annuities ncs mathematics dvd series resources... · future value of annuity capital is accumulated...

1

NCS Mathematics

DVD Series

Annuities

2

Outcomes for this DVD

In this DVD you will:

• Focus on the future value of an annuity.

LESSON 1

• Focus on the present value of an annuity.

LESSON 2

• Solve problems related to annuities.

LESSON 3

3

NCS Mathematics

DVD Series

Lesson 1

Future Value

of an

Annuity

4

You should know.

Compound Depr 1 eciation:n

F P i

Future Value

Present Value

Interest Rate

Period

F

P

i

n

11n n

P FF P ii

Compound App 1 reciat on: in

F P i

5

Future Value of Annuity

Capital is accumulated (to a future value) by means of

regular payments into a savings account or an investment

fund. Compound interest is paid on the money in the fund.

1 1

Future V

Formula to ca

alue

lculate future value is

Regular Payment (Instalment)

Interest Rate

Period

n

F

x

i

n

x iF

i

This formula assumes that the regular payment is made

at the of each payment period (e.g. end of month).end

6

Deducing Future Value Formula

1

period

1

1

s

n

n

a rS

r

a x

r i

n

2 1

Value after periods is given by the sum of the GS:

1 1 1n n

n F

x x i x i x i

1st

2nd

0th

1 1 1

2 2 1

0 1

n

n

n x i

n x i

n x i x

No. of Payment No. of interest periods Contribution to final value

1 11 1

1 1

nnx i

i

x iF

i

7

Calculate Future Value of Annuity

1) At the end of each month Cosmo invests R500 at 11% p.a.,

compounded monthly. Calculate the value of the investment

after 9 years (i.e. the future value of the annuity).

12 9

1 1

0,11500 1 1

12 R91 588,61

0,11

12

nx i

Fi

1 1n

x iF

i

500

0,11

12

9 12

?

x

i

n

F

Know :

8

Related Formulae

1 1n

x iF

i

1 1n

F ix

i

1 1n F i

ix

1 1n F i

ix

log 1 log 1F i

n ix

log 1

log 1

F i

xn

i

9

Calculate Regular Payment In 5 years' time Cosmo needs R20 000 to buy

a new car. How much must he invest at the end

of each month in a savings account that accrues

interest at 10,5% p.a., compounded monthly?

1 1n

F ix

i

20 000

0,105

12

5 12

?

F

i

n

x

Know :

60

0,10520 000

12 0,105

1 112

254,88R

Check!

10

Calculate the Period At the end of each quarter Cosmo invests R1 000 at 11% p.a.,

compounded quarterly in an annuity. After how many years

will his annuity be worth R26 197,40?

log 1

log 1

F i

xn

i

26 197,40

0,11

4

1 000

?

F

i

x

n

Know :

0,1126 197,4

4log 11 000

40,11

log 14

n

20

5 yearsn

11

Tutorial 1: Future Value Annuities

1 At the end of each quarter Merlyn invests R5 000 at 11.5% p.a.,

compounded quarterly, for 15 years in an annuity.

Calculate the future value of her annuity.

(2) At the end of each month Merlyn invests R350 at 10,5% p.a.,

compounded monthly in an annuity. After how many years

will her annuity be worth R73 785,18?

PAUSE DVD

• Do Tutorial 1

• Then View Solutions

12

Tutorial 1 Problem 1: Suggested Solution

60

1 1

0,1155 000 1 1

4

0,115

4

nx i

Fi

1 1n

x iF

i

1 At the end of each quarter Merlyn invests R5 000 at 11.5% p.a.,

compounded quarterly, for 15 years in an annuity.

Calculate the future value of her annuity.

R778 714,31

5 000

0,115

4

4 15

?

x

i

n

F

Know :

13


log 1

log 1

F i

xn

i

0,10573 785,18

12log 1350

120,105

log 112

n

119,9999952 120

10 yearsn

(2) At the end of each month Merlyn invests R350 at 10,5% p.a.,

compounded monthly in an annuity. After how many years

will her annuity be worth R73 785,18?

73 785,18

0,105

12

350

?

F

i

x

n

Know :

14

NCS Mathematics

DVD Series

Lesson 2

Present Value

of an

Annuity

15

Present Value of Annuity

A loan (the present value), plus interest payable on loan, is

repaid by equal, regular payments, or an amount of money is

available to make regular payments for a specific period of time.

1 1

Present Valu

Formula to calcul

e

at

Regular Payment (Instalment)

Interest R

e present valu

ate

Pe

e

rio

is

d

n

P

x

i

n

x iP

i

This formula assumes that the regular payment is made

at the of each payment period (e.g. end of month).end

16

Deducing Present Value Formula

You should know that:

(1) 1

2

1

1 1n

n nF P P F ii

x iF

i

1 1 1

1 11

n n

n nx i x i

P Fi

ii

i

17

Calculate Present Value of a Bond

Pragashni wants to buy a house. She can afford a monthly

instalment of R4 000. Calculate the amount she can

borrow, repayable over 25 years at an interest of 13% p.a.,

compounded monthly.

1 1n

x iP

i

12 250.13

4 000 1 112

0.13

12

R354 661,71

4 000

0,13

12

12 25 300

?

x

i

n

P

Know :

18

Calculate Monthly Bond Instalment

Kenneth wants to buy a flat. He must take out a mortage

bond of R425 360, repayable over 20 years at 14% p.a.,

compounded monthly.

How much will his monthly instalment be?

1 1

1 1

n

nx i

Pi

P ix

i

240

0,14425 360

12

0,141 1

12

x

R5 289,44

425 360

0,14

12

20 12 240

?

P

i

n

x

Know :

19

Life Annuity

1 1n

x iP

i

When Nico retires he wants a lump sum to be available

from which a monthly payment of R12 000 can be made to

him for 15 years. The lump sum will earn 11% p.a. interest,

compounded monthly. Calculate the lump sum.

R1 055 783,25

12 000

0,11

12

15 12 180

?

x

i

n

P

Know :

1800,11

12 000 1 112

0,11

12

20

Tutorial 2: Present Value Annuities

1 Julius secures a home loan of R2 850 000, repayable over

30 years. If the interest rate is 13.75% p.a., compounded

monthly, what is his monthly instalment?

(2) A student took out a loan from a bank in order to

purchase a home computer at a cost of R5 870,80.

The loan is to be repaid in instalments of R121,87

at the end of each month. The bank charge an annual

interest rate of 9% compounded monthly.

How long will it take to repay this loan?

PAUSE DVD Do Tut 2


21


1 Julius secures a home loan of R2 850 000, repayable over

30 years. If the interest rate is 13.75% p.a., compounded

monthly, what is his monthly instalment?

2 850 000

0,1375

12

30 12 360

?

P

i

n

x

Know :

1 1

1 1

n

n

x i P iP x

i i

360

0,13752 850 000

12

0,13751 1

12

x

R33 205,71

22


(2) A student took out a loan from a bank in order to

purchase a home computer at a cost of R5 870,80.

The loan is to be repaid in instalments of R121,87

at the end of each month. The bank charge an annual

interest rate of 9% compounded monthly.

How long will it take to repay this loan?

log 11 1

log 1

nP i

x i xP n

i i

5 870,80

0,09

12

121,87

?

P

i

x

n

Know :

0,095 870.8

12log 1121,87

120,09

log 112

n

59,99885 60

It will take 5 years.

Left as exercise!

23

NCS Mathematics

DVD Series

Lesson 3

Problems

related to

Annuities

24

Balance on Loans

Balance of Loan

[(1 ) 1]1

nn x i

P ii

The balance on a loan after a certain period

(Loan amount, with interest accrued) (Repayments to date, with interest)

Monthly Instalment:

1 1n

P ix

i

25

Tutorial 3: Balance of Account

A home loan of R650 000 is secured at 11,5% p.a.,

compounded monthly, repayable over 25 years.

(a) Calculate the monthly instalment.

(b) Calculate the loan amount with interest accrued after 10 years.

(c) Repayments with interest earned after 10 years.

(d) Calculate the balance (amount outstanding) after

10 years.

PAUSE DVD

• Do Tutorial 3


26

Tutorial 3 Problem 1a: Suggested Solution

300

0,115650 000

Monthly I

12 R6 607,05

nsta

1 1 0,1151 1

men

2

l t:

1

n

P ix

i



(a) Calculate the monthly instalment.

27

Tutorial 3 Problem 1b: Suggested Solution

1n

P i

Loan amount, with interest accrued

1200,115

650 000 112

R2 041 615,94



(b) Calculate the loan amount with interest accrued after 10 years.

28

Tutorial 3 Problem 1c: Suggested Solution

[(1 ) 1]nx i

i

Repayments plus interest earned

1200,115

6 607,05 1 112

0,115

12

R1 476 036,30



(c) Repayments with interest earned after 10 years.

29

Tutorial 3 Problem 1d: Suggested Solution

[(1 ) 1]

1n

n x iP i

i

The balance on a loan after a certain period

R2 041 615,94 R1 476 036,30

R565 579,64



(d) Calculate the balance after 10 years.

Loan amount plus

interest accrued

Repayments plus

interest earned

30

Sinking Funds A sinking fund is a fund that is set up to replace an asset at the

end of its useful life, by making regular, equal payments into a

fund. It is therefore based on the same principle as a future

value annuity.

Sinking Fund Inflated Value Depreciated Value

1 1

1 1

n n

Inflation Depreciation

n

Investment

Investment

SF P i P i

x i

i

Present Value of A ssetP

31

Tutorial 4: Sinking Fund

Swop Shop needs to replace their truck in

5 years time. Their current truck is valued

at R235 000 and depreciates at 15% p.a.

compounded annually on a reducing balance.

The price of a replacement truck increases

by 20% p.a., compounded quarterly.

Calculate:

(a) Trade-in value of current truck in 5 years time.

(b) The price of the new truck.

(c) The value of the sinking fund needed to replace

the truck.

(d) The monthly payments into a sinking fund, if the

interest rate is 9.5% p.a., compounded monthly.

PAUSE DVD

• Do Tutorial 4


32

Tutorial 4 Problem 1a: Suggested Solution



at R235 000 and depreciates at 15% p.a.

compounded annually on a reducing balance.

Calculate:

(a) Trade-in value of current truck in 5 years time.

Depreciated Value

Trade-in value

1 n

DepreciationP i

235 000

0,15

5

P

i

n

Know :

5

235 000 1 0,15

R104 270,75

33

Tutorial 4 Problem 1b: Suggested Solution

Inflated Value

New price of truck

1 n

InflationP i

235 000

0,20,05

4

4 5 20

P

i

n

Know :

20

235 000 1 0,05

R623 524,96



at R235 000 and the price of a replacement

truck increases by 20% p.a., compounded

quarterly.

Calculate:

(b) The price of the new truck.

34

Tutorial 4 Problem 1c: Suggested Solution


5 years time. Their current truck is valued at R235 000

and depreciates at 15% p.a. compounded annually on a

reducing balance. The price of a replacement truck increases

by 20% p.a., compounded quarterly. Calculate:

(c) The value of the sinking fund needed to replace the truck.

Sinking Fund Inflated Value Depreciated Value

R623 524,96 R104 270,75

R519 254,21

35

Tutorial 4 Problem 1d: Suggested Solution

519 254,21

0,095

12

12 5 60

?

SF

i

n

x

Know :Swop Shop needs to replace their truck in 5 years time.

Their current truck is valued at R235 000 and depreciates

at 15% p.a. compounded annually on a reducing balance.

The price of a replacement truck increases by 20% p.a.,

compounded quarterly. Calculate:

(d) The monthly payments into a sinking fund, if the

interest rate is 9.5% p.a., compounded monthly.

1 1

1 1

n

InvestmentInvestment

n

Investment Investment

x i SF iSF x

i i

60

0,095519 254,21

12

0,0951 1

12

x

R6 794,54

36

Deferred Annuities

• First payment is deferred to some later date

than the first interest period.

• Formula used to calculate present value is:

1 1m m n

Def

x i iP

i

is the number of periods the payments are deferred.

is the number of payments.

m

n

37

Analysis of Deferred Annuities

1 1 1m n

x i iP

i

Compound Interest earned/payed over deferred periodsm

1m

DefP P i

Present Value Annuity over payment periodsn

1 1n

Def

x iP

i

1 11

n

mx i

P ii

1 1m m n

x i iP

i

38

Determine the present value of an annuity of R500 each year for

10 years that is deferred 5 years if the interest rate is 6% p.a.

1 1m m n

Def

i iP R

i

5 5 10500 1 0.06 1 0.06

0.06

R2 749.94Alternative Solution

5

1 1.06m

DefP P i P

Compound interest earned over 5 years

present value for annuity with 10 paymentsDefP

10

51 1 500 1 1.06

1.060.06

nx i

Pi

5 10500 1.06 1 1.06

0.06P

2 749.94R

Present Value of Deferred Annuities

39

In a "Buy Now, Pay Later" scheme a man buys

a car of R100 000 with the provision that his

repayments are deferred for 6 months.

Calculate his monthly repayments if the interest

rate is 12% compounded monthly and the

repayment period is 60 months.

Tutorial 5: Deferred Annuities

PAUSE DVD

• Do Tutorial 5


40

1 1

1 1

m m n

Def

Def

m m n

x i iP

i

i Px

i i

In a "Buy Now, Pay Later" scheme a man buys a car of R100 000

with the provision that his repayments are deferred for 6 months.

Calculate his monthly repayments if the interest rate is 12%

compounded monthly and the repayment period is 60 months.

6

60

0,120,01

12

100 000

?

Def

m

n

i

P

x

Tutorial 5: Suggested Solution

6 66

0,01 100 000

1,01 1,01x

R2 361,29

Redo by means of

alternative method!

41

End of the DVD on Annuities

REMEMBER!

•Consult text-books for additional examples.

•Attempt as many as possible other similar examples

on your own.

•Compare your methods with those that were

discussed in the DVD.

•Repeat this procedure until you are confident.

•Do not forget:

Practice makes perfect!

annuities ncs mathematics dvd series resources... · future value of annuity capital is accumulated...

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