1MATH 1053 â€“

Quantitative Methods for Business

Week 2 â€“

Time Value of Money

Annuities and their

Applications

1

2Course outline

Time value

of money

Annuities &

Net Present Value

Linear

programming

Making Good

Business Decisions

Mathematics for Business

Simple &

compound

interest

Percentages

and

proportions

Break-even

analysis

Sampling,

data displays

& elementary

probability

Correlation

& Linear

Regression

Hypothesis

TestingStatistics for BusinessCLT &

confidence

intervalsDescriptive Statistics

of a sample

Statistical Inference

from a sampleNormal

distributionSummary

measures

School of Info. Tech. & Mathematical Sciences 2Week 2

2

3Topics to be covered

ï° Working with multiple

cash flows:

ï® Annuities

ï° Present value

ï° Future value

ï° Payment size

ï® Sinking funds

ï® Amortisation

ï® Net Present Value

ï° Case Studies

School of Info. Tech & Mathematical Sciences 33

4School of Info. Tech & Mathematical Sciences

Case Study 1

Set for Life!

â€œImagine waking up one day finding

that you were Set for Life, and that youâ€™d

won $20,000 each and every month for

20 years. Imagine the possibilities â€¦â€

4Would you be better off with $20,000 every

month? How much would you really win?

ï° Youâ€™ve just won Set for Life!

ï° You can take holidays, drive luxury cars,

start your own business or invest the

income and grow your wealth!

4

5Case study 2

Whatâ€™s the best option?

School of Info. Tech & Mathematical Sciences

Samsung 65â€ Full HD LED LCD

3D Capable SMART TV

NO DEPOSIT

12 MONTHS INTEREST FREE*

with monthly repayments *Offer available on advertised or ticketed price. Minimum financed amount $825.

Establishment fee ($35 for new accounts), account service fee (currently $2.95

per month) and other fees and charges are payable.

55

6Annuity

ï° Series of equal payments

ï® often made under contract

ï® paid at equal intervals

(e.g. quarterly or monthly)

ï® from an agreed date

ï® for a specified period of time.

ï° Also the sum of money that

makes a periodic payment.

ï° Examples:

ï® Mortgage or home loan

repayments

ï® Hire purchase repayments

ï® Insurance premiums

ï® Body corporate sinking

fund

ï® Lease payments on cars.

School of Info. Tech & Mathematical Sciences

ï° In this course, all annuities are simple and ordinary :

ï® Interest is compounded at the same times as the annuity

payments.

ï® The first payment is made at the end of the first period.

66

7Lecture example 1:

Simple ordinary annuity

ï° Suppose you set aside $100 at the end of each year for

three years. What will the accumulated amount be at the

end of three years?

ï° Assume compound interest at 10% per annum. ï° Using what you learned last week:

0 1 2 3

$100 $100 $100

Use with each payment:

n ï€½1

n ï€½ 2 100 ï€¨1 1ï€© $121 2 ï‚´ . ï€½

100 ï€¨1 1ï€© $110 1 ï‚´ . ï€½

100 ï€¨1 1ï€© $100 0 ï‚´ . ï€½

S = $331

ï€¨ ï€©

n M ï€½ P ï‚´ 1ï€« i

School of Info. Tech & Mathematical Sciences 77

8School of Info. Tech & Mathematical Sciences

Future value of an ordinary annuity

ï° where:

S = future or accumulated value

R = annuity payment per period

i = interest rate per period

n = number of payments

0 1 2 â€¦ n â€“ 1 n

R R â€¦ R R

S ï€½ ?

88

9School of Info. Tech & Mathematical Sciences

Lecture example 1 revisited

ï€¨ ï€© ï€¨ ï€© 100 3.31 $331

0.10

1.10 1

100 1 1

3

10%

$100

3

ïƒº ï€½ ï‚´ ï€½

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€

ïƒº ï€½ ï‚´

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€« ï€

ï€½ ï‚´

ï€½

ï€½

ï€½

i

i

S R

n

i

R

n

9Same answer!

ï° Suppose you set aside $100 at the end of each year for

three years. What will the accumulated amount be at the

end of three years?

ï° Assume compound interest at 10% per annum. ï° Using the future value formula:

ð‘† = ð‘… Ã—

1 + ð‘– à¯¡ âˆ’ 1

ð‘– = 100 Ã—

1.10 à¬· âˆ’ 1

0.10 = 100 Ã— 3.31 = $3319

10Summary: Future Value

ï€¨ ï€©

n M ï€½P ï‚´ 1ï€« i

Compound Interest Annuities

M = future value

P = present value

i = interest rate per period

n = number of compounding periods

S = future value

R = annuity payment per periodi = interest rate per period

n = number of payments

0 1 2 â€¦ n

P M

0 1 2 â€¦ nS

10A

R R … R

à¯¡

10

11School of Info. Tech & Mathematical Sciences

ï° Suppose you set aside $100 at the end of the year for

three years. What is the value today of those three

deposits?

ï® Assume compound interest at 10% per annum.

0 1 2 3

$100 $100 $100

Use with each payment: P ï€½ M ï‚´ 1ï€«i ï€¨ ï€©ï€n

n ï€½1

n ï€½ 2

100 ï€¨1 1ï€© $75.13 3

ï‚´ ï€½

ï€

.

100 ï€¨1 1ï€© $82.64 2

ï‚´ ï€½

ï€

.

100 ï€¨1 1ï€© $90.91 1

ï‚´ ï€½

ï€

. A = $248.69

n ï€½ 3

11Lecture example 2:

Simple ordinary annuity

11

12School of Info. Tech & Mathematical Sciences

Present (discounted) value of an

ordinary annuity

ï° where:

A = present value

R = annuity payment per period

i = interest rate per period

n = number of payments

0 1 2 â€¦ n â€“ 1 n

R R â€¦ R R

12A ï€½ ?

12

13School of Info. Tech & Mathematical Sciences

Lecture example 2 revisited

ï° Using the present value formula:

ï€¨ ï€© ï€¨ ï€© 100 2.48 $248.69

0.10

1 1.10 100 1 1

3

10%

$100

3

ïƒº ï€½ ï‚´ ï€½

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€

ïƒº ï€½ ï‚´

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€ ï€«

ï€½ ï‚´

ï€½

ï€½

ï€½

ï€ ï€

i

i

A R

n

i

R

n

13Same answer!

ï° Suppose you set aside $100 at the end of the year for three

years. What is the value today of those three deposits?

ï® Assume compound interest at 10% per annum.

( ) ( )

ð´ = ð‘… Ã—

1 âˆ’ 1 + ð‘– à¬¿à¯¡

ð‘–

= ð‘… Ã—

1 âˆ’ 1.10 à¬¿à¬·

0.10 = 100 Ã— 2.48 = $248.6913

14Summary: Present Value

P = present value

M = future value

i = interest rate per period

n = number of compounding periods

A = present value

R = annuity payment per periodi = interest rate per period

n = number of payments

Compound Interest Annuities

0 1 2 â€¦ n

P

R R â€¦ RS

0 1 2 â€¦ nA

P ï€½ M ï‚´ 1ï€«i ï€¨ ï€©ï€n

M

14à¬¿à¯¡14

15Lecture exercise 1

ï° An investment pays $50 at the end of every 6 months for 15 years. ï° What is the value of all the payments today if money can earn

8.5% per year compounded semi-annually?

School of Mathematics and Statistics 15The value today of all the payments is

15

16Have a go!

ï° Josh deposits $300 every three months into an investment account

that pays interest at 6% per year compounded quarterly.

ï° How much will Josh have in his account at the end of four years?School of Mathematics and Statistics 16Josh will have in his account at the end of four years.

16

17Case study 1

Set for Life!

ï° Channel 7â€™s Sunrise asked exactly how much a winner will be paid,

over the 20 year horizon.

ï° Kochie said â€œIâ€™ve done the mathsâ€ and her face changed as below.

What was discussed that had her so worried?

School of Info. Tech & Mathematical Sciences 1717

18Case study 1

Set for Life!

ï° If $20,000 in 20 yearsâ€™ time is worth $11,000 in todayâ€™s terms, this

suggests an interest rate of about 3% p.a. compounded monthly.

ï° How much do you win in total over 20 years? Is it $4.8 million

(20,000 x 12 x 20)?

R ï€½ $20,000

i ï€½ 3% = 0.03/12 ï€½ 0.0025

n ï€½12 ï‚´ 20 ï€½ 240

Interpretation:

ï° If the winner received

a single equivalent

payoff today, it would

be $3.6 million.

ï° The Lottery

Commission needs to

set aside only $3.6

million today to pay

the prize money!

School of Info. Tech & Mathematical Sciences 18ï€¨ ï€©

ï€¨ ï€©

20,000 180.3109 $3,606,218

0.0025

1 1.0025 20,000

(1 1 )

240

ï€½ ï‚´ ï€½

ï€

ï€½ ï‚´

ïƒº

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€ ï€«

ï€½ ï‚´

ï€

ï€

i

i

A R

n

( )

18

19Saving up or borrowing â€“ how much

is enough?

School of Info. Tech & Mathematical Sciences

ï° You want to have $12,000 one year from now.

ï° Should you set aside $1,000 per month?

ï° You borrow $12,000 today for a year.

ï° Will $1,000 per month be enough to pay it off?

1919

20Sinking Funds vs Amortisation

Sinking Funds Amortisation

Want to accumulate a

nominated amount of money

Want to discharge a debt

School of Info. Tech & Mathematical Sciences 2020

21School of Info. Tech & Mathematical Sciences

Sinking fund

ï° Periodic payments made so as to accumulate a

nominated amount of money in a specified period.

ï° where:

S = future value (target amount)

R = annuity payment per period

i = interest rate per period

n = number of payments

0 1 2 â€¦ n â€“ 1 n

R R â€¦ R R ?21S

21

22School of Info. Tech & Mathematical Sciences

Amortisation

ï° A steady stream of even payments (constant

dollar amount) over the life of a loan.

ï° Each period you pay interest and repay some of

the principal.

ï® E.g. car loans, home loans.

2222

23School of Info. Tech & Mathematical Sciences

Amortisation

ï° Periodic payments that will discharge a debt :

ï° where:

A = present value (borrowed amount)

R = annuity payment per period

i = interest rate per period

n = number of payments

0 1 2 â€¦ n â€“ 1 n

R R â€¦ R R ?23A

23

24School of Info. Tech & Mathematical Sciences

Lecture example 5

ï° If you take out a $55,000 car

loan for 10 years, what will

the monthly repayments be?

ï° The interest rate on this loan

is set at 8.5% per annum

compounded monthly.

ï° How much will you still owe at the end of two

years?

2424

25School of Info. Tech & Mathematical Sciences

Lecture example 5 solution

The monthly repayment will be $681.92.

2525

26School of Info. Tech & Mathematical Sciences

Lecture example 5 continued

Outstanding Balance = PV(remaining payments)

8 12 96

0 007083

12

0 085

$681 91

ï€½ ï‚´ ï€½

ï€½ ï€½

ï€½

n

i

R

.

.

.

So there will be $47,381.44 still owing at the end of two years.

26à¬¿à¯¡ à¬¿à¬½à¬º26

27Amortisation schedule

ï° Table detailing the effect of each periodic payment

on the loan balance.

Calculations:

ï° Interest Amount = Interest rate per period x Opening Balanceï° Principal Amount = Payment Amount â€“ Interest Amount

ï° Closing Balance = Opening Balance â€“ Principal Amount

ï° New Opening Balance = Previous Closing Balance

School of Info. Tech & Mathematical Sciences 2727

28Amortisation schedule:

Our own â€˜loan calculatorâ€™

Total = $55,000

28

29Our own â€˜loan calculatorâ€™

School of Info. Tech & Mathematical Sciences

Outstanding balance decreases slowly over time.

29$-

$10,000.00

$20,000.00

$30,000.00

$40,000.00

$50,000.00

$60,000.00

0 10 20 30 40 50 60 70 80 90 100 110 120 Dollar amount

Month

Outstanding Balance

29

30Our own â€˜loan calculatorâ€™

School of Info. Tech & Mathematical Sciences

Initially, repayments go almost entirely towards interest.

3030

31Case study 2

How much is that TV?

ï° No deposit

ï° 12 months interest free

ï° Monthly repayments

ï° $35 establishment fee

ï° $2.95 monthly account keeping feeSchool of Info. Tech & Mathematical Sciences

ï° What cash amount should the store be willing to accept

instead of the interest-free plan on this TV?

ï° The store can use surplus cash to pay down the balance

on its operating loan on which interest accrues at 11.5%

per year compounded monthly.

3131

32Case study 2

How much is that TV?

ï° Monthly repayment = $4,999/12 = $416.58

School of Info. Tech & Mathematical Sciences

0 1 2 â€¦ 11 12 (months)

$35 $416.58 $416.58 â€¦ $416.58 $416.58$2.95 $2.95 â€¦ $2.95 $2.95Now

No-interest plan

This is an annuity.

How much is it worth?

3232

33Case study 2

How much is that TV?

ï° The cash amount the store should be willing to accept is

the present value of all the no-interest plan cash flows:

School of Info. Tech & Mathematical Sciences

12 1 12

0 009583

12

0 115

416 58 2 95 $419 53

ï€½ ï‚´ ï€½

ï€½ ï€½

ï€½ ï€« ï€½

n

.

.

i

R . . .

PV = Establishment fee + A = $4,734.31 + $35

= $4,769.31 Equivalent cash price

33à¬¿à¯¡ à¬¿à¬µà¬¶

33

34School of Info. Tech & Mathematical Sciences 3434

35Net Present Value (NPV)

NPV Investment Criterion:

ï® Accept the investment if NPV >0

ï® Reject the investment if NPV < 0

School of Info. Tech & Mathematical Sciences

NPV = ï€ Present value of

cash inflows

Present value of

cash outflows

3535

36Lecture example 6

ï° A firm is contemplating the purchase of a

$10,000 machine that would reduce labour costs

by $3,000 each year for the next 4 years.

ï° The firm expects to sell the machine for $1,000

at the end of four years.

ï° Should the machine be purchased if the firmâ€™s

cost of capital is 15% compounded annually?

School of Info. Tech & Mathematical Sciences 3636

37Lecture example 6 solution

School of Info. Tech & Mathematical Sciences

0 1 2 3 4

($10,000)

$1,000

$3,000 $3,000 $3,000 $3,000

37$1000

4

0 15

$4 000

S ,

n

i .

R ,

ï€½

ï€½

ï€½

ï€½

M

$3,000

ï€¨ ï€©

ï€¨ ï€©n

n

i

S

i

i NPV R

ï€«

ïƒº ï€«

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€ ï€«

ï€½ ï€ ï€« ï‚´

ï€

1

1 1

Initial investment ( ) MNPV ï€½ ï€10,000 ï€«3, 000 ï‚´

1ï€ï€¨ ï€© 1.15 ï€4

0.15

ïƒ©

ïƒ«

ïƒª

ïƒ¹

ïƒ»

ïƒºï€«

1, 000

ï€¨ ï€© 1.15 4

ï€½ ï€10, 000 ï€«8,564.9350 ï€« 571.7532

ï€½ ï€$863.31

( )

37

38Lecture example 6 solution

ï° Decision:

ï® Since the NPV < 0, the machine should not be

purchased.

ï° Interpretation:

ï® Investing in the machine will cost the firm $863.31 in

the long run, and so the purchase will not pay for itself.

School of Info. Tech & Mathematical Sciences 3838

39NPV function in Excel â€“ Itâ€™s wrong!!!

School of Info. Tech & Mathematical Sciences 3939

40Case study 3

Should MotionMedia invest in an App?ï° MotionMedia want to invest in developing a

new iPhone app that streams live TV and is

designed to â€˜Take TV With Youâ€™.

ï° The initial outlay for design and filming is $80,000

now and a second payment of $50,000, in one yearâ€™s time. ï° Annual operating profits of $28,000 per year will be generated for

the next 10 years, through sponsorship.

ï° Technology-related updates costing ,000 will be required after

5 years. In 10 years time, the product resale value would be $70,000. School of Info. Tech & Mathematical Sciences 40Should MotionMedia invest in developing the iPhone app if its

cost of capital is 15% compounded annually?

40

41Case study 3

Should MotionMedia invest in an App?School of Info. Tech & Mathematical Sciences

0 1 2 3 4 5 6 7 8 9 10 (yrs)

(80) (50)

28 28 28 28 28 28 28 28 28 28 ($â€™000)

70 ($â€™000)

(12) ($â€™000)

ï€¨ ï€©

ï€¨ ï€¨ ï€© ï€©

ï€¨ ï€©

$28,384.07

80,000 43,478.2609 5,966.1208 140,525.5000 17,302.9294

1.15

70,000

0.15

1 1.15 28,000

1.15

12,000

1.15

50,000 80,000 10

10

5

ï€½

ï€½ ï€ ï€ ï€ ï€« ï€«

ïƒº ï€«

ïƒ»

ïƒ¹

ïƒª

ïƒ«

ïƒ© ï€

ï€½ ï€ ï€ ï€ ï€« ï‚´

ï€

NPV

4141

42ï° Decision:

ï® Since the NPV > 0, MotionMedia should invest in the

iPhone app.

ï° Interpretation:

ï® Additional profits from producing the iPhone app will

be more than enough to repay the initial outlay and

subsequent technology update.

ï® Given MotionMediaâ€™s cost of capital, the investment

produces an operating surplus of $28,384.

School of Info. Tech & Mathematical Sciences 42Case study 3

Should MotionMedia invest in an App?42

43School of Info. Tech & Mathematical Sciences

Next week

ï° Algebra in business

ï° Making good business decisions (linear equations)

ï° Break-even analysis

4343

## Get Professional Assignment Help Cheaply

Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

## Why Choose Our Academic Writing Service?

- Plagiarism free papers
- Timely delivery
- Any deadline
- Skilled, Experienced Native English Writers
- Subject-relevant academic writer
- Adherence to paper instructions
- Ability to tackle bulk assignments
- Reasonable prices
- 24/7 Customer Support
- Get superb grades consistently

## Online Academic Help With Different Subjects

### Literature

Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.

### Finance

Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.

### Computer science

Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!

### Psychology

While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.

### Engineering

Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.

### Nursing

In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.

### Sociology

Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.

### Business

We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!

### Statistics

We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.

### Law

Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.

## What discipline/subjects do you deal in?

We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.

## Are your writers competent enough to handle my paper?

Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.

## What if I don’t like the paper?

There is a very low likelihood that you won’t like the paper.

### Reasons being:

- When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
- We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.

### In the event that you don’t like your paper:

- The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
- We will have a different writer write the paper from scratch.
- Last resort, if the above does not work, we will refund your money.

## Will the professor find out I didn’t write the paper myself?

Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.

## What if the paper is plagiarized?

We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.

## When will I get my paper?

You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.

## Will anyone find out that I used your services?

We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. **Noone will ever know that you used our assignment help services. ** It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.

## How our Assignment Help Service Works

#### 1. Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

#### 2. Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

#### 3. Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

#### 4. Download the paper

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

** PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A PERFECT SCORE!!! **