Business Mathematics Jerome Chapter 10

Ordinary

Annuities

Ordinary

Annuities

10101010

10-1

McGraw-Hill Ryerson©

Chapter 10

Ordinary AnnuitiesOO AA


Ordinary

Annuities

Ordinary

Annuities

10101010

10-2


Calculate the…

Define and distinguish between…

Learning ObjectivesLearning ObjectivesAfter completing this chapter, you will be able to:

… Future Value and Present Value 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

LO-3LO-3

Ordinary

Annuities

Ordinary

Annuities

10101010

10-3


Calculate the…

Learning ObjectivesLearning Objectives

… Present Value of and period of deferral of a deferred annuity

… principal balance owed on a loan immediately after any payment

… Future Value and Present Value of ordinary general annuities

LO-4LO-4

LO-5LO-5

LO-6LO-6

Ordinary

Annuities

Ordinary

Annuities

10101010

10-4


TerminologyTerminology

- A series of equal payments at regular intervals

Term of the Annuity

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

Future ValuePresent Valuethe 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

AnnuityLO-1LO-1

Ordinary

Annuities

Ordinary

Annuities

10101010

10-5



… is the amount of each payment in an annuityPMTPMT

… is the number of payments in the annuitynpayment 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

Ordinary

Annuities

Ordinary

Annuities

10101010

10-6



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

Ordinary

Annuities

Ordinary

Annuities

10101010

10-7



PMT

0 1 2 3 nn-1 Intervalnumber

Term of the annuity

Payment interval

… for an n-payment Ordinary Annuity

PMT PMT PMTPMT

Ordinary

Annuities

Ordinary

Annuities

10101010

10-8


Ordinary Annuity

Ordinary

Simple Annuities

Ordinary

Simple AnnuitiesOrdinary

General Annuities

Ordinary

General Annuities

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


The payment interval

=

the compounding interval


=



differs from



differs from


Ordinary

Annuities

Ordinary

Annuities

10101010

10-9


$1000

$1000 (1.04)1n = 1

Sum = FV of annuity

0 1 2 3 4 Intervalnumber

$1000 $1000 $1000

$1000 (1.04)2n = 2

$1000 (1.04)3n = 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


LO-2LO-2

Ordinary

Annuities

Ordinary

Annuities

10101010

10-10


= $4246.46

= $1000 +FV of annuity

$1000$1000 (1.04)1n = 1

Sum = FV of annuity

0 1 2 3 4 Intervalnumber

$1000 $1000 $1000

$1000 (1.04)2n = 2

$1000 (1.04)3n = 3


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

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

Future Value of an


Future Value of an


Ordinary

Annuities

Ordinary

Annuities

10101010

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

$ 500.00501.25502.50503.76

$2,007.51

Future Value of an


Future Value of an


Ordinary

Annuities

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Annuities

10101010

10-12


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

Future Value of an


Future Value of an


Using the …

Ordinary

Annuities

Ordinary

Annuities

10101010

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

Future Value of an


Future Value of an


012

Using the formulaUsing the formula

NoteNote

Keys direction

P/Y= 120FV = 18810.28

Ordinary

Annuities

Ordinary

Annuities

10101010

10-14


…the sum of the future values of all the payments

Future Value of an


Future Value of an


FV = PMT (1+ i)n - 1[ i ]Formula Formula

Ordinary

Annuities

Ordinary

Annuities

10101010

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


Future Value of an


0.0025[FV = PMT (1+ i)n - 1i ] 1.0025 1.09410.094137.620618810.28

12.03

500

361

1

Ordinary

Annuities

Ordinary

Annuities

10101010

10-16


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 paymentsPMT = -500

Solving earlier Question using Annuities

Solving earlier Question using Annuities

Ordinary

Annuities

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Annuities

10101010

10-17


Cash FlowsCash Flows

… payments received e.g. receipts

Treated as:Treated as:Positives

+Positives

+Negatives -Negatives -

..a term that refers to payments that can be either …

..a term that refers to payments that can be either …

… payments madee.g. cheques

Therefore…Therefore…

Ordinary

Annuities

Ordinary

Annuities

10101010

10-18


Therefore…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

Using the …

Ordinary

Annuities

Ordinary

Annuities

10101010

10-19


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


Future Value of an


4

3

500

012 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!

Formula solutionFormula solution

Ordinary

Annuities

Ordinary

Annuities

10101010

10-20


12.03

500

41

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 - 1i ]

PMT = $500

n = 4

i = .03/12 = 0.0025

0.00251.0025 1.01000.01004.01502007.51

Ordinary

Annuities

Ordinary

Annuities

10101010

10-21


Not seeing the total picture!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

NOT apparent!

Ordinary

Annuities

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Annuities

10101010

10-22


10% Compounded Annually10% Compounded Annually

$10.00$10.00

YearsYears0 1 2 3 4 5

14.64

13.31

12.10

11.00

10.00

Contribution$

$61.05$61.05

FV ContributionsFV Contributions

$10.00$10.00

$10.00$10.00

$10.00$10.00

$10.00$10.00

FVFV

Ordinary

Annuities

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Annuities

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10-23


Future Value of an


Future Value of an


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 = = 7 n = 4 * 12 = 48 - $75

PV = 0 FV = ?

Solve…

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

= 12

Ordinary

Annuities

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Annuities

10101010

10-24


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








earned

487

75012


FV……….. $4,140.69

Interest Earned = $ 540.69Deposits…... 3,600.00

P/Y = 12FV = 4140.69

Ordinary

Annuities

Ordinary

Annuities

10101010

10-25


FV $4,140.69= Interest Earned $540.69

- Deposits 3,600.00


0.005833 1.0058331.322050.32205

12.07

75

481

1

55.209244140.6927You decide to save $75/month for the







earned








earned

Ordinary

Annuities

Ordinary

Annuities

10101010

10-26


…the sum of the present values of all the payments

PV = PMT 1-(1+ i)-n[ i ]

PresentValue of an


PresentValue of an


Formula Formula

Ordinary

Annuities

Ordinary

Annuities

10101010

10-27


$1000

Sum = PV of annuity

$1000 $1000 $1000

…the sum of the present values of all the payments


Present Value of an


Present Value of an


$1000 (1.04)-1 n = 1

$1000 (1.04)-2n = 2

$1000 (1.04)-3n = 3

$1000 (1.04)-4n = 4

0 1 2 3 4 Interval Number

Ordinary

Annuities

Ordinary

Annuities

10101010

10-28


= $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


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 Number

$1000 (1.04)-4

Sum = PV of annuity

Ordinary

Annuities

Ordinary

Annuities

10101010

10-29


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, not receiving them, PMT = -450…Since you are making payments, not receiving them, PMT = -450

Solve…

… Repaid 9 payments at $450 = $4,050

Ordinary

Annuities

Ordinary

Annuities

10101010

10-30

McGraw-Hill Ryerson©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.





involved.

98

450

012

PV = 3,918.24

Amount Borrowed (PV) $ 3,918.24

Interest Paid =

Repaid.…………………. 4,050.00

$ 131.76

Ordinary

Annuities

Ordinary

Annuities

10101010

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

91

1

0.0066671.0066670.94195-0.05804793,918.24 You overhear your friend saying the he is repaying a




involved.





involved.

Ordinary

Annuities

Ordinary

Annuities

10101010

10-32


Contribution of Each Payment to an Annuity’s

Present Value

Ordinary

Annuities

Ordinary

Annuities

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10-33


$10.00$10.00

YearsYears0 1 2 3 4 5

Contribution$

9.09

8.20

7.51

6.83

6.21

$37.91$37.91

PV ContributionsPV Contributions

$10.00$10.00

$10.00$10.00

$10.00$10.00

$10.00$10.00

$10.00$10.00

PVPV

Ordinary

Annuities

Ordinary

Annuities

10101010

10-34


…of a cash flow stream that includes an annuity

Ordinary

Annuities

Ordinary

Annuities

10101010

LO-3LO-3

Ordinary

Annuities

Ordinary

Annuities

10101010

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

Ordinary

Annuities

Ordinary

Annuities

10101010

10-36


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

focus?

On what information should we

focus?

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

Back to Offer Comparison Back to Offer Comparison

Ordinary

Annuities

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Annuities

10101010

10-37

McGraw-Hill Ryerson©Preparing Time Lines

Mr. BelcherMs. Armstrong

$20,000 down

plus $5000 every quarter for five years

$25,000 down

plus a $100,000 lump sum payment five

years from nowFocal 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.

Ordinary

Annuities

Ordinary

Annuities

10101010

10-38


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

$20,000

Years0 1 2 3 4

Time Lines

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

$25,000 down plus a $100,000 lump sum payment five years from nowAA

BB

$25,000

$20,000

Ms. Armstrong

Mr.Belcher$5000 every quarter

5

$100,000

Ordinary

Annuities

Ordinary

Annuities

10101010

10-39


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.


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

1 5

25,000

PV= 78352.692 103,352.62 today’s value of Ms. A’s

total offer

today’s value of Ms. A’s

total offer

Step 2…Step 2…

0

Ordinary

Annuities

Ordinary

Annuities

10101010

10-40


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

4

1

5 0

4500

20

P/Y = 4 C/Y = 1 0PV = 79,376.93

20000

99,376.93 today’s value of

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.


Ordinary

Annuities

Ordinary

Annuities

10101010

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

Difference in Offers

Ordinary

Annuities

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10101010

10-42


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!


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


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

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

Calculating the Original Loan

and a Subsequent Balance

Calculating the Original Loan

and a Subsequent Balance

LO-4LO-4

Ordinary

Annuities

Ordinary

Annuities

10101010

10-43


Original Principal = PV of all 60 payments

PMT = FV = n = i = c =249.10 0 5*12 = 60 .08/12 1

12

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,


a) What was the original principal

amount of the loan?

b) What is the balance owed just after the twentieth

payment?

The required





amount of the loan?


payment?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-44


Balance after 20 payments = PV

of 40 payments leftPMT = FV = n = i =249.10 0 60 - 20 = 40 .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





amount of the loan?


payment?

The required





amount of the loan?


payment?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-45


A Deferred Annuity may be viewed as an ordinary annuity that does not begin until a time interval (named the period of deferral)

has passed

LO-5LO-5

Ordinary

Annuities

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10101010

10-46


Deferred AnnuitiesDeferred Annuities

A Deferred Annuity may be viewed as

an ordinary annuity that does not begin

until a time interval (named the period

of deferral) has passed

A Deferred Annuity may be viewed as

an ordinary annuity 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:Two-step procedure to find PV:

Calculate the present value, PV1,

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

Ordinary

Annuities

Ordinary

Annuities

10101010

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.

Solve…

Ordinary

Annuities

Ordinary

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10-48


$500 $500 $500 $500

…of the Annuity

Present Value of a

Deferred Annuity

Present Value of a

Deferred Annuity

10 11 12 13 14 Months0

PVPV

Step 1 – Determine PV of Annuity 10 months from now

Hint: (Use Compound Discount) Step 2 - Discount for 10 months to get today’s Loan ValueStep 2 - Discount for 10 months to get today’s Loan Value

Ordinary

Annuities

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10-49


…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.

12

0

0 8

4

10

PV = 1967.11 FV = - 1967.11PV = 1840.65 value 10 months from now

value 10 months from now

loan value today

loan value today

500

Ordinary

Annuities

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10101010

10-50



differs from



differs from


e.g. A typical Canadian mortgage has Monthly payments, but the interest is


Using calculators…Using calculators…

LO-6LO-6

Ordinary

Annuities

Ordinary

Annuities

10101010

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

Ordinary

Annuities

Ordinary

Annuities

10101010

10-52


We can input the number of compoundings per year into the

financial calculator. This can be performed by using

the symbolTo access this symbol use:

…and you will see

Ordinary

Annuities

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10101010

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 Payments per Year

… represents the number of Compoundings per Year

To access use:

Note: You can override these values by entering new ones!

…Example…Example

Appearsautomatically

Appearsautomatically

Ordinary

Annuities

Ordinary

Annuities

10101010

10-54


12

2

P/Y = 12.00C/Y = 12.00

UsingC/Y = 2.00

Adding New Formulae

Typical Canadian mortgageInterest is


and payments are each month.

Typical Canadian mortgageInterest is



Ordinary

Annuities

Ordinary

Annuities

10101010

10-55


to calculate the equivalent periodic rate that matches the payment interval

C =

number of interest compoundings per yearnumber of payments per year

Use c to determine i2 Step 2Step 2

Use i2 = (1+i)c - 1

Use this equivalent periodic rate as the value for “i”

in the appropriate simple annuity formula

Step 3

Step 3

…Example…Example

Step 1Step 1 Determine the number of Interest periods per compounding interval

Adding New FormulaeAdding New Formulae

Ordinary

Annuities

Ordinary

Annuities

10101010

10-56


Typical Canadian mortgage

6% Interest is compounded

semi-annually and

payments are each month.

Find C and i2.



semi-annually and

payments are each month.

Find C and i2.

C =


2 12

0.166666

Step 1Step 1 To determine the number of Interest periods per compounding interval

= C


Ordinary

Annuities

Ordinary

Annuities

10101010

10-57



i2 = (1+i)c - 1

i2 = (1+ .06/2).16666 -1Typical

Canadian mortgage


semi-annually


Find C and i2.



semi-annually


Find C and i2.

1.03

1

0.166666 = i2 1.0049 0.0049

…another example…another example

Ordinary

Annuities

Ordinary

Annuities

10101010

10-58


5% interest is

compounded monthly

and payments

are each week

5% interest is

compounded monthly

and payments

are each week

MortgageMortgage

Step 1Step 1 To determine the number of compoundings

C =


12 52

0.23076 = C


Ordinary

Annuities

Ordinary

Annuities

10101010

10-59


1


i2 = (1+i)c - 1

i2 = (1+ .05/12).2308 -1

1

= i2

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

MortgageMortgage

0.230769 1.00096 0.00096

…another example…another example

Ordinary

Annuities

Ordinary

Annuities

10101010

10-60


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 payment interval differs from


CriteriaCriteria

As the Criteria have been met, therefore,

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

we need to determine C

Ordinary

Annuities

Ordinary

Annuities

10101010

10-61


Find i2 Step 2Step 2 i2 = (1+i)c - 1

i2 =

1.035

1

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.

i2 =

Step 1Step 1 Find c

Use i2Step 3

Step 3

1.00575 0.00575

2 12

Ordinary

Annuities

Ordinary

Annuities

10101010

10-62



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%



3 years.

PMT = PV = n = i = c = i2 =

50 0 3*12 = 36.07/2 2/12 = .16666 0.00575

1

50

36

1

0.00575 1.00575 1.229255

Use i2 in the appropriate formulaStep 3

Step 3

0.229255 39.8702 1993.51

Solve…

Ordinary

Annuities

Ordinary

Annuities

10101010

10-63


P/Y = 12 C/Y = 12C/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%



3 years.

12

2

50

0

36

7

0FV = 1993.51

Ordinary

Annuities

Ordinary

Annuities

10101010

10-64


…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 i2 should be saved in

memory as soon you calculate it! it later!

Ordinary

Annuities

Ordinary

Annuities

10101010

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?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-66


…of the Annuity

1 2 3 14 40

FV1FV1

Step 2 – Determine FV using compound interest

FV2FV2

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

10 years after the last deposit?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-67


Step 1 – Determine FV1 of Annuity 10 years from now

1

1

6 0

4

P/Y = 1.00C/Y = 1.00 value at end of 4 years

value at end of 4 years

Step 2…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?



interest





last deposit?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-68


0

10


Step 2 – Determine FV2 using compound interest

FV = 4374.62 FV = 7834.27 value 14 years from now

value 14 years from now



interest





last deposit?



interest





last deposit?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-69



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



Step 2…Step 2…



interest





last deposit?



interest





last deposit?

Ordinary

Annuities

Ordinary

Annuities

10101010

10-70


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


What will be the balance in the account 10 years after the last deposit?

n = i =4374.62 0.06PV = 10

1.262477 0.262477 4374.62 value 14 years from now

value 14 years from now 1.1708477 7834.27

FV = PV(1 + i)nFormula Formula

Ordinary

Annuities

Ordinary

Annuities

10101010

10-71


How much more interest will Reid David accumulate over the 14 years if his

account earns 6%

compounded daily?

1

365

1000

0

4

6

P/Y = 10 value at end of 4 years

value at end of 4 yearsC/Y = 1 C/Y = 365

Step 1 – Determine FV of Annuity 4 years from now

0FV = 4386.52

Ordinary

Annuities

Ordinary

Annuities

10101010

10-72


0 3650365

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 nowP/Y = 1P/Y = 3650FV = 7992.37

Step 2 – Determine FV in 10 years using compound interest

Ordinary

Annuities

Ordinary

Annuities

10101010

10-73


InterestInterest

$7,992.37$7,992.37 $7,834.27$7,834.27

Ordinary

Annuities

Ordinary

Annuities

10101010

10-74


This completes Chapter 10This completes Chapter 10

Education

Business Mathematics Jerome Chapter 10