AM015 20212022

7/7/2022

Solution : Solution :

Example 2: Solution :
Shiela plans to save RM500,000 for her retirement. She plans
to invest a fix amount every month into a fund that pays an
annual rate 5% compounded monthly. If she plans to retire
within 20 years, how much she should save every year? What
is the amount of interest will she earned?

Solution : TOPIC 7

MATHEMATICS OF FINANCE
7.4 ANNUITY

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PRESENT VALUE OF THE ANNUITY PRESENT ANNUITY

Payments R R R ….. RR R Therefore, the formula for the present annuity is:
n–2 n–1 n
Period 01 2 3

R(1 + i)-1  1 (1  i)n 
An  R  
R(1 + i)-2  i 

R(1 + i)-3 An = present value of annuity (the amount of money that has the same value as
: the sum of a series of payments at the end of the period)

R(1 + i)-(n-2) R = the regular or periodic payment

R(1 + i)-(n-1) i = the interest rate per compounding period
n = number of compounding periods
R(1 + i)-n

An

PRESENT ANNUITY Example 1:
Shirley has to pay RM300 every month for 30 months to
Interest earned, I  Rtm  An settle a loan at 12% compounded monthly. What is the
original value of the loan? What is the total interest that she
An = present value of annuity has to pay?
R = the regular or periodic payment
T = time or term in years
m = number of compounding periods per year

Solution : Solution :

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Example 2: Solution :

Rudy buys a land for RM110000. 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 effective rate.

Solution : Solution :

Example 3: Solution :

Three years ago, En Abu deposited RM5000 in a saving account at 12%
interest rate compounded quarterly. At the same time, he deposited
RM300 at the end of every month in another account at 7.5% interest rate
compounded monthly. Today, he intends to withdraw all his money from
both accounts.
(a) Calculate the total withdraws from both accounts.
(b) Hence, calculate the total amount of interest will he earned?

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Solution : Solution :

15

TOPIC 7.0: MATHEMATICS OF FINANCE

7.1 Simple Interest

1. RM10,000 is invested in a savings bank account that earned 6% simple interest.
(a) Find the interest earned if the amount was kept in the bank for 4 years.
(b) Hence, calculate the amount of money at the end of four years?

2. Mohan takes loan of RM12000 to buy a used car at the simple interest of 5.5%
per annum for 3 years.
(a) Calculate the amount he has to pay at the end of the three years?
(b) How much is the interest charged?

3. RM1000 is invested for 1095 days in a bank earning an exact interest of 4.2% per
annum.
(a) Find the simple amount at the end of the investment period.
(b) Find the interest earned.

4. Naufal borrows RM3000 from a bank that charge an ordinary rate of 6% per
annum for 720 days. Find the sum he has to repay to the bank and interest
charged?

5. How long does it take for RM10000 to accumulate to RM13500 at a simple
interest rate of 5% per annum?

6. Ilyani kept RM25000 in a bank account. After 30 months the amount in the
account is RM 29,000.50. Calculate the interest rate if the bank gives him a
simple interest rate.

7. Daniel took a loan from a bank at the simple interest rate of 10% per annum.
After three years, he had to pay RM5400 interest only for the period. Determine
the principal amount borrowed by him.

8. Umar deposits RM5500 in Bank Y that gives a simple interest rate of 8% per
year. After 6 months, he withdraws RM5500 from his account for Hari Raya
Aidilfitri. Find the balance of his account.

7.2 Simple Discount

1. Adam borrows RM800 at a simple discount of 9% per year for 3 months. Find
(a) the simple discount
(b) the proceed
(c) the simple interest rate that is equivalent to the discount rate.

2. Mr Chong borrows RM8000 from a bank that charges a simple discount rate of
6.6% per year. If the proceed he received is RM7208, find how long he borrows
the money?

3. Zainudin borrows RM4000 from a bank that charged a bank discount of d % for 6
months. The proceeds he received was RM3915. Find
(a) the simple discount
(b) the simple discount rate, d % that the bank charged.
(c) the simple interest rate that is equivalent to the discount rate?

4. Hanisah needs RM3000 in cash. A bank offers him a 4.5% simple discount for 9
months. Find,
(a) the amount she borrowed from the bank.
(b) the discount
(c) effective rate

5. Ehsan borrowed RMA from a bank that charged a bank discount of 7% for 300
days. If the bank discount was RM495.83, find RMA and the proceed he obtained.
Hence, determine the simple interest rate that equivalent to the given discount rate

7.3 Compound Interest

1. A principal of RM1000 is invested at 12% interest for one year. Determine the
future value if the interest is compounded
(a) annually
(b) semi-annually
(c) quarterly
(d) monthly

2. Izwan invests RM4800 in a company which promises 9% interest compounded
quarterly. What is the amount after 5 1 years? Hence, calculate the interest
2
earned.

3. What is the nominal rate compounded semi-annually that will make the
investment of RM6000 to become RM9000 in three years?

4. Mr Yeoh earned RM1729.34 after investing RM5000 for 5 years. Determine the
nominal interest rate if the interest is compounded every 4 months.

5. How long does it take from a sum of RM4000 to become RM7000 at 8%
compounded quarterly.

6. An account earning 7% annual interest rate compounded semi-annually was
initiated 4 years ago. Today, the account is worth RM10534.47. What was the
initial amount in the account? If the sum is invested for another two more years,
what is the future value?

7. Puan Zainab intends to borrow some money to renovate her house. She has

received two different quotes:

Bank D : charges 8% compounded semi-annually

Bank E : charges 7.5% compounded quarterly

Which bank provides a better deal?

8. Which of the following saving accounts offer the greater return?

Account A : 8.05% compounded quarterly

Account B : 7.95% compounded monthly

9. Bank Z offers a simple interest of 6% for the first 3 years and 7.5% compounded

quarterly for next 31 years. If Abu invest RM5000, calculate the amount after
2

61 years.
2

10. Three years ago, Mr Lee had save RM20000 in an account that pays 8%
compounded monthly. Now, he intends to add another RMX in the account.
Determine the value of RMX if the account will amount to RM61867.02 in seven
years’ time.

7.4 Annuity

1. Fatini invests RM320 at the end of every month which offers an interest 6%
compounded monthly.

(a) Determine the amount accumulated at the end of three years?
(b) Calculate the interest earned from her saving.
(c) Find the effective rate which is equivalent to the above nominal rate.

2. Aisyah plans to save RM100000 for her retirement. She plans to invest a fix
amount every year into a fund that pays an annual rate 8% compounded annually.
(a) If she plans to retire within 15 years, how much she should save every
year?
(b) What is the amount of interest will she earned?

3. Yusuf invests RM200 at the end of every month that pays an interest 10%
compounded monthly. After 3 years he saves RM260 every month. Determine the
amount in the account after 6 years? Hence, calculate the amount of interest will
she earned.

4. Alicia plans to accumulate RM100787 in N years. She deposits RM500 from the
first year to the 10th year and made no deposit from 11th year to Nth. If the interest
is 9% compounded annually, calculate the value of N.

5. At the end of each quarter, a 45-year-old woman puts RM2000 in a retirement
account that pays 5.95% interest compounded quarterly. When she reaches 55,
she withdraws the entire amount and places it in a mutual fund that pays 6.5%
interest compounded monthly. How much is in the account when she reaches age
60?

6. Lyana borrows RM35000 for buying low-cost house. Bank M offers him 8%
interest compounded monthly. The loan must be settled in 5 years.
(a) Find his monthly payment.
(b) How much interest paid?
(c) Calculate the effective rate which is equivalent to the above nominal rate.

7. Marina has to pay RM788 every month for 7 years to settle her car loan at interest
rate of 4.25% compounded monthly. What is the original value of the loan and the
total interest that she has to pay?

8. Nadia has inherited RM100000 from her grandfather’s estate. She deposits the
money in an account offering 6.75% interest compounded annually. She wants to
make equal annual withdrawals from the account so that the money lasts exactly
30 years.
(a) Find the amount of each withdrawal.
(a) Calculate the interest earned from her saving.

9. Suzana plans to buy a semi-detached house which cost him RM200000. She made
10% down payment and settle the balance by taking a loan for 20 years with
annual interest 4% compounded monthly.
(a) Calculate her monthly payment.
(b) What will be the total amount paid by Suzana for the house?
(c) If Suzana has not paid her first two monthly payment, how much should
he pay on her third payment to settle all the outstanding arrears?
(d) Immediately after making the 200th repayment, Suzana decided to settle
the balance of the loan. What is the amount must be paid?
(e) Calculate the effective rate which is equivalent to the above nominal rate.

10. The cash price of a condominium is RM204 600.00. It can be purchase through an
instalment payments by making down of 10% followed by 360 equal monthly
payments.
The interest charged is 6.8% compounded monthly.
(b) Find the monthly payment.
(c) Calculate the amount of interest charge.
(d) If immediately after 15 years, the buyer decided to settle the loan by a
single payment,
(i) determine the value of this single payment.
(ii) Calculate the total interest paid by the buyer.

ANSWER

7.1 Simple Interest (b) RM12400
1. (a) RM2400 (b) RM1980
2. (a) RM13980 (b) RM126
3. (a) RM1126
4. RM3360, RM230
5. 7 years
6. 6.40%
7. RM18000
8. RM220

7.2 Simple discount

1. (a) RM18 (b) RM782 (c) 9.21%

2. 1.5 years or 18 months (c) 4.25%
(c) 4.66%
3. (a) RM85 (b) 4.34%

4. (a) RM3104.79 (b) RM104.79

5. (a) RM8499.94, RM8004.11, 7.43%

7.3 Compound Interest

1. (a) RM1120 (b) RM1123.60 (c) RM1125.57

(d) RM1126.83

2. RM7831.31, RM3031.31

3. 13.98%

4. 6.00%

5. 7.065 years

6. RM8000, RM9834.04

7. Bank E provides a better deal as it charges a lower effective rate.

8. Account A gives greater return.

9. RM7652.42

10. RM10000

7.4 Annuity

1. (a) RM12587.55 (b) RM1067.55 (c) 6.17%

2. (a) RM3682.95 (b) RM44755.75

3. RM22129.17, RM5569.71

4. 40 years

5. (a) RM709.67 (b) RM7580.20 (c) 8.30%

6. RM149688.94

7. RM76934.21, RM10742.21

8. (a) RM7857.22 (b) RM135716.60

9. (a) RM1090.76 (b) RM281782.40 (c) RM3283.20

(d) RM613672.72 (e) 4.07%

10. (a) RM1200.46 (b) RM248025.60

(c) (i) RM135235.07 (ii) RM167177.87