AMORTIZATIONAmortizationAmortization ScheduleOutstanding Balance

AmortizationAmortization method: repay a loan by means of installment payments at periodic intervalsThis is an example of annuityWe already know how to calculate the amount of each paymentOur goal: find the outstanding principal (balance)Two methods to compute it:prospectiveretrospective

Two MethodsProspective method: outstanding principal at any point in time is equal to the present value at that date of all remaining paymentsRetrospective method: outstanding principal is equal to the original principal accumulated to that point in time minus the accumulated value of all payments previously madeNote: of course, this two methods are equivalent. However, sometimes one is more convenient than the other

Examples(prospective) A loan is being paid off with payments of 500 at the end of each year for the next 10 years. If i = .14, find the outstanding principal, P, immediately after the payment at the end of year 6.(retrospective) A 7000 loan is being paid off with payments of 1000 at the end of each year for as long as necessary, plus a smaller payment one year after the last regular payment. If i = 0.11 and the first payment is due one year after the loan is taken out, find the outstanding principal, P, immediately after the 9th payment.

Amortization ScheduleGoal: divide each payment (of annuity) into two parts interest and principalAmortization schedule table, containing the following columns:paymentsinterest part of a paymentprincipal part of a paymentoutstanding principal

Example:5000 at 12 % per year repaid by 5 annual paymentsAmortization schedule:

Duration (Period)PeriodicPaymentInterest PaidPrincipal RepaidOutstanding Principal05000.0011387.05600.00787.054212.9521387.05505.55881.503331.4531387.05399.77987.282344.1741387.05281.301105.751238.4251387.05148.611238.440

Examplet - 1tPayment XOutstanding principal P Interest earned during interval (t-1,t) is iP Therefore interest portion of payment X is iP and principal portion is X - iPA 1000 loan is repaid by annual payments of 150, plus a smaller final payment. If i = .11, and the first payment is made one year after the time of the loan, find the amount of principal and interest contained in the third paymentRecall: in practical problems, the outstanding principal P can be found by prospective or retrospective methods

Example 1A loan of 5000 at 12% per year is to be repaid by 5 annual payments, the first due one year hence. Construct an amortization schedule.Given: Required: R and Amortization schedule Formula:

General rules to obtain an amortization scheduleTake the entry from Outs. Principal of the previous row, multiply it by i, and enter the result in InterestPayment Interest = Principal RepaidOuts. Principal of prev. row - Principal Repaid = Outs. PrincipalContinuei = 12 %

DurationPaymentInterestPrincipal RepaidOutstanding Principal05000.0011387.0521387.0531387.0541387.0551387.05

Example 2A 1000 loan is repaid by annual payments of 150, plus a smaller final payment. The first payment is made one year after the time of the loan and i = .11. Construct an amortization scheduleGiven: Required: n , amortization schedule , concluding paymentFormula:

Sheet1

DurationPaymentInterestPrincipal RepaidOutstanding Principal

01000

1150110.0040.00960.00

2

3

4

5

6

7

8

9

10

11

12

Example 3A P20,000 loan at 18% compounded quarterly is to be amortized every 3 months for two years. Find the quarterly payment and construct an amortization schedule.Given: Required: R and Amortization schedule Formula:

Example 4A debt of P30,000 with interest at 23% compounded quarterly will be discharged; interest included, by payments of P5,000 at the end of each three months for as long as it is necessary. Given: Required: n, Amortization schedule , concluding paymentFormula:

Two MethodsProspective method:

Outstanding Balance (Remaining Liabilities) after the kth paymentInterest paid after the kth paymentPrincipal repaid after the kth payment

Example 3A P20,000 loan at 18% compounded quarterly is to be amortized every 3 months for two years. Find the following: The remaining liabilities just after the 2nd paymentThe outstanding balance after one year. Solution:

Example 3A P20,000 loan at 18% compounded quarterly is to be amortized every 3 months for two years. How much of the 6th payment goes to interest and how much goes to the principal?Solution:

Two MethodsRetrospective method:

Outstanding Balance (Remaining Liabilities) after the kth payment

Example 4A debt of P30,000 with interest at 23% compounded quarterly will be discharged; interest included, by payments of P5,000 at the end of each three months for as long as it is necessary. Find the following: Outstanding principal at the end of 1.5 years

Example 4A debt of P30,000 with interest at 23% compounded quarterly will be discharged; interest included, by payments of P5,000 at the end of each three months for as long as it is necessary. Find the following: b) Outstanding balance just after the 5th payment

Example 4A debt of P30,000 with interest at 23% compounded quarterly will be discharged; interest included, by payments of P5,000 at the end of each three months for as long as it is necessary. Find the following: c) Concluding payment

SINKING FUNDSThis is an annuity that is invested for a specific purpose and is continued for a predefined period.Examples:Childs college fundTo buy a new computer in 3 years time.

SINKING FUNDSIn creating a fund it is important to know the periodic deposit and the amount to be put up. amount in the fund after the kth depositperiodic depositincrease in fund on the kth depositinterest earned in fund on the kth deposit

Example 1A man needs P30,000 at the end of 3 years. He decides to put his savings every six months that earns 10% converted semiannually. Construct the sinking fund schedule. Given: Required: R and Sinking Fund schedule Formula:

General rules to obtain a sinking fund scheduleTake the entry from Amount in Fund of the previous row, multiply it by i, and enter the result in Interest in FundDeposit + Interest = Increase in FundAmount in Fund of prev. row + Increase in Fund = Amount in FundContinuei = 0.05

DurationPeriodic DepositInterest in FundIncrease in FundAmount in Fund14,410.5224,410.5234,410.5244,410.5254,410.5264,410.52

Example 1... continuationGiven: a. Find the amount in fund after the 4th depositb. Find the interest earned in the 6th depositc. Find the increase in fund in the 5th deposit

Sinking FundsAlternative way to repay a loan sinking fund method:Pay interest as it comes due keeping the amount of the loan (i.e. outstanding principal) constantRepay the principal by a single lump-sum payment at some point in the future

012ninterest iLiLiL..lump-sum payment LLump-sum payment L is accumulated by periodic deposits into a separate fund, called the sinking fundSinking fund has rate of interest j usually different from (and usually smaller than) iIf (and only if) j is greater than i then sinking fund method is better (for borrower) than amortization method

ExamplesJohn borrows 15,000 at 17% effective annually. He agrees to pay the interest annually, and to build up a sinking fund which will repay the loan at the end of 15 years. If the sinking fund accumulates at 12% annually, find the annual interest paymentthe annual sinking fund paymenthis total annual outlaythe annual amortization payment which would pay off this loan in 15 yearsHelen wishes to borrow 7000. One lender offers a loan in which the principal is to be repaid at the end of 5 years. In the mean-time, interest at 11% effective is to be paid on the loan, and the borrower is to accumulate her principal by means of annual payments into a sinking fund earning 8% effective. Another lender offers a loan for 5 years in which the amortization method will be used to repay the loan, with the first of the annual payments due in one year. Find the rate of interest, i, that this second lender can charge in order that Helen finds the two offers equally attractive.

************************