rosscorporatefinance_cdn6e_chapter5

7/30/2019 RossCorporateFinance_Cdn6e_Chapter5

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Prepared by

Professor Wei Wang

Queens University

2011 McGrawHi ll Ryerson Limited

The Time Value of

Money

Chapter Five


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5-1

2011 McGrawHi ll Ryerson Limited 2011 McGrawHi ll Ryerson Limited

Chapter Outline

5.1 The One-Period Case

5.2 The Multiperiod Case

5.3 Compounding Periods

5.4 Simplifications

5.5 What Is a Firm Worth?

5.6 Summary and Conclusions


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5-2


The One-Period Case: Future Value LO5.1

If you were to invest $10,000 at 5-percent interest forone year, your investment would grow to $10,500

$500 would be interest ($10,000 .05)

$10,000 is the principal repayment ($10,000 1)$10,500 is the total due. It can be calculated as:

$10,500 = $10,000(1.05).

The total amount due at the end of the investment is

called theFuture Value (FV).


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5-3


The One-Period Case: Future Value LO5.1

In the one-period case, the formula forFVcanbe written as:

FV= C0(1 + r)

Where C0 is cash flow at date 0 and ris the

appropriate interest rate.

FV= $10,500

Year 0 1

C0 = $10,000$10,000 1.05

C0(1 + r)


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5-4


The One-Period Case: Present Value LO5.1

If you were to be promised $10,000 due in one yearwhen interest rates are at 5-percent, your investment

would be worth $9,523.81 in todays dollars.

05.1000,10$81.523,9$

The amount that a borrower would need to set aside

today to be able to meet the promised payment of

$10,000 in one year is call thePresent Value (PV) of

$10,000.

Note that $10,000 = $9,523.81(1.05).


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The One-Period Case: Present Value LO5.1

In the one-period case, the formula forPVcanbe written as:

r

CPV

11

Where C1 is cash flow at date 1 and ris

the appropriate interest rate.

C1 = $10,000

Year 0 1

PV= $9,523.81$10,000/1.05

C1/(1 + r)


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The One-Period Case: Net Present ValueLO5.1

The Net Present Value (NPV) of an investmentis the present value of the expected cash flows,

less the cost of the investment.

Suppose an investment that promises to pay$10,000 in one year is offered for sale for

$9,500. Your interest rate is 5%. Should you

buy?

81.23$

81.523,9$500,9$

05.1

000,10$

500,9$

NPV

NPV

NPV

Yes!


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5-7


The One-Period Case: Net Present Value LO5.1

In the one-period case, the formula forNPVcan be

written as:

PVCostNPV

If we had notundertaken the positiveNPVprojectconsidered on the last slide, and instead invested our

$9,500 elsewhere at 5-percent, ourFVwould be less

than the $10,000 the investment promised and we

would be unambiguously worse off inFVterms aswell:

$9,500(1.05) = $9,975 < $10,000.


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5-8


The Multiperiod Case: Future Value LO5.2

The general formula for the future value of aninvestment over many periods can be writtenas:

FV= C0(1 + r)T

Where

C0 is cash flow at date 0,

ris the appropriate interest rate, and

Tis the number of periods over which the cash isinvested.


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5-9


The Multiperiod Case: Future ValueLO5.2

Suppose that Jay Ritter invested in the initialpublic offering of the Modigliani company.

Modigliani pays a current dividend of $1.10,

which is expected to grow at 40-percent peryear for the next five years.

What will the dividend be in five years?

FV= C0(1 + r)T

$5.92 = $1.10(1.40)5


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Future Value and Compounding LO5.2

Notice that the dividend in year five, $5.92, isconsiderably higher than the sum of the

original dividend plus five increases of 40-

percent on the original $1.10 dividend:

$5.92 > $1.10 + 5[$1.10.40] = $3.30

This is due to compounding.


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Future Value and Compounding LO5.2

0 1 2 3 4 5

10.1$

3)40.1(10.1$

02.3$

)40.1(10.1$

54.1$

2)40.1(10.1$

16.2$

5)40.1(10.1$

92.5$

4)40.1(10.1$

23.4$


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5-12


Present Value and Discounting LO5.2

How much would an investor have to setaside today in order to have $20,000 five

years from now if the current rate is 15%?

0 1 2 3 4 5

$20,000PV

5)15.1(

000,20$53.943,9$


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5-13


How Long is the Wait? LO5.2

If we deposit $5,000 today in an account paying 10%,how long does it take to grow to $10,000?

TrCFV )1(0 T)10.1(000,5$000,10$

2000,5$

000,10$)10.1( T

2ln)10.1ln( T

years27.7

0953.0

6931.0

)10.1ln(

2lnT


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5-14


What Rate Is Enough? LO5.2

Assume the total cost of a university education will be

$50,000 when your child enters university in 12 years.

You have $5,000 to invest today. What rate of interest

must you earn on your investment to cover the cost of

your childs education? About 21.15%.TrCFV )1(0

12)1(000,5$000,50$ r

10000,5$000,50$)1( 12 r 12110)1( r

2115.12115.1110 121 r


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Calculator Keys LO5.2

Texas Instruments BA-II Plus

FV = future value

PV = present value

I/Y = periodic interest rate

P/Y must equal 1 for the I/Y to be the periodicrate

Interest is entered as a percent, not a decimal

N = number of periods Remember to clear the registers (CLR TVM) after

each problem


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Calculator Keys LO5.2

Hewlett Packard 10B

FV = future value

PV = present value

I/Y = periodic interest rate

P/Y must equal 1 for the I/Y to be the periodicrate

Interest is entered as a percent, not a decimal

N = number of periods Remember to clear the registers (Shift C ALL)

after each problem

Other calculators are similar in format


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5-17


Multiple Cash Flows LO5.2

Consider an investment that pays $200 one

year from now, with cash flows increasingby $200 per year through year 4. If the

interest rate is 12%, what is the present value

of this stream of cash flows? If the issuer offers this investment for

$1,500, should you purchase it?


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Multiple Cash Flows LO5.2

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Present Value < Cost Do Not Purchase


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Valuing Lumpy Cash Flows LO5.2

First, set your calculator to 1 payment per year.

Then, use the cash flow menu:

CF2

CF1

F2

F1

CF0

1

200

1

1,432.93

0

400

I

NPV

12

CF4

CF3

F4

F3 1

600

1

800


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Compounding Periods LO5.3

Compounding an investment m times a year forTyears provides for future value of wealth:

Tm

m

rCFV

10

For example, if you invest $50 for 3 years at

12% compounded semi-annually, your

investment will grow to

93.70$)06.1(50$2

12.150$ 6

32

FV


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5-21


Effective Annual Interest Rates LO5.3

A reasonable question to ask in the aboveexample is what is the effective annualrate of

interest on that investment?

The Effective Annual Interest Rate (EAR) is the

annual rate that would give us the same end-of-investment wealth after 3 years:

93.70$)06.1(50$)212.1(50$ 632 FV

93.70$)1(50$ 3 EAR


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Effective Annual Interest Rates (cont.)LO5.3

So, investing at 12.36% compounded annually

is the same as investing at 12% compounded

semiannually.

93.70$)1(50$3

EARFV

50$

93.70$)1( 3 EAR

1236.150$

93.70$31

EAR


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Effective Annual Interest Rates (cont) LO5.3

Find the Effective Annual Rate (EAR) of aloan with an annual percentage rate (APR) of

18% compounded monthly.

What we have is a loan with a monthlyinterest rate rate of 1 percent.

This is equivalent to a loan with an annual

interest rate of 19.56 percent

19561817.1)015.1(12

18.11 12

12

mn

m

r


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5-24


EAR on a financial Calculator LO5.3

keys: display: description:

12 [gold] [P/YR] 12.00 Sets 12 P/YR.

[gold] [EFF%] 19.56

Hewlett Packard 10B

18 [gold] [NOM%] 18.00 Sets 18 APR.

keys: description:

[2nd] [ICONV] Opens interest rate conversion menu

[] [EFF=] [CPT] 19.56

Texas Instruments BAII Plus

[][NOM=] 18[ENTER] Sets 18 APR.[] [C/Y=] 12 Sets 12 payments per year

2


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5-25


Continuous Compounding (Advanced) LO5.3

The general formula for the future value of aninvestment compounded continuously over many

periods can be written as:

FV= C0erT

WhereC0 is cash flow at date 0,

ris the stated annual interest rate,

Tis the number of periods over which the cash isinvested, and

e is a transcendental number approximately equalto 2.718. ex is a key on your calculator.

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5.4 Simplifications LO5.4

Perpetuity A constant stream of cash flows that lasts forever.

Growing perpetuity

A stream of cash flows that grows at a constant rate

forever.

Annuity

A stream of constant cash flows that lasts for a fixed

number of periods. Growing annuity

A stream of cash flows that grows at a constant rate for a

fixed number of periods.

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Perpetuity LO5.4

A constant stream of cash flows that lasts forever.

0

1

C

2

C

3

C

The formula for the present value of a perpetuity is:

32 )1()1()1( r

C

r

C

r

CPV

r

CPV

5 28


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Perpetuity: Example LO5.4

What is the value of a British consol that promises to

pay 15 each year, every year until the sun turns

into a red giant and burns the planet to a crisp?

The interest rate is 10-percent.

0

1

15

2

15

3

15

15010.

15PV

5 29


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Growing Perpetuity LO5.4

A growing stream of cash flows that lasts forever.

0

1

C

2

C(1+g)

3

C(1+g)2

The formula for the present value of a growing perpetuity is:

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

CPV

gr

CPV

5 30


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Growing Perpetuity: Example LO5.4

The expected dividend next year is $1.30 and

dividends are expected to grow at 5% forever.

If the discount rate is 10%, what is the value of thispromised dividend stream?

0

1

$1.30

2

$1.30(1.05)

3

$1.30 (1.05)2

00.26$

05.10.

30.1$

PV

5 31


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Annuity LO5.4

A constant stream of cash flows with a fixed maturity.

0 1

C

2

C

3

C

The formula for the present value of an annuity is:

Tr

C

r

C

r

C

r

CPV

)1()1()1()1( 32

T

rr

CPV

)1(

11

T

C

5 32


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Annuity Intuition LO5.4

An annuity is valued as the difference betweentwo perpetuities:

one perpetuity that starts at time 1

less a perpetuity that starts at time T+ 1

0 1

C

2

C

3

C

T

C

Tr

r

C

r

CPV

)1(

5 33


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Annuity: Example LO5.4

If you can afford a $400 monthly car payment, how

much car can you afford if interest rates are 7% on36-month loans?

0 1

$400

2

$400

3

$400

59.954,12$)1207.1(

11

12/07.

400$36

PV

36

$400

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How to Value Annuities with a Calculator LO5.4

First, set your calculator to 12 payments per year.

PMT

I/Y

FV

PV

N

400

7

0

12,954.59

36

PV

Then enter what you know and

solve for what you want.

5 35


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5-35


97.327$)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$4321

4

1

1 t

tPV

0 1 2 3 4 5

$100 $100 $100 $100$323.97$297.22

22.297$

09.1

97.327$

0

PV

What is the present value of a four-year annuity of $100

per year that makes its first payment two years from

today if the discount rate is 9%?

Annuity: Example for Lumpy Cash FlowsLO5.4

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5-36


How to Value Lumpy Cash Flows LO5.4



CF2

CF1

F2

F1

CF0

1

0

4

297.22

0

100

I

NPV

9

5-37


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Annuity: Canadian Mortgages LO5.4

What is special about Canadian Mortgages?

Canadian banks quote the annual interestcompounded semi-annually for mortgages, althoughinterest is calculated (compounded) every month.

The terms of the mortgage are usually renegotiatedduring the term of the mortgage. For example, the

interest of a 25-year mortgage can be negotiated 5years after the initiation of the mortgage.

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5 38


Canadian Mortgages: Example LO5.4

You have negotiated a 25-year, $100,000 mortgage at a

rate of 7.4% per year compounded semi-annually with

the Toronto-Dominion Bank. What is your monthly

payment?

To answer this question, we first have to convert the

quoted mortgage rate, to the effective interest rate

charged each month.

We have:

%5369.7075369.12

074.01

2

EAR

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5 39


We use the present value of annuity formula to findthe monthly payment (PMT):

Canadian Mortgages: Example (Cont.) LO5.4

The effective interest rate charged each month is

given by:

%607369.00607369.1075369.01 121

300)00607369.1(

1

100607369.000,100

PMT

28.725$PMT

5-40


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5 40


Growing Annuity LO5.4

A growing stream of cash flows with a fixed maturity.

0 1

C

The formula for the present value of a growing annuity:

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

2

C(1+g)

3

C(1+g)2

T

C(1+g)T-1

5-41


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5 41


PV of Growing Annuity: Example LO5.4

You are evaluating an income property that is providing

increasing rents. Net rent is received at the end of each year.The first year's rent is expected to be $8,500 and rent is

expected to increase 7% each year. Each payment occur at the

end of the year. What is the present value of the estimated

income stream over the first 5 years if the discount rate is12%?

0 1 2 3 4 5

500,8$

)07.1(500,8$ 2)07.1(500,8$

095,9$ 65.731,9$

3)07.1(500,8$

87.412,10$

4)07.1(500,8$

77.141,11$

$34,706.26

5-42


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5 42


PV of Growing Annuity: Cash Flow Keys LO5.4



$34,706.26

I

NPV

12CF0

CF1

CF2

CF3

CF4

CF5

8,500.00

9,095.00

9,731.65

10,412.87

11,141.77

0

5-43


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PV of Growing Annuity Using TVM Keys LO5.4


PMT

I/Y

FV

PV

N

7,973.93

4.67

0

34,706.26

5

PV

100107.112.1

07.1

500,8

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Why it works LO5.4

The Time Value of Money Keys use thefollowing formula:

T

r

g

gr

CPV

1

11

NN

YI

FV

YIYI

PMTPV

)/1()/1(

11

/

SinceFV= 0, we can ignore the last term.

We want to get to this equation:

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N

g

r

g

r

g

PMT

PV

)11

11(

11

11

1

1

We begin by substituting forPMTand forr111

gr

gPMT1

Nrr

PMT

PV )1(

1

1becomes

Why it works (continued) LO5.4

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N

gr

gr

g

PMT

PV

)1111(

11

111

1

We can now simplify terms:

N

g

r

g

rg

PMTPV

1

1

11

11

1)1(


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N

g

r

g

rg

PMTPV

1

111

11

1)1(

N

r

g

g

g

g

rg

PMTPV

1

11

1

1

1

1)1(

g

g

1

11We continue to simplify terms. Note that:


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N

r

g

g

grg

PMTPV

1

11

1)1(

N

rg

grPMTPV

111

Finally, note that: (1 + r)(1 +g) = rg


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The Result of our Algebrations:

We have proved that we can value growingannuities with our calculator using the followingmodifications:

PMT

I/Y

FV

PV

N

0

PV

100111

gr

g

PMT

1

5-50


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Growing Annuity: Example LO5.4

A retirement plan offers to pay $20,000 per year for 40

years and increase the annual payment by 3-percenteach year. What is the present value at retirement if

the discount rate is 10-percent?

0 1

$20,000

57.121,265$10.1

03.11

03.10.

000,20$40

PV

2

$20,000(1.03)

40

$20,000(1.03)39

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PV of Growing Annuity: BAII Plus

PMT

I/Y

FV

PV

N

19,417.48 =

6.80 =

0

265,121.57

40

PV

A retirement plan offers to pay $20,000 per year for 40 years

and increase the annual payment by 3-percent each year.What is the present value at retirement if the discount rate is10-percent?

20,000

1.03

1.10

1.031 100

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PV of a delayed growing annuity LO5.4

Your firm is about to make its initial public offering of stock

and your job is to estimate the correct offering price.Forecast dividends are as follows.

Year: 1 2 3 4

Dividends pershare

$1.50 $1.65 $1.82 5% growththereafter

If investors demand a 10% return on investments of this risk

level, what price will they be willing to pay?

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Year 0 1 2 3

Cashflow $1.50 $1.65 $1.82

4

$1.821.05

The first step is to draw a timeline.

The second step is to decide on what we know and

what it is we are trying to find.

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Year 0 1 2 3

Cashflow $1.50 $1.65 $1.82 dividend +P

3

PV

of

cashflow

$32.81

22.38$05.10.

05.182.13

P

81.32$)10.1(

22.38$82.1$

)10.1(

65.1$

)10.1(

50.1$320

P

= $1.82 + $38.22

5-55


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What Is a Firm Worth? LO5.5

Conceptually, a firm should be worth thepresent value of the firms cash flows.

The tricky part is determining the size,

timing, and risk of those cash flows.

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Summary and Conclusions LO5.6

Two basic concepts, future value and present valueare introduced in this chapter.

Interest rates are commonly expressed on an annual

basis, but semi-annual, quarterly, monthly and even

continuously compounded interest ratearrangements exist.

The formula for the net present value of an

investment that pays $CforNperiods is:

N

ttN

r

CC

r

C

r

C

r

CCNPV

1

020 )1()1()1()1(

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Summary and Conclusions (continued) LO5.6

We presented four simplifying formulae:

r

CPV :Perpetuity

gr

CPV :PerpetuityGrowing

T

rr

CPV

)1(

11:Annuity

T

r

g

gr

CPV

)1(

11:AnnuityGrowing

5-58


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How do you get to Bay Street?

Practice, practice, practice. Its easy to watch Olympic gymnasts and

convince yourself that you are a leotardpurchase away from a triple back flip.

Its also easy to watch your finance professordo time value of money problems andconvince yourself that you can do them too.

There is no substitute for getting out thecalculator and flogging the keys until you cando these correctly and quickly.

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This is my calculator. This is my friend!

Your financial calculator has two majormenus that you mustbecome familiar with:

The time value of money keys:

N; I/YR; PV; PMT; FV Use this menu to value things with level cash flows,

like annuities e.g. student loans.

It can even be used to value growing annuities.

The cash flow menu CFjet cetera

Use the cash flow menu to value lumpy cash flow

streams.

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Problems (1)

You have $30,000 in student loans that call

for monthly payments over 10 years.$15,000 is financed at an annual percentage rate

(APR) of 7% compounded monthly

$8,000 is financed at an APR of 8% compounded

monthly and

$7,000 at 15% APR compounded monthly

What is the interest rate on your portfolio of

debt?

15,000

30,0007%

8,000

30,0008%

7,000

30,00015%

Hint: dont even think about doing this:

5-61


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Problems (1 Continued)

Find the payment on each loan, add the payments to get your

total monthly payment: $384.16.Set PV = $30,000 and solve for I/YR = 9.25%

PMT

I/Y

FV

PV

N

PV

0

7

120

15,000

174.16

0

8

120

8,000

97.06

0

15

120

7,000

112.93 384.16

30,000

120

0

9.25

+

+

+

+

=

=

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Problems (2)

You are considering the purchase of a prepaid tuition plan

for your 8-year old daughter. She will start university inexactly 10 years, with the first tuition payment of $12,500due at the start of the year. Second year tuition will be$15,000; third year tuition $18,000, and fourth year tuition$22,000. How much money will you have to pay today to

fully fund her tuition expenses? The discount rate is 14%

CF2

CF1

F2

F1

CF0

9

$12,500

1 $14,662.65

0

0

I

NPV

14CF4

CF3

F4

F3 1

$18,000

1

15,000 CF4

F4

$22,000

1

5-63

bl (3)


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Problems (3)

You are thinking of buying a new car. You bought your current

car exactly 3 years ago for $25,000 and financed it an annualpercentage rate of 7% compounded monthly for 60 months.

You need to estimate how much you owe on the loan to

make sure that you can pay it off when you sell the old car.

PMT

I/Y

FV

PV

N

PV

0

7

60

25,000

495.03 PMT

I/Y

FV

PV

N

PV

0

7

24

11,056

495.03 PMT

I/Y

FV

PV

N

25,000

7

36

11,056

495.03

5-64

P bl (4)


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Problems (4)

You have just landed a job and are going to start

saving for a down-payment on a house. You wantto save 20 percent of the purchase price and thenborrow the rest from a bank.

You have an investment that pays 10 percent per yearcompounded monthly. Houses that you like andcan afford currently cost $100,000. Real estate has

been appreciating in price at 5 percent per yearand you expect this trend to continue.

How much should you save every month in order tohave a down payment saved five years fromtoday?

5-65

P bl (4 C i d)


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Problems (4 Continued)

First we estimate that in 5 years, a house that costs

$100,000 today will cost $127,628.16 Next we estimate the monthly payment required to

save up that much in 60 months.

PMT

I/Y

FV

PV

N

100,000

5

5

127,628.16

0 PMT

I/Y

FV

PV

N

0

10

60

$25,525.63 = 0.20$127,628.16

329.63

5-66

Q i k Q i


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Quick Quiz

How is the future value of a single cash flowcomputed?

How is the present value of a series of cash flows

computed.

What is the Net Present Value of an investment?

What is an EAR, and how is it computed?

What is a perpetuity? An annuity?

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