effective interest rate

Theeffectiveinterestrate,effectiveannualinterestrate,annualequ Theeffectiveinterestrate,effectiveannual interestrate,annualequivalentr ivalentr ate(AER)orsimplyeffectiverateistheinte ate(AER)orsimplyef fectiverateistheinterestrateonaloanorfi restrateonaloanorfinancialp nancialp roductrestatedfromthenominalinterestrate roductrestatedfromt henominalinterestrateasaninterestratewith asaninterestratewithannualc annualc ompoundinterestpayableinarrears.[1] Itisusedtocomparetheannualinterestbetw Itisusedtocompare theannualinterestbetweenloanswithdifferent eenloanswithdifferentcompoundi compoundi ngterms(daily,monthly,annually,orother). ngterms(daily,month ly,annually,orother).Theeffectiveinterestr Theeffectiveinterestratediffe atediffe rsintwoimportantrespectsfromtheannualp rsintwoimportantre spectsfromtheannualpercentagerate(APR):[2] ercentagerate(APR):[2] 1. theeffect theeffectivei iveintere nterestra stratege tegeneral nerallydo lydoesno esnotinc tincorpor orporateo ateone-ti ne-timech mechar ar gessuchasfront-endfees; 2. theeffect theeffectivei iveintere nterestra strateis teis(gen (generall erally)no y)notdef tdefined inedbyle bylegalo galorreg rregul ul atoryauthorities(asAPRisinmanyjurisdictions).[3] Bycontrast,theeffectiveAPRisusedasale Bycontrast,theeffec tiveAPRisusedasalegalterm,wherefront-fee galterm,wherefront-feesandoth sandoth ercostscanbeincluded,asdefinedbylocallaw.[2][3] Annualpercentageyieldoreffectiveannualyi Annualpercentageyiel doreffectiveannualyieldistheanalogousconc eldistheanalogousconceptused eptused forsavingsorinvestmentproducts,suchasa forsavingsorinvestm entproducts,suchasacertificateofdeposit.S certificateofdeposit.Sinceany inceany loanisaninvestmentproductforthelender, loanisaninvestment productforthelender,thetermsmaybeusedto thetermsmaybeusedtoapplyto applyto thesametransaction,dependingonthepointofview. Effectiveannualinterestoryieldmaybecalc Effectiveannualinter estoryieldmaybecalculatedorapplieddiffere ulatedorapplieddifferentlydepe ntlydepe ndingonthecircumstances,andthedefinition ndingonthecircumsta nces,andthedefinitionshouldbestudiedcarefu shouldbestudiedcarefully.For lly.For example,abankmayrefertotheyieldonalo example,abankmayre fertotheyieldonaloanportfolioafterexpect anportfolioafterexpectedlosses edlosses asitseffectiveyieldandincludeincomefro asitseffectiveyiel dandincludeincomefromotherfees,meaningtha motherfees,meaningthattheint ttheint erestpaidbyeachborrowermaydiffersubstan erestpaidbyeachbor rowermaydiffersubstantiallyfromthebank'sef tiallyfromthebank'seffectivey fectivey ield. Theeffectiveinterestrateiscalculatedasi Theeffectiveinterest rateiscalculatedasifcompoundedannually.Th fcompoundedannually.Theeffecti eeffecti verateiscalculatedinthefollowingway,wh verateiscalculated inthefollowingway,whereristheeffectivean ereristheeffectiveannualrate nualrate ,ithenominalrate,andnthenumberofcomp ,ithenominalrate, andnthenumberofcompoundingperiodsperyear oundingperiodsperyear(forexam (forexam ple,12formonthlycompounding): Forexample,anominalinterestrateof6%compoundedmonthlyisequiva Forexample,anominalinterestrateof6%com poundedmonthlyisequivalenttoa lenttoa neffectiveinterestrateof6.17%.6%compoun neffectiveinterestr ateof6.17%.6%compoundedmonthlyiscrediteda dedmonthlyiscreditedas6%/12= s6%/12= 0.005everymonth.Afteroneyear,theinitia 0.005everymonth.Af teroneyear,theinitialcapitalisincreasedby lcapitalisincreasedbythefact thefact or(1+0.005)12≈1.0617. Whenthefrequencyofcompoundingisincreased Whenthefrequencyof compoundingisincreaseduptoinfinitythecalcu uptoinfinitythecalculationwi lationwi llbe: Theyielddependsonthefrequencyofcompounding: EffectiveAnnualRateBasedonFrequencyofCompounding N o m i na l R a t e Semi-Annual Q ua r t er l y M o n t h ly D ai l y C on t i n u o u s 1% 1.002% 1.004% 1.005% 1.005% 1.005% 5% 5.062% 5.095% 5.116% 5.127% 5.127% 10% 10% 10.2 10.250 50% % 10.3 10.381 81% % 10.4 10.471 71% % 10.5 10.516 16% % 10.5 10.517 17% % 15% 15% 15.5 15.563 63% % 15.8 15.865 65% % 16.0 16.075 75% % 16.1 16.180 80% % 16.1 16.183 83% % 20% 20% 21.0 21.000 00% % 21.5 21.551 51% % 21.9 21.939 39% % 22.1 22.134 34% % 22.1 22.140 40% % 30% 30% 32.2 32.250 50% % 33.5 33.547 47% % 34.4 34.489 89% % 34.9 34.969 69% % 34.9 34.986 86% % 40% 40% 44.0 44.000 00% % 46.4 46.410 10% % 48.2 48.213 13% % 49.1 49.150 50% % 49.1 49.182 82% % 50% 50% 56.2 56.250 50% % 60.1 60.181 81% % 63.2 63.209 09% % 64.8 64.816 16% % 64.8 64.872 72% % Theeffectiveinterestrateisaspecialcase Theeffectiveinterest rateisaspecialcaseoftheinternalrateofr oftheinternalrateofreturn. eturn. Ifthemonthlyinterestratejisknownandre Ifthemonthlyinteres tratejisknownandremainsconstantthroughout mainsconstantthroughouttheyear theyear ,theeffectiveannualratecanbecalculatedasfollows: Beforeyoutakeoutabankloan,youneedtoknowhowyourinterestrat Beforeyoutakeoutabankloan,youneedtok nowhowyourinterestrateiscalc eiscalc ulated.Therearemanymethodsbanksusetoca ulated.Therearemany methodsbanksusetocalculateinterestratesan lculateinterestratesandeachme deachme thodwillchangetheamountofinterestyoupa thodwillchangethea mountofinterestyoupay.Ifyouknowhowtocal y.Ifyouknowhowtocalculatein culatein terestrates,youwillbetterunderstandyour terestrates,youwill betterunderstandyourloancontractwithyourb loancontractwithyourbank.You ank.You arealsoinabetterpositiontonegotiateyou arealsoinabetterp ositiontonegotiateyourinterestratewithyour rinterestratewithyourbank.Ba bank.Ba nkswillquoteyoutheeffectiverateofinter nkswillquoteyouthe effectiverateofinterest.Theeffectiverateo est.Theeffectiverateofinteres finteres tisalsoknownastheannualpercentagerate tisalsoknownasthe annualpercentagerate(APR).TheAPRoreffecti (APR).TheAPRoreffectiverateo verateo finterestisdifferentthanthestatedrateo finterestisdifferen tthanthestatedrateofinterest.Banksalsoti finterest.Banksalsotieyourin eyourin terestratetoabenchmark,usuallytheprime terestratetoabench mark,usuallytheprimerateofinterest. rateofinterest. EffectiveInterestRateonaoneYearLoan

Ifyouborrow$1000fromabankforoneyearandhavetopay$60ininterestfor thatyear,yourstatedinterestrateis6%.Hereisthecalculation: EffectiveRateonaSimpleInterestLoan=Interest/Principal=$60/$1000=6% YourannualpercentagerateorAPRisthesameasthestatedrateinthisexampl ebecausethereisnocompoundinteresttoconsider.Thisisasimpleinterestl oan. EffectiveInterestRateonaLoanWithaTermofLessThanoneYear Ifyouborrow$1000fromabankfor120daysandtheinterestrateis6%,whati stheeffectiveinterestrate? Effectiverate=Interest/PrincipalXDaysintheYear(360)/DaysLoanisOutsta nding EffectiverateonaLoanwithaTermofLessThanoneYear=$60/$1000X360/120= 18% Theeffectiverateofinterestis18%sinceyouonlyhaveuseofthefundsfor1 20daysinsteadof360days. EffectiveInterestRateonaDiscountedLoan Somebanksofferdiscountedloans.Discountedloansareloansthathavetheinte restpaymentsubtractedfromtheprincipalbeforetheloanisdisbursed. Effectiverateonadiscountedloan=Interest/Principal-InterestXDaysinth eYear(360)/DaysLoanisOutstanding Effectiverateonadiscountedloan=$60/$1,000-$60X360/360=6.38% Asyoucansee,theeffectiverateofinterestishigheronadiscountedloanth anonasimpleinterestloan. EffectiveInterestRatewithCompensatingBalances Somebanksrequirethatthesmallbusinessfirmapplyingforabusinessbankloa nholdabalance,calledacompensatingbalance,withtheirbankbeforetheywil lapprovealoan.Thisrequirementmakestheeffectiverateofinteresthigher. Effectiveratewithcompensatingbalances(c)=Interest/(1-c) Effectiveratecompensatingbalance=6%/(1-0.2)=7.5%(ifcisa20%compensa tingbalance) EffectiveInterestRateonInstallmentLoans Oneofthemostconfusinginterestratesthatyouwillhearquotedonabankloa nisthatonaninstallmentloan.Installmentloaninterestratesaregenerally thehighestinterestratesyouwillencounter.Usingtheexamplefromabove: Effectiverateoninstallmentloan=2XAnnual#ofpaymentsXInterest/(Total no.ofpayments+1)XPrincipal Effectiverate/installmentloan=2X12X$60/13X$1,000=11.08% Theinterestrateonthisinstallmentloanis11.08%ascomparedto7.5%onthe loanwithcompensatingbalances. 4.Effectiveannualinterestrateoninstallmentloans Aninstallment(amortized)loanisaloanthatisperiodicallypaidoffinequal installments.Examplesmayincludecarloans,commercialloans,andmortgages. Therearefourmethodsusedtocalculatetheeffectiveannualinterestrateoni nstallmentloans(refertothetablebelow). Illustration2:Effectiveinterestratesoninstallmentloans Actuarial method • Most accurate method • Used by lenders • Complicated formulas Constant-ratio method • Simple formula • Overstated EAR • Higherquotedrate,moreoverstatedEAR EAR=2xMxC÷[Px(N+1)] Misthenumberofpaymentperiodsperyear Cisthecostofcredit(financecharges) Pistheoriginalproceeds Nisthenumberofscheduledpayments Direct-ratio method • Simple formula • Morecomplicatedthanconstant-ratiomethodbutlesscomplicatedthanactuarial

method •

Slightlyunderstateseffectiveannualinterestrate EAR=6xMxC÷[3xPx(N+1)+Cx(N+1)] Misthenumberofpaymentperiodsperyear Cisthecostofcredit(financecharges) Pistheoriginalproceeds Nisthenumberofscheduledpayments N-ratiomethod • Moreaccuratethanconstant-ratioordirect-ratiomethods • Effectiveannualinterestrateisslightlyoverstatedorunderstateddependingo nthenominalrateandthematurityoftheloan EAR=MxCx(95xN+9)÷[12xNx(N+1)x(4P+C)] Misthenumberofpaymentperiodsperyear Cisthecostofcredit(financecharges) Pistheoriginalproceeds Nisthenumberofscheduledpayments Iftheamountofpaymentortimebetweenpaymentsvariesfromperiodtoperiod( e.g.,balloonpayments),theconstant-ratio,direct-ratio,andN-ratiomethodsc annotbeused.Ifalenderchargesacreditinvestigation,loanapplication,or lifeinsurancefee,suchacostshouldbeaddedtothecostofcredit(financec harge). Letuslookatasimpleexampletoseehowtheeffectiveannualinterestrateis calculatedoninstallmentloans. InstallmentLoans–Example1: CompanyABCborrows$12,000toberepaidin12months.Themonthlyinstallments are$1,116each.Thefinancechargeis$1,400.Whatistheapproximatevalueof effectiveannualinterestrate? Constantratiomethod: EAR= 2x12x$1,400 =0.2154 12,000x(12+1) Direct-ratiomethod: EAR= 6x12x$1,400 =0.2073 3x12,000x(12+1)+1,400x(12+1) N-ratiomethod: EAR= 12x$1,400(95x12+9) =0.2104 12x12(12+1)(4x12,000+1,400) Usingtheactuarialmethod,theeffectiveannualinterestrateislikelytobec loseto21.04%. Aswecanseefromthisexample,theconstant-ratiomethodoverstatedtheeffect iveannualinterestrate,whilethedirect-ratiomethodslightlyunderstatedthe effectiveannualinterestrateontheinstallmentloan. Thereisanothermethodusedtoapproximatethisrateonone-yearinstallmentlo anstobepaidinequalmonthlyinstallments.Theeffectiveinterestrateisdet erminedbydividingtheinterestbytheaverageamountoutstandingfortheyear. Iftheloanisdiscounted,theaverageloanbalanceequalstheaverageofproce eds(i.e.,principallessinterest). InstallmentLoans–Example2: CompanyABCborrows$10,000ata10%interestratetobepaidin12monthlyinst allments. Inthisexample,theEARcouldbeapproximatedasfollows: Interest=$12,000x0.10=$1,200 AverageLoanBalance=$12,000÷2=$6,000 EffectiveAnnualInterestRate=$1,200÷$6,000=0.20 Ifthisloanisdiscounted,theeffectiveannualinterestratewillbecalculate dasfollows: Interest=$1,200 Proceeds=$12,000-$1,200=10,800 AverageLoanBalance:$10,800÷2=$5,400

EffectiveAnnualInterestRate=$1,200÷$5,400=0.2222