McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 4-0 CHAPTER 4 Net Present Value.

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McGraw-Hill/IrwinCorporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights

Reserved.

CHAPTER

4Net Present

Value

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Chapter Outline

4.1 The One-Period Case

4.2 The Multiperiod Case

4.3 Compounding Periods

4.4 Simplifications

4.5 What Is a Firm Worth?

4.6 Summary and Conclusions

4.1 The One-Period Case

4.2 The Multiperiod Case


4.4 Simplifications



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4.1 The One-Period Case:Future Value

If you were to invest $10,000 at 5-percent interest for one 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 call the Future Value (FV).

If you were to invest $10,000 at 5-percent interest for one 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 call the Future Value (FV).

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4.1 The One-Period Case:Future Value

In the one-period case, the formula for FV can be written as:

FV = C0×(1 + r)T

Where C0 is cash flow today (time zero) and

r is the appropriate interest rate.

In the one-period case, the formula for FV can be written as:

FV = C0×(1 + r)T

Where C0 is cash flow today (time zero) and


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4.1 The One-Period Case: Present Value

If you were to be promised $10,000 due in one year when interest rates are at 5-percent, your investment be worth $9,523.81 in today’s dollars.

If you were to be promised $10,000 due in one year when interest rates are at 5-percent, your investment be worth $9,523.81 in today’s dollars.

05.1

000,10$81.523,9$

The amount that a borrower would need to set aside today to to able to meet the promised payment of $10,000 in one year is call the Present Value (PV) of $10,000.Note that $10,000 = $9,523.81×(1.05).

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4.1 The One-Period Case:Present Value

In the one-period case, the formula for PV can be written as:In the one-period case, the formula for PV can be written as:

r

CPV

11

Where C1 is cash flow at date 1 and


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4.1 The One-Period Case:Net Present Value

The Net Present Value (NPV) of an investment is 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?

The Net Present Value (NPV) of an investment is 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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4.1 The One-Period Case: Net Present Value

In the one-period case, the formula for NPV can be written as:

NPV = –Cost + PV

In the one-period case, the formula for NPV can be written as:

NPV = –Cost + PV

If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5-percent, our FV would be less than the $10,000 the investment promised and we would be unambiguously worse off in FV terms as well:

$9,500×(1.05) = $9,975 < $10,000.

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4.2 The Multiperiod Case:Future Value

The general formula for the future value of an investment over many periods can be written as:

FV = C0×(1 + r)T

Where

C0 is cash flow at date 0,

r is the appropriate interest rate, and

T is the number of periods over which the cash is invested.

The general formula for the future value of an investment over many periods can be written as:

FV = C0×(1 + r)T

Where


r is the appropriate interest rate, and

T is the number of periods over which the cash is invested.

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4.2 The Multiperiod Case:Future Value

Suppose that Jay Ritter invested in the initial public offering of the Modigliani company. Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year 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

Suppose that Jay Ritter invested in the initial public offering of the Modigliani company. Modigliani pays a current dividend of $1.10, which is expected to grow at 40-percent per year 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

Notice that the dividend in year five, $5.92, is considerably 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.

Notice that the dividend in year five, $5.92, is considerably 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

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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Present Value and Compounding

How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?

How much would an investor have to set aside 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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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

years 27.70953.0

6931.0

)10.1ln(

2lnT

How Long is the Wait?

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Assume the total cost of a college education will be $50,000 when your child enters college 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 child’s education?

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

About 21.15%.

What Rate Is Enough?

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

Compounding an investment m times a year for T years 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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Effective Annual Interest Rates

A reasonable question to ask in the above example is what is the effective annual rate of interest on that investment?

A reasonable question to ask in the above example is what is the effective annual rate 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$)06.1(50$)2

12.1(50$ 632 FV

93.70$)1(50$ 3 EAR

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Effective Annual Interest Rates (continued)

So, investing at 12.36% compounded annually is the same as investing at 12% compounded semiannually.

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 (continued)

Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.

What we have is a loan with a monthly interest rate rate of 1½ percent.

This is equivalent to a loan with an annual interest rate of 19.56 percent

Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.

What we have is a loan with a monthly interest rate rate of 1½ 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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EAR on a financial Calculator

keys: display: description:12 [gold] [P/YR] 12.00 Sets 12 P/YR.

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

[gold] [EFF%] 19.56

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Continuous Compounding (Advanced)

The general formula for the future value of an investment compounded continuously over many periods can be written as:

FV = C0×erT

Where


r is the stated annual interest rate,

T is the number of periods over which the cash is invested, and

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

The general formula for the future value of an investment compounded continuously over many periods can be written as:

FV = C0×erT

Where


r is the stated annual interest rate,

T is the number of periods over which the cash is invested, and

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

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4.4 Simplifications

PerpetuityA constant stream of cash flows that lasts forever.

Growing perpetuityA stream of cash flows that grows at a constant rate forever.

AnnuityA stream of constant cash flows that lasts for a fixed number of periods.

Growing annuityA stream of cash flows that grows at a constant rate for a fixed number of periods.

PerpetuityA constant stream of cash flows that lasts forever.

Growing perpetuityA stream of cash flows that grows at a constant rate forever.

AnnuityA stream of constant cash flows that lasts for a fixed number of periods.

Growing annuityA stream of cash flows that grows at a constant rate for a fixed number of periods.

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Perpetuity

A constant stream of cash flows that lasts forever.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

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Perpetuity: Example

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.

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

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Growing Perpetuity

A growing stream of cash flows that lasts forever.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

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Growing Perpetuity: Example

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 this promised dividend stream?

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 this promised dividend stream?

0

…1

$1.30

2

$1.30×(1.05)

3

$1.30 ×(1.05)2

00.26$05.10.

30.1$

PV

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AnnuityA constant stream of cash flows with a fixed maturity.A constant stream of cash flows with a fixed maturity.

The formula for the present value of an annuity is:

Tr

C

r

C

r

C

r

CPV

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

Trr

CPV

)1(

11

0 1

C

2

C

3

C

T

C

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Annuity Intuition

An annuity is valued as the difference between two perpetuities:

one perpetuity that starts at time 1 less a perpetuity that starts at time T + 1

An annuity is valued as the difference between two 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

TrrC

r

CPV

)1(

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Annuity: Example

If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?

If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?

59.954,12$)1207.1(

11

12/07.

400$36

PV

0 1

$400

2

$400

3

$400

36

$400

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How to Value Annuities with a Calculator

First, set your calculator to 12 payments per year.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 whatyou want.

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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%?

22.297$09.1

97.327$0

PV0 1 2 3 4 5

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

97.327$)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$4321

4

11

tt

PV

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How to Value “Lumpy” Cash FlowsFirst, set your calculator to 1 payment per year.Then, use the cash flow menu:First, set your calculator to 1 payment per year.Then, use the cash flow menu:

CF2

CF1

F2

F1

CF0

1

0

4

297.22

0

100

I

NPV

9

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Growing AnnuityA growing stream of cash flows with a fixed maturity.A growing stream of cash flows with a fixed maturity.

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

0 1

C

2

C×(1+g)

3

C ×(1+g)2

T

C×(1+g)T-1

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PV of Growing AnnuityYou 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 is 12%?

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

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PV of Growing Annuity: Cash Flow Keys

First, set your calculator to 1 payment per year.

Then, use the cash flow menu:

First, set your calculator to 1 payment per year.

Then, use the cash flow menu:

$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

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PV of Growing AnnuityUsing TVM Keys

First, set your calculator to 1 payment per year.First, set your calculator to 1 payment per year.

PMT

I/Y

FV

PV

N

7,973.93

4.67

0

– 34,706.26

5

PV

100107.1

12.1

07.1

500,8

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Why it worksThe Time Value of Money Keys use the following formula:The Time Value of Money Keys use the following formula:

T

r

g

gr

CPV

1

11

NN YI

FV

YIYI

PMTPV

)/1()/1(

11

/

Since FV = 0, we can ignore the last term.

We want to get to this equation:

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N

gr

gr

gPMT

PV

)111

1(

11

111

1

for PMT and for r11

1

g

rg

PMT

1

Nrr

PMTPV

)1(

11 becomes

We begin by substituting

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Reserved.

N

gr

gr

gPMT

PV

)111

1(

11

111

1

We can now simplify terms:

N

gr

gr

g

PMTPV

11

11

111

)1(

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N

gr

gr

g

PMTPV

11

11

111

)1(

N

r

g

gg

gr

g

PMTPV

1

11

11

11

)1(

g

g

1

11We continue

to simplify terms. Note that:

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N

r

g

ggr

g

PMTPV

1

11

1)1(

N

r

g

gr

PMTPV

1

11

We continue to simplify terms.

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

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The Result of our Algebrations:We have proved that we can value growing annuities with our calculator using the following modifications:We have proved that we can value growing annuities with our calculator using the following modifications:

PMT

I/Y

FV

PV

N

0

PV

10011

1

g

r

g

PMT

1

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Growing AnnuityA defined-benefit retirement plan offers to pay $20,000 per

year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?

A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?

57.121,265$10.1

03.11

03.10.

000,20$40

PV

0 1

$20,000

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

20,0001.03

1.101.03

–1 ×100

A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent per annum?

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PV of a delayed growing annuityYour 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.

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 per share

$1.50 $1.65 $1.82 5% growth thereafter

If investors demand a 10% return on investments of this risk level, what price will they be willing to pay?

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PV of a delayed growing annuity

Year 0 1 2 3

Cash flow

$1.50 $1.65 $1.82

4

$1.82×1.05

…

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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PV of a delayed growing annuity

Year 0 1 2 3

Cash flow

$1.50 $1.65 $1.82 dividend + P3

PV of cash flow

$32.81 22.38$

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

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Conceptually, a firm should be worth the present value of the firm’s cash flows.

The tricky part is determining the size, timing and risk of those cash flows.

Conceptually, a firm should be worth the present value of the firm’s cash flows.

The tricky part is determining the size, timing and risk of those cash flows.

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

Interest rates are commonly expressed on an annual basis, but semi-annual, quarterly, monthly and even continuously compounded interest rate arrangements exist.

The formula for the net present value of an investment that pays $C for N periods is:

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

Interest rates are commonly expressed on an annual basis, but semi-annual, quarterly, monthly and even continuously compounded interest rate arrangements exist.

The formula for the net present value of an investment that pays $C for N periods is:

N

ttN r

CC

r

C

r

C

r

CCNPV

1020 )1()1()1()1(

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4.6 Summary and Conclusions (continued)

We presented four simplifying formulae:We presented four simplifying formulae:

r

CPV :Perpetuity

gr

CPV

:Perpetuity Growing

Trr

CPV

)1(

11:Annuity

T

r

g

gr

CPV

)1(

11 :Annuity Growing

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How do you get to Carnegie Hall?Practice, practice, practice.It’s easy to watch Olympic gymnasts and convince yourself that you are a leotard purchase away from a triple back flip.It’s also easy to watch your finance professor do time value of money problems and convince yourself that you can do them too.There is no substitute for getting out the calculator and flogging the keys until you can do these correctly and quickly.

Practice, practice, practice.It’s easy to watch Olympic gymnasts and convince yourself that you are a leotard purchase away from a triple back flip.It’s also easy to watch your finance professor do time value of money problems and convince yourself that you can do them too.There is no substitute for getting out the calculator and flogging the keys until you can do these correctly and quickly.

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This is my calculator. This is my friend!

Your financial calculator has two major menus that you must become familiar with:

The time value of money keys: N; I/YR; PV; PMT; FVUse 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 menuCFj et ceteraUse the cash flow menu to value “lumpy” cash flow streams.

Your financial calculator has two major menus that you must become familiar with:

The time value of money keys: N; I/YR; PV; PMT; FVUse 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 menuCFj et ceteraUse the cash flow menu to value “lumpy” cash flow streams.

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Problems

You have $30,000 in student loans that call for monthly payments over 10 years.

$15,000 is financed at seven percent APR $8,000 is financed at eight percent APR and $7,000 at 15 percent APR

What is the interest rate on your portfolio of debt?

You have $30,000 in student loans that call for monthly payments over 10 years.

$15,000 is financed at seven percent APR $8,000 is financed at eight percent APR and $7,000 at 15 percent APR

What is the interest rate on your portfolio of debt?

15,000 30,000

× 7% 8,00030,000

× 8% 7,00030,000

× 15%

Hint: don’t even think about doing this:

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ProblemsFind 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%

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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ProblemsYou are considering the purchase of a prepaid tuition plan for your 8-year old daughter. She will start college in exactly 10 years, with the first tuition payment of $12,500 due at the start of the year. Sophomore year tuition will be $15,000; junior year tuition $18,000, and senior year tuition $22,000. How much money will you have to pay today to fully fund her tuition expenses? The discount rate is 14%

You are considering the purchase of a prepaid tuition plan for your 8-year old daughter. She will start college in exactly 10 years, with the first tuition payment of $12,500 due at the start of the year. Sophomore year tuition will be $15,000; junior year tuition $18,000, and senior 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 9

$18,000

1

15,000 CF4

F4

$22,000

1

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ProblemsYou are thinking of buying a new car. You bought you current car

exactly 3 years ago for $25,000 and financed it at 7% APR 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.

You are thinking of buying a new car. You bought you current car exactly 3 years ago for $25,000 and financed it at 7% APR 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

4-57


Reserved.

Problems

You have just landed a job and are going to start saving for a down-payment on a house. You want to save 20 percent of the purchase price and then borrow the rest from a bank.

You have an investment that pays 10 percent APR. Houses that you like and can afford currently cost $100,000. Real estate has been appreciating in price at 5 percent per year and you expect this trend to continue.

How much should you save every month in order to have a down payment saved five years from today?

You have just landed a job and are going to start saving for a down-payment on a house. You want to save 20 percent of the purchase price and then borrow the rest from a bank.

You have an investment that pays 10 percent APR. Houses that you like and can afford currently cost $100,000. Real estate has been appreciating in price at 5 percent per year and you expect this trend to continue.

How much should you save every month in order to have a down payment saved five years from today?

4-58


Reserved.

ProblemsFirst 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.

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

McGraw-Hill/Irwin Corporate Finance, 7/e © 2005 The McGraw-Hill Companies, Inc. All Rights Reserved. 4-0 CHAPTER 4 Net Present Value.

Documents

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net present value slide

present value pv

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net present value npv

mcgrawhill companies

fv terms

expected cash flows