1. QMT 3301 BUSINESS MATHEMATICS
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CHAPTER 10
ANNUITY
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2. 10.1 Future Value of Ordinary Annuity Certain
Future value (accumulated value) of an ordinary annuity certain is the sum of all the future values of the periodic payments.
The derivation of the formula of future value of ordinary annuity certain are as follow:
Periodic payments = RM R
Interest rate per interest period = i%
Term of investment = n interest periods
Future value of annuity at end of n interest periods = RM S
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3. QMT 3301 BUSINESS MATHEMATICS
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Formula:
S = R[(1 + i)n – 1] i
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4. QMT 3301 BUSINESS MATHEMATICS
Example :
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RM100 is deposited every month for 2 years 7
months at 12% compounded monthly. What is the
future value of this annuity at the end of the
investment period? How much interest is earned?
Solution:
months … S
100
100
100
100
100
100
No deposit
Annuity begins
Last deposit
Annuity ends
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5. QMT 3301 BUSINESS MATHEMATICS
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From , we get
Interest earned, I = S – nR
= – (31 x 100)
= RM
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6. 10.2 Present Value of Ordinary Annuity Certain
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Consist of the sum of all the present values of periodic payments.
The deviation of the formula of present value of ordinary annuity certain is illustrated in the following:
Periodic payments = RM R
Interest rate per interest period = i%
Term of investment = n interest periods
Future value of annuity at end of n interest periods = RM A
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7. QMT 3301 BUSINESS MATHEMATICS
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Formula:
A = R[1 - (1 + i)-n] i
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8. QMT 3301 BUSINESS MATHEMATICS
Example :
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Raymond has to pay RM 300 every month for 24
months to settle a loan at 12% compounded
monthly.
a) What is the original value of the loan?
b) What is the total interest that he has to pay?
Solution:
a) payments … A
300
300
300
300
300
300
No payment
Last payment
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9. QMT 3301 BUSINESS MATHEMATICS
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From , we get
b) Total interest = (300 x 24) –
= RM
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10. QMT 3301 BUSINESS MATHEMATICS
10.3 Amortization
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An interest bearing a debt is said to be amortized when all the principal and interest are discharged by a sequence of equal payments at equal intervals of time.
A table showing the distribution of principal and interest payments for the various periodic payments.
10.4 Amortization Schedule
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11. QMT 3301 BUSINESS MATHEMATICS
Example :
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A loan of RM 1000 at 12% compounded monthly is
to be amortized by 8 monthly payments.
a) Calculate the monthly payment.
b) Construct an amortization schedule.
Solution:
a) From , we get
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12. b) Amortization Schedule
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Period
Beginning balance (RM)
Ending balance (RM)
Monthly payment (RM)
Total paid (RM)
Total principal paid (RM)
Total interest paid (RM) 1
1000
879.31
130.69
120.69 10 2
757.41
261.38
242.59
18.79 3
634.29
392.07
365.71
26.36 4
509.94
522.76
490.06
32.70 5
384.35
653.45
615.65
37.80 6
257.50
784.14
742.50
41.64 7
129.39
914.83
870.61
44.22 8
0.00
45.51
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13. QMT 3301 BUSINESS MATHEMATICS
10.5 Sinking Fund
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When a loan is settled by the sinking fund method, the 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 the face value, debtor will create a separate fund in which he will make periodic deposits over the term of the loan.
The series of deposits made will amount to the original loan.
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14. QMT 3301 BUSINESS MATHEMATICS
Example :
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A debt of RM 1000 bearing interest at 10%
compounded annually is to be discharged by the
sinking method. If five annual deposits are made
into a fund which pays 8% compounded annually,
a) Find the annual interest payment,
b) Find the size of the annual deposit into the sinking fund,
c) What is the annual cost of this debt,
d) Construct the sinking fund schedule.
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15. QMT 3301 BUSINESS MATHEMATICS
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Solution:
a) Annual interest payment = RM 1000 x 10%
= RM 100
b) From , we get
c) Annual cost = Annual interest payment Annual deposit
= RM RM = RM
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16. d) Sinking Fund Schedule
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End of period (year)
Interest earned (RM)
Annual deposit (RM)
Amount at the end of period (RM) 1
170.46 2
13.64
354.56 3
28.36
553.38 4
44.27
768.11 5
61.45
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17. 10.6 Annuity With Continuous Compounding
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Future value of annuity:
Present value of annuity:
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18. QMT 3301 BUSINESS MATHEMATICS
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Where:
S = Future value of annuity
A = Present value of annuity
R = Periodic payment or deposit
e = Natural logarithm
k = Annual continuous compounding rate
t = Time in years
p = Number of payments in 1 year
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19. QMT 3301 BUSINESS MATHEMATICS
Example :
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James wins an annuity that pays RM 1000 at the
end of every six months for four years. If money is
worth 10% per annum continuous compounding,
what is
a) The future value of this annuity at the end of four years,
b) The present value of this annuity?
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20. QMT 3301 BUSINESS MATHEMATICS
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Solution:
a) From , we get
b) From , we get
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