Miss Soo

Annuity

At some point in your life, you may have had to make a series of fixed payments over a period of time - such as rent or car payments - or have received a series of payments over a period of time, such as bond coupons. These are called annuities.

Annuity concepts enable people to plan for the future in terms of investments and savings. Financial consultants use annuity calculations to advise people on investment whereas a property agent uses it to estimate the repayments of housing or car loan. Annuity calculations can also help people plan for their retirement.

Objective

Students are able to calculate the present and future value of an annuity.

Solve for annuity payment, R, the number of payments, n, and the interest rate, i.

Produce an amortization table.

Introduction

Annuity is a series of (usually) equal payments made at (usually) equal intervals of time. Some examples of annuity are insurance policy premiums, annual dividends received and installment payments. In each case, equal payments are deposited or paid periodically over time.

An Ordinary Annuity is a series of payments having the following three characteristic:All payments are the same amount.

All payments are made at the same intervals of time.

All payments are made at the end of each period.

Future Value

Future value (or accumulated value) of an ordinary annuity is the sum of all the periodic payments plus all interest earned at the end of every period at a specified date in the future. The derivation of the formula of the future value is shown below.To find the amount of annuity, we will consider each periodic payments R separately by using the compound amount formula,

A = R(1 + i)nWhere periodic payments = R,interest rate per interest period = i %,term of investment = n interest periods andfuture value of annuity at the end of n interest period = A

Diagram 1

A = R + R(1+i)1 + R(1+i)2 + R(1+i)3 + + R(1+i)n-1A = R[1 + (1+i)1 + (1+i)2 + (1+i)3 + + (1+i)n-1]Note that the series in the bracket is a geometric series with 1 as the first term and (1+i) as the common ratio. Therefore the future value which is also the accumulated value is the sum of this series.

Example 1.Jason deposits RM500 into an account at the end of every month for 4 years at 6% interest compounded monthly.Find the future value of the annuity at the end of 4 years.

How much interest will be earned during the 4 years?

Total deposit = RM500 X 48 = RM 24000.00

Total interest earned = RM (27048.92 24000.00) = RM 3048.92

Present Value

The value of an annuity at the present time is called the present value of an annuity. It represent the current value of a set of cash flows, given a specified rate of return or discount rate.

The present value of an ordinary annuity is the sum of all present values of the periodic payments.

Let periodic payments = R,interest rate per interest period = i %,term of investment = n interest periods andpresent value of annuity at the end of n interest period = P

Diagram 2

From the diagram above, we obtain

P = R(1+i)-1 + R(1+i)-2 + R(1+i)-3 + + R(1+i)-nP = R[(1+i)-1 + (1+i)-2 + (1+i)-3 + + (1+i)-n]

Note that the series in the bracket is a geometric series with (1+i)-1 as the first term and

(1+i)-1 as the common ratio. Summing up the terms, we obtain

Example 2

Ravi has to pay RM300.00 every month for 24 months to settle a loan at 12% compounded monthly. What is the original value of the loan?

R = 300

n = 24

Original value of the loan is the present value = P

Amortisation

An interest bearing debt is said to be amortised when all the principal and interest are discharged by a sequence of equal payments at equal intervals of time.

An amortisation schedule is a table showing the distribution of principal and interest payments for the various periodic payments.

Example 3

Harjit borrows RM5000 for refurbishing kitchen cabinets. He will pay off the loan in 2 years. The money borrowed at 8% per annum, compounded semiannually.

Calculate the monthly payment.

Contruct an amortization schedule.

P = 5000 n = 2 X 2 = 4 payments

Payment numberPayment amount (RM)Interest for period (RM)Principal repayment (RM)Outstanding Principal (RM)

05000.00

11377.45200.001177.453822.55

21377.45152.901224.552598.00

31377.45103.921273.531324.47

41377.4552.981324.470.00

Sinking Fund

A sinking fund is a fund established by a person or company where they can deposit money on a regular basis to fund a future capital expense, or repayment of a long-term debt that will come due in the future.

The money set aside for the periodic deposits or payments plus interest on them will equal the future obligations or debts.

Formula for future value of an annuity is used to derive the sinking fund.

Where periodic payments = R,interest rate per interest period = i %,term of investment = n interest periods andfuture value of annuity at the end of n interest period = A

When a loan is settled by sinking fund method, creditor will only receive the periodic interest due. The face value of the loan will only be settled at the end of the term. In order to pay this face value, the debtor will create a separate account in which he will make a periodic deposit over the term of the loan. The series of deposits made will amount to the original loan.

Example 4

A debt of RM1000.00 bearing interest at 10% compounded annually is to be discharged by sinking fund method. If five annual deposits are made into a fund which pays 8% compounded annually,

find the annual interest payment,

find the size of the annual deposit into the sinking fund.

what is the annual cost of this debt?

i = 8%

A = RM 1000.00 n = 5

Total annual cost (or payments) = RM100.00 + RM170.46 = RM270.46

Past years questions

STPM 2013 Q7

A company purchased an equipment at a price of RM368 000 and makes a down payment of 10% of the purchase price. The company amortises the balance with 360 monthly payments at an annual interest rate of 4.5% which is compounded monthly.

Calculate

The monthly repayment,

[3 marks]

The total interest payable.

[2 marks]

If the company does not settle the first six monthly payments, determine the total amount payable on the seventh month. Assume no extra charge on late payment.

[5 marks]

If the monthly payment is RM1996.40, calculate the number of monthly payments required to amortise the loan.

[5 marks]

Answer:

a. i) RM 1678.14.

ii) RM 272 930.40

b. RM 11 841.85

c. 260 payments.

STPM 2014 Q1

A company is offered two alternative method to repay a loan in three years. In the first method, the company has to pay RM200 000 at the end of the third year. In the second method, the company has to pay RM x, RM 2x and RM 3x at the end of the half year, at the end of one and a half years and at the end of the third year, respectively. The annual interest rate is 8% compounded semi-annually.

Find the present value of payment in the first method.

[3 marks]

Determine the value of x.

[5 marks]

Answers:

RM 158 062.91

b. RM 30 929.20

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