business math chapter 5

CHAPTER 5 :CHAPTER 5 : ANNUITY

5.0 Introduction5.0 Introduction5.1 Future & Present Value of Ordinary 5.1 Future & Present Value of Ordinary Annuity CertainAnnuity Certain5.2 Amortization 5.2 Amortization 5.3 Sinking Fund5.3 Sinking Fund5.4 Annuity with Continuous 5.4 Annuity with Continuous Compounding Compounding

5.0 INTRODUCTION

Annuity is a series of (usually) equal payments made at (usually) equal intervals of time. Examples of annuity:

Shop rentals Insurance policy premium Regular deposits to saving accounts Installment payments

Annuity – Definition

5.0 INTRODUCTION

Annuity can be classified into many classes: Annuity certain – payment are made at the endend of each payment period. Annuity due – payment are made at the beginningbeginning of each period. General annuity Perpetuity & others.

In this chapter we shall mainly discuss ordinary ordinary annuity certainannuity certain where payment are made at the the endend of each payment periods & the interest of each payment periods & the interest and payment periods are of the same interval.and payment periods are of the same interval.

Annuity – Classes

The formula to calculate the future value of the annuity at the end of investment periods is given by

where : R = Periodic paymenti = Interest rate per interest

period n = Term of investment

5.1 FUTURE & PRESENT VALUES ORDINARY ANNUITY CERTAIN

i

iRS

n 11

i

iRS

n 11

Future Value of Ordinary Annuity Certain

m

ri mr

i

mtn mtn The sum of The sum of all future all future values of values of

the periodic the periodic paymentspayments

The sum of The sum of all future all future values of values of


The expressions, is the future value of annuity of 1 per payment for n intervals. Its read as “s angle n at i ” & its value can be found for certain i and n in the tables.


in

n

Rsi

iRS

11 in

n

Rsi

iRS

11

Future Value of Ordinary Annuity Certain

ins

1. RM 100 is deposited every month for 2 years 7 months at 12% compounded monthly. What is the futures value of this annuity at the end of the investment?

EXAMPLE 1

Solution

27.6133

01.0

101.01100

31

RM

S

3112

3112

%112

%1212

31

12

72

;12%;12

100

n

i

t

mr

R

EXAMPLE 12. RM 100 is deposited every 3 months for 2 years 9

months at 8% compounded quarterly. What is the futures value of this annuity at the end of the investment?

Solution

1112

334

%24

%812

33

12

92

;4%;8

100

n

i

t

mr

R

EXAMPLE 13. Find the amount to be invested every 3 months at 10%

compounded quarterly to accumulate RM 10,000 in 3 years.

Solution

1234

%5.24

%10

3

;4%;10

00010

n

i

t

mr

S

4. RM 100 was invested every month in an account that pays 12% compounded monthly for two years. After the two years, no more deposits was made. Calculate the amount of the account at the end of five years.

EXAMPLE 1

Solution

35.6972

01.0

101.01100

24

2

RM

S

36312

24212

%112

%12

3;2

;12%;12

100

2

1

21

n

n

i

tt

mr

R

M: 0

100

12% monthly

No depositM: 24

M: 60

Amount in the account at the end of 5 years:

28.8593

01.0135.6972 365

RM

S

Amount in the account just after 2 years:

5. Julia invested RM 100 every month for 5 years in an investment scheme. She was offered 5% compounded monthly for the first 3 years & 9% compounded monthly for the rest of the period. Determine the accumulated amount at the end of 5 years.

EXAMPLE 1

Solution

33.8753

11100

12%5

36

12%5

3

RM

S

24212

36312

%75.012

%9;

12

%5

2;3

%;9%;5

12;100

2

1

21

21

21

n

n

ii

tt

rr

mR

Amount of annuity just after 3rd years:

M: 0

100

5% monthly 9% monthlyM: 36

M: 60

100 100

Total amount at the end of 5 years

= RM 4636.50 + RM 2618.85 = RM 7255.35

Total amount at the end of 5 years

= RM 4636.50 + RM 2618.85 = RM 7255.35

EXAMPLE 1Solution

85.6182

%75.0

1%75.01100

24

2

RM

S

24212

36312

%75.012

%9;

12

%5

2;3

%;9%;5

12;100

2

1

21

21

21

n

n

ii

tt

rr

mRAmount of annuity at the end of 5 years:

M: 0

100

5% monthly 9% monthlyM: 36

M: 60

100 100

50.6364

%75.0133.8753 245

RM

S

Amount of annuity for another the 2 years:

The formula to calculate the present value of the annuity at the end of investment periods is given by

where : R = Periodic paymenti = Interest rate per interest

period n = Term of investment


i

iRA

n11

i

iRA

n11

Present Value of Ordinary Annuity Certain

m

ri mr

i

mtn mtn The sum of The sum of all present all present values of values of


The sum of The sum of all present all present values of values of


1. Lisa 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 she has to pay?

EXAMPLE 2

Solution

02.3736

01.0

01.011300a)

24

RM

A

24212

%112

%12

2

;12%;12

300

n

i

t

mr

R

98.826

02.373624300b)

RM

I

2. Johan won an annuity that pays RM1000 every 3 months for 3 years. What is the present value of this annuity if the money worth 16% compounded quarterly?

EXAMPLE 2

Solution

07.3859

04.0

04.0111000

12

RM

A

1234

%44

%16

3

;4%;16

1000

n

i

t

mr

R

EXAMPLE 23. Mimi intends to give scholarship worth RM 7500 every

year for 6 years. How much must she deposit now into an account that pays 7% per annum to provide this scholarship?

Solution

661

%71

%7

6

;1%;7

7500

n

i

t

mr

R

EXAMPLE 24. Mariam invests RM 12 000 in an account that pays 6%

compounded monthly. She intends to withdraw her account every month for 2 years and when she makes her last withdrawal her account will zero. Determine the size of these withdrawal.

Solution

24212

%5.012

%6

2

;12%;6

00012

n

i

t

mr

A

EXAMPLE 25. Firdaus borrowed RM 80 000 at 12% compounded monthly for 3

years.a) Calculate his monthly payment.b) If he has not paid his first 5 monthly payments, how much

should he pay on his 6th payment to settle all outstanding arrears?

c) If immediately after paying the first 5 monthly payments, he wants to settle all the loan. How much additional payments does he has to make?

d) If he has made the first 5 monthly payments & wants to settle all the loan in the sixth payment, how much should he has to pay? How much interest was paid?

Solution

36312%;112

%12

3;12%;12

00080

ni

tmr

A

i

iRA

n11a)

EXAMPLE 2Solution

36312%;112

%12

3;12%;12

00080

ni

tmr

A

b) Outstanding arrears (5 months), S =

c) Outstanding loan (after 5 months) A =

d) 6th payment (used answer in c) as P), S = P(1+i)n

Total interest paid = R(5) + 6th payment – A

PRACTICE 1PRACTICE 11. Find the future values and the present values of

the following annuities:a)RM6000 every year for 8 years at 12%

compounded annually.b)RM800 every month for 2 years 5 months at

5% compounded monthly.c) RM950 every 3 months for 3 years 9 months

at 6% compounded quarterly.2. Serena invested RM300 every three months for

four years. She was offered 5% compounded quarterly for the first two years and 8% compounded quarterly for the rest of the period. Find the accumulated amount at the end of four years.

PRACTICE 1PRACTICE 13. RM500 was invested every month for twenty

months in an account that pays 5% compounded annually. After the twenty months, no more deposit was made. Find the amount in the account at the end of four years.

4. Find the amount that must be deposited at the end of each month at 5.5% compounded monthly for two years so as to accumulate RM2000.

5. Mariana borrowed RM 100 000 at 4% compounded monthly. She has to repay the loan by making 60 monthly payments.a)Find her monthly payments.b)If she has not paid her first 15 monthly

payments, how much should she pay on her 16th payment to settle all outstanding arrears?

It’s a table showing the distribution distribution of principal & interest payments of principal & interest payments for the various of periodic payments.

5.2 AMORTIZATIONAmortization Schedule

EXAMPLE 3A loan of RM 1000 at 12% compounded monthly is to be Amortized by 18 monthly payments.a) Calculate the monthly payment.b) Construct an amortization schedule.

Solution

185.112

%112

%12

5.112

18

;12%;12

0001

n

i

t

mr

A

98.60

3983.161000

01.0

01.0111000

11a)

18

RMR

R

R

i

iRA

n

EXAMPLE 3b) Amortization scheduleSolution

PeriodBeginning

balanceEnding balance

Monthly payment Total paid

Total principal paid

Total interest paid

1 MYR 1,000.00 MYR 949.02 MYR 60.98 MYR 60.98 MYR 50.98 MYR 10.00

2 MYR 949.02 MYR 897.53 MYR 60.98 MYR 121.96 MYR 102.47 MYR 19.49















17 MYR 120.16 MYR 60.38 MYR 60.98 MYR 1,036.69 MYR 939.62 MYR 97.07

18 MYR 60.38 MYR 0.00 MYR 60.98 MYR 1,097.68 MYR 1,000.00 MYR 97.68

Sinking fund is an account that is set account that is set up for a specific purpose at some up for a specific purpose at some future datefuture date.For example:

An individual might establish a sinking fund for the purpose of discharging a debt discharging a debt at a future date.at a future date.A company might establish a sinking fund in order to accumulate the sufficient accumulate the sufficient capitalcapital to replace equipment that is expected to obsolete at some future date.

5.3 SINKING FUNDSinking Fund – Definition

EXAMPLE 4A debt of RM 1000 bearing interest at 10% compounded annually is to be discharged by the sinking fund method. If 5 annual deposits are made into a fund which pays 8% compounded annually,a) calculate the annual interest payment.b) Determine the size of the annual deposit into sinking

fund.c) What is the annual cost of this debt?d) Construct the sinking fund schedule.

Solution

551

%81

%8

5;1%;8

5;1%;10

;0001

2

1

n

i

tmr

tmr

S

45.170

8667.51000

08.0

108.011000

11b)

100%101000a)

5

RMR

R

R

i

iRS

RMRMIn

EXAMPLE 4

c) Annual cost = Annual interest payment + annual deposit = RM 100 + RM 170.45 = RM 270.45

Solution

End of period (year)

Interest earned

Annual deposit

Amount at the end of period

1 MYR 0.00 MYR 170.46 MYR 170.46

2 MYR 13.64 MYR 170.46 MYR 354.55

3 MYR 28.36 MYR 170.46 MYR 553.37

4 MYR 44.27 MYR 170.46 MYR 768.10

5 MYR 61.45 MYR 170.46 MYR 1,000.00

d)

170.46 x 8%170.46 x 8%

170.46 + 13.64 +170.46170.46 + 13.64 +170.46

354.55 x 8%354.55 x 8%

354.55 + 28.36 +170.46354.55 + 28.36 +170.46

5.4 ANNUITY WITH CONTINUOUS COMPOUNDING

Future value of the annuity:

where : R = Periodic paymentk = annual continuous compounding

rate t = time in yearsp = number of payments in 1 year

1

1pk

e

eRS

kt

1

1pk

e

eRS

kt

Present value of the annuity:

1

1p

k

e

eRA

kt

1

1p

k

e

eRA

kt

EXAMPLE 5Zainal wins an annuity that pays RM1000 at the end of every 6 months for 4 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?

Solution

2

4%;10

;0001

p

tk

R

13.4306

1

11000b)

63.5929

1

11000a)

21.0

21.0

41.0

41.0

RMA

e

eA

RMS

e

eS

EXAMPLE 6Ah Chong won an annuity that pays RM5000 at the end of every 3 months for 5 years. If money is worth 6% per annum continuous compounding, calculate a) the future value of this annuity at the end of 5 years.b) the present value of this annuity.

Solution

business math chapter 5

Education