REV 00 CHAPTER 10 ANNUITY QMT 3301 BUSINESS MATHEMATICS 1

REV 00 CHAPTER 10 ANNUITY QMT 3301 BUSINESS MATHEMATICS 1

10.1 Future Value of Ordinary Annuity Certain REV 00 Future value (accumulated value) of an ordinary annuity certain is the sum of all the future values of the periodic payments. The derivation of the formula of future value of ordinary annuity certain are as follow: Periodic payments RM R Interest rate per interest period i% Term of investment n interest periods Future value of annuity at end of n interest periods RM S QMT 3301 BUSINESS MATHEMATICS 2

REV 00 Formula: S R[(1 i)n – 1] i QMT 3301 BUSINESS MATHEMATICS 3

REV 00 Example : RM100 is deposited every month for 2 years 7 months at 12% compounded monthly. What is the future value of this annuity at the end of the investment period? How much interest is earned? Solution: 0 1 2 3 29 30 100 100 100 31 months S 100 100 100 No deposit Last deposit Annuity begins Annuity ends QMT 3301 BUSINESS MATHEMATICS 4

REV 00 1 i n 1 From S R , we get i 1 0.01 31 1 S 100 0 . 01 RM3613.27 Interest earned, I S – nR 3613.27 – (31 x 100) RM 513.27 QMT 3301 BUSINESS MATHEMATICS 5

10.2 Present Value of Ordinary Annuity Certain REV 00 Consist of the sum of all the present values of periodic payments. The deviation of the formula of present value of ordinary annuity certain is illustrated in the following: Periodic payments RM R Interest rate per interest period i% Term of investment n interest periods Future value of annuity at end of n interest periods RM A QMT 3301 BUSINESS MATHEMATICS 6

REV 00 Formula: A R[1 - (1 i)-n] i QMT 3301 BUSINESS MATHEMATICS 7

REV 00 Example : Raymond has to pay RM 300 every month for 24 months to settle a loan at 12% compounded monthly. a) What is the original value of the loan? b) What is the total interest that he has to pay? Solution: a) 0 1 2 3 300 300 300 A 22 300 23 300 24 payments 300 No payment Last payment QMT 3301 BUSINESS MATHEMATICS 8

REV 00 1 1 i n From A R , we get i 1 1 0.01 24 A 300 0.01 RM 6373.02 b) Total interest (300 x 24) – 6373.02 RM 826.98 QMT 3301 BUSINESS MATHEMATICS 9

10.3 Amortization REV 00 An interest bearing a debt is said to be amortized when all the principal and interest are discharged by a sequence of equal payments at equal intervals of time. 10.4 Amortization Schedule A table showing the distribution of principal and interest payments for the various periodic payments. QMT 3301 BUSINESS MATHEMATICS 10

REV 00 Example : A loan of RM 1000 at 12% compounded monthly is to be amortized by 8 monthly payments. a) Calculate the monthly payment. b) Construct an amortization schedule. Solution: 1 1 i n a) From A R , we get i 1 1 0.01 8 1000 R 0 . 01 R RM130.69 QMT 3301 BUSINESS MATHEMATICS 11

REV 00 b) Amortization Schedule Period Beginning balance (RM) Ending balance (RM) Monthly payment (RM) Total paid (RM) Total principal paid (RM) Total interest paid (RM) 1 1000 879.31 130.69 130.69 120.69 10 2 879.31 757.41 130.69 261.38 242.59 18.79 3 757.41 634.29 130.69 392.07 365.71 26.36 4 634.29 509.94 130.69 522.76 490.06 32.70 5 509.94 384.35 130.69 653.45 615.65 37.80 6 384.35 257.50 130.69 784.14 742.50 41.64 7 257.50 129.39 130.69 914.83 870.61 44.22 8 129.39 0.00 130.69 1045.52 1000.00 45.51 QMT 3301 BUSINESS MATHEMATICS 12

10.5 Sinking Fund REV 00 When a loan is settled by the sinking fund method, the creditor will only receive the periodic interest due. The face value of the loan will only be settled at the end of the term. In order to pay the face value, debtor will create a separate fund in which he will make periodic deposits over the term of the loan. The series of deposits made will amount to the original loan. QMT 3301 BUSINESS MATHEMATICS 13

REV 00 Example : A debt of RM 1000 bearing interest at 10% compounded annually is to be discharged by the sinking method. If five annual deposits are made into a fund which pays 8% compounded annually, a) Find the annual interest payment, b) Find the size of the annual deposit into the sinking fund, c) What is the annual cost of this debt, d) Construct the sinking fund schedule. QMT 3301 BUSINESS MATHEMATICS 14

Solution: a) Annual interest payment RM 1000 x 10% RM 100 n 1 i 1 b) From S R , we get i REV 00 1 0.08 5 1 1000 R 0 . 08 R RM170.46 c) Annual cost Annual interest payment Annual deposit RM 100 RM 170.46 RM 270.46 QMT 3301 BUSINESS MATHEMATICS 15

REV 00 d) Sinking Fund Schedule End of period (year) Interest earned (RM) Annual deposit (RM) Amount at the end of period (RM) 1 0 170.46 170.46 2 13.64 170.46 354.56 3 28.36 170.46 553.38 4 44.27 170.46 768.11 5 61.45 170.46 1000.02 QMT 3301 BUSINESS MATHEMATICS 16

10.6 Annuity With Continuous Compounding Future value of annuity: REV 00 kt e 1 S R kp e 1 Present value of annuity: QMT 3301 BUSINESS MATHEMATICS kt 1 e A R kp e 1 17

REV 00 Where: S Future value of annuity A Present value of annuity R Periodic payment or deposit e Natural logarithm k Annual continuous compounding rate t Time in years p Number of payments in 1 year QMT 3301 BUSINESS MATHEMATICS 18

REV 00 Example : James wins an annuity that pays RM 1000 at the end of every six months for four years. If money is worth 10% per annum continuous compounding, what is a) The future value of this annuity at the end of four years, b) The present value of this annuity? QMT 3301 BUSINESS MATHEMATICS 19

REV 00 Solution: a) From b) From kt e 1 , we get S R k p e 1 10%( 4 ) e 1 S 1000 10% 2 1 e RM9592.63 kt 1 e A R k p e 1 , we get 10%( 4 ) 1 e A 1000 10% 2 1 e RM 6430.13 QMT 3301 BUSINESS MATHEMATICS 20