4 MateriTerbuka Ordinay Anuity

(1)

Ordinary Annuities

Ordinary

Annuities

10

101010

10-1

Chapter 10

O

rdinary

A

nnuities

(2)

AnnuitiesAnnuities

10

101010

Calculate the…

Define and distinguish between…

Learning Objectives

Learning

Objectives

After completing this chapter, you will be able to:

…

F

uture

V

alue and

P

resent

V

alue of

ordinary simple annuities

… ordinary

simple

annuities

and ordinary

general

annuities

…

fair market value

of a cash flow stream

that includes an annuity

LO-1LO-1

LO-2LO-2

(3)

Ordinary

Annuities

10

101010

10-3

Calculate the…

Learning Objectives

…

P

resent

V

alue of and

period of

d

eferral

of a

d

eferred annuity

…

principal

balance owed on a loan

immediately after any payment

…

F

uture

V

alue and

P

resent

V

alue of

ordinary general annuities

LO-4LO-4

LO-5LO-5

(4)

10

101010

Terminology

- A series of equal payments at regular intervals

Term of the

A

nnuity

- the time from the beginning of the first payment period to the end of the last payment period

F

uture

V

alue

P

resent

V

alue

the future dollar amount of a series of payments plus interest

the amount of money needed to invest today in order to

receive a series of payments for a given number of years

in the future

(5)

Ordinary

Annuities

10

101010

10-5

Terminology

_Terminology

… is the amount of each payment

in an annuity

PMT

… is the number of payments

in the annuity

n

payment interval

ordinary annuities

… is the time between

successive payments

in an annuity

… are ones in which payments

are made

(6)

10

101010

Terminology

Suppose you obtain

a personal loan

to be

repaid by

payment interval

Term

ordinary annuities

48 equal monthly payments

48 months or 4years.

1 month

first payment will be due 1 month after you receive the loan,

(7)

Ordinary

Annuities

10

101010

10-7

Terminology

_Terminology

PMT

0

1

2

3 n

-1

n

Interval

_number

Term

of the annuity

Payment interval

… for an

n

-payment

O

rdinary

A

nnuity

(8)

10

101010

Ordinary Annuity

Ordinary

S

imple

A

nnuities

Ordinary

S

imple

A

nnuities

Ordinary

G

eneral

A

nnuities

Ordinary

G

eneral

A

nnuities

(9)

Ordinary

Annuities

10

101010

10-9

$1000

$1000 (1.04)1

n = 1

Sum

=

FV

of annuity

0

1

2

3

4 Interval

number

$1000 $1000 $1000

$1000 (1.04)2

n = 2

$1000 (1.04)3

n = 3

…the sum of the future values of all the payments

Assume that there are four(4) annual $1000 payments with interest at 4%

Future Value of an

Ordinary Simple Annuity

Future Value of an

(10)

10

101010

= $4246.46

= $1000 +

FV of annuity

$1000

$1000 (1.04)1

n = 1

Sum

=

FV

of annuity

0 1 2 3 4 Interval

number

$1000 $1000 $1000

$1000 (1.04)2

n = 2

$1000 (1.04)3

n = 3

Assume that there are four(4) annual $1000 payments with interest at 4%

$1000(1.04) + $1000(1.04)2 + $1000(1.04)3

= $1000 +$1040+ $1081.60+$1124.86

of an Ordinary Simple Annuity

of an

(11)

Ordinary

Annuities

10

101010

10-11

ResultResult

$500

$500(1+.03/12)

Sum = FV of annuity

0 1 2 3 _{4 Month}

$500 $500 $500

$500(1+.03/12)3

Suppose that you vow to save $500 a month for the next four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now.

$500(1+.03/12)2

$ 500.00 501.25

502.50

503.76 $2,007.51

(12)

10

101010

Now imagine that you save $500 every month for the next three years. Although the same logic applies, I

certainly don’t want to do it this way!

Since your ‘account’ was empty when you began…

PV = 0

n = 3 yrs * 12 payments per year =

36 payments

of an

(13)

Ordinary

Annuities

10

101010

10-13

36

You save $500 every month for the next three years. Assume your savings can earn 3% converted monthly.

Determine the total in your account three years from now.

3

500

0

12

Using the formulaUsing the formula NoteNote

Keys direction

(14)

10

101010

…

the sum of the

future values

of all the

payments

of an

FV

=

PMT

[

(1+

i

)

n

- 1

i

]

(15)

Ordinary

Annuities

10

101010

10-15

You save $500 every month for the next three years. Assume your savings can earn 3% converted monthly.

Determine the total in your account three years from now.

0.0025

[

FV

=

PMT

(1+

i

)

n

- 1

i

]

18810.28

37.6206

1.0025

0.0941

1.0941

12 .03

500

36

1

(16)

10

101010

You vow to save $500/month for the next four months, with your first deposit one month from today.

If your savings can earn 3% converted monthly, determine the total in your account four months from now.

Since your ‘account’ was empty when you began…

PV

=

0 n =

4 payments

PMT

=

-500

(17)

Ordinary

Annuities

10

101010

10-17

Cash Flows

… payments received e.g. receipts

Treated as:

Positives

+

Positives

+

Negatives

-

-..a term that refers to

payments

that can be either …

..a term that refers to

payments

that can be either …

… payments made e.g. cheques

(18)

10

101010

Therefore…

…when you are making payments,

or

even making

deposits to

savings

,

Really payments to

the bank!

Really payments to

the bank!

these are

cash outflows

,

and therefore

the values must be negative!

Cash Flow Sign Convention

(19)

Ordinary these from before, so we don’t have to enter

them again!

We already have these from before, so we don’t have to enter

them again!

(20)

10

101010

12 .03

500

4

1

You vow to save $500/month for the next four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now. You vow to save $500/month for the next

four months, with your first deposit one month from today. If your savings can earn 3% converted monthly, determine the total in your account four months from now.

Formula Formula

FV

=

PMT

[

(1+

i

)

n

- 1

i

]

PMT = $500

n =

4 i

=

.

03/

12 =

0 .

0025

0.0025

1.0025

(21)

Ordinary

Annuities

10

101010

10-21

Not seeing the total picture!

When you use formula or a calculator’s financial functions to calculate an annuity’s Future Value,

the amount

each payment

contributes to the future value

is

(22)

10

101010

10% Compounded Annually10% Compounded Annually

$10.00

YearsYears

0 1 2 3 4 5

14.64

13.31

12.10

11.00

10.00

C

ontribution $

$61.05

FV Contributions

$10.00$10.00

(23)

Ordinary

Annuities

10

101010

10-23

You decide to save $75/month for the next four years. If you invest all of these savings in an account which will pay you 7% compounded monthly,

determine:

a) the total in the account after 4 years

b) the amount you deposited

c) the amount of interest earned

Extract

necessary

data...

PMT = - $75 = 7 n = 4 * 12 =

48

PV = 0 FV = ?

Solve…

Total Deposits = $75* 48 = $3,600

(24)

10

101010

You decide to save $75/month for the next four years.

If you invest all of these savings in an account which

will pay you 7%

compounded monthly, determine:

a) the total in the account after 4 years

b) the amount you deposited

c) the amount of interest earned

You decide to save

$75/month for the next four years.

will pay you 7%

(25)

Ordinary You decide to save

will pay you 7%

c) the amount of interest earned

You decide to save

will pay you 7%

(26)

10

101010

…

the sum of the

present values

of all the

payments

PV

=

PMT

[

1-(1+

i

)

-n

i

]

of an

(27)

Ordinary

Annuities

10

101010

10-27

$1000

Sum = PV of annuity

$1000 $1000 $1000

…the sum of the present values of all the

payments

Assume that there are four(4) annual $1000

payments with interest at 4%

Present Value of an

Present Value of an

$1000 (1.04)-1 n = 1

$1000 (1.04)-2 n = 2

$1000 (1.04)-3 n = 3

$1000 (1.04)-4 n = 4

0

1

2

3

4

Interval

(28)

10

101010

= $3629.90

PV of annuity

= $1000(1.04)-1 + $1000(1.04)-2 + $1000(1.04)-3 +

= $961.54 + $924.56 + $889.00+ $854.80

$1000 $1000 $1000 $1000

Assume that there are four(4) annual $1000

payments with interest at 4%

of an

$1000 (1.04)-1 n = 1

$1000 (1.04)-2 n = 2

$1000 (1.04)-3 n = 3

$1000 (1.04)-4 n = 4

0

1

2

3

4

Interval

Number

$1000 (1.04)-4

(29)

Ordinary

Annuities

10

101010

10-29

McGraw-Hill Ryerson©

Present Value of an

Present Value of an

You overhear your friend saying the he is repaying a loan at $450 every month for the next nine months.

The interest rate he has been charged is 12%

compounded monthly. Calculate the amount of the loan, and the amount of interest involved.

… Interest - use 12, not .12 when using financial calculator

… Interest - use

12

, not .12 when using financial calculator

… At the end of the loan, you don’t owe any money, so

FV

= 0

… n =

9

payments

…Since you are making payments…Since you are making payments, , not receiving themnot receiving them, PMT = ,

PMT

=

-

450

450 Solve…

(30)

AnnuitiesAnnuities The interest rate he has been charged is

8% compounded monthly. Calculate

the amount of the The interest rate he has been charged is

the amount of the

Amount Borrowed (PV) $ 3,918.24

Interest Paid =

Repaid.………. 4,050.00

(31)

Ordinary The interest rate he has been charged is

the amount of the The interest rate he has been charged is

the amount of the

loan, and the

amount of interest

(32)

10

101010

Contribution of

Each

Payment

to an

Annuity’s

(33)

Ordinary

Annuities

10

101010

10-33

$10.00

YearsYears

0 1 2 3 4 5

C

ontribution $

9.09

8.20

7.51

6.83

6.21

$37.91

PV Contributions

$10.00$10.00

(34)

10

101010

…of a

cash flow

stream

that

includes an annuity

10

101010

(35)

Ordinary

Annuities

10

101010

10-35

You have received two offers on a

building lot that you want to sell.

Ms. Armstrong’s offer is

$25,000 down

plus a

$100,000 lump sum payment

five years from now.

Mr. Belcher has offered

$20,000 down

plus

$5000

every quarter for

five years.

Compare the economic values of the two offers

if money can earn 5%

compounded annually

.

(36)

10

101010

The economic value of a payment stream on a particular date (focal date) refers to a single amount that is an

economic substitute for the payment stream

On what information

should we

f

ocus?

On what information

should we

f

ocus?

WE need to choose a focal date, and determine the

values of the two offers at that focal date.

(Obvious choices would be today, the date of the offers, or the end of the term i.e. 5 years from now.)

ocu

(37)

Ordinary

Annuities

10

101010

10-37

Preparing Time Lines Mr. Belcher

Ms. Armstrong

$20,000 down

plus $5000 every quarter for five years

$25,000 down

plus a $100,000 lump sum payment

five years from now

Focal Date: TodayFocal Date: Today

You have received two offers on a building lot that you want to sell. Ms. Armstrong’s offer is $25,000

down plus a $100,000 lump sum payment five years from now. Mr. Belcher has offered $20,000

down plus $5000 every quarter for five years. Compare the economic values of the two offers if money can earn 5% compounded annually.

(38)

10

101010

$20,000 $20,000 $20,000 $20,000 $20,000

Years

0

1

2

3

4

$20,000 down plus $5,000 every quarter for five years

$25,000 down plus a $100,000 lump sum payment

five years from now

AA

BB

$25,000

$20,000

Ms. Armstrong

Mr.Belcher

$5000 every quarter

5

(39)

Ordinary

building lot that you want to sell. Ms.

has offered $20,000 down plus $5000 every

quarter for five years. Compare the economic values of the two offers if money can earn 5%

compounded annually.

Step 1–Determine today’s value of Ms. Armstrong’s offer

(40)

10

101010

Step 2 – Determine today’s value of Mr. Belcher’s offer.

4 building lot that you

want to sell. Ms.

has offered $20,000 down plus $5000 every

quarter for five years. Compare the economic values of the two offers if money can earn 5%

compounded annually.

(41)

Ordinary

Annuities

10

101010

10-41

$103,352.62

99,376.93

$

3,975.69

Better off accepting Ms. Armstrong’s offer!

Ms. Armstrong

Mr.Belcher

Total Value

of each offer

Total Value

of each offer

(42)

10

101010

The required

monthly payment

on

a

five-year loan

, bearing

8%

interest

,

compounded monthly

, is $

249.10 .

Since you are “borrowing” money, you are looking for PV … and FV = 0 once you have repaid the loan!

n = 5 yrs * 12 payments per year = 60 payments

Since you are “borrowing” money, you are looking for PV … and FV = 0 once you have repaid the loan!

n = 5 yrs * 12 payments per year =

60 payments

a) What was the original principal amount of the loan? b) What is the balance owed just after the twentieth payment?

Original Loan

and a

Subsequent

Balance

O

riginal

L

oan

and a

S

ubsequent

B

alance

(43)

Ordinary on a five-year loan, bearing 8% interest,

compounded monthly, is $249.10.

a) What was the

original principal

amount of the loan? b) What is the

balance owed just after the twentieth

payment?

amount of the loan?

b) What is the

balance owed just

(44)

AnnuitiesAnnuities on a five-year loan, bearing 8% interest,

compounded monthly, is $249.10.

a) What was the

original principal

amount of the loan? b) What is the

balance owed just after the twentieth

payment?

amount of the loan?

b) What is the

balance owed just

(45)

Ordinary

Annuities

10

101010

10-45

A

D

eferred

A

nnuity

may be viewed as an

o

rdinary

a

nnuity

that does not begin until

a time interval

(named the period of deferral

)

(46)

10

101010

D

Deferred Annuities

eferred

A

nnuities

A

Deferred

Annuity

may be viewed as

an

ordinary

annuity

that does not begin until a time

Two-step

procedure to find PV: procedure to find PV: Calculate the present value,

PV

₁,

of the payments at the end of the period of deferral — this is just the

PV of an ordinary annuity Calculate the present value,

PV₂, of the STEP 1 amount

(47)

Ordinary

Annuities

10

101010

10-47

… your friend saying the he is repaying a loan at $450 every month for four months. The interest rate he has been charged is 8% compounded monthly. Calculate the amount of the loan, and

the amount of interest involved.

… your friend saying the he is repaying a loan at $450 every month for four months. The interest rate he has been charged is 8% compounded monthly. Calculate the amount of the loan, and

the amount of interest involved.

…this same friend doesn’t begin to repay his loan for another 11 months, at a rate $500 every month for four months. The interest

rate is still 8% compounded monthly. Determine the

size of the loan

.

…this same friend doesn’t begin to repay his loan for another 11 months, at a rate $500 every month for four months. The interest

rate is still 8% compounded monthly. Determine the

size of the loan

.

(48)

10

101010

$500 $500 $500 $500

…of the Annuity

of a Deferred Annuity

of a

Deferred Annuity

10

11

12

13

14

Months

0 PV

PV

Step 1 – Determine PV of Annuity 10 months from now

Hint: (Use Compound Discount)

(49)

Ordinary another 11 months,

at a rate $500

every month for four

months. The interest rate is still

8%

(50)

10

101010

The payment interval

differs from

the compounding interval

The payment interval

differs from

e.g.

A typical Canadian mortgage has

Monthly payments

,

but the

interest

is

compounded semi-annually

(51)

Ordinary

Annuities

10

101010

10-51

For those who are using this type of calculator,

the

C/Y

worksheet

will now be used

For those who are using this type of calculator,

the

C/Y

worksheet

will now be used

See following REVIEW

For those who are using a non-financial calculator,

new formulae

will be added to find

the solution

For those who are using a non-financial calculator,

new formulae

will be added to find

the solution

(52)

10

101010

We can input the number of compoundings per year into the

financial calculator. This can be performed by using

the symbol To access this symbol use:

(53)

Ordinary

Annuities

10

101010

10-53

The 12 is a default

setting

The 12 is a default

setting

This display is referred to as “the worksheet”. … represents the number of

P

ayments per

Y

ear

… represents the number of

C

ompoundings per

Y

ear To access use:

Note:

You can override these values by entering new ones!

…Example

Appears

automatically

(54)

10

101010

12

2 P/Y =

12.00 C/Y =

12.00 Using

C/Y

=

2.00 Adding

New Formulae

Typical

Canadian

mortgage

Interest is compounded semi-annually

and

payments are each month.

Typical

Canadian

mortgage

Interest is compounded semi-annually

and

(55)

Ordinary

Annuities

10

101010

10-55

to calculate the equivalent

periodic rate that matches the payment interval

C

=

number of

interest

compoundings per year

number of

payments per year

Use

c

to determine

i

₂

Step 2Step 2

Use i

₂

= (1+i)

c

- 1

Use this equivalent periodic rate as the value for “

i

”

in the appropriate simple annuity formula

Step 3

…Example

Step 1Step 1

D

etermine

the number of

Interest

periods per

c

ompounding interval

(56)

number of interest compoundings per year

number of payments per year

2

12 0.166666

Step 1Step 1 To

determine

the number of

Interest

periods per

c

ompounding interval

=

C

Use

c

to determine

i

₂

(57)

Ordinary

(58)

12

52

0.23076 =

C

Use

c

to determine

i

₂

(59)

Ordinary

(60)

10

101010

You decide to save $50/month for the next three years. If you invest all of these savings in an account which will pay you 7% compounded semi-annually,

determine the total in the account after 3 years. Is the following a

General Annuity?

The payment interval differs from

Criteria

As the Criteria have been met, therefore, we need to determine

C

(61)

(62)

(63)

(64)

10

101010

…your calculator retains at least two more digits than you see displayed!

Improving the

Accuracy of

Calculated Results

C =

the value for

c

can be a repeating decimal

SAVE

c

in memory…

when you need the exponent for

Simply the

c

value from memory!

The value for

i

₂

should be saved in

(65)

Ordinary

Annuities

10

101010

10-65

Reid David made annual deposits

of $1,000

to Fleet Bank, which pays

6% interest

compounded annually

.

After 4 years

, Reid makes

no more

deposits

.

(66)

10

101010

…of the Annuity

1

2

3

4

14

0 FV

FV

₁1

Step 2 – Determine FV using compound interest

FV

₂2

Step 1 – Determine FV₁ of Annuity 10 years from now

Years

$1000 $1000 $1000 $1000

Reid David made annual deposits of $1,000 to Fleet Bank, which pays 6% interest compounded

annually. After 4 years, Reid makes no more

deposits. What will be the balance in the account

(67)

Ordinary After 4 years, Reid

makes no more

(68)

Step 2 – Determine FV₂ using compound interest

FV = 4374.62

After 4 years, Reid

makes no more

(69)

Ordinary After 4 years, Reid

makes no more

(70)

10

101010

1.06 10

Step 2 – Determine FV using compound interest

Reid David made annual deposits of $1,000 to Fleet Bank, which pays

6% interest

compounded

annually.

After 4 years, Reid makes no more

deposits.

What will be the balance in the account

10 years

after the last deposit?

n =

i =

4374.62

0 .

06 PV =

10 1.262477

0.262477

4374.62

value 14 years value _{from now}_{from now}14 years

1.1708477

7834.27

FV = PV(1 +

i

)

n

(71)

(72)

10

101010

0 3650

365 FV

= 4386.52

How

much more interest will Reid David accumulate over the 14

years if his account earns

6%

compounded daily?

value 14 years from now value 14 years

from now

P/Y =

1 P/Y

FV =

=

7992.37

3650

Step 2 – Determine FV in 10 years

(73)

Ordinary

Annuities

10

101010

10-73

Interest

(74)

10

101010