Chap 005

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McGraw Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights

Fundamentals

of Corporate

Finance

Sixth Edition

Richard A. Brealey

Stewart C. Myers

Alan J. Marcus

Slides by

Matthew Will

Chapter 5

McGraw Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights

The Time Value of Money



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Topics Covered

Future Values and Compound Interest Present Values

Multiple Cash Flows

Level Cash Flows Perpetuities and Annuities

Effective Annual Interest Rates

Inflation & Time Value



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Future Values

Future Value - Amount to which an investmentwill grow after earning interest.

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the

original investment.



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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Interest Earned Per Year = 100 x .06 = $ 6



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Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years1 2 3 4 5

Interest Earned

Value 100

Future Values

6

106

6

112

6

118

6

124

6

130

Value at the end of Year 5 = $130



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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the

previous year’s balance.

Interest Earned Per Year =Prior Year Balance x .06



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Example - Compound Interest

Interest earned at a rate of 6% for five years onthe previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned

Value 100

Future Values

6

106

6.36

112.36

6.74

119.10

7.15

126.25

7.57

133.82

Value at the end of Year 5 = $133.82



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Future Values

Future Value of $100 = FV

FV r t

= × +$100 ( )1



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Future Values

FV r t

= × +$100 ( )1

Example - FV

What is the future value of $100 if interest is

compounded annually at a rate of 6% for five years?

82.133$)06.1(100$ 5=+×= FV



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0

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years

F V

o f $ 1 0 0

0%

5%

10%

15%

Future Values with Compounding

Interest Rates



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Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1626.Was this a good deal?

trillion

FV

63.140$

)08.1(24$ 382

=

+×=

To answer, determine $24 is worth in the year 2008,

compounded at 8%.

FYI - The value of Manhattan Island land is FYI - The value of Manhattan Island land is

well below this figure.well below this figure.



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Present Values

Present Value

Value today of

a future cash

flow.

Discount Rate

Interest rate usedto compute

present values of

future cash flows.

Discount Factor

Present value of

a $1 future

payment.



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Present Values

Present Va lue = PV

PV =Future Val ue after t periods

(1+r)

t



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Present Values

ExampleYou just bought a new computer for $3,000. The payment

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

money, how much money should you set aside today in order

to make the payment when due in two years?

572,2$2)08.1(

3000

== PV



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Present Values

Discount Factor = DF = PV of $1

Discount Factors can be used to compute the present value of any cash flow.

DF r t =

+1

1( )

Ti V l f M



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The PV formula has many applications.Given any variables in the equation, you can

solve for the remaining variable.

PV FV r

t = ×

+

1

1( )

Time Value of Money(applications)



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0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years

P V

o f $ 1 0 0

0%

5%

10%

15%

Present Values with Compounding

Interest Rates

Ti V l f M



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Value of Free Credit

Implied Interest Rates

Internal Rate of Return

Time necessary to accumulate funds

Time Value of Money(applications)



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PV of Multiple Cash Flows

ExampleYour auto dealer gives you the choice to pay $15,500 cash

now, or make three payments: $8,000 now and $4,000 at

the end of the following two years. If your cost of money is

8%, which do you prefer?

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

paymentImmediate

=

==

==

+

+

PV

PV



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Present Values

Present Value

Year 0

4000/1.08

4000/1.082

Total

= $3,703.70

= $3,429.36

= $15,133.06

$4,000

$8,000

Year 0 1 2

$ 4,000

$8,000



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Perpetuities & Annuities

Finding the present value of multiple cash flows by using a spreadshee

Time until CF Cash flow Present value Formula in Column C

0 8000 $8,000.00 =PV($B$11,A4,0,-B4)

1 4000 $3,703.70 =PV($B$11,A5,0,-B5)

2 4000 $3,429.36 =PV($B$11,A6,0,-B6)

SUM: $15,133.06 =SUM(C4:C6)

Discount rate: 0.08



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PV of Multiple Cash Flows

PVs can be added together to evaluatemultiple cash flows.

PV C

r

C

r = + +

+ +

1

1

2

21 1( ) ( )....



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Perpetuity

A stream of level cash payments thatnever ends.

Annuity

Equally spaced level stream of cash

flows for a limited period of time.



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PV of Perpetuity Formula

C = cash payment

r = interest rate

PV C

r =



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Example - Perpetuity In order to create an endowment, which pays

$100,000 per year, forever, how much money must

be set aside today in the rate of interest is 10%?

PV = =100 000

10

000 000,

.

$1, ,



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Example - continued If the first perpetuity payment will not be received

until three years from today, how much money needs

to be set aside today?

PV = =+

1 000 000

1 103 315, ,

( . )

$751,



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PV of Annuity Formula

C = cash payment

r = interest ratet = Number of years cash payment is received

[ ] PV C

r r r t

= −+

1 1

1( )



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PV Annuity Factor (PVAF) - The present valueof $1 a year for each of t years.

[ ] PVAF

r r r

t = −

+

1 1

1( )

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Example - AnnuityYou are purchasing a car. You are scheduled to

make 3 annual installments of $4,000 per year.

Given a rate of interest of 10%, what is the price you

are paying for the car (i.e. what is the PV)?

[ ] PV

PV

= −

=

+

4 000

947 41

1

10

1

10 1 10 3

,

$9, .

. . ( . )

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Applications Value of payments

Implied interest rate for an annuity

Calculation of periodic payments Mortgage payment

Annual income from an investment payout

Future Value of annual payments

[ ] FV C PVAF r t

= × × +( )1

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Example - Future Value of annual paymentsYou plan to save $4,000 every year for 20 years and

then retire. Given a 10% rate of interest, what will

be the FV of your retirement account?

[ ] FV

FV

= − × +

=

+

4 000 1 10

100

110

1

10 1 10

2020, ( . )

$229,

. . ( . )

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Effective Interest Rates

Annual Percentage Rate - Interest rate that is

annualized using simple interest.

Effective Annual Interest Rate - Interest rate

that is annualized using compound interest.

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exampleGiven a monthly rate of 1%, what is the Effective

Annual Rate(EAR)? What is the Annual Percentage

Rate (APR)?

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5- 34


exampleGiven a monthly rate of 1%, what is the Effective

Annual Rate(EAR)? What is the Annual Percentage

Rate (APR)?

12.00%or .12=12x.01=APR

12.68%or .1268=1-.01)+(1=EAR

r =1-.01)+(1=EAR

12

12

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Inflation

Inflation - Rate at which prices as a whole are

increasing.

Nominal Interest Rate - Rate at which moneyinvested grows.

Real Interest Rate - Rate at which the

purchasing power of an investment increases.

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5- 36

Inflation

AnnualInfla

tion,%

Annual U.S. Inflation Rates from 1900 - 2007

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5- 37

Inflation

1 + real inter est rate =1+nominal in terest rat e

1+inflation rate

approximation formula

Real int. rate nominal in t. rate - inflation rate≈

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Inflation

Example

If the interest rate on one year govt. bonds is 6.0%

and the inflation rate is 2.0%, what is the real

interest rate?

4.0%or .04=.02-.06=ionApproximat

3.9%or .039=rateinterestreal

1.039=rateinterestreal1

=rateinterestreal1+.021+.061

+

+ Savings

Bond

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Inflation

Remember: Current dollar cash flows must bediscounted by the nominal interest rate; real

cash flows must be discounted by the real

interest rate.

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