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QUANTITATIVE INVESTMENT ANALYSIS

CFAInstitute isthepremierassociationforinvestmentprofessionalsaroundtheworld,with over160,000membersin165countriesandterritories.Since1963theorganizationhas developedandadministeredtherenownedCharteredFinancialAnalyst®Program.Witha richhistoryofleadingtheinvestmentprofession,CFAInstitutehassetthehigheststandards inethics,education,andprofessionalexcellencewithintheglobalinvestmentcommunity andistheforemostauthorityoninvestmentprofessionconductandpractice.Eachbook intheCFAInstituteInvestmentSeriesisgearedtowardindustrypractitionersalongwith graduate-levelfinancestudentsandcoversthemostimportanttopicsintheindustry.The authorsofthesecutting-edgebooksarethemselvesindustryprofessionalsandacademics andbringtheirwealthofknowledgeandexpertisetothisseries.

QUANTITATIVE INVESTMENT ANALYSIS

FourthEdition

RichardA.DeFusco,CFA

DennisW.McLeavey,CFA

JeraldE.Pinto,CFA

DavidE.Runkle,CFA

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CHAPTER1

TheTimeValueofMoney1 LearningOutcomes1

1.Introduction1

2.InterestRates:Interpretation2

3.TheFutureValueofaSingleCashFlow4

3.1.TheFrequencyofCompounding9

3.2.ContinuousCompounding11

3.3.StatedandEffectiveRates12

4.TheFutureValueofaSeriesofCashFlows13

4.1.EqualCashFlows OrdinaryAnnuity14

4.2.UnequalCashFlows15

5.ThePresentValueofaSingleCashFlow16

5.1.FindingthePresentValueofaSingleCashFlow16

5.2.TheFrequencyofCompounding18

6.ThePresentValueofaSeriesofCashFlows20

6.1.ThePresentValueofaSeriesofEqualCashFlows20

6.2.ThePresentValueofanInfiniteSeriesofEqualCashFlows Perpetuity24

6.3.PresentValuesIndexedatTimesOtherthant=025

6.4.ThePresentValueofaSeriesofUnequalCashFlows27

7.SolvingforRates,NumberofPeriods,orSizeofAnnuityPayments27

7.1.SolvingforInterestRatesandGrowthRates28

7.2.SolvingfortheNumberofPeriods30

7.3.SolvingfortheSizeofAnnuityPayments31

7.4.ReviewofPresentandFutureValueEquivalence35

7.5.TheCashFlowAdditivityPrinciple37

8.Summary38 PracticeProblems39

CHAPTER2

Organizing,Visualizing,andDescribingData45

LearningOutcomes45 1.Introduction45

2.DataTypes46

2.1.NumericalversusCategoricalData46

2.2.Cross-SectionalversusTime-SeriesversusPanelData49

2.3.StructuredversusUnstructuredData50

3.DataSummarization54

3.1.OrganizingDataforQuantitativeAnalysis54

3.2.SummarizingDataUsingFrequencyDistributions57

3.3.SummarizingDataUsingaContingencyTable63

4.DataVisualization68

4.1.HistogramandFrequencyPolygon68

4.2.BarChart69

4.3.Tree-Map73

4.4.WordCloud73

4.5.LineChart75

4.6.ScatterPlot77

4.7.HeatMap81

4.8.GuidetoSelectingamongVisualizationTypes82

5.MeasuresofCentralTendency85

5.1.TheArithmeticMean85

5.2.TheMedian90

5.3.TheMode92

5.4.OtherConceptsofMean92

6.OtherMeasuresofLocation:Quantiles102

6.1.Quartiles,Quintiles,Deciles,andPercentiles103

6.2.QuantilesinInvestmentPractice108

7.MeasuresofDispersion109

7.1.TheRange109

7.2.TheMeanAbsoluteDeviation109

7.3.SampleVarianceandSampleStandardDeviation111

7.4.TargetDownsideDeviation114

7.5.CoefficientofVariation117

8.TheShapeoftheDistributions:Skewness119

9.TheShapeoftheDistributions:Kurtosis121

10.CorrelationbetweenTwoVariables125

10.1.PropertiesofCorrelation126

10.2.LimitationsofCorrelationAnalysis129 11.Summary132 PracticeProblems135

CHAPTER3 ProbabilityConcepts147

LearningOutcomes147

1.Introduction148

2.Probability,ExpectedValue,andVariance148

3.PortfolioExpectedReturnandVarianceofReturn171

4.TopicsinProbability180

4.1.Bayes’ Formula180

4.2.PrinciplesofCounting184

5.Summary188 References190 PracticeProblem190

CHAPTER4 CommonProbabilityDistributions195

LearningOutcomes195

1.IntroductiontoCommonProbabilityDistributions196

2.DiscreteRandomVariables196

2.1.TheDiscreteUniformDistribution198

2.2.TheBinomialDistribution200

3.ContinuousRandomVariables210

3.1.ContinuousUniformDistribution210

3.2.TheNormalDistribution214

3.3.ApplicationsoftheNormalDistribution220

3.4.TheLognormalDistribution222

4.IntroductiontoMonteCarloSimulation228

5.Summary231 References233 PracticeProblems234

CHAPTER5 SamplingandEstimation241

LearningOutcomes241

1.Introduction242

2.Sampling242

2.1.SimpleRandomSampling242

2.2.StratifiedRandomSampling244

2.3.Time-SeriesandCross-SectionalData245

3.DistributionoftheSampleMean248

3.1.TheCentralLimitTheorem248

4.PointandIntervalEstimatesofthePopulationMean251

4.1.PointEstimators252

4.2.ConfidenceIntervalsforthePopulationMean253

4.3.SelectionofSampleSize259

5.MoreonSampling261

5.1.Data-MiningBias261

5.2.SampleSelectionBias264

5.3.Look-AheadBias265

5.4.Time-PeriodBias266

6.Summary267 References269 PracticeProblems270

CHAPTER6

HypothesisTesting275

LearningOutcomes275

1.Introduction276

2.HypothesisTesting277

3.HypothesisTestsConcerningtheMean287

3.1.TestsConcerningaSingleMean287

3.2.TestsConcerningDifferencesbetweenMeans294

3.3.TestsConcerningMeanDifferences299

4.HypothesisTestsConcerningVarianceandCorrelation303

4.1.TestsConcerningaSingleVariance303

4.2.TestsConcerningtheEquality(Inequality)ofTwoVariances305

4.3.TestsConcerningCorrelation308

5.OtherIssues:NonparametricInference310

5.1.NonparametricTestsConcerningCorrelation:TheSpearman RankCorrelationCoefficient312

5.2.NonparametricInference:Summary313

6.Summary314 References317 PracticeProblems317

CHAPTER7

IntroductiontoLinearRegression327

LearningOutcomes327

1.Introduction328

2.LinearRegression328

2.1.LinearRegressionwithOneIndependentVariable328

3.AssumptionsoftheLinearRegressionModel332

4.TheStandardErrorofEstimate335

5.TheCoefficientofDetermination337

6.HypothesisTesting339

7.AnalysisofVarianceinaRegressionwithOneIndependentVariable347

8.PredictionIntervals350

9.Summary353 References354 PracticeProblems354

CHAPTER8

MultipleRegression365

LearningOutcomes365

1.Introduction366

2.MultipleLinearRegression366

2.1.AssumptionsoftheMultipleLinearRegressionModel372

2.2.PredictingtheDependentVariableinaMultiple RegressionModel376

2.3.TestingWhetherAllPopulationRegressionCoefficients EqualZero378

2.4.Adjusted R2 380

3.UsingDummyVariablesinRegressions381

3.1.DefiningaDummyVariable381

3.2.VisualizingandInterpretingDummyVariables382

3.3.TestingforStatisticalSignificance384

4.ViolationsofRegressionAssumptions387

4.1.Heteroskedasticity388

4.2.SerialCorrelation394

4.3.Multicollinearity398

4.4.Heteroskedasticity,SerialCorrelation,Multicollinearity: SummarizingtheIssues401

5.ModelSpecificationandErrorsinSpecification401

5.1.PrinciplesofModelSpecification402

5.2.MisspecifiedFunctionalForm402

5.3.Time-SeriesMisspecification(IndependentVariables CorrelatedwithErrors)410

5.4.OtherTypesofTime-SeriesMisspecification414

6.ModelswithQualitativeDependentVariables414

6.1.ModelswithQualitativeDependentVariables414

7.Summary422 References425 PracticeProblems426

CHAPTER9

Time-SeriesAnalysis451

LearningOutcomes451

1.IntroductiontoTime-SeriesAnalysis452

2.ChallengesofWorkingwithTimeSeries454

3.TrendModels454

3.1.LinearTrendModels455

3.2.Log-LinearTrendModels458

3.3.TrendModelsandTestingforCorrelatedErrors463

4.Autoregressive(AR)Time-SeriesModels464

4.1.Covariance-StationarySeries465

4.2.DetectingSeriallyCorrelatedErrorsinanAutoregressiveModel466

4.3.MeanReversion469

4.4.MultiperiodForecastsandtheChainRuleofForecasting470

4.5.ComparingForecastModelPerformance473

4.6.InstabilityofRegressionCoefficients475

5.RandomWalksandUnitRoots478

5.1.RandomWalks478

5.2.TheUnitRootTestofNonstationarity482

6.Moving-AverageTime-SeriesModels486

6.1.SmoothingPastValueswithan n-PeriodMovingAverage486

6.2.Moving-AverageTime-SeriesModelsforForecasting489

7.SeasonalityinTime-SeriesModels491

8.AutoregressiveMoving-AverageModels496

9.AutoregressiveConditionalHeteroskedasticityModels497

10.RegressionswithMorethanOneTimeSeries500

11.OtherIssuesinTimeSeries504

12.SuggestedStepsinTime-SeriesForecasting505 13.Summary507 References508 PracticeProblems509

CHAPTER10

MachineLearning527

LearningOutcomes527

1.Introduction527

2.MachineLearningandInvestmentManagement528

3.WhatisMachineLearning?529

3.1.DefiningMachineLearning529

3.2.SupervisedLearning529

3.3.UnsupervisedLearning531

3.4.DeepLearningandReinforcementLearning531

3.5.SummaryofMLAlgorithmsandHowtoChooseamongThem532

4.OverviewofEvaluatingMLAlgorithmPerformance533

4.1.GeneralizationandOverfitting534

4.2.ErrorsandOverfitting534

4.3.PreventingOverfittinginSupervisedMachineLearning537

5.SupervisedMachineLearningAlgorithms539

5.1.PenalizedRegression539

5.2.SupportVectorMachine541

5.3. K-NearestNeighbor542

5.4.ClassificationandRegressionTree544

5.5.EnsembleLearningandRandomForest547

6.UnsupervisedMachineLearningAlgorithms559

6.1.PrincipalComponentsAnalysis560

6.2.Clustering563

7.NeuralNetworks,DeepLearningNets,andReinforcementLearning575

7.1.NeuralNetworks575

7.2.DeepLearningNeuralNetworks578

7.3.ReinforcementLearning579

8.ChoosinganAppropriateMLAlgorithm589 9.Summary590 References593 PracticeProblems593

CHAPTER11

BigDataProjects597

LearningOutcomes597

1.Introduction597

2.BigDatainInvestmentManagement598

3.StepsinExecutingaDataAnalysisProject:FinancialForecasting withBigData599

4.DataPreparationandWrangling603

4.1.StructuredData604

4.2.Unstructured(Text)Data610

5.DataExplorationObjectivesandMethods617

5.1.StructuredData618

5.2.UnstructuredData:TextExploration622

6.ModelTraining629

6.1.StructuredandUnstructuredData630

7.FinancialForecastingProject:ClassifyingandPredicting SentimentforStocks639

7.1.TextCuration,Preparation,andWrangling640

7.2.DataExploration644

7.3.ModelTraining654

7.4.ResultsandInterpretation658

8.Summary664 PracticeProblems665

CHAPTER12

UsingMultifactorModels675

LearningOutcomes675

1.Introduction675

2.MultifactorModelsandModernPortfolioTheory676

3.ArbitragePricingTheory677

4.MultifactorModels:Types683

4.1.FactorsandTypesofMultifactorModels683

4.2.TheStructureofMacroeconomicFactorModels684

4.3.TheStructureofFundamentalFactorModels687

4.4.Fixed-IncomeMultifactorModels691

5.MultifactorModels:SelectedApplications695

5.1.FactorModelsinReturnAttribution696

5.2.FactorModelsinRiskAttribution698

5.3.FactorModelsinPortfolioConstruction703

5.4.HowFactorConsiderationsCanBeUsefulinStrategic PortfolioDecisions705 6.Summary706 References707 PracticeProblems708

CHAPTER13

MeasuringandManagingMarketRisk713

LearningOutcomes713

1.Introduction714

2.UnderstandingValueatRisk714

2.1.ValueatRisk:FormalDefinition715

2.2.EstimatingVaR718

2.3.AdvantagesandLimitationsofVaR730

2.4.ExtensionsofVaR733

3.OtherKeyRiskMeasures SensitivityandScenarioMeasures735

3.1.SensitivityRiskMeasures736

3.2.ScenarioRiskMeasures740

3.3.SensitivityandScenarioRiskMeasuresandVaR746

4.UsingConstraintsinMarketRiskManagement750

4.1.RiskBudgeting751

4.2.PositionLimits752

4.3.ScenarioLimits752

4.4.Stop-LossLimits753

4.5.RiskMeasuresandCapitalAllocation753

5.ApplicationsofRiskMeasures755

5.1.MarketParticipantsandtheDifferentRiskMeasuresTheyUse755

6.Summary764 References766 PracticeProblems766

CHAPTER14

BacktestingandSimulation775

LearningOutcomes775

1.Introduction775

2.TheObjectivesofBacktesting776

3.TheBacktestingProcess776

3.1.StrategyDesign777

3.2.RollingWindowBacktesting778

3.3.KeyParametersinBacktesting779

3.4.Long/ShortHedgedPortfolioApproach781

3.5.PearsonandSpearmanRankIC785

3.6.UnivariateRegression789

3.7.DoDifferentBacktestingMethodologiesTelltheSameStory?789

4.MetricsandVisualsUsedinBacktesting792

4.1.Coverage792

4.2.Distribution794

4.3.PerformanceDecay,StructuralBreaks,andDownsideRisk797

4.4.FactorTurnoverandDecay797

5.CommonProblemsinBacktesting801

5.1.SurvivorshipBias801

5.2.Look-AheadBias804

6.BacktestingFactorAllocationStrategies807

6.1.SettingtheScene808

6.2.BacktestingtheBenchmarkandRiskParityStrategies808

7.ComparingMethodsofModelingRandomness813

7.1.FactorPortfoliosandBMandRPAllocationStrategies814

7.2.FactorReturnStatisticalProperties815

7.3.PerformanceMeasurementandDownsideRisk819

7.4.MethodstoAccountforRandomness821

8.ScenarioAnalysis824

9.HistoricalSimulationversusMonteCarloSimulation828

10.HistoricalSimulation830

11.MonteCarloSimulation835

12.SensitivityAnalysis840

13.Summary848 References849 PracticeProblems849

PREFACE

Wearepleasedtobringyou QuantitativeInvestmentAnalysis,FourthEdition,whichfocuses onkeytoolsthatareneededfortoday’sprofessionalinvestor.Inadditiontoclassicareas suchasthetimevalueofmoneyandprobabilityandstatistics,thetextcoversadvancedconceptsinregression,timeseries,machinelearning,andbigdataprojects.Thetextteaches criticalskillsthatchallengemanyprofessionals,andshowshowthesetechniquescanbe appliedtoareassuchasfactormodeling,riskmanagement,andbacktestingandsimulation.

Thecontentwasdevelopedinpartnershipbyateamofdistinguishedacademicsand practitioners,chosenfortheiracknowledgedexpertiseinthefield,andguidedbyCFA Institute.Itiswrittenspecificallywiththeinvestmentpractitionerinmindandisreplete withexamplesandpracticeproblemsthatreinforcethelearningoutcomesanddemonstrate real-worldapplicability.

TheCFAProgramCurriculum,fromwhichthecontentofthisbookwasdrawn,issubjectedtoarigorousreviewprocesstoassurethatitis:

• Faithfultothefindingsofourongoingindustrypracticeanalysis

• Valuabletomembers,employers,andinvestors

• Globallyrelevant

• Generalist(asopposedtospecialist)innature

• Repletewithsufficientexamplesandpracticeopportunities

• Pedagogicallysound

TheaccompanyingworkbookisausefulreferencethatprovidesLearningOutcome Statementsthatdescribeexactlywhatreaderswilllearnandbeabletodemonstrateafter masteringtheaccompanyingmaterial.Additionally,theworkbookhassummaryoverviews andpracticeproblemsforeachchapter.

WeareconfidentthatyouwillfindthisandotherbooksintheCFAInstitute InvestmentSerieshelpfulinyoureffortstogrowyourinvestmentknowledge,whetheryou arearelativelynewentrantoranexperiencedveteranstrivingtokeepuptodateinthe ever-changingmarketenvironment.CFAInstitute,asalong-termcommittedparticipant intheinvestmentprofessionandanot-for-profitglobalmembershipassociation,ispleased toprovideyouwiththisopportunity.

ACKNOWLEDGMENTS

Specialthankstoallthereviewers,advisors,andquestionwriterswhohelpedtoensurehigh practicalrelevance,technicalcorrectness,andunderstandabilityofthematerialpresented here.

Wewouldliketothankthemanyotherswhoplayedaroleintheconceptionandproductionofthisbook:theCurriculumandLearningExperienceteamatCFAInstitute,with specialthankstothecurriculumdirectors,pastandpresent,whoworkedwiththeauthors andreviewerstoproducethechaptersinthisbook;thePracticeAnalysisteamatCFA Institute;andthePublishingandTechnologyteamforbringingthisbooktoproduction.

ABOUTTHECFAINSTITUTE INVESTMENTSERIES

CFAInstituteispleasedtoprovidetheCFAInstituteInvestmentSeries,whichcoversmajor areasinthefieldofinvestments.Weprovidethisbest-in-classseriesforthesamereasonwe havebeencharteringinvestmentprofessionalsformorethan45years:toleadtheinvestment professiongloballybysettingthehigheststandardsofethics,education,andprofessional excellence.

ThebooksintheCFAInstituteInvestmentSeriescontainpractical,globallyrelevant material.Theyareintendedbothforthosecontemplatingentryintotheextremelycompetitivefieldofinvestmentmanagementaswellasforthoseseekingameansofkeepingtheir knowledgefreshanduptodate.Thisserieswasdesignedtobeuserfriendlyandhighly relevant.

Wehopeyoufindthisserieshelpfulinyoureffortstogrowyourinvestmentknowledge,whetheryouarearelativelynewentrantoranexperiencedveteranethicallybound tokeepuptodateintheever-changingmarketenvironment.Asalong-term,committed participantintheinvestmentprofessionandanot-for-profitglobalmembershipassociation, CFAInstituteispleasedtoprovideyouwiththisopportunity.

THETEXTS

CorporateFinance:APracticalApproach isasolidfoundationforthoselookingtoachieve lastingbusinessgrowth.Intoday’scompetitivebusinessenvironment,companiesmustfind innovativewaystoenablerapidandsustainablegrowth.Thistextequipsreaderswiththe foundationalknowledgeandtoolsformakingsmartbusinessdecisionsandformulatingstrategiestomaximizecompanyvalue.Itcoverseverythingfrommanagingrelationshipsbetween stakeholderstoevaluatingmergerandacquisitionbids,aswellasthecompaniesbehind them.Throughextensiveuseofreal-worldexamples,readerswillgaincriticalperspective intointerpretingcorporatefinancialdata,evaluatingprojects,andallocatingfundsinways thatincreasecorporatevalue.Readerswillgaininsightsintothetoolsandstrategiesused inmoderncorporatefinancialmanagement.

EquityAssetValuation isaparticularlycogentandimportantresourceforanyone involvedinestimatingthevalueofsecuritiesandunderstandingsecuritypricing.Awellinformedprofessionalknowsthatthecommonformsofequityvaluation dividenddiscountmodeling,freecashflowmodeling,price/earningsmodeling,andresidualincome modeling canallbereconciledwithoneanotherundercertainassumptions.Withadeep understandingoftheunderlyingassumptions,theprofessionalinvestorcanbetter

understandwhatotherinvestorsassumewhencalculatingtheirvaluationestimates.Thistext hasaglobalorientation,includingemergingmarkets.

FixedIncomeAnalysis hasbeenattheforefrontofnewconceptsinrecentyears,andthis particulartextofferssomeofthemostrecentmaterialfortheseasonedprofessionalwhois notafixed-incomespecialist.Theapplicationofoptionandderivativetechnologytothe oncestaidprovinceoffixedincomehashelpedcontributetoanexplosionofthoughtinthis area.Professionalshavebeenchallengedtostayuptospeedwithcreditderivatives,swaptions,collateralizedmortgagesecurities,mortgage-backedsecurities,andothervehicles,and thisexplosionofproductshasstrainedtheworld’sfinancialmarketsandtestedcentralbanks toprovidesufficientoversight.Armedwithathoroughgraspofthenewexposures,theprofessionalinvestorismuchbetterabletoanticipateandunderstandthechallengesourcentral bankersandmarketsface.

InternationalFinancialStatementAnalysis isdesignedtoaddresstheever-increasingneed forinvestmentprofessionalsandstudentstothinkaboutfinancialstatementanalysisfroma globalperspective.Thetextisapracticallyorientedintroductiontofinancialstatementanalysisthatisdistinguishedbyitscombinationofatrueinternationalorientation,astructured presentationstyle,andabundantillustrationsandtoolscoveringconceptsastheyareintroducedinthetext.Theauthorscoverthisdisciplinecomprehensivelyandwithaneyeto ensuringthereader’ssuccessatalllevelsinthecomplexworldoffinancialstatementanalysis.

Investments:PrinciplesofPortfolioandEquityAnalysis providesanaccessibleyetrigorous introductiontoportfolioandequityanalysis.Portfolioplanningandportfoliomanagement arepresentedwithinacontextofup-to-date,globalcoverageofsecuritymarkets,trading, andmarket-relatedconceptsandproducts.Theessentialsofequityanalysisandvaluation areexplainedindetailandprofuselyillustrated.Thebookincludescoverageofpractitionerimportantbutoftenneglectedtopics,suchasindustryanalysis.Throughout,thefocusison thepracticalapplicationofkeyconceptswithexamplesdrawnfrombothemergingand developedmarkets.Eachchapteraffordsthereadermanyopportunitiestoself-checkhisor herunderstandingoftopics.

AllbooksintheCFAInstituteInvestmentSeriesareavailablethroughallmajorbooksellers.And,alltitlesareavailableontheWileyCustomSelectplatformathttp://customselect. wiley.com/whereindividualchaptersforallthebooksmaybemixedandmatchedtocreate customtextbooksfortheclassroom.

CHAPTER 1 THETIMEVALUEOFMONEY

RichardA.DeFusco,PhD,CFA

DennisW.McLeavey,DBA,CFA

JeraldE.Pinto,PhD,CFA

DavidE.Runkle,PhD,CFA

LEARNINGOUTCOMES

Thecandidateshouldbeableto:

• interpretinterestratesasrequiredratesofreturn,discountrates,oropportunitycosts;

• explainaninterestrateasthesumofarealrisk-freerateandpremiumsthatcompensate investorsforbearingdistincttypesofrisk;

• calculateandinterprettheeffectiveannualrate,giventhestatedannualinterestrateand thefrequencyofcompounding;

• solvetimevalueofmoneyproblemsfordifferentfrequenciesofcompounding;

• calculateandinterpretthefuturevalue(FV)andpresentvalue(PV)ofasinglesumof money,anordinaryannuity,anannuitydue,aperpetuity(PVonly),andaseriesof unequalcashflows;

• demonstratetheuseofatimelineinmodelingandsolvingtimevalueofmoneyproblems.

1.INTRODUCTION

Asindividuals,weoftenfacedecisionsthatinvolvesavingmoneyforafutureuse,or borrowingmoneyforcurrentconsumption.Wethenneedtodeterminetheamountweneed toinvest,ifwearesaving,orthecostofborrowing,ifweareshoppingforaloan.As investmentanalysts,muchofourworkalsoinvolvesevaluatingtransactionswithpresentand futurecashflows.Whenweplaceavalueonanysecurity,forexample,weareattemptingto determinetheworthofastreamoffuturecashflows.Tocarryoutalltheabovetasks

2QuantitativeInvestmentAnalysis

accurately,wemustunderstandthemathematicsoftimevalueofmoneyproblems.Money hastimevalueinthatindividualsvalueagivenamountofmoneymorehighlytheearlieritis received.Therefore,asmalleramountofmoneynowmaybeequivalentinvaluetoalarger amountreceivedatafuturedate.The timevalueofmoney asatopicininvestment mathematicsdealswithequivalencerelationshipsbetweencashflowswithdifferentdates. Masteryoftimevalueofmoneyconceptsandtechniquesisessentialforinvestmentanalysts.

Thechapter1 isorganizedasfollows:Section2introducessometerminologyused throughoutthechapterandsuppliessomeeconomicintuitionforthevariableswewill discuss.Section3tacklestheproblemofdeterminingtheworthatafuturepointintimeofan amountinvestedtoday.Section4addressesthefutureworthofaseriesofcashflows.These twosectionsprovidethetoolsforcalculatingtheequivalentvalueatafuturedateofasingle cashfloworseriesofcashflows.Sections5and6discusstheequivalentvaluetodayofasingle futurecashflowandaseriesoffuturecashflows,respectively.InSection7,weexplorehowto determineotherquantitiesofinterestintimevalueofmoneyproblems.

2.INTERESTRATES:INTERPRETATION

Inthischapter,wewillcontinuallyrefertointerestrates.Insomecases,weassumea particularvaluefortheinterestrate;inothercases,theinterestratewillbetheunknown quantityweseektodetermine.Beforeturningtothemechanicsoftimevalueofmoney problems,wemustillustratetheunderlyingeconomicconcepts.Inthissection,webriefly explainthemeaningandinterpretationofinterestrates.

Timevalueofmoneyconcernsequivalencerelationshipsbetweencashflowsoccurringon differentdates.Theideaofequivalencerelationshipsisrelativelysimple.Considerthe followingexchange:Youpay$10,000todayandinreturnreceive$9,500today.Wouldyou acceptthisarrangement?Notlikely.Butwhatifyoureceivedthe$9,500todayandpaidthe $10,000oneyearfromnow?Cantheseamountsbeconsideredequivalent?Possibly,because apaymentof$10,000ayearfromnowwouldprobablybeworthlesstoyouthanapayment of$10,000today.Itwouldbefair,therefore,to discount the$10,000receivedinoneyear; thatis,tocutitsvaluebasedonhowmuchtimepassesbeforethemoneyispaid.An interest rate,denoted r,isarateofreturnthatreflectstherelationshipbetweendifferentlydatedcash flows.If$9,500todayand$10,000inoneyearareequivalentinvalue,then$10,000 $9,500 ¼ $500istherequiredcompensationforreceiving$10,000inoneyearratherthan now.Theinterestrate therequiredcompensationstatedasarateofreturn is$500/ $9,500 ¼ 0.0526or5.26percent.

Interestratescanbethoughtofinthreeways.First,theycanbeconsideredrequiredrates ofreturn thatis,theminimumrateofreturnaninvestormustreceiveinordertoacceptthe investment.Second,interestratescanbeconsidereddiscountrates.Intheexampleabove, 5.26percentisthatrateatwhichwediscountedthe$10,000futureamounttofinditsvalue today.Thus,weusetheterms “interestrate” and “discountrate” almostinterchangeably. Third,interestratescanbeconsideredopportunitycosts.An opportunitycost isthevalue thatinvestorsforgobychoosingaparticularcourseofaction.Intheexample,ifthepartywho supplied$9,500hadinsteaddecidedtospendittoday,hewouldhaveforgoneearning

1Examplesinthis,andotherchapters,inthetextwereupdatedbyProfessorSanjivSabherwalofthe UniversityofTexas,Arlington.

5.26percentonthemoney.Sowecanview5.26percentastheopportunitycostofcurrent consumption.

Economicstellsusthatinterestratesaresetinthemarketplacebytheforcesofsupply anddemand,whereinvestorsaresuppliersoffundsandborrowersaredemandersoffunds. Takingtheperspectiveofinvestorsinanalyzingmarket-determinedinterestrates,wecanview aninterestrate r asbeingcomposedofarealrisk-freeinterestrateplusasetoffourpremiums thatarerequiredreturnsorcompensationforbearingdistincttypesofrisk:

r ¼ Realrisk-freeinterestrate þ Inflationpremium þ Defaultriskpremium þ Liquiditypremium þ Maturitypremium

• The realrisk-freeinterestrate isthesingle-periodinterestrateforacompletelyrisk-free securityifnoinflationwereexpected.Ineconomictheory,therealrisk-freeratereflectsthe timepreferencesofindividualsforcurrentversusfuturerealconsumption.

• The inflationpremium compensatesinvestorsforexpectedinflationandreflectstheaverage inflationrateexpectedoverthematurityofthedebt.Inflationreducesthepurchasingpower ofaunitofcurrency theamountofgoodsandservicesonecanbuywithit.Thesumof therealrisk-freeinterestrateandtheinflationpremiumisthe nominalrisk-freeinterest rate 2 Manycountrieshavegovernmentalshort-termdebtwhoseinterestratecanbe consideredtorepresentthenominalrisk-freeinterestrateinthatcountry.Theinterestrate ona90-dayUSTreasurybill(T-bill),forexample,representsthenominalrisk-freeinterest rateoverthattimehorizon.3 UST-billscanbeboughtandsoldinlargequantitieswith minimaltransactioncostsandarebackedbythefullfaithandcreditoftheUSgovernment.

• The defaultriskpremium compensatesinvestorsforthepossibilitythattheborrowerwill failtomakeapromisedpaymentatthecontractedtimeandinthecontractedamount.

• The liquiditypremium compensatesinvestorsfortheriskoflossrelativetoaninvestment’ s fairvalueiftheinvestmentneedstobeconvertedtocashquickly.UST-bills,forexample, donotbearaliquiditypremiumbecauselargeamountscanbeboughtandsoldwithout affectingtheirmarketprice.Manybondsofsmallissuers,bycontrast,tradeinfrequentlyafter theyareissued;theinterestrateonsuchbondsincludesaliquiditypremiumreflectingthe relativelyhighcosts(includingtheimpactonprice)ofsellingaposition.

• The maturitypremium compensatesinvestorsfortheincreasedsensitivityofthemarket valueofdebttoachangeinmarketinterestratesasmaturityisextended,ingeneral (holdingallelseequal).Thedifferencebetweentheinterestrateonlonger-maturity,liquid Treasurydebtandthatonshort-termTreasurydebtreflectsapositivematuritypremium forthelonger-termdebt(andpossiblydifferentinflationpremiumsaswell).

2Technically,1plusthenominalrateequalstheproductof1plustherealrateand1plustheinflationrate. Asaquickapproximation,however,thenominalrateisequaltotherealrateplusaninflationpremium.In thisdiscussionwefocusonapproximateadditiverelationshipstohighlighttheunderlyingconcepts.

3OtherdevelopedcountriesissuesecuritiessimilartoUSTreasurybills.TheFrenchgovernmentissues BTFsornegotiablefixed-ratediscountTreasurybills(BonsduTrésoràtauxfixeetàintérêtsprécomptés)with maturitiesofuptooneyear.TheJapanesegovernmentissuesashort-termTreasurybillwithmaturitiesof6 and12months.TheGermangovernmentissuesatdiscountbothTreasuryfinancingpaper (FinanzierungsschätzedesBundes or,forshort, Schätze)andTreasurydiscountpaper(Bubills)with maturitiesupto24months.IntheUnitedKingdom,theBritishgovernmentissuesgilt-edgedTreasury billswithmaturitiesrangingfrom1to364days.TheCanadiangovernmentbondmarketiscloselyrelated totheUSmarket;CanadianTreasurybillshavematuritiesof3,6,and12months.

Usingthisinsightintotheeconomicmeaningofinterestrates,wenowturntoadiscussionof solvingtimevalueofmoneyproblems,startingwiththefuturevalueofasinglecashflow.

3.THEFUTUREVALUEOFASINGLECASHFLOW

Inthissection,weintroducetimevalueassociatedwithasinglecashfloworlump-sum investment.Wedescribetherelationshipbetweenaninitialinvestmentor presentvalue (PV),whichearnsarateofreturn(theinterestrateperperiod)denotedas r, andits future value(FV),whichwillbereceived N yearsorperiodsfromtoday.

Thefollowingexampleillustratesthisconcept.Supposeyouinvest$100(PV ¼ $100)in aninterest-bearingbankaccountpaying5percentannually.Attheendofthefirstyear,you willhavethe$100plustheinterestearned,0.05 $100 ¼ $5,foratotalof$105.To formalizethisone-periodexample,wedefinethefollowingterms:

PV ¼ presentvalueoftheinvestment

FVN ¼ futurevalueoftheinvestment N periodsfromtoday r ¼ rateofinterestperperiod

For N ¼ 1,theexpressionforthefuturevalueofamountPVis

¼ PV(1 þ r)(1)

Forthisexample,wecalculatethefuturevalueoneyearfromtodayasFV1 ¼ $100(1.05) ¼ $105.

Nowsupposeyoudecidetoinvesttheinitial$100fortwoyearswithinterestearnedand creditedtoyouraccountannually(annualcompounding).Attheendofthefirstyear(the beginningofthesecondyear),youraccountwillhave$105,whichyouwillleaveinthebank foranotheryear.Thus,withabeginningamountof$105(PV ¼ $105),theamountatthe endofthesecondyearwillbe$105(1.05) ¼ $110.25.Notethatthe$5.25interestearned duringthesecondyearis5percentoftheamountinvestedatthebeginningofYear2.

Anotherwaytounderstandthisexampleistonotethattheamountinvestedatthe beginningofYear2iscomposedoftheoriginal$100thatyouinvestedplusthe$5interest earnedduringthefirstyear.Duringthesecondyear,theoriginalprincipalagainearnsinterest,as doestheinterestthatwasearnedduringYear1.Youcanseehowtheoriginalinvestmentgrows: The$5interestthatyouearnedeachperiodonthe$100originalinvestmentisknownas simple interest (theinterestratetimestheprincipal). Principal istheamountoffundsoriginally Originalinvestment$100.00

Interestforthefirstyear($100 0.05)5.00

Interestforthesecondyearbasedonoriginalinvestment($100 0.05)5.00

Interestforthesecondyearbasedoninterestearnedinthefirstyear(0.05 $5.00 interestoninterest) 0.25

Total $110.25

FV1

invested.Duringthetwo-yearperiod,youearn $10ofsimpleinterest.Theextra$0.25thatyou haveattheendofYear2istheinterestyouearnedontheYear1interestof$5thatyou reinvested.

Theinterestearnedoninterestprovidesthefirstglimpseofthephenomenonknownas compounding.Althoughtheinterestearnedontheinitialinvestmentisimportant,fora giveninterestrateitisfixedinsizefromperiodtoperiod.Thecompoundedinterestearned onreinvestedinterestisafarmorepowerfulforcebecause,foragiveninterestrate,itgrowsin sizeeachperiod.Theimportanceofcompoundingincreaseswiththemagnitudeofthe interestrate.Forexample,$100investedtodaywouldbeworthabout$13,150after 100yearsifcompoundedannuallyat5percent,butworthmorethan$20millionif compoundedannuallyoverthesametimeperiodatarateof13percent.

Toverifythe$20millionfigure,weneedageneralformulatohandlecompoundingfor anynumberofperiods.Thefollowinggeneralformularelatesthepresentvalueofaninitial investmenttoitsfuturevalueafter N periods:

(2) where r isthestatedinterestrateperperiodand N isthenumberofcompoundingperiods.In thebankexample,FV2 ¼ $100(1 þ 0.05)2 ¼ $110.25.Inthe13percentinvestment example,FV100 ¼ $100(1.13)100 ¼ $20,316,287.42.

Themostimportantpointtorememberaboutusingthefuturevalueequationisthatthe statedinterestrate, r,andthenumberofcompoundingperiods, N,mustbecompatible.Both variablesmustbedefinedinthesametimeunits.Forexample,if N isstatedinmonths,then r shouldbetheone-monthinterestrate,unannualized.

Atimelinehelpsustokeeptrackofthecompatibilityoftimeunitsandtheinterestrate pertimeperiod.Inthetimeline,weusethetimeindex t torepresentapointintimeastated numberofperiodsfromtoday.Thusthepresentvalueistheamountavailableforinvestment today,indexedas t ¼ 0.Wecannowrefertoatime N periodsfromtodayas t ¼ N.Thetime lineinFigure1showsthisrelationship.InFigure1,wehavepositionedtheinitial investment,PV,at t ¼ 0.UsingEquation2,wemovethepresentvalue,PV,forwardto t ¼ N bythefactor(1 þ r)N.Thisfactoriscalledafuturevaluefactor.Wedenotethefuture valueonthetimelineasFVandpositionitat t ¼ N.Supposethefuturevalueistobe receivedexactly10periodsfromtoday’sdate(N ¼ 10).Thepresentvalue,PV,andthefuture value,FV,areseparatedintimethroughthefactor(1 þ r)10.

Thefactthatthepresentvalueandthefuturevalueareseparatedintimehasimportant consequences:

FIGURE1 TheRelationshipbetweenanInitialInvestment,PV,andItsFutureValue,FV

• Wecanaddamountsofmoneyonlyiftheyareindexedatthesamepointintime.

• Foragiveninterestrate,thefuturevalueincreaseswiththenumberofperiods.

• Foragivennumberofperiods,thefuturevalueincreaseswiththeinterestrate.

Tobetterunderstandtheseconcepts,considerthreeexamplesthatillustratehowtoapplythe futurevalueformula.

EXAMPLE1TheFutureValueofaLumpSumwithInterimCash

ReinvestedattheSameRate

Youaretheluckywinnerofyourstate’slotteryof$5millionaftertaxes.Youinvest yourwinningsinafive-yearcertificateofdeposit(CD)atalocalfinancialinstitution. TheCDpromisestopay7percentperyearcompoundedannually.Thisinstitution alsoletsyoureinvesttheinterestatthatrateforthedurationoftheCD.Howmuch willyouhaveattheendoffiveyearsifyourmoneyremainsinvestedat7percentfor fiveyearswithnowithdrawals?

Solution: Tosolvethisproblem,computethefuturevalueofthe$5millioninvestment usingthefollowingvaluesinEquation2:

$7,012,758 65

Attheendoffiveyears,youwillhave$7,012,758.65ifyourmoneyremainsinvestedat 7percentwithnowithdrawals.

Notethatinthisandmostexamplesinthischapter,thefactorsarereportedatsixdecimal placesbutthecalculationsmayactuallyreflectgreaterprecision. Forexample,thereported 1.402552hasbeenroundedupfrom1.40255173(thecalculationisactuallycarriedoutwith morethaneightdecimalplacesofprecisionbythecalculatororspreadsheet).Ourfinalresult reflectsthehighernumberofdecimalplacescarriedbythecalculatororspreadsheet.4

4Wecouldalsosolvetimevalueofmoneyproblemsusingtablesofinterestratefactors.Solutionsusing tabledvaluesofinterestratefactorsaregenerallylessaccuratethansolutionsobtainedusingcalculatorsor spreadsheets,sopractitionersprefercalculatorsorspreadsheets.

EXAMPLE2TheFutureValueofaLumpSumwithNoInterimCash

Aninstitutionoffersyouthefollowingtermsforacontract:Foraninvestmentof ¥2,500,000,theinstitutionpromisestopayyoualumpsumsixyearsfromnowatan 8percentannualinterestrate.Whatfutureamountcanyouexpect?

Solution: UsethefollowingdatainEquation2tofindthefuturevalue:

186

Youcanexpecttoreceive¥3,967,186sixyearsfromnow.

Ourthirdexampleisamorecomplicatedfuturevalueproblemthatillustratesthe importanceofkeepingtrackofactualcalendartime.

EXAMPLE3TheFutureValueofaLumpSum

Apensionfundmanagerestimatesthathiscorporatesponsorwillmakea$10million contributionfiveyearsfromnow.Therateofreturnonplanassetshasbeenestimated at9percentperyear.Thepensionfundmanagerwantstocalculatethefuturevalueof thiscontribution15yearsfromnow,whichisthedateatwhichthefundswillbe distributedtoretirees.Whatisthatfuturevalue?

Solution: Bypositioningtheinitialinvestment,PV,at t ¼ 5,wecancalculatethe futurevalueofthecontributionusingthefollowingdatainEquation2:

¼ $10 million

¼ 9% ¼ 0 09 N ¼ 10

$10,000,000ð1 09Þ10 ¼ $10,000,000ð2 367364Þ ¼ $23,673,636 75

Thisproblemlooksmuchliketheprevioustwo,butitdiffersinoneimportant respect:itstiming.Fromthestandpointoftoday(t ¼ 0),thefutureamountof $23,673,636.75is15yearsintothefuture.Althoughthefuturevalueis10yearsfrom

FIGURE2 TheFutureValueofaLumpSum,InitialInvestmentNotat t ¼ 0

itspresentvalue,thepresentvalueof$10millionwillnotbereceivedforanotherfive years.

AsFigure2shows,wehavefollowedtheconventionofindexingtodayas t ¼ 0 andindexingsubsequenttimesbyadding1foreachperiod.Theadditional contributionof$10millionistobereceivedinfiveyears,soitisindexedas t ¼ 5 andappearsassuchinthefigure.Thefutu revalueoftheinvestmentin10yearsis thenindexedat t ¼ 15;thatis,10yearsfollowingthereceiptofthe$10million contributionat t ¼ 5.Timelineslikethisonecanbeextremelyusefulwhendealing withmore-complicatedproblems,especiallythoseinvolvingmorethanonecash flow.

Inalatersectionofthischapter,wewilldiscusshowtocalculatethevaluetodayofthe $10milliontobereceivedfiveyearsfromnow.Forthemoment,wecanuseEquation2. Supposethepensionfundmanagerin Example3 aboveweretoreceive$6,499,313.86today fromthecorporatesponsor.Howmuchwillthatsumbeworthattheendoffiveyears?How muchwillitbeworthattheendof15years? PV ¼ $6,499,313 86

r ¼ 9% ¼ 0:09

N ¼ 5

¼ $6,499,313 86ð1 09Þ5

¼ $6,499,313:86ð1:538624Þ

¼ $10,000,000 at the five-year mark and PV ¼ $6,499,313:86

r ¼ 9% ¼ 0:09

N ¼ 15