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5.3ANNUITY

Define ordinary and simple annuity

Find the future and present value

Find the regular periodic payment

Find the interest

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If we wish to invest in “Amanah Saham Bumiputera” with a fixed installment of RM100 monthly. If the scheme gives 10% interest monthly, try and calculate the amount accumulated after 10 years.

At the end of the first month, we will have 100(0.1) + 100 = RM A

At the end of the second month, we will haveA (0.1) + 100 = RM B

At the end of the third month, we will haveB (0.1) + 100 = RM C

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Definition - Annuity

•An annuity is a sequence of equal payments made at equal intervals of time.

•The payments are computed by the compound interest method such as annually, semiannually, quarterly or monthly.

•Assume that the first payment is made at the end of the first interest period.

•Annuities in which payments are made at the same time the interest is compounded are called ordinary and simple annuities.

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Future Value of an Ordinary Annuity

The future value of an annuity of R ringgit per period for n period when the interest rate is i per period is given by:

1 1n

n

iS R

i

Sn = future value R = regular or periodic paymenti = interest rate per compounding periodn = number of annuity payments

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Example 1 – future value

Ali has made equal payments of RM100 every 6 months at an interest rate of 5% compounded semiannually for 5 years. The future value which is the amount he gets after 5 years is

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Example 2

Lim decides to save RM1000 per month in her saving account that pays 8% interest p.a compounded monthly. After making 8 deposits, how much money does Lim have?

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Example 3

A teenager plans to deposit RM50 in a savings account at the end of each quarter for the next 6 years. Interest is earned at a rate of 8 percent per year-compounded quarterly. What should her account balance be 6 years from now? How much interest will she earn?

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Example 4 Finding saving amount to achieve future goal

Suppose you want to buy a house 5 years from now and you want to estimate that an initial down payment of RM20,000 will be required at that time. Suppose a saving account paying annual interest rate of 6% p.a compounded annually. How much do you need to make equal annual end-of-year deposit into the saving acount to accumulate the RM20,000 at the end of year 5?

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1 1n

n

iA R

i

An = present value

R = regular or periodic payment

i = interest rate per compounding period

n = number of annuity payments

Present value of an Ordinary Annuity

The present value of an ordinary annuity of R Ringgit per period for n period when the rate of return or interest is i per period is given by :

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Example 5 Finding present value

Find the present value which is the amount to be invested now in order to receive equal payments of RM100 every 6 months for 5 years.

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Example 6 Finding the amount of payment of a loan

Lim plans to start up a new business and he needs to borrow RM100,000. You propose to pay off the loan quickly by making 5 equal annual payments. If the interest rate is 10%p.a compounded annually, how much is the amount of each payment?

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Example 7 Finding loan amount and interest paid

Veni agrees to pay RM 300 per month for 48 months to pay off a car loan. If interest of 12% per annum is charged monthly, find

a) how much did the car originally cost?b) how much interest was paid?

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Example 8

Rudy buys a land for RM110,000. He makes 20% down payment and the balance he takes a loan for 25 years that charges an annual interest rate of 5% compounded monthly. Find

(a) the monthly payments.(b) the total amount of interest that will be paid.(c) the amount of the loan that will have already paid after 10 years.

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b) Monthly payment of RM514.44 for 25 years yield a total payment 514.44(25)(12)= RM154,331.77

Thus the total amount of interest = RM154,331.77- 88,000 = RM66,331.77

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1 1n

niA R

i

c)

= 72,444.94The amount of loan that will have already paid after 10 years is RM88,000- RM72,444.94 = RM15,555.05

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05.0

)15(12

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05.011

.An=

Hisham is 20 years away from retiring and starts saving RM100 a month in an account paying 6% p.a compounded monthly. When he retires, he wishes to withdraw a fixed amount each month for 25 years. What will the fixed amount be?

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Example 10

Mariam purchased a house by paying RM5,750.32 down and promised to pay RM811.41 every months for next 15 years. The interest charged is at the rate of 9% compounded monthly. a) What was the cash value of the house?b) If Mariam missed the first 10 payments, what must she pay at the time the 11th payment is due to bring herself up to date?c) After paying for the first 5 years, Mariam wished to discharge her remaining debt by making single payment at time when the 61st regular payment was due. What must she pay in addition to the regular payment then due?

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Example 11

At the end of every month, Mr. Zaki saves RM200 in an account that pays an annual rate of 10% compounded monthly. After 3 years, he adds RM60 to his savings per month. Show that the total amount after 6 years is RM22,129.17

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