10,and40yearsintothefuture99100Thereisalotgoingonhere,sowewillfurtherconstrainourviewFirstlookatstockpricesoveraperiodof10yearsThepricesaredistributedaccordingtothelognormaldistribution101102Notethescaleis$0to$800thedistributiondiffusesanddriftstowardshigherpriceswithtimethediffusionismorepronouncedintheearlieryearsthaninthelateryearsyoumayseethatthemode,median,andmeanappeartodriftapartwithtime103BondinTimeYouwillrecallthatifyouinvestina5-yeardefault-freepurediscountbondfor5years,thereturnisknownwithcertaintyToavoidthiseffect,assumeweinvestinshorttermbonds,androllthemoverastheymature104105Notethescaleisnow$0to$400(not$0to$800asinthecaseofthestock)weobservethesamekindofdiffusionanddriftbehavior,andthereislessofeach(remembertoadjustforthescale)106ContrastofTrajectoriesandDistributionsThepricedistributionsandthetrajectoriesweregeneratedfromthesamedistribution.ButTheydonotseemtoagreeThedistributionsappeartoproducemuchloweraverages(expectedreturns)thanthetrajectories107MeatyTailsTheresolutionisthatthedistributionshavemuchmeatiertailsthanyourintuitionallows,pushingthemedianandmeanfurtherandfurtherfromthemodewithtimeTheregionwherethelefttailappearstohavedriftedintoinsignificancehasaprofoundaffectonthemean108StockandBondsDistributionsComparedattheSameTimesThenextsequenceofslidescontraststhedistributionofstockandbondpricesat1,2,5,10,and40intothefutureSomeoftheslideshavedifferentmeasuresofcentraltendencyindicatedNotethebehaviorofthesestatisticsastimeincreases109Mode=104Mode=106Median=104Mean=104Median=111Mean=113110111Mode=122Mode=135Median=126Mean=128Median=165Mean=182112113Mode=503Mode=1,102Median=650Mean=739Median=5,460Mean=12,151114SlideSequenceSummaryThenexttablesummarizesthedriftsofthemeasuresofcentraltendencyNotethatthemeansdoinfacttiebacktothetrajectoriesThelast(anomalous?)trajectorynotanuncommonoccurrence,andIwaspfahwithyou115116ImplicationforInvestorsIfyouareolder,theaverageremaininglifeoftheinvestmentisrelativelyshort,andthereisalargerprobabilitythataninvestmentintheriskysecuritywillresultinalossThisisnotseriousifyouhavesubstantialassets,inwhichcaseyoucanaffordtotaketherisk,andenjoyhigherexpectedreturns117ImplicationforInvestorsIfyouareyounger,theaverageremaininglifeofretirementinvestmentislonger,andthereisonlyasmallprobabilitythataninvestmentintheriskysecuritywillbelessthanthe“safer”oneInvestinginthelessriskysecuritywillalmostalwaysresultinasignificantlysmallerretirementincome118ImplicationforInvestorsRelativelyearlyduringatypicallifecycle,theremaybeaneedtoliquidatesomeinvestedfunds,perhapsforahousedeposit,achild’seducation,oranuninsuredmedicalemergencyInthecasewhereliquidatinganinvestmentearlymaydamagelong-termgoals,someprecautionaryfundsshouldbekeptinlower-risksecurities119TimeHorizonsPlanninghorizonThetotallengthoftimeforwhichoneplansDecisionhorizonThelengthoftimebetweendecisionstoreviseaportfolioTradinghorizonTheshortestpossibletimeintervaloverwhichinvestorsmayrevisetheirportfolios120ComputingLifeExpectancyMortalitytablesmaybeorganizedasthreecolumns:actuaryage,deaths/yearper1000livebirths,andremaininglifeexpectation.Note:ifyousurvivefrom60to65,forexample,theexpecteddate