01 Chap 1 Time Value of Money

8/8/2019 01 Chap 1 Time Value of Money

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Chapter 1PrinciplesPrinciplesofof

CorporateCorporate

FinanceFinance

Time Value of Money

Slides by

Matthew Will

Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reservedMcGraw Hill/Irwin


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Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved

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McGraw Hill/Irwin

Topics Covered

Introductionto Present Value and FutureValue

Valuing Long-Lived Assets

Looking for Shortcuts Perpetuities andAnnuities

More Shortcuts Growing Perpetuities

and AnnuitiesCompound Interest & Present Values


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McGraw Hill/Irwin

Present and Future Value

Present Value

Value today of a

future cashflow.

Future Value

Amountto which an

investment will grow

after earning interest


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McGraw Hill/Irwin

Simple and Compound Interest

Compound interest: When money isinvested at compound interest, each interest

payment is reinvested to earn more interest

in subsequent periods.Simple interest: The opportunity to earn

interest on interest is not provided.


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McGraw Hill/Irwin

Simple and Compound Interest

The value of a $100 investment earning 10% annually.


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McGraw Hill/Irwin

Compound Interest

Compound interest versus simple interest. The longerthe funds are invested,thegreaterthe advantage with compound interest. The bottom line shows that

$38.55 must be invested now to obtain $100 after 10 periods. Conversely,the

present value of $100 to be received after 10 years is $38.55.


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McGraw Hill/Irwin

Future Values

Future Value of $100 = FV

FV r t! v $100 ( )1


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McGraw Hill/Irwin

Future Values

FV rt

! v $100 ( )1

Example -FV

What is the future value of $100 if interestis

compounded annually ata rate of 6% for five years?

82.133$)06.1(100$ 5 !v!FV


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McGraw Hill/Irwin

Future Values

FV rt

! v $100 ( )1

Example -FV

What is the future value of $400,000 if interestis

compounded annually ata rate of 5% forone year?

000,4 0$)01(000,400$ 1 !v!FV


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McGraw Hill/Irwin

Discount Factors and Rates

Discount Rate

Interest rate used

to compute

present values offuture cash flows. Discount Factor

Present value of

a $1 futurepayment.


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McGraw Hill/Irwin

Present Value

1actordiscountPV

PVValuePresent

Cv


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McGraw Hill/Irwin

Present Value

Discount Factor = DF = PV of $1

Discount Factors can be used to compute the present value of

any cash flow.

DF r t! 11( )


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McGraw Hill/Irwin

Present Values

Discount Factors can be used to compute

the present value of any cash flow.

DFr

t!

1

1( )

1

11

1 r

CCDFPV

!v!


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McGraw Hill/Irwin

Present Values

ExampleYou justboughta new computer for $3,000. The payment

terms are 2 years same as cash. If you can earn 8% on

yourmoney, how much money should you set aside today

in orde

rto make

the paymen

tw

hen due in

two yea

rs?

572,2$2)08.1(3000

!!PV


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McGraw Hill/Irwin

Present Values

ExampleYou have the opportunity to purchase the baseballhit by

Barry Bonds to break Hank Arrons home run record

(home run # 756). You estimate this baseball will be worth

$2,000,000 wh

en youre

tire a

tth

e end oftwen

ty yea

rs. Ifyou expecta 12% return on your investment, how much

will you pay forthe baseball ?

33,207$20)12.1(000,000,2

!!PV


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McGraw Hill/Irwin

Present Values

Replacing 1 with t allows the formula

to be used for cash flows that exist at any

point intime

t

t

t

r

C

CDFPV )1( !v!


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McGraw Hill/Irwin

Present Values

Example

You willreceive $200 in two years. Ifthe annualrate of

interest is 7.7%, what is the present value ofthe $200?

42.172$2)077.1(200 !!PV


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McGraw Hill/Irwin

Present Values

PVs can be added togetherto evaluate

multiple cash flows.

PVC

r

C

r!

1

1

2

21 1( ) ( )....


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McGraw Hill/Irwin

Present Values

PVs can be added togetherto evaluate

multiple cash flows.

15.26821 )07.1(200

)07.1(

100!!

PV


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McGraw Hill/Irwin

Present Values

Present Value

Year 0

100/1.07

200/1.0772

Total

= $93. 6

= $172. 2

= $265.88

$100

$200

Year

0 1 2


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McGraw Hill/Irwin

Present Values

Giventwo dollars, one received a year from nowand the othertwo years from now,the value of

each is commonly called the Discount Factor.

Assume r1 = 20% and r2 = 7%.

7.

.

2

1

07.1

00.12

20.1

00.11

!!

!!

DF

DF


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McGraw Hill/Irwin

Present Values

Example

Assume thatthe cash flows

from the construction and sale

of an office building is as

follows. Given a 5% required

rate ofreturn, create a present

value worksheet

000,320000,100000,170

2ear1ear0ear


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McGraw Hill/Irwin

Present Values

Example - continued

Assume thatthe cash flows from the construction and sale of an office

building is as follows. Given a 5% requiredrate ofreturn, create a

presentvalue worksheet.

011,25$

2 9,290000,320907.2

238,95000,100952.1

000,170000,1700.10Value

Present

Flow

Cash

Factor

DiscountPeriod

205.1

1

05.11

!

!!

Total


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McGraw Hill/Irwin

Present Values

Present Value

Year 0

-170,000

-100,000/1.05

320,000/1.052

Total

-$170,000

= -$170,000

= $95,238

= $290,2 9

= $25,011

-$100,000

+$320,000

Year

0 1 2

Ex

ample - continuedAssume thatthe cash flows from the construction and sale of an office

building is as follows. Given a 5% requiredrate ofreturn, create a

presentvalue worksheet.


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McGraw Hill/Irwin

Short Cuts

Sometimes there are shortcuts that make itvery easy to calculate the present value of

an assetthat pays off in different periods.

These tools allow us to cutthrough thecalculations quickly.


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McGraw Hill/Irwin

Short Cuts

Perpetuity - Financial concept in which a constantcash flow is theoretically received forever.

r

CPV1

0

ratediscount

flowcashFlowCashofPV

!

!


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McGraw Hill/Irwin

Present Values

Example

What is the presentvalue of $1 billion every year, for all

eternity, if you estimate the perpetual discountrate to be

10%??

billion10$10.0

bil$1!!PV


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McGraw Hill/Irwin

Short Cuts

Annuity - An assetthat pays a fixed sum each year fora specified number of years.

r

CPerpetuity (firstpayment in year 1)

Perpetuity (first payment

in yeart + 1)

Annuity from year

1 to yeart

Asset Year of Payment

1 2..t t + 1

Present Value

trr

C

)1(

1

t

rr

C

r

C

)1(

1


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McGraw Hill/Irwin

Ex

ampleTiburon Autos offers you easy payments of $5,000 per year, atthe end

of each year for5 years. If interestrates are 7%, per year, whatis the

costofthe car?

Present Values

5,000

Year0 1 2 3 5

5,000 5,000 5,000 5,000

20,501NPVTotal

565,307.1/000,5

81,307.1/000,5

081,407.1/000,5

367,407.1/000,5

673,407.1/000,5

5

4

3

2

!

!

!

!

!

!

Present Value at

year 0


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McGraw Hill/Irwin

Short Cuts

Annuity - An assetthat pays a fixed sum eachyear for a specified number of years.

-

v!

trrr

C1

11annuityofPV


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McGraw Hill/Irwin

Annuity Short Cut

Example

You agree to lease a car for 4 years at $300 permonth.

You are notrequiredto pay any money up frontor atthe

end of youragreement. If youropportunity costof capital

is 0.5% permonth, what is the costofthe lease?


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McGraw Hill/Irwin

Annuity Short Cut

Example - continued

You agree to lease a car for4 years at $300 per

month. You are notrequiredto pay any money up

frontor atthe end of youragreement. If your

opportunity cost of capital is 0.5% per month,whatis the cost ofthe lease?

10.774,12$

005.1005.

1

005.

1300CostLease

48

!

-

v!

Cost


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McGraw Hill/Irwin

Annuity Short Cut

Example

The state lottery advertises a jackpotprize of $295.7

million, paid in 25 installments over25 years of $11.828

million peryear, atthe end of each year. If interestrates

are 5.9% what is the true value ofthe lottery prize?

000,600,152$ 059.1059.

1

059.

1828.11ValueLottery

25

!

-

v!

Value


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McGraw Hill/Irwin

FV Annuity Short Cut

Future Value of an Annuity The future value ofan assetthat pays a fixed sum each year for a

specified number of years.

-

v!

r

rC

t

11annuityofFV


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McGraw Hill/Irwin

Annuity Short Cut

Example

What is the future value of $20,000 paid atthe end of each

ofthe following 5 years, assuming your investmentreturns

8% peryear?

332,117$

08.

108.1000,20FV

5

!

-

v!


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McGraw Hill/Irwin

Constant Growth Perpetuity

gr

V

1

0

g = the annual growth rate ofthe

cash flow


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McGraw Hill/Irwin


gr

CPV

!1

0

NOTE: This formula can be used

to value a perpetuity at any point

in time.

gr

CPV tt

!1


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McGraw Hill/Irwin


Example

What is the presentvalue of $1 billion paid atthe end of

every year in perpetuity, assuming a rate ofreturn of 10%

and a constantgrowthrate of 4%?

billion667.16$04.10.

1

0

!

!PV


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McGraw Hill/Irwin

Perpetuities

A three-year stream of cash flows thatgrows atthe rate g is

equal to the difference betweentwo growing perpetuities.


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McGraw Hill/Irwin

Compound Interest

A note on compounding intervals: Aninvestment of $1 at a rate of r per annum

compounded m times a year amounts by the

end ofthe yearto [1 + (r/m)]m

, and theequivalent annually compounded rate of

interest is [1 + (r/m)]m - 1


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McGraw Hill/Irwin

Compound Interest

i ii iii iv v

Periods Interest Value Annually

per per APR after compounded

year period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183


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McGraw Hill/Irwin

Compound Interest

02

4

6

8

1012

14

16

18

0 3 6 912 15 18 21 24 27 30

Number of Years

FV

of$1

10% Simple

10% Compound


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McGraw Hill/Irwin

Compound Interest

Example

Suppose you are offered an automobile loan atan APR of

6% peryear. What does that mean, and what is the true

rate of interest, given monthly payments?


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Copyright 2008 by The McGraw Hill Companies Inc All rights reserved

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McGraw Hill/Irwin

Compound Interest

Example - continued

Suppose you are offered an

automobile loan at an APR of 6% per

year. What does thatmean, and what

is the true rate of interest, given

monthly payments? Assume $10,000loan amount.

%1678.6

78.616,10

)005.1(000,10PmtLoan 12

!

!

v!

APR

Documents

01 Chap 1 Time Value of Money