Ordinary Annuity

Ordinary Annuity

S.Y.Tan

Ordinary Annuity

Annuitya sequence of equal payments made at equal time intervals

– Examples: daily wages, periodic payments of installment purchases, monthly rent, annualinsurance 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

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Ordinary Annuity

• The time between two consecutive payments of an

annuity is called payment interval or payment period.

R

Last payment intervalFirst payment interval

Term

RR … R

• Payments maybe made monthly, quarterly, semi-

annually, annually or every 2 months or every 4

months.

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Ordinary Annuity

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

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Ordinary Annuity

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

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Ordinary Annuity

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

Annuity due – an annuity where payments are made

at the beginning of each payment interval

Annuity due of n payments

R

nn-1n-243210

R R R R R R

R

nn-1n-243210

R R R R RR

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Ordinary Annuity

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.

d+nd+(n-1)d+(n-2)d-1d-2210

R R R R R

d d+1 d+2

NO payment for d periods

Deferred annuity of n payments

1st payment starts on the (d+1)th period

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Ordinary Annuity

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

R

nn-1n-243210

R R R R R R

F

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Ordinary Annuity

R

nn-1n-243210

R R R R R R

SUM=F

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Ordinary Annuity

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

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Ordinary Annuity

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.

3000

40393843210

3000 3000 3000 3000 3000 3000

= 232,457.71F

(semi-annual periods)

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Ordinary Annuity

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?

5000

24232243210

5000 5000 5000 5000 5000 5000

= 124,714.44F

(months)

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Ordinary Annuity

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

R

nn-1n-243210

R R R R R R

P

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Ordinary Annuity

R

nn-1n-243210

R R R R R R

P= SUM

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Ordinary Annuity

R

nn-1n-243210

R R R R R R

FP value of annuity on

last payment date

value of annuity at

one period before

1st payment date

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

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Ordinary Annuity

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.

3000

40393843210

3000 3000 3000 3000 3000 3000

= 232,457.71F

(semi-annual periods)

P68,018.62 =

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fr Ex 1

Ordinary Annuity

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)

4624.5

0

24232243210

4624.50 4624.50 4624.50 4624.50 4624.50

P

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Ordinary Annuity

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?

1000

0

12111043210

10000 10000 10000 10000 10000 10000

P

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Ordinary Annuity

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?

1400

0

16151443210

14000 14000 14000 14000 14000 14000

F

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Ordinary Annuity

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?

2800

0

43210

28000 28000 28000

P

7040.1

5

12111043210

7040.15 7040.15 7040.15

P

cheaper,

1st offer better

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Ordinary Annuity

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

60161543210

2400 2400 2400 2400 2400 2400

59

2400

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Ordinary Annuity

Ordinary Annuity

Finding periodic payment (R) of an Ordinary annuity

R

nn-1n-243210

R R R R R R

P

Formulas for the amount F, present value P of an ordinary annuity:

F

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Ordinary Annuity

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

40393843210

R R R R R R

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Ordinary Annuity

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

6543210

R R R RR

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Ordinary Annuity

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

100999843210

R R R R R R

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Ordinary Annuity

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

109843210

R R R R R R

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Ordinary Annuity

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

24232243210

R R R R R R

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