Compound Interest Formula With Examples - The Calculator Site

If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, with additional deposits of ... Popupcalculator Copyalinktothispage Printthispage Emailalinktothispage Scrolluptoform Whatdoesthismean? 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TheformulaforcompoundinterestisP(1+r/n)^(nt),wherePistheinitialprincipalbalance,ristheinterestrate,nisthenumberoftimesinterestiscompoundedpertimeperiodandtisthenumberoftimeperiods. Advertisement Theconceptofcompoundinterestisthatinterestisaddedbacktotheprincipalsumsothatinterestisgainedonthatalready-accumulatedinterestduringthenextcompoundingperiod.Howimportantisit?JustaskWarrenBuffett,oneoftheworld'smostsuccessfulinvestors: "MywealthhascomefromacombinationoflivinginAmerica,someluckygenes,andcompoundinterest." WarrenBuffett,2010 Inthisarticle,we'lltakealookatthecompoundinterestformulainmoredepth.We'llalsogothroughanexampleanddiscussothervariationsoftheformulathatcanhelpyoutocalculatetheinterestrateandtimefactorortoincorporateregularcontributions.Shouldyouwishtotrysomecalculationsusingyourownfigures,youcanuseourpopularcompoundinterestcalculator. Skiptosection: Howtousetheformula Formulatocalculateinterestrate(%) Formulatocalculateprincipal Formulatocalculatetimefactor Regularcontributionsformulae Howtousethecompoundinterestformula Tousethecompoundinterestformulayouwillneedfiguresforprincipalamount,annualinterestrate,timefactorandthenumberofcompoundperiods.Onceyouhavethose,youcangothroughtheprocessofcalculatingcompoundinterest. Theformulaforcompoundinterest,includingprincipalsum,is: A=P(1+r/n)(nt) Where: A=thefuturevalueoftheinvestment/loan,includinginterest P=theprincipalinvestmentamount(theinitialdepositorloanamount) r=theannualinterestrate(decimal) n=thenumberoftimesthatinterestiscompoundedperunitt t=thetimethemoneyisinvestedorborrowedfor It'sworthnotingthatthisformulagivesyouthefuturevalueofaninvestmentorloan,whichiscompoundinterestplustheprincipal.Shouldyouwishtocalculatethecompoundinterestonly,youneedtodeducttheprincipalfromtheresult.So,yourformulalookslikethis: Compoundedinterestonly(withoutprincipal):P(1+r/n)(nt)-P Let'slookatanexample Ifanamountof$5,000isdepositedintoasavingsaccountatanannualinterestrateof5%,compoundedmonthly,thevalueoftheinvestmentafter10yearscanbecalculatedasfollows... P=5000. r=5/100=0.05(decimal). n=12. t=10. Ifweplugthosefiguresintotheformula,wegetthefollowing: A=5000(1+0.05/12)(12*10)=8235.05. So,theinvestmentbalanceafter10yearsis$8,235.05. Methodology Afewpeoplehavewrittentomeaskingmetoexplainstep-by-stephowwegetthe8235.05.ThisallrevolvesaroundBODMAS/PEMDASandtheorderofoperations.Let'sgothroughit: A=5000(1+0.05/12)^(12(10)) (notethat^means'tothepowerof') Usingtheorderofoperationsweworkoutthetotalsinthebracketsfirst.Withinthefirstsetofbrackets,youneedtodothedivisionfirstandthentheaddition(divisionandmultiplicationshouldbecarriedoutbeforeadditionandsubtraction).Wecanalsoworkoutthe12(10).Thisgivesus... A=5000(1+0.00416)^120 (notethattheover-lineinthecalculationsignifiesadecimalthatrepeatstoinfinity.So,0.00416666666...) Then: A=5000(1.00416)^120 Theexponentgoesnext.So,wecalculate(1.00416)^120. Thismeansweendupwith: 5000×1.6470095042509848 =8235.0475. YoumayhaveseensomeexamplesgivingaformulaofA=P(1+r)t.Thissimplifiedformulaassumesthatinterestiscompoundedonceperperiod,ratherthanmultipletimesperperiod. Advertisement Thebenefitofcompoundinterest Ithinkit'sworthtakingamomenttoexaminethebenefitofcompoundinterestusingourexample.ThebenefithopefullybecomesclearwhenItellyouthatwithoutcompoundinterest,yourinvestmentbalanceintheaboveexamplewouldbeonly$7,500($250peryearfor10years,plustheoriginal$5000)bytheendoftheterm.So,thankstothewonderofcompoundinterest,youstandtogainanadditional$735.05. Togiveagraphicalexample,thegraphbelowshowstheresultof$1000investedover20yearsataninterestrateof10%.Theprincipalfigureisingreen.Thebluepartofthegraphshowstheresultof10%interestwithoutcompounding.Finally,thepurplepartdemonstratesthebenefitofcompoundinterestoverthose20years. Youcanreadmoreaboutthebenefitsofcompoundinterestinourarticle,whatiscompoundinterest? Interactivecompoundinterestformula Ihavecreatedthecalculatorbelowtoshowyoutheformulaandresultingaccruedinvestment/loanvalue(A)forthefiguresthatyouenter.NotethatthiscalculatorrequiresJavaScripttobeenabledinyourbrowser. Principalamount(P): Annualrate(r): decimal % Compoundsperyear(n): Years(t): calculate P ( 1+ r n ) ( n t ) = A Foracomprehensivesetoftoolsforcalculatingcompoundinterestonyoursavings,pleaseusethecompoundinterestcalculator. Formulaetofindcompoundinterestrate,timeandprincipal ItmaybethatyouwanttomanipulatethecompoundinterestformulatoworkouttheinterestrateforIRRorCAGR,oraprincipalinvestment/loanfigure.Herearetheformulaeyouneed. Formulaforinterestrate(r) Shouldyouwishtoworkouttheyearlyinterestrateyou'regettingonyoursavings,investment,personalloanorcarloan,thisformulacanhelp.Notethatyoushouldmultiplyyourresultby100togetapercentagefigure(%) Formulaforprincipal(P) Thisformulaisusefulifyouwanttoworkbackwardsandfindouthowmuchyouwouldneedtostartwithinordertoachieveachosenfuturevalue. Example:Let'ssayyourgoalistoendupwith$10,000in5years,andyoucangetan8%interestrateonyoursavings,compoundedmonthly.Yourcalculationwouldbe:P=10000/(1+0.08/12)(12×5)=$6712.10.So,youwouldneedtostartoffwith$6712.10toachieveyourgoal. Formulafortime(t) Thisvariationoftheformulaworksforcalculatingtime(t),byusingnaturallogarithms.Youcanseehowthisformulawasworkedoutbyreadingthisexplanationonalgebra.com. t=ln(A/P)/n[ln(1+r/n)] Where: A=thevalueoftheaccruedinvestment/loan P=theprincipalamount r=theannualinterestrate(decimal) n=thenumberoftimesthatinterestiscompoundedperunitt t=thetimethemoneyisinvestedorborrowedfor Advertisement Compoundinterestformula(withregularcontributions) Alotofpeoplehaveaskedmetoincludeasingleformulaforcompoundinterestwithmonthlyadditions.BelievemewhenItellyouthatitisn'tquiteassimpleasitsounds.Inordertoworkoutcalculationsinvolvingmonthlyadditions,youwillneedtousetwoformulae-ouroriginalone,listedabove,plusthe'futurevalueofaseries'formulaforthemonthlyadditions. Attherequestofreaders,I'veadaptedtheformulaexplanationtoallowyoutocalculateperiodicadditions,notjustmonthly(addedMay2016).Theseformulaeassumethatyourfrequencyofcompoundingisthesameastheperiodicpaymentinterval(monthlycompounding,monthlycontributions,etc).Ifyouwouldliketotryaversionoftheformulathatallowsyoutohaveadifferentperiodicpaymentintervaltothecompoundingfrequency,pleaseseethe' periodicpayments'sectionbelow. IftheadditionaldepositsaremadeattheENDoftheperiod(endofmonth,year,etc),herearethetwoformulaeyouwillneed: Compoundinterestforprincipal: P(1+r/n)(nt) Futurevalueofaseries: PMT×{[(1+r/n)(nt)-1]/(r/n)} IftheadditionaldepositsaremadeattheBEGINNINGoftheperiod(beginningofyear,etc),herearethetwoformulaeyouwillneed: Compoundinterestforprincipal: P(1+r/n)(nt) Futurevalueofaseries: PMT×{[(1+r/n)(nt)-1]/(r/n)}×(1+r/n) Where: A=thefuturevalueoftheinvestment/loan,includinginterest P=theprincipalinvestmentamount(theinitialdepositorloanamount) PMT=themonthlypayment r=theannualinterestrate(decimal) n=thenumberoftimesthatinterestiscompoundedperunitt t=thetime(months,years,etc)themoneyisinvestedorborrowedfor Example Ifanamountof$5,000isdepositedintoasavingsaccountatanannualinterestrateof5%,compoundedmonthly,withadditionaldepositsof$100permonth(madeattheendofeachmonth).Thevalueoftheinvestmentafter10yearscanbecalculatedasfollows... P=5000.PMT=100.r=5/100=0.05(decimal).n=12.t=10. Ifweplugthosefiguresintotheformulae,weget: Total=[Compoundinterestforprincipal]+[Futurevalueofaseries] Total=[P(1+r/n)^(nt)]+[PMT×(((1+r/n)^(nt)-1)/(r/n))] Total=[5000(1+0.05/12)^(12×10)]+[100×(((1+0.00416)^(12×10)-1)/(0.00416))] Total=[5000(1.00416)^(120)]+[100×(((1.00416^120)-1)/0.00416)] Total=[8235.05]+[100×(0.647009497690848/0.00416)] Total=[8235.05]+[15528.23] Total=[$23,763.28] So,theinvestmentbalanceafter10yearsis$23,763.28. Youcanlearnmoreaboutthisfuturevalueofaseriesformula,anduseaninteractiveformulaforit,inthefuturevalueformulaarticle. Differentperiodicpayments Afewpeoplehaverequestedaversionoftheaboveformulathattakesintoaccountthenumberofperiodicpayments(bothformulaeaboveassumeyourperiodicpaymentsmatchthefrequencyofcompounding).Forexample,yourmoneymaybecompoundedquarterlybutyou'remakingcontributionsmonthly.Inthiscase,youmaywishtotrythisversionoftheformula,originallysuggestedbyDarinthDouglas,andthenexpandeduponbyJean-BaptisteDelaroche.I'mmostgratefulfortheirinput.Thisformulaassumesthatregulardepositsarepaidatthebeginningratherthanattheendoftheperiod. Compoundinterestforprincipal: P(1+r/n)(nt) Futurevalueofaseries: PMT×p{[(1+r/n)(nt)-1]/(r/n)} (With'p'beingthenumberofperiodicpaymentsinthecompoundingperiod,dividedbyn) Formoreinformationaboutwhattodowiththeformulawhencalendarintervalsareirregular,seethisusefulpagefromJonWittwer. Example Anamountof$100isdepositedquarterlyintoasavingsaccountatanannualinterestrateof10%,compoundedmonthly.Thevalueoftheinvestmentafter12monthscanbecalculatedasfollows... PMT=100.r=0.1(decimal). n=12.p=4/n=4/12=0.3333333. Ifweplugthosefiguresintotheformula,wegetthefollowing: Total=PMT×p{[(1+r/n)(nt)-1]/(r/n)} Total=100×0.3333333×{[(1+0.1/12)^(12×1)-1]/(0.1/12)} Total=100×0.3333333×{[1.008333^(12)-1]/0.008333} Total=100×0.3333333×{0.104709/0.008333} Total=100×0.3333333×12.565583 Total=418.85 So,theinvestmentbalanceafter12monthsis$418.85(or$418.84ifyouroundthenumbersduringthecalculation). Insummary Thisarticleaboutthecompoundinterestformulabeganfairlyshortandhasexpandedandevolvedbaseduponyourrequestsforadaptedformulaeandexamples.So,Iappreciateit'snowquitealotlongerandmoredetailed.Thatsaid,Ihopeyou'vefoundithelpful.Ifyouhave,Iwouldbeverygratefulifyouwouldconsidersharingitonsocialmediaoronyourwebsite/blog.Thankyou.Shouldyouwishtoleaveacomment,pleasedosobelow. ArticlewrittenbyAlastairHazellandreviewedbyJamesWhittington. Advertisement Ratethisarticle Pleaseratethisarticlebelow.Ifyouhaveanyfeedbackonit, pleasecontactme. 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