deriving present value

1. RemainingBalance on Loan
The formulafor the remainingbalance onaloancan be usedto calculate the remainingbalance ata
giventime(time n),whetherata future date or at present.The remainingbalance onaloanformula
shownisonlyusedfora loanthat isamortized,meaningthatthe portionof interestandprincipal
appliedtoeachpaymentispredetermined.
The term "future value"inthe remainingbalance formulamayseemconfusing,butthe balance atany
time afterpaymentsare beingmade isthe future value inrespecttothe originationof the loan.
It isimportant,as withall financial formulas,thatthe interestrate perperiodandtermrelate toone
anotherand to whenthe paymentsare made.Withmonthlypayments,the rate wouldneedtobe the
monthlyrate and notthe annual rate.
Use of RemainingBalance Formula
Usingthis formula,asimple interestloanwillreturnanincorrectanswerinmostcasesbecause the
portionof principal andinterestisdeterminedbythe date the paymentwasmade on.
It isalso importanttonote that the remainingbalance onaloan formulashownisonlyusedfor
conventional loanswhere the payment,rate,andtermare fixed.Specializedloanslike graduating
payment,option,andnegativeamortizationwill require special calculationstodeterminethe loan
balance at a particulartime.
The remainingbalance ona loanformulacan be usedwithmortgages,consumerloans,andcommercial
loans.Where the moneyisspentdoesnothave an affectonhow to calculate the remainingbalance on
a loan.
How is the RemainingBalance Formula derived?
The remainingbalance of a loanformulacanbe separatedintotwosections,the future valueof the
original loanamountandthe future value of the annuity.

2. The future value of the original loanamountinthe firstsectionof the formulawill determinewhatthe
value wouldbe attime n if no paymentswere made.
A loanisan annuity,inthat itis a seriesof periodicpayments.Giventhis,the secondsectionof the
formulawouldbe the future value of the paymentsmade upuntil time n.
The remainingbalance canthenbe calculatedbysubtractingthe future value of the paymentsmade
fromthe future value of the original balance attime n
Loan Payment
The loan paymentformulaisused tocalculate the paymentsona loan.The formulausedtocalculate
loanpaymentsisexactlythe same asthe formulausedto calculate paymentsonanordinaryannuity.A
loan,bydefinition,isanannuity,inthatit consistsof a seriesof future periodic payments.
The PV,or presentvalue,portionof the loanpaymentformulausesthe original loanamount.The
original loanamountisessentiallythe presentvalueof the future paymentsonthe loan,muchlike the
presentvalue of anannuity.
It isimportantto keepthe rate per periodandnumberof periodsconsistentwithone anotherinthe
formula.If the loanpaymentsare made monthly,thenthe rate perperiodneedstobe adjustedtothe
monthlyrate and the numberof periodswouldbe the numberof monthsonthe loan.If paymentsare
quarterly,the termsof the loanpaymentformulawouldbe adjustedaccordingly.
Standard Loan Payment
The loan paymentformulashownisusedforastandard loanamortizedfora specificperiodof time with
a fixedrate.Examples of specializedloansthatdonotapplyto thisformulainclude graduatedpayment,
negativelyamortized,interestonly,option,andballoonloans.

3. An adjustable rate loanwill use the formulashownbutwill needtobe recalculatedbasedonthe
remainingbalance andremainingtermforeachnew rate change.
Use of Loan PaymentFormula
The loan paymentformulacanbe usedto calculate anytype of conventional loanincludingmortgage,
consumer,andbusinessloans.The formuladoesnotdifferbasedonwhatthe moneyisspenton,but
onlywhenthe termsof repaymentdeviate fromastandardfixedamortization.
Simple interestandamortizedloanswill generallyhave the same payment.The termsamortizedand
simple interestrelate tohowmuchof the paymentisappliedtoprincipal andhow muchisappliedto
interestforeachpayment.
Simple interestloansrelyonthe date of paymenttodetermine the amountof interestpaidwiththe
remainingamountgoingtoprincipal.If apaymentismade early,the interestportionof the payment
will be lessthanif paidlater.Lessinterestaccrueswhenthe amountispaidearlybecause the loan
balance will be lessdue tothe extraprincipal payments.
On the otherhand,an amortizedloanhasa predeterminedamountof interestpaidperpaymentsoan
earlierpaymenthasnoaffectonloweringthe principal balancesooner.Differentcompaniesandtheir
loanswill have differentpoliciesonhowtheyare amortized.Anexample of how acompanymay
amortize theirloan,isbyre-amortizingeveryyearsothatextraprincipal paymentstothe loanwill only
go ineffectaftera yearto lowerthe monthlyinterestportionsof the payment.
Alternative Loan Payment Formula
The paymenton a loancan also be calculatedbydividingthe original loan amount(PV)bythe present
value interestfactorof anannuitybasedonthe termand interestrate of the loan.Thisformulais
conceptuallythe same withonlythe PVIFA replacingthe variablesinthe formulathatPVIFA is
comprisedof