Simple Annuities

Ordinary Annuity: Amount and Present Value

Annuity

- is a sequence of payments (usually of equal size) made at equal intervals of time. Examples of which are:- monthly house rent payment

- Annual premiums of life insurance policy

- Installment payments in purchasing a house

- monthly retirement benefits (pension plan)

Simple Annuity

- An annuity in which the payment period is same as the interest period (conversion period)

Payment period – time between successive period of annuity.

Term – time from the beginning of the first payment period to the last payment period.

Periodic Payment (R) – size of each annuity payment

Ordinary Annuity

- An annuity in which the payments are made at the end of each payment period.

Amount of Ordinary Annuity (S) – sum of accumulated values of the payments at the end of its terms.

Present Value of O. Annuity (A) – sum of the present values of the payments.

Amount and Present Value of Ordinary Annuity

Amount Present Value

jim

n tmand

,

1 (1 )

nli

n

nli

A Ra

where

ia

i

,

(1 ) 1

nli

n

nli

S Rs

where

is

i

Problem Set 3.3

1. Find the Present Value and the amount of an P8,000 ordinary annuity payable quarterly for 10 years if the money is worth 12% converted quarterly.

Problem Set 3.3

8. The purchaser of a portable DVD player with USB will pay P1,500 cash and P565.80 at the end of each month for 12 months to discharge all principal and interest at 9% compounded monthly. Find the cash price of the DVD player.

Problem Set 3.3

11. Linda deposits P10,000 every 3 months for 7 years in an account paying 4% compounded quarterly. How much will she save in her account at the end of 7 years assuming no withdrawals were made?

Problem Set 3.3

17. Find the amount of a 4- year ordinary annuity whose present value is P8,300 if the money is worth 7% m = 4.

Problem Set 3.3

18. Find the present value of a 7- year ordinary annuity whose amount is P72,900 if the money is worth 6% m = 2.

Periodic Payment

Given the Amount of Annuity

Given the Present Value of Annuity

nli

nli

SR

S

AR

A

Problem Set 3.4

1. How much should be invested in a fund at the end of each year in order to accumulate to P150,000 at the end of 10 years if the fund is earns 6.5% effective?

Problem Set 3.4

1. The borrower of a P500,000 loan plans to repay the loan by making equal payments at the end of each three months for 10 years. If the rate of the interest is 17% compounded quarterly, find the quarterlt payment.

Problem Set 3.4

6. A video camera is worth P35,950. Nancee bought one by paying P5,000 and an equal payment at the end of each month for 18 months. Find the equal payments if the interest rate is 18% m = 12.

Problem Set 3.4

9. The Citizen Cottage Industry has a high-speed sewing machine that will retire in 5 years. How much must be set aside each 3 months in order to by a new sewing machine that costs P720,000 to replace the old one if the fund is invested at 8% compounded quarterly?

Finding the nominal rate using (A)

2 2

2

( 1) 6( 1) 12 1 0

4

2

nRn i n i

A

b b aci

a

2 2

2

6( 1) (6( 1)) 4( 1)(12(1 ))

2( 1)

nRn n n

Ain

j im

n tm

a b c

Given A

Finding the nominal rate using (S)

2 2

2

( 1) 6( 1) 12 1 0

4

2

nRn i n i

S

b b aci

a

2 2

2

6( 1) ( 6( 1)) 4( 1)(12(1 ))

2( 1)

nRn n n

Sin

j im

n tm

Given S

a b c

Problem Set 3.51. At what nominal rate compound

semiannually is P5,000 the present value of P1,000 ordinary annuity payable semiannually for 3 years?

A = 5000R = 1000m = 2 (semiannually)t = 3 yearsn = (3)(2) = 6

2 2

2

1

2

2

1

( 1) 6 1 35

6( 1) 6(6 1) 42

6(1000)12 1 12 1 2.4

5000

42 42 4(35)( 2.4)0.05465...

2(42)

42 42 4(35)( 2.4)1.2547

2(42)

(2) 0.1093 10.93%

a n

b n

nRc

A

i

i

j im i

Problem Set 3.5

4. A man invests P10,000 at the end of every 3 months. If he has P390,000 in 7 years, at what rate compounded quarterly did his investment earn interest?

S = 390000R = 10000m = 4 (quarterly)t = 7 yearsn = (7)(4) = 28

2 2

2

1

2

2

1 1

( 1) 28 1 783

6( 1) 6(28 1) 162

28(10000)12 1 12 1 3.3846

390000

( 162) ( 162) 4(783)(3.3846)0.18331...

2(783)

( 162) ( 162) 4(783)(3.3846)0.02358...

2(783)

(4)

a n

b n

nRc

S

i

i

j im i

2 2

0.73324 13.32%

(4) 0.09432 9.43%j im i

Problem Set 3.5

6. A 31-inch LCD television costs P61,990. Stanley bought one by making a down payment of P7,000 and paying P5,040.75 at the end of every 3 months for 3 years. At what rate converted quarterly was the interest charged?

Problem Set 3.5

10. Marian invests P11,600 at the end of each year in a fund. If she wants to have P246,500 in the fund in 14 years, at what rate compounded annually should the money be invested?