4 MateriTerbuka Ordinay Anuity

(1)

Ordinary Annuities

101010

10-1

Chapter 10

O

rdinary

A

nnuities

(2)

AnnuitiesAnnuities

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

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)

AnnuitiesAnnuities

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

A

nnuity

LO-1LO-1

(5)

Ordinary Annuities

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

at the end of each payment

interval

(6)

AnnuitiesAnnuities

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,

i.e. at the end of the first payment interval

(7)

Ordinary Annuities

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

PMT PMT PMT

PMT

(8)

AnnuitiesAnnuities

101010

Ordinary Annuity

Ordinary

S

imple

A

nnuities

Ordinary

S

imple

A

nnuities

Ordinary

G

eneral

A

nnuities

Ordinary

G

eneral

A

nnuities

Monthly payments,

and interest is

compounded monthly Monthly payments,

and interest is

compounded monthly

Monthly payments, but interest is

compounded semi-annually Monthly payments,

but interest is

compounded semi-annually The payment interval

=

the compounding

interval

The payment interval

=

the compounding

interval

The payment interval

differs from

the compounding interval

The payment interval

differs from

(9)

Ordinary Annuities

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

Ordinary Simple Annuity LO-2LO-2

(10)

AnnuitiesAnnuities

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

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

Future Value of an

Ordinary Simple Annuity

Future Value of an

(12)

AnnuitiesAnnuities

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

Ordinary Simple Annuity

of an

Ordinary Simple Annuity

(13)

Ordinary Annuities

101010

10-13

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.

500

Future Value of an

Ordinary Simple Annuity

Future Value of an

Ordinary Simple Annuity

Using the formulaUsing the formula NoteNote

Keys direction

P/Y= 120 FV =

18810.28

(14)

AnnuitiesAnnuities

101010

…

the sum of the

future values

of all the

payments

of an Ordinary Simple Annuity

of an

Ordinary Simple Annuity

FV

=

PMT

[

(1+

i

)

- 1

i

]

(15)

Ordinary Annuities

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.

Future Value of an

Ordinary Simple Annuity

Future Value of an

Ordinary Simple Annuity

0.0025

[

FV

=

PMT

(1+

i

)

- 1

i

]

18810.28

37.6206

1.0025

0.0941

1.0941

12 .03

500

1

(16)

AnnuitiesAnnuities

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

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)

AnnuitiesAnnuities

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 Annuities

101010

10-19

McGraw-Hill Ryerson©

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.

PV = 0 n =

4

payments PMT -500

Future Value of an

Ordinary Simple Annuity

Future Value of an

Ordinary Simple Annuity

4

500 0

_{FV = 2007.51}

We already have 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)

AnnuitiesAnnuities

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

)

- 1

i

]

PMT = $500

n =

4 i

=

.

03/

12 =

0 .

0025

0.0025

1.0025

(21)

Ordinary Annuities

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)

AnnuitiesAnnuities

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

FV

(23)

Ordinary Annuities

101010

10-23

Future Value of an

Ordinary Simple Annuity

Future Value of an

Ordinary Simple Annuity

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)

AnnuitiesAnnuities

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.

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

48 7

75 0

Formula solutionFormula solution FV……….. $4,140.69

Interest Earned = $ 540.69

Deposits…... 3,600.00

P/Y =

12

(25)

Ordinary Annuities

101010

10-25

McGraw-Hill Ryerson©

FV $4,140.69

=

Interest Earned $540.69

- Deposits 3,600.00

Formula Formula

FV

=

PMT

[

(1+

i

)

- 1

i

]

0.005833

1.005833

1.32205

0.32205

12 .07

75

1 55.20924

4140.6927

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.

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

(26)

AnnuitiesAnnuities

101010

…

the sum of the

present values

of all the

payments

PV

=

PMT

[

1-(1+

i

)

-n

i

]

of an Ordinary Simple Annuity

of an

Ordinary Simple Annuity

(27)

Ordinary Annuities

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

Ordinary Simple Annuity

Present Value of an

Ordinary Simple Annuity

$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)

AnnuitiesAnnuities

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 Ordinary Simple Annuity

of an

Ordinary Simple Annuity

$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

101010

10-29

McGraw-Hill Ryerson©

Present Value of an

Ordinary Simple Annuity

Present Value of an

Ordinary Simple Annuity

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 Solve…

(30)

AnnuitiesAnnuities

101010

Formula solutionFormula solution 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

8% compounded monthly. Calculate

the amount of the loan, and the

amount of interest

involved.

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

8% compounded monthly. Calculate

the amount of the

loan, and the

amount of interest

involved. 9 8 450 0 12

PV =

3,918.24

Amount Borrowed (PV) $ 3,918.24

Interest Paid =

Repaid.………. 4,050.00

(31)

Ordinary Annuities

101010

10-31

Formula Formula

i

PV

=

PMT

1-(1+

i

)

-n

[

]

- Borrowed $3,918.24

=

Interest Charged $131.76

Repaid $4,050.00 12 .08

450

1 0.006667

1.006667

0.94195

-0.0580479

3,918.24

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

8% compounded monthly. Calculate

the amount of the loan, and the

amount of interest

involved.

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

8% compounded monthly. Calculate

the amount of the

loan, and the

amount of interest

(32)

AnnuitiesAnnuities

101010

Contribution of

Each

Payment

to an

Annuity’s

(33)

Ordinary Annuities

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

PV

(34)

AnnuitiesAnnuities

101010

…of a

cash flow

stream

that

includes an annuity

AnnuitiesAnnuities

101010

(35)

Ordinary Annuities

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

.

LO-3LO-3

(36)

AnnuitiesAnnuities

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

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)

AnnuitiesAnnuities

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 Annuities

101010

10-39

McGraw-Hill Ryerson©

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.

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

today’s value of lump sum today’s value of lump sum

5 100,000

5 25,000

PV= 78352.692

103,352.62

_{of Ms. A’s} today’s value total offer

today’s value of Ms. A’s total offer

Step 2…

0

(40)

AnnuitiesAnnuities

101010

McGraw-Hill Ryerson©

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

5

0 4500

20 P/Y

=

4 C/Y

PV = 79,376.93

=

1

0

20000

99,376.93

value of today’s lump sum

today’s value of lump sum today’s value

of Mr. B’s total

offer

today’s value of Mr. B’s

total

offer 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.

(41)

Ordinary Annuities

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)

AnnuitiesAnnuities

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 Annuities

101010

10-43

Original Principal

=

PV

of

all

60 payments

PMT =_249.10 FV = 0 n = 5*12 = 60 i = .08/12

c

= 1

0

8

60

0 PV = 12,285.22

Original loan

value

Original loan

value

249.10

The required monthly payment 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?

The required monthly payment 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?

(44)

AnnuitiesAnnuities

101010

₌

_PV

of

40 payments left

PMT =249.10 FV = 0 n = 60 - 20 = 40 i = .08

40 PV = 8,720.75

New loan

balance

New loan

balance

We will leave it to you to do

the algebraic solution…!

We will leave it to you to do

the algebraic solution…!

The required monthly payment 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?

The required monthly payment 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?

(45)

Ordinary Annuities

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

)

has passed

LO-5LO-5

(46)

AnnuitiesAnnuities

101010

D

Deferred Annuities

eferred

A

nnuities

Deferred

Annuity

may be viewed as

ordinary

annuity

that does not begin until a time

interval (named the period of deferral)

has passed

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)

has passed

d

= Number of

payment

intervals

in the period of deferral

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

at the beginning of the period of deferral

(47)

Ordinary Annuities

101010

10-47

McGraw-Hill Ryerson©

… 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)

AnnuitiesAnnuities

101010

$500 $500 $500 $500

…of the Annuity

of a Deferred Annuity

of a

Deferred Annuity

10

11

12

13

14

Months

0 PV

Step 1 – Determine PV of Annuity 10 months from now

Hint: (Use Compound Discount)

(49)

Ordinary Annuities

101010

10-49

McGraw-Hill Ryerson©

…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

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.

0

8

4

10 PV = 1967.11

FV = - 1967.11

PV =

1840.65

_{value 10 months}

from now

value 10 months

from now

loan value

today

loan value

today

500

(50)

AnnuitiesAnnuities

101010

The payment interval

differs from

the compounding interval

The payment interval

differs from

the compounding interval

e.g.

A typical Canadian mortgage has

Monthly payments

,

but the

interest

is

compounded semi-annually

Using calculators

…

LO-6LO-6

(51)

Ordinary Annuities

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

See following

(52)

AnnuitiesAnnuities

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

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

Appears automatically

(54)

AnnuitiesAnnuities

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

payments are each month.

(55)

Ordinary Annuities

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)

- 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)

AnnuitiesAnnuities

101010

Typical

Canadian

mortgage

6% Interest is

compounded semi-annually and

payments are each month. Find

C

and

i

₂.

Typical

Canadian

mortgage

6% Interest is

compounded semi-annually and

payments are each month. Find

C

and

i

₂.

C

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 Annuities

101010

10-57

McGraw-Hill Ryerson©

Use

c

to determine

i

₂

Step 2Step 2

i

₂

(1+i)

- 1

i

₂

(1+

.06

/2)

.16666

-1

Typical

Canadian

mortgage

6% Interest is

compounded semi-annually

and

payments are each month. Find

C

and

i

₂.

Typical

Canadian

mortgage

6% Interest is

compounded semi-annually

and

payments are each month. Find

C

and

i

₂.

1.03

0.166666 =

1.0049

0.0049

i

₂

(58)

AnnuitiesAnnuities

101010

5% interest

is

compounded

monthly

and

payments

are each

week

5% interest

is

compounded

monthly

and

payments

are each

week

Mortgage

Step 1Step 1 To

determine

the number of

c

ompoundings

C =

number of interest compoundings per year

number of payments per year

12

52

0.23076 =

C

Use

c

to determine

i

₂

(59)

Ordinary Annuities

101010

10-59

1

Use

c

to determine

i

₂

Step 2Step 2

i

₂

(1+i)

- 1

i

₂

(1+

.05

/12)

.2308

-1

1 i

₂

0.05

12 0.0041667

1.0041667

5% interest

is

compounded

monthly

and

payments

are each

week

5% interest

is

compounded

monthly

and

payments

are each

week

Mortgage

0.230769

1.00096

0.00096

(60)

AnnuitiesAnnuities

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

the compounding interval

The payment interval differs from

the compounding interval

Criteria

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

C

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

C

(61)

Ordinary Annuities

101010

10-61

Find

i

₂

Step 2Step 2

i

2 = (1+i)c - 1

i

₂ = 1.035

0.1666

(1+

.07

/2)

.1666

-1

0.00575 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.

i

₂

=

Step 1Step 1 Find

c

Use

i

₂

Step 3

1.00575

0.00575

(62)

AnnuitiesAnnuities

101010

Formula Formula

FV

=

PMT

[

(1+

i

)

- 1

i

]

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.

PMT = PV = n =

i = .07/250

c

=2/12 = 0 .166663*12i₂ = = 36_0.00575

36 1

0.00575

1.00575

1.229255

i

₂ Step

0.229255

39.8702

1993.51

(63)

Ordinary Annuities

101010

10-63

P/Y =

12 C/Y =

12 C/Y

=

2

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.

12

2

50

0

36

7

0 FV = 1993.51

(64)

AnnuitiesAnnuities

101010

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

Improving the

Accuracy of

Calculated Results

C =

number of interest compoundings per year

number of payments per year

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

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

.

What will be

the balance in the account

10 years

after the last deposit?

(66)

AnnuitiesAnnuities

101010

…of the Annuity

1

2

3

4

14

0 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 Annuities

101010

10-67

Step 1 – Determine FV₁ of Annuity 10 years from now

6

0

4 P/Y

=

1.00 C/Y

=

1.00

value at end of 4 years value at end of 4 years

Step 2…

0 1000

FV = 4374.62

Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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? Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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?

(68)

AnnuitiesAnnuities

101010

0

10

Formula solutionFormula solution

Step 2 – Determine FV₂ using compound interest

FV = 4374.62

FV = 7834.27

value 14 years from now value 14 years

from now Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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? Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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?

(69)

Ordinary Annuities

101010

10-69

Formula Formula

FV

=

PMT

[

(1+

i

)

- 1

i

]

n =

i =

c

=

1000

0 .

06

1.06 1000

4

1 0.06

PMT =

1.262477

0.262477

4374.62

4

1

Step 1 – Determine FV of Annuity 4 years from now

value at end of 4 years value at end

of 4 years

Step 2…

Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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? Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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?

(70)

AnnuitiesAnnuities

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

)

(71)

Ordinary Annuities

101010

10-71

How much more interest will Reid David accumulate over the 14

years if his account earns

compounded daily?

1

365 1000

0

4

6 P/Y =

10

value at end of 4 years value at end

of 4 years

C/Y

C/Y =

=

1

365

Step 1 – Determine FV of Annuity 4 years from now

0

(72)

AnnuitiesAnnuities

101010

0 3650

365 FV

= 4386.52

How

much more interest will Reid David accumulate over the 14

years if his account earns

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

101010

10-73

Interest

(74)

AnnuitiesAnnuities

101010

(1)

AnnuitiesAnnuities

10 101010

Formula Formula

FV

=

PMT

[

(1+

i

)

- 1

i

]

n =

i

c

=

1000

0.06

1.06 1000

4

1 0.06

PMT =

1.262477

0.262477

4374.62

4

1

Step 1 – Determine FV of Annuity 4 years from now

value at end of 4 years value at end

of 4 years

Step 2…

Reid David made annual

deposits of $1,000 to Fleet Bank, that 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? Reid David

made annual

deposits of $1,000 to Fleet Bank, that 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?

(2)

1.06 10

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 7834.27

1708477

FV

=

PV

(1 +

i

)

(3)

AnnuitiesAnnuities

10 101010

How much more interest will Reid David accumulate over the 14

years if his account earns

compounded daily?

1

365 1000

0

4

6 P/Y =

10

value at end

of 4 years value at end

of 4 years

C/Y

C/Y =

=

1

365

Step 1 – Determine FV of Annuity 4 years from now

0

(4)

0 3650

365 FV

= 4386.52

How much more interest will Reid David accumulate over the 14

years if his account earns

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

(5)

AnnuitiesAnnuities

10 101010

Interest

(6)