Continuous Compounding Definition & Formula - Investopedia

Example of How to Use Continuous Compounding · Annual Compounding: FV = $10,000 x (1 + (15% / 1)) (1 x 1) = $11,500 · Semi-Annual Compounding: FV = $10,000 x (1 + ... Stocks DividendStocks DividendStocks GuidetoDividendInvesting WhatIsContinuousCompounding? Continuouscompoundingisthemathematicallimitthatcompoundinterestcanreachif it'scalculatedandreinvestedintoanaccount'sbalanceoveratheoreticallyinfinitenumberofperiods. Whilethisisnotpossibleinpractice,theconceptofcontinuouslycompoundedinterestisimportantinfinance.Itisanextremecaseofcompounding,asmostinterestiscompoundedonamonthly,quarterly,orsemiannualbasis. FormulaandCalculationofContinuousCompounding Insteadofcalculatinginterestonafinitenumberofperiods,suchasyearlyormonthly,continuouscompoundingcalculatesinterestassumingconstantcompoundingoveraninfinitenumberofperiods.Theformulaforcompoundinterestoverfiniteperiodsoftimetakesintoaccountfourvariables: PV=thepresentvalueoftheinvestment i=thestatedinterestrate n=thenumberofcompoundingperiods t=thetimeinyears Theformulaforcontinuouscompoundingisderivedfromtheformulaforthefuturevalueofaninterest-bearinginvestment: FutureValue(FV)=PVx[1+(i/n)](nxt) Calculatingthelimitofthisformulaasnapproachesinfinity(perthedefinitionofcontinuouscompounding) resultsintheformulaforcontinuouslycompoundedinterest: FV=PVxe (ixt),whereeisthemathematicalconstant approximatedas2.7183. KeyTakeaways Mostinterestiscompoundedonasemiannually,quarterly,ormonthlybasis.Continuouslycompoundedinterestassumesinterestiscompoundedandaddedbackintothebalanceaninfinitenumberoftimes.Theformulatocomputecontinuouslycompoundedinteresttakesintoaccountfourvariables.Theconceptofcontinuouslycompoundedinterestisimportantinfinanceeventhoughit’snotpossibleinpractice. WhatContinuousCompoundingCanTellYou Intheory,continuouslycompoundedinterestmeansthatanaccountbalanceisconstantlyearninginterest,aswellasrefeedingthatinterestbackintothebalancesothatit,too,earnsinterest. Continuouscompoundingcalculatesinterestundertheassumptionthatinterestwillbecompoundingoveraninfinitenumberofperiods. Althoughcontinuouscompoundingisanessentialconcept,it'snotpossibleintherealworldtohaveaninfinitenumberofperiodsforinteresttobecalculatedandpaid.Asaresult,interestistypicallycompoundedbasedonafixedterm,suchasmonthly,quarterly,orannually.  Evenwithverylargeinvestmentamounts,thedifferenceinthetotalinterestearnedthroughcontinuouscompoundingisnotveryhighwhencomparedtotraditionalcompoundingperiods. ExampleofHowtoUseContinuousCompounding Asanexample,assumea$10,000investmentearns15%interestoverthenextyear.Thefollowingexamplesshowtheendingvalueoftheinvestmentwhentheinterestiscompoundedannually,semiannually,quarterly,monthly,daily,andcontinuously. AnnualCompounding:FV=$10,000x(1+(15%/1))(1x1)=$11,500Semi-AnnualCompounding:FV=$10,000x(1+(15%/2))(2x1)=$11,556.25QuarterlyCompounding:FV=$10,000x(1+(15%/4))(4x1)=$11,586.50MonthlyCompounding:FV=$10,000x(1+(15%/12)) (12x1)=$11,607.55DailyCompounding:FV=$10,000x(1+(15%/365))(365x1)=$11,617.98ContinuousCompounding:FV=$10,000x2.7183(15%x1)=$11,618.34 Withdailycompounding,thetotalinterestearnedis$1,617.98,whilewithcontinuouscompoundingthetotalinterestearnedis$1,618.34,amarginaldifference. CompareAccounts AdvertiserDisclosure × TheoffersthatappearinthistablearefrompartnershipsfromwhichInvestopediareceivescompensation.Thiscompensationmayimpacthowandwherelistingsappear.Investopediadoesnotincludealloffersavailableinthemarketplace. Provider Name Description RelatedTerms TimeValueofMoney(TVM) Thetimevalueofmoney(TVM)istheconceptthatasumofmoneyhasgreatervaluenowthanitwillinthefutureduetoitsearningspotential. more WhattheEffectiveAnnualInterestRateTellsUs Theeffectiveannualinterestrateistherealreturnonaninvestment,accountingfortheeffectofcompoundingoveragivenperiodoftime. more WhatIsEuler'sConstant? Euler'sconstantisamathematicalconstantrecurringinanalysisandnumbertheory,usuallydenotedbythelowercaseGreeklettergamma(γ). more WhatIsDiscreteCompounding? Discretecompoundingreferstothemethodbywhichinterestiscalculatedandaddedtotheprincipalatcertainsetpointsintime. more AnnualPercentageYield(APY) Theannualpercentageyield(APY)istheeffectiverateofreturnonaninvestmentforoneyeartakingintoaccounttheeffectofcompoundinginterest. more LearnAboutCompounding Compoundingistheprocessinwhichanasset'searnings,fromeithercapitalgainsorinterest,arereinvestedtogenerateadditionalearnings. more PartnerLinks RelatedArticles InterestRates DiscreteCompoundingvs.ContinuousCompounding:What'stheDifference? InvestingEssentials HowDoIUsetheRuleof72toCalculateContinuousCompounding? InterestRates ContinuousCompoundInterest CorporateFinance&Accounting LearnAboutSimpleInterestandCompoundInterest StudentLoans HowtoCalculateStudentLoanInterest InterestRates WhatFormulaCalculatesInterest-on-Interest?