Compound Interest Definition, Formula, and Calculation

Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus ... PersonalFinance Banking Banking BankReviews PersonalLoanGuide PrepaidDebitCards GuidetoCheckingAccounts CheckingAccounts SavingsAccounts CertificateofDeposits(CDs) MoneyMarketAccount TableofContents Expand WhatIsCompoundInterest? HowCompoundInterestWorks CompoundInterestSchedules SpecialConsiderations CompoundAnnualGrowthRate ProsandConsofCompounding CompoundInterestInvestments CalculatingCompoundInterest HowtoSpotCompoundInterest SimpleDefinition WhoBenefitsFromCompoundInterest? CanCompoundInterestMakeYouRich? WhatIsCompoundInterest? Compoundinterest(orcompoundinginterest)istheinterestonaloanordepositcalculatedbasedonboththeinitialprincipalandtheaccumulatedinterestfrompreviousperiods.Thoughttohaveoriginatedin17th-centuryItaly,compoundinterestcanbethoughtofas"interestoninterest,"andwillmakeasumgrowatafasterratethansimpleinterest,whichiscalculatedonlyontheprincipalamount. Therateatwhichcompoundinterestaccruesdependsonthefrequencyofcompounding,suchthatthehigherthenumberofcompoundingperiods,thegreaterthecompoundinterest.Thus,theamountofcompoundinterestaccruedon$100compoundedat10%annuallywillbelowerthanthaton$100compoundedat5%semi-annuallyoverthesametimeperiod.Becausetheinterest-on-interesteffectcangenerateincreasinglypositivereturnsbasedontheinitialprincipalamount,compoundinghassometimesbeenreferredtoasthe"miracleofcompoundinterest." KeyTakeaways Compoundinterest(orcompoundinginterest)isinterestcalculatedontheinitialprincipal,whichalsoincludesalloftheaccumulatedinterestfrompreviousperiodsonadepositorloan. Compoundinterestiscalculatedbymultiplyingtheinitialprincipalamountbyoneplustheannualinterestrateraisedtothenumberofcompoundperiodsminusone. Interestcanbecompoundedonanygivenfrequencyschedule,fromcontinuoustodailytoannually. Whencalculatingcompoundinterest,thenumberofcompoundingperiodsmakesasignificantdifference. 1:59 UnderstandingCompoundInterest HowCompoundInterestWorks Compoundinterestiscalculatedbymultiplyingtheinitialprincipal amountbyoneplustheannualinterestrateraisedtothenumberofcompoundperiodsminusone.Thetotalinitialamountoftheloanisthensubtractedfromtheresultingvalue. KatieKerpel{Copyright}Investopedia,2019. Theformulaforcalculatingtheamountofcompoundinterestisasfollows: Compoundinterest=totalamountofprincipalandinterestinfuture(orfuturevalue)less principalamountatpresent(orpresentvalue) =[P(1+i)n]–P =P[(1+i)n–1] Where: P=principal i=nominalannualinterestrateinpercentageterms n=numberofcompoundingperiods Takeathree-yearloanof$10,000ataninterestrateof5%thatcompoundsannually.Whatwouldbetheamountofinterest? Inthiscase,itwouldbe: $10,000[(1+0.05)3–1]=$10,000[1.157625–1]=$1,576.25 HowCompoundInterestGrows Becausecompoundinterestincludesinterestaccumulatedinpreviousperiods,itgrowsatanever-acceleratingrate.Intheexampleabove,thoughthetotalinterestpayableoverthethree-yearperiodofthisloanis$1,576.25,theinterestamountisnotthesameforallthreeyears,asitwouldbewithsimpleinterest.Theinterestpayableattheendofeachyearisshowninthetablebelow. Compoundinterestcansignificantlyboostinvestmentreturnsoverthelongterm.Whilea$100,000depositthatreceives5%simpleannualinterestwouldearn$50,000intotalinterestover10years,theannualcompoundinterestof5%on$10,000wouldamountto$62,889.46overthesameperiod.Ifthecompoundingperiodwereinsteadpaidmonthlyoverthesame10-yearperiodat5%compoundinterest,thetotalinterestwouldinsteadgrowto$64,700.95. CompoundInterestSchedules Interestcanbecompoundedonanygivenfrequencyschedule,fromdailytoannually.Therearestandardcompoundingfrequencyschedulesthatareusuallyappliedtofinancialinstruments. Thecommonlyusedcompoundingscheduleforsavingsaccountsatbanksisdaily.Foracertificateofdeposit(CD),typicalcompoundingfrequencyschedulesaredaily,monthly,orsemiannually;formoneymarketaccounts,it'softendaily.Forhomemortgageloans,homeequityloans,personalbusinessloans,orcreditcardaccounts,themostcommonlyappliedcompoundingscheduleismonthly. Therecanalsobevariationsinthetimeframeinwhichtheaccruedinterestisactuallycreditedtotheexistingbalance.Interestonanaccountmaybecompoundeddailybutonlycreditedmonthly.Itisonlywhentheinterestisactuallycredited,oraddedtotheexistingbalance,thatitbeginstoearnadditionalinterestintheaccount. Somebanksalsooffersomethingcalledcontinuouslycompoundinginterest,whichaddsinteresttotheprincipalateverypossibleinstant.Forpracticalpurposes,itdoesn'taccruethatmuchmorethandailycompoundinginterestunlessyouwanttoputmoneyinandtakeitoutthesameday. Morefrequentcompoundingofinterestisbeneficialtotheinvestororcreditor.Foraborrower,theoppositeistrue. CompoundingPeriods Whencalculatingcompoundinterest,thenumberofcompoundingperiodsmakesasignificantdifference.Thebasicruleisthatthehigherthenumberofcompoundingperiods,thegreatertheamountofcompoundinterest. Thefollowingtabledemonstratesthedifferencethatthenumberofcompoundingperiodscanmakefora$10,000loanwithanannual10%interestrateovera10-yearperiod. SpecialConsiderations CompoundinterestiscloselytiedtothetimevalueofmoneyandtheRuleof72,bothimportantconceptsininvesting. TimeValueofMoneyConsideration Understandingthetimevalueofmoneyandtheexponentialgrowthcreatedbycompoundingisessentialforinvestorslookingtooptimizetheirincomeandwealthallocation. Theformulaforobtainingthefuturevalue(FV)andpresentvalue(PV)areasfollows: FV=PV(1+i)n andPV=FV/(1+i)n Forexample,thefuturevalueof$10,000compoundedat5%annuallyforthreeyears: =$10,000(1+0.05)3 =$10,000(1.157625) =$11,576.25 Thepresentvalueof$11,576.25discountedat5%forthreeyears: =$11,576.25/(1+0.05)3 =$11,576.25/1.157625 =$10,000 Thereciprocalof1.157625,whichequals0.8638376,isthediscountfactorinthisinstance. Ruleof72Consideration Theso-calledRuleof72calculatestheapproximatetimeoverwhichaninvestmentwilldoubleatagivenrateofreturnorinterest"i,"andisgivenby(72/i).Itcanonlybeusedforannualcompounding. Asanexample,aninvestmentthathasa6%annualrateofreturnwilldoublein12years.Aninvestmentwithan8%annualrateofreturnwillthusdoubleinnineyears. CompoundAnnualGrowthRate(CAGR) Thecompoundannualgrowthrate(CAGR)isusedformostfinancialapplicationsthatrequirethecalculationofasinglegrowthrateoveraperiodoftime. Let'ssayyourinvestmentportfoliohasgrownfrom$10,000to$16,000overfiveyears;whatistheCAGR?Essentially,thismeansthatPV=-$10,000,FV=$16,000,andt=5,sothevariable“i”hastobecalculated.UsingafinancialcalculatororExcel,itcanbeshownthati=9.86%. Accordingtothecash-flowconvention,yourinitialinvestment(PV)of$10,000isshownwithanegativesignbecauseitrepresentsanoutflowoffunds.PVandFVmustnecessarilyhaveoppositesignstosolvefor“i”intheaboveequation. CAGRReal-LifeApplications TheCAGRisextensivelyusedtocalculatereturnsoverperiodsoftimeforstock,mutualfunds,andinvestmentportfolios.TheCAGRisalsousedtoascertainwhetheramutualfundmanagerorportfoliomanagerhasexceededthemarket’srateofreturnoveraperiodoftime.If,forexample,amarketindexhasprovidedtotalreturnsof10%overafive-yearperiod,butafundmanagerhasonlygeneratedannualreturnsof9%overthesameperiod,themanagerhasunderperformedthemarket. TheCAGRcanalsobeusedtocalculatetheexpectedgrowthrateofinvestmentportfoliosoverlongperiodsoftime,whichisusefulforpurposessuchassavingforretirement.Considerthefollowingexamples: Example1: Arisk-averseinvestorishappywithamodest3%annualrateofreturnonherportfolio.Herpresent$100,000portfoliowould,therefore,growto$180,611after20years.Incontrast,arisk-tolerantinvestorwhoexpectsanannualreturnof6%onherportfoliowouldsee$100,000growto$320,714after20years. Example2: TheCAGRcanbeusedtoestimatehowmuchneedstobestowedawaytosaveforaspecificobjective.Acouplewhowouldliketosave$50,000over10yearstowardadownpaymentonacondowouldneedtosave$4,165peryeariftheyassumeanannualreturn(CAGR)of4%ontheirsavings.IftheyarepreparedtotakealittleextrariskandexpectaCAGRof5%,theywouldneedtosave$3,975annually. Example3: TheCAGRcanalsodemonstratethevirtuesofinvestingearlierratherthanlaterinlife.Iftheobjectiveistosave$1millionbyretirementatage65,basedonaCAGRof6%,a25-yearoldwouldneedtosave$6,462peryeartoattainthisgoal.A40-yearold,ontheotherhand,wouldneedtosave$18,227,oralmostthreetimesthatamount,toattainthesamegoal. CAGRsalsocropupfrequentlyineconomicdata.Hereisanexample:China’spercapitaGDPincreasedfrom$193in1980to$6,091in2012.WhatistheannualgrowthinpercapitaGDPoverthis32-yearperiod?Thegrowthrate“i”inthiscaseworksouttobeanimpressive11.4%. ProsandConsofCompounding ThoughthemiracleofcompoundinghasledtotheapocryphalstoryofAlbertEinsteincallingittheeighthwonderoftheworldorman’sgreatestinvention,compoundingcanalsoworkagainstconsumerswhohaveloansthatcarryveryhighinterest rates,suchascreditcarddebt.Acreditcardbalanceof$20,000carriedataninterestrateof20%compoundedmonthlywouldresultinatotalcompoundinterestof$4,388overoneyearorabout$365permonth. Onthepositiveside,compoundingcanworktoyouradvantagewhenitcomestoyourinvestmentsandcanbeapotentfactorinwealthcreation.Exponentialgrowthfromcompoundinginterestisalsoimportantinmitigatingwealth-erodingfactors,suchasincreasesinthecostofliving,inflation,andreducedpurchasingpower. Mutualfundsofferoneoftheeasiestwaysforinvestorstoreapthebenefitsofcompoundinterest.Optingtoreinvestdividendsderivedfromthemutualfundresultsinpurchasingmoresharesofthefund.Morecompoundinterestaccumulatesovertime,andthecycleofpurchasingmoreshareswillcontinuetohelptheinvestmentinthefundgrowinvalue. Consideramutualfundinvestmentopenedwithaninitial$5,000andanannualadditionof$2,400.Withanaverageannualreturnof12%over30years,thefuturevalueofthefundis$798,500.Thecompoundinterestisthedifferencebetweenthecashcontributedtoaninvestmentandtheactualfuturevalueoftheinvestment.Inthiscase,bycontributing$77,000,oracumulativecontributionofjust$200permonth,over30years,compoundinterestis$721,500ofthefuturebalance. Ofcourse,earningsfromcompoundinterestaretaxable,unlessthemoneyisinatax-shelteredaccount;it'sordinarilytaxedatthestandardrateassociatedwiththetaxpayer'staxbracket. CompoundInterestInvestments Aninvestorwhooptsforareinvestmentplanwithinabrokerageaccountisessentiallyusingthepowerofcompoundinginwhatevertheyinvest. Investorscanalsoexperiencecompoundinginterestwiththepurchaseofazero-couponbond.Traditionalbondissuesprovideinvestorswithperiodicinterestpaymentsbasedontheoriginaltermsofthebondissue,andbecausethesearepaidouttotheinvestorintheformofacheck,interestdoesnotcompound.Zero-couponbondsdonotsendinterestcheckstoinvestors;instead,thistypeofbondispurchasedatadiscounttoitsoriginalvalueandgrowsovertime.Zero-couponbondissuersusethepowerofcompoundingtoincreasethevalueofthebondsoitreachesitsfullpriceatmaturity. Compoundingcanalsoworkforyouwhenmakingloanrepayments.Makinghalfyourmortgagepaymenttwiceamonth,forexample,ratherthanmakingthefullpaymentonceamonth,willendupcuttingdownyouramortizationperiodandsavingyouasubstantialamountofinterest. HowtoCalculateCompoundInterest Ifit'sbeenawhilesinceyourmathclassdays,fearnot:Therearehandytoolsforfiguringoutcompounding.Manycalculators(bothhandheldandcomputer-based)haveexponentfunctionsyoucanutilizeforthesepurposes. CalculatingCompoundInterestinExcel Ifmorecomplicatedcompoundingtasksarise,youcanperformtheminMicrosoftExcel—inthreedifferentways. Thefirstwaytocalculatecompoundinterestistomultiplyeachyear'snewbalancebytheinterestrate.Supposeyoudeposit$1,000intoasavingsaccountwitha5%interestratethatcompoundsannually,andyouwanttocalculatethebalanceinfiveyears.InMicrosoftExcel,enter"Year"intocellA1and"Balance"intocellB1.Enteryears0to5intocellsA2throughA7.Thebalanceforyear0is$1,000,soyouwouldenter"1000"intocellB2.Next,enter"=B2*1.05"intocellB3.Thenenter"=B3*1.05"intocellB4andcontinuetodothisuntilyougettocellB7.IncellB7,thecalculationis"=B6*1.05".Finally,thecalculatedvalueincellB7—$1,276.28—isthebalanceinyoursavingsaccountafterfiveyears.Tofindthecompoundinterestvalue,subtract$1,000from$1,276.28;thisgivesyouavalueof$276.28.Thesecondwaytocalculatecompoundinterestistouseafixedformula.Thecompoundinterestformulais((P*(1+i)^n)-P),wherePistheprincipal,iistheannualinterestrate,andnisthenumberofperiods.Usingthesameinformationabove,enter"Principalvalue"intocellA1and1000intocellB1.Next,enter"Interestrate"intocellA2and".05"intocellB2.Enter"Compoundperiods"intocellA3and"5"intocellB3.NowyoucancalculatethecompoundinterestincellB4byentering"=(B1*(1+B2)^B3)-B1",whichgivesyou$276.28.Athirdwaytocalculatecompoundinterestistocreateamacrofunction.FirststarttheVisualBasicEditor,whichislocatedinthedevelopertab.ClicktheInsertmenu,andclickonModule.Thentype"FunctionCompound_Interest(PAsDouble,IAsDouble,NAsDouble)AsDouble"inthefirstline.Onthesecondline,hitthetabkeyandtypein"Compound_Interest=(P*(1+i)^n)-P."Onthethirdlineofthemodule,enter"EndFunction."Youhavecreatedafunctionmacrotocalculatethecompoundinterestrate.ContinuingfromthesameExcelworksheetabove,enter"Compoundinterest"intocellA6andenter"=Compound_Interest(B1,B2,B3)."Thisgivesyouavalueof$276.28,whichisconsistentwiththefirsttwovalues. OtherCompoundInterestCalculators Anumberoffreecompoundinterestcalculatorsareofferedonline,andmanyhandheldcalculatorscancarryoutthesetasksaswell. ThefreecompoundinterestcalculatorofferedthroughFinancial-Calculators.comissimpletooperateandofferstocompoundfrequencychoicesfromdailythroughannually.It includesanoptiontoselectcontinuouscompoundingand alsoallowsinputofactualcalendarstartandenddates.Afterinputtingthenecessarycalculationdata,theresultsshowinterestearned,futurevalue,annualpercentageyieldorAPY)(ameasurethatincludescompounding),anddailyinterest. Investor.gov,awebsiteoperatedbytheU.S.SecuritiesandExchangeCommission(SEC),offersafreeonlinecompoundinterestcalculator.Thecalculatorisfairlysimple,butitdoesallowinputsofmonthlyadditionaldepositstotheprincipal,whichishelpfulforcalculatingearningswhereadditionalmonthlysavingsarebeingdeposited. AfreeonlineinterestcalculatorwithafewmorefeaturesisavailableatTheCalculatorSite.com.Thiscalculatorallowscalculationsfordifferentcurrencies,theabilitytofactorinmonthlydepositsorwithdrawals,andtheoptiontohaveinflation-adjustedincreasestomonthlydepositsorwithdrawalsautomaticallycalculatedaswell. HowCanITellifInterestIsCompounded? TheTruthinLendingAct(TILA)requiresthatlendersdiscloseloantermstopotentialborrowers,includingthetotaldollaramountofinteresttoberepaidoverthelifeoftheloanandwhetherinterestaccruessimplyoriscompounded.Anothermethodistocomparealoan'sinterestratetoitsannualpercentagerate(APR),whichtheTILAalsorequireslenderstodisclose.TheAPRconvertsthefinancechargesofyourloan,whichincludeallinterestandfees,toasimpleinterestrate.AsubstantialdifferencebetweentheinterestrateandAPRmeansoneorbothoftwoscenarios:Yourloanusescompoundinterest,oritincludesheftyloanfeesinadditiontointerest.Evenwhenitcomestothesametypeofloan,theAPRrangecanvarywildlybetweenlendersdependingonthefinancialinstitution'sfeesandothercosts.You'llnotethattheinterestrateyouarechargedalsodependsonyourcredit.Loansofferedtothosewithexcellentcreditcarrysignificantlylowerinterestratesthanthosechargedtoborrowerswithpoorcredit. WhatisaSimpleDefinitionofCompoundInterest? Compoundinterestreferstothephenomenonwherebytheinterestassociatedwithabankaccount,loan,orinvestmentincreasesexponentially—ratherthanlinearly—overtime.Thekeytounderstandingtheconceptistheword“compound.”Supposeyoumakea$100investmentinabusinessthatpaysyoua10%dividendeveryyear.Youhavethechoiceofeitherpocketingthosedividendpaymentslikecashorreinvestingthosepaymentsintoadditionalshares.Ifyouchoosethesecondoption,reinvestingthedividendsandcompoundingthemtogetherwithyourinitial$100investment,thenthereturnsyougeneratewillstarttogrowovertime. WhoBenefitsFromCompoundInterest? Simplyput,compoundinterestbenefitsinvestors,butthemeaningof“investors”canbequitebroad.Banks,forinstance,benefitfromcompoundinterestwhentheylendmoneyandreinvesttheinteresttheyreceiveintogivingoutadditionalloans.Depositorsalsobenefitfromcompoundinterestwhentheyreceiveinterestontheirbankaccounts,bonds,orotherinvestments.Itisimportanttonotethatalthoughtheterm“compoundinterest”includestheword“interest,”theconceptappliesbeyondsituationsforwhichthewordinterestistypicallyused,suchasbankaccountsandloans. CanCompoundInterestMakeYouRich? Yes.Infact,compoundinterestisarguablythemostpowerfulforceforgeneratingwealtheverconceived.Therearerecordsofmerchants,lenders,andvariousbusinesspeopleusingcompoundinteresttobecomerichforliterallythousandsofyears.IntheancientcityofBabylon,forexample,claytabletswereusedover4,000yearsagotoinstructstudentsonthemathematicsofcompoundinterest. Inmoderntimes,WarrenBuffettbecameoneoftherichestpeopleintheworldthroughabusinessstrategythatinvolveddiligentlyandpatientlycompoundinghisinvestmentreturnsoverlongperiodsoftime.Itislikelythat,inoneformoranother,peoplewillbeusingcompoundinteresttogeneratewealthfortheforeseeablefuture. ArticleSources Investopediarequireswriterstouseprimarysourcestosupporttheirwork.Theseincludewhitepapers,governmentdata,originalreporting,andinterviewswithindustryexperts.Wealsoreferenceoriginalresearchfromotherreputablepublisherswhereappropriate.Youcanlearnmoreaboutthestandardswefollowinproducingaccurate,unbiasedcontentinour editorialpolicy. TheCalculatorSite."CompoundInterestCalculator."AccessedAug.19,2021. Financial-Calculators.com."CompoundInterestCalculator."AccessedAug.19,2021.  Investor.gov."CompoundInterestCalculator."AccessedAug.19,2021. RelatedTerms WhatIsDiscreteCompounding? Discretecompoundingreferstothemethodbywhichinterestiscalculatedandaddedtotheprincipalatcertainsetpointsintime. more WhattheEffectiveAnnualInterestRateTellsUs Theeffectiveannualinterestrateistherealreturnonaninvestment,accountingfortheeffectofcompoundingoveragivenperiodoftime. more CompoundAnnualGrowthRate(CAGR) Thecompoundannualgrowthrate(CAGR)istherateofreturnthatwouldberequiredforaninvestmenttogrowfromitsbeginningbalancetoitsendingone. more TimeValueofMoney(TVM) Thetimevalueofmoney(TVM)istheconceptthatasumofmoneyhasgreatervaluenowthanitwillinthefutureduetoitsearningspotential. more StatedAnnualInterestRateDefinition Astatedannualinterestrateisthereturnonaninvestment(ROI)thatisexpressedasaper-yearpercentage. more AnnualPercentageYield(APY) Theannualpercentageyield(APY)istheeffectiverateofreturnonaninvestmentforoneyeartakingintoaccounttheeffectofcompoundinginterest. more PartnerLinks RelatedArticles CorporateFinance&Accounting LearnAboutSimpleInterestandCompoundInterest InterestRates DiscreteCompoundingvs.ContinuousCompounding:What'stheDifference? ToolsforFundamentalAnalysis UnderstandingtheTimeValueofMoney TechnicalAnalysisBasicEducation DetermineWhatThingsWillBeWorthviatheTimeValueOfMoney InterestRates WhatFormulaCalculatesInterest-on-Interest? StudentLoans HowtoCalculateStudentLoanInterest