Finance 2009 Spring

Chapter 4Discounted Cash Flow Valuation

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Chapter Outline

• Valuation: The One-Period Case• The Multiperiod Case• Compounding Periods• Simplifications (Annuities & Perpetuities)• What Is a Firm Worth?

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Basic Definitions

• Time Line

• PV (Present Value): earlier money on a time line• FV (Future Value): later money on a time line• r (Interest rate): “exchange rate” between earlier money and later

money Discount rate, Cost of capital Opportunity cost of capital, Required return

0 1 2 t

PV FV

…

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Future Values

0 1

$1,000 FV=?

r = 5%

•Interest = 1000 x .05 = 50•Value in one year = principal + interest = 1,000 + 50 = 1,050•Future Value (FV) = 1,000 x (1 + .05) = 1,050

• Example: 1 year

• Example: 2 year0 2

$1,000 FV=?

r = 5% 1

•FV = 1,000 x 1.05 x 1.05 = 1,102.50

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Future Values: General Formula

• FV = PV(1 + r)t

FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods

• Future value interest factor = (1 + r)t

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Effects of Compounding

• Simple interest FV with simple interest

= 1000 + 50 + 50 = 1100

• Compound interest FV with compound interest

= 1000 + 50 + 52.50 = 1102.50 The extra 2.50 comes from the interest

= .05 x 50 = 2.50

0 2

$1,000 FV=?

r = 5% 1

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Future Values

• Example: 5 year

•FV = 1,000 x (1.05)5 = 1,276.28

• The effect of compounding

is small for a small number of periods increases as the number of periods increases FV with simple interest = $1,250

0 1 2 5

$1,000 FV=?

…r = 5%

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Future Values: compound effect

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Future Values

• Example: 200 year

•FV = 10 x (1.055)200 = 447,189.84

0 1 2 200

$10 FV=?

…r = 5.5%

• The effect of compounding Simple interest = 10 + 200(10)(.055) = 120.00 Compounding added $447,069.84 to the value of the investment

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Future Values

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FV as a General Growth

• Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?

•FV = 3,000,000(1.15)5 = 6,034,072

0 1 2 5

3,000,000 FV=?

…r = 15%

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Present Values

• How much do I have to invest today to have some amount in the future? FV = PV(1 + r)t

PV = FV / (1 + r)t

• Discounting mean finding the present value of some future amount.

• Value the present value unless we specifically indicate that we want the

future value.

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Present Values

• Example1: need $10,000 for a new car, 1 yr, 7%

• Example2: prepare daughter’s college tuition $150,000 , 17 yr, 8%

0 1

PV=? $10,000

r = 7%

•PV = 10,000 / (1.07)1 = 9,345.79

0 1 2 17

PV=? $150,000

…r = 8%

•PV = 150,000 / (1.08)17 = 40,540.34

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PV – Important Relationship

Present Value of $1 for Different Periods and Rates

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1 2 3 4 5 6 7 8 9 10

Time(years)

Prese

nt va

lue of

$1 ($

)

r = 0%

r = 5%

r = 20%

r = 15%

r = 10%

• For a given interest ratethe longer the time period, the lower the present value

• For a given time periodthe higher the interest rate, the smaller the present value

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Discount Rate

• What is the implied interest rate on an investment?

• Rearrange the basic PV equation and solve for r FV = PV(1 + r)t

r = (FV / PV)1/t – 1

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Discount Rate

• Example1: invest $1000 today, pay $1200 in 5 years

• Example2: invest $5,000 today, double in 6 years

0 5

$1,000 $1,200

r = ?

•r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%

0 6

$5,000 $10,000

r = ?

•r = (10,000 / 5,000)1/6 – 1 = .1225 = 12.25%

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Number of Periods

• Start with basic equation and solve for t • FV = PV(1 + r)t

• t = ln(FV / PV) / ln(1 + r)

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Number of Periods

• Example:want to buy a new car (price $20,000)have $15,000 todaycan invest (r = 10%)How long?

0 t = ?

$15,000 $20,000

r = 10%

•t = ln(20,000 / 15,000) / ln(1 + 0.1) = 3.02 years

4-19

Spreadsheet Example

• Use the following formulas for TVM calculations FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv)

• Click on the Excel icon to open a spreadsheet containing four different examples.

4-20

Formula

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FV - Multiple Cash Flows

• Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?

0 1 2

$500

r = 9%

$600 FV=?

•FV = 500(1.09)2 + 600(1.09) = 1,248.05

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FV - Multiple Cash Flows

• How much will you have in 5 years if you make no further deposits?

0 1 2 5

$500

r = 9% 3

$600

4

FV=?

•FV = 500(1.09)5 + 600(1.09)4 = 1,616.26

• Second way – use value at year 2:

• First way:

•FV = 1,248.05(1.09)3 = 1,616.26

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FV - Multiple Cash Flows

0 1 2 4

₤7,000

r = 8% 3

₤4,000 ₤4,000 ₤4,000

•Today (year 0): FV = 7000(1.08)3 = 8,817.98•Year 1: FV = 4,000(1.08)2 = 4,665.60•Year 2: FV = 4,000(1.08) = 4,320•Year 3: value = 4,000•Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000 = 21,803.58

• the value at year 3

• the value at year 4

•Value at year 4 = 21,803.58(1.08) = 23,547.87

Example 6.1

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FV - Multiple Cash Flows

• to invest $2,000 at the end of each of the next 5 years. (Figure 6.4)

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PV - Multiple Cash Flows

• Example 6.3

Year 1 CF: 200 / (1.12)1 = 178.57 Year 2 CF: 400 / (1.12)2 = 318.88 Year 3 CF: 600 / (1.12)3 = 427.07 Year 4 CF: 800 / (1.12)4 = 508.41 Total PV = 1432.93

0 1 2 3 4

200 400 600 800

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PV - Multiple Cash Flows

• to pay $1,000 at the end of every year for the next 5 years. (Figure 6.5)

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Using a Spreadsheet

• You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows

• Setting the data up is half the battle – if it is set up properly, then you can just copy the formulas

• Click on the Excel icon for an example

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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.

• NPV = –Cost + PV

• The Net Present Value is positive, so the investment should be purchased.

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NPV Decisions• Your broker calls you and tells you a investment opportunity.

invest $100 today, receive $40 in one year and $75 in two years. require a 15% return on investments of this risk

• Should you take the investment?

0 1 2

-$100

r = 15%

$40 $75

•Year 1 CF: 40 / (1.15)1 = 34.78•Year 2 CF: 75 / (1.15)2 = 56.71•PV = 34.78 + 56.71 = 91.49

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Saving For Retirement

• You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%?

0 1 2 … 39 40 41 42 43 44

0 0 0 … 0 25,000 25,000 25,000 25,000 25,000

Investment today = $1,084.71

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Annuities & Perpetuities• Annuity

finite series of equal payments that occur at regular intervals ordinary annuity: the 1st payment occurs at the end of the period

annuity due: the 1st payment occurs at the beginning of the period

• Perpetuity infinite series of equal payments PV = C / r

PV C1

1

(1 r)t

r

, FV C(1 r)t 1

r

0 1 2 t…

C …C C

0 1 2 t…

C …CC

PV C 1 r 1

1

1 r tr

, FV C 1 r 1 r t 1

r

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Annuities & Perpetuities

Example: Buying a new Italian sports car

• Ordinary annuity• t = 48 months, r = 0.01 / month, C = $632• Formula:

Annuity PV factor = (1 - PV factor) / r• Spreadsheet

PV 6321

1

(1.01)48

.01

$24,000

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Growing Annuities & Perpetuities• Growing Annuity

A growing stream of cash flows with a fixed maturity

• Growing Perpetuity A growing stream of cash flows that lasts forever

1

11

t

C gPV

r g r

0 1 2 t…

C …C(1+g) C(1+g)t-1

CPV

r g

0 1 2 …

C …C(1+g)

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Buying a House• You are ready to buy a house and you have $20,000 for a down

payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?

• Bank loan Monthly income = 36,000 / 12 = 3,000 Maximum payment = .28(3,000) = 840 PV = 840[1 – 1/1.005360] / .005 = 140,105

• Total Price Closing costs = .04(140,105) = 5,604 Down payment = 20,000 – 5,604 = 14,396 Total Price = 140,105 + 14,396 = 154,501

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Finding the Payment

• Borrow $20,000 for a new car r = 8% / year, compounded monthly 4 year loan what is your monthly payment?

• Spreadsheet PMT(rate,nper,pv,fv)

20,000 C

1 1

10.08

12

48

0.08

12

C $488.26

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Finding the Number of Payments

• Example: spring break vacation put $1,000 on your credit card afford to $20 / month r = 1.5% / month

1,000 20

1 1

1 0.015 t

0.015

1

1 0.015 t 0.25

ln 1 / 0.25 t ln 1.015 t 93.11 months 7.76 years

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Finding the Rate

• Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? t = 60 PV = 10,000 C = 207.58

10,000 207.581

1

(1 r)60

r

r 0.0075

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• Without a Financial Calculator

• Trial and Error Process Choose an interest rate and compute the PV of the payments based

on this rate Compare the computed PV with the actual loan amount If the computed PV > loan amount, then the interest rate is too low If the computed PV < loan amount, then the interest rate is too high Adjust the rate and repeat the process until the computed PV and the

loan amount are equal

Finding the Rate

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Annuity Due

• You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?

FV 10,000 1 0.08 (1 0.08)3 1

r

$35,061

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Perpetuity

Example: stock price = $ 40 dividend = $1 / quarter perpetuity formula: PV = C / r

• Current required return: 40 = 1 / r r = .025 per quarter

• want to sell preferred stock at $100 per share, dividend ? 100 = C / .025 C = $2.50 per quarter

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Formula

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APR & EAR

• the actual rate paid

• use for comparison to alternatives

• the annual rate quoted by law

• APR = period rate x # of periods

• period rate = APR / # of periods

• NEVER divide the effective rate by the number of periods per year

Effective Annual Rate (EAR)Annual Percentage Rate (APR)

1 m

APR 1 EAR

m

1 - EAR) (1 m APR m

1

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Computing

• to earn 1% per month on $1 invested today.

APR? 1 x 12 = 12%effectively earning?

FV = 1(1.01)12 = 1.1268rate = (1.1268 – 1) / 1 = .1268 =12.68%

• to earn 3% per quarter.APR? 3 x 4 = 12%effectively earning?

FV = 1(1.03)4 = 1.1255rate = (1.1255 – 1) / 1 = .1255 = 12.55%

• if the monthly rate is .5%, APR? 5 x 12 = 6%

• if the semiannual rate is .5%, APR?

.5 x 2 = 1%

• if the APR is 12% with monthly compounding, monthly rate?

12 / 12 = 1%

• need to make sure that the interest rate and the time period match.

Effective Annual Rate (EAR)Annual Percentage Rate (APR)

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Decisions II

Suppose you invest $100 in each account.How much will you have in each account in one year?

• pays 5.3%, with semiannual compounding

• EAR = (1 + .053/2)2 – 1 = 5.37%

Semiannual rate = .0539 / 2 = .0265FV = 100(1.0265)2 = 105.37

• pays 5.25%, with daily compounding

• EAR = (1 + .0525/365)365 – 1 = 5.39%

Daily rate = .0525 / 365 = .000144FV = 100(1.000144)365

= 105.39

Saving account 2Saving account 1

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Computing Payments with APRs

• Suppose you want to buy a new computer system • price = $3,500, monthly payments, loan period = 2 years,

r = 16.9% with monthly compounding.• What is your monthly payment?

3,500 C

1 1

10.169

12

24

0.169

12

C $172.88

Monthly rate = .014083

Number of months = 2 x 12

4-46

FV with Monthly Compounding

• Suppose you deposit $50 a month into an account • APR = 9%, monthly compounding • How much will you have in the account in 35 years?

FV 501

0.09

12

420

1

0.09

12

$147,089.22

4-47

PV with Daily Compounding

• need $15,000 in 3 years for a new car• deposit money into an account, APR = 5.5%, daily compounding• how much would you need to deposit?

PV 15,000

10.055

365

1095

$12, 718.56

4-48

Continuous Compounding

• Sometimes investments or loans are figured based on continuous compounding

• EAR = eq – 1 The e is the exponential function.

q = r x t

• Example: What is the effective annual rate of 7% compounded continuously? EAR = e.07 – 1 = .0725 or 7.25%

exp x ex limn

1x

n

n

xn

n!n0

1 x x2

2!

x3

3!,

e 2.71828183

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What Is a Firm Worth?

• 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.