Chapter

Brealey, Myers, and Allen

Principles of Corporate Finance

11th Global Edition

HOW TO CALCULATE PRESENT VALUES

2

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2-1 FUTURE VALUES AND PRESENT VALUES

•Calculating Future Values•Future Value

• Amount to which investment will grow after earning interest

•Present Value• Value today of future cash flow

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2-1 FUTURE VALUES AND PRESENT VALUES

• Future Value of $100 =

• Example: FV

• What is the future value of $100 if interest is compounded annually at a rate of 7% for two years?

tr)1(100$FV

49.114$)07.1(100$FV

49.114$)07.1()07.1(100$FV2

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FIGURE 2.1 FUTURE VALUES WITH COMPOUNDING

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2-1 FUTURE VALUES AND PRESENT VALUES

1factordiscount =PV

PV=luePresent va

C

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2-1 FUTURE VALUES AND PRESENT VALUES

•Discount factor = DF = PV of $1

• Discount factors can be used to compute present value of any cash flow

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2-1 FUTURE VALUES AND PRESENT VALUES

• Given any variables in the equation, one can solve for the remaining variable

• Prior example can be reversed

10049.114PV

DFPV

2)07.1(1

22

C

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FIGURE 2.2 PRESENT VALUES WITH COMPOUNDING

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2-1 FUTURE VALUES AND PRESENT VALUES

•Valuing an Office Building• Step 1: Forecast Cash Flows

• Cost of building = C0 = $700,000

• Sale price in year 1 = C1 = $800,000

• Step 2: Estimate Opportunity Cost of Capital

• If equally risky investments in the capital market offer a return of 7%, then cost of capital = r = 7%

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2-1 FUTURE VALUES AND PRESENT VALUES

• Valuing an Office Building• Step 3: Discount future cash flows

• Step 4: Go ahead if PV of payoff exceeds investment

664,747$PV )07.1(000,800$

)1(1 rC

664,47$

000700$664,747$NPV

,

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2-1 FUTURE VALUES AND PRESENT VALUES

r

CC

1=NPV

investment requiredPV=NPV

10

•Net Present Value

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2-1 FUTURE VALUES AND PRESENT VALUES

• Risk and Present Value• Higher risk projects require a higher rate of return

• Higher required rates of return cause lower PVs

664,747$.071

$800,000PV

7%at $800,000 of PV 1

C

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2-1 FUTURE VALUES AND PRESENT VALUES

•Risk and Net Present Value

$47,664

$700,00047,6647$=NPV

investment requiredPV=NPV

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2-1 FUTURE VALUES AND PRESENT VALUES

• Net Present Value Rule• Accept investments that have positive net present value

• Using the original example: Should one accept the project given a 10% expected return?

273,27$1.1

$800,000+$700,000=NPV

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2-1 FUTURE VALUES AND PRESENT VALUES

• Rate of Return Rule• Accept investments that offer rates of return in excess of their opportunity cost of capital

• In the project listed below, the opportunity cost of capital is 12%. Is the project a wise investment?

14.3%or .143,$700,000

0,00070$000,00$8

investment

profitReturn

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2-1 FUTURE VALUES AND PRESENT VALUES

• Multiple Cash Flows• Discounted Cash Flow (DCF) formula:

tt

r

C

r

C

r

C

)1()1()1(0 ....PV 22

11

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FIGURE 2.5 NET PRESENT VALUES

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2-2 PERPETUITIES AND ANNUITIES

•Perpetuity• Financial concept in which cash flow is theoretically received forever

PV

luepresent va

flowcash Return

Cr

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2-2 PERPETUITIES AND ANNUITIES

•Perpetuity

r

C10PV

ratediscount

flowcash flowcash of PV

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2-2 PERPETUITIES AND ANNUITIES

•Present Value of Perpetuities• What is the present value of $1 billion every year, for eternity, if the perpetual discount rate is 10%?

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2-2 PERPETUITIES AND ANNUITIES

•Present Value of Perpetuities• What if the investment does not start making money for 3 years?

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2-2 PERPETUITIES AND ANNUITIES

• Annuity• Asset that pays fixed sum each year for specified number of years

Perpetuity (first payment in year 1)

Perpetuity (first payment in year t + 1)

Annuity from year 1 to year t

Asset Year of Payment

1 2…..t t + 1

Present Value

trr

C

r

C

)1(

1

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2-2 PERPETUITIES AND ANNUITIES

• Example: Tiburon Autos offers payments of $5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car?

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2-2 PERPETUITIES AND ANNUITIES

trrr

C1

11annuity of PV

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2-2 PERPETUITIES AND ANNUITIES

• Annuity• Example: The state lottery advertises a jackpot prize of $365 million, paid in 30 yearly installments of $12.167 million, at the end of each year. Find the true value of the lottery prize if interest rates are 6%.

000,500,167$

06.106.

1

06.

1167.12ValueLottery 30

Value

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2-2 PERPETUITIES AND ANNUITIES

•Future Value of an Annuity

r

rC

t 11annuity of FV

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2-2 PERPETUITIES AND ANNUITIES

• Future Value of an Annuity• What is the future value of $20,000 paid at the end of each of the following 5 years, assuming investment returns of 8% per year?

332,117$

08.

108.1000,20 FV

5

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2-3 GROWING PERPETUITIES AND ANNUITIES

• Constant Growth Perpetuity

• This formula can be used to value a perpetuity at any point in time

gr

C

1

0PV g = the annual growth rate of the cash flow

gr

Ctt 1PV

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2-3 GROWING PERPETUITIES AND ANNUITIES

• Constant Growth Perpetuity• What is the present value of $1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and constant growth rate of 4%?

billion 667.16$04.10.

1PV0

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2-3 GROWING PERPETUITIES AND ANNUITIES

• Growing Annuities• Golf club membership is $5,000 for 1 year, or $12,750 for three years. Find the better deal given payment due at the end of the year and 6% expected annual price increase, discount rate 10%.

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2-4 HOW INTEREST IS PAID AND QUOTED

•Effective Annual Interest Rate (EAR)• Interest rate annualized using compound interest

•Annual Percentage Rate (APR)• Interest rate annualized using simple interest

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2-4 HOW INTEREST IS PAID AND OUTLAID

• Given a monthly rate of 1%, what is the (EAR)? What is the (APR)?

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

12.68%or .1268,=1.01)+(1=EAR

=1.01)+(1=EAR12

12

r