04120802 Chap 004

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

CHAPTER

4Discounted Cash Flow

Valuation

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

Key Concepts and Skills

• Be able to compute the future value and/orpresent value of a single cash flow orseries of cash flows

• Be able to compute the return on aninvestment

• Be able to use a financial calculator and/orspreadsheet to solve time value problems

• Understand perpetuities and annuities

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Chapter Outline

4.1 Valuation: The One-Period Case

4.2 The Multiperiod Case

4.3 Compounding Periods4.4 Simplifications

4.5 What Is a Firm Worth?

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4.1 The One-Period Case

If you were to invest $10,000 at 5-percentinterest for one year, your investment wouldgrow 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 theinvestment is call the Future Value (FV ).

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Future Value

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

FV = C 0×(1 + r )

Where C 0 is cash flow today (time zero), and

r is the appropriate interest rate.

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Present Value

• If you were to be promised $10,000 due in oneyear when interest rates are 5-percent, yourinvestment would be worth $9,523.81 in today’sdollars.

05.1

000,10$81.523,9$

• The amount that a borrower would need to setaside today to be able to meet the promised

payment of $10,000 in one year is called thePresent Value (PV ).

Note that $10,000 = $9,523.81×(1.05).

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Present Value

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

r

C PV

1

1

Where C 1 is cash flow at date 1, and

r is the appropriate interest rate.

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Net Present Value

• The Net Present Value (NPV ) of aninvestment is the present value of the

expected cash flows, less the cost of theinvestment.

• Suppose an investment that promises to

pay $10,000 in one year is offered for salefor $9,500. Your interest rate is 5%.Should you buy?

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Net Present Value

81.23$81.523,9$500,9$

05.1

000,10$500,9$

NPV

NPV

NPV

The present value of the cash inflow is greaterthan the cost. In other words, the Net PresentValue is positive, so the investment should bepurchased.

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Net Present Value

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

NPV = –Cost + PV

If we had not undertaken the positive NPV projectconsidered 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 wewould be worse off in FV terms :

$9,500×(1.05) = $9,975 < $10,000

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4.2 The Multiperiod Case

• The general formula for the future value ofan investment over many periods can bewritten as:

FV = C 0×(1 + r )T

Where

C 0 is cash flow at date 0,

r is the appropriate interest rate, andT is the number of periods over which the cashis invested.

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Future Value

• Suppose a stock currently pays adividend of $1.10, which is expected to

grow at 40% per year for the next fiveyears.

• What will the dividend be in five years?

FV = C 0×(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 fiveincreases 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 Discounting

• How much would an investor have to setaside today in order to have $20,000 fiveyears from now if the current rate is 15%?

0 1 2 3 4 5

$20,000 PV

5)15.1(

000,20$53.943,9$

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How Long is the Wait?

If we deposit $5,000 today in an account paying10%, how long does it take to grow to $10,000?

T r C FV

)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(

)2ln(T

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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 ofinterest must you earn on your investment to coverthe cost of your child’s education?

What Rate Is Enough?

T r C FV )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%.

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Calculator Keys

• 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 periodic rate

• Interest is entered as a percent, not a decimal

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

after each problem

– Other calculators are similar in format

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Multiple Cash Flows

• Consider an investment that pays $200one year from now, with cash flowsincreasing by $200 per year through year

4. If the interest rate is 12%, what is thepresent 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

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

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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4.3 Compounding Periods

Compounding an investment m times ayear for T years provides for future value ofwealth:

T m

m

r C FV

10

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Compounding Periods

For example, if you invest $50 for 3years 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 Rates ofInterest

A reasonable question to ask in the aboveexample is “what is the effective annual

rate of interest on that investment?”

The Effective Annual Rate (EAR) of interest 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 Rates ofInterest

So, investing at 12.36% compoundedannually is the same as investing at 12%compounded semi-annually.

93.70$)1(50$ 3 EAR FV

50$

93.70$

)1( 3

EAR

1236.150$

93.70$ 31

EAR

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Effective Annual Rates of Interest

• Find the Effective Annual Rate (EAR) ofan 18% APR loan that is compoundedmonthly.

• What we have is a loan with a monthlyinterest rate rate of 1½%.

• This is equivalent to a loan with an annual

interest rate of 19.56%.

1956.1)015.1(12

18.11 12

12

mn

m

r

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EAR on a Financial Calculator

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 [ENTER] Sets 12 payments per year

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Continuous Compounding

• The general formula for the future value of aninvestment compounded continuously overmany periods can be written as:

FV = C 0×erT

WhereC 0 is cash flow at date 0,

r is the stated annual interest rate,

T is the number of years, ande is a transcendental number approximately

equal to 2.718. e x is a key on yourcalculator.

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4.4 Simplifications

• Perpetuity – A constant stream of cash flows that lasts forever

• Growing perpetuity

– A stream of cash flows that grows at a constant rateforever

• 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 ratefor a fixed number of periods

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Perpetuity

A constant stream of cash flows that lasts forever

0…

1

C

2

C

3

C

32 )1()1()1( r

C

r

C

r

C

PV

r

C PV

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Perpetuity: Example

What is the value of a British consol thatpromises to pay £15 every year forever?

The interest rate is 10-percent.

0

…

1

£15

2

£15

3

£15

£150

10.

£15 PV

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Growing Perpetuity

A growing stream of cash flows that lasts forever

0…

1

C

2

C ×(1+ g )

3

C ×(1+ g )2

3

2

2 )1(

)1(

)1(

)1(

)1( r

g C

r

g C

r

C

PV

g r

C PV

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Growing Perpetuity: Example

The expected dividend next year is $1.30, anddividends are expected to grow at 5% forever.

If the discount rate is 10%, what is the value ofthis 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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Annuity A constant stream of cash flows with a fixed

maturity

0 1

C

2

C

3

C

T

r

C

r

C

r

C

r

C PV

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

T r r

C PV

)1(

11

T

C

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Annuity: Example

If you can afford a $400 monthly car payment,how much car can you afford if interest rates are7% on 36-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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What is the present value of a four-year annuity of$100 per year that makes its first payment two years fromtoday if the discount rate is 9%?

22.297$09.1

97.327$

0 PV

0 1 2 3 4 5

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

97.323$)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$4321

4

1

1 t

t PV

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Growing Annuity

A growing stream of cash flows with a fixedmaturity

0 1

C

T

T

r

g C

r

g C

r

C PV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

g r

C PV

)1(

11

2

C ×(1+ g )

3

C ×(1+ g )2

T

C×(1+ g )T -1

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Growing Annuity: Example A defined-benefit retirement plan offers to pay $20,000

per year for 40 years and increase the annual payment by3% each year. What is the present value at retirement ifthe discount rate is 10%?

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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Growing Annuity: Example

You are evaluating an income generating property. Net rent isreceived 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. 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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4.5 What Is a Firm Worth?

• Conceptually, a firm should be worth thepresent 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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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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04120802 Chap 004