《金融学》（第二版） 课件 （英文） -Chapter4-6 Allocating Resources Over Time；Household Saving；nvestment Projects

Chapter4:

AllocatingResources OverTimeObjectiveExplaintheconceptofcompoundinganddiscountingandtoprovideexamplesofreallifeapplications1Note:ThisslideshowdoesnotfollowthebookascloselyasshowsfortheotherchaptersStudentsgettingtimevalueofmoneyforthefirsttimeneedtodoubleortripleexposuretothisbasicskillTherearemanyexamplesattheendofthiscollectionorganizedintosubsectionsstartingatslide95of2632Introduction:TimeValueofMoney(TVM)

$20todayisworthmorethantheexpectationof$20tomorrowbecause:abankwouldpayinterestonthe$20 inflationmakestomorrows$20lessvaluablethantoday’suncertaintyofreceivingtomorrow’s$2034.1CompoundingAssumethattheinterestrateis10%p.a.Whatthismeansisthatifyouinvest$1foroneyear,youhavebeenpromised$1*(1+10/100)or$1.10nextyearInvesting$1foryetanotheryearpromisestoproduce1.10*(1+10/100)or$1.21in2-years4ValueofInvesting$1Continuinginthismanneryouwillfindthatthefollowingamountswillbeearned:5Valueof$5InvestedMoregenerally,withaninvestmentof$5at10%weobtain6GeneralizingthemethodGeneralizingthemethodrequiressomedefinitions.LetibetheinterestratenbethelifeofthelumpsuminvestmentPVbethepresentvalueFVbethefuturevalue7FutureValueandCompoundInterest8FutureValueofaLumpSum9Example:FutureValueofaLumpSumYourbankoffersaCDwithaninterestrateof3%fora5yearinvestment.Youwishtoinvest$1,500for5years,howmuchwillyourinvestmentbeworth?10

RULEOF72

Thisrulesaysthatthenumberofyearsittakesforasumofmoneytodoubleinvalue(“thedoublingtime”)isapproximatelyequaltothenumber72dividedbytheinterestrateexpressedinpercentperyearDoublingTime= 72

InterestRate11Hint:Roundingiscommonsense“plus”traditionItsimportanttoroundappropriatelyIna$7billionsproject,roundingmightbetothenearest$’000,000Yourcheckbookshouldberoundedto$0.01Inanaccountingsituation,anyunexpectederror,howeversmallcouldbetheresultoftwolargercompensatingerrors.Accordingly,theyneedtoberesolvedAvoidanytruncationwithinacalculation12Hint:AvoidanytruncationwithinacalculationEngineersstudynumericalanalysis.Infact,itissoimportant,theymaytakeseveralcourses.Fewfinancefolkhaveanyideaofcomputationaldangers.FornowyoushouldbesafeifyouAvoidremovingintermediateresultsfromyourcalculator.Storetheminamemoryregister.ThisavoidsinputandoutputcopyingerrorsLearntousethe“stack”orbracketsprovidedbyyourfinancialcalculator.YourcalculatorprobablykeepsamoreaccurateversionofdisplayednumbersinternallyInnocaseshouldyouevertruncateanintermediatecomputationunlessyoufullyunderstandtheaffectonaccuracy(Youprobablydon’t!)13PresentValueofaLumpSum14Example:PresentValueofaLumpSumYouhavebeenoffered$40,000foryourprintingbusiness,payablein2years.Giventherisk,yourequireareturnof8%.Whatisthepresentvalueoftheoffer?15LumpSumsFormulaeYouhavesolvedapresentvalueandafuturevalueofalumpsum.Thereremainstwoothervariablesthatmaybesolvedforinterest,inumberofperiods,n16SolvingLumpSumCashFlowforInterestRate17Example:InterestRateonaLumpSumInvestmentIfyouinvest$15,000fortenyears,youreceive$30,000.Whatisyourannualreturn?18ReviewofLogarithmsThenextthreeslidesareaquickreviewoflogarithmsIknowthatyouprobablylearnedthisineighthgrade,butthoseofuswhodonotusethemfrequentlyforgetthebasicrules19Logarithmsareimportantinfinancebecausegrowthisrelatedtothetheexponential,andtheexponentialistheinversefunctionofthelogarithmLogarithmsmayhavedifferentbases,butinfinanceweneedonlythenaturallogarithm,thatisthelogarithmofbasee.Theeistheirrationalnumberthatmaybeapproximatedas2.718281828.Itiseasytorememberbecauseitstartstorepeat,butdon’tbefooled,itdoesn't,anditis

irrational20ReviewofLogarithmsThebasicpropertiesoflogarithmsthatareusedbyfinanceare:21ReviewofLogarithmsThefollowingpropertiesareeasytoprovefromthelastones,andareusefulinfinance22SolvingLumpSumCashFlowforNumberofPeriods234.2TheFrequencyofCompoundingYouhaveacreditcardthatcarriesarateofinterestof18%peryearcompoundedmonthly.Whatistheinterestratecompoundedannually?Thatis,ifyouborrowed$1withthecard,whatwouldyouoweattheendofayear?24TheFrequencyofCompounding18%peryearcompoundedmonthlyisjustcodefor18%/12=1.5%permonthAllcalculationmustbeexpressedintermsofconsistentunitsArawrateofinterestexpressedintermsofyearsandmonthsmayneverbeusedinacalculation25TheFrequencyofCompoundingTheannualratecompoundedmonthlyiscodeforonetwelfthofthestatedratepermonthcompoundedmonthlyTheyearisthemacroperiod,andthemonthisthemicroperiodInthiscasethereare12microperiodsinonemacroperiod26TheFrequencyofCompoundingWhenarateisexpressedintermsofamacroperiodcompoundedwithadifferentmicroperiod,thenitisanominalorannualpercentagerate(APR)Ifmacroperiod=microperiodthentherateisreferredtoasatherealoreffectiveratebasedonthatperiod27TheFrequencyofCompoundingAssumemmicroperiodsinamicroperiodandanominalratekpermacroperiodcompoundedmicro-periodically.Thatistheeffectiverateisk/mpermicroperiod.Invest$1foronemacroperiodtoobtain$1*(1+k/n)n,producinganeffectiverateoverthemacroperiodof($1*(1+k/n)n-$1)/$1=(1+k/n)n-128CreditCardIfthecreditcardpaysanAPRof18%peryearcompoundedmonthly.The(real)monthlyrateis18%/12=1.5%sotherealannualrateis(1+0.015)12-1=19.56%ThetwoequalAPRwithdifferentfrequencyofcompoundinghavedifferenteffectiveannualrates:29EffectiveAnnualRatesofanAPRof18%30TheFrequencyofCompoundingNotethatasthefrequencyofcompoundingincreases,sodoestheannualeffectiverateWhatoccursasthefrequencyofcompoundingrisestoinfinity?31TheFrequencyofCompoundingTheeffectiveannualratethat’sequivalenttoanannualpercentagerateof18%isthene0.18-1=19.72%Moreprecisionshowsthatmovingfromdailycompoundingtocontinuouscompoundinggains0.53ofonebasispoint32TheFrequencyofCompoundingAbankdeterminesthatitneedsaneffectiverateof12%oncarloanstomediumriskborrowersWhatannualpercentageratesmayitoffer?33TheFrequencyofCompounding34TheFrequencyofCompounding35TheFrequencyofCompoundingManylendersandborrowersdonothaveaclearunderstandingofAPRs,butinstitutionallendersandborrowersdoInstitutionsarethereforeabletoextractafewbasispointsfromconsumers,butwhybother?36TheFrequencyofCompoundingFinancialintermediariesprofitfromdifferencesinthelendingandborrowingrates.Overheads,badloansandcompetitionresultsinanarrowmargin.SmallrategainsthereforeresultinalargeincreasesininstitutionalprofitsInthelongterm,ill-informedconsumerslosebecauseofcompounding374.5MultipleCashFlowsTimeLinesFutureValueofaStreamofCashFlowPresentValueofaStreamofCashFlowsInvestingwithMultipleCashFlows38TimeLine39PresentValueofMultipleCashFlows404.6AnnuitiesFinancialanalystsuseseveralannuitieswithdifferingassumptionsaboutthefirstpayment.Wewillexaminejusttwo:regularannuitywithitsfirstcoupononeperiodfromnow,(detaillook)annuityduewithitsfirstcoupontoday,(cursorylook)41CashFlowDiagramofAnnuities42RationaleforAnnuityFormulaasequenceofequallyspacedidenticalcashflowsisacommonoccurrence,soautomationpaysoffatypicalannuityisamortgagewhichmayhave360monthlypayments,alotofworkforusingelementarymethods43AssumptionsRegularAnnuitythefirstcashflowwilloccurexactlyoneperiodformnowallsubsequentcashflowsareseparatedbyexactlyoneperiodallperiodsareofequallengththetermstructureofinterestisflatallcashflowshavethesame(nominal)valuethepresentvalueofasumofpresentvaluesisthesumofthepresentvalues44AnnuityFormulaNotationPV=thepresentvalueoftheannuityi=interestratetobeearnedoverthelifeoftheannuityn=thenumberofpaymentspmt=theperiodicpayment45DerivationofPVofAnnuityFormula:Algebra.1of546DerivationofPVofAnnuityFormula:Algebra.2of547DerivationofPVofAnnuityFormula:Algebra.3of548DerivationofPVofAnnuityFormula:Algebra.4of549DerivationofPVofAnnuityFormula:Algebra.5of550PVofAnnuityFormula51PVAnnuityFormula:Payment52PVAnnuityFormula:NumberofPayments53PVAnnuityFormula:ReturnThereisnotranscendentalsolutiontothePVofanannuityequationintermsoftheinterestrate.StudentsinterestedinthereasonwhyarereferredtoGaloisTheory,2nd.EdI.Stewart.Studentswithastrongersenseoffashionare“seen”carryingMichioKuga’spoison-ivy-green-coloredbook“GaloisDream.”54AnnuityFormula:PVAnnuityDue55DerivationofFVofAnnuityFormula:Algebra56FVAnnuityFormula:Payment57FVAnnuityFormula:NumberofPayments58FVAnnuityFormula:ReturnThereisnotranscendentalsolutionNumericalmethodshavetobeemployed594.7PerpetualAnnuitiesRecalltheannuityformula:Letn->infinitywithi>0:60GrowingAnnuitiesGrowingannuitiessolvethesuper-normalgrowthproblemTheyareoftenmoreappropriateinday-to-daysituationsthanannuities614.8LoanAmortization:Mortgageearlyrepaymentpermittedatanytimeduringmortgage’s360monthlypaymentsmarketinterestratesmayfluctuate,buttheloan’srateisaconstant1/2%permonththemortgagerequires10%equityand“threepoints”assumea$500,000houseprice62Mortgage:ThepaymentWewillexaminethisproblemusingafinancialcalculatorThefirstquantitytodetermineistheamountoftheloanandthepoints63CalculatorSolutionThisisthemonthlyrepayment64Mortgage:EarlyRepaymentAssumethatthefamilyplanstosellthehouseafterexactly60payments,whatwillbetheoutstandingprinciple?65MortgageRepayment:IssuesTheoutstandingprincipleisthepresentvalue(atrepaymentdate)oftheremainingpaymentsonthemortgageThereareinthiscase360-60=300remainingpayments,startingwiththeone1-monthfromnow66CalculatorSolutionOutstanding@60Months67SummaryofPaymentsThefamilyhasmade60payments=$2687.98*12*5=$161,878.64Theirmortgagerepayment= 450,000-418,744.61=$31,255.39Interest=payments-principlereduction=161,878.64-31,255.39=$130,623.2568AvoidAddingCashFlowsFromDifferentPeriodsIntheaboveslide,webrokeoneofthecardinalrulesoffinance:Webundledthecashflowsfor5-yearsbyaddingthemtogetherThiskindofanalysiscanleadtoinappropriatefinancialdecisions,suchasearlyrepaymentofamortgage69AResultofBreakingtheRuleGiventhetaxadvantagesofamortgage,andthefactitcollateralized,theirinterestratesarequitelowSomefinancialpunditsrecommendadding(say10%)tomonthlypaymentstoreducethemortgagelifeby5-to10-yearsAtyourage,investingthatextra10%inamutualfundmaybemoreappropriate70AResultofBreakingtheRuleThepunditsmaketheirargumentbyadding(withoutdiscounting!)thedifferenceinthecashflowsbetweenthescenarios.Thisistypicallyahugesumofmoney,andthisiswhatis“saved”Whendiscountedappropriately,therearenosignificantsavings.Therearehugeopportunitylossesforthosewillingtoaccepttheriskofastockmutualfund71OutstandingBalanceasaFunctionofTimeThefollowinggraphsillustratethatintheearlyyears,monthlypaymentaremostlyinterest.Inlatteryears,thepaymentsaremostlyprincipleRecallthatonlytheinterestportionistax-deductible,sothetaxshelterdecays72737475764.9ExchangeRatesandTimeValueofMoney

Youareconsideringthechoice:Investing$10,000indollar-denominatedbondsoffering10%/yearInvesting$10,000inyen-denominatedbondsoffering3%/yearAssumeanexchangerateof0.0177$10,000$11,000¥1,000,000¥1,030,000¥Time10%$/$(direct)0.01$/¥3%¥/¥?$/¥U.S.A.Japan78ExchangeRateDiagramReviewofthediagramindicatesthatyouwillendtheyearwitheither$11,000or¥1,030,000Ifthe$priceoftheyenrisesby8%/yearthentheyear-endexchangeratewillbe$0.0108/¥79$10,000$11,124$11,000¥1,000,000¥1,030,000¥Time10%$/$(direct)0.01$/¥3%¥/¥0.0108$/¥U.S.A.Japan80InterpretationandAnotherScenarioInthecaseofthe$priceof¥risingby8%yougain$124onyourinvestmentNow,ifthe$priceof¥risesby6%,theexchangerateinoneyearwillbe$0.010681$10,000$10,918¥$11,000¥1,000,000¥1,030,000¥Time10%$/$(direct)0.01$/¥3%¥/¥0.0106$/¥U.S.A.Japan82InterpretationInthiscase,youwilllose$82byinvestingintheJapanesebondIfyoudivideproceedsoftheUSinvestmentbythoseoftheJapaneseinvestment,youobtaintheexchangerateatwhichyouareindifferent$11,000/¥1,030,000=0.1068$/¥83$10,000$11,000¥$11,000¥1,000,000¥1,030,000¥Time10%$/$(direct)0.01$/¥3%¥/¥0.01068$/¥U.S.A.Japan84ConclusionIftheyenpriceactuallyrisesbymorethan6.8%duringthecomingyearthentheyenbondisabetterinvestment85FinancialDecisioninanInternationalContextInternationalcurrencyinvestorsborrowandlendinTheirowncurrencyThecurrencyofcountrieswithwhichtheydobusinessbutwishtohedgeCurrenciesthatappeartoofferabetterdealExchangeratefluctuationscanresultinunexpectedgainsandlosses86ComputingNPVinDifferentCurrenciesInanytime-value-of-moneycalculation,thecashflowsandinterestratesmustbedenominatedinthesamecurrencyUSAprojectUrequiresaninvestmentof$10,000,asdoesaJapaneseprojectJ.Ugenerates$6,000/yearfor5years,andprojectJgenerates¥575,000/yearfor5yearsTheUSinterestis6%,theJapaneseinterestis4%,andthecurrentexchangerateis0.0187SolutionUsingyourfinancialcalculatorDeterminethepresentvalueofUin$bydiscountingthe5paymentsat6%,andsubtracttheinitialinvestmentof$10,000DeterminethepresentvalueofJin¥bydiscountingthe5paymentsat4%,andsubtracttheinitialinvestmentof¥1,000,000Obtain$15,274&¥1,5599,798respectively88SolutionConvertthe¥1,5599,798to$usingthecurrentexchangeratetoobtain$15,600TheJapaneseNPVof¥of$15,600ishigherthantheUSANPVor$15,274,soinvestintheJapaneseproject894.10InflationandDiscountedCashFlowAnalysisWewillusethenotationIntherateofinterestinnominaltermsIrtherateofinterestinrealtermsRtherateofinflationFromchapter2wehavetherelationship90IllustrationWhatistherealrateofinterestifthenominalrateis8%andinflationis5%?TherealrateorreturndeterminesthespendingpowerofyoursavingsThenominalvalueofyourwealthisonlyasimportantasitspurchasingpower91InvestinginInflation-protectedCD’sYouhavedecidedtoinvest$10,000forthenext12-months.YouareofferedtwochoicesAnominalCDpayinga8%returnArealCDpaying3%+inflationrateIfyouanticipatetheinflationbeingBelow5%investinthenominalsecurityAbove5%investintherealsecurityEqualto5%investineither92WhyDebtorsGainFromUnanticipatedInflation

Youborrow$10,000at8%interest.Thetoday’sspendingpoweroftherepaymentis$10,000*1.08/(1+inflation)Iftheactualinflationistheexpected6%,thentherealcostoftheloanintoday’smoneyis$10,188.68Iftheactualinflationis10%,thentheloan’srealcost(intoday’svalues)is$9,818.18Unexpectedinflationbenefitsborrower93InflationandPresentValue

AcommonplanningsituationisdetermininghowlongittakestosaveforsomethingTheproblemisthatthethingbeingsavedforincreasesin(nominal)priceduetoinflationUsingarealapproachsolvesthisissue94InflationandPresentValue

IllustrationAssumethataboatcosts$20,000todayGeneralinflationisexpectedtobe3%Attoday’svalues,youcansaveataninflationadjustedrateof$3,000/year,makingthefirstdeposit1-yearhenceYouareabletoearn12%loansatHonestJoe’sPawnEmporium

®Whenistheboatyours?95BoatIllustrationContinuedSolutionTheboatisalreadyatnominalvalueToconvertthenominalratetotherealrateIreal=(Inominal-inflation)/(1+inflation)=(0.12-0.03)/1.03=8.7378641%UsingyourcalculatorN->?;I->8.7378641;PV->0;PMT->3000;FV->20000“=/-”;Result:n=5.48years,(6yearsw/change)96BoatIllustrationContinuedConclusionGivenboatermakesdepositsattheendofeachyear,theboatwillnotbehersforsixyearsLookattheproblemfromanominalvantage:97BoatIllustration(Nominal)98InflationandSavingsPlansWehaveseenhowtocomputethenumberofyearsittakestosaveforsomethingusingbothrealandnominalmethodsAnotherimportantquestionisHowmuchmustIsaveeachyearinordertoachieveasavingsgoal?Wewillreusetheboatproblem,butwiththeassumptionthattheboateriswillingtowait8-years,butwishestominimizeannual(inflationadjusted)payments994.11TaxesandInvestmentDecisionsRule:Investsoastomaximizeyourafter-taxrateofreturnThisisnotatallthesamethingas Minimizethetaxyoupay(False)100InvestinginTax-ExemptBondsIntheUSA,municipalbondsareexemptfromincometaxesUnderwhatcircumstanceswouldyoubeindifferenttoinvestinginanidenticalbondthatpaystaxifyourmarginalrateoftaxis(say)20%?101AdditionalSolvedProblemsLumpSumFutureValue102TheProblemYou'vereceiveda$40,000legalsettlement.Yourgreat-unclerecommendsinvestingitforretirementin27-yearsby“rollingover”one-yearcertificatesofdeposit(CDs)Yourlocalbankhas3%1-yearCDsHowmuchwillyourinvestmentbeworth?Comment.103CategorizationYourcapitalgainswillbereinvested.Thereisnocash-flowfromthesettlementfor27years,sothisisalumpsumproblem.Thereissomeuncertaintyinthecashflowsbecauseinterestratearestaticforjustthefirstyear,butweassumethatitwillbe3%untilyouretireIfyouareunabletoshelteryourearnings,theIRSwillwanttheircut104DataExtractionPV=$40,000i=3%(or3%*(1-marginaltaxrate)?)n=27-yearsFV=?105SolutionbyEquation106CalculatorSolution107CommentsYourgreatuncle'safinancialidiotGivena27-yearinvestment,youshouldeitherInvestthemoneymoreaggressivelytoaccumulatethemoneyyouneedtosurvive,orLive!Blowthemoneyonthatredconvertible!1083AdditionalSolvedProblemsLumpSumInterestRate109Problem1Ifyouhavefiveyearstoincreaseyourmoneyfrom$3,287to$4,583,atwhatinterestrateshouldyouinvest?110AlgebraicSolution111Problem2Aninvestmentyoumade12-yearsagoistodayworthitspurchaseprice.Ithasneverpaidadividend.Closerinspectionrevealsthatthesharepricehasbeenhighlyperiodic,movingfrom$150whenpurchased,to$300inthenextyear,to$75inthenext,backto$150,beforerepeating11211312-YearandAverageReturnsComparewithAverageHPR114CommentsHerewehavetheaverageholdingperiodreturnbeing41.67%peryear,whilethesecurityhasreturnedyounothingoverthewholeperiod!AveragesseduceuswiththeirintuitivenessThecorrectaveragetohaveusedwasthegeometricaverageofreturnfactors,notthearithmeticaverageofreturnrates115AveragesMustbeMeaningful1Youwalk1mileat2mphandanotherat3mph.Whatwasyouraveragespeed?(2+3)/2=2.5mph.NO!Thefirstleglasts1/2hour,andthesecondleglasts1/3hours,total5/6hours.Soaveragespeedis2/(5/6)=2.4mph.116AveragesMustbeMeaningful2AlittleanalysisshowsthatthecorrectmeanforthewalkeristheharmonicmeanThecorrectmeanforthereturnproblemmaybeshowntobethegeometricmeanofthe(1+return)’sTheappropriatemeanrequiresthought117Problem3In1066theFirstDukeofOxbridgewasawardedasquaremileofLondonforhisservicesinassistingtheconquesttheEngland.The30thDukewishedtoliveafasterpacedlife,andsoldhisholdingin1966for£5,000,000,000.Examinationoforiginalproject’scostshowedonlytheentry“1066a.d.:torepairarmor,£5”Whatwasrateofcapitalappreciation?118CategorizationWemayassumethattheDukeslivedquitewellfromleasinglandtotheirtenants,butwearenotinterestedintherevenuecashflowshere,justthecapitalcashflowsThereisapresentcashflow,afuturecashflow,andnoannuitypayments,sotheproblemisthereturnonalump-suminvestedforanumberofperiods119DataExtractionPV=10FV=5,000,000,000n=(1966-1066)=1900i=?120SolutionbyEquation121SolutionbyCalculator122CommentsNotethatacapitalgainofonly1.1%peryearresultsinahugevalueovertimeTimeplusreturnisverypotentTherealissuehereiswhatismissing,namelytherevenuestreams123AdditionalSolvedProblemsLumpSumNumberofperiods124TheProblemHowmanyyearswouldittakeforaninvestmentof$9,284togrowto$22,450iftheinterestrateis7%p.a.?p.a.=perannum=peryear125CategorizationThisisalumpsumproblemaskingforasolutionintermsoftime.Mostoftheseproblemsareusefulmodelsofrealityifexpressedinrealterms,notnominaltermsInanynominalsituation,theterminal$22,450willnotbeaconstant,butwilldependontheunknowntimeWewillassumethatthenumbersandratesareinrealterms126DataExtractionPV=$9,284FV=$22,450i=7%p.a.n=?127SolutionbyEquation128AdditionalSolvedProblemsLumpSumPresentValue129TheProblemIfinvestmentratesare1%permonth,andyouhaveaninvestmentthatwillproduce$6,000onehundredmonthsfromnow,howmuchisyourinvestmentworthtoday?130CategorizationThisisthemostbasicoffinancialsituations,andinvolvesfindingthepresentvalueofafuturepaymentgivennoperiodicpaymentsTheissueofriskisalittlefuzzy.Itisassumedthattherategivenisfortheproject’sriskcategory131DataExtractionFV=$6000PV=?n=100monthsi=1%132SolutionbyEquation133CalculatorSolution134AdditionalSolvedProblemsLumpSumSpecialCase:DoublingRuleof72135TheProblemConsiderthefollowingsimpleexample:SolCooperInvestmentshaveofferedyouadeal.Investwiththemandtheywilldoubleyourinvestmentin10years.Whatinterestratearetheyofferingyou?Wecouldsolvethisusing butthisisover-kill136DataExtractionDoublingn=10i=?137SomeAlgebra138SolutionbyEquation139TheSecretReveledNowyouhaveseenthederivationoftheruleof72,youarenowabletoproduceyourownpersonalrules.Example:“TheRuleofaMagnitude”Toincreaseyourwealthby10times,theproductofinterestandtimeis240,thatisabout(2.08/2)*ln(10)Example,howlongwillittaketoincreaseyourmoneytentimes,giveninterestratesof10%?N=240/10=24years,realansweris24.16years140HowgoodistheRuleof72?Wehavederivedaruleusingapproximationmethods,buthavenoideahowaccurateitisTherearetwotestswecouldapplywecouldtakesomerange,anddeterminetheabsolutemaximumerroroftheruleinthatrangewecouldsimplygraphtheerrorGraphsarefun:141142143GraphofRuleof72ErrorThehigherrorinapartofthegraphthatdoesnotinterestusishidingtheerrorinthepartthatdoes.Wehavetwochoicesplotabsoluteerroronalogscaletruncatethegraphandre-scaleTruncationisfun144145AnotherExampleYouareastockbrokerwishingtopersuadeayoungclienttoreconsiderher$50,000investedin3%-CDs.Yourclientbelievesthatstockmutualfundswillreturnabout12%fortheforeseeablefuture,butisaversetothevolatilityrisks.Hermoneywillremainfullyinvestedforthenext48years.146Step1ThefirststeprequiresthecalculationofhowlongisrequiredtoobtainasingledoublingCDs:72/3=24yearstodoubleMutualfund:72/12=6yearstodouble147Step2ThesecondsteprequiresthecalculationofhowmanydoublingswilloccurduringthelivesoftheinvestmentsCDs:48/24=2doublingsMutualfund:48/6=8doublings148Step3Thethirdstepcalculatesthevalueoftheinvestmentin48yearsCDs:2doublingsof$50,000=$200,000Mutualfund:8doublingof$50,000=256*$50,000=$12,800,000in48years149ConclusionWeshalldiscoverthatherriskissmallerthansheimagines,butshewillbeabout64timesmorewealthyifsheacceptsthatriskUsingtheaccuratemethod,herrespectivewealthsare$206,613and$11,519,539,Thelessonistostarttoinvestearly,andacceptsomerisk150Growthat3and12%Thefollowinggraphshowsherwealthincreasesover10yearsata3%and12%Thegraphwascutat10yearsbecausethe12%rateofgrowthissolargethatitdwarfsthe3%growth,makingthegraphmeaningless151Growthof$50,000for10Years@3%and12%152LogTransformationofY-AxisAcommonwaytoplottwosuchcashflowsonthesamegraphistouseasemi-loggraph.ThispreventsscaleproblemsfromhidingoneofthegraphsNotethatthetwographsappeartobestraightlines,andthisisinfactthecase153Growthof$50,000at3%and12%for48Years(LogScale)154WhatistheuseoftheRule?Asignificantsourceofavoidableerrorinfinancialcalculationsresultsfromblindly“runningthenumbers”withoutreviewingthemforempiricalreasonablenessItisagoodpracticetoestimatevaluesbeforecomputingthemTheruleof72isonetoolthatsometimesgivesyou“numericalfeel”ofaproblem155Yourreactiontolearningtheruleof72is“Whybother,I’vegotthelatestandbestHPfinancialcalculator.”Inabusinessmeeting,theunilateraldrawingofafinancialcalculatorhasachillingeffectonyouropponentsflexibilityinanegotiationItisamazinghowmanyrealproblemsyoucansolveinyourheadusingtheruleof72156AdditionalSolvedProblemsIrregularCashFlowsBackwards