Ordinary Annuity. S.Y.Tan. Ordinary Annuity

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2 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 () the amount of each payment

3 First payment interval Last payment interval Term The time between two consecutive payments of an annuity is called payment interval or payment period. Payments maybe made monthly, quarterly, semiannually, annually or every months or every months.

4 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

5 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

6 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 n- n- n Annuity due an annuity where payments are made at the beginning of each payment interval Annuity due of n payments n- n- n

7 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- d- d d+ d+ d+(n-) d+(n-) d+n NO payment for d periods st payment starts on the (d+)th period

8 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 n- n- n

9 SUM=F n- n- n

10 F is the amount of an ordinary annuity of n payments is the periodic payment of the annuity i is the interest rate per period n is the total number of payments or periods

11 Ex Find the amount of an annuity of P paid at the end of every 6 months for years if money is worth 6.% converted semi-annually. F =, (semi-annual periods)

12 Ex In order to create a fund for his forth coming business venture, Matthew decides to deposit P5 in a fund at the end of each month. If the bank pays % compounded monthly on his deposits, how much is in the fund at the end of years? F =, (months)

13 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 st payment date sum of the discounted values of all payments at the beginning of the term or at one period before the st payment date P n- n- n

14 P= SUM n- n- n

15 value of annuity at one period before st payment date P F value of annuity on last payment date n- n- n P is the present value of an ordinary annuity of n payments is the periodic payment of the annuity i is the interest rate per period n is the total number of payments or periods

16 Ex Find the present value of an annuity of P paid at the end of every 6 months for years if money is worth 6.% converted semi-annually. 68,8.6 = P F =,57.7 fr Ex 8 9 (semi-annual periods)

17 Ex 5. An LCD TV is purchased with down payment of P, and P6.5 at the end of each month for two years to discharge all principal and interest at 5% converted monthly. Find the cash price of the TV set. Cash value (CV) = Down payment (D) + Present Value (P) P

18 Ex 6. Linda is paying P, every months for years for a loan she acquired. If she is being charged an interest of 5% converted quarterly, how much was her original loan? P

19 Ex 7. Jane deposits P, every 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 years? F 5 6

20 Ex 8 A multimedia workstation is for sale at Php8, every six months for two years at % compounded semi-annually or at Php, down and Php7,.5 each month for the next months at 5% compounded monthly. Which terms should you choose? cheaper, st offer better P P

21 Ex 9. In preparation for the college education of his son, Mr Yanga will deposit P at the end of each month for 5 years in a fund earning.5% compounded monthly. How much is in the fund (a) just after 5 th deposit? (b) just after the last deposit?

22 Finding periodic payment () of an Ordinary annuity Formulas for the amount F, present value P of an ordinary annuity: P F n- n- n

23 Ex A newly-formed business bought a property worth P.5M. They paid a downpayment of PM with an agreement to pay the balance in years at % compounded quarterly. How much is the quarterly payment?,5, = P 8 9

24 Ex Jim and his partners want to have P.M in years. They make semi-annual deposits in an account which pays interest at 7% compounded semi-annually. Find their semi-annual deposit. F =,, 5 6

25 Ex On his retirement at age 6, ic receives P8, as a share of a pension fund. His heir invests this sum at 6.6% compounded quarterly. How much could ic or his heir regularly withdraw at the end of each months for the next 5 years? 8, = P 98 99

26 Ex At the end of each year for years, a corporation will deposit equal sums in a depreciation fund to provide for the replacement of machinery worth P5,. If the fund accumulates at 8% effective, how much must each deposit be? F = 5, 8 9

27 Ex 5 A loan of P5, with interest at 5% compounded every months is to be repaid by equal payments made at the end of every months. Find the size of each payment. 5, = P