Ordinary Annuity S.Y.Tan

Annuity a sequence of equal payments made at equal time intervals Examples: daily wages, periodic payments of installment purchases, monthly rent, annual insurance premiums Payment interval – the time between successive payments Term – the time between the first and last payment intervals Periodic payment (R) – the amount of each payment S.Y.Tan

R Last payment interval First payment interval Term … The time between two consecutive payments of an annuity is called payment interval or payment period. Payments maybe made monthly, quarterly, semi-annually, annually or every 2 months or every 4 months. S.Y.Tan

Types of Annuity Simple annuity – an annuity for which the payment period is the same as the interest period Example: An annuity for which the interest rate is compounded monthly and payments are also made monthly General annuity – interest and payment periods do not coincide with one another - Example: An annuity for which the interest rate is compounded quarterly while payments are made monthly S.Y.Tan

Classification of Annuity Annuity certain – an annuity where payments begin and end at fixed times. Example: installment payments for certain purchase made Contingent annuity – payments are dependent on an event that can not be foretold - Example: premium on life insurance policy S.Y.Tan

3 kinds of Annuity Certain (Simple Annuity) Ordinary annuity – an annuity where payments are made at the end of each payment interval Ordinary annuity of n payments R n n-1 n-2 4 3 2 1 Annuity due – an annuity where payments are made at the beginning of each payment interval Annuity due of n payments R n n-1 n-2 4 3 2 1 S.Y.Tan

Deferred annuity - an annuity wherein the first payment interval does not coincide with the first interest period. The first payment is put off to some later date. Deferred annuity of n payments d+n d+(n-1) d+(n-2) d-1 d-2 2 1 R d d+1 d+2 1st payment starts on the (d+1)th period NO payment for d periods S.Y.Tan

Ordinary Annuity Amount of an Ordinary annuity (F) – value of annuity in one lump sum amount at the end of the term – value of annuity on the last payment date – sum of the accumulated values of all payments at the end of the term or on the last payment date F R n n-1 n-2 4 3 2 1 S.Y.Tan

SUM=F R n n-1 n-2 4 3 2 1 S.Y.Tan

F is the amount of an ordinary annuity of n payments R is the periodic payment of the annuity i is the interest rate per period n is the total number of payments or periods S.Y.Tan

Ex 1 Find the amount of an annuity of P3000 paid at the end of every 6 months for 20 years if money is worth 6.24% converted semi-annually. F = 232,457.71 3000 40 39 38 4 3 2 1 (semi-annual periods) S.Y.Tan

Ex 2 In order to create a fund for his forth coming business venture, Matthew decides to deposit P5000 in a fund at the end of each month. If the bank pays 4% compounded monthly on his deposits, how much is in the fund at the end of 2 years? F = 124,714.44 5000 24 23 22 4 3 2 1 (months) S.Y.Tan

Ordinary Annuity Present Value of an Ordinary annuity (P) – value of annuity in one lump sum amount at the beginning of the term – value of annuity at one period before 1st payment date – sum of the discounted values of all payments at the beginning of the term or at one period before the 1st payment date P R n n-1 n-2 4 3 2 1 S.Y.Tan

P= SUM R n n-1 n-2 4 3 2 1 S.Y.Tan

P F P is the present value of an ordinary annuity of n payments value of annuity at one period before 1st payment date P F value of annuity on last payment date R n n-1 n-2 4 3 2 1 P is the present value of an ordinary annuity of n payments R is the periodic payment of the annuity i is the interest rate per period n is the total number of payments or periods S.Y.Tan

Ex 4 Find the present value of an annuity of P3000 paid at the end of every 6 months for 20 years if money is worth 6.24% converted semi-annually. 68,018.62 = P F = 232,457.71 fr Ex 1 3000 40 39 38 4 3 2 1 (semi-annual periods) S.Y.Tan

P Cash value (CV) = Down payment (D) + Present Value (P) Ex 5. An LCD TV is purchased with down payment of P30,000 and P4624.50 at the end of each month for two years to discharge all principal and interest at 15% converted monthly. Find the cash price of the TV set. Cash value (CV) = Down payment (D) + Present Value (P) P 4624.50 24 23 22 4 3 2 1 S.Y.Tan

Ex 6. Linda is paying P10,000 every 3 months for 3 years for a loan she acquired. If she is being charged an interest of 5% converted quarterly, how much was her original loan? P 10000 12 11 10 4 3 2 1 S.Y.Tan

Ex 7 . Jane deposits P14,000 every 3 months in a savings account that pays 6% compounded quarterly. Assuming that she does not withdraw any amount, how much would she have in her account at the end of 4 years? F 14000 16 15 14 4 3 2 1 S.Y.Tan

Ex 8 A multimedia workstation is for sale at Php28,000 every six months for two years at 12% compounded semi-annually or at Php20,000 down and Php7,040.15 each month for the next 12 months at 15% compounded monthly. Which terms should you choose? cheaper, 1st offer better P P 7040.15 12 11 10 4 3 2 1 28000 4 3 2 1 S.Y.Tan

Ex 9 . In preparation for the college education of his son, Mr Yanga will deposit P2400 at the end of each month for 5 years in a fund earning 4.5% compounded monthly. How much is in the fund (a) just after 15th deposit ? (b) just after the last deposit? 2400 60 16 15 4 3 2 1 2400 59 S.Y.Tan

Ordinary Annuity Finding periodic payment (R) of an Ordinary annuity P Formulas for the amount F, present value P of an ordinary annuity: P F R n n-1 n-2 4 3 2 1 S.Y.Tan

Ex 1 A newly-formed business bought a property worth P14. 5M Ex 1 A newly-formed business bought a property worth P14.5M. They paid a downpayment of P3M with an agreement to pay the balance in 10 years at 12% compounded quarterly. How much is the quarterly payment? 11,500,000 = P R 40 39 38 4 3 2 1 S.Y.Tan

Ex 2 Jim and his partners want to have P4. 2M in 3 years Ex 2 Jim and his partners want to have P4.2M in 3 years. They make semi-annual deposits in an account which pays interest at 7% compounded semi-annually. Find their semi-annual deposit. F = 4,200,000 R 6 5 4 3 2 1 S.Y.Tan

Ex 3 On his retirement at age 60, Ric receives P800,000 as a share of a pension fund. His heir invests this sum at 6.36% compounded quarterly. How much could Ric or his heir regularly withdraw at the end of each 3 months for the next 25 years? 800,000 = P R 100 99 98 4 3 2 1 S.Y.Tan

Ex 4 At the end of each year for 10 years, a corporation will deposit equal sums in a depreciation fund to provide for the replacement of machinery worth P500,000. If the fund accumulates at 8% effective, how much must each deposit be? F = 500,000 R 10 9 8 4 3 2 1 S.Y.Tan

Ex 5 A loan of P50,000 with interest at 15% compounded every 4 months is to be repaid by 24 equal payments made at the end of every 4 months. Find the size of each payment. 50,000 = P R 24 23 22 4 3 2 1 S.Y.Tan