Compound Interest | Formulas, Derivation & Solved Examples

The formula used to calculate compound interest is CI = P( 1 + r/100)n - P. Here in this formula the amount is calculated and then the principal is subtracted ... LearnPracticeDownload  Compound Interest Compoundinterestisaninterestaccumulated ontheprincipalandinteresttogetheroveragiventimeperiod.Theinterestaccumulated onaprincipaloveraperiodoftimeisalsoaccountedunder theprincipal.Further,theinterestcalculation forthenexttimeperiodisontheaccumulatedprincipalvalue.Compoundinterestisthenewmethodofcalculationofinterestusedfor allfinancialandbusinesstransactionsacrosstheworld.Thepowerofcompoundingcaneasilybeunderstood,whenweobservethecompoundinterestvaluesaccumulatedacross successivetimeperiods. Asumofmoneyof$100investedoveraperiodoftimefora10%rate wouldgiveasimpleinterestof$10,$10,$10...oversuccessivetimeperiodsof1year,butwouldgiveacompoundinterestof  $10,$11,$12.1,$13.31...Letusunderstandmoreaboutthis,andthecalculationsofcompoundinterestinthebelowcontent. 1. WhatismeantbyCompoundInterest? 2. CompoundInterestFormula 3. DerivationofCompoundInterestFormula 4. CompoundInterestFormulaforDifferentTimePeriods 5. FAQsonCompoundInterest WhatismeantbyCompoundInterest? Compoundinterestistheinterestpaidonbothprincipalandinterest,compoundedat regularintervals. Atregularintervals,theinterestsofaraccumulatedisclubbedwiththeexistingprincipalamountandthentheinterestiscalculatedforthenewprincipal.Thenewprincipalisequaltothesumofthe Initialprincipal,andtheinterestaccumulatedsofar. CompoundInterest=InterestonPrincipal+CompoundedInterestatRegularIntervals Thecompoundinterestis calculatedatregularintervalslikeannually(yearly),semi-annually,quarterly,monthly,etc;Itislike,re-investingthe interestincomefromaninvestmentmakesthe moneygrowfasterovertime!Itisexactlywhatthecompoundinterestdoestothemoney.Banksoranyfinancialorganizationcalculatetheamountbasedoncompoundinterestonly.  CompoundInterestFormula Thecompoundinterestiscalculated,aftercalculatingthetotalamountoveraperiodoftime,basedontherateofinterest,andtheinitialprincipal.ForaninitialprincipalofP,rateofinterestperannumofr,timeperiodtinyears,frequencyof the numberoftimes theinterestiscompoundedannuallyn,theformulaforcalculationofamountisasfollows.  Theaboveformularepresentsthetotalamountattheendofthetimeperiodandincludesthecompoundedinterestandtheprincipal.Further,wecancalculatethecompoundinterestbysubtractingtheprincipalfromthisamount.Theformulaforcalculatingthecompoundinterestisasfollows  Intheaboveexpression, Pistheprincipalamount ristherateofinterest(decimal) nisfrequencyorno.of timestheinterestiscompoundedannually tistheoveralltenure. Itistobenotedthattheabove-givenformulaisthegeneralformulawhentheprincipaliscompoundednnumberoftimesinayear.Ifthegivenprincipaliscompoundedannually,theamountafterthetimeperiodatpercentrateofinterest,r,isgivenas: A=P(1+r/100)t,andC.I.wouldbe: P(1+r/100)t -P. Derivationof CompoundInterestFormula The formulafor compoundinterestcanbederivedfromtheformulaforsimpleinterest.Theformulaforsimpleinterestistheproductoftheprincipal,timeperiod,andrateofinterest(SI=ptr/100). Beforelookingintotoderivationoftheformulaforcompoundinterest,letusunderstandthebasic differencebetweensimpleinterest,compoundinterestcomputation.Theprincipalremainsconstantoveraperiodoftime,forsimpleinternetcomputation,butforcompoundinterestcomputationtheinterestisaddedto the principal,forcompoundinterestcomputation. Derivation:  LettheprincipalisPandtherateofinterestber.Attheendofthefirstcompoundingperiod,thesimpleinterestontheprincipalisP× r/100.Andhence,theamountisP+P×r/100=P(1+r/100).Theamountistakenastheprincipalforthesecondcomputationperiod.   Attheendofthesecondcompoundingperiod, thesimpleinterestontheprincipalis: P(1+r/100) ×r/100,andhencetheamountis:P(1+r/100) ×r/100+P(1+r/100) ×r/100=P(1+r/100)2. Continuinginthismannerforncompoundingperiods,theamountattheendofthe nth compoundingperiodisA= P(1+r/100)n.  Fromtheaboveformulas andcomputations,youcanobservethatthecompoundinterestisthesameassimpleinterestforthefirstinterval.But,overaperiodoftime,thereisaremarkabledifferenceinreturns.  Thesimpleinterestvalue foreachoftheyearsis thesame,astheprincipalonwhichitiscalculated isconstant.Butthecompoundinterestisvaryingandincreasingacrosstheyears.Becausetheprincipalonwhichthecompoundinterestiscalculatedisincreasing.Theprincipal foraparticular yearisequaltothesumoftheinitialprincipalvalue,andthe accumulatedinterestof thepastyears.  Forexample,asumof$10,000isdepositedatarateof10%.Thebelowtableexplainsthedifferencebetweensimpleinterestandcompoundinterestcomputationonthisprincipal: SimpleInterestCalculation(r=10%) CompoundInterestCalculation(r=10%) For1st year: P=10,000 Time=1year Interest=1000 For1st year: P=10,000 Time=1year Interest=1000 For2nd year: P=10,000 Time=1year Interest=1000 For2nd year: P=11000 Time=1year Interest=1100 For3rd year: P=10,000 Time=1year Interest=1000 For3rd year: P=12100 Time=1year Interest=1210 For4th year: P=10,000 Time=1year Interest=1000 For4thyear: P=13310 Time=1year Interest=1331 For5th year: P=10,000 Time=1year Interest=1000 For5th year: P=14641 Time=1year Interest=1464.1 TotalSimpleInterest=5000 TotalCompountInterest=6105.1 TotalAmount=1000+5000=6000 TotalAmount=1000+6105.1=7105.1 CompoundInterestFormulafor DifferentTimePeriods Compoundinterestforagivenprincipalcanbecalculatedfordifferenttimeperiodsusingdifferentformulas. CompoundInterestFormula- HalfYearly Theinterestinthecaseofcompoundinterestvaries basedontheperiodofcomputation. Ifthetimeperiodforthecalculationofinterestishalf-yearly,theinterestiscalculatedeverysixmonths,andtheamountiscompoundedtwiceayear.  Theformulatocalculatethecompoundinterestwhentheprincipaliscompoundedsemi-annuallyorhalf-yearlyisgivenas: Herethecompoundinterestiscalculatedforthehalf-yearlyperiod,andhence therateofinterestr,isdividedby2 andthetimeperiodisdoubled. Theformulatocalculatetheamountwhentheprincipaliscompoundedsemi-annuallyorhalf-yearlyisgivenby:  Intheaboveexpression, Aistheamountattheendofthetimeperiod Pistheinitialprincipalvalue,ristherateofinterestperannum tisthetimeperiod C.I.isthecompoundinterest. CompoundInterestFormula-Quarterly  Ifthetimeperiodforthecalculationofinterest isquarterly,theinterestiscalculatedforeverythree months,andtheamountiscompounded4 times ayear. Theformulatocalculatethecompoundinterestwhentheprincipaliscompoundedquarterlyisgivenas: Herethecompoundinterestiscalculatedforthequarterlytimeperiod,andhence therateofinterestr,isdividedby4 andthetimeperiodisquadrupled.Theformulatocalculatetheamountwhentheprincipaliscompoundedquarterlyisgivenby:  Intheaboveexpression, Aistheamountattheendofthetimeperiod Pistheinitialprincipalvalue,ristherateofinterestperannum tisthetimeperiod C.I.isthecompoundinterest. Monthly CompoundInterestFormula Themonthlycompoundinterestformulaisalsoknownastheinterestcalculatedpermonthi.e.,n=12.Totalcompoundinterestisthefinalamountexcludingtheprincipalamount. Themonthly compoundinterestformulaisexpressedas: CI=P(1+r/12)12t -P DailyCompoundInterestFormula Whentheamountcompoundsdaily,itmeansthattheamountcompounds365 timesinayear.i.e.,n=365.Thedailycompoundinterestformulaisexpressedas: CI=P(1+r/365)365t-P ImportantNotes Compoundinterestdependsontheamountaccumulatedattheendofthe previoustenurebutnotontheoriginalprincipal.  Banks,insurancecompanies,etc.generallylevycompoundinterest. Iftheinterestiscompoundedquarterly,theformulaofamountisgivenby:\begin{equation}A=P\left(1+\frac{r/4}{100}\right)^{4n}\end{equation}  Whilecalculatingthecompoundinterest,therateofinterest,andeachtimeperiodmustbeofthesameduration. Tips&Tricks Theruleof 72:Itisaquickmethodtoknowhowlongitwilltakeforyourmoneytodouble. DoublingTime=72/InterestRate  Usingtheruleof 72,wecanfindthenumberofyearstodoubleyourmoneybysimplydividing 72bytherateofinterest.Forexample,atan8%compoundedinterestrateyourmoneywilldoublein 72÷ 8=9 Thetimedurationoverwhichaninterestrateisapplicableisreferredtoinmanydifferentterms. Sometimesitiscalled“perannum”or“annual”or“peryear”.Allofthesemeanyou’llgetthegivenrateofinterestoveraperiodof1year. Semi-annualis 6 months. Whilequarterlyis 3 monthsduration.   SimpleInterestFormula   SimpleInterestCalculator   CompoundInterestCalculator   CommercialMath   InterestFormula   MonthlyCompundInterestFormula   DailyCompoundInterestFormula     Example1:Noahlends$4000 toEmma ataninterestrateof 10%perannum,compoundedhalf-yearlyforaperiodof 2 years.Canyouhelphimfindouthowmuchamounthegetsafteraperiodof2yearsfromEmma? Solution: Letusidentifythedatagiventous:Theprincipalamount'P'is$4000.Therateofinterest,r'is 10%perannum.Conversionperiod= Half-year,  Rateofinterestperhalf-year=10/2%=5%.Thetimeperiod't'is2years. TimePeriod AmountCalculation 1sthalfyear Principal=$4000 Interest=5%× $4000=(5/100)×4000=$200 Amount=$4000+$200=$4200 2ndhalfyear Principal=$4200 Interest=5%×$4200=5/100×4200=$210  Amount=$4200+$210=$4410 3rdhalfyear Principal=$4410 Interest=5%×$4410=$220.5 Amount=$4410+$220.5=$4630.5 4thhalfyear Principal=$4630.5 Interest=5%×$4630.5= $231.53  Amount=$4630.5+$231.53=$4862.03 Thetotalinteresttobepaidover 2years 200 +$210 +$220.5+$231.53  =$862.03.TotalAmount =P+I=$4000 +$862.03 =$4862.03.Thereforethetotalamountis$4862.03.   Example2: Solvetheabove-givenproblemusingthecompoundinterestformula. Solution: Theprincipalamount'P'is$4000.Therateofinterest'r'is 10%perannum.Conversionperiod=Half-year,  Rateofinterestperhalf-year =10/2%=5%.Thetimeperiod't'is2years.Thecompoundingfrequency'n'is2.  Letussubstitutethegivendatainthecompoundinterestformula:A=P(1+{r/2}/100)2n=4000(1+{10/2}/100)2(2)= $4862.03 Thereforethefinalamountis$4862.03,andthecompoundinterestformulamakesthesolutionsimple.   ShowSolution> gotoslidegotoslide       Breakdowntoughconceptsthroughsimplevisuals. Mathwillnolongerbeatoughsubject,especiallywhenyouunderstandtheconceptsthroughvisualizations.     BookaFreeTrialClass           CheckAnwser> gotoslidegotoslidegotoslidegotoslide   FAQsonCompound Interest HowtoCalculateCompoundInterest? TheformulausedtocalculatecompoundinterestisCI=P(1+r/100)n -P. Hereinthisformulatheamountiscalculatedandthentheprincipalissubtractedfromit,toobtainthecompoundinterestvalue. WhatIstheDifferenceBetweenSimpleandCompoundInterest? Simpleinterestistheinterestpaidonlyontheprincipal,whereas,compoundinterestis theinterestpaidonbothprincipalandinterestcompoundedat regularintervals. HowtoCalculateAmountUsingCompoundInterest? Thereisadirectformulaforthecalculationofcompoundinterest.A=P(1+r/100)n.Hereweneedtodefinetherateofinterestandthetimeintervalatwhichthecompoundinterestiscalculated.  HowToCalculateAmountUsingCompoundInterestFormula?  Thereisageneralcompoundinterestformulaforthecalculationofcompoundinteresti.e., CI=Amount-Principal where,Amount=P(1+r/100)t.Bysubstitutingthegivenparameterssuchas P(principalamount),r(rateofinterest),andt(time)amountcanbeeasilycalculated.  WhatIstheMonthlyCompoundInterestFormula?  Themonthlycompoundinterestformulaisgivenas  CI=P(1+(r/12))12t -P.Where,Pistheprincipalamount,ristheinterestrateindecimalform,n=12(itmeansthattheamountcompounded12 timesinayear),andtisthetime. WhatIstheDailyCompoundInterestFormula?  Thedaily compoundinterestformulaisgivenas  A=P(1+r/365)365t,wherePistheprincipalamount,ristheinterestrateofinterestindecimalform,n=365(itmeansthattheamountcompounded 365timesinayear),andtisthetime. WhatIstheFutureValue CompoundInterest Formula? ThefuturevaluecompoundinterestformulaisexpressedasFV =PV(1+r/n)nt.Here,PV=PresentValue(Initialinvestment),r=rateofinterest,n=numberoftimestheamountiscompounding,andt=timeinyears. IsInterestCompoundedDailyBetterthanMonthly? Theinterestcompoundeddaily has 365compoundingcycles ayear.Itwillgeneratemoremoneycomparedtointerestcompoundedmonthly,whichhasonly12compoundingcyclesperyear. WhatAretheMainDisadvantagesofCompoundInterest? Ifwemissapaymentbyadayalso,towardstheendoftenureit mayincurahugeloss.Theinterestcalculationisforthenextcycleandforahighervalue.Compoundinterestisactuallydesignedtohelpthelendersbutnottheborrowers. HowDoesCompoundInterestDependonTimePeriod? Thecompoundinterestdependsonthetimeintervalofcalculationofinterest.Thetimeintervalforthecalculationofinterestcanbeaday,aweek,month,quarterly,half-yearly.Fortheshortertimeperiodofcalculation,thenetaccumulatedcompoundinterestishigher.  HowMuchisCompoundInterestGreaterthanSimpleInterest? Thecompoundinterestcanbegreaterthanthesimpleinterest.Thecompoundinterestvaluevariesandincreasesforsuccessivetimeperiods.  Aninitialprincipalof$100investedoveraperiodoftimewouldgiveasimpleinterestof$10,$10,%10...oversuccessivetimeperiodsof1year,butwouldgiveacompoundinterestof  $10,$11,$12.1,$13.31.....Thusthecompoundinterestisgreaterthanthesimpleinterest.Onlyforthefirstyear,orforthefirstcycleofcalculation,thecompoundinterest,andthesimpleinterestvaluesareequal. CanCompoundInterestbeGreaterthanPrincipal? Thecompoundinterestcanbegreaterthantheprincipal.Thecompoundinterestvaluevariesandincreasesforsuccessivetimeperiods.Aninitialprincipalof$100investedoveraperiodoftimewouldgiveacompoundinterestof  $10,$11,$12.1,$13.31....oversuccessivetimeperiodsof1yeareach.Thusthe compoundinterestincreasesoveraperiodoftimeandcanbegreaterthantheinitialprincipalvalue.  HowDoyouCalculateCompoundInterestforHalfYear? TheformulaforcalculationofcompoundinterestforhalfyearisCI =p(1+{r/2}/100)2t.- p.Hereinthisformula'A'isthefinalamount,'p'istheprincipal,and't'isthetimeinyears.Intheformulawecanobservethattherateofinterestishalvedandthetimeisdoubled,toaccountforthecalculationofcompoundinterestforhalfayear. WhatIstheInformationRequiredtoCalculateCompoundInterest? Thecalculationofcompoundinterestrequiresustoknowtheprincipal,rateofinterest,andthetimeperiod.Also,weneedtoknowthetimeintervalforwhichtheinterestistobecalculated.  WhatAretheUnitsofCompoundInterest? Theunitsofcompoundinterestaretheunitofcurrencyandare thesameastheunitusedfortheprincipalvalue.Iftheprincipalisindollars,oryen,thecompoundinterestwouldalsobeindollarsoryen. ExploremathprogramExplorecodingprogramDownloadFREEStudyMaterialsCommercialMathWorksheetExploremathprogramExplorecodingprogramMakeyourchildnaturallymathmindedBookAFreeClass