Amortization. Amortization

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2 If a loan (debt) is repaid on installments (usually in equal amount), then the loan is said to be repaid by amortization method. a debt-repayment scheme wherein the original amount borrowed is repaid by making equal payments periodically Each installment payment consists of payment of interest and repayment of principal. All payments form an annuity whose present value is the original loan L. original loan = L P R R 3 4 n- n- n

3 Ex A man buys a color TV worth P48,. It is to be amortized through 6 monthly installments starting the next month. If interest is charged at 4% compounded monthly, find the monthly payment. 48, = P R

4 Any remaining debt after k payments is called outstanding balance ( OB k ) or outstanding liability or remaining liability ( RL k ) or outstanding principal. Outstanding balance (OB) or Remaining Liability (RL) refers to the amount of debt still unpaid Methods of finding or (i) Prospective Method (n is known) (ii) Retrospective Method (n is not known)

5 If a loan L is to be repaid by n payments of R, the outstanding balance just after the k th payment denoted by or, is the present value of the (n-k) payments still to be made. original loan = L P n-k remaining (n-k) payments still to be made k- k- k k+ k+ (n-) (n-) n k payments already made

6 If total number of payments (n) is NOT KNOWN, then we use retrospective method to find the outstanding balance or remaining liability after k payments. obligations = payments remaining debt on the kth period Value of k payments on the kth period k payments already made F k remaining unknown number of payments R R k- k- k k+ k+ CD original loan = L

7 Ex A loan is to be amortized via equal payments of P, each at the end of six months for 9 years. If the interest is based on % compounded semi-annually, find a) the original amount of the loan b) outstanding principal after the 8 th payment c) outstanding principal after the 8 th year. original loan = L = P

8 Ex A loan is to be amortized via equal payments of P, each at the end of six months for 9 years. If the interest is based on % compounded semi-annually, find b) outstanding principal after the 8 th payment P remaining (8-8) payments still to be made ,68, = L 8 payments already made

9 Ex A loan is to be amortized via equal payments of P, each at the end of six months for 9 years. If the interest is based on % compounded semi-annually, find c) outstanding principal after the 8 th year P remaining (8-6) payments ,68, = L 6 payments already made

10 Ex 3 A debt is being repaid at P5 every 6 months. The interest is 8% compounded semi-annually. If the outstanding liability after the 5 th payment is P4,, find the original loan. 5 payments already made F original loan = L = A

11 Ex 4 Goriotik obtains a P3M bank loan at % interest compounded semi-annually to construct another studio. The company repays the loan by paying P.5M every 6 months. What is the outstanding principal after the th payment? payments already made F.5M.5M.5M.5M.5M.5M.5M P3,, = L = A

12 If a loan (debt) is amortized, each installment payment is broken down into interest payment and repayment of principal. A table showing the breakdown of each periodic payment into interest payment (I) and repayment of principal (RP) is known as an amortization schedule.

13 Periodic payment Outstanding balance/principal Interest payment on the kth period Repayment of Principal on the kth period Outstanding balance at the end of kth period

14 5. Construct an amortization schedule for a loan of P to be amortized through 5 semi-annual payments if interest is charged at 6% compounded semi-annually. PERIOD R

15 Construct an amortization schedule for a loan of P to be amortized through 5 semi-annual payments if interest is charged at 6% compounded semi-annually. PERIOD R

16 Construct an amortization schedule for a loan of P to be amortized through 5 semi-annual payments if interest is charged at 6% compounded semi-annually. PERIOD R

17 Construct an amortization schedule for a loan of P to be amortized through 5 semi-annual payments if interest is charged at 6% compounded semi-annually. PERIOD R

18 Construct an amortization schedule for a loan of P to be amortized through 5 semi-annual payments if interest is charged at 6% compounded semi-annually. PERIOD R

19 Construct an amortization schedule for a loan of P to be amortized through 5 semi-annual payments if interest is charged at 6% compounded semi-annually. PERIOD R

20 PERIOD R TOTAL Interest payment decreases as repayment of principal increases.

21 PERIOD R TOTAL

22 Ex6. A loan of P5, with interest at 8% is payable quarterly for 6.5 years. a) Find the amount of each quarterly payment. b) How much of the 6 th payment goes to interest payment? c) By how much will principal be lessened by 6 th quarter? d) How much is the total interest paid? 5, = L= P

23 Ex6. A loan of P5, with interest at 8% is payable quarterly for 6.5 years. b) How much of the 6 th payment goes to interest payment? P remaining (6-5) payments still to be made , = L 5 payments already made

24 Ex6. A loan of P5, with interest at 8% is payable quarterly for 6.5 years. c) By how much will principal be lessened by 6 th quarter? d) How much is the total interest paid? , = L

25 Ex7. A P5, loan is amortized by 5 installments made every 6 months. The interest rate is 5% converted semi-annually. a) What is the semi-annual payment? b) What part of the th payment is used to pay interest? c) How much of the principal is repaid on the th period? d) What is the total interest paid in discharging the debt? 5, = L= P

26 Ex7. A P5, loan is amortized by 5 installments made every 6 months. The interest rate is 5% converted semi-annually. b) What part of the th payment is used to pay interest? P 4 remaining (5-) payments still to be made , = L

27 Ex7. A P5, loan is amortized by 5 installments made every 6 months. The interest rate is 5% converted semi-annually. c) How much of the principal is repaid on the th period? d) What is the total interest paid in discharging the debt? , = L

28 Ex8. A loan of P,5 is amortized at P every 6 months and a final payment, with interest at 6% compounded semi-annually. a) Find outstanding liability at the end of years. b) How much of the 5 th payment is interest payment and how much is the principal repaid?

29 Ex8. A loan of P,5 is amortized at P every 6 months and a final payment, with interest at 6% compounded semi-annually. b) How much of the 5 th payment is interest payment and how much is the principal repaid?

30 AMORTIZATION WITH FINAL IRREGULAR PAYMENT Ex. A P75 loan is to be amortized at P5 each quarter and a final irregular payment made 3 months after last regular payment. If interest is % converted quarterly. a) How many regular payments are needed? b) When is the final payment due? c) How much is the final payment? d) Construct an amortization schedule for it.

31 Ex. A P75 loan is to be amortized at P5 each quarter and a final irregular payment made 3 months after last regular payment. If interest is % converted quarterly. b) When is the final payment due? c) How much is the final payment? a) There should be 5 regular quarterly payments of P5. b) He will complete his full payment (final irregular payment x ) on the 6 th quarter. Remaining balance after last regular payment of P5 (end of 5 th quarter) Final irregular payment if it is to made one period after the last regular payment of P5. 75=L

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39 Ex. A loan of P,5 is amortized at P every 6 months and a final payment, with interest at 6% compounded semi-annually. a) Find the final payment and when is it due? b) What is the total interest paid in discharging the loan?

40 Ex. A loan of P,5 is amortized at P every 6 months and a final payment, with interest at 6% compounded semi-annually. a) Find the final payment and when is it due? b) What is the total interest paid in discharging the loan? Final irregular payment is due at the end of 3 th semi-annual period.

41 Ex. A loan of P,5 is amortized at P every 6 months and a final payment, with interest at 6% compounded semi-annually. b) What is the total interest paid in discharging the loan?