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

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